The Coherence Length (Pippard's Equation)

The London equations whose derivation appears elsewhere (see [13] for instance), are local equations (i.e. they relate the current density at a point $\vec{r}$ to the vector potential $\vec{A}$ at the same point), and hence define the superconducting properties as such. However, early discrepancies between experimental estimations of $\lambda_{L}$(0) (the penetration depth at zero temperature) for certain conventional superconductors, and those predicted by Eq. (2.1) led Pippard [13,15] in 1950 to introduce non-local effects into the London equations. Spatial changes of quantities such as n_s in a superconductor may only occur on a finite length scale, the coherence length $\xi_{\circ}$, and not over arbitrarily small distances. That is to say, whenever n_s is varying in space its value may change significantly over distances of order $\xi_{\circ}$. Thus the coherence length defines the intrinsic nonlocality of the superconducting state. Using the uncertainty principle, Pippard estimated the coherence length for a pure metal to be [17,18]:

$$\xi_{\circ}\approx \frac{\hbar v_{F}}{\pi \Delta(0)} \approx 0.18 \frac{\hbar v_{F}}{k_{B} T_{c}}$$ (4)

where k_B is Boltzmann's constant, and $\Delta (0)$ is the energy gap that forms at the Fermi surface in the superconducting state at absolute zero. This will be discussed in more detail later.

In the London model, the local value of the magnetic flux density $B(\vec{r})$and the supercurrent density $J(\vec{r})$ are assumed to vary slowly in space on a length scale $\lambda_{L}$, and have negligible variation over distances $\sim \xi_{\circ}$. In Fig. 2.2, the superconducting carrier density n_sjumps discontinuously from zero to a maximum value at the sample surface. As already mentioned, n_s is expected to make a significant change like this only over a distance on the order of $\xi_{\circ}$. Thus in Fig. 2.2, $\xi_{\circ}= 0$, so that one would anticipate an exponential screening of the magnetic field in a real superconductor only when $\lambda_{L}\gg \xi_{\circ}$. Indeed, the London model is valid only for $\lambda_{L}\gg \xi_{\circ}$.

In a type-I superconductor, the magnetic penetration depth $\lambda_{L}$ is much less than the coherence length $\xi_{\circ}$, so that n_s does not reach its maximum value near the surface. That is to say, not all of the electrons within a thickness $\xi_{\circ}$ from the surface of the superconductor contribute to the screening currents. Consequently, the penetration of the magnetic field does not follow the exponential form of Eq. (2.2) until one ventures an appreciable distance $x \sim \xi_{\circ}$ into the superconductor. Thus the usual London model is inadequate in describing type-I superconductors where, $\xi_{\circ}\gg \lambda_{L}$. A modification of the London equations which gives the magnetic flux and supercurrent densities a more rapid variation in space, predicts the actual penetration depth $\lambda$ in a type-I superconductor to be [14]:

$$\lambda = (0.62 \lambda_{L}^{2} \xi_{\circ})^{1/3},\ \ \ (\xi_{\circ}\gg \lambda_{L})$$ (5)

where $\lambda$ is larger than $\lambda_{L}$, but much smaller than $\xi_{\circ}$.