Thermoelectric transport properties of ferromagnetic graphene with CT-invariant quantum spin Hall effect

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 Sc, spin Seebeck coefficient Ss, charge Nernst coefficient Nc, and spin Nernst coefficient Ns 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 νσ. The nth peak heights of the QH-type Nc and Sc satisfy |Sc,n|=Nc,n=ln2/(|n|+12), exhibiting the half-integer QH effect. However, the mth peak height of the QSH-type Nc satisfies Nc,m=ln2/|m|, similar to the integer QH effect. The peak height of Ns remains Ns=sgn(s)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 Ss remains 2ln2. In the clean system, the QSH-type Sc and Ss are zero, while the QSH-type Nc and Ns appear at the zeroth Landau levels, which is different from the zero Nc and Ns 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, Sc and Ss are more robust than Nc and Ns. Notably, the QSH-type Sc and Ss are no longer zero in dirty systems.