Chemical Physics Letters 583 (2013) 109–113

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Chemical Physics Letters

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Ab-initio study of anisotropy and nonuniaxial anisotropy coefficients inPd nanochains

0009-2614/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cplett.2013.07.069

⇑ Corresponding author. Fax: +91 1905 237945.E-mail address: arti@iitmandi.ac.in (A. Kashyap).

Pankaj Kumar a, Ralph Skomski b, Priyanka Manchanda a, Arti Kashyap a,⇑a School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh 175001, Indiab Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience, University of Nebraska, Lincoln, NE 68588, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 13 May 2013In final form 26 July 2013Available online 3 August 2013

Orbital moments and magnetocrystalline anisotropies of linear chains, ladder and zigzag belts of Pd areinvestigated by first-principle calculations. The second- and fourth-order anisotropy coefficients havebeen determined, and there are generally strong nonuniaxial contributions, yielding a saddle point inthe energy landscape of the zigzag nanobelt. For some directions, spin (S) and orbital moment (L) arestrongly noncollinear. In the ladder and zigzag belts, the maximum angle between spin and orbitalmoment is about 23�, but for the monatomic chain, we find an unexpected continuous change in theangle, covering the whole range between 0� and 180�.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

The relation between spin and orbital moment is undoubtedlyone of the most intriguing relationships in solid-state physics[1,2]. It indirectly determines the magnetic an isotropy of perma-nent magnets, data storage media, and soft magnets [3,4] and isalso important in other areas of science and technology, such asspin electronics and magnetic dichroism [5–13]. For example, theanisotropic magnetoresistance is an orbital moment effect. Re-newed interest in the topic has also been sparked by the tighteningrare-earth supply [3]. Magnetocrystalline anisotropies are well-investigated for many bulk and nanostructured systems includingnanowires [12,14–22]. Less attention has been devoted to theunderlying orbital moment, although the relationship betweenorbital moments and anisotropy has long been known andexploited in circular dichroism (XMCD) [7,8].

Magnetocrystalline anisotropy is defined as the dependence ofthe magnetic energy on the magnetization direction, that is, asthe angular variation of the magneto crystalline an isotropy energy(MAE). Spin–orbit coupling (SOC), parameterized by the couplingconstant n, creates an orbital moment with an electric charge dis-tribution, and the corresponding ‘crystal-field’ interaction energydepends on the spin direction [6,23]. Concerning the relationshipbetween spin, orbital moment and anisotropy, two limits havebeen well understood for a long time. First, iron-series transition-metal elements have a rather small SOC (n � 0.05 eV) and can betreated in fair approximation by perturbation theory [1,2]. Inlowest (second) order the corresponding SOC induced MAEexpression is

dEðh;/Þ ¼ �n2X0;u

j < oðkÞjS � LjuðkÞ > j2

Eu � E0ð1Þ

Where S = ½ ⁄r is the spin angular momentum operator, containingthe vector r of the three Pauli matrices, and L is the orbital-momentoperator. The states hoðkÞjandjuðkÞi are the occupied and un occu-pied k-states, respectively [24–25]. This approximation correspondsto the nearly completely quenched limit of small spin induced orbi-tal moments n=W , where W is the band width, whereas the anisot-ropy scales as n2=W [1,26]. However, degeneracies, Eo � Eu in Eq.(1), require particular care and nonperturbative diagonalizationmethods are preferable even for 3d systems [27]. The second limitis very heavy elements, most notably rare-earth elements and com-pounds [4,23,28]. In this limit, the orbital moment is large but rig-idly coupled to the spin by Hund’s rules and therefore somewhatless interesting from the perspective of solid-state physics.

Our focus is on the intermediate case. Such 4d/5d structurescombine itinerant features with a relatively strong SOC, of the or-der of 0.2 eV (Pd) to 0.5 eV (Pt). We consider Pd, which can be con-sidered as a typical element with intermediate SOC, and study Pdnanowires. Palladium is a well-known exchange-enhanced Pauliparamagnet in its elemental bulk form [4,26] but gets ferromag-netic in some nanostructures.

In general, the net spin–orbit coupling may have either sign,corresponding to ferromagnetic and antiferromagnetic alignmentsbetween spin and orbital moments [4,29]. Together with the pos-sible existence of two or more sublattices (or atomic species), thisgives rises to a variety of spin-reorientation transitions. Typically,these transitions are of the first-order spin-reorientation type, thatis, realized by jumps in the moment orientation [4]. Aside fromthis, it is well-known that spin and orbital moments are somewhatnoncollinear unless they point along the principal axes, but this is

110 P. Kumar et al. / Chemical Physics Letters 583 (2013) 109–113

generally believed to be a relatively small effect, even in 4d and 5dsystems. In terms of (1), < S > and < L > are not necessarily paral-lel or antiparallel. This leads to the question how L and S behave inintermediate regime, with respect to both noncollinearity andfirst-order phase transitions.

