Thermoelectric transport properties of ferromagnetic graphene with C T -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 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 ν_σ. The n th peak heights of the QH-type N_c and S_c satisfy | S_{c ,n}|= N_{c ,n}=ln2 /(|n | +1/2 ) , exhibiting the half-integer QH effect. However, the m th peak height of the QSH-type N_c satisfies N_{c ,m}=ln2 /|m | , similar to the integer QH effect. The peak height of N_s remains N_s=sgn (s ) 2 ln2 , 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 2 ln2 . 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.