ELECTRONIC PROPERTIES OF NOVEL ...siba.unipv.it/fisica/ScientificaActa/Tesi ottobre 2007...Electronic properties of novel phthalocyanine based molecular crystals Marta Filibian PhD

UNIVERSITÀ DEGLI STUDI DI PAVIA DOTTORATO DI RICERCA IN FISICA – XX CICLO

ELECTRONIC PROPERTIES OF

NOVEL PHTHALOCYANINE BASED

MOLECULAR CRYSTALS

Marta Filibian

Tesi per il conseguimento del titolo

DOTTORATO DI RICERCA IN FISICA – XX CICLO

ELECTRONIC PROPERTIES OF

NOVEL PHTHALOCYANINE BASED

MOLECULAR CRYSTALS

dissertation submitted by

Marta Filibian

to obtain the degree of

DOTTORE DI RICERCA IN FISICA

Supervisor: Prof. Pietro Carretta

Referee: Prof. Stephen J. Blundell

Università degli Studi di Pavia

Dipartimento di Fisica

“A. Volta”

Cover:

Electronic properties of novel phthalocyanine based molecular crystals Marta Filibian

PhD thesis – University of Pavia Printed in Pavia, Italy, November 2007 ISBN 978-88-95767-07-9

Schematic representation of the crystal structure of K2.75CuPc- eprint arXiv:cond-mat/0609405

Ad Armen e Massimo

”...Vedi, in questi silenzi in cui le coses’abbandonano e sembrano vicine

a tradire il loro ultimo segreto,talora ci si aspetta

di scoprire uno sbaglio di Natura,il punto morto del mondo, l’anello che non tiene,

il filo da disbrogliare che finalmente ci mettanel mezzo di una verita.

Lo sguardo fruga d’intorno,la mente indaga accorda disunisce

nel profumo che dilagaquando il giorno piu languisce.

Sono i silenzi in cui si vedein ogni ombra umana che si allontana

qualche disturbata Divinita...”

Eugenio Montale, I limoni

Preface

In this work I present the main results of my Ph.D. activity at the Univer-sity of Pavia focused on the experimental study of the electronic properties ofthe bulk alkali-doped transition metal phthalocyanines LixZnPc and LixMnPcand of the rare-earth metal double decker phthalocyanines [TbPc2][TBA]. Theultimate improvement of the synthesis method of these compounds has beenattained very recently, thus the results illustrated hereafter represent a pioneer-ing investigation of the undisclosed phenomenology of novel molecular crystals.The research on the phthalocyanine-based materials covers many areas of phys-ical concern, ranging from the low-dimensional magnetism and conduction, tothe metal-insulator transitions, to the strongly correlated metallicity and su-perconductivity. Primarily the Nuclear Magnetic Resonance (NMR) techniquewas used as it is an optimal tool to obtain valuable information on the staticand dynamic electronic properties of these systems over a wide range of tem-perature and magnetic field. In addition also SQUID magnetometry, MuonSpin Rotation (µSR) and Electronic Paramagnetic Resonance (EPR) exper-iments were performed to enrich and sometimes clarify the physical pictureoutlined by the result of the NMR measurements.In chapter I I will briefly review the state of art of the research concerningtransition and rare-earth metal phthalocyanines, focusing in particular on thepristine compounds to provide a reference for the subsequent discussion relatedto the doped compounds. In chapter II I will describe the principal theoreticaland practical aspects involved in magnetometry, NMR and µSR. In chapter IIIthe synthesis method, the structural characterization and the estimate of thelithium doping in transition metal phthalocyanines will be illustrated. Chap-ters IV,V,VI will be dedicated to the presentation and the discussion of theexperimental results for LixZnPc, LixMnPc and [TbPc2][TBA], respectively,and finally in chapter VII the concluding remarks will be summarized.

I

Contents

1 Transition metal and rare-earth metal phthalocyanines 11.1 Transition metal phthalocyanine crystals and their alkali-doped

derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Rare-earth phthalocyanine based single-ion magnets . . . . . . . 12

2 Experimental techniques 192.1 DC SQUID magnetometry . . . . . . . . . . . . . . . . . . . . . 20

2.1.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . 202.1.2 Basic aspects of SQUID magnetometry . . . . . . . . . . 21

2.2 Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . . . . . 232.2.1 The resonance condition . . . . . . . . . . . . . . . . . . 232.2.2 The Bloch equations and nuclear relaxation times . . . . 252.2.3 The NMR signal and spectra . . . . . . . . . . . . . . . 272.2.4 The spin-echo technique . . . . . . . . . . . . . . . . . . 282.2.5 The microscopic approach to NMR . . . . . . . . . . . . 312.2.6 The recovery laws in NMR . . . . . . . . . . . . . . . . . 33

2.3 Muon spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 352.3.1 The muon decays and the asymmetry function . . . . . . 352.3.2 Longitudinal and transverse geometry experiments . . . 362.3.3 Muonium . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.4 The study of the static properties by means of magnetometry,NMR and µSR . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.4.1 The analysis of magnetization data . . . . . . . . . . . . 402.4.2 The NMR spectra . . . . . . . . . . . . . . . . . . . . . . 412.4.3 The static relaxation in µSR . . . . . . . . . . . . . . . . 44

2.5 The study of the dynamical properties by means of NMR andµSR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.5.1 Longitudinal relaxation in NMR and µSR . . . . . . . . 462.5.2 The effects of spin fluctuations on spectrum and signal

in NMR and µSR . . . . . . . . . . . . . . . . . . . . . . 492.5.3 Some model systems: from localized spins to

metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

III

IV CONTENTS

2.6 Summarizing remarks . . . . . . . . . . . . . . . . . . . . . . . . 53

3 Synthesis, structure and stoichiometric analysis of phthalocya-nine based compounds 553.1 Synthesis of AxMPcs . . . . . . . . . . . . . . . . . . . . . . . . 563.2 Synthesis and structure of TbPc2[TBA] . . . . . . . . . . . . . . 603.3 Stoichiometric analysis of AxMPcs via NMR measurements . . . 623.4 Structure of LixMnPc . . . . . . . . . . . . . . . . . . . . . . . 643.5 Muons as probes of doping in AxMPcs . . . . . . . . . . . . . . 67

4 Magnetization and NMR measurements in LixZnPc 754.1 One-dimensional correlated metals . . . . . . . . . . . . . . . . 764.2 NMR in one-dimensional conductors . . . . . . . . . . . . . . . 814.3 The magnetic susceptibility of LixZnPc . . . . . . . . . . . . . . 844.4 LixZnPc susceptibility under high hydrostatic pressure . . . . . 914.5 NMR measurements in LixZnPc . . . . . . . . . . . . . . . . . . 97

4.5.1 NMR spectra in LixZnPc . . . . . . . . . . . . . . . . . . 974.5.2 Recovery laws for nuclear magnetization in

LixZnPc . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.5.3 Nuclear spin-lattice relaxation rates in LixZnPc . . . . . 107

4.6 Final discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5 Magnetization and NMR measurements in LixMnPc 1215.1 The susceptibility of LixMnPc . . . . . . . . . . . . . . . . . . . 1225.2 NMR spectra of LixMnPc . . . . . . . . . . . . . . . . . . . . . 1305.3 The recovery laws in LixMnPc . . . . . . . . . . . . . . . . . . . 1355.4 The temperature dependence of nuclear relaxation in LixMnPc . 139

5.4.1 Analysis of nuclear relaxation in Li0.5MnPc . . . . . . . . 1435.4.2 Analysis of nuclear relaxation in Li2MnPc . . . . . . . . 147

5.5 The field dependence of nuclear relaxation in LixMnPc . . . . . 1505.6 Summarizing remarks . . . . . . . . . . . . . . . . . . . . . . . . 157

6 Magnetization and NMR measurements in TbPc2[TBA] molec-ular magnets 1616.1 Magnetization measurements in TbPc2[TBA] . . . . . . . . . . 1626.2 NMR spectra in TbPc2[TBA] compounds . . . . . . . . . . . . . 1666.3 NMR in molecular nanomagnets . . . . . . . . . . . . . . . . . . 1696.4 Nuclear spin-lattice relaxation rates in

TbPc2[TBA] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1746.5 Summarizing remarks . . . . . . . . . . . . . . . . . . . . . . . . 179

7 Conclusions 183

A Instrumental details and pressure cell for magnetization mea-surements 189

CONTENTS V

B The experimental apparatus of NMR 193

C The muon beam lines at the Rutherford AppletonLaboratory 197

List of publications 201

Bibliography 203

Acknowledgements 211

Ringraziamenti 214

VI CONTENTS

Chapter 1Transition metal and rare-earthmetal phthalocyanines

Phthalocyanines are metallorganic compounds, not found in nature, whosefirst accidental synthesis dates back to the early years of the ’900 [1, 2]. Sincetheir discovery, owing to their high thermal and chemical stability, the prop-erties of these compounds were intensively investigated in view of their po-tential in many fields of science and technology [3], as in non linear optics[4, 5], photovoltaic and liquid crystals devices [6], and in view of their medicaland biological applications in the photodynamic therapy of cancer [7] and ascontrast agents for MRI [8]. Phthalocyanines are commonly known for theirlarge use as dyes [9, 10] and, recently, as photoreceptor devices in laser-beamprinters and photocopiers [11, 12]. However, beyond the appealing perspectiveof the practical large-scale industrial employment, the fundamental study ofphthalocyanine-based bulk materials has been carried out for decades owing tofascinating pronounced molecular character of their electro-optical, transportand magnetic properties.The basic structure of phthalocyanines, in brief Pc (Pc=C32H16N8), consistsin a planar macrocycle formed by four pyrrole units, which are linked in acircular manner by azamethine bridges. Basically, phthalocyanines are verysimilar to porphyrins [13] (Fig. 1.1), molecules which are fundamental in biol-ogy since their iron compounds are contained in the core of hemoproteins. Thecavity of the organic ring can accommodate either metals (e.g., K, Ti, Fe, Pb),semimetals (e.g., Si, Sb) or two hydrogens. Phthalocyanines with a metal ora semimetal in the center are normally called metallophthalocyanines, MPcshereafter. The coordination number of the square-planar phthalocyanines isfour and the divalent transition metals Mn2+,Fe2+,... stabilize in the moleculeforming two bonds of partial ionic character and two of covalent characterwith the four isoindole nitrogens. Monovalent metals, as alkali metals, forminstead negatively charged radicals, for example LiPc−. The cavity size of theunsubstituted metal-free phthalocyanine H2Pc is large about 3.93 A, thereforemany types of metals having a sufficiently small ionic radius are caged without

1

2 1. Transition metal and rare-earth metal phthalocyanines

A

B

C

D

Figure 1.1: A. Molecular structure of metalloporphirines. B. Molecular structure ofMPc: Nbr, Ni, Cα and Cβ denote bridging azamethine nitrogens, isoindole nitro-gens, α-pyrrole and β-pyrrole carbons, respectively [15]. C. Molecular structure ofLuPc2 [16]; D. Projection of the LuPc2 molecule along the D4d axis.

perturbing the overall planar D4h molecular symmetry [14]. In particular, theaverage Ni-M-Ni distance varies from 3.66 A in NiPc to 3.96 A in PtPc, indi-cating that the cavity size is modified in order to fit the host metal, contractingor expanding with respect to the one of H2Pc. MnPc and ZnPc retain to agood approximation a planar structure, while heavier metals as tin and leadare too large to be accommodated in the cavity and must locate out-of-plane.In this case, in order to achieve the bonding between the isoindole nitrogensand the metal, all the molecule deforms assuming a slight dome-like curvature.These complexes are clearly characterized by the lower symmetry C4v relatedto the four isoindole units. Metals requiring a higher coordination, as the lan-thanides, compose together with the Pc macrocycle more complex structures,the so called bisphthalocyanines or double decker phthalocyanines. Here themetal is sandwiched between two phthalocyanine macrocycles, which are ro-tated by a staggering angle α [15] (Fig. 1.1). The central metal is coordinatedto all of the isoindole nitrogens in each phthalocyanine ring. TbPc2, for ex-ample, exhibits an antiprismatic coordination geometry, having α = 45 (Fig.1.1).The crystal structure of phthalocyanines is typically polymorphic (Fig. 1.2).The stable polymorph of the bulk crystals is indicated with β, while films usu-

3

HERRINGBONE BRICKSTONE

Figure 1.2: Stacking of LiPc molecules in the x, α, β polymorphs of LiPc [34] andherringbone and brickstone molecular arrangements [15].

ally adopt the so called α polymorph. The flat phthalocyanine molecules pileup forming typical slipped-stacked columns; the main differences between thepolymorphs consist in different stacking angles of the molecules along the stack-ing axis, with consequent changes in the overlap of the orbitals of neighbouringmolecules. Then, since the electronic properties of phtalocyanine-based crys-tals depend strongly on the symmetry of the packing, they can change in thevarious polymorphs. Slipped-stacked columns can be arranged in two manners,a) a herringbone type or b) a brickstone type (Fig. 1.2). In the herringbonearrangement the molecules from neighboring chains form a non-zero angle withthe ones in adjacent chains. This is for example the case of MnPc and ZnPc(more details will be given in chapter 3). Differently, in the brickstone arrange-ment the molecules are parallel.The next sections will focus on the properties of MnPc and ZnPc, belongingto the family of the transition metal phthalocyanines and of TbPc2, belongingto the family of the double decker phthalocyanines.


1.1 Transition metal phthalocyanine crystals andtheir alkali-doped derivatives

Pristine MPc crystals are in general p-type semiconductors. Experimentalworks found in literature focus mainly on the DC and AC conductivity offilms, due to their potential application in the technology of organic field ef-fect transistors [17] and solar cells [18]. The thermally activated transportprocesses of these materials are typically governed by the hopping of holes,depend on the concentration of defect centers and traps and show a Debye-like frequency dependence of the dielectric constant, owing to the formationof polarons [19, 20]. The DC conductivity is also affected by the absorptionof atmospheric gases, such as oxygen and nitrogen, and humidity [19, 21].Therefore, singularly, the transport properties of ZnPc can be exploited bothin optics and photovoltaic applications, where these materials are manipulatedunder vacuum to avoid contaminations, and for the employment in the sensorindustry as gas detectors [21]. The high reactivity and easy contamination ofphthalocyanines compounds represents a severe limit to their manageability,requiring handling in vacuum and several purification steps during the synthe-sis.Magnetism in the pristine phthalocyanines is determined by the central metalion. The compounds including a non magnetic ion like Zn2+ are diamagnetic,on the contrary the other transition metal phthalocyanines have a non zero,usually intermediate, spin ground state. The phthalocyanines magnetism hasbeen studied since early ’70s by means of magnetometry, EPR and MossbauerSpectroscopy. FePc shows a ferromagnetic behavior below 7 K, consistent witha doublet S = 1 ground state [22]. MnPc was reported in early works as a weakferromagnet with a moderate anisotropy, having an S = 3/2 4A2g ground state[23]. The authors ascribed the low temperature rise in the single crystal sus-ceptibility to superexchange mechanisms among adjacent molecules belongingto the same stack (Fig. 1.3.A). In particular, they argued that the ferromag-netic superexchange is mediated by the π molecular orbitals, granted that theslipped stacking of MnPc molecules places each metal atom directly above orbelow an azamethine bridging nitrogen and the arrangement Mn-N-Mn formsan angle very close to 90. This is supported by comparative studies of α-Mnand β-Mn polymorphs which demonstrate that the different coordination ofthe metals of adjacent molecules in the two forms affects sizeably the magneticproperties, as it modifies the superexchange pathways. Fig. 1.3.A shows theorbital overlaps determining the antiferromagnetic superexchange in α-MnPcand the ferromagnetic superexchange in β-MnPc [24].In the light of subsequent calculations, the authors of Ref. [23] suggested thepossibility of a tridimensional ordering at very low temperature due to weakinterchain antiferromagnetic interactions [25]. Later publications define MnPcas a molecular canted ferromagnet where a spontaneous magnetization is gen-erated by the partial cancellation of the molecular moments of adjacent chains,

1.1. Transition metal phthalocyanine crystals and their alkali-doped derivatives 5

BA

b form a form

Figure 1.3: A. Scheme of the molecular orbitals possibly involved in the superex-change between adjacent molecules in α and β MnPc [24]. B. Effect of the field onthe canted MnPc moments, as depicted in [27].

aligned along the anisotropy axis of each molecule but being perpendicular oneto the other (Fig. 1.3.B) [26, 27]. Recent works however have revisited thisinterpretation and are strongly oriented to the hypothesis of one-dimensionalmolecular magnetism in phthalocyanines [28, 29] without the achievement ofa tridimensional long range order. On the other hand, the AC susceptibilitymeasurements in these compounds remarkably show a frequency dependenceof the maximum in the real and complex components, caused by the gradualslowing down of the spin fluctuations with temperature. In particular, in thecase of α-Fe this temperature dependent spin excitations were attributed tosingle-kink and pair-kink solitons within finite ferromagnetic chains [22, 29].A better understanding of the electronic properties of transition metal phthalo-cyanines emerged after detailed DFT calculations of the molecular energy levels[30, 31], reported in Fig. 1.4. In this scheme, owing to the considered D4h pointgroup, the metal 3d-orbitals are classified as b2g (dxy), eg (dπ, i.e. dxz and dyz),a1g (dz2), and b1g (dx2−y2). One notices that going from MnPc towards ZnPcall the metallic d-orbitals, in particular b1g, are gradually lowered. In ZnPcb1g is lower than the Highest Occupied Molecular Orbital (HOMO) a1u whichis mainly of ligand nature, together with the Lowest Unoccupied MolecularOrbital (LUMO) 2eg. This corresponds to the occupation of all the d-orbitalsin ZnPc and excludes the d electrons from oxidation and reduction processes,


1eg

MnPc

(84%

(

Figure 1.4: Molecular orbitals of MPc compounds adapted from [30] and [31].

concerning respectively the HOMO and the LUMO. The other transition metalphthalocyanines are evidently open shell molecules, having some of the innermolecular orbitals partially occupied. The ground state of MnPc is calculatedto have a 4Eg ground state, in agreement with the recent magnetic circulardichroism (MCD) and UV-vis measurements [32] of the molecule in an ar-gon matrix but different from the early magnetic measurements of solid MnPc[23, 25]. This suggests that the crystal packing shifts the molecular orbitals ofthe free molecule changing the electronic ground state. As concerns the reduc-tion of MnPc, experimental observation of absorption spectra indicate that thereduction pattern for MnPc follows the same sequence of MgPc or ZnPc, withall electrons assigned to the Pc 2eg-orbitals [33]. However, according to theDFT calculations upon the first reduction the electron enters a Mn d-orbital(dπ) and only the subsequent reduction occurs into the Pc 2eg-orbital.The interesting feature of the energy levels of MPcs is the double degener-ate organic 2eg LUMO of exquisite π nature, well separated from the HOMOby a gap of 1÷2 eV. This represents a similarity with C60 where the sp2-likeelectron energy levels are deeper in energy and completely occupied, while the


Figure 1.5: On the left, molecular structure of C60 and the diagram of the HOMO,LUMO and LUMO+1 levels [34]. On the right Huckel calculations of the bands inC60 crystals [35].

60 pz-like orbitals partially overlap, giving rise to the bonds and the higherenergy levels (Fig. 1.5). The LUMO in C60 has t1u symmetry and is triplydegenerate, while the HOMO-LUMO gap is of 1.7 eV [34] (Fig. 1.5). Theband structure of C60 retains a highly molecular character due to the weakintermolecular overlap. In particular, the bandwidth W is of the order of 0.5eV, smaller than the HOMO-LUMO separation. In this respect also phthalo-cyanines form molecular crystals with narrow bands, in analogy with otherone-dimensional molecular compounds, as (TMTTF)2X and (TMTSF)2X [36].The combination of a narrow band and orbital degeneracy is known to play akey role in the physics of intercalated fullerides. The alkali-doped AxC60, inparticular, become metallic when the LUMO is half filled, for x = 3, and ex-hibit a transition to the superconducting state at low temperatures (Tc=40 Kis the highest transition temperature in Cs3C60 under pressure) [37]. In thesecompounds the steric hindrance of the alkali metal is reduced upon dopingto allow an easier accommodation into the lattice spacings. Then, the alkalielectrons are completely donated to the LUMO of the C60 molecules and canpossibly delocalize. The phenomenology of alkali-doped fullerides shows manyparticular features related to the crystal symmetry, phase separation, molecu-lar and lattice deformation and dynamics, whose illustration goes beyond thescope of this work (see the excellent reviews [38, 39]). However, it must beunderlined that the analogies between MPcs and C60 undoubtedly catalyzedthe research on the novel alkali-doped phtalocyanine compounds at the aim to


create new organic conductors and, hopefully, new superconductors.A rich variety of molecular phthalocianine conductors have been synthesizedin order to produce hole conductivity, achieved oxidizing the Pc ligands, forexample the phthalocyanine iodide compounds NiPcI, CuPcI, and H2PcI de-noted generally as, MPcI [40, 41]. In these materials well-ordered metal-on-metal columnar stacks form and the Pc ligands, surrounded by parallel chainsof I−3 ions, are partially (1/3) oxidized. Regardless of the spin state of thecentral ion, electron transport is associated with the conduction through the5/6 th-filled π band of the overlapping Pc rings with conductivities as highas 2×104 Ω−1· cm−1. In alkali-doped phthalocyanines conduction should relyon the opposite mechanism, the charge transfer from the alkali metal to thePc 2eg LUMO, namely by electron doping. Alkali doping has been recentlyachieved in phthalocyanine thin films, as reported in recent works by Craciunet al. [42, 43, 44, 45]. They measured the film conductivity versus doping usingpotassium vapours (Fig. 1.6). For all these systems, insulating in the pristinestate, the electrical conductivity was found to increase to a value in excess of100 S·cm−1 upon potassium intercalation (quite similar to the ones observedin fulleride thin films). In this state the electrical conductivity of all com-pounds remains high at cryogenic temperatures, indicating the occurrence ofmetallic behavior. The high conductivity state (optimally doped state) occursin a broad interval up to approximately three potassium atoms per molecule.Upon reaching the 4 electrons transferred per molecule, the system becomesan insulator. Robust differences in the doping dependence of the conductancefor the different MPcs were noticed (visible in Fig. 1.6) and were correlatedto the different possible reduction processes in which the alkali electron can bedonated either to the ligands (ZnPc, CuPc and NiPc) or to the metal atoms(CoPc, FePc and MnPc) [45]. These results point out that it could be possibleto tune the electrical conductivity by doping and that, like in fullerenes, themolecular level scheme seems to govern also the behaviour of the solid. Cer-tainly, these observations must find a counterproof through the investigationof bulk crystals of novel AxMPcs.Recent theoretical works encouraged the research in AxMPcs also in view ofa strikingly non conventional phenomenology possibly arising at the half-bandfilling in non magnetic phthalocyanines, Zn and MgPc [46]. If A2ZnPc andA2MgPc were superconductors as the A3C60, they could represent model sys-tems to study the role of strong electron-electron correlation in the pairingmechanism in organic superconductors. In both fullerides and phthalocya-nines the intrinsic band-width W is narrow and the intermolecular hoppingcan be in principle strongly depressed by the large on-site Coulomb repulsion.This issue plays an important role in the superconductivity of the fullerideswhere U ∼ 0.8÷1.3 eV, yielding a U/W>1. Thus, the value of the U/W ratioin the fullerides family is around 1.5-2.5 and these systems should be Mott-Hubbard insulators. However, Gunnarson et al. [38, 47] pointed out that theorbital degeneracy d of the partially filled LUMO rises the U/W critical value,


Figure 1.6: Conductance of K-doped MPc thin films as a function of potassiumdoping and temperature [42, 43, 44, 45].

(U/W )c, for the metal-to-insulator transition from 1 to√

d, since the electronsof the LUMO find additional hopping channels without costs in Coulomb en-ergy. This renormalization of U/W should reasonably bring A3C60 on themetallic side of a Mott-Hubbard transition. Remarkably, also the LUMO ofthe metal phthalocyanines is degenerate. In addition to the orbital degeneracyfavouring delocalization, the other determinant ingredient of superconductivityin fullerides and, possibly, in phthalocyanines is expected to be the Jahn-Teller(JT) effect, induced by a strong electron coupling with the molecular phononmodes [48, 46]. Normally the energy scale of electrons is much larger thanthe energy scale of phonons, but in fullerides the single particle bandwidthW is very narrow so that the two energies can become comparable and theelectron-phonon interaction can have a dramatic effect. In the pure JT effectthe molecular system lowers its symmetry and the energy levels are split. Animportant consequence of the JT effect is that of partial filling: if the LUMObands are appreciably split, Hund’s rule is not obeyed and a low spin state isexpected (Fig. 1.7). Moreover, if the effective exchange J − EJT /4 (J is theHund’s coupling, EJT is the energy related to the the Jahn-Teller distortions)is positive and of the order of the effective bandwidth W ∗, even in presence ofa high Coulomb repulsion, an attractive singlet pairing between electrons canarise in a strongly correlated narrow-band metal [48] and a transition to su-perconductivity can occur with a gap even 1000 times larger than the BCS one


Figure 1.7: (Top) Splitting of the t1u LUMO in Na2C60 for the two most stable distor-tions (called JTD 1 and JTD 2) without spin degeneracy (thick lines) and with spindegeneracy (thin lines) [53]. (Bottom) Intramolecular Hg modes in A3C60 responsi-ble for Jahn-Teller coupling [54].

Figure 1.8: On the left, atomic vibrations of ZnPc in the B mode [14]. On the right,molecular distortions and electronic density calculated of the MgPc− molecule [50].


[48, 49]. This model can explain the high-Tc superconductivity of fullerenes.In fullerene-based compounds the JT effect is ascribed to the electron-phononcoupling with the intramolecular Hg modes. In phthalocyanines an analoguecoupling should take place with the B modes, being the most intense in theRaman spectra with excitation frequencies from 457.9 to 1064 nm [14]. Theintramolecular B1 mode is uniquely sensitive to the central metal ion becausein this vibration there are large displacements on the C-N-C bridges betweenthe isoindole groups (Fig. 1.8). Recent calculations confirm that a monore-duced MgPc− should distort according to the B1g + A1g symmetry (Fig. 1.8)with a JT energy gain of EJT = 50.4 meV [50]. This estimate of the staticdistortions of the molecule is consistent with values found for C60 [51].In metal phthalocyanines the estimate of the quantity EJT −4|J | ∼ 0.06÷0.07eV suggests that at half band filling the molecular ground state is a nonde-generate spin singlet accompanied by a dynamical Jahn-Teller effect [46]. IfU/W > (U/W )c the system would turn to a Mott-Jahn-Teller insulator, withthe lower 2eg level fully occupied and well separated by the upper level. IfU/W < (U/W )c, on the contrary, if the system was unstable against particle-particle instabilities, it could possibly be an s-wave superconductor. However,it has to be considered that, if AxMPcs are highly one-dimensional, the su-perconductivity cannot be achieved. In this case the precursors of supercon-ductivity should emerge, with the formation of electron-electron pairs but thematerial should rather behave as a spin-gapped metal. The diagram in Fig.1.9 shows the possible phases arising close to the half filling in a system onthe edge of a metal-insulator transition as a function of the doping, defined asδn = n− 2 [52].To summarize, in this section the principal properties of metal phthalocyaninescrystals were reviewed in the light of the doping with alkali metals. MPcsform highly anisotropic molecular crystals in the solid state, where the weakintermolecular interactions are expected to yield a narrow bandwidth. MPcsolids are semiconductors and can have a non-zero spin ground state when-ever some inner orbitals of the molecule are incomplete. The magnetism isof one-dimensional nature, as the dominant exchange interaction is betweenadjacent molecules stacked in chains. Since the onsite Coulomb repulsion isprobably comparable to the single particle bandwidth W , this opens the pos-sibility to consider the metal phthalocyanines as strongly correlated materials.A complex phase diagram is envisaged for these systems upon the filling of thedouble degenerate LUMO. At the half filling, in presence of a sizeable Jahn-Teller coupling with the higher frequency intramolecular vibrations, AxMPcscould be become instable against particle-particle pairing and become s-wavesuperconductors, in analogy with fullerides. In the subsequent chapters an ex-perimental investigation by means of magnetometry and NMR techniques willbe presented with the purpose of clarifying tentatively some issues suggestedby these theoretical predictions.


Figure 1.9: Phase diagram as a function of U/W and doping δ = n − 2 at J =0.05W . The thick vertical line marks the singlet Mott insulator. The inset shows,for U = 0.92W , the superconducting gap ∆ divided by a factor 10−3 and the Drudeweight D (normalized to the noninteracting value) as functions of doping [52].

1.2 Rare-earth phthalocyanine based single-ionmagnets

Double decker lanthanide phthalocyanines crystals show properties of interestmainly in the field of low dimensional magnetism. The magnetic behaviour ofseveral f-metal (e.g. Pr, Gd, Tb, Dy, Tm, Er, Ho, Yb, Lu) bisphthalocyaninecomplexes have been intensively investigated from the 80’s, the results sug-gesting that f-electrons in most of these complexes behave as in nearly isolatedions [16, 55, 56]. The properties of the molecules vary with the oxidation stateof the Pc ligands. The double decker π system binding to the trivalent Ln[III]cation can form either a neutral (green form) or a negatively charged LnPc−

compound (blue form). The green form is an open shell π system since oneof the two Pc rings is oxidized, i.e. Pc−; therefore, a strong antiferromagneticinteraction arises between the lanthanide f-electrons and the phthalocyaninatoligand radical electron in these compounds [55]. On the contrary, the blueform has a closed shell π system and the magnetic properties are governed bythe Ln f-electrons only. Very recent publications [57, 58] focused on the pe-culiar features of the blue form derivatives [TbPc2][TBA], [DyPc2][TBA] and[HoPc2][TBA], in which the organic Tertbuthilammonium [TBA]+ is intro-duced to stabilize the TbPc−2 anion. NMR, DC and AC susceptibility depictedan appealing scenario in [LnPc2][TBA] compounds, due to the presence of a

1.2. Rare-earth phthalocyanine based single-ion magnets 13

Figure 1.10: Energy diagram of the sublevels of the ground-state multiplets of[LnPc2]−TBA+ (Ln = Tb, Dy, Ho, Er, Tm or Yb). The Jz value of each state isindicated to the right of the corresponding energy level [57].

very large separation among the Ln[III] levels, owing to the pronounced uniax-ial crystal field anisotropy. In particular, in [TbPc2][TBA] Tb[III] is in the J=6ground state and the lowest sub-states mJ = ±6 and mJ = ±5 doublets areexpected to be separated by a large energy gap ∼ 436 cm−1, corresponding to628 K (Fig. 1.10) [57]. Such a high anisotropy barrier is also responsible for thetemperature dependence of the Tb[III] spin dynamics, which was preliminarilyanalyzed observing the χ′′ out of phase susceptibility at different frequencies.In [TbPc2][TBA] χ′′(ω) displays a peak which shifts at lower temperatures onlowering the AC magnetic field frequency, manifestly related to the slowing-down of the Tb[III] spin dynamics on cooling (Fig. 1.11) [58]. The activatedlaw of the peak frequency yields an energy barrier of the order of the groundstate intramultiplet splitting, indicating that the detected dynamics is inherentto the thermally activated relaxation of the Tb[III] magnetic moment betweenstates with different mJ , split by the crystal field. This phenomenology is veryclose to the one observed in the Single Molecule Magnets (SMM), finite sizedsystems containing a small number of correlated magnetic ions [59, 60, 61].In these molecules the intramolecular exchange interactions are dominant overthe intermolecular interactions, so that the bulk magnetism actually mirrorsthe single molecule properties. The most notorious representatives of this classof complexes are indeed Mn12 (Fig. 1.12) and Fe8 (all the names are abbrevia-tions). The ions can be arranged according to various geometries by means of


Figure 1.11: (a) χ′MT , (b) χ′′M/χM vs. temperature T , where χ′M , χ′′M and χM arein-phase ac, out-of-phase ac and dc molar magnetic susceptibilities, respectively,for a powder sample of [TbPc2][TBA] , measured in zero dc magnetic field with a3.5 G ac field oscillating at the indicated frequencies [58].

organic ligands, which provide also the intramolecular superexchange pathwaysbetween the magnetic centers. The transition metal complexes, in particular,have a quenched orbital moment, so that the molecular angular momentumcoincides with the total spin ~S =

∑i~Si, where i runs over the metal ions.

Mn12 and Fe8 are giant spin complexes, having an S = 10 ground state, owingto ferromagnetic superexchanges. Meaningfully, the degeneracy of the S man-ifold is partially removed in these complexes by the single-ion anisotropy (Fig.1.12). The effect of the anisotropy is generally described phenomenologicallyby two terms expressed on the basis of the total spin components, adding tothe main Heisenberg exchange Hamiltonian [61]:

H =∑ij

Jij~Si · ~Sj −DS2

z − E(S2x − S2

y) (1.1)

(higher order terms are neglected for brevity). D and E are the constantsquantifying respectively the energy gain for the total spin lying along z andthe the energy gain for the total spin lying in the plane xy. Mn12 exhibits a highuniaxial anisotropy among the SMM with D = 0.6 K and E = 0 [62]. Now,similarly to what observed for the terbium double decker phthalocyanines, theχ′′ in this compound features maxima shifting at lower temperatures on low-ering the AC field frequency [63, 61]. This is commonly explained by means


Mn12

Fe8

Figure 1.12: (Top) Sketch of the molecular structure of Mn12-ac (left) and Fe8 (right).(Bottom) Structure of the ground state S = 10 multiplet in Mn12-ac.


Mn12

Figure 1.13: (Left) Hysteresis cycle in Mn12-ac measured with the field parallel tothe anisotropy axis [66] and (right) modification of the anisotropy barrier under theapplication of the field; at the level crossing QTM is active.

of the relaxation of the total magnetization of the molecule which at low tem-peratures is frozen along the anisotropy axis, i.e the molecule is in the groundstate mS = ±10, while on increasing temperature can gradually change orien-tation due to thermal excitations between the sublevels of the ground S = 10manifold. Then, at very low temperatures T ¿ D the Mn12 molecules behaveas giant spin superparamagnets. Surprisingly, below the freezing temperatureTf , the relaxation of magnetization is extremely long (days), with a relaxationtime τ = τ0 exp(∆/kBT ) where τ0 = 2.1×10−7 s and ∆/kB=62 K. For T < Tf

the molecules are characterized by magnetic hysteresis as permanent magnets.Accordingly they have been commonly named nanomagnets, a term which un-derlines the quasi 0-dimensional nature of these finite size complexes. Thehysteresis cycle of Mn12 and Fe8 displays a series of steps [64]. The rapid vari-ation of the M(H) in the intermediate regions between adjacent plateaux wasrecognized as the effect of Quantum Tunneling of the Magnetization (QTM)between states at different mS lying on the opposite sides of the anisotropybarrier, becoming nearly degenerate at certain field values called ’crossing field’(Fig. 1.13). It should be underlined that a stair-case like magnetization hys-teresis loop was observed also in [HoPc2][TBA] single crystals, isostructural to[TbPc2][TBA] , were the QTM is attained due to the crossing of the mJ = ±5levels split by the hyperfine interaction with the I = 7/2 holmium nucleus(Fig. 1.14) [65].To summarize, in a nanomagnet the magnetization spin dynamics is driven by


Figure 1.14: (Left) Hysteresis loops at several temperatures for a single crystal of[Pc2Ho0.02Y0.98][TBA] and (right) Zeeman energy diagrams as a function of longi-tudinal magnetic field for the lowest mJ = ±5 substates with the holmium I = 7/2nucleus [65].

thermal excitations and in addition, only at very low temperatures, by quan-tum fluctuations. [TbPc2][TBA] is seemingly the smallest achievable nano-magnet constituted by a single magnetic center and by the largest anisotropybarrier ever observed, much larger than the one of Mn12. In view of thegreat impact of this material especially in the technology of magnetic storageand quantum computing, the ultimate research has been addressed towardsthe [TbPc2][TBA] diluted crystal, in which the average distance between theTb[III] ions is increased by means of intercalation with diamagnetic [TBA]Brunits. Magnetic dilution is typically operated also in other rare-earth basedcompounds as LiYxHo1−xF [67], to decouple the single ion from interatomicexchanges. The study of compounds with variable magnetic dilution is ex-pected to yield useful information on the role of the intermolecular distance onthe static and dynamic properties of the ground state. One one hand, dilutiondiminishes the intermolecular exchange and dipolar interactions, on the otherhand it affects the crystal field changing the molecular neighbourhood. Theresults of the magnetization and NMR measurements in these systems will beillustrated and discussed in chapter 6.


Chapter 2Experimental techniques

The experimental techniques used in this thesis are presented hereafter: mag-netometry, Nuclear Magnetic Resonance (NMR) and Muon Spin Rotation/Relaxation/ Resonance (µSR). Magnetometry characterizes the bulk averagestatic magnetic behaviour of materials, while NMR and µSR rather probe thelocal properties as they use microscopic probes, the nuclei and the muons, sus-ceptible mainly to local interactions. Also the NMR and µSR measurementsrequire a macroscopic amount of samples as they collect an experimental signalgenerated from a statistical ensemble of nuclei and muons. However, since thesignal is the sum of the ones coming from each nucleus and muon, one canreconstruct the local properties inside the sample by means of specific models.The NMR and µSR signals are time dependent and their shape is related by aFourier transform to the frequency spectrum of the microscopic interactions.Both static and dynamic interactions are probed by the two techniques, whichare suitably applied to investigate the low frequency range of the electronicexcitations in materials.A typical quantity observed in NMR is the relaxation of nuclear magnetizationtowards equilibrium in an external magnetic field, occurring after the nucleihave been excited by a radio frequency (r.f.) pulse tuned to the resonancefrequency. In this respect NMR investigations can be very selective, since onecan irradiate one or simultaneously more nuclear species, simply establishingthe proper experimental conditions. The muons, differently from the nuclei,must be implanted in materials by means of a dedicated beamline at largescale facilities. Since they have a magnetic moment, they precess around localfields and dynamically relax owing to the field distribution and fluctuations.Accordingly, NMR and µSR can give either comparable or complementary in-formation on the microscopic processes occurring in matter, depending on therespective positions of nuclei and muons and their couplings to the local envi-ronment.Both NMR and µSR have become widely used techniques in many scientificdisciplines (physics, chemistry, biology etc..). In particular, in the field of mag-netism they are extremely successful in the detection of structural and phase

19

20 2. Experimental techniques

transitions, disorder and low temperature quantum effects etc., thus beingsuitable to study the compounds addressed in this work.

2.1 DC SQUID magnetometry

2.1.1 Introductory remarksDC SQUID magnetometry measures the magnetic moment µ of bulk samplesas a function of temperature (T ), magnetic field (H), time (t) or pressure(P ). Before describing the experimental aspects, some remarks are to be madeabout the physical quantities derived by means of this technique.The response of a medium to a magnetic field is described by the generalizedrelation [68]

Mν(~k, Ω) =∑

~q

∫dω

∑µ

χνµ(~k, ~q; Ω, ω)Hµ(~q, ω) (2.1)

where M is the magnetization (magnetic moment per unit volume), χ is thesusceptibility, H the applied field and ν, µ = x, y, z. Thus, in general, thesusceptibility is a tensor and is complex, having an in-phase χ′ and an outof phase χ′′ component. If the field is uniform and static, only the long-wavelength ~q = 0 and ω = 0 modes contribute to (2.1). Assuming a stationary andtranslationally invariant medium with a linear response, one has

Mν(0, 0) =∑

µ

χνµ(0, 0)Hµ. (2.2)

Nevertheless, the macroscopic magnetization and the susceptibility are actu-ally the outcome of several microscopic interactions occurring at the atomicand molecular level. In presence of an external field H the magnetization ofa real system is related to the energy variation by M = −∂E/∂H, where Eis calculated on the basis of the Hamiltonian H. From the point of view ofthe magnetic properties, the field dependence of magnetization M = f(H) canbe non linear, for example in presence of saturation or non equilibrium phe-nomena, as hysteresis or relaxation. Accordingly, the susceptibility is correctlydefined by

χ(H ′) =∂M

∂H

)

H′. (2.3)

Since DC magnetometry measures the magnetic moment µ, the magnetic sus-ceptibility is a derived quantity. In this thesis the molar susceptibility will bedefined as

χ =µ

H

1

Nm

, (2.4)

where Nm is the number of moles of a sample per formula unit. When present-ing the measurements of χ as a function of temperature at constant field, also

2.1. DC SQUID magnetometry 21

the behaviour of µ(H) for fields close to the chosen one will be specified. Incase of a linear µ(H) the quantity (2.4) will represent the true susceptibility,while non for linear M(H) (2.4) will be considered as an approximated quan-tity.Finally, as already mentioned, the tensor χ can be anisotropic. Then the re-sult of a magnetization measurement depends on the angles formed by theexternal field with the principal axes of the susceptibility tensor. Providedthat measurements are carried just along the field direction (z), the principalsusceptibilities of a crystal are directly accessed only if the principal axes canbe oriented along z. If only powders are available, as the samples subject ofthis work, then experimental susceptibility is the average

(χ)av = (χ1 + χ2 + χ3)/3. (2.5)

2.1.2 Basic aspects of SQUID magnetometry

SQUID magnetometry is based on the induction method and measures themagnetic moment of a sample observing the voltage variation originated byits movement through a detection circuit. While this is a simple and generaloperating principle, the sensitivity of the single instrument is essentially deter-mined by the circuitry design and it is incredibly enhanced by the introductionof a Superconducting Quantum Interference Device. The SQUID, in brief, isa superconducting ring interrupted by weak links, called Josephson junctions[69, 70]. It acts as a flux converter with an output a voltage which is a periodicfunction of the enclosed flux with a period proportional to Φ0 (Φ0 = 2.07×10−7

G/cm2), allowing to detect even very low values of the generating magneticmoment with a great accuracy. More details about the specific superconduct-ing device configuration of the single junction r.f. SQUID can be found inexcellent texts [69, 70]. Hereafter an overview of the basic principles of thetechnique will be illustrated.The samples are mounted on a rod and inserted in a cavity where a staticfield is supplied by a superconducting magnet. The measurement is performedby moving the sample with a stepping motor through a closed loop supercon-ducting pick-up system, sketched in Fig. 2.1, to detect a flux variation. Thefour coils are configured as a second-order gradiometer in which the currentflowing in the two internal double loops is opposite to the one flowing in theexternal loops, to cancel out background and drift contributions due to thefield fluctuation or relaxation. The sample is displaced along the axis of thedetection circuit, lying along the external field (z axis), to measure the longi-tudinal component of the magnetization. After the sample is moved in a newposition it can be stopped thanks to the fact that the current variation in thesuperconducting detection circuit doesn’t decay. The change of magnetic flux4Φ induces a current variation in the NP primary coils, with inductance LP .The SQUID, with loop inductance L, put in a shielded area, senses it via amutual coupling of constant M = κS

√LSL with a secondary inductor LS. In


LP

LS

L

Figure 2.1: SQUID detection system: from the left, the second order gradiometer,the SQUID loop and the tank circuit.

conclusion, the flux sensed by the SQUID is

∆φe =κS

√LSL

LP + LS

NP ∆Φ (2.6)

The r.f. SQUID is coupled inductively to a tank resistive circuit designed toacquire the output voltage of the measurement. The tank circuit is driven bya current oscillating at or near his resonating frequency ω0 and has a qualityfactor Q = ω0LT /RT . Thus the current circulating in the inductor coupled tothe SQUID is IT sin(ωrf t) = QIrf sin(ωrf t), corresponding the output voltageVT sin(ωrf t). Finally, a direct flux-to-voltage conversion is achieved as theamplitude VT is periodic in Φe, the flux entering the SQUID loop, with periodΦ0 (for details see the appendix A). The example of a scan along the axis ofthe gradiometer, stopping the sample at discrete steps is shown in Fig. 2.2.If the sample is small with respect to the total length of the coil set, it givesrise to a signal close to the one arising from a point-like dipole. The curvereaches a maximum when the sample is placed at the central double loop and,on the contrary, it reaches symmetrically two minima when it passes throughthe external loops. Definitely the determination of the magnetic moment ismade fitting the theoretical signal of a dipole moving through the gradiometerto the collected data by linear or iterative regression methods.A more detailed description of the experimental SQUID apparatus will begiven in appendix A.

2.2. Nuclear Magnetic Resonance 23

Figure 2.2: Typical voltage vs. position curve from a point-like positive magneticdipole moving across the second-order gradiometer.

