PHYSICAL REVIEW B VOLUME 57, NUMBER 18 1 MAY 1998-II ARTICLES Resonant inelastic x-ray scattering P. M. Platzman and E. D. Isaacs Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey 07974 Received 24 October 1997 We analyze the physics of resonant inelastic x-ray scattering particularly from the 3d transition-metal series. We discuss what types and with what intensity we may expect to observe various final states, by considering an array of many particle dynamical effects. We conclude with the results of an experiment on NiS1.5Se0.5. S0163-1829 98 01217-X INTRODUCTION transfer. They will compliment experiments such as resonant light scattering4 which are confined to nearly zero momen- Inelastic x-ray scattering from electrons in condensed tum and inelastic electron scattering5 which is confined to matter systems is a rapidly developing field which promises small momentum transfer and microscopically thin samples. to give us detailed information about the excited states of In this paper, we will focus on a discussion and analysis these systems.1 When the incident x-ray energy is far from of the various physical phenomena which arise in such reso- any atomic absorption edges in the sample, inelastic scatter- nant scattering experiments particularly from the 3d ing measures the dynamic structure factor of the electronic transition-metal series. Since the resonant process is very excitation spectrum. In some materials the low-lying elec- local, exciting electron-hole pairs at a single atomic site, we tronic charge excitation spectrum consists of, for example, will consider the role of the strong Coulomb interactions in collective features such as plasmons, spin waves, excitons, these strongly correlated systems in accessing final excited and a single-particle-like continuum related to the band states which, for example, involve a hole on one site and an structure. The excitation energies of these spectral features electron on a neighboring site such as in a charge transfer or and their momentum dependence can tell us a great deal exciton. about the role of electronic correlations, as well as the be- We will not try to give a complete treatment of a particu- havior of the material. lar sophisticated model problem since in a real solid there are Because 10 KeV hard x rays have a wave vector q1 too many diverse phenomena to consider. Instead, we will 2 / 1 5 Å 1, they are particularly well matched to emphasize the order of magnitude of the various effects and studying the excitation spectrum over the entire Brillouin stay away from detailed calculations. We will also address zone. However, because the scattering of x rays from elec- the important issue of momentum transfer and conservation trons is weak, diffuse, and spread out in energy, and because in the context of the resonant inelastic cross section. the absolute energy resolution is so small ( 1 / 1 We will then present the results of an experiment in the 10 4), most inelastic studies have been restricted to sys- classic Mott-Hubbard system NiS1.5Se0.5 Ref. 6 which un- tems with low x-ray absorption in order to keep the scatter- dergoes a metal-insulator transition at Tc 80 K in our ing volume high. sample.7 Our discussion will center on the momentum de- Recently, it has been demonstrated that large enhance- pendence of the position and intensity of a well-defined fea- ments in the scattering cross section can be achieved when ture in the inelastic scattering spectra with an energy loss of the incident x-ray energy is tuned near to an atomic absorp- about 5.5 eV in the insulating phase. We associate this fea- tion edge of one of the atomic species in the sample.2 Much ture with the creation of an exciton involving the excitation as resonant enhancements have made it possible to study of an electron from a sulfur state to the upper Hubbard 3d magnetic structure in a broad range of interesting condensed band associated with the nickel. The momentum dependence matter systems,3 resonance effects are now making it pos- of this feature clearly demonstrates that the momentum sible to study interesting electronic excitations previously in- transferred to the system is carried, at least in part, by the accessible to inelastic x-ray scattering. In many electron sys- final excited state as it is in the simpler nonresonant excita- tems interactions between electrons makes the possible tion processes. excited states very interesting and the coupling to them ex- In a typical scattering experiment, an x ray of energy 1 , tremely difficult to analyze even for the case of nonresonant polarization 1 , and momentum q1(h 1) scatters weakly scattering. On resonance, because of coupling to the deep from the electronic system in an initial ground many-body atomic core hole, the analysis is even more difficult and pos- state i to a final state ( 2 , 2 ,q2). This leaves the system sibly more interesting. Experiments with energy resolutions in an electronically excited state f with momentum q q2 of 100 meV, which