Magnetic, magneto-optical, and structural properties of URhAl from first-principles calculations

PHYSICAL REVIEW B, VOLUME 63, 205111

Magnetic, magneto-optical, and structural properties of URhAl from first-principles calculations

J. Kunesˇ and P. Nova´kInstitute of Physics, Czech Academy of Sciences, Cukrovarnicka´ 10, CZ-162 53 Prague, Czech Republic

M. DivisDepartment of Electronic Structures, Charles University, Ke Karlovu 5, CZ-121 16 Prague, Czech Republic

P. M. OppeneerInstitute of Solid State and Materials Research, P.O. Box 270016, D-01171 Dresden, Germany

~Received 10 October 2000; published 2 May 2001!

We present a first-principles investigation of the electronic properties of the intermetallic uranium compoundURhAl. Two band-structure methods are employed in our study, the full-potential augmented plane-wave~FLAPW! method, in which the spin-orbit interaction was recently implemented, and the relativistic, non-full-potential, augmented-spherical-wave method. To scrutinize the relativistic implementation of the FLAPWmethod, we compare the spin and orbital moments on each atom, as well as the magneto-optical Kerr spectra,as calculated with both methods. The computed quantities are remarkably consistent. With the FLAPW methodwe further investigate the magnetocrystalline anisotropy energy, the x-ray magnetic circular dichroism at theuraniumM4,5 edge, the equilibrium lattice volume, and the bulk modulus. The magnetocrystalline anisotropyenergy is computed to be huge, 34 meV per formula unit. The calculated uranium moments exhibit an Ising-like behavior—they almost vanish when the magnetization direction is forced to lie in the uranium planes. Theorigin of this behavior is analyzed. The calculated optical and magneto-optical spectra, and also the equilibriumlattice parameter and bulk modulus, are found to compare well to the available experimental data, whichemphasizes the itinerant character of the 5f ’s in URhAl.

DOI: 10.1103/PhysRevB.63.205111 PACS number~s!: 71.20.2b, 71.28.1d, 75.30.Gw, 78.20.Ls

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I. INTRODUCTION

The group of ternary uranium intermetallics with compsition UTX, whereT is a transition metal andX a p element,has recently attracted attention~for a review, see Ref. 1!.Most of the intermetallics of this composition crystallizethe hexagonal ZrNiAl structure~sometimes called the Fe2Pstructure!, which contains three formula units per unit ceThe ZrNiAl structure has a layered structure, consistingplanes of uranium atoms admixed with one-third of theTatoms, that are stacked consecutively along thec axis, whiletwo adjacent uranium planes are separated from one anoby a layer consisting of the remainingT atoms and theXatoms; see Fig. 1. The uranium interlayer exchange coupis relatively weak and depends sensitively on the specifiTand X elements, which gives rise to a variety of magnebehaviors observed in the UTX compounds.1 Some of theUTX intermetallics are ferromagnets, such as, e.g., UP~Ref. 2!, while others exhibit unusual antiferromagnestructures, such as, e.g., UNiGa~Ref. 3!. A metamagnetictransition from a paramagnetic state to a ferromagnetic shas been observed4,5 in UCoAl in magnetic fields of only0.5 T. Also URhAl, which becomes ferromagnetic atTC527 K has received attention.1,6–10

One of the key questions to be addressed when discusactinide compounds is the degree of localization of thefelectrons, which may range from nearly localized to pracally itinerant, depending on the specific compound.11,12

Since the 5f electrons are simultaneously involved in thchemical bonding and magnetism, a broad variety of phcal properties may emerge from the degree of 5f localiza-

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tion. Also URhAl has been considered in this respect. Inpolarized neutron study, magnetization-density profiles wmeasured which supplied evidence of a high degree ofisotropic hybridization between the 5f and the Rh 4dorbitals.8,9 A sizable induced moment of 0.28mB on the Rhatom within the basal uranium plane was detected, wherinterestingly, only a very small induced moment of 0.03mB

was detected on the equally close Rh site out of the pla8

Later inelastic neutron-scattering experiments, howevfound a peak at 380 meV, which was interpreted as thenature of an intermultiplet transition,10 promoting thus thelocalized picture. The 380-meV peak occurred at the saenergy where a uranium intermultiplet transition wobserved13 in UPd3, which is one of the uranium compoundwhere the 5f electrons are undoubtedly localized.

