Seebeck effect of Dirac electrons

We study the Seebeck effect in the three-dimensional Dirac electron system based on the linear response theory with Luttinger's gravitational potential. The Seebeck coefficient $S$ is defined by $S = L_{12} / L_{11} T$, where $T$ is the temperature, and $L_{11}$ and $L_{12}$ are the longitudinal response coefficients of the charge current to the electric field and to the temperature gradient, respectively; $L_{11}$ is the electric conductivity and $L_{12}$ is the thermo-electric conductivity. We consider randomly-distributed impurity potentials as the source of the momentum relaxation of electrons and microscopically calculate the relaxation rate and the vertex corrections of $L_{11}$ and $L_{12}$ due to the impurities. It is confirmed that $L_{11}$ and $L_{12}$ are related through Mott's formula in low temperatures when the chemical potential lies above the gap ($|\mu| > \Delta$), irrespective of the linear dispersion of the Dirac electrons and unconventional energy dependence of the lifetime of electrons. On the other hand, when the chemical potential lies in the band gap ($|\mu| < \Delta$), Seebeck coefficient behaves just as in conventional semiconductors: Its dependences on the chemical potential $\mu$ and the temperature $T$ are partially captured by $S \propto (\Delta - \mu) / \kB T$ for $\mu > 0$. The Seebeck coefficient takes the relatively large value $|S| \simeq 1.7 \,\mathrm{m V/K}$ at $T \simeq 8.7\,\mathrm{K}$ for $\Delta = 15 \,\mathrm{m eV}$ by assuming doped bismuth.