Importance of van der Waals interactions and cation-anion coupling in an organic quantum spin liquid

The Mott insulator ${\beta }^{\prime }\text{−}{\mathrm{EtMe}}_3\mathrm{Sb}{\left[\mathrm{Pd}{\left(\mathrm{dmit}\right)}_2\right]}_2$ belongs to a class of charge transfer solids with highly frustrated triangular lattice of $S=1/2$ molecular dimers and a quantum-spin-liquid ground state. Our experimental and ab initio theoretical studies show the fingerprints of strong correlations and disorder, the important role of cation-dimer bonding in charge redistribution, no sign of intra- and interdimer dipoles, and the decisive van der Waals contribution to interdimer interactions and the ground-state structure. The latter consists of quasidegenerate electronic states related to the different configurations of cation moieties, which permit two different equally probable orientations. Upon reducing the temperature, the low-energy excitations slow down, indicating glassy signatures as the cation motion freezes out.

Figure 1

(a) Side view of extended unit cell in $C2/c$ symmetry of ${\beta }^{\prime }\text{−}{\mathrm{EtMe}}_3\mathrm{Sb}$. The molecule $\mathrm{Pd}{\left(\mathrm{dmit}\right)}_2$ is shown within the shaded area. Yellow, grey, green, light grey, and violet circles denote sulfur, carbon, palladium, hydrogen, and antimony atoms, respectively. H-bonds between the terminal S atoms of $\mathrm{Pd}{\left(\mathrm{dmit}\right)}_2$ molecules and H of Et (${\mathrm{CH}}_2\text{−}{\mathrm{CH}}_3$) or Me (${\mathrm{CH}}_3$) groups of cations are marked by green lines. (b) Relaxed ground-state structure in the $ac$ plane projected along the $b$ axis as obtained from DFT calculations. Et and the Me sites are in the proximity of S atoms of dimers in the central and side layer, respectively. There are eight short contacts (green lines) between two S atoms at one end of the dimer and cations in the central layer, while there are only six contacts (blue lines) in the side layer. Yellow and white lines denote large and small S-S dimer openings near the Et and Me groups, respectively.

Figure 2

Lower panel: Cation-anion interaction-induced electronic charge redistribution in the relaxed ground state calculated as the difference between the charge density of the complete system and its constituents, i.e., cations and anions. Two isosurfaces are shown: electron accumulation (red) and depletion (blue). Balls of different color denote different atoms in the structure. The largest electron accumulation is concentrated close to the end S sites of [Pd(dmit)${}_2{]}_2$ dimers; it is much more enhanced near Et than Me sites. Upper panel: the averaged charge density in the $ac$ plane (see Appendix pp3). The color bar shows charge transfer ($e/{\AA }^2$).

Figure 3

Calculated band structure in the relaxed ground state of ${\beta }^{\prime }\text{−}{\mathrm{EtMe}}_3\mathrm{Sb}$ plotted along the high-symmetry directions. The segments $\mathrm{\Gamma }X$, $\mathrm{\Gamma }K$, and $\mathrm{\Gamma }Y$ in the first Brillouin zone correspond to the $a$, $b$, and $c$ directions, respectively. Weakly dispersive bands are identified at about 0.5 eV, $-0.4\mathrm{eV}$, and $-1.1\mathrm{eV}$, while strongly dispersive bands cross the Fermi level and accommodate two electrons (red shaded area in the density of states).

Figure 4

DC resistivity of ${\beta }^{\prime }\text{−}{\mathrm{EtMe}}_3\mathrm{Sb}$ versus inverse temperature indicating nearest-neighbor hopping at higher temperatures. Inset displays normalized dc conductivity within dimer planes as a function of $T^{-1/2}$, demonstrating the Efros-Shklovskii hopping due to strong electron correlations for $T\lesssim 100\mathrm{K}$. Data for $\mathbf{E}\parallel b$ are upshifted for clarity.

Figure 5

(a), (b), (c) Real part of the dielectric function ${\varepsilon }^{\prime }$ of ${\beta }^{\prime }\text{−}{\mathrm{EtMe}}_3\mathrm{Sb}$ versus temperature demonstrates the relaxorlike ferroelectric behavior for ac electric field applied within ($\mathbf{E}\parallel a$, $\mathbf{E}\parallel b$) and perpendicular ($\mathbf{E}\parallel c$) to the dimers plane. Dielectric strength $\mathrm{\Delta }\varepsilon$ (d), relaxation time distribution ($1-\alpha$) (e), mean relaxation time ${\tau }_0$ (f) as function of inverse temperature. Inset in (f) shows double logarithmic plot of frequency dependence of real ${\varepsilon }^{\prime }$ and imaginary ${\varepsilon }^{″}$ parts of dielectric function for $\mathbf{E}\parallel a$ at 36 K. Full lines are fits to generalized Debye function $\varepsilon \left(\omega \right)-{\varepsilon }_{\mathrm{HF}}=\mathrm{\Delta }\varepsilon /\left[1+{\left(\mathrm{i}\omega {\tau }_0\right)}^{1-\alpha }\right]$, ${\varepsilon }_{\mathrm{HF}}$ is high-frequency dielectric constant. Full lines in (f) show the Arrhenius slowing down of ${\tau }_0$ at the rate of 36 meV along all three crystallographic axes.

