Uncovering selective excitations using the resonant profile of indirect inelastic x-ray scattering in correlated materials: Observing two-magnon scattering and relation to the dynamical structure factor
C. J. Jia, C.-C. Chen, A. P. Sorini, B. Moritz, T. P. Devereaux
UUncovering selective excitations using the resonantprofile of indirect inelastic x-ray scattering incorrelated materials: Observing two-magnonscattering and relation to the dynamical structurefactor
C. J. Jia , , C.-C. Chen , A. P. Sorini , B. Moritz , , T. P.Devereaux , SIMES, SLAC National Accelerator Laboratory, Menlo Park, California 94025,USA Department of Applied Physics, Stanford University, Stanford, California94305, USA Lawrence Livermore National Laboratory, Livermore, CA 94550, USA Department of Physics and Astrophysics, University of North Dakota, GrandForks, ND 58202, USA Geballe Laboratory for Advanced Materials, Stanford University, Stanford,California 94305, USA
Abstract.
Resonant inelastic x-ray scattering (RIXS) is a spectroscopictechnique which has been widely used to study various elementary excitationsin correlated and other condensed matter systems. For strongly correlatedmaterials, besides boosting the overall signal the dependence of the resonantprofile on incident photon energy is still not fully understood. Previous endeavorsin connecting indirect RIXS, such as Cu K -edge for example where scatteringtakes place only via the core-hole created as an intermediate state, with thecharge dynamical structure factor S ( q, ω ) neglected complicated dependence onthe intermediate state configuration. To resolve this issue, we performed an exactdiagonalization study of the RIXS cross-section using the single-band Hubbardmodel by fully addressing the intermediate state contribution. Our results arerelevant to indirect RIXS in correlated materials, such as high Tc cuprates.We demonstrate that RIXS spectra can be reduced to S ( q, ω ) when there is noscreening channel for the core-hole potential in the intermediate state. We alsoshow that two-magnon excitations are highlighted at the resonant photon energywhen the core-hole potential in the corresponding intermediate state is poorlyscreened. Our results demonstrate that different elementary excitations can beemphasized at different intermediate states, such that selecting the exact incidentenergy is critical when trying to capture a particular elementary excitation.PACS numbers: 78.70.Ck, 74.72 -h, 78.20 Bh a r X i v : . [ c ond - m a t . s t r- e l ] D ec ncovering selective excitations using RIXS in correlated materials
1. Introduction
Resonant inelastic x-ray scattering (RIXS) is a widely used technique for studyingvarious condensed matter materials, including semiconductors, metals, oxides, andtransition-metal compounds[1, 2]. Because of its many advantages, including bulksensitivity, orbital and elemental specificity and polarization control, RIXS has movedto the forefront of important spectroscopic techniques in recent years[3]. While ingeneral, the probability of x-rays to be scattered from a solid is small, the cross sectioncan be enhanced by orders of magnitude when tuning the incoming x-ray energy to aresonance of the system where an electron from a deep lying core state is excited intothe valence shell. The resulting intermediate core-hole state can connect the many-body ground and final excited states through pathways that may be inaccessible usingother non-resonant x-ray probes, thus providing a wealth of new information aboutcharge, spin, lattice and orbital degrees of freedom. This is particularly importantin strongly correlated transition-metal oxides where these degrees of freedom areheavily intertwined. In this way, RIXS can offer complementary investigation ofelementary excitations along with resonant diffraction, neutron scattering, angleresolved photoemission spectroscopy(ARPES) and scanning tunneling microscopy.In particular, as resonant diffraction yields understanding of the ground state