Spin-orbit coupling in a half-filled $t_{2g}$ shell: The case of $5d^3$ ${\mathrm{K}}_2{\mathrm{ReCl}}_6$

The half-filled $t_{2g}$ shell of the $t_{2g}^3$ configuration usually, in $LS$ coupling, hosts a $S=3/2$ ground state with quenched orbital moment. This state is not Jahn-Teller active. Sufficiently large spin-orbit coupling $\zeta$ has been predicted to change this picture by mixing in orbital moment, giving rise to a sizable Jahn-Teller distortion. In $5d^3{\mathrm{K}}_2{\mathrm{ReCl}}_6$ we study the electronic excitations using resonant inelastic x-ray scattering and optical spectroscopy. We observe on-site intra-$t_{2g}$ excitations below 2 eV and corresponding overtones with two intra-$t_{2g}$ excitations on adjacent sites, the Mott gap at 2.7 eV, $t_{2g}$-to-$e_g$ excitations above 3 eV, and charge-transfer excitations at still higher energy. The intra-$t_{2g}$ excitation energies are a sensitive measure of $\zeta$ and Hund's coupling $J_{\mathrm{H}}$. The sizable value of $\zeta \approx 0.29\mathrm{eV}$ places ${\mathrm{K}}_2{\mathrm{ReCl}}_6$ into the intermediate coupling regime, but $\zeta /J_{\mathrm{H}}\approx 0.6$ is not sufficiently large to drive a pronounced Jahn-Teller effect. We discuss the ground state wave function in a Kanamori picture and find that the $S=3/2$ multiplet still carries about 97% of the weight. However, the finite admixture of orbital moment allows for subtle effects. We discuss small temperature-induced changes of the optical data and find evidence for a lowering of the ground state by about 3 meV below the structural phase transitions.

Figure 1

(a) Sketch of the room-temperature crystal structure of ${\mathrm{K}}_2{\mathrm{ReCl}}_6$. The ${\mathrm{Re}}^{4+}$ ions form an fcc lattice. The ${\mathrm{ReCl}}_6$ octahedra are depicted in green and the ${\mathrm{K}}^{+}$ ions are gray. (b) Jahn-Teller distortion of a $t_{2g}^3$ configuration as a function of ${\zeta }^{\mathrm{eff}}/J_{\mathrm{H}}^{\mathrm{eff}}$, normalized to the value for ${\zeta }^{\mathrm{eff}}/J_{\mathrm{H}}^{\mathrm{eff}}\to \infty$ (adapted from [4]). The calculation assumes $J_{\mathrm{H}}B/g^2=1/2$, where $g$ and $B$ are electron-phonon coupling parameters of the Jahn-Teller Hamiltonian [4]. Red symbol marks the position of ${\mathrm{K}}_2{\mathrm{ReCl}}_6$ according to our analysis.

Figure 2

RIXS data of ${\mathrm{K}}_2{\mathrm{ReCl}}_6$ at $T=300\mathrm{K}$. (a) Resonance map of the RIXS intensity based on spectra measured with different incident energy $E_{\mathrm{in}}$ for transferred momentum $\mathbf{q}\approx \left(7.47.45.9\right)$. (b) RIXS spectra for $E_{\mathrm{in}}=10.534\mathrm{keV}$ and 10.5375 keV, i.e., at $t_{2g}$ and $e_g$ resonance, respectively. Inset: sketch of the scattering geometry.

Figure 3

Optical data of ${\mathrm{K}}_2{\mathrm{ReCl}}_6$. (a) Transmittance $T\left(\omega \right)$ for a sample thickness $d=471\mu \mathrm{m}$ at 18 and 300 K. The data reveal weak excitations within the $t_{2g}^3$ subshell, while the onset of excitations across the Mott gap at ${\mathrm{\Delta }}_{\exp }=2.7\mathrm{eV}$ at 18 K marks the upper energy limit of the transparency range. The inset highlights Fabry-Pérot interference fringes. (b) Refractive index $n\left(\omega \right)$ extracted from Fabry-Pérot interference fringes in $T\left(\omega \right)$ (symbols). Note that $n\left(\omega \right)$ is nearly constant below the gap. Dashed: empirical fit using the Cauchy model. Lines: Kramers-Kronig-consistent fit using an oscillator model with an infrared-active phonon at $E_{ph}=39\mathrm{meV}$ and a Tauc-Lorentz oscillator with the measured gap ${\mathrm{\Delta }}_{\exp }=2.7\mathrm{eV}$.

Figure 4

Optical conductivity ${\sigma }_1\left(\omega \right)$ of ${\mathrm{K}}_2{\mathrm{ReCl}}_6$ below the Mott gap for several temperatures. The onset of excitations across the Mott gap is at 2.7 eV at 18 K. Below the gap, there are phonon-assisted intra-$t_{2g}$ excitations in the range from 0.9 to 2 eV. Above 2 eV, we observe weak overtones, i.e., intersite pair excitations that correspond to two intra-$t_{2g}$ excitations on neighboring sites. Inset: comparison of intra-$t_{2g}$ excitations of ${\mathrm{K}}_2{\mathrm{ReCl}}_6$ and its $d^4$ sister compound ${\mathrm{K}}_2{\mathrm{OsCl}}_6$ [23]. From the latter, a small constant offset has been subtracted.