will soon be possible, should enable us to q1 and energy 1 2 . In the nonrelativistic limit study relevant excited states as a function of momentum ( 1 mc2 5 105 eV), the matrix element for scattering, 0163-1829/98/57 18 /11107 8 /$15.00 57 11 107 © 1998 The American Physical Society 11 108 P. M. PLATZMAN AND E. D. ISAACS 57 to second order in the electromagnetic field is given by e2 * 2 n n pq * 1 i M f pq2 1 mc2 f 2* 1 q i 1m En Ei 1 i f pq * 1 n n pq * 2 i 1 2 E , 1 n Ei 2 here q jeiq*r is the density operator, pq jpjeiq*r is the momentum operator. The energies Ei(En) are the energy of the ground intermediate state of the interacting many-body system with correlated wave functions ( i , n ). When 1 is not near the binding energy of an atomic core state, Eq. 1 is dominated by the first term on the right-hand FIG. 1. Perturbation theory diagrams for the interaction of x-ray side and the scattering cross section at T 0 can be written photons of momentum q as8 1 ,q2 with band electrons. d 2 In order to better understand many of the interesting as- pects of Eq. 3 , we now turn to a discussion of some of the d d 1 2 2 e2 mc2 physics contained in the resonance process. f q i 2 Ef Ei , 2 THEORETICAL CONSIDERATIONS f which is only a function of q and . Since (e2/mc2)2 To establish a framework for discussing the various 10 26 cm2 is small and since the number of interesting physical processes involved and to include Coulomb interac- valence electrons, n 1022 cm 3, the total scattering is tions we choose to represent the scattering processes by a set rather weak. This is why, as mentioned above, even with the of time ordered Feynman diagrams which are nothing more best synchrotron sources, inelastic scattering rates are low then pictorial representations of the various terms in the per- and it is thus only possible to do nonresonant inelastic scat- turbation expansion of the full many-body problem.11 The tering experiments on materials with small absorption. perturbation terms include the coupling of the x rays to the When the incident x-ray energy is tuned near to the bind- system, which is weak, as well as the coupling of the elec- ing energy of a deep core level of an atom in the system, the trons to each other and to the nuclei via their Coulomb in- second term in Eq. 1 dominates the cross section. In this teractions, which is not. Since the various Coulomb cou- case the energy denominator can vanish and the cross section plings are not weak we will often have to sum many terms in can become large. However, the cross section also becomes the perturbation series in the Coulomb interactions between more complicated than Eq. 2 , in that the nature of the cou- electrons to display a given effect. In many cases this can be pling to the excited state f depends on the presence of the easily represented graphically. intermediate state n which contains an almost real, strongly The x rays primarily interact with the system to make perturbing core hole. Nevertheless the cross section can still electron holes pairs. In Fig. 1 a , an x ray of momentum q1 be written as in Eq. 2 . It is wiggly line is annihilated and an electron is excited from a filled band state n to an empty band state n. Energy is not d 2 conserved if the electron-hole pair is an intermediate state, however, crystal momentum, i.e., momentum plus or minus d d 1 2 e2 mc2 f Oq i 2 some reciprocal lattice vector Kn is conserved at each vertex. E The electrons and holes empty states propagating in the f Ei . 3 valence bands of the material are represented by solid lines. The finite q resonant Raman operator Oq conserves momen- Those lines going up forward in time are electrons while tum and gets large when 1 is near an absorption edge. In those going down backward in time are holes. Since the principle Oq is a function of 1 , q1 , q2 , 1 , 2 . The tensor hole in one of the inner shells for example a hole in the K character of O q arises from the momentum operator in the shell plays a unique role, we will designate it by a double matrix elements. As we shall see, this implies that we can solid line, and label it with a c. The fact that the solid lines couple to transverse and spin excitations as well as longitu- represents a band state means that we have already implicitly dinal excitations. included all the multiple elastic scattering of the electrons No one has successfully given a complete many-body de- from the nuclei and from the mean in the local density scription of the operator O q .9 However, making the as- sense charge of the other electrons exchange included in sumption that the intermediate state energy denominator can the details of these states. The scattering due to the q term be replaced by some average energy allows one to sum over in Eq. 1 , can also create an electron-hole pair at a single intermediate states and reduce the problem to the calculation vertex where the initial photon is destroyed and the final one of an autocorrelation function, as in the nonresonant case, of