Thermoelectric transport of type-I, II, and III massless Dirac fermions in a two-dimensional lattice model

We study longitudinal electric and thermoelectric transport coefficients of Dirac fermions on a simple lattice model where tuning of a single parameter enables us to change the type of Dirac cones from type I to type II. We pay particular attention to the behavior of the critical situation, i.e., the type-III Dirac cone. We find that the transport coefficients of the type-III Dirac fermions behave as the limiting case of neither type I nor type II. On the one hand, the qualitative behaviors of the type-III case are similar to those of the type-I case. On the other hand, the transport coefficients do not change monotonically upon increasing the tilting; namely, the largest thermoelectric response is obtained not for the type-III case but for the optimally tilted type-I case. For the optimal case, sizable transport coefficients are obtained; for example, the dimensionless figure of merit is 0.18.

Figure 1

Schematic figure of the tight-binding model of Eq. (3). (a) The lattice structures and (b) the hopping processes in the $x$ and $y$ directions. The red and blue dots denote orbitals 1 and 2, respectively.

Figure 2

(a) The first Brillouin zone. The green lines correspond to the high-symmetry lines on which we plot the band structure in (b)–(d). Band structures for the Hamiltonian in Eq. (3) with $t=-1$ eV, $t^{\prime }=-0.3$ eV, and (b) $\lambda =0.5$, (c) $\lambda =1$, and (d) $\lambda =1.5$. The horizontal axis denotes $\mathit{k}$. The Dirac cones at $\mathit{k}=(\frac{\pi }{2a_0},\frac{\pi }{2a_0})$ are encircled by cyan circles. The Fermi surface at $\mu =0$ eV for (e) $\lambda =0.5$, (f) $\lambda =1$, and (g) $\lambda =1.5$. The green dots represent the Dirac points. Red and blue lines are the electron-type surface (i.e., ${\varepsilon }_{\mathit{k},+}=\mu$) and the hole-type surface (i.e., ${\varepsilon }_{\mathit{k},-}=\mu$), respectively.

Figure 3

Density of states for the model in Eq. (3).

Figure 4

Longitudinal conductivity for (a) $\mathrm{\Gamma }=0.02$ eV, (b) $\mathrm{\Gamma }=0.05$ eV, and (c) $\mathrm{\Gamma }=0.1$ eV.

Figure 5

Temperature dependence of (a) the conductivity, (b) $L_{12}$, (c) $L_{22}$, (d) the Seebeck coefficient, (e) the power factor, and (f) the dimensionless figure of merit. We set $\mathrm{\Gamma }=0.02$ eV.

Figure 6

$\lambda$ dependence of (a) the Seebeck coefficient, (b) the power factor, and (c) the dimensionless figure of merit. We set $\mathrm{\Gamma }=0.02$ eV.

Figure 7

Seebeck coefficient at $T=1.16\mathrm{K}$ for (a) $\mathrm{\Gamma }=0.02$ eV, (b) $\mathrm{\Gamma }=0.05$ eV, and (c) $\mathrm{\Gamma }=0.1$ eV.

Figure 8

$\lambda$ dependence of the Seebeck coefficient at $T=1.16\mathrm{K}$ and $\mu =0.0125$ eV.

Figure 9

$\mu$ dependence of the power factor at $T=1.16$ K for (a) $\mathrm{\Gamma }=0.02$ eV, (b) $\mathrm{\Gamma }=0.05$ eV, and (c) $\mathrm{\Gamma }=0.1$ eV.

Figure 10

(a) $X\left(w_0\right)$ from Eq. (B8) and (b) $Y\left(w_0\right)$ from Eq. (B9) as a function of $-w_0$. Note that $w_0$ is negative when $\mu$ is positive.

Figure 11

(a) Schematic figure of the division of the Brillouin zone into the two regions, (A) and (B). Region (A) is composed of four circles whose radii are $\frac{\pi }{4a_0}$ and centers are the Dirac points. The gray lines are the Fermi surface for $\mu =0$ eV. (b) ${\alpha }_{xx}^{\left(\mathrm{A}\right)}\left(\epsilon \right), {\alpha }_{xx}^{\left(\mathrm{B}\right)}\left(\epsilon \right)$, and ${\alpha }_{xx}\left(\epsilon \right)$ as functions of $\epsilon$. (c) Temperature dependence of $S^{\left(\mathrm{A}\right)}, S^{\left(\mathrm{B}\right)}$, and $S$. For (b) and (c), we set $\mathrm{\Gamma }=0.02$ eV.