Physica Scripta in press (2013)

µSR and inelastic neutron scattering investigations of the caged type Kondo semimetals: CeT2Al10 (T=Fe, Ru and Os)

D.T. Adroja1,4$, A.D. Hillier1, Y. Muro2, T. Takabatake3, A.M. Strydom4, A. Bhattacharyya1,4, A. Daoud-Aladin1 and J. W. Taylor1

1ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot Oxon, OX11 0QX, UK 2Liberal Arts and Sciences, Faculty of Engineering, Toyama Prefectural University, Imizu 939-

0398, Japan 3Department of Quantum matter, ADSM, and IAMR, Hiroshima University, Higashi-Hiroshima,

739-8530, Japan 4Physics Department, University of Johannesburg, PO Box 524, Auckland Park 2006, South

Africa (Dated:25th Nov. 2013)

Recently the Ce-based caged-type compounds having general formula CeT2Al10 (T=Fe, Ru and Os) have generated considerable interest due to the Kondo semiconducting paramagnetic ground state (down to 40 mK) observed in CeFe2Al10 and anomalously high magnetic ordering temperature with a spin gap formation at low temperatures in Kondo semimetals CeRu2Al10 and CeOs2Al10. The formation of long-range magnetic ordering out of the Kondo semiconducting/semimetallic state itself is extraordinary and these are the first examples of this enigmatic coexistence of electronic ground states. These compounds also exhibit strong anisotropy in the magnetic and transport properties, which has been explained on the basis of single ion crystal electric field (CEF) anisotropy in the presence of strongly anisotropic hybridization between localised 4f-electron and conduction electrons. Further they also exhibit a remarkable modification of magnetic and transport properties with doping on Ce, or T or Al sites. In this review we briefly discuss the bulk properties of these compounds, giving a detailed discussion on our muon-spin-relaxation (µSR) investigations and inelastic neutron scattering (INS) results. We present µSR and INS results of Ce(Ru1-xFex)2Al10 and CeOs2Al10 as well as the µSR results of NdFe2Al10, NdOs2Al10 and YFe2Al10 for comparison. The zero-field µSR spectra clearly reveal coherent two-frequency oscillations at low temperatures in CeT2Al10 (T=Ru and Os) and Ce(Ru1-xFex)2Al10 (x=0.3 to 0.5), which confirms the long-range magnetic ordering with a reduced moment of the Ce. On the other hand the µSR spectra of Ce(Ru1-xFex)2Al10 (x=0.8 and x=1) down to 1.2 K and 0.04 K, respectively exhibit a temperature independent Kubo-Toyabe (KT) term confirming a paramagnetic ground state. INS measurements on CeT2Al10 (T=Ru and Os) exhibit sharp inelastic excitations at 8 and 11 meV at 5 K due to an opening of a gap in the spin excitation spectrum. A spin gap of 8-12 meV at 7 K, with a strong Q-dependent intensity, is observed in the magnetic ordered state of Ce(Ru1-xFex)2Al10 with x=0.3 and 0.5 which remarkably extends into the paramagnetic state of x=0.8 and 1. The observation of a spin

1

gap in the paramagnetic samples (x=0.8 and 1) is an interesting finding in this study and it challenges our understanding of the origin of the semiconducting energy gap in CeT2Al10 (T=Ru and Os) in terms of a hybridization gap opening only a small part of the Fermi surface, gapped spin waves, or a spin-dimer gap. Further, the µSR study of NdFe2Al10 below TN exhibits a clear sign of two frequency oscillations, which are absent in NdOs2Al10. Moreover the µSR study of YFe2Al10, which has been proposed as a compound exhibiting ferromagnetic critical fluctuations did not reveal any clear sign of critical magnetic fluctuations down to 60 mK, within ISIS µSR time window, which is unexpected for a T→0 quantum phase transition (QPT).

PACS No: 71.27.+a , 75.30.Mb, 75.20.Hr, 25.40.Fq

$E-mail address: devashibhai.adroja@stfc.ac.uk

Keywords: Magnetic excitations, Spin gap, Kondo semiconductor, Antiferromagnetic phase transition, CeT2Al10 (T=Fe, Ru, Os), Ce(Ru1-xFex)2Al10, NdFe2Al10, NdOs2Al10, YFe2Al10, Muon spin relaxation, Reduced moment magnetic ordering, Inelastic neutron scattering, Quantum phase transition.

Contents I. Introduction A. General aspects of caged-type structures II. Experimental techniques III. Results and discussion A. Physical properties of CeT2Al10 (T=Fe, Ru and Os) B. µSR investigations 1. µSR investigations on Ce(Ru1-xFex)2Al10 (x=0 to 1) 2. µSR investigations on CeO2Al10 3. µSR investigations on NdFe2Al10

4. µSR investigations on NdOs2Al10

5. µSR investigations of a quantum critical behaviour in YFe2Al10

C. Inelastic neutron scattering (INS) investigations 1. INS study on Ce(Ru1-xFex)2Al10 (0≤x≤ 1) 2. Spin gap above the magnetic ordering in CeOs2Al10

2

IV. Conclusions and future work

3

I. Introduction In recent years the investigation of intermetallic compounds of rare-earth elements (with localised 4fn electrons) and transition metals (with itinerant dn electrons) has been a frontier topic of research in magnetism due to the fact that several compounds of this family can be used as permanent magnets of outstanding quality, giant magnetocaloric materials, memory storage, optical sensor devices, and solid state thermoelectric coolers [1-7]. Not only the magnetic attributes of rare-earth metals and alloys, but also a further diversity of thermodynamic and structural features have attracted attention to some of their exceptional properties with respect to reversible absorption of hydrogen gas at room temperature and nearly atmospheric pressures providing ideal candidates for energy storage materials [7]. Besides the technological applications of intermetallic compounds, Ce and Yb-based

compounds exhibit a rich fundamental physics due to localised 4fn (n=1 for Ce and 13 for Yb)

electrons, which are often predisposed towards a localised versus delocalised or itinerant

behaviour [8-12]. Due to this unique property the compounds exhibit interesting behaviour such

as the Kondo effect [13], in which local moment is screened by the conduction electrons cloud,

heavy electron behaviour (or heavy fermion, HF) [14], in which the effective mass (m*) of an

electron becomes 1000 times or more heavier than the mass of a bare electron (m0), mixed

valence behaviour [15], in which the 4fn-electrons gain energy from the hybridization between

conduction electrons to jump into the conduction band and vice-versa, reduced magnetic moment

ordering [16], Kondo insulator or Kondo semiconductor [17-19], spin and charge gap formation

[20], charge and spin density waves, metal-insulator transition, unconventional superconductivity

[21-25], spin-dimer formation, non-Fermi-liquid (NFL) behavior and quantum criticality

associated with quantum phase transitions (QPT) [26-33]. These phenomena arise due to the

presence of strong hybridization between localised 4f-electrons and conduction electrons and are

present mainly in Ce and Yb systems, due to the proximity of 4f-level close to the Fermi level,

and occasionally also in Pr, Sm, Eu and Tm based compounds. It is interesting to note that these

compounds can have two static valence states (4fn and 4fn+1) or dynamic valence fluctuations

between the two valence states with a characteristic time scale, tvf. Thus the experimental

techniques, which have probing time faster than tvf will see two valence states separately (i.e. X-

ray photoelectron spectroscopy (XPS) and X-ray absorption near edge structure (XANES)) [34].

While neutron scattering as well as most other bulk physical properties having probing time

slower than tvf will give information about an average valence state [35].

4

In heavy fermion systems the existence of strong electron-electron correlations gives a huge

electronic density of states near the Fermi level (EF) (Fig.1a), which is responsible for the origin of heavy

effective mass. Among the novel properties mentioned above, the fascinating the Kondo insulator

behaviour observed in d- and f-electron systems has been of recurring interest to this community and has

recently attracted considerable interest in theoretical and experimental condensed matter physics [17-19,

36-42]. Some of these materials exhibit a very small energy gap (Fig.1b), called a hybridization gap, near

EF and it is believed that the gap arises in the lattice, from the hybridization between the localized

electrons (d- or f-electrons) and the conduction electrons [28-29, 34-36]. The main theoretical interest in

these materials is due to the existence of large many-body renormalizations [43].

The application of pressure, magnetic field or doping that acts an adjustable tuning

parameter (δ) in HF systems, as there is a strong competition between electronic localisation and

itinerant behaviour, which provides an opportunity to continuously modify the properties of HF

systems (Fig.2). Generally, the localization of the electrons leads to a fundamental state of

antiferromagnetic (or in some cases ferromagnetic) nature, while their delocalization is

accompanied by a paramagnetic Fermi liquid (FL) state, which is a key ingredient of our

fundamental understanding of metals, and founded upon the Landau FL theory [44]. At low

temperatures FL systems show characteristic behaviour with resistivity, ρ(T) ∝ Tn with n=2,

linear heat capacity with temperature Cv(T) ∝ γeleT and temperature independent susceptibility,

χ(T) [44]. The FL model is the correct description of the low-temperature measurable parameters

of a metal provided that the electron interactions as T→0 become temperature independent and

are short ranged in both space and time. Deviations from the FL behaviour has been observed at

low temperatures in many HF systems, near a critical value of the tuning parameter (δc), which is

referred to as non-Fermi-liquid behaviour (NFL) [26, 27, 30, 31, 45]. NFL systems at low

temperatures exhibit ρ(T) ∝Tn with n~1, Cv(T) ∝-log(T) and χ(T) ∝–log(T)/T, which are

expected for two-dimensional antiferromagnetic spin fluctuations [26, 45,46]. The phase diagram

of HF systems, shown in Fig. 2, originates from the competition between the onsite Kondo

interaction (TKondo), which destabilizes magnetism and support formation of nonmagnetic singlet,

and intersite Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions (TRKKY), which favors a

magnetic ground state [47]. The Kondo temperature varies exponentially with the product of the

electron density of state D(EF) at EF and the exchange coupling Jsf between conduction electrons

and local moments, TKondo ~ (D(EF)* Jsf )1/2*exp(-1/|D(EF)* Jsf|), while magnetic RKKY

interaction energy varies as square of this product, TRKKY ~ (D(EF)* Jsf)2. Doniach [47] made the

first theoretical study of a Kondo lattice using a one dimensional chain of spins and this result is

still used to classify many HF compounds on the phase diagram. It can be controlled via an

5

adjustable parameter δ (called lattice density), which allows us to tune systems from an

antiferromagnetic phase to a paramagnetic and Fermi-liquid regime. These two regimes are

separated by a T= 0 quantum phase transition (QPT), also called quantum critical point (QCP)

[26, 46, 48]. The phase space near quantum phase transitions has been a source of new and

unexpected behaviour in physical properties of solids. Among these are collective NFL behaviour

and the development of an unconventional superconductivity. The notion of critical magnetic

fluctuations is at the heart of HF superconductivity. The existence of magnetically mediated

superconductivity in HF compounds could help to shed light on the question of whether magnetic

interactions are relevant for describing the superconducting and normal-state properties of other

strongly correlated electron systems, perhaps including the high-temperature copper oxide

superconductors (HTSC) [49-50]. Although superconducting transitions temperatures (Tc) are

remarkably different in HFSC (below 2-3K) and in HTSC (50-150K), these two families of

materials have surprisingly similar phase diagrams and thus also the likelihood of a common

origin of the superconductivity. The common behaviour can be understood by considering the

ratio between TC and the electronic band width or Fermi temperatures.

A. General aspects of caged- type structures

Compounds having a caged type crystal structure have been of enduring interest for example clathrates [51], filled skuttrudites [5, 52, 53], β-pyrochore [54], RT2X20 (R=rare earths T=transition metals and X=Al and Zn) [55], R3T4Sn3 and many more [56-58]. Their crystal structure commonly possesses three-dimensional skeletons containing large atomic cages, inside of which relatively small atoms may naturally be located or otherwise may often be injected and these can rattle with large atomic excursions owing to either a virtual size mismatch or due to very weak chemical bonding as a consequence of large interatomic spacing. The rattling atoms thus have weak structural coupling, and strong electron–phonon (rattler) coupling, leading to a significant anharmonicity for rattling vibration for the central atom. These compounds have attracted a huge interest because rattling vibration may suppress the phonon part of the thermal conductivity, resulting in conditions favourable for the enhancement of thermoelectric efficiency [5, 52, 58]. Besides their suitability for exploiting the thermoelectric effect, the caged compounds of Ce, Pr and Yb are deemed to be especially suitable for thermoelectric power generation due to the high Seebeck coefficient found in strongly correlated compounds of these elements (the Seebeck coefficient features in the thermoelectric figure of merit as a squared quantity). These compounds are important moreover to investigate superconductivity, Kondo effect and anisotropic hybridization, quadrupolar ordering and many more exotic properties [55, 57, 58].

