Enhanced Seebeck and Nernst effects in the complex honeycomb lattice

We study the Seebeck and Nernst effects in the complex honeycomb (CHC) model, where $m$ atoms are attached to each edge of the hexagon, with $m=0$ corresponding to graphene. When $m=0$ and $m=2$, the energy band structure exhibits a sombrerolike structure near the Fermi energy (${E}_{F}$), and the transmission coefficient plateaus are $1,3,5,\ensuremath{\cdots}$. The peak height of the Seebeck coefficient satisfies the relation $ln2/n(n=\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}2,)$. In contrast, for $m=1$ and $m=3$, the energy band structure forms a Dirac cone near the Fermi energy, with a flat band at ${E}_{F}=0$, while the transmission coefficient plateaus are $0,1,2,\ensuremath{\cdots}$ and the system exhibits insulation. Here, the peak height of the Seebeck coefficient follows the relation $ln2/(n\ifmmode\pm\else\textpm\fi{}1/2)$. We find that larger Seebeck coefficients and Nernst coefficients are obtained when $m$ is odd. At the same time, the Seebeck coefficient exhibits two additional positive and two additional negative peaks. Due to the flat band, the peak near ${E}_{F}=0$ has an opposite sign compared to the other peaks. Unlike the lattice structure when $m$ is even, the Nernst coefficient for odd $m$ does not exhibit a peak at the Fermi energy due to the presence of the band gap.