Spin-Orbit Interaction and Isotropic Electronic Transport in Graphene

Broken symmetries in graphene affect the massless nature of its charge carriers. We present an analysis of scattering by defects in graphene in the presence of spin-orbit interactions (SOIs). A characteristic constant ratio ($\simeq 2$) of the transport to elastic times for massless electrons signals the anisotropy of the scattering. We show that SOIs lead to a drastic decrease of this ratio, especially at low carrier concentrations, while the scattering becomes increasingly isotropic. As the strength of the SOI determines the energy (carrier concentration) where this drop is more evident, this effect could help evaluate these interactions through transport measurements in graphene systems with enhanced spin-orbit coupling.

Figure 1

Polar plots of the differential cross section, normalized to its maximum, for different values of the intrinsic spin-orbit interaction, ${\mathrm{\Delta }}_{\mathrm{SO}}$—here $ER/(\hslash v_F)=8\times 10^{-3}$ and $VR/(\hslash v_F)=1.5$. Top inset: ${\sigma }_{\text{max}}(\theta )$, which increases as $1/[k{\ln }^2(kR)]$ for ${\mathrm{\Delta }}_{\mathrm{SO}}/E\approx 1$, sets the scale used in the polar plots. Bottom inset: Dependence of $\xi ={\sigma }_t/{\sigma }_{\text{tr}}={\tau }_{\text{tr}}/{\tau }_e$ vs ${\mathrm{\Delta }}_{\mathrm{SO}}/E$. Notice that $\sigma (\theta )/{\sigma }_{\max }$ and $\xi$ do not depend on the value of $V$ in this regime; $V$ only determines the amplitude of ${\sigma }_{\max }$ in the top inset.

Figure 2

(a) Total cross section for spin-preserving processes as a function of the scattering potential shift $V$, for different values of the Rashba SOI, with $kR=1.5\times 10^{-3}$ and ${\mathrm{\Delta }}_{\mathrm{SO}}=0$ (for ${\mathrm{\Delta }}_{\mathrm{SO}}\ne 0$ see [37]). Inset: Total cross section for spin-flip processes. (b) The ratio ${\xi }_R=({\sigma }_{t,\uparrow \uparrow }+{\sigma }_{t,\uparrow \downarrow })/({\sigma }_{\text{tr,↑↑}}+{\sigma }_{\text{tr,↑↓}})$ for different values of Rashba SOI [legend as in (a)]. Notice ${\xi }_R=1$ at ${\sigma }_t$ resonances.

Figure 3

The ratio ${\xi }_R$ for 500 randomly sized impurities in the range of $5\AA \le R\le 8\AA$, for different values of the Rashba coupling and $V=2\mathrm{eV}$, as a function of carrier energy. Notice a clear drop of ${\xi }_R$ from 2 for $E<{\lambda }_R/2$. Inset: ${\xi }_R$ as a function of Rashba coupling for different energies.