Effects of local exchange field in different directions on spin transport of stanene

Abstract

Topological insulator is a new quantum state of matter in which spin-orbit coupling gives rise to topologically protected gapless edge or surface states. The nondissipation transport properties of the edge or surface state make the topological device a promising candidate for ultra-low-power consumption electronics. Stanene is a type of two-dimensional topological insulator consisting of Sn atoms arranged similarly to graphene and silicene in a hexagonal structure. In this paper, the effects of various combinations of local exchange fields on the spin transport of stanene nanoribbons are studied theoretically by using the non-equilibrium Green's function method. The results show that the spin-dependent conductance, edge states, and bulk bands of stanene are significantly dependent on the direction and strength of the exchange field in different regions. Under the joint action of the exchange fields in [I: <inline-formula><tex-math id="M12">\begin{document}$ \pm Y $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M12.png"/></alternatives></inline-formula>, II: <inline-formula><tex-math id="M13">\begin{document}$ +Z $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M13.png"/></alternatives></inline-formula>, III: <inline-formula><tex-math id="M14">\begin{document}$ \pm Y $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M14.png"/></alternatives></inline-formula>] direction, the edge states form a band-gap under the influence of the <i>Y</i>-direction exchange field. The band-gap width is directly proportional to the exchange field strength <i>M</i>, and the conductance is zero in an energy range of <inline-formula><tex-math id="M15">\begin{document}$ -M<E<M $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M15.png"/></alternatives></inline-formula>. When the exchange fields in the direction of <inline-formula><tex-math id="M16">\begin{document}$ +Z $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M16.png"/></alternatives></inline-formula> or <inline-formula><tex-math id="M17">\begin{document}$ -Z $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M17.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M17.png"/></alternatives></inline-formula> are applied, respectively, to the upper edge region and the lower edge region at the same time, the spin-up energy band and the spin-down energy band move to a high energy region in opposite directions, and strong spin splitting occurs in the edge state and bulk bands. Increasing the strength of the exchange field, the range of spin polarization of conductance spreads from the high energy region to the low energy region. When the directions of the exchange field are [I: <inline-formula><tex-math id="M18">\begin{document}$ \mp Z $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M18.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M18.png"/></alternatives></inline-formula>, II: <inline-formula><tex-math id="M19">\begin{document}$ \pm Y $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M19.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M19.png"/></alternatives></inline-formula>, III: <inline-formula><tex-math id="M20">\begin{document}$ \pm Z $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M20.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M20.png"/></alternatives></inline-formula>], the edge states are spin degenerate, but the weak spin splitting occurs in the bulk bands. Under the condition of different exchange field strengths, the spin-dependent conductance maintains a conductance platform of <inline-formula><tex-math id="M21">\begin{document}$ G_\sigma=e^2/h $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M21.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M21.png"/></alternatives></inline-formula> in the same energy range of <inline-formula><tex-math id="M22">\begin{document}$ -\lambda_{\rm so} <E<\lambda_{\rm so} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M22.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20220277_M22.png"/></alternatives></inline-formula>.