ofcorrelated materials, we show that RIXS can provide the same sort of understanding ofexcited state properties. By making explicit use of the resonance profile and associatedpathways, one can understand simpler density response in even the most stronglycorrelated systems and the more complicated shake-up processes involved in RIXS.RIXS spectra display both incident photon energy and energy loss dependence,where the underlying multi-particle process makes it hard to interpret the experimentalcross-section[4]. One of the main efforts to better understand this cross-section forcorrelated materials has been devoted to disentangling the resonant profile and studyits response function, which only depends on the energy loss. As proposed undercertain approximations involving intermediate states for indirect RIXS, (the caseas the Cu K -edge for example where scattering takes place only via the core-holecreated as an intermediate state)[5, 6, 7], the cross-section could be decomposed intoa Lorentzian resonant profile and a response function, which is directly connectedto the charge dynamical structure factor S ( q , ω ). Experimentally, in most of theendeavors to interpret the response function of RIXS spectra for correlated materials,either a particular incident photon energy has been considered[8, 9, 10] or the incidentenergy dependence was treated using this Lorentzian profile derived from theoreticalapproximations[5]. However, these experiments do not provide definitive proof thatthe response function can be directly related to S ( q , ω ), and the validity of the aboveapproximations still need to be discussed for particular cases. In particular, for theCu K -edge measurements on various cooper oxides, RIXS cast in terms of the non-resonant response applies for some materials, but strong deviations were found forother materials[11]. This suggests, at least in correlated materials, that the validityof treating the RIXS response function as S ( q , ω ) needs to be carefully investigated.Furthermore, RIXS is known to display other elementary excitations for correlatedmaterials, such as magnon excitations[12, 13, 14], which are beyond the simple chargeresponse provided by S ( q , ω ). Thus, in concert with boosting the overall signal,the dependence of the resonant profile on incident photon energy is an importantconsideration in strongly correlated materials.In this work, we performed an exact diagonalization study of the single-band ncovering selective excitations using RIXS in correlated materials K -edge RIXS for high Tc cuprates. In two examples, we show how theincident energy dependent RIXS response can be related to the charge response S ( q , ω )and how the incident photon energy may be used to highlight two-magnon scattering.Our results show that the RIXS spectra display strong resonances at multipleincident photon energies, each of which corresponds to a different intermediate stateelectronic configuration. We find that only at particular intermediate states wherethe screening of the core-hole potential and correlation effect are not important, oneobserves a RIXS spectrum similar to S ( q , ω ). However, RIXS behaves very differentlywhen compared to S ( q , ω ) at other incident energies where the screening of thecore-hole potential is important in the corresponding intermediate state. Moreover,different elementary excitations are highlighted at different incident photon energies,in which, strong intensities for two-magnon excitations are found at the so calledpoorly-screened intermediate states. The two-magnon excitations are not shown at theintermediate state where the core-hole potential screening is absent. The significanceof emphasized two-magnon excitation at one intermediate state compared to the othersprovides a focus to the fact that tuning to a proper incident photon energy dictated bythe character of the intermediate state is critical to study the elementary excitationsin RIXS.
2. Methods