6

Recently the CeT2Al10 (T=Fe, Ru and Os) compounds, having caged-type orthorhombic crystal structure (Fig.3), have attracted considerable interest in strongly correlated electron systems’ community, both experimentally and theoretically, due to the remarkable physical properties they exhibit [59-70]. Among the explanations forwarded to date to capture the perplexing physics found in these compounds, are for example the opening of a spin and charge gap, anisotropic hybridization and charge density modulation [59-60, 65-68]. The Ru and Os compounds order antiferromagnetically at TN = 27 and 28.5 K, respectively, while the Fe compound remains paramagnetic down to 50 mK [59-61]. It is interesting to note that the ordering temperature of these compounds is extraordinarily high compared to their isostructural Gd-compound, GdRu2Al10 (TN = 16 K) and GdOs2Al10 (TN = 18 K) (See Fig.4). This inconsistency becomes even more significant when noting that TN of CeRu2Al10 and CeOs2Al10 compounds is higher than most magnetically ordered Ce-compounds, with only a few exceptions (see Table. I). For example CeRh3B2 orders ferromagnetically (FM) with TC=120 K [71, 72], while isostructural GdRh3B2 orders FM with TC=90 [73]. This unusual behaviour is not expected based on the de Gennes scaling [74], which shows that the ordering temperature of an isostructural family of rare-earth compounds with different rare-earth ions across the series from Ce to Yb should scale with ~ (gJ - 1)2 J(J+1), where gJ is the Lande g-factor and J is the total angular momentum. Thereby, the family member containing Gd should have the highest ordering temperature since it has the highest values of the de Gennes factor. Hence, the ordering temperature of the Ce-compounds should be a factor 0.0113 smaller than that of the Gd-compounds, which is not the case for most of the ordered Ce-compounds (see Table-I). This discrepancy can be partly understood by considering two facts. One is the exchange interactions, J(Q), where Q is the wave vector for which J(Q) is maximum, not having a constant values across the series (TC (or TN) ~ J(Q)*(gJ - 1)2 J(J+1)). The other is the crystal field effect, which may change the ground state magnetic moment [75-76]. Usually rare-earth compounds are classified into exchange

dominated or crystal-field-dominated systems. This classification is clearly not valid for CeRh3B2 where all energy scales (i.e. spin-orbit, crystal field, and exchange) must be taken into account to understand the ground state and low-lying magnetic excitations [77]. The spin wave excitation spectrum evidences high exchange interaction along the c-axis about two orders of magnitude higher than the ones in the basal plane of CeRh3B2. The easy axis is in the basal plane of the hexagonal structure and the saturation magnetization 0.4µB is strongly reduced compared to free cerium ion value 2.14 µB. The electronic structure calculations of CeRh3B2 reveal that the unconventional ground state is stabilized by the strong 4f–4f direct mixing between the neighbored Ce atoms along the extremely small distance along the c-axis in the hexagonal crystal cell [77]. In this review we first discuss the structural properties and then magnetic and transport properties of CeT2Al10 and doped compounds. We then introduce µSR results on Ce(Ru1-

xFex)2Al10, CeOs2Al10 followed by data on the local 4f-moment systems NdT2Al10 (T=Fe and Os), and we conclude with µSR findings on quantum critical system YFe2Al0. We present the inelastic neutron scattering investigations on Ce(Ru1-xFex)2Al10 and CeOs2Al10, compared with published

7

results. Finally, we summarize the present work with important conclusions and propose future investigations to be carried out for further understanding the complex behaviour of CeT2Al10 compounds. II. Experimental details

The polycrystalline samples investigated in the present work were prepared by argon arc melting and the details can be found in refs [59-61]. For the zero-field (ZF) µSR experiments, the powdered samples (thickness ~1.5mm) were mounted onto a 99.995+% pure silver plate using GE-varnish and were covered with 18 micron silver foil. We used the MuSR spectrometer in longitudinal geometry at the ISIS Pulsed Neutron and Muon Source, UK. At the ISIS facility, a pulse of spin polarised muons is produced every 20 ms and has a full-width at half-maximum (FWHM) of ~70 ns. The polarized muons are implanted into the sample and decay with a half-life of 2.2 µs into a positron which is emitted preferentially in the direction of the muon spin axis. Due to the magnetic moment of muons, the polarized muons precesses around a local (non-zero) magnetic field Bloc, with a precession frequency ωµ proportional to Bloc, ωµ = 2πfµ=γµBloc, where fµ is the muon precession frequency and γµ (=851.6 Mrad/s/T or 135.6 MHz/T) is the gyromagnetic ratio (γµ) of muon: 1MHz = 73.8 G. In a constant local field, therefore, moment of muon rotates by =γµBloc* t in the elapsed time t. As the typical data collection time window at the ISIS muon facility is 15µs, this permits detection of a local field as small as 0.04mT (or 0.4G). The emitted positrons are detected and time stamped in the detectors which are positioned before, F, and after, B, the sample. From the measured positron counts in the F and B detectors, NF(t) and NB(t), respectively, the asymmetry of the muon decay, Gz(t) is determined using

Gz(t)=(NF(t)-αNB(t))/(NF(t)+ αNB(t)) (1)

where α is a calibration coefficient.

The inelastic neutron scattering measurements on the polycrystalline samples were carried out using the MARI/MERLIN time-of-flight (TOF) chopper spectrometers at the ISIS Facility. The powder samples (mass ~20g) were wrapped in a thin Al-foil and mounted inside a thin-walled cylindrical Al-can, which was cooled down to 4.5 K inside a closed-cycle refrigerator with He-exchange gas around the samples. The measurements were performed with various selected incident neutron energies (Ei) between 20 meV and 100 meV. Inelastic neutron scattering investigations are ideal for the present study as they give direct information of spin-spin correlations as well as to as well as to enable a direct estimation of spin gap value, its wavevector and temperature dependent behaviour. High resolution neutron powder diffraction measurements on CeOs2Al10 sample were carried out between 4 K and 80 K using the HRPD diffractometer at ISIS facility.

III. Results and discussions 8

The CeT2Al10 (T=Fe, Ru and Os) compounds crystallize in the formation of a caged orthorhombic YbFe2Al10-type structure (space group Cmcm, No. 63) (Fig. 3 bottom). In this caged-type structure the Ce atom is surrounded by a polyhedron formed by 4 T (=Ru/Fe/Os) and 16 Al atoms and forms a zigzag chain along the orthorhombic c-axis [78]. The lattice parameters, unit cell volume and the selected Ce-Ce, Ce-T and Ce-Al interatomic distances are given in Table II. One can see from Table-II that the lattice parameters (a, b, c) are nearly similar for Os and Ru, but decrease noticeably when going from Os and Ru to the member comprised of the smaller Fe. The lattice parameters a and c contract more than b when going from Os or Ru to Fe. This observation is in agreement with the charge density distribution study on CeT2Al10 (T=Ru, Fe) and the crystal structure parameters of RT2Al10 [79-80]. This study shows that the lattice parameters of a-, b-, and c-axes exhibit an anisotropic contraction when Ru is replaced by Fe in RT2Al10, in contrast to the isotropic contraction simply expected from the smaller ionic radius of Fe compared to Ru. This anisotropic contraction of the YbFe2Al10-type crystal structure originates in the deviation from linearity of the zigzag chain formed by T and Al bond along the a- and c-axes that is larger than that along the b-axis [79]. The anisotropic contraction of the lattice parameters are expected to be the origin of the anisotropic c–f hybridization in CeT2Al10. The lattice parameters (especially a- and c-axes) of CeT2Al10 (T= Ru, Fe) exhibit the anisotropic deviation from the lanthanide contraction of RT2Al10 series [80]. This deviation is largest in the a-axis and is very small in the b-axis. Both the characteristic YbFe2Al10-type crystal structure and the anisotropic deviation towards the intermediate valence (for T=Fe) indicate that the largest c–f hybridization is along the a-axis and plays a dominant role which is associated with the unusual antiferromagnetic order in CeT2Al10 (T= Ru, Os).

The high-energy synchrotron x-ray powder diffraction study has been reported on LaRu2Al10 [79]. The data have been analysed using the maximum entropy method/Rietveld technique, which shows that the charge density between Ru and surrounding Al atoms is large, but that between La and surrounding Al atoms is small. It has been proposed that the Ru-Al10 polyhedron is the fundamental component of the crystal and the two-dimensional layer is constructed by these polyhedra in the ac-plane and is stacked along the b-axis by way of the Al5 atom [79].

The unit-cell volume is 861.7435Å3 for T=Os, 863.635 Å3 for T=Ru and 839.203 Å3 for T=Fe. The small unit cell volume of T=Fe is due to the small ionic radius of Fe (3d) atoms as well as mixed valence nature of the Ce ions in this compound. An unstable valence such as this is the consequence of very strong c-f hybridization that may cause a partial transformation of the 4f electron of Ce to the conduction band. This has been also supported through the plot of lattice parameters of rare earth series, RRu2Al10 and RFe2Al10 [80]. The change in the unit cell volume is about ~3% while going from Ru to Fe. Furthermore, the nearest neighbour Ce-Ce and Ce-T distances are nearly the same for T=Ru and Os, but smaller for T=Fe. It is unusual to have high magnetic ordering temperatures in T=Ru and Os compunds as they have a Ce-Ce separation

9

distances as large as ~5.2 Å. This is greater than 3.25-3.4 Å, the Hill limit beyond which direct 4f-4f interaction should cease [81]. Therefore, direct f-f interactions can be ruled out as being responsible for the high magnetic ordering temperature in these compounds. It is interesting to note that for CeRh3B2 the Ce–Ce distance is 3:09 Å (along c-axis), which is shorter than the Hill limit, and hence large 4f–4f direct mixing is expected in this direction. These results show that the unusually high ordering temperature of CeT2Al10 (T=Ru and Os) probably has a different origin than that of CeRh3B2.

We have recently carried out temperature dependent lattice parameters measurements on CeOs2Al10 using the HRPD diffractometer at ISIS. Our study shows that the lattice parameters a and c decrease monotonically, while the b lattice parameter reveals small change in the slope near 30 K (Fig. 5). When compared to the normalised lattice parameters (normalised to 300 K), the lattice parameter b showed the weakest temperature dependent compared to the a and c lattice parameters. Both the lattice parameters a and c revealed similar temperature dependence. The different behaviour of the b-axis lattice parameter is thought to be connected to the degree of hybridization, as the b-axis exhibits the smallest value of the magnetic susceptibility [60]. Considering temperature dependent atomic distances, it was noticed that Ce-Ce and Ce-Os2 distances reveal a slope change near 45 K (Fig.5), which may be related to the opening of the spin gap or hybridization gap and which has been proposed from the magnetic susceptibility and optical conductivity measurements [66-68]. We also examined the temperature dependence of the Ce-Al (all 5-distances) and Ru-Al distances, but did not find any noticeable anomaly.

A. Physical Properties of CeT2Al10 (T=Fe, Ru and Os)

The scope and diversity of reported studies into the magnetic and transport properties of CeT2Al10 (T=Fe, Ru and Os) in polycrystalline form as well as in the single crystalline form at ambient pressure as well as in applied pressure including study at high magnetic fields [59-65] attest to the wide-spread interest in this class of compounds and the significance of their unusual behaviour. Here we confine ourselves to a discussion of the single crystal magnetic susceptibility and resistivity measurements at ambient pressure on CeT2Al10 (Fig. 6) [60-62]. The magnetic susceptibility of all three compounds exhibits considerable anisotropy with χa>χc>χb. Of first importance is the absolute values of the overall susceptibility for all three directions, which is largest for T=Ru and decreases slightly for T=Os. On the other hand the susceptibility of T=Fe is almost factor of 3 (for a-axis) smaller than that of T=Ru. This marked difference in the susceptibility between T=Ru and T=Fe, indicates that the origin of the susceptibility behaviours (especially anisotropy) is different in these compounds (i.e. in CeRu2Al10 and CeFe2Al10). The magnetic susceptibility of CeFe2Al10 exhibits a peak at 75 K for χa, a broad maximum at 105 K

10

for χb and possibility of a broad maximum above 300 K for χc. As Ce ions are in the valence fluctuating state as evident from L3-absorption measurements [82], it is surprising to find such pronounced anisotropy in the magnetic susceptibility: valence fluctuation (VF) phenomenon is a single ion property and one would generally expect isotropic properties or very weak anisotropy [83]. Thus the observed strong anisotropy in CeFe2Al10 has been attributed to strong anisotropic hybridization between 4f-electron and conduction electrons [61]. Further from the peak position, the Kondo scale of on-site interaction, as measured through the Kondo temperature TK=3*Tmaxχ, is similarly anisotropic along the three directions [84]. Although the susceptibility data of CeFe2Al10 have been analysed on the basis of CEF model, this approach is found to show poor agreement between the data and calculated susceptibility [85], which suggests that one need is to include anisotropic hybridization effect in the CEF calculations. It is interesting to compare this situation with the tetragonal intermediate valence (IV) or valence fluctuating system YbB4 which exhibits a similarly strong anisotropy in the physical properties [86]. The c-axis susceptibility of YbB4 follows Curie-Weiss behavior with smaller crystalline electric field splitting, but the ab-plane susceptibility shows a typical temperature dependence as observed in IV or VF compounds [87]. This implies that YbB4 and CeFe2Al10 are rare examples of anisotropic behaviour in IV/VF state. In general the anisotropic susceptibility behaviour has not been noticed in cubic Kondo semiconductors such as Ce3Pt3Bi4, SmB6, U3Ni3Sb4 and YbB12, and hence DC-susceptibility measurements along the cubic-axes do not provide any information on the anisotropic hybridization. It is interesting to note that inelastic neutron scattering investigation on the cubic VF system CePd3 has provided direct evidence on the k-dependent, hybridization [88]. In an orthorhombic Kondo semimetal CeNiSn, the anisotropic gap has also been observed in the inelastic neutron scattering study [89], which has been explained theoretically by considering k-dependence of the hybridization matrix element between f- and conduction electrons that can give rise to an anisotropic hybridization gap of heavy fermions if the filling of electrons corresponds to that of the band insulator [90-91]. These theories show that the most interesting case occurs when the hybridization vanishes along some symmetry axis of the crystal reflecting a particular symmetry of the crystal field and hence the wave functions of the CEF ground state are very important ingredients [90-91]. χa(T) of CeOs2Al10 exhibits a broad peak near 45 K, which is well above the ordering temperature TN=28.5 K (Fig.6). This peak is associated with an opening of the spin gap well above TN, which has been supported through the optical conductivity study [65-68] and our inelastic neutron scattering study [92] also discussed in Section-C below. A very similar behaviour may exist for χa(T) for CeRu2Al10 compound, however in this case TN and this susceptibility peak are at similar temperatures and therefore masks the effect (Fig.6). It is agreed that Ce ions in CeRu2Al10 are in Ce3+ ionic state (4f1), but CeOs2Al10 shows a strong hybridization effect with valence close to Ce3+ [60]. This has been supported through the Ce L3-edge measurements near room temperature [82]. For the CeOs2Al10 and CeRu2Al10 compounds

11

the magnetic moment in the ordered state is aligned along the c-axis (and not along a-axis, which is an easy axis of the magnetization in the paramagnetic state and supported from the CEF analysis [85, 93]) with µord = 0.42 and 0.29 μB/Ce for T = Ru and Os, respectively [94, 95]. Strigari et al. have shown that the crystal electric field (CEF) scheme obtained from soft X-ray absorption spectroscopy can explain the easy axis and the small magnetic moment on CeRu2Al10 and CeOs2Al10 [85, 93], which are in agreement with our inelastic neutron scattering data [92]. Kunimori et al. also performed mean-field calculations and showed that the CEF scheme by itself is not enough to account for the magnetic order below TN and pointed out the importance of the conduction- and f-electron (c-f) hybridization for the unusual magnetic ordered state [96]. Kondo et al. speculated that the strong c-f hybridization suppresses the spin degrees of freedom along the a-axis and forces the ordered moment µord to be along the c-axis in place of the a-axis [97]. By applying a magnetic field along the c-axis, a spin-flop transition is observed for µord from c-axis to b-axis, even though the a-axis susceptibility is the highest in the paramagnetic state, when the magnetic field is beyond a characteristic value of about 4 T for T=Ru, [97]. In spite of the unusually high ordering temperatures, it has been demonstrated that the magnetic ground state of CeRu2Al10 and CeOs2Al10 is very unstable and changes dramatically with very small electronic perturbation [98, 99]. It has been shown that the direction of the ordered state magnetic moment is along a-axis (as expected, based upon on CEF analysis) for very small Ir and Rh doped systems i.e. Ce(Os1-xIrx)2Al10 (x=0.08) [98] and Ce(Ru1-xRhx)2Al10 (x=0.03-0.1) [99]. The temperature dependent resistivity of CeT2Al10 (T=Fe, Ru and Os) single crystals exhibits strong anisotropy that is larger at low temperature for T=Fe (see Fig.6). The temperature dependent resistivity of T=Fe passes through a broad peak near 50 K for current along the a-axis and near 100 K for c-axis, which is due to an onset of coherence among the 4f-electrons which scatter conduction electrons. No clear distinct coherence anomaly was observed in the high temperature for T=Ru and Os, which is due to the proximity of coherence and magnetic ordering in these two compounds. Furthermore the resistivity of CeFe2Al10 exhibits Kondo semiconducting behavior with an activation energy of 15 K, while NMR and heat capacity studies reveal a larger value of the gap, 125 K and 100 K, respectively [61, 69, 70]. The Kondo semiconductor behavior observed in CeFe2Al10 bears similarity with that of the well-known Kondo semiconductors CeNiSn and CeRhSb [100, 101]. Although the ground state of CeRu2Al10 is metallic (Fig.6), its resistivity increases abruptly at TN and exhibits a maximum at 23 K and then decreases with temperature, but with a visible inflection point at TN. On the other hand the resistivity of T=Os exhibits semiconducting or semimetallic behaviour even below TN, which is opposite to that observed in T=Ru, and rather qualitatively similar to that observed in T=Fe. Furthermore, a pressure study has shown that the resistivity of T=Ru at low temperatures increases and the ground state became semiconducting at 2 GPa [102]. The increase in the resistivity of T=Ru becomes suppressed above 3 GPa and the ground state again becomes metallic above 5 GPa. The pressure range of semimetallic behaviour in T=Ru is probably