Figure 6

Left panel: Temperature evolution of the dimeric ${\nu }_1$ and intramolecular vibration ${\nu }_2$ of ${\beta }^{\prime }\text{−}{\mathrm{EtMe}}_3\mathrm{Sb}$ measured for $\mathbf{E}\parallel c$. Except the 5 K data, all curves are upshifted vertically for clarity. No splitting of the mode occurs. Temperature dependence of the resonance frequency (upper right panel) and damping (lower right panel) obtained by Fano function fits of the modes.

Figure 7

(a) Sketch of the molecule $\mathrm{Pd}{\left(\mathrm{dmit}\right)}_2$, where dmit = 1,3-dithiole-2-thione-4,5-dithiolate: yellow, grey, and green circles denote sulfur, carbon, and palladium atoms, respectively. (b), (c) View of $\mathrm{Pd}{\left(\mathrm{dmit}\right)}_2$ dimers in the $ab$ plane projected along the direction tilted ${17}^{\circ }$ away from the $c$ axis. (b) and (c) panels show dimers in two neighboring layers which have different stacking directions $(a+b)$ and $(a-b)$, respectively. An almost isotropic triangular lattice is denoted by gray lines, the interdimer transfer integrals are labeled by $t_b$ (thick gray) and $t_{+}$ (thin gray), and $t_{-}$ (dashed gray), while the intradimer transfer integral is labeled by $t_d$. (d) Side view of extended unit cell in $C2/c$ symmetry. Light grey and violet circles denote hydrogen and antimony atoms, respectively. Possible hydrogen bonds between the end S ions of the $\mathrm{Pd}{\left(\mathrm{dmit}\right)}_2$ molecules and H of Et (${\mathrm{CH}}_2\text{−}{\mathrm{CH}}_3$) or Me (${\mathrm{CH}}_3$) groups of the cations are indicated by full green lines.

Figure 8

Two relaxed configurations of ${\beta }^{\prime }\text{−}{\mathrm{EtMe}}_3\mathrm{Sb}$, the state with minimum energy (upper panel) and the state with energy higher by 28 meV (lower panel) in the $ac$ plane projected along the $b$ axis. In both configurations, two ends of each dimer are in same environment, while the environment is different in the center and side layer. In the former configuration, the dimer ends are near to Et in the center layer, while they are near to the Me groups in the side layer. It is just the opposite in the latter, i.e., dimers' ends are near Me in the center layer, while they are near Et groups in the side layer. Yellow, grey, light grey, and violet circles denote sulfur, carbon, hydrogen, and antimony atoms, respectively. The unit cell is marked as a parallelogram.

Figure 9

Four relaxed configurations of ${\beta }^{\prime }\text{−}{\mathrm{EtMe}}_3\mathrm{Sb}$, higher in energy by 45–79 meV from the ground state, in the $ac$ plane projected along the $b$ axis. In two configurations displayed in the left column, two ends of each dimer are in the same environment, but the environment differs within the layer, as well as between the center and side layers. In two configurations displayed in the right column, two ends of each dimer are in different environment. The environment differs within the layer, but is the same in the center and side layers. Yellow, grey, light grey, and violet circles denote sulfur, carbon, hydrogen, and antimony atoms, respectively. The unit cell is marked as a parallelogram.

Figure 10

Two relaxed configurations of ${\beta }^{\prime }\text{−}{\mathrm{EtMe}}_3\mathrm{Sb}$, at 146 meV above the ground state, in the $ac$ plane projected along the $b$ axis. In both configurations, two ends of each dimer are in different environments: one end is near Et, while the other end is near the Me group. The environment of dimer ends is the same in the center and side layers. However, note, that in contrast to the configurations in Figs. 8 and 9, 50%–50% occupancy of the two Et sites of the original average structure is not preserved. Yellow, grey, light grey, and violet circles denote sulfur, carbon, hydrogen, and antimony atoms, respectively. The unit cell is marked as a parallelogram.

