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 condensed-matter 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.

近年来，量子系统中的热电效应一直受到关注。量子霍尔绝缘体和量子自旋霍尔绝缘体等边缘态系统的热电能量转换研究是材料科学和凝聚态物理领域最重要的前沿课题之一。基于之前的论文（Gresta in Phys Rev Lett 123:186801, 2019），我们进一步研究了量子自旋霍尔绝缘体螺旋边缘态与双纳米磁体耦合的线性和非线性热电输运性质，计算了塞贝克系数\(S_c \)和热电品质因数ZT，讨论了纳米磁体的长度和磁化垂直分量的相对倾斜角对热电系数（\(S_c\)和ZT）的影响，并总结了线性响应机制中的一些有意义的结论。在非线性状态下，我们计算了纳米磁体不同长度的等效品质因数\(ZT_M\)和发电效率\(\eta\)，得到了实现最佳热电的温差。本文的结果进一步证实了该装置确实可以用作实现高性能热电的装置。