created see Fig. 1 b . a simple operator such as q .10 Such approaches are not The photons can also couple to phonons by two quite generally valid and we will discuss their limitations later in distinct mechanisms. The first and most commonly accepted the article. way is to first excite an electron hole pair and then have 57 RESONANT INELASTIC X-RAY SCATTERING 11 109 FIG. 3. The lowest order noninteracting matrix element for reso- nant inelastic x-ray scattering. FIG. 2. a Electron Coulomb scattering diagram; b screening is of order one and v,p refers to some good one electron of the bare Coulomb interaction dotted line in a by valence band approximation to a valence band wave function with a crys- electron hole pairs. tal momentum p. either the electron or hole scatter quasielastically from the In the absence of Coulombic effects the energy denomi- nucleus or equivalently the deeply bound electrons creating a nator in Eq. 4 has a real singularity. The singularity means phonon and leaving an electron hole pair. The second is to that the second order matrix element describing scattering, couple directly to the center of mass motion as discussed by has become a single photon absorption process, followed by Platzman, Tzoar,10 and Sette.12 In either case, processes in- a single photon emission. The divergence is nothing more volving phonons lead to momentum nonconserving effects. than a statement of the fact that the time available for ab- As far as the electronic excitations are concerned, such ef- sorption under steady state conditions is infinite.2 Of course fects will lead in many cases to a broad featureless back- in a real system with Coulomb interactions the core hole ground which can often be ignored. In any event they will decays predominantly nonradiatively with a lifetime 1. To not be of primary concern to us here, although we will dis- take this into account it is acceptable to replace by . For cuss some aspects of them. scattering near a K edge comes primarily from the Auger In addition to coupling to the electromagnetic EM field decay of the 1s core hole to a 2s, 2p hole, i.e., it is nonra- the electrons can couple to each other by direct Coulomb diative. In transition metals such as Ni and Cu is a few eV interactions dashed line , see Fig. 2 a . Each dashed line and the particle hole pairs which are excited have energies corresponds to a matrix element n(k) 4 e2/(k EAug E1s E2p which are nearly a kilovolt. Kn)2 F(Kn), where k k1 k2 . The form factor F(Kn) When the incident and emitted photons are much closer in for the Coulomb matrix element for different reciprocal lat- energy than EAug , the lifetime is a good way of phenom- tice vectors Kn depends on the Fourier transform of the enologically including a host of many-body effects that do Bloch parts of the scattering electrons wave functions. For not interest us. It correctly limits the size of the resonant the electron gas, i.e., electrons in a uniform positive back- enhancement, and gives us a rough estimate of the amount of ground, F(Kn) 0,K . x rays which are scattered and which still conserve energy n In order to include cooperative effects in a mean field and momentum. In addition it tells us correctly that a range random-phase approximation RPA sense, e.g., plasmons of states off the energy shell of the order of fix the resonant for free electronlike metals, we generally screen matrix element. n(k) by the dielectric function. More precisely we replace At the intermediate state energies for hard x rays con- n(k) by densed matter systems have a continuum of energies. Thus n(k)/ (k, ). The replacement is equivalent to changing the bare dashed Coulomb line in Fig. 2 a into a dressed no single intermediate state dominates the scattering process braided Coulomb line pictorially represented by the infinite and most of the intermediate states which contribute are off set of diagrams shown in Fig. 2 b . When there are signifi- the energy shell by an amount . However, independent of cant band structure effects k, is a tensor, e.g., the many electron origins of , it is always true that if there n,n (k, ) with n,n band indices and the situation is more compli- is one photon in and one photon out the many-body system is cated, i.e., the various interband terms represented by left in an excited state with momentum q and energy . The lifetime of the core hole will not contribute to the width of n,n (k, ) with n n must be included.7,13 Near resonance and in the absence of any interaction ef- features in the spectrum. For example, in a semiconductor fects the time ordered diagram which dominates the reso- the band edge will be sharp, i.e., spectrometer resolution lim- nance cross section is shown in Fig. 3. This process leads to ited. An excitonic feature will be there with a width deter- a matrix element 0 , mined by its decay. Moreover, it should be possible to ob- serve sharp many-body features, provided they have large 1s p enough matrix elements. In general though, there are almost