Abstract
Eliashberg's foundational theory of superconductivity is based on the application of Migdal's approximation, which states that vertex corrections to first order electron-phonon scattering are negligible if the ratio between phonon and electron energy scales is small. The resulting theory incorporates the first Feynman diagrams for electron and phonon self-energies. However, the latter is most commonly neglected in numerical analyses. Here we provide an extensive study of full-bandwidth Eliashberg theory in two and three dimensions, where we include the full back reaction of electrons onto the phonon spectrum. We unravel the complex interplay between nesting properties, Fermi surface density of states, renormalized electron-phonon coupling, phonon softening, and superconductivity. We propose furthermore a scaling law for the maximally possible critical temperature $T_c^{\textrm{max}}\propto\lambda (\Omega ) \sqrt{\Omega_0^2-\Omega^2}$ in 2D and 3D systems, which embodies both the renormalized electron-phonon coupling strength $\lambda(\Omega)$ and softened phonon spectrum $\Omega$. Also, we analyze for which electronic structure properties a maximal $T_c$ enhancement can be achieved.
Highlights
The current state-of-the-art description of superconductors is Eliashberg theory [1], which is especially applied in cases where the more simplified BCS (Bardeen-Cooper-Schrieffer) treatment [2] cannot capture the main characteristics of a given system
One of the key aspects to the success of Eliashberg theory is the applicability of Migdal’s approximation [3], which states that higher-order Feynman diagrams for electronphonon scattering can be neglected if the ratio of phonon to electron energy scale is a small number
In this paper we have investigated the details of Eliashberg theory including self-consistent phonon renormalization on a model basis
Summary
The current state-of-the-art description of superconductors is Eliashberg theory [1], which is especially applied in cases where the more simplified BCS (Bardeen-Cooper-Schrieffer) treatment [2] cannot capture the main characteristics of a given system. The most commonly studied system in these works is the two-dimensional (2D) Holstein model [8,11,12,13,14,15,16], often with additional constraints such as half filling, while three-dimensional (3D) systems are rarely considered in numerical calculations, presumably due to the large computational complexity Another way of checking the validity of the commonly employed Migdal approximation, and of the resulting Eliashberg theory, is to compute vertex corrections corresponding to additional Feynman diagrams. The first vertex-corrected self-consistent Eliashberg theory without further simplifications has been recently proposed by the current authors [27] All these works have in common that one or more additional Feynman diagrams to the electron-phonon interaction are studied and compared to the commonly employed Eliashberg formalism, that does not include a finite phonon self-energy. VI with a brief discussion on related works, possible extensions to our theory, and potential future directions
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