High-resolution Resonant Inelastic soft X-ray Scattering (RIXS)

1 Introduction: Charge, orbital, spin and lattice degrees of freedom in correlated solids

Crystalline solids are at the core of electronics and information technology. Their thermal and charge conductivity and their magnetic properties are exploited in devices, where they are manipulated or measured for transmitting, processing or storing information. Mature technology found in commercial devices is mostly based on band semiconductors, metallic conductors, insulators and magnetic alloys whose intrinsic electronic and magnetic structures are well known experimentally and described theoretically by powerful though relatively simple models. Miniaturization, i.e. nanotechnology, has opened new opportunities by entering length scales where quantum effects become relevant, e.g., in quantum dots. The identification of graphene is also revolutionizing materials science and technology by offering a versatile playground for real two-dimensional (2D) phenomena in several fields. A bit farther from the world of applications, the physics of solids is looking at other classes of materials with prospective interest for applications. Those are materials where the single-particle approach fails in predicting or explaining the electronic structure because the electron-electron interaction is so strong that it has to be accounted for beyond the perturbative corrections. These materials have been recently referred to as quantum materials. They show innumerable examples of thermodynamic, transport, dielectric, magnetic properties potentially interesting for the technological devices of the future. Their relevance is also, if not mainly, of fundamental breadth: understanding quantum materials means refining our comprehension of the physics of solids in general.

The complexity and richness of quantum materials lies in the tight entanglement among the different degrees of freedom characterizing solids: the charge density due to valence electrons, the local symmetry of electronic states (orbitals), the atomic spin moment and the actual crystalline structure, where the local coordination plays an essential role. As shown in fig. 1, these properties are all mutually intertwined in quantum materials, and the respective energy scales are comparable. The structure is determined by the orbital symmetry, transport is influenced by lattice periodicity, which is affected by the magnetic order, which is the result of exchange interactions and hopping integrals that depend on orbitals, etc. We can thus understand the existence of several phase transitions related to different orders appearing or melting. And we can see the opportunity of changing, e.g., the magnetic order by applying strain or electric fields. Spectacular phenomena appear, such as high critical temperature superconductivity in layered copper oxides (the so-called high T_c superconductors, HTcS) and the colossal magneto-resistance in manganese perovskites (manganites). In this context the systems based on 3d transition metals (3dTM) are playing a major role, because the open 3d shell is sufficiently atomic-like to lead to strong electronic correlation and localized magnetic momenta; but also radially extended enough to form highly covalent bonds with partnering anions, resulting in large hopping integrals and strong superexchange interaction. This explains the very rich phenomenology of 3dTM systems (alloys, oxides, pnictides, calchogenides), and the large effort developed in spectroscopies that are especially suitable for the study of 3dTMs. Below, we will focus on 3dTM oxides as examples.

Besides the macroscopic phenomenology, the goal is gaining a better understanding of the microscopic origin of this complex landscape. We need to determine the actual order parameters: not only atomic positions, but also charge disproportionation and density modulations, orbital ordering, exchange interaction among atoms in the lattice, electronphonon coupling. The experimental task is twofold: structural information is obtained from direct space (microscopy) or reciprocal space (diffraction) imaging with atomic resolution; and energy ruling over the relevant interactions is measured by spectroscopy. Over the years, an increasing number of experimental techniques have entered the game, most of them relying on one of three main particle/wave probes: electrons, neutrons and photons. Electron beams are easy to produce and control; they can be used for microscopy down to single-atom resolution, for surface diffraction and for energy loss spectroscopy. But their interaction with matter is extremely strong and they cannot penetrate below the first few atomic layers unless very strongly accelerated. Conversely, neutrons interact very weakly with electrons of the sample because they are neutral (but quite directly with nuclei), so they are very bulk sensitive. They are ideal for diffraction and for low-energy loss spectroscopy and have become the probe of choice in magnetism since many decades, thanks to their spin that interacts with atomic moments in the sample. Unfortunately, neutron beams are expensive to produce, are hard to focus on small samples and the eventual counting rate is often the limiting factor for experiments. Photons are the most versatile probes. In the optical regime laser sources are offering a panoply of opportunities for spectroscopy (absorption, reflectivity, inelastic Raman scattering, etc.). And diffraction can be performed with short-wavelength photons, i.e. X-rays.