A 2* 2eiq2*r2 v,p q2 v,p q1 p1* 1eiq1*r1 1s 0 m . E always a rather broad continuum of states even at excitation v,p q E 1 1s 1 i energies of 10 eV and the presence of a peak implies some- 4 thing more subtle about the many electron system. Here, A0 is dimensionless and its contribution to the cross The most obvious and common examples are a plasmon section is given relative to the leading term in Eq. 1 which or a spin wave collective state. For the case of a simple metal 11 110 P. M. PLATZMAN AND E. D. ISAACS 57 such as aluminum we will predict that the plasmon excitation The noninteracting expression forms the basis for all the will be present at low q and that it will disperse exactly as in early discussions of resonant x-ray scattering. Carlisle the nonresonant case. et al.16 have applied a one electron single particle description Now that we have digressed a bit, discussing at least to a set of experiments in graphite. They where able to show qualitatively how some Coulomb effects modify our inter- a correlation between the noninteracting picture and the band pretation of Eq. 4 even in the noninteracting approxima- structure of Graphite. Veenendal et al.9 developed a theory tion, let's go back to Eq. 4 for some more discussion of the of resonant x-ray scattering for the rare earths which was one physics. The matrix elements in Eq. 4 are clearly very lo- electron in character and made the additional assumption that cal. They involve matrix elements of the single particle mo- the energy En in the denominator could be replaced by some mentum operator p sandwiched between a K shell (1s) core average or typical En. This assumption is in their case not wave function which is very confined compared to the in- quantitatively accurate because is smaller than separation coming x rays wavelength, and a partially filled valence band between various bands. For example, in Ni the dominant VB electronic wave function which is spread out. This transition at resonance is to the 4p level. The 5p levels, etc., means we need the atomic part of the VB wave function. The are roughly 10 V away, so that with 2 eV these transi- evaluation of such matrix elements has been carried out for tions are down by almost one order of magnitude from the the L shell (2p) core wave function in rare-earth compounds energy denominator alone and another factor of 2 or so from by Carra et al.9 and at the soft x-ray edges such as in the matrix element. Replacing the denominator by an aver- graphite10 and CaF2.14 age En means we take in an atomic picture all p levels 4p, When we excite near the K edge of a transition metal ion 5p, continuum, etc., weighted only by their matrix element. the matrix elements are even more local. However, the best A better approximation for small is to take one state, i.e., one electron estimate of the local part of the wave function in our case only 4p. Nevertheless, in Ref. 9 they do show involves one additional bit of physical intuition. Suppose that that their theoretical noninteracting average energy approxi- the valence state of nickel in our transition-metal compound mation does reproduce some symmetry features of the ex- is approximately Ni , i.e., it has a 3d8 configuration. The periments by Hamalainen et al.,12 who probed excitations dipole allowed transition to the first empty state, which is a near the Dysprosium L3 edge with inelastic x-ray scattering. valence band state in the solid, is to a VB state, which has at Despite all the complications associated with explicitly small distance from the Ni 4p atomic character. However, evaluating Eq. 3 it is quite clear from Fig. 3 that this because the transition is so high in energy ( E 9 keV) it is resonant coupling leads to a single electron-hole pair with a sudden. Since the core hole is slow to relax it is much better particular weighting which depends primarily on the energy to think of the 4p atomic wave function as the 4p in a Cu denominator in the intermediate state, and which conserves with (3d8, 4p configuration, i.e., the Z 1 atom describes crystal momentum. Since crystal momentum is conserved, the frozen hole in the atom with charge Z. this resonant process is a bit similar to having the operator q We have used a local density approximation LDA atom acting on the ground state just as in the nonresonant case. program15 to evaluate matrix elements relevant to Ni There is, of course, an enhanced magnitude. However, there (S1.5 /Se0.5). For Ni in the frozen hole approximations as is an important distinction which is related to the general discussed above the important matrix elements are properties of the resonant operator. For the first term in Eq. 4p r 1s 0.6 a.u. and 3d r( / r) 1s 1s 0.01 a.u. Us- 1 the density operator is precisely q , where q ing these matrix elements, a 2 eV and a q 1 2 a.u., we p,n,n ap q,n ap,n creates an electron at momentum p q find that the resonant cross section is roughly a factor of 100 and a hole with momentum p. In contrast, for the resonant larger