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

Resonant inelastic x-ray scattering (RIXS) is a widely used technique for studying various condensed matter materials, including semiconductors, metals, oxides and transition-metal compounds [1, 2]. Because of its many advantages, including bulk sensitivity, orbital and elemental specificity and polarization control, RIXS has moved to the forefront of important spectroscopic techniques in recent years [3]. While in general the probability of x-rays to be scattered from a solid is small, the cross section can be enhanced by orders of magnitude when tuning the incoming x-ray energy to a resonance of the system where an electron from a deep-lying core state is excited into the valence shell. The resulting intermediate core–hole state can connect the many-body ground and final excited states through pathways that may be inaccessible using other non-resonant x-ray probes, thus providing a wealth of new information about charge, spin, lattice and orbital degrees of freedom. This is particularly important in strongly correlated transition-metal oxides where these degrees of freedom are heavily intertwined. In this way, RIXS can offer a complementary investigation of elementary excitations along with resonant diffraction, neutron scattering, angle-resolved photoemission spectroscopy and scanning tunneling microscopy. In particular, as resonant diffraction yields an understanding of the ground state of correlated materials, we show that RIXS can provide the same sort of understanding of excited state properties. By making explicit use of the resonance profile and the associated pathways, one can understand simpler density response in even the most strongly correlated 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 experimental cross-section [4]. One of the main efforts at understanding this cross section for correlated materials better has been devoted to disentangling the resonant profile and studying its response function, which only depends on the energy loss. As proposed under certain approximations involving intermediate states for indirect RIXS (the case of the Cu K-edge, for example, where scattering takes place only via the core–hole created as an intermediate state) [5–7], the cross section could be decomposed into a Lorentzian resonant profile and a response function, which is directly connected to the charge dynamical structure factor S(q,ω). Experimentally, in most of the endeavors to interpret the response function of RIXS spectra for correlated materials, either a particular incident photon energy has been considered [8–10] or the incident energy dependence was treated using this Lorentzian profile derived from theoretical approximations [5]. However, these experiments do not provide definitive proof that the response function can be directly related to S(q,ω), and the validity of the above approximations still need to be discussed for particular cases. In particular, for the Cu 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 for other materials [11]. This suggests, at least in correlated materials, that the validity of treating the RIXS response function as S(q,ω) needs to be carefully investigated. Furthermore, RIXS is known to display other elementary excitations for correlated materials, such as magnon excitations [12–14], which are beyond the simple charge response 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 important consideration in strongly correlated materials.

In this work, we performed an exact diagonalization study of the single-band Hubbard model to understand the indirect RIXS cross-section in strongly correlated materials. The intermediate state contribution has been fully considered, so that the incident energy-dependent resonant profile could be investigated. This goes beyond previous theoretical treatments using approximations in which the complicated intermediate state dependence has been ignored [5–7]. The result can be connected to the Cu K-edge RIXS for high-T_c cuprates. In two examples, we show how the incident 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 multiple incident photon energies, each of which corresponds to a different intermediate state electronic configuration. We find that only at particular intermediate states where the screening of the core–hole potential and the correlation effect are not important, one observes an RIXS spectrum similar to S(q,ω). However, RIXS behaves very differently when compared to S(q,ω) at other incident energies where the screening of the core–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-called poorly screened intermediate states. The two-magnon excitations are not shown at the intermediate state where the core–hole potential screening is absent. The significance of emphasized two-magnon excitation at one intermediate state compared to the others points to the fact that tuning to a proper incident photon energy dictated by the character of the intermediate state is critical for studying the elementary excitations in RIXS.

We employ the single-band Hubbard Hamiltonian that serves as an effective low-energy model for studying correlated materials, written as

$$\begin{equation} \mathit{H}= -\sum_{\mathbf{i},\mathbf{j},\sigma} t_{\mathbf{i},\mathbf{j}} d_{\mathbf{i},\sigma}^{\dagger}d_{\mathbf{j},\sigma} +\sum_{\mathbf{i}} U n_{\mathbf{i},\uparrow}^d n_{\mathbf{i},\downarrow}^d, \end{equation} \tag{ 1 }$$

where d^†_i,σ (d_i,σ) creates (annihilates) a fermion with spin σ on lattice site i and t_i,j is the hopping integral restricted to the nearest- and next-nearest-neighbor, denoted as t and t'. 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 using the Kramers–Heisenberg formula [15]

