A rule is derived to use x-ray magnetic circular dichroism spectra to extract the magnetic moment of the conduction-band states with j= l -1/2 separately from those with j= l + 1/2 as a function of energy. This quantity is straightforward to determine from the electronic band structure. The rule is illustrated with an application to pure iron and to the random substitutional alloy Fe_{80}CO_{20}

B. L. Gyorffy

H. Ebert

M. E. Rose

P. Strange

R. Wu

English

Strange, Paul

Gyorffy, Balazs L.

Kent Academic Repository

PHYSICAL REVIEW 8 VOLUME 52, NUMBER 18 1 NOVEMBER 1995-IIInterpretation of x-ray-absorption dichroism experimentsP. StrangeDepartment of Physics, Keele University, Keele, Staffordshire ST55BG, United KingdomB. L. GyorffyDepartment of Physics, Bristol University, Tyndall Avenue, Bristol BS81TL, United Kingdom(Received 15 June 1995)A rule is derived to use x-ray magnetic circular dichroism spectra to extract the magnetic moment of theconduction-band states with j= l —2 separately from those with j= l + 2 as a function of energy. This quantityis straightforward to determine from the electronic band structure. The rule is illustrated with an application topure iron and to the random substitutional alloy Fe8pCO20.The observation of magnetic circular dichroism in x-rayabsorption (MXCD) by Schiitz et al. ' and subsequent experi-mental work has led to MXCD becoming a very popularand unique probe of magnetism in transition and rare-earthmetallic systems. Theoretical understanding of MXCD hasalso made significant progress in recent years starting fromboth a localized and itinerant ' view of magnetism. This hasculminated in the recent exceedingly powerful, yet simple,MXCD sum rules which have shown that MXCD can beused to measure the spin and orbital contributions to themagnetic moment separately. The sum rules are derived froma single ion model and it has been shown that, althoughapproximate, they are often accurate to better than 5%. 'More recently, Ankudinov and Rehr" have rederived thesum rules using Fermi's golden rule within the independentelectron approximation. There is some uncertainty in such acalculation as the sum rule requires an energy integrationwhich is cut off at some judiciously chosen, but ultimatelyarbitrary value. Recently it has been shown that there aresome circumstances where the sum rules are of limited value.A particular example is on some transition metal surfaceswhere errors of up to 50% occur. ' However, even in suchcases the ratio of the spin to orbital moments can be deter-mined to an accuracy of about 10%. One limitation of thesum rules derived so far is that they are based on nonrelativ-istic quantum theory with relativistic corrections to order1/c, not on Dirac s relativistic quantum theory. In this paperwe derive an alternative rule for interpreting MXCD experi-ments based on an itinerant, rather than localized approachwithin a fully relativistic quantum mechanical framework.The rule enables us to determine directly the total magneticmoment of the conduction band j=l ——,' states separatelyfrom the magnetic moment of the j=l+ —,' states where l isdefined by dipolar transitions. Furthermore this can be deter-mined at every energy, so in principle no energy integrationis required. In principle this is a significant advance becausedichroism curves often contain considerable structure as afunction of energy, which is not described by the sum rules,however, broadening effects may well lead to the rule beingmore applicable in integral form where the total magneticmoment of the occupied j= l ——,' states can be determined.Our starting point for this calculation is the golden rulefor the transition probability per unit time:tKc Kc 1I g.'., ( )I tf ', ( ) x '„,( )C ' C C(3a)and the final state wave function can be written as a linearcombination of atomic orbitalswhereP/(r)= g c t/t '(r),K, m~(3b)t/ J(r)= g.'., (r) X.,'(r) l'=,— -t I, if '.(r) x '., (r)) (3c)Here the X '(r) are the usual spin angular functions, and wework in the (ic,m, ) representation throughout. ' The n areodd matrices and so couple the large and small parts of thewhere the summation is over all possible final states, and8(E) is zero for E(0 preventing the filling of states belowthe Fermi energy EF. E, is the energy of the core electronbeing excited and co is the frequency of the incident photon.P; and P/ are the wave function describing the system in itsinitial and final states, respectively. In a density functionalapproach these are commonly replaced by single particlelikewave functions. ' The photon-electron interaction operatorXqz is defined asX~i,(r) —= —en'A(r) = —en e1 1e'q',where A(r) is the vector potential associated with the inci-dent radiation and u are Dirac matrices in the standard rep-resentation. It is in the e 1 that we distinguish between dif-ferent polarizations of the incident radiation. The core wavefunction can be separated into radial and angular parts0163-1829/95/52(18)/13091(4)/$06. 