than its nonresonant counterpart. process the O q in Eq. 3 may be written schematically, While the matrix elements are local, the coherent effect of adding up many matrix elements involving tightly bound electrons at different lattice sites should for a noninteracting Oq A0 p,q1, 1,q2, 2 ap q,n ap,n , 5 system lead to overall crystal momentum conservation.8 p,n,n However, in our case, in the presence of interaction effects where the subscript 0 on A means no Coulomb interaction. momentum conservation for a K shell hole intermediate state To next order first in the Coulomb interactions among is a bit subtle and interesting. Since the Auger width is the electrons and between the virtual core hole and the elec- E1s , where E1s is the band width of the 1s core hole, trons we must consider the scattering processes depicted by the deep core hole has no idea it is in a crystal lattice. Dif- the diagrams shown in Fig. 4.10 While it is possible to evalu- ferent matrix elements from different lattice sites will not ate such terms in some detail it is really unnecessary, since interfere. Instead the initial x-ray photon gives its momentum we really want to know their approximate size and possibly to the outgoing electron and to a recoiling transition metal what new kinds of final excited states can be reached. Ulti- ion. Since D , the lattice Debye frequency, the momen- mately, we might also want to consider how the details of the tum given the single transition metal ion is returned to the matrix element might influence the line shape. electronic system when the final x-ray photon is emitted. The Because the perturbation diagrams in Fig. 4 all have one emission of a real phonon in this process leads to momentum resonant denominator and one extra Coulomb interaction breaking. The size of such a momentum breaking process they will generally be of order / 1 relative to the zeroth caused by the intermediate state recoil will be small, i.e., of order diagram shown in Fig. 3. Diagrams I­IV in Fig. 4 as order D / relative to the momentum conserving piece for Fig. 3 lead to a single pair final state. Insofar as the pair since most intermediate states are off the energy shell by an is on the same transition metal site, this process is simply a amount . modification of the amplitude A0 Eq. 5 . However, often the 57 RESONANT INELASTIC X-RAY SCATTERING 11 111 has a 1/q2 piece and is of order / 1 as discussed earlier. So we conclude that diagrams I­III in Fig. 4 are roughly equal and an order of magnitude larger than the noninteract- ing expression Eq. 4 in most transition-metal compounds. In addition they can lead to charge transfer excitons. In the charge transfer case the Coulomb matrix element has an overlap factor in it which probably makes the matrix element somewhat smaller than the noninteracting overall one see Fig. 3 . Diagram III Fig. 4 also describes the coupling between a virtual absorption process and a true nonresonant Raman process. As we have already discussed the off energy shell electron-hole pair present just before the Coulomb interac- tion could be the pair produced in a ``real'' luminescence process. Since for our final state it is virtual, i.e., off the energy shell, it lives a short time decaying into the Raman- like final state by means of a Coulomb coupling. If the real final state is characterized by a sharp peak, e.g., the pair is an exciton or a plasmon as discussed below, then this type of lowest order virtual coupling to the continuum of lumines- cence states can change the line shape leading to a so-called Fano line shape.17 If the final state is a continuum, then the line shape will also be distorted in a different way which we discuss in some detail.18 Diagram IV is an interesting one. The dipole matrix ele- ment at the first vertex produces a p-like electron. Since for the transition metal sulfides and oxides this state is about 15 eV above the occupied 3d states,7 it is quite extended. This p state can scatter from the core hole falling easily into a 3d state on a neighboring transition-metal site. The annihilation FIG. 4. The first order Coulomb interaction horizontal dotted of the occupied 3d implies we have transferred a 3d electron line corrections to resonant inelastic x-ray scattering. from one site to another. However, it is important to note that this diagram does not have the 1/q2 dependence of the other important low-lying excitations involve the excitation of an three diagrams, and it is reduced by one order of magnitude electron from the cation e.g., oxygen to the anion e.g., by the quadropole matrix element involved in the final d copper on a neighboring lattice site, a so-called charge s transition, but there is no overlap as in the charge trans- transfer excitation. To zeroth order Fig. 3 in the Coulomb fer case. interaction such excitations are very weak since there is little Diagram V in Fig. 4, which is also lowest order in the overlap in the wave function, for example, of an electron on Coulomb interaction, leads directly to two pair final states, a neighboring oxygen with the localized core states of a i.e., it breaks momentum conservation and it behaves a bit Ni to create such an extended excitation. However, in dia- like19 grams I­III in Fig. 4 such excitations are present, i.e., the Auger-like pair produced by the dashed Coulomb line can be on different sites because of the substantial overlap of the k k q i . 