12

connected with proximity of the sharp 4f-electron resonance to the Fermi energy, admixed with effects of partial gapping in the conduction band. The magnetic contribution to the resistivity of T=Ru above 4 GPa shows a maximum (at 100 K), which seems to be attributed to Kondo coherence. The systematic changes in the temperature dependent resistivity were also observed in applied pressure for T=Fe and Os. At an ambient pressure CeOs2Al10 has a Kondo semimetallic ground state [102], but with increasing pressure at 2 GPa and above its ground state becomes metallic, and also exhibits a maximum in the resistivity near 100 K, which is opposite to that observed in T=Ru. Further the ground state of CeFe2Al10 changes to metallic at a very small pressure of 0.8GPa. Interestingly, the long-range magnetic ordering suddenly disappears under pressures at 2 GPa for T = Os and at 4 GPa for T = Ru [102]. The phase transition at TN exhibits maximum (TN=33 K) at 2 GPa for T=Ru, while for T=Os it does not change much with pressure up to 1.5 GPa and the suddenly decreases to zero at 2 GPa [102]. This behaviour of T=Ru and Os allows us to position them accordingly on the Doniach phase diagram, see Fig.2. In order to investigate a structural change induced by pressure, where the magnetic order disappears, synchrotron X-ray diffraction study at room temperature under high pressure up to 10 GPa for T=Fe, Ru and Os has been reported [103]. This study shows that there is no detectable structural phase transition up to 10 GPa at 300K. The lattice parameters a, b, c decrease linearly with pressure for all three compounds. This suggests that the pressure induced nonmagnetic ground state in CeRu2Al10 and CeOs2Al10 is not due to any structural phase transition [103] B. µSR measurements µSR is an ideal local probe to investigate small moment magnetism in strongly correlated electron systems, for a review see refs. [105, 106]. In fact, µSR technique is so sensitive to small magnetism that the distribution of fields arising from nuclear moments is easily investigated. The high sensitivity of µSR technique, compared to neutron diffraction, is due to the large gyromagnetic ratio (γµ =851.6 Mrad/s/T) of the muon. From µSR study the distribution and dynamics of the internal field at muon sites can be probed. Due to being a local probe, µSR study provides information on volume fraction of different phases as well as different spin dynamic information for multi sites systems. There are many examples in heavy fermion systems where the ordered state moment is very small, due to screen of the local moment (f or d-moment) by conduction electrons through the Kondo coupling [105]. Further, µSR technique has been also used to investigate spin gap formation in many strongly correlated electron systems [107] as well as quantum criticality and NFL-behaviour in heavy fermion systems [105]. In this section we will discuss the µSR measurements on selected compounds of interest among Ce(Ru1-xFex)2Al10 (x=0 to 1) and CeOs2Al10, which exhibit very small moment ordering and also on spin gap formation in non-ordered ground state (x=0.8 to 1). For comparison we will also present results on stable 4f-moment systems, NdT2Al10 (T=Fe and Os) as well as the 3d-moment system YFe2Al10.

13

1. µSR study on Ce(Ru1-xFex)2Al10 x=0 to 1 A detailed µSR investigations on Ce(Ru1-xFex)2Al10 and CeOs2Al10 are reported in refs. [82, 92, 94, 108]. Figure 7 (a-h) shows the zero-field (ZF) µSR spectra at various temperatures of Ce(Ru1-xFex)2All0 (x=0, 0.3, 0.5, 0.8 and 1) from refs. [82]. At 30-35 K a strong damping at shorter time (Fig.7d-h) and the recovery at longer times have been observed, which is a typical muon response to nuclear moments, described by the Kubo-Toyabe (KT) formalism [109], arising from a static distribution of the nuclear dipole moments. Above the anomaly at 28 K, i.e. in the paramagnetic state, the µSR spectra have been analysed by the following equation (see Figs. 7d-h):

Gz(t) = 𝐴0 �13

+ 23

(1 − (σ𝐾𝑇t)2)exp�−(σ𝐾𝑇t)2

2�� exp(−λt) + BG (2)

where A0 is the initial asymmetry, σKT is nuclear depolarization rate, σKT/γµ =Δ is the local Gaussian field distribution width, γµ is the gyromagnetic ratio of the muon, λ is the electronic relaxation rate and BG is a constant background. The value of σKT was found to be 0.32-0.36 µs-1 (depending on x) from fitting the spectra of 35/30 K to Eq.(1) and was found to be temperature independent above 35 K. It is to be noted that using a similar value of the σΚΤ Kambe et al [108] have suggested 4a as the muon stopping site in CeRu2Al10, while for CeOs2Al10 the muon stopping site was assigned to the (0.5, 0, 0.25) position [92]. Recently Guo et al [99] have investigated Ce(Ru1-xRhx)2Al10 (x=0.05 and 1) using µSR and their dipolar fields calculation supports the (0.5, 0, 0.25) site for muons.

Interestingly the µSR spectra of Ce(Ru1-xFex)2Al10 for x=0 to 0.5 below 27 K reveal coherent frequency oscillations and they have been described by two oscillatory terms and an exponential decay, as given by the following equation

𝐺𝑧(𝑡) = �� 𝐴𝑖𝑐𝑜𝑠(𝜔𝑖 𝑡 + 𝜑)2

𝑖=1exp�−

(σit)2

2�� + 𝐴3exp(−λt) + BG (3)

where ωi=γµ Hiint are the muon precession frequencies (Hi

int is the internal field at the muon site), σi is the muon depolarization rate (arising from the distribution of the internal field) and φ is the phase. Although, one might expect that there are two exponential decays each corresponding to a precession frequency. In our data, we only observed one, indicating that the two values must be very close to each other.

Fig. 8 (a-c) from ref. [82] shows the plot of internal fields (or muon precession frequencies) at the muon sites as a function of temperature for x=0, 0.3 and 0.5. This shows that the internal fields appear just below 27 K for x=0, below 26 K in x=0.3 and below 22 K in x=0.5.

14

Given the amplitudes of the oscillations this shows clear evidence for long-range magnetic ordering of the Ce moments. Further it is very important to see that the asymmetry A3 drops nearly 2/3 and the relaxation rate exhibits small drops at TN for x=0, 0.3 and 0.5 (see inset Fig.8c left bottom), which confirms that the magnetic ordering is observed in the full volume of the samples and hence is bulk in nature. This is also in agreement with the phase diagram, x vs TN proposed from the magnetic susceptibility and heat capacity measurements as shown in Fig. 9. Further the decrease in the high internal field values with x may indicate that the ordered state Ce moment is reducing with x in agreement with the observed susceptibility behaviour [110]. The small value of the internal fields observed in x=0 to 0.5 are in agreement with the small ordered state magnetic moment of the Ce3+ ion observed through the neutron diffraction for x=0 and x=0.5 [ 82, 94, 95].

The occurrence of the dip in x=0, which has also been observed in the µSR study of

CeOs2Al10 (discussed below) at 10-15 K [92], may have some relation with a super lattice formation observed in the recent electron diffraction study of CeOs2Al10 [60]. In principle this could originate from various phenomena related to a change in the distribution of internal fields associated with a small change in the moment values or modulation. Further, for x=0.3 and 0.5 the observed anisotropy of the depolarization rates (observed in x=0) becomes smaller with increasing x. It is to be noted that although the µSR measurements on unaligned single crystals of CeRu2Al10 are more or less in agreement with the present study [99], however, there are small differences in the temperatures dependent behaviour of the relaxation rate σ (Fig. 8d), which require further investigations on well aligned high quality single crystal using µSR and neutron diffraction studies. Further it is to be noted that the observed two frequencies/internal fields in the present study is attributed to the possibility of two muon sites, while in the single crystal study the low frequency/internal field has been attributed to the Fermi contact field from the polarized electrons at the muon site [99].

2. µSR study on CeOs2Al10 A detailed µSR study of CeOs2Al10 has been reported in ref. [92], which also revealed the presence of two frequencies oscillations (or internal fields) below 29 K (Figs. 10 and 11) as seen in CeRu2Al10. The presence of internal field at the muon stopping site in zero-field indicates unambiguously long-range magnetic ordering of the Ce3+ moments, which was later confirmed through the neutron diffraction investigation on the single crystal of CeOs2Al10 [95]. The magnetic structure proposed from the single crystal study of CeOs2Al10 is with propagation vector k=[0 1 0], which is equivalent to [1 0 0] for the orthorhombic Cmcm structure, with the moment of 0.3 µB along the c-axis [94, 95]. The c-axis moment direction is not expected based on the single ion CEF anisotropy [85], which indicates that the moment direction is governed by anisotropic two ions exchange interactions. The support of anisotropic two ions exchange

15

interactions also comes from the spin wave measurements on the single crystal of CeOs2Al10 [111], which reveals that nearest neighbour exchange parameter is highly anisotropic and strongest along the c-axis [111, 112]. The maximum internal field observed is 50 G at 5 K in CeOs2Al10 which is smaller than 120 G at 15 K for CeRu2Al10. The temperature dependence of the internal fields of CeOs2Al10 (Fig.11), show that there is a dip in the internal field (see Fig. 11 top left), which occurs around 15 K. The occurrence of the dip coincides with both a structural distortion observed in the recent electron diffraction study and with the onset of semiconducting behaviour in the resistivity below 15 K [61]. Moreover, below 15 K the first and the second component of the depolarisation rates also increase (Fig. 11 right). In principle this could originate from various phenomena related to a change in distribution of internal fields, but a structural transition is a likely candidate in view of the structural instability reported on this system [62]. It is to be noted that the third depolarization rate associated with zero-frequency also exhibits dramatic changes with temperature (see the inset Fig. 10 bottom-right). 3. µSR study on NdFe2Al10

It is interesting to note that among CeT2Al10 (T=Fe, Ru and Os), the compound with T=Fe does

not order magnetically down to 40 mK, while T=Ru and Os show ordering with very small

ordered state moments 0.4 and 0.3µB, respectively. Hence it is of interest to study magnetic

properties of other members of the RT2Al10 series, where 4f-moment is very localised compared

with the Ce-compounds. The magnetic properties of RFe2Al10 compounds with R=Pr, Nd, Sm,

Tb, Dy, Er, Ho and Yb have been investigated [113]. It has been found that the magnetic

ordering transitions TN1=3.9 K (TN2=1.5 K) for R=Nd, 16.5 K for T=Tb and 7.5 for R=Dy are

very low despite a large, stable magnetic moment on the rare earth atoms (i.e 10.63µB for Dy),

while the compound with T=Ho does not order magnetically down to 1.5 K even though the

magnetic moment of Ho is 10.60 µB. This indicates the important of crystal field ground state on

the magnetic ordering of RFe2Al10 compounds. On the other hand YbFe2Al0 shows a mixed

valence behaviour as in CeFe2Al10 and SmFe2Al10 is Van Vleck paramagnetic [114]. Here we

have carried out µSR measurements on NdFe2Al10 to investigate the observed two magnetic

phase transitions in the heat capacity [96]. It is to be noted that heat capacity of NdFe2Al10

exhibits very sharp jump at TN and very broad peak around 1.8 K.

Our ZF µSR measurements of NdFe2Al10 reveal interesting behaviour with temperature (see Fig.12). At high temperature above (TN) µSR spectra reveal KT-type behaviour, but below

16

3.8 K a clear sign of two frequency oscillations have been seen down to 1.2 K. However, at 1.2 K, frequency oscillations are visible only below 2µs time, which is most probably due to a larger values of internal fields, with broad distribution of the internal fields (which gives stronger damping) at the muon sites. We have used maximum entropy method to determine the internal fields and number of frequencies at 1.2 K and then µSR data of NdFe2Al10, were analysed in real time as we did for CeT2Al10 (T=Ru and Os). Above the phase transition at TN and in the high temperature range the data for NdFe2Al10

were fitted with using KT function times the exponential decay and a constant BG term. We were

able to fit the data very well with temperature independent BG term. Further the σKT and

asymmetry parameters were also temperature independent with values 0.33 (µS-1), indicating

very similar muon sites as in CeT2Al10 as discussed above.

The µSR data at 1.2 K were fitted using two frequency terms with a Lorentzian envelope

(i.e. exp(-λt)), rather than a Gaussian envelope used for CeT2Al10, and keeping the same BG as

it was used in the high temperature range above TN. From the analysis it was obvious that there

are two frequencies present in the data at 1.2 K (Fig. 12). The value of the higher internal field

(B1) is much higher than that observed in the CeT2Al10 compounds, which is expected as the

magnetic moment of Nd ion is larger than that of the Ce ion as well as 4f-electrons are localised

on the former than the latter. On the other hand the value of lower internal field (B2) is very close

to the highest internal field observed in CeRu2Al10. The temperatures dependent internal

fields/frequencies of NdFe2Al10 are shown in Fig. 13, which reveals sharp rise in the high

field/frequency (first component) just below TN1 and saturation below 1.8 K. The lower

field/frequency (second component) also increases with decreasing temperature below TN1 and

exhibits weak anomaly near 2.2 K. The µSR data are consistent with the heat capacity with a very

sharp jump at TN and very broad peak around 1.8K [96].

4. µSR study on NdOs2Al10

The magnetic susceptibility, electrical resistivity, and specific heat of ROs2Al10 (R=Pr,

Nd, Sm, and Gd) compounds, which are isostructural with a Kondo semiconductor CeOs2Al10,

have been reported [115-117]. The compound with R=Pr does not order magnetically down to 0.4 K, whereas the compounds with R=Nd, Sm, and Gd undergo antiferromagnetic transitions at TN1= 2.2 (TN2 = 1.1 K), 12.5 (TN2 = 9.1 K, TN3 = 5.6 K), and 18 K (TN2 = 15 K), respectively. The magnetic susceptibility of NdOs2Al10 exhibits Curie-Weiss behaviour with an effective magnetic moment, µeff =3.3 µB and Curie-Weiss temperature, θP=-6.0 K. The small value of θP indicates a weak antiferromagnetic interactions between Nd moments, which is in agreement with smaller ordering temperatures.

