Resonant inelastic x-ray scattering studies of elementary excitations

Luuk J. P. Ament, Michel van Veenendaal, Thomas P. Devereaux, John P. Hill, and Jeroen van den Brink

Rev. Mod. Phys. 83, 705 – Published 24 June 2011

Abstract

In the past decade, resonant inelastic x-ray scattering (RIXS) has made remarkable progress as a spectroscopic technique. This is a direct result of the availability of high-brilliance synchrotron x-ray radiation sources and of advanced photon detection instrumentation. The technique’s unique capability to probe elementary excitations in complex materials by measuring their energy, momentum, and polarization dependence has brought RIXS to the forefront of experimental photon science. Both the experimental and theoretical RIXS investigations of the past decade are reviewed, focusing on those determining the low-energy charge, spin, orbital, and lattice excitations of solids. The fundamentals of RIXS as an experimental method are presented and then the theoretical state of affairs, its recent developments, and the different (approximate) methods to compute the dynamical RIXS response are reviewed. The last decade’s body of experimental RIXS data and its interpretation is surveyed, with an emphasis on RIXS studies of correlated electron systems, especially transition-metal compounds. Finally, the promise that RIXS holds for the near future is discussed, particularly in view of the advent of x-ray laser photon sources.

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Received 13 April 2010

DOI:https://doi.org/10.1103/RevModPhys.83.705

Authors & Affiliations

Luuk J. P. Ament

Institute-Lorentz for Theoretical Physics, Universiteit Leiden, 2300 RA Leiden, The Netherlands

Michel van Veenendaal

Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA and Department of Physics, Northern Illinois University, De Kalb, Illinois 60115, USA

Thomas P. Devereaux

Stanford Institute for Materials and Energy Sciences, Stanford University and SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA

John P. Hill

Department of Condensed Matter Physics and Materials Science, Brookhaven National Laboratory, Upton, New York 11973, USA

Jeroen van den Brink

Institute for Theoretical Solid State Physics, IFW Dresden, 01069 Dresden, Germany

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Vol. 83, Iss. 2 — April - June 2011