Although optical and X-ray microscopy is still far from reaching atomic resolution, it has been progressing very fast. X-ray spectroscopy is well established too, in particular in absorption; and in more recent years the progress of synchrotron storage rings as sources of brilliant and powerful X-ray beams has led to the development of energy loss techniques based on X-rays. The advantage, there, is that high-energy photons carry a momentum significant for the study of collective excitations in solids, whose energy depends on the wave vector (i.e., the momentum) of the excitation (the quasiparticle). In summary, electrons couple with the charge, the spin and the lattice degrees of freedom in solids, but they are very surface sensitive which limits the range of applications. Neutrons are ideal for diffraction and for very low-energy excitation spectroscopy, but they do not couple to charge, unless very indirectly, and they often suffer from signal weakness and require large samples. Optical photons interact mainly with the valence charge and can indirectly couple to phonons, but they do not carry significant momentum. X-ray photons benefit from sizable momentum (indeed they can be used in diffraction experiments), are sensitive to the orbital symmetry and can couple to the atomic spin thanks to the spin-orbit interaction. X-rays are the most versatile probes, also because they are compatible with non-standard sample physical conditions (pressure, volume, temperature).

In this context the development of resonant inelastic X-ray scattering (RIXS) can be seen as the accumulation of most of the wishes listed above for the ideal probe for quantum materials. RIXS is an energy loss spectroscopy, realized with X-rays tuned at a specific energy to match an absorption edge of one atomic species in the sample. As we explain below, RIXS can access a variety of excitations, with momentum resolution and with scattering cross sections usually larger than for neutrons, allowing experiments on thin films and nanopatterned samples. The development of RIXS has been driven in the last 20 years mostly by the improvement of the instrumentation that has led to a gain in energy resolution by almost 2 orders of magnitude. On the theoretical side the progress has also been remarkable, but the full description of cross sections is not available in general, and much remains to be done to reach the level of neutron scattering in that sense.

2 The mechanisms of RIXS

RIXS is a sort of Raman spectroscopy performed with X-rays, which are tuned to match the binding energy of a core level. As shown in fig. 2, the fact that we use X-rays adds the momentum conservation to the energy conservation law applied in

usual Raman experiments: the energy difference between the incident and the scattered photons provide the energy loss $\hbar \Omega = \hbar \omega - \hbar \omega^{\prime}$; the vector difference between the scattered and incident photon wave vectors provide the momentum transfer q = k' - k.

If the sample is a single crystal the momentum q is a significant vector in the solid’s reciprocal space and can be taken by electrons, magnons, phonons and other quasiparticles or combinations of them. Inelastic X-ray scattering can be also performed non-resonantly, resulting into two main families of experiments: X-ray Raman spectroscopy, when the energy loss is of several eV or even hundreds of eV, corresponding to plasmon or core level electronic excitations. And high-resolution inelastic X-ray scattering (IXS), when $\Omega$ is of few meV and corresponding to vibrational and lattice excitations. In both cases the scattering can be seen as a firstorder process, with no intermediate steps between the initial and final configurations.

