Vertex corrections to the dc conductivity in anisotropic multiband systems

For an isotropic single-band system, it is well known that the semiclassical Boltzmann transport theory within the relaxation time approximation and the Kubo formula with the vertex corrections provide the same result with the $(1-cos\theta )$ factor in the inverse transport relaxation time. In anisotropic multiband systems, the semiclassical Boltzmann transport equation is generalized to coupled integral equations relating transport relaxation times at different angles in different bands. Using the Kubo formula, we study the vertex corrections to the dc conductivity in anisotropic multiband systems, and we derive the relation satisfied by the transport relaxation time for both elastic and inelastic scatterings, verifying that the result is consistent with the semiclassical approach.

Figure 1

Schematic diagrams for (a) Drude conductivity without the $(1-cos\theta )$ factor, and (b) ladder diagrams giving the $(1-cos\theta )$ factor.

Figure 2

Diagrams for the ladder vertex corrections for elastic scattering.

Figure 3

Contour $C$ used for the contour integral in Eq. (23) having two branch cuts along the axes $z=0$ and $z=-i{\nu }_m$.

Figure 4

Dyson's equation for the ladder vertex corrections for inelastic scattering.

Figure 5

Contour $C^{\prime }$ used for the contour integral in Eq. (34) having two branch cuts along the axes $z=-i{\omega }_n$ and $z=-i{\omega }_n-i{\nu }_m$.

Figure 6

Self-energy diagrams for an impurity-scattered electron.

Figure 7

Self-energy diagrams for a phonon-scattered electron.