Controllable spin Nernst effect of surface states in three-dimensional topological insulator film

We systematically investigate the spin Nernst effect (SNE) of surface states in three-dimensional topological insulator film under a perpendicular magnetic field. The spin Nernst coefficient $N_s$ of surface states, which are lie in the quantum spin Hall regime (QSH), quantum anomalous Hall regime (QAH), and quantum pseudospin Hall regime (QPH), is theoretically calculated by using the Non-equilibrium Green’s function method combined with the square lattice model. Regardless of the presence of a magnetic field, SNE will occur in the system because it primarily arises from spin–orbit coupling. When the Fermi energy $E_F$ crosses the discrete transverse channels, $N_s$ exhibits a pronounced peak. The height of peak strongly depends on the temperature, decreasing with increasing temperature. In the QPH regime, $N_s$ is an even function of $E_F$ with $N_s(-E_F)=N_s\left(E_F\right)$ without applying a magnetic field. However, this the symmetrical property is destroyed when a magnetic field is applied. In the QSH and QAH regimes, $N_s$ is also an even function of $E_F$ in magnetic field $\varphi =0.0$. When a magnetic field is applied, $N_s$ of QSH regime retains this symmetrical property due to protection from time-reversal symmetry. But the symmetrical property $N_s(-E_F)=N_s\left(E_F\right)$ is broken in the QAH regime. This is because that the combined influence of the exchange field and magnetic field breaks time-reversal symmetry, leading to the peak structure of $N_s$ to reverse around $E_F=0$. These results indicate that SNE is closely related to the topological characteristics, magnetic field, and temperature, which may provide valuable assistance for the control and modulation of spin currents.