Induced spin-orbit coupling in twisted graphene–transition metal dichalcogenide heterobilayers: Twistronics meets spintronics

We propose an interband tunneling picture to explain and predict the interlayer twist-angle dependence of the induced spin-orbit coupling in heterostructures of graphene and monolayer transition metal dichalcogenides (TMDCs). We obtain a compact analytic formula for the induced valley Zeeman and Rashba spin-orbit coupling in terms of the TMDC band structure parameters and interlayer tunneling matrix elements. We parametrize the tunneling matrix elements with few parameters, which in our formalism are independent of the twist angle between the layers. We estimate the value of the tunneling parameters from existing density functional theory calculations at zero twist angle and we use them to predict the induced spin-orbit coupling at nonzero angles. Provided that the energy of the Dirac point of graphene is close to the TMDC conduction band, we expect a sharp increase of the induced spin-orbit coupling around a twist angle of ${18}^{\circ }$.

Figure 1

3D view of graphene on top of monolayer TMDC. Here, $\theta$ is the twist angle between graphene and the TMDC layer, while $d_{\perp }$ is the perpendicular distance between graphene and the upper (closest) chalcogen layer of the TMDC.

Figure 2

Backfolded TMDC BZ vectors satisfying the quasimomentum conservation of Eq. (4) for the rotated Dirac point of graphene ${\mathit{K}}^{\theta }$. The dashed lines indicate the full paths of the backfolded vectors in the range of twist angles $\theta \in [0,\pi /3]$. Moreover, ${\mathit{G}}_{1,2}^{\theta }$ are rotated reciprocal lattice vectors of graphene, while ${\mathit{G}}_{1,2,3}^{\prime }$ are reciprocal lattice vectors of the TMDC. As an example, here we have shown in orange the BZ of ${\mathrm{MoS}}_2$ (with lattice constant $a_T=3.15\AA$).

Figure 3

(a), (b) Spin splitting in conduction (a) and valence band (b) of the TMDC. Blue (orange) arcs indicate the paths of the three backfolded vectors ${\mathit{k}}_j^{\prime }$ ($-{\mathit{k}}_j^{\prime }$) for Dirac point $\mathit{K}$ ($-\mathit{K}$). (c) Valley Zeeman spin-orbit strength induced in graphene when the Dirac point energy is close to the TMDC conduction band edge ($f_G=1$). The blue (orange) line shows the result of second-order perturbation theory for Dirac point $\mathit{K}$ ($-\mathit{K}$), as derived in Eq. (12). The dashed black line is obtained from the exact diagonalization of the bilayer Hamiltonian (6) for $\mathit{K}$. (d) Same as (c) but in the case when the Dirac point energy is in the middle of the TMDC band gap ($f_G=0.55$) and with a larger TMDC band gap of $E_G=2.0$ eV in order to reproduce the case of Ref. [35]. (e) Spin-orbit splitting in TMDC encountered by the backfolded vectors of $\mathit{K}$ along the paths in (a) (green line) and (b) (purple line). (f) Tunneling strength squared for a tunneling process from graphene to the conduction (green line) or the valence band (purple line) of the TMDC. The gray vertical lines in (c), (e), and (f) highlight the angles where the backfolded vectors ${\mathit{k}}_j^{\prime }$ get as close as possible to the maximum of the spin splitting in the conduction band of the TMDC.

Figure 4

Induced valley Zeeman SOC as a function of the twist angle $\theta$ and the parameter $f_G$ that indicates how close the Dirac point lies to the conduction ($f_G=1$) or to the valence band ($f_G=0$). The dashed black lines indicate the two values of $f_G=1$ and $f_G=0.55$ used in Figs. 3 and 3, respectively.

Figure 5

Magnitude of the induced Rashba type SOC, ${\lambda }_{\text{R}}$, as a function of the twist angle $\theta$ for $f_G=1$ (a) and $f_G=0.55$ (b). In (b) we have also set the TMDC band gap to $E_G=2.0$ eV in order to reproduce the case of Ref. [35]. The purple lines shows the total Rashba type SOC, the gray lines indicate separately the contribution related to two asymmetric bands above the conduction band and an asymmetric band below the valence band, respectively.

Figure 6

Estimation of $t_{\parallel }$ and $t_{\perp }$. (a) The blue ellipse indicates the possible values of $t_{\parallel }$ and $t_{\perp }$ that give a valley Zeeman spin-orbit strength of $-0.26$ meV at $\theta =0^{\circ }$ for a corresponding value of $f_G=0.95$. (b) Magnification of (a). The red rectangle indicates the window of values where $|t_{\parallel }|,|t_{\perp }|\le 100$ meV.