Considerations on the Mechanisms and Transition Temperatures of Superconductors

TL;DR

This paper reviews the mechanisms and factors influencing superconducting transition temperatures across various materials, emphasizing electron interactions, quantum critical fluctuations, and empirical data to understand and potentially enhance $T_c$.

Contribution

It provides a comprehensive overview of electron interaction mechanisms, quantum fluctuations, and empirical data to explain and compare $T_c$ across different superconductors.

Findings

Universal $T_c$ to Fermi-energy ratio at unitarity is about 0.15.

Maximum $T_c$ to phonon frequency ratio is of the same order in phonon-induced superconductors.

In cuprates and heavy fermions, the $T_c$ to Fermi-energy ratio is around 1%.

Abstract

An overview of the momentum and frequency dependence of effective electron-electron interactions which favor electronic instability to a superconducting state in the angular-momentum channel $\ell$ and the properties of the interactions which determine $T_c$ is provided. Both interactions induced through exchange of phonons as well as purely electronic fluctuations of spin density, charge density or current density are considered. Special attention is paid to the role of quantum critical fluctuations including pairing due to their virtual exchange as well as de-pairing due to inelastic scattering. In light of the above, empirical data and theory specific to phonon induced superconductivity, in cold atoms, superfluidity in liquid $He^3$, superconductivity in some of the heavy fermion compounds, in Cuprates, in pncitides and the valence skipping compound, is reviewed. The physical basis for the following observation is provided: The universal ratio of s-wave $T_c$ to Fermi-energy for fermions at the unitarity limit with attractive interactions is about 0.15, the ratio of the maximum $T_c$ to the typical phonon frequency in phonon induced s-wave superconductivity is of the same order; the ratio of p-wave $T_c$ to the renormalized Fermi-energy in liquid $He^3$, a very strongly correlated Fermi-liquid near its melting pressure, is only $O(10^{-3})$; in the Cuprates and the heavy-fermions where d-wave superconductivity occurs in a region governed by a special class of quantum-critical fluctuations, this ratio rises to $O(10^{-2})$. These discussions also suggest factors important for obtaining higher $T_c$.