6 outer electrons at different sites. k Diagrams I and II are shakeup process produced by the Of course the coefficient weighting the two pair shakeup intermediate state 4p electron 1s hole. The pair produced is operator is not simply unity but depends in detail as in Eq. in the final state so the Coulomb matrix element contains a 4 , on the initial photon energy as well as the exact single 1/q2 factor. In addition the final vertex is again dipole al- particle states which are excited in the final state. Such a two lowed, i.e., the 4p electron falls back into the 1s hole. Dia- pair process breaks the single pair momentum conservation gram III for Ni the initial photon resonantly creates the 4p and allows us to observe states which because of some se- state and a 1s hole. The final photon is emitted at the next lection rule are almost orthogonal to q i . It is well known vertex and the 1s core hole changes into a dipole allowed by now that resonant Raman light scattering spectra from 2p core hole. The matrix element is dipole allowed and carriers in the fractional quantum Hall regime are dominated about one to two orders of magnitude larger than the 4p by this kind of two pair shakeup process.19 More precisely, matrix element which dominates diagrams II and III. Never- such experiments rather conclusively show the presence of a theless, it has an intermediate state which is off the energy two magneto-Roton bound state. The sharp, feature is very shell by the binding of the 2p 1 keV which means that it is specific to the quantum Hall systems but the finite intensity suppressed by roughly /(E2p E1s) 10 2. The final near q 0 Å 1 only regime accessible with light is related vertex is Coulomb-like, i.e., the 2p hole annihilates creating to Fig. 4, diagram V. The lowest order two pair shakeup the particle hole pair in the valence band. The matrix element coupling mechanism is the same in both systems. 11 112 P. M. PLATZMAN AND E. D. ISAACS 57 The vertex q is shorthand for the sum of diagrams I­IV in Fig. 4, where the final state is now on the same transition- metal site. The nonresonant process has a similar structure except there A0 1 and Q0(q, ), where 0 q, 1 qQ0 q, 8 and 0 is the mean field Lindhard function. Screening the Coulomb interaction this way sums all of the terms with powers of 1/q2, i.e., it gives us a physically meaningful re- sult at low momentum transfers. The form of Eq. 7 tells us that in resonant scattering, as in nonresonant, there will be a peak at a zero of 0(q, ), i.e., a collective plasmon mode in simple metal appears with a modified strength. Since such a mode in any real material has a width, the complicated form of the vertex distorts the FIG. 5. A schematic picture of a possible two magnon inelastic shape. In addition since A0 and 1, as in the nonresonant x-ray scattering process. case, there will be incomplete screening of the single particle continuum. In a simple metal, at low q all the weight in the For strongly correlated Mott-Hubbard insulators such a spectrum is in the plasmon.7 For resonant scattering a finite two pair processes could enable us to observe a two-magnon fraction of the scattering will be in the particle hole con- process.5 More specifically two electrons are excited on tinuum. In a real sense the resonant process couples to trans- neighboring sites, with at least one in an empty p state. Then verse currents which are unscreened, and the weight in the in perturbation theory a simple Coulomb exchange process low-energy particle hole continuum is for simple metals which has quite a large matrix element allows the p electron roughly proportional to assumed to be on site 1 and have for example spin up to wind up in a d-like state on site 2 while the d electron on site 2 2 with spin down winds up in a d state on site 1 see Fig. 5 . W Q0 q, Q . 9 0 q, The two holes left in the figure reflect the fact that we have annihilated the electrons which were occupying the orbital In many instances Q0 and all of the weight is in the originally and replaced them with the electrons of opposite single particle continuum even at low momentum transfers. spin. The exchange of the two spins in the antiferromagnetic In transition-metal complexes such as NiS1.5Se0.5 the Mott-Hubbard insulator is a two magnon process whose de- screening problem is obviously much more complex. There tailed shape depends on the initial x-ray energy and on the is no well defined zero of the real part q, primarily be- momentum transferred. In the case under discussion it will cause interband transitions are strong and overlap with any