2.2 Nuclear Magnetic Resonance

2.2.1 The resonance condition

In this section a simple vectorial model will be introduced to explain the phe-nomenon of the nuclear magnetic resonance. As aforementioned, the nucleuspossesses a magnetic moment ~µ = γ~I parallel to the spin angular momentum~I. The gyromagnetic ratio γ is a scalar determined by the subnuclear structureand it is of the order of 1÷ 50 of MHz/Telsa. In presence of a magnetic field~H0 = H0z the magnetic moment µ undergoes a torque and precesses aroundz. In the laboratory frame the motion is described by

d~µ

dt= ~µ× γ ~H0. (2.7)

This precession around H0 occurs at the Larmor frequency ω0 = γH0 (some-times indicated also as ωL or ωN) and conserves the longitudinal componentµz along the field. A real sample contains rather a large quantity of nuclei,however, since they don’t correlate magnetically, also the motion of the totalmagnetization ~M =

∑i ~µi/V is governed by the Eq. (2.7). ~M rotates gener-

ating a cone of fixed angle θ around H0. The component Mz and θ are two


Figure 2.3: (a) The effective field Heff and (b) the motion of the moment µ in therotating frame [71].

statistical quantities: for a given equilibrium temperature T and a static fieldH, Mz follows the Curie law for paramagnets and < cos θ >= L(µH/kBT ) isthe Langevin function.The equilibrium magnetization Mz = M0 is constant in time but in a NMRexperiment it is prepared in a new initial state Mz 6= M0 and the subsequenttime evolution M(t) towards the restore of the equilibrium value M0 is trackedto get useful information about the microscopic mechanisms coupling the nu-clei to the local environment. The most simple way to describe the preparationof the system out of equilibrium consists in making a transformation to a ref-erence frame rotating at a pulse frequency ω around z′ ≡ z. An additionalfield ~H1 must be also applied to deflect the magnetization from z. In the newframe Eq. (2.7) becomes

d ~M

dt= ~M × γ

[k

(H0 − ω

γ

)+ ~H1

]= ~M × γ ~Heff (2.8)

where ~ω/γ is a fictitious field. The magnetization now makes a complicated

nutation around a new effective field ~Heff (see Fig. 2.3). In general, for ~H1 not

lying along z, the angle θ and Mz become time dependent and ~M periodicallydeparts and approaches to z (Fig. 2.3). But if the pulse frequency of therotating frame is ω = γH0 = ω0 and the perturbative field is chosen of theform

~H1(t) = H1(i cos(ω0t) + j sin(ω0t)) (2.9)

perpendicular to ~H0, ~M simply precesses around ~H1, at rest in the x-y plane,without sensing the torque of the large static field. This represents the strikingrequest for the phenomena of the nuclear magnetic resonance. At resonancethe nuclei absorb the maximum power from the oscillating field, no matter itsintensity with respect to ~H0. On the contrary, as ~H1 departs from (2.9), even

a strong ~H1 cannot tilt ~M efficiently. The time variation of magnetization is


ROTATING FRAME LABORATORY

FRAME

Figure 2.4: Motion of ~M under the application of a π/2 pulse in the rotating coordi-nate system with ω = ω0 and in the laboratory frame.

reasonably related to the imaginary part of the the nuclear magnetic complexsusceptibility χ′′(ω) which has a characteristic frequency width and is peakedat ω = ω0 (see later).

If ~H1(t) is applied for a finite time τ the experiment is pulsed and the magne-tization is rotated by an angle θ = γH1τ (Fig. 2.4). In this case to attain atransfer of power to the nuclear ensemble over a short period of time H1 mustbe moderately intense. In practice, linearly polarized fields are applied to thesamples by means of a r.f. coil. The resonance condition is identically satisfiedbecause ~H1(t) = 21i cos(ω0t) is the sum of two counter rotating fields of theform (2.9) at the frequencies ω0 and −ω0, but the last component is negligible.

2.2.2 The Bloch equations and nuclear relaxation times

In the vectorial model the time evolution is described by the Bloch equations,which in the general case of the laboratory frame are [72, 71]

dMz

dt= γ( ~M × ~Heff )z +

M0 −Mz

T1

(2.10)

dMx,y

dt= γ( ~M × ~Heff )x,y − Mx,y

T2

. (2.11)

This equations can be solved with several procedures depending on the initialconditions. It is noticeable that the time evolution includes a rotational termand two terms which take in account relaxation. First, the simple case in whichthe r.f. pulse has been already applied and switched off can be treated. Inthe frame rotating at ω0, the equations don’t contain the term of precessionγ( ~M × ~H0). If at time t = 0 the magnetization is in the xy plane, namely after


a π/2 r.f. pulse, the solutions of Eq. (2.10) and Eq. (2.11) are

Mz(t) = M0

(1− exp

(− t

T1

))(2.12)

Mx,y(t) = M0 exp

(− t

T2

). (2.13)

Eq. (2.12) describes the built up of Mz along the external field. Supposingthat the sample is initially unmagnetized, when H0 is applied the magneticmoments must minimize their magnetic energy giving up excess energy to theenvironment and aligning along the field. The hole process of approach toequilibrium is described by an exponential rise of Mz to M0 and occurs ona time scale determined by T1, called the ”spin-lattice” relaxation time. In asolid T1 relaxation can be very fast, even in the range of tenth-hundreds µs.Aside T1, another phenomenological parameter T2 appears in Eq. (2.13). Thisis called ”spin-spin” relaxation time and corresponds to the time scale duringwhich the in-plane components Mx,y vanish with a typical decay process. Thisoccurs because each nucleus probes a slightly different local field produced bydifferent couplings with other nuclei or electrons. Hence each one rotates ata different ω0 and the initial relative phase differences grow as time goes by.The spreading of the in-plane spin components definitely leads to Mx,y → 0 ina time of the order of T2. The so-called T2 processes are distinguished by theT1 processes and don’t involve energy exchanges with the bath; so they are notresponsible for the regrowth of Mz. T2 in solids can be often one/two or moreorders of magnitude shorter than T1. Therefore an NMR experiments can beset up to measure distinctly T1 or T2.If the Bloch equations are solved under irradiation with a small r.f. field~H1 = 2H1i cos(ωt), a direct calculus of the nuclear dynamic complex r.f. sus-ceptibility χ(ω) is possible. In fact, considering

Mx(t) = 2H1(χ′(ω) cos(ωt) + χ′′(ω) sin(ωt))

= Mx cos(ωt)− My sin(ωt), (2.14)

where Mx, y are evaluated in a system rotating at pulse frequency ω, the rela-tions χ′ = Mx/2H1 and χ′′ = My/2H1 hold. Solving Eq. (2.10) and Eq. (2.11)in the steady state, one has [72]

χ′(ω) =χ0

2

ω0(ω − ω0)T22

1 + (T2(ω − ω0))2 + γ2H21T1T2

(2.15)

χ′′(ω) =χ0

2

ω0T2

1 + (T2(ω − ω0))2 + γ2H21T1T2

. (2.16)

If γ2H21T1T2 ¿ 1, the response to H1 is linear and the susceptibilities (2.15) and

(2.16) verify the Kramers-Kronig relations. In this model χ′′ is a Lorentzianfunction peaked at ω0 (Fig. 2.5) and can be expressed in terms of the normal-ized shape function of the frequency distribution f(ω) with

χ′′(ω) =π

2ω0χ0

T2

π

1

1 + (T2(ω − ω0))2=

π

2ω0χ0f(ω). (2.17)


Figure 2.5: χ′ and χ′′ from the Bloch equations plotted versus x = (ω0 − ω)T2 [71].

f(ω) has half width at half intensity ∆ω/2 = 1/T2, which is reasonable inthe light of the previous treatment as the decay of Mx,y is the more rapid themore nuclear resonance frequencies are spread. In the next chapter the relationbetween the NMR signal and f(ω) will be formalized.

2.2.3 The NMR signal and spectraThe most simple way to observe M(t) is to exploit the induction law ∆V ∼−LdΦ/dT . Since a coil with the axis perpendicular to z is used to irradiatethe r.f. field, the same can pick-up the flux variations originated by the timeevolution of the in-plane components Mx,y when the exciting pulse is off. Then,the NMR signal is a voltage V (t) proportional to Mx,y(t). In the laboratoryframe the magnetization precedes freely around H0. If all the nuclei sensed thesame ω0, the transverse component after an r.f. pulse of length τ should be

M⊥(τ) = M0 sin(ω1τ) exp(iω0τ). (2.18)

Since in reality the nuclear resonances are described by f(ω), all nuclei aretilted by the angle ω1τ only if the spectral width of the pulse is larger thanthe width of the distribution, γH1 À 2/T2. Then, if this holds, M⊥(τ) reachesthe maximum M0 after a τπ/2 pulse.For t > τπ/2, integrating over all the possible resonance frequencies, one has[71]

M⊥(t) = M0

∫f(ω) exp(iωt)dω = M0 exp(iω0t)

∫ +∞

−∞f(ω0 + u) exp(iut)du.

(2.19)The measured V (t), the so called Free Induction Decay (briefly FID), is V (t) ∝G(t) exp(iω0t) with G(t) = M0

∫ +∞−∞ f(ω0 + u) exp(iut)du. Nevertheless, G(t)

is the NMR signal and represents only the slowly decaying time-envelope of


Figure 2.6: Schematic illustration of the correspondence between the FID signaland the NMR spectrum.

the rapid component at exp(iω0t). From the experimental point of view, it isobtained through a mixer which allows to detect V (t) in the frame of referencerotating at ω0. G(t) is exactly the Fourier transform of the shape functionf(ω), which represents the NMR spectrum (Fig. 2.6).A deeper physical insight on the concept of NMR signal, leads to the use ofcorrelation functions. This is clear by means of the fluctuation-dissipationtheorem [73]

f(ω) ∝ χ′′(ω) = ω/kBT

∫ +∞

0

〈Mx,y(t)Mx,y(0)〉dt (2.20)

implying thatG(t) = 〈Mx,y(t)Mx,y(0)〉 (2.21)

where the symbols 〈...〉 stay for a statistical average. The crucial informationsupplied by Eq. (2.21) is how the nuclear ensemble keeps memory of the initialcondition Mx,y(0).By means of the simple π/2 pulse experiment a parameter T2 can be estimated.However, the observed decay is the consequence of destructive interferences dueto both the intrinsic interactions within the sample and the spatial inhomo-geneity ∆H of the external field H0. Therefore, the FID decays with a timeT ∗

2 shorter than the real T2. To extrapolate correctly the nuclear relaxationtimes, the extrinsic sources of spatial field variation must be certainly mini-mized through the optimization of the magnet homogeneity and, additionally,a smart solution is offered by the application of the spin-echo technique.

2.2.4 The spin-echo techniqueThe spin-echo technique adopts a particular sequence alternating pulses andwaiting times to eliminate most of the spurious decays from the NMR mea-surements. Considering the magnetization as a vector in the rotating frame


Figure 2.7: (Top) Vectorial representation of the spin echo (a-f). (Bottom) Echosignals for different waiting times τ .

(x′y′z′), a typical spin-echo sequence can be easily illustrated in Fig. 2.7.At time t = 0 a π/2 r.f. pulse is applied along x′; at the end of the r.f. ir-radiation the magnetization lies along y′ (Fig. 2.7(a)). After a time τ thespins sensing different local fields have dephased (Fig. 2.7(b)) and a portion ofthe signal is lost. The nuclear momenta having ω > ω0 have precessed fasteraround z′, acquiring a positive phase +θ, while those having ω < ω0 haveprecessed slower and lost a phase −θ with respect one at θ(τπ/2). If at timeτ1 = τπ/2 + τ a second pulse π is applied the spins make a second rotationaround x′ and turn again to the x′y′ plane (Fig. 2.7(c)). Provided that thetime τπ ¿ T2, now each spin goes on precessing around z′ keeping the sameverse but with an opposite phase (Fig. 2.7(d)). In brief, θ(τ1) → −θ(τ1 + τπ).Hence, after a time 2τ1 + τπ/2 the spins refocus along −y′ (Fig. 2.7(e)), grant-ing the dephasing probed after the π/2 pulse is totally reversible. In otherwords, applying a second pulse, one gets an echo of the time evolution of thespins after the first pulse. In particular, this holds when the field spatial dis-tribution is stationary or its relaxation and fluctuations occur in a time muchlonger than the total time of the NMR experiment and when the nuclei are at


rest. In fact, the motion of spins causes an irreversible damping of the signalin a inhomogeneous field since during the π pulse the distribution of resonancefrequency changes with the change of position. Therefore the case of diffusionmust be treated separately.While the refocusing is in course, the modulus of the voltage induced throughthe r.f. coil grows and reaches a maximum at the time 2τ1 + τπ/2 (bottomof Fig.2.7). Finally, after the instant of refocusing, the nuclear spins restartdephasing (Fig. 2.7(f)) and the signal decays as a FID. After the magneti-zation is completely restored along z′ over a time T1, a new sequence can bestarted with different values of τ . The time spent to wait for the reaching ofM0 along z′ is the Repetition time and it is of the order of several T1 times.When τ À τπ/2 one can refer the maximum of the echo amplitude at time 2τ .If the experiment is performed varying τ , the experimental dependence of theecho amplitude E(2τ) ≈ |My(2τ)| on the waiting time is obtained (bottomof Fig. 2.7). It decays effectively with the time T2 related to the irreversiblemicroscopic phenomena shifting the resonance frequencies of nuclei.The spin echo technique is also implemented in the measurements of T1. In thiscase the experiment must measure a quantity proportional to Mz(τ). Since thepickup coil can detect only the time-variations of the Mx,y components, slightlymore complex sequences than the π/2-τ -π are adopted. One of these is thesaturation sequence (Fig. 2.8). In this case M0 is first completely destroyedby means of one or more pulses, whose length is nearly τπ/2. The necessityto combine several pulses arises when the line width of the resonance ∆ω isvery large with respect of the spectral width of a single radio frequency pulse,having [74]

τπ2

>1

4∆ω. (2.22)

In this case only a portion of the spins can be rotated in the xy plane. Anywaythe perturbation of the entire spin ensemble is obtained through conjugatedspin reversals induced by the bilinear dipolar couplings between excited andnon excited spins, the so called spin flip-flops, that occur on a time scale ofthe order of T2. Therefore if several saturation pulses are repeated with a timeseparation T2 ¿ τsat ¿ T1, Mz is destroyed, even in case of large lines. Afterthis, to measure the component Mz of magnetization, it is necessary to wait a

Figure 2.8: Example of saturation sequence.


time τ during which it can rebuilt and subsequently a π/2-τ -π must be appliedto bring back Mz(τ) to the xy plane. In this case the variable time to measureMz(τ) is the one separating the train of saturation pulses and the spin echosequence, which has the role of a ”reading sequence”.Definitely, the phenomenological Bloch equations are not sufficient to interpretthe experimental Mx,y,z(τ) in the general case. The nature of the microscopicinteractions determining the relaxation processes must be necessarily treatedin the light of a quantum mechanical approach.

2.2.5 The microscopic approach to NMR

The magnetic moment of a nucleus is related to its spin by ~µ = γ~~I and~I can assume the integer or semi integer values m = −I, I + 1, ..., I alongthe quantization axis. In presence of an external field one has the ZeemanHamiltonian

HZ = −γ~ ~H · ~I (2.23)

while the equation of motion of the angular momentum ~~I is [71, 72]

~d~I

dt= i[HZ , ~I], (2.24)

analogous to Eq. (2.7). Assuming the field ~H = H0z, the eigenvalues of Eq.(2.23) are E = −γ~H0m, the expectation value 〈Iz〉 is a constant of motionwhile 〈Ix〉 = 〈Iy〉 = 0. Figure 2.9 shows an illustrative scheme of the energylevel for I = 3/2.

Nucleus-electroninteraction

Nucleus-nucleusinteraction

Quadrupolarinteraction

Figure 2.9: Schematic illustration of the modifications in the hyperfine levels ofI = 3/2 nuclei, due to the different terms of the hyperfine Hamiltonian (2.31).

The nuclear resonance occurs when magnetic dipole transitions between the


Zeeman level are induced by a time-dependent perturbation. In the light ofthe energy conservation principle, the stimulating field H1(t) coupling to thenuclear moment must oscillate at the frequency matching the energy level sep-aration ω0 = γH0. Furthermore, for the selection rules H1 must be circularlypolarized in the xy plane in the form (2.9) and only ∆Iz = ±1 are allowed.The transition probability under irradiation Pnm is given by the Fermi goldenrule, taking into account a certain energy distribution peaked at the Zeemanenergies ρ(~ω0) with a width ~∆ω.The nuclei with I > 1/2 can have also a quadrupole moment tensor Q couplingwith the Electrical Field Gradient (EFG) tensor V through the Hamiltonian[71, 72, 75]

HEFG =∑

i

eQVzz

4I(2I − 1)

(3(I i

z)2 − I(I + 1) + η/2[(I i

+)2 + (I i−)2]

), (2.25)

expressed in the reference frame of the principal axes of V , with η the asymme-try parameter. If the Zeeman interaction is very strong, then the interaction(2.25) corrects the Zeeman levels by an amount depending on m2 (Fig. 2.9).As regards the transitions ∆Iz = ±1, in addition to the central resonance fre-quency ω0, the system is characterized now by other 2I−1 resonances centeredat

ωm =Em−1 − Em

~= ω0 +

e2qQ

~4I(2I − 1)3(2m + 1)

(3 cos θ2 − 1)

2, (2.26)

with θ the angle between H0 and Z in the considered reference frame andtaking η = 0 for simplicity. Eq. (2.26) gives a first order perturbation form > 1/2. When H = 0, the 2m degeneracy of the nuclear levels is partiallylifted by the term (2.25) and one has

ωQ =3e2qQ

~4I(2I − 1)[2m + 1] (2.27)

for the transitions |m| → |m + 1| with 0 ≤ m ≤ I − 1.When a great number of non interacting spins is considered, statistical mechan-ics enters the problem. Now the expectation values of the total spin momentof the ensemble is the statistical average

〈Ix,y,z〉 =∑m

〈m|Ix,y,z|m〉e−Em/kBT

Z, (2.28)

with Z the partition function. The magnetization, thus, is an operator withexpectation value 〈M〉 = γ~〈I〉. Its time evolution is described starting fromthe density matrix ρ = exp(−βH/Z)

〈M(t)〉 = Trρ(t)M, (2.29)


where ρ(t) is governed by

dρ(t)

dt=

i

~[ρ(t),H]. (2.30)

In the quantum approach the total Hamiltonian of the system H encloses allthe microscopic interactions involving nuclei. A general expression of the totalnuclear Hamiltonian is the following

Hint = −γN~∑

i

~Ii~H0

︸︷︷︸Zeeman

+∑

i

∑

k

~IiAik~Sk

︸︷︷︸Electron−Nucleus

+∑i<j

~IiDij~Ij

︸︷︷︸Dipolar

+∑

i

~IiV ~Ii

︸︷︷︸Quadrupolar

.

(2.31)where the contribution of the electrons’ angular momenta has been neglected,as it is often quenched in the solid state. In addition to the Zeeman term, thehyperfine electron-nucleus interactions and the dipolar nucleus-nucleus inter-actions are included.The tensor Aij is composed by

Aij = −γeγN~2

(r2ikI − 3~rik · ~rik

r5ik

)− A′

ik (2.32)

where the first represents the dipolar term and the second one the contactterm and γe,N are respectively the electronic and the nuclear gyromagneticratios. Starting from specific cases, the static and dynamic interactions ofHamiltonian (2.31) will be analyzed in the sections 2.4 and 2.5 to explain theirinfluence on the character of the NMR spectra and the relaxation processes.

2.2.6 The recovery laws in NMR

In the light of the quantum mechanical treatment introduced in the previouschapter, some remarks can be made about the general nature of the recoverylaws of Mz.In the framework of time evolution naturally one can assume

〈Mz(t)〉 = γ~∑m

Nm(t)〈m|Iz|m〉, (2.33)

having that the variation of populations Nm(t) is achieved by means of nuclearspin transitions between levels. If Wnm are the transition probabilities fortime unit, then a system of coupled differential equations, called the masterequations, rules the system:

dNm

dt=

∑n

(WnmNn −WmnNm) (2.34)


The resolution of the master equations must be made for specific initial con-ditions. In the simple case of I = 1/2, assuming that the r.f. field has estab-lished N1/2 = N−1/2, the recovery to the Boltzmann populations with the ratioN+1/2/N−1/2 = ∆ = exp(γH/kbT ) at the thermal equilibrium is given by

〈Mz(t)〉 = M0(1− exp(−2W )) (2.35)

where W is the transition probability between states | − 12〉 and | + 1

2〉. In

conclusion for spins I = 1/2, recalling (2.12), the following relation holds

1

T1

= 2W. (2.36)

The transition probabilities Wmn given by the energy exchanges with the latticeare different from the stimulated transition probabilities Pmn for the system indynamical equilibrium with the r.f. field. In general P À W . Besides, whileP− 1

2−→ 1

2= P+ 1

2−→− 1

2, W = W− 1

2−→+ 1

2= (1 + ∆)W+ 1

2−→− 1

2. To accomplish

the regrowth of M , the W downward transitions are more favoured than theupward transitions. During irradiation both stimulated transitions and relax-ation are effective, but if the stimulating field is strong enough to saturate thesystem, roughly speaking if P > 1/2T1, the r.f.induced transitions overcomerelaxation.In case I > 1/2 [76], if the nuclear levels splitting is affected by the nuclearquadrupole interactions according to Eq. (2.26), Wm−→m−1 depend on m andthe general case is described by 2I relaxation rates λi:

〈Mz(t)〉 = M0

[1−

∑i

ai exp(−λit)

]. (2.37)

Moreover, also the transition probabilities Wm+2−→m and Wm−→m+2 == Wm+2−→m(1 + 2∆) of quadrupolar relaxation enter the master equations.Two particular initial conditions are presented for sake of illustration: 1) thecomplete saturation of the 2I + 1 levels and 2) the saturation of the centralline only. Case 1) is satisfied when the quadrupolar splitting of the satellitesis very subtle, so the r.f. pulse can excite all transitions but simultaneousspin-exchanges, tending to restore the Boltzmann populations through dipolarinteractions, are suppressed. In this case, if dipolar and quadrupolar transitionprobabilities are equal, one recovers Eq. (2.36). In Case 2), on the contrary,one has a sum of exponentials. For I = 3/2 the recovery law has two terms

〈Mz(t)〉 = M0[1− 0.9 exp(−12Wt)− 0.1 exp(−2Wt)] (2.38)

where W = W− 12−→+ 1

2.

2.3. Muon spectroscopy 35

νe

Figure 2.10: (Top) The µ+ decay. (Bottom) Angular distribution of probability ofpositron emission for the maximum energy (a) and integrated over all the energies(b).

2.3 Muon spectroscopy

2.3.1 The muon decays and the asymmetry functionThe positive muons are the microscopic probes of the µSR technique describedhereafter. They are leptons with charge +e, spin 1

2, rest mass mµ ' 207 me

and gyromagnetic ratio γµ/2π = 13.5 kHz· G−1 and are originated in the decayof positive pions

π+ −→ µ+ + νµ. (2.39)

The pions used to produce µ+ are generated by high energy nuclear collisions(see Appendix III) and at the instant of decay can be considered at rest on thesurface of the source, generally a graphite target. For the conservation laws,the muon and the neutrino are emitted with opposite momentum and spin andeach with negative helicity. Thus, the muons generated in a certain directionconstitute a 100% polarized beam with a momentum of about 29.8 MeV/c,corresponding to 4.2 MeV, an energy low enough to ensure that muons willstop besides few hundreds microns from the sample surface. To perform µSRexperiment, the beam must be then bended, focalized and decontaminated byspurious decay products and finally directed on the sample under investigation.At the impact with the sample surface the muons thermalize in a few picosec-onds, undergoing Coulomb interactions as electronic scattering, ionization orforming muonium, the bound system of a muon and one electron. All these


processes, however, don’t affect the initial muon polarization, so that the timeevolution of the muon spins is related solely to magnetic interactions probedafter the definitive implantation.The muons anyway are not long lived particles, having a mean lifetime ofτµ = 2.2 µs. Their polarization is subject to the local static and fluctuatingfields, inducing a change in the initial polarization and after a time of the orderof τµ they decay (Fig. 2.10):

µ+ −→ e+ + νe + νµ. (2.40)

For the properties of the weak interaction, this decay violates parity. Theprobability of emission of the positron with energy ε = E/Emax, Emax ' 52MeV, in a direction forming the angle θ with the initial muon polarization isgiven by

W+(θ, ε) = 1 + A+(ε)cosθ. (2.41)

A+ is called the asymmetry parameter and has the form

A+(ε) =2ε− 1

3− 2ε. (2.42)

The integrated asymmetry over all the energy range is 1/3. However, thepositrons with maximum energy have the maximum probability of emission inthe direction of the muon spin at the moment of decay and zero probability inthe opposite direction (Fig. 2.10), therefore they can be counted to follow thetime evolution of the muon polarization over a time domain of some tens ofµs. In the following some experimental configurations will be illustrated andthe nature of the experimental µSR signal will be discussed.

2.3.2 Longitudinal and transverse geometry experiments

The muon has a magnetic moment that precedes in presence of an appliedfield at the Larmor frequency ω = γµH. Thus, once the muons are im-planted in matter, their magnetic moments rotate about the effective fieldbeing ~Heff (r) = ~Hext + ~Hloc(r). If the external field is parallel to the ini-

tial polarization ~P (0) ‖ z, the measurements probe the depolarization alongz caused by the precessions about the local field or by energy exchanges withthe lattice. Two sets of detectors are consequently used, backward and for-ward, along z in the longitudinal configuration (Fig. 2.11). On the contrary,

if ~Hext ⊥ ~P (0) the muons precede about the external field. Therefore theP (t) is a function oscillating at ωµ = γHext shaped by a slower decay. In thistransverse configuration the detectors are divided in four groups around thefield direction (Fig. 2.11). Now, starting from the general formula giving thenumber of positrons emitted in the solid angle around θ for unity of energyand time,

dN = W (θ, ε)N(t)dσdtdε, (2.43)


2

1 µµµµ++++ detector

e++++ backwarddetector

e++++ forwarddetector

sample

H

Figure 2.11: (1) µSR experiments in longitudinal geometry (left) and number ofpositron counts of the forward and backward detectors (right). (2-a) µSR experi-ments in transverse geometry, (2-b) number of positron counts of the forward andbackward detectors and (2-c) the asymmetry function [78].

where N(t) = N0 exp(−t/τµ) includes the spontaneous muon decay, the inte-gration over the energy distribution leads to

dN(θ, t) = k exp(−t/τµ)(1 + |P | cos θ)dtdσ. (2.44)

|P | = AP (0) is now the initial polarization, rescaled by the asymmetry inte-grated over the detector efficiency D(ε). Since the polarization is a function oftime, in the previous it must be P (t) = |P |G(t), where G(t) is the depolariza-tion or asymmetry function.P (t) is actually the µSR signal. It is extracted from the experimental detectorcounts which are statistical averages over a huge number of experiments, nor-mally million of events. Since their direct interaction is negligible, the muonscan be implanted in bunches of hundreds by means of a pulsed source and thetime trigger of the experiment is set at the center of the pulse, whose lengthis 80 ns. In the longitudinal geometry, the experimental P (t) is given by

P (t) = |P |G‖(t) =NB(0, t)− αNF (180, t)NB(0, t) + αNF (180, t)

, (2.45)

where α is introduced to weight the counts by the different sensitivity of theB e F groups. At very short times, one has Pt→0 ' 0.25. In the transversegeometry, the experimental P (t) is given by

P (t) = |P |G⊥(t) cos(ωµt). (2.46)


P (t) relaxation behaves differently depending on the distribution of static anddynamic fields at the muon site. Since the muon has a zero quadrupole mo-ment, these effective fields arise from the dipolar interactions with the sur-rounding nuclear spins and from the hyperfine interaction with the electronicspins. The couplings can be very different depending on the muon sites, espe-cially in magnetically ordered materials. Therefore the total relaxation func-tion,

P (t) =∑

i

Pi(t) + Bg, (2.47)

is in general a sum of partial asymmetries over all the i inequivalent sites and ofbackground contributions Bg coming from the sample holder (see appendix C),which are set to be constant or have a negligible relaxation on the time scale ofthe muon experiment. Sometimes the initial polarization can be P (0) < 0.23.This happens in transverse field experiments where the finite length of themuon pulse introduces an intrinsic dephasement. In addition, it happens whena fraction of the muons are already depolarized when the acquisition begins,as a consequence of muons bonding in muonium or muoniated compounds.

2.3.3 Muonium

Since the muon is a charged particle, it can be involved in several bonds. Inthe most simple case, a positive muon picks up one electron to form a neutralhydrogen like system, called muonium [77, 78, 79]. This is effectively verifiedwhen the system is in vacuum or when in semiconductors it is nearly isolatedin some interstitial sites. In many cases instead the muon attaches to paramag-netic species, such as free radicals having unpaired electrons, or to diamagneticmolecules. Here the molecule and the muon share two electrons, forming a sat-urated molecular orbital. In the isolated muonium, the muon spin ~I and theelectron spin ~S couple through the hyperfine interaction Hhyp = ~IA~S, which in

case of isotropy reduces toHhyp = A~I · ~S. The total spin of muonium F = I+Sis F = 0 in the ground state |0〉 = | + −〉 − | − +〉 and F = 1 in the highertriply degenerate state, with |1〉 = |+−〉+ | −+〉, |+ +〉 or | −−〉 (Fig. 2.12).The hyperfine constant in vacuum when the the electron is in the 1s atomicorbital is due to the contact term and is about A=4.4 GHz.In absence of an external field the muon spin is depolarized by the rapid fluc-tuations at frequency A and in a µSR experiment its signal is partially lost.An external field H > 2πA/γe, however, can split the muonium hyperfinelevels and quench the muon-electron spin coupling, since Zeeman interactionsγe~~Se

~H and γµ~~Sµ~H increase. This is the so called Paschen-Back regime

where the eigenstates are labelled with the electron and the muon spin quan-tum numbers (Fig. 2.12). Consequently, it is possible to align the muon ofthe Mu-e system efficiently increasing the field. In the longitudinal geometrythe repolarization of the initial asymmetry is the experimental signature of theformation of muonium in a material (Fig. 2.13). In isotropic muonium this is


E

F=1

F=0

S I

Figure 2.12: The Breit-Rabi diagram in case of isotropic hyperfine interaction [75].

Figure 2.13: Longitudinal repolarization as a function of the applied field in isotropicmuonium and anisotropic muonium (Mu∗) in silicon [77].

described by the function

P‖ =1 + 2(H/H0)

2

21 + (H/H0)2 (2.48)

with H0 = A/(γµ + γe).In presence of a transverse field, the muon spins of muonium can precess aroundH, which corresponds to ∆mµ = 1 transitions between the levels having thesame electronic quantum number me. This two precessions take places at


frequencies ν ¿ A and can be resolved with the µSR time resolution. Thisimplies that the asymmetry function will be a sum of harmonically oscillatingsignals, generated by free precessing µ+ and muonium. In matter the muoniumhyperfine coupling is usually smaller then A because of the delocalization of themuonium electron over the surrounding atoms and the presence of anisotropicbonds. A Fourier transform of the time spectra can supply the frequenciesof the aforementioned transitions and therefore the hyperfine couplings canbe evaluated. This analysis gives information on the structural, chemical andmagnetic microscopic surrounding of the muonium, offering the possibility touse muons as substitutes of other atoms to predict the behaviour of dopants,defects or adduct radicals.

2.4 The study of the static properties by meansof magnetometry, NMR and µSR

2.4.1 The analysis of magnetization dataThe static magnetization measured with the DC SQUID is a sum of severalcontributions. As already explained, the sample is mounted on a rod andmoved through the coils of the gradiometer. A plastic can is used as holderand the sample is suspended at the inside in a position corresponding to thedouble wounded central loop, as every scan is executed taking this as the centerof the measuring scan length. The samples are generally wrapped in a piece ofteflon or parafilm or closed under inert atmosphere in glass tubes. Thus, onemeasures the sum of the magnetic moment of the sample, the sample holderand the sample cover. The contribution of the sample holder and cover is usu-ally smaller than 10−6 emu and is generally diamagnetic and constant over thehole temperature range. Therefore, if the sample signal is at least of the orderof 10−5 ÷ 10−4 emu, this background can be neglected or taken in to accountdirectly in the data fitting procedures. On the contrary, if the sample origi-nates a poor signal, a separate measurement of the magnetic moment of thesample holder can be performed and subtracted point by point from the totalmeasurement. This can’t be avoided also if the sample holder gives a temper-ature dependent contribution to the signal. In some cases the sample holder isnot homogeneous. For example, a glass tube containing powders can be muchlonger than the sample, which is deposited on the bottom, or, in alternative,the endings of a pressure cell cause abrupt local changes of susceptibility. Inthis cases the signal can be deformed by spurious oscillations. However, if thelocal inhomogeneity is far from the sample position, as is the case of the pres-sure cell, the two signals are well separated and resolved over the total scanlength and the measurement can be done by resorting to a simple backgroundsubtraction.Once the data are corrected by the undesired contributions, the measurementof the magnetic moment of the sample is obtained. At this point, the SQUID

2.4. The study of the static properties by means of magnetometry, NMR and µSR 41

measurement relies on the quality of the sample itself. It is recalled thatby means of the SQUID DC measurement one measures the bulk quantityM =

∫M(V )dV ; to get direct information on the microscopic electronic sus-

ceptibility Electronic Paramagnetic Resonance (EPR) and NMR are generallypreferred.

2.4.2 The NMR spectraAs already explained, the NMR spectrum is derived from the Fourier transformof the FID signal. Remembering Eq. (2.29), the signal is described by

G(t) = cos(ω0t)Tre−iH′(t)P t/~MxeiH′(t)P t/~Mx (2.49)

where H′(t)P is the perturbative Hamiltonian in the rotating frame. Thespectrum then encloses the spectral character of the interactions containedin H′(t)P . To a first approximation only the secular parts of this Hamiltoniandetermine the nature of the line, as they commute with the principal ZeemanHamiltonian. In the solid state, since each nucleus couples directly and in-directly with many other nuclei and electrons and the nature of the couplingdepends on the positions they occupy in the crystal lattice, the NMR line canbe in general broad and structured.The first basic information extracted by the experimental line shape is the thetotal number N of nuclei I resonating in the sample by means of [71]

∫ +∞

0

f(ω)dω =1

2~γ2~2 I(I + 1)

3N(2I + 1)N (2.50)

which is useful for chemical calibration measurements. Moreover, the methodof the moments based on the estimate of the integrals [71, 72, 73]

Mn =

∫ +∞0

ωnf(ω)dω∫ +∞0

f(ω)dω(2.51)

gives valuable indications. In particular, the second moment is M2 ∝ ∆ω2,where ∆ω is the bandwidth. This is useful when M2 can be calculated fromfirst principles: for example in case of dipolar static interactions between iden-tical nuclei in a cubic lattice the Van Vleck formula gives [72]

M2 = 〈∆ω2〉 =3

4γ4~2I(I + 1)

∑

k

1− 3 cos2 ϑk

r6k

. (2.52)

Therefore, one can establish if the experimentally observed broadening is ofdipolar nature and, precisely, deduce the internuclear distances rjk. The staticnuclear dipolar interactions in general gives rise to a Gaussian decay

G(t) ∝ exp(−t2

2T 22dip

). (2.53)


and, conversely, f(ω) is a Gaussian distribution with Full Width at Half Max-imum (FWHM) ∆ω ∝ 1/T2dip and centered at ω0 = 0 (in the rotating frame).This is a case of homogeneous broadening, usually of the order of 1÷ 10 kHz,without a shift of the average frequency of the line from ω0.As regards the interaction with the electrons spins ~Se, neglecting the orbitalcontribution, the nuclei probe a paramagnetic shift. The average local fieldinduced by ~H0 is 〈 ~Hloc〉 = A〈~Se〉 = V χpA ~H0, where A is the hyperfine tensor

and χp the electronic susceptibility. 〈~Se〉 indicates the value at the thermalequilibrium. If the nuclei are coupled to localized spins in many cases the hy-perfine coupling A is dipolar.In metals the paramagnetic shift of the mean frequency of the line ω0 canbe sizeable, as all nuclei see the same effective average field generated by thedelocalized electrons. In case of s-electrons and scalar hyperfine coupling, therelative shift, called the Knight shift is

K =∆ν

ν0

=8π

3NA

〈|ψk(0)|2〉F χP (2.54)

where χP is the Pauli molar susceptibility. The Knight shift is also probedby muons implanted in metals and in case of muon spin rotation affects theprecessional frequency. The nuclear dipolar broadening of the line and theKnight shift are depicted in Fig. 2.14

Nucleus-electroninteraction

Spectra

Nucleus-nucleusinteraction

Quadrupolarinteraction

Figure 2.14: Effect of the interactions of Hamiltonian (2.31) on the NMR line.

The calculation of the shift and the linewidth are useful when compared withthe results of susceptibility measurements. If ∆K is the total shift due to theaforementioned effects [73],

∆K =

∑k Ak〈~Sk〉H0

=∑

k

Akχ(~q = 0, ω = 0) (2.55)

where k runs over all the couplings, if the static macroscopic χ is known, thenthe average hyperfine coupling is deduced.


Figure 2.15: Powder spectrum of spin 23Na I = 3/2 in a powdered sample ofNaNO3 recorded by means of several techniques (c is acquired with Magic AngleSpinning) (see [80]).

In addition to positive shifts, the nuclear frequencies can be affected by nega-tive shifts, as the core electrons and the electrons paired in molecular saturatedbonds give rise to a field induced diamagnetism. This shift depends on thesymmetry of the electronic density involved in the formation of the chemicalbonds. Hence, generally, the tensor of chemical shift ~Hloc = (1 + σ) ~H0 is in-troduced. In the solid state this effects are completely shielded by the dipolarbroadening.Lines can also have inhomogeneous broadenings and complex structures, for ex-ample in case of quadrupolar couplings and ordered magnetism of the electronicspins. When the experiment is performed on powders an additional asymmet-ric broadening increasing with the field is observed in case of anisotropy. Theeigenvalues of the interactions and therefore the resonance frequencies ν de-pend on the orientation of the magnetic field with respect to the principalaxis of the EFG or the Knight Shift tensor K. If θ is the angle between theprincipal axis OX of the tensors and the field, the powder spectrum f(ν) willbe proportional to the number of nuclei resonating at frequency ν, given byf(ν) = sin(θ(ν))(dν/dθ)−1. The spectra display singularities for dν/dθ = 0. Incase of quadrupolar interactions, for I = 3/2 and η = 0, the spectrum corre-sponding to the transitions 3/2 → 1/2 with frequencies ∆ν = CνQ(3 cos2 θ−1)is singular for θ = 0, 90, 180. Considering both the transitions 3/2 → 1/2


and −1/2 → −3/2 one obtains the spectrum illustrated in Fig. 2.15.In case of an uniaxial Knight shift the resonance frequency is

ν =1

2K‖ν0(3 cos2 θ). (2.56)

The resulting line shape is a convolution of the function [71]

g(ν) ∼ (1 +2ν

K‖ν0

)−1/2, (2.57)

defined between ν⊥ and ν‖, with the symmetrical broadening due to the otherinteractions.

2.4.3 The static relaxation in µSRµSR is a very powerful tool to study the distribution of internal fields in ma-terials, having the advantage over other techniques to be performed also asa zero field experiment. In this case, if the earth’s field is compensated byan opposite magnetic field, the time evolution of the muon spin reduces to aprecession about the local field. If ~Hloc forms an angle θ with the initial ~P (0),then G(t) follows

G(t) = cos2 θ + sin2 θ cos(γµHloct). (2.58)

Let first consider the static case. If the sample has an internal uniform field~Hloc = ~H, the asymmetry clearly shows coherent spin precession oscillationscoming from all the implanted muons at the Larmor muon frequency. Then

G(t) = cos(γµHt). (2.59)

This is classically the signature of the long-range order in ferromagnets andantiferromagnets. On the contrary, if the microscopic fields are randomly ori-ented and their intensity has a Gaussian distribution p(H) centered at 0 andwidth ∆, integrating the (2.59) over the angles and field distribution, one ob-tains the Kubo-Toyabe function [79] (Fig. 2.16)

G(t) =1

3+

2

3(1− γ2

µ〈∆H2〉t2) exp

(−1

2γ2

µ〈∆H2〉t2)

. (2.60)

This is analogous to the T2 experiment performed by means of NMR, as it isthe consequence of the decoherence of the muon spin precessions under theaction of different magnetic environments. Eq. (2.60) is overdamped at shorttimes, showing a marked minimum, while recovers the value 1/3, the averagefraction of random field component along the direction z, at long times. Thedistribution p(H) can also be Lorentzian, from which follows the LorentzianKubo-Toyabe [81]

G(t) =1

3[1 + 2(1− γµ∆Ht) exp(−γµ∆Ht)]. (2.61)


Figure 2.16: Kubo-Toyabe function (2.60) in red compared with functions (2.59)plotted in grey for several values of H.

Fitting directly the zero field µSR time spectra, one reconstructs H or p(H)for the material under study. Dynamic effects generally affect the shape of thedecay function and will be illustrated later in a dedicated chapter.When an external uniform static field ~Hext is applied, each muon precedesabout ~Hext + ~Hloc. When γ| ~Hext| >> γµ∆H, the strong field limit, and~Hext ⊥ ~P (0), one recovers the (2.59), enveloped with a slower decaying Fouriertransform of p(H)

G(t) = FT (p(H)) cos(ωµt), (2.62)

while if ~Hext ‖ ~P (0) the expression is slightly complex and is reviewed in ded-icated works [82]. Anyway, in brief, in a strong longitudinal field the initialmuon polarization is conserved and the short-time damping induced by thefield distribution is sizeably overcome when Hext/∆H > 1.Even if the µSR data are typically treated in the time frame, also µSR fre-quency spectra can be obtained by means of a Fourier transform. Several issuesmust be considered in the analysis of spectral information gained by pulsedµSR, since the frequency response is limited by the time resolution, the finitelength of the muon burst and the poor statistics at long times leading to noise.To avoid this troubles, the ’maximum entropy method’ is often used. µSRfrequency spectrum is particularly useful to display the precessional frequen-cies of muonium species in materials and when the precession signal of µ+ is afinite sum of signals oscillating at different frequencies. This happens in case


of inequivalent muon sites giving G(t) =∑

i Ai cos(γµHit).

