X-ray Faraday effect at the L 2, 3 edges of Fe, Co, and Ni: theory and experiment

PHYSICAL REVIEW B, VOLUME 64, 174417

X-ray Faraday effect at the L 2,3 edges of Fe, Co, and Ni: Theory and experiment

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

H.-Ch. Mertins, F. Scha¨fers, A. Gaupp, and W. GudatBESSY GmbH, Albert-Einstein-Str. 15, D-12489 Berlin, Germany

P. NovakInstitute of Physics, Academy of Sciences, Cukrovarnicka´ 10, CZ-162 53 Prague, Czech Republic

~Received 19 April 2001; published 11 October 2001!

The x-ray Faraday effect at theL2,3 edges of the 3d ferromagnets Fe, Co, Ni and of Fe0.5Ni0.5 alloy is studiedboth theoretically and experimentally. We performab initio calculations of the x-ray Faraday effect on the basisof the local spin-density approximation and we adopt the linear-response formalism to describe the material’sresponse to the incident light. Experimental x-ray Faraday rotation and ellipticity spectra are measured withlinearly polarized soft-x-ray synchrotron radiation at BESSY, Berlin. The measured x-ray Faraday rotations areremarkably large, up to 2.83105 deg/mm, which is more than one order of magnitude larger than thoseobserved in the visible range. From the measured Faraday spectra we determine the intrinsic dichroic contri-butions to the dispersive and absorptive parts of the refractive index, and compare these toab initio calculatedcounterparts. The theoretical dichroic spectra are in good qualitative agreement with the experimental data. Theinclusion of the spin polarization of the core states leads to a small, yet non-negligible, improvement of thetheoretical dichroic spectra. Our results illustrate that the many-particle x-ray excitation spectrum can besufficiently well approximated by the Kohn-Sham single-particle spectrum. From the computed magneto-x-rayspectra we determine, using the sum rules, the orbital moments, which we compare to the exact orbitalmoments.

DOI: 10.1103/PhysRevB.64.174417 PACS number~s!: 78.70.Dm, 78.20.Ls

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I. INTRODUCTION

X-ray magneto-optical spectroscopies have develorapidly over the last few years, during which they have bcome established as valuable tools for the investigationmagnetic properties of materials.1–11 Among the variousavailable magneto-x-ray spectroscopies there are commtwo distinctions made. The first distinction is with regardthe polarization state of the incident x-ray light being usi.e., whether it is circularly or linearly polarized, and thsecond one is whether the intensity or the polarization oflight beam after its interaction with the magnetic materiameasured. Most frequently applied are circularly polarizerays, which are employed to measure the x-ray magncircular dichroism~XMCD!. The XMCD has gained appreciable importance due to the sum rules,12,13 which allow oneto evaluate the spin and orbital moment of a specific elemin the material from a spectral integral over the XMCD spetrum ~see, e.g., Refs. 14–17!. Less frequently applied arlinearly polarized x rays, which are employed to measureresonant magnetic x-ray scattering,5 the x-ray magneticlinear dichroism ~XMLD ! ~e.g., Refs. 18 and 19!, andthe x-ray Faraday effect~XFE!.4,7–10The resonant magnetix-ray scattering, the XMCD, and XMLD are intensity mesurements, whereas the x-ray Faraday effect demanpolarization-state analysis. Both the Faraday rotation, whis the rotation of the polarization plane upon transmissiand the Faraday ellipticity, which is the amount by which ttransmitted light has become elliptically polarized, are todetermined. While it is in itself already a complicated task

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make a polarization analysis, this is even more so insoft-x-ray energy region, where suitable analyzers are rSo far only a few experimental studies of the XFE wereported.4,7–11The x-ray Faraday effect at theK edge of Cowas investigated in Ref. 4. More recently, the Faraday effat theL2,3 edges of Fe~Ref. 7!, of Pt in Fe3Pt ~Ref. 8!, and ofFe, Co, and Ni~Refs. 9–11! were measured. These recemeasurements have demonstrated that XFE experimentsbe employed as an alternative to XMCD measurementsdetermining the dichroic spectra.8,9 The advantages over thXMCD technique are that the dichroic part in both the asorptive and dispersive spectrum can be obtained fromsingle XFE measurement and the wider availability of x-rsources for linearly polarized light.