Figure 11

View of the layers of cations connected to the end sulfur atoms of the [Pd(dmit)${}_2{]}_2$ dimers in the $ab$ plane projected along the $c$ axis. For simplicity, only one layer of cations with terminal S atoms in one neighboring anion layer is shown. The unit cell is marked as a rectangle. Gray thick line denotes the phase boundary. Yellow, grey, light grey, and violet circles denote sulfur, carbon, hydrogen, and antimony atoms, respectively. Yellow and white full lines denote small and large S-S dimer openings in the vicinity of Me and Et cation groups, respectively. Thick full line stands for the carbon-carbon bond within the Et group. Assumed local inversion symmetry breaking promotes formation of an antiphase boundary between two neighboring domains, in which overall 50%–50% occupancy of the two Et sites in the original average structure is preserved.

Figure 12

Band structure of the relaxed ground state of ${\beta }^{\prime }\text{−}{\mathrm{EtMe}}_3\mathrm{Sb}$ plotted along the high-symmetry directions. Left panel: If only the molecular self-standing subsystem is calculated, the generic anion bands (blue lines) are obtained. These bands exhibit a gap which exceeds those calculated for $\kappa$-CuCN and $\kappa$-AgCN [7, 8]. Right panel: Shifting the generic anion bands by 250 meV (red lines) results in a fair coincidence with the band structure calculated for the complete system (black), thereby revealing the dominantly anion-derived band character around the Fermi energy.

Figure 13

Band structure of ${\beta }^{\prime }\text{−}{\mathrm{EtMe}}_3\mathrm{Sb}$ at ambient pressure (black) and at $p=11\mathrm{kbar}$ (orange) plotted along the high-symmetry points. The band structure remains almost unchanged under pressure although the volume is reduced by 5%. This indicates that the structure is dominantly stabilized by van der Waals forces.

Figure 14

Double logarithmic plot of the frequency dependence of the real (${\varepsilon }^{\prime }$) and imaginary (${\varepsilon }^{″}$) parts of the dielectric function in representative single crystals of ${\beta }^{\prime }\text{−}{\mathrm{EtMe}}_3\mathrm{Sb}$ at $T=36\mathrm{K}$. Data for $\mathbf{E}\parallel a$, for $\mathbf{E}\parallel b$, and for $\mathbf{E}\parallel c$ are shown in upper, center, and lower panels, respectively. The full lines are fits to a generalized Debye function (see text).

Figure 15

DC conductivity, $\sigma$ (full line) and the product ${\epsilon }_0\mathrm{\Delta }\varepsilon {\omega }_0$ of ${\epsilon }_0$, dielectric strength $\mathrm{\Delta }\varepsilon$, and inverse of the mean relaxation time $1/{\tau }_0$ (full circles) of ${\beta }^{\prime }\text{−}{\mathrm{EtMe}}_3\mathrm{Sb}$ versus inverse temperature. Data for the measurements with the applied ac field $\mathbf{E}\parallel a$, $\mathbf{E}\parallel b$, and $\mathbf{E}\parallel c$ are shown by black squares, blue circles, and red diamonds, respectively.

Figure 16

Polarization-dependent temperature evolution of the in-plane optical conductivity ($\mathbf{E}\parallel a$, $\mathbf{E}\parallel b$) of ${\beta }^{\prime }\text{−}{\mathrm{EtMe}}_3\mathrm{Sb}$ in the range of the out-of-phase combination of the ${\nu }_1$ vibration and in-phase combination of the ${\nu }_2$ mode in a ${\left[\mathrm{Pd}{\left(\mathrm{dmit}\right)}_2\right]}_2$ dimer. Apart from the smaller intensity and more pronounced asymmetry along the crystallographic $b$-direction, the resonance frequency and temperature dependence of the in-plane modes are similar. In general, there is a typical blue-shift upon cooling down from the incoherent, highly conducting regime. When entering the Mott insulating regime, i.e., upon crossing the quantum Widom line at 120 K [4], the shift comes to a halt. This behavior is distinct from the out-of-plane response ($\mathbf{E}\parallel c$), where the ${\nu }_2$ mode shows an unusual red shift at low temperatures. The differences between in- and out-of-plane response stem from the influence of the electronic background; for $\mathbf{E}\parallel c$ we probe the ${\nu }_2$ vibration undisturbed by the electronic background, so it provides a clean probe of the molecular charge. Except the 295 K data, all curves are shifted vertically by a constant value.

Figure 17

The overlap of the $\mathrm{Sb}{\left({\mathrm{CH}}_3\right)}_3{\left({\mathrm{CH}}_2{\mathrm{CH}}_3\right)}^{+}$ in the gas phase (red) and in a crystal (blue).

Figure 18

Potential energy curve (in eV) for the methyl rotation in the solid state ${\beta }^{\prime }\text{−}{\mathrm{EtMe}}_3\mathrm{Sb}{\left[\mathrm{Pd}{\left(\mathrm{dmit}\right)}_2\right]}_2$ as computed using vdW-DF with optB88. The energies of the ground (red) and first excited (green) vibrational states are shown.