Why magnesium diboride is not described by anisotropic Ginzburg-Landau theory

TL;DR

This paper explains why the anisotropic Ginzburg-Landau theory fails for MgB₂ due to its unique band structure, and derives simplified equations for dirty $\pi$-band superconductors near the critical temperature.

Contribution

It demonstrates the breakdown of anisotropic Ginzburg-Landau theory in MgB₂ and derives new equations for dirty $\pi$-band superconductors near $T_c$.

Findings

Anisotropic Ginzburg-Landau theory is not applicable to MgB₂.

Derived simplified equations for dirty $\pi$-band superconductors near $T_c$.

Identified the limitations of gradient expansion in the $c$ direction.

Abstract

It is well established that the superconductivity in the recently discovered superconducting compound MgB$_{2}$ resides in the quasi-two-dimensional band ($\sigma$-band) and three-dimensional band ($\pi$-band). We demonstrate that, due to such band structure, the anisotropic Ginzburg-Landau theory practically does not have region of applicability, because gradient expansion in the $c$ direction breaks down. In the case of dirty $\pi$-band we derive the simplest equations which describe properties of such superconductors near $T_{c}$ and explore some consequences of these equations.