In the recent XFE measurements performed by someus, the full magneto-x-ray spectra at theL2 (2p1/2) and L3(2p3/2) edges of Co, Fe, and Ni in Fe0.5Ni0.5 were obtained.9

From these recent experiments and from new experimenbe presented here, we have obtained the complete dichspectraDd(v), Db(v), to the dispersive and absorptivparts of the refractive indices. The refractive indices canwritten asn6512(d06Dd)1 i (b06Db), which is validfor the so-called polar geometry where the magnetizatand propagation direction of the light are parallel. The indent linearly polarized light is decomposed into two circlarly polarized waves of opposite helicity. The indices6refer to the parallel or antiparallel orientation of photon hlicity and magnetization, and thed0 (b0) stands for the dis-persive~absorptive! component, of the unmagnetized matrial. Db and Dd are the intrinsic material’s magneti

©2001 The American Physical Society17-1

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J. KUNESet al. PHYSICAL REVIEW B 64 174417

quantities, which comprise information on the resonant trsitions from the 2p1/2 and 2p3/2 core states to the emptspin-polarizedd states, and therefore are highly relevantthe 3d ferromagnetism of Fe, Co, and Ni.

Concurrent to our XFE experiments, we have performab initio calculations of the magneto-x-ray effects. Preously, several calculations of the XMCD, but also of the XFwere reported.20–26 However, since precise XFE data at thL2,3 edges of Fe, Co, and Ni, were not available, a fuconclusive comparison with experiment could not be maThis comparison is one of the goals of the present paper.calculations, ours as well as previous, are based ondensity-functional theory in the local spin-density appromation~LSDA!.27 This implies that the many-body quasipaticle excitation spectrum is approximated by the Kohn-Shsingle-particle spectrum. We apply furthermore the Kulinear-response theory for evaluating the material’s respoto the incident x-ray beam. As we shall show, theab initiocalculations provide a good description of the experimendichroic x-ray spectra. Consequently, the XFE and relamagneto-x-ray effects can be investigated theoreticallythe basis of first-principles calculations. Previously, velittle attention was devoted to the influence that the exchasplitting of the 2p core states has on the XFE spectra. Wfind that taking the exchange splitting of the core statesaccount does improve the agreement between theab initiocalculated and measured dichroic spectraDd, Db. Further-more, using the sum rules, we determine from the calculamagneto-x-ray spectra the spin and orbital moments, whwe compare to directly calculated moments.

In the following section we first describe schematicathe experimental technique. Expressions for the XFE inexperimental geometry are derived in Sec. III, and in Secwe outline the computational scheme. Results and consions are presented in the Secs. V and VI.

II. EXPERIMENTAL SETUP

The magneto-x-ray experiments were performed atBerlin synchrotron radiation source BESSY I using the plagrating Petersen-type~SX700! monochromator28 PM3 and atBESSY II using the elliptical undulator UE56-PGM,29 withspectral resolutionE/DE5700 andE/DE52500, respec-tively. Our samples consisted of 50-nm-thick amorphofilms that were either magnetron-sputter deposited onnm Si3N4 foils ~Fe, Co! or electron-beam evaporated o1 mm Mylar (Fe0.5Ni0.5). Our experimental setup is showschematically in Fig. 1. The transmission sample canplaced at any incident anglef i between 90°~grazing! and0° ~normal incidence!. The polarization of the incident x-rabeam was in the plane of incidence. A magnetic coil systsupplies variable magnetic fields of2500 Oe<H<1500 Oe lying in the sample’s plane and in the planeincidence; i.e., the measurements were carried out in thegitudinal geometry. In the visible spectral range Faradayfect measurements are normally performed in the polarometry, where the magnetization is perpendicular tofilms. We chose the longitudinal geometry to assure thatmeasurements could be performed with magnetically s

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rated films. The linear polarization analysis was performby rotating a W/B4C reflection multilayer~300 periods, pe-riod 1.2 nm, angle of incidence close to the Brewster ang!around the beam by the anglea ~analyzer scan! while re-cording the transmitted intensity.30 The experiment is de-scribed in detail in Ref. 9.