Figure 5

Energy level diagram for the $d^3$ configuration. (a)–(c) Using quanty [47, 48], the cubic multiplet energies were calculated as a function of cubic crystal-field splitting $10Dq$, spin-orbit coupling $\zeta$, and interelectronic Coulomb interaction in terms of the Slater integrals $F^2$ and $F^4$. The latter is captured by $J_{\mathrm{H}}=1/14(F^2+F^4)$. Panel (a) uses $J_{\mathrm{H}}=\zeta =0$ and shows the $t_{2g}^3$ states at $E=0$ and $t_{2g}^{3-n}{e}_g^n$ states at higher energy. (b) Coulomb interactions lift the degeneracy of the $t_{2g}^3$ states. (c) Effect of $\zeta$. Orange symbols: intra-$t_{2g}$ energies taken from ${\sigma }_1\left(\omega \right)$ and the $t_{2g}$-to-$e_g$ excitation energy of 3.3 eV observed in RIXS. Their position on the $\zeta$ axis marks the value obtained from a fit (see Sec. 3c2). For the $e_g$ states, the maximum of the broadened RIXS response was chosen, lying between the second and third energy levels. (d) Effect of a tetragonal crystal field. Circles highlight the splitting of the lowest and highest intra-$t_{2g}$ excitations.

Figure 6

Temperature dependence of intra-$t_{2g}$ excitations. Both panels depict ${\sigma }_1\left(\omega \right)$ over a range of 0.5 eV. The multipeak structure and the temperature-driven increase of spectral weight indicate the phonon-assisted character; see Fig. 7. Vertical dashed lines denote the bare electronic energies $E_{0,i}$ for the excited states ${\mathrm{\Gamma }}_8$ at 948 meV, ${\mathrm{\Gamma }}_8$ at 1107 meV, ${\mathrm{\Gamma }}_6$ at 1164 meV, ${\mathrm{\Gamma }}_7$ at 1723 meV, and ${\mathrm{\Gamma }}_8$ at 1901 meV. The tiny feature at 1959 meV (633 nm) is an artifact from a HeNe laser used to calibrate the energy of the Fourier spectrometer.

Figure 7

Common peak structure of intra-$t_{2g}$ excitations at 18 K. Data of the five absorption bands $i=1$–5, see Fig. 6, are plotted as a function of $E-E_{0,i}$ to highlight the common energies of the symmetry-breaking phonons with $E_{\mathrm{ph}}=5$, 16, 20, and 39 meV (vertical dotted lines) that yield peaks at $E_{0,i}+E_{\mathrm{ph}}$. Moreover, the plot shows the Franck-Condon phonon sidebands at $E_{0,i}+E_{\mathrm{ph}}+E_{a_{1g}}$ with $E_{a_{1g}}\approx 44\mathrm{meV}$ (arrows). Note that the excitations at $E_{0,2}=1107$ meV and $E_{0,3}=1164\mathrm{meV}$ are close in energy (light blue and orange). To disentangle the corresponding features, the lines have been cut at 1159 meV, i.e., at 52 meV for $E_{0,2}$ and $-5$ meV for $E_{0,3}$. The corresponding plot for 6 K, below $T_N$, is given in Appendix pp1.

Figure 8

Temperature dependence of ${\sigma }_1\left(\omega \right)$ around 1.7 eV. (a) Closeup of ${\sigma }_1\left(\omega \right)$ highlighting the thermal population of the phonon mode with $E_{\mathrm{ph}}=39\mathrm{meV}$ that yields an intra-$t_{2g}$ excitation at $E_{0,4}-E_{\mathrm{ph}}$ for $E_{0,4}=1723\mathrm{meV}$. For each temperature, an offset has been subtracted that was fixed outside the absorption band, as described for Fig. 9. Vertical black lines denote the energy range used for the integration to calculate the spectral weight. (b) Spectral weight of the phonon-assisted feature shown in (a) (symbols). Red line: fit using a Bose occupation factor with $E_{\mathrm{ph}}=39\mathrm{meV}$.

Figure 9

Spectral weight as function of temperature. (a) Spectral weight (symbols) of the absorption bands with $E_{0,i}$ within the given integration ranges. Note that the data of the close lying bands with $E_{0,2}$ and $E_{0,3}$ have been integrated together (blue). The data for $E_{0,1}=948\mathrm{meV}$ (black) and $E_{0,5}=1901\mathrm{meV}$ (green) nearly fall on top of each other. In each case, the data show the characteristic temperature dependence of a phonon-assisted process, as shown by the fits (red lines) following Eq. (1) with four phonon modes at $E_{\mathrm{ph},k}=5$, 16, 20, and 39 meV. For each temperature, a linear offset has been subtracted from ${\sigma }_1\left(\omega \right)$ that was fixed outside the absorption bands, i.e., at 755 and 1275 meV for $E_{0,1}$ to $E_{0,3}$ and at 1400 and 2040 meV for the upper two bands. Vertical dashed lines denote the structural transition temperatures. (b) The normalized spectral weights of the four integration ranges plotted in (a) collapse to a single curve, corroborating the common phonon energies.