17

In order to shed light on the nature of magnetic ground state in NdOs2Al10, and also for

comparison with other compounds reported here, especially CeOs2Al10, we have carried out µSR measurements on NdOs2Al0 and the results are shown in Figs.14 and 15. As can be seen from Fig.14 that above the magnetic ordering temperature (i.e at 4.15 K), the µSR spectra reveal KT type behaviour, similarly to what is seen in other RT2Al10 compounds [82, 94, 99]. Below TN loss of asymmetry was observed indicating a long range magnetic ground state below 2.2 K, which is in agreement with the heat capacity and magnetic susceptibility results [115]. The absence of clear frequency oscillations below TN1 indicates that internal fields at the muon sites are large and are outside the maximum observable field limit (about 800 gauss) of the MUSR spectrometer at ISIS. The observable maximum field is limited due to finite pulsed width of muon pulse at ISIS. The data were fitted with KT*exp(-λ*t)+BG and the BG was estimated from 4.2 K and was kept fixed. The data fit very well to this function (see Fig. 14). Interestingly below 2.24 K the muon initial asymmetry drops to 2/3 to its high temperature value (Fig.15), which indicates that bulk nature of long range magnetic ordering of Nd-moments in NdOs2Al10. The electronic relaxation rate (λ) rate increases below 2.24 K (i.e at TN1) and exhibits a peak near 2 K, while Gaussian KT relaxation rate (σKT) remains temperature independent and drops to zero below 2.24 K, which we expect as in the presence of internal field KT term becomes unity. This is due to the fact that nuclear magnetism is a factor of 103 smaller than that of electronic magnetism and hence muons respond predominantly to the electronic magnetism. There is a possibility to see second phase transition at TN2 = 1.1 K in the λ(T), however due to limitation on the minimum sample temperature during this experiment, 1.2 K, we could not investigate the second magnetic transition in our µSR study.

5. µSR investigations of a quantum critical behaviour in YFe2Al10

There have been extensive investigation on NFL and QCP behaviour on compounds having 4f or 5f-local moment using µSR measurements, for example CeCu6-xAux [118], UCu5-xPdx [119], Y1-

xUxPd3 [120], CeRh0.85Pd0.15 [121], CeInPt4 [122] and may more for review see refs. [105]. The µSR relaxation rate exhibits power law or logarithmic divergence at low temperatures in a manner similar to what is exhibited in χ(T). The NFL and QCP behaviour in these systems have been explained using various theoretical scenarios [26]: local quantum critical point, in which Kondo temperature breaks down (TK=0) at QCP [26, 123], spin density wave (SDW) model [26,46], in which TK remains finite for either sides of QCP, and valence induced QCP, in which one observes line of quantum criticality instead of a quantum critical point [124].

There are only a few examples among d-electron systems showing QCP and NFL behaviour, for example ferromagnetic quantum criticality in Ni1-xPdx [125], V-doped Cr [126] pressured induced NFL in ferromagnetic ZrZn2 [127], and NFL behaviour near AFM instability CaRuO3 [128]. A subclass with exotic d-electron magnetism can be formed by those compounds

18

in which the rare-earth (or actinide) atom bears no magnetic moment (4f0, La, Lu, Y and tetravalent Ce and divalent Yb). The compounds belonging to this subclass have received much attention since they offer a good opportunity to study the origin and the nature of the d-electron magnetism to compare with isostructural f-electron systems. In these systems the d-atoms do not occupy the random crystallographic sites, but are fixed in well-defined crystallographic positions. Rather conclusive information on the d-electron magnetism can therefore be obtained by means of advanced techniques such as nuclear magnetic resonance, µSR, Mössbauer effect and diffraction with polarized neutrons

Recently YFe2Al10 has been extensively investigated using magnetization, specific heat, and NMR over a wide range of temperature and magnetic field and zero field NQR measurements [129]. Magnetic susceptibility, specific heat, and spin-lattice relaxation rate divided by T (1/T1T ) follow a weak power law (∼T−α with α=0.35 to 0.5) temperature dependence below 4 K, which is a signature of the critical fluctuations of the Fe moments (see Fig. 16). The value of the Sommerfeld-Wilson ratio and the linear relation between 1/T1T and χ suggest the existence of ferromagnetic correlations in this system. The absence of a long range magnetic ordering down to 50 mK was confirmed through various studies [129]. Further the unusual T and H scaling of the bulk and NMR data are associated with a magnetic instability which drives the system to quantum criticality. The χ(T) of YFe2Al10 obeys a Curie-Weiss behavior and the effective moment (μeff = 0.52μB/f.u.) is rather small. The field induced moment at 6T (at 2K) μ6T ∼ 0.02μB/f.u. [129] leads to μeff/μ6T ≈ 26, which classifies YFe2Al10 as being a weak itinerant FM in the Rhodes Wohlfarth plot [129]. However, a clue about the importance of magnetocrystalline anisotropy even in this d-electron system is found in the fact that the saturation magnetization for field perpendicular to the b-axis at 2K amounts to only 0.02 µB/fu [131]. Indeed, the findings of Park et al. indicate that the susceptibility in this compound is governed largely by critical fluctuations confined to the ac-plane and that the b-axis may be considered as decoupled from the magnetic fluctuations. This arrangement of the anisotropy is, interestingly, quite similar to that of the local-moment, f-electron CeT2Al10 members of the series (see Section B). The enhancement of ac and dc magnetic susceptibility accompanied by χ(T ) ∼ T−0.5 divergence below 10 K in a weak magnetic fields could be due to exchange enhanced q = 0 excitations and suggests the presence of spin correlations [129]. The magnetic properties of the system are tuned by applying magnetic fields wherein ferromagnetic fluctuations are suppressed and a crossover from quantum critical to Fermi-liquid behaviour is observed [129]. To shed light on the natures of critical magnetic fluctuations in YFe2Al10, we have carried out ZF and longitudinal applied field (LF) µSR measurements on this compound down to 60 mK. Fig.17a and b show µSR spectra in ZF and LF of 50 G at 0.06K and 3 K (2 K for 50 G). The ZF µSR spectra show a typical KT type temperature independent behaviour, which indicates that the

19

response seen by muons is mainly dominated by the nuclear magnetism (from Fe and Al) and very small or weak contribution from d-electronic magnetism of the Fe atoms. The latter has been supported through 50 G LF data (Fig.17b), which show nearly time independent and also temperature independent behaviour of µSR spectra. In order to gain more information on the weak magnetism we also carried out field dependent study at 0.06 K (Fig.17c) and 3 K (not shown here) up to 2500 G LF. The field response at 0.06 K and 3 K, were very similar. It can be seen from Figs. 17 and 18 that between 50 G and 2500 G the electronic relaxation rate (λ) remains nearly field independent, but the initial asymmetry show weak field dependent and increases linearly with applied field. Temperature dependent data in ZF, 50 G and 2500 G LF were analysed using KT*exp(-λ*t)+BG form of the relaxation function. The fit parameters are shown in Fig.18. It is again clear that nuclear depolarization rate σKT(T) is temperature independent. The ZF λ(T) exhibits very weak temperature dependent, change from 0.02 µs-1 at 3 K to 0.033µs-1 at 0.06 K. Further in 50 G and 2500 G LF the value of λ drops considerably and again it is nearly temperature independent. The observed nearly temperature independent behaviour of λ(T) of YFe2Al10 was unexpected because the magnetization, heat capacity and NMR/NQR measurements reveal the presence of a power law diversion due to the presence of critical fluctuations [129]. The µSR measurements on f-electron based NFL and quantum critical systems exhibit very strong temperature and field dependent behaviour of λ [105, 121]. For example FM QCP and NFL 4f-electron system CePd0.15Rh0.85, λ(T) exhibits a power-law behavior, λ(T) ~T−n with n~0.8, while the field dependence at 0.1 K reveals a time-field scaling of the muon relaxation function, Pz(t , H)=Pz(t /Hγ) with γ=1.0±0.1 [121]. Furthermore, the exponent derived from the ZF-µSR data agrees well with the power-law behavior of the temperature-dependent susceptibility, χT)~T-α (α=0.6), the E/T scaling of the neutron dynamical susceptibility, as well as the magnetization-field-temperature scaling γm=0.8±0.1. In order to check whether we can observe magnetic fluctuations in YFe2Al10 through neutron dynamic susceptibility, we have also carried out inelastic neutron scattering measurements using the MARI time-of-flight spectrometer between 4 K and 100 K range with incident energies between 6 and 25 meV. Our INS study did not reveal any clear sign of magnetic signal in YFe2Al10. This may suggest that the probing time scale (also energy scale for INS study) of the experimental techniques is very important to see critical magnetic fluctuations in YFe2Al10. Further µSR and high resolution low energy INS studies on a good quality single crystal of YFe2Al10 are highly desirable to unravel the nature of the magnetic fluctuations in this interesting d-electron system. C. Inelastic neutron scattering study

20

Inelastic neutron scattering (INS) is the only method which allows us to determine both the spatial and time correlations of the magnetic excitations through the dipolar coupling of the magnetic moment of the neutron with the spin-correlation function of the sample [132134]. The magnetic moment of the neutron acts as a wavevector and frequency dependent magnetic field that probes the dynamic magnetic response of the sample. The energy of thermal neutrons is comparable to the energy scale of magnetic excitations and lattice vibration (or phonons) in solid state materials and hence INS is suitably disposed to provide vital information on magnetic exchanges as well as lattice dynamics. INS study has been extensively used to understand the origin of superconductivity in Cu-based and Fe-based materials as well as unconventional superconductivity in HF by investigating high- and low-energy dispersive spin excitations, spin gap formation and spin resonances and phonon dispersions [135-137]. Further INS study has been extensively used recently to investigate low dimensional quantum magnets in which the ground state forms nonmagnetic singlet state (total spin S=0 with formation of short rage AFM dimers) down to the lowest energy [138]. INS study directly probes the energy scale of singlet (S=0) to triplet (S=1) state, by flipping neutron spin from -1/2 to +1/2 (total spin change of neutron is 1), and hence provides information on the interdimer and intradimer exchange interactions [138, 139]. In addition to this the INS study has been used to investigate magnons condensation in the Bose-Einstein condensate (BEC) state of many oxides based magnets [140] as well as to investigate quantum fluctuations near QCP and NFL state in HF systems [141]. In the present systems, CeT2Al10 (T=Ru and Os), INS technique was used to unravel the contribution of gapped spin wave ground state, arising from single ion CEF anisotropy, and the gap formation due to hybridization between 4f-electrons and conduction electrons or spin dimer formation. The latter two contributions are also investigated in Ce(Ru1-xFex)2Al10 with x=0.8 and CeFe2Al10, which have nonmagnetic ground state down to 1.2 K and 40 mK, respectively [82]. 1. INS study on Ce(Ru1-xFex)2Al10 (x=0 to 1) To date, two groups have investigated and reported on powder samples of CeRu2Al10 using inelastic neutron scattering and both have confirmed the formation of a spin gap of 8 meV in the inelastic response, together with its strongly temperature dependent character [82, 142]. On the other hand, the Q-dependent response following Ce3+ magnetic form factor in the powder sample may indicates the importance of single-ion interactions or strong intensity of magnon near the zone boundary arising from powder averaging effect [82, 142]. There is also a possibility of existence of a gap type response above TN up to 34 K in agreement with the optical conductivity study [82, 66-68]. The gap is nearly temperature independent very close to TN, but then abruptly transforms into a broad quasi-elastic/inelastic response above TN [82, 142]. By raising the temperature still further (above 35-40 K), the INS response becomes very broad, with quasi-elastic character [82, 142]. The observation of a spin gap in these compounds is in good agreement with predictions based on a theoretical model for a spin-dimer formation pertinent to

21

this class of compounds which has recently been put forward by Hanzawa [143, 143]. However, recent spin wave studies by Robert et al on single crystals of CeRu2Al10 [112] using a triple axis spectrometer revealed the gapped spin wave excitations (gap ~ 4-5 meV at AFM zone centre). Near the AFM zone centre there are two spin waves modes, which are expected considering that there are two spin per primitive unit cell. The zone boundary energy is about 8 meV in all directions. These observations are in full agreement with our spin waves investigation on CeRu2Al10 single crystal using a time-off-flight neutron scattering technique on the MERLIN spectrometer [111]. The analysis of spin waves by Robert et al and also by us indicates that the nearest neighbour (NN) interactions are highly anisotropic, i.e. exchange along the c-axis is the strongest [112]. This result supports the observation of the magnetic structure with magnetic moment along the c-axis and not along the a-axis as expected from the CEF anisotropy in the paramagnetic state [93]. Even though we have a clear sign that the INS at 4.5 K is due to spin wave in CeRu2Al10, it is still an open question to find direct evidence of INS scattering arising from the hybridization gap or dimer formation. In order to unravel these two contributions it was necessary to investigate Ce(Ru1-xFex)2Al10 alloys with x=0.8 and 1 where magnetic ordering disappears and hence we can directly study the INS response arising from the hybridization gap formation or dimer formation. A detailed report on this has been published by some of us [82] and here we will give a brief discussion on this.

Figure 19 shows INS scattering at low-Q from Ce(Ru1-xFex)2Al10 with x=0, 0.3, 0.5 ,0.8 and 1. There is a clear sign of magnetic excitation centred around 8 meV in x=0 and around 10-12 meV for x=0.8. The value of the peak position can be taken as a measure of the spin gap energy in these compounds [145]. It was found that the temperature dependence of the observed spin gap in CeRu2Al12 and theoretically predicted spin gap by the model of Hanzawa [143, 144] have similar behaviour just below TN, but there is clear evidence in the experimental data to support the existence of the spin gap just above TN (possibly up to 33 K) in x=0 and also in x=0.3 (up to 35K) [82]. The optical study on CeRu2Al10 also shows the existence of a energy gap above TN through the effective electron number Neff, which is related to the gap Neff ~ ∆2

opt [67]. A very similar situation has also been observed for CeOs2Al10 through an optical study [68], where a CDW gap (or ∆opt) exists up to 39 K, and also from our recent INS study [92] discussed below. If we take the value of the quasi-elastic linewidth as a measure of Kondo temperature TK (just above TN, ideally one takes the value at T=0) then it shows that TK increases from 52(3) K in x=0, 83(5) K in x=0.3, to 110(10) K in x=0.5.