III. FORMULAS FOR X-RAY MO EFFECTS

Expressions for magneto-x-ray effects can be derivfrom the Fresnel equation for the magnetic refractive indiof the material and the continuity requirements for electmagnetic fields at the interface. These expressions depenboth the magnetization direction and the propagation dirtion of the beam. The simplest configuration for which exaexpressions for MO effects can be derived is the polarometry where the magnetization and incident light are nmal at the material’s surface. The electric field eigenmodethe material are left- and right-circularly polarized wavewhich would each match continuously to an incident circlarly polarized beam. The incident linearly polarized x-rbeam can be written as the sum of equal amounts of left-right-circularly polarized light. The Faraday rotationuF andthe Faraday ellipticity«F can be related to the refractivindicesn6 for left- and right-circularly polarized light by theexact equation31

S 12tan«F

11tan«FDe2iuF5eivdt(n12n2)/c, ~1!

wheredt is the transmitted thickness of the sample,dt5d0 atnormal incidence. The influence of the magnetization onindex of refraction is given by the difference ofn1 andn2 ,which is normally small. From the Fresnel equation one otains forn6 in the present geometry

n62 5exx6 i exy , ~2!

FIG. 1. Schematic setup for Faraday measurements on magfilms with angle of incidencef i and magnetic fieldH. The Faradayrotation angleuF and Faraday ellipticity«F are detected by anazimuthal rotation of the analyzer and detector around the beThe rotationuF and ellipticity «F of the transmitted light are indi-cated in the polarization ellipse with main axesa and b, wheretan«F5b/a.

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X-RAY FARADAY EFFECT AT THE L2,3 EDGES OF . . . PHYSICAL REVIEW B 64 174417

where exx and exy and are the diagonal and off-diagonelements, respectively, of the permittivity tensor. Since Meffects are small compared to the nonmagnetic response~1! can be expanded to first order in the small quantitiwhich gives

uF1 i tan«F'vd0

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The off-diagonal permittivityexy is the magnetically activecomponent, which is antisymmetric in the magnetizatioexy(2M)52exy(M).

In the case that the incoming light is not normal, boblique at an incident anglef i ~see Fig. 1!, it becomes rathercomplicated to derive exact expressions for the MO quaties. However, it is always possible to give approximaequations to lowest order in the small quantities. At obliqincidence the actual transmitted thicknessdt of the sampledepends on the refraction anglef t , i.e., dt5d0 /cosft . Inthe longitudinal geometry at oblique incidence one can msure the Faraday effect provided the magnetization hanonzero component parallel to the propagation directionthe beam. The approximate solutions to the Fresnel equaare in this geometry,32 to first order inexy ,

n62 'exx6 i exy sinf t

6 , ~4!

wheref t6 are the refraction angles of the two eigenmod

To first order inexy one can replacef t6 by f t in Eq. ~4!,

where f t is the angle of refraction of the nonmagnetizmaterial. It can be shown that the approximate expressionthe Faraday effect in this case becomes31

uF1 i tan«F'vd0

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Here tanf t can be rewritten further using Snell’s law. Fox-ray light one may replace tanf t'tanf i to a good approxi-mation. From this expression it is obvious that potentiallylarge Faraday rotation can be achieved from an in-plmagnetized material for a largef t of 70° or more, i.e., atgrazing incidence, as was indeed observed experimenta9

As written in the Introduction, the intrinsic dichroic material’s quantities of interest areDd andDb. From Eq.~5! itis evident that these quantities can readily be obtained fthe measured quantitiesuF , «F , in the longitudinal geom-etry through

uF52vd0

cDd tanf t , tan«F5

vd0

cDb tanf t . ~6!

The XFE is measured with linearly polarized x rays, busimilar magneto-optical quantity can be measured withcularly polarized light as well. In XMCD experiments onmeasures the dichroism, which is given by the asymmeparameterA and defined as

A5~T12T2!/~T11T2!, ~7!

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whereT6 are the transmission coefficients of left- and righhanded circularly polarized x rays. It can be shown thatXMCD asymmetry parameter is directly proportional to tFaraday ellipticity~see Refs. 8, 9, and 33!. In XMCD experi-ments, however, one measures only one of the dichroic qutities (Db), while the other one has to be derived fromKramers-Kronig transformation. In XFE experiments, on tother hand, one measuresbothdichroic quantitiesDb, Dd ina single experiment. However, at present the XMCD canmeasured more precisely than the Faraday ellipticity.

Magneto-optical effects can also be measured in theflection of linearly polarized x rays off a magnetized marial, in which case the effect is called the magneto-optiKerr effect~MOKE!. Both in the longitudinal and polar configuration the detection of MOKE requires a polarizatiostate analysis. So far no MOKE spectra have been measin the x-ray energy range, but the existence of the x-MOKE could be proved at a single soft-x-ray frequency.7,34

Although experimental spectra for the x-ray MOKE atherefore not yet available, theoretical predictions canready be made for the purpose of comparison with futexperiments. In the polar geometry at normal incidenceKerr rotation uK and Kerr ellipticity «K are given by the~exact! expression31

S 12tan«K

11tan«KDe2iuK5

11n2

12n2

12n1

11n1. ~8!

