Spin and orbital dynamics through the metal-to-insulator transition in Cd$_2$Os$_2$O$_7$ probed with high-resolution RIXS

TL;DR

This study uses high-resolution RIXS to investigate the magnetic and electronic excitations in Cd$_2$Os$_2$O$_7$ across its metal-insulator transition, revealing a continuous magnetic mode, hybridization effects, and itinerant behavior.

Contribution

It provides detailed RIXS measurements across the MIT, confirming magnetic ground state and highlighting electronic hybridization and itinerant characteristics in Cd$_2$Os$_2$O$_7$.

Findings

Magnetic excitation at 125 meV persists through the MIT.

Evidence of strong hybridization between Os 5d and O 2p states.

Indications of itinerant electronic behavior and breakdown of the free-ion model.

Abstract

High-resolution resonant inelastic x-ray scattering (RIXS) measurements ($\Delta$E = 46 meV) have been performed on Cd$_2$Os$_2$O$_7$ through the metal-to-insulator transition (MIT). A magnetic excitation at 125 meV evolves continuously through the MIT, in agreement with recent Raman scattering results, and provides further confirmation for an all-in, all-out magnetic ground state. Asymmetry of this feature is likely a result of coupling between the electronic and magnetic degrees of freedom. We also observe a broad continuum of interband excitations centered at 0.3 eV energy loss. This is indicative of significant hybridization between Os 5$d$ and O 2$p$ states, and concurrent itinerant nature of the system. In turn, this suggests a possible break down of the free-ion model for Cd$_2$Os$_2$O$_7$.

Tables

Table 1. Table 1: Comparison of exchange parameters between different studies.

	$J$	$\left|𝒅\right|$	$𝖠$	$J/\left|𝒅\right|$	$J/𝖠$
DFT ($U_{\text{eff}}=\text{1.25 eV}$) Shinaoka et al. (2012)	14	4	-24	3.5	-0.6
Quantum chemistry Bogdanov et al. (2013)	6.43	1.65	-6.77	3.9	-1.0
Raman Nguyen et al. (2017) (from two-magnon)	5.1	1.7	-5.3	3.0	-1.0
RIXS (from raw data)	13.1	6.8	-12.9	1.9	-1.0
RIXS (Effective $\Delta S=1$)	6.5	3.4	-6.5	1.9	-1.0

Table 2. Table 2: MRCI and MRCI+SOC relative energies (eV) for the Os 5 ​ d 4 5 superscript 𝑑 4 5d^{4} multiplet structure in Cd 2 Os 2 O 7 , given to three significant figures. The lowest 18 MRCI+SOC states are shown for brevity. Note that the dominant configuration label only corresonds to the states computed using MRCI.

dominant config.	MRCI	MRCI+SOC
$S=1$ $t_{2g}^4$	0.0; 0.192; 0.196	0.000; 0.363; 0.364;
$S=0$ $t_{2g}^4$	1.096; 1.097; 1.291;	0.521; 0.758; 0.776;
	1.296; 1.322; 2.481	0.778; 0.800; 0.806;
$S=2$ $t_{2g}^3{e}_g^1$	2.770; 2.776	1.74; 1.74; 1.89;
$S=1$ $t_{2g}^3{e}_g^1$	4.144; 4.15; 4.514;	1.94; 1.97; 3.19;
	4.517; 4.520; 4.659;	3.37; 3.39; 3.40…
	4.683

Table 3. Table 3: MRCI and MRCI+SOC relative energies (eV) for the Os 5+ 5 ​ d 3 5 superscript 𝑑 3 5d^{3} multiplet structure in Cd 2 Os 2 O 7 . Since cubic symmetry is lifted, the T 𝑇 T states are split even without SOC. Each MRCI+SOC value stands for a spin-orbit doublet; for the T 4 superscript 𝑇 4 {}^{4}T states, only the lowest and highest components are given. Reproduced from Ref.  Bogdanov  et al. , 2013 .

$5d^3$ splittings	MRCI	MRCI+SOC
${}_{}{}^{4}A_2^{}\left(t_{2g}^3\right)$	0.00;	0.000; 13.5$\times$10-3
${}_{}{}^{2}E\left(t_{2g}^3\right)$	1.51; 1.51	1.40; 1.53
${}_{}{}^{2}T_1^{}\left(t_{2g}^3\right)$	1.61; 1.62; 1.62	1.63; 1.66; 1.76
${}_{}{}^{2}T_2^{}\left(t_{2g}^3\right)$	2.46; 2.49; 2.49	2.63; 2.76; 2.87
${}_{}{}^{4}T_2^{}\left(t_{2g}^2{e}_g\right)$	5.08; 5.20; 5.20	5.14; …; 5.45
${}_{}{}^{4}T_1^{}\left(t_{2g}^2{e}_g\right)$	5.89; 6.01; 6.01	6.02; …; 6.33
${}_{}{}^{4}T_1^{}\left(t_{2g}^2{e}_g^2\right)$	10.29; 10.63; 10.63	10.41; …; 11.00

