# Structural phases of elemental Gallium: universal relations in type-I superconductors

## TL;DR

This paper investigates the thermodynamic properties of elemental Gallium's phases, revealing universal scaling relations in type-I superconductors that connect critical fields, specific heat, and coupling strength, supported by strong-coupling theory.

## Contribution

It establishes simple, analytical scaling relations between key superconducting parameters across elemental and binary phonon-mediated type-I superconductors, incorporating strong-coupling effects.

## Key findings

- Identified universal scaling relations in type-I superconductors.
- Explained the linear relation between critical field and transition temperature.
- Supported relations with strong-coupling corrections to BCS theory.

## Abstract

The temperature dependent measurements of the thermodynamic critical field and the specific heat for the pressure stabilized Ga-II phase of elemental Gallium are presented. The discussion of these and other Ga phases data in the context of elemental and binary phonon-mediated type-I superconductors allowed to establish simple scaling relations between BCS quantities such as $B_{\rm c}(0)/T_{\rm c}\sqrt{\gamma_{\rm e}}$ and the specific heat jump at $T_{\rm c}$ versus the coupling strength $2\Delta/k_{\rm B} T_{\rm c}$ [$\Delta$ and $B_{\rm c}(0)$ are the zero-temperature values of the superconducting energy gap and the thermodynamic critical field, respectively, $T_{\rm c}$ is the transition temperature, and $\gamma_{\rm e}$ is the electronic specific heat]. The scaling relations can be analytically expressed by taking into account strong-coupling corrections to BCS theory. Such correlations can naturally explain the linear relation between $B_{\rm c}(0)$ and $T_{\rm c}$, which holds for type-I superconducting materials.