00 52 R13 091 1995 The American Physical SocietyII) ~ ~ iilr l l I II gl II li II jiR13 092 P. STRANGE AND B. L. GYORFFY 520.3 the wave function can b 1 b 1 gl b-e a eed with ust a0.2R1 r g. (r)f.(r)dr, (4a)0. 1E0.00—0.1C303 I I I I—8 —6 —4(b)0.25—g) 0.20—C:E0.15—0 00.10—0.05—2 0 2 4 6Energy above Edge (eV)8 10R2 r f.,(r)g.(r)dr. (4b)we find thatWJc JcW +W +WJc Jc JcI ~m I cj m.l™jEquation (5) is the central result of this a er. Itan alternative interpr t t f o'sm experiments on th=,——, edge in transition and t dFrom an experiment at the ' ed e oac ini e metal systems.j, ge nly, the total magneticR2 i=R2 „=—Rl „=—R1 (4c)and in the cases we have examined these are at lea'tud 1 fo th dany other terms. E ur e ipolar matrix elements than forpps. q ation (4c) is an a ro 'as a nonre ativistic a roximatioapproximationions in our evaluation of E . 1 fi ti n. Making theseh ' ' 1onsi era y and we can calcub bl' fs a e. we restrict ourselves to transitions of the typeis is not a great restriction in rlarge majority of exp actice as theperiments work on ~dtions), we find the following.p or d~f transi-0.00—1I I 1 I1 2 3 4Energy above edge (eV)0.30.2FIG. l. (a) Illustration of Eq. (5) for pure iron.t}1 i ht-h nd 'd of E '5'i e q. ~5g calculated for the '=-The points are the left-h dj= z d-electrons.q. ( ) calculated at the L- an side of E . ~5e u one here is the ri ht-hand2j= 2 -e ectronic states above F. . T 'i a aussian of width 1.2 eV to simh ' d h bs an as een inverted to men . e points are evaluated directlt l lt o th L de 2 e ge of iron in Ref. 2.0. 1E0.00—0.1C5wave functions only. Substituting E s. ~2 aq. ( i o(ing e angular integrals leads to thesorption selection rules wh' hs usual ab-of the incident photon. "Th d' ' ees ic depend upon the olto be evaluated. Ine radial matrix lements also havea e . common with Refs. 6—8 we fine K, = —K, —1 and K = K 1 terms inc ea to radial matrix clemmuch small th her an t e terms with K'=K ane ents which areg ecte or our purposes. Hence the radial parts of0 2 8 10 12 14Energy above Edge (eV)FIG. 2. Illustration of E (5) fq. or the randome u ine is the right-handfor the j= —d- 1 — an side of Eq. (5) calculated-e ectrons on the iron site. Thside of Eq. (5) Ica culated at the iron L d. e points are the left-hand2e ge.INTERPRETATION OF X-RAY-ABSORPTION DICHROISM. . . R13 0930.30.2—0. 1E00.0E0—0. 1C5—0.2—0.30 2 4 6 8I J I I10 12 14 16 18Energy above Edge (eV)FIG. 3. Illustration of Eq. (5) for the random substitutional alloyFesoCo2O. The full line is the right-hand side of Eq. (5) calculatedfor the j= z d electrons on the cobalt site. The points are the left-hand side of Eq. (5) calculated at the cobalt L2 edge.moment per electron of the j=l ——,' states in the conductionband can be found energy by energy. Furthermore, in anal-ogy with the already known sum rules we can integrateEq. (5) from the Fermi energy up to the top of the band tofind the total magnetic moment of the j=I——,' electronicstates in that region. As a completely full band can have nonet magnetic moment the result of the integration is just thenegative of the total magnetic moment per electron of thefilled part of the band. Equation (5) is dimensionless and sothe absorption rate can be determined in arbitrary units. For(N, ) we can simply use the density of states and for~c~~mj we use the mj quantum number multiplied by themj decomposed contribution to the j= l —2 part of the den-sity of states. Both these quantities are easily obtainable froma standard band theory calculation. However, in the pastthere has been no way of determining them independently ofthe j=l+-,' contributions experimentally. Equation (5) hasthe minor advantage over the earlier sum rules ' that it onlyrequires the approximation (4c) over one edge, not over boththe j,=l, ~-,' edges. Finally, it does not seem to be sostraightforward to find a similar rule using excitations fromthe j=l, + —,' edge. A consequence of this is that we are re-stricted to materials where the width of the empty part of theconduction band is smaller than the spin-orbit splitting of thecore level. In practice this is not a severe restriction. Even inpure iron this is easily satisfied.To illustrate the use of Eq. (5) we have applied it to theL2 edge in pure iron using the theories of Strange andEbert. ' ' Figure 1(a) shows the result. The full line is theright-hand side of Eq. (5) and the points are the left-handside. Here, for illustrative purposes we have applied the ruleto the whole band structure, although, obviously, an MXCDexperiment could only probe the region above zero on thisfigure. Clearly there is excellent quantitative agreement be-tween both sides of Eq. (5), even in the low-energy regionwhere the d-electron contribution to the density of states issmall. In Fig. l(b) we show the right-hand side of Eq. (5)compared directly with experiment for the L2 edge in Fe. Tomake this comparison we fitted energy scale such that theedges in both theory and experiment coincided and per-formed a Gaussian broadening of width 1.2 eV on the calcu-lated values to take account of the various broadeningmechanisms