Impurity scattering in superconductors revisited: Diagrammatic formulation of the supercurrent-supercurrent correlation and Higgs-mode damping

Abstract

The diagrammatic formalism and transport equation are conventionally considered as separate but complementary basic techniques to tackle the impurity scattering effect. To compare with previous studies from the gauge-invariant kinetic equation approach [F. Yang and M. W. Wu, Phys. Rev. B 98, 094507 (2018); F. Yang and M. W. Wu, Phys. Rev. B 102, 144508 (2020)], we analytically perform a diagrammatic formulation of the impurity scattering in superconductors, with both transport and collective Higgs modes studied, to fill the long missing calculation of the Kubo current-current correlation in superconductors with impurity scattering and resolve the controversy (whether the impurity scattering can lead to the damping of Higgs mode) between the gauge-invariant kinetic equation and Eilenberger equation. For transport behavior, through a special unitary transformation that is equivalent to the Wilson-line technique for the diamagnetic response, we derive the Meissner-supercurrent vertex. Then, by formulating the supercurrent-supercurrent correlation with Born and vertex corrections from impurity scattering, we recover the previously revealed microscopic momentum-relaxation rate of superfluid by gauge-invariant kinetic equation. This rate is finite only when the superconducting velocity is larger than a threshold, at which the normal fluid emerges and causes friction with the superfluid current, similar to Landau's superfluid theory of liquid helium. This derivation also provides a physical understanding of the relaxation-time approximation in the previous diagrammatic formulation in the literature, which leads to the friction resistance of the Meissner supercurrent. For the collective Higgs mode, we calculate the amplitude-amplitude correlation with Born and vertex corrections from impurity scattering. The vertex correction, which only emerges at the nonequilibrium case, leads to a Higgs-mode damping, whereas the Born correction that is equivalent to equilibrium self-energy makes no contribution due to the Anderson theorem. This induced damping agrees with the analysis through the Heisenberg equation of motion and is also exactly the same as the one obtained from the gauge-invariant kinetic equation.

Similar papers