This expression can be approximated again for smalluK ,«K , andn12n2 , and, similar to the XFE@Eq. ~3!#, it can beformulated in terms ofexy andexx .

The above expressions exemplify that the theorymagneto-x-ray effects can be developed on the basissingle quantity, the permittivity tensor. As the permittiviand conductivity tensorsab are related througheab5dab14p isab /v, one may equivalently compute the conductiity tensor. Its evaluation from first principles is the primtask to be performed, which is outlined in the following.

IV. COMPUTATIONAL METHOD

A. Calculation of the permittivity tensor

The near-edge x-ray and magneto-x-ray effects originfrom excitations of the core electrons to unoccupied valestates close above the Fermi level. The response of the sto an external electromagnetic field is to linear order infield described by the dielectric or conductivity tensor, eaof which is sufficient to describe all experimentally accesible quantities. We start from the Kubo linear-responsepression forsab in a single-particle formulation~see, e.g.,Ref. 31!, in the limit of zero lifetime broadening, which wrewrite in several steps. First, the occurring current operamatrix elements containing the relativistic impulse operaare replaced by the matrix elements of the nonrelativisimpulse operator,p52 i\“, which holds to a very goodapproximation.35 Second, we use the fact that the core staare localized and therefore the impulse matrix elementsbe replaced by dipolar moment matrix elements. The res

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x,y(k) are the dipolar matrix elements. Thpositionsx, y, refer to the coordinate system centered atatom of interest. In a practical evaluation of the full permtivity tensor, first a convolution with a Lorentzian to accoufor lifetime broadening and subsequently a Kramers-Krotransformation are performed to obtain the dispersive paThe dipolar matrix elements can be rewritten in termsp-type spherical harmonicsY1m . A nice feature of Eq.~9! isthat the transition energy, which can be obtained onlyproximately within a one-particle formulation, enters onthed function and can therefore easily be renormalized acoincide with the experimental edge energy.

In the present numerical implementation of Eq.~9! we usethe full-potential, linearized augmented plane-wave meth~FLAPW! method as implemented in theWIEN97 code.36 Thewave functions are expanded into products of sphericalmonicsYlm and radial wave functions within a given atomsphere. When we neglect the effect of the exchange fieldthe core states, then these can be expressed as produradial solutions of the Kohn-Sham-Dirac equation and anlar functions having the relativistic symmetry. Furthermowe adopt a spherical approximation for the core potentThe relativistic valence states are computed within the fnonspherical potential, using the second variational approto self-consistently include the spin-orbit interaction37

Among the valence states there are three kinds of radial futions that are used for the expansion at a given orbital qutum number. These are the initial solutions of the scarelativistic approximation to the Kohn-Sham-Dirac equatiin a spherical potential at a given expansion energy, itsergy derivative, and optionally local orbitals that couldincluded as well in the basis.38 The dipolar matrix elementsin the FLAPW basis read

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s , andclms are the expansion coefficients of th

wave function inside a given atomic sphere with radiusRMTcorresponding to the three types of radial functions u

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2 , j , j z)are vector-coupling coefficients. The angular integralsthree spherical harmonicsYl cmc

uY1mYlm& are the standardGaunt coefficients. Due to the symmetry properties ofspherical harmonics, which result in the so-called selectrules, the summations in Eq.~10! reduce tol 5 l c61, m5mc21,0,11, and to a single summation over the spin vaable. The latter summation reduction comes about becawe restrict ourselves in the present approach to the nonrtivistic dipolar approximation for the impulse operator.

In our computational scheme we pay attention to the rof the exchange splitting of the core levels. We employ firorder perturbation theory to account for the spin splittingthe core states. In particular, we solve the Dirac equawith the averaged potential12 @V↑(r )1V↓(r )# and treat thetermD5 1

2 @V↑(r )2V↓(r )#sz ~where the quantization localzaxis was chosen along the net-magnetization direction! as aperturbation. Neglecting the hybridization between the staof different total momentum and using the fact that the mtrix elements ofD between states of different magnetic quatum numberj z are zero, we obtain a first-order shift of theigenenergies, whereby the energy degeneracy of thestates is removed.