$$\begin{equation} I(\mathbf{q}, \Omega, \omega_{\mathrm{i}}) = \frac{1}{\pi} \mathrm{Im} \left\langle \Psi\left | \frac{1}{{H} - {E}_0 -\Omega - \mathrm{i} 0^+} \right| \Psi \right\rangle \end{equation} \tag{ 2 }$$

and

$$\begin{equation} | \Psi \rangle = \sum_{\mathbf{i},\sigma} \mathrm{e}^{\mathrm{i}\mathbf{q}\cdot\mathbf{r}_{\mathbf{i}}} \mathit{D}^{\dagger} \frac{1}{\mathit{H}^{\prime}-\mathit{E}_0-\omega_{\mathrm{i}}-\mathrm{i}\Gamma} \mathit{D} | 0 \rangle, \end{equation} \tag{ 3 }$$

where q is the momentum transfer; ω_i and Ω = ω_i − ω_f are the incident energy and energy transfer, respectively; Γ is the inverse core–hole lifetime; E₀ is the ground state energy of the system absent in the core–hole; |0〉 is the ground state wave function; $\mathit{H}^{\prime}=\mathit{H}-\mathit{U}_{\mathrm{c}} \sum _{\mathbf {i}, \sigma , \sigma^{\prime}} n_{\mathbf {i}\sigma }^d n_{\mathbf {i}, \sigma ^{\prime }}^s$ represents the intermediate state Hamiltonian, which includes the core–hole potential with strength U_c, with n^s_i,σ denoting the core–hole number operator; D is the dipole transition operator, which for example excites a 1s core electron up to 4p level and is formulated as $\mathit{D}=p_{\mathbf{i}, \sigma}^{\dagger}s_{\mathbf{i}, \sigma}$ in transition-metal K-edge indirect RIXS. The wave function overlap of the intermediate state with the final/ground state is fully considered and thus the whole resonant profile can be investigated using this method without further approximation. Elastic lines are trivial and have been removed from the results in the following sections.

We also investigate the x-ray absorption spectroscopy (XAS) given by

$$\begin{equation} B(\omega_{\mathrm{i}})= \frac{1}{\pi} \mathrm{Im} \left\langle 0 \left| \mathit{D}^{\dagger} \frac{1}{\mathit{H}^{\prime}-\mathit{E}_0-\omega_{\mathrm{i}}-\mathrm{i}\Gamma} \mathit{D} \right| 0 \right\rangle, \end{equation} \noindent \tag{ 4 }$$

in which D and H' are the same as in RIXS's definition. The XAS peaks can be connected with the resonant energies in the RIXS spectra.

The charge dynamical structure factor to be compared to RIXS is

$$\begin{equation} S(\mathbf{q},\omega)= \frac{1}{\pi} \mathrm{Im} \left\langle 0 \left| \rho_{-\mathbf{q}}^d \frac{1}{H - \mathit{E}_0 - \omega - \mathrm{i}0^+} \rho_{\mathbf{q}}^d \right| 0 \right\rangle, \end{equation} \noindent \tag{ 5 }$$

where $\rho _{\mathbf {q}}^d=\sum _{\mathbf {k},\sigma }d_{\mathbf {k+q},\sigma }^{\dagger } d_{\mathbf {k},\sigma }=\sum _{\mathbf {i},\sigma }\mathrm {e}^{\mathrm {i}\mathbf {q}\cdot \mathbf {R}_{\mathbf {i}}}n_{\mathbf {i},\sigma }^d$ is the density operator. We note that the charge dynamical structure factor S(q,ω), which represents the non-resonant response, is an effective two-particle process; however, for the resonant response the intermediate core–hole state increases the complexity of the problem to an effective four-particle process, making its interpretation more difficult [4].