In RIXS the resonant absorption of the incident photon is added, implying the promotion of one core electron into a bound state just above the Fermi level; subsequently the core hole left after the absorption is filled back by the same or by a different electron, with the emission of another photon. The process implies thus 3 states (initial, intermediate, final). However, the intermediate state is not observed: only the initial and the final state are fully known from the measurement. Therefore, we have a quantum interference among all the paths that can bring from the initial to the final state via different intermediate states. This interference is governed by the energy overlap of the intermediate states, via their energy splitting and lifetime Lorentzian broadening. Consequently, the RIXS process has to be described by the Kramers- Heisenberg equation for a second-order process, where the absorption and emission amplitudes are multiplied before taking the squared modulus to evaluate the scattering probability. Another consequence of being a second-order process is that there is no core hole in the initial and final states: the associated lifetime broadening does not smear out the spectral details in the RIXS spectra, as opposite to what happens in core level photoemission, X-ray absorption and X-ray emission spectroscopies.

In fig. 2 the RIXS process is exemplified for a 3d transition metal ion. In 3dTM systems the electronic and magnetic properties are determined by the 3d shell configuration, so it is ideal to perform RIXS at the 2p core level absorption resonance: the 2p-3d transition is allowed for electric dipole radiative transitions, resulting in large resonance, i.e., in large enhancement of the scattering cross section. Moreover, the corresponding X-ray absorption spectroscopy (XAS, at the L_2,3 edge) is a well-established technique and its polarization dependence is currently exploited to gain insight into the orbital symmetry and the spin polarization of the 3dTM ion. So the first step consists in the absorption of the incident photon and the promotion of one 2p electron into one of the empty 3d states. The intermediate state is highly excited (450 eV to 950 eV going from Ti to Cu), its spin state is mixed (due to the large spin-orbit interaction in the 2p level, notoriously split into the 2p_1/2 and 2p_3/2 levels, 7 eV to 20 eV apart) and very short lived (few femtoseconds, mostly due to the non-radiative Auger decay processes). So, before the solid has the time to fully readjust its lattice and magnetic configuration under the effect of the core-hole and the extra valence electron, there is the possibility for a 3d electron to fill the 2p hole with the emission of a photon of similar energy as the incident one. The other decay processes (mostly Auger) are disregarded here (i.e., not observed), but still they determine the lifetime of the intermediate state.

The final state differs from the ground state by a change in energy and momentum of the X-ray photon and the creation of one excitation in the solid. The direct involvement of the 3d shell in the two radiative transitions and the large spin-orbit interaction in the 2p state conjure in allowing a full set of possible excitations: electron-hole (e-h) pairs, charge transfer from the anion to the 3dTM ion, spin excitations (see below for more details), phonons (thanks to the electron-phonon interaction at play in the intermediate state). The energy of the photons is imposed by the 2p level binding energy, and lies in the soft X-ray range ($\hbar \omega > 1$ keV, $k < 0.5$ Å^-1), resulting in some limitation in the maximum momentum transfer (typical reciprocal lattice units in 3dTM systems are as large as $\sim 1.6$ Å⁻¹). We can see that soft X-ray RIXS has a huge potential for its ability to probe, in a momentum-resolved way, a variety of excitations that can carry valuable information on the intertwined degrees of freedom of 3dTM quantum materials. The key for successful RIXS lies in resolving power and detection efficiency, i.e. in instrumentation.

3 Instrumentation for high-resolution RIXS

High energy resolution and count rate on the detector are usually in competition and a compromise has to be found compatibly with the current technological limits. Due to the relatively long wavelength, soft X-rays cannot be monochromatized nor analyzed by Bragg reflection from perfect crystals, because of the scarcity of high-quality crystals with large enough lattice units and of the strong absorption, which limits the number of crystalline planes contributing to the reflection, thus resulting in broad intrinsic width of the rocking curves. Consequently, diffraction gratings, ruled on high-quality X-ray mirrors and used at very grazing incidence (1.5° to 2.2° from the surface) are at the core of soft X-ray monochromators (for the incident beam) and spectrometers (for the scattered photons). The incident beam is generated at dedicated beam lines by long (4–6 m) undulators of 3rd generation synchrotron radiation storage rings. Usually, the undulator can provide 100% polarized radiation with linear (σ or π with respect to the horizontal scattering plane) or circular polarization, fully tunable in the relevant range. The beam line mirrors and grating monochromator cut the band width to 2–8 $\times$ 10^-5 of the photon energy (10–80 meV, typically in the 450–950 eV range) and focalize it onto the sample surface to few microns in height and some tens of microns in horizontal. The intensity on the sample is $\sim$ 10¹¹-10¹² photons s⁻¹. The illuminated spot on the sample is the source for the spectrometer. In the simplest cases a single optical element, a grating with spherical shape (radius of curvature $\sim$ 100 m), disperses the scattered photons in the vertical plane and refocuses the different energies onto a focal plane where a 2D detector, usually a Si CCD cooled to reduce thermal noise, collects the intensity over a few eV energy range, in parallel. The spectrum is then easily obtained by converting the detector coordinate into energy (see fig. 3).

