Charge and Spin Hall Conductivity in Metallic Graphene

Abstract

Graphene has an unusual low-energy band structure with four chiral bands and half-quantized and quantized Hall effects that have recently attracted theoretical and experimental attention. We study the Fermi energy and disorder dependence of its spin Hall conductivity σ SH xy . In the metallic regime we find that vertex corrections enhance the intrinsic spin Hall conductivity and that skew scattering can lead to σ SH xy values that exceed the quantized ones expected when the chemical potential is inside the spin-orbit induced energy gap. We predict that large spin Hall conductivities will be observable in graphene even when the spin-orbit gap does not survive disorder.Introduction-The low-energy band structure of graphene consists of four chiral bands that realize (2+1)-dimensional relativistic field theory models with parity anomalies. The anomalies imply unusual spectra in an external magnetic field and quantized and halfquantized Hall effects 1,2 . Theoretical interest 3 in these unusual electronic systems has increased 4 recently because of experimental progress 5 , including measurements of the anticipated half-quantized quantum Hall effect. One particularly interesting observation, due to Kane and Mele 6,7 , is that because of a gap produced by spinorbit interactions, the spin Hall conductivity σ SH of undoped graphene is quantized in the absence of a magnetic field. This suggestion is related to recent work on the anomalous Hall effect in ferromagnetic metals 8 and on its paramagnetic cousin, the spin Hall effect 9 , in which it was suggested that these transport coefficients can be dominated by an intrinsic momentum-space Berry phase contribution that reduces to quantized values when the Fermi level is in a gap. Here we examine how the quantized spin Hall effect is altered when the Fermi energy in the graphene plane is gated into the metallic regime. We find that the intrinsic spin Hall effect is no longer quantized, that it is enhanced by disorder vertex corrections, and that in the metallic regime skew scattering can potentially lead to parametrically larger spin Hall conductivities. Because the Bloch state disorder broadening in current samples is (according to our estimates) much larger than the clean system spin-orbit gap, these results are necessary for the interpretation of experiment. SpinHall effects should be observable even when the spin-orbit gap does not survive disorder. Disordered Graphene Model-When spin-orbit interactions are included, 6 the low-energy physics of a clean undoped graphene crystal is described by an eight-band envelope function Hamiltonian

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