B. Details of the calculation

Self-consistent band-structure calculations were pformed for bcc Fe, hcp and fcc Co, fcc Ni, and an Fe0.5Ni0.5alloy. For all these materials the exchange field was cstrained to the@001# direction. The number ofk points in theirreducible wedge of the Brillouin zone, used in the calcution of the components of the dielectric tensor, exceeded 5The LSDA exchange-correlation potential of Perdew aWang39 was adopted. The present implementation ofFLAPW method36 uses the spin-orbit coupling for the valence states included self-consistently in the second vational step. For a detailed description of the implementatof the spin-orbit coupling we refer to Ref. 37.

V. RESULTS

A. X-ray Faraday effect of Fe, Co, and Ni

Using the above given approach, we calculatedab initiothe components of the dielectric tensors of bcc Fe, hcp

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and fcc Ni, from which we computed their polar x-ray Faaday rotations. These are shown in Fig. 2. The depictedaday rotations were all computed with the spin polarizatof the core states included. We mention that since the mnitude of the studied magneto-x-ray effects is expected tomuch larger than their counterparts in the visible region,used exact expressions instead of the approximate onesmonly employed in the visible region~cf. Ref. 31!. To ac-count for the experimental resolution and the lifetime broening, a Lorentzian spectral broadening of 0.9 and 1.4was applied at theL3 andL2 edges, respectively.40 The fea-tures of the x-ray Faraday rotations can be understood qtatively: The overall width of the spectra is determinedthe spin-orbit splitting of the 2p1/2 and 2p3/2 states. Thissplitting is the largest for Ni and smallest for Fe. The manitude of the x-ray Faraday rotation, on the other hand,sically follows the amount of spin polarization of the 3dvalence states. This quantity is the largest for Fe andsmallest for Ni, and, likewise, the calculated Faraday rotatis the largest for Fe. In Fig. 2 we have shifted the respecpositions of the absorption edges to coincide with oneother. Note that the polar x-ray Faraday rotations at theL2,3edges are an order of magnitude larger than the polar Fday rotations measured in the visible energy range.41 Ourmeasurements of the x-ray Faraday rotations9 performed inthe longitudinal geometry yielded also values much larthan those known from the visible range.

B. Intrinsic dichroic quantities

From the measured longitudinal XFE spectra we havetermined the intrinsic dichroic partsDb and Dd of the re-fractive indices. Since the XFE experiments were performon a thin film of amorphous Co and of Fe0.5Ni0.5 alloy, wehave calculated the dichroic quantities for fcc Co and forordered Fe0.5Ni0.5 alloy derived from bcc Fe by replacing thcentral atom with Ni. These systems were chosen in ordemimic optimally the experimental materials. In the orderFe0.5Ni0.5 alloy the average nearest-neighbor distance ofNi and of bcc Fe was used for the Fe-Ni separation.

The calculated and experimental dichroic spectra at thLedge of Fe in Fe0.5Ni0.5 are shown in Fig. 3. The quantitie

FIG. 2. Ab initio calculated polar x-ray Faraday rotations at tL edges of bcc Fe, hcp Co, and fcc Ni.

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have been scaled by a factor of 2, because iron only contutes half of the sample thickness. It is immediately seenthere exists a very good agreement between experimenttheory. The two theoretical curves correspond to calculatieither with~solid curves! or without ~dashed curves! includ-ing the spin splitting of the 2p core states. The calculatiowhich takes the 2p cores splitting into account reproducethe measured spectra better. In particular at theL3 edge thecore polarization induces a shift of 1 eV in theDb spectrumto lower energies, which yields a better correspondencethe experiment. TheL2 edge is not affected. A similar shifoccurs in theDd spectrum at theL3 edge, where the measured data points at 710–712 eV are better reproduced bycalculation which includes the core polarization. The reasfor the influence of the core polarization is explained in dtail below. We remark that the measured dichroic spectraFe in Fe0.5Ni0.5 can be reasonably well approximated byDb,Dd spectra computed for pure Fe, because these spectrdominated by the resonant 2p to 3d optical transitions.

Figure 4 shows the corresponding dichroic spectra forParticularly the experimental and calculatedDb spectraagree very well. While the shape of the experimental

FIG. 3. Experimental and calculated dichroic spectraDb andDd at theL edge of Fe in Fe0.5Ni0.5. The two calculated spectraillustrate the influence of the exchange splitting of the 2p corestates on the dichroic spectra. The dashed curves give the dicspectra computed without exchange splitting of the core states

FIG. 4. As Fig. 3, but for theL edge of Co. The experimentadichroic spectra were measured for amorphous Co, and the calated spectra were obtained for fcc Co.