A second aspect of this Letter is the real-space symmetry of thewires. Much work has been devoted to the anisotropies of 4d and5d systems with uniaxial anisotropy, such as thin films [30] andlinear monatomic wires [17–18,29–35]. However, such linearwires are scientifically interesting but difficult to realize in practice[36], while real atomic nanowires are typically supported, forexample by a substrate [22,37–40]. The main difference is the re-duced symmetry of the wires, which leads to qualitative changesin orbital moment and anisotropy. These changes can already beseen by considering freestanding wires with reduced symmetry.Note that the symmetry breaking investigated in the Letter persistson substrates even if quantitative predictions are substrate-depen-dent. In a slightly different context, this question has been dis-cussed by Smogunov et al. [32].

Figure 1 shows the uniaxial and nonuniaxial Pd nanowires con-sidered in this Letter. By going from linear nanowires to belt andzigzag nanowires, we break the axial symmetry, with the corre-sponding far-reaching consequences for orbital moment, spin andanisotropy. The wires are cut from bulk palladium, which gives riseto the factor

ffiffiffi2p

in Figure 1, and their periodicities are then relaxedwithout additional geometrical changes. We have also calculatedthe behavior for some distorted nanowires ðdh–

ffiffiffi2p

dtÞ but did notfind any qualitative changes.

In this Letter, we investigate the magnetic energy as a functionof the magnetization angle (magnetocrystalline anisotropy) andthe underlying relationship between spin and orbital-momentdirections.

2. Method

Our first-principle calculations use an accurate frozen-core pro-jector augmented plane-wave method within the framework ofdensity functional theory, as implemented in the Vienna Ab-InitioSimulation Package (VASP) [41,42]. Relativistic pseudopotentialswere employed, with exchange and correlation described by Per-dew, Burke and Ernzerhof (PBE), using a generalized gradientapproximation (GGA) [43]. Compared to LDA, the GGA somewhatoverestimates the lattice volume, which may lead to incorrect pre-

Figure 1. The free-standing Pd structures considered in this Letter: (a) linear chain, (b) laffiffiffi2p

.

dictions of the magnetism, for example in bulk Pd close to the on-set of ferromagnetism [44–46]. However, here we investigatestructures having substrate-related geometrical constraints, andeven for bulk Pd, small changes in lattice constants yield ferromag-netism in the LDA.

The chains are modeled by a standard supercell approach,ensuring that neighboring chains do not interact [16]. The spin–or-bit coupling, which must been added to determine orbital momentand anisotropy, is implemented in a noncollinear mode [47–48]. Toobtain sufficiently accurate energy values, a convergence limit of10�7 eV was used, and 801 k-points were generated in the irreduc-ible part of Brillouin zone, using the Monkhorst–Pack scheme [49].

3. Results and discussion

The lattice constants have been obtained by optimizing thebond lengths of the chains (Figure 1) and correspond to periodici-ties of dh = 2.47 Å (a), 3.20 Å (b) and 3.48 Å (c). To fix the spin an-gles h and /, we use a VASP penalty function, ensuring that thepenalty contribution to the total energy is negligible (<10�12 eV).We have considered 72 different angles (h, /) for each of thechains, although some of the angles are equivalent by symmetry,especially for the linear chains.

Figure 2 shows the corresponding orbital-projected densities ofstates (DOS). Comparing linear wires (a) with ladders and zigzagwires (b–c), we see that the breaking of the axial symmetry affectsthe electronic structure of the wires by removing the degeneracy ofthe m > 0 pairs of 3d orbitals, namely dxz and dyz (m = 1) and dxy

and dx2

–y2 (m = 2). Since the orbital moment and the anisotropy

are caused by spin–orbit matrix elements connecting 3d orbitalsabove and below the Fermi level, this creates a rather complicatednonuniaxial anisotropy.

First, we have calculated the magnetocrystalline anisotropyenergy (MAE). Figure 3(a) defines the angles h and / with respectto the nanowire orientation. The corresponding magnetocrystal-line anisotropy is often parameterized in terms of anisotropy con-stants KðmÞn , which refer to energy contributions of the typesin2n h cosð2m/Þ. From symmetry considerations [4,18,26], oneexpects not only uniaxial anisotropy constants, such as K1 and K2

(m = 0) but also sig nificant nonuniaxial contributions, for exampleK01 (m = 1).

The anisotropy constants jðmÞn are convenient in micromagne-tism but are non-orthogonal and even mix anisotropy contribu-

dder and (c) zigzag belt. The distances dh are the lengths of the unit cells and dt = dh/

Figure 2. Orbital-projected spin-polarized density of states for (a) linear chains, (b) ladder and (c) zigzag chains.