Equations

$$\mathcal{H}=J\sum _{ij}{\mathbf{S}}_i\cdot {\mathbf{S}}_j+\mathbf{d}\sum _{ij}{\mathbf{S}}_i\times {\mathbf{S}}_j+\sum _i{\mathbf{S}}_i\cdot \mathsf{A}\cdot {\mathbf{S}}_i,$$

$$I(\omega )=I_0\frac{{\left[1+(\omega -{\omega }_0)/q\Gamma \right]}^2}{1+{\left[(\omega -{\omega }_0)/\Gamma \right]}^2}.$$

Full text

I Introduction

A number of osmates undergo unconventional metal-to-insulator transitions (MITs) which appear to be driven to some degree by the onset of antiferromagnetic order, notably NaOsO3 and Cd2Os2O7. The former has been proposed to be a rare example of a Slater insulator in three dimensions Shi et al. (2009); Calder et al. (2012); Du et al. (2012); Jung et al. (2013); Vecchio et al. (2013); Middey et al. (2014); Calder et al. (2015), albeit with some controversy Kim et al. (2016). Like NaOsO3, bulk measurements on Cd2Os2O7 show a direct correspondence between the MIT and onset of antiferromagnetism at $T_{N}=\text{227~{}K}$ Sleight et al. (1974); Mandrus et al. (2001); Yamaura et al. (2012); Hiroi et al. (2015). Resistivity data indicate that a BCS-like indirect charge gap ($2\Delta_{C}=\text{80~{}meV}$) opens continuously below the MIT.Mandrus et al. (2001); Hiroi et al. (2015) Optical conductivity measurements reveal similar behavior for the direct optical gap ($2\Delta_{O}=\text{158~{}meV}$ Padilla et al. (2002); Sohn et al. (2015), with the most recent study suggesting a weak decoupling of the two phenomena Sohn et al. (2015). This manifests as the formation of an apparent antiferromagnetic metallic phase between 210 K and $T_{N}$. 111Whether this is representative of the bulk behavior, or an effect limited close to the sample surface, remains an open question. These observations, along with density functional theory (DFT) calculations Shinaoka et al. (2012), led to the suggestion that Cd2Os2O7 undergoes a Lifshitz transition (change in topology of the Fermi surface), which is partly driven by the onset of antiferromagnetic order.

Yet there have been comparatively few studies of the magnetic behavior of Cd2Os2O7. Yamaura et al. proposed an all-in, all-out (AIAO) $\bm{k}=0$ magnetic structure for Cd2Os2O7, based upon resonant elastic x-ray scattering (REXS) data and representation analysis Yamaura et al. (2012). This was confirmed by the authors of the present article using neutron powder diffraction Calder et al. (2016). Such a ground state has also been unambiguously found for the isostructural R2Ir2O7 (R = Sm, Nd) Donnerer et al. (2016); Guo et al. (2016), and is a direct manifestation of the significant spin-orbit coupling (SOC) present in these systems. Recent measurements have shown that the Néel temperature $T_{N}$ is continuously suppressed as a function of applied hydrostatic pressure, going to zero temperature at 36 GPa Wang et al. (2018). Such behavior is consistent with the expectations for a Lifshitz transition and its underlying $T=0$ quantum critical point.

The effect of SOC also extends to the magnetic excitations. Previous low resolution ($\Delta E=\text{130~{}meV}$) resonant inelastic x-ray scattering (RIXS) results presented a dispersionless feature in the excitation spectra for Cd2Os2O7 at 170 meV, which could not be explained in terms of intra-t2g spin-flip excitations or a simple spin wave picture Calder et al. (2016). Quantum chemistry (QC) calculations using a nearest-neighbor Heisenberg Hamiltonian,Bogdanov et al. (2013) which included significant single-ion anisotropy (SIA), and the Dzyaloshinskii-Moriya (DM) interaction:

[TABLE]

had determined the antiferromagnetic exchange parameter $J=6.3$ meV, considerably smaller than the energy scale of the new peak. Based on exact diagonalization (ED) calculations using the same Hamiltonian, it was concluded that the new feature (henceforth referred to as peak A) was a signature of a combination of $\Delta S\!=\!1,2,3$ excitations: $S_{z}=3/2\rightarrow 1/2$ ($\Delta S_{z}=1$), $S_{z}=3/2\rightarrow-1/2$ ($\Delta S_{z}=2$), and $S_{z}=3/2\rightarrow-3/2$ ($\Delta S_{z}=3$). The latter of these was found to be the dominant process in the density of states and previously unobserved by RIXS 222Note that the density of states is inequivalent to the RIXS cross-section; the argument is that the single- and bi-magnon processes should be dominant in the RIXS cross-section.. Excitations with $\Delta S=3$ are usually forbidden in the RIXS process; however it was argued that due to strong spin-orbit coupling in the intermediate state, $S$ was no longer a good quantum number. Hence the excited state wavefunction is comprised of a superposition of different spin-orbital states, all of which have a finite overlap with the $S=3/2$ ground state. Recent Raman scattering measurements meanwhile reveal two broad features at 130 and 150 meV Nguyen et al. (2017), proposed to arise from $\Delta S=0$ scattering from the two-magnon density of states. It was found that the coupling parameters were generally in good agreement with the previous QC calculations Bogdanov et al. (2013).

In this manuscript, we present high resolution RIXS measurements ($\Delta E=\text{46~{}meV}$) on Cd2Os2O7. Our results provide new information about the electronic and magnetic excitations in this material as a function of momentum transfer. We determine that the magnetic feature (peak A) has a lower energy than previously reported (125 meV), but has minimal dispersion within experimental uncertainty. By modelling the magnetic excitations well below $T_{N}$, we confirm that peak A includes higher order processes in $\Delta S$, and unambiguously establish that the magnetic ground state is AIAO. At higher temperatures, the excitation becomes progressively more damped, potentially abating above the MIT. Notably an asymmetry can be observed at all temperatures, which was previously unresolved in the low-resolution RIXS measurements. It is proposed that this results from coupling between the electronic and magnetic degrees of freedom, further highlighting the interplay of these phenomena in this system. We also find that the orbital excitations exhibit a degree of $5d^{4}$ character, despite the nominal electronic ground state being $5d^{3}$. This is likely due to significant hybridization and long-ranged electronic interactions, which go beyond a simple free ion model for Cd2Os2O7.

II Experimental details

RIXS measurements at the Os L3 edge (10.871 keV) were performed on a single crystal of Cd2Os2O7 (ca. 0.5 mm across) on the ID20 spectrometer at the ESRF, Grenoble Sala et al. (2018). The sample was oriented such that the $\left[1,\,1,\,1\right]$ direction was perpendicular to a copper sample mount, fixed with GE varnish and placed in a closed-flow (Dynaflow) He cryostat. The scattering plane and incident photon polarisation were both horizontal, i.e. $\pi$–incident polarization, with the incident beam focussed to a size of 20 $\times$ 10 $\mu$m2 (H$\times$V) at the sample position.

A Si $(6,\,6,\,4)$ four-bounce secondary monochromator was used to define the incident energy, with a Si $(6,\,6,\,4)$ diced spherical analyser (2 m radius, 40 mm diameter mask) used to reflect the scattered photons towards a Medipix CCD detector (pixel size 55 $\mathrm{\mu}$m). Total energy resolution was determined to be $\Delta$E = 46 meV based on scattering from a structural Bragg reflection.

III Low temperature behavior

We first examine the low-energy behavior deep in the antiferromagnetic insulating phase (30 K). This permits detailed study of the spin Hamiltonian towards the localized limit.

III.1 RIXS

RIXS spectra collected in the $(6,\,7,\,7)$ Brillouin zone are plotted in Fig. 1(a) as a function of momentum transfer. At first glance they appear similar to those presented in Ref. Calder et al., 2016. The main distinction, however, is that peak A exhibits a degree of asymmetry not observed in the previous study. In order to quantify our observations further, the data were modelled with Gaussians to represent the elastic line and high-energy intra-$t_{2g}$ excitations. A suitable lineshape was added to describe the asymmetric magnetic feature (peak A), and the resultant function was convoluted with the experimental resolution function prior to fitting. A number of models were considered which take this asymmetry into account, with the best found to be a Breit-Wigner-Fano (BWF) peakshape, given by:

[TABLE]

Such a model successfully described the effect of spin-phonon coupling observed in Raman scattering Sohn et al. (2017); Nguyen et al. (2017). The asymmetry is a direct manifestation of coupling between a resonant mode (with lifetime $\propto 1/\Gamma$, where $\Gamma$ is a damping parameter) and a continuum of states, the magnitude of which is parametrized by the coefficient $|1/q|$. Note that in the subsequent analysis, it has been assumed that $q\approx 7$ – indicative of moderate coupling between the resonance and continuum – is globally fixed for all temperatures and momentum transfers. This is due to insufficient data between the peak maximum and the broad elastic line. In this region the functional form of the lineshape is especially sensitive to the magnitude of $q$. It was found that the quality of the fit was consistently improved at all temperatures and momentum transfers by the inclusion of a broad, weakly dispersive feature centred at approximately 300 meV [Fig. 1(d)]. This feature – henceforth referred to as peak A2 – was assumed to have a Gaussian lineshape; potential origins are discussed later.