We employ the single-band Hubbard Hamiltonian that serves as an effective low energymodel for studying correlated materials, written as H = − (cid:88) i , j ,σ t i , j d † i ,σ d j ,σ + (cid:88) i U n d i , ↑ n d i , ↓ , (1)where d † i ,σ ( d i ,σ ) creates (annihilates) a fermion with spin σ on lattice site i , t i , j isthe hopping integral restricted to the nearest- and next-nearest-neighbor, denoted as t and t (cid:48) . U is the on-site Coulomb repulsion, and n d i ,σ is the fermion number operator.The momentum-dependent indirect RIXS cross section can be expressed usingthe Kramers-Heisenberg formula[15] I ( q , Ω , ω i ) = 1 π Im (cid:104) Ψ | H − E − Ω − i + | Ψ (cid:105) (2)and | Ψ (cid:105) = (cid:88) i ,σ e i q · r i D † H (cid:48) − E − ω i − i Γ D | (cid:105) , (3)where q is the momentum transfer; ω i and Ω = ω i − ω f are the incident energyand energy transfer, respectively; Γ is the inverse core-hole lifetime; E is the groundstate energy of the system absent the core-hole; | (cid:105) is the ground state wave function; H (cid:48) = H − U c (cid:80) i ,σ,σ (cid:48) n d i σ n s i ,σ (cid:48) represents the intermediate state Hamiltonian, whichincludes the core-hole potential with strength U c , with n s i ,σ denoting the core-holenumber operator; D is the dipole transition operator, which for example excites a ncovering selective excitations using RIXS in correlated materials s core-electron up to 4 p level and is formulated as D = p † i ,σ s i ,σ in transition-metal K -edge indirect RIXS. The wave function overlap of the intermediate state with thefinal/ground state is fully considered, and thus the whole resonant profile can beinvestigated using this method without further approximation. Elastic lines are trivialand have been removed in the results in the following sections.We also investigate the x-ray absorption spectroscopy (XAS) given by B ( ω i ) = 1 π Im (cid:104) | D † H (cid:48) − E − ω i − i Γ D | (cid:105) (4)in which D and H (cid:48) are the same as in the RIXS’s definition. The XAS peaks can beconnected with the resonant energies in the RIXS spectra.The charge dynamical structure factor to be compared to RIXS is S ( q , ω ) = 1 π Im (cid:104) | ρ d − q H − E − ω − i + ρ d q | (cid:105) , (5)where ρ d q = (cid:80) k ,σ d † k + q ,σ d k ,σ = (cid:80) i ,σ e i q · R i n d i ,σ is the density operator. We notethat the charge dynamical structure factor S ( q , ω ), which represents the non-resonantresponse, is an effective two-particle process; however for resonant response theintermediate core-hole state increases the complexity of the problem to an effectivefour-particle process making its interpretation more difficult[4].Our calculations are done on the 16 B Betts cluster (16-site square cluster) withperiodic boundary conditions. The ground state wave function is obtained usingthe exact diagonalization (ED) technique utilized by PARPACK(Parallel ArnoldiPACKage)[16]. This algorithm is based on the implicitly restarted arnoldi method,which takes into account properly orthogonalized eigenvectors at each step of theiteration, avoiding the problem of degeneracy collapse common in most other Lanczosmethods[17], such as those used in Ref. [18, 19] This is crucial to understandingintensities of the RIXS cross section since all intermediate states must be retainedin a properly orthonormalized way. | Ψ (cid:105) from Eq. 3 is calculated using the bi-conjugate gradient stabilized method (BiCGSTAB)[20], a variant of the bi-conjugategradient method (BiCG). In BiCGSTAB, generalized minimal residual method(usually abbreviated GMRES) is applied after each step of BiCG in order to get rid ofthe irregular convergence behavior, thus obtaining faster and smoother convergencethan regular BiCG. For obtaining the final spectrum, we also employed the continuedfraction expansion method.We use the model parameters taken from Cu K -edge RIXS in cuprate (La CuO )as an example[21]: U = 10 t , t (cid:48) = − . t , U c = 15 t and Γ = t . Our results willbe presented in units of t , with t = 0 .