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Figure 1
Kinetic energy and momentum carried by the different elementary particles that are often used for inelastic scattering experiments. These determine the scattering phase space—the range of energies and momenta that can be transferred in a scattering event—of x rays, electrons, and neutrons.Reuse & Permissions
Figure 2
Energy of $K$ , $L_{1}$ , $L_{3}$ , $M_{1}$ , and $M_{5}$ x-ray absorption edges as a function of atomic number $Z$ . X-ray energies below 1 keV are referred to as soft, above as hard.Reuse & Permissions
Figure 3
Hard x-ray regime: Development of RIXS resolution, count rate, and resulting orders of magnitude increase in the figure of merit (count rate and resolution) as measured by the 6 eV feature in ${CuGeO}_{3}$ at the Cu $K$ edge (8990 eV).Reuse & Permissions
Figure 4
Progress in soft x-ray RIXS resolution at the Cu $L$ edge at 931 eV (a) (105), BLBB at the Photon Factory; (b) I511-3 at MAX II (56); (c) AXES at ID08, ESRF (72); (d) AXES at ID08, ESRF (22); (e) SAXES at SLS (71). Figure courtesy of 71.Reuse & Permissions
Figure 5
Resonant inelastic x-ray scattering (RIXS). Left: Number of publications on RIXS in 5 yr periods. Right: Number of citations to RIXS publications in 5 yr periods. From ISI Web of Knowledge.Reuse & Permissions
Figure 6
Different elementary excitations in condensed matter systems and their approximate energy scales in strongly correlated electron materials such as transition-metal oxides.Reuse & Permissions
Figure 7
In a direct RIXS process the incoming x rays excite an electron from a deep-lying core level into the empty valence band. The empty core state is then filled by an electron from the occupied states under the emission of an x ray. This RIXS process creates a valence excitation with momentum $ℏ k^{'} - ℏ k$ and energy $ℏ ω_{k} - ℏ ω_{k^{'}}$ .Reuse & Permissions
Figure 8
In an indirect RIXS process, an electron is excited from a deep-lying core level into the valence shell. Excitations are created through the Coulomb interaction $U_{c}$ between the core hole (and in some cases the excited electron) and the valence electrons. The photoelectron subsequently decays leaving behind an excitation in the valence band with momentum $ℏ k^{'} - ℏ k$ and energy $ℏ ω_{k} - ℏ ω_{k^{'}}$ .Reuse & Permissions
Figure 9
Two classes of inelastic energy-loss features in the RIXS spectrum of ${CuB}_{2} O_{4}$ . The RIXS intensity is represented on a color scale vs transferred energy (energy loss) $ℏ ω$ and incident energy $ℏ ω_{k}$ . The zero-loss line is the vertical line at zero transferred energy. X-ray Raman features are parallel to that line and tend to resonate strongly at a specific incident energy, for instance, the $ℏ ω = 2 eV$ loss feature resonating at $ℏ ω_{k} = 8992 eV$ . Emission lines appear as diagonal features in this $ℏ ω − ℏ ω_{k}$ plot because in this case energy of the emitted photon $ℏ ω_{k^{'}}$ is roughly constant. From 87.Reuse & Permissions
Figure 10
Relative intensity of charge-transfer spectral weight in NiO (defined as all intensity between 5 and 20 eV energy loss) compared to the whole RIXS spectrum. The charge-transfer processes dominate for incident energies more than 4 eV above the peak in the absorption. From 74.Reuse & Permissions
Figure 11
Comparison of RIXS spectra on ${La}_{2} {CuO}_{4}$ taken at different total $q$ positions but with an equivalent reduced wave vector $κ$ for seven different $q$ ’s corresponding to $κ = (0, 0)$ , $(π, 0)$ , and $(π, π)$ , from top to bottom. The two-dimensional reciprocal space net is shown in the top panel. The same symbols are used both to denote the $q$ position in reciprocal space and to show the spectra. All data were obtained with $ℏ ω_{k} = 8991 eV$ . From 130.Reuse & Permissions
Figure 12
The schematic layout of the SAXES spectrometer, at the Swiss Light Source. It is based on a variable-line-spacing spherical grating and a two-dimensional position sensitive CCD detector. (a) The main optical parameter symbols are defined. (b) The main mechanical components are sketched together with their degrees of freedom. From 75.Reuse & Permissions