The energy and spectral weight of peak A are plotted in Fig. 1(b,c). Within experimental uncertainty, peak A exhibits minimal dispersion throughout the Brillouin zone, with little concurrent weak variation in the spectral weight. Using the AIAO magnetic ground state and Eqn. 1 as a starting point, we were able to successfully model the observed behavior within linear spin wave theory (LSWT). This implies that this minimal Hamiltonian is sufficient to describe the magnetic interactions in Cd2Os2O7. The coupling parameters we extract, however, are a factor of two larger than determined from quantum chemistry calculations and Raman scattering (Table 1). This discrepancy is rather puzzling, and we considered alternative explanations.

III.2 Verification of magnetic ground state and Hamiltonian

We first ascertained how the magnitude of different coupling parameters influence the magnetic ground state, and hence, the validity of the putative magnetic Hamiltonian. An isotropic Heisenberg antiferromagnet on the pyrochlore lattice exhibits geometric frustration. Due to the lattice topology, it is not possible to simultaneously satisfy all of the pairwise interactions between spins. Consequently no long-ranged order is observed down to $T\rightarrow 0$, with the resulting ground state being highly degenerate. Introducing anisotropy (or further neighbour interactions) breaks this degeneracy, leading to a unique magnetic ground state. In the case of zero SIA, the magnetic ground state depends on the sign of the DM vector. If the DM vector is negative (direct DM interaction), then one obtains the AIAO arrangement of spins (irreducible representation $\Gamma_{3}$). This situation exactly corresponds to that determined for R2Ir2O7 Donnerer et al. (2016). Meanwhile if the DM vector is positive (indirect DM interaction), then the spins are confined to the $xy$-plane ($\Gamma_{5}$). Adding uniaxial SIA along the local $\langle 111\rangle$ direction (compressive trigonal distortion of the OsO${}_{\text{6}}$ octahedra) actually helps to stabilize the AIAO ground state. Bogdanov et al. argue that this is the dominant driving force for putative AIAO order in Cd2Os2O7, with the DM interaction playing a more minor role Bogdanov et al. (2013).

The results of a semiclassical energy minimization of a Heisenberg antiferromagnet on the pyrochlore lattice are plotted in Fig. 2(e) for different values of SIA, and the DM interaction. Note that negative SIA corresponds to a uniaxial anisotropy along the local $\langle 111\rangle$ direction for each moment. We find that the resulting ground state spin configuration is highly dependent upon the choice of coupling parameters, with four different ground states occurring in different regions of the phase diagram. These states correspond to the possible magnetic representations $\Gamma_{\text{mag}}=\Gamma_{3}+\Gamma_{5}+\Gamma_{7}+2\Gamma_{9}$ for $\bm{k}=0$ order on the pyrochlore lattice. It is possible to go further and calculate the expected spin wave dispersion for a given set of coupling parameters and magnetic structure. In Fig. 2(a–d), we plot the results of least-squares fits to the experimental dispersion and intensity, which were performed in SpinW Tóth and Lake (2015). It is clear that an AIAO spin configuration ($\Gamma_{3}$) provides a much better description of the data for sensible values of the coupling parameters.

Further support for an AIAO ground state can be garnered from results of simulated annealing runs. These were performed on system sizes of $L\times L\times L$ unit cells, where $L$ ranges from 1 to 5 (16 – 2000 spins in total). The energy of the system and its interactions were determined via the Hamiltonian given by Equation 1. Coupling parameters used were the same as those obtained from the fits to the spin wave dispersion in Figs. 2(a)–(d), albeit divided by two 333We provide a natural explanation for this later in the manuscript.. The simulated annealing was performed with a standard Metropolis spin-flip algorithm, with $5\times 10^{4}$ Monte Carlo steps (MCS) per spin to equilibrate the system for a given temperature, followed by a further $5\times 10^{4}$ MCS per spin to evaluate thermodynamic parameters. Two sets of runs were performed for each system size: one with a relatively coarse temperature step ($T_{\text{new}}=0.92\,T$) initialised at 2000 K ($T/J\approx 25$) with the spins oriented randomly, and a dataset focussing on the critical region. The latter was initialised at 320 K (experimentally $T_{N}+93$ K) with the state obtained from the coarse dataset at the same temperature, and run with a step-size of 5 K.