Our calculations are performed on the 16B Betts cluster (16-site square cluster) with periodic boundary conditions. The ground state wave function is obtained using the exact diagonalization technique utilized by PARPACK (Parallel Arnoldi PACKage) [16]. This algorithm is based on the implicitly restarted Arnoldi method, which takes into account properly orthogonalized eigenvectors at each step of the iteration, avoiding the problem of degeneracy collapse common in most other Lanczos methods [17], such as those used in [18, 19]. This is crucial for understanding intensities of the RIXS cross-section since all intermediate states must be retained in a properly orthonormalized way. |Ψ〉 from equation (3) is calculated using the bi-conjugate gradient stabilized method (BiCGSTAB) [20], a variant of the bi-conjugate gradient method (BiCG). In BiCGSTAB, the generalized minimal residual method (usually abbreviated GMRES) is applied after each step of BiCG in order to get rid of the irregular convergence behavior, thus obtaining faster and smoother convergence than regular BiCG. For obtaining the final spectrum, we also employed the continued fraction expansion method.

We use the model parameters taken from Cu K-edge RIXS in cuprate (La₂CuO₄) as an example [21]: U = 10t, t' = −0.34t, U_c = 15t and Γ = t. Our results will be presented in units of t, with t = 0.35 eV. For the indirect RIXS processes in other correlated materials, the parameter values can vary, but the key feature from the correlation effect remains similar.

3.1. Resonant inelastic x-ray scattering (RIXS)

Figure 1 shows the momentum-dependent XAS and indirect RIXS spectra for the half-filled, 12.5% hole-doped and 12.5% electron-doped single-band Hubbard model. Zero incident energy is defined as the ground state energy E₀ plus E_edge, the energy difference between the two levels in the dipole transition process. For example, E_edge equals E_4p − E_1s for Cu K-edge RIXS. For the half-filling and dopings, RIXS displays strong resonances at the peaks of XAS, with the energies corresponding to the difference between the ground state and the intermediate state energy in the RIXS process.

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The nature of the intermediate state can be understood in terms of how well the valence electrons screen the core–hole potential. The XAS peak at an energy ∼ − 2U_c + U (∼20t for half-filling, ∼22t for 12.5% hole-doping and ∼20t for 12.5% electron doping) corresponds to the intermediate state with two electrons bound to the core–hole site that screen the core–hole potential. This intermediate state exhibits strong screening and is usually termed the 'well-screened' state [22, 23]. The XAS peak at ∼ − U_c (∼13.7t for half-filling, ∼14t for 12.5% hole-doping and ∼13.5t for 12.5% electron doping) corresponds to the intermediate state where only a single electron is bound at the core–hole site, weakly screening the core–hole potential, usually termed the 'poorly screened' state [22, 23]. In both processes, the electrons experience a strong 'shake-up' by the core–hole potential via screening. On the other hand, the intermediate state at an incident energy ∼3t for the hole-doped or ∼ − 28t for the electron-doped system represents the state in which the core–hole is created at an empty 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 in figure 1. For half-filling (figures 1(b)–(d)), RIXS displays a high-energy spectrum ranging from ∼5t up to ∼15t in energy loss for both well-screened and poorly screened intermediate states, signaling excitations from the lower Hubbard band (LHB) to the upper Hubbard band (UHB), which is mainly controlled by Coulomb U. The main peaks have strong momentum dependence with the shift to higher loss energy at larger momentum transfer, due to the increase in phase space available for the creation of excitations in the particle–hole continuum [24]. The main peaks are located at ∼6–8t for momentum (0,0), and shifted to ∼8–9t at momentum (0,π), ∼12–14t at momentum (π,π). This dispersion with increasing momentum at poorly and well-screened intermediate states can also be seen in figure 2. In the electron- and hole-doped cases, the same charge transfer dispersion on UHB is observed also for the well- and poorly screened intermediate states. These charge transfer features are consistent with various experiments [5, 11, 23, 25] and calculations [18, 19, 26] performed for the charge transfer excitations.

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3.2. Two-magnon excitations

The RIXS spectra also exhibit distinctive structures at low energy (∼1–1.2t), highlighted by blue arrows in figure 1, emphasizing excitations of the spin degree of freedom—two-magnon excitations [12–14, 27–29]. This structure is found both for the poorly screened (figure 2(b)) and the well-screened intermediate states (figure 2(a)) for the undoped system, but is significantly stronger at the poorly screened intermediate state. In the doped system, strong intensities at low energy are also shown for the poorly screened intermediate state (figures 3(b) and (e)). Only one electron is bound at the core–hole site in the poorly screened intermediate state; thus the intermediate state wave function may have a big overlap with the final state wave functions carrying two-magnon excitations, while for the well-screened intermediate state, the two bound electrons at the core–hole site make a smaller overlap with the final states for two-magnon excitations. Previous intermediate density of states calculations have also shown that the poorly screened intermediate state has a larger weight connected to the low-energy (magnetic) excitations [30].