2.5 The study of the dynamical properties bymeans of NMR and µSR

2.5.1 Longitudinal relaxation in NMR and µSRIn this section an outline of the dynamical response of the nuclei and the muonsto time dependent microscopic fields will be developed. In the case of nuclei,the energy exchanges with the bath lead to the recovery of the equilibriummagnetization, while in the case of muons lead to the decay of initial polar-ization. The dynamical relaxation of muons is observed in longitudinal fieldexperiments. The Zeeman interaction splits the ground states |1/2〉 and the| − 1/2〉 state by the amount ∆E = γµ~H, opening the possibility of mag-netic dipolar transitions between the two at the frequency ωµ = γµH. Thus,the muons can be treated as I = 1/2 nuclei interested by T1 processes. If Wis the probability of transition, in absence of statical relaxation the generalasymmetry function for muons is expressed by

G(t) =∑

i

Ai exp(−λit) =∑

i

Ai exp(−2Wt) (2.63)

as for a pure decay phenomena.In the frame of a perturbative treatment, considering the Hamiltonian Hp(t),the transition probabilities among the nuclear (muon) levels k and m are ex-pressed by [71]

Wkm =1

~2

∫ +∞

−∞Gmk(τ)e−iωmkτdτ (2.64)

where Gmk(τ) is the correlation function of the local field fluctuations

Gmk(τ) = (k|Hp(t)|m)(m|Hp(t + τ)|k) (2.65)

averaged on several statistical ensembles. This is the analogous of the Fermigolden rule applied to a system with sharp levels k,m and a time dependentperturbation with a broad spectra. The restriction on this assumption is thatseveral collisions are necessary to induce a transition (weak collision regime).Then, if one defines the correlation time τc

τc =

∫ ∞

0

Gmk(τ)dτ (2.66)

so that for τ > τc Gmk(τ) is almost completely decayed, Eq. (2.64) is validwhen T1 À τc. The spectral density Jmk(ω) is defined by

Jmk(ω) = ~2Wkm =

∫ +∞

−∞Gmk(τ)e−iωmkτdτ. (2.67)