It is clear from the INS study that a spin gap exists at 7 K and the response becomes

quasi-elastic at 94 K in Ce(Ru1-xFex)2Al10 with x=0.8 and 1 [82]. A notable feature of the spin gap energies of these compounds measured through INS is their universal scaling relationship with the Kondo energy (TK) derived from the maximum in the susceptibility [145, 146, 147]. According to the single impurity model [145, 148], we can estimate the high temperature Kondo

22

temperature TK through the maximum Tmax(χ) in the bulk susceptibility as TK= 3*Tmax(χ)=150 K (12.92 meV) for x=0.8. This shows that the spin gap of 10 meV observed through the INS study is in agreement with the scaling behavior.

The Q-dependent intensity of the 10 meV excitation in x=0.8 exhibits a clear peak near Q

= 0.8 Å-1, which is different from that observed for many spin gap systems that do not exhibit long-range magnetic order [145]. The fit to the dimer structure factor I(Q) ~ sin(Q d)//(Q d)/, where d is the Ce-Ce distance, gave d=5.07(4)Å, which is close to d=5.21(4)Å estimated through neutron diffraction study at 300 K [82], and which in turn is supportive of dimer formation. Another possible interpretation of the observed spin gap in x=0.8 and 1 could be an anisotropic spin gap opening only on a small part of the Fermi surface or along a specific direction in Q-space. This is somewhat similar to the spin gap observed only along [0 0 l] direction in the Kondo semimetal CeNiSn [89].

2. Spin gap above magnetic ordering in CeOs2Al10 On CeOs2Al10 a preliminary study of a spin gap formation using INS study has been reported by some of us [92]. Here we present a detailed analysis of temperature dependence of the spin gap formation in CeOs2Al10. Besides our previous INS study, we have also obtained additional INS data using the high flux neutron spectrometer MERLIN (see Fig. 20) and carried out detailed analysis (see Fig. 21). The phonon contribution was subtracted using the data of nonmagnetic reference compound LaOs2Al10. From Fig. 20 it is clear now that we have strong magnetic scattering centred near 11 meV with dispersion coming out from Q = 0.6-0.7Å-1, which is the AFM zone centre [0 1 0], which indicates that a part of INS response is associated with the gapped spin waves [111, 112]. The presence of clear dispersive gapped spin wave (with gap of ~5 meV) has also been observed in our single crystals INS study of CeOs2Al10 [111]. Considering in particular the temperature dependent scattering (see Fig. 21 bottom), it is clear that magnetic scattering remains practically temperature independent up to 25 K and then decreases with increasing temperature, but remains inelastic up to 35-38 K (i.e. above TN = 28.5 K) and from 40 K and above the scattering becomes quasi-elastic. A very similar behavior has been observed in CeRu2Al10 as discussed above [82]. It is to be noted that the optical study reveals the presence of charged density wave (CDW) gap above TN in CeOs2Al10 [66, 67]. The value of the optical gap observed at 4 K is Δopt = 20 meV, which is almost double that of the spin gap (powder average gap) observed through INS study.

Figure 21 shows the temperature dependent parameters estimated from the fit to the INS data (filled circles are inelastic peak fits and open circles are quasi-elastic fits) of CeOs2Al10. A detailed account of the data analysis and fitting procedure are given in ref. [82]. Fig. 20(a) shows the estimated magnetic susceptibility from INS fit (symbols). Therefore, we assumed that van Vleck contribution from the high energy CEF is small at low temperature. The estimated

23

susceptibility is close to that measured using a SQUID magnetometer on a powder sample of CeOs2Al10 shown by a green solid line in Fig. 21(a). Fig.21(b) shows the temperature dependent linewidth, Γ(T), and Fig. 21(c) shows the temperature dependent peak position, ∆(T), (i.e. spin gap). It is clear that Γ(T) decreases below TN. We have analysed Γ(T) using two models: (1) Γ(T),~T2 (see solid line in Fig.21b) and (2) exponential behavior, Γ(T)~exp(-∆(0)/kBT) (see dotted line in Fig. 21b), where kB is Boltzmann’s constant. The exponential relation for Γ(T) is found to describe the data much more reliably with gap value of ∆(0)~10.2(1.0) meV, which is in good agreement with the peak position observed at 4.5 K. For comparison we have also plotted temperature dependent optical gap, ∆opt/2 (divided by factor 2), in Fig. 21(c) (left y-axis) [67]. A very similar temperature dependent behaviour has been observed for both INS gap and optical gap, i.e. both gaps exist up to 39 K. This may indicate either the presence of spin waves above TN or the involvement of a hybridization gap (or dimer gap) above TN. Considering that gapping is also observed in the optical study above TN and also a broad peak in the DC-susceptibility near 45 K (see Fig.6), it is likely that the observed gap above TN is associated with the hybridization (or dimer) gap. Support for the existence of a gap above TN can also be found through Ce-Ce and Ce-Os2 distances (see Fig. 5). If we take the value of the quasi-elastic linewidth as a measure of Kondo temperature TK (as we did for CeRu2Al10) then estimated TK is 92(5) K for CeOs2Al10, which is almost double that of CeRu2Al10. This indicates that the hybridization between 4f-electrons and conduction electrons is stronger in CeOs2Al10 than in CeRu2Al10. IV. Conclusions and future work

We have carried out a suite of comprehensive µSR and inelastic neutron scattering measurements on Ce(Ru1-xFex)2Al10 (x=0, 0.3, 0.5, 0.8 and 1), CeOs2Al10 and reference compounds, NdFe2Al10, NdOs2Al10 as well as on the 3d-electron quantum magnet YFe2Al10 to understand the unusual magnetic phase transition and spin gap formation. Confirming the long-range antiferromagnetic nature of the phase transition observed in the two compounds CeRu2Al10 and CeOs2Al10 using these techniques was a major milestone and a step forward in advancing our understanding of correlated phenomena in these systems. Our µSR studies reveal the presence of two frequency oscillations below the magnetic ordering temperatures for the Ce-based compounds as well as in NdFe2Al10, which provides direct evidence of the long-range magnetic ordering. On the other hand the µSR spectra of NdOs2Al10 reveal strong damping and loss of initial asymmetry without a clear sign of frequency oscillations below TN, indicating that the internal fields at the muon stopping sites are larger than 800 G (maximum field that can be investigated at ISIS facility. Further no clear sign of magnetic critical fluctuations was observed down to 60 mK in µSR study of YFe2Al10. For the Ce compounds exhibiting long range magnetic ordering, the temperature dependence of the µSR frequencies and the muon depolarization rates follow an unusual behavior with cooling of the sample below TN, pointing to the possibility of another phase transition at low temperatures. Further, the µSR spectra of x=0.8 and 1 down to the lowest temperature confirm the paramagnetic ground state. The inelastic neutron scattering (INS) study has established the

24

formation of a spin energy-gap with an energy scale of ~8 meV in the magnetic ordered compounds x=0, 0.3 and 0.5 as well as a spin gap of 11 meV in CeOs2Al10.

Our INS study reveals the possibility of an INS peak existing above TN in CeRu2Al10 and CeOs2Al10, which may indicate that the origin of the spin gap is associated with the hybridization gap or spin dimer formation. Moreover, the INS results of paramagnetic compounds x=0.8 and 1 reveal the presence of inelastic peaks (or spin gap) at 10 and 12.5 meV, respectively and at high temperature the response transforms into a quasi-elastic line. Further the investigations of very small Rh doping in CeRu2Al10 and Ir doping in CeOs2Al10 reveal that the magnetic moment orients along the a-axis as expect based on the single ion crystal field anisotropy [98, 99], which suggests that magnetic ground state and 4f-electrons and conduction electrons hybridization of these compounds are highly sensitive to electron doping. Further support of the presence of the anisotropic hybridization in CeOs2Al10 comes from a lightly hall doping (i.e 3% Re doping on Os site), which indicates a considerable reduction in the ordered state Ce moment ~0.18 µB with reduce TN = 20 K [ 180].

We believe the coexistence of the Kondo semiconducting state with spin-gap formation and magnetic order in CeT2Al10 (T=Ru, Os) to be unique among 4f-electron systems and it poses a perplexing new ground state for the strongly correlated class of materials. Considering the observation of a spin gap above TN in powder sample of CeT2Al10 (T=Ru and Os) and well defined spin wave at 4.5 K in the single crystal samples, it is highly desirable to investigate CeT2Al10 single crystals using INS above TN. Further it would be very interesting to investigate spin waves, spin gap and crystal field excitations in Ce(Ru1-xRhx)2Al10 and Ce(Os1-xIrx)2Al10 single crystals, where the ordered state magnetic moment is along the a-axis as expected base on CEF anisotropy, and anisotropic exchange and hybridization seem to have weaken, to unravel the role of hybridization on the magnetic properties of the parent compounds.

Acknowledgement:

We acknowledge interesting discussion with Andrea Severing, Dimitry Khalyavin, Peter Baker, Stephen Cottrell, Pascale Deen, Ross Stewart, Pascale Manuel, Amir Murani, Peter Riseborough, Qimiao Si and Piers Coleman. DTA and ADH would like to thank CMPC-STFC, grant number CMPC-09108, for financial support. The work at Hiroshima University was supported by a Grant-in-Aid for Scientific Research on Innovative Area “Heavy Electrons” (20102004) of MEXT, Japan. AMS thanks the SA-NRF (Grant 78832) and UJ Research Committee for financial support. AB thanks the FRC of UJ and ISIS-STFC for funding support.

.

Figure captions 25

Fig. 1 (color online) (a) Single impurity Kondo effect, a schematic view of the conduction

electron band and localised 4f-electron, and resonant density of states (DOS) showing build up of

the fermionic resonance near the Fermi level. The width of the resonance peak gives an

estimation of Kondo temperature, TK. (b) Kondo lattice case, hybridized band picture showing

renormalized bands, lower (EK-) and upper (EK

+) hybridized bands, and direct gap, ∆dir (we called

charged gap, ∆char) at q = 0 and indirect gap, ∆ind (we called spin gap, ∆spin) at q � 0 and the gap in

resonant density of states (DOS). (c) The 4f-weight factors of the upper (EK+) and lower (EK

-)

bands as a function of wave vector, taken from Riseborough [18].

Fig.2 (color online) Generic heavy fermion phase diagram (originated from Doniach’s model [47]): coupling strength (δ) versus the characteristic temperature (from ref. [149]). The proposed position of CeT2Al10 (T=Fe, Ru and Os) compounds on the phase diagram are obtained from the pressure dependent data on T0/TN [102].

Fig.3 (color online) (bottom) The caged type orthorhombic unit cell of CeT2All0 (T=Fe, Ru and Os) and (top) magnetic structure of CeRu2Al10 from ref. [94]

Fig.4 (color online) de Gennes factor versus ordering temperature of RRu2Al10 (R=rare earths) (top) and ROs2Al10 (bottom).

Fig.5 Temperature dependence of the lattice parameters and unit cell volume (left) and selected interatomic distances of CeOs2Al10 investigated using the high resolution neutron powder (HRPD) diffractometer at ISIS facility.

Fig.6 (a-c) Temperature dependence of the magnetic susceptibility of CeT2Al10 (T=Fe, Ru and Os) and (d-f) temperature dependence of the electrical resistivity of CeT2Al10 (T=Fe, Ru and Os) from refs. [60-62]

Fig. 7 (color online) Zero-field (ZF) µSR spectra of Ce(Ru1-xFex)2Al10 (x=0 to 1) at various temperatures from ref [82]. The solid lines show a fit as discussed in the text.

Fig. 8 (color online) Fit parameters of zero-field (ZF) µSR spectra of Ce(Ru1-xFex)2Al10 (x=0 to 1), internal fields vs temperature (left), and depolarization rate vs temperature (right) from ref.[82]. Two distinct frequencies have been found, leading also to two depolarization rates which are plotted in red and black symbols. The inset in (c) shows the temperature dependence of the muon initial asymmetry for three components from Eq.(2).

Fig. 9 (color online) Phase diagram of Ce(Ru1-xFex)2Al10 [from Nishioka et al [46]] and Ce(Os1-

xFex)2Al10. The phase transition disappears sharply at xC = 0.8 in Ce(Ru1-xFex)2Al10 and at xC = 0.6 in Ce(Os1-xFex)2Al10. The jump in the heat capacity, Δ(C/T), at TN decreases gradually with

26

increasing x. Note that the phase diagram of Ce(Os1-xFex)2Al10 overlaps that of Ce(Ru1-

xFex)2Al10 when the x value is shifted by 0.25.

Fig. 10 (color online) Zero-field (ZF) µSR spectra of CeOs2Al10 at various temperatures from ref. [92]. The solid lines show the fit (see text).

Fig. 11 (color online) Fit parameters of zero-field (ZF) µSR spectra of CeOs2Al10, internal fields vs temperature (left), and depolarization rate vs temperature (right) from ref. [92]. Two distinct internal fields (or frequencies) have been found, leading also to two depolarization rates which are plotted in red and black symbols.

Fig. 12 (color online) Zero-field (ZF) µSR spectra of NdFe2Al10 at various temperatures. The solid lines show a fit as discussed in the text.

Fig. 13 (color online) Temperature dependence of the internal fields at the muons’ stopping sites of NdFe2Al10.

Fig. 14 (color online) Zero-field (ZF) µSR spectra of NdOs2Al10 at various temperatures. The solid lines show the fit (see text) and the dotted line shows the background.

Fig. 15 (color online) (a) Temperature dependence of the muon initial asymmetry and, (b) KT-depolarization rate (σKT, red open circles) and electronic relaxation rate ( λ, filled squares) of NdOs2Al10.

Fig. 16 (a) Magnetic susceptibility as function of temperature (from ref. [129]) and, (b) heat capacity divided by temperature as function of temperature of YFe2Al10. The solid line shows the power law (T-n) behaviour.

Fig.17 (a) Zero-field (ZF) µSR spectra at 0.06 K and 3 K, (b) µSR spectra measured in 50 G longitudinal field (LF) at 0.06 K and 2 K and, (c) field (LF) dependence of µSR spectra at 0.06 K.

Fig. 18 (color online) Temperature dependence of the muon initial asymmetry (top) and electronic relaxation rate ( λ) of YFe2Al10 measured in 0 G (ZF), 50 G and 2500 G and and KT-depolarization rate (σKT, left y-axis) measured in 0 G (ZF).

Fig. 19 (color online) Q-integrated inelastic scattering intensity versus energy transfer of Ce(Ru1-

xFex)2Al10 for x=0, 0.3, 0.5 0.8 and 1 at 4.5 (7 ) K at Q=1.27 (2.64) Å-1 (from ref. [82]). The open squares in (d-e) are from Ei=100 meV and open circles in (e) are from Ei=40 meV measured on MERLIN. The broad peak centred near 50 meV is the response across the hybridized bands, which is in agreement with the 55 meV charge gap observed through optical conductivity [67, 68 ].

27

Fig. 20 (color online) Colour coded inelastic neutron scattering intensity of CeOs2Al10 at 5 K and 38 K measured with incident energy of Ei=25 meV on the MERLIN spectrometer. The phonon contribution was subtracted using the data of non-magnetic reference compound LaOs2Al10.