Figure 13
The mechanical design of SAXES. The sample is placed at the far right. The diffraction grating is in the vacuum chamber $\sim 1 m$ from the sample. The CCD is inclined at an angle of 20° and is 5.1 m from the sample. The whole spectrometer is supported by a single steel girder and can rotate around a vertical axis about the sample on air cushions. This allows the momentum transfer to be varied. From 78.Reuse & Permissions
Figure 14
The calculated resolving power and energy resolution of SAXES. Top panel: The expected energy resolution for various openings of the entrance slit. Bottom panel: The corresponding values of the resolving power, $E / Δ E$ . From 78.Reuse & Permissions
Figure 15
Schematic of optical components in the 30 IDB inelastic x-ray scattering beam line at the Advanced Photon Source, USA. Figure courtesy of D. Casa (APS).Reuse & Permissions
Figure 16
The MERIX spectrometer for hard x-ray inelastic x-ray scattering at 30 IDB, Advanced Photon Source, USA. The sample is located at the center of the instrument and the beam is incident from the right-hand side of the figure. The red cones show the path of the scattered light onto the analyzer and back onto the detector. There are 3 degrees of freedom on the sample and the detector arm may move in both the vertical and the horizontal planes. The detector is typically a position sensitive detector. Figure courtesy of S. Coburn (BNL).Reuse & Permissions
Figure 17
In the theory of RIXS, the scattering amplitude $F_{f g}$ occurring in the Kramers-Heisenberg equation is separated into several pieces. One can split off the angular and polarization dependence $T^{x}$ , leaving fundamental scattering amplitudes $F^{x}$ . Several approximation schemes then break down these scattering amplitudes into a resonance function $P$ and effective transition operators $W^{x}$ .Reuse & Permissions
Figure 18
Theoretical approach to the intermediate-state propagator, classifying direct and indirect RIXS processes and common approximations to the propagator.Reuse & Permissions
Figure 19
Feynman diagram for the transition probability in an indirect RIXS process in the Born approximation. Green’s functions for Cu $1 s$ , $4 p$ , and $3 d$ electrons correspond to the solid lines labeled $1 s$ , $4 p$ , and $3 d$ , respectively. The wavy and broken lines represent the photon propagator and core-hole potential $U_{1 s, 3 d}$ , respectively. The shaded triangles are the effective scattering vertex renormalized interactions between the valence electrons in the $3 d$ shell. From 164.Reuse & Permissions
Figure 20
Indirect RIXS scattering process for observing charge-transfer excitations at a metal $K$ edge, illustrated for the case of a copper oxide. From 98.Reuse & Permissions
Figure 21
Experimental Ni $L_{3}$ RIXS spectra measured with 750 meV combined resolution. The lower left panel shows the Ni $L_{3}$ absorption spectrum measured with the same incident energy resolution and the labels of the excitation energies used for the RIXS spectra. The polarization of the incident photons was V pol, i.e., perpendicular to the scattering plane. From 74.Reuse & Permissions
Figure 22
The RIXS and absorption spectra calculated with the Anderson-impurity model at the same excitation energies used in Fig. 21. Symbols: Dashed lines are V pol; solid lines are H pol. The spectra were broadened with a 500 meV FWHM Guassian in the final state. From 74.Reuse & Permissions
Figure 23
Pseudocolor plot of Cu $K$ -edge RIXS intensity as a function of momentum and energy transfer ( $ℏ ω_{k} = 8982 eV$ ) as measured in the 1D corner-sharing cuprate, ${SrCuO}_{2}$ . The highly dispersive feature has been interpreted as a holon-antiholon continuum. From 126.Reuse & Permissions
Figure 24
Dispersion of CT excitations for optimally doped Hg1201 (squares), compared with previous results below 6 eV for ${Ca}_{2} {CuO}_{2} {Cl}_{2}$ (triangles) (91) and ${La}_{2} {CuO}_{4}$ (circles) (127), and new data for ${La}_{2} {CuO}_{4}$ (crosses). From 148.Reuse & Permissions
Figure 25
