Dynamical spin Hall conductivity in a disordered two-dimensional system calculated in a self-consistent Born approximation

The dynamical spin Hall conductivity is calculated in a two-dimensional system with spin-orbit interaction within a self-consistent Born approximation. The wave-vector dependence of the spin splitting plays an important role in the high-frequency region corresponding to inter-spin-band transitions while being unimportant in the low-frequency region. The relaxation time characterizing the low-frequency behavior of the spin Hall conductivity is twice as large as the transport relaxation time and is independent of the electron concentration in the weak-scattering limit. The cancellation of positive interband and negative intraband contributions in the imaginary part leads to vanishing static conductivity.

Figure 1

A schematic illustration of the energy dispersion and vertical transitions from the band ${ϵ}_{\mathbf{k}-}$ to ${ϵ}_{\mathbf{k}+}$. (a) The Fermi energy ${ϵ}_F$ for $n∕n_0=20$. (b) ${ϵ}_F$ for $n∕n_0=0.5$.

Figure 2

The ladder approximation for the velocity vertex part in the self-consistent Born approximation.

Figure 3

The density of states for different disorder parameters $\gamma =\hslash ∕2\tau$. The solid lines are the results in the disordered system and the dotted line is that in the clean system. The vertical lines show the Fermi energy for $n∕n_0=3$.

Figure 4

Calculated dynamical spin Hall conductivity at zero temperature for the density $n∕n_0=3$. The solid lines are that in the disordered system, the dotted lines are that in the clean system, and the dotted-dashed line is Eq. (2.25) with $\gamma ∕{ϵ}_0=0.1$. (a) The real part. (b) The imaginary part.

Figure 5

Calculated dynamical spin Hall conductivity at zero temperature for $n∕n_0=20$. (a) The real part. (b) The imaginary part.

Figure 6

Calculated dynamical spin Hall conductivity at zero temperature for the electric density $n∕n_0=1$. (a) The real part. (b) The imaginary part.

Figure 7

Calculated dynamical spin Hall conductivity at zero temperature for the electric density $n∕n_0=0.5$. (a) The real part. (b) The imaginary part.

Figure 8

Calculated real part of $\sigma \left(\omega \right)$ at zero temperature. The conductivity becomes independent of the frequency corresponding to interband transitions.

Figure 9

Calculated spin Hall conductivity at zero temperature for the electric density $n∕n_0=3$. (a) The real part. (b) The imaginary part.