Tight-binding theory of the spin-orbit coupling in graphene

The spin-orbit coupling in graphene induces spectral gaps at the high-symmetry points. The relevant gap at the $\Gamma$ point is similar to the splitting of the $p$ orbitals in the carbon atom, being roughly 8.5 meV. The splitting at the $\mathrm{K}$ point is orders of magnitude smaller. Earlier tight-binding theories indicated the value of this intrinsic gap of $1\mu \text{eV}$, based on the $\sigma \text{−}\pi$ coupling. All-electron first-principles calculations give much higher values, between 25 and $50\mu \text{eV}$, due to the presence of the orbitals of the $d$ symmetry in the Bloch states at $\mathrm{K}$. A realistic multiband tight-binding model is presented to explain the effects the $d$ orbitals play in the spin-orbit coupling at $\mathrm{K}$. The $\pi \text{−}\sigma$ coupling is found irrelevant to the value of the intrinsic spin-orbit-induced gap. On the other hand, the extrinsic spin-orbit coupling (of the Bychkov-Rashba type), appearing in the presence of a transverse electric field, is dominated by the $\pi \text{−}\sigma$ hybridization, in agreement with previous theories. Tight-binding parameters are obtained by fitting to first-principles calculations, which also provide qualitative support for the model when considering the trends in the spin-orbit-induced gap in graphene under strain. Finally, an effective single-orbital next-nearest-neighbor hopping model accounting for the spin-orbit effects is derived.

Figure 1

(Color online) Sketch of the Slater-Koster hopping parameters (a) $V_{pp\pi }$, (b) $V_{pd\pi }$, (c) $V_{dd\pi }$, and (d) $V_{dd\delta }$, needed to calculate the contributions of the $d$ orbitals to the $\pi$ band.

Figure 2

Results of the first-principles (circles), analytical (solid lines) and numerical (squares) tight-binding calculations of the SOC intrinsic gap in graphene as a function of the artificial lattice constant ratio. Those dependences originate from the hopping parameters. The inset shows the dominance of the $p$ orbitals for larger values of the lattices constant ratio.

Figure 3

(Color online) Calculated band structure of graphene obtained from first-principles calculation (symbols) and tight-binding model (solid lines) using the parameters presented in Table . The size of the symbols reflects the contribution of the orbitals to the corresponding eigenstates.

Figure 4

(Color online) Calculated Bychkov-Rashba constant as a function of the artificial lattice constant ratio: first-principles calculations (circles), numerical diagonalization of the $p$-orbital part of TB Hamiltonian including overlap (squares), and the analytical calculations (solid line). Those dependences on the lattice constant arise from the hopping parameter $V_{sp\sigma }$.

Figure 5

(Color online) The zig-zag chains in the graphene lattice are defined in the way that the corresponding hopping path goes horizontally, alternating in the direction of the unit vectors ${\vec{n}}_2$ and ${\vec{n}}_3$. The hopping between the chains is possible only in the direction of the unit vector ${\vec{n}}_1$.

Figure 6

Calculated spectrum in the vicinity of the $\mathrm{K}$ point, plotted along the high-symmetry lines in the $\mathrm{\Gamma }\text{−}\mathrm{K}\text{−}\mathrm{M}$ direction. Circles correspond to the numerical diagonalization of the multiorbital TB Hamiltonian (neglecting the $d$ orbitals which do not play any role here) and solid lines give the eigenenergies of the effective hopping Hamiltonian. The artificially large spin-orbit parameters for this plot are given in the text. The effective hopping model describes well the spin-orbit physics in graphene at the $\mathrm{K}$ point.

Figure 7

(Color online) Two of the possible nnn hopping paths through the $s,p$ orbitals, (black) arrows, with a corresponding spin, shown by (yellow) arrows on the orbitals. The opposite sign for the clock-wise (a) and the anti-clock-wise (b) effective hoppings is determined by the signs of the two SOC of the $p$ orbitals.

Figure 8

(Color online) Two of the possible nnn hopping paths through $d$ orbitals, (black) arrows. Their spin is shown by (yellow) arrows on the orbitals. The opposite sign for clock-wise (a) and anti-clock-wise (b) hopping is given by the opposite sign in the SOC of the $d$ orbitals.

Figure 9

(Color online) A representative leading hopping path, (black) arrows, which is responsible for the Bychkov-Rashba SOC effect, by coupling states of different spins, illustrated by (yellow) arrows on the orbitals. The effective hopping is between nearest neighbors. (a) The dominant $p$ orbital contribution. (b) The negligible $d$ orbital contribution. For clarity the orbitals of the same atoms are separated vertically, according to their contribution either to the $\sigma$ bands (bottom) or to the $\pi$ bands (top).