Summary: Resonant inelastic x-ray scattering (RIXS) is a fast developing experimental technique in which one scatters x-ray photons inelastically off matter. It is a photon-in, photon-out spectroscopy for which one can, in principle, measure the energy, momentum, and polarization change of the scattered photon. The change in energy, momentum, and polarization of the photon are transferred to intrinsic excitations of the material under study and thus RIXS provides information about those excitations.

RIXS is a resonant technique in which the energy of the incident photon is chosen such that it coincides with, and hence resonates with, one of the atomic x-ray transitions of the system. The resonance can greatly enhance the inelastic scattering cross section, sometimes by many orders of magnitude, and orders a unique way to probe charge, magnetic, and orbital degrees of freedom on selective atomic sites in a crystal.

The microscopic picture of the resonant inelastic x-ray scattering process is most easily explained in terms of an example as shown in Fig 1. In a copper-oxide material, for instance, one can tune the incoming photon energy to resonate with the copper K, L or M absorption edges, where in each case the incident photon promotes a different type of core electron into an empty valence shell. The electronic configuration of Cu²⁺ is 1s²2s²2p⁶3s²3p⁶3d⁹, with the partially filled 3d valence shell characteristic of transition metal ions. The copper K-edge transition 1s -> 4p, is around 9000 eV and in the hard x-ray regime. The L_2;3-edge 2p -> 3d (~ 900 eV) and M_2;3-edge 3p -> 3d (~ 80 eV) are soft x-ray transitions. Alternatively, by tuning to the Oxygen K- edge, one can choose to promote an O 1s to an empty 2p valence state, which takes ~ 500 eV. After absorbing a soft or hard x-ray photon, the system is in a highly energetic, unstable state: a hole deep in the electronic core is present. The system quickly decays from this intermediate state, typically within 1-2 femtoseconds. The net result is a final state with an electron-hole excitation, since an electron was created in an empty valence band and a hole in the filled conduction band. The electron-hole excitation can propagate through the material, carrying momentum and energy.

Fig. 1. Two different types of resonant inelastic x-ray scattering, occurring via direct or indirect light scattering involving the valence electrons. In a direct RIXS process (left panel) the incoming x-rays excite an electron from a deep-lying core level into the empty valence. 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 definite momentum and energy. In an indirect RIXS process (right panel), an electron is excited from a deep-lying core level into the valence shell. Excitations are created through the Coulomb interaction Uc between the core hole (and in some cases the excited electron) and the valence electrons.

The main limitation faced previously that has limited its usefulness is that the RIXS process is photon hungry, that is, it requires a substantial incident photon flux to obtain enough scattered photons to collect spectra with a high enough resolution in energy and momentum in a reasonable time. With a required resolving power (defined as the incident photon energy divided by the energy resolution) of four orders of magnitude, RIXS has been a real challenge. Up until a few years ago this has limited RIXS experiments to measuring energy losses on the order of half an electron volt or greater. Thus neutron scattering and angle-resolved photoemission spectroscopy offered a more direct examination of the low energy excitations near the Fermi level. However, recent progress in RIXS instrumentation has been dramatic and this situation is now changing.

In the past decade, 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 advancement in resolution is shown in Fig. 2. For the first time, single magnon excitations on the order of hundreds of millielectron volts can be viewed directly with RIXS, to be complementary to neutron scattering investigations.

Fig. 2. Progress in soft x-ray RIXS resolution at the Cu Ledgeat 931 eV. The broad peak around 2 eV constitute d-d excitations, while the lower energy peak represent phonons, magnons and bi-magnons. Figure courtesy of G. Ghiringhelli and L. Braicovich.

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. In a recent review article submitted to Reviews of Modern Physics by L. J. P. Ament, Michel van Veenendaal, Thomas P. Devereaux, John P. Hill, and Jeroen van den Brink, all of the CMSN RIXS CRT, 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, with a strong focus on strongly correlated materials.

The bulk of the interesting transition metal oxide compounds, including the cuprates, nickelates and manganites are all in the charge transfer limit. This means the lowest lying excitations across the optical gap are charge transfer excitations and therefore these are of central importance in these materials. Key questions include the size of the gap (typically on the order of a few eV) and the nature of the excitations - do they form bound exciton states? Are these localized or can they propagate through the lattice? What are their lifetimes, symmetries and temperature dependence, etc. While some studies have been performed using other techniques, notably EELS and optical conductivity measurements, RIXS offers a powerful probe for many of these questions and has been applied extensively.

The review article presents the fundamentals of RIXS as an experimental method and then reviews the theoretical state of affairs, its recent developments and discusses the different (approximate) methods to compute the dynamical RIXS response. 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 article discusses the promise that RIXS holds for the near future, particularly in view of the advent of x-ray laser photon sources. With the current steady progress in instrumentation, combined with the increasing theoretical expertise, the use of resonant inelastic x-ray scattering techniques is becoming increasingly prevalent

RIXS provides an alternative and complementary look at the charge dynamics of a material. In the coming years, the experimental RIXS work in the L and M edge regions likely will be focused on establishing itself as a viable technique to measure elementary excitations at low energies, and investigate in detail the momentum dependent elementary charge response, viz magnons, orbitons and phonons, of correlated and uncorrelated materials. At the K edge, the majority of the work has been performed on cuprates and this research is likely to continue. However, more detailed extensions to other transition-metal compounds, such as manganites, pnictides, and ferroelectric materials, can undoubtedly be expected. Furthermore, the work on other strongly correlated systems, such as the edges of the rare-earths, is currently extremely limited and is expected to grow. In addition, with the advent of new synchrotron sources, there will be a push to improving the experimental resolution towards the 10 meV level. As has been demonstrated many times to date, each improvement in resolution brings with it the discovery of new details in the excitation spectrum of these materials. The future for RIXS as a characterization tool able to probe elementary excitations in real materials under real conditions looks bright.

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