Single and bilayer graphene on the topological insulator ${\mathrm{Bi}}_2{\mathrm{Se}}_3$: Electronic and spin-orbit properties from first principles

We present a detailed study of the electronic and spin-orbit properties of single and bilayer graphene in proximity to the topological insulator ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. Our approach is based on first-principles calculations, combined with symmetry derived model Hamiltonians that capture the low-energy band properties. We consider single and bilayer graphene on 1–3 quintuple layers of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ and extract orbital and proximity induced spin-orbit coupling (SOC) parameters. We find that graphene gets significantly hole doped (350 meV), but the linear dispersion is preserved. The proximity induced SOC parameters are about 1 meV in magnitude, and are of valley-Zeeman type. The induced SOC depends weakly on the number of quintuple layers of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. We also study the effect of a transverse electric field that is applied across heterostructures of single and bilayer graphene above 1 quintuple layer of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. Our results show that band offsets, as well as proximity induced SOC parameters, can be tuned by the field. Most interesting is the case of bilayer graphene, in which the band gap, originating from the intrinsic dipole of the heterostructure, can be closed and reopened again, with inverted band character. The switching of the strong proximity SOC from the conduction to the valence band realizes a spin-orbit valve. Additionally, we find a giant increase of the proximity induced SOC of about 200% when we decrease the interlayer distance between graphene and ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ by only 10%. Finally, for a different substrate material ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$, band offsets are significantly different, with the graphene Dirac point located at the Fermi level, while the induced SOC strength stays similar in magnitude compared to the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ substrate.

Figure 1

Geometry of SLG or BLG above 1–3 QLs of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. We highlighted the SLG/1 QL structure, which is the minimal geometry we consider. Different colors correspond to different atomic species.

Figure 2

Calculated band structures of SLG on (a) one, (b) two, and (c) three QLs of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. The color is the $s_z$ expectation value. In (a) we define the Dirac point energy $E_{\mathrm{D}}$ and the doping energy of the topological insulator $E_{\mathrm{TI}}$.

Figure 3

Calculated low energy band properties (symbols) for SLG/${\mathrm{Bi}}_2{\mathrm{Se}}_3$ for 1 QL, with a fit to the model Hamiltonian ${\mathcal{H}}_{\text{SLG}}$ (solid lines). (a)–(d) The spin expectation values of the four low energy bands. (e) The low energy band structure of proximitized SLG. The color is the $s_z$ expectation value. Letters A and B indicate the pseudospin of the bands at the $K$ point. (f) The splitting of the valence (conduction) band in blue (red).

Figure 4

First-principles calculated spin-orbit fields around the $K$ point of the bands (a) ${\varepsilon }_1^{\mathrm{CB}}$, (b) ${\varepsilon }_2^{\mathrm{CB}}$, (c) ${\varepsilon }_2^{\mathrm{VB}}$, and (d) ${\varepsilon }_1^{\mathrm{VB}}$, for the SLG/${\mathrm{Bi}}_2{\mathrm{Se}}_3$ stack with 1 QL, corresponding to the four low energy bands in Fig. 3. The dashed white lines represent the edge of the Brillouin zone.

Figure 5

First-principles calculated spin-orbit fields around the $K$ point of the bands (a) ${\varepsilon }_1^{\mathrm{CB}}$, (b) ${\varepsilon }_2^{\mathrm{CB}}$, (c) ${\varepsilon }_2^{\mathrm{VB}}$, and (d) ${\varepsilon }_1^{\mathrm{VB}}$, for the SLG/${\mathrm{Bi}}_2{\mathrm{Se}}_3$ stack with 3 QLs. The dashed white lines represent the edge of the Brillouin zone.

Figure 6

Calculated band structures of BLG on (a) one, (b) two, and (c) three QLs of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. The color is the $s_z$ expectation value. In (a) we define the Dirac point energy $E_{\mathrm{D}}$ and the doping energy of the topological insulator $E_{\mathrm{TI}}$.

Figure 7

Calculated low energy band properties (symbols) for BLG/${\mathrm{Bi}}_2{\mathrm{Se}}_3$ for 1 QL, with a fit to the model Hamiltonian ${\mathcal{H}}_{\text{BLG}}$ (solid lines). (a)–(d) The spin expectation values of the four low energy bands. (e) The low energy band structure of proximitized BLG. The color is the $s_z$ expectation value. (f) The splitting of the valence (conduction) band in blue (red).