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chroic Dd spectrum is well reproduced by theab initio cal-culation, its measured magnitude is smaller in the regbetween theL2 and L3 edges. At the moment we have nexplanation for the magnitude difference, but we suspectit arises from the measurement technique~see also Ref. 11!.A second difference between theoretical and measured stra is recognizable in theDb spectrum at the high-energside of theL2 edge, where the theoretical curve falls osteeper than the experiment. This also happens for Fe~seeFig. 3!. The overall correspondence between theory andperiment is none the less very good.

The calculated and experimental dichroic spectra at theL edge of Ni in Fe0.5Ni0.5 alloy are shown in Fig. 5. Onceagain, the experimental and theoretical spectra compareisfactorily. There are two experimental results given for tDb spectrum of Ni. OneDb spectrum was measured withlower energy resolution~at BESSY I,s in Fig. 5! than thatof the other measurements, which explains why thisDbcurve is smoother and less peaked. The otherDb data points

FIG. 5. As Fig. 3, but for theL edge of Ni in Fe0.5Ni0.5. TheexperimentalDb data points were measured with two differespectral resolutions:E/DE5700 ~s! andE/DE52500 ~L!.

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~L! were measured at theL3 edge with the higher spectraresolution of BESSY II. It could be that the measuredDbspectra contain a weak shoulder at 855 eV. It could bethe measuredDb spectrum contains a weak shoulder at 8eV. Previously a weak shoulder~labeledB) was observed inthe XMCD spectrum of Ni at this energy.42,43 The origin ofthis weak shoulder is not understood, but it could be a maparticle excitation feature. This interpretation is corroboraby the absence of such shoulder in the calculated sinparticle spectrum. Nonetheless, our whole set of resultsphasizes that the full x-ray Faraday spectra of transition mals can be quite well described by the single-particle KoSham approach. Although previously calculations of tXFE were reported,21,22,24,25the agreement to the availabexperimental data could not be considered as such good.new measurements of the XFE provide accurate dichspectra, which are highly suited for a conclusive comparisto first-principles theoretical spectra.

C. Core-state exchange splitting

To exemplify the role of the exchange splitting of the costates we have calculated the spectra both with and withthe exchange splitting of the core states. We present hresults for a model study of Fe in Fe0.5Ni0.5, because theinfluence on Fe is expected to be the largest of the studmaterials. The computed spin splitting of thej z523/2 and13/2 sublevels is almost 1 eV for Fe in Fe0.5Ni0.5 andsmaller for Co and Ni. It is instructive to consider the asorption coefficientsa6 for left- and right-circularly polar-ized in the polar geometry, which are related to the refractindicesn6 by a652v Im n6 /c. In Fig. 6 we show the totaabsorption spectruma11a2 as well as the XMCD spectrum a12a2 as obtained with or without the exchangsplitting of the 2p core states taken into account. The inclsion of the exchange splitting of the core states modifies

f-

-

FIG. 6. Study of the influenceof the spin polarization of the 2pcore states on the absorption coeficientsa1 anda2 for right- andleft-circularly polarized light, re-spectively. The absorption coefficients are calculated for theLedge of Fe in the Fe0.5Ni0.5 alloy.The total absorption is given bya11a2, the dichroic asymmetryby a12a2. The inset shows anexpanded view ofa1 and a2

~shown with negative sign forclarity! at theL3 edge.

7-6

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FIG. 7. Ab initio calculatedpolar x-ray MOKE rotationuK

and ellipticity«K at theL edges ofbcc Fe, hcp Co, and fcc Ni. Thespectra have been shifted relativto the energy offset of the edgand are shown with a minus sign

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theoretical XMCD spectrum in two ways: first, the spaciof the L2,3 peaks is increased by approximately 1 eV, asecond, the slope of the low-energy side of theL3 peak isincreased, which results in a more pronounced asymmetrthe shapes of theL2 andL3 peaks. Both features improve thagreement with the experimental data~see Fig. 3!. We furthernote that from a comparison of the spectra computed withcore polarization and experiment, one could draw the wroconclusion that the spin-orbit splitting of the 2p core state istoo small, where it is actually the core-state exchange sting that is responsible.

