Characteristics of superconducting state in vanadium: the Eliashberg equations and semi-analytical formulas

TL;DR

This paper investigates the superconducting properties of vanadium using Eliashberg equations and semi-analytical formulas, revealing high Coulomb pseudopotential values and confirming BCS-like ratios despite strong coupling.

Contribution

It provides a detailed analysis of vanadium's superconducting state, including high Coulomb pseudopotential values, and demonstrates the effectiveness of semi-analytical formulas in reproducing Eliashberg results.

Findings

Critical temperature $T_c$ is 5.3 K.

High Coulomb pseudopotential values ($^* eq 0.259$).

Dimensionless ratios close to BCS predictions.

Abstract

The superconducting state in vanadium characterizes with the critical temperature ($T_{c}$) equal to $5.3$~K. The Coulomb pseudopotential, calculated with the help of the Eliashberg equations, possesses anomalously high value $\mu^{\star}\left(3\Omega_{\rm max}\right)=0.259$ or $\mu^{\star}\left(10\Omega_{\rm max}\right)=0.368$ ($\Omega_{\rm max}$ denotes the maximum phonon frequency). Despite the relatively large electron-phonon coupling constant ($\lambda=0.91$), the quantities such as: the order parameter ($\Delta$), the specific heat ($C$), and the thermodynamic critical field ($H_{c}$) determine the values of the dimensionless ratios not deviating much from the predictions of the BCS theory: $R_{\Delta}=2\Delta\left(0\right)/ k_{B}T_{c}=3.68$, $R_{C}=\Delta C\left(T_{c}\right)/ C^{N}\left(T_{c}\right)=1.69$, and $R_{H}=T_{c}C^{N}\left(T_{c}\right)/ H^{2}_{c}\left(0\right)=0.171$. This result is associated with the reduction of the strong-coupling and the retardation effects by the high value of the Coulomb pseudopotential. It has been shown that the results of the Eliashberg formalism can be relatively precisely reproduced with the help of the semi-analytical formulas, if the value of $\mu^{\star}$ is determined on the basis of the $T_{c}$-Allen-Dynes expression ($\mu^{\star}_{AD}=0.198$). The attention should be paid to the fact that in the numerical and in the semi-analytical approach the comparable values of the thermodynamic parameters for the same $\mu^{\star}$ have been obtained only in the vicinity of the point $\mu^{\star}=0.1$.