A very unusual property that was reported for URhAlthe enormous magnetocrystalline anisotropy.1,6 The suscep-

FIG. 1. The ZrNiAl-type unit cell, which is adopted by numeous UTX intermetallic compounds, withT a transition-metal ele-ment andX a p element.

©2001 The American Physical Society11-1

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J. KUNES, P. NOVAK, M. DIVIS , AND P. M. OPPENEER PHYSICAL REVIEW B63 205111

tibility measured at 4.2 K for a field in the basal hexagonplane was very different from that for a field along thec axis.Typically the former was identical to that of paramagneticcompounds of the same structural group, such as,URuAl.1 Even for fields up to 35 T only a moment of abo0.1mB per formula unit could be induced in the basal planA linear extrapolation of the measured induced moment4.2 K would lead to an estimated field of 1500 T neededrotate the moment into the basal plane. At elevated temptures the magnetization can be forced to lie in the basal pwith much smaller fields. However, it was observed thatmagnetization vanished when it was forced to lie in-plane14

The large anisotropy in the induced Rh moments that wobserved in the polarized neutron study8,9 clearly witnessesthe anisotropy of the U(5f )-Rh(4d) hybridization: A stronghybridization occurs between the valence orbitals of theand Rh atoms within the basal plane, but the hybridizatbetween the valence orbitals of the U atom and those ofequally close Rh atom in the adjacent plane is much sma

The aim of the present study is to investigate the eltronic properties of URhAl on the basis of first-principlecalculations. For delocalized 5f electrons, the band-structurapproach, based on the local-spin-density approxima~LSDA!, is expected to provide an appropriate descriptiwhereasf-electron materials containing localizedf electronstend to be better explained by specifically adaptedproaches, such as, e.g., the LSDA1U approach.15,16 Twoother electronic structure calculations for URhAl were cried out recently.15,17 These indicated, first, that the bondinand magnetism are governed by the U 5f –Rh 4dhybridization,17 and, second, that the calculated magneoptical Kerr spectrum15—based on the assumption of delcalized 5f ’s—compares reasonably well to the experimenKerr spectrum. In the present paper we address, in particthose electronic properties which have not been explaipreviously. These are the magnetic moments that were msured with polarized neutrons8,9 and with x-ray magnetic cir-cular dichroism,14 the enormous magnetocrystalline anisropy energy~MAE! of approximately 41 meV per formulaunit ~cf. Ref. 1!, the measured x-ray magnetic circular dchroism~XMCD! spectrum14 at the uraniumM4,5 edges, andthe equilibrium lattice parameter and bulk modulus.18

In the present electronic structure investigation we eploy the full-potential linearized-augmented-plane-wa~FLAPW! method, as implemented inWIEN97 code,19 whichhas recently been extended to include the relativistic sorbit coupling by one of us.20 As the spin-orbit~SO! imple-mentation is rather new, we compare our results for tespurposes to those obtained by the relativistic augmenspherical-wave~ASW! method,21 which is based on thespherical potential approximation. The moments as coputed with both methods are remarkably consistent, asthe magneto-optical Kerr spectra.

As we will point out in detail below, the comparison omost of the calculated and experimental properties poundoubtedly to relatively delocalized 5f states in URhAl.One of the most interesting findings of our study is thatmagnetism of URhAl is computed to be highly anisotropThe magnetic moment of uranium, which is certainly pres

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for magnetization along the hexagonal axis, behaves inIsing-like manner,22 i.e., it practically disappears when it iforced to lie perpendicular to it, in accordance with expemental observations.14 One of the consequences is a hucomputed MAE, amounting up to 34 meV per formula unExperimentally such an anisotropic magnetic behavior wfrequently observed for UTX compounds crystalyzing in theZrNiAl structure ~cf. Ref. 1!, but to our knowledge it hasnever been studied theoretically until now.

In the following, in Sec. II we briefly describe the implementation of SO interaction in the FLAPW scheme. We thcompare the spin and orbital moments, as well asmagneto-optical Kerr effect~MOKE! spectra, as computewith the FLAPW and ASW schemes. Subsequent to the ting of the relativistic FLAPW code, in Sec. III we preseour results for the equilibrium lattice parameter, the bumodulus, the XMCD spectrum, and the MAE. Conclusioare formulated in Sec. IV.

