We investigate the proximity-induced spin-orbit coupling in heterostructures of twisted graphene and monolayers of transition metal dichalcogenides (TMDCs) ${\mathrm{MoS}}_2, {\mathrm{WS}}_2, {\mathrm{MoSe}}_2$, and ${\mathrm{WSe}}_2$ from first principles. We identify strain, which is necessary to define commensurate supercells, as the key factor affecting the band offsets and thus magnitudes of the proximity couplings. We establish that for biaxially strained graphene the band offsets between the Dirac point and conduction (valence) TMDC bands vary linearly with strain, regardless of the twist angle. This relation allows us to identify the apparent zero-strain band offsets and find a compensating transverse electric field correcting for the strain. The resulting corrected band structure is then fitted around the Dirac point to an established spin-orbit Hamiltonian. This procedure yields the dominant, valley-Zeeman, and Rashba spin-orbit couplings. The magnitudes of these couplings do not vary much with the twist angle, although the valley-Zeeman coupling vanishes for ${30}^{\circ }$ and Mo-based heterostructures exhibit a maximum of the coupling at around ${20}^{\circ }$. The maximum for W-based stacks is at $0^{\circ }$. The Rashba coupling is in general weaker than the valley-Zeeman coupling, except at angles close to ${30}^{\circ }$. We also identify the Rashba phase angle which measures the deviation of the in-plane spin texture from tangential, and find that this angle is very sensitive to the applied transverse electric field. We further discuss the reliability of the supercell approach with respect to atomic relaxation (rippling of graphene), relative lateral shifts of the atomic layers, and transverse electric field.

中文翻译：

---

石墨烯/过渡金属二硫属化物异质结构中的扭曲角相关邻近诱导自旋轨道耦合

我们研究了扭曲石墨烯异质结构和过渡金属二硫属化物 (TMDC) 单层中的邻近诱导自旋轨道耦合 ${\mathrm{硫化钼}}_2, {\mathrm{WS}}_2, {钼}_2$， 和 ${\mathrm{WSe}}_2$从第一原则。我们确定应变，这是定义相称的超级单元所必需的，是影响带偏移的关键因素，从而影响邻近耦合的大小。我们确定，对于双轴应变石墨烯，狄拉克点和传导（价）TMDC 带之间的带偏移随应变呈线性变化，而与扭曲角无关。这种关系使我们能够识别明显的零应变带偏移并找到校正应变的补偿横向电场。然后将得到的修正带结构围绕狄拉克点拟合到已建立的自旋轨道哈密顿量。这个过程产生了主要的、谷塞曼和拉什巴自旋轨道耦合。这些耦合的大小随扭转角变化不大，尽管谷塞曼耦合消失了${30}^{\circ }$ 和 Mo 基异质结构在大约 ${20}^{\circ }$. 基于 W 的堆栈的最大值为$0^{\circ }$. Rashba 耦合一般比 Valley-Zeeman 耦合弱，除非角度接近${30}^{\circ }$. 我们还确定了 Rashba 相位角，它测量平面内自旋纹理与切线的偏差，并发现该角度对施加的横向电场非常敏感。我们进一步讨论了超级单元方法在原子弛豫（石墨烯的波纹）、原子层的相对横向位移和横向电场方面的可靠性。