We investigate thermoelectric transport properties of ferromagnetic graphene with CT-invariant quantum spin Hall (CT-QSH) effect. Considering a strong magnetic field, we calculate the charge Seebeck coefficient $S_c$, spin Seebeck coefficient $S_s$, charge Nernst coefficient $N_c$, and spin Nernst coefficient $N_s$ based on the nonequilibrium Green's function and Landauer-Büttiker formula. Due to the coexistence of the CT-QSH and quantum Hall (QH) effects in ferromagnetic graphene, thermoelectric coefficients are divided into the QSH and QH types appearing at the zeroth and nonzero Landau levels, respectively. We find both the charge thermoelectric coefficients are determined by the filling factor ${\nu }_{\sigma }$. The $n\mathrm{th}$ peak heights of the QH-type $N_c$ and $S_c$ satisfy $|S_{c,n}|=N_{c,n}=ln2/\left(\right|n|+\frac{1}{2})$, exhibiting the half-integer QH effect. However, the $m\mathrm{th}$ peak height of the QSH-type $N_c$ satisfies $N_{c,m}=ln2/\left|m\right|$, similar to the integer QH effect. The peak height of $N_s$ remains $N_s=\mathrm{sgn}\left(s\right)2ln2$, and its sign depends on the spin of Landau level, either the QH or QSH type. In addition, the peak height of the QH-type $S_s$ remains $2ln2$. In the clean system, the QSH-type $S_c$ and $S_s$ are zero, while the QSH-type $N_c$ and $N_s$ appear at the zeroth Landau levels, which is different from the zero $N_c$ and $N_s$ in the conventional QSH system. In the presence of disorders, the QH-type thermoelectric coefficients are more robust than the QSH. For the QH-type thermoelectric coefficients, $S_c$ and $S_s$ are more robust than $N_c$ and $N_s$. Notably, the QSH-type $S_c$ and $S_s$ are no longer zero in dirty systems.

• Received 7 February 2020
• Revised 19 June 2020
• Accepted 27 July 2020

DOI:https://doi.org/10.1103/PhysRevB.102.075432