3. Results
Figure 1 shows the momentum-dependent XAS and indirect RIXS spectra for thehalf-filled, 12.5% hole-doped and 12.5% electron-doped single-band Hubbard model.Zero incident energy is defined as the ground state energy E plus E edge , the energydifference between the two levels in the dipole transition process. For example, E edge equals E p − E s for Cu K -edge RIXS. For the half-filling and dopings, RIXS ncovering selective excitations using RIXS in correlated materials Figure 1. (Color online) (a)-(d) XAS and momentum-dependent indirect RIXScross-section at momentum (0,0), ( π, π ) and (0 , π ) at half-filling in the single-band Hubbard model. The filled blue arrows in (b) and (d) highlight two-magnonpeaks in the RIXS cross-section. (e)-(h) the same for 12.5% hole-doping and(i)-(l) 12.5% electron-doping, respectively. The elastic lines of RIXS have beenremoved. The incident energy window is chosen so that the main XAS peaksand RIXS features display for each doping. Zero incident energy is defined as thesum of ground state energy E and E edge . E edge is the energy difference betweenthe two levels in the dipole transition process, such as in Cu K -edge RIXS caseequaling E p − E s . displays strong resonances at the peaks of XAS, with the energies corresponding tothe difference between the ground state and intermediate state energy in the RIXS ncovering selective excitations using RIXS in correlated materials Figure 2. (Color online) Momentum-dependent RIXS and S ( q , ω ) in the half-filled single-band Hubbard model highlighting the (a) well-screened intermediatestate at incident energy E in = − . t and (b) poorly-screened intermediate stateat E in = − . t . (c) RIXS at the well-screened intermediate state incidentenergy [from (a)] vs. S ( q , ω ) multiplied by a Lorentzian prefactor given by(( ω f − ω res ) +Γ f ) − (( ω i − ω res −| U | ) +Γ i ) − , where ω res is a free parameter.[11]We consider equal incident/outgoing photon lifetimes (Γ i = Γ f ) for fitting. Notethat the elastic lines of RIXS and S ( q = , ω ) have been removed. process.The nature of the intermediate state can be understood in terms of how wellthe valence electrons screen the core-hole potential. The XAS peak at an energy ∼ − U c + U ( ∼ t for half-filling, ∼ t for 12.5% hole-doping and ∼ t for 12.5%electron-doping) corresponds to the intermediate state with two electrons bound tothe core-hole site that screen the core-hole potential. This intermediate state exhibitsstrong screening and is usually termed the “well-screened” state[22, 23]. The XASpeak at ∼ − U c ( ∼ . t for half-filling, ∼ t for 12.5% hole-doping and ∼ . t for 12.5% electron-doping) corresponds to the intermediate state where only a singleelectron is bound at the core-hole site, weakly screening the core-hole potential, usuallytermed the “poorly-screened” state[22, 23]. In both processes, the electrons experiencea strong “shake-up” by the core-hole potential via screening. On the other hand, theintermediate state at an incident energy ∼ t for the hole-doped or ∼ − t for theelectron-doped system represents the state in which the core-hole is created at anempty or a doubly-occupied site, respectively. These two states can be denoted as“unscreened” states, where the core-hole potential does not appreciably “shake-up”the ground state electronic system.We next examine the energy loss dependence of RIXS spectra on resonances inFig. 1. For half-filling (Fig. 1(b)-(d)), RIXS displays a high energy spectrum rangingfrom ∼ t up to ∼ t in energy loss for both well-screened and poorly-screenedintermediate states, signaling excitations from the lower Hubbard band (LHB) to theupper Hubbard band (UHB), which is mainly controlled by Coulomb U . The mainpeaks have strong momentum dependence with the shift to higher loss energy atlarger momentum transfer, due to the increase in phase space available for creation of ncovering selective excitations using RIXS in correlated materials ∼ − t for momentum (0 , ∼ − t at momentum (0 , π ), ∼ − t atmomentum ( π, π ). This dispersion with increasing momentum at poorly- and well-screened intermediate states can be also seen in Fig. 2. In the electron- and hole-dopedcases, the same charge transfer dispersion on UHB is observed also for the well- andpoorly-screened intermediate states. These charge transfer features are consistent withvarious experiments[5, 11, 25, 23] and calculations[18, 19, 26] carried out for the chargetransfer excitations. The RIXS spectra also exhibit distinctive structures at low energy ( ∼ − . t ),highlighted by blue arrows in Fig. 1, emphasizing excitations of the spin degree offreedom - two-magnon excitations[14, 12, 27, 13, 28, 29]. This structure is foundboth for the poorly-screened (Fig. 2(b)) and the well-screened intermediate states(Fig. 2(a)) for the undoped system, but is significantly stronger at the poorly-screenedintermediate state. In the doped system, strong intensities at low energy are alsoshown for the poorly-screened intermediate state (Fig. 3(b) and (e)). Only oneelectron is bound at the core-hole site in the poorly-screened intermediate state,thus the intermediate state wavefuncion may have a big overlap with the finalstate wavefunctions carrying two-magnon excitations; while for the well-screenedintermediate state the two bound electrons at the core-hole site make a smaller overlapwith the final states for two-magnon excitations. Previous intermediate density ofstates calculations also shown that the poorly-screened intermediate state has a largerweight connected to the low energy (magnetic) excitations[30].Our data shows a strong two-magnon peak at ( π,