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Our data show a strong two-magnon peak at (π,0) momentum transfer and no peak associated with two-magnon excitation at (π,π) [12, 14]. We also obtain two-magnon excitations at momentum transfer (0,0), which are symmetry allowed by the core–hole intermediate states. Similar findings at momentum transfer (0,0) have also been reported in early calculations [18, 27]. We show the two-magnon peaks at momentum transfer (0,0) located at ∼2.7J (the effective exchange coupling J = 4t²/U and is 0.4t in our calculation), which is in agreement with the calculation taking into account the magnon–magnon interaction [27]. The two-magnon excitations exhibited in RIXS can be connected to the excitations revealed by the spin dynamical structure factor [7]. Single-magnon excitations are forbidden in this case due to spin conservation, as we have neglected spin–orbit coupling.

Our calculations are consistent with the Cu K-edge experiments of the magnetic excitations taken on La₂CuO₄ for the observation of strong peak at momentum transfer (π,0) and no weight at momentum transfer (π,π) [12, 14]. No experimental confirmation of the two-magnon feature at momentum (0,0) has been reported, although recent improvements in resolution may resolve this issue. The two-magnon excitation is significant at the poorly screened intermediate state compared to the others, pointing to the fact that a particular elementary excitation may be highlighted when tuning to a proper incident photon energy dictated by the character of the intermediate state.

3.3. RIXS versus S(q,ω)

In summary, we have employed exact diagonalization to study the resonance profile for indirect RIXS over a wide range of dopings and incident energies in a simple correlated model. The RIXS spectra display complex incident energy dependence in the resonant profile that offers rich information besides a simple boost of intensity. Contrary to the previous understanding that RIXS is S(q,ω) in the charge transfer channel, we found 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 a poorly screened intermediate state. We conclude that at different incident energies, certain subdominant characteristics of final excitations can therefore be identified with a corresponding resonant enhancement in the RIXS profile [26]. One could thus explore particular elementary excitations by focusing on certain incident energies in the resonant profile.

We note that to make a better comparison to experimental Cu K-edge RIXS or XAS data, one needs to do a convolution with the Cu 4p density of states for the incident photon energy [26]. For experimental indirect RIXS data, the spectra on one incident energy may not be related to a single intermediate state configuration, since the convolution may mix the contributions from different intermediate states. Thus, for studying a certain elementary excitation in RIXS, such as the two-magnon or charge dynamical structure factor, one can focus on the incident phonon energy, which has the biggest weight connecting the specific intermediate state through the convolution. One may also do a deconvolution for the analysis of the experimental data. We emphasise that the purpose of our single-band Hubbard model study was to investigate how the 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 elementary excitations such as the dd excitations, although less momentum points are accessible since we can only study smaller clusters as more bands are included in the Hamiltonian. For cuprate materials in reality, the intermediate state also has ligand character which is beyond our down-folded model, but the intermediate state contribution to RIXS spectra in terms of the core–hole potential screening remains an important observation [11]. Our model also addresses well the two-magnon excitations and charge transfer dispersion as in experiments for cuprates.

As the single-band Hubbard model carries the key character of correlated materials, our conclusion is not limited to cuprates but could also be generalized to all correlated materials in understanding the indirect RIXS response on general forms. When the core–hole potential is comparable to other energy scales, the importance of the intermediate state configuration cannot be ignored and the incident energy dependence must be considered. The screening of the core–hole potential is a key factor in selecting the best incident photon energy to study certain elementary excitations.

We thank J P Hill for valuable discussions. This work was supported at SLAC and Stanford University by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering, under contract no. DE-AC02-76SF00515 and by the Computational Materials and Chemical Sciences Network (CMCSN) under contract no. DE-FG02-08ER46540. C J J was also supported by the Stanford Graduate Fellows in Science and Engineering. A portion of the computational work was performed using the resources of the National Energy Research Scientific Computing Center supported by the US Department of Energy, Office of Science, under contract no. DE-AC02-05CH11231.