Proximity Effect in Graphene–Topological-Insulator Heterostructures

We formulate a continuum model to study the low-energy electronic structure of heterostructures formed by graphene on a strong three-dimensional topological insulator (TI) for the cases of both commensurate and incommensurate stacking. The incommensurability can be due to a twist angle between graphene and the TI surface or a lattice mismatch between the two systems. We find that the proximity of the TI induces in graphene a strong enhancement of the spin-orbit coupling that can be tuned via the twist angle.

Figure 1

(a) Schematics of the $\sqrt{3}\times \sqrt{3}$ stacked graphene BZ (red or dark) and TIS BZ (green or light) in the repeated zone scheme without tunneling. (b) Folded BZ after turning on tunneling. (c) Renormalized bands of SLG-TIS for ${\mu }_1={\mu }_2=0$. Here, $k_0\equiv 830\text{meV}/(\hslash v_2)$. (d) Spin texture on the bands at $E=80\mathrm{meV}$. The arrows indicate spin directions. (e) Texture of the in-plane component of the pseudospin at $E=80\mathrm{meV}$ and (f) full pseudospin orientation on the three Fermi surfaces closest to the $\overline{\mathrm{\Gamma }}$ point. (g) Renormalized bands of BLG-TIS. (h) Renormalized bands of SLG-TIS for ${\mu }_1=0$, ${\mu }_2=100\mathrm{meV}$. (i) Rashba-like splitting ${\mathrm{\Delta }}_R$ in SLG-TIS and BLG-TIS as a function of $t$.

Figure 2

Schematics of the graphene and TIS BZs in an incommensurate structure formed from (a) a twist and (b) a lattice mismatch, with the corresponding ${\mathbf{q}}_j$ vectors at the $K$ and $K^{\prime }$ points.

Figure 3

(a) The band structure along the path $A\text{−}B\text{−}C\text{−}D\text{−}A$ indicated in Fig. 2. [(b),(c)] Spin texture on the bands at different energies; $E_0\equiv \hslash v_{2}q=t^{\prime }/\gamma$. (d) Splitting ($\mathrm{\Delta }$) of the low-energy bands as a function of twist angle for $t^{\prime }=30\mathrm{meV}$ and $t^{\prime }=15\mathrm{meV}$.

Figure 4

(a) Schematics of the induced spin texture on graphene (right) from the TIS spin helix (left). (b) Renormalized graphene bands (solid lines) for $t_0=100\mathrm{meV}$, $\sigma =2k_0$. Spin-split gap ($\mathrm{\Delta }$) as a function of (c) $t_0$ and (d) $\sigma$.