2.5. The study of the dynamical properties by means of NMR and µSR 47

Spectral Density

~~~~ 1

Frequency

Figure 2.17: Spectral density J(ω) for different values of τcωL.

Jmk(ω) has a cutoff frequency Γc = 1/τc (see J(ω) for different values of τc inFig. 2.17), but since Gmk(0) = 1/2π

∫ +∞−∞ Jmk(ω)dω is constant, the area of

Jmk(ω) remains fixed even if τc varies.

Now, having the local effective field ~H(t), in the approximation that the i =x, y, z components fluctuate independently,

jimk(ω) = γ2

n,µ~2|(m|In,µi |k)|2

∫ +∞

−∞Hi(t)Hi(t + τ)e−iωτdτ (2.68)

and

Wmk(ω) =1

~2

∑i

jimk(ωmk) (2.69)

(n and µ are related to nuclei and muons respectively). Since longitudinalrelaxation is driven by the transverse components of the field entering theWkm, recalling the result for I = 1/2, 1/T1 = 2W1/2,−1/2, follows

1

T1

= λµ =γ2

n,µ

2

∫ ∞

−∞〈H+(t)H−(0)〉eiωL,µtdt =

γ2n,µ

2J±(ωL,µ) (2.70)

where ωL = γnH. Often one has Hi(t)Hi(t + τ) = (∆H2i )e−τ/τc , then (Fig.

2.18)1

T1

= λµ =γ2

n,µ

2〈∆H2

i 〉2τc

1 + τ 2c ω2

L,µ

. (2.71)

This model (BPP) is applied to many cases and was originally developed


FAST MOTIONSSLOW MOTIONS

Figure 2.18: 2W=1/T1 (expressed in a.u.=arbitrary units) as a function of the corre-lation time τc according to Eq. (2.71), with ωL = 1 MHz.

by Bloembergen, Purcell e Pound for liquids. It is a Lorentzian reaching themaximum for ωL,µτc = 1 (Fig. 2.18):

(1

T1

)

max

=γ2

n,µ

2ωL,µ

〈∆H2⊥〉; (2.72)

moreover, it tends to the following approximations in the two regimes of slowand fast fluctuations (Fig. 2.18):

SLOW MOTIONS when ωL,µτc À 1 1T1∼ γ2

n,µ〈∆h2⊥〉

τcω2L,µ

FAST MOTIONS when ωL,µτc ¿ 1 1T1∼ γ2

n,µ〈∆h2⊥〉τc.

The field fluctuations can be generated by various phenomena of differentorigin. The approximation of a random distribution of fields is adopted totake into account the modulations caused by incoherent processes as Brow-nian motion in liquids, diffusion, vibrations and rotations of molecular com-plexes, exchange interactions. Since each process determining the local fieldcorrelations has a typical τc temperature dependence, it can be recognized an-alyzing 1/T1(T ) at fixed field, if models are available. Furthermore, 1/T1(H)measurements at fixed temperature directly probe the spectral density of thefluctuations and, thus, the character of the correlation function.


Dipolar broadening,

Gaussian Line

Motional narrowing,

Lorentzian Line

Figure 2.19: (Top) Gaussian line broadened by dipolar interactions in the slow mo-tions regime. (Bottom) Narrowed Lorentzian line in the fast motion regime.

2.5.2 The effects of spin fluctuations on spectrum andsignal in NMR and µSR

When the local fields probed by the nuclei and the muons have a time depen-dent component

~h(t) = ~H(t)− 〈 ~H(t)〉 (2.73)

the NMR and µSR signal, and conversely the spectra, can have a differentshape from the static case. This in particular occurs when the zz componentof the correlation function of the local fields

g(t′) =〈hz(t)hz(t− t′)〉

〈∆h2z〉

(2.74)

rapidly decays for effect of rapid fluctuations.The starting point is the definition of NMR signal. When the local fields hi

z(t)at the i-th site inducing the shift ∆ωi = γhi

z(t) are uncorrelated, the signalhas the form [72]

G(t) = G(0)〈ei〈∆ω〉t〉 = G(0)〈exp

(−iγ

∫ t

0

hz(t′)dt′

)〉, (2.75)

Furthermore if hz(t) is Gaussian, stationary and 〈∆h2z〉 is the second moment

of the static distribution, the adiabatic approximation holds

G(t) = G(0) exp

[−γ2〈∆h2

z〉∫ t

0

(t− t′)g(t′)dt′]

. (2.76)


Without specifying the shape of g(t), introducing τc, one considers

SLOW MOTIONS when γ2〈∆h2z〉τc À 1

FAST MOTIONS when γ2〈∆h2z〉τc ¿ 1.

In the first case τc À t, the decay time of the signal G(t); so g(t) ∼= 1 underthe integral and the Gaussian shape of the static case is recovered (Fig. 2.19)

G(t) ∝ G(0) exp

(−γ2〈∆h2

z〉t2

2

). (2.77)

In the second case τc ¿ t, so that g(t) at time t is almost completely decayed.Then, in the limit

∫ t

0

(t− t′)g(t′)dt′ = t

∫ ∞

0

g(t′)dt′ = tτc, (2.78)

the decay is exponential

G(t) ∝ G(0) exp(−γ2〈∆h2z〉τct) (2.79)

and the line shape is Lorentzian with FWHM γ2〈∆h2z〉τc (Fig. 2.19). Definitely,

for effect of rapid modulation of the local field, the nuclei probe only the averagecomponent of the local field. Consequently, the decay of the signal is slowerand the spectra is narrowed.The effect on the µSR signal is similar and can be discussed also in the light ofthe previous chapter (Fig. 2.20). The striking difference of muons with respectto nuclei is that T1 and T2 processes drive simultaneously the decay. Whenboth transversal and longitudinal components of the local fields fluctuate fast,the measured decay is a exponential and λ is given by Eq. (2.71). On thecontrary, when slow fluctuations of the transverse fields are present, the signalis a product

P (t) = P (0)GDKT (t)e−λt (2.80)

where GDKT is the dynamical Kubo-Toyabe function. For very slow fluc-tuations it reduces to a static Kubo-Toyabe with the 1/3 tail decaying as1/3 exp(−(2t/3τc)) (in zero field). Is it evident that if the longitudinal relax-ation is much more faster then transverse relaxation, then only the exponentialdecay of function (2.80) is detected (Fig. 2.20).The intermediate regime between very fast and very slow fluctuations is com-plex. It is formally described in the approximation of a diffusive correlationfunction of the local fields (the same assumed for BPP) by the Abragam func-tion

G(t) = G(0) exp

−〈∆h2

z〉τ 2c

[exp

(− t

τc

)− 1 +

t

τc

], (2.81)


Figure 2.20: Muon relaxation function for several muon hopping rates ν. ∆/γµ

is the width of the Gaussian distribution of the local fields. The curve for ν = 0corresponds to the zero-field Kubo-Toyabe relaxation function of Eq. (2.60) [78].

fitting both the NMR signal and the µSR transverse decay.Following the previous description, it is straightforward that the measuredtransverse relaxation time T2 cannot be univoquely considered as the decaytime of a simple-shaped signal. In the case of the FID, the signal dimin-ishes for the contemporary action of field inhomogeneities, of intrinsic staticand dynamical processes and of the recovery of Mz. However, neglecting thecontributions of the extrinsic inhomogeneities and restricting to the fast fluctu-ation limit decaying with T

′2, in the frame of the previous chapter, T2 is defined

by means of the spectral densities

1

T2

=1

T ′2

+1

2T1

=γ2

2Jzz(0) +

γ2

4J±(ωL). (2.82)

2.5.3 Some model systems: from localized spins tometals

When the random field approximation cannot be applied, the correlation func-tion must be specialized. It happens for example when the local fields aregenerated by localized spins correlated through collective excitations. If thenuclei and the electron spins interact by means of the hyperfine interaction,then the effective fields are

~Hj(t) =∑

i

Aji~Si(t), (2.83)


where j refers to nuclei and i to electron spins. Introducing the harmonicexpansion over the first Brillouin zone

~Si(t) =1√N

∑

~q

ei~q·~Ri ~S~q(t) (2.84)

with Ri the position of ~Si in the direct space and N the total number of spins,after some calculations one obtains [83]

〈Hα(t)Hα(0)〉 =1√N

∑

~q

|∑

i

(Aαxji + Aαy

ji + Aαzji )ei~q·~Ri|2〈Sα(~q, t)Sα(−~q, 0)〉.

(2.85)Definitely, defining the dynamical structure factor

Sαβ(~q, ω) =

∫ ∞

−∞〈Sα(~q, t)Sβ(−~q, 0)〉eiωtdt, (2.86)

the spin-lattice relaxation becomes

1

T1

=γ2

2N

∑

~q

[|A~q|2~S(~q, ωL)]⊥. (2.87)

|A~q|2 is called the form factor and depends on the symmetry of both the hy-perfine tensor and the lattice; in some cases it reduces to a constant A2.For the fluctuation-dissipation theorem the dynamical structure factor is pro-portional to the imaginary part of the electronic susceptibility

Sαβ(~q, ω) =χ”

αα(~q, ω)

1− exp(− ~ωkBT

). (2.88)

and in the high temperature limit

1

T1

=γ2

2N

kBT

~A2

∑

~q

[χ”(~q, ωL)]⊥ωL

. (2.89)

Since the previous is a general expression, it can be used to model the behaviourof many systems.For example in a 3-d Fermi gas, having an s-state contact interaction betweenthe nuclear and electron spins, 1/T1 is linear in T according to the Korringalaw

1

T1

=

(16

3

)2

π3~2γ2µ2B(|ψ(0)|2)FSD2(EF )kBT (2.90)

and is proportional to the density of states at the Fermi level D(EF ).

2.6. Summarizing remarks 53

2.6 Summarizing remarksThe suitability of magnetometry, NMR and µSR in studying the static anddynamic electronic properties of materials has been reviewed in this chapter.As concerns the static properties, primarily DC magnetometry measures thestatic bulk susceptibility χ(~q = 0, ω = 0) while analyzing the NMR spectraone estimates the nature and the strength of the local static microscopic in-teractions between the nuclei and the environment. The nuclei can interactwith other nuclei, electrons and probe a quadrupolar interaction with the localelectric field gradient. Interactions among nuclei give rise to a homogeneousbroadening of the NMR lines of the order of some kHz. Nucleus-electron hy-perfine interactions can induce sizeable local fields, introducing a shift of thelines or possibly an asymmetric broadening. The both are proportional to theelectronic susceptibility, therefore the results of the SQUID measurements canbe used in the study of the NMR spectra to obtain an estimate of the effectivehyperfine coupling between nuclei and electrons. The quadrupolar interactionfor nuclei with I > 1/2 originates 2I − 1 lines due to satellite transitions.From the frequency splitting of the lines one can estimate the value and thesymmetry of the EFG probed by the nuclei. In addition, observing the staticrelaxation of muons studied in µSR zero field experiments one can evaluate thewidth and the shape of the local field distribution, which can be compared tothe one probed by the nuclei and deduced by the study of the NMR spectra.As concerns the dynamic properties, measurements of the NMR spin-latticenuclear relaxation rate 1/T1 yield directly the observation of important quan-tities. 1/T1 is proportional to the q-integrated imaginary electronic suscep-tibility (2.87-2.89), therefore its behaviour as a function of temperature andfield allows to recognize the nature of the spin dynamics of the investigatedsystems, recurring to appropriate models (itinerant electrons, paramagnets,long-range order magnetism...). The measurements of muon relaxation in lon-gitudinal geometry experiments is in principle analogous to the measurementof NMR 1/T1. However, T1 and the spin-spin nuclear relaxation time T2 canbe measured separately by means of various spin echo-techniques, while bothtransverse relaxation and longitudinal relaxation are probed by muons in lon-gitudinal geometry experiments. On the other hand, µSR in transverse geom-etry, where the muon polarization precesses about an external field, can beperformed to acquire unique information on the formation of muonium, veryuseful to perform spectroscopic and dynamic studies of molecular compounds.Moreover, both NMR and µSR signal and spectra are very sensitive to the rel-ative motion between the local field sources are nuclei (muons). In presence offast motions as diffusion, rotations and vibrations nuclear and muon relaxationcan become very slow, as the consequence of a narrowing of the distribution oflocal fields. From the analysis of NMR spectra and muon relaxation one canextract for example the correlation times of the these motions as a function oftemperature.


Chapter 3Synthesis, structure andstoichiometric analysis ofphthalocyanine basedcompounds

The synthesis of alkali-doped metal phthalocyanines is an arduous task, stillat a preliminary stage. While the technique of ionic implantation has beensuccessful in the past years for the doping of metal phthalocyanine thin films[42, 43, 44, 45], the crystallization of stable and homogeneous bulk compoundshas been attained only recently by means of liquid phase reaction methods.Through these methods the pristine phthalocyanine combines with the ionscoming from the dissociation of alkali salts or from the pure alkali metal in so-lution. The practical execution of the synthesis is not trivial for several reasons:phthalocyanines and their derived compounds are very sensitive to humidityand oxidizing agents, therefore they must be handled under constant controlledatmosphere; furthermore, the choice of the materials is tricky since phthalo-cyanine is insoluble in most of the common solvents and the final product canbe degraded on heating and during separation. However, the unquestionableadvantage offered by the liquid phase reaction is the achievement of homoge-neous samples and the precise knowledge of the alkali content introduced inthe solution, with the consequent possibility to decide the target stoichiome-try before the reaction. This makes this method more reliable with respectto solid state reactions, based on the exposition of the bulk phthalocyanine toalkali vapours under inert atmosphere and controlled gradient temperatures,reported in several works [42, 84]. In fact, in the samples produced by meansof these methods the arising of concentration gradients and the coexistence ofdifferent phases seem hardly avoidable and the control of alkali diffusion andconcentration during the reaction is critical as well. In this chapter some detailsabout the synthesis methods will be presented, together with structural and

55

56 3. Synthesis, structure and stoichiometric analysis of phthalocyanine based compounds

morphological information concerning both AxMPcs and TbPc2[TBA] com-pounds. In addition, some structural aspects of the doped AxMPcs will bediscussed in the light of µSR measurements performed at ISIS on NaxZnPccompounds. This study relies on the observation of muonium in these ma-terials, a bound system of a muon and an electron often generating radicalswhen reacting with molecular compounds [79]. The comparison between themuonium states in pristine ZnPc and in the doped compound will supply use-ful indications for the evaluation of the dopant sites. Remarkably, both X-raystructural studies of LixMnPc and µSR measurements in NaxZnPc suggest theintercalation of alkali ions for x=0..4 in intrastack positions, strongly boundto the pyrrole-bridging nitrogens of the Pc rings.

3.1 Synthesis of AxMPcs

β-LixMnPc , β-LixZnPc and β-NaxZnPc powders were synthesized by meansof liquid phase reaction and deposition methods. In the following the syn-thesis methods adopted for the production of the different compounds will bedescribed. It must be mentioned that the properties of NaxZnPc compoundswill not be explicitly treated in this thesis. However, the observation of someinteresting µSR and AFM results in the NaxZnPc compounds (section 3.5) hassuggested to extend this illustration to the entire family of AxMPcs, in orderto provide a complete morphological and structural characterization.β-LixMnPc were synthesized at the Institute for Materials Research of the To-hoku University in Japan, following the protocol published by Taguchi et al.[28]. The liquid-phase reaction was executed in an Ar-filled glovebox using stoi-chiometric amounts of n-butyllithium and MnPc. As-purchased MnPc powderssingly sublimed at 450-500 C (ca. 30 mg) were dispersed in n-BuLi/hexanesolutions (ca. 3 mL) of the appropriate molarity for the intended doping valuex (0 < x < 4) for 1 day. All solutions used were obtained by dilution of a 1.6 Mn-BuLi/hexane solution. Samples doped to saturation instead were preparedby means of the undiluted solution. The powders were collected by filtrationand were subjected to annealing procedures in vacuum at 200-300 C for 2days.Phase purity was confirmed by laboratory powder X-ray diffraction. Li con-centrations in the products were determined by Inductively Coupled Plasma(ICP) spectroscopy. The excellent agreement between the measured and nom-inal doping values confirmed that all Li from the reactant n-BuLi solutionsis essentially intercalated in the course of the reactions into the MnPc hostlattice. For the LixMnPc samples prepared from undiluted solutions the ICPanalysis showed a Li concentration of about 4, implying that the saturationcomposition of these phases has the stoichiometry Li4MnPc.β-LixZnPc were prepared following the same protocol, with some modificationsreported here by courtesy of Giorgio Zoppellaro, working at the Institute ofNanotechnology, Forschungszentrum, Karlsruhe, Germany. ZnPc was freshly

3.1. Synthesis of AxMPcs 57

synthesized and washed with CH2Cl2 (10 mL) followed by n-hexane (300 mL).This allowed the separation of the ZnPc by several by-products of the ini-tial synthesis. After sonication and cooling at 4 C, a fine bluish precipitateformed slowly, leaving a brown-greenish solution which was filtered off whilethe residue was collected. The washing procedure was then repeated fourtimes until the washing solution became almost colourless. The bluish pow-der collected (∼600 mg), which contained the ZnPc product plus unmetalledphthalocyanine ring, was initially dried under reduced pressure overnight inmoderate vacuum (70C, 40 mbar); then ZnPc was purified by sequential vac-uum sublimation, collected and stored under nitrogen. After this proceduresthe final fraction of unmetalled phthalocyanine ring can be estimated as nearly4% of the total product introduced in the n-BuLi solution.The sublimed ZnPc (∼ 100 mg, 0.173 mmol, 1 eq) was charged in a Schlenkglassware (25 mL) and suspended in 10 mL of dry and nitrogen saturated n-hexane. A solution of n-BuLi (∼5 mL) of the appropriate molarity for thetarget doping value x (1.5≤ x ≤4) was carefully inserted by a syringe keptunder nitrogen into the vessel. All the solutions employed were prepared bydilution of a stock solution of commercial 1.6 M n-BuLi in hexane, which hasbeen previously titrated with 1.00 mmol dipheynylacetic acid in 8 mL dryTHF to calculate the effective concentration of n-BuLi as [(mmols DPAA) /(ml n-BuLi)]. The so formed suspension was stirred for one day and then leftwithout stirring for 3-4 days. The bluish residue was collected using a frit andwashed on frit with dried and oxygen free hexane (∼10 mL). The so formeddoped LixZnPc powders were dried under moderate vacuum (52 mbar) at 100C for one day and then transferred quickly under nitrogen into NMR tubes(∼ 3 - 3.5 cm length) which contained 60 - 90 mg of the doped LixZnPc.The doped compounds inside the tubes were then subjected to annealing pro-cedure with heating at 240 C under vacuum (6.8× 10−2 mbar) for 3-4 days.Longer annealing (4 days) was required for the fully doped sample (x = 3.8).This heating procedure ensures formation of the β polymorph. Upon satu-ration with nitrogen, the tubes were quickly frozen (77 K) in N2 liquid bathand by careful keeping N2 stream on the quartz walls the tubes were finallysealed with acetilenic flame, avoiding sample heating. Li concentrations in theproducts were determined by means of both ICP spectroscopy and NMR mea-surements, as illustrated in section 3.3.β-NaxZnPc were prepared according to a protocol established by A. Bertasaand P. Ghigna at the Department of Physical Chemistry ’M. Rolla’ of the Uni-versity of Pavia. As purchased ZnPc powders (0.1083 g, Aldrich, yield 97%)and metallic Na (0.5 g, Aldrich, stored under mineral oil, yield 99.95%) werepoured in a glass flask containing dry THF (300 mL, Aldrich, yield 99.9%),operating in a glovebox (MBRAUN Unilab, H2O< 10 ppm, O2<1 ppm). Nahad been previously modelled and reduced to thin foils to increase the surfaceexposed to the solution. THF had been distilled from sodium/benzophenoneketyl under nitrogen prior to use and dried in CaH2 (Aldrich, 95%). The so-


lution was kept under magnetic stirring for 4 days in the glove box and wassubsequently mounted on a Rotovapor, working under nitrogen flux and keep-ing the bath temperature under 40. This precaution was adopted to preservethe integrity of the product, since at higher temperatures the final residue wasobserved to turn to violet, very similar to the color of the pristine ZnPc. Afterdrying, a blue-green residue was collected and transferred quickly under vac-uum into NMR tubes to prevent oxidation.Several methods were tried to separate the product from the spurious residues.The product resulted hardly soluble and it decomposed under contact withseveral supports, manifestly changing color. However, it was concluded fromthermogravimetric analysis that on heating up to 120C in argon atmospherethe residues of THF can be removed from the sample without evident mor-phologic variations. The effective formation of NaxZnPc phase was verified bymeans of UV-absorption measurements. The comparison between UV spec-tra of the pristine ZnPc and of the reaction product, both solved in distilledTHF, showed that in the latter absorption is shifted to higher wavelengths.Analogue results are reported in literature for closed-shell reduced metal ph-thalocyanines [33] where this shift is ascribed to the electronic excitations ofthe 2eg LUMO electrons. Hence, since the typical absorption bands of isolatedmono and dinegative ions ZnPc−,2− are not detected in the recorded spectra,this low-energy absorption must be due to the formation of NaxZnPc , wherethe reduction of ZnPc is achieved by charge transfer from Na. The Na contentin NaxZnPc compound was calculated by means of Flame Atomic Absorptionspectrometry. From the reduction of the light intensity due to absorption bythe atomized Na, an amount of Na equivalent to 8.6% of the total reactionproduct was estimated. This corresponds to a molar fraction per ZnPc unit ofnearly x ' 2, yielding Na2ZnPc.AxMPc produced by means of the aforementioned protocols have colours rang-ing from black, as in the case of LixMnPc, to blue, in case of LixZnPc andNaxZnPc. NaxZnPc were inspected by SEM and AFM microscopy performedrespectively at the Institute of Chemical Physics ’M.Rolla’ of the Universityof Pavia and at the Institute of General Physiology and Biological Chemistryof the University of Milan. The images acquired by the two techniques atdifferent resolutions are shown in Fig. 3.1 and 3.2. It can be noticed thatthis compound crystallizes as extremely thin needles having length of the or-der of some µm, width of 40-60 nm and thickness of 1-2 nm. This shape is acommon feature of metal phthalocyanines crystals (see for example images ofpowder α-FePc in Ref. [22]) and other one-dimensional molecular compoundsand fascinates particularly in the perspective of the technological applicabilityof these systems as nanowires.

3.1. Synthesis of AxMPcs 59

Figure 3.1: SEM image of Na2ZnPc grains. Enlargement: 14.1kX.

Figure 3.2: AFM image of Na2ZnPc grains.


3.2 Synthesis and structure of TbPc2[TBA]

TbPc2[TBA] powders were prepared at the Institute of Nanotechnology,Forschungszentrum, Karlsruhe, Germany. The synthesis of TbPc2[TBA] wasperformed introducing several modifications of the standard protocol and willbe reported in future publications [85]. In a round bottomed flask were mixeddicyanobenzene (32.0 mmol, 4.1 g), terbium acetoacetate (2.0 mmol, 0.64 g) inhexanol (50 mL), DBU (16.0 mmol, 2.4 mL), and a catalytic amount of benzyl-triethylamoniumchloride (30 mg, 0.13 mmol). The reaction was performed ina microwave oven using an applied microwave power of 450 W for 10 min fol-lowed by longer irradiation (15 min) at lower energy (300 W). By employingmicrowaves, side reactions and especially formation of large amount of emptyphthalocyanine ring appeared markedly reduced. The blue mixture was cooleddown to room temperature (r.t.) and kept in cold (4C) overnight. The densesolution was filtered using a glass frit and the residue collected. In order toremove as much n-hexanol as possible, the precipitate was further washed onfrit with portions of cold n-eptane (5×50 mL). Then the product TbPc−2 (blue)and TbPc2 neutral (green) were extracted from the residue using mixture ofMeOH/ CH2Cl2 (200 mL, 50 mL, 4C), while most of the phthalocyanine ring(violet, very shining needles) remained in this way on the filter. The solutionwas evaporated to dryness under reduced pressure which afforded formationof an oily blue-green residue. The final product was washed by addition ofCH2Cl2 (50 mL) followed by hexane (400 mL), as described for the ZnPc syn-thesis in section 3.1.Purification of TbPc2

− was performed by nitrogen saturated flash-chromatogra-phy on Al2O3 (basic) column, where at first CH2Cl2 was used as eluent toremove remaining yellowish side products followed by CH2Cl2/MeOH (2/1,v/v). The product TbPc−2 was then eluted by addition of sodium methanolate(NaOMe) using only MeOH as eluting solvent. The blue fraction collected wasquickly evaporated to dryness under reduced pressure and kept under nitrogen.Further purification of TbPc−2 was carried out by size-exclusion chromatogra-phy using Sephadex LH20 with MeOH/CH2Cl2 as eluent mixture (1/1, v/v).Residual bluish compounds remained trapped in the column. The blue solu-tion was quickly evaporated under reduced pressure affording the negativelycharged product in form of very hygroscopic dark-blue powder (0.56 g, 0.47mmol, yield 6%) which still contained one solvent molecule (n-hexanol) andwhere most probably one acid water molecule acts as counter-cation. Thenstabilization of TbPc−2 was achieved by addition of the cation (TBA)+.Tetrabutylammonium-bromide (0.47 mmol, 0.153 g) was added to the bluesolid TbPc−2 together with 50 mL of MeOH; the resulting deep blue solutionwas then stirred vigorously under nitrogen for 5 min. Most of the solvent (∼45 mL) was removed under vacuum and cold hexane was added (100 mL). Ablue precipitate immediately formed, which was collected and dried in vacuum(0.1 mbar) at room temperature for one day and then stored under nitrogen.Magnetically diluted TbPc2[TBA] samples (Br concentration is omitted for

3.2. Synthesis and structure of TbPc2[TBA] 61

Figure 3.3: Local structure of TbPc2[TBA]×3 [TBA]Br × 3 H2O and crystal packingof the complex [85].

simplicity), indicated as TbPc2[TBA]10 and TbPc2[TBA]144, were prepared byaddition of a solution of tetrabutylammonium-bromide, previously dissolvedin MeOH with the desired stoichiometry, to the undiluted TbPc2[TBA] pow-der sample, followed by sonication of the solution in a water bath (5 min) atr.t., solvent evaporation under reduced pressure and high-vacuum drying (10-2mbar). The powders used for NMR and SQUID measurements were trans-ferred in NMR tubes, that could contain about 70-90 mg of compound, andwere closed following the same method explained for the preparation LixZnPcsample holders. The samples were always kept in fridge to prevent possibledegradation at ambient temperature.The structure of TbPc2[TBA] was determined by means of x-ray diffraction oncrystals of complete stoichiometry TbPc2[TBA] ×3 [TBA]Br × 3 H2O. Thisdetails will be published in [85]. The system crystallizes in the monoclinic P21/c

space group, in analogy with transition metal phthalocyanines, but with fourTbPc−2 per unit cell. The calculated cell parameters were a = 13.807(2) A, b= 37.015(5) A, c = 25.127(5) A and β = 100.764. The [TBA]+ complexes andthe TbPc−2 anion alternate in stacks along the b axis. The minimum distancebetween neighbouring Tb3+ ions, d = 12.353 A, occurs along the c axis. Fig.3.3 shows the structure prospected along several directions [85].


3.3 Stoichiometric analysis of AxMPcs via NMRmeasurements

NMR measurements can be suitably performed to evaluate the content of al-kali atoms per phthalocyanine molecule in the doped AxMPcs compounds.The method essentially consists in the comparison between the intensity of theNMR signal of nuclei of known molar fraction with the NMR signal comingfrom the dopants in the same sample. In particular, the phthalocyanine macro-cycle contains 16 peripheral protons, therefore, having 16 protons per formulaunit, the intensity of their signal can be measured to solve the proportion

IH : IA = 16 : x, (3.1)

where I are the intensities, and infer x, the unknown alkali content. In thefollowing the experimental procedure adopted to estimate the lithium contentin LixZnPc will be described.First, the NMR signals of the two nuclei can be compared whenever they areacquired at the same frequency or at the same field. Under one of these twoconstraints the ratio between the intensities will be proportional to the relativesensitivity R of the two isotopes, calculated at constant frequency or constantfield. Since 1H and 7Li have different gyromagnetic ratios with a relative ratioγH/γLi = 42.576 MHz·Tesla−1/16.455 MHz·Tesla−1=2.58, when working atfixed field the measurements must be performed in a different frequency range.Then, since the properties of the r.f. field H1 supplied by the NMR apparatuscan change with frequency, the signals were acquired at the same frequencyin order to attain a comparable irradiation of the two nuclear species. Inparticular most of the measurements were performed with an electromagnetreaching the maximum field H = 1.4 ÷ 1.6 Tesla. Therefore, as the field waskept at the highest achievable intensity to enhance as possible the signal of7Li, the working frequencies, corresponding to ω = γLiH, were about 23-27MHz. Accordingly, to achieve the same irradiation frequency on 1H , the fieldwas rescaled to ω/γH ' 5400÷6300 Gauss.Now, provided the two nuclei are irradiated at the same frequency, their signalswill depend on

• the number of nuclei in the sample

• the number of scans of the measurement, since the signals are added andaveraged after each scan

• the sensitivity R

• the spin-spin relaxation, reducing the intensity at finite times

It is evident that the signal amplitude should be evaluated for t → 0 in orderto avoid underestimates of the signal caused by transverse relaxation. How-ever, the FID is affected at short times by the instrumental dead times, ringing

3.3. Stoichiometric analysis of AxMPcs via NMR measurements 63

0 20 40 60 80 100 120 140 160 180 200

104

Ech

o am

plitu

de (

arb.

units

)

2τ (µs)

Sample 1 ν= 26.392 MHz

7Li, 600 scans 1H, 100 scans

Figure 3.4: Echo amplitude as a function of t = 2τ in sample 1 for 1H and 7Li ,obtained by means of conventional spin echo techniques.

and fast spurious decays, so that a simple 90 pulse sequence is not an afford-able method to deduce the signal amplitude. Therefore, for each nucleus theexperimental decay of the echo amplitude E(2τ) was measured by means ofstandard spin echo techniques and then the value at t = 0 was extrapolatedby fitting methods.An example of these measurements on 1H and 7Li in a sample of LixZnPc areshown in Fig. 3.4. It can be noticed that the 1H echo amplitude has a Gaus-sian decay with T2=40 µs, while 7Li echo amplitude has a slower exponentialdecay with T2=110 µs. From the fits shown by the solid lines the E(0) valueswere obtained, E(0)H = 30080±1284 for 1H and E(0)Li = 39123±977 for 7Li.Now, the complete expression for the proportion (3.1) between the 1H and 7Lisignals will be given by

IH : IA = 16 : x = (E(0)H/NH) : (E(0)Li/NLi ∗R), (3.2)

where NH and NLi are the total scans for each experiment and R=1.94. Sub-stituting the experimental values in the previous expression, for this sampleone obtains x=1.9 ±0.1. This is close to the value obtained by ICP analysis.All the samples were analyzed with the same procedure and an overall agree-ment was achieved with the results obtained by ICP or from the Li titolationon the residues of the synthesis. This comparison is summarized in table 1.The samples LixZnPc will be labeled from now with the x average values re-ported in the Table 3.1. An analogue procedure was executed for the LixMnPc


samples yielding x=0.5±0.01 and x=2±0.1.

NMR ESTIMATE ICP ESTIMATE AVERAGE xor Li Titolation

Sample 1 1.37± 0.1 1.6± 0.12 1.5±0.1Sample 2 1.9± 0.1 1.8± 0.12 1.85± 0.1Sample 3 2.45± 0.2 2.2÷ 2.3 2.35± 0.1Sample 4 2.7± 0.1 2.6÷ 2.8 2.7± 0.1Sample 5 3.62±0.3 3.8÷ 3.9 3.75± 0.3

Table 3.1: Comparison between the Li contents in several samples of LixZnPc eval-uated by means of NMR and ICP.

3.4 Structure of LixMnPcThe structure on LixMnPc was carefully analyzed by Taguchi et al. in Ref.[28]. In this section the important experimental results illustrated by this workwill be summarized in order to give a useful basis for the discussion of the prop-erties of LixMnPc and of the other doped compounds, investigated with theaim of magnetization, NMR and µSR measurements.In Ref. [28] the cell structure of LixMnPc was determined by means of high-resolution synchrotron X-ray powder diffraction measurements, treated withRietveld analysis programs. X-ray diffraction profiles readily revealed remark-able similarities with that of pristine β-MnPc for x varying between 0 and 4,implying that these materials are single phase and form an isosymmetric struc-tural series. The β-form is usually adopted in the bulk MPc crystalline solidsinstead of the α-form, which in turn is frequently encountered in thin films.The β-polymorph of MnPc has a monoclinic structure (a = 14.764 A, b = 4.764A, c = 19.422 A and β = 120.70; space group P21/c) in which the planar met-allomacrocycles form slip-stacked one-dimensional columns along the short baxis. In Fig. 3.5.a [28] the intrastack distance d1 (3.16 A), the slipping dis-tance d2 (3.55 A) and the slipping angle φ (48.2) in MnPc are marked. Theslipped packing motif of MnPc molecules is such that two pyrrole-bridging Nbr

atoms of each Pc unit lie at 3.16 A almost directly above or below the Mn(II)ions of its two nearest neighbors. This gives rise to angles (Mn-Nbr...Mn)∼93.4 which determine a superexchange pathway for the intrachain correlationbetween nearest neighbor Mn ions. Adjacent stacks are inclined by symmetryat an angle of 90 and are well separated from each other at a distance of 10.00A, sensibly larger than the typical intrachain separations.While the overall symmetry of the cell is preserved, the unit cell volume andparameters were observed to vary continuously upon doping. The unit cell vol-ume first increases marginally up to x ∼1 but then continuously and rapidly

3.4. Structure of LixMnPc 65

a

Figure 3.5: The slip stack arrangement of MnPc molecules in the pristine (a) anddoped Li1.5MnPc compound (b). The green spheres indicate the possible sitesoccupied by Li+ [28].

inflates, eventually approaching an overall increase of 4.5% for x ∼ 4. Thisprovides an unambiguous signature of Li intercalation into empty spaces ofthe structure of pristine β-MnPc without significantly perturbing the basicstacking architecture. From the Fourier analysis at different stoichiometriesand from structural refinements, it was inferred that, for x between 0 and 2,4e (0.09, -0.18, 0.88) sites of the unit cell are populated by the dopants. Thesepositions are located in the intracolumnar spacing between successively stackedMnPc units and correspond to interstitial sites, lying 2.0 A directly above (orbelow) the same pyrrole-bridging Na atoms of the Pc units that also coordinateto Mn(II) ions of neighboring MnPc molecules. In Fig. 3.5.b the slip stackedmotif is shown for Li1.5MnPc with the modified parameters d1 (3.18 A), d2

(3.84 A) and the slipping angle φ (50.3). It can be noticed that the occupa-tion of the green sites implies mainly the increase of the slipping distance d2

along the stacking direction and φ, while d1 results nearly unperturbed. Fornon integer doping the Li+ ions are disordered and reside in the intrastackspace, bonding strongly to one of the two pyrrole-bridging Na atoms (red dot-ted lines). Upon doping also the intermolecular Mn...Na distance is increasedto 3.21 A and Mn-N...Mn angle opens up to 98. Conversely, while the slippingdistances are enlarged to allow the accommodation of the dopants, the lack ofoff-axis counterions leads to a contraction of the separation of adjacent MnPcchains.For x between 3 and 4 Rietveld refinement of the diffraction data pointed tothe population of additional intrachain sites at the 4e (0.68, 0.22, 0.27) po-sitions of the unit cell. In this geometry Li+ is strongly coordinated at 2.0


Figure 3.6: (a) The slip stack arrangement of MnPc molecules in Li3MnPc and(b) in Li4MnPc. The green, pink and dark violet spheres indicate respectively Li+

bonding to Na, Nb sites and Li+ occupying interchain interstitial sites at saturationdoping [28].

A to one of the other pyrrole-bridging Nb atoms (at 90 to the Na atoms) ofthe Pc macrocycles (inset of Fig. 3.6). Thus, for x > 2 the Li+ defects arealmost uniformly disordered over the corners of a [Li4] rectangle defined bythe 4 pyrrole-bridging atoms, sketched in Fig. 3.6.a. The distances for x = 3evolve to d1=3.25 A, d2=3.87 Aand the slipping angle φ ∼ 50, denoting aslight modification of the intrastack distance to allow a larger number of Li inthe intrastack spaces.At saturation doping of MnPc, for x ∼ 4, diffraction profile evidences a strongrenormalization of the unit cell constants, despite retaining the same mon-oclinic crystal structure. In this phase, however, the separation of adjacentchains with increasing Li content is suddenly increased to 9.99 A, comparableto that in pristine β-MnPc. This provides the signature that Li is now alsoaccommodated at interstitial holes between the stacks (Fig. 3.6.b). The Ri-etveld refinements identified this position as the high-symmetry 2b (1/2, 0, 0)site, which presents no sterical crowding and is fully occupied.From this analysis it is deduced that in LixMnPc compounds the dopants al-locate preferably in positions allowing a strong bond to the pyrrole-bridging Natoms of the MnPc rings. The accommodation in the interstitial sites betweenthe stacks is attained only after the complete population of the sites located atthe corners of the [Li4] square in the space defined by adjacent planar MnPc

3.5. Muons as probes of doping in AxMPcs 67

molecules (Figure Fig. 3.6.a). The occupation of the interchain interstitialsites is reasonably unfavoured at low doping, since in this configuration anydirect bonding interaction between the dopants and the macrocycles is stronglyinhibited.The response of the one-dimensional MnPc packing architecture to the Li+ in-sertion is very different from that ubiquitously encountered in hole-doped MPcphases. In the latter compounds the pristine material changes in such a waythat the MPc units form nonslipped staggered stacks and separate chains ofcounteranions, such as I−, extending parallely. Instead, in LixMnPc the MnPcunits allow for the accommodation of the Li+ ions by increasing their slippingdistance, so that no independent columnar structure is formed by the dopants.At the moment, no structural analysis of the LixZnPc compounds is available.However, in this work they will be considered as isostructural compounds ofLixMnPc. ZnPc crystallizes in the β form with cell parameters a = 19.274(5)A, b = 4.8538 (15) A, c = 14.553 (4) A, β = 120.48, labeled within the sym-metry space group P21/a [86] (corresponding to P21/c for exchange of the aand c axes). One sees that the proportion between the different lengths is sim-ilar to the one found in MnPc, suggesting a similar pattern of intermolecularcontacts. The variation of the absolute values of a, b, c is expected to accom-pany the overall expansion yielded in the phthalocyanine ring to accomplishthe introduction of the Zn+2 ion, having a larger closed shell core. The struc-tural analogy of the pristine MnPc and ZnPc compounds suggests that a lightdopant as lithium should easily intercalate in the same way in the intrastackinterstices and set in very analog cell positions. Turning instead to heavieralkali dopants, as Na or K, the situation could be different from the case ofLi, since the larger core of Na or K should require larger deformation of thepristine structure to afford the insertion of these atoms along the chain. Evenif an X-ray study of these compounds is not yet available, some µSR mea-surements on NaxZnPc compounds provided some hints on this issue. Thesewill be discussed in the following section and it will be claimed that also inNaxZnPc Na likely occupies intrastack positions.

3.5 Muons as probes of doping in AxMPcs

As already illustrated in chapter 2, the muons act as local probes for bothspectroscopic and dynamics studies in a large variety of materials, from in-organic semiconductors to polymers. In particular, in organic materials asphthalocyanines they can form muonium since, during the process of ther-malization after implantation, they can easily ”pick up” an electron from themolecules. This process generally cannot be reviewed as a simple ionizationbut involves a reaction between muonium and the molecule which originates anew compound, a muoniated radical. These radicals are paramagnetic and thehyperfine interaction between the muon and the electron is generally stronglyanisotropic and much smaller than the one of the muonium in vacuum. This


allows to observe distinctly the signal of muonium in a typical TF (transversefield) µSR time spectrum, since the precessional frequencies range from tens tohundreds MHz and are thus resolvable by a common µSR spectrometer. Thehyperfine interaction of the radical is determined by the properties of the localcharge distribution and by the nature of the molecular bonds. The muon likelyattaches to specific sites where the molecular bonds are unsaturated; doublecarbon-carbon bonds are typically broken to allow the addition of the µ+ andone electron remains unpaired. For example, in the reaction with C6H6 thecyclohexadienyl radical C6H6Mu is formed and the unpaired electron is delo-calized over the benzene ring. In other compounds, like CH3, the added muonis more covalently bonded to a single saturated carbon; then, in CH2Mu theunpaired electron is more localized on the unlabelled carbon and the radicalhas an electronic doublet ground state [79]. Hence, muonium is a very versatilespectroscopic probe. Each muonium and radical state can be characterized,having a typical hyperfine constant and, thus, a typical precessional frequency,detectable, for example, by means of a Fourier transform spectrum of the µSRsignal. In addition, muonium also probes local spin dynamics of localized anddelocalized spins, diffusion, molecular motions, scattering with phonons andcharges etc.. Hence, the study of the radical states forming in organic mate-rials is a powerful tool to envisage the location of dopants and to study theirpossible interaction with the environment at the same time.These investigations have been already performed in the pristine phthalocya-nines [87, 88, 89] and identified three muoniated radical states in ZnPc, H2Pcand CuPc by means of TF experiments in the high field regime, where thetotal muon polarization is achieved (this will be explained more clearly later).In presence of a field perpendicular to the initial muon polarization, the muonspins precess at two different frequencies ν12 and ν34, corresponding to the tran-sitions between the energy levels of the muonium, split by both the hyperfineand Zeeman interactions, conserving the component of the electron spin alongz (Fig. 3.7, see also section 3.3 in chapter 2). One has A = ν34 ± ν12, wherethe ± takes in two account the reversal of the levels E1 −E2 above their levelcrossing. Fig. 3.8 shows the frequency spectrum measured in ZnPc, showingdistinctly three pairs of precession frequencies corresponding to three differentmuonium states. I-II are two paramagnetic states with hyperfine interactionsof the order of 110-150 MHz while III has a smaller hyperfine interaction (about25 MHz). The line at 54.2 MHz corresponds to muons in a diamagnetic envi-ronment and it is broadened by a distribution of spectral density correspondingto paramagnetic states with a very low hyperfine interaction. The overall for-mation probability of these states in ZnPc is very high, nearly 90%, with mostof the radicals forming in the state III.The assignment of the muonium sites was made by means of DFT calculations,yielding the results in Fig. 3.9. The most stable states I and II originate fromthe addition of muonium to the outer benzene rings of the molecules, whilestate III was tentatively ascribed to the muonium anchored to the bridging-


x=B/B0 x=B/B0

Figure 3.7: The hyperfine energy level diagram for anisotropic muonium, for thefield parallel (θ = 0) and perpendicular (θ = 90) to the symmetry axis respectively.The dashed lines are high-field asymptotes. In the used notation B0 = ~ω0/(gµµµ−geµB) [77].

Figure 3.8: Fourier power spectrum of ZnPc obtained in transverse field geometryat 0.4 T, 500 K, taken from Ref. [89]. One distinctly sees three pairs of lines markedby the roman numerals I,II,III. The sharp line at 54.2 MHz corresponds to muons ina diamagnetic environment.


Figure 3.9: Molecular structure of ZnPc with the indication of the possible muonaddition sites from Ref. [89]. The electronic calculations yield stable positions forsites I and II only.

nitrogens of the Pc ring. Authors also consider that the small hyperfine con-stant relative to all these radicals, in particular III, indicates a strong electrontransfer to the molecule, leaving a small residual electron density (1-4%) at themuon sites. These evaluations are interesting in relation to the X-ray struc-tural studies conducted on LixMnPc by Taguchi et al. [28] reviewed in section3.4. They suggest that for low to moderate doping, Li+ intercalates withinthe molecular stacks directly above (or below) the pyrrole-bridging N atoms ofthe Pc units, while at saturation doping lithium accommodates in interstitialsites between adjacent chains (Fig. 3.5 and 3.6). In the light of the analogyof the two studies, the muonium sites with the maximum probability of for-mation should be actually the same sites occupied by the dopants and thusµSR in metal phthalocyanines could give indications about several issues. Onone hand, at ground level, if the aforementioned evaluations were correct, thecomponent of the µSR signal ascribed to muonium should decrease in dopedcompounds due to the occupation of muon sites by the dopants themselves,giving strict evidence of alkali doping. On the other hand, the study of localdynamics at the muonium radicals should supply information enriching the ob-servations coming from NMR on 7Li nuclei. The presentation of the followingexperimental data will clarify these aspects.NaxZnPc powders were prepared by means of the liquid phase method de-scribed in section 3.1. The powders were rapidly compacted, put on silver


sample holders and kept under controlled atmosphere in the days precedingthe experiment to avoid air contamination. The measurements were performedon the ARGUS and MuSR beamlines at ISIS with the Variox cryostat. A ten-tative estimate of the sodium content prior to the experiment indicated x ' 2.Two cycles of measurements were performed on the ARGUS beamline, thefirst at T = 200 K in longitudinal fields varying between 0-2000 Gauss, thesecond at H = 1000 Gauss between 10-270 K. The µ+ depolarization showedan exponential decay over all the explored field and temperature range and thebackground B=5.8, due to the silver sample holder, was found to be weaklytemperature and field dependent. Then, the data were fitted to the function

A(t) = A exp(−λt) + B (3.3)

where the initial asymmetry A and the relaxation rate λ were free parameters.The results of these fits are shown in Fig. 3.10 and 3.11. The experimentalcurve of the asymmetry as a function of the field in Fig. 3.10 manifestly in-dicates the presence of muonium in the sample. When muonium is formed,the initial experimental asymmetry is lower than the total expected asymme-try. This occurs since the hyperfine coupling with the electron induces mutualspin flip-flops, destroying the initial polarization of the muon beam. However,on increasing field the Zeeman energy progressively overcomes the hyperfinecoupling and definitely decouples the muon and the electron spins, with theconsequent recovery of the total muon polarization at short times. Now, oneshould notice in Fig. 3.10 that the initial asymmetry in zero field is very low,about 20%. This residual quantity possibly represents the approximate frac-tion of diamagnetic µ+ not involved in muonium formation. Furthermore thetotal asymmetry is nearly recovered at moderately high fields, H = 0.2 Tesla.The shape of a repolarization curve attained on increasing field depends onthe strength and on the symmetry of the hyperfine tensor. The low valueof the asymmetry at zero field is different from A=0.5 expected for isotropicmuonium. Therefore one may argue that the muonium hyperfine coupling inNaxZnPc should be strongly anisotropic. On the other hand, Piroto Duarte etal. in Ref. [87] suggest that for very low fields in TF experiments nearly halfof the asymmetry ascribed to paramagnetic muonium anchoring to the ben-zene rings of ZnPc is observed in the precession signal. This denotes that thiscouplings are instead nearly isotropic. Now, as concerns the muonium statebonded to the nitrogen, if this was present in NaxZnPc , one would expect toobserve possibly dips due to level crossing resonances in the polarization curve,since the hyperfine coupling in this state is retained to be highly anisotropic.The resonance should be attained at a field Bres = πAµ/γµ for which the cross-ing of energy levels is avoided if the muon electron hyperfine interaction is notperfectly isotropic, in practice if the degeneracy is lifted. In a powder thisshould be observed as a broad peak centered at Bres and for Aµ=30 MHz, asexpected from [89], this field should be Bres=0.12 Tesla; however, no evidenceof this effect is supplied by the curve in Fig. 3.10, which on the contrary is


0 500 1000 1500 20000

20

40

60

80

100

300

% o

f sam

ple

Asy

mm

etry

H(Gauss)

T= 200 K

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.500 50 100 150 200 250

Na2ZnPc Zero Field

Musr

T(K)

λ (µ

s-1)

Figure 3.10: Repolarization curve of the µ+ asymmetry in Na2ZnPc as a function ofthe magnetic field in longitudinal geometry, measured with the ARGUS spectrome-ter at T = 200 K. The solid line is the fit according to Eq. (3.5). The inset shows themuon relaxation in Na2ZnPc in zero-field, acquired with the MuSR spectrometer atISIS.

perfectly monotonic.One can then ascribe the hard reduction of the asymmetry at low fields dueto rapidly fluctuating fields at the muonium due to hyperfine interactions withnuclei [90]. In the inset of Fig. 3.10 this hypothesis is supported by the pre-sentation of relaxation measurements in zero-field configuration on the MuSRbeamline. The relaxation is of the order of 0.4 µs−1 at the lower temperaturesand decreases on heating. The asymmetry functions were found to be expo-nentials, indicating a pure dynamic relaxation. This relaxation is completelydifferent from the one observed in LF experiments reported in Fig. 3.11. Onesees that at low temperatures the relaxation in zero field is two orders of mag-nitude larger and the two curves follow an opposite temperature dependence.In Ref. [88] λ is reported as a temperature activated quantity with an energyactivation Ea=220 meV, corresponding to ∼880 K. These activation processesis attributed to the scattering of the muon spin with the diffusing charge carri-ers. Also in NaxZnPc the relaxation rate is consistently temperature activatedand fitted by the function

λ(T ) = λ0 exp(−EA/T ) + C (3.4)

where C is a small constant offset. Fitting the curve in Fig. 3.11 with Eq.(3.4), one obtains Ea = 660 K, which is smaller than the one of the pristinecompound but of the same order of magnitude. One might associate this gap


0 50 100 150 200 250 3000.000

0.005

0.010

0.015

0.020

0.025

λ(µs

-1)

T(K)

Na2ZnPc H=1000 Gauss

Argus

Figure 3.11: Muon longitudinal relaxation in Na2ZnPc for H = 1000 Gauss, ac-quired with the ARGUS spectrometer at ISIS. The solid line is the fit according toEq. (3.4).

with an activated hopping of the electrons injected in the LUMO band, namelyNaxZnPc still behaves like a semiconductor. On the other hand, as it will beshown in chapter 4, AxZnPc for x ' 2 rather displays a metallic behaviour.Therefore it is likely that this activated process is described by another mech-anism. The most probable one is muon or muonium diffusion. Since the muondepolarization in zero-field is exponential, the local fields probed by the µ+

are rapidly fluctuating. In particular, if muons diffuse, they probe effectivelocal fields fluctuating with a temperature dependent correlation time τµ+

c (T )and muon relaxation is given by λ(T ) = γ2〈∆h2〉τµ+

c (T ), where 〈∆h2〉 is thesecond moment of the rigid lattice distribution of the local fields. τc for diffu-sion processes has the typical form τc(T ) = τ0 exp(∆E/kBT ), thus diminishingon heating. On the other hand, the longitudinal relaxation is a different phe-nomenon, excited by energy exchanges with the lattice by means of transitionsbetween the muon states split by the Zeeman interaction. Differently fromthe zero-field depolarization, it reflects the spectral density of the spin fluctua-tions J(ω) for ω = ωµ = γµH. In presence of diffusion longitudinal relaxationtypically follows a BPP function (see chapter 2), which for ωµτc À 1 is approx-imated by λ = γ2

µ〈∆h2⊥〉/τcω

2µ, coincident to the function (3.4) for an activated

τc. Now, since in H = 1000 Gauss, 80% of the asymmetry is recovered (Fig.3.10), one observes the relaxation of both µ+ and muonium. Then, if alsomuonium diffuses probing field fluctuations with the correlation time τMu

c (T )and (γ2〈∆h2〉)−1 À τµ+

c , τMuc À ω−1

µ , the zero-field depolarization decreases


on increasing temperature, while, on the contrary, LF relaxation rises.In principle, other phenomena inducing relative motion among the muon spinand the sources of the local field could induce these results. However, molecularmotions for example can be neglected as NMR measurements don’t show anyevidence of their activation in the other alkali-doped phthalocyanines. More-over, one could explain the behavior of relaxation on heating in zero-field eitherconsidering a simple hyperfine interaction with the nuclei or a super-exchangebetween the muon and Na, owing to an overlap between the Na(1s) electronand the muonium orbitals [90]. The activated hopping of electrons with tem-perature would gradually diminish this overlap and accordingly the relaxationwould become slower. This process would regard a little fraction of the formedmuonium since in zero field the polarization of muonium is heavily quenched.Anyway, both these interactions are expected to be quenched on increasingfield.Now, putting aside the hypothesis of anisotropy, one can tentatively fit therepolarization curve using the expression (see chapter 2 or Ref.[79, 77])

Pz(H) =1 + 2(H/H0)

2

21 + (H/H0)2 (3.5)

for isotropic muonium and consider the high field region. The result of the fitis the solid line in Fig. 3.10; one sees that the fit is rather good above H = 200Gauss. The fit yields H0=563 Gauss and from the relation H0 = 2πAµ/γe

one obtains Aµ=1.57 GHz. Evidently the result is quite different from the oneexpected in the light of Ref. [88, 89]. This clearly indicates that the muoniumstate forming in NaxZnPc is different from the muonium states observed inZnPc. The high value of the contact term suggests that this state is quasi-atomic, and therefore the muonium is not added to the Pc molecule as a radical.The effect of doping in the change of the muonium states in the material is thuscrucial. The intercalation of Na between the molecules prevents the bonding ofmuonium to the Pc ring . Muonium is seemingly confined to interstitial sitesamong adjacent chains as expected by Taguchi et al. for high alkali doping. Inaddition, less muonium is formed as the fraction of diamagnetic µ+ is doublewith respect to the one observed in ZnPc. From these observations one infersthat definitely Na must anchor the Pc molecules along the chain substitutingthe muonium formed in the pristine compound.

Chapter 4Magnetization and NMRmeasurements in LixZnPc

