Superconducting coherence length

The superconducting coherence length is one of two parameters in the Ginzburg–Landau theory of superconductivity. It is given by:

${\displaystyle \xi =\sqrt{\frac{{\hslash }^2}{2m|\alpha (T)|}}}$

where ${\displaystyle \alpha (T)}$ is a parameter in the Ginzburg–Landau equation for ${\displaystyle \psi }$ with the form ${\displaystyle {\alpha }_0(T-T_c)}$, where ${\displaystyle {\alpha }_0}$ is a constant.

In Landau mean-field theory, at temperatures ${\displaystyle T}$ near the superconducting critical temperature ${\displaystyle T_c}$, ${\displaystyle \xi (T)\propto (1-T/T_c{)}^{-\frac{1}{2}}}$. Up to a factor of ${\displaystyle \sqrt{2}}$, it is equivalent to the characteristic exponent describing a recovery of the order parameter away from a perturbation in the theory of the second order phase transitions.

In some special limiting cases, for example in the weak-coupling BCS theory of isotropic s-wave superconductor it is related to characteristic Cooper pair size:

${\displaystyle {\xi }_{BCS}=\frac{\hslash v_f}{\pi \mathrm{\Delta }}}$

where ${\displaystyle \hslash }$ is the reduced Planck constant, ${\displaystyle m}$ is the mass of a Cooper pair (twice the electron mass), ${\displaystyle v_f}$ is the Fermi velocity, and ${\displaystyle \mathrm{\Delta }}$ is the superconducting energy gap. The superconducting coherence length is a measure of the size of a Cooper pair (distance between the two electrons) and is of the order of ${\displaystyle {10}^{-4}}$ cm. The electron near or at the Fermi surface moving through the lattice of a metal produces behind itself an attractive potential of range of the order of ${\displaystyle 3\times {10}^{-6}}$ cm, the lattice distance being of order ${\displaystyle {10}^{-8}}$ cm. For a very authoritative explanation based on physical intuition see the CERN article by V.F. Weisskopf.

The ratio ${\displaystyle \kappa =\lambda /\xi }$, where ${\displaystyle \lambda }$ is the London penetration depth, is known as the Ginzburg–Landau parameter. Type-I superconductors are those with ${\displaystyle 0<\kappa <1/\sqrt{2}}$, and type-II superconductors are those with ${\displaystyle \kappa >1/\sqrt{2}}$.

In strong-coupling, anisotropic and multi-component theories these expressions are modified.