II. COMPUTATIONAL METHOD

A. Implementation of the SO interaction

Here we briefly outline the treatment of the SO interactiin the FLAPW scheme, which is a crucial element in ostudy of URhAl. The FLAPW method, as implementedthe WIEN97 code,19 was used in the present work. The stadard LAPW basis with wave functions expanded into sphecal harmonics inside the atomic spheres and plane wavethe interstitial space, is employed~see, e.g., Ref. 23!. Thestarting radial basis functions inside the atomic spheresobtained as scalar-relativistic solutions of the Dirac equatin the spherical part of the effective potential. The SO copling is included subsequently via the second variational sapproach.24 In the first step of this approach, the scalarelativistic part of the Hamiltonian is diagonalized on a baadopted for each of the spin projections separately. Insecond step the full Hamiltonian matrix is constructed onbasis of eigenfunctions of the first step Hamiltonian. Onlylimited number of eigenstates of the first step Hamiltoniannecessary to construct the basis used in the second stethe present implementation the size of the second step bis controlled by an energy cutoff, so that its influence upthe results can be checked. The main approximation ofapproach comes from the fact that the initial basis functioare constructed from the scalar-relativistic Hamiltonianstead of the fully relativistic Hamiltonian. This does not leto substantial errors, because the second variational apprhas been shown to yield results that are in good agreemwith those of fully relativistic calculations.25

We neglect the SO coupling in the interstitial space. Alfor the SO coupling we consider only the spherical partthe effective potentialV in the atomic spheres. The SO paof the Hamiltonian in the spheres is then given by

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MAGNETIC, MAGNETO-OPTICAL, AND STRUCTURAL . . . PHYSICAL REVIEW B 63 205111

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The contribution of a given atomic sphere to the SO maelement is given by

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almsk( i ) are the expansion coefficients of the functioni inthe spherical harmonics basis. Indicesl , m, and m8 corre-spond to the orbital quantum numbers,s ands8 run over thetwo spin projections, andk andk8 denote the radial functionfor a given expansion energy, the energy derivative ofradial function and the local orbital radial function~if used!.Rls is the large component of the corresponding radfunction.24 For further details, we refer to Ref. 20.

For magnetic systems the direction of the magnetizawith respect to the crystal coordinate system is treated ainput parameter. It is assumed that the exchange field isdirectional, i.e., the arrangement of the spin momentseach atomic site is collinear. The selectable magnetizadirection is facilitated by rewriting the scalar productl•sas26

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whereu and f are azimuthal and polar angles of the manetization in the rectangular crystal coordinate system. Tfeature allows us to study the dependence of the total enon the exchange-field direction, which is a common methto determine the easy magnetization axis and MAE.27

B. Symmetry considerations

In magnetic systems the spin-orbit coupling causes rtions in the coordinate and the spinor space to be no lonindependent as they are for the scalar-relativistic Hamtonian. The symmetry is thereby reduced to at mosuniaxial one along the exchange-field direction, which mpossibly increase the number of crystallographic nonequlent atomic positions. A second consequence is that theinversion and operations inverting the exchange field belno longer to the symmetry group of the Hamiltonian, hoever, their combinations still do. This must properly be takinto account when symmetry is employed to generate eigfunctions in the symmetry coupledk points. In the case oURhAl, whose Fe2P crystal structure has no inversion symmetry, this aspect is of particular relevance. An immedi

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outcome of the symmetry considerations is that, in systewithout inversion symmetry, thek-space symmetry groupdiffers from the point group of the effective potential. Thisillustrated by the example

Rteik•rfk~r !5e2 iRk•rfk* ~R21r !5e2 iRk•rf2Rk~r !, ~5!

whereRt5Rst is a combined operation of a space rotatiR and the time inversiont, and the spinor part is not showfor the sake of simplicity. The consequence of Eq.~5!—forsystems without inversion symmetry—is that, whileR be-longs to the symmetry group of the charge density andpotential, it is 2R that belongs to thek-space symmetrygroup.