Figure 3. Moment angles with respect to the orientation of the nanobelts. For the linear chain, the X- and Y-directions are equivalent by symmetry. The path I-II-III is used toplot spin and orbital moments. (b) Energy landscape E(h, /) for the zigzag nanobelt in eV/atom. Cold (bluish) and warm (brownish) colors indicate energy minima andmaxima, respectively. Note the saddle point at h = 0�, that is, parallel to the wire axis. (For interpretation of the references to colour in this figure legend, the reader is referredto the web version of this article.)

Table 1Anisotropy coefficients for linear, ladder and zigzag chains of Pd.

Anisotropy coefficients Linear chain(leV)

Ladder(leV)

Zigzag belt(leV)

j20 2874 2359 �313.6

j22 0 �2906 �977.6

j40 �125.9 �34.52 �16.52

j42 0 �37.53 12.37

j44 0 �88.9 �98.5

P. Kumar et al. / Chemical Physics Letters 583 (2013) 109–113 111

tions of different orders [4]. Physically, it is more natural to useanisotropycoefficients jm

1 [4–14], obtained by expanding the anisot-ropy-energy density g in terms of real combinations the sphericalharmonics

Yml ðh;/Þ ¼ ð�1Þm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2lþ 1ðl�mÞ!

4pðlþmÞ!

sPm

l ðcos hÞeim/ ð2Þ

Using normalization factors that maintain maximum similaritywith the anisotropy constants [14], this yields the second- andfourth-order anisotropy-energy expressions

g2 ¼13j0

2ð3 cos2 h� 1Þ þ j22 sin2 h cos 2/ ð3aÞ

and

g4 ¼15j0

4ð35 cos4 h� 30 cos2 hþ 3Þ þ j24ð1� 7 cos2 hÞ

� sin2 h cos 2/þ j24ðsin4 h cos 4/Þ ð3bÞ

To determine the jm1 , we have performed least-square fits using

g = g2 + g4 and the 72 first-principle data points per chain. Table 1shows the extracted anisotropy coefficients. Note that 2nd-and 4th-order anisotropy coefficients, Eq. (3), are sufficient to

reproduce the energy landscape very accurately. The inclusion of6th-order terms does not yield further improvements, partiallydue to the limited accuracy of the first-principle calculations(about 20 leV). This value, which should not be confused withthe numerical convergence limit of Section 2, also serves as a mea-sure for the accuracy of the fitting. No E(h, /) singularities due toquantum–mechanical level crossings have been detected, but thismay be due to the relatively low angular resolution (72 datapoints). Experimentally, nonuniaxial anisotropy constants such asj0

2 and j22 can be determined, in principle, from hysteresis loops

along the three principal anisotropy axes.

112 P. Kumar et al. / Chemical Physics Letters 583 (2013) 109–113

Figure 3(b) shows a typical nonuniaxial energy landscape,namely that of the zigzag belt. The easiest magnetization directionis in the X-direction, that is, in the belt plane but perpendicular tothe chain. The intermediate axis is in the Z-direction parallel to thechain, which gives rise to a saddle point at h = 0�. The ladder beltexhibits the same hierarchy (easy X, intermediate Z, and heavy Yaxes), whereas the linear chain has easy-plane anisotropy, the pref-erential magnetization direction being in the X–Y plane perpendic-ular to the wire axis.

Let us now return to the relation between spin and orbital mo-ment. For all three nanowire geometries, we have calculated themagnitudes of the spin and orbital moment and the orbital-mo-ment direction. The spin and orbital moments as well as the orbi-tal-moment direction vary continuously as a function of the spindirection. Figure 4 visualizes the numerical results along the pathI-II-III of Figure 3(a).

The top row (a–c) shows the magnitudes of the spin and orbitalmoments, whereas the bottom row (d–f) displays the misalign-ment angle between spin and orbital moment. For the linearchains, the curves are flat between the X and Y directions, that is,for the physically equivalent spin directions perpendicular to thechain. Quantum–mechanically, this corresponds to the degeneracybetween the dxy, dx

2–y

2 and dyz, dxz orbitals, Figure 2(a), which getslifted in the ladder and zigzag belts, Figure ure2(b–c). The inclusionof spin orbit coupling (SOC) suppresses the spin moment. The sup-pression of the spin moment is directionally dependent. Withoutspin–orbit coupling, the respective spin moments are 0.67, 0.59and 0.39 lB/atom for the linear, ladder and zigzag chains. Notethat, the suppression of the spin moment is much larger for the lin-ear chain along the chain direction as compared to the ladder andzigzag ones as can be seen from Figure 4. However, the spin valuesdon’t get much affected in a direction perpendicular to the chains.Inclusion of SOC, gives the orbital moment of 0.126, 0.214 and0.217 lB/atom for linear chain, ladder and zigzag chains, respec-

Figure 4. Spin and orbital moments of the Pd chains as a function of the spin angle: (a–c)orbital moment. The points X, Y, and Z, as well as the path I-II-III, are explained in Figur

tively. Large orbital moment in case of zigzag chains can be ac-counted to the lowest symmetry in zigzag chains followed byladder and linear ones.