present in the experiment. This shows clearlythe utility of the method either for predicting dichroism spec-tra from theory, or for using the experimental results toevaluate features of the band structure accurately.We have made two further applications of the rule, to theLz edge of both Fe and Co in the random substitutional alloyFe8oCo2o. The results are shown in Figs. 2 and 3 respec-tively. Clearly the agreement here is at least as good as in thepure iron case, and many details are well reproduced by therule. For example, the peak in the dichroism at about 8 eVabove the Fermi energy in pure iron has split into two whenalloyed with 20%%uo Co. The magnetic moment per j=l——,'electron clearly reproduces this.The power and utility of Eq. (5) can only be proven bycomparison between experiment and the theory, and as withall such rules is subject to the limitations of the experiment.In any real experiment much of the detail on figures such asthose shown is washed out by the various mechanisms suchas core-hole relaxation and instrumental broadening. There-fore only some broadened form of the right-hand side of Eq.(5) can be obtained directly from experiment. Nonetheless ifwe integrate both sides of (5), the total moment per j= l-—,' electron can be obtained, and we believe this should beobtainable from experiment. On the other hand, as we haveshown, the right-hand side of Eq. (5) can be obtained rathereasily from band theory. Broadening this quantity by convo-luting it with a suitable function should reproduce the energydependence of the dichroism curve.Some approximations have gone into our deviation. Inparticular we have ignored dipole transitions from l~l —1and we have approximated the initial core state as a pureeigenstate defined by quantum numbers (I~, m, ), which isequivalent to ignoring the polarization of the core hole. Fur-thermore we have not gone beyond the dipole approximationin evaluating our matrix elements. The extent to which theexperimental spectra do not agree with Eq. (5) may be usedto investigate their validity of these assumptions.In summary then, we have derived a rule for the interpre-tation of MXCD experiments which relates the dichroic sig-nal to the magnetic moment of the j=l ——,' electrons in thesolid, separately from the magnetic moment of the j=l+ —,'electrons at every energy. The rule means that it is possibleto separate these two quantities experimentally. Hence thisrule offers a deeper comparison between experiment andtheory than has been possible previously. This rule is appli-cable where an itinerant view of the magnetism is appropri-ate. It is complementary to the previously derived sumrules, ' and essentially offers an alternative representation ofthe ground state magnetic properties.Now that we have derived the relation (5) the next step isto test its range of validity by applying it to a wide range ofmaterials and in more general circumstances, such as at mag-netic surfaces. One further possibility is that this rule willalso be applicable to materials which exhibit dichroism, be-cause the symmetry is broken not by magnetism, but by acrystal structure without a center of inversion. We hope thatthis paper will stimulate such investigations.R13 094 P. STRANGE AND B. L. GYORFFY 52'G. Schutz, W. Wagner, W. Wilhelm, P. Kienle, R. Zeller, R.Frahm, and C. Materlik, Phys. Rev. Lett. 58, 737 (1987).N. V. Smith, C. T. Chen, F. Sette, and L. F. Mattheiss, Phys. Rev.B 46, 1023 (1992).B. T. Thole, G. van der Laan, and G. A. Sawatzky, Phys. Rev.Lett. 55, 2086 (1985).H. Ebert, P. Strange, and B. L. Gyorffy, Z. Phys. B Condens.Matter 73, 77 (1988).See, for example, H. Ebert, B. Drittler, P. Strange, R. Zeller, andB. L. Gyorffy, in The Effects of Relativity in Atoms, Moleculesand the Solid State, edited by S. Wilson, I. P. Grant, and B. L.Gyorffy (Plenum, New York, 1991).B. T. Thole, P. Carra, F. Sette, and G. van der Laan, Phys. Rev.Lett. 6$, 1943 (1992).P. Carra, B. T. Thole, M. Altarelli, and X.-D. Wang, Phys. Rev.Lett. 70, 694 (1993).M. Altarelli, Phys. Rev. B 47, 597 (1993).R. Wu, D. Wang, and A. J. Freeman, Phys. Rev. Lett. 71, 3581(1993).' P. Strange, J. Phys. C 6, L491 (1994)."A. Ankudinov and J. J. Rehr, Phys. Rev. B 51, 1282 (1995).' R. Wu and A. J. Freeman, Phys. Rev. Lett. 73, 1994 (1994).' P. J. Durham, in The Electronic Structure of Complex Systems,edited by P. Phariseau and W. M. Temmerman (Plenum, NewYork, 1984).' P. Strange, J. B. Staunton, and B.L. Gyorffy, J. Phys. C 17, 3355(1984).M. E. Rose, Relativistic Electron Theory (Wiley, New York,1961).' P. Strange, H. Ebert, J. B. Staunton, and B. L. Gyorffy, J. Phys.Condens. Matter 1, 2959 (1989).

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Interpretation of x-ray-absorption dichroism experiments

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Interpretation of x-ray absorption dichroism experiments

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Interpretation of x-ray-absorption dichroism experiments

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