Charge-transfer excitations in NiO as a function of pressure. Data were taken in a powder sample at $q = 6 Å^{- 1}$ with the incident energy tuned to the preedge resonance. Bottom dashed line: nonresonant background. As the pressure is increased the resonant intensity decreases and the 5.3 and 8.5 eV structures are seen to smear out due to the increasing electronic bandwidth. From 196.Reuse & Permissions
Figure 26
(a) RIXS dispersion in ${La}_{2} {NiO}_{4}$ along the $[100] = [010]$ direction. The spectra have been shifted vertically; the baseline is indicated for each spectrum. Solid line: three-point smoothed guide for the eye; dotted line: Brillouin zone (BZ) edge spectrum; open circles label crystal-field excitations. (b) Differences between smoothed spectra at various points in the BZ and the BZ edge spectrum from (a). (c) Dispersion of the exciton. From 44.Reuse & Permissions
Figure 27
Comparison between the calculated Anderson model results and the experiment in the charge-transfer excitation region as measured by $L$ -edge RIXS in MnO. (A)–(C) Incident energies in the vicinity of the peak of the $L_{3}$ resonance. (H) The peak of the $L_{2}$ resonance. From 75.Reuse & Permissions
Figure 28
Temperature dependence of the $K$ -edge RIXS intensity at fixed energy transfer of 2.25 eV in ${La}_{0.8} {Sr}_{0.2} {MnO}_{3}$ . The arrow indicates the insulator to ferromagnetic metal transition temperature ( $T_{C}$ ). From 113.Reuse & Permissions
Figure 29
The result of dividing the energy denominator, Eq. (54), out from a series of incident energy spectra for the molecular orbital charge-transfer excitation at 6 eV in ${La}_{2} {CuO}_{4}$ . The thick line is a Gaussian fit. From 1.Reuse & Permissions
Figure 30
Comparison of $S (ω)$ with the measured $Im [- 1 / ϵ (ω)]$ for various cuprates. Different symbols are used to denote RIXS spectra obtained with different incident energies. EELS data from the literature are also shown for comparison. These incident photon energies (in eV) are (a) 8997.5 (▪), 8998.5 (◯), 8999.5 (▵); (b) 8988 (▪), 8990 (◯), 8992 (▵), 8994 (□); (c) 8992 (▪), 8993 (◯), 8994 (□); (d) 8991 (▪), 8992 (◯), 8993 (▵), 8994 (◯); and (e) 8993 (▪), 8994 (◯), 8995 (□). The EELS data for ${CuGeO}_{3}$ , ${La}_{2} {CuO}_{4}$ , and ${Sr}_{2} {CuO}_{2} {Cl}_{2}$ are from 210, 12, and 161, respectively. Although there is no EELS spectrum for ${Sr}_{2} {Cu}_{3} O_{4} {Cl}_{2}$ available in the literature, that of ${Ba}_{2} {Cu}_{3} O_{4} {Cl}_{2}$ is shown (133). From 124.Reuse & Permissions
Figure 31
Polarization dependent soft x-ray Raman scattering spectra of ${TiO}_{2}$ plotted relative to the elastic peaks located at zero. Five vertical dotted lines between 5 and 10 eV indicate Raman scattering to the multiple splitting $3 d^{1} L^{- 1}$ nonbonding states, whereas a higher energy shift line around 14 eV indicates Raman scattering to the antibonding state. Solid bars on each spectrum indicate peak positions of the fluorescence lines. From 88.Reuse & Permissions
Figure 32
Upper panel: RIXS spectra of ${Sr}_{2} {CuO}_{2} {Cl}_{2}$ measured at grazing and normal incidence with V and H polarizations ([GI,V], [NI,V], [NI,H]). Bottom panel: Calculated spectra for the same scattering geometries. From 72.Reuse & Permissions
Figure 33
RIXS intensity for scattering from a single ${Cu}^{2 +}$ ion with initial state $3 d_{x^{2} - y^{2}}$ and spin up along the $z$ axis. From 228.Reuse & Permissions
Figure 34
Schematic illustration of the orbital wave. Top: the electron cloud in the ground state. Bottom: a snapshot of the electron cloud when an orbital wave is excited. From 181.Reuse & Permissions
Figure 35
RIXS spectra at the Ti $L_{3}$ edge of ${LaTiO}_{3}$ and ${YTiO}_{3}$ . For the La compound, $q$ is directed along the [110] direction, while for the Y compound, $q$ is directed along the [100] direction. The incident photons were $π$ polarized. (a), (b) The temperature dependence at scattering angle $2 θ = 90 °$ . (c), (d) The dependence on scattering angle $2 θ$ . From 220.Reuse & Permissions
Figure 36
$K$ -edge RIXS spectrum of orbiton excitations in the $A$ -type antiferromagnetic phase of a 2D $e_{g}$ system. From 67.Reuse & Permissions
Figure 37