Figure 10

Intersite $d\text{−}d$ overtones in ${\sigma }_1\left(\omega \right)$ of ${\mathrm{K}}_2{\mathrm{ReCl}}_6$. Above the highest intra-$t_{2g}$ excitation with $E_{0,5}=1901\mathrm{meV}$, we observe peaks at energies that (nearly) coincide with the sum $E_{0,i}+E_{0,j}$ of two intra-$t_{2g}$ excitation energies. Hence these features are not phonon assisted but directly infrared active, in agreement with their temperature dependence. Still also these higher-order excitations show a Franck-Condon sideband of the $a_{1g}$ breathing mode with $E_{a_{1g}}=44\mathrm{meV}$ (arrows; cf. Fig. 7).

Figure 11

Temperature dependence of the refractive index $n\left(\omega \right)$ and of the intra-$t_{2g}$ energies. (a) $n\left(\omega \right)$ extracted from the energies of two Fabry-Pérot interference fringes at around 0.75 and 1.6 eV with order $m_{\mathrm{fringe}}=979$ and 2131, respectively. The curve of $n$ of the latter has been shifted down by 0.021 to facilitate comparison. (b)–(d) Peak energies $E_{0,i}+E_{\mathrm{ph}}$ for $i=2$, 3, and 4 and two different phonon modes. Symbols have been shifted as indicated in the panels. (e) Peak energies of the overtones at $E_{0,i}+E_{0,j}$ for $(i,j)=(1,2)$, (2,2), and (2,3). In all panels, vertical dashed lines indicate the temperatures of the structural phase transitions, i.e., 77, 103, and 111 K.

Figure 12

Value of ${\sigma }_1\left(\omega \right)$ at the peak maximum for the band around $E_{0,1}=948\mathrm{meV}$. Left: zoom in on ${\sigma }_1\left(\omega \right)$ around the peak at $E_{0,1}+E_{\mathrm{ph}}$. A background has been subtracted, as discussed for Fig. 9. Right: the increase of the peak maximum at high temperature reflects the increase of the spectral weight; see Fig. 9. In view of the spectral weight, the opposite behavior of the peak maximum below 77 K indicates a change of the linewidth. Vertical dashed lines mark the temperatures of the structural phase transitions.

Figure 13

Absolute energies of the $t_{2g}^3$ states and ground state wave function for $10Dq=\infty$. (a) To obtain excitation energies, the ground state energy $E_0$ (lowest line) has to be subtracted; see Eq. (2). For ${\zeta }^{\mathrm{eff}}=0$, the ground state is at zero while the excited states are at $3J_{\mathrm{H}}^{\mathrm{eff}}$ and $5J_{\mathrm{H}}^{\mathrm{eff}}$. Spin-orbit coupling mixes some of the states (solid lines) while other states remain unaffected (dashed), giving rise to, in total, four excitation energies. The best agreement with experiment (symbols) is obtained for ${\zeta }^{\mathrm{eff}}/J_{\mathrm{H}}^{\mathrm{eff}}=0.82$ with $J_{\mathrm{H}}^{\mathrm{eff}}=343\mathrm{meV}$. Numbers in the right panel denote the number of states. Note the change in scale of the horizontal axis from ${\zeta }^{\mathrm{eff}}/J_{\mathrm{H}}^{\mathrm{eff}}$ in the left panel to $J_{\mathrm{H}}^{\mathrm{eff}}/{\zeta }^{\mathrm{eff}}$ in the right panel. This is accompanied by a change of the energy scale, which is given in units of $J_{\mathrm{H}}^{\mathrm{eff}}$ on the left and ${\zeta }^{\mathrm{eff}}$ on the right. (b) Squared amplitudes of the three contributions to the ground state wave function corresponding to the solid lines in (a).

Figure 14

Comparison of experimental and simulated RIXS data. Red symbols: RIXS result for $E_{\mathrm{in}}=10.534\mathrm{keV}$. Solid blue line: RIXS spectrum simulated using quanty for $10Dq=3.25$ eV, $\zeta =290\mathrm{meV}$, and $J_{\mathrm{H}}=464\mathrm{meV}$ for Voigt oscillators with 10 meV width. Dashed blue line: same data using a Voigt line shape where the width of the Gaussian part is set to the experimental resolution of 295 meV, while the width of the Lorentzian contribution is 10 meV for the intra-$t_{2g}$ excitations and 300 meV for excitations to $e_g$ states above 3 eV. The total intensity is scaled to agree with the experimental data at the maximum. Note that the charge-transfer excitations at higher energies are not included in the simulation.

Figure 15

Peak structure of intra-$t_{2g}$ excitations below $T_N$. Top: comparison of the line shape of the five absorption bands at 6 K. Bottom: temperature dependence of the band at $E_{0,4}=1723\mathrm{meV}$.