In order to understand in more detail the role of the echange splitting of the core states we show theL3 absorptioncoefficients for the two circular polarizations in the insetFig. 6 ~for clarity’s sake the absorption coefficienta2 isshown with a minus sign!. The behavior of the absorptiocoefficients when the exchange splitting of the core state‘‘switched on’’ can be understood by considering the weigwith which the transitions from the individual core statcontribute to the absorption spectra. These weights are dmined by the angular parts of the dipolar matrix elemenwhich increase with increasingj z for negative helicity anddecrease for positive helicity, and by the occupation offinal d states, which favors transitions to unoccupied minity (↓) spin states. The major contribution to thea2 peakstems from thej z523/2 level, which eigenenergy is increased by the exchange interaction. This leads to the shthe absorption peak towards lower energies. The major ctribution to thea1 peak comes with approximately equweight from thej z521/2 and 1/2 sublevels. Since the echange interaction increases and decreases, respectiveleigenenergies of these states by the same amount, it mleads to a broadening of the absorption peak without shifits center of gravity. At theL2 the influence of the spin splitting of the 2p1/2 states is much smaller, but it can qualittively be discussed in the same manner. Thus, althoughspin splitting of the 2p core states is small, it neverthele

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leads to an improvement of the theoretical spectra. Thefluence of the spin splitting of the core states was previoucomputed for theL2,3 edge of Co in CoPt alloys,23 for whichsimilar modifications were found, but no details of their ogin were given.

D. X-ray polar MOKE

Although so far no x-ray polar MOKE spectra have bemeasured, for the aim of comparing with future experimewe have computed these. To this end, we have employedexact expression~8! for the magneto-x-ray rotationuK andellipticity «K . It is, however, for the analysis of the computed x-ray MOKE spectra convenient to note that the exexpression can be approximated by31

uK1 i«K' in12n2

n1n221'

2exy

exx1/2~12exx!

. ~12!

The calculated polar MOKE spectra of bcc Fe, hcp Co, afcc Ni are shown in Fig. 7. Unlike in the visible range whethe shape of the MOKE spectra in most cases reflectsshape of the off-diagonal conductivity spectrum,44 for thex-ray regime the denominator in Eq.~12! is found to play acrucial role. Contrary to the x-ray Faraday spectrum,x-ray MOKE spectrum displays a large maximumin betweenthe two absorption edges. A closer look at Eqs.~3! and ~12!clarifies the origin of this spectral difference. Both the Faday and polar Kerr effects contain as numeratorsn12n2 ,but the Kerr effect is additionally modified by the denomnatorn1n221. Since in the soft-x-ray rangen is close to 1,a profound influence of the denominator can be anticipaTo illustrate the influence of the denominator we showFig. 8 the x-ray MOKE of fcc Co, as well as the separaspectra of«xy and @12«xx#

21. The x-ray MOKE quantitiesare approximately composed of the product of the latter tspectra, where 12«xx is the dominating part of«xx

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2«xx). The denominator affects the x-ray Kerr rotation suthat a broad maximum occurs between theL2 andL3 edges,which is caused by the peak in the imaginary part ofinverse~see Fig. 8!. The marked impact of the denominatimplies a high sensitivity of polar MOKE measurementssurface oxidation effects. Already a surface layer wochange the denominator ton1n22ns

2 , wherens is the re-fractive index of the surface layer.31

E. Spin and orbital moments

From the above-reported dichroic spectra it is evident tthe experimental spectra can be very well described byab initio calculations. It has become customary in magnex-ray experiments to exploit the measured dichroic spectrextract the spin and orbital moment of an element, usingsum rules.12,13 From the computed dichroic spectra we ca

FIG. 8. Analysis of the x-ray polar MOKE spectrum at theLedge of fcc Co. In the top panel the real~i.e., uK) and imaginary(«K) part of the complex Kerr response are depicted. The midpanel shows the real and imaginary parts of«xy , which is the nu-merator of Eq.~12!. The bottom panel shows the real and imaginaparts of the inverse of@12«xx#, which is the dominating term in thedenominator of Eq.~12!. The product of the quantities in the middand bottom panels leads to the x-ray Kerr rotation and ellipticdepicted in the top panel.