III. TESTS AND RESULTS

A. Numerical aspects

In all FLAPW calculations, a regular sampling of the Brilouin zone with 838k points was used while the sets oirreduciblek points were chosen according to the symmeimposed by the crystal structure and the exchange-fieldrection. For the exchange field along thec axis, an irreduc-ible part of 1/12th Brillouin zone was used, which was elarged to the 1/4th Brillouin zone when the exchange fiwas along thea axis, corresponding to the reduced symmeof the system. The LSDA exchange-correlation paramterization of Perdew-Wang~Ref. 28! was adopted. We usedtotal amount of about 1260 basis functions. The uraniumsand 6p states and rhodium 4p and aluminum 2p states weretreated as local orbitals. Approximately 200 functions weused as a basis for the second variational step HamiltonWe found that an energy cutoff, which determines the sizethe second step basis, of about 1 Ry above the Fermi land the orbital momentum cutoffl 54 in Eq. ~3! are suffi-cient for a calculation of the ground-state properties in msystems.

B. Magnetic moments

To probe the SO implementation, we compute the sand orbital moments on each atom in URhAl, and compthese to moments computed with the ASW scheme. Theculated moments on the uranium site and on the twoequivalent rhodium sites for the magnetization along thcaxis, as obtained by the two methods, are shown in TabIn the ASW calculation 288k points in the 1/12th irreduciblepart of the Brillouin zone were used. The SO couplingtreated in a second variational manner in the ASW scheFor the exchange-correlation potential the parametrizatiovon Barth and Hedin29 has been applied. Theab initio com-puted moments are found to be remarkably consistentspite of the quite different construction of the FLAPW anASW basis sets and the spherical-potential versus fpotential approach. The difference per computed individmoment is less than 0.044mB . The spin moments obtaineby the FLAPW method are slightly less than those of tASW method, which can be explained by different spheradii being used. The FLAPW atomic spheres do not overand there is an interstitial contribution to the spin moment

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J. KUNES, P. NOVAK, M. DIVIS , AND P. M. OPPENEER PHYSICAL REVIEW B63 205111

TABLE I. The spin magnetic moments (Ms) and orbital moments (Ml) of the individual atoms in URhAlin mB , as calculated by the FLAPW and ASW methods. For comparison the experimental momeobtained from neutron diffraction experiments~Ref. 8! and the XMCD spectrum~Ref. 14! are also given.

ASW FLAPW Neutr. Expt. XMCDMs Ml Ms Ml Ms Ml Ml

U 1.238 21.629 1.215 21.585 1.16 22.10 21.6360.14Rh~I! 20.032 0.014 20.048 0.009 0.28Rh~II ! 20.043 20.010 20.074 20.012 0.03

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0.379mB per unit cell, while the ASW atomic spheres fill thspace entirely and the whole spin density is distributedtween them. The total spin moments per formula unit otained by the two methods differ by 0.077mB . The goodagreement obtained for the moments supports the faultlness of the SO implementation in the FLAPW code. Tfurther illustrates that, for this relatively closed packed mterial, the nonspherical potential does not have a significinfluence upon the moments.

A previous calculation17 of the spin and orbital momentof URhAl yielded very different values: a 5f spin ~orbital!moment of 0.26mB (20.10mB , respectively! on uraniumwas obtained, and an averaged spin ~orbital! moment of20.02mB (20.003mB , respectively! on Rh. These momentdo not compare favorably with our calculated moments,with the experimental moments. It was therefore proposeRef. 17 that a high-moment state exists in URhAl, which ha total energy close to the low-moment state.

In Table I we also list the experimental moments astermined from neutron-diffraction experiments8 and from theXMCD spectrum,14 using the XMCD sum rule for the orbitamoment.30 While the two calculations are in close correspodence with one another, the same cannot be said of theexperiments, for which the orbital moments differ by 0mB . The origin of this discrepancy in the uranium orbitmoment is unknown. However, it should be mentioned tthe XMCD sum rule is based on an atomic model, in wha theoretical 5f occupation number enters by its applicatioFrom the XMCD spectrum a high ratio of the orbital anspin moment was deduced, which was put forward asdence of fairly localized 5f states.14 The neutron-diffractionexperiments, on the other hand, yielded a smaller ratio,consequently, it was concluded that there is substantiafhybridization.8 The calculated spin moment on U comparwell to that obtained from neutron scattering, but not to twhich follows from the XMCD spectrum. From the macrscopic total moment7 of 0.94mB a spin moment of only(0.6960.14)mB was obtained in Ref. 14, which is consideably smaller than the computed uranium spin moment. Tcalculated orbital moment seems, at first sight, to be closthe XMCD orbital moment, but the common experiencetablished by LSDA calculations for actinide materials is ththe LSDA underestimates the orbital moment.12

To correct the underestimation of the orbital momentthe LSDA, the orbital polarization~OP! correction has beenproposed.31 The additional OP term in the Kohn-Sham eqution enlarges the orbital moment,12 and we anticipate thaapplication of the OP to URhAl would bring the computed

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orbital moment in better agreement with that measured inpolarized neutron study. Future investigations with a prently being developed OP code are envisaged.

