Linear and nonlinear thermoelectric transport in a quantum spin Hall insulators coupled with a nanomagnet

Thermoelectric effects in quantum systems have been focused in recent years. Thermoelectric energy conversion study of systems with edge states, such as quantum Hall insulators and quantum spin Hall insulators, is one of the most important frontier topics in material science and condensedmatter physics. Based on the previous paper (Gresta in Phys Rev Lett 123:186801, 2019), we further investigated the linear and nonlinear thermoelectric transport properties of helical edge states of the quantum spin Hall insulators coupled with double nanomagnet, calculated the Seebeck coefficients S-c and the thermoelectrical figure of merit ZT, discussed the influence of the length of the nanomagnet and the relative tilt angle of component of the magnetization perpendicular on the thermoelectric coefficients (S-c and ZT), and summarized some meaningful conclusions in the linear response regime. In the nonlinear regime, we calculated the equivalent figure of merit ZT(M) and the power-generation efficiency eta in different length of the nanomagnet, obtain the temperature difference of achieving optimal thermoelectricity. The results of this paper further confirm that the setup can indeed be used as a device for achieving high performance thermoelectric.