Figure 8

First-principles calculated spin-orbit fields around the $K$ point of the bands (a) ${\varepsilon }_1^{\mathrm{CB}}$, (b) ${\varepsilon }_2^{\mathrm{CB}}$, (c) ${\varepsilon }_2^{\mathrm{VB}}$, and (d) ${\varepsilon }_1^{\mathrm{VB}}$, for the BLG/${\mathrm{Bi}}_2{\mathrm{Se}}_3$ stack with 1 QL, corresponding to the four low energy bands in Fig. 7. The dashed white lines represent the edge of the Brillouin zone.

Figure 9

Fit parameters of Hamiltonian ${\mathcal{H}}_{\text{SLG}}$ for the SLG/${\mathrm{Bi}}_2{\mathrm{Se}}_3$ stack with 1 QL as a function of a transverse electric field. (a) The gap parameter $\mathrm{\Delta }$, (b) the Fermi velocity $v_{\mathrm{F}}$, (c) the dipole of the structure, (d) Rashba SOC parameter ${\lambda }_{\mathrm{R}}$, intrinsic SOC parameters ${\lambda }_{\mathrm{I}}^{\mathrm{A}}$ and ${\lambda }_{\mathrm{I}}^{\mathrm{B}}$, (e) PIA SOC parameters ${\lambda }_{\mathrm{PIA}}^{\mathrm{A}}$ and ${\lambda }_{\mathrm{PIA}}^{\mathrm{B}}$, and (f) the Dirac point energy $E_{\mathrm{D}}$ and the doping energy of the topological insulator $E_{\mathrm{TI}}$, as defined in Fig. 2.

Figure 10

Fit parameters of Hamiltonian ${\mathcal{H}}_{\text{BLG}}$ for the BLG/${\mathrm{Bi}}_2{\mathrm{Se}}_3$ stack with 1 QL as a function of a transverse electric field. (a) The nearest neighbor hopping parameter ${\gamma }_0$, the asymmetry in the energy shift of the bonding and antibonding states $\mathrm{\Delta }$, and the SLG layer potential $V$. (b) The intra- and interlayer hoppings ${\gamma }_1$, ${\gamma }_3$, and ${\gamma }_4$. (c) The dipole of the structure. (d) The two proximity modified intrinsic SOC parameters ${\lambda }_{\mathrm{I}}^{\mathrm{A}}1$ and ${\lambda }_{\mathrm{I}}^{\mathrm{B}}1$. (e) The global and local space inversion symmetry breaking SOC parameters ${\lambda }_0$ and ${\lambda }_{\mathrm{R}}$. (f) The Dirac point energy $E_{\mathrm{D}}$ and the doping energy of the topological insulator $E_{\mathrm{TI}}$, as defined in Fig. 6.

Figure 11

Calculated low energy bands for BLG/${\mathrm{Bi}}_2{\mathrm{Se}}_3$ for 1 QL, with applied transverse electric field of (a) 2 V/nm and (b) 2.3 V/nm.

Figure 12

Fit parameters of Hamiltonian ${\mathcal{H}}_{\text{SLG}}$ for the SLG/${\mathrm{Bi}}_2{\mathrm{Se}}_3$ stack with 1 QL as a function of the interlayer distance. (a) The gap parameter $\mathrm{\Delta }$, (b) the Fermi velocity $v_{\mathrm{F}}$, (c) the total energy with respect to the minimal energy $E_0$, (d) Rashba SOC parameter ${\lambda }_{\mathrm{R}}$, intrinsic SOC parameters ${\lambda }_{\mathrm{I}}^{\mathrm{A}}$ and ${\lambda }_{\mathrm{I}}^{\mathrm{B}}$, (e) PIA SOC parameters ${\lambda }_{\mathrm{PIA}}^{\mathrm{A}}$ and ${\lambda }_{\mathrm{PIA}}^{\mathrm{B}}$, and (f) the Dirac point energy $E_{\mathrm{D}}$ and the doping energy of the topological insulator $E_{\mathrm{TI}}$, as defined in Fig. 2.

Figure 13

Calculated band structures of SLG on (a) one, (b) two, and (c) three QLs of ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$. The color corresponds to the $s_z$ expectation value. In (a) we define the Dirac point energy $E_{\mathrm{D}}$ and the doping energy of the topological insulator $E_{\mathrm{TI}}$.