The neutron-diffraction experiments8,9 allocated a ratherlarge induced magnetic moment on the Rh~I!, which is theRh atom within the uranium plane~see Fig. 1!. The largeinduced moment of 0.28mB was considered as evidenceU 5 f –Rh 4d hybridization.8 A similarly large Rh~I! momentdoes not follow from our calculations. Instead, both Rhoms are computed to be only slightly polarized with a sppolarization opposite to that of the uranium~see Table I!. Wecan not explain this difference at present. One aspectdeserves to be mentioned is that an additional negativemoment contribution of20.11mB has been observed expermentally, which was attributed to conduction electron8

While apparently further studies of URhAl are desirable, opresent calculations at least provide consistent values formoments given by LSDA band-structure theory.

C. Magneto-optical Kerr effect

A second, experimentally accessible quantity that depecrucially on the SO interaction is the MOKE.32 Themagneto-optical Kerr rotationuK and Kerr ellipticity«K arerelated to the optical conductivity tensorsab via the expres-sion

uK1 i eK'2sxy

sxxS 11

4p i

vsxxD 21/2

, ~6!

which is valid for Kerr angles and Kerr ellipticities of up ta few degrees.33 The conductivity tensor can conveniently bcalculated from its linear-response theory expression:

sab~v!5ie2

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MAGNETIC, MAGNETO-OPTICAL, AND STRUCTURAL . . . PHYSICAL REVIEW B 63 205111

to the optical interband conductivity spectra as given by~7!, an intraband contribution of the Drude form shouldadded to the diagonal elements of the conductivity ten~see, e.g., Ref. 34!. For technical details of the evaluation othe conductivity tensor, we refer to Refs. 34 and 35.

The MOKE spectra, as obtained by the FLAPW and ASmethods, are compared to the experimental data36 in Fig. 2.In the FLAPW calculation we used the same number okpoints as in the self-consistency procedure, whereassecond-variational-step energy cutoff was increased2.5 Ry above the Fermi level, in order to describe betterrelevant high-energy states. We checked that increasingnumber ofk points by a factor of 4, and raising the enercutoff to 4 Ry above the Fermi level, left the spectra virtuaunchanged. The intraband Drude contribution with plasfrequency\vP53.1 eV ~calculated! was added to the diagonal elements of the conductivity tensor. Lifetime broadeings d50.4 and 0.6 eV were used for the interband aintraband transitions, respectively. Figure 2 shows thagood agreement between the MOKE spectra of the twoferent computational schemes is present over the wholeergy range. Consequently, the studied SO-sensitive quais identically reproduced by bothab initio schemes. Below4-eV photon energy, the theoretical MOKE spectra satisftory describe both the shape and magnitude of the measspectra. Above 4 eV there occurs a deviation of the calated spectra from the experimental ones of Kucˇera et al.36

The gratifying correspondence of experimental MOKE aLSDA calculations applicable for delocalized 5f ’s, suggestsa relatively delocalized nature of the 5f ’s in URhAl. Opticalspectra that are computed adopting localizedf states are gen

FIG. 2. The polar Kerr rotationuK and Kerr ellipticity «K formagnetization direction along thec axis of URhAl. The experimen-tal data~Ref. 36! are given by the symbols, the theoretical specas calculated by the FLAPW scheme by the solid line, and thcalculated by the ASW scheme by the dashed line.

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erally quite different from spectra computed for the samaterial, but assuming delocalizedf states.37

