Scattering theory of spin-orbit active adatoms on graphene

The scattering of two-dimensional massless Dirac fermions from local spin-orbit interactions with an origin in dilute concentrations of physisorbed atomic species on graphene is theoretically investigated. The hybridization between graphene and the adatoms' orbitals lifts spin and valley degeneracies of the pristine host material, giving rise to rich spin-orbit coupling mechanisms with features determined by the exact adsorption position on the honeycomb lattice—bridge, hollow, or top position—and the adatoms' outer-shell orbital type. Effective graphene-only Hamiltonians are derived from symmetry considerations, while a microscopic tight-binding approach connects effective low-energy couplings and graphene-adatom hybridization parameters. Within the $T$-matrix formalism, a theory for (spin-dependent) scattering events involving graphene's charge carriers, and the spin-orbit active adatoms is developed. Spin currents associated with intravalley and intervalley scattering are found to tend to oppose each other. We establish that under certain conditions, hollow-position adatoms give rise to the spin Hall effect, through skew scattering, while top-position adatoms induce transverse charge currents via trigonal potential scattering. We also identify the critical Fermi energy range where the spin Hall effect is dramatically enhanced, and the associated transverse spin currents can be reversed.

Figure 1

Schematic picture of an adatom (pink sphere in the left panel) in the hollow position. $A$-sublattice ($B$-sublattice) carbon atoms are represented by blue (red). Right panels show the modulus of wave functions ${\psi }_m(x,y)$ created by operators ${\Omega }_m^{†}$, for $m=0,\pm 1,\pm 2,3$. Space coordinates $(x,y)$ have the adatom as origin, and verify $(x,y)\in {[-3.5,3.5]}^2$ in units of $a\approx 1.43\text{Å}$. The color scale is linear and represents $|{\psi }_m|$, from dark blue (lowest values) to red (highest values).

Figure 2

Interpretation of the effective Hamiltonian ${\mathcal{H}}_{\mathrm{hollow}}$ in terms of the hopping between graphene's $p_z$ orbitals. Given the short-range nature of the adatom-induced effective potential, all atoms around the adatom mutually interact. Intervalley terms arise from hopping between adjacent atoms. Although they were neglected in previous works [15, 16], these terms play a central role in scattering from dilute random ensembles of adatoms.

Figure 3

Schematic picture of adatoms (pink spheres) in the top position, on an $A$-sublattice (blue) and $B$-sublattice (red) carbon atom. Site numbering used in the main text is shown for both the $A$ and $B$ sublattice. Right panels show the modulus of wave functions ${\varphi }_m(x,y)$ created around the $A$-sublattice adatom by operators ${\Gamma }_m^{†}$, for $m=0,\pm 1$. Space coordinates $(x,y)$ have this adatom as origin, and verify $(x,y)\in {[-3.5,3.5]}^2$ in units of $a\approx 1.43\text{Å}$. The color scale is linear and represents $|{\varphi }_m|$, from dark blue (lowest values) to red (highest values).

Figure 4

Interpretation of effective Hamiltonian ${\mathcal{H}}_{\mathrm{top}}^A$ in terms of hopping between graphene's $p_z$ orbitals.

Figure 5

Adatom (pink sphere) in the bridge position. $A$- and $B$-sublattice carbon atoms are shown in blue and red, respectively. Relevant atoms are numbered as in the main text.

Figure 6

Schematic representation of skew-scattering induced by adatoms in the hollow position. The black arrow represents the momentum ${\mathbf{k}}_{\mathrm{in}}$ of an incoming quasiparticle in the $K$ valley. Blue (red) half-circles correspond to the region of the Fermi line where the outgoing momentum ${\mathbf{k}}_{\mathrm{out}}$ is most likely to be after a scattering event, if the incoming charge carrier has spin down (spin up). Resulting pure spin currents are depicted as blue and red planar arrows. Spin currents in $K$ and $K^{\prime }$ valleys associated with intra- and intervalley scattering tend to oppose each other.

Figure 7

Resonant features of spin transverse currents generated by hollow-position physisorbed adatoms on graphene. (a) $J_S^{\perp }/J_C^{\parallel }$ (in %) against Fermi energy (in eV), for adatoms with $E_{1/2}^{\pm }=-1$ eV, $E_{3/2}^{\pm }=-1.5$ eV, and $E_{5/2}^{\pm }=-2$ eV and different $({{\rm Y}}_1,{{\rm Y}}_2)$, corresponding to points $A$, $B$, $C$, and $D$ shown in the lower panel. (b) Maximum of $|J_S^{\perp }/J_C^{\parallel }|$ for $|E_F|\le 0.5$ eV, against ${{\rm Y}}_1$ and ${{\rm Y}}_2$. $\left|E\right|/E_{\mathrm{c}}=0.1$ lines are shown for $E=E_{x_{\pm },d_{\pm }},E_{\mathrm{inv}}$. Each line partitions the $({{\rm Y}}_1,{{\rm Y}}_2)$ space into two parts: a region containing the origin ${{\rm Y}}_1={{\rm Y}}_2=0$, characterized by high-energy resonances, $\left|E\right|/E_{\mathrm{c}}>0.1$, and a region away from the origin, characterized by low-energy resonances, $\left|E\right|/E_{\mathrm{c}}<0.1$.

Figure 8

Resonant features of charge transverse currents generated by top-position physisorbed adatoms on graphene. (a) $J_C^{A\perp }/J_C^{A\parallel }$ (in %) against Fermi energy (in eV), for top-position adatoms with fixed $E_{1/2}^{\pm }=-1$ eV and $E_{3/2}^{\pm }=-1.5$ eV and different $(\gamma ,{\theta }_1)$, corresponding to points $A$, $B$, $C$, $D$, and $E$ shown in the lower panel. (b) Maximum of $|J_C^{A\perp }/J_C^{A\parallel }|$ for $|E_F|\le 0.5$ eV, against $\gamma$ and ${\theta }_1$. $\left|E\right|/E_{\mathrm{c}}=0.1$ lines are shown for $E={\mathcal{E}}_{1,2,3,4},{\mathcal{E}}_{\mathrm{inv}}$. Similarly to Fig. 7, each line partitions $(\gamma ,{\theta }_1)$ space into regions whose farthest point from the origin corresponds to $\left|E\right|/E_{\mathrm{c}}<0.1$.

Figure 9

Illustration of typical spin-flip (red) and spin-conserving (blue) processes induced by a $p$-orbital adatom (gray) on graphene (light blue). Energy levels ${\epsilon }_0$ and ${\epsilon }_1$ of the adatom's $p$ orbitals $m=0$ and $m=\pm 1$ are represented as gray solid lines. Core orbitals are depicted as a black ball. The shaded region corresponds to the adatom's immediate vicinity, where carbon atom $p_z$ orbitals couple strongly to the adatom valence $p$ orbital. Red (blue) straight vertical arrows represent the spin of an electron transiting between graphene and the adatom while flipping (conserving) its spin. Partial waves $|{\psi }_{\infty }\rangle$, $|{\psi }_{\mathcal{N}}\rangle$, and $|{\psi }_{\text{ad}}\rangle$ introduced in the Appendix are associated with the blue area, the dashed area, and the adatom's valence orbital, respectively.