The results of these simulations are plotted in Fig. 2(f) for $L=3$. In each case, the final ground state was the same as that obtained from the semiclassical energy minimization. As expected, the magnetic specific heat $C_{m}$ diverges at the Néel temperature, consistent with a second-order phase transition. For $\Gamma_{3}$, we find a remarkable agreement between the maximum of $C_{m}$, and the experimental Néel temperature ($T_{\text{N}}=\text{227~{}K}$). The other magnetic representations, however, appear to significantly underestimate $T_{\text{N}}$. This suggests that the AIAO spin configuration is indeed the correct one for Cd2Os2O7.

We tested this further by examining the effect of the finite-sized lattice upon the thermodynamic properties (Fig. 3). The continuous phase transition observed in the magnetic specific heat and susceptibility appears to sharpen and converge towards the experimental Néel temperature $T_{\text{N}}=227$ K with increasing system sizes. This would be expected as one progresses towards the thermodynamic limit.

There are, however, a number of limitations with the simulations as presented here. Firstly there is a fundamental problem in the Metropolis spin-flip algorithm of ‘critical slowing-down’. In the vicinity of the critical point, equilibration is slow as both the length and time-scales involved diverge. Thus there are frequently problems with accurate determination of transition temperatures and critical exponents within this approach. Improvements could be made by utilising a cluster-flipping algorithm where critical slowing down is dramatically reduced. Second, the number of equilibration and averaging steps per spin is relatively low. Ideally one would want to perform the simulations with a greater number of steps to ensure equilibration and accurately determine the thermodynamic parameters with appropriate statistical errorbars. The number of MCS performed per spin was sufficient to ensure equilibration for each temperature and lattice size, however, as determined from the temporal dependence of the internal energy. Sharp fluctuations can also be observed in the internal energy $\langle U\rangle$ and magnetization at low temperature ($T\!<\!\text{20~{}K}$). These fluctuations occur over a single temperature step, and may be due to some long timescale behavior which has been incorrectly compensated for in the averaging procedure. Performing a full autocorrelation analysis is likely to improve this.

Finally a complete finite-size scaling analysis has not been performed. Transition temperatures determined from the magnetization, for example, for a given lattice size frequently do not represent the true behavior of the system in the thermodynamic limit. This is why the maxima of the magnetic specific heat and susceptibility curves have been plotted in Fig. 3(d) instead of the calculated Néel temperature. Practically what one would do within a finite-size scaling analysis is to calculate the Binder cumulant for each temperature and lattice size, and determine their crossing point via a data collapse. Such an analysis is somewhat subjective, however, since it requires some fine-tuning of the critical temperature and critical exponents to obtain the correct crossing-point.

Despite these limitations, the simulations appear to provide further evidence for an AIAO magnetic ground state in Cd2Os2O7. In particular, we obtain remarkable agreement between the calculated and experimental Néel temperatures for a set of coupling parameters (divided by two relative to the values obtained experimentally) that best fits the observed spin wave dispersion. This validates the choice of Hamiltonian, although the factor of two discrepancy with the RIXS results remains puzzling.

IV Temperature dependence

Now that the low-energy excitations have been characterised deep in the antiferromagnetic insulating phase, we establish how they vary through the MIT. RIXS spectra were collected at the $(7,\,7,\,8)$ Brillouin zone centre $\Gamma$ as a function of temperature. Note that this is in a different Brillouin zone from the momentum dependence data presented in the previous section; the advantage is that the quasielastic peak is noticeably suppressed due to the scattering angle $2\theta$ being closer to the ideal condition of $2\theta=90^{\circ}$.

The results are shown in Fig. 4. Utilizing the same fitting model as used previously, we observe a clear evolution of the lineshape of peak A from 30 K to 300 K [Figs. 4(b-d)]. The intensity of this peak decreases continuously as a function of increasing temperature, abating at ca. 230 K [Fig. 4(e)]. This coincides with the Néel temperature, and is consistent with the previous RIXS and Raman scattering results Calder et al. (2016); Nguyen et al. (2017). Meanwhile the peak energy appears approximately constant through the MIT (within experimental resolution) [Fig. 4(f)]. This contrasts with the Raman scattering data, which shows that the two-magnon peaks $\mathrm{M_{1}}$ and $\mathrm{M_{2}}$ are weakly renormalized with temperature. Even so, the general energy scale is consistent.