D. Equilibrium volume and bulk modulus

Ground-state properties, such as the lattice parameterbulk modulus, are generally expected to be adequatelyscribed by the LSDA density-functional formalism. Also foactinide compounds this is expected, provided the itinerapproach to the 5f ’s is warranted. We have determined thequilibrium volume and bulk modulus of URhAl by minimizing the total energy. For URhAl this is a computationademanding task, because the unit cell contains natoms and there are three free internal parame@c/a,x(U),x(Al) #. In order to reduce the numerical efforthe total energies for a given unit-cell volume have becalculated at the experimental lattice parameters as obtafrom pressure measurements.18 The structural parameterx(U) and x(Al) have also been fixed at the experimenatomic positions, but we checked that the calculated atoforces are quite small and thus the positions are presumnot influenced much by the volume change. The calculatotal energy vs unit-cell volume has been fitted with the stdard Murnaghan equation of state, from which we obtainequilibrium volume and the bulk modulusB. This procedurehas been carried out both for scalar-relativistic and relativtic FLAPW calculations; see Fig. 3. The scalar-relativisFLAPW calculation yields an equilibrium volume oVth /Vexpt50.957 and a bulk modulusB5157 GPa. The lat-ter value should be compared to the experimental valuB5175 GPa. When SO coupling is included, the respecvalues areVth /Vexpt50.962 andB5181 GPa~see Fig. 3!,which compare favorably with the experimental values18

The inclusion of SO coupling obviously has a non-negligibinfluence on the cohesive properties. We also mentionwe have found that the orbital moment decreases fasterthe spin moment, with an applied pressure up to 8 GPa.total uranium moment at 8 GPa is thereby reduced to mthan 50% of the ambient pressure value. It would be ofterest to test this finding experimentally.

E. Magnetocrystalline anisotropy

One of the interesting results of our study is that the mnetism of URhAl is calculated to be Ising-like, i.e., the manetic moment practically disappears when the magnetizais forced to lie in the basal plane. A similar looking behaviexists for some rare-earth ions~e.g., Dy31 in C3h symmetry!,where the combination of SO coupling and the axial crysfield selects, as the ground state, a Kramers doubletuJ,MJ56J&, which is split when the magnetic field is along thsymmetry axis, but remains nonmagnetic when the fieldperpendicular to the axis.22

In our calculations we started from the magnetic solutioand constrained the exchange field along thea axis. Theiterative procedure leads to a practically nonmagnetic sconsistent solution with all spin and orbital moments lethan 0.02mB . The very same behavior was observed expementally on a URhAl single crystal in an external field.14 Inorder to analyze the reason for such strongly anisotropic

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J. KUNES, P. NOVAK, M. DIVIS , AND P. M. OPPENEER PHYSICAL REVIEW B63 205111

havior, we first consider the uranium site projected densiof the 5f states. The densities of states projected onrelativistic j 55/2 and 7/2 basis are shown in Fig. 4. Trelativistic SO splitting of about 1 eV is clearly visible foboth orientations of the exchange field. This suggests thastrength of the SO interaction is superior to that of thechange interaction. Next we consider the densities of stprojected on theYlmxs basis for both orientations of thexchange field, which are shown in Fig. 5. The sizes ofSO and exchange splitting can be estimated from the sration of 5f 23↑ and 5f 3↑ states and 5f 23↑ and 5f 3↓ states,respectively. It is apparent from Fig. 5~b! that the peak in thedensity of states located just below the Fermi level is formpredominantly by 5f 23↑ and 5f 3↓ states located in the uranium planes. Their orbital counterparts 5f 3↑ and 5f 23↓ arepushed up by the SO interaction to energies that are appmately 1 eV higher. If we now consider the stability of thnonmagnetic state in Fig. 5~b! with respect to the perturbation by an exchange field, we see readily that in a firstproximation the 5f 23↑ and 5f 3↓ states are split by the exchange field collinearly with thec axis, while no splitting isinduced by the exchange field perpendicular to thec axis. Inthe latter case an exchange-field-induced mixing of5 f 23↑ and 5f 3↓ states with the 5f 3↑ and 5f 23↓ states, orwith states of different magnetic quantum numbers, is nessary for any splitting. The former possibility is prevent

FIG. 3. The LSDA total energy of URhAl as a function of thratio V/Vexpt. Total energies obtained from scalar-relativistic cculations are shown in~a!, and those obtained upon including thSO coupling are shown in~b!. The lines are guides to the eyes.

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by the strong SO coupling. The latter would require a chanin the hybridization. The fact that we obtain a nonmagnesolution for the exchange field constrained along thea axisshows that exchange splitting can prevail over neither oftwo above-mentioned possibilities.