Fig.21 (color online) The fit parameters, susceptibility, linewidth and peak position versus temperature obtained from fitting the magnetic scattering intensity of CeOs2Al10. The closed circles represent the fit using an inelastic peak and open circles represent the fit using a quasi-elastic peak. The green solid line in (a) shows the measured dc-susceptibility of the polycrystalline sample of CeOs2Al10 and in (b) the dotted and solid line represents the fits using exponential and T2 behavior respectively (see text). In (c) the solid line represents the power law simulation ∆(T)=∆0 (1-T/TN)β with ∆0 =11 meV, β=0.1 and TN=28.5 K and the filled green diamonds are the optical gap (∆op) from ref. [66].

Refeneces [1] W.E. Walace, Rare-earth intermettalics, Academic Press New York (1973) [2] P. Campbell, Permanent Magnet Materials and Their Aplication, Cambridge University Press (1994) [3] D. Haskel, Y.B. Lee, B.N. Harmon, Z. Islam, J.C. Lang, G. Srajer, Ya. Mudryk, K.A. Gschneidner, Jr., and V.K. Pecharsky, Phys. Rev. Lett., 98, 247205 (2007). [4] S.J. Poon, Seminod. Semimet. 70, 37 (2001) [5] T. Takabatake,, E. Matsuoka, S. Narazu, K. Hayashi, S. Morimoto,T. Sasakawa, K. Umeo, M. Sera, Physica B 383. 93 (2006); B. C. Sales, D. Mandrus, and R. K. Williams: Science 272, 1325 (1996). [6] F. Yuan, Q. Gu, Y. Guo, W. Sun, X. Chena and X. Yu, J. Mater. Chem., 22, 1061 (2012) [7] K. H. J. Buschow, Rep. Prog. Phys. 40, 1179 (1977) [8] O. Stockert, S. Kirchneri, F. Steglich and Q. Si, J. Phys. Soci. Jpn. 81, 011001 (2012) [9] N. D. Mathur, F. M. Grosche, S. R. Julian, I. R. Walker, D. M. Freye, R. K. W. Haselwimmer, and G. G. Lonzarich: Nature 394, 39 (1998) [10] S. Paschen, et al, Nature 432, 881 (2004); S.Friedemann, S. et al, Proc. Natl Acad. Sci. USA 107, 14547 (2010). [11] S. Nakatsuji et al., Nature Phys. 4, 603 (2008); A.H. Nevidomskyy and P. Coleman, Phys. Rev. Lett., 102, 077202 (2009); E. Bauer, et al., Phys. Rev. Lett. 92, 027003 (2004). [12] M. B. Maple, R.E. Baumbach, N. P. Butch, J.J. Hamlin and M.Janoschek, J. Low Temp. Phys. 161, 4 (2010) [13] J. Kondo, Progr. Theor. Phys. Osaka, 32, 37 (1964) [14] G.R. Stewart, Rev. Mode. Phys. 56, 755 (1984); J. Flouquet and H. Harima, arXiv:0910.3110 (2009); K. Andres, J.E. Graebner and H.R. Ott, Phys. Rev. Lett. (APS) 35: 1779 (1975)

28

[15] C.M. Varma, Rev. Mod. Phys. 48, 219 (1976); J M Lawrence, P S Riseborough and R D Parks, Rep. Prog. Phys. 44, 1 (1981) [16] E. Bauer, Adv. Phys. 40, 417 (1991) [17] A. Georges, G. Kotliar, W. Krauth and M.J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996) [18] P.S. Riseborough, Adv. Phys. 49, 257 (2000); P.S. Riseborough, Phys. Rev. B 45, 13984 (1992);T. Takabatake et al., J. M.M.M 177-181, 277, 1998. [19] P. Coleman, in “Handbook of Magnetism and Advanced Magnetic Materials,” ed. by H. Knoemuller and S. Parkin, John Wiley and Sons, Vol.1, 95 (2007) [20] B. Bucher and Z. Schlesinger, P. C. Canfield and Z. Fisk, Phys. Rev. Lett. 72, 522 (1994) [21] G. Gruner, Density Waves in Solids (Addison-Wesley, Reading, MA) (1994): H. Fröhlich, Proc. R. Soc. London, Ser. A 223, 296 (1955): M. Kumar, V. K. Anand, C. Geibel, M. Nicklas, and Z. Hossain, Phys. Rev. B 81, 125107 (2010) [22] H. S. Jeevan, D. T. Adroja, A. D. Hillier, Z. Hossain, C. Ritter, C. Geibel, Phys. Rev. B 84, 184405 (2011) [23] S. H. Curnoe, H. Harima, K. Takegahara, and K. Ueda, Phys. Rev. B 70, 245112 (2004) [24] F. Steglich et al., Phys. Rev. Lett. 43, 1892 (1979). [25] Frontiers of Novel Superconductivity in HFC, J. Phys. Soc. Jpn. 76, 051001 (2007) edt. Y. Onuki and Y. Kitaoka [26] G.R. Stewart, Rev. Mod. Phys. 73, 797 (2001); C.M. Varma,, Z. Nussinov, W. van Saarloos, Physics Reports 361, 267 (2002) [27] H. v. Löhneysen, A. Rosch, M. Vojta, and P. Woelfle, Rev. Mod. Phys. 79, 1015 (2007) [28] K. Takegahara, H. Harima, Y. Kaneta, A. Yanase, J. Phys. Soc. Jpn. 62, 2103 (1993) [29] L.F. Matheiss, D.R. Hamman, Phys. Rev. B 47, 13114 (1993) [30] H.v. Löhneysen, et al, Phys. Rev. Lett. 72, 3262 (1994). [31] M.C. Aronson et al, Phys. Rev. Lett. 75, 725 (1995). [32] D.T. Adroja, J.-G. Park, K.H. Jang, H.C. Walker, K.A. McEwen and T. Takabatake. Magn. Magn. Mater. 310, 858, (2007); J.-G. Park, et al., J. Phys. Condens. Matter 14, 3865 (2002) [33] Q. Si, Phys. Stat. Solidi 247, 476 (2010) [34] D. Wohlleben, “Valence Fluctuations in Solids,” L.M. Falicov, W. Hanke, M.B. Maple (Eds.),North-Holland, Amsterdam, p. 1 (1981) [35] A.P. Murani, A. Severing, W.G. Marshall, Phys. Rev. B 53, 2641 (1996). [36] A. Menth, E. Bucher and T. H. Geballe, Phys. Rev. Lett. 22, 295 (1969). [37] J.-M. Mignot, P. A. Alekseev, K. S. Nemkovski, L.-P. Regnault, F. Iga, and T. Takabatake, Phys. Rev. Lett. 94, 247204 (2005) [38] P. Wachter, Physica B 300, 105 (2001) [39] P. Wachter, Intermediate valence and heavy fermions, in: K.A. Gschneidner, L. Eyring Jr., G.H. Lander, G.R. Choppin (Eds.), Handbook on the Physics and Chemistry of Rare Earths, Vol. 19, Lanthanides/Actinides: Physics II, Elsevier Science, Amsterdam, 1994, p. 177 (Chapter 132).

29

[40] P.C. Canfield, J.D. Thompson, Z. Fisk, M.F. Hundley, A. Lacerda, J. Magn. Magn. Mater. 108, 217 (1992) [41] Z. Fisk, G. Aeppli, Comm. Condens. Matter Phys. 16, 155 (1992) [42] V. Alexandrov, M. Dzero, P. Coleman, arXiv:1303.7224 (2013) [43] A.C. Hewson, The Kondo Problem to Heavy Feermions, Cambridge University Press (1993) [44] L.D. Landau, Sov. Phys. JETP 6, 920 (1957). [45] L. Degiorgi, Rev. Mod. Phys., 71, 687, (1999) [46] J.A. Hertz, Phys. Rev. B 14, 1165 (1976) ; A.J. Millis, Phys. Rev. B 48, 7183 (1993) ; T. Moriya, and T. Takimoto, J. Phys. Soc. Japan 64, 960 (1995) ; T. Moriya and K. Ueda, Adv. Phys. 49, 555 (2000); U. Zülicke, and A. J. Millis, Phys. Rev. B 51, 8996 (1995); P. Coleman, Physica B 259-261, 353 (1999). [47] S. Doniach, Physica B 91, 231 (1977). [48] Q. Si, S. Rabello, K. Ingersent and J.L. Smith, Nature 413, 804 (2001); A. Schröder, et al, Nature 407, 351 (2000).; S. Sachdev, Quantum Phase transitions (Cambridge University Press, New York) (1999); P. Coleman and A. Schofield, Nature 433, 226 (2005) [49] S. Sachdev, arXiv:1002.3823v1 (2010); T. Moriya and K. Ueda, Rep. Prog. Phys. 66, 1299 (2003) [50] C. Pfleiderer: Rev. Mod. Phys. 81, 1551 (2009) [51] G. S. Nolas, J. L. Cohn, G. A. Slack, and S. B. Schujman: Appl. Phys. Lett. 73, 178. (1998) [52] V. Keppens, D. Mandrus, B. C. Sales, B. C. Chakoumakos, P. Dai, R. Coldea, M. B. Maple, D. A. Gajewski, E. J. Freeman, and S. Bennington: Nature 395 (1998) 876. [53] M. M. Koza, M. R. Johnson, R. Viennois, H. Mutka, L. Girard, and D. Ravot: Nat. Mater. 7 , 805 (2008) [54] Z. Hiroi, J. Yamaura, and K. Hattori, J. Phys. Soci. Jpn, 81, 011012 (2012) [55] T. Onimaru, K. T. Matsumoto, Y. F. Inoue, K. Umeo, T. Sakakibara, Y. Karaki, M. Kubota, and T. Takabatake, Phys. Rev. Lett. 106,177001 (2011) [56] D.T. Adroja, A.M. Strydom, A.P. Murani, W.A. Kockelmann, A. Fraile, Physica B 403, 898 (2008) [57] Z. Hiroi, J. Yamura and K. Hattori, J. Phys. Scoi. Jpn, 81, 011012 (2012) [58] J. Custers, K-A. Lorenzer, M. Müller, A. Prokofiev, A. Sidorenko, H.Winkler, A. M. Strydom,Y. Shimura, T. Sakakibara, R. Yu, Q. Si and S. Paschen1, Nature Mate. 11, 189 (2012) [59] A.M. Strydom, Physica B 404, 2981 (2009) [60] Y. Muro. J. Kajino, K. Umeo, K. Nishimoto and R. Tamura and T. Takabatake, Phys. Rev. B 81, 214401(2010); K. Yutani, Y. Muro, J. Kajino, T. J. Sato and T. Takabatake, J.Phys. Conf. Ser. 391, 012070 (2012) [61] Y. Muro, K. Motoya, Y. Saiga and T. Takabatake, J. Phys.: Conf. Ser. 200, 012136 (2010); Y. Muro, K. Motoya, Y. Saiga, and T. Takabatake, J. Phys. Soc. Jpn. 78, 083707 (2009) [62] T. Takasaka, K. Oe, R. Kobayashi, Y. Kawamura, T. Nishioka, H. Kato, M. Matsumura, and K. Kodama, J. Phys.: Conf. Ser. 200, 012201 (2010)

30

[63] M. Matsumura, Y. Kawamura, S. Edamoto, T. Takesaka, H. Kato, T. Nishioka, Y. Tokunaga, S. Kambe, and H. Yasuoka, J. Phys. Soc. Jpn. 78 123713 (2009) [64] T. Nishioka, Y. Kawamura, T. Takesaka, R. Kobayashi, H. Kato, M. Matsumura, K. Kodama, K. Matsubayashi, and Y. Uwatoko: J. Phys. Soc. Jpn. 78 123705 (2009) [65] H. Tanida, D. Tanaka, M. Sera, C. Moriyoshi, Y. Kuroiwa, T. Nishioka, H. Kato, and M. Matsumura, J. Phys. Soc. Jpn. 79, 043708 (2010) [66] S. Kimura, T. Iizuka, H. Miyazaki A. Irizawa, Y. Muro and T. Takabatake, Phys. Rev. Lett. 106, 056404 (2011) [67] S. Kimura,T. Iizuka, H. Miyazaki,T. Hajiri, M. Matsunami, T. Mori, A. Irizawa, Y. Muro J. Kajino, and T. Takabatake, Phys. Rev. B 84, 165125 (2011) [68] S. Kimura, Y. Muro, and T. Takabatake, J. Phys. Soc. Jpn. 80, 033702 (2011). [69] S.C. Chen and Lue, Phys. Rev. B 81, 075113 (2010) [70] Y. Kawamura, S. Edamoto, T. Takesaka, T. Nishioka, H. Kato, M. Matsumura, Y. Tokunaga, S. Kambe, and H. Yasuoka: J. Phys. Soc. Jpn. 79, 103701 (2010) [71] S. K.Dhar, S. K. Malik, and R. Vijayaraghavan, J. Phys. C 14, L321 (1981). [72] A. Galatanu, E. Yamamoto, T. Okubo, M. Yamada, A. Thamizhavel, T. Takeuchi, K. Sugiyama, Y. Inada, and Y. Ōnuki, J. Phys.: Condens. Matter 15, S2187 (2003). [73] S. K. Malik, S. K.Dhar, R. Vijayaraghavan and W.R. Wallace, J. Appl. Phys. 53, 8074 (1982). [74] H.R. Kirchmayer and C.A. Poldy, in Handbook on the physics and chemistry of rare-earths, Vol.2, eds K.A. Gschneider Jr. and L. Eyring (North-Holland, Amsterdam), p.55, (1979) [75] R.M. White, “ Quantum Theory of Magnetism, Springer-Verlag, Berin, 1983) [76] D.R. Noakes and G.K. Shenoy, Phys. Lett., 91A, 35 (1982);D. T. Adroja and S. K. Malik, Phys. Rev. B 45 779 (1992) [77] S. Raymond, J. Panarin, F. Givord, A. P. Murani, J. X. Boucherle, and P. Lejay, Phys. Rev. B 82 094416 (2010); K.Yamauchi, A. Yanase, and H. Harima, J. Phys. Soci. Jpn., 79, 044717 (2010) [78] V.M. T. Thiede, T. Ebel, and W. Jeitschko, J. Mater. Chemistry, 8, 125 (1998); A.I. Tursina, S. N. Nesterenko, E. V. Murashova, H. V. Chernyshev, H. Nöel, and Y. D. Seropegin,, Acta Crystallogr. Sect. E61, 112 (2005) [79] H. Tanida, D. Tanaka, M. Sera, S. Tanimoto, T. Nishioka, M. Matsumura, M. Ogawa, C. Moriyoshi, Y. Kuroiwa, J. E. Kim, N. Tsuji, and M. Takata, Phys. Rev. B 84, 115128 (2011) [80] M. Sera, D. Tanaka, H. Tanida, C. Moriyoshi, M. Ogawa, Y. Kuroiwa, T. Nishioka, M. Matsumura, J. Kim, N. Tsuji and M. Takata, J. Phys. Soci. Jpn 82, 024603 (2013) [81] H.H. Hill, Plutonium and Other Actinides, 2nd ed.; Miner, W. N., Ed.; Metallurgical Society of the AIME: New York, (1970) [82] D. T. Adroja, A. D. Hillier, Y. Muro, J. Kajino, T. Takabatake, P. Peratheepan, A. M. Strydom, P. P. Deen, F. Demmel, J. R. Stewart, J. W. Taylor, R. I. Smith, S. Ramos, and M. A. Adams, Phys. Rev. B 87, 224415 (2013)