As outlined in the previous chapters, recent experimental works on thin filmsand theoretical predictions have inspired the investigation of the evolution fromthe semiconducting state of pristine MPc to a strongly correlated metallic statein bulk AxMPc. SQUID DC magnetometry and NMR measurements were per-formed in LixZnPc and LixMnPc in order to find evidences of these differentphases. The comparison between the results achieved by the two techniquesgives an indication both on the magnetic and electronic properties, whereasdirect conductivity measurements cannot be performed for technical reasons.On one hand, DC magnetometry supplies information about the nature ofthe magnetic ground state, the dimensionality of the exchange couplings, thepresence of long range order and, possibly, of non equilibrium phenomena inmaterials through the analysis of the temperature, field and time dependence ofmagnetization. On the other hand, the NMR Nuclear Spin-Lattice RelaxationRate (NSLRR) 1/T1 acts as a probe of the dynamic electronic correlations atall wave vectors ~q and of their dimensionality.From the analysis of the temperature dependence of the spin excitations ofthe electron system, one can infer the degree of localization and correlationbetween the electrons. In the case of a Fermi gas, i.e. uncorrelated electrons,1/T1 ∝ T , while upon increasing the correlations one generally expects that1/T1 would deviate from linearity and possibly evidence a large variety of phasetransitions. Eventually for fully localized electrons 1/T1 should stay constantor rather increase on cooling. One-dimensional (1-d) metals can undergo sev-

eral instabilities due to the nesting of the Fermi surface at the wave vector 2~kF

(Peierls instability, spin density waves, etc...). Hence, NMR 1/T1 is an optimaltool to detect ~q ≈ 0 uniform excitations (paramagnons), which are dominant in

all weakly correlated systems, as well as ~q ≈ 2~kF excitations arising in stronglycorrelated systems.The study of nuclear relaxation and magnetic susceptibility has solved impor-

75

76 4. Magnetization and NMR measurements in LixZnPc

tant issues in the field of one-dimensional organic conductors as the Bechgaardsalts (TMTSF)2X and the sulphur analogs (TMTTF)2X, in which electroniccorrelations play a key role. A brief review of the experimental evidences col-lected in these systems will be outlined in the theoretical framework of the 1-dLuttinger liquid model. Then, the work carried out on LixZnPc and LixMnPcwill be presented in the light of the analogies and of the differences with theBechgaard salts. Moreover, a possible interpretation of the interplay betweenCoulomb repulsion, delocalization, Jahn-Teller distortions and magnetism inthe two different families of compounds will be developed, in order to con-sider the possibility of strongly correlated superconductivity foreshadowed byTosatti et al. [46].

4.1 One-dimensional correlated metals

One-dimensional organic conductors as (TMTSF)2X and (TMTTF)2X havebeen extensively studied in the past decades because of their unconventionaland complex phase diagram. The large literature produced on these systemscan be helpful in order to understand the unknown behaviour of the novelalkali-doped phtalocyanines. The similarities between these compounds pri-marily pertain to the conduction of the electrons across the materials. In(TMTSF)2X and (TMTTF)2X salts the planar TTF molecules stack along thea axis, pictorially forming columns of superimposed pancakes (Fig. 4.1). Alongthis direction the antibonding π∗ orbitals have the strongest overlap, thereforethe resulting band is nearly one-dimensional. The charges are transferred to itfrom negative anions (PF−6 , ClO−

4 , Br−...) segregated in side separate chains.In AxMPcs the situation is expected to be similar, even if the donating atomsare rather intercalated between the molecules without ordering in separate ar-rays (see chapter 3.4).All these compounds are molecular crystals in which the intermolecular hop-ping generally gives rise to narrow bands. Hence, to a first approximation, thecharge diffusion is the outcome of the competition between on site Coulombrepulsion U and the hopping t, granted by the overlap of molecular orbitals ofadjacent molecules. This competition, in general, is expressed in the Hubbardform [92], which includes both the terms

H = t∑i,σ

(c†i,σci+1,σ + c†i+1,σci,σ) + U∑

i

c†i,↑ci,↑c†i,↓ci,↓, (4.1)

the sum running over the sites i and spin states σ. The first is the kinetic term,while the second accounts for the repulsion between electrons of opposite spinon the same site. In the tight-binding approximation the band dispersion canbe expressed as Ek = 2t cos(ak) and the total width is W = 4t. In gen-eral, when U/W ¿ 1 delocalization is favored and the electrons hop betweenmolecules, while when U/W À 1 electrons are localized on each site and the

4.1. One-dimensional correlated metals 77

A

B C

A

B C

Figure 4.1: (A) Molecular structure and (B-C) crystal structure along different direc-tions of (TMTSF)2X salts [91].

system minimizes the energy developing intersite exchange spin couplings (an-tiferromagnetic or ferromagnetic depending on the molecular orbital overlaps).In the Bechgaard salts approximately W ≈ 0.4÷ 1 eV along the main overlapdirection and U ≈ 1÷ 5 eV, which lowers to U ≈ 0.3÷ 0.5 eV in considerationof the dielectric screening of the environment [36]. One can notice that thesevalues are close in magnitude to the ones find in fullerides (W ≈ 0.5 eV andU ≈ 1 ÷ 1.5 eV [38]) and also to the ones evaluated for the alkali-doped Pcs(W ≈ 0.3 eV and U ≈ 1 ÷ 1.5 eV [46]) and collocate all these compounds inthe intermediate regime U/W ∼ 1.The (TMTSF)2X and (TMTTF)2X series are metals at ambient temperature,but the sizeable electron correlations can lead a complex behaviour on cooling,depending also on the applied pressure and field. The phase diagram of thesecompounds goes beyond the scope of this work, but the crucial point is thatthe majority of these phase transitions is well described and even predictableconsidering these systems as a 1-d Luttinger liquids. In this model the mainpart of the Hamiltonian is the kinetic energy of free electrons with energiesclose to the Fermi level [93]

H =∑

k,σ

vF (|k| − kF )c†i,σci,σ. (4.2)


π/a-π/a -kF k

F

2k02k0

Ek

Figure 4.2: Dispersion relation of a 1-d electron gas with the indication of the band-width cut-off k0. The straight dotted lines represent the linearized dispersion relationof the Luttinger model.

In some models the linearized dispersion Ek = vF (|k| − kF ) approximates the

band of the tight-binding calculation in the region |~k − ~kF | < k0 where k0

is a cut-off for the allowed states. In the Luttinger model [94], instead, thelinear description is defined for every k (comparison shown in Fig. 4.2). This

approximation is powerful as long as only electrons with energies close to ~kF

participate to low-energy processes at low temperatures (kBT ¿ EF ). Inparticular the spectral density of electron excitations will be strongly peakedaround ~q = 0 and ~q = 2~kF . Two distinct branches describing electrons movingin opposite directions are defined, one containing ~kF and the other containing−~kF . Defining the operators ak,σ and bk,σ distinctly acting on the two branches,one has

H =∑

k,σ

vF [(k − kF )a†k,σak,σ + (−k − kF )b†k,σbk,σ] (4.3)

The electron-electron interactions are then introduced as perturbations of theHamiltonian (4.3). As long as scattering is concerned, four type of processesare allowed with respective coupling strengths g1, g2, g3, g4 (Fig. 4.3). For-ward scattering, involving electrons in different branches (g2) or in the samebranch (g4), occurs with the minimum momentum transfer ~q = 0. On thecontrary, backward scattering between moments of different branches (g1) re-

quires ~q = 2~kF . Finally, the Umklapp process (g3), allowed only in the case

of half filled band between moments of the same branch, requires ~q = 4~kF . In

4.1. One-dimensional correlated metals 79

g1

g3 g4

g2

Figure 4.3: Possible scattering processes in a 1-d correlated metal. The solid linesrepresent the electrons of the branch containing ~kF , the dashed lines the ones ofthe branch containing −~kF .

all the scattering processes the spin dependence must also be considered. Theinteraction Hamiltonian is then expressed on the basis of the operators ak,σ

and bk,σ. For example for the back-scattering term [36]

Hint =

Q,σ,σ′∑

k1,k2

g1[a†k1,σb

†k2,σ′ak2+2kF +Q,σ′bk1−2kF−Q,σ]. (4.4)

With some substitutions one can demonstrate that the interaction Hamiltonianis a weighted spectral sum over the Fourier components of charge density andspin density (boson representations [93, 95]). In particular, in the 1-d systemsthe response functions of charge and spin density have singularities for thewave vector ~q = 2~kF . The divergence of the susceptibilities at the wave vector~q = 2~kF is due to the peculiar structure of the Fermi surface in one-dimensionalmetals, made of of two planar sheets [91]. Namely, a great number of degener-ate states belonging to these two branches can be coupled by the wave vector~q = 2~kF , for which the system becomes unstable against Peierls distortion orordering. In addition, also the response functions of the triplet and singletCooper pairs can have singularities. It is dare to say that such excitationsare not single-particle excitations but rather boson-like independent collectivephenomena. Spin excitations (spinons) and charge excitations (holons) canpropagate separately with different velocities vσ 6= vρ 6= vF (spin-charge sepa-ration [96]). Thus, the physical properties of the system get renormalized withrespect to a conventional Fermi liquid.In the Luttinger model the renormalization coefficients are vσ, vρ, Kρ, where


TS SDW

g1=2g2

A B

Figure 4.4: A. The most divergent correlations of the one-dimensional Luttinger liq-uid (neglecting Umklapp scattering). SDW, CDW, SS and TS indicate Spin DensityWaves, Charge Density Waves, Singlet Superconductivity, Triplet Superconductiv-ity. B. Generalized phase diagram for the (TM)2X series. The notations SP, SCand CL refer to Spin-Peierls, superconducting ground state and charge localizedbehaviour respectively. The names of the compounds indicate their location at at-mospheric pressure in the generalized diagram [97, 36].

[95]

Kρ =

√2πvF + 2g4 + (g1 − 2g2)

2πvF + 2g4 − (g1 − 2g2). (4.5)

Then, the static spin susceptibility, for instance, becomes χ = χ0vF /vσ. Fornon-interacting systems Kρ = 1, while for interacting systems one has, forg1−2g2 > 0, an overall attractive correction with Kρ > 1 and, for g1−2g2 < 0,a repulsive correction with Kρ < 1. The aforementioned response functionsat low temperature behave as power laws with a coefficient depending on Kρ

with logarithmic corrections [95]

χCDW ≈ ( TEF

)Kρ−1|ln( TEF

)|−3/2 χSDW ≈ ( TEF

)Kρ−1|ln( TEF

)|1/2

χSS ≈ ( TEF

)1/Kρ−1|ln( TEF

)|−3/2 χTS ≈ ( TEF

)1/Kρ−1|ln( TEF

)|1/2

The frequency dependence of these quantities can be obtained replacingT with ω, since in logarithmic approximation ln(T/EF ) can be replaced by

4.2. NMR in one-dimensional conductors 81

ln[max(ω, T )/EF ]. Then, these susceptibilities for small ω, T have a leadingterm in the form of a simple power law [93].In the phase diagram in Fig. 4.4-A the line g1−2g2 = 0 separates the region ofdivergency for the density fluctuations (CDW and SDW) at 2~kF , g1− 2g2 < 0,from the regions of divergency of the pairing fluctuations (singlet and tripletsuperconductivity), g1−2g2 > 0. In addition singlet superconductivity requiresg1 + g2 < 0, while triplet superconductivity requires g1 − g2 > 0.The (TMTSF)2X and (TMTTF)2X compounds were ideal systems to test thedevelopment of these phases in presence of strong correlations. In Fig. 4.4-Bthe temperature-pressure phase diagram of (TMTSF)2X is shown [97, 36]. Onecan see that these salts exhibit a SDW transition at ∼12 K at ambient pressurewhile the application of pressure suppresses antiferromagnetic correlations andbrings out superconductivity as, naively, it pushes the systems towards tridi-mensionality. Most of these phase diagrams for the different compounds of thefamily were successfully reconstructed by means of NMR measurements. Inthe next section some experimental works on (TMTSF)2X will be recalled toexplain what has to be expected in alkali-doped MPcs.

4.2 NMR in one-dimensional conductors

The characteristic temperature dependence of 1/T1 in Bechgaard salts is shownin Fig. 4.5. Two different temperature domains can be distinguished. At hightemperatures 1/T1 has an upward curvature indicating an enhancement withrespect to the normal metal Korringa law, 1/T1T [~q ∼ 0] ∼ const. At lowtemperatures, on the contrary, a singularity can occur at TN below 10 K. Again,at high temperatures 1/T1 ∝ χ2

ST (Fig. 4.6), where the χS = χ(~q = 0, ω = 0)is the uniform and static spin susceptibility measured by EPR. Therefore, theplot of 1/T1 as a function of χ2

ST yields a straight line which in some casesis followed down to the lower temperatures and intercepts the y axis close tozero. In other cases an additive contribution emerges on cooling, giving rise tothe low-T singularity. One can summarize this profile in the Eq.

1/T1(T ) = C0χ2ST + C1(T ) (4.6)

where C0 is constant.A consistent number of papers have investigated and explained this behaviour[99, 100, 101, 36]. Thanks to the validity of dynamical scaling considerations,the first term of the above equation is attributed to the 1-d ~q=0 magneticexcitations (paramagnons) which are dominant over a large temperature in-terval. At variance with a normal metal, for which the Pauli susceptibility istemperature independent, here one finds

χS(T ) = χS(0)/(1− g1(T )/2πv∗F ) (4.7)

g1(T ) = g1/[1− g1/πv∗F ln(T/EF )] (4.8)


Figure 4.5: 77 Se 1/T1 as function of temperature at different fields in (TMTSF)2ClO4

single crystals [98].

where g1(T ) describes the effective coupling between the backward and forwardmoving electrons. This justifies the positive curvature of 1/T1, deviating fromthe Korringa law.The second term of Eq. (4.6), instead, is ascribed to the ~q = 2~kF antifer-romagnetic fluctuations expected for the 1-d correlated systems, leading toa Spin Density Wave (SDW). NMR 1/T1 can detect both these dynamics asit is proportional to the q-integrated dynamical structure factor, namely to∑

~q |A~q|2χ”(~q, ωN). Recalling the dynamical scaling hypothesis, one has that

the low temperature 1/T1 divergence is driven by χSDW (2~kF ) ∝ ( TEF

)Kρ−1

(remember the susceptibilities of the Luttinger model). In brief, at low tem-

perature the spectral weight of ~q = 0 fluctuations diminishes and ~q = 2~kF

AF-like correlations increase. The complete expression for 1/T1, specializingthe (4.6), becomes [99, 100, 101]

1

T1

(T ) = C0χS(T )2T + C1(2~kF )T χ1D(2~kF , ωL) (4.9)

For Kρ → 0 (strong repulsive correlations) the previous equation reduces tothe expression (4.6) with C1(T ) = C1. This is the case of the Sulphur series,which show a Mott-Hubbard localization at a temperature Tρ >> TN , i.e a

correlation gap is opened. For 0 < Kρ < 1, 1/T1(2~kF ) ∝ C1(2~kF )TKρ . Thisimplies that in the plot 1/T1(χ

2ST ) one could see the antiferromagnetic corre-

lations emerging even at temperatures T >> TN . This occurs in the Selenideseries in which Tρ ' TN , hence localization is hindered and electrons itineratein spite of the strong correlations.

4.2. NMR in one-dimensional conductors 83

Figure 4.6: 77 Se 1/T1 vs χ2ST in several (TMTSF)2X and (TMTTF)2X single crystals

[99].

In the light of this presentation, the expectations from the NMR measurementson AxMPcs could be the following. In LixZnPcs, first of all, one supposes thatin absence of localized moments nuclei should couple only to the x electrons in-jected in the 2eg band. For light doping, let’s say for x ' 1, on site correlationsshould be weak and the material should tend to a nearly uncorrelated metal,following a Korringa law. For x → 2 the correlations should raise significantly.If A2ZnPc was the one-dimensional counterpart of A3C60 [38], at the LUMOhalf filling a strongly correlated metal is expected with a hypothetic transi-tion to superconductivity at low temperature. Such a material should show ahigh temperature dependence described by 1/T1 ≈ C0χ

2S(T ), with the T de-

pendence of χ2S(T ) determined by the g1 term where, in the limit U/W À 1,

g1 ≈ U .In the strong correlation limit Capone et al. [48] suggest that there’s a criticalregion 0.8 < U/W < 0.9 in which isotropic organic conductors display super-conductivity at the crossover between metallic and a Mott insulating state atthe half band filling. This work developed in the frame of Dynamical MeanField Theory (DMFT) proposes a model in which in the critical region the freefermion-like quasiparticle residue Z → 0 and the effective metallic band widthgets renormalized to W ∗ = ZW . Moreover, whenever a strong Jahn-Teller cou-pling to molecular vibrations is present, the Hund’s exchange is reverted givingrise to an an effective weak coupling J ≈ |JH − 3/4EJT |. If W ∗ ∼ ZW ∼ J , inparticular, in spite of the Coulomb repulsion the electrons are slowed down, so


that they can experience a pair attraction λ = 10D(EF )|JH |/3, where D(EF ) isthe bare density of states at the Fermi level. Granted that repulsion is stronglyrenormalized while the pairing attraction is not, this has been recognized as themechanism responsible for the singlet superconductivity in non conventionalsuperconductors such as fullerides, where Tc, the temperature of the super-conducting transition, is found to increase with increasing U/W , contrarilyto what expected from a BCS superconductor. Since a similar phenomenol-ogy is expected in alkali-doped phthalocyanines in reason of the competitionbetween W , U , JH and EJT [46], the primary purpose of this work must beto analyze the doping dependence of D(EF ) derived from susceptibility and1/T1(T ) measurements. The estimate of D(EF ) is crucial as D(EF ) ∼ 1/W .An enhancement of this quantity for x ' 2 would indicate a narrowing of thequasi-particle bandwidth and, in presence of metallicity, the scenario of thestrongly correlated superconductivity in alkali-doped phthalocyanines envis-aged by Tosatti et al. could be possibly observed.In LixMnPc the situation is more complicated due to the presence of the lo-calized Mn2+ magnetic moments coupled via intrachain exchange [28]. Herenuclei probe both the dipolar hyperfine interaction with the transition metalelectrons and the contact hyperfine interaction with the 2eg electrons. The 1/T1

contribution of the 2eg electrons could be sizeably masked by the manganesespin dynamics and it could be hard to gain information on delocalization andcorrelation. Nevertheless, one expects NMR measurements to be useful in thestudy of the structure of the LUMO. If Jahn-Teller distortions removed the de-generacy of the 2eg orbitals, a gap would open in the LUMO and 1/T1 shouldbe activated. Remarkably, the gap arising from the competition among Jahn-Teller distortions and Hund exchange, analogue to the one found in fullerides[53], corresponds to the superconducting coupling in the strongly correlatedlimit.

4.3 The magnetic susceptibility of LixZnPc

Magnetization (M) measurements were performed on LixZnPc by means of theMPMS-XL SQUID magnetometer of Quantum Design. The working principlesof the apparatus are illustrated in detail in chapter 2 and appendix A.The magnetization was measured cooling the samples from room temperaturedown to 2 K in an applied field (field cooling procedure). The dependenceof the magnetization on the magnetothermal history was verified comparingmeasurements at different cooling rates, between 10 K/min and 0.1 K/min. Insome samples it was noticed that rapid cooling as well as short waiting timesafter temperature changes could affect the results leading to an underestimateof the magnetization. This was not attributed to intrinsic processes but to aslow thermalization of the system due to the low thermal conductivity of theglass sample holder and the usage of powder samples. Hence, cooling rates of1 K/min and long waiting times were set, if necessary, to attain the thermal

4.3. The magnetic susceptibility of LixZnPc 85

10 1000.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

χ(em

u/m

ole)

T(K)

LixZnPc x=3.75 x=2.35 x=1.850 2000 4000 6000 8000

01x10-3

2x10-3

3x10-3

4x10-3

5x10-3

6x10-3

7x10-3

8x10-3

9x10-3 Li1.85ZnPc T=3 K

µ (

emu*

G)

H(Gauss)

Figure 4.7: The magnetic susceptibility of several LixZnPc compounds for H = 1000Gauss as a function of temperature. The diamagnetic contribution of the ZnPc plusthe core diamagnetism of the lithium atom have already been subtracted from thedata. In the inset the magnetic moment of Li1.85ZnPc powders as a function of thefield is reported for T = 3 K.

equilibrium of the system. Unfortunately it was not possible to measure sepa-rately the magnetization of the sample holder. However, it is expected that theglass tube contribution to the total magnetic moment adds a constant diamag-netic correction to the total magnetic moment. In particular this is expectedto be smaller or at least of the order of the diamagnetic contribution due tothe closed shell organic ligands, present in all phtalocyanines (see below).The magnetization of all the samples as a function of temperature was mea-sured at H = 1000 Gauss. As long as the field dependence of the magneticmoment is linear in this field range, even down to 3 K, one can obtain the sus-ceptibility from the ratio χ = M/H. The molar susceptibility of Li1.85ZnPc,Li2.35ZnPc and Li3.75ZnPc is sketched in Fig. 4.7. In the inset the linearityof the magnetic moment as a function of the field at T=3 K is shown. Thesusceptibility of all LixZnPc can be described by the following expression:

χ = χ0 +C

T −Θ. (4.10)

The temperature dependence of the total susceptibility is given by the Curie-Weiss (CW) term. In the presented data the diamagnetic contribution of theZnPc reported in literature [102], plus the core diamagnetism of the lithiumatom (−1 · 10−6 emu/mole) have already been subtracted, corresponding to a


0 50 100 150 200 250 3004.2x10-4

4.3x10-4

4.4x10-4

4.5x10-4

4.6x10-4

4.7x10-4

4.8x10-4

4.9x10-4

5.0x10-4

T(K)

χ(em

u/m

ole)

T(K)

Li1.85

ZnPc FC 1000 G Curie-Weiss Curie-Weiss

+ Bleaney-Bowers

50 100 150 200 250

3.0x10-6

4.0x10-6

5.0x10-6

χ-CW fit Bleaney-Bowers

Figure 4.8: Blow up of the magnetic susceptibility of Li1.85ZnPc for H = 1000 Gaussas a function of temperature. The dashed line is the best fit according to Eq. (4.10).The solid line is the best fit according to Eq. (4.10)+(4.11). In the inset the differencebetween χ and the CW fit is presented (data have been shifted due to a negativebackground at high temperatures depending on the choice of C in Eq. (4.10)). Thedata are fit with Eq. (4.11).

total correction of nearly −4.2 · 10−4emu/mole. A fit of the susceptibility forthe sample Li1.85ZnPc with the function (4.10) is shown in Fig. 4.8. One cannotice the different scale on the y axis with respect to Fig. 4.7. A slight misfitof the CW law with respect to experimental data should be noted (dashed linein Fig. 4.8). In particular, it has been verified that the fit of susceptibilitydata is markedly improved (solid line in Fig. 4.8) adding to Eq. (4.10) a termof the form

χ∆ =D

T

exp(−∆T)

1 + 3 exp(−∆T), (4.11)

characteristic of systems with a gap between singlet and triplet excitations, asdimers [103]. In the inset of Fig. 4.8 this weak additional contribution, peakedaround T = 40 K, is obtained through the subtraction of the CW fit from theexperimental susceptibility. It is noticed that the Eq. (4.11) suitably fits thedata.The magnetic behaviour of LixZnPc undergoes dramatic changes with doping(Fig. 4.7). Close to the half-band filling of the LUMO, i.e. for x ' 2, theCurie-Weiss term is nearly quenched and the susceptibility turns to be veryweakly temperature dependent. In fact, the steep rise of susceptibility at lowtemperatures shown in Fig. 4.8 is likely due to impurities or defects and corre-


1.5 2.0 2.5 3.0 3.5 4.00.000

0.015

0.030

0.045

0.060

0.00

0.04

0.08

0.12

0.16

Num

ber of spin 1/2 per molecule

C (

emu*

K/m

ole)

x (Lithium/ZnPc molecule)

Figure 4.9: Curie constant of the several LixZnPc compounds obtained by meansof the fits with Eq. (4.10) as a function of lithium doping.

sponds to a vanishing number of Bohr magnetons. On the contrary, increasingx beyond the half band filling, the CW term is enhanced. The same increaseof the CW contribution is noted on diminishing x below 2, as in Li1.5ZnPcthe susceptibility is close to the one of Li3.75ZnPc (it was not plotted to avoidvisualization problems). The values of the Curie Constant obtained by the fit(4.10) are reported in Fig. 4.9 as a function of x. Coherently with the pre-vious considerations, C has a minimum for x = 1.85 and in general assumeslow values corresponding to a very tiny fraction of S = 1/2 per molecule, lessthan 0.15 over the whole range. The values of Θ extracted by the same fitare not reported. They don’t show a marked dependence on x and are allnegative with an average of about |Θ| <2 K. The inclusion of the term (4.11)in the fit does not substantially affect the values of C and Θ obtained from theprevious analysis. The gap parameter ∆ varies between 22 K÷ 44 K, yieldingan average ∆ ' 30 K. The constant D is of the same order of C and has thesame temperature dependence. It is difficult to attribute a definite origin tothe term (4.11).The CW term characterizes the magnetism of localized spins. In principle thiscan arise from either a slight concentration of paramagnetic impurities in thesample or from a percentage of S = 1/2 2eg electron spins that eventuallylocalize due to disorder. Hence, C is an average quantity. Isolated impuritiesshould follow a simple Curie law (Θ → 0), which is in agreement with thepresent finding of a very small Θ. They can be simply dirt deposited on thesurface of the sample holder, or residues of the synthesis nested in the powders.


Despite the concentration of such impurities can change from sample to sam-ple, one expects their fraction to be low, as C follows a well defined trend. Inparticular, this fraction can be of the order of the S = 1/2 spins per moleculecalculated in Fig. 4.9 for the sample Li2.35ZnPc, n ∼ 8%. In fact, x is alwaysan average value and a substantial symmetry in the magnetic behaviour isexpected for a similar value δx = |x− 2| below or above the half band filling.Therefore, Li2.35ZnPc and Li1.85ZnPc should have a comparable C. n ∼ 8%for Li2.35ZnPc is an overestimate, because it is calculated considering S = 1/2spins from the CW law with the expression n = 3kBC/NAg2µ2

BS(S+1). Keep-ing g = 2 for S = 1 the percentage drops to n ∼ 3%.Impurities can arise from the formation of LiPc, occurring when excess lithiumoccupies vacant Pc2− rings during the synthesis. Isolated LiPc radicals cou-ple ferromagnetically in S = 1 dimers in the β crystallographic phase [104].Alternatively LiPc form AF S = 1/2 chains whose susceptibility is fit by theBonner-Fisher law with J=40 K [105]. One hypothesis could be the reversal ofthe situation prospected in the aforementioned Ref. [104] and the formationof S = 0 LiPc-LiPc dimers, where the change of the S ground state would beinduced by the crystal packing. This could certainly justify the presence ofthe additional contribution (4.11) in the magnetic susceptibility. The constantD obtained by the fit (4.11), proportional to the number of spins couplingin dimers, gives values of the order of C of the CW law. The estimate ofvacant Pc2− rings in solution after several purification steps described in chap-ter 3.1 is nearly 4% [106]. Correspondingly, a maximum fraction of 4% LiPcshould form, whereas one obtains approximately 8% from D ≈ C for S = 1/2.Nevertheless it is hard to distinguish and evaluate the ratio between the con-tributions coming from the different possible LiPc impurities, dimers or chainsegments.Now, for x far from 2, the Curie Constant increases again. On the other hand,the high temperature paramagnetic background, χ0, is higher for x ' 2, whileit is evidently depressed toward zero in the other compounds. Observing thatthe susceptibility of Li1.85ZnPc is nearly temperature independent and posi-tive, one recognizes χ0 as a Pauli-like susceptibility, χP . It arises from theelectrons injected in the LUMO which are delocalized as in normal metals.In brief, when χP has a maximum, C reaches a minimum and viceversa. Asthe high temperature background in the SQUID measurements is affected bya systematic error due to the contribution of the sample holder, separate EPRmeasurements were performed on the entire LixZnPc series (the details of theEPR measurements will be reported elsewhere). The microscopic static uni-form susceptibility for T = 300 K, shown in Fig. 4.10, can be obtained throughthe integration of the EPR line intensity over the whole field range. All themeasurements were performed under the same conditions, so that one can com-pare the relative intensity of the signal in all samples. Nevertheless, since theywere not rescaled by the intensity of a reference signal, all the quantities willbe expressed in arbitrary units. As derived from the SQUID data, the Pauli


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00

50

100

150

200

250

300

350

400

450

χ P (

Arb

. Uni

ts)

x (Lithium/ZnPc molecule)

LixZnPc , T=300 K

EPR susceptibility

Figure 4.10: Pauli susceptibility estimated from EPR measurements in LixZnPccompounds at T = 300 K.

3150 3200 3250 3300 3350 3400 3450 3500 3550 3600

-16

-12

-8

-4

0

4

8

12

16

Der

ivat

ive

EP

R S

igna

ls (

arb.

uni

ts)

H (Gauss)

Li1.5

ZnPc Li

1.85ZnPc

Li2.7

ZnPc

3000 3200 3400 360002468

10121416182022242628

Li2.35

ZnPc, T=50 K

Inte

rgra

l of t

he E

PR

line

(a.u

.)

H(Gauss)

Figure 4.11: Comparison of the derivative EPR signals measured in severalLixZnPc compounds at T = 300 K. In the inset an example of χ′′ in Li2.35ZnPc atT = 50 K, obtained from the integration of the derivative EPR signals, is reported.


0 20 40 60 80 100 120 140 160 180 200 220 2400

2000

4000

6000

8000

10000

12000

14000

16000

18000

EP

R a

rea

(arb

. uni

ts)

T(K)

LixZnPc x=2.35 x=2.7 x=3.75

Figure 4.12: EPR susceptibility, as the integral of the EPR line, in LixZnPc com-pounds as a function of temperature.

susceptibility reaches the maximum for x ' 2 and a specular decrease of themeasured values is noticed moving from x = 2 towards x = 0 or x = 4.EPR measurements were also performed on Li2.35ZnPc, Li2.7ZnPc and Li3.75ZnPcbetween 300 K and 4 K. The EPR line showed two components (Fig. 4.11); onenarrow component corresponding to g = 2 and second broad component witha shifted average g. Integrating separately the two components, it is observedthat in Li2.35ZnPc and Li2.7ZnPc the relative ratio of the narrow line with re-spect to the integral on the total scan window is of a few percents. Thereforeit is likely that in these samples the reported susceptibility is mainly relatedto the broad component, while the narrow signal originates from impurities.On the contrary, in Li3.75ZnPc this ratio grows to almost 50%.The susceptibility of Li2.35ZnPc and Li2.7ZnPc is approximately constant downto low temperatures (Fig. 4.12). Differently from the SQUID data, no raise atthe low temperature is measured, however, superimposed to the constant back-ground, a weak broad maximum emerges in the susceptibility around T = 50K. This strongly reminds the term (4.11) contained in the SQUID suscep-tibility, tentatively attributed to dimer excitations. The same temperaturedependence is observed for Li2.7ZnPc except for a renormalization of the back-ground χP . In turn, Li3.75ZnPc susceptibility follows a marked gradual rise oncooling below 100 K. As the presence of impurities is low in this samples, thisrise is likely the consequence of localization of electrons in the 2eg band. TheCW fit of this data gives Θ ' −9 K (solid line in Fig. 4.12).To summarize, both SQUID and EPR measurements suggest that the suscep-

4.4. LixZnPc susceptibility under high hydrostatic pressure 91

tibility resembles the one of a normal metal for x ' 2, while a tendency tolocalization is evident for x ' 3.75, on approaching the total filling of the2eg band. For x ' 2, a very slight increase in the susceptibility on cooling isnoticed and, subsequently, a decrease below T ' 50 K. The susceptibility isweakly temperature dependent for x approaching the half band filling and itsintensity is strongly peaked at x ' 2.

4.4 LixZnPc susceptibility under high hydrostaticpressure

Magnetization measurements at variable pressure were performed on Li2.35ZnPcand Li3.75ZnPc. The system described in appendix A was used to apply atorque between 10 N·m and 27 N·m to the piston which transfers the pressureto the cell, corresponding to P between 3 and 7.7 kBar. The powders weretransferred in the teflon sample holder inside a dry box to prevent contami-nation with oxygen and humidity. Due to the reduced capacity of the sampleholder, only few milligrams of powder (5-6 mg) could be inserted in it.Several centering scans were run prior to the measurement to assure the iden-tification of the sample signal. This was necessary as long as the distributionof inhomogeneities, related principally to the cell borders, introduces strongoscillations in the measured voltage. Since the sample position is reasonablyat the center of the cell and the total length of the cell is about 8 cm, the sam-ple signal is well separated by a distance of about 4 cm from these spurioussignals. In addition, the intensity of the signal due to the cell border has apoor temperature dependence. Then, low fields H < 50 Gauss and T < 40 Kwere chosen as optimal ranges to perform affordable measurements, becausein this condition the signal/background ratio is improved owing to the growthof the Curie tail on cooling.The magnetization of the cell could not be measured separately due to the dif-ficulty in keeping the centering in the same position of the sample on the wholetemperature range. The cell background was rather evaluated by subtractingthe magnetization of the sample from its measurement inside the cell at ambi-ent pressure. The sample magnetization was calculated starting from the dataobtained at H = 1000 G, rescaled to the actual weight and field applied to thesample. The cell magnetization was then subtracted from the data recordedunder pressure.The set of measurements in Li2.35ZnPc at H = 8 Gauss and in Li3.75ZnPc atH = 18 Gauss between 2 K and 35 K is depicted in Fig. 4.13 and Fig. 4.14.The data were taken with a field cooling sequence. The indicated fields havebeen estimated as the effective fields inside the cell from a separate measure-ment at fixed temperature as a function of the field. M(H) was linear witha small positive intercept, showing that the real field inside the cell is smallerthan the nominal set field. This is due to the demagnetization factor of the


0 5 10 15 20 25 30 351E-3

0.01

0.1

χ mol(e

mu/

mol

e)

T(K)

1 atm 3.7 kBar 6.2 kBar

Figure 4.13: SQUID susceptibility of Li2.35ZnPc for H = 8 Gauss at different appliedpressures.

cell and the residual trapped fields in the superconducting coils of the magne-tometer. In Fig. 4.13 the solid line is the susceptibility of the sample outsidethe cell at ambient pressure. Under pressure the high temperature backgroundχP evidently grows in Li2.35ZnPc, while it decreases in Li3.75ZnPc. These op-posite trends are illustrated in Fig. 4.15 where the values of χP calculated bymeans of the Eq. (4.10) with χ0 = χP are sketched, neglecting the diamagneticcorrections. In Li2.35ZnPc χP varies linearly with the pressure until 4 kBar,then rapidly rises between 4 kBar and 6 kBar. In Li3.75ZnPc the pressure de-pendence of χP is much less marked. In Fig. 4.16 also the calculated Curieconstants are shown. In Li2.35ZnPc C slightly decreases between 0 and 4 kBar,then it has an upturn to a value 8 times higher for P = 6.2 kBar. On thecontrary, C in Li3.75ZnPc remains roughly constant under pressure.The simultaneous growth of χP and of the Curie constant cannot be inter-preted on a simple framework of localization or delocalization. Thus, othermechanisms must be invoked to explain this behavior. The primary effect ofpressure on a crystal is the variation of cell parameters. The system adoptedfor these measurements is designed to exert an hydrostatic pressure. There-fore the response of the grains to the pressure are solely determined by thecompressibility and not by the properties of the external stress. Since ph-thalocyanines are strongly anisotropic systems, built up of long chains, thecompressibility is reasonably an anisotropic tensor. This was already observedwith dedicated studies in phthalocyanine conductors doped with charge trans-


0 5 10 15 20 25 30 35

1E-3

0.01

χ mol (

emu/

mol

e)

T(K)

Li3.75

ZnPc 1 atm 6.2 kBar 7.7 kBar

Figure 4.14: SQUID susceptibility of Li3.75ZnPc for H = 18 Gauss at different ap-plied pressures.

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.00.000

0.005

0.010

0.015

0.020

0.025

0.030

χ P (

emu/

mol

e)

Pressure (kBar)

Li2.35

ZnPC H=8 G Li

3.75ZnPC H=18 G

Figure 4.15: χP = χ0 obtained fitting the susceptibility of Li2.35ZnPc and Li3.75ZnPcunder pressure with Eq. (4.10). The dotted lines are guides to the eye.


0 1 2 3 4 5 6 7 8 9

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

Cur

ie C

onst

ant (

emu*

K/m

ole)

Pressure (kBar)

Li2.35

ZnPc H=8 Gauss Li

3.75ZnPc H=18 Gauss

Figure 4.16: Curie constant C obtained fitting the susceptibility of Li2.35ZnPc andLi3.75ZnPc under pressure with Eq. (4.10). The dotted lines are guides to the eye.

fer salts [107] and in the Bechgaard salts [108]. It has been evidenced in [107]that the crystal symmetry is maintained till very-high pressures between 0-40kBar and works on MnPc under pressure assume linearity between the axialstrains and pressure until 10 kBar [27] (constant compressibility). The authorsof [108] indicate that the contraction of the intermolecular distances along thechain axis is 0.5%/kbar and it is reduced by at least an order of magnitude inthe normal directions. Similar values are reported for several aromatic organiccrystals, leading to the conclusion that the compressibility is enhanced in thedirection of the π − π overlap characterizing all these systems.The effect of external pressure can be put in relation with the chemical pres-sure induced by lithium doping, which modifies the cell parameters as reportedin Fig. 4.17. In the light of this structural study [28], it should be noted thatfor 1<x<3 compressing the system along the chain axis one recovers the in-termolecular distances observed at lower doping. On the contrary, for 3<x<4,under compression one recovers the intermolecular distances observed at higherdoping. Therefore, under pressure Li2.35ZnPc should become equivalent toLi2ZnPc, while Li3.75ZnPc to Li4ZnPc. Moreover, from Fig. 4.17 one deducesthat between x = 2 and x = 2.5 the relative variation of the cell parameter bis less than 1%; hence, if the contraction along the chain axis in LixZnPc wasactually of the order of 0.5%/kbar and the strain varied linearly with pressure,Li2.35ZnPc could be effectively be reduced to Li2ZnPc in few kBar. Contrarily,between x = 3.5 and x = 4 ∆b/b0 changes of approximately -2%. A twice


Figure 4.17: Evolution of the change in the monoclinic lattice constants, a, b, c inLixMnPc with Li concentration x [28].

larger pressure is required to drive Li3.75ZnPc towards the parameters typicalof Li4ZnPc. Namely, this system is less compressible than Li2.35ZnPc.Now, compressing LixZnPc one fundamentally observes the modification of theelectronic properties induced by a stronger orbital overlap along the chain axis,which is expected to yield an increase of the bare bandwidth W . It must beevidenced that χP is proportional to the density of states at the Fermi levelD(EF ) through the relation

χP = µ2BD(EF )NA. (4.12)

Conversely, in a one-dimensional Fermi Gas, D(EF ) = 1/2EF and close to thehalf band filling W ' 2EF . Hence, D(EF ) ' 1/W and χP = NAµ2

B/W . χP

and D(EF ) should decrease on increasing W in a one-dimensional metal, i.e.namely increasing the pressure. However, the opposite behaviour is observedin the pressure dependence of χP in Li2.35ZnPc, which is rather consistent withthe increase of the EPR spin susceptibility for x → 2 reported in Fig. 4.10.The same agreement is found between the decrease of the Curie constant onincreasing pressure between P = 0 and P = 4 kBar, and the decrease of theCurie constant for x → 2.LixZnPc can’t be described as simple metals and the effect of the correlationshould be considered in order to explain the variation of χP , i.e. D(EF ), upondoping or under pressure. It is clear that applying pressure to Li2.35ZnPc, notonly the bare badwidth W but also the ratio U/W is changed. Therefore, onone hand, on increasing W , U/W is lowered and delocalization is enhanced at


the on site repulsion’s cost. If the system is weakly correlated, the susceptibilityincreases on lowering U/W according to

χ ∝ χP1

1 + U/W. (4.13)

On the other hand, if the system is strongly correlated and in particular U/Wis maximum for x ' 2 one can remind the model developed by Capone etal. [48] for U/W between 0.8 ÷ 0.9 and ascribe the growth of D(EF ) to therenormalization of the bare bandwidth W to a narrow W ∗. Such stronglycorrelated materials can be viewed as insulators with only a residue Z of single-particles, available for conduction and possibly to coupling. Z → 1 for U/W →0 and Z → 0 for U/W → 0.9. Approaching the Metal-Insulator Transition(MIT) from the metallic phase, for U/W → 0.8 Z is already very low, Z ' 0.06.All the quantities get renormalized, having W ∗ = ZW , D(EF )∗ = D(EF )/Zand for the susceptibility

χ(F ) =χP

Z

1

1 + F(4.14)

where F is proportional to the Landau parameter fS(A) by means of the relationF = 6D(EF )fS(A)/Z within the Landau Fermi-liquid theory [109]. F → ∞at the metal-insulator transition. Then, if LixZnPc is strongly correlated forx ' 2, the increase of the susceptibility under pressure in Li2.35ZnPc can beinterpreted as the signal of an increase of U/W ∗ possibly towards the criticalvalue for the MIT, due to the renormalization of Z. However, this picture isonly a starting point since it cannot explain for example the further abruptincrease of both χP and C for P = 6.2 kBar. A more articulated discussionwill be proposed in section 4.8 on the basis of the complementary informationobtained by means NMR investigation.As concerns the effects of pressure in Li3.75ZnPc, Fig. 4.15 shows only a slightdecrease of χP on increasing pressure. This is consistent with the fact thatχP reaches a minimum for x → 4. It should be recalled that, on the basis ofsusceptibility measurements at ambient pressure, this system is already local-ized, as it is close the complete band filling for x = 4. Consequently, it mustbe much less compressible then Li2.35ZnPc. The subtle decrease of the hightemperature background as well as the independence of the Curie constant onpressure confirm the proximity of Li3.75ZnPc to the insulating state.In conclusion, it appears that pressure changes the electronic properties of theLixZnPc compounds towards the one expected for compressed compounds ac-cording to the structural study of Taguchi et al. [28]. Since Li2.35ZnPc is veryclose to x ' 2, this could be very useful to explore the behaviour at the halfband filling even in the absence of stoichiometric Li2ZnPc samples. This is alsofascinating in view of a possible unified phase diagram of the family LixZnPcunder pressure as was discovered for example in (TMTSF)2 and (TMTTF)2X.

4.5. NMR measurements in LixZnPc 97

4.5 NMR measurements in LixZnPc

The NMR measurements presented in this thesis were performed with ApolloTecMag and Bruker MSL 200 spectrometers. Their working principles and thetechnical aspects of the cryogenic apparatus were described in Appendix B.Standard radio-frequency (rf) pulse sequences were used. 7Li and 1H powderspectra were obtained from the Fourier transform of half of the echo signalafter a π/2− τ − π/2 pulse sequence. When the spectra became broader thanthe pulse frequency width, they were reconstructed measuring the modulus ofthe echo as a function of frequency at a fixed field H0. The nuclear spin-spinrelaxation time T2 was derived from the decay of the echo intensity E(2τ)following a spin echo pulse sequence. The Nuclear Spin-Lattice RelaxationRate (NSLRR) 1/T1 was estimated from the recovery of nuclear magnetizationm(τ) after a saturating rf pulse sequence.

4.5.1 NMR spectra in LixZnPc

In this section the shape of the NMR powder spectra and the temperaturedependence of the linewidth for 7Li and 1H in LixZnPc will be illustrated anddiscussed.The comparison between 7Li high temperature spectra in the different com-pounds is shown in Fig. 4.18. The shape is characteristic for all the familyLixZnPc: one notices a central line and two shoulders centered at a frequencyν = νL ± ∆νQ with ∆νQ ' 22 kHz. The ratio between the intensity of thecentral line Ic and the intensity of the shoulders Is is R = Ic/Is ' 6.5 in allthe compounds. The shape doesn’t depend on the applied field, as evidencedin Fig. 4.19, and R is independent on temperature.The independence of ∆νQ on the field intensity demonstrates that the shoul-ders are not due to inequivalent 7Li nuclei with different hyperfine couplings. Infact, if this was the case, ∆νQ should increase with the field as ∆νQ = γ∆KHwhere ∆K = K + σ, with K the paramagnetic shift tensor and σ the chemicalshift tensor. In addition, the position of the shoulders is perfectly symmetricwith respect to the central frequency νL. Therefore, the lineshape is likely de-termined by the quadrupolar term of the Hamiltonian, as it could be expectedfor I = 3/2 of 7Li. The shoulders must be the satellite lines corresponding to| ± 3/2〉 ↔ | ± 1/2〉 transitions. 2∆νQ is the quadrupole coupling frequency.In particular, in a powder the maxima of the broad satellites correspond toan orientation θ = 90 between the magnetic field and the principal axis ofthe electric field gradient. This happens since the powders have a the highestprobability of orientation of the anisotropy axis in the plane perpendicular toH0 and the lowest along H0. Therefore, for θ = 90 in Eq.(2.26), one has forI = 3/2

2∆νQ =e2Qq

2h(1 + η2/2), (4.15)


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FT

Pow

er (

a.u.

)

Frequency (Herz)

7Li NMR Spectra Li

1.85ZnPc

Li2.7

ZnPc Li

3.75ZnPc

-80000 -40000 0 40000 800000

40

80

120

FT

Pow

er (

a.u.

)

Figure 4.18: Comparison between the 7Li NMR spectra in the different LixZnPccompounds (a.u.=arbitrary units).The inset shows the best fit of the line (see text).

-80 -60 -40 -20 0 20 40 60 800

8000

16000

Spe

ctru

m P

ower

(a.

u.)

Frequency (kHz)

7Li NMR Spectra Li

1.85ZnPc T=30K

5T 3T 2T

Figure 4.19: 7Li NMR spectra in Li1.85ZnPc powders at different fields (a.u.=arbitraryunits).


-100000 -50000 0 50000 1000000

7

14

21

FT

Pow

er (

a.u.

)

ν (Hz)

Li2.7

ZnPc, 7Li Spectra ν = 82.7 Mhz

r.t. 90 K 20 K 10 K 5 K

Figure 4.20: 7Li NMR spectra in Li2.7ZnPc powders at different temperatures(a.u.=arbitrary units).

0 50 100 150 200 250 300

12000

14000

16000

18000

20000

22000

FW

HM

(H

z)

T(K)

7Li NMR Linewidth Li

1.85ZnPc

Li2.7

ZnPc Li

3.75ZnPc

Figure 4.21: FWHM of the 7Li NMR spectra in LixZnPc powders as a function oftemperature. The dotted line is a guide to the eye.


The actual electric field gradient experienced by the nuclei is amplified by theSternheimer antishielding factor γ(∞) which takes in account the polarizabilityof the core electrons of Li+ in the external field gradient. Definitely, one hasVzz = eq = V 0

zz[1− γ(∞)] where V 0zz is the gradient due to the external charge

alone. γ(∞) is nearly compound independent and it is tabulated in literaturefor alkali metals. For 7Li γ(∞) = 0.255 and Q = −0.0366×10−8 A2. Reversingthe previous equation, the value of the electric field gradient at the lithiumnuclei can be derived. Since the asymmetry η of the local electric field gradientcould not be determined, assuming η = 0 (axial symmetry) for simplicity, for∆νQ = 22 kHz one has V 0

zz = 4.634×1012 Hz/e A2. The experimental splittingis of the order of others found in literature for chain systems with intercalatedlithium [110, 111].Now, one can notice that the high intensity ratio R=6.5 between the centraland the satellite lines is sizeably increased with respect to the one, R = 4/3,expected for a single crystal. This is possibly caused by the fact that the π/2pulse maximizing the signal of the satellite transitions can be different fromthe one maximizing the signal of the central transition. The LixZnPc spectrawere fitted for simplicity to a sum of 3 independent Lorentzian or Gaussianlines with the fixed intensity ratio R and the same linewidth. An example ofthis fit is plot in the inset of Fig. 4.18. The Lorentzian shape fits well thespectra of Li1.85ZnPc and Li3.75ZnPc in all the temperature range, while inLi2.75ZnPc, for T ≤ 50 K the experimental curves are better described by aGaussian shape (Fig. 4.20).The temperature dependence of the Full Width at Half Maximum (FWHM)obtained from the fit are shown in Fig. 4.21. In all the analyzed samplesthe linewidths have a comparable value, ∆ν ' 12 kHz, at room temperature,then broaden on cooling. The line of Li2.7ZnPc starts broadening below T '150 K and its FWHM doubles at the lowest temperature. On the contrary, inthe other two samples the FWHM is approximately temperature independentdown to T ' 100 K and grows of only 30% at the lowest temperature. Thebroadening extracted by the fit at high temperature can be interpreted as thecombination of a nuclear dipolar broadening and the powder distribution of theparamagnetic shift tensor. At room temperature in all the samples the decayof the echo intensity was exponential with a T2 = 120 ÷ 140 µs. Thus, thetemperature independent nucleus-nucleus dipolar broadening 1/T2 ∼ 1 kHz istoo small to describe the effective FWHM. Indeed, the line broadening mustbe affected by the electron-nucleus interactions. The chemical shift of the coreand bond electrons σ is expected to be anisotropic due to the nature of thearomatic π orbitals of the molecule [102]. However, it should be negligible incomparison with the paramagnetic shift tensor due to the conduction electrons.This should be composed of an isotropic and an anisotropic part, as the LUMOhas a strong π component. The isotropic Kiso is the shift given by s-electronsand depends on the contact hyperfine coupling Aiso by means of


-80000 -60000 -40000 -20000 0 20000 40000 60000 80000

0

200

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600

800

1000

1200

1400

1600

1800

2000

2200

FT

Pow

er (

a.u.

)

T(K)

1H NMR Spectra ν=127.73 MHz, H=3 Tesla

Li1.85

ZnPc Li

3.75ZnPc

Figure 4.22: 1H NMR spectra in LixZnPc powders for H = 3 Tesla (a.u.=arbitraryunits).

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 30025000

30000

35000

40000

45000

FW

HM

(H

z)

T(K)

1H NMR Linewidth H=3 Tesla, ν=127.728 Mhz

Li1.85

ZnPc Li

3.75ZnPc

Figure 4.23: FWHM of the 1H NMR spectra in LixZnPc powders as a function oftemperature.


Kiso =Aiso

~γe

χS. (4.16)

The anisotropic shift, conversely, is the tensor related to the pseudodipolarinteraction with conduction electrons. For its components Ki (i=x,y,z) holds arelation analogous to the (4.16) with the hyperfine dipolar couplings di. Now,no shift of the line center-of-mass, due to the isotropic Kiso, could be estimatedowing to the intrinsic broadening. In turn, the powder linewidth is clearly re-lated to the angular distribution of the anisotropic shift tensor with respect tothe external field. This explains the behaviour of FWHM on cooling, followingthe one observed for the susceptibility in the light of Eq. (4.16).Now, one can verify these conclusions analyzing the 1H spectra. These areshown in Fig. 4.22 for the two samples Li1.85ZnPc and Li3.75ZnPc. Since 1Hhas no quadrupole moment, the spectra are single lines with a Gaussian shapeover all the explored temperature range. At ambient temperature the FWHMis about 25 − 27 kHz, about twice the one observed in 7Li, then, similarly, itbroadens on cooling (Fig. 4.23). A larger linewidth is expected on the pro-ton as effect of a shorter T2 ' 40 µs at room temperature, corresponding to1/T2 ' 4 kHz. Actually, the dependence of the linewidth on temperature witha trend very similar to the one observed for 7Li in the corresponding com-pounds, suggests that hyperfine couplings of the same nature are responsiblefor the line width, i.e. for the shift tensor anisotropy.The dependence of the FWMH on susceptibility reveals some interesting phe-nomena. In Fig. 4.24 and 4.25 the fractional linewidth ∆ν/νL as a function ofthe susceptibility is reported for Li1.85ZnPc and Li2.7ZnPc. It must be specifiedthat SQUID χ were used. In Li2.7ZnPc two different regions are visible: forsmall values of the susceptibility (high temperatures), ∆ν/νL has a steep rise;then, above the value of χ corresponding to T = 50 K the curve bends andfollows a line with a very low slope on increasing χ (lowering the temperature).Differently, in Li1.85ZnPc the relative linewidth is linear in the susceptibilityover all the T range. The figures also show that the same dependence of ∆ν/νL

on χ occurs both in 7Li and 1H. To interpret this behavior, one identifies theobserved ∆ν/νL with the powder averaged paramagnetic shift ∆ν/νL ∝ 〈K〉and estimates an average hyperfine constant 〈A〉 from the Eq. (4.16).The linear fit of the data in Li1.85ZnPc gives 〈A〉 of nearly 10 kGauss for both7Li and 1H. These values are of the order of hyperfine contact coupling with theconduction electrons in metals. On the contrary, in Li2.7ZnPc and Li3.75ZnPcthe low temperature linear dependence of ∆ν/νL on χ yields from the samefit 〈A〉 ' 100− 200 Gauss. These values are of the typical order of magnitudefor dipolar interactions with localized spins. Similarly, the higher temperatureregions of ∆ν/νL as a function of χ, were independently fit with a line. Thefits indicated the increasing of the effective coupling constant by one order ofmagnitude 〈A〉 ' 1 kGauss. This values must be considered as a rough es-timate as in Li2.7ZnPc and Li3.75ZnPc an additional intrinsic contribution tothe macroscopic χ possibly comes from impurities. More reliable values will


0.00042 0.00043 0.00044 0.00045 0.00046

1.5x10-4

2.0x10-4

2.5x10-4

3.0x10-4

DECREASING T

∆ω/ω

L

χ (emu/mole)

Li1.85

ZnPc

1H, H=3 Tesla 7Li, H=5 Tesla

Figure 4.24: Linewidth of the 7Li NMR lines in Li1.85ZnPc powders as a function ofthe susceptibility χ. The lines are fits (see text).

0.000 0.002 0.004 0.006

1.5x10-4

1.6x10-4

1.7x10-4

1.8x10-4

1.9x10-4

2.0x10-4

2.1x10-4

2.2x10-4

2.3x10-4

2.4x10-4

2.5x10-4

2.6x10-4

INCREASING T

T=50 K

T=90 K

∆ω/ω

L

χ (emu/mole)

7Li H=5 Tesla Li

2.7ZnPc

Figure 4.25: Linewidth of the 7Li NMR lines in Li2.7ZnPc powders as a function ofthe susceptibility χ. The lines are fits (see text).


be derived by the analysis of the NSLRR. The synthesis of all these resultssuggests that the sizeable lowering of 〈A〉 in the low temperature region inLi2.7ZnPc and Li3.75ZnPc means a gradual electron localization.Looking at the Fig.4.25 one could consider that such deviation of the linewidthfrom the linear dependence on χ at high temperature, recalls the phenomenonof narrowing originated by activated molecular motions, as it is common inmolecular materials. However, in this case one would expect sizeable modifi-cations of the line broadening with temperature, whereas the FWHM variesweakly with temperatures in the whole temperature range. Furthermore, asthe doping should not affect molecular motions, it would be expected to observethe same effect in all the LixZnPc series at similar temperatures. Meaningfully,also ionic diffusion of lithium cannot take place, since 7Li spectra retain thesame shape over all the T range. As ∆νQ is temperature independent, lithiummust be rather strongly bond to the Pc molecule. The information collectedby the NSLRR 1/T1 will be helpful in the interpretation of the true scenario.

4.5.2 Recovery laws for nuclear magnetization inLixZnPc

The recovery laws for nuclear magnetization m(τ), defined as y(τ) = 1 −m(τ)/m(∞) in LixZnPc, for both proton and 7Li are presented in Fig. 4.26 and4.27. The values of m(τ) were derived from the integration of the echo signalaround the peak. This method was used as it supplies a value proportional tothe echo intensity E(2τ), if performed over the same range for all the echoes,and, at the same time, averages the noise affecting the signal.It is evident that in all the compounds the data follow the same law