Although the anisotropic ferromagnetism in URhAl looksimilar to the above-mentioned Ising behavior22 of rare-earthions, its physical origin is different. For URhAl both thstrong SO coupling and the hybridization prevent thechange splitting for the field along thea axis. While the SOcoupling is equally large for all U compounds, the strohybridization is typical for the Fe2P structure. This very anisotropic crystal structure~see Fig. 1! imposes a very anisotropic hybridization, where the U-U and U-Rh bonding in thbasal plane is much stronger than the U-Rh bonding betwadjacent planes. Thus the anisotropic hybridization of5 f ’s is essential for an explanation of the unusual magnfeature. An equivalent anisotropic magnetization behavhas, to our knowledge, never been computed for a ferrom

FIG. 4. The projected densities of states in the relativistic bafor c-axis ~a! anda-axis ~b! orientations of the exchange field. Thsolid line denotes the total 5f 5/2 density of states and the dotted linthe total 5f 7/2 density of states. Note that, for both cases, the qutization axis is along the crystallographicc axis.

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MAGNETIC, MAGNETO-OPTICAL, AND STRUCTURAL . . . PHYSICAL REVIEW B 63 205111

netic material. We note, for comparison, that simicalculations38 were performed for the layered transitiometal compounds CoPt and FePt, which belong to the mnetically hardest transition-metal materials. The spin momin these materials was computed to be completely isotrowhereas the orbital moment exhibited only a slight anisropy, and a MAE of up to about 3 meV per formula unit wcomputed.38 While such a value is commonly regarded asvery large MAE, for URhAl we compute an enormous MAof 34 meV per formula unit. The strong SO coupling of urnium obviously contributes decisively to the huge MAE. Aexperimental value for the MAE can be obtained from tmagnetization curves, which were measured for appfields of up to 35 T along thec and a axes, respectively.1

Assuming the magnetization curve to be linear in the appfield above 35 T, we have deduced an experimental MAE41 meV per formula unit. In spite of the fact that no externmagnetic field is introduced in the calculation, and so

FIG. 5. Densities of states projected on the nonrelativisu l ,m,s& basis forc-axis ~a! and a-axis ~b! orientations of the ex-change field. The arrows denote the spin projections. The 5f 23

states are depicted by the shaded area, the 5f 3 states by the thickline, and the total 5f densities by the thin line, each for a given spprojection. The quantization axis is along the crystallograpc-axis.

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theoretically studied system is not in the experimental cditions, the computed and experimental MAE’s agree faiwell with one another.

F. X-ray magnetic circular dichroism

XMCD is a relatively new magneto-optical tool for thinvestigation of ferromagnetic materials. Its theoretical fomulation is identical to that of the Faraday~or Kerr! effect,except that in XMCD core-valence excitations are creatwhereas in the optical Faraday or Kerr effect valence-bexcitations are involved. In x-ray-absorption and XMCD eperiments, one measures the absorption coefficientsa6 forthe two circular polarizations of the x rays, parallel or anparallel to the magnetization.a6 are related to the opticaconductivity tensor:

a6'4p

cRe~sxx6 isxy!. ~8!

Due to the localized, non-dispersive character of the cstates, we use different techniques for the evaluation ofoptical conductivity@Eq. ~7!# in the valence-band spectrarange and in the x-ray spectral range.40

The measured and calculated x-ray absorptionXMCD spectra at the uraniumM4,5 edges are shown in Fig6. TheM4,5 peaks arise dominantly from transitions from th

c

c

FIG. 6. Isotropic absorption spectrum~curves in the top figures!and XMCD spectrum~bottom figures! of the uraniumM4,5 edges.Theab initio calculated spectra are shown by the solid line, andexperimental spectra~Ref. 14! by the dotted line. Note that theXMCD spectra at theM5 and M4 edges have been multiplied bfactors of 10 and 4, respectively.

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J. KUNES, P. NOVAK, M. DIVIS , AND P. M. OPPENEER PHYSICAL REVIEW B63 205111

