Spin-Orbit Coupling Effect on the Seebeck Coefficient in Dirac Electron Systems in $α$-(BETS)$_2$I$_3$

TL;DR

This work investigates the Seebeck coefficient in the two-dimensional Dirac-electron system of ambient-pressure alpha-(BETS)2I3 by constructing an ab initio tight-binding model that naturally includes spin-orbit coupling (SOC) and scattering from impurities and phonons. The Seebeck response is computed from linear-response theory via spectral conductivity, with the chemical potential $mu$ fixed by a three-quarter filling condition. Key findings are that $S_x<0$ and $S_y<0$ at low $T$, while $S_y$ undergoes a sign change at higher $T$, and SOC enhances the magnitude of $S$ in the low-$T$ regime due to asymmetries in the conduction/valence band contributions governed by DOS features and velocity anisotropy. These results, tied to DOS shoulders and Van Hove singularities, elucidate the anisotropic thermoelectric behavior and distinguish BETS from its BEDT-TTF counterpart, with implications for interpreting experiments and guiding future studies on SOC and transport in organic Dirac systems.

Abstract

The Seebeck coefficient, $S=L_{12}/(TL_{11})$, which is proportional to a ratio of the thermoelectric conductivity $L_{12}$ to the electric conductivity $L_{11}$ with $T$ being temperature is examined for two-dimensional Dirac electrons in the three-quarter filled organic conductor, $α$-(BETS)$_2$I$_3$, [BETS = BEDT-TSeF = bis(ethylenedithio)tetraselenafulvalene] at ambient pressure.Using a tight-binding model obtained with the first-principles relativistic density-functional theory method [Tsumuraya and Suzumura, Eur. Phys. J. B 94, 17 (2021)], we calculate $S$ in the presence of the impurity and electron--phonon scatterings. We show that $S_x < 0$ and $S_y >0$ at high temperatures, where $S_x$ ($S_y$) denotes $S$ perpendicular (parallel) to the molecular stacking axis. There is a sign change of $S_y$ with increasing $T$. We find that, at low temperatures the absolute value of $S$ is enhanced by the spin-orbit coupling. The Seebeck coefficient is examined by dividing it into components of the conduction and valence bands; we find that the electron and hole contributions compete with each other. Such $T$ dependence of $S$ is clarified using the spectral conductivity, which determines $L_{12}$ and $L_{11}$

Figures (6)

• Figure 1: (Color online) Crystal structure of $\alpha$-(BETS)$_2$I$_3$. There are four molecules, $A$, $A'$, $B$ and $C$ in the unit cell (dot-dashed line), which forms a square lattice. The transfer energies with the same spin (opposite spin ) are shown in Table \ref{['table_1']} (Table \ref{['table_2']}); the energies for the nearest-neighbor (NN) sites are given by $a1, \cdots, b4$. Those for the next-nearest-neighbor (NNN) sites with the same molecules are given by $a1'$, $a3'$, and $a4'$ along the $k_y$ direction and $s1, \cdots, s4$ (not shown here) along the $k_x$ direction. Those for NNN sites with different kinds of molecules are given by $c1, \cdots, c4$ and $d0, \cdots, d3$, which are not shown here, but we refer readers to our previous paper Tsumuraya_Suzumura. The cross denotes an inversion center between $A$ and $A'$. $x$ ($y$) denotes a coordinate perpendicular to (along) the molecular stacking direction.
• Figure 2: (Color online ) DOS per spin as a function of $\omega - \mu_0$, where $\mu_0=0.1687$ denotes $\mu$ at $T = 0$. The solid line (SOC I) shows DOSs with the same spin ($s = s'$), while the dotted line (SOC III) denotes DOSs with only Re[$w_h(s = s')$]. The dash-dotted line (SOC II) shows DOSs in the presence of SOC with both the same and opposite spins ($s = s'$ and $s = -s'$). The energy at the Dirac point is shown by $D$ and those of the Van Hove singularity are shown by $A$ and $B$, which are explained in the text. The inset denotes $\mu$ as a function of $T$.
• Figure 3: (Color online ) (a) Two bands, conduction and valence bands, given by $E_1(\bm{k})$ (upper band) and $E_2(\bm{k})$ (lower band), which contact at a Dirac point. TRIMs Y and M' denote $(k_x,k_y)/\pi$ = (0,-1) and (1,-1) respectively. The cross in $E_2(\bm{k})$ shows a saddle point on a line connecting Y and the Dirac point. (b) Magnified two bands $E_1(\bm{k})$ and $E_2(\bm{k})$ showing a tilted Dirac cone around the Dirac point $\bm{k}_{\rm D}$, where $\delta \bm{k} = \bm{k} - \bm{k}_{\rm D}$ and $\bm{k}_{\rm D} = -(0.729,-0.577)\pi$. (c) Contour plots of $E_1(\bm{k}) - E_2(\bm{k})$ on the plane of $\delta \bm{k}$, where the red line corresponds to $E_1(\bm{k}) - E_2(\bm{k}) = 0.02$. (d) Contour plots of conduction band $E_1(\bm{k})$ near the M' point $(k_x, k_y) = (\pi, -\pi)$. The yellow to red contours show the region with $0 \le E_1 - \mu_0 < 0.0074$. The Dirac points are located at $(k_x/\pi -1, k_y/\pi+1) = \pm(-0.271,0.423)$, and the pinch point indicates the saddle point at M'. (e) Contour plots of valence band $E_2(\bm{k})$ near the Y point $(k_x, k_y) = (0, -\pi)$. The yellow to red contours show the region with $-0.014 \le E_2 - \mu_0 \le 0$. The Dirac points are located at $(k_x/\pi, k_y/\pi+1) = \pm(0.729,0.423)$. The energy at Y point has a maximum and there are two saddle points, which are represented by two pinch points.
• Figure 4: (Color online ) $T$ dependence of the components of the Seebeck coefficient $S_\nu^{\gamma \gamma '}$ given by Eq. (\ref{['eq:S_comp']}). The normalized e--p interaction is taken as $R$ = 1, which corresponds to weak coupling. The solid and dashed lines correspond to $\nu = x$ and $y$, respectively.
• Figure 5: (Color online ) $T$ dependence of the Seebeck coefficients $S_x$ and $S_y$, which is calculated using Eqs. (\ref{['eq:S']}) and (\ref{['eq:S_comp']}) for $R$ = 1 (solid line), $R$ = 0.5 (double-dot-dashed line) and $R$ = 0 (dashed line). The chemical potential $\mu$ is estimated from the inset of Fig. \ref{['fig2']}. $S_y$ exhibis a sign change, while $S_x < 0$ for $T < 0.02$. The dotted line is obtained for transfer energy with only Re $w_h$.