y(t) = exp

(−

(τ

T1

)β)

, (4.17)

a stretched exponential function. The dotted curves in Fig. 4.26 and 4.27 arefits performed with the previous function, keeping T1 and β as free parameters.β was found to be constant at high temperature and nearly field independent.Fig. 4.28, presenting the results from the fits of 7Li recovery laws, indicates avariation on cooling below T = 100 K. At high temperatures in Li1.85ZnPc it isfound β ' 0.55, then this value continuously rises up to 0.75 below T = 30 K.In Li2.7ZnPc and Li3.75ZnPc, instead, a decrease is noticed from β ' 0.7 at hightemperature down to β ' 0.55 ÷ 0.6. Nevertheless, these can be consideredas negligible changes, yielding an average 0.65 for the entire series LixZnPc.Analogous observations hold for the recovery laws of 1H. So the weak temper-ature dependence of β won’t be explicitly interpreted and taken into accountin the discussion below.The stretched character of the recovery laws indicates a distribution of relax-ation rates 1/T i

1 with an average relaxation 1/TA1 given by


0 1 2 3 4

0.01

0.1

1

1H NMR, H=3 Tesla Li

1.85ZnPc T=75K

Li3.75

ZnPc T=79K <β>=0.65

y(τ)

τ (s)

Figure 4.26: 1H recovery laws in different LixZnPc compounds.

0 10 20 30 40 50

0.01

0.1

1

y(τ)

τ (s)

7Li NMR, T=40 K Li

1.85ZnPc

Li2.7

ZnPc Li

3.75ZnPc

<β>=0.65

Figure 4.27: 7Li recovery laws in different LixZnPc compounds.


0 50 100 150 200 250 3000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

7 Li NMR Li

1.85ZnPc in 5 Tesla

Li2.7

ZnPc in 5 Tesla Li

3.75ZnPc in 3 Tesla

β

T(K)

Figure 4.28: Temperature dependence of β obtained by the fit of the 7Li recoverylaws in LixZnPc compounds by means of Eq. (4.17).

(1/T1)A =

∫f(1/T i

1)1/Ti1di (4.18)

where f(1/T i1) is a distribution function. A distribution of relaxation rates can

originate from a distribution of hyperfine couplings (in other cases of correla-tion times of the spin fluctuations). In the present case this is likely originatedby the random distribution of the powders, causing a distribution of polar an-gles between the hyperfine dipolar tensor and the external field ~H0. Similarfindings were discussed to justify the line broadening in the previous section. Inaddition, it can be supposed that some nuclei could be magnetically inequiv-alent, with different hyperfine couplings to the conduction electrons. Thisscenario is very difficult to be evaluated since it would require to know pre-cisely the amplitude of the molecular orbitals at each nuclear site from bandcalculations. In the light of the discussion in section 4.3, lithium in LiPc shouldin principle probe different hyperfine couplings, since it would be placed in thecenter of the organic macrocycle along the chain direction. Differently, thedopants in the range x = 0 ÷ 4 should stay outside the ring, bonding to aza-nitrogens. However, no clear evidences of different dynamics emerge from 1/T1

measurements to support this idea.


4.5.3 Nuclear spin-lattice relaxation rates in LixZnPc

The (1/T1)A of 7Li and 1H in the compounds LixZnPc were obtained from thefit (4.17). The data were collected varying temperature at fixed field. A firstobservation of a deviation between data recorded with different thermal cyclesindicated that long thermalization times were necessary after every tempera-ture change. Thus, the measurements were performed in two cycles. In thefirst cycle data were collected on lowering the temperature continuously downto the minimum with a slow cooling rate, while in the second they were col-lected on heating up to room temperature. This afforded a better control ofthe uncertainty of the measured temperature and, effectively, the reproducibil-ity of the values of (1/T1)A.First the temperature dependence of 7Li (1/T1)A, indicated as 1/T1 for brevity,will be shown. In Fig. 4.29 the measurements performed at fixed field in therange 4 K-300 K are presented. It is strikingly evident that the curves 1/T1(T )show significant changes upon varying the lithium doping. 7Li 1/T1(T ) inLi1.85ZnPc at H = 5 Tesla is linear, except for a small departure at ambienttemperature. It is better evidenced in Fig. 4.30 where the solid line is a linearfit. Differently, 7Li 1/T1(T ) in Li2.7ZnPc at H = 5 Tesla is linear above 200 K;at lower temperatures, between 130 K-200 K it flattens, then below 100 K itrises. 1/T1(T ) reaches two distinct maxima at T ' 70 K and T ' 10 K. Finally,for T < 10 K it rapidly drops. At last, 7Li 1/T1(T ) in Li3.75ZnPc at H = 3Tesla is weakly temperature dependent between 150 K-250 K; then below 150

0 50 100 150 200 250 3000.00

0.08

0.16

0.24

0.32

0.40

0.48

0.56

0.64

0.72

7Li 1/T1

Li2.7

ZnPc in 5 Tesla Li

1.85ZnPc in 5 Tesla

Li3.75

ZnPc in 3 Tesla

1/T

1 (s-1

)

T(K)

Figure 4.29: 7Li 1/T1 in LixZnPc as a function of temperature.


0 50 100 150 200 250 3000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1/T

1(s-1

)

T(K)

7Li NMR Li1.85

ZnPc 5 Tesla

Figure 4.30: 7Li 1/T1 in Li1.85ZnPc as a function of temperature for H = 5 Tesla.The solid line is the fit according to Eq.(4.19)

K it smoothly increases until, below 50 K, it starts diverging. The divergenceoccurs at T '10 K, the same temperature for which also 1/T1(T ) in Li2.7ZnPcshows a peak. In Li3.75ZnPc some experimental points were collected belowT = 70 K also at a lower field, H = 1.24 Tesla. The figure 4.31 shows the dataat the two fields for comparison. For T > 30 K the two curves coincide, whilethe low temperature peak is enhanced at the lower field by a factor ∼ 1.3. Theposition of the divergence doesn’t seem to change significantly at lower field.The 7Li 1/T1(T ) behavior will now be discussed. In agreement with the suscep-tibility, 1/T1(T ) in Li1.85ZnPc confirms that the electron system is delocalized.In fact, it follows the Korringa law, typical of a Fermi liquid

1

T1T= C =

γ2N

2A2kB~D(EF )2, (4.19)

where A is the effective hyperfine constant. From a linear fit, the constantC = 0.00193 rad/s·K was evaluated. The hyperfine coupling A can be esti-mated by substituting in Eq. (4.19) the D(EF ) derived from the susceptibilitydata through the relation χP = µ2

BD(EF ). Since EPR susceptibility is ex-pressed in arbitrary units, the value of the high temperature background ofSQUID measurement χ0 = χP was used. One then finds A ' 2 kG. Since thesame trend is observed also in Li2.75ZnPc for T > 200 K, it is suggested thatalso in this system the electrons are delocalized at high temperature. How-ever, lowering temperature electron correlations visibly enhance. In particular,


0 20 40 60 80 100 120 140 160 180 2000.0

0.2

0.4

0.6

1/T

1(s-1

)

T(K)

Li3.75

ZnPc, 7 Li NMR 3 Tesla 1.24 Tesla

Figure 4.31: 7Li 1/T1 in Li3.75ZnPc as a function of temperature for H = 1.24 Teslaand H = 3 Tesla.

the rise below T = 100 K recalls the one observed in the Bechgaard salts inpresence of ~q = 2~kF antiferromagnetic fluctuations. This similarity is evenenforced for 1/T1(T ) in Li3.75ZnPc.To test the validity of this observation a useful method is to analyze the de-pendence of (1/T1) on χ2T . As already explained in the introductory chapters,this allows to verify the linear dependence of the NSLRR on the ~q = 0 excita-tions at high temperature while evidences the low temperature deviations dueto the growth of the ~q = 2~kF excitations. First 1/T1T was calculated for all thesamples (Fig. 4.32). At high temperatures the curves of all the samples tendsto a constant value in all the compounds. As expected, in Li1.85ZnPc 1/T1T isconstant and has a higher value which indicates a larger D(EF ). Then the hightemperature value of 1/T1T diminishes gradually in Li2.7ZnPc and reaches theminimum in Li3.75ZnPc. This is in perfect agreement with the trend derivedfrom susceptibility measurements reported in Fig. 4.10. It is important to no-tice that if Li3.75ZnPc was completely localized one would expect this functionto vanish on increasing T as 1/T1 is constant for a localized paramagnet. Now,recalling the expression (4.9) and dividing by T one has

1

T1T(T ) = C0χS(T )2 + C1(2~kF )χ1D(~kF , ωL). (4.20)

It is clear that in general both the ~q = 0 and ~q = 2~kF susceptibility de-termine the temperature dependence of this function. In particular it wasexplained that in Bechgaard salts χ(~q = 0) susceptibility was enhanced by the


0 50 100 150 200 250 300

0.00

0.02

0.04

0.06

0.08 7Li NMR Li

3.75ZnPc

Li2.7

ZnPc Li

1.85ZnPc

1/T

1T(s

-1/K

)

T(K)

180 200 220 240 2600.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

T(K)

Figure 4.32: 7Li 1/T1T in LixZnPc as functions of T . The solid line is a fit accordingto a power law Tα. The dashed-dotted line is a guide to the eye. The inset showsthe high temperature 7Li 1/T1T values (the dotted lines are guides to the eye).

temperature dependence of the backscattering amplitude g1(T ). This termfundamentally induces a growth of the susceptibility on increasing tempera-ture because it depends logarithmically on T/EF . However, in the case ofLi1.85ZnPc and Li2.7ZnPc it has been shown that susceptibility is weakly tem-perature dependent and in Li3.75ZnPc it rather grows on cooling. Therefore,the g1(T ) dependence in the explored temperature range must be very weakand substantially negligible.It can be noticed that 1/T1T is nearly temperature independent in all thethree compounds down to 100 K, then in Li2.7ZnPc and Li3.75ZnPc it divergesat lower temperatures. This indicates that the ~q = 2~kF correlations are en-hanced and accordingly the spectral components of the susceptibility become

χSDW (2~kF ) ∝ ( TEF

)Kρ−1 À χ(q ' 0). Fitting the 1/T1T data with a power law,

one derives that Kρ → 0 for x → 4, as expected for U/W À 1. This confirmsthat in Li3.75ZnPc the electrons tend to localize [36]. In particular, the secondterm of Eq. (4.20) is nearly constant C1(T ) ' C1, then AF correlations cansurvive up to high temperatures. This material shows a strong similarity withthe Sulphurides (TMTTF)2X which develop a charge localization at Tρ > TN .Kρ seems to remain higher in Li2.7ZnPc. This compound would better resem-ble the (TMTSF)2X with a localization occurring at lower temperature.These analogies indicate that the divergence noted in 1/T1 in both Li2.7ZnPcand Li3.75ZnPc at about T = 10 K could be caused by the inset of a SDW. Thelow temperature region between T = 10 K÷50 K was then fitted to a power


0 10 20 30 40 500.0

0.1

0.2

0.3

0.4

0.5

7Li NMR Li

3.75ZnPc

1/T

1(s-1

)

T(K)

Figure 4.33: 7Li 1/T1 as a function of temperature fitted by means of Eq. (4.21) withTN=10 K and α = −0.3

law1

T1

(T ) ≈ (T − TN)α. (4.21)

This rough estimate gave TN=10 K and α ' −0.3 (Fig. 4.33), smaller thenthe one expected in case of 3D critical fluctuations and possibly explained bythe fact that these systems are strongly one-dimensional. The observation oflow temperature antiferromagnetic-like excitations is also a striking signal ofthe one-dimensionality of the system LixZnPc. In fact, ~q = 2~kF excitationsget strongly depressed as the dimensionality is raised, since the Fermi surfacebecomes more and more curved and its nesting is prevented.Now 1H NMR in LixZnPc will be discussed. The measured 1/T1 are sum-marized in Fig 4.34. All the measurements were performed at H = 3 Tesla.In Li1.85ZnPc 1/T1 rapidly increases upon increasing temperature, reaching amax at T = 40 K. Then it flattens and it starts increasing again above 80K. A straight line marks that above 200 K 1/T1 is linear. On the contrary, inLi3.75ZnPc 1/T1 shows a broad maximum at about T = 120 K, then it decreasesand again increases above 240 K. Thus 1H and 7Li clearly probe different spindynamics. Similar discrepancies are also found in the isostructural LixMnPccompounds, as illustrated in the next chapter. Hence, it seems that the twonuclear species have different hyperfine couplings, no matter if the electronsare localized.The comparison between 1H and 7Li in Li1.85ZnPc is instructive. In Fig. 4.35


0 40 80 120 160 200 240 2800

1

2

3

4

5

1/T

1 (s

-1)

T(K)

1H NMR Li

1.85ZnPc

Li3.75

ZnPc

Figure 4.34: 1H 1/T1T as a function of temperature in Li1.85ZnPc and Li3.75ZnPc forH = 3 Tesla.

0 50 100 150 200 250 3000.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

1/T

1*T

(s-1

)

T(K)

Li1.85

ZnPc

1/T1*T 1H H=3 Tesla 1/T1*T 7Li H=5 Tesla

Figure 4.35: 1H and 7Li 1/T1T as a function of temperature in Li1.85ZnPc. Thedotted lines are guides to the eye.


0 50 100 1500.0

0.5

1.0

1.5

2.0

2.5

Li1.85

ZnPc (1/T

1)1H

-(γ1H/γ7Li

)2(1/T1)7Li

)*3/5

1/T

1(s-1

)

T(K)

Figure 4.36: The result of the subtraction of 7Li [(γ21H/γ2

7Li)1/T1] by 1H 1/T1 inLi1.85ZnPc. The dotted line is the fit according to 1/T1 ∝ C/T exp(−∆/T ), yielding∆= 48 K.

1/T1T is plotted for the two nuclei. As already discussed, 7Li 1/T1T is con-stant, while 1H 1/T1T grows sizeably below T = 150 K, has a peak at aboutT = 30 K and then decreases on cooling further. The high temperature behav-ior indicates that a Korringa law is observed by both nuclei. In particular, for1H the proportionality constant C ' 0.15 (sec·K)−1 is 7.5 times the one foundfor 7Li, which is consistent with the ratio γ2

H/γ2Li = 6.7 (see Eq. 4.19). At high

temperature, then, the two nuclei probe the same dynamics. The deviationof 1H 1/T1 was then estimated subtracting the Korringa term from the data.Fig. 4.36 exhibits the result of this operation. The residual contribution hasan activated behaviour fitted by 1/T1 ∝ 1/T exp(−∆/T ), yielding an activa-tion energy ∆= 48 K. Thus, 1H 1/T1 in Li1.85ZnPc probes both the couplingwith delocalized 2eg electrons and an additional contribution originated eitherby localized or delocalized electrons. If the same operation is performed for1H 1/T1 in Li3.75ZnPc, subtracting 7Li 1/T1 measured in that sample, a broadmaximum with negative values at low temperature is obtained. Therefore ingeneral 1H 1/T1 cannot be described as a sum of the dynamics probed by 7Liand of an independent one.To summarize, 1/T1 measurements in LixZnPc showed that for x ' 2 the sys-tem is delocalized over all the explored temperature range. For x ' 2.75 andx ' 3.75 the system becomes more and more correlated and 1/T1 divergesat TN '10 K, which can possibly be ascribed to SDW. In analogy with the


(TMTTF)X2 Li3.75ZnPc could have a Mott-Hubbard transition at temperatureTρ > TN . Proton NMR always shows additional contributions with respect toLi NMR. The excitations underlying this difference arise from activated pro-cesses that are not completely understood.

4.6. Final discussion 115

4.6 Final discussion

The results obtained by means of the different experimental techniques will becombined to obtain a more detailed overview of the complex phenomenologydeveloping in LixZnPc.The first fundamental evidence arising from this work is that these compoundsbecome metallic upon lithium doping. The alkali electrons are injected in theZnPc LUMO and at high temperature are delocalized in all the LixZnPc com-pounds. In Li1.85ZnPc they remain delocalized down to liquid helium tempera-ture, while in Li2.35ZnPc and Li3.75ZnPc NMR NSLRR meaurements evidencethat localization is possibly achieved. In the case of Li3.75ZnPc, in particu-lar, this is supported by the low temperature Curie-Weiss like behaviour ofthe spin susceptibility. NMR relaxation also shows that LixZnPc compoundsare strongly correlated itinerant electron systems. At low temperature 1/T1

in Li2.35ZnPc and Li3.75ZnPc deviates from the Korringa law, owing to theenhancement of 2~kF -like excitations typical of one-dimensional strongly corre-lated metals. In Li3.75ZnPc a 1/T1 divergence at a low temperature analogousto the one found in Bechgaard salts, where a SDW takes place, is observed. Itshould be remarked that the NMR measurements indicate that the Li donorsare strongly bound to the molecules and no Li diffusion takes place in LixZnPc.Therefore, all the described phenomena are related to charge and spin dynam-ics of the electrons in the 2eg LUMO band.However, there are still some open issues that have to be clarified. In particularthe different temperature dependence of 1H and 7Li 1/T1. On one hand, 7Li1/T1 probes the spin dynamics of the 2eg electrons, on the other hand 1H 1/T1

shows an additional contribution tentatively ascribable to localized spins. Thisterm is markedly large in Li3.75ZnPc 1H 1/T1 and its temperature dependencefollows an activated law, typical of dimers. Localized spin excitation also seemto contribute to the static susceptibility, both the one measured by the SQUIDand the one measured by EPR, which shows a slight maximum at T = 50 K.However, it is not yet clear if the phenomenon detected by NMR relaxationand susceptibility is substantially the same. The presence of 4% LiPc defects,forming dimers, was reported. In this regard, it cannot be assured that LiPcspin dynamics is the source of the additional component detected by 1H 1/T1

since, in principle, also 7Li 1/T1 should detect it. Hence, LixZnPc appear ascomplex systems in which delocalization and localization can coexist possiblydue to some disorder. Nevertheless, it is still possible to derive significant in-formation on the correlated delocalized electrons from the combined analysis of1/T1 and χ(~q = 0, ω = 0) data derived from EPR and SQUID magnetometry.The evaluation of the doping dependence of the density of states at the Fermilevel D(EF ) will be now presented. D(EF ) can be directly deduced from χP

and 1/T1, since χ2P appears in the Korringa relation. However, χP measured

by means of EPR is expressed in arbitrary units. To establish the conversionfactor necessary to convert the χP in emu/mole units the following procedurewas followed. First, the EPR χP expressed in arbitrary units was used to ob-


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

1.0x10-4

2.0x10-4

3.0x10-4

4.0x10-4

5.0x10-4

χ S (

emu/

mol

e)

x (Lithium/molecule)

Figure 4.37: EPR spin susceptibility in LixZnPc in absolute units.

tain the constant C0 = 1/T1χ2P . Then, the variation of this constant gives an

estimate of the variation of the hyperfine coupling A in the samples Li1.85ZnPc,Li2.7ZnPc and Li3.75ZnPc. C0 was found to increase from x=1.85 to x=3.75,giving A(x = 2.7) = 1.27A(x = 1.85) and A(x = 3.75) = 1.675A(x = 1.85).A(x = 1.85) was calculated by means of the Korringa relation. To make thisinitial evaluation the value of χ = 4.24 · 10−4 emu/mole of the room temper-ature SQUID susceptibility was adopted, giving A(x = 1.85) = 2 kGauss.The corresponding density of states for x=1.85, assuming these values, isD(EF ) = 8.19 · 1012erg−1. Then, χP and D(EF ) were calculated also in theother samples.χP in Li3.75ZnPc was found to coincide with the high T SQUID susceptibil-ity, χPauli = 1.338 · 10−4emu/mole, while in Li2.7ZnPc it was lower, namelyχP = 2.617 · 10−4emu/mole. Nevertheless, this discrepancy is solely causedby the systematic error affecting the value of the SQUID χ at high temper-atures, associated with the contribution of the sample holder. Probably inLi1.85ZnPc and Li3.75ZnPc the sample holder contribution is comparable, whilein Li2.7ZnPc it is not. Then, considering the perfect consistency of the valuesfound in the former samples, the initial value A(x = 1.85) = 2kGauss usedto make all the calculations can be considered as a good value. In conclusion,the conversion factor of the EPR susceptibility has been estimated 1 arb.unit=1.174 · 10−6 emu/mole.The final calculations of χP and D(EF ) per electron are presented in Fig. 4.37


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

D(EF)

D(E

F)(s

tate

s/eV

)

x

0.0

0.2

0.4

0.6

0.8

1.0

Z(x)/Z(x~4)Z

(x)/Z(x~

4)

Figure 4.38: Density of states in LixZnPc at the Fermi level per electron, obtainedfrom susceptibility data, and the quasi-particle residue Z(x), normalized to thevalue Z(x ∼ 4). The lines are guides to the eye.

and 4.38. D(EF ) has been estimated dividing the D(EF ) obtained by theaforementioned method by the number of electrons per molecule, namely thedoping x. D(EF ) reaches a peak for x = 2, while diminishes for x → 0 andx → 4, describing a bell shape. Anyway, D(EF ) is finite for every x 6= 0 or 4,indicating that electron injection always drives the system towards metallicity,when the 2eg band is partially occupied. From the experimental point of view,these are the first results achieved in bulk compounds of alkali doped MPcs,indicating an insulator to metal crossover upon doping the pristine compound.Up to now the onset of metallicity in doped thin films is controversial, sincerecent PES measurements affirm that KxZnPc films are insulating [112], atvariance with the transport measurements [42]. The different results obtainedby means of the two techniques have been ascribed by the authors of Ref. [112]to inhomogeneity of the K distribution at the film surfaces or to the formationof K clusters or layers during doping, which can cause a surface conduction.On the contrary, Craciun et al. [42] suggest that the suppression of conductiv-ity on approaching the complete filling for x = 4 excludes that it is generatedin superficial K conducting layers. In this case, on increasing doping, the con-ductivity shouldn’t disappear but rather enhance or at least be kept constant.Remarkably, they found that conductivity is particularly enhanced for x ' 2,


a result that is consistent with the doping dependence of D(EF ) presented inthis work. Anyway, it must be considered that in one-dimensional conductioncan be strongly dependent on the concentration of traps and breaks of chains.The formation and the quantity of these defects can be hardly controlled underfast ionic implantation, adopted in [112] and [42]. On the contrary, the liquidphase synthesis of bulk materials is a low energy and slow processes and it isexpected to grant a better homogeneity and to minimize the quantity of defectsin the sample by several purification and annealing processes. Therefore, thebulk samples examined in this work should be the optimal probe to analyzethe true phase diagram of bulk LixZnPc.The combination of metallicity and strong correlations in LixZnPc opens anew discussion on the onset of strongly correlated superconductivity in thesesystems. The available data suggest that no transition to a superconduct-ing state takes place down to liquid helium temperature. This is probably theconsequence of the pronounced one-dimensionality of LixZnPc, whereas the su-perconducting transition would require the onset of 3D fluctuations. A loweranisotropy, for example, is known to be responsible of the onset of supercon-ductivity in (TMTSF)2ClO4. Here the interchain hopping t⊥ is larger than inthe other (TMTSF)2X compounds, which instead evolve towards localizationand SDW state at low temperatures. These compounds can become supercon-ducting only under high pressure, by increasing the hopping integral t⊥ andtherefore increasing dimensionality. The susceptibility of LixZnPc under pres-sure was measured and it was observed that in Li2.35ZnPc, χP increases withpressure. It was discussed that if LixZnPc are strongly correlated the increaseof D(EF ) close to the half filling can be described by a renormalization ofthe bandwith W ∗ = ZW ∼ 4ZEF = 4E∗

F in the proximity of a MIT, whichaccompanies the reduction of the quasi-particle residue Z(x) (Fig. 4.38).The scenario can now be better discussed in the light of the D(EF ) final calcula-tion and 1/T1 measurements. Now, since 1/T1 and susceptibility in Li1.85ZnPcindicate an almost completely delocalized system, the same is expected forx ' 2. On the other hand, 1/T1 and SQUID measurements for x ' 2.7 in-dicate an increase of the correlations and localization. Then if Li2.35ZnPc ishalfway between the two and pressure should drive the system towards the con-dition expected for x → 2, pressure should lower the ratio U/W in Li2.35ZnPcpowders. However for P > 4 kBar, χP and D(EF ) abruptly increase with P .This increase could be associated with the suppression of a pseudogap.The occurring of a pseudogap has been claimed by Capone et al. [52] to de-scribe the phase diagram of strongly correlated systems near a Mott transition.This diagram is shown in Fig. 4.39. The pseudogap phase is expected to appearbetween the Fermi liquid and the Mott insulating phase near the half band fill-ing for U/W > 1. In the case of LixZnPc this gap should be the consequenceof a pairing of itinerant electrons in singlets, as in singlet superconductors.However this pairs should not be able to give rise to superconductivity due tothe large U/W ratio and strong anisotropy. This diagram indicates that for


Figure 4.39: Phase diagram as a function of U = W and doping δ = n − 2 atJ = 0.05W The thick vertical line marks the singlet Mott insulator. The inset shows,for U = 0.92W , the superconducting gap ∆ divided by a factor 10−3 and the Drudeweight D (normalized to the noninteracting value) as functions of doping [52].

values 0.8 < U/W < 1 the pseudogap can arise aside superconductivity verynear to the value x = 2. Finally, it is suppressed for U < 0.8. Then, one canpossibly ascribe the enhancement of χP in Li2.35ZnPc at high pressures to thesuppression of this pseudogap, while driving the system to a lower U/W ratio.On the other hand, it has to be observed that upon increasing P the sys-tem seems to be more strongly correlated since the Curie Constant C ' 0.15emu/mole·K deduced by susceptibility measurements is 3 orders of magnitudelarger than in the uncompressed Li1.85ZnPc and at least one order of magni-tude larger than in all the other LixZnPc compounds. This value of C wouldcorrespond to 40% of the molecules bearing a localized spin S = 1/2 or 15%of the molecules bearing a S = 1 spin. This is surely an intrinsic effect andcannot be ascribed to impurities since these values are very high. It is notclear if the increase of C is related to a localization or to a modification in theT dependence of χ close to a MIT. The understanding of the effect of high Pclose to x=2 will be the purpose of the future research.


Chapter 5Magnetization and NMRmeasurements in LixMnPc

Magnetic measurement in LixMnPc compounds are expected to depict a phe-nomenology even more complex than the one characterizing LixZnPc. While inLixZnPc systems magnetism is determined solely by the electrons injected inthe 2eg LUMO, in LixMnPc Mn2+ ions have a non-zero S ground state. In thepristine compound MnPc S = 3/2, these localized spins correlate ferromag-netically by means of superexchange pathways created by the superposition ofatomic orbitals along the chains [23]. Basically, alkali-doping could modify themagnetic properties in different ways. The electrons donated by the dopants tothe MnPc molecules could fill the Mn2+ vacant orbitals and remain localized.Alternatively, they could fill the 2eg LUMO band and possibly delocalize overthe MnPc chains.Recent publications demonstrated a change in the Mn2+ spin ground state fromS = 3/2 towards S = 5/2 for x →2 and the transformation of intermolecularcorrelations from ferromagnetic to antiferromagnetic (AF) [28]. This impliesthe transfer of the alkali electrons to the 2eg LUMO but it isn’t enough yet topredict their fate in view of the delicate balance between several competing on-site interactions in MPcs, ranging from Coulomb repulsion, Hund’s exchange toJahn-Teller couplings [46]. Indeed, the magnetization measurements have thefundamental purpose to determine the spin ground state of the doped LixMnPccompounds. Nevertheless, NMR measurements should be crucial in determin-ing the spin dynamics associated either with the localized Mn2+ electrons orwith the 2eg electrons.

121

122 5. Magnetization and NMR measurements in LixMnPc

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000.0

0.2

0.4

0.6

0.8

µ(em

u*G

auss

)

H (Gauss)

Li0.5

MnPc 15K 7K 3,5K 2,5K

Figure 5.1: The magnetic moment of Li0.5MnPc powders as a function of the field,measured at different temperatures.

5.1 The susceptibility of LixMnPc

Magnetization measurements were performed in LixMnPc by using the samemethods explained in chapter 4 for LixZnPc. Also in the case of LixMnPc itwas not possible to obtain the magnetization of the sample holder separately.Some measurements were performed on the pure MnPc compound wrappingsome milligrams of as-purchased powder in a piece of teflon. The magnitudeof the sample signal was high enough to neglect the contribution from Teflon.The magnetization measurements in Li0.5MnPc will be presented first. Themagnetic moment µ as a function of the field H was measured at several tem-peratures, as shown in Fig. 5.1. In the range 0-1 Tesla the curve is linear downto 15 K, while for lower temperatures it is linear up to low fields, H ' 500Gauss. At higher fields, the data deviate from linearity, assuming a downwardcurvature. The magnetic moment µ of Li0.5MnPc at T = 3 K in the full rangeH = −6÷ 6 Tesla is shown in Fig. 5.2. The cycle is closed, thus no long rangeferromagnetic order is expected for this system at this temperature. One alsonotices that the value of µ is not completely saturated at the maximum appliedfield H = 6 Tesla at T = 3 K.The magnetization was also measured at constant field H = 1000 Gauss in thetemperature range T = 2÷300 K. Due to the non linearity of M(H), the rela-tion χ = M/H must be considered as an approximation, but this notation willbe used hereafter to indicate the quantity M/H. χ in Li0.5MnPc is reportedin Fig. 5.3 together with the one measured in the pristine MnPc in the same

5.1. The susceptibility of LixMnPc 123

-60000 -40000 -20000 0 20000 40000 60000-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

µ(em

u*G

auss

)

H(Gauss)

Li0.5

MnPc 3K

Figure 5.2: The magnetic moment of Li0.5MnPc powders as a function of the fieldat T = 3 K.

1 10 1000.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

M/H

(em

u/m

ole)

T(K)

H=1000 Gauss

Li0.5MnPc

MnPc

0 50 100 150 200 250 3000

200

400

600

800

1000

1200 Li0.5

MnPc

1/χ

(mol

e/em

u)

Figure 5.3: Temperature dependence of the susceptibility, defined as χ = M/H, inMnPc and Li0.5MnPc for H = 1 kG. (Inset) Temperature dependence of the inverseof the susceptibility in Li0.5MnPc. One can notice that for T > 15 K the Curie-Weisslaw is obeyed [113].


0 10 20 30 40 50 60 70 802

4

6

8

10

12

14

16

Li0.5MnPc

100 Gauss

2000 Gauss

χT(e

mu*

K/m

ole)

T(K)

Figure 5.4: Temperature dependence of χT in Li0.5MnPc for H = 100 G and H = 2kG.

experimental conditions. It has the same shape in both the compounds; in par-ticular, it should be noticed that below T ' 10 K the value of χ progressivelysaturates on cooling. In addition, the magnetization of MnPc for T < 30 Kis clearly depressed in comparison with the magnetization of Li0.5MnPc at thesame field. Hence, the built-up of M along H conditions seems to be differentin the two samples for effect of different magnetic couplings or anisotropy. Theinset of Fig. 5.3 shows also the inverse χ−1 as a function of T . The function islinear in T above T ' 15 K, therefore, in this region it is conveniently fit bythe Curie-Weiss law

χ = χ0 +C

T −Θ, (5.1)

yielding C = 2.34 emu·K/mole and Θ = 7.5 K [113]. χ0 is the sum of Van-Vleck and diamagnetic contributions, which are assumed weakly temperaturedependent. The susceptibility of the Pc organic macrocycle and of the ionicclosed shells sums up totally to χ = −2.9 · 10−4 emu/mole [23]. The positiveΘ implies a ferromagnetic exchange coupling between the nearest Mn2+ ionsalong the chains. From the value of C, assuming a Lande factor close to 2,one estimates a spin per molecule slightly larger than 3/2. As S = 3/2 is thevalue of the spin ground state of the pristine compound MnPc (see chapter 1),this increase has to be associated with the electron doping. Taking z = 2 forthe number of nearest neighbors, from the formula J = 3Θ/[2zS(S + 1)], oneevaluates J = 1.5 K.The change of the temperature dependence of the susceptibility on cooling is


well evidenced by plotting the χT as a function of temperature in Fig. 5.4 fordifferent fields. On lowering temperature it increases as expected for Θ > 0.For different fields the temperature dependence is the same on cooling untilT ' 10 K, then at lower temperature χT taken at the highest field, H = 2000Gauss, deviates, reaches a maximum at T ' 7 K and subsequently decreases.The curve at H = 100 Gauss, on the contrary, increases again and then reachesthe maximum at T = 5 K. The change in the slope of χT at low temperaturescannot be ascribed to the onset of an antiferromagnetic coupling among adja-cent chains. This can be excluded in the light of recent studies in the dopedsystems [28], which are retained magnetic one-dimensional systems, and of theanalysis of the nuclear relaxation as a function of the field presented in thiswork (see chapter 5.5). Basically the onset of antiferromagnetism at thesetemperature would require a coupling of the same order of the ferromagneticintrachain coupling and the system would turn to bidimensional. Then, ex-cluding this possibility, the observed shift at higher temperatures of the χT onincreasing field points to anisotropy.The magnetic anisotropy is taken into account adding a term to the Heisenbergisotropic Hamiltonian of the form

H =∑ij

J ~Si · ~Sj −DS2z , (5.2)

where ij are the set of site indices restricted to the nearest neighbors alongthe chain. If positive, D indicates the energy gain of spins aligned along z,which is determined by the symmetry of the local crystal field probed by themagnetic ions. Then, on cooling the system one expects that for DS2

z ≥ kBTthe magnetic Mn2+ will be freezed along the anisotropy axis z. When a mag-netic field is applied along a direction different from z, the anisotropy acts asa barrier to the total alignment of the spins along the field and the suscep-tibility is depressed. Anisotropy can then explain the saturation of χ at lowtemperatures in both powders of Li0.5MnPc and MnPc where the distributionof angles between the anisotropy tensor and the field is random. Furthermore,in order to explain the shift of the maximum of χT at higher temperatureon increasing the field (Fig. 5.4), one can consider that at a higher field theratio gµBH/DS2 increases, then the Zeeman energy and anisotropy competeat higher temperatures. This competition is signaled by a change of slope ofthe χT function.This phenomenology was already known in MnPc, defined as a one dimen-sional canted ferromagnet in old papers [23, 25]. Recently, AC susceptibilitystudies have lead to a different interpretation of the magnetism in MnPc, aslow frequency dynamics were observed, similar to the ones of glassy systems[28]. These are due to the slow relaxation of magnetization at low temperatureacross the anisotropy barrier EA = DS2

z with an activation time τ followingτ = τ0 exp(EA/T ). This dynamics was recognized observing the shift of thepeak of the χ”(ω) at high temperature on increasing the frequency, i.e. onreducing τ = ω−1. More in detail, it was also concluded that the relaxation of


0 1000 2000 3000 4000 50000.695

0.700

0.705

0.710

0.715

0.720

µ (e

mu*

Gau

ss)

Elapsed Time (seconds)

Li0.5

MnPc 3K, H=1 Tesla

after ZFC

Figure 5.5: Recovery of µ along the field after cooling in H = 0 Gauss and suddenlyapplying a field of H = 1 Tesla at T = 3 K.

magnetization is the result of a local freezing of the Mn2+ magnetic moments,owing to the fact that EA À J (J the exchange coupling along adjacentmolecules). Due to disorder the system can be seen as a collection of chainswith a different local anisotropy barrier and the slow dynamics of the systemis in turn described in terms of local relaxation in a time varying from chainto chain. This gives rise to a relaxation of the macroscopic magnetization in atime τav average over the distribution of the relaxation times. The phenomenonwas then verified also in the doped Li0.5MnPc by means of DC magnetometry.After a procedure of cooling in H = 0 down to T = 3 K and the subsequentapplication of H = 1 Tesla, the magnetization was measured as a functionof the elapsed time. The acquired data are plotted in Fig. 5.5, fitted by thefunction

M(t) = M(∞)[1− exp(−t/τ)]] (5.3)

with a single relaxation time τ , giving τ = 1050±90 s [113]. This is comparablewith the average relaxation time measured by means of AC susceptibility at thesame temperature, τ = 1070 s. The relation between the quantities measuredby means of the two techniques can be clarified by the following considerations.By applying a unit step-like varying field, H(t) = H0 = 1 at t = 0, themagnetization must evolve as [71]

M(t) =

∫ t

0

m(τ)dτ =1

2π

∫ t

0

[∫ +∞

−∞χ(ω)eiωτdω

]dτ. (5.4)


0 10000 20000 30000 40000 50000 600000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

µ (e

mu*

Gau

ss)

H (Gauss)

T=3 K Li0.5MnPc Li2MnPc

Figure 5.6: A comparison between µ(H) in Li0.5MnPc and Li2MnPc at T = 3 K. Thepowder samples have nearly the same weight.

Now, in MnPc and Li0.5MnPc at a fixed temperature the bell shaped dynamicsusceptibility χ”(ω) is strongly peaked at a value ωav, the central value of thedistribution of frequencies of the microscopic relaxation processes. Converselyχ′(ω) is an odd function with respect to ωav. Therefore, in a simplified picture,the Fourier transform of χ(ω)is a function decaying with the time 1/ωav = τav.In particular with m(τ) = χ0

τave−t/τav , integrating according to Eq. (5.4), one

obtains Eq. (5.3) with τ = τav. Certainly, in a complete description one shouldconsider the effects of the width of the χ”(ω), 〈∆ω2〉. Taguchi et al. also de-duce from the experiments an average barrier EA = 90 K in MnPc. Looking atthe experimental data, it could be inferred that this barrier is possibly higherin MnPc than in Li0.5MnPc, significantly attenuating the susceptibility. Apossible explanation of this behavior will be given by means of the analysis ofNMR 1/T1 in this system.The magnetization measurements in Li2MnPc will be presented hereafter. Asin Li0.5MnPc, M as a function of the field was measured at several temper-atures. In Fig. 5.6 the comparison between the data acquired in the twosamples is presented. M(H) in Li2MnPc displays a different field dependence,having a weak, non linear trend at low fields and, on the contrary, a nearlylinear dependence for H > 4 Tesla. It was noticed that the susceptibility atH = 1000 Gauss shows a steeply rising Curie tail at low temperature maybecaused by undesired ferromagnetic contributions. Thus it was decided to ana-lyze the high field region and acquire separately the susceptibilities at H = 4Tesla and H = 6 Tesla. The ratio M(H = 6 Tesla) −M(H = 4 Tesla)/∆H


0 10 20 30 40 50 60 70 80 90 100

0.1

1

χ (e

mu/

mol

e)

T (K)

Li0.5MnPc

Li2MnPc

Figure 5.7: Susceptibility of Li0.5MnPc powders for H = 1000 Gauss and of Li2MnPcpowders, defined as the slope of M(H), for H = 4 Tesla [114].

was then evaluated; it is reported in Fig. 5.7. This operation should subtractthe spurious contributions and exalt the susceptibility given by the effectivemagnetic couplings between the Mn2+ ions.The susceptibility in Li2MnPc has the form of the Fisher susceptibility of one-dimensional antiferromagnets for large spins [115, 114]

χN(T ) =Ng2µ2

B

12KBT

1 + u(K)

1− u(K)(5.5)

where u(K) = coth(K)− 1/K and K = 2JS2/kBT . The parameters obtainedby the fit were S = 2.07 ± 0.01 and θ = −11.32 ± 0.05 K, comparable withthe evaluations of Taguchi et al. [28] who report a value S = 5/2 in Li2MnPc.At this point, to briefly summarize, from the results obtained by susceptibil-ity measurements it is inferred that, on doping MnPc, the in-chain couplingschange from ferromagnetic to antiferromagnetic. The spin of the molecule alsovaries from S = 3/2 → 5/2 for x → 2.A discussion of the data shall now be presented in the light of the configurationof MnPc molecular orbitals [113]. β-MnPc in its ground-state has a completelyfilled a1u level, a half-filled a1g level and two degenerate half-filled 1eg levels[30, 23]. Differently form ZnPc, MnPc is an open shell molecule, with severalof the molecular orbitals containing the atomic orbitals of the transition metalstill unoccupied. Upon doping the modification of the magnetic properties ofthe pristine molecule is determined by the injection of the electrons in the low-energy vacant orbitals. The spin configuration will depend on which orbitals


a1ua1g1eg2eg

b1g

S=5/2S=5/2S=5/2S=5/2

a1ua1g1eg2egA B

Figure 5.8: The two possible electronic spin configurations of the S = 5/2 groundstate in Li2MnPc.

are filled, taking also into account that doping modifies the crystal field probedby the Mn2+. In particular, the addition of ions in the structure changes thesymmetry of the field and the atomic levels of the transition metal must belowered or raised accordingly to attain a more favorable distribution of theelectronic density. The magnetization data show that in MnPc the upper en-ergy levels, as the two-fold degenerate 2eg(π) LUMO and the b1(dx2−y2) states,are filled upon doping. This scenario differs from the one of MPc containingtransition metal ions with more d-electrons than Mn2+. In fact, in FePc andCoPc Hund rule is not obeyed and the other electrons occupy the 1eg levels,so that one has an S = 1 and an S = 1/2 configuration, respectively. On thecontrary, the filling of the LUMO in the doped MnPc justifies the change ofsign of Curie-Weiss temperature with increasing the lithium content: previ-ous works report that the filling of the LUMO, which overlaps with the a1g

orbitals of adjacent molecules along the chain, can induce antiferromagneticcorrelations [22]. From the levels diagram one also notices that the configura-tion S = 2 upon the injection of two electrons in the molecular orbitals is notobtainable since an odd number of electrons occupies the outermost molecularorbitals. Hence, the achievement of the S = 5/2 state for x → 2 will be ratherconsidered, being the parameter extracted by the Fisher law S = 2.07 an un-derestimate.Taking into account the crystal-field modifications induced by Li+, the S = 5/2state could in principle result from two different configurations, shown in Fig.5.8. On one hand, if Li+ gives rise to a more isotropic crystal-field and the en-ergy of the LUMO and b1(dx2−y2) is lowered, then an electron can be promoted


from the a1u to the b1(dx2−y2) level, in accordance to Hund’s rule (scheme A inFig. 5.8). In this case the S = 5/2 state would result from the occupancy ofall d-character orbitals. The electrons injected by Li doping, which fill 2eg(π)LUMO, would not contribute to the total molecular spin once they form asinglet state, a situation similar to the one observed in the fullerides [116].Singlet formation can occur if Jahn-Teller coupling for the 2eg(π) state withthe B1g and B2g β-MnPc vibrational modes is larger than its Hund coupling[46]. On the other hand, an S = 5/2 state is also consistent with a tripletstate for the electrons on the LUMO, provided that b1(dx2−y2) energy is notsignificantly lowered by Li doping and this latter level remains empty (schemeB in Fig. 5.8).The configuration with two Li+ ions close to an MnPc molecule gives rise to acrystal-field which decreases the energy of the b1 states and also of the LUMO.Accordingly, one should expect that the energy of S = 5/2 state is possiblylower than the one of S = 2 and S = 3 states, associated with one unpairedelectron on the LUMO. For this reason it is expected that for non integer dop-ing a separation between x = 0 and x = 2 chain segments might occur. Besidesit is likely that the progressive increase in the Curie constant, upon increasingLi content from x = 0 to x = 2, actually originates from the contribution ofS = 3/2 and of S = 5/2 chain segments. Although from magnetization dataalone one cannot say if both spin configurations coexist at the microscopiclevel, it has been noticed that the LixMnPc susceptibility data can be fit withχ = (x/2)χLi2 + [(2 − x)/2]χLi0 , for T À 10 K. This interpretation is cor-roborated by the observation of two different dynamics in nuclear spin-latticerelaxation measurements, as will be discussed hereafter.

5.2 NMR spectra of LixMnPc1H and 7Li NMR spectra of LixMnPc were obtained with the same methodsexplained in chapter 4 for LixZnPc. In 7Li spectra in LixMnPc don’t displaya structure. Figure 5.9 shows an example of two 7Li lines in Li0.5MnPc atdifferent temperatures for H = 1.3 Tesla. At T = 50 K the line was obtainedby the Fourier transform of half of the echo, while at T = 11 K it was recon-structed measuring the echo intensity at different frequencies, keeping the fieldconstant. The intensity of the two spectra were suitably rescaled to allow thecomparison. At the higher temperature, the spectrum is Lorentzian, while atlow temperature it turns to Gaussian. The same shape was observed also inLi2MnPc.1H spectra in LixMnPc are Gaussian in the whole temperature range. In Fig.5.10 two 1H spectra in Li0.5MnPc at different temperatures are reported. Apartfrom a consistent broadening, it can be noticed that the ”center of mass” ofthe line shifts to lower frequencies on cooling.The temperature dependence of the Full Width at Half Maximum (FWHM)for 7Li and 1H in the two compounds are presented respectively in Fig. 5.11

5.2. NMR spectra of LixMnPc 131

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.80

200

400

600

800

1000

1200

1400

1600

1800

Ech

o am

plitu

de o

r F

T P

ower

(a.u

.)

Frequency (MHz)

Li0.5MnPc ν0=21.7 MHz 11.5 K 50 K

Figure 5.9: 7Li NMR lines in Li0.5MnPc powders at different temperatures(a.u.=arbitrary units). The zero frequency on the x axis corresponds to ν=21.7MHz, the Larmor frequency for H = 1.3 Tesla.

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.40

10000

20000

30000

40000

50000

60000

FT

Pow

er (

a.u.

)

Frequency (MHz)

Li0.5

MnPc

1H NMR ν= 63.864 MHz

180 K 40 K

Figure 5.10: 1H NMR lines in Li0.5MnPc powders at different temperatures(a.u.=arbitrary units). The zero frequency on the x axis corresponds to ν=63.86MHz, the Larmor frequency for H = 1.5 Tesla.


0 50 100 150 200 250 3000

50

100

150

200

250

300

350

400

FW

HM

(kH

z)

T(K)

7Li NMR Li

0.5MnPc, H=1.35 Tesla

Li2MnPc, H=7 Tesla

Figure 5.11: FWHM of the 7Li NMR lines in LixMnPc powders as a function oftemperature. The dotted line is a guide to the eye. The solid line is a Curie-Weisslaw reproducing the susceptibility data in Li2MnPc.

0 20 40 60 800

100

200

300

400

500

600

700

FW

HM

(kH

z)

T(K)

1H NMR Linewidth Li

0.5MnPc, H=6500 Gauss

Li2MnPc, H=6100 Gauss

Figure 5.12: FWHM of the 1H NMR lines in LixMnPc powders as a function oftemperature. The dotted line is a guide to the eye. The solid line is a Curie-Weisslaw reproducing the susceptibility data in Li2MnPc.

5.2. NMR spectra of LixMnPc 133

20 40 60 80 100 120 140 160 180 200 220 240 260 280 300-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

∆ν(M

Hz)

T(K)

1H NMR Li0.5MnPc

3 Tesla

Figure 5.13: Shift of the 1H NMR lines in Li0.5MnPc powders as a function of tem-perature for H = 3 Tesla.

and Fig. 5.12. The lines have been drawn to mark the shape of the experimen-tal curves: the dotted line, which is a guide to the eye, follows a shape similarto the one of the susceptibility in the Li0.5MnPc. The solid lines instead areCurie-Weiss functions which describe the susceptibility in Li2MnPc obtainedwithout the subtraction explained in chapter 5.1. Clearly the FWHM in boththe samples grows on cooling, proportionally to the susceptibility. In both thesamples, the echo intensity E(2τ) was observed to decay with τ following anearly Gaussian law, with a characteristic decay time TG

2 = 165 ± 10 µs for7Li and TG

2 = 45 ± 3 µs for 1H, around 100 K. Therefore, the nuclear dipolarbroadening is of the order of 1 kHz for 7Li and 3.5 kHz for 1H.As discussed for LixZnPc (see chapter 4.6), the broadening is then induced bythe anisotropic component of the paramagnetic shift tensor, related to the hy-perfine interaction with the electrons. This magnetic broadening in LixMnPcis larger than the splitting between the central 7 Li line and the satellite lines∆νQ ' 22 kHz estimated in the LixZnPc series. The difference is clearly as-cribed to the presence of manganese. Furthermore, the shift of the position ofthe 1H spectra is due to the isotropic part of the paramagnetic shift tensor.The temperature dependence of this shift is reported for H = 3 Tesla in Fig.5.13. This was detected only for the 1H lines of Li0.5MnPc. Now, relating theshift and the FWHM to the susceptibilities, as already done in the analysis ofLixZnPc spectra, one obtains a linear relation. This is depicted in Fig. 5.14,where both the fractional shift ∆K and fractional line broadening ∆ω/ωN of1H in Li0.5MnPc are illustrated. For both nuclei one finds an effective coupling


0.00 0.02 0.04 0.06 0.08

0.0000

0.0005

0.0010

0.0015

0.0020

∆K, ∆

ω/ω

N

χ(emu/mole)

Li0.5

MnPc, 1H NMR H= 3 Tesla

∆K ∆ω/ω

N

Figure 5.14: Fractional linewidth and shift of the 1H NMR lines in Li0.5MnPc powdersas a function of the susceptibility χ [113]. The lines are linear fits.

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.180.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

0.0045

0.0050

0.0055

0.0060

0.0065

0.0070

0.0075

∆ωN/ω

N

χ(emu/mole)

Li2MnPc

7Li, H=7 Tesla

1H, H=6500 Gauss

Figure 5.15: Fractional linewidth of the 1H and 7Li NMR lines in Li2MnPc powdersas a function of the susceptibility χ. The lines are linear fits.

5.3. The recovery laws in LixMnPc 135

A ' 100 Gauss, which is of the order of magnitude of the dipolar hyperfinefield generated by localized electron spins. Performing the same operation forthe FWHM of the 7Li and 1H spectra, measured at H = 7 Tesla and H = 6500Gauss respectively, in Li2MnPc one finds again a linear relation between thefractional linewidth ∆ω/ωN and the susceptibility, shown in Fig. 5.15. Theeffective hyperfine coupling of 7Li results A = 170 Gauss, comparable to theone found in Li0.5MnPc. The effective hyperfine coupling of proton is larger,A = 440 Gauss. However, it should be specified that this value represents arough estimate.To summarize, in the LixMnPc in 7Li spectra show no signature of the satel-lite transitions, since the magnetic broadening proportional to χ is much largerthan the splitting ∆νQ ' 22 kHz estimated in the LixZnPc series. 1H spectrahave a Gaussian broadening without any structure. A shift of the line is ob-served for effect of the isotropic component of the electron-nucleus interaction.The effective coupling of 7Li and 1H in Li0.5MnPc is of the order of A ' 100Gauss, which is typical of dipolar interactions with localized spins. A couplingof the same order is evaluated for 7Li in Li2MnPc, while A = 440 Gauss of 1His increased with respect to the lightly doped sample.

5.3 The recovery laws in LixMnPc1H and 7Li NMR recovery laws in LixMnPc were obtained with the samemethods explained in chapter 4.5 and 4.7 for LixZnPc.In the LixMnPc family the recovery laws were found to be the sum of twocontributions, a stretched exponential and a simple exponential, for both 1Hand 7Li, yielding the general law [113]

y(τ) = Ae−(τ/T ′1)β

+ (1− A)e−(τ/T ′′1 ). (5.6)

In Li0.5MnPc 7Li recovery law was observed to be a stretched exponential(A = 1) with β ' 0.45 over all the temperature range. On the contrary, 1Hrecovery law was of the form (5.6) with A ' 0.8 and β ' 0.5 over the exploredtemperature range. Examples of the different recovery laws are presented inFig. 5.16 and 5.17.For 1H T ′

1 = T s1 was the shortest relaxation time, giving 1/T s

1 of the order of500 ÷ 5000 s−1, while T ′′

1 = T l1 was the longest, giving 1/T l

1 of the order of10 ÷ 100 s−1. In Li2MnPc, instead, both 1H and 7Li recovery laws, shownin Fig. 5.18 and 5.19, were of the form (5.6). In Fig. 5.19 the dotted lineshows the deviation from a single stretched exponential function. In the caseof 1H it was found A ' 0.3 and β ' 0.4, T ′

1 = T l1 was the longest relaxation

time and T ′′1 = T s

1 the shortest. Nevertheless, in the case of 7Li it was foundA ' 0.7 and β ' 0.4; T ′

1 = T l1 was the longest relaxation time and T ′′

1 = T s1 the

shortest. 1/T s1 of both 1H and 7Li are respectively of the order of thousands

and hundreds of s−1 in all the temperature range, indicating a strong enhance-ment with respect to 1H 1/T s

1 and 7Li 1/T1 in Li0.5MnPc. Instead, 1/T l1 of


0 200 400 600 800

0.1

1

y(τ)

τ (ms)

Li0.5MnPc, 7Li NMR

1.3 Tesla 3 Tesla 7.5 Tesla

Figure 5.16: Examples of recovery laws for 7Li nuclear magnetization in Li0.5MnPcafter a saturating pulse sequence. The dashed line are the best fits according toy(τ) = exp(−(τ/T1)β).

0 10 20 30 40 500.01

0.1

1

Li0.5MnPc, 1H NMR

T=15 K, H=1.35 Tesla

y(τ)

τ (ms)

Figure 5.17: Recovery law for 1H nuclear magnetization in Li0.5MnPc after a satu-rating pulse sequence. The solid line is the best fit according to Eq. (5.6).

5.3. The recovery laws in LixMnPc 137

0 20 40 60 80 100

0.1

1

y(τ)

τ (ms)

Li2MnPc, 1H NMR

T=90 K, H=1.45 Tesla

Figure 5.18: Recovery law for 1H nuclear magnetization in Li2MnPc after a saturat-ing pulse sequence. The solid line is the best fit according to Eq. (5.6).

0 1000 2000 3000 40000.01

0.1

1

y(τ)

τ (ms)

Li2MnPc, 7Li Recovery

T=70 K, H=7 Tesla

Figure 5.19: Recovery law for 7Li nuclear magnetization in Li2MnPc after a saturat-ing pulse sequence. The solid line is the best fit according to Eq. (5.6). The dottedline is a fit with a stretched exponential.


10 1000.0

0.2

0.4

0.6

0.8

β

T (K)

Li0.5MnPc

1H, H=1.5 Tesla

7Li, H=1.3 Tesla

Figure 5.20: Temperature dependence of the stretched exponential exponent for7Li and for the fast component of 1H recovery laws.

both 1H and 7Li are of the order of magnitude of 1/T l1 in Li0.5MnPc. It must

be also underlined that β were found to be weakly dependent on temperaturein the two compounds. This is shown for Li0.5MnPc in Fig. 5.20 In particularthe data taken on Li2MnPc were fit keeping both β and A fixed in the entiretemperature range.It was explained in chapter 4.7 that a stretched exponential recovery indicatesa distribution of relaxation rates 1/T i

1 and the average value 1/TA1 is described

analytically by means of the Laplace transformation (4.18).The T1 distributioncould arise either from different hyperfine couplings or from a distribution ofcorrelation times for the spin fluctuations. Inequivalent couplings concern nu-clei located in nonequivalent sites. In the case of 1H this is inherent to theirposition in the Pc molecule while in the case of 7Li this could originate fromthe disorder achieved by doping. The disorder can be also be responsible for adistribution of correlation times. This consideration in particular was useful tointerpret the phenomenology of Li0.5MnPc, in which the fractional stoichiom-etry x = 0.5 implies that there will be a distribution of the number of lithiumatoms per Pc molecule. That is, the material can be seen as a collection of al-ternating Li-rich and Li-depleted chain segments in which distinct microscopicprocesses can take place [113].It was already observed in the analysis of the susceptibility measurements thatthe total susceptibility of Li0.5MnPc seems to arise from the weighted sumof the two susceptibilities χLi=0 and χLi=2, due two the existence of the twoenergetically stable configurations MnPc and Li2MnPc. Then, the arising of

5.4. The temperature dependence of nuclear relaxation in LixMnPc 139

dynamics developing on two different time scales T s1 and T l

1 detected by the1H recovery laws was recognized to be generated by the presence of 1H in bothLi-rich and Li-depleted chain segments. In particular, for the very low dop-ing x=0.5, the high weight of the fast component A = 0.8 should representquantitatively the spatial fraction of protons residing in the pristine chainsMnPc. Moreover, the fact that T l

