Extrinsic spin Hall effect from anisotropic Rashba spin-orbit coupling in graphene

We study the effect of anisotropy of the Rashba coupling on the extrinsic spin Hall effect due to spin-orbit active adatoms on graphene. In addition to the intrinsic spin-orbit coupling, a generalized anisotropic Rashba coupling arising from the breakdown of both mirror and hexagonal symmetries of pristine graphene is considered. We find that Rashba anisotropy can strongly modify the dependence of the spin Hall angle on carrier concentration. Our model provides a simple and general description of the skew scattering mechanism due to the spin-orbit coupling that is induced by proximity to large adatom clusters.

Figure 1

(a) Spin Hall angle $\gamma$ vs energy of the carrier $\left(E\right)$ for different values of the strength of scalar potential $v_0={\lambda }_0/R^2$. We set $R=20$ nm, $T=0$ K, ${\mathrm{\Delta }}_I={\lambda }_I/R^2=15$ meV, $\beta =-\frac{\pi }{8}$, and $\frac{{\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2}{2}=\frac{({\lambda }_1+{\lambda }_2)}{2R^2}=15$ meV. The location of resonances gradually approaches the Dirac point when $v_0$ increases, illustrating the general feature of resonant scattering even for the anisotropic case. (b) Corresponding starred transport cross sections of each case. See Appendix pp2 for the definition of starred transport cross section.

Figure 2

(a) Spin Hall angle $\gamma$ vs the energy of the scattered electron $E$ for different anisotropy values of the Rashba SOC anisotropy parameter $\beta ={tan}^{-1}\left({\lambda }_2/{\lambda }_1\right)$. We have chosen the strength of the scalar part of the potential, $v_0={\lambda }_0/R^2=0.1$ eV, and other parameters are the same as in Fig. 1. The magnitude of the spin Hall angle $\gamma$ is enhanced near the position of the scattering resonance, which is signaled by a peak in the transport scattering cross section. (b) Corresponding starred transport cross sections ${\mathrm{\Sigma }}_{\text{tr}}^{*}$ of each case. The curves for the transport cross section and the spin Hall angle are asymmetric about $E=0$ because the Rashba SOC breaks the particle-hole symmetry of the $\mathbf{k}·\mathbf{p}$ Hamiltonian.

Figure 3

(a)–(d) Temperature dependence of $\gamma$ vs $E_F$ plot at ${\mathrm{\Delta }}_I=0\mathrm{meV}$ and four different $\beta$. (e)–(h) Temperature dependence of $\gamma$ vs $E_F$ plot at ${\mathrm{\Delta }}_I=15\mathrm{meV}$ and four different $\beta$. $v_0=0.1\mathrm{eV}$ in all eight plots. Other parameters are the same as in Figs. 1 and 2. (Especially, $\frac{{\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2}{2}=\frac{({\lambda }_1+{\lambda }_2}{2R^2}=15\mathrm{meV})$. Note that, as mentioned in the Introduction, our semiclassical approach is not accurate near the $E_F\approx 0$ regime.

Figure 4

(a) Comparison between the spin Hall angle $\left(\gamma \right)$ of a standard Rashba (SR) and nonstandard Rashba (NS) SOC for a circular (i.e., “pill-box”) scattering potential. (b) Transport cross section. We have assumed the radius of the scatterer $R=4$ nm, $v_0=0.1\mathrm{eV}$, and ${\mathrm{\Delta }}_{\text{SR}}={\mathrm{\Delta }}_{\text{NR}}=25\mathrm{meV}$. The same asymmetric resonant property as that discussed in the main context recurs in the finite-disk modeling.