31

[83] H. Fujii, T. Inoue, Y. Andoh, T. Takabatake, K. Satoh, Y. Maeno, T. Fujita, J. Sakurai, and Y. Yamaguchi, Phys. Rev. B 39, 6840 (1989) [84] D. T. Adroja, J.-G. Park, E. A. Goremychkin, K. A. McEwen, N. Takeda, B. D. Rainford, K. S. Knight, J. W. Taylor, J. Park, H. C. Walker, R. Osborn, and P. S. Riseborough, Phys. Rev. B 75, 014418 (2007) ;D.T. Adroja, J.-G. Park, K.A. McEwen, N. Takeda, M. Ishikawa, J.-Y. So, Phys. Rev. B 68, 094425 (2003) [85] F. Strigari, T. Willers, Y. Muro, K. Yutani, T. Takabatake,Z. Hu, S. Agrestini, C.-Y. Kuo, Y.-Y. Chin, H.-J. Lin, T. W. Pi, C. T. Chen, E. Weschke, E. Schierle, A. Tanaka, M. W. Haverkort, L. H. Tjeng, and A. Severing, Phys. Rev. B 87, 125119 (2013) [86] J.Y. Kim, B.G. Cho, H.J. Lee and H.-C. Kim, J. Appl. Phys. 101, 09D501 (2007) [87] S. K. Malik et al., Solid State Commun. 43, 243-245 (1982) [88] V. R. Fanelli, J. M. Lawrence, C H Wang, A. D. Christianson, E. D. Bauer, K. J. McClellan, E A Goremychkin, R Osborn, arXiv:0908.4378; J. M. Lawrence, V. R. Fanelli, E. A. Goremychkin, R. Osborn, E. D. Bauer, K. J. McClellan, A. D. Christianson, Physica B 403, 783 (2008). [89] T.E. Mason, G. Aeppli, A.P. Ramirez, K.N. Clausen, C. Broholm, N. St¨ucheli, E. Bucher and T.I.M. Palstra, Phys. Rev. Lett. 69, 490 (1992); T.J. Sato, H. Kadowaki, H. Yoshizawa, T. Ekino, T. Takabatake, H. Fuji, L.P. Regnault and Y. Isikawa, J. Phys. Condens. Matter 7, 8009 (1995) [90] H. Ikeda and K. Miyake ,J. Phys. Soc. Jpn. 1769 (1996) [91] J. Moreno and P. Coleman, Phys. Rev. Lett. 84, 342 (2000) [92] D. T. Adroja, A. D. Hillier, P. P. Deen, A. M. Strydom, Y. Muro, J. Kajino, W. A. Kockelmann, T. Takabatake, V. K. Anand, J. R. Stewart, and J. Taylor, Phys. Rev. B 82 104055 (2010); D. T. Adroja et al unpublished (2013) [93] F. Strigari, T. Willers, Y. Muro, K. Yutani, T. Takabatake, Z. Hu, Y.-Y. Chin, S. Agrestini, H.-J. Lin, C. Chen, A. Tanaka, M. W. Haverkort, L.-H. Tjeng, and A. Severing, Phys. Rev. B 81, 081105(R) (2012). [94] D. Khalyavin, A. D. Hillier, D. T. Adroja, A. M. Strydom, P.Manuel, L. C. Chapon, P. Peratheepan, K. Knight, P. Deen, C. Ritter, Y. Muro, and T. Takabatake: Phys. Rev. B 82, 100405(R) (2010) [95] H. Kato, R. Kobayashi, T. Takesaka, T. Nishioka, M. Matsumura, K. Kaneko, and N. Metoki, J. Phys. Soc. Jpn, Suppl. 80, 073701 (2011). [96] K. Kunimori, M. Nakamura, H. Nohara, H. Tanida, M. Sera, T. Nishioka, and M. Matsumura, Phys. Rev. B 86, 245106 (2012) [97] A. Kondo, K. Kindo, K. Kunimori, H. Nohara, H. Tanida, M. Sera, R. Kobayashi, T. Nishioka, and M. Matsumura, J. Phys. Soc. Jpn. 82, 054709 (2013). A. Kondo, J. Wang, K. Kindo, Y. Ogane, Y. Kawamura arXiv:1308.3285v (2011); A. Kondo, J. Wang, K. Kindo, T. Takesaka, Y. Kawwamura, T. Nishioka, D. Tanaka, H. Tanida, and M. Sera, J. Phys. Soc. Jpn. 79, 073709 (2010).

32

[98] D. D. Khalyavin, D. T. Adroja, P. Manuel, J. Kawabata, K. Umeo, T. Takabatake, and A. M. Strydom, arXiv:1306.5269v1 (2013), Phys. Rev. B 88, 060403(R) (2013) [99] H. Guo, H. Tanida, R. Kobayashi, I. Kawasaki, M. Sera, T. Nishioka, M. Matsumura, I. Watanabe, and Z. Xu, arXiv:1308.3285v1 (2013); Phys. Rev. B 88, 115206 (2013) [100] T. Takabatake, F. Teshima, H. Fujii, S. Nisjhigori, T. Suzuki, T. Fujita, Y. Yamaguchi, J. Sakurai, and D. Jaccard, Phys. Rev. B 41, 9607 (1990) [101] S.K. Malik and D.T. Adroja, Phys. Rev. B 43, 6277 (1991) [102] T. Nishioka, D. Hirai, Y. Kuwamura, H. Kato, M. Matsumurata, H. Tanida, M. Sera, K. Matsubayashi and Y. Uwatoko, .J. Phys.: Conf. Ser. 273, 012046 (2011); Y. Kawamura1, Y. Ogane, T. Nishioka, H. Kato, M. Matsumura, K. Matsubayashi and Y. Uwatoko, Journal of Physics: Conference Series 273, 012038 (2011) [103] Y. Kawamura, K. Matsui, T. Kuwayama, T. Kawaai, S. Yamaguchi, Y. Nishijima, J.Hayashi, K.Takeda, C. Sekine and T. Nishioka, Photon Factory Activity Report, B, 18C/2011G555 (2012) [104] C. S. Lue, H. F. Liu, B. D. Ingale, J. N. Li, and Y. K. Kuo Phys. Rev. B 85, 245116 (2012) [105] A. Amato, Rev. Mod. Phys. 69, 1119 (1997) [106] P. Dalmas de Réotier and A. Yaouanc, J. Phys. Condens. Matter, 9 9113 (1997); S. Blundell, arXive:cond-mat/0207699 (2002) [107] V.H. Tran, A. D. Hillier, D. T. Adroja, Z. Bukowski and W. Miiller, J. Phys.: Condens. Matter 21 485701 (2009) [108] S. Kambe, H. Chudo, Y. Tokunaga, T. Koyama, H. Sakai,U. Ito, K. Ninomiya,W. Higemoto, T. Takesaka, T. Nishioka and Y. Miyake, J. Phys. Soc. Jpn, 79, 053708 (2010) [109] R. Kubo, T. Toyabe, in “Magnetic Resonanceand Relaxation,” ed. by R. Blinc, North-Holland, Amsterdan, p. 810 (1966); R.S. Hayano, Y.J. Uemura, J. Imazato, N. Nishida, T. Yamazaki and R. Kubo, Phys. Rev. B 20, 850 (1979) [110]] T. Nishioka, D. Hirai, Y. Kuwamura, H. Kato, M. Matsumurata, H. Tanida, M. Sera, K. Matsubayashi and Y. Uwatoko, .J. Phys.: Conf. Ser. 273, 012046 (2011) [111] D.T. Adroja, T. Takabatake and Y.Muro, ISIS Facility experimental report, RB1110535 (2012) [112] J. Robert, J-M Mignot, S. Petit, P. Steffens,T. Nishioka, R. Kobayashi,M. Matsumura, H. Tanida, D. Tanaka, and M. Sera, Phys. Rev. Lett. 109, 267208 (2012) [113] M. Reehuis, B. Fehrmann, M.W. Wol, W. Jeitschko, M. Hofmann, Physica B 276-278, 594 (2000); S. Niemann, W. Jeitschko, Z. Kristallogr. 210, 338 (1995) [114] A.M. Strydom and P. Peratheepan, Phys. Status Solidi RRL 4,356 (2010) [115] Y.Muro, J. Kajino, T. Onimaru and T. Takabatake, J. Phys. Soci. Japan, 80, SA021, (2011) [116] H. Tanida, Y. Nonaka, D. Tanaka, M. Sera, Y. Kawamura, Y. Uwatoko, T. Nishioka, and M. Matsumura, Phys. Rev. B 85, 205208 (2012); G. Morrison, N. Haldoaarachchige, D. P Young and J. Y. Chan, J. Phys.: Condens. Matter 24, 356002 (2012)

33

[117]. Tanida, D. Tanaka, M. Sera, S. Tanimoto, T. Nishioka, M. Matsumura, M. Ogawa, C. Moriyoshi, Y. Kuroiwa, J. E. Kim, N. Tsuji, and M. Takata, Phys. Rev. B 84, 115128 (2011) [118] A. Amato, R. Feyerherm, F. N. Gygax, A. Schenck, H.v. Löhneysen, and H. G. Schlager ,

Phys. Rev. B 52, 54 (1995)

[119] O. O. Bernal, D. E. MacLaughlin, H. G. Lukefahr, and B. Andraka, Phys. Rev. Lett. 75,

2023 (1995); D. E. MacLaughlin, O. O. Bernal, R. H. Heffner, G. J. Nieuwenhuys, M. S. Rose, J.

E. Sonier, B. Andraka, R. Chau, and M. B. Maple, Phys. Rev. Lett. 87, 066402 (2001) [120] W.D. Wu, A. Keren, L. P. Le, G. M. Luke, B. J. Sternlieb, Y. J. Uemura, C. L. Seaman, Y. Dalichaouch, and M. B. Maple, Phys. Rev. Lett. 72, 3722 (1994) [121] D. T. Adroja, A. D. Hillier, J.-G. Park, W. Kockelmann, K. A. McEwen, B. D. Rainford, K.H. Jang, C. Geibel, and T. Takabatake, Phys. Rev. B 78, 014412 (2008) [122] A. D. Hillier, D. T. Adroja, S. R. Giblin, and W. Kockelmann, B. D. Rainford and S.K. Malik, Phys. Rev. B 76, 174439 (2007) [123] P. Gegenawart, Q. Si and F. Steglich, Nature physics 4, 186 (2008) [124] S. Watanabe and K. Miyake, Phys. Rev. Lett. 105, 186403 (2010) [125] M. Nicklas, M. Brando, G. Knebel, F. Mayr, W. Trinkl, and A. Loidl, Phys. Rev. Lett. 82, 4268 (1999) [126] A. Yeh , Y.-A. Soh, J. Brooke, G. Aeppli, T.F. Rosenbaum and S.M. Hayden, Nature, 419 (2002) 459: E. Fawcett, Rev. Mod. Phys. 60 (1988) 209. [127] F.M. Grosche, C. Pfleiderer, G. J. McMullan, G. G. Lonzarich, and N. R. Bernhoeft, Physica B 206-207, 20 (1995) [128] P. Kostic et al., Phys. Rev. Lett. 81, 2498 (1998). [129] P. Khuntia, A. M. Strydom, L. S. Wu, M. C. Aronson, F. Steglich, and M. Baenitz, Phys. Rev. B 86, 220401(R) (2012) [130] A.M. Strydom, P. Peratheepan, R. Sarkar, M. Baenitz and F. Steglich, J. Phys. Soc. Jpn 80 SA043 (2011) [131] K. Park, L.S. Wu, Y. Janssen, M.S. Kim, C. Marques, and M.C. Aronson, Phys Rev B 84, 094425 (2011) [132] G.L.Squires, Introduction to the theory of thermal neutron scattering, Cambridge University Press (1978) [133] S.W. Lovesey, Theory of neutron scattering from condensed matter, Clarendon Press, Oxford (1984) [134] W.G. Williams, Polarized Neutrons, Clarendon Press, Oxford (1988). [135] S. M. Hayden, H. A. Mook, V. Dai, T. G. Perring, and F. Dogan, Nature (London) 429, 531 (2004). [136] J. M. Tranquada et al., Nature (London) 429, 534 (2004) [137] V. Hinkov et al., Nature (London) 430, 650 (2004). [138] Ch. Rüegg, B. Normand, M. Matsumoto, Ch. Niedermayer, A. Furrer, K.W. Krämer, H.-U. Güdel, Ph. Bourges, Y. Sidis, and H. Mutka, Phys. Rev. Lett. 95, 267201 (2005)

