Spin-Orbit Coupling in Hydrogenated Graphene

First-principles calculations of the spin-orbit coupling in graphene with hydrogen adatoms in dense and dilute limits are presented. The chemisorbed hydrogen induces a giant local enhancement of spin-orbit coupling due to $s{p}^3$ hybridization which depends strongly on the local lattice distortion. Guided by the reduced symmetry and the local structure of the induced dipole moments, we use group theory to propose realistic minimal Hamiltonians that reproduce the relevant spin-orbit effects for both single-side semihydrogenated graphene (graphone) and for a single hydrogen adatom in a large supercell. The principal linear spin-orbit band splittings are driven by the breaking of the local pseudospin inversion symmetry and the emergence of spin flips on the same sublattice.

Figure 1

Top: Calculated electronic band structure of single-side semihydrogenated graphene. (a) Sublattice resolved band structure for the distortion $\Delta /a=14\%$. The filled (red) circles correspond to sublattice $A$, whereas the open (blue) circles correspond to sublattice $B$. The circles’ radii correspond to the carbon atom charge densities. The inset shows the structure and the probability density of the flat band at the $K$ point. (b) Orbital resolved local density of states (LDOS). Bottom: Extracted spin-orbit coupling parameters for the ${\pi }^{*}$ band at $K$ and $\Gamma$ as functions of $\Delta /a$. (c) Intrinsic spin-orbit coupling splitting $2{\lambda }_{\mathrm{I}}$ at $K$. The inset shows the band splitting. (d) Adatom-induced SOC splitting ${\lambda }_{\mathrm{BR}}^K$ at $K$. The inset shows the spin texture around $K$ for the lower spin-orbit split band. The in-plane components are shown by the arrows, while the $z$ component is shown by the color map. (e) Adatom-induced SOC splitting ${\lambda }_{\mathrm{BR}}^{\Gamma }$ at $\Gamma$. The inset shows the band splitting around $\Gamma$ with the identified Bychkov-Rashba energy $E_{\mathrm{BR}}=0.87\mu \mathrm{eV}$ for $\Delta /a=14\%$.

Figure 2

First-principles results and their tight-binding model fits for a hydrogen adatom on a $5\times 5$ supercell. (a) Electronic band structure around the Fermi level. The dots are first-principles results, and the solid lines are tight-binding model fits. The band spin-orbit splitting around $K$ is sketched for the valence band, indicating the out-of-plane $z$ components and the rotation directions of the in-plane spin components. (b) Broadened total density of states per atom (gray areas) and $p_z$ projected local densities for atoms in the vicinity of the adatom: ${\mathrm{C}}_{\mathrm{H}}$ the hydrogenated carbon atom (shown in green), ${\mathrm{C}}_{\mathrm{nn}}$ its nearest neighbor (in dark blue), ${\mathrm{C}}_{\mathrm{nnn}}$ the next-nearest neighbor (in light blue), and the total density on hydrogen H (in red). The projected densities of states are normalized to the corresponding number of atoms in the set. (c) Conduction, (d) impurity, and (e) valence band spin-orbit splittings along the high-symmetry lines; the symbols are as in (a). Tight-binding model least-squares fits are performed within the shaded regions around $K$. (f) Spin-expectation values around $K$ for the spin-orbit split valence and conduction bands closer to the Fermi level.

Figure 3

(a) First-principles calculations of electric dipole moments induced by hydrogen adatoms on a $5\times 5$ supercell. The directions of the dipole moments are shown by arrows; the sphere radii correspond to the dipole magnitudes. (b) Hopping scheme of the tight-binding model showing the relevant orbital and spin-orbit coupling parameters.