1 is of the order of T1 probed by 7Li rendersthe hypothesis that they are both relative to dynamics attained in the Li-richLi2MnPc chains. The 7Li recovery displays only one component granted that7Li ions, differently from protons, are present only in the Li-rich chains.The character of the recovery laws in Li2MnPc appears more complex. Sincethe material is homogeneous, 7Li should equally populate the Pc sites and theeffects of disorder should disappear. Disorder is not the sole origin of the devi-ation from simple exponential recovery. For example the recovery is naturallya sum of exponentials if the 7Li satellite lines are only partially saturated. Inthe case of Li0.5MnPc, in particular, it was noticed that the π/2 pulses max-imizing the echo are nearly half of the π/2 pulses maximizing the signal in aLiCl solution. Then, as in LiCl all the lines are irradiated, in Li0.5MnPc onlythe central line is rather irradiated. In addition, since the hyperfine couplinghas an anisotropic component, the powder distribution can lead to a distribu-tion of hyperfine couplings. Now, in the case of Li2MnPc one cannot postulatethe presence of different dynamics occurring on different type of chains. Bet-ter, the complementarity between the relative weights of the fast relaxationA = 0.7 for 1H and A = 0.3 for 7Li in the recovery laws could descend fromtwo different dynamics occurring on the single MnPc site. This recalls the sce-nario observed in LixZnPc, for which it was noticed that probably 1H is moreeffectively coupled to the localized electrons while 7Li probes mainly chargedelocalization on the 2eg orbitals.To summarize, in the LixMnPc the recovery laws show that 1H and 7Li canconveniently probe distinct dynamics, described by T s

1 and T l1. In the case

of Li0.5MnPc this is caused by the spatial disorder of Li ions, while in thehomogeneous Li2MnPc it rather descends from an intrinsic different couplingof 1H and 7Li to the electron spins. Only the analysis of 1/T1 temperaturedependence can reveal what phenomena are responsible of the diversificationof T s

1 and T l1 in the different LixMnPc compounds.

5.4 The temperature dependence of nuclear re-laxation in LixMnPc

The temperature dependence for 7Li and 1H 1/T s1 and 1/T l

1 in the compoundsLixMnPc was obtained fitting the recovery laws with Eq. (5.6). The resultsfor Li0.5MnPc are shown in Fig. 5.21 and 5.22 [113]. Above 40 K, one observesthat 1H 1/T s

1 is practically temperature independent. On the other hand, uponcooling below 20 K 1/T s

1 rapidly increases and then shows a broad maximum


0 10 20 30 40 50 60 70 800

1000

2000

3000

4000

5000

1/T

1s (s-1)

T(K)

Li0.5MnPc, 1H NMR

6000 Gauss 1.5 Tesla 3 Tesla

Figure 5.21: Temperature dependence of 1H 1/T s1 at different magnetic fields. The

solid lines are the best fit according to Eq. (5.9), for 〈EA〉 = ∆ = 25± 5 K [113].

0 50 100 150 200 250 300 3500

10

20

30

40

50

60

70

80

90

100

7Li H=1.3 Tesla

7Li H=7.5 Tesla

1H H=1.5 Tesla

T(K)

1/T

1 (s

-1)

0

30

60

90

120

150

180

210

240

1/T1 l (s

-1)

Figure 5.22: Temperature dependence of 7Li 1/T1 (squares) and of 1H 1/T l1 (circles

and right vertical scale), the relaxation rate of the slowly relaxing component of 1Hmagnetization. The solid lines are the best fit according to Eq. (5.9), for 〈Eg〉 =∆ = 410± 46 K [113].


0 50 100 150 200 2500

1000

2000

3000

4000

5000

6000

Li2MnPc

1H, H=1.45 Tesla

7Li, H=7 Tesla

(rescaled)

1/T

1(s-1

)

T(K)

Figure 5.23: 1H and 7Li (multiplied by a factor of 200) 1/T1(T ) in Li2MnPc [114].The lines are fits (see text).

around 10 K (Fig. 5.21) with an amplitude of the maximum strongly field de-pendent. Turning to 1H 1/T l

1 and 7Li 1/T1 in Fig. 5.22, however, a remarkablydifferent temperature dependence is noted. Primarily the relaxation rates ofthe two nuclei follow a similar behavior. They both do not show any peakaround 10 K; instead, they decrease upon cooling and are much smaller than1/T s

1 . The comparison between 7Li 1/T1 at different fields also indicates thatthe relaxation rates strongly depend on the field. Furthermore, 7Li 1/T1 mea-sured at H = 1.3 Tesla shows a broad maximum at high temperature around180 K. The different temperature range at which the maxima in 1H 1/T s

1 and7Li 1/T1 occur suggests that they are the signature of different dynamics atfrequencies of the order of ωN .Before analyzing in detail these processes, it is worth to observe the completelydifferent temperature dependence of 1/T s

1 and 1/T l1 of 1H and 7Li respectively

in Li2MnPc reported in Fig. 5.23 and 5.24. 1/T s1 is the dominant relaxing

component in the recovery of 1H. For T > 100 K it is nearly constant; then,for T < 100 K, the curve bends marking a drop of the value of 1/T s

1 on cool-ing. Strikingly, one notices that in the high temperature limit it is an order ofmagnitude larger than 1/T s

1 of 1H in Li0.5MnPc. 1/T l1, the dominant relaxing

component in the recovery of 7Li, is reported only for the temperature rangeT < 160 K. It must be noticed in Fig. 5.23 that it was multiplied by a factor200 to become comparable with the magnitude of 1/T s

1 of 1H. One sees clearlythat it continuously decreases upon cooling and it doesn’t flatten for T > 100


0 20 40 60 80 100 120 140 160 1800

2

4

6

8

10

12

14

16

1/T

1(s-1

)

T(K)

7Li NMR Li

2MnPc H=7 Tesla

Li0.5

MnPc H=7.5 Tesla

Figure 5.24: 7Li 1/T1(T ) in Li0.5MnPc and Li2MnPc . The lines are fits (see text).

K. In Fig. 5.24 7Li 1/T l1 of Li2MnPc was plotted again in its natural scale

together with 7Li 1/T l1 of Li0.5MnPc. The two quantities measured at similar

fields H = 7 Tesla and H = 7.5 Tesla, are of the same order and show a similartrend.Now, one should point out the role of the minor contributions 1/T l

1 of 1H and1/T s

1 of 7Li in Li2MnPc. It was not possible to extract a definite tempera-ture dependence for both of them. The exponential fast component noticedin the 7Li recovery laws could be related to spin diffusion processes. Roomtemperature measurements of the spin-spin nuclear relaxation time indicatea value T2 ' 200 µs. Then, since the initial rapid relaxation of the nuclearmagnetization of 7Li is attained in a few ms, this corresponds to the time scalein which the nuclei can reach a common spin temperature by means of rapidexchange processes. This should be expected since the line is partially irradi-ated. In fact, the saturation sequences used to measure T1 are optimized toirradiate the central line of 7Li with pulses having a different length from theones required to saturate the satellite transitions. One can minimize the effectof spin diffusion stimulating all the nuclei relaxing with the central transitionby means of the saturation sequence, but definitely can’t avoid the exchangewith the other nuclei. Probably the spin diffusion is enhanced in Li2MnPc withrespect to Li0.5MnPc because the distribution of Li atoms is uniform, allowinga more effective exchange among them. In the light of these considerations,only the temperature dependence of 7Li 1/T l

1 will be analyzed and it will beassumed from now as the effective 7Li 1/T1.


As regards 1/T l1 of 1H one could consider that, since the main component of

relaxation is exponential, the long tail could arise from the presence of in-equivalent 1H, as already explained. A deviation is commonly detected inmolecular systems having many protons, as for example the single moleculemagnets [129]. Furthermore, in view of the arise of a low-temperature Curietail in the susceptibility measurements, one could hypothesize the presence ofsome defects in the material where the protons would relax slowly. This can bePc broken rings, H2Pc or LiPc impurities. Their presence could be expectedsince the synthesis method of LixMnPc was carried out with less purificationcycles than for LixZnPc. In broken rings and H2Pc 1H should relax very slowlyin comparison with the ones located Li2MnPc sites as the coupling with thelocalized Mn2+ lacks. In LiPc instead the protons can couple with Li spinswhich are expected to correlate generating a temperature dependent dynamicswhen forming dimers or chains [104, 105]. This was discussed also for LixZnPc.However, it is difficult to estimate the fraction of these defects from the mag-netization measurements. In fact the signal coming from the MnPc moleculesis very high, masking the contribution of small signals coming from the possi-ble defects and impurities, except in the region of very low temperatures. Inaddition, the temperature dependence of 7Li 1/T1 doesn’t exhibit anomaliesindicating ad additive contribution coming from LiPc defects. One can onlynotice a little enhancement of 1/T l

1 at T = 50 K recalling the peaks observed in7Li 1/T1 of Li2.7ZnPc, 1H 1/T1 of Li1.85ZnPc and in the EPR susceptibility forthe same samples. However, the growth of this peak is a very subtle effect tobe interpreted unambiguously and to confirm the presence of LiPc impurities.Hence, in conclusion, since the origin of 1H 1/T s

1 is not clear and it doesn’tdisplay a definite temperature dependence, only 1H 1/T s

1 will be consideredand it will be assumed from now on as the effective 1H 1/T1 of Li2MnPc.The scenario is substantially different from the one found in Li0.5MnPc wherethe fast component of nuclear relaxation had an exquisite stretched charac-ter due to disorder induced by Li doping, as already anticipated and betterexplained later. This manifestly indicates that in Li2MnPc the effects of dis-order are suppressed, since the distribution of Li is homogeneous. ActuallyLi doping is clearly responsible of the different behavior of 1/T1 as a functionof temperature in Li0.5MnPc and Li2MnPc. The measurements performed onthe two samples will be now discussed separately and a critical comparisonbetween the two will be developed.

5.4.1 Analysis of nuclear relaxation in Li0.5MnPcIt has been already anticipated that Li disorder is crucial in Li0.5MnPc . It hasbeen also explained that two configurations are expected to form in this ma-terial at the microscopic level, x=0 Li-depleted chains and x=2 Li-rich chains.In addition it was observed how 1H 1/T s

1 should be related to the dynamicsof Li-depleted chains, while 1H 1/T l

1 and 7Li 1/T l1 to the dynamics of Li-rich

chains. The following discussion will validate this scenario.


The temperature dependence of 1H 1/T s1 will be first analyzed [113]. The

dynamics probed by these protons, relaxing fast, should arise from the mag-netism of the pristine MnPc molecules. As illustrated in chapter 5.1, the MnPcmolecules ferromagnetically correlate at low temperature with J ' 1.5 K andthe molecular spins generate a low frequency dynamics due to the presence ofmagnetic anisotropy. Now, in the high temperature limit T >> J the spinsare uncorrelated and, in the absence of diffusive processes, one would expecta temperature and field independent 1/T1 with [117]

(1

T1

)

T>>J

= γ2A2S(S + 1)

3

√2π

ωH

, (5.7)

where ωH = (J/~)√

2zS(S + 1)/3 is the Heisenberg exchange frequency. In-deed, above 40 K one observes that 1H 1/T s

1 is practically temperature inde-pendent. However, on the basis of Eq. (5.7) one would estimate 1/T s

1 around23 s−1 at least an order of magnitude lower than the experimental value (Fig.5.21). Moreover, 1/T s

1 is field dependent. This marked field dependence of1/T s

1 for T À J is associated with the diffusive character of the spin correla-tions, which in one-dimension gives rise to a 1/

√H low-frequency divergence

in the spectral density [117, 118] and will be illustrated in the chapter 5.5.Turning to the low temperature region, it was noticed that upon cooling below20 K 1/T s

1 rapidly increases and then shows a broad maximum around 10 K.This maximum should not be confused with the one due to soliton excitationsobserved in some one-dimensional ferromagnets for T ¿ J , which would causea completely different magnetic field dependence of 1/T1 [119]. This shouldrather be ascribed to the progressive freezing of the molecular spins as detectedby means of AC susceptibility in the pristine β-MnPc [28]. The nuclear 1/T1

likely grows on cooling because the Mn spins progressively fluctuate at frequen-cies of the order of ω ' ωN and then it drops to a low value when ω ¿ ωN .It should be noticed that this requires frequencies of the order of ωN ¿ ωH .A slowing down of the frequency of the spin fluctuations to the MHz range atT > Θ can occur only if the magnetic anisotropy barrier EA = DS2

z ≥ Θ andit overcomes the thermal energy kBT , a situation analogous to the one of sin-gle molecule magnetic clusters [129] and which is also observed in α-FePc [22].Thus, one should distinguish two regimes, a high temperature one (kBT > EA)where Eq. (5.7) applies and a low temperature one (kBT < EA) characterizedby a very low-frequency activated dynamics.Now, the peaks of 1H 1/T s

1 at low temperatures recall the shape of the BBPfunction, introduced in chapter (2.5),

1

T1

=γ2

2〈∆h2

⊥〉2τc

1 + ω2Nτ 2

c

(5.8)

which holds when the correlation function of the field fluctuations at thenucleus decays with a unique correlation time τc, namely 〈h+(t)h−(0)〉 =


〈∆h2⊥〉 exp(−t/τc). Then, one could take into account an effective energy bar-

rier EA and write τc = τ0 exp(EA/T ) (EA in Kelvin). However, by means ofthis assumption a poor fit of the data is obtained. Hence, the broad maximumin 1/T s

1 indicates rather a distribution of correlation times generated by a dis-tribution of activation energies.In the light of Eq. (5.8), considering a rectangular distribution p(EA) of acti-vation energies from 〈EA〉 −∆ to 〈EA〉+ ∆, the average relaxation time turnsout [120]

(1

T1

)

A

=

∫ +∆

−∆

p(EA)1/T1(EA)dEA =

γ2

2

〈∆h2⊥〉T

ωN∆

[arctan

(ωNτoe

〈EA〉+∆

T

)−

− arctan

(ωNτoe

〈EA〉−∆

T

)]. (5.9)

A good fit of 1/T s1 data for T ≤ 40 K, shown in Fig. 5.21, is obtained for

τ0 = (1 ± 0.4) · 10−10 s and 〈EA〉 = ∆ = 25 ± 5 K > Θ, as expected for afreezing of MnPc spins. A distribution of energy barriers and correlation timesis also consistent with what found by Taguchi et al. for the pristine MnPcand ascribed to the natural fragmentation of the MnPc chains in segments. Inthe case of the doped material the effect could be emphasized by the disorderinduced by Li doping. However, it should be remarked that this activationenergy is quite different from the one (about 90 K) derived from the analysisof AC susceptibility data. Since NMR 1/T1 probes the q-integrated spin sus-ceptibility [121], while the AC susceptibility the ~q = 0 modes, this observationwould indicate a more rapid softening of the latter ones with respect to theother ~q modes, as expected for a dominant ferromagnetic character of the spincorrelations.Now the discussion will focus on 7Li 1/T1 and 1H 1/T l

1. In the previous chapterit was outlined that these relaxation rates should follow processes occurring inthe x=2 Li-rich chains. One expects that the freezing dynamics would changealong with the modification of the crystal field brought by the dopants. In par-ticular, the on-site magnetic anisotropy at the Mn-ions could be suppressed,since the environment becomes more isotropic. In addition, it must be re-called that susceptibility measurements in Li2MnPc denoted a change of theexchange couplings to antiferromagnetic and the spin-crossover of the Mn ionstowards the S = 5/2 ground state. Now, if the spin freezing was present alsoin the Li-rich chains, one would expect a peak also in 7Li 1/T1 around 10 K,as the order of magnitude of the hyperfine coupling of 7Li, derived from theline broadening, is the same of 1H. However, it was underlined that low tem-perature peaks are absent in 7Li 1/T1 and 1H 1/T l

1. This cannot be due toa filtering of the spin excitations at the critical wave-vector, as Li is not in asymmetric position with respect to Mn ions [121]. Then the dynamics inherentto the Mn spins must have been perturbed by Li insertion.


At high temperature, around 180 K, 7Li 1/T1 shows a broad maximum. Onecould argue that this maximum is analogous to the one observed in 1/T s

1 at lowtemperature and that it could arise from a spin freezing. However, at variancewith what is found at T ' 10 K, no evidence of a spin freezing around 200K is present in the magnetization data (see Fig. 5.3). Another mechanismthat could give rise to a maximum in 7Li 1/T1 is Li diffusion, with a hoppingfrequency reaching the MHz range around 200 K. However, if this was thecase 7Li NMR linewidth should be much narrower and should not follow thesame temperature dependence of the macroscopic susceptibility (Fig. 5.14).Moreover, if Li was diffusing one should not observe a stretched exponentialrecovery of nuclear magnetization with a temperature-independent exponentβ (Fig. 5.20). Therefore, 7Li relaxation has to originate from a different mech-anism [113].As already explained for LixZnPc compounds, in Li-rich chain segments theelectrons should start filling 2eg(π) orbitals, which overlap with the orbitalsof adjacent molecules to form a one-dimensional band [46, 30]. However, theelectron delocalization can be prevented both by a strong Coulomb repulsion,as well as by the disorder associated with the non-uniform Li distribution. Ac-cordingly, an effective gap between localized and itinerant states develops. Inthis scenario one can ascribe to the electron a phenomenological [111] hoppingtime τe = τ 0

e exp(Eg/T ) with τ 0e renormalized with respect to its bare value

τ 0e ' ~/W (W the bandwidth) owing to spin-polaron formation and other

effects [122], while Eg is an effective gap between localized and delocalizedstates. The disorder, already evidenced by the stretched exponential recovery,causes a distribution of Eg’s. Assuming a rectangular distribution of Eg’s oneobtains again an expression for 1/T1 as in Eq. (5.9). It was estimated thatthe temperature dependence of 7Li 1/T1 can be accounted for Eq. (5.9) with〈Eg〉 = ∆g = 410± 46 K and a τ 0

e = (3 ± 0.8) · 10−10 s (see fits in Fig. 5.22).This model interestingly explains also the field dependence of 1H 1/T l

1, whichwill be treated in chapter 5.5.To conclude, the analysis of the 1/T1 temperature dependence in Li0.5MnPchave enlightened the presence of two different dynamics coexisting at the mi-croscopic level. One arises from the freezing of MnPc spins at low tempera-tures, which is similar to the one observed in the undoped compound and isassociated with Li depleted regions. The other one, characteristic of Li-richregions, is associated with the hopping of the electrons along the chain, whosedelocalization is hindered by the disorder, the Coulomb and spin interaction.A distribution of correlation times was found to be necessary to analyze theexperimental data relating both the dynamics because of a sizeable effect ofdisorder. Now, these results will be compared with the ones obtained by theanalysis of Li2MnPc experimental data, where the role of disorder is minimizedby achieving an homogeneous Li distribution.


5.4.2 Analysis of nuclear relaxation in Li2MnPcIn the light of the previous discussion one could expect Li2MnPc to show ametallic character, since the local disorder should be quenched and the delocal-ization of 2eg electrons favored. In accordance with the description introducedto explain the phenomenology in LixZnPc, then, in this case 1/T1 should followa Korringa law and, in case of strong correlation, display a possible instabil-ity at low temperature arising from ~q = 2~kF spin excitations typical of one-dimensional metals. However, the experimental 1H and 7Li 1/T1 in Li2MnPcindicate a different scenario. It was pointed out that Li2MnPc is more homo-geneous, so that the two nuclei must probe excitations generated in the samechain segments. However, it was noticed that 7Li 1/T1 and 1H 1/T1 in Li2MnPcshow a different temperature dependence, as observed also in LixZnPc.The temperature dependence of 1H 1/T1 will be first analyzed. For T > 100 K1H 1/T1 is flat (trianlges in Fig. 5.23). This is a signature of the coupling withlocalized spins in the paramagnetic regime for T À J . In fact, in Li2MnPcJ ' 3 K, therefore for T > 100 K the Mn spins are clearly uncorrelated.However, the spin dynamics of the Mn ions don’t drive the 1/T1 temperaturedependence on the entire range. In fact, on cooling at T ∼ J the antifer-romagnetic correlations should induce a slow divergence of 1/T1, as expectedfor a S = 5/2 AF chain [123], whereas it rather decreases below T = 100 K.Since this cannot be due to a filtering of the spin excitations at the criticalwave-vector [121], as Li is not in a symmetric position with respect to Mn ions,the drop of the NSLRR must be explained in terms of a different coupling. Inparticular it has been noticed that 1H 1/T1 can be fit by an activated law

1

T1

∝ exp(−∆

T), (5.10)

(dotted curve in Fig. 5.23) which yields a small energy barrier ∆ ' 7 K. Thepresence of a gap is also suggested by the low temperature behavior of 7Li1/T1 for which the fit with the same law yields a value ∆ ' 125 K. It canbe noticed from the solid line in Fig. 5.24 that the quality of the fit is poor.The possibility of a distribution of barriers and of correlation times has alsobeen taken into account to improve the quality of the fit, by means of theEq.(5.9) used for Li0.5MnPc. The result is represented by the dotted line inFig. 5.24; the enhancement is negligible in comparison to the fit with function5.10. Furthermore, it can be observed that 7Li 1/T1 in Li2MnPc grows muchfaster on heating with respect to 7Li 1/T1 in Li0.5MnPc. To better visualizethis difference the fit of the latter with Eq. (5.9), corresponding to the dashedcurve in Fig. 5.24, is also plot. It should be remarked that the average barrierestimated by the fit (5.9) for these data was about 〈Eg〉 = ∆g = 410 ± 46 K.Hence, even if ∆ ' 125 K is an approximated value it accounts for more rapidactivation of 1/T1 in Li2MnPc. The two gaps detected by 1H 1/T1 and 7Li 1/T1

are apparently different, however one should consider that the fluctuations dueto the dynamics of paramagnetic Mn ions are present to some extent below


Figure 5.25: 13C NMR 1/T1 in Na2C60 and K4C60 from 10 to 700 K. Solid lines arefitted to an activated law below room temperature [53]

T = 100 K. Therefore in the intermediate region the two contributions mightadd up to produce a more complicated temperature dependence in 1H 1/T1

which cannot be fit a priori with a simple activated law.This low temperature gap strongly resembles the one detected with NMR inthe doped fullerides, reported by Brouet et al. [114, 53]. Fig. 5.25 showsthe temperature dependence of 13C NMR 1/T1 in Na2C60 and K4C60. Belowthe room temperature the data are fitted to the activated law (5.10), whileat higher temperature 1/T1 is constant in K4C60 and slightly increasing inNa2C60. The low temperature activated 1/T1 was attributed in fullerides to agap related to singlet-triplet transitions between two different Jahn-Teller (JT)distorted states of the C2−

60 and C4−60 . Conversely, the deviation from the acti-

vated law for T > 300 K is explained with the onset of structural modificationsin the materials leading to a different local symmetry and, correspondingly, toa different JT distortion. The situation in fullerides is slightly different andmore complicated for the presence of a triply degenerate t1u orbital, while inphthalocyanines the 2eg LUMO is only doubly degenerate. However, the anal-ogy is striking as also the electrons injected by alkali doping in the LUMOassume a spin configuration which depends on the separation of the levels.Now, if the levels of the LUMO are weakly separated, the Hund rule imposesthe maximum spin multiplicity. On the contrary, if the levels are sensibly splitby Jahn-Teller distortions electron occupy first the low lying energy levels at-taining the minimum spin multiplicity. In Li2MnPc this would lead to theformation of an S = 0 ground state for the LUMO electrons, as shown in Fig.5.26. The activation with temperature of 1/T1 in Li2MnPc, thus, resembles


a1ua1g1eg2egb1g

LiLiLiLi2222MnPcMnPcMnPcMnPcS=5/2S=5/2S=5/2S=5/2

MgPcMgPcMgPcMgPc----MgPcMgPcMgPcMgPc

Figure 5.26: (Left) S = 5/2 ground state configuration in Li2MnPc expected in pres-ence of a Jahn-Teller splitting of the LUMO. (Right) Relevant ground state electroniclevels of neutral MgPc and of the JT distorted, spin polarized MgPc monoreducedmolecular ion. Occupation and spin polarization of the levels are indicated by ar-rows. Empty orbitals are indicated by dashed arrow with empty head, occupied withfull line arrow and filled arrow head [50].

the one typical of excitations between an S = 0 and an S = 1 excited states.No spin cross-over is detected by the susceptibility measurements on varyingtemperature and the data were rather well fitted to the Fisher law for antiferro-magnetic chains. Thus, the spin ground state doesn’t change with temperatureas expected if the two 2eg electrons were excited to a different spin state. Theelectrons, when excited, should better hop from site to site. In the light ofthe similarity with the model proposed for Li0.5MnPc, it is clear that 7Li 1/T1

probes such charge dynamics also in Li2MnPc . Here the effective Eg gap be-tween localized and delocalized states is sensibly reduced thanks to a betterhomogeneity, however it survives due to the Jahn-Teller effect.As long as a strong Jahn-Teller distortion was already predicted by the stud-ies of Tosatti et al. , density functional theory (DFT) methods were recentlyapplied to predict the changes of the electronic states in view of the full vibra-tional spectrum attained upon the addition of one electron to the molecular eg

level of a neutral MgPc molecule [50]. Under the assumption that half a nom-inal electron is added to each of the two degenerate eg sublevels, so that theoriginal molecular D4h symmetry is preserved, this study predicts an energysplitting of 126 meV between occupied spin-up b3g and unoccupied spin-up b2g

orbitals (Fig. 5.26). b3g and b2g are the molecular levels 2eg split by the A1g

+ B1g modes, leading to a reduced D2h symmetry. This splitting correspondsto roughly 1400 K which is much larger than the barrier detected by NMR in


Li0.5MnPc. However, this calculation assumes sharp molecular levels whereasin the bulk state these are rather spread by a bandwidth W ' ~/τe. Secondary,only one donated electron per molecule was assumed, while at higher dopingthe structural deformations of the molecules could change and, together withthem, the gap. Also, the values given here are closer to the hypothetical JTsplitting of the eg orbital of MgPc− anion measured by photo absorption spec-tra analyzed in Ref. [124], of about 37 meV (430 K).While a gap is detected at low temperature by both 1H and 7Li 1/T1, at hightemperatures 1H seems decoupled by the dynamics inherent electron hoppingin the LUMO, and rather probes the fluctuations of the paramagnetic Mn ions.On the contrary 7Li at variance with 1H is decoupled from the paramagneticMn ions. Another important observation regards the order of magnitude of 1H1/T1 in Li2MnPc. It is at least two orders of magnitude larger than 1H 1/T s

1

in Li0.5MnPc in the high temperature region. If this was due solely to differenthyperfine couplings, one should have Ax=2 ' 12Ax=0. However, since it wasalready observed that the couplings are of the same order, this enhancementshould be better ascribed to the increase of the spectral density of the spinfluctuations J(ω). This quantity is reconstructed by means of measurementsof 1/T1 varying the field which will be presented in the next section. In thelight of these data, the role of the J(ω) in the singular enhancement of the 1H1/T1 in Li2MnPc will be estimated.Definitely, from the previous analysis it was argued that, contrarily to the ex-pectations, Li2MnPc is not a metal but still a semiconductor from the point ofview of the conduction properties. This behavior is possibly caused by a split-ting of the LUMO in an upper and a lower band for the effect of Jahn-Tellerdistortions. This is a spin gap between the S = 0 and S = 1 spin states formedby the electrons injected by the alkali atoms. The estimate of the gap was ofthe order of T = 120 K and it was detected by both 7Li and 1H NMR. In ad-dition, it was noticed that at high temperatures 7Li 1/T1 probes the dynamicsof the activated hopping of the electrons across the gap, whereas 1H 1/T1 isentirely driven by the paramagnetic fluctuations of the localized Mn spins.

5.5 The field dependence of nuclear relaxationin LixMnPc

The field dependence of 1/T1 was measured to obtain information on the spec-tral density of the spin fluctuations J(ω) in LixMnPc. This is a very usefulapproach since it allows to derive the shape of J(ω) which is determined bythe dimensionality of the system.The spectral density of the fluctuations is the Fourier transform of the spin-spinpair correlation function. This was already explained in chapter 2, introducingthe general definition of the dynamical structure factor. More in detail, one

5.5. The field dependence of nuclear relaxation in LixMnPc 151

has [129]

1

T1

= 2(γNγe~)2S(S + 1)

[ ∑ij

αij

∫ ∞

−∞〈s±i (t)s±j (0)〉ei(ωN±ωe)tdt +

+∑ij

βij

∫ ∞

−∞〈sz

i (t)szj(0)〉eiωN tdt

](5.11)

and by substituting the spectral density functions:

1

T1

= 2(γNγe~)2S(S + 1)

(∑ij

αijJij± (ωe) + βijJ

ijz (ωN)

)(5.12)

where i,j number the electron spins, ωe and ωN are the Larmor frequenciesof the electron and the nucleus respectively, αij and βij are the geometricalfactors associated with the hyperfine couplings.In a system with nearest neighbour correlations J , in the high temperaturelimit T À J , the spins rapidly fluctuate at the exchange frequency ωex =J/~

√2zS(S + 1)/3. In 3d systems, the correlation function of the spin fluc-

tuations would be a Gaussian 〈s±(t)s±(0)〉 = 〈sz(t)sz(0)〉 = 2/3S(S + 1)×× exp(−ω2

ext2) and, accordingly, the spectral density would be a Gaussian with

second moment ω2ex. For long times t or ω ¿ ωex this functions are nearly con-

stant; therefore, in an NMR experiment, for ωe, ωN ¿ ωex, one expects nofield dependence of the relaxation rate. Moreover, as concerns the wavelenghtdependence of the pair correlation functions, in this regime, since no shortor long-range order is present, the correlation must have a dominant ~q ' 0character. Thus, at long-times and long-wavelenght the spin fluctuations aregoverned by a diffusion equation, in order to attain the conservation of the to-tal spin of the system [125]. The diffusive terms of the pair-correlation (~q ' 0)are of the form

〈Sαq (t)Sα

q (0)〉 ∝ e−Dq2r2t (5.13)

where D is the spin diffusion coefficient. Now, a diffusive process gives a de-crease of the spin correlation function as t−d/2, where d is the dimensionalityof the system. The reduction of dimensionality induces a very very slow decayof the correlations in both space and time since propagation can be attainedalong less directions. In 1d systems this has a dramatic effect since the spec-tral densities J(ω) diverge as ω−1/2. In brief, this means that an NMR 1/T1

proportional to 1/√

H is an experimental evidence of the one-dimensionalityof a system.It must be pointed out that in this model both J ij

± (ωe) and J ijz (ωN) grow

as 1/√

ω under the constraint that ωe, ωN ¿ ωex. However, in real systemsJ ij

z (ωN) usually displays a cut-off term, expressing a truncation of the functionat a low frequency value ωc. This can arise from interchain dipolar or exchangeinteractions, introducing a 3d diffusion which depresses the long time tail ofthe correlation function. In alternative the cut-off can arise by intrachain pro-cesses which don’t conserve the total spin or by the dissipative interactions


0.008 0.010 0.012 0.014 0.016 0.018200

300

400

500

600

700

800

900

1000

1100

1200

Li0.5

MnPc, 1H NMRT=45 K

1/T

1s (s-1)

1/H1/2 (1/Gauss1/2)

Figure 5.27: 1H nuclear spin-lattice relaxation rate 1/T s1 for Li0.5MnPc at T = 45 K,

reported as a function of 1/√

H in order to evidence the diffusive nature of the spincorrelations. The dotted line is the best fit according to Eq. (5.14) in the text. Thedashed line is a guide to the eye.

with defects or impurities. Now, as long as ωc ¿ ωe and ωN ¿ ωe the effectof the cut-off is usually negligible on J ij

± (ωe), while is important on J ijz (ωN).

Moreover, if the weight of J ij± (ωe) in the nuclear relaxation is negligible, since

the processes ωe À ωN involving electronic spin flips are not directly excited,one finds the experimental field dependence [125]

(1

T1

)

T>>J

=P√H

+ Q = γ2A2S(S + 1)

3

1√2D

1√ωN

+ Q, (5.14)

where Q is a constant. This scenario was then verified in LixMnPc [113].In Li0.5MnPc the measurements of 1/T1 as a function of the field in Li0.5MnPcwere carried out at T = 45 K, in the paramagnetic regime. The experimentaldata are reported in Fig. 5.27 and 5.28. The former graph is relative to the1H 1/T s

1 component, while the latter regards 7Li 1/T1. Now, it is evident that1H 1/T s

1 is a function of 1/√

H, while 7Li 1/T1 is a function of 1/H. Thisdifference will now be explained. Neglecting low-frequency cutoffs, the datain Fig. 5.27 are fit with the above equation (see the solid line in the picture).One estimates a diffusion constant D ' (1.2 ± 0.1)ωH , of the right order ofmagnitude. In the light of the previous considerations, this confirms that inLi0.5MnPc the spin correlations are characterized by a diffusive dynamics inthe Li-depleted segments, namely MnPc segments. This results are consistent


0.0 0.2 0.4 0.6 0.8 1.0 1.20

5

10

15

20

25

30

T=45 K

1/T1 (s

-1)

1/H (1/Tesla)

Figure 5.28: 7Li nuclear spin-lattice relaxation rate for Li0.5MnPc at T = 45 K,reported as a function of 1/H. The solid line shows the behaviour expected fromEq. (5.9) for < Eg >= ∆g = 410 K .

with observations reported in an early work on CuPc [118] and lead to animportant observation. The protons in the Li-depleted segments have a largemean average distance from boundaries. In fact, the Eq. (5.14) ideally worksfor infinite chains or at least chains extending over a large number of latticesteps. Therefore the suitability of Eq. (5.14) to describe the 1/T s

1 (H) is asignature of the fact that the chain average length of Li-depleted segments isstill very long. This further confirms that most of the material is in the pristinestate, as already estimated from the weight A = 0.8 of the fast component inthe recovery laws. Only the small fraction ∼ 0.2 of the material is built ofLi-rich segments and it must be pointed out that the two configurations mustbe randomly distributed at the microscopic level. This is suggested by high-resolution x-ray diffraction (XRD) [28] which does not show any evidence ofa macroscopic segregation in Li-rich and Li-depleted phases. XRD line shapeand width also enforce the observation of the absence of finite-size effects on thepristine segments, which could arise if very short chain segments were present.On the basis of these considerations it is meaningful to show also that NMR cangive an estimate of the lower limit for the average length of Li-rich segments.Since no peak in 7Li 1/T1 is detected and since its low-temperature value ismore than two orders of magnitude smaller than 1/T s

1 (see Figs. 5.21 and


0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 0.022 0.024 0.0261500

2000

2500

3000

3500

4000

4500

5000

5500

6000

1/T

1(s-1

)

1/H0.5(1/Gauss0.5)

Li2MnPc, 1H NMR

T=73 K T=20K T=4.2K

Figure 5.29: 1H nuclear spin-lattice relaxation rate in Li2MnPc at different tempera-tures, reported as a function of 1/

√H.

5.22), the average dipolar hyperfine coupling of 7Li nuclei with the Mn spins inLi-depleted segments must be more than an order of magnitude smaller thanthe one of protons in the same segments. Then, taking into account that thedipolar coupling decreases as 1/r3, one can roughly estimate 12 lattice stepsas a lower limit for the length of the Li-rich segments and 4 lattice steps forthe separation among adjacent Li-rich chain segments.Turning to 7Li 1/T1, it is clear that different processes must be introduced toexplain the field dependence. The model of phenomenological hopping impliedin the interpretation of the temperature dependence of 7Li 1/T1 in the Li-richchains must be considered. If the activated electron hopping along the chainis the origin of nuclear relaxation, then it can be observed that in the lowtemperature limit, from Eq. (5.9), 1/T1 goes as 1/ωN ∝ 1/H. Indeed, inthis model the low temperature region is accessed for T ¿ Eg, which wasestimated to be 〈Eg〉 = ∆g = 410± 46 K from the fit of Fig. 5.22. Therefore,since 7L 1/T1 was measured at T = 45 K, it is effectively treatable in the lowtemperature limit. The good agreement with these assumptions is shown bythe solid line in Fig 5.27, which is a linear fit on the data taken as a functionon the variable 1/H.Now, the behavior of 1/T1 upon increasing field in Li2MnPc will be considered.In Fig. 5.29 one sees 1H 1/T1 measured at three different constant temperaturesT = 73 K, T = 20 K and T = 4.2 K. The points have been reported as afunction of 1/

√H. One sees an interesting modification of the behavior of


1/T1 on decreasing temperature. At the higher temperatures 1H 1/T1 fielddependence is described by Eq. (5.14). The dotted lines in the figure showthe fits in the high field region, yielding straight lines. One notices that,assuming A = 440 Gauss, a hyperfine coupling estimated in section 5.2, andtaking ω = 5.55 · 1012 rad/s, the value D = 31.5ωH = 1.75 · 1013 rad/s isextracted. This is about 15 times higher than D evaluated from the 1H 1/T s

1

field dependence in Li0.5MnPc. It should be pointed out that the value ofD is strongly affected by the choice of the coupling A, which is only roughlyestimated. Nevertheless, this order of magnitude of D in Li2MnPc is stillcompatible with a diffusive model. Furthermore, it should be noticed that aflattening of 1/T1 around a value ' 5000 s−1 is attained on increasing 1/

√H,

namely lowering the field below H = 3500 Gauss (see the dashed line in Fig.5.29). This field correspond to ωN ∼ ωc = 93.63 × 106 rad/s and representsthe low-frequency cut-off. A similar departure from the linear fit was observedalso for 1H 1/T s

1 in Li0.5MnPc (dashed line in Fig. 5.27), although it could notbe commented due to the lack of experimental points.Now, one notices that ωc/ωex ' 1.7 × 10−4, which is of the same order ofmagnitude found in TMMC. If this cut-off was due to interchain exchangeinteractions, one would have [125]

ω′x =[ ωc

0.53(8D)1/3

]3/4

, (5.15)

with ω′x the interchain exchange frequency. Then from the extracted values,ω′x = 5.271× 109 rad/s and J ′ = 1.4× 10−2 K, with J ′ the exchange constantbetween the chains. This is a very low value typical of one-dimensional systemswith weak interchain interactions. At lower temperature, T = 20 K, 1H 1/T1

manifests an analogous field dependence but the fit with Eq. (5.14) suppliesan increasing value of D, denoting that the diffusive regime is progressivelyabandoned as T → J .Finally, for T = 4.2 K the relaxation rate is weakly field dependent but displaysa peak around H = 1 Tesla. An analogous deviation from linearity is noticedfor the curve measured at T = 20 K for the same applied field. The occurrenceof divergencies has been noticed also for 7Li 1/T1 measured at various field atT = 4.2 K, shown in Fig. 5.30. In this figure the data have been reportedas a function of the Larmor frequencies ω = γNH, and the 1H 1/T1 has beenadded in order to underline the similarity of the two curves. One noticesthat rescaling 1H 1/T1, the two curves merge following the same frequencydependence. Moreover 7Li 1/T1 undergoes a sizeable enhancement aroundω = 90 MHz corresponding to H = 5.5 Tesla. This resonant behavior isnot explained yet. It could arise from the activation of electronic transitionsdepending on the splitting of the levels acted by the external field; howeverthis scenario is still controversial.The comparison between the field dependence of 7Li and 1H 1/T1 points outthat in both samples 1H probes a diffusive behavior of the spin correlations athigh temperature typical of one dimensional systems, while 7Li 1/T1 probes


0 10 20 30 40 50 60 70 80 90 100 110 120

1.0

1.5

2.0

1/T

1(s-1

)

ν(MHz)

Li2MnPc

1H 1/T1(s-1)/1400

7Li 1/T1(s-1)

Figure 5.30: 7Li and 1H nuclear spin-lattice relaxation rates in Li2MnPc at T = 4.2K, reported as a function of ν = γNH.

the spectral character of the charge diffusion processes. This is consistentwith the observations reported about the different temperature dependence of7Li and 1H 1/T1. The diffusion constant D of the spin correlations extractedby experimental 1H 1/T1 is of the order of the exchange frequency ωe, inaccordance with results obtained in analogous systems. A cut-off frequencyin Li2MnPc is also observed, due possibly to interchain interactions with anexchange constant of the order of J ′ = 1.4 · 10−2 K. It should be remindedthat in Li0.5MnPc the field dependence of 1/T1 changes as a consequence ofthe microscopic coexistence of the two subsystems of Li-rich and Li-depletedchains. In Li-rich chains 1/T1 ∝ 1/H is explained in the light of the model ofphenomenological hopping of electrons across the activation barrier betweenlocalized and delocalized states.


5.6 Summarizing remarks

Magnetization and NMR 1/T1 measurements in LixMnPc clarified the natureof the intermolecular couplings and electron dynamics both for light alkalidoping and at half filling of the 2eg LUMO (x=2). The analysis of the NMRlinewidth indicates that lithium doesn’t diffuse. It is accommodated betweenadjacent molecules in intrastack positions, strongly bound to the pyrrole bridg-ing nitrogens of the Pc ring, as suggested by structural studies [28]. Thus, thepristine MnPc structure is deformed by alkali intercalation affecting the molec-ular orbital overlap and the superexchange interactions.For x → 2 the intrachain correlation changes from ferromagnetic to antifer-romagnetic due to the increase of the angle of the superexchange pathwayMn-N..-Mn to values greater well above 90. Moreover, the doping rendersthe crystal field probed by Mn[II] ions more symmetric and some 3-d orbitalsof the metal are lowered. Accordingly for x → 2 the Mn[II] turns from theintermediate spin state S = 3/2 to the S = 5/2 state, in which all the the3-d orbitals are occupied. This observation arises from the comparison be-tween the results of DC magnetometry and NMR measurements. From theanalysis of the susceptibility alone one evaluates the spin ground state and theexchange coupling, but cannot establish in principle the spin configuration.On the other hand, both 1H and 7Li 1/T1 measurements in Li2MnPc follow anactivated behaviour owing to a gap corresponding reasonably to the splittingbetween the 2eg LUMO bands [114]. Hence, at low temperature the injectedalkali electrons should occupy the lower LUMO band and couple in a singletS = 0. Furthermore, the system doesn’t displey any crossover to a higher spinstate on increasing temperature, possibly due to spin excitations in the LUMOband. Li2MnPc remains in the S = 5/2 ground state formed solely by 3-delectrons, which fill the b1g orbital lower than the LUMO.Lithium doping also affects the dynamic magnetism of the pristine MnPc whereslow relaxation of the magnetization occurs at low temperature due to thefreezing of the Mn[II] moments along a direction determined by the singleion anisotropy [28, 113]. The anisotropy is weakly affected for low doping,∆E = 90 K in Li0.5MnPc, where the relaxation processes are still present.This activated spin dynamics is evidenced by the time dependent magnetiza-tion, building along an external magnetic field in a very long time, of the orderof a thousand seconds at T = 3 K. On the other hand, for high doping x = 2,there’s no experimental evidence for low temperature Mn[II] freezing. Thissuggests the reduction of the local anisotropy in agreement with the achieve-ment of a more symmetric environment upon Li intercalation assumed in theprevious discussion.At variance with LixZnPc, in LixMnPc compounds spin dynamics involvesboth the 2eg electrons and the Mn[II] electrons. The coexistence of two dis-tinct dynamics in LixMnPc is evidenced by the hyperfine interaction, differentfor the protons in the peripheral benzene rings of the Pc macrocycle and forthe lithium allocated in intrastack positions, between the Pc molecules. In


particular, it was observed that 7Li 1/T1 probes the dynamics of the LUMOelectrons, whereas 1H 1/T1 can be driven by the fluctuations of the localizedMn spins as well.7Li 1/T1 shows an activated behaviour in LixMnPc , owing to the hopping ofthe LUMO electrons across an energy gap between localized and delocalizedstates. Thus, in Li2MnPc the electrons are not delocalized as in Li2ZnPc, asexpected from the results of resistivity measurements reported in Ref. [28], andthe bands retain a highly molecular character with a very narrow bandwidth.However, the energy barrier of the activated hopping is reduced with increasingdoping from 〈Eg = 410〉 K [113] in Li0.5MnPc K to Eg = 120 K Li2MnPc. Thisis attained as long as the configurational disorder of the Li dopants decreasesgoing towards the x=2 integer stoichiometry. In fact, highly disordered Liarrangements generate a large number of defects in regards of the conductingproperties of the material, modulating locally the bandwidth and the gap ac-cording to a random distribution. This distribution is absent or at least verynarrow in the more homogenous Li2MnPc , since, differently from 7Li 1/T1 ofLi0.5MnPc , 7Li 1/T1 of Li2MnPc can be described by a single activation energy.Then the gap in the LUMO in Li2MnPc is not generated by disorder but hasa different origin, tentatively identified with the Jahn-Teller distortions of themolecules, a phenomenon commonly observed in alkali-doped fullerenes [53]. Ifdue to Jahn-Teller coupling with the intramolecular B vibrational modes, thissplitting ∆E = 120 K of the LUMO is much smaller than the one expectedfor a reduced MgPc− molecule ∆E = 1400 K from theoretical calculations[50]. However, it should be remarked that these calculations are performedassuming sharp molecular levels, while in a crystal the molecular orbitals gen-erate bands of width W . The bandwidth W is then expected to reduce theeffective Jahn-Teller splitting of the LUMO in bulk compounds. Furthermore,a theoretical prediction concerning specifically MnPc still lacks, whereas it isknown that the structural deformations undergone by the Pc molecules vary indifferent MPc compounds. Therefore one expects in principle a different Jahn-Teller coupling in different AxMPc compounds. The different phenomenologyfound in the isostructural Li2ZnPc compounds indicates that the properties ofthe LUMO are the result of a complex competition between localization, on-site repulsion, Jahn-Teller coupling and Hund coupling, all strictly compounddependent parameters.Li disorder for 0<x<2 induces also interesting effects in the NMR proton re-laxation. For non integer stoichiometries it is governed by a slow and a fastrelaxation rates 1/T1 with a different temperature dependence. Since NMRis a local probe of the microscopic environment, it was recognized that inLi0.5MnPc these two relaxations occur in separate chains with stoichiometriesclose to x=0 and x=2, randomly admixed at the microscopic level. The lowtemperature peaks of fast proton 1/T1 are related to the low temperature freez-ing of the Mn[II] ions occurring in the pristine MnPc chains; on the contrary,the slow 1/T1 is smoothly temperature activated in analogy with the 7Li 1/T1,


owing to the fact that in the x=2 chains the spin dynamics is governed bythe phenomenological electron hopping across the high barrier between local-ized and delocalized states [113, 122]. Accordingly, 1/T1 in the two chainshas a different field dependence: in the x=0 chains 1H 1/T1 is proportionalto 1/

√H, indicating a diffusive nature of the spin correlation typical of one-

dimensional systems; in the x=2 7Li 1/T1 is proportional to 1/H owing to thepresence of the charge hopping processes. Also in Li2MnPc 1H 1/T1 is pro-portional to 1/

√H and displays evidently a low-frequency cutoff, tentatively

attributed to the interchain molecular exchange interactions. These are twoorders of magnitude smaller then the intrachain exchanges among the Mn[II]ions, demonstrating that the physics of LixMnPc is of highly one-dimensionalnature.At present there are still some open issues in the interpretation of the NMRmeasurements in LixMnPc compounds. First, the different temperature de-pendence of 7Li and 1H 1/T1 in Li2MnPc cannot be ascribed to configura-tional disorder of Li and the segmentation of the system in chains with dif-ferent stoichiometries. Therefore, 7Li and 1H belonging to the same MnPcunits probe different dynamics, the former ones the LUMO electrons dynam-ics, while the latter the Mn[II] localized electrons dynamics. Remarkably, 1H1/T1 in Li2MnPc is activated as 7Li 1/T1 at low temperatures, while in thetemperature region where the Mn[II] are uncorrelated it flattens about a veryhigh value of the order of thousands s−1, implying that the spectral densityJ(ω) is strongly enhanced with respect to the one present in Li0.5MnPc. Inaddition, both the 1H and 7Li 1/T1 display clear peaks as a function of thefield at low temperatures, i.e. temperatures at which both 1H and 7Li 1/T1 areactivated. The origin of these peaks is still undisclosed and will be hopefullyexplained of the basis of a more detailed study of the 1/T1 for other values ofthe applied field at fixed or variable temperatures.


Chapter 6Magnetization and NMRmeasurements in TbPc2[TBA]molecular magnets

In the following sections the magnetic properties of the TbPc2[TBA] com-pounds will be illustrated. Previous publications [57, 58] suggest that thesesystems should be characterized by a strong anisotropy inducing a large split-ting between the sublevels of the Tb[III] ground state J=6 manifold. On onehand, the static magnetization M is expected to give indications about theenergy separations between the discrete electronic levels. On the other hand,NMR measurements on protons, allow to clarify the nature of spin excitationsof Tb[III] magnetic moments over different temperature and field ranges. Thecomplex phenomenology observed in molecular nanomagnets, ranging fromthermally activated low-frequency processes to low temperature Quantum Tun-neling of Magnetization (QTM), will be presented in section 6.3 and the ex-perimental data discussed on the basis of the literature results. In the conclud-ing remarks the applicability of the magnetic properties of TbPc2[TBA] com-pounds in several fields will be suggested and the importance of the magneticdilution in the achievement of possible technological improvements consideredin the light of the experimental results.

161

162 6. Magnetization and NMR measurements in TbPc2[TBA] molecular magnets

6.1 Magnetization measurements in TbPc2[TBA]

Magnetization measurements in TbPc2[TBA] were performed according to aprocedure already illustrated in chapter 3, 4, 5, handling the powders closed inglass sample holders. These experimental investigations focused mainly on thetemperature dependence of the susceptibility in samples with different [TBA]concentration and on the effect of the magnetothermal history on the measuredmagnetization. This aspect was readily clarified performing measurements atseveral fields with different cooling rates.Figure 6.1 shows the inverse of the molar susceptibility χ−1 = H/M of TbPc2××[TBA]144 for H = 4500 Gauss, collected by means of two distinct cooling pro-cedures. The data of the blue curve were acquired measuring the magnetizationon heating between 10 K and 320 K in H = 4500 Gauss, after a rapid coolingat 10 K/min in H = 5 Tesla from room temperature down to 10 K. The dataof the red curve were acquired identically on heating but after a slow coolingat 1 K/min in H = 5 Tesla. The two data sets manifestly diverge, the redcurve following approximately a Curie-like temperature dependence χ−1 ∝ Tand the blue curve showing a marked downward curvature in the region be-tween T = 10 K and T = 200 K. This possibly indicates that magnetothermalhistory affects the molecular configuration, which in turn affects the magneticproperties.As illustrated in the following sections, the molecules rapidly rotate for theexcitation of molecular motions above T '200 K. As long as the system is

Cooling 1 K/min in H=5 Tesla

Cooling 10 K/min in H=5 Tesla

Figure 6.1: Molar 1/χ of TbPc2[TBA]10 as a function of temperature after differentcooling procedures, 1 K/min (red line) and 10 K/min (blue line) [126].

6.1. Magnetization measurements in TbPc2[TBA] 163

0 50 100 150 200 250 300 3500

4

8

12

Theory Powder

H=4.5 kGauss

TbPc2TBA10

χT(e

mu.

K/T

b m

ole)

T(K)

TbPc2TBA144

Figure 6.2: Experimental χT vs T of TbPc2[TBA]10 (red) and of TbPc2[TBA]144

(blue) for H = 4500 kGauss and χT calculated from the energy level scheme pro-posed by Ishikawa (green) [126].

cooled, the motions gradually freeze. Now, if the cooling is rapid, they rea-sonably block in a random configuration. The magnetization doesn’t grow oncooling according to a Curie law but it rather follows a complex temperaturedependence governed by the crystal field splitting of the J=6 multiplet, with aquantization axis determined by the anisotropy. The anisotropy tends clearlyto block the molecules in their initial random configuration, hindering theiralignment along the field. Since the molecules can attain a different blockingconfiguration at each cooling cycle, rapid cooling cannot afford reproduciblemeasurements. However, one notices that, if the cooling rate is very slow,their blocking configuration can be recreated in different measurements andthe same susceptibility is deduced from different measurements. On the basisof these observations, it was established that the optimal procedure to obtaincomparable results in these systems, whenever cooling in field from room tem-perature, was 1 K/min in a field above H = 1 Tesla.The susceptibilities of the two samples TbPc2[TBA]10 and TbPc2[TBA]144, de-fined as χ = M/H, were then measured adopting the slow cooling procedure.Figure 6.2 compares the experimental and the theoretical χT , per mole of ter-bium ions, as a function of temperature. The magnetization was calculatedconsidering all the complete J=6 multiplet substructure, having


Mz(T ) =

∑+6i=−6 NAgµB〈µi

z〉 e−Ei

zT

∑+6i=−6 e−

Eiz

T

(6.1)

and, under assumption of uniaxial symmetry,

Mx(T ) = My(T ) =

∑+6i=−6 NAgµB〈µi

x〉 e−Ei

xT

∑+6i=−6 e−

Eix

T

. (6.2)

In this expression 〈µiz〉 and 〈µi

x〉 are the expectation values of the componentsof Tb[III] magnetic moments under the application of H, calculated on thebasis of the eigenstates of the total Hamiltonian of the system

H = H0 + gµBJ ·H + AhypI · J (6.3)

constituted, from the left, by the crystal-field term H0, the Zeeman interactionand the hyperfine interaction between the angular moment J of the Tb(III) ionand its nuclear spin I. One should notice that this Hamiltonian concerns a sin-gle ion, nevertheless it is expected to describe a macroscopic sample, containingat least 1017 TbPc2[TBA] units. This holds as far as these systems behave as acollection of quasi 0-D uncorrelated magnets, the single ion TbPc2[TBA] mo-ments. H0 is the main part of the Hamiltonian. When the external field is notapplied the quantization axis of TbPc2[TBA] coincides with the anisotropy axisand, therefore, the eigenstates and the eigenvalues of H0 are to be estimatedunder this constraint. Then, in this calculation the energy levels evaluated byIshikawa et al. [57] for the undiluted TbPc2[TBA] , shown in Fig . 6.3, wereintroduced as eigenvalues of the crystal field Hamiltonian. Since the separationbetween the ground state and the first excited state is large, ∆E '628 K, itis expected that at room temperature most of magnetic moments will be inthe ground state and that the system could be fairly described by these twolow-energy levels alone.Considering that the molecules of the powder are oriented in a random configu-ration, the probability to achieve the alignment of the anisotropy axis along thefield is 1/3, while 2/3 is the probability that they are orthogonal. For the lattermolecules the magnetization along the field is actually the transverse compo-nent of the magnetization with respect to the molecular anisotropy axis. Theseconsiderations lead to the expression

M(T ) =2

3Mx(T ) +

1

3Mz(T ), (6.4)

for the powder magnetization of [TbPc2][TBA]. The results are reported inFig. 6.2. On one hand, the χT of TbPc2[TBA]10 and the theoretical curve areconsistent up to T = 200 K, then at higher temperature the experimental curve

6.1. Magnetization measurements in TbPc2[TBA] 165

Figure 6.3: Energy diagram of the sublevels of the ground-state multiplets of[LnPc2]−TBA+ (Ln = Tb, Dy, Ho, Er, Tm or Yb), calculated by Ishikawa et al. [57].

