Abstract
We solve the anisotropic, full-bandwidth and non-adiabatic Eliashberg equations for phonon-mediated superconductivity by fully including the first vertex correction in the electronic self-energy. The non-adiabatic equations are solved numerically here without further approximations, for a one-band model system. We compare the results to those that we obtain by adiabatic full-bandwidth, as well as Fermi-surface restricted Eliashberg-theory calculations. We find that non-adiabatic contributions to the superconducting gap can be positive, negative or negligible, depending on the dimensionality of the considered system, the degree of non-adiabaticity, and the coupling strength. We further examine non-adiabatic effects on the transition temperature and the electron-phonon coupling constant. Our treatment emphasizes the importance of overcoming previously employed approximations in estimating the impact of vertex corrections on superconductivity and opens a pathway to systematically study vertex correction effects in systems such as high-$T_c$, flat band and low-carrier density superconductors.
Highlights
The foundation for establishing the microscopic description of phonon-mediated superconductors was laid by the pioneering work of Migdal on the electron-phonon interaction in metals, where his famous theorem was introduced [1]
We examine the possibility of applying a Fermi surface restricted (FSR) isotropic approximation to the nonadiabatic Eliashberg equations as an attempt to considerably decrease the high computational complexity of the problem
We investigated the influence of vertex corrections to the electronic self-energy on electron-phonon mediated, anisotropic, and full-bandwidth Eliashberg theory
Summary
The foundation for establishing the microscopic description of phonon-mediated superconductors was laid by the pioneering work of Migdal on the electron-phonon interaction in metals, where his famous theorem was introduced [1]. For typical metals, where the degree of nonadiabaticity α ≡ / F ∼ 10−2, such an approximation to the electron self-energy (below referred to as Migdal’s approximation), is commonly believed to be valid even in materials characterized by strong-coupling λ O(1). Eliashberg generalized this formalism to the superconducting state [2] and thereby laid the foundation for the -far most successful description of a vast amount of superconductors that deviate from the weak-coupling limit of the Bardeen-Cooper-Schrieffer (BCS) theory [3,4,5,6]
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