Indirect, magnetic RIXS mechanism for the ${Cu}^{2 +}$ $K$ edge. In the intermediate state, the $3 d$ holes either screen or do not screen the core hole. The Coulomb repulsion $U$ is effectively replaced by $U^{'}$ , incorporating the effect of the core-hole potential $U_{c}$ . Note that the open circles represent holes, and the arrows indicate the spin direction. From 66.Reuse & Permissions
Figure 38
RIXS data after subtraction of the elastic signal. The open diamonds indicate the center of mass, obtained by fitting a Gaussian to the spectra. Theoretical one-triplon (dashed line) and two-triplon (full lines) dispersions are also shown. The right axis is scaled in units of joule along the rungs, extracted from a single-band Hubbard model. From 186.Reuse & Permissions
Figure 39
Single-magnon RIXS data at the copper $L_{3}$ edge of ${La}_{2} {CuO}_{4}$ with an energy resolution of 140 meV. The transferred momentum (projected on the copper-oxide plane) is along the $(π, 0)$ direction. The decomposition of the low-energy spectrum is shown, and it is found that the maximum follows the single-magnon dispersion. From 23.Reuse & Permissions
Figure 40
Comparing single-magnon neutron data and RIXS measurements at the copper $L_{3}$ edge of ${La}_{2} {CuO}_{4}$ . Details as in Fig. 39. From 23.Reuse & Permissions
Figure 41
The RIXS spectra of the underdoped cuprate ${La}_{1.92} {Sr}_{0.08} {CuO}_{4}$ shows two dispersing branches at low temperature (a), which merge at room temperature (b). The lines in (c) display the corresponding neutron data. The data were taken at the copper $L_{3}$ edge with an energy resolution of 140 meV. The transferred momentum (projected on the copper-oxide plane) is along the $(π, 0)$ direction. From 23.Reuse & Permissions
Figure 42
Theoretical RIXS spectrum of ${Sr}_{2} {IrO}_{4}$ at the Ir $L$ edge for $π$ and $σ$ polarization in 90° scattering geometry, and where $q$ is in the first Brillouin zone. The broad peak at 10 J is due to transitions between the spin-orbital multiplets. From this peak one can determine the spin-orbit coupling $λ$ . At low energy, strongly dispersive single- and double-magnon features are seen. From 9.Reuse & Permissions
Figure 43
RIXS spectra at the copper $1 s \to 4 p$ edge for a few points in the two-dimensional Brillouin zone of ${La}_{2} {CuO}_{4}$ (97; 63). (a) is taken with $π$ polarization at room temperature, (b) with $σ$ polarization at 45 K. The excitation energy is 8994 eV. The solid lines are Lorentzians fitted to the data. The elastic peak has been subtracted from these data. Note that in the figure the reduced momentum transfer ( $κ$ in the text) is denoted by $q$ . From 63.Reuse & Permissions
Figure 44
Bimagnon RIXS spectra obtained with interacting spin wave theory for a nearest-neighbor $S = 1 / 2$ Heisenberg model. The vertical axis indicates transferred momentum projected to the ${CuO}_{2}$ plane and $ω$ is the energy loss. For the 2D square lattice, $z = 4$ . From 162.Reuse & Permissions
Figure 45
Experimental (dots) and theoretical (solid lines) RIXS spectra of CoO at the Co $L$ edges (154). The theoretical curve is a broadening of the vertical bars. The excitation energy runs from 778.1 eV (a) to 785.2 eV (e) at the $L_{3}$ edge, and 795.0 eV (f) is at the $L_{2}$ edge. The work of 39 includes similar spectra with a slightly better combined energy resolution. From 154.Reuse & Permissions
Figure 46
Left panel: Peak brightness; right panel: average brightness of present and future x-ray sources. Present third generation synchrotron light sources average brightnesses are in the $10^{20}$ $phot / s / 0.1 % BW / {mm}^{2} / {mrad}^{2}$ range. A brilliance enhancement by 2 orders of magnitude is envisaged for ultimate x-ray storage ring-based light sources and energy recovery linacs (ERLs). Free-electron lasers (FELs) based on self-amplification of spontaneous emission (SASE) are pulsed and can enhance the peak brightness by $\sim 8$ orders of magnitude [see 184, and references therein]. Feasibility studies show that the x-ray FEL oscillator can greatly enhance the average x-ray brightness (125). Figure courtesy of K.-J. Kim and R. Lindberg.Reuse & Permissions

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