Superconducting coherence length

In superconductivity, the superconducting coherence length, usually denoted as ${\displaystyle \xi }$ (Greek lowercase xi), is the characteristic exponent of the variations of the density of superconducting component.

The superconducting coherence length is one of two parameters in the Ginzburg–Landau theory of superconductivity. It is given by:[1]

${\displaystyle \xi ={\sqrt {\frac {\hbar ^{2}}{2m|\alpha |}}}}$

where ${\displaystyle \alpha }$ is a constant in the Ginzburg–Landau equation for ${\displaystyle \psi }$ with the form ${\displaystyle \alpha _{0}(T-T_{c})}$.

In Landau mean-field theory, at temperatures T near the superconducting critical temperature T_c , ξ(T) ∝ (1-T/T_c)^−1/2. Up to a factor of ${\displaystyle {\sqrt {2}}}$, it is equivalent 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:[2]

${\displaystyle \xi _{BCS}={\frac {\hbar v_{f}}{\pi \Delta }}}$

where ${\displaystyle \hbar }$ 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 ${\displaystyle \Delta }$ is the superconducting energy gap.

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 [3].