First-principles study of the effective Hamiltonian for Dirac fermions with spin-orbit coupling in two-dimensional molecular conductor $\alpha$-(BETS)$_2 \hbox {I}_3$

Abstract

We employed first-principles density-functional theory (DFT) calculations to characterize Dirac electrons in quasi-two-dimensional molecular conductor $\alpha$-(BETS)$_2 \hbox {I}_3$ [= $\alpha$-(BEDT–TSeF)$_2 \hbox {I}_3$] at a low temperature of 30 K. We provide a tight-binding model with intermolecular transfer energies evaluated from maximally localized Wannier functions, where the number of relevant transfer integrals is relatively large due to the delocalized character of Se p orbitals. The spin–orbit coupling gives rise to an exotic insulating state with an indirect band gap of about 2 meV. We analyzed the energy spectrum with a Dirac cone close to the Fermi level to develop an effective Hamiltonian with site potentials, which reproduces the spectrum obtained by the DFT band structure.

Graphic abstract

Appendices

Site-energy potentials

We define site potentials acting on B and C sites, $\varDelta V_\mathrm{B}$ and $\varDelta V_\mathrm{C}$, which are measured from site energy at A ($A^\prime$) site, $V_\mathrm{A}$ [37]:

$$\begin{aligned} \varDelta V_\mathrm{B}= & {} V_\mathrm{B} - V_\mathrm{A},\\ \varDelta V_\mathrm{C}= & {} V_\mathrm{C} - V_\mathrm{A}, \end{aligned}$$

where $V_\mathrm{A}$, $V_\mathrm{B}$, and $V_\mathrm{C}$ are the site energies at each molecule that were calculated using MLWFs $|\phi _{\alpha ,0} \rangle$:

$$\begin{aligned} V_{\alpha } =\langle \phi _{\alpha , \sigma ,0}| H |\phi _{\alpha , \sigma ^{\prime },0} \rangle , \end{aligned}$$

where ${\alpha }$ indicates A (= $A^{\prime }$), B, and C molecules. These site potentials are referred as to $\varDelta V_\mathrm{B}^{\mathrm{DFT}}$ and $\varDelta V_\mathrm{C}^{\mathrm{DFT}}$ in the present study, and listed in Table 1.

Matrix elements

In terms of Eq. (1b) with $X= \mathrm{e}^{ik_x}$, $\bar{X}= \mathrm{e}^{-ik_x}$, $Y= \mathrm{e}^{ik_y}$, and $\bar{Y}= \mathrm{e}^{-ik_y}$, matrix elements, $t_{ij} = ({\hat{H}})_{ij}$, are given by:

$$\begin{aligned} t_{11}= & {} t_{22} = t_{55} =t_{66} =a_{1d}(Y+\bar{Y}) +s1(X+\bar{X}) \; , \\ t_{33}= & {} = t_{77}= a_{3d}(Y+\bar{Y}) + s3(X + \bar{X})+ \varDelta V_\mathrm{B} \; , \\ t_{44}= & {} t_{88} = a_{4d}(Y+\bar{Y}) + s4(X + \bar{X})+ \varDelta V_\mathrm{C} \; , \\ t_{12}= & {} a_3+a_2Y+d_0\bar{X}+ d_1XY \; , \\ t_{13}= & {} b_3+b_2\bar{X}+ c_2 \bar{X} Y + c_4\bar{X}\bar{Y} \; , \\ t_{14}= & {} b_4Y+b_1\bar{X}Y+c_1\bar{X} + c_3 \; , \\ t_{23}= & {} b_2+b_3\bar{X}+c_2 \bar{Y} + c_4 {Y} \; , \\ t_{24}= & {} b_1+b_4\bar{X}+c_1Y + c_3\bar{X}Y \; , \\ t_{34}= & {} a_1+a_1Y + d_2\bar{X}+d_3X + d_2 XY+d_3\bar{X}Y \\ t_{17}= & {} b2_\mathrm{so1} \bar{X} + c2_\mathrm{so1}\bar{X} Y + c4_\mathrm{so1}\bar{X} \bar{Y} \; , \\ t_{18}= & {} b1_\mathrm{so1}\bar{X}Y + b4_{so1} Y + c1_\mathrm{so1} \bar{X}\; , \\ t_{27}= & {} b2_\mathrm{so1} + c2_\mathrm{so1} \bar{Y}+ c4_\mathrm{so1} Y \; , \\ t_{28}= & {} b1_\mathrm{so1} + c1_\mathrm{so1}Y + c3_{so1}\bar{X}Y \; , \\ t_{35}= & {} b2_\mathrm{so2} X + c2_\mathrm{so2}X \bar{Y}+ c4_\mathrm{so2}X Y \; , \\ t_{36}= & {} b2_\mathrm{so2} + c2_\mathrm{so2} Y + c4_\mathrm{so2} \bar{Y}\; , \\ t_{45}= & {} b1_\mathrm{so2}X \bar{Y}+ b4_\mathrm{so2}\bar{Y}+ c1_\mathrm{so2}X \; , \\ t_{46}= & {} b1_\mathrm{so2} + c1_\mathrm{so2} \bar{Y}+ c3_\mathrm{so2}X \bar{Y}\; , \end{aligned}$$

and $t_{15} = t_{16} = t_{25} = t_{26} =t_{37} = t_{38} = t_{47} = t_{48} = 0$, and $t_{ji} = t_{ij}^*$.