Functional derivatives of Tc for a two-band superconductor: application to MgB2

TL;DR

This paper develops a method to analyze how phonons of different frequencies influence the superconducting transition temperature in multi-band superconductors, with a specific application to MgB2.

Contribution

A general scheme for computing functional derivatives of Tc with respect to electron-phonon interactions in multi-band superconductors is introduced, including analytical proofs and numerical application to MgB2.

Findings

Diagonal functional derivatives are linear in phonon energy at small energies.

Off-diagonal derivatives diverge as 1/Ω at zero energy.

Numerical results for MgB2 illustrate the theoretical framework.

Abstract

We address the question of how phonons of various frequencies contribute to the superconducting transition temperature $T_c$ of a multi-band superconductor by considering the functional derivatives of $T_c$ with respect to various intraband and interband electron-phonon coupling functions. A general scheme for computing such functional derivatives is developed. The proofs are given that the functional derivatives which are diagonal in band indices are linear in phonon energy $\Omega$ at small $\Omega$, while the functional derivatives which are off-diagonal in band indices diverge at $\Omega$ = 0 as 1/$\Omega$. The case of a two-band model for MgB$_{2}$ is treated numerically.