Finally the magnitude of the damping parameter $\Gamma$ is approximately resolution-limited at low temperatures and increases rapidly above 150 K [Fig. 4(g)]. Note that it is unclear within experimental uncertainty whether $\Gamma$ saturates above the MIT, or simply increases monotonically 444The increase in width (and intensity) of the elastic line through the MIT is likely due to low-energy phonon excitations that cannot be discriminated due to the finite energy resolution.. There are two possible origins for the observed behavior. The first is that the magnetic excitations become Landau damped by intraband particle-hole excitations. At $T=0$, optical conductivity measurements have determined the optical gap $2\Delta_{O}=\text{158~{}meV}$ Sohn et al. (2015). The magnitude of $2\Delta_{O}$ decreases continuously with increasing temperature, and becomes comparable to the energy of the magnetic excitations at ca. 150 K. Hence one would expect the damping to increase accordingly above this temperature. This scenario is broadly consistent with the data. An alternative proposition is that the damping is driven by spin-phonon coupling; as already observed by Raman scattering Nguyen et al. (2017) and optical spectroscopy measurements Sohn et al. (2017). From fitting the data above 100 K to an Arrhenius function: $\Gamma=A\exp{\left(-\Delta/k_{B}T\right)}$, we find that the critical energy scale $\Delta=\text{25(3)~{}meV}$. This is in good agreement with the lowest energy Os phonon modes, both infrared-active [Os-Os stretch, 24.6(5) meV] and Raman-active [Os-O(1) stretch, 28 meV]. At present it is not possible to disentangle these two pictures.

V Discussion

Thus far we have determined that the magnetic excitations in Cd2Os2O7 are indicative of coupling between a number of different degrees of freedom. Furthermore, we have unambiguously confirmed the AIAO magnetic ground state and relevant leading order terms in the Hamiltonian. There are, however, some outstanding questions. Raman scattering measurements, quantum chemistry calculations, and classical Monte Carlo simulations have all obtained values of the exchange parameters which broadly agree with one another (Table 1). Yet the momentum dependence of the feature observed by RIXS appears to be best described by a Hamiltonian dictated by significantly larger coupling parameters ($J=\text{13.1~{}meV}$, $|\mathbf{d}|=\text{6.8~{}meV}$, $\mathsf{A}=-\text{12.9~{}meV}$). We suggest a possible origin for this discrepancy. In the semiclassical limit (LSWT), and assuming no magnon damping, one would expect magnon branches ($\Delta S=1$) at 65 and 75 meV, and a two-magnon ($\Delta S=0,2$) continuum between 130–150 meV. The kinematic $\Delta S=3$ continuum would be between 195 and 225 meV [Fig. 5(a)]. If mixing occurs between these different $\Delta S$ terms, then presumably they would move closer together in energy. The corresponding peak in the imaginary part of the dynamic susceptibility $\chi^{\prime\prime}(\mathbf{Q},\omega)$ would consequently be shifted towards the nominal $\Delta S=2$ value. We show this heuristic picture by the dashed line in Fig. 5(a). Such a scenario would qualitatively agree with the results of the ED calculations as presented in Ref. Calder et al., 2016, which equate to the quantum limit.

Even so, the ED calculations cannot fully describe all of the details observed in the experimental data. In order to better compare with theory, all of the data presented in Fig. 1(a) have been summed together, and the respective elastic and d-d components subtracted off. This acts as a crude approximation for the magnetic part of the RIXS DOS. We compare this in Fig. 5(b) with the theoretical RIXS intensity calculated by ED, which is shown without the effect of instrumental broadening. Whilst the energy scale of the excitations agree, there are discrepancies in the lineshape between experiment and theory, especially above 200 meV.

There are a number of factors which are likely to contribute. Firstly, the intensity observed experimentally actually corresponds to the RIXS scattering cross-section, which is significantly more difficult to calculate than the RIXS DOS. It is likely that the RIXS cross-section will be dominated by the $\Delta S=1$ component, as opposed to $\Delta S=3$. Fano coupling will also remove spectral weight at low energies and shift it to higher energies. Finally the ED calculations used coupling parameters determined from quantum chemistry calculations Bogdanov et al. (2013); Nguyen et al. have already shown that the experimental values obtained from Raman scattering appear to be slightly lower Nguyen et al. (2017).

