Graphene on two-dimensional hexagonal BN, AlN, and GaN: Electronic, spin-orbit, and spin relaxation properties

We investigate the electronic band structure of graphene on a series of two-dimensional hexagonal nitride insulators $\mathrm{h}X\mathrm{N}, X=\text{B}$, Al, and Ga, with first-principles calculations. A symmetry-based model Hamiltonian is employed to extract orbital parameters and spin-orbit coupling (SOC) from the low-energy Dirac bands of the proximitized graphene. While commensurate hBN induces a staggered potential of about 10 meV into the Dirac band structure, less lattice-matched hAlN and hGaN disrupt the Dirac point much less, giving a staggered gap below 100 $\mu \mathrm{eV}$. Proximitized intrinsic SOC surprisingly does not increase much above the pristine graphene value of 12 $\mu \mathrm{eV}$; it stays in the window of 1–16 $\mu \mathrm{eV}$, depending strongly on stacking. However, Rashba SOC increases sharply when increasing the atomic number of the boron group, with calculated maximal values of 8, 15, and 65 $\mu \mathrm{eV}$ for B-, Al-, and Ga-based nitrides, respectively. The individual Rashba couplings also depend strongly on stacking, vanishing in symmetrically sandwiched structures, and can be tuned by a transverse electric field. The extracted spin-orbit parameters were used as input for spin transport simulations based on Chebyshev expansion of the time-evolution of the spin expectation values, yielding interesting predictions for the electron spin relaxation. Spin lifetime magnitudes and anisotropies depend strongly on the specific ($\mathrm{h}X\mathrm{N}$)/graphene/$\mathrm{h}X\mathrm{N}$ system, and they can be efficiently tuned by an applied external electric field as well as the carrier density in the graphene layer. A particularly interesting case for experiments is graphene/hGaN, in which the giant Rashba coupling is predicted to induce spin lifetimes of 1–10 ns, short enough to dominate over other mechanisms, and lead to the same spin relaxation anisotropy as that observed in conventional semiconductor heterostructures: 50%, meaning that out-of-plane spins relax twice as fast as in-plane spins.

Figure 1

Top view of the supercell geometry of (a) graphene on hBN ($4\times 4$), (b) graphene on hAlN, and (c) graphene on hGaN. Different colors correspond to different atom types.

Figure 2

Calculated band structures of graphene on (a) hBN, (b) hAlN, and (c) hGaN. Bands corresponding to different layers (graphene, hBN, hAlN, hGaN) are plotted in different colors (black, green, purple, red). The inset in (a) shows a sketch of the low-energy dispersion close to the $K$ point. Due to the presence of the substrate, graphene's low-energy bands are split into four states, ${\varepsilon }_{1/2}^{\text{CB/VB}}$, with a band gap.

Figure 3

Calculated low-energy band properties of graphene on hAlN in the vicinity of the $K$ point with an interlayer distance of $3.48\AA$. (a) First-principles band structure (symbols) with a fit to the model Hamiltonian (solid line). (b) The splitting of the conduction band $\mathrm{\Delta }{E}_{\text{CB}}$ (blue) and the valence band $\mathrm{\Delta }{E}_{\text{VB}}$ (red) close to the $K$ point and calculated model results. (c)–(e) The spin expectation values of the bands ${\varepsilon }_2^{\text{VB}}$ and ${\varepsilon }_1^{\text{CB}}$ and comparison to the model results.

Figure 4

Calculated low-energy band properties of graphene on hGaN in the vicinity of the $K$ point with an interlayer distance of $3.48\AA$. (a) First-principles band structure (symbols) with a fit to the model Hamiltonian (solid line). (b) The splitting of the conduction band $\mathrm{\Delta }{E}_{\text{CB}}$ (blue) and the valence band $\mathrm{\Delta }{E}_{\text{VB}}$ (red) close to the $K$ point and calculated model results. (c)–(e) The spin expectation values of the bands ${\varepsilon }_2^{\text{VB}}$ and ${\varepsilon }_1^{\text{CB}}$ and comparison to the model results.