It is apparent that both for the monochromator and the spectrometer the optical arm lengths are crucially influencing the final resolving power: longer instruments can reach higher resolution. Therefore, if the first RIXS spectrometers in the beginning of 1990’s were $\sim$ 50 cm long from the sample to the detector, 10 years later the 2.2 m long AXES spectrometer at ID08 of the ESRF could reach 500 meV band width, the 5 m long SAXES at the ADRESS beam line of the PSI/SLS breached the 100 meV band pass and in 2015 the 11 m long ERIXS at ID32 of the ESRF established a new record around 35 meV at the 930 eV of the Cu L₃ edge. Other important parameters are the vertical spot size on the sample (and thus the quality of the refocusing by the beam line optics), the spatial resolution of the detector (to get the best performances single-photon detection with hit-position reconstruction has to be used) and overall thermal and mechanical stability. Consequently, RIXS beam lines are becoming extremely complex and expensive, and require large surfaces, so that dedicated buildings outside the main experimental halls of storage rings are put in place.

In fig. 3 the scheme of the ERIXS instrument (the European RIXS facility) of the ESRF is shown. In addition to the high resolution already mentioned, this instrument has been equipped for the first time with a continuous rotation of the spectrometer from 50° to 150° with respect to the incident beam, and with a high-quality 6 axis diffractometer to align and orient the sample, both in ultra-high-vacuum operation conditions. Moreover, the luminosity is preserved, despite the extreme distance of the detector from the sample, thanks to a 60 cm long parabolic mirror collimating the scattered radiation in the horizontal direction over 20 mrad. Finally, an originally designed multilayer mirror allows for the determination of the linear degree of polarization of the scattered photons while keeping the very high resolving power unaffected, though with a non-negligible loss in detection efficiency. With ERIXS a new generation of RIXS experiments has become possible, where all the excitation energy scales are resolvable and with a full access to the reciprocal space. ERIXS has set a new standard and is being followed by equally or even more ambitious projects at storage rings (Diamond Light Source, NSLS in Brookhaven, MAX IV, TPS, Sirius) and free electron lasers (XFEL, LCLS).

To illustrate the opportunities offered by RIXS we provide two examples, one on ligand field excitations and one on spin excitations. More recently RIXS has been used also to study phonons, plasmons and charge excitations in general, also in combination with incipient charge density order, but those subjects would require a longer and more specialized presentation, unsuitable for the spirit of this introductory overview.

4 Ligand field excitations

As mentioned in the introduction, the orbital symmetry of the 3d occupied and empty states is linked to the crystalline structure, to the magnetic atomic moment residing on the 3dTM ion and on its coupling to the other ions via the superexchange interaction. RIXS can be used to determine this orbital symmetry and its stability, by the measurement of the energy and point symmetry of the dd excitations. Those correspond to the reshuffling of occupied and empty orbitals during the 2-step RIXS process. In the simplest view, a dd excitation is the transfer of one electron from one orbital to another. We note that 3d to 3d transitions are not allowed upon absorption of a single photon (they are forbidden by electric dipole selection rule), and are thus formally forbidden in optical absorption spectroscopy, high-energy electron energy loss spectroscopy (subject to the same rules) and neutron scattering (neutrons do not couple to charge), although in some cases they have been measured by those techniques thanks to a lowering of the local symmetry. For an isolated ion (spherical symmetry) the set of dd excitations is relatively limited and their energy is dictated by the intra-atomic exchange interaction, Coulomb repulsion (both related to Hund’s rules) and to the spin-orbit interaction.