Figure 4(d–f) shows the angle between the spin and the orbitalmoment. Along the principal axes (X, Y, and Z), the spin and orbitalmoments are parallel or antiparallel to each other, but for arbitraryspin directions the spin and the orbital moment are noncollinear.Table 1 shows the spin and orbital moments of the chains alongthe principal axes. There are two basic scenarios. When the spinaxis is parallel to the chain axis, the orbital moment aligns antipar-allel with the spin moment for the linear chains and parallel forladder and zigzag belts. For the spin axis being perpendicular tothe chain axis, the orbital moments align parallel to the spin mo-ment for all three configurations. For intermediate spin directions,away from the principal axes, the spin and the orbital moment arenoncollinear. This noncollinearity can be inferred from the fact thatbonding is anisotropic and an anisotropic charge cloud would giveorbital moments which will be different for different quantizationdirections determined by the balance between the spin–orbitinteractions and chain/belt field effects. The linear chain, Figure4(d), exhibits an antiparallel spin–orbit alignment in its easy-planestate (X–Y plane), but forcing the spin into the direction of the wireaxis (Z axis) changes the alignment to parallel. Interestingly, thischange is not realized in form of an abrupt switching but occursgradually, so that the misalignment angle covers the whole rangefrom 0� to 180�. The zigzag and ladder belts exhibit parallelspin–orbital alignment along all principal axes, but for inter medi-ate directions, the orbital and spin moments are noncollinear, withmaximum misalignment angles of about 23�.

Our result goes beyond previous work [18] by focusing onthe angular dependence of the orbital and spin magnetic moments,as contrasted to a focus on structural angular relaxation. It isinstructive to keep in mind that the belt chains considered hereand in Ref. [18] may be difficult to prepare experimentally, but

magnitude of spin and orbital moment and (d–f) lead angle of spin moment from thee 3(a).

P. Kumar et al. / Chemical Physics Letters 583 (2013) 109–113 113

they are more realistic than linear chains. In fact, experimentalrealizations with fixed bond angle may be possible by depositionon a substrate, and these wires will have symmetries similar tothose considered here.

4. Conclusions

In summary, we have investigated the orbital and spin mo-ments of linear and nonuniaxial (zigzag and ladder) chains of Pdatoms. Aside from calculating higher-order and non uniaxialanisotropy coefficients, our emphasis has been on the relation be-tween spin (S) and orbital moment (L). When the spin is parallel toa principal axis, then S and L are parallel or antiparallel but collin-ear. For intermediate directions, S and L are noncollinear. For theladder and zigzag chains, the maximum misalignment angle isclose to 23�, but in the linear monatomic chain, the angle betweenL and S changes continuously rather than abruptly between 0� and180�.

The present research shows how nanostructuring can be used tocontrol and tune the orbital moment and its relationship to magne-tization and magnetic anisotropy. This is important for variousapplications, such as permanent magnetism and spin electronics.An important practical challenge in permanent magnetism is thatabout 50% of all uniaxial magnetic materials are of the easy-planetype and therefore not suitable for permanent-magnet application,even if the magnitude of the anisotropy is very high [4]. Lowest-or-der nonuniaxial anisotropies, such as those considered in this Let-ter, effectively ‘kill’ this easy-plane feature and potentially enhancethe range of suitable permanent-magnet materials. However, mag-netic anisotropy is a necessary but not sufficient condition forinformation storage, because one-dimensional structures of atom-ic-scale cross-sections are very susceptible to thermal excitations.The practical realization of the effect requires broader nanobeltsand substrates, such as copper. However, this yields moderatequantitative changes only but leaves the core physics of this Letter(symmetry breaking) unaffected. [50].

In spintronics, the present mechanism allows the discreteswitching of the magnetization direction in fairly low magneticfields. This makes it possible to tune the conductance, becausethe anisotropic magnetoresistance is pro portional to the orbitalmagnetic moment.

Acknowledgments

This work has been supported by DST (PKS, AK), NSF-MRSEC(RS), and NCMN. Thanks are due to S.S. Jaswal, A.K. Solanki, A. En-ders, and D.J. Sellmyer for discussing various details.

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