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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
| ![\begin{displaymath}
{\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} \, ,\end{displaymath}](img100.gif) |
(29) |
where
,
is the coherence length and
is an effective
coherence length related to the electron mean-free path l
through the equation
| ![\begin{displaymath}
\frac{1}{\xi} = \frac{1}{\xi_0} + \frac{1}{\alpha l} \, ,\end{displaymath}](img103.gif) |
(30) |
where
is a constant on the order of unity. For a pure
superconductor,
.The response of the superconductor to the applied magnetic field is
nonlocal, in the sense that the value of
measured at a point
depends on the value of
throughout a volume of radius
surrounding the point
.The Pippard kernel relating the kth Fourier component of
to
the vector potential
can be determined
from Eq. (2.30), and is given by
| ![\begin{displaymath}
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\} \, .\end{displaymath}](img107.gif) |
(31) |
QP (k) is always smaller than the London kernel QL (k). As a
result, substituting the expression for QP (k) into Eq. (2.5)
will always yield a value for
which is larger than
.In particular, for
| ![\begin{displaymath}
\lambda = \lambda_L \left( \frac{\xi_0}{\xi} \right)^{1/2} \, ,\end{displaymath}](img109.gif) |
(32) |
and for
| ![\begin{displaymath}
\lambda = \left( \frac{\sqrt{3}}{2 \pi} \xi_0 \lambda_L^2 \right)^{1/3} \, .\end{displaymath}](img111.gif) |
(33) |
Note that in the first limiting case,
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
| ![\begin{displaymath}
\xi_0 = a \frac{\hbar v_f}{k_B T_c} \, ,\end{displaymath}](img112.gif) |
(34) |
where vf is the Fermi velocity and
.In BCS theory, the response kernel
is similar to
that in Pippard theory. The BCS coherence length
, is the range of the Fourier transform of
and is defined as
| ![\begin{displaymath}
\xi_0 = \frac{\hbar v_f}{\pi \Delta_0} \, ,\end{displaymath}](img114.gif) |
(35) |
where
is the uniform energy gap at
. Since
, the Pippard estimate is close to the
BCS coherence length.
In a conventional superconductor, vf and hence
is large.
The bound electron pairs which make up the superfluid have a
spatial extent on the order of
.
The high-Tc 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-Tc compounds are generally in the clean limit, where
. Furthermore, both
fluctuation and boundary effects are much stronger in these short
superconductors.
In a dx2-y2-wave superconductor where the energy gap is
anisotropic, one must define an angle dependent coherence length
| ![\begin{displaymath}
\xi_0 ({\bf \hat{k}}) = \frac{\hbar v_f}{\pi \Delta_{{\bf \hat{k}}}} \, ,\end{displaymath}](img117.gif) |
(36) |
where
and
is the maximum value of the energy gap.