As shown in fig. 4, the situation is more interesting when the 3dTM ion is coordinated with negatively charged ligands, usually with bonds having mixed ionic and covalent nature. As a result, the Coulomb potential felt by the 3d electrons is not spherical anymore and the resulting wave function must be symmetry-adjusted. One of the most common cases (e.g., in perovskites) is a 6-fold coordination, where the 3dTM cation is at the center of an octahedron and the ligands lie at the vertices. The octahedron can be regular or distorted. If a pair of opposite ligands is farther or closer to the center than the other four, a tetragonal distortion is present. In such a case the 5 textbook d orbitals (labelled as b₁, a₁, b₂, e_g for $x^2$-$y^2$, $3z^2$-$r^2$, $xy$, $xz/yz$ symmetry) are not degenerate anymore: in the example of the figure we assume an elongated octahedron. In these conditions the dd excitation spectrum becomes more complex and the energy of the peaks can be directly tied to the parameters describing the ligand field that lowers the spherical symmetry.

In the central panel of fig. 4 we show the simplest of all cases, Cu²⁺ where the 3d⁹ configuration in the ground state implies a single empty state. In the tetragonally distorted octahedral symmetry (most common in high T_c superconducting cuprates) the 3d hole has b₁ symmetry. In the RIXS intermediate state then, the 3d¹⁰configuration is unique, as one can appreciate from the single Lorentzian line shape in the XAS spectrum. Then any of the ten 3d electrons can decay into the 2p hole, leading either to elastic, spin flip or dd excitations, if coming from b₁ with same spin, b₁ with opposite spin, or any of the other orbitals (with same or reversed spin). Therefore, the energy loss in the RIXS spectrum would correspond directly to the ligand field splitting of the 3d states. This type of measurement has been exploited to study the actual influence of the local symmetry and coordination on the 3d state splitting in various families of cuprates and compare them to quantum chemistry results.

When the cation ground state has multiple holes (or electrons) like in Ni²⁺(3d⁸) and Mn²⁺(3d⁵) the set of dd excitations is wider, leading to RIXS spectra with many more peaks distributed over a broader energy range of several eV (see fig. 4). The assignment of the peaks is relatively straightforward in the ligand field model: already from just Sugano-Tanabe diagrams one can get an initial and very good estimate of the ligand field parameters, and adding the calculation of the RIXS cross sections, which depend on the polarization of the incident photons and sample orientation, the confidence in the results can be excellent and fine effects from, e.g., strain, can be evaluated. We underline here that the RIXS cross sections for dd excitations are relatively easy to calculate and are well understood, at variance from other techniques where the ligand field excitations are less direct and cross sections are subject to multiple approximations and assumptions. It is also worth mentioning that, usually, dd excitations have local character: the momentum q is not relevant and there is no dispersion (dependence of the peak position in energy on q) detected in the experiments, apart from some interesting cases, mostly in the quasi-1D materials.

5 Spin excitations in high T_c superconducting cuprates

The pure spin-flip excitation is always possible in L₃ RIXS and can be viewed, as mentioned above, as a special dd excitation. However, it is known that in an ordered spin lattice the reversal of a single spin at one site is not an eigenstate of the magnetic Hamiltonian. The actual excited state corresponding to a change by one quantum unit of the total spin of the solid is described by the creation of a magnon, a.k.a. spin wave, a linear combination of local spin flip excitations with complex weight given by a site-to-site relative phase proportional to the momentum q. In fig. 5 we show a semi-classical representation of a spin wave for a section of antiferromagnetic (AF) chain: the atomic spins, treated here as a classical magnetic dipole, precess around the ground state alignment direction, with the constant dephasing from site to site proportional to q. The magnon energy can be shown to depend on $q$ as $4JS |sin qa|$ for the antiferromagnetic chain, whereas in the ferromagnetic 1D chain the magnon energy is proportional to $4JS ( 1 - cos qa )$.

