Crossover to the anomalous quantum regime in the extrinsic spin Hall effect of graphene

Recent reports of spin-orbit coupling enhancement in chemically modified graphene have opened doors to studies of the spin Hall effect with massless chiral fermions. Here, we theoretically investigate the interaction and impurity density dependence of the extrinsic spin Hall effect in spin-orbit coupled graphene. We present a nonperturbative quantum diagrammatic calculation of the spin Hall response function in the strong-coupling regime that incorporates skew scattering and anomalous impurity density-independent contributions on equal footing. The spin Hall conductivity dependence on Fermi energy and electron-impurity interaction strength reveals the existence of experimentally accessible regions where anomalous quantum processes dominate. Our findings suggest that spin-orbit-coupled graphene is an ideal model system for probing the competition between semiclassical and bona fide quantum scattering mechanisms underlying the spin Hall effect.

Figure 1

Kubo-Streda diagrams. (a) Response bubble for the SH conductivity with dressed charge vertex ${\tilde{v}}_x=v_x+\delta v_x$. (b) Bethe-Salpeter equation for the vertex correction $\delta v_x$.

Figure 2

$T$ Matrix ladder. Skeleton expansion of the ladder diagram in terms of an infinite series of two particle, noncrossing diagrams. On the left side, a full (open) square interaction vertex denotes a $T \left(T^{*}\right)$ matrix insertion, while on the right the $T$ matrix is expanded in its bare components ($M$ insertions). The red $\times$ represents an impurity density insertion.

Figure 3

SH conductivity. The semiclassical SS and anomalous contributions to ${\sigma }_{\mathrm{SH}}$ are shown for different values of the Fermi energy in solid and dotted lines, respectively. ${\sigma }_{\mathrm{SS}} \left({\sigma }_Q\right)$ increases (decreases) with $\epsilon$, and both conductivities decrease at increasing scalar potential magnitude, in agreement with the unitary limit result. Note that ${\sigma }_Q$ has been scaled by a factor of 10. We have used ${\alpha }_3=0.01$ eV, $R=4$ nm, and $n=4\times {10}^{12}{\mathrm{cm}}^{-2}$, typical parameters for physisorbed metal nanoparticles [10, 18]. The inset shows the regions $(\epsilon ,n)$ dominated by the semiclassical and anomalous contributions (${\alpha }_0=0.05$ eV, other parameters as in main figure).

Figure 4

Phase diagram of the SH conductivity in our model. The diagram shows the parameter regions in which either ${\sigma }_Q$ or ${\sigma }_{\mathrm{SS}}$ is dominant. The black line is the phase boundary and the different colors represent the absolute value of ${\sigma }_{\mathrm{SH}}$. We have used ${\alpha }_3=0.01$ eV, $R=4$ nm, and $n=4\times {10}^{12} {\mathrm{cm}}^{-2}$.