0) momentum transfer, andno peak associated with two-magnon excitation at ( π, π )[14, 12]. We also obtaintwo-magnon excitations at momentum transfer (0 , , ,
0) located at ∼ . J (the effective exchange coupling J = 4 t /U and is 0 . t in our calculation), which is in agreement with the calculationtaking into account the magnon-magnon interaction[27]. The two-magnon excitationsexhibited in RIXS can be connected to the excitations revealed by the spin dynamicalstructure factor[7]. Single-magnon excitations are forbidden in this case due to spinconservation, as we have neglected spin-orbit coupling.Our calculations are consistent with the Cu K -edge experiments of the magneticexcitations taken on La CuO for the observation of strong peak at momentumtransfer ( π,
0) and no weight at momentum transfer ( π, π )[14, 12]. No experimentalconfirmation of the two-magnon feature at momentum (0 ,
0) has been reported,although recent improvements in resolution may resolve this issue. The two-magnonexcitation is significant at the poorly-screened intermediate state compared to theothers providing a focus to the fact that particular elementary excitation may behighlighted when tuning to a proper incident photon energy dictated by the characterof the intermediate state. ncovering selective excitations using RIXS in correlated materials Figure 3. (Color online) Momentum dependent RIXS and S ( q , ω ) at 12.5%hole-doping [(a) E in = − . t (b) E in = − . t (c) E in = 3 . t ] and 12.5%electron-doping [(d) E in = − . t (e) E in = − . t (f) E in = − . t ] for single-band Hubbard model. We note that the elastic lines of RIXS and S ( q = , ω )have been removed. Blue, red and green lines represent RIXS spectra with theintermediate state having the core-hole well-screened, poorly-screened and createdat an empty/doubly-occupied site, respectively. S ( q , ω )To make comparisons between RIXS and S ( q , ω ), in Fig. 2 we plot the RIXS spectrafor the well- and poorly-screened intermediate states, compared to S ( q , ω ), at half-filling. When ω > t , S ( q , ω ) (black dotted) and the RIXS spectra (colored solid)generally span the same energy range at a given momentum transfer, and both shiftto higher energy at larger momentum transfer. Nonetheless, their overall intensities arevery different: at momentum (0 , π/ , π ) and ( π, π/
2) the RIXS spectra have moreweight at lower energy and a lower threshold; at momentum ( π, π ) and ( π/ , π/
2) theRIXS spectra and S ( q , ω ) exhibit different main peaks and weights. Another strikingdifference lies in the (0 ,
0) momentum transfer. For S ( q , ω ) no energy loss excitationsare allowed, since the charge operator ρ d q at q = (see Eq. 3) becomes the total ncovering selective excitations using RIXS in correlated materials S ( q , ω ) can be connected by a simpleresonant prefactor, we compare RIXS at the well-screened intermediate state with S ( q , ω ) multiplied by a fitted Lorentzian prefactor as shown in Fig. 2(c). These resultsshow that these two spectra only agree at momentum (0 , π ) and ( π/ , π/ S ( q , ω ) agree well with one another, specificallyfor charge excitations within the LHB. This happens for the “unscreened” intermediatestates [Fig. 3(c) and (f)], in which the core-hole potential does not significantly “shake-up” the electrons, and the effect of the core-hole potential can be neglected or isless prominent. In this case, the complicated four-particle RIXS process could besimplified to the two-particle S ( q , ω ) process. Our results show that if the core-hole“shakes-up” the electrons when U c is comparable to the other energy scales, RIXSdeviates from S ( q , ω ). If there is a lack of this “shake-up” or core-hole potentialscreening, RIXS may be approximated by S ( q , ω ) with a resonant prefactor. Previoustheoretical treatment of connecting indirect RIXS the same as S ( q , ω ) works at weak U c for the series expansion[5] and strong and weak U c for the ultrashort core-holelife expansion[6, 7]. These schemes do not describe the system with an intermediate U c and fail to address the RIXS cross-section difference associated with the differentintermediate state configuration. We note that typically U c is comparable to otherenergy scales in the problem mostly connected with the Coulomb energy scale for d -electrons, and therefore an expansion may have no small parameter. However, if theintermediate state does not substantially involve these energy scales, then the approachused in Refs 6,7 would be appropriate. This is in agreement with our results: whenthe intermediate state involves a core-hole created at an empty state, the core-holeinteraction is ineffective, meaning that the RIXS cross section can be then related tothe charge response. In general however, the role of the core hole interaction must beretained for a faithful representation of the RIXS response.Experimental evidences of the core-hole potential screening on the indirect RIXSand the comparison to S ( q , ω ) have been found on various cuprate materials[11]. Forthose cuprates with less screening channels, such as 1D cuprates, RIXS shows similarresponse as S ( q , ω ); while for 2D cuprates with more screening channels the deviationsof the two spectra look more obvious.
