2.2 The Superconducting Coherence Length

To explain numerous experimental results which deviated from the predictions of the London theory, Pippard [28] proposed the following general nonlocal relation for the supercurrent response  

$${\bf J_s}({\bf r}) = -\frac{3}{\xi_0} \left( \frac{c}{4 \pi... ...f A} ({\bf r}^{\prime})]}{R^4} \exp(-R/\xi) d^3 r^{\prime} \, ,$$ (29)

where ${\bf R} \! = \! {\bf r} - {\bf r}^{\prime}$,$\xi_0$ is the coherence length and $\xi$ is an effective coherence length related to the electron mean-free path l through the equation

$$\frac{1}{\xi} = \frac{1}{\xi_0} + \frac{1}{\alpha l} \, ,$$ (30)

where $\alpha$ is a constant on the order of unity. For a pure superconductor, $\xi \! = \! \xi_0$.The response of the superconductor to the applied magnetic field is nonlocal, in the sense that the value of ${\bf J_s}$measured at a point ${\bf r}$depends on the value of ${\bf A}$ throughout a volume of radius $\xi$surrounding the point ${\bf r}$.The Pippard kernel relating the kth Fourier component of ${\bf J_s}$ to the vector potential ${\bf A}(k)$ can be determined from Eq. (2.30), and is given by

$$Q_P (k) = \frac{\xi}{\xi_0 \lambda_L^2} \left\{ \frac{3}{2(k \xi)^3} [(1+k^2 \xi^2) \tan^{-1} (k \xi) - k \xi] \right\} \, .$$ (31)

Q_P (k) is always smaller than the London kernel Q_L (k). As a result, substituting the expression for Q_P (k) into Eq. (2.5) will always yield a value for $\lambda$ which is larger than $\lambda_L$.In particular, for $\xi \! \ll \! \lambda$

$$\lambda = \lambda_L \left( \frac{\xi_0}{\xi} \right)^{1/2} \, ,$$ (32)

and for $\xi \! \gg \! \lambda$

$$\lambda = \left( \frac{\sqrt{3}}{2 \pi} \xi_0 \lambda_L^2 \right)^{1/3} \, .$$ (33)

Note that in the first limiting case, $\lambda$ agrees with the London prediction in a pure superconductor. On the other hand, the second limiting case is completely independent of the electron mean-free path. A superconductor described by the first equation, is called a ``type-II superconductor'', whereas one that is described by the second equation is a ``type-I superconductor''.

Using an argument based on the uncertainty principle, Pippard estimated that the coherence length in a pure metallic superconductor is

$$\xi_0 = a \frac{\hbar v_f}{k_B T_c} \, ,$$ (34)

where v_f is the Fermi velocity and $a \! = \! 0.15$.In BCS theory, the response kernel $Q ({\bf k})$ is similar to that in Pippard theory. The BCS coherence length $\xi_0$, is the range of the Fourier transform of $Q ({\bf k})$and is defined as

$$\xi_0 = \frac{\hbar v_f}{\pi \Delta_0} \, ,$$ (35)

where $\Delta_0$ is the uniform energy gap at $T \! = \! 0$. Since $\Delta_0 \! \sim \! k_B T_c$, the Pippard estimate is close to the BCS coherence length. In a conventional superconductor, v_f and hence $\xi_0$ is large. The bound electron pairs which make up the superfluid have a spatial extent on the order of $\xi_0$.

The high-T_c materials differ markedly from conventional superconductors in that they have much smaller coherence lengths. Consequently, these materials are in the extreme type-II limit. The small value of the coherence length also means that the high-T_c compounds are generally in the clean limit, where $\xi \! \ll \! l$. Furthermore, both fluctuation and boundary effects are much stronger in these short $\xi$ superconductors. In a d_x²-y²-wave superconductor where the energy gap is anisotropic, one must define an angle dependent coherence length  

$$\xi_0 ({\bf \hat{k}}) = \frac{\hbar v_f}{\pi \Delta_{{\bf \hat{k}}}} \, ,$$ (36)

where $\Delta_{{\bf \hat{k}}} \! = \! \Delta_0 (\hat{k}_x^2 - \hat{k}_y^2)$and $\Delta_0$ is the maximum value of the energy gap. The divergence of Eq. (2.37) along the node directions $\vert\hat{k}_x\vert \! = \! \vert\hat{k}_y\vert$ means that the extreme nonlocal limit is obtained near the nodes (i.e. $\xi_0 \! \rightarrow \! \infty$). Recently, Kosztin and Legget [29] determined that nonlocal electrodynamics leads to a crossover from a T to a T² dependence for the penetration depth in the Meissner state at extremely low temperatures. To observe this experimentally, one must distinguish this effect from the T² dependence expected from sample impurities. Both the temperature and magnetic field dependence of $\xi$ will be discussed later in the thesis.