Recall that the fit to the data was significantly improved at all temperatures and momentum transfers upon the inclusion of a broad, weakly dispersive feature (peak A2) centered at 0.3 eV energy loss [Fig. 1(d)]. No significant temperature dependence was observed [Figs. 4(c,d)], which implies that it is likely to have an electronic, rather than magnetic, origin. A similar feature observed in NaOsO3 was suggested to result from inter-band transitions based on calculations of the electronic band structure from density functional theory (DFT) Vale et al. (2018a); *vale2018_prb. We performed a similar analysis for Cd2Os2O7  using the previously published results by Shinaoka et al calculated in the local spin density approximation Shinaoka et al. (2012), including spin-orbit coupling and on-site Coulomb repulsion $U$ ($LSDA+SO+U$) [Fig. 6(a)]. The effective coupling $U_{\text{eff}}=U-J$ was taken to be 1.25 eV, which the authors of Ref. Shinaoka et al., 2012 found gave a ground state which was close to the MIT, but still in the antiferromagnetic insulating phase. For simplicity, the two electronic bands closest to the Fermi level were fitted in the parabolic approximation, using the band energies at high symmetry points to constrain the fits, and assuming that the band minimum is at $\Gamma$. The results are shown in Fig. 6(b), with all the main features of the two electronic bands reasonably reproduced.

Dipole-allowed inter-band transitions were calculated using a Monte Carlo approach. Two wavevectors in the Brillouin zone were randomly selected; these correspond to states in the valence and conduction bands which each have some particular energy. The difference in energy and wavevector between these states was computed, and mapped back onto the first Brillouin zone. This was repeated $4\times 10^{8}$ times, with the result shown in Fig. 6(c). A broad feature can be seen in the data which is centred around 0.55 eV. Along the $\Gamma$-$L$ and $\Gamma$-$X$ directions, however, a continuum of states can be clearly seen which extends to lower energies. Comparing the results at $\Gamma$ with experimental data taken at 200 K [Fig. 6(d)], we find a weak peak at 0.6 eV which is consistent with the energy scale of our excitations. Moreover, the low-energy continuum appears to be centered at around 0.25 eV. This suggests that the broad continuum observed previously [Fig. 4(d)] does indeed manifest from inter-band transitions.

It should be noted that our conclusions come with a number of limitations. First, our calculations only considered the two closest bands to the Fermi level, and assumed they were parabolic. Additional contributions from neighbouring bands will modify the lineshape somewhat, especially at higher energy loss [Fig. 6(a,b)]. The parabolic band approximation also averages over a degree of fine structure which is clearly present from the DFT calculations. Furthermore, RIXS at the $L_{3}$ edge is not a direct probe of the band structure, even in weakly correlated systems. Finally, we have neglected likely $\bm{Q}$-dependent matrix element effects, which enter the RIXS cross-section. This may explain the difference in the fitted width of peak A2 for different Brillouin zones at the same high-symmetry point [Figs. 1(f), 4(c)]. Nevertheless, the presence of inter-band transitions appears to provide an excellent description of the experimental data.

Our results clearly show the itinerant (non-local) nature of Cd2Os2O7. Osmium containing transition metal oxides (TMOs) are typically characterized by significant hybridization between the extended 5$d$ orbitals on the osmium site, and the 2$p$ orbitals on the oxygen sites Singh et al. (2002); Irizawa et al. (2006); Gangopadhyay and Pickett (2015); Taylor et al. (2016); Xu et al. (2016); Calder et al. (2017). For instance, in NaOsO3, it was determined by DFT that there were on average 4.3 $d$-electrons per Os site, instead of the nominal three Jung et al. (2013). Quantum chemistry calculations on a small cluster find that the orbital excitations in Cd2Os2O7 are generally better described by a Os $5d^{4}$ electronic configuration rather than the nominal $5d^{3}$ (Appendix A). Yet, it has been previously demonstrated that the exchange parameters obtained using a $5d^{3}$ are in excellent agreement with experiment (Table 1). The complexity of these calculations means that – by necessity – they neglect the effect of longer-ranged electronic interactions which may be important. Even so, it highlights that a free-ion description is inadequate for Cd2Os2O7.

This is even more apparent when comparing a representative RIXS spectrum of Cd2Os2O7 with that for the nominally isoelectronic (5$d^{3}$) Ca${}_{\text{3}}$LiOsO${}_{\text{6}}$ Taylor et al. (2017); Calder et al. (2017). This insulating double perovskite consists of isolated, almost ideal OsO${}_{\text{6}}$ octahedra separated by Li cations, and consequently acts as a good approximation to a free-ion model. This is in spite of strong hybridization between Os 5$d$ and O 2$p$ orbitals Calder et al. (2017). Four resolution-limited ($\Delta E=\text{150~{}meV}$) excitations can be observed for Ca3LiOsO6. This contrasts markedly with the situation for Cd2Os2O7, where the features are considerably broader despite the improved resolution ($\Delta E=\text{46~{}meV}$). Note that for $L$-edge RIXS – in particular on 5d TMOs – the intrinsic width (in energy) of an arbitrary excitation is essentially proportional to the inverse lifetime of the final state. This contrasts with x-ray absorption spectroscopy (XAS), in which the spectra are broadened by the core-hole in the intermediate state. What we observe experimentally is a convolution of this excitation with the spectrometer resolution function. Clearly the intrinsic width of these excitations is significantly broader for Cd2Os2O7 than the experimental resolution. This is indicative of the final state having a short lifetime. We posit this is due to longer-ranged electronic interactions between neighboring OsO6 octahedra through the shared oxygen site (absent in Ca3LiOsO6), which, in turn, leads to greater itinerancy.