also depend on the initial frequency zero of the real part. In such materials will partially screen 1 . Such x-ray experi- ments should clearly display the two magnon piece in anti- the effects of the 1/q2 matrix element and ultimately soften ferromagnetically ordered insulators. the dependence of the cross section of q, particularly at mod- Our discussion has been based on simple arguments about est (q 1 a.u.) momentum transfers. the nature of a few low order terms in a perturbation expan- sion. Such arguments make it very clear that an important RESONANT INELASTIC X-RAY SCATTERING aspect of the resonant x-ray scattering process, is that be- IN Ni S/Se...2 cause the excited p state is so high in energy, a resonant scattering in transition metal oxides can easily access all the As we have tried to make clear, the processes leading to electronic excited states including those which involve near resonant inelastic x-ray scattering are somewhat more com- or next-near neighbor overlap and those which have mag- plex and interesting than the nonresonant one. A complete netic character. Our discussion emphasizes the fact that theoretical understanding is at best very difficult. Therefore, simple energy conservation implies that the nonradiative in- in this section we will attempt to shed some light on these termediate state lifetime 1 does not limit the resolution resonant processes by describing them in the context of a but does limit the size of enhancement which is still very recent experimental result. significant. We will focus on inelastic scattering measurements in the In order to include many important higher order Coulomb classic Mott-Hubbard system Ni S/Se 2. NiS2 is an insulator effects as discussed we simply screen the bare dashed Cou- and NiSe2 is a metal. The trinary alloy Ni S1 xSex)2 has a lomb interaction by the dielectric constant tensor phase diagram which includes an insulator-metal transition and antiferromagnetism. For instance, the alloy NiS n,n (k, ), which amounts to summing bubbles as in Fig. 3. 1.5Se0.5 is For the resonant single pair scattering this implies that the a nonmagnetic insulator at room temperature and becomes amplitude is given neglecting local field effects schemati- metallic upon cooling below Tc 85 K. The main valence cally by the operator band features are the Ni(3d) electrons and the S/Se 3p states. The highest occupied state, according to LDA,20 is a half-filled Ni e g band (3d) which via strong Coulomb cor- O q relations splits into an upper and lower Hubbard band with a 1C A0 ap qaq . 7 0 q, gap of about U 5 eV.21 The antibonding S pp *, between 57 RESONANT INELASTIC X-RAY SCATTERING 11 113 FIG. 6. The absorption profile of x rays near the K edge of nickel in NiS2. the upper and lower Hubbard bands, then becomes the high- est occupied state.22 In this picture, NiS2 is a charge transfer insulator with a gap of about 2 eV Refs. 14 and 18 between FIG. 7. The resonant inelastic spectrum of NiS1.5Se0.5 for 1 the S pp * and the upper Hubbard band. In the inelastic 8344 eV. x-ray scattering measurements of Ref. 23, with an energy resolution 1 eV, we focus on electronic excitations among The well defined feature at 5 eV tracks the incident energy, these valence states at intermediate energy transfers.23 i.e., it has a roughly constant energy loss. The peak intensity We start our discussion of experimental results with a also decreases rapidly as the incident energy is varied from description of the x-ray absorption, which is directly related 8.344 keV and cannot be detected above background when to the first vertex in Fig. 3. All of the data we show were 1 is / 5 from 8.344 keV. measured at the dedicated inelastic scattering beamline X21 Figure 8 shows the spectrum taken with an energy of at the National Synchrotron Light Source. Figure 6 shows 8.344 keV plotted versus E for four different momentum the absorption profile near the Ni K edge in NiS2 measured transfers. The data is shown out to E 15 eV in order to by monitoring the Ni K emission lines 7.478 and 7.461 emphasize the peak at 5 eV. Here, the quasielastic scattering keV with an energy dispersive detector. As discussed above, is centered at E 0 eV and the large positive slope at the the feature at 8.344 keV can be thought of as a dipole tran- higher energy-loss is the tail of the 3d-4p excitation de- sition from a Ni 1s core state to an unoccupied scribed in the previous paragraph see Fig 7 . In addition to Cu (Z 1) 4p band state. We can also see the weaker the peak at 5 eV we also note a weaker peak at 9.5 eV. The feature at 8.3325 keV, which can be ascribed to the quadru- momentum transfer dependence of the spectra in Fig. 8 is polar 1s to 3d transition. The ratio in intensities of these seen as both a variation in the position and intensity of the two features of roughly a factor of 50 is consistent with our peak near 5 eV. At q 1.5 Å 1 the peak position is E LDA estimate of the matrix elements. The resonance en- 5.5 eV and at q 4.2 Å 1 the peak