4. Summary and discussions
In summary, we have employed exact diagonalization to study the resonance profile forindirect RIXS over a wide range of dopings and incident energies in a simple correlatedmodel. The RIXS spectra display complex incident energy dependence in the resonantprofile that offers rich information besides a simple boost of intensity. Controversialto previous understanding that RIXS is S ( q , ω ) in the charge transfer channel, wefound that RIXS could only be considered as S ( q , ω ) when the screening of core-hole potential at the corresponding intermediate state is not important. Moreover,our results show strong two-magnon excitations for poorly-screened intermediate state. ncovering selective excitations using RIXS in correlated materials K -edge RIXS orXAS data, one needs to do a convolution with the Cu 4p density of states for theincident photon energy[26]. For experimental indirect RIXS data, the spectra on oneincident energy may not be related to a single intermediate state configuration, sincethe convolution may mix the contributions from different intermediate states. Thusfor studying a certain elementary excitation in RIXS, such as two-magnon or chargedynamical structure factor, one can focus on the incident phonon energy which has thebiggest weight connecting the specific intermediate state through the convolution. Onemay also do a deconvolution for the analysis of the experimental data. We emphasisthat our purpose for this single-band Hubbard model study was to investigate howthe nature of a particular intermediate electronic structure affects the RIXS spectra,and look into the momentum dependent RIXS spectra at different intermediate states.More complicated multi-band models may also be needed to address more elementaryexcitations such as the dd excitations, although less momentum points are accessiblesince we can only study smaller cluster as more bands are included in the Hamiltonian.For cuprate materials in reality, the intermediate state also has ligand characterwhich beyond our down-folded model, but the intermediate state contribution toRIXS spectra in terms of the core-hole potential screening remains an importantobservation[11]. Our model also well addresses the two-magnon excitaions and chargetransfer dispersion as in experiments for cuprates.As the single-band Hubbard model carries the key character of correlatedmaterials, our conclusion is not limited in cuprates but could also be generalizedto all correlated materials in understanding indirect RIXS response on general forms.When the core-hole potential is comparable to other energy scales, the importanceof the intermediate state configuration cannot be ignored and the incident energydependence must be considered. The screening of the core-hole potential is a key factorfor selecting the best incident photon energy to study certain elementary excitations. Acknowledgments
We would like to thank J. P. Hill for valuable discussions. This work was supported atSLAC and Stanford University by the U.S. Department of Energy, Office of BasicEnergy Sciences, Division of Materials Science and Engineering, under ContractNo. DE-AC02-76SF00515 and by the Computational Materials and Chemical SciencesNetwork (CMCSN) under Contract No. DE-FG02-08ER46540. C. J. Jia is alsosupported by the Stanford Graduate Fellows in Science and Engineering. A portionof the computational work was performed using the resources of the National EnergyResearch Scientific Computing Center (NERSC) supported by the U.S. Departmentof Energy, Office of Science, under Contract No. DE-AC02-05CH11231.
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