17441

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using the orbital moment sum rule, now compute the orbmoment, which we can compare to the exactly compuorbital moment.45 The question is how well these two compare. In Table I we give the exactly calculated orbital mments as well as those obtained from the theoretical speemploying the orbital moment sum rule. There exists reasably good agreement between the orbital moments obtaby the two different approaches. Note that in all casesorbital moment sum rule leads to an orbital momentsmallerthan the straightforwardly calculated one. A similar result hbeen found too in a previous computational study.23 Al-though the agreement appears reasonable, it should betioned that considerable uncertainties enter the evaluatiothe orbital moment sum rule. First, the number of holes isclearly defined for solids, which becomes critical in the caof almost filled final states. Second, there exists the enecutoff for the integration~or subtraction of the background!that has to be fixed in the integration of the total absorptiThis procedure is arbitrary even in the theoretical calcution, since the accessible final states ofd symmetry originatenot only from the considered atom~in a tight-binding sense!.This difficulty is easily demonstrated by integrating thedpartial density of states up to high energies~more than 40 eVabove the Fermi energy! yielding a number ofd states largerthan 10. On account of this difficulty, an uncertainty in tdetermination of the orbital moments from the sum rule cbe estimated to be 10%-20% for the here-considered maals. Such uncertainty would appear to limit an accuratetermination of orbital moments from measured dichrospectra with the sum rule. Considering, however, the factthe sum rules are derived only for isolated atoms12,13 andinvolve a large number of approximations,23 it is remarkablehow reasonable the obtained orbital moments nevertheare. Larger deviations~up to 50%) were previously obtainein a theoretical investigation of the sum rule for the spmoment.46 To end with, we remark that recently an exasum rule for the orbital moment of solids has been reporby two of us,47 which, however, requires an integration ovan unrestricted magneto-optical spectrum.

VI. CONCLUSIONS

We have carried outab initio calculations and experiments of the x-ray Faraday effect at theL2,3 edges of Fe, Co,

le

y

TABLE I. The orbital moments obtained for various ferromanetic 3d materials. The exactly calculated orbital moment is dnoted byMl ~calc.!, whereasMl (s rule! denotes the orbital mo-ment derived from the sum rule.Ms is the computed spin momenandnd the calculated number of holes in the 3d band.

Ml ~calc.! Ml (s rule! nd Ms

Fe ~FeNi! 0.072 0.067 6.04 2.76Ni ~FeNi! 0.056 0.052 8.26 0.71bcc Fe 0.047 0.041 6.04 2.23fcc Ni 0.052 0.043 8.18 0.62fcc Co 0.075 0.070 7.20 1.64hcp Co 0.081 0.074 7.24 1.63

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and Ni. Our investigations demonstrate that the measuand calculated dichroic spectra at theL edges of the 3d fer-romagnets are in good agreement. Since the theory of xmagneto-optical effects can be formulated in terms of coponents of the dielectric tensor, we can anticipate that aother x-ray magneto-optical effects as the x-ray MOKE cbe explained by first-principles theory. For comparison wfuture experiments we have computed the polar x-MOKE of Fe, Co, and Ni.

Our ab initio calculations show explicitly that the manyparticle x-ray excitation spectrum can to a good approximtion be replaced by the single-particle Kohn-Sham spectrThe good agreement between the calculated and experimtal x-ray magneto-optical spectra suggests that for the invtigated materials the role of many-particle contributionsthe excitation spectra, such as, e.g., core-hole effects, isited to a possible renormalization of the edge transitionergy, without any major impact upon the shape of the sptra. An exception could be the weak shoulder in theDbspectrum at theL3 edge of Ni, which deserved further studApart from this weak feature, on the basis of the presinvestigation we foresee no direct demand to invoke supmental Coulomb correlation termsU for calculations of

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XFE, x-ray MOKE, or XMCD spectra. The need for Coulomb correlation effects has previously been argued forexplanation of photoemission spectra obtained from thepstates. We furthermore find no direct evidence that multiptransitions are required to explain the magneto-x-ray specat least not at the spectral resolutionsE/DE52500 and 700used here. It could be, however, that signatures of multiptransitions become visible at higher resolutions. Also,might well be that multiplet excitations must be taken inaccount to describe the magneto-x-ray spectra of, e.g.,fmaterials.

Our calculations reveal the influence of the exchansplitting of the core states on the dichroic spectra. Its inflence is small, yet the computed dichroic spectra areproved compared to experiment when the core polarizatioaccounted for.

ACKNOWLEDGMENTS

This work was supported financially by the Sonderforchungsbereich 463, Dresden, Germany, and by the EuropCommunity under Project No. ERB FMG ECT 980105.

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*Permanent address: Institute of Physics, Academy of ScienCukrovarnicka´ 10, CZ-162 53 Prague, Czech Republic.

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X-ray Faraday effect at the L 2, 3 edges of Fe, Co, and Ni: theory and experiment

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