progressively diminishes. On the other hand, the χT of TbPc2[TBA]144 is al-ways lower than the other two curves by an amount of ∼ 1 emu·K/mole, nearlyconstant for 10 K< T <200 K and reducing to approximately 0.2 emu·K/moleat room temperature. The measured susceptibility of the TbPc2[TBA] samplesis the sum of several contributions

χTot = χTb3+ + χPc4−2+ χ[TBA]x + χ0, (6.5)

where the first term is the dominant contribution of the rare earth Tb(III)and the others are diamagnetic corrections coming from: the Pc2 rings, of theorder of (−3.44 · 2) · 10−4 emu/mole= 6.88 · 10−4 emu/mole [102]; the complex[TBA]Br, χ[TBA]Brx = xχ[TBA]Br; the sample holder, χ0 ∼ 10−4 emu/mole.The susceptibility of [TBA]Br should be of the order of −1 · 10−4 emu/moleat room temperature [127]. In principle, also a temperature independent Van-Vleck paramagnetic correction of the rare earth metal susceptibility should beconsidered, if the non-diagonal terms 〈J,mJ |β((~L + 2~S)|J ′,mJ ′〉 due to crossexcitation between the ground and the higher energy multiplets are non-zero.However, since the separation between the J=6 ground and J=5 first excitedmanifolds should be larger than the overall splitting of the J=6 manifold,nearly 850 K, this contribution can be reasonably omitted. This also excludesthe possibility to observe effects arising from the J-mixing of states belongingto different J multiplets in TbPc2[TBA] in the field and temperature ranges


explored by the presented measurements.Now, no clear explanation can be afforded at present to explain the smallervalue of the TbPc2[TBA]144 χT with respect to the TbPc2[TBA]10 χT . Thehypothesis of the change of the crystal field splittings in the J=6 manifold fordifferent [TBA] dilutions could be advanced. The proximity of TbPc2[TBA]groups could create a path for intermolecular interactions influencing both thestatic and the dynamic magnetic properties of TbPc2[TBA] . Furthermore, asconcerns the high temperature range, it should be argued that the decreaseof χT in both samples for T > 200 K, should be ascribed to the arise of theseveral diamagnetic terms of the susceptibility. In this temperature region,where instead the χTb3+T varies slowly on increasing temperature approachingχT ' 11.8 expected for a free Tb(III) ion [128], the corrective terms appearingin Eq. (6.5) sum up in χdiaT ∝ CT , with C' −1×10−3, implying χdiaT ' −0.3emu/mole at T = 300 K.To conclude, the magnetization measurements in TbPc2[TBA]10 and TbPc2××[TBA]144 show that the static susceptibility of these systems can be repro-duced in terms of the model proposed by [57], taking into account a sizeablecrystal field splitting in the J=6 ground manifold of Tb(III). However, thepossible achievement of a molecular ordering, modulated by fast molecularmotions, induces a strong dependence of the magnetic measurements on themagnetothermal treatment of the samples, avoidable by means of very slowthermal treatments in applied field. The dilution by means of [TBA]Br inter-calation seems to affect the static and dynamics properties of the TbPc2[TBA]compound. The magnetization data alone cannot supply a straightforwardinterpretation and the validity of these argumentations must be verified in thelight of NMR measurements.

6.2 NMR spectra in TbPc2[TBA] compounds1H NMR spectra in TbPc2[TBA]10 and TbPc2[TBA]144 samples will be pre-sented and discussed in the following. The spectra were obtained from theFourier transform of half of the echo signal after a π/2-τ -π pulse sequence. Inthe temperature range 200 K< T <320 K, the spectra were acquired whilecontinuously cooling at the constant rate of 1 K/min under the applicationof the magnetic field H = 1 Tesla. At lower temperatures the spectra wereacquired during several sessions, provided that the sample had been kept atT < 200 K in the meantime. In both the samples the line shape was observedto change from Lorentzian-like to Gaussian like on cooling. Furthermore, inTbPc2[TBA]10 the high temperature spectra exhibited an additional narrowcomponent, appearing on top of a broader one, with an intensity growing withincreasing temperature. Figures 6.4, 6.5 and 6.6 clearly present the describedfeatures for both the samples at different temperatures. Associated withthe shape transformation, a broadening of the lines was observed; the FWHMincreases from few tens kHz up to about 70 kHz at the lower temperatures

6.2. NMR spectra in TbPc2[TBA] compounds 167

Inte

nsi

ty(a

.u.)

Frequency (MHz)

Figure 6.4: 1H NMR spectra of TbPc2[TBA]144 , acquired at H = 1 Tesla and T =309 K (a.u.=arbitrary units). The solid line is a Lorentzian function [126].

Inte

nsi

ty(a

.u.)

Frequency (MHz)

Figure 6.5: 1H NMR spectra of TbPc2[TBA]144 , acquired at H = 1 Tesla and T =90.5 K (a.u.=arbitrary units). The solid line is a Gaussian function [126].


Inte

nsit

y(a

.u.)

Frequency (MHz)

Figure 6.6: 1H NMR spectra of TbPc2[TBA]10 , acquired at H = 1 Tesla and T =310 K (a.u.=arbitrary units). The solid lines are Lorentzian functions with FWHMindicated in figure [126].

in TbPc2[TBA]144. Correspondingly also in TbPc2[TBA]10 the narrow andthe broad lines, with ∆ω1 = 28 kHz and ∆ω2 = 7 kHz at room temperaturerespectively, undergo an analogous broadening on cooling, the narrow com-ponent rapidly vanishing. The temperature dependence of the FWHM of 1HNMR spectra in TbPc2[TBA]144 is reported in Figure 6.7.As already explained in chapters 4 and 5 with regard to alkali-doped phthalo-cyanines, a line broadening is observed on cooling in powder samples wheneverthe hyperfine couplings between electrons and nuclei are anisotropic. Depend-ing on the orientation of each crystallite with respect to the magnetic fieldone probes a different component of the paramagnetic shift tensor. In addi-tion, in TbPc2[TBA] compounds each unit cell contains several inequivalentprotons, each of them characterized by a different hyperfine coupling. In pres-ence of a distribution of hyperfine constants of width ∆A, arising from powderrandom orientation and site dependent couplings, the overall proportionality∆ω/ω0 ∝ ∆K ∝ ∆Aχmol/NA holds. This dependence is shown in figure 6.8.From the linear fits indicated with the solid lines, one derives an effective hy-perfine coupling in the range of a few tens of Gauss. It should be marked thatthis calculation gives just an estimate of the order of magnitude of the hyper-fine coupling. However, this is useful in order to observe that the hyperfinecoupling must be dominated by the dipolar interaction.For low values of the susceptibility in the plots, corresponding to high temper-

6.3. NMR in molecular nanomagnets 169

0 50 100 150 200 250 300

30

40

50

60

∆ν∆ν ∆ν∆ν (

kHz)

T (K)

Figure 6.7: FWHM of 1H NMR spectra as a function of temperature inTbPc2[TBA]144 , measured in H = 1 Tesla. The dotted line reproduces the tem-perature dependence of the susceptibility [85].

atures, one observes a departure from the linear fit. This can be associatedwith the onset of fast rotational motions of TBA molecules at a frequencyhigher than the rigid-lattice linewidth, providing a narrowing of the NMR spec-trum. In particular in the fast motion regime the effective linewidth is given by∆ω ' 〈∆ω2〉τc where 〈∆ω2〉 is the mean square rigid lattice linewidth and τc isthe correlation time of the molecular motions. The temperature dependence ofthe Full Width ar Half Maximum (FWHM) of the NMR line indicates that themolecular motions appear to be frozen below 200 K in TbPc2[TBA]144 . On thecontrary, in TbPc2[TBA]10 the analysis is complicated by the presence of twodistinct components. However, the insurgence of a narrow component on topof a broader one could be explained considering the coexistence of molecularmotions characterized by distinct correlation times. Hence, at a given temper-ature some protons possibly probe slow motions and other fast motions. Onthe other hand, as the temperature is raised more and more molecules reachthe motional narrowing condition and the intensity of the narrow componentof the NMR spectrum increases.

6.3 NMR in molecular nanomagnetsIn the introductory chapter 1, several issues concerning Single Molecule Mag-nets have been introduced to stress the analogy between these quasi 0-D sys-


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.745

50

55

60

65

Pc2Tb[TBA]

144

H= 1 Tesla

∆ν (

kHz)

χ (emu/mole)

Figure 6.8: 1H NMR spectra FWHM as a function of the molar magnetic suscepti-bility. The fit is linear for T > 200 K [85].

tems and the TbPc2[TBA] compounds. NMR can be suitable to investigatespin dynamics in molecular nanomagnets, over a wide field and temperaturerange. In particular, proton NMR can be performed, owing to the fact thatthese metallorganic complexes usually contain a large number of protons. 1Hbelonging to organic ligands and peripheral groups, as counteranions or dia-magnetic stabilizing shells, are coupled to the localized metal ions by meansof weak dipolar hyperfine interactions. Therefore, generally the spin correla-tion functions are described in the frame of the weak collision approximation,considering that the nuclei probe the random fluctuation of an average fieldgenerated by the several ions of the molecule. Having the local hyperfine fieldsH(t) = A · ~S(t) and the average square of their fluctuating amplitude 〈H2

eff〉,the correlation of the local hyperfine fields at the nuclei are described by an ex-ponential decay 〈H2

eff〉e−t/τc . The correlation time τc is a crucial phenomeno-logical parameter, whose temperature and field dependence strictly indicatethe nature of the physical processes responsible of the spin decoherence.In molecular nanomagnets similar to TbPc2[TBA], as Mn12 and Fe8, whichhave a high-spin S = 10 ground state, proton NMR has clearly detected theonset of distinct dynamics on cooling temperature. In the temperature depen-dence of 1/T1 a pronounced maximum is observed in an intermediate regionT ' ∆E, where ∆E corresponds to the splitting of the ground multipletS = 10. Conversely, for T ¿ ∆E, 1/T1 becomes temperature independent.Such a change in 1/T1 has been interpreted as the signature of the evolution


of these systems from a regime of uniform collective thermally excited fluctua-tions of the localized spins, at frequencies matching the nuclear ωn, to a regimeof temperature independent quantum fluctuations of the total magnetizationof the molecule, namely the magnetization tunneling.For sake of a concise illustration, the NMR 1/T1 in these systems must becalculated on the basis of the discrete electronic spin levels |S, ms〉, labeled interms of the total spin S of the molecule. If the collective quantum spin statesare well defined and underdamped, the nuclear relaxation induced by directelectronic transitions can be expressed on the basis of the matrix elements ofthe local spins operators whose correlation functions are [129]

〈Si(t)Sj(0)〉 =Trexp(−βH) exp(iHt/h)Sj exp(−iHt/h)Si

Trexp(−βt) , (6.6)

yielding

1

T1

=

∑i,j

∑n,m exp(−βEn)[A〈n|Sj|m〉〈m|Si|n〉δ(En − Em − ωN)]∑

n exp(−βEn). (6.7)

When the separation between different levels is ∆mn = En − Em À ~ωN ,the nuclear relaxation can’t be induced by direct transitions between differentms states. Nevertheless, these processes, corresponding to magnetic dipoletransitions, regard only the transverse components of the spin operators S±i ,while the longitudinal Sz

i components can be independently involved in low-frequency processes at ω ' ωN . Therefore, in most cases the nuclear 1/T1 inthese systems is driven by the spectral density of the zz components of thespin fluctuations,

1

T1

= 2(γNγe~)2S(S + 1)∑ij

αijJijz (ωN). (6.8)

In many SMM at temperatures T ' ∆E, this nuclear spins relax via ~q ' 0Raman-like electron spin-phonon scatterings. Phenomenologically these low-energy interactions broaden the electronic levels by an amount Γm ¿ ∆mn,so that the levels are still well-separated. 1/Γm also corresponds to a finitelifetime τm of the spin states. Then, identifying the correlation time of thespin fluctuations τc with an unique average life time in the random weak-collisions approximation 〈Sz

0(t)Sz0(0)〉 = |Szz

q=0|2e−t/τc = S(S + 1)e−t/τc , oneobtains a Lorentzian spectral density Jzz(ω). From (6.8), under the assumptionof an average hyperfine coupling A, 1/T1 can be expressed in several equivalentexpressions

1

T1

=γ2

N〈∆h2⊥〉

2

τc

1 + ω2Nτ 2

c

= Aχ(~q = 0)Tτc

1 + ω2Nτ 2

c

. (6.9)

This BPP form well describes the maximum observed in a large variety ofantiferromagnetic rings and ferrimagnetic SMM (Mn12, Fe8, etc...) under the


Mn12-ac

Figure 6.9: Experimental 1H 1/T1 (T) curves for Mn12-ac [131] (Left) and Fe8 [132](Right) at different fields.

assumption of a single correlation time τc, which is temperature dependentaccording to a power-law τc ∝ T−n or an activated law and is field indepen-dent. 1/T1 displays a maximum at the temperature T0 at which ωNτc(T ) = 1,shifting at higher temperature on increasing the field and scaling with the fieldaccording to 1/T1χT = A/2ωN = A/2γNH, where A is constant. It should benoticed that a single Lorentzian spectral density Jzz(ωN) is valid strictly whenthe average life time is equivalent for all the spin levels or just essentially theground state is populated.Generally, in presence of spin excitations the average broadening of the m-thlevel and the transition probabilities pm,m±n are related. Considering only thetransition between mS = m and mS = m± 1 spin levels one has

1

τm

= pm,m−1 + pm,m+1. (6.10)

In molecular nanomagnets as Mn12 and Fe8, it was found that the spin phononcoupling is the main responsible of the level broadening and it can be takeninto account in the expression of pm,m±1 through a phenomenological constantC in the form [130]

Pm,m−1 = C(Em−1 − Em)3

eEm−1−Em

kBT − 1e Pm,m+1 = C

(Em − Em+1)3

1− e−Em−Em+1

kBT

, (6.11)

Accordingly, owing to the existence of several τm lifetimes, 1/T1 is a sum overthe spectral contributions Jzz

m (ωL) of each m-th level weighted by Boltzmann


Figure 6.10: Low temperature 1/T1 in Fe8 measured in zero-field detecting the QTMin the ground state [129].

statistics:1

T1

=A

Z

∑m

τme−Em/kBT

1 + ω2Lτ 2

m

. (6.12)

For sake of illustration, the experimental NMR 1H 1/T1 measured in Mn12-ac[131] and Fe8 [132] interpreted on the basis of Eq. (6.12) are reported in Fig.6.9.In the low temperature regime, kBT ¿ ∆E, only the ground state is popu-lated and these systems behave as superparamagnetic particles. In presence ofanisotropy, in this regime the single molecule magnetization can relax in a verylong time only by means of the quantum tunneling between the low-lying stateshaving a residual Kramers degeneracy. In Mn12 these are the ms = ±10 levels,separated by an energy barrier ∆E = 60 K. The tunneling occurs as long as themolecule spin eigenstates are not pure |S,mS〉 states but are admixed due toperturbative terms of the Hamiltonian containing the transverse componentsof the total spin Sx, Sy. The two low-lying states are then normally separatedin energy by an amount defined tunneling splitting ∆T of pure quantum ori-gin. However, this splitting is hardly observable owing to the spread of theground levels by other perturbations, as intermolecular magnetic interactionsand hyperfine nuclear interactions.The splitting can be greatly enhanced under the application of an external mag-netic field in such a way to be experimentally detected, for example by meansof NMR. If the external field is transverse with respect of the anisotropy axis,the levels are still nearly degenerate and the transitions between the two occur


at a frequency ∆T /~. In presence of a longitudinal field, on the contrary, thelevels are no longer degenerate due to the Zeeman splitting shifting the energylevels according to

Em = Em(0) + gµBHm, (6.13)

where Em(0) are the energy values determined by anisotropy only. Remarkably,1/T1 is proportional to the transition probability between the two states andreaches a maximum when the tunneling rate matches the Larmor frequency.The tunneling probability is determined by the tunneling splitting ∆T , bythe broadening of the levels W and the possible splitting ∆ introduced byadditional perturbations by means of the relation [129]

Γ =∆2

T W

∆2 + W 2. (6.14)

This implies that, if all these quantities are temperature independent, then1/T1 ∼ Γ displays a low temperature plateau. This can be observed in Fig.6.10 representing 1/T1 measured in zero field by means of both proton and 57FeNMR. On the other hand, if a sizeable splitting ∆ is introduced, for examplea Zeeman splitting, at constant temperature the tunneling rate is depressed(for H ¿ Hc, the crossing field between different m-th levels on the oppositeside of the anisotropy barrier, see for example [64]). For this reason the mea-surement of the tunneling splitting is particularly successful in single crystalswhere a precise orientation of the anisotropy axis with respect to the field canbe achieved.These scenario will be now compared with the one emerging from 1/T1 mea-surements in TbPc2[TBA]10 and TbPc2[TBA]144 compounds.

6.4 Nuclear spin-lattice relaxation rates inTbPc2[TBA]

Nuclear spin-lattice relaxation rate 1/T1 was estimated from the recovery ofnuclear magnetization M(τ) after a saturation recovery pulse sequence. Therecovery law y(τ) = 1−m(τ)/m(∞) was found to be a stretched exponential,namely y(τ) = exp(−(τ/T1)

β) (Fig. 6.11), with an exponent β ' 0.5 prac-tically temperature independent. This form of the recovery law indicates adistribution of relaxation rates for different1H nuclei, possibly associated witha distribution of hyperfine couplings. The 1/T1 derived from the fit of therecovery laws has the meaning of an average relaxation rate over the wholedistribution (see for example sections concerning AxMPcs). In analogy withthe magnetization measurements, 1/T1 strongly depends on the cooling rateand on the intensity of the magnetic field applied during the cooling procedure.The measurements were found to be reproducible if the samples were left for30 minutes at 320 K and then cooled down to 200 K at a rate of 1 K/minute(or less) in a 1 Tesla magnetic field.

6.4. Nuclear spin-lattice relaxation rates inTbPc2[TBA] 175

Figure 6.11: Recovery law of 1H nuclear magnetization after a saturating RF pulsesequence. The solid line is the best fit according to y(τ) = exp(−(τ/T1)β) withβ ' 0.5 [85].

The temperature dependence of 1/T1 below 200 K in TbPc2[TBA]10 and TbPc2··[TBA]144 after the aforementioned cooling procedure are reported in Figs. 6.12and 6.13. Both samples show a peak in 1/T1 around 130 K and a low temper-ature shoulder, under T = 60 K. The 1/T1 peak is observed to move to highertemperature on increasing the external field and to be inversely proportionalto the field intensity. TbPc2[TBA] systems constitute actually the simplestachievable molecular nanomagnet, bearing a single magnetic center. Clearly,in this case the Tb[III] electronic levels split by the crystal field, rather thanthe collective quantum spin levels, determine the magnetic properties of thecompound. The scheme of the single ion energy levels proposed by Ishikawaet al. (Fig. 6.3) indicates that at temperatures T = 125 K for which 1/T1

peaks are observed, kBT < ∆E, with ∆E=628 K the difference between theground state and the first excited level. Therefore mainly the ground doubletmJ = ±6 of the multiplet J=6 should be populated. Accordingly, a single BPPfunction (6.9) satisfactorily fits the experimental 1H 1/T1 of TbPc2[TBA]10 andTbPc2[TBA]144 with

τc = τ0e∆ET = 1/(C∆E3)e

∆ET , (6.15)

as shown by the solid line in Fig 6.13. However, from the fit one estimatesfor TbPc2[TBA]144 ∆E=920 K , τ0=4.4·10−12 s, A = 7.3 · 109 (rad/sec)2 andC = 280 (Hz· rad)/ K3. For TbPc2[TBA]10 a smaller ∆E=833 K is found.In both samples the splitting is much larger than the one derived by Ishikawa


Figure 6.12: 1/T1 as a function of temperature in TbPc2[TBA]10 powders for H0 =4.5 kGauss. The solid line shows the behaviour expected by assuming the J=6manifold splitting proposed by Ishikawa et al. [57, 85].

et al. and 1/T1 calculated according to Eq. (6.12) on the basis of the Ishikawascheme of the ground multiplet (red curve in Fig. 6.12) cannot reproduce theexperimental data. Remarkably, ∆E constitutes the anisotropy barrier, whichis the largest ever observed in molecular nanomagnets. Furthermore, the cou-pling constant C here is about two orders of magnitude smaller than the oneof Mn12 [130]. It should be underlined that the high-temperature peaks of1/T1 cannot be induced by the fast rotations of the group [TbPc2][TBA], sincethe analysis of the linewidth temperature dependence indicates a substantialrestore of the rigid lattice linewidth for temperatures below T = 200 K. Infact, the subsequent broadening on cooling is solely proportional to the in-crease of the susceptibility, as already discussed in section 6.2. This implies〈∆ω2〉τ 2

c > 1, where τc is the correlation time of the fluctuations induced bymolecular motions; then, having 1/τc = ωc <

√〈ω2〉 ' 45 kHz around T = 200

K, the spectral contribution of these fluctuations must be almost absent forω ' ωN below T = 200 K. Definitely, the 1/T1 peaks observed in TbPc2[TBA]must be the consequence of low-frequency spin dynamics produced by spin-phonon scattering.Below 80 K 1/T1 in both TbPc2[TBA]10 and TbPc2[TBA]144 departs from theBPP law, which would in turn monotonously tend to zero on cooling below thepeak temperature. 1/T1 is nearly temperature independent for 40 K< T <80K, therefore the nuclear relaxation is not driven by temperature dependentspin fluctuations. It seems that the process responsible for nuclear relaxation

6.4. Nuclear spin-lattice relaxation rates inTbPc2[TBA] 177

10 1000.01

0.1

1

10

Pc2Tb[TBA]

144

H=4.5 kG H=1.6 kG

1/T

1 (s

-1)

T (K)

Figure 6.13: 1/T1 as a function of temperature in TbPc2[TBA]144 powders for H0 =4.5 kGauss (circles) and H0 = 1.6 kGauss (squares). The solid line is the best fitaccording to Eq. (6.9) in the text [126].

1 2 3 4 5 6 7 8 9 10 11 120.02

0.04

0.06

0.08

Pc2Tb[TBA]

144

T=4.2 K

1/T

1 (s

-1)

H (kG)

Figure 6.14: 1/T1 as a function of the applied field at T = 4.2 K [126].


Figure 6.15: Temperature dependence of the characteristic correlation times ofthe spin fluctuations as derived from 1/T1 data. The solid line is the best fit toTbPc2[TBA]10 data according to Eq. (6.16) and (6.17) [85].

is the tunneling among the mJ = +6 and -6 states of the ground doublet.The transition probability is determined by the mixing of the |±mJ〉 states,which is sizeably affected by the crystal field symmetry, namely by the ar-rangement of TBA molecules around the TbPc−2 units. It must be consideredthat in a powder some molecules will be orientated with the anisotropy axisperpendicular to the field, while in the remaining molecules the anisotropyaxis and the field will form an angle θ 6= 90. For the former ones, thetunneling will actually occur between the nearly degenerate mJ = +6 and-6 states at a temperature independent rate Γ ' ∆T /W , while in the othermolecules the levels will be split by an amount ∆Zeeman = 12gµBH cos θ, ac-cording to Eq. (6.13). Therefore in these molecules, at low temperatures whenkBT ¿ 12gµBH0, the transition among the mJ = ±6 levels becomes a temper-ature activated process with a gap ∆Zeeman. The plateau in the intermediateregion 40 K< T <80 K corresponds to the region where the thermally acti-vated tunneling rate Γ(T ) = Γ(∞) exp(−∆Zeeman/kBT ) approaches the hightemperature value Γ(∞) > ∆Zeeman. This is consistent with a gap of the orderof 10-20 K, to be compared with the Zeeman splitting gµBH ' 7 K. It shouldbe underlined that 1/T1 measurements at T = 4.2 K as a function of the fieldstrongly support this interpretation, since 1/T1 increases on reducing the fieldintensity (see Fig. 6.14). This is consistent with the increase of the tunnelingrate predicted by Eq. (6.14) on reducing the Zeeman splitting ∆Zeeman = ∆.The temperature dependence of the correlation time over all the explored tem-


perature range was deduced by the 1/T1 data inverting Eq. (6.9), introducingthe constant value of 〈h2

⊥〉 estimated from the 1/T1 peak. One notices in Fig6.15 that at high temperature τc shows practically the same T -dependencefor both samples with the aforementioned activated trend, arising from spin-phonon scattering. Below 100 K a plateau is present and the estimated valuefor τc in the TbPc2[TBA]144 sample is larger than the one in the TbPc2[TBA]10.This difference should be ascribed to the different tunneling rate for the twosystems, which can be associated with a different symmetry of the crystal fieldaround TbPc−2 and a different state mixing for the two samples. Below 30 Kthe activated increase of τc, associated with the Zeeman splitting, is finallydetected. The total correlation time for the hyperfine field fluctuations can beexpressed in terms of the high temperature activated contribution and of thetunneling processes among the mJ = ±6 states:

1

τc

= (1

τc

)att + (1

τc

)tun . (6.16)

The correlation time regarding tunneling, estimated in the low temperaturerange, corresponds to an average over all possible orientations, namely [85]

(τc)tun(T ) = (τc)tun(∞)

∫ π

0

sin θe12gµBH0 cos θ/kBT dθ =

= (τc)tun(∞) sinh(e12gµBH0 cos θ

kBT)/(

e12gµBH0 cos θ

kBT). (6.17)

The τc values reported in Fig. 6.15 can be fit with Eq. (6.16) with (τc)tun givenby Eq. (6.17). The agreement is good down to T = 25 K. On the contrary, atlower temperature a discrepancy between the experimental and the expectedvalues could be possibly associated with a partial orientation of TbPc2[TBA]molecules during the cooling procedure.To summarize, TbPc2[TBA]10 and TbPc2[TBA]144

1H 1/T1 displays low tem-perature peaks associated with indirect spin-phonon couplings, as observed inmany molecular nanomagnets. The experimental data can be fit with a singleBPP law, since just the low-lying energy levels are occupied. The correlationtime of the spin fluctuations follows an activated behavior with an activationenergy ∆E of the order of 900 K in both samples, sizeably larger than thesplitting between the ground and the first excited levels estimated by Ishikawaet al.. At low temperatures 1/T1 probes a temperature independent quantumtunneling between the Zeeman split levels mJ = ± 6 of the ground doublet.The correlation frequency over the whole temperature range is therefore a sumof a temperature independent term and a temperature activated term.

6.5 Summarizing remarksMagnetization and NMR spin-lattice relaxation measurements have allowedto unravel several of the static and dynamic properties of TbPc2[TBA]10 and


TbPc2 [TBA]144 compounds. Both the temperature dependence of the sus-ceptibility and 1H NMR 1/T1 clearly suggest that up to room temperaturethe magnetic properties of these systems are determined by the ground statemJ = ±6 of the J=6 manifold of the Tb[III] rare earth ion. This is consistentwith previous estimates of Ishikawa et al., who indicate a large energy splittingbetween the mJ = ±6 state and the first excited state mJ = ±5 ∆E = 628K, induced by the crystal field anisotropy. However, while the susceptibilityis consistent with this energy separation, the analysis of the 1/T1 temperaturesuggest a sizeably higher activation energy of the temperature induced spinexcitations between the lowest energy levels, ∆E ' 900 K.In TbPc2[TBA] compounds NMR 1/T1 exhibits the classical features found inthe molecular nanomagnets, a pronounced maximum in an intermediate tem-perature region, for T < ∆E, and a shoulder in the low temperature region,for T ¿ ∆E, the first ascribed to thermal fluctuations of the Tb[III] magne-tization due to spin-phonon interactions, the second to QTM. The tunnelingis observed in both samples and it reduces on increasing dilution by meansof [TBA]Br intercalation between different TbPc2[TBA] units. Meaningfully,these complexes behave as superparamagnetic particles having the magneticmoment frozen along the anisotropy axis at temperatures of the order of liquidnitrogen. The same condition in the analogous molecular nanomagnets canbe attained only at very low temperatures of the order of few K, since theanisotropy barrier is reduced to some tens of K at most.This represents a major attractive of TbPc2[TBA] compounds in view of theirimplementation as molecular bits in memory storage applications. In partic-ular, the value of the spin-phonon coupling C=280 (Hz·rad)/K3, two ordersof magnitude smaller than the ones evaluated in the other molecular nano-magnets, indicates a modest decoherence of the spin correlations for effect ofthe coupling with the thermal bath. Therefore, by increasing the dilution andimproving the uniaxial symmetry of the crystal field by means of appropriatechoice of ligands, it is expected that the single molecule magnetization wouldbe weakly affected by both tunneling transitions and quasi-elastic scatterings,being able to retain the same orientation, i.e. the memory of the initial orien-tation, for long times.The preparation of these these systems in the ground state, however, seemscomplicated by the dependence on the magnetothermal history of the molec-ular orientation along an external field. In particular, if molecules are cooledrapidly in an applied field, the low temperature physics shows non reproduciblefeatures, owing to the fact that the molecule become orientationally orderedin different configurations. Even in presence of a single crystal this problemcould be persistent for the presence of molecular motions above T = 200 K,randomly reorienting the molecules. These problem is actually avoided bymeans of slow cooling in an intense field but the modality of the molecular or-dering is not yet understood. Nevertheless, the exploitation of the interestingproperties of TbPc2[TBA] compounds seems promising for the field of contrast


agents for MRI (Magnetic Resonance Imaging), due to their biocompatibility.In T1 weighted experiments the high value of the Tb magnetic moment andthe slow spin dynamics of the TbPc2[TBA] complex could slow down the T1

relaxation of tissues affected by pathologies and sizeably enhance their signalwith respect to the one of normal tissues.


Chapter 7Conclusions

The results presented in this thesis evidence that metal phthalocyanine com-pounds are low dimensional materials with tunable electronic properties. Tran-sition metal phthalocyanines are one-dimensional systems which are semicon-ductors in the pristine state but can become metals upon doping by means ofalkali intercalation. The alkali-doped phthalocyanines are actually new com-pounds in which the cell parameters of the pure form are modified in orderto accomplish the accommodation of alkali ions between adjacent molecules[28]. In this sense alkali-doped phthalocyanines are then different from othermolecular conductors where the counteranions yielding the charge transfer aresegregated in separate columns. Lithium in both LixZnPc and LixMnPc isstrongly bound to the Pc ring and does not diffuse up to room temperature.Furthermore, its (1s) electron is completely transferred to MPc 2eg LUMO.Upon the charge transfer the alkali-doped phthalocyanines can become metal-lic if the overlap among LUMOs of adjacent molecules produces a bandwidthW at least of the order of the onsite repulsion U . The delocalization was ob-served in the LixZnPc compounds which have a finite density of states at theFermi level D(EF ) for 0<x<4, reaching a maximum at the exact half band fill-ing. On the contrary, LixMnPc are localized electron systems in which chargediffusion occurs via an activated hopping. While in weakly doped materials asLi0.5MnPc hopping is affected by disorder, spin-polarons and other local prop-erties, in the more homogeneous Li2MnPc its activated behaviour is ratherdetermined by an intrinsic structure of the LUMO, split by a gap ∆E in alower and an upper orbital. Then, since in Li2MnPc possibly ∆E > W , itcannot become a metal like Li2ZnPc.The splitting of the LUMO can be the consequence of a Jahn-Teller couplingwhich reasonably varies in the different AxMPc compounds owing to differentstatic and dynamic deformations of the molecules. It is important to remarkthat in AxMPc the Jahn-Teller interaction and the Hund’s exchange competeand the 2eg electrons probe an effective coupling Jeff = J − EJT /4. It islikely that in Li2ZnPc Jeff is of the order of the reduced bandwidth W ∗ andU/W < (U/W )c, the critical value for a metal insulator transition. Then the

183

184 7. Conclusions

2eg electrons are delocalized and can possibly probe a pair attraction [48]. Onthe contrary, in Li2MnPc Jeff > W ∗ and the the material resembles a Mott-Jahn-Teller like insulator with a small gap.In strongly correlated materials on the edge of a metal-insulator transition,Jeff ∼ W ∗ induces an attractive pairing, expected to generate superconduc-tivity at low temperatures [48]. Interestingly, NMR 1/T1 measurements inLixZnPc indicated that these materials are strongly correlated and possiblyexperience a low temperature localization on cooling for non integer x → 4,in analogy with the Beechgard salts[36]. Correlations and localization increaseon increasing doping for x>2, as the NMR 1/T1 at low temperature signals

the onset of ~q = 2~kF instabilities typical of strongly correlated one-dimensionalmaterials for T < Tρ the localization temperature [36]. On the other hand, inLi2ZnPc the electrons are strongly correlated and delocalized down to liquidhelium temperature and no evidence for a superconducting transition emergesfrom the experimental data. This is probably the consequence of the pro-nounced one-dimensionality of LixZnPc, whereas this transition would requirethe onset of 3D correlations. In this respect it has been observed that theapplication of pressure for x ∼ 2 (Li2.35ZnPc) induces a steep increase of theD(EF ) and suggests that LixZnPc could be a pseudogapped metal close to thehalf-filling, where the pseudogap is originated by the pairing in singlets of theitinerant electrons.While in LixZnPc the electron delocalization significantly changes upon alkali-doping, in LixMnPc compounds the main modifications concern the intrisicmagnetic properties. Doping enhances the crystal field symmetry in the MnPcmolecules, so that the 3d Mn[II] orbitals are lowered and are equally popu-lated, attaining a spin crossover from S = 3/2 to S = 5/2 spin ground statefor x=2. Simultaneously the cell deformations induced by alkali intercalationturn the superexchange interaction between adjacent molecules from ferromag-netic to antiferromagnetic. The 2eg electrons, due to the Jahn-Teller splittingof the LUMO at low temperature, fill the lower LUMO orbital, forming anS = 0 ground state. However, on rising temperature the spin excitations ofthe LUMO don’t induce any spin-crossover in the compounds, since the 2eg

electron dimers are broken by the activation of intermolecular hopping. Upondoping also the low temperature slow relaxation of the MnPc moments in-duced by anisotropy in the pristine compounds is affected. While 1H 1/T1 inLi0.5MnPc detects this process since a large amount of material still in thepristine state for this low Li content, 1H 1/T1 in Li2MnPc doesn’t indicate anylow temperature freezing of the Mn[II] magnetic moments. This is consistentwith the reduction of local anisotropy at the MnPc molecules upon the inter-calation with the alkali ions.Remarkably, a fascinating aspect of the NMR investigations in LixMnPc andLixZnPc is the different temperature and field dependence of the 1/T1 re-laxation rate of 7Li and 1H. In LixZnPc this was tentatively ascribed to thepresence of defects more sensitively coupled to the protons, but in LixZnPc

185

this is a clear consequence of the interaction with the Mn[II] ions. In Li2MnPcthe strong coupling of 1H to the Mn[II] paramagnetic fluctuations is visiblydifferent from the one probed by 7Li whose 1/T1 is two orders of magnitudelower and activated with a larger gap [114]. It was then concluded that 7Li and1H belonging to the same Pc units in LixMnPc and LixZnPc probe differentdynamics, the former ones the LUMO electrons dynamics, while the latter onesboth the 2eg electrons and the Mn[II] localized electrons dynamics. The originof this different behaviour, possibly involving different hyperfine couplings, stillhas to be clarified and constitutes an issue of interest for the future research.While transition metal phthalocyanines are one-dimensional systems, the rare-earth phthalocyanines form exquisitely quasi 0-D systems exhibiting the prop-erties of the single-molecule magnets [59, 60, 61, 129]. These single-ion com-plexes behave as superparamagnets on cooling at moderately low temperatures,owing to a very high splitting in the ground state multiplet of the rare-earthmetal, induced by crystal field anisotropy. In [TbPc2][TBA] compounds thesplitting between the low lying J = 6 sublevels is ∆ = 900 K according toproton NMR 1/T1 measurements, therefore the system is frozen in the groundstate mJ = ±6 of the J = 6 manifold for temperatures that can be reachedcooling with liquid nitrogen. This certainly constitutes a major advantage inview of the technological implementation of these materials as memory units.In addition, the achievement of the superparamagnetic state at the highertemperature ever observed for a single-molecule magnet allows to easily in-vestigate the Quantum Tunneling of the Magnetization (QTM) phenomenon,which in compounds as Fe8 is active only at temperatures of the order of1 K, thus requiring a demanding experimental effort for its observation. In[TbPc2][TBA] the study of the tunneling between mJ = ±6 levels of the groundstate gives precise indications about the processes inducing spin decoherence in[TbPc2][TBA]. While the activated reversal of Tb[III] at high temperatures isinduced by the spin-phonon coupling, the weaker observed for a single-moleculemagnet, the temperature independent QTM can be induced by intermolecularexchanges and dipolar interactions. The research on the diluted systems willthen be crucial in the establishment of the optimal molecular configurationof the [TbPc2][TBA] compounds in order to improve the uniaxial asymmetryof the crystal field and minimize the decoherence processes. This would en-hance the capability of the [TbPc2][TBA] molecules to be frozen in the samespin state over times sufficiently long for the usage in several intriguing futureapplications, for example as units of RAM memories in quantum computers.

186 7. Conclusions

Appendices

187

Appendix AInstrumental details andpressure cell for magnetizationmeasurements

Fig. A.1 and A.2 show the interior of the apparatus employed for the magneti-zation measurements, the Quantum Design MPMS-XL SQUID magnetometer.The samples are mounted on a rod and inserted in a cavity where a static fieldis supplied by a superconducting magnet. The chamber is sealed and main-tained at a low pressure with static helium gas. The temperature control ofthe sample area is provided by heaters and low temperature pumps. The mea-surement is performed by moving the sample with a stepping motor throughthe closed loop superconducting pick-up system, as described in chapter 1.DC magnetization measurements can be performed also at variable pressureintroducing the sample in a dedicated pressure cell. A little amount of sample(usually some mg) is put in the cavity of a cylindrical teflon sample holder andsealed with a cap. The sample is immersed in a liquid, the Fluorinert, inertto chemical reaction but affording an isotropic transmission of the pressure,and it is placed inside the cell. The scheme of the pressure cell designed forthe MPMS-XL SQUID is shown in Fig A.3. The pressure is exerted by meansof a transmission line. It is composed, starting symmetrically from the sides,by Cu-Be pieces, two Cu sealing discs, an additional centering teflon cylinderand the sample holder. All the pieces are contained in the cavity of a Cu-Becylinder with two Cu-Be screw plugs having a central hole.Once the hole system is mounted, it is blocked in a supplementary holderwhich prevents rotations and movements of the cell. The pressure is applieddefinitely by means of a large bolt turning in it, which pushes down a piston onthe top of the transmission line. While the pressure is increased, also the topbolt of the pressure cell is gradually turned to subsequently keep the internalpressure at the desired value when the external press is lifted. The calibrationof the cell is done by means of magnetization measurements in samples having

189

190 A. Instrumental details and pressure cell for magnetization measurements

properties with a known pressure dependence, for example transition temper-atures between different magnetic phases (as in MnSi). So, as the externaltorque necessary to turn the pressing bolt is measured by a dynamometric key,the proportionality between the applied torque and the internal pressure canbe deduced.

1.1.1.1. SampleSampleSampleSample RodRodRodRod

2.2.2.2. SampleSampleSampleSample RotatorRotatorRotatorRotator

3.3.3.3. SampleSampleSampleSampleTransportTransportTransportTransport

4.4.4.4. Probe Probe Probe Probe AssemblyAssemblyAssemblyAssembly

5.5.5.5. HeliumHeliumHeliumHelium LevelLevelLevelLevelSensorSensorSensorSensor

6.6.6.6. SuperconductingSuperconductingSuperconductingSuperconductingSolenoidSolenoidSolenoidSolenoid

7777 FlowFlowFlowFlow ImpedanceImpedanceImpedanceImpedance

8888 SQUID Capsule SQUID Capsule SQUID Capsule SQUID Capsule withwithwithwithMagneticMagneticMagneticMagnetic ShieldShieldShieldShield

8

6

Figure A.1: MPMS system components [133].

191

Figure A.2: Location of the detection second-order gradiometer superconductingcoils inside the sample chamber. [133]

192 A. Instrumental details and pressure cell for magnetization measurements

PTFESample holder

Cu ring

Cu-Becylinders

Figure A.3: Scheme of the pressure cell fabricated for the MPMS-XL SQUID mag-netometer.

Appendix BThe experimental apparatus ofNMR

The heart of the NMR experimental apparatus is the circuit constituted bythe probe and the spectrometer. The schematics of the NMR spectrometeris reported in Fig. B.1. The spectrometer produces the pulse sequences bymeans of a synthesizer, a pulse programmer and a power amplifier. The syn-thesizer generates a reference wave with fixed frequency and phase and thepulse programmer mixes it to four possible carriers at frequency νi and with aphase difference of 0, 90, 180, 270 from the reference. Cycling the phase ofthe emitted signal also affords the elimination of spurious periodic oscillationsin the signal arising for example from the mechanical vibration of the coil, i.e.the ’ringing’. The final working frequency νL is obtained when the referencefrequency is ν = νi + νL; other high frequency products of the mixing areopportunely filtered. The intensity of the r.f. field H1 can be between tensand hundreds gauss. The τπ/2 are normally between 0.1 µs a 10 µs, their timeseparation between µs and minutes and the repetition time after each entiresequence between tens of ms and tens of minutes. The pulse shape can beopportunely chosen; the most common is the rectangular shape in the timedomain corresponding to a sinc(ω) in the frequency domain.The radio frequency signal is transmitted to the probe, a LRC resonant circuitwith variable inductance, which is tuned to the desired resonance frequencyand adapted (50 Ω) to the output power of the amplifier. The sample is in-serted in a resistive coil, used for both transmission and reception. To ensurethe maximum homogeneity of H1, the coil is fabricated, when possible, in or-der to contain the entire volume of the sample. The inductances of the circuitmust be regulated to enhance the Q factor (>70-90)

Q =R

ωLL. (B.1)

All the in and out coming electric signals are picked up from the coil by means

193

194 B. The experimental apparatus of NMRPULSEPROG.DEPHASERSYNTHESIZER

DIGITALIZER

PROBE

Figure B.1: Sketch of an NMR spectrometer.

of a coaxial line and are directed to a decoupler. This is a bridge of diodesthat deviates the detected signal to the receiver of the spectrometer, blockingthe way back to the radio frequency amplifier.The received signal is preamplified and demodulated at the reference frequencyν = νi + νL with a phase sensitive lock-in system. The voltage measurement issampled with a sampling frequency which is the inverse of the time resolutionof the experiment and digitalized. A computer with dedicated software is usedto establish the experimental parameters, visualize and treat the measuredvoltage as a function of time. The two Fourier transform NMR spectrometersApollo Tecmag and Bruker MSL200 were used to perform the NMR measure-ments presented in this thesis. They both can reach a working frequency of300 MHz.The static external field is generated by an electromagnet or a superconductormagnet. The first is used to cover a field range of 1000−17000 Gauss, while thesecond can achieve much stronger fields of 2− 9 Tesla. The field homogeneityreached in a superconducting coil is generally much higher than in an electro-

195

magnet. The estimate of the line broadening due to the magnet is performedbefore the experiments, measuring the NMR line width in a solution. In fact,the intrinsic line broadening in a liquid is negligible because the moleculesare free to move and the local magnetic interactions are rapidly modulated.Therefore this measurement supplies the effective width of the field distribu-tion in the magnet over the sample volume. Certainly, the solution must havea volume comparable with the sample and put in the same position.The cryogenic systems used to vary the sample temperature are flux cryostatsand bath cryostats. In the case of the first ones, a cold flow of nitrogen orhelium gas directly laps and cools the sample. The gases are collected by anexternal transfer line immersed in the liquid and flow towards the sample areaon the bottom of the cryostat through a capillary. Before reaching the sample,the desired temperature of the gas is regulated by a heater ruled by an externaltemperature controller. After passing the sample area the gas is then recoveredby means of a pump and the flux is manually controlled. The temperature ismeasured both in the heater and in the sample area by thermocouples or ther-moresistors. The minimum temperature reached with these systems, when theheater is disabled and the flux is high, is the temperature of the liquid at am-bient pressure, 77 K for N2 and 4.2 K for 4He. In the bath cryostats, instead,the sample is directly immersed in liquid helium and therefore it is kept toits constant temperature. To lower the liquid temperature, then, a pumpingis exerted over the liquid surface and the minimum temperature reached withthis method is about 1.2 K.Clearly the cryostats are connected to external pumps which keep ultra lowvacuum in the interspace between the sample chamber and the ambient tominimize thermal exchanges. In the case of the bath cryostats, in addition, anintermediate internal layer is filled with liquid nitrogen to pre-refrigerate thesample space before the liquid helium transfer.

196 B. The experimental apparatus of NMR

Appendix CThe muon beam lines at theRutherford AppletonLaboratory

In this appendix the production of muons, the properties of the muon beamand the spectrometers implied at the Rutherford Appleton Laboratory will beillustrated. The positive muons are produced through several decays. High en-ergy protons (∼ 800 MeV), coming from the synchrotron, collide on a graphitetarget. They are so involved in the processes

p + p −→ p + n + π+ (C.1)

p + n −→ n + n + π+, (C.2)

and generate pions. The pions have a short lifetime of about 26 ns and decayaccording to

π+ −→ µ+ + νµ, (C.3)

finally generating the muon, as it is been already described in section (2.3.1).At ISIS the synchrotron emits 50 double proton pulses per second, each witha 80 ns duration (width at half maximum) and separated by 300 ns. As thepions decay very early, the muons beam retains the same pulse structure of theproton beam. The pion decay just introduces a slight asymmetry in the pulseshape. Thanks to the double pulse beam, different lines can work in parallel,as will be illustrated later.The muons are first focused by quadrupolar magnets and directed by dipolarmagnets; only the muons with a momentum of 26.5 MeV/c proceed in thebeam line towards the spectrometers. The products of spurious decays areseparated on fly by means of two mutually orthogonal magnetic and electricfields. This combination deflects the positrons because discriminates the ve-locity of particles, 0.24 c for the muons, c for the positrons. Afterwards anelectrostatic ”kicker” feeds three experiments, addressing half of the first pulse

197

198C. The muon beam lines at the Rutherford Appleton

Laboratory

Figure C.1: The MuSR spectrometer at RAL[134]

to the spectrometer EMU and half to DEVA and the second pulse to MuSR.The beam is collimated by means of slits to adjust the section to the samplesize and moderate the energy of the beam, as required. The sample holderis generally a silver plate, giving rise to a low background, mounted at theend of a long rod to be inserted in cryostats or furnaces. Powders must bepacked in aluminium or Mylar foils and contained between the silver plate anda silver mask, which catches and stops the muons missing the sample in theirtrajectory.The cryogenic system or the furnace, carrying the sample in the cavity, arepositioned in the magnetic field produced by the spectrometer. The geometryof the experiment varies with the instrument. For example EMU is suitable forlongitudinal field measurements; in this case a field between 0 and 4 kGaussis produced by Helmotz coils and additional coils can apply a transverse fieldof 20 Gauss. This is used to calibrate the instrumental α factor appearing inexpression (2.45) every time a sample is changed. For the zero field experi-ments, other coils produce a field which compensates the earth field. MuSRis a 64-detector spectrometer (Fig. C.1) which can be rotated through 90 toenable both longitudinal and transverse measurements to be made. Fields ofup to 2500 G can be applied. The positron detectors are composed in practiceby plastic scintillators, lightguides and photomultipliers and are segmented toenhance the instantaneous responsivity to high count rates. They cover a widesolid angle but are usually grouped in two or four groups. The photomultipliertubes are connected to a digital converter to handle data with computers. Thetime trigger is referenced to the center of the muon pulse and the entire ac-

199

quisition lasts 32 µs. The bin resolution is 16 ns, thus for the pulse width andthe instrumental deadtimes usually a certain amount of time bins immediatelyafter the time zero must be neglected. Statistical averages are performed overseveral million of events to obtain the final µSR time spectra.

List of Publications

Marta FilibianDipartimento di fisica ”A. Volta”,University of PaviaVia Bassi 6, 27100 Pavia - Italyemail: [email protected]

1. M. Corti, M. Filibian, P. Carretta, L. Zhao and L. K. Thompson, ”LowEnergy spin excitations in Mn3x3 molecular nanomagnets”, Phys. Rev.B 72, 064402 (2005)

2. P. Santini, S. Carretta, E. Liviotti, G. Amoretti, F. Borsa, P. Carretta,M. Filibian, A. Lascialfari and E. Micotti, ”NMR as a probe of the re-laxation of the magnetization in magnetic molecules”, Phys. Rev. Lett.94, 077203 (2005)

3. M. Filibian, P. Carretta, T. Miyake, Y. Taguchi and Y. Iwasa, ”Low-energy excitations in the electron-doped metal phthalocyanine Li0.5MnPcfrom 7Li and 1H NMR”, Phys. Rev. B 75, 085107 (2007)

4. M. Corti, F. Carbone, M. Filibian, Th. Jarlborg, A. A. Nugroho and P.Carretta, ”Spin dynamics in a weakly itinerant magnet from 29Si NMRin MnSi”, Phys. Rev. B 75, 115111 (2007)

5. M. Filibian, P. Carretta, T. Miyake, Y. Taguchi, Y. Iwasa, G. Zoppellaroand M. Ruben, ”Low-energy excitations in electron-doped metal phthalo-cyanines”, Physica B, in press (2007)

6. M. Filibian, ”Electronic properties of novel Alkali-doped Metal Phtalo-cyanines molecular crystals”, Scientifica Acta, N1, Issue 1 pag. 133-136(2007)

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Acknowledgements

I thank my supervisor Pietro Carretta for the supervision of the experimentalactivity and for the enlightening discussions and all the NMR group of theUniversity of Pavia for the support, Prof. Alessandro Lascialfari, Prof. Maur-izio Corti, Prof. Attilio Rigamonti, Prof. Ferdinando Borsa, Sergio Aldrovandiand the Ph.D. students who have been working or are currently working withme, Dr. Ileana Zucca, Dr. Nico Papinutto, Dr. Edoardo Micotti, Dr. ManuelMariani, Ettore Bernardi and Francesca Branzoli.

Thanks to Dr. Cristina Mozzati for the precious help she gave in solvingvery important issues by means of EPR measurements.

I also acknowledge the fundamental collaboration with chemists, who workedhard on the synthesis and preparation of several samples, and sincerely thankthem: Prof. Paolo Ghigna and Anna Maria Bertasa of the Department ofPhysical Chemistry ”M. Rolla” of the University of Pavia; Giorgio Zoppel-laro and Mario Ruben of the Institut fur Nanotechnologie, ForschungszentrumKarlsruhe (Germany); Toshiaki Miyake, Yasujiro Taguchi and Yoshihiro Iwasaof the Institute for Materials Research, Tohoku University, Sendai (Japan).

Finally I thank Stephen J. Blundell for his encouraging comments about thescientific value of this work.

Ringraziamenti

Eccoci dunque alla fine di un’altra avventura.

Scrivere questa tesi non e stato facile per me, tuttavia man mano che le ideehanno preso forma e le parole hanno cominciato a comporre un senso, le pauree le insicurezze hanno cominciato a lasciare spazio al piacere della riflessionee ad una ricerca che ha anche avuto quasi una valenza introspettiva. E possodire che questo ha costituito un piccolo traguardo gia di per se.

Ringrazio Pietro Carretta, in quanto mi ha trasmesso una passione per questolavoro che mi ha fatto crescere dal punto di vista scientifico e umano. Loringrazio, oltre che per i suoi i suoi insegnamenti e la sua guida in questocammino, anche per la sua pazienza, per la sua autenticita, per la sua fiducia.Spero che qualunque sia il destino di questo viaggio, continueremo a collabo-rare e a confrontarci in uno scambio sempre piu ricco, nella fisica e nella vita.

I miei compagni di stanza in ordine di comparsa, Ileana Zucca, Nico Papinutto,Edoardo Micotti, Manuel Mariani, Ettore Bernardi, Moreno Pasin, FrancescaBranzoli. Non pecco di adulazione dicendo che lavorare assieme a loro e statauna fortuna e sono diventati gli irrinunciabili punti di riferimento delle miegiornate, settimane, dei mesi e poi degli anni.

Sergio Aldrovandi, sempre generoso e comprensivo e i Prof. Alessandro Las-cialfari, Maurizio Corti, Attilio Rigamonti, Paolo Ghigna e Giorgio Zoppellaro.

Ringrazio la mia famiglia. Loro lo sanno.

I compagni di universita per la presenza in tanti momenti della vita fuori edentro il dipartimento: Macina, Conci, Fede, il Faraone, Gemi, Laura, il Ne-gri, Ilaria, Marco G, Zinna, Pedro, Angela.

Ringrazio Mori, anche lei lo sa.

Ringrazio poi le persone che collaborano con me nella musica e determi-nano i miei colori.I Cumbajanca per le tinte forti e per i primi segnali di rincitrullimento: EnricoNegri, Gianfranco Brigada, Americo Costantino, Marco Sannella, Massimo Ar-tiglia, Moreno Rizzato.Il pamTrio per le tinte tenui e il calore intimista di tre calici di vino rosso:Ambrogio Dalo e Paolo Malusardi.I SenzaPazienza, la mia seconda famiglia hippy, per le porpore e l’alto livellodi sgamo: Bruno Cerutti, Marco Sannella, Ambrogio Dalo, Paolo Malusardi,Americo Costantino. Il produttore Armandin Chambier de Richelieu, dispen-satore di ordine musicale e morale a forza di esercizi e stangate. E poi la dolceMariariosa Delleani, l’indaffarata Cristina Magistri, la vulcanica Sciortino, ilpiccolo Francy.

In ordine sparso: Stefania, Wainer, Pietro, Mike, Lo, Teo, Cinzia, Alice,Savino, Massj, Gio.

Ringrazio la luce pulviscolare dalla finestra sul blu di quella casa di Bogliasco.E un cigolio distante che pareva cantare tra il fragore ciclico delle onde ”I amthe walrus”. Ringrazio la spiaggetta, le lucciole, i tramonti del Canarazzo.

Ringrazio i Beatles, Antonio Carlos Jobim, Caetano Veloso, Richard Galliano,Bireli Lagrene, Carmen McRae, Bruce Springsteen, Otis Redding, i Police, gliSkiantos, Lou Reed, Peter Gabriel, Paul Simon, Vinicio Capossela, Enzo Jan-nacci, Giorgio Gaber.

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