VI Conclusion

We have presented high-resolution RIXS measurements on Cd2Os2O7. Our results highlight the complementary information that RIXS and Raman scattering provide. Raman scattering has superior energy resolution and count rates, however it is only sensitive to zone-center $\Delta S=0$ excitations as a consequence of selection rules. Meanwhile RIXS is a momentum resolved, orbital and element-specific technique, which can probe $\Delta S\neq 0$ excitations due to spin-orbit coupling in the intermediate state. We conclude that the magnetic excitations previously observed by RIXS and Raman scattering essentially correspond to different parts of the same feature. The asymmetry of peak A is proposed to result from coupling between a resonant magnetic process and inter-band electronic excitations. This further highlights the interplay between magnetism and the electronic behavior of Cd2Os2O7. Moreover, the presence of inter-band transitions is due to significant hybridization and itinerant behavior, and is indicative of a breakdown of the free-ion model frequently used to describe RIXS spectra. This would have ramifications for the analysis of orbital excitations in other systems proximate to a metal-insulator transition, and ties in with the description of magnetic excitations in the itinerant and localized regimes Vale et al. (2018a); *vale2018_prb.

Acknowledgements.

Research in London was supported by the Engineering and Physical Sciences Research Council (Grants No. EP/N027671/1, EP/N034872/1). Work at ORNL’s High Flux Isotope Reactor was supported by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy (DOE). J.G.V. would like to thank University College London (UCL) and École Polytechnique Fédérale de Lausanne (EPFL) for financial support through a UCL Impact award. A.D.C. was partially supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division.

Appendix A Quantum chemistry calculations

New quantum chemistry calculations were performed for Cd2Os2O7 following the protocol described in Ref. Bogdanov et al., 2013 for calculating the intra-site excitation energies Werner et al. (2012); Figgen et al. (2009); Schautz et al. (1998); Dunning (1989); Pierloot et al. (1995). These calculations simulate the effect of adding one extra electron to the finite cluster, giving rise to a $5d^{4}$ electronic configuration. The embedded cluster consists of one reference OsO6 octahedron, six adjacent Cd sites, and six nearest neighbor OsO6 octahedra in which Os5+ were represented by closed-shell Ta5+ $d^{0}$ species. The farther environment was modeled as an one-electron effective potential, which in a ionic picture reproduces the Madelung field in the cluster region. We performed complete active space self-consistent-field (CASSCF) calculations with 4 electrons in 5 orbitals averaging over lowest six singlet, ten triplet and two quintet states. In the following multireference configuration interaction (MRCI) computation, we included on top of the CASSCF wave functions all single and double excitations from the Os $5d$ and O $2p$ orbitals at the central octahedron. Finally, the SOC Hamiltonian Berning et al. (2000) between all spin components of the 18 spin-free MRCI states is computed and diagonalized resulting in 46 SO states. The splittings between the lowest 18 states are listed in Table 2.

We compared the results of these calculations (and those presented in Ref. Bogdanov et al., 2013, which are reproduced in Table 3) with the experimental data. We find that the agreement with experiment is generally better for a $5d^{4}$ configuration than $5d^{3}$. There are two main observations which lead to this conclusion. For instance, previous high-resolution RIXS measurements show a broad excitation centred at 4.5 eV (peak B). This was proposed by the authors to result from multiple overlapping $t_{2g}^{3}\rightarrow t_{2g}^{2}e_{g}$ transitions Calder et al. (2016). Yet the previous QC calculations (Table 3) determine the lowest energy $t_{2g}^{2}e_{g}$ multiplet to be at 5.1 eV. This overestimate of the energy scales consistently occurs both within MRCI and MRCI+SOC. Meanwhile, the energy scale of the $S=1$ $t_{2g}^{4}\rightarrow t_{2g}^{3}e_{g}$ excitations appears to match rather well with the experimental results (Table 2). We also note that the MRCI+SOC calculations predict an excitation at 0.36 eV; again comparable to the energy of peak A2. This suggests that the electronic behavior of Cd2Os2O7 may indeed exhibit some $5d^{4}$ character.