position has decreased hancement occurs at the 8.344 keV feature. to E 5 eV or by about one-half a linewidth. Perhaps more Figure 7 shows inelastic scattering of x rays in NiS1.5Se0.5 with the incident energy tuned to the peak of the absorption at 8.344 keV. The energy of the scattered photons was ana- lyzed with a spherically bent 1 m radius Si 553 crystal placed 1 m from the sample. Since the intrinsic resolution of the Si 553 is approximately 50 meV, the energy resolution of 1 eV was determined by a combination of the incident energy resolution 0.7 eV at 8.4 keV as determined by the Si 220 monochromator and the size of the x-ray spot on the sample. The very bright peak at zero energy loss ( E E E0 0) is quasielastic scattering from the sample, which includes contributions from phonons whose energies 100 meV are too small to resolve.7 The remaining features are resonant inelastic x-ray scattering. The rather broad peak centered at an energy loss of 15 eV we ascribe to the excitation of a 3d electron on the nickel to the empty 4p state. This broad peak has only a weak q dependence which would indicate that it is dominated by the direct process Fig. 3 . In addition it is so broad that it is difficult to determine how it moves with 1 . FIG. 8. The q dependence of the charge transfer exciton in The spectrum below the energy loss is more interesting. NiS1.5Se0.5. 11 114 P. M. PLATZMAN AND E. D. ISAACS 57 dramatic, is that over the same momentum-transfer range, of enhanced cross sections. Relaxed selection rules at finite the integrated intensity drops by roughly an order of magni- momentum transfer and the presence of a core hole in the tude. The peak at 9.5 eV is too weak to make quantitative intermediate state can lead to very interesting final states statements regarding its q dependence, but does seem to de- including, for example, the charge-transfer excitations ob- crease in intensity as q increases. The rapid decrease of in- served in NiS1.5Se0.5. We have described how the dispersion tensity with q ties these features in the Raman spectrum to of such excited states arises in a perturbation expansion of the Coulomb shakeup process described in Fig. 4, diagrams the Coulomb interactions among electrons. That this is the I­III. Diagram I, for instance, would give rise to a 1/qn case is made clear with the measurements in NiS1.5Se0.5, behavior where 2 n 4. The exact value of n will clearly where the dispersion of a charge-transfer-like excitation is depend on screening effects and on the range of q relative to observed and the spectral weight of that excitation decreases the screening length. In NiS strongly with increasing q. Such a decrease is consistent 1.5Se0.5 it was observed that n 3. with the Coulomb coupling picture. We tentatively identify the feature at 5.5 eV with a charge We have stressed the fact, that independent of complex transfer exciton, associated with the complex consisting of many-body interactions, that if there is one photon in and Ni and a S/Se Ni . This object in the phenomenology one photon out, whether on or off resonance, the many-body system being probed is left in an excited state with momen- of the Hubbard model, has a energy electron notation E tum q and energy . This leads us to the conclusion that the D P UD UP . Here D and P are the one electron width of the features observed in the elastic spectrum are energies and UD and UP the Hubbard on site repulsive en- independent of the effects that lead to the core hole lifetime ergies of an added carrier. Zhang et al.7,24 have analyzed . We predict that the spectral width of any excitation such dispersion properties of such excitons for Sr2CuO2Cl2. The as a plasmon or an exciton at a band edge in a semiconduc- feature at 9.5 eV we tentatively identify with a transition tor, for instance, will be determined only by its own decay from a S/Se pp or bonding orbital to an unoccupied S/Se time and not the core hole lifetime. pp * antibonding orbital. This peak is not seen in pure NiS2 possibly due to the fact that the splitting between the S bond- ACKNOWLEDGMENTS ing and antibonding states is larger than for the Ni S/Se 2 and therefore it is lost in the relatively stronger 3d 4p The authors would like to thank Fu-Chun Zhang for many excitation. interesting discussions about excitons in transition-metal compounds. We also thank D. R. Hamann for making his CONCLUSION LDA atomic wave function program available to us and for a discussion of momentum conservation and core hole band Resonant inelastic x-ray scattering makes it possible to widths. Thanks is also due to Chi-Chang Kao and Clem probe electronic excitations of the outer electrons in a broad Burns for informative discussions of his inelastic Raman range of materials previously inaccessible to x rays because data and for his help at beamline X-21 at the NSLS. 1 E. D. Isaacs and P. M. Platzman, Phys. Today 49 2 , 40 1996 . 9 P. Nozieres and E. Abrahams, Phys. Rev. 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