In layered cuprates, the insulating parent compounds of high T_c superconductors host CuO₂ planes where the spin 1/2Cu²⁺ ions form a 2D antiferromagnetic square lattice, as sketched in fig. 5. Cu L₃ RIXS can there be used to determine the dispersion relation of magnons in AF cuprates. Two examples are shown in the figure, where q is actually a vector in the 2D reciprocal space, that is made exploring all the significant positions in the first magnetic Brillouin zone (BZ). The possibility of measuring magnon dispersion with RIXS has been acclaimed as a significant novelty about a decade ago, because previously this type of measurement had been possible only using inelastic neutron scattering, with severe limitations due to the need for very large single crystals and to the difficulty in reaching loss energies of few hundreds of meV using thermal neutron sources. RIXS, on the contrary, is limited at low energies by resolution (at present around 35 meV at best at the Cu L₃ edge) but can be performed on very small crystals and ultrathin films. What is the interest in measuring the spin wave dispersion in cuprates? Upon doping with positive or negative charges the CuO₂ planes change from Mott insulators to unconventional metals and superconductors. Although a general theory for high T_c superconductivity is still missing, consensus is growing that spin fluctuations play a crucial role in Cooper pair formation, similarly to what is done by phonons in conventional BCS superconductors. Therefore, studying spin excitations in cuprates is very useful. In particular, it has been shown that in the doped superconducting materials, even in the superconducting state, the spin excitations preserve their energy and dispersion properties, although with a strong damping due to the much shorter spin correlation length determined by the presence of doping charges that cancel, randomly, some of the spin moments in the 2D lattice.

6 Perspectives for high-resolution RIXS

In this overview we have presented the basic concepts of RIXS, with a couple of examples of the best known results published in the last years. We have restricted the discussion to the soft X-ray range and the 3d transition metals for clarity and simplicity. But we would like to remind that all the considerations can be extended straightforwardly to the 4dTM and 5dTM systems, e.g., for ruthenates and iridates, respectively. The main differences are of technical nature: the 2p levels there are more tightly bound and the L₃ edges fall in the tender (2–3.5 keV) and hard (9–12 keV) X-ray range for 4d and 5d elements, respectively, implying Bragg crystal spectrometer and monochromator technology. Important results have been published in particular for Ir compounds, and very recently on Ru. Somehow different is the case of 3p levels (M_2,3 absorption edge) of 3dTMs, which suffer from the much smaller absorption cross sections and too small photon momentum. Finally, it is important to mention that, still in the soft X-ray range, one can find the M_4,5 edges (3d levels) of lanthanides: RIXS can thus be used to study ff excitations.

High-resolution RIXS is still in its pioneering phase, mostly due to the limited number of facilities available around the world as yet: we can estimate the number of experiments made every year with state-of-the-art instruments in less than 30. Exploring new scientific opportunities is hard under the pressure of competition for beam time allocation. The most recent developments are in the study of phonons to measure the electron phonon coupling and in the study of the charge order (charge density waves and charge density fluctuations) and its coupling to other degrees of freedom. The perspective of further improvements in the energy resolution, approaching the 10 meV limit, is extremely exciting and will open new opportunities in the study of low-energy magnetic e-h pair excitations.

Acknowledgements

We gratefully acknowledge insightful discussions with Marco Moretti Sala, Matteo Rossi, Nicholas B. Brookes, Sergio Caprara, Tom Devereaux, Carlo Di Castro, Marco Grilli, Jeroen van den Brink, and the support of the ERIXS project by Francesco Sette and Harald Richert.