Figure 5

Fit parameters as a function of distance between graphene and hAlN or hGaN. (a) Total energy, (b) Fermi velocity $v_{\text{F}}$, (c) gap parameter $\mathrm{\Delta }$, (d) Rashba SOC parameter ${\lambda }_{\text{R}}$, (e) intrinsic SOC parameter ${\lambda }_{\text{I}}^{\text{A}}$ for sublattice A, (f) intrinsic SOC parameter ${\lambda }_{\text{I}}^{\text{B}}$ for sublattice B, (g) PIA SOC parameter ${\lambda }_{\text{PIA}}^{\text{A}}$ for sublattice A, and (h) PIA SOC parameter ${\lambda }_{\text{PIA}}^{\text{B}}$ for sublattice B.

Figure 6

Fit parameters as a function of the transverse electric field for graphene/hAlN and graphene/hGaN at fixed interlayer distances of 3.48 Å. (a) The valence-band edge with respect to the Fermi level, (b) the Fermi velocity $v_{\text{F}}$, (c) gap parameter $\mathrm{\Delta }$, (d) Rashba SOC parameter ${\lambda }_{\text{R}}$, (e) intrinsic SOC parameter ${\lambda }_{\text{I}}^{\text{A}}$ for sublattice A, (f) intrinsic SOC parameter ${\lambda }_{\text{I}}^{\text{B}}$ for sublattice B, (g) PIA SOC parameter ${\lambda }_{\text{PIA}}^{\text{A}}$ for sublattice A, and (h) PIA SOC parameter ${\lambda }_{\text{PIA}}^{\text{B}}$ for sublattice B.

Figure 7

Predicted spin lifetime in graphene/$\mathrm{h}X\mathrm{N}$ heterostructures. The solid (dashed) lines show the out-of-plane (in-plane) spin lifetimes calculated from Eqs. (7) to (10). The symbols show the spin lifetime from numerical simulations of graphene/hGaN. The left inset shows the spin lifetime anisotropy for each heterostructure, and the right inset shows the momentum relaxation time and intervalley scattering time calculated from numerical simulations.

Figure 8

Spin lifetime anisotropy in graphene/hGaN and graphene/hAlN heterostructures. Results for graphene/hGaN were calculated numerically, while those for graphene/hAlN were derived from Eq. (9).

Figure 9

Predicted spin lifetime in (a) hBN/graphene/$\mathrm{h}X\mathrm{N}$ and (b) $\mathrm{h}X\mathrm{N}$/graphene/$\mathrm{h}X\mathrm{N}$ heterostructures. The solid (dashed) lines show the out-of-plane (in-plane) spin lifetimes calculated from Eqs. (7) to (11). The insets show the corresponding spin lifetime anisotropy.

Figure 10

(a) Calculated valence charge density of the graphene/hBN heterostructure considering only states in an energy window of $\pm 100\mathrm{meV}$ around the Fermi level. The colors correspond to isovalues between $2\times {10}^{-5}$ (blue) and $5\times {10}^{-8}$ (red) $e/{\AA }^3$, while the isolines range from $1\times {10}^{-3}$ to $1\times {10}^{-7}e/{\AA }^3$. (b) The atom and orbital resolved DOS. The DOS of B and N atoms is multiplied by a factor of ${10}^3$ for comparative reasons, while for C atoms only the $d$-orbital contribution is multiplied by a factor of 10.

Figure 11

(a) Calculated valence charge density of the graphene/hAlN heterostructure considering only states in an energy window of $\pm 100\mathrm{meV}$ around the Fermi level. The colors correspond to isovalues between $2\times {10}^{-5}$ (blue) and $1\times {10}^{-7}$ (red) $e/{\AA }^3$, while the isolines range from $1\times {10}^{-3}$ to $1\times {10}^{-7}e/{\AA }^3$. (b) The atom and orbital resolved DOS. The DOS of Al (N) atoms is multiplied by a factor of ${10}^3$ (${10}^2$) for comparative reasons, while for C atoms only the $d$-orbital contribution is multiplied by a factor of 10.

Figure 12

(a) Calculated valence charge density of the graphene/hGaN heterostructure considering only states in an energy window of $\pm 100\mathrm{meV}$ around the Fermi level. The colors correspond to isovalues between $1\times {10}^{-5}$ (blue) and $1\times {10}^{-7}$ (red) $e/{\AA }^3$, while the isolines range from $1\times {10}^{-3}$ to $1\times {10}^{-7}e/{\AA }^3$. (b) The atom and orbital resolved DOS. The DOS of Ga (N) atoms is multiplied by a factor of ${10}^3$ (${10}^2$) for comparative reasons, while for C atoms only the $d$-orbital contribution is multiplied by a factor of 10.