34

[139] N. Cavadini, G. Heigold, W. Henggeler, A. Furrer, H.-U. Güdel, K. Krämer and H. Mutka, Phys. Rev. B 63, 172414 (2001) [140] Ch. Ruegg et al. Nature 423 62 (2003) [141] B. Fåk, D. F. McMorrow, P. G. Niklowitz, S. Raymond, E. Ressouche, J. Flouquet, P. C. Canfield, S. L. Bud’ko, Y. Janssen, and M. J. Gutmann, J. Phys.: Cond. Matt. 17, 301 (2005). [142] J. Robert, J.M. Mignot, G. André, T. Nishioka, R. Kobayashi, M. Matsumura, H. Tanida, D. Tanaka and M. Sera, Phys. Rev. B 82 100404 R (2010); J. M. Mignot, J. Robert, G. Andre, A. M. Bataille, T. Nishioka, R. Kobayashi, M. Matsumura, H. Tanida, D. Tanaka, and M. Sera, J. Phys. Soc. Jpn. 80, SA022 (2011) [143] K. Hanzawa, J. Phys. Soc. Jpn. 79, 043710 (2010) [144] K. Hanzawa, J. Phys. Soc. Jpn 80, 113701 (2011) [145] D. T. Adroja, K.A. McEwen. J.-G. Park, A.D. Hillier, N. Takeda, P.S. Riseborough and T. Takabatake, J. Opto. Adv. Mate., 10, 1564 (2008) [146] D. T. Adroja, J.-G. Park, K. A. McEwen, N. Takeda,M. Ishikawa, J.-Y. So, Phys. Rev. B 68, 094425 (2003); [147] R.Viennois, L. Girard, L.C. Chapon, D.T. Adroja, R.I. Bewley, D. Ravot, P. S. Riseborough, and S. Paschen, Phys. Rev. B 76, 174438 (2007) [148] A.J. Fedro and S.K. Sinha, in Valence Fluctuations in Solids, edited by L.M. Falicov, W. Hanke, and M.B. Maple (North-Holland, Amsterdam, p. 329 (1981) [149] http://www.toulouse.lncmi.cnrs.fr/spip.php?rubrique67&lang=en [150] B. M. Sondezi-Mhlungu, D. T. Adroja, A. M. Strydom, W. Kockelmann, and E. A. Goremychkin, J. Phys.: Conf. Ser. 200, 012190 (2010) [151] S. Rayaprol, C. P. Sebastian, and R. Pöttgen, J. Solid State Chem. 179, 2041 (2006) [152] S. K. Malik, and D. T. Adroja, Phys. Rev. B 43, 6295 (1991) [153] B.D. Rainford, and D.T. Adroja, Physica B 194-196, 365 (1994) [154] K. Katoh, A. Ochiai, and T. Suzuki, Physica B 223, 340 (1996) [155] J. Goraus, A. Slebarski, and M. Fija1kowski, Intermetallics 32, 219 (2013) [156] K. Prokěs, J. A. Mydosh, O. Prokhnenko, W.-D. Stein, S. Landsgesell, W. Hermes, R. Feyerherm, and R. Pöttgen, Phys. Rev. B 87, 094421 (2013) [157] K. Satoh, Y. Maeno, T. Fujita, Y. Uwatoko, and H. Fujii, J. Phys. Colloques 49, C8-779-C8-780 (1988) [158] S. K. Malik, D. T. Adroja, S. K. Dhar, R. Vijayaraghavan, and B.D. Padalia, Phys. Rev. B 40, 2414 (1989). [159] G. M. Kalvius, A. Kratzer, D. R. Noakes, K. H. Münch, R. Wäppling, H. Tanaka, T. Takabatake, and R. F. Kiefl, Europhys. Lett. 29, 501 (1995) [160] V. K. Pecharsky, L. L. Miller, and K. A. Gschneidner, Phys. Rev. B 58, 497 (1998) [161] V. N. Nikiforov, M. Baran, A. Jedrzejczak, V. Yu. Irkhin, arXiv:1204.4903 [162] A. Gil, A. Szytula, Z. Tomkowicz, K. Wojciechowski, and A. Zygmunt, J. Magn. Magn. Mater. 129, 271 (1994)

35

[163] A.Thamizhavel, H. Nakashima, T. Shiromoto, Y. Obiraki1, T. D Matsuda, Y. Haga, S. Ramakrishnan, T. Takeuchi, R. Settai1, and Y. Ōnuki, J. Phys. Soc. Jpn. 74 2617 (2005) [164] M.J. Besnus, A. Essaihi, N. Hamdaoui, G. Fischera, J.P. Kappler, A. Meyer, J. Pierre, P. Haenc, and P. Lejay, Physica B 171, 350 (1991) [165] A. Prasad, V. K. Anand, U. B. Paramanik, Z. Hossain, R. Sarkar, N. Oeschler, M. Baenitz, and C. Geibel, Phys. Rev. B 86, 014414 (2012) [166] C. Godart and L. C. Gupta, J. Less-Comm. Metals 94, 187 (1983) [167] N. G. Patil and S. Ramakrishnan, Phys. Rev. B 56, 3360 (1997) [168] M. Gamża, W. Schnelle, R. Gumeniuk, Y. Prots, A. Ślebarski, H. Rosner and Y. Grin, J. Phys.: Condens. Matter 21, 325601 (2009) [169] Y. Oduchi, C. Tonohiro, A. Thamizhavel, H. Nakashima, S. Morimoto, T. D. Matsuda, Y. Haga, K. Sugiyama, T. Takeuchi, R. Settai, M. Hagiwara, Y. Ōnuki, J. Magn. Magn. Mater. 310, 249 (2007) [170] M. F. Hundley, J. L. Sarrao, J. D. Thompson, R. Movshovich, M. Jaime, C. Petrovic, and Z. Fisk, Phys. Rev. B 65, 024401 (2001) [171] M. Szlawska and D. Kaczorowski, Phys. Rev. B 85, 134423 (2012) [172] D. Gout, E. Benbow, O. Gourdon, and G. J. Miller, J. Solid State Chem. 174, 471–481 (2003) [173] D. Kaczorowski, and A. Ślebarski, Phys. Rev. B 81, 214411 (2010) [174] E. Bauer, G. Hilscher, H. Michor, Ch. Paul, E. W. Scheidt, A. Gribanov, Y. Seropegin, H. Noël, M. Sigrist, and P. Rogl, Phys. Rev. Lett. 92, 027003 (2004) [175] T. U. Ito, W. Higemoto, K. Ohishi, K. Satoh, Y. Aoki, S. Toda, D. Kikuchi, H. Sato, and C. Baines, Phys. Rev. B 82, 014420 (2010) [176] V. Eyert, E.-W. Scheidt, W. Scherer, W. Hermes, and R. Pöttgen, Phys. Rev. B 78, 214420 (2008) [177] O. Sichevych, C. Krellner, Y. Prots, Y. Grin, and F. Steglich, J. Phys.: Condens. Matter 24 256006 (2012) [178] S. Singh, S. K. Dhar, C. Mitra, P. Paulose, P. Manfrinetti and A. Palenzona, J. Phys.: Condens. Matter 13, 3753 (2001) [179] V. K. Pecharsky, O.-B. Hyun, and K. A. Gschneidner, Jr, Phys. Rev. B 47, 11 839 (993) [180] D. D. Khalyavin, D. T. Adroja, A. Bhattacharyya, A. D Hillier, P. Manuel, A. M. Strydom,

J. Kawabata, T. Takabatake, arXiv:1311.2904 (2013)

36

Table-I

The crystal structure type, magnetic properties, effective magnetic moment (µeff),

paramagnetic Curie-Weiss temperature (θP) and ordering temperature (TC/TN) of the

magnetically ordered ternary Ce intermetallic compounds. For comparison, we have also

presented the magnetic ordering temperature of isostructural Gd compounds.

Compound Structure type TMO TN,C [K] µeff

[µB]

p

[K]

TMO, TC,N of

homolog Gd

compound

References

CeRh3B2 hexagonal F 120.8 3 -373 F, 90 [71,22]

CeCuGe hexagonal F 10 2.52 -7.92 AF, 14 [150,151

]

CePdSb hexagonal F 17 2.6 10 F, 17 [152]

CePtSb hexagonal F 4.5 2.62 -34.4 - [153]

CePdAs hexagonal F 4 2.6 -0.73 - [154]+

CeCoGa monoclinic AF 4.3 1.8 -80.5 - [155]

CeRuSn monoclinic AF 2.8 2.1 -1.9 - [156]

CePdIn hexagonal AF 1.8 2.56 -15 - [157]

CePdSn orthorhombic AF 7 2.50 -40 - [158]

CePtSn orthorhombic AF 7 2.7 -40 - [159]

CeNiC2 orthorhombic AF 20,10,2.2 2.47 -18.3 AF, 20.0 [160]

CeRuSi2 monoclinic F 11.7 1.7 -40 - [161]

CeNiGe2 orthorhombic AF 3.2,3.9 2.5 -20.8 AF, 24.5 [162]

CePdSb3

CeCoGe3

orthorhombic

tetragonal

AF

AF

3.1

21.0

2.54

2.43

-

-30.4

- [163]

[179]

CeRu2Ge2 tetragonal F 8 2.83 40.3 AF, 32, 29 [164]

CeIr2B2 orthorhombic F 5.1 2.5 -13.4 [165]

CeRh2Si2 tetragonal AF 36 2.9 72 AF, 106 [166]

CeRu2Al10 orthorhombic AF 27 3.03 -44 AF, 16 [59]

CeOs2Al10 orthorhombic AF 29 2.7 -30 AF, 18 [61]

Ce5Rh4Sn10 tetragonal AF 4.4 2.34 -8.04 - [167]

Ce5Ir4Sn10 tetragonal AF 4.2 2.09 -11.88 - [167]

CeRh2Sn4 orthorhombic AF 3.2 2.47 -22 - [168]

Ce3Rh4Sn13 cubic AF 1.2,2 2.53 -29 - [169]+

Ce3Pt4In13 cubic AF 0.95 2.64 -36 - [170]

Ce2NiSi3 hexagonal AF 3.2 2.56 -10 - [171]

Ce4Ni6Al23 monoclinic AF,F 3, 6 2.60 -225 - [172]

CeRh2Si orthorhombic AF 1.65 2.45 -65 - [173]

CePt3Si tetragonal AF 2.2 2.54 -46 AF, 15.1 [174]

*TMO: type of magnetic ordering; AF: antiferromagnetic; F: ferromagnetic

+ Single crystal data

CeOs4Sb12 cubic AF 1.6 2.41 -28 - [175]

Ce2RuZn4 tetragonal AF 2 2.57 -2.6 - [176]

Ce2Ga12Pt tetragonal AF 7.3, 5.5 2.46 -20 - [177]

CeScSi

CeScGe

Tetragonal

tetragonal

AF

AF

26.0

46.0

2.59

15.7

[178]

[178]

Table II. Lattice parameters, unit cell volume and selected interatomic distances for CeT2Al10

(T=Fe, Ru and Os) at 300K.

Compounds a (Å) b (Å) c (Å) V (Å3) Ce-Ce

(Å)

Ce-T

(Å)

Ce-

Al(Å)

(1,2,3)

CeFe2Al10 9.0159 10.2419 9.0882 839.204 5.2032 3.4467 3.1726

3.1545

3,2155

CeRu2Al10 9.1320 10.2871 9.1933 863.635 5.2604 3.4828

3.1971

3.1947

3.2486

CeOs2Al10 9.1386 10.2662 9.1852 861.744 5.2500 3.4711 3.1469

3.1867

3.2324

Fig.1 Adroja et al

EK-

EK

+

TRKKY TKondo

Fig.2 Adroja et al

Valence fluctuation

Ru Os

Fe

(Quantum critical point)

T

Al

Ce

Fig.3 Adroja et al

Fig.4 Adroja et al

RRu2Al10

ROs2Al10

CeOs2Al10

Vol

ume

(Å3 )

856.3

856.4

856.5

856.6

856.7

856.8

Temperature (K)

0 20 40 60 80

Lat

tice

para

met

ers

0.99975

0.99980

0.99985

0.99990

0.99995

1.00000

abc

Ce-

Ce

(Å)

5.2555.2605.2655.270

Ce-

Os 1 (

Å)

3.4663.4683.4703.4723.4743.4763.478

Temperature (K)

0 20 40 60 80

Ce-

Os 2 (

Å)

5.020

5.025

5.030

5.035

5.040

Fig. 5 Adroja et al

Fig. 6 Adroja et al

0 1 2 3 4

0.0

0.1

0.2 1.4 K11 K23 K

0 4 8 12

0.0

0.1

0.2

Asy

mm

etry

0.0

0.1

0.21.4 K10 K20 K

0.0

0.1

0.2

0.0

0.1

0.21.4 K10 K20 K

0.0

0.1

0.2

(a)

(b)

(c)

(e)

(f)

(g)

Ce(Ru1-xFex)2Al10

Time (µs)0 4 8 12

0.0

0.1

0.21.4 K35 K

0 4 8 120.0

0.1

0.20.04 K30.0 K

35 K

35 K

35 K

(d) (h)

x = 0

x = 0.3

x = 0.5x = 0.8 x = 1

Time (µs)

Fig. 7 Adroja et al

0 10 20 3001234

0

30

60

90

120

012345

Inte

rnal

fiel

d (G

)

0

30

60

90

120

Sigm

a ( µ

s-1)

01234

Temperature (K)0 10 20 30

0

30

60

90

120

(a)

(b)

(c)

(d)

(e)

(f)

Ce(Ru1-xFex)2Al10

Fig.8 Adroja et al

x = 0

x = 0.3

T (K)0 10 20 30

Asy

m.

0.000.050.100.15 x = 0.5

[64]

Fig. 9 Adroja et al

Fig. 10 Adroja et al

CeOs2Al10

Fig. 11 Adroja et al

CeOs2Al10

Fig. 12 Adroja et al

NdFe2Al10

Fig. 13 Adroja et al

NdFe2Al10

NdOs2Al10

Time (S)0 2 4 6 8

Asy

mm

etry

0.10

0.15

0.20

0.251.16 K1.87 K2.07 K2.27 K4.15 K

Fig. 14 Adroja et al

Fig.15 Adroja et al

NdOs2Al10A

sym

met

ry

0.00

0.03

0.06

0.09

0.12

Temperature (K)0 1 2 3 4 5

(

S-1) /

(

S-1)

0.0

0.1

0.2

0.3

0.4

(a)

(b)

Fig. 16 Adroja et al

YFe2Al10

χ(T) ~ T-0.5

C(T) ~ T-0.35

Fig. 17 Adroja et al

0.06 K

Time (S)0 2 4 6 8 10 12 14

0.0

0.1

0.2 0 G50 G1000 G2500 G

YFe2Al10ZF

0.0

0.1

0.2

0.30.06 K3.0 K

50G (LF)

Asy

mm

etry

0.0

0.1

0.2

0.06 K2.0 K

(a)

(b)

(c)

Fig. 18 Adroja et al

YFe2Al10

Asy

mm

etry

0.0

0.1

0.2

0.3

0 G50 G2500 G

Temperature (K)0 1 2 3

(

S-1)

0.00

0.01

0.02

0.03

0.04

KT(

S-1)

0.0

0.1

0.2

0.3

0.4

0 G0 G50 G2500 G(a)

(b) KT(ZF)

S M (Q

, ) (

mb/

sr/m

eV/f.

u.)

0

1

2

3Ei = 100 meV7 KQ =2.64 Å-1

x = 0.8

Energy transfer (meV)0 20 40 60 80

0

1

2

x = 1.0

(d)

(e)

0

10

20

30

-10 -5 0 5 10 15012340

2

4

6

(a)

(b)

(c)

Ce(Ru1-xFex)2Al10

Q=1.27 Å--1

Ei = 20 meV

Ei = 25 meV

Ei = 25 meV

x=0

x=0.3

x=0.5

4.5 K

4.5 K

4.5 K

Fig. 19 Adroja et al

0

5

10

15

204.5 K

(a)

0 2 4 60

5

10

15

20

|Q| (Å-1)

38 K

(b)

0

1

2

3

4

5

Ene

rgy

tran

sfer

(meV

)

Fig. 20 Adroja et al

CeOs2Al10

Γ (m

eV)

0

4

8

12

Temperature (K)0 10 20 30 40 50 60 70

Cen

ter

(meV

)

0

3

6

9

12∆

opt (

meV

)

0

10

20

30

χ (1

0-3 em

u/m

ole)

0

2

4

6

INSQEχdcINS MERLINQE MERLIN

CeOs2Al10

.... exponential____ T2

∆

Fig. 21 Adroja et al

(a)

(b)

(c)