3d3/2 and 3d5/2 core states to the 5f valence states. While thsplitting of theM4,5 edges follows from the FLAPW calculation, the onset energy of theM5 edge has been adjusted bhand. The theoretical spectra have been broadened wLorentzian of 3-eV width. The calculated spectra qualitively reproduce the experimental ones. However, sevdifferences can be observed. While the ratios of theM4,5absorption peaks are very similar, the experimental pedeviate from the Lorentzian shape, which is likely causedan energy dependence of the quasiparticle lifetime, whicnot taken into account in the calculation. The major discrancy between the calculation and experiment is the siztheM4 XMCD peak. To quantify this discrepancy we calclated the difference between theoretical and experimentategrated XMCD peaks. For theM4 edge this difference is18% of the experimental XMCD spectra integrated over bedges, while in the case of theM5 edge it is only 8%. As theintegrated XMCD signal is proportional to the orbitmoment30 this discrepancy could be related to an underemation of the orbital moment by LSDA-based computatiomethods. This would imply that the orbital moment obtainusing the XMCD sum rule is underestimated. In orderconfirm this we applied the XMCD sum rule to the theorical spectra using the calculated 5f occupation of 2.5. Whilethe calculated uranium orbital moment is21.59mB , theXMCD sum rule yields an orbital moment of21.36mB .This discrepancy is too large to be explained by a variatof the occupation number or the contribution of thed statesto the orbital moment. It thus appears that an applicationthe sum rule to URhAl leads to an underestimation oforbital moment.

The XMCD spectra can be understood qualitatively frothe partial densities of states; see Fig. 5. The major conbution to the absorption at theM4 edge stems from opticatransitions to 5f 3↓ states~the 5f 23↑ states are occupied!,resulting in a single-peak structure of this part of the XMCspectrum. The major contribution to the absorption at theM5edge stems from transitions to 5f 3↑ and 5f 23↓ states, whichare both unoccupied, resulting in two peaks of an opposign in the XMCD spectrum. As the separation of the peis smaller than the typical lifetime broadening, the peacancel each other to a large extent, thus leading to a msmaller signal than obtained at theM4 edge.

IV. CONCLUSIONS

We have probed the recent implementation of the SOteraction in the FLAPW scheme by computing the momeand magneto-optical Kerr spectra of URhAl, and comparthese quantities to results obtained with the ASW scheThe obtained agreement between the quantities compwith the two schemes can be regarded as outstanding.

Second, we have calculated several electronic propeof URhAl, in order to investigate the degree of 5f electron

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localization in URhAl. Two previous experimentainvestigations8,9 suggested rather delocalized 5f ’s, but twomore recent experiments10,14 put forward evidence of a highdegree of 5f localization. Our study exemplifies that thelectronic properties of URhAl are very well explained bLSDA-based calculations, in which the 5f ’s are treated asitinerant. In particular, the magneto-optical Kerr effect, eqlibrium volume, bulk modulus, MAE, and magnetocrystaline anisotropy of the uranium moment are satisfactoryscribed. Somewhat less well explained are the XMCspectrum and the uranium orbital moment, which mightconnected to the insufficient treatment of OP within tLSDA.12 The delocalized description furthermore reproducthe anomalous magnetic anisotropy that has been obsefor URhAl ~Ref. 14! and for other uranium intermetalliccompounds that belong to the same structural group.1 Con-nected with the anomalous magnetic Ising-like behavior isenormous computed MAE of 34 meV/formula unit, compared to an experimental MAE of 41 meV/formula unThese values considerably outrange all the MAE’s thatknown for transition-metal compounds, and are also thtimes larger than the MAE’s that were recently computedthe cubic uranium monochalcogenides.39 The origin of thelarge magnetic anisotropy rests in the strong SO interacof uranium and the particular hybridization, which is typicof the very anisotropic crystal structure, that cannot be ovcome by an exchange field along thea axis. To understandbetter the detailed relationship between the crystal strucand the anisotropic hybridization we are presently perforing calculations for other isostructural UTX compounds.

Note added in proof.After this paper was accepted fopublication, we performed new calculations of the electrostructure of various UTX compounds. For URhAl we foundthat beside the self-consistent solution forM i@100#, as dis-cussed above, a second self-consistent solution for this dtion of M exists, which possesses a lower total energy. Tsolution has a spin moment which is only slightly reducrelative to the moment of the@001# orientation. The magnetocrystalline anisotropy energy remains large, with thec-axisbeing the easy axis. A full description of this finding will baddressed in a subsequent paper.

ACKNOWLEDGMENTS

The authors gratefully acknowledge discussions wA. V. Andreev and V. Sechovsky´. This work was supportedfinancially by the German Sonderforschungsbereich 4Dresden, Germany. The work done at Prague was suppoby the GACR ~Project No. 202/99/0184!, GAUK ~ProjectNo. 145/2000! and GAAV ~Project No. A1010715/1997!,and partially by Grant No. LB98202 within the Project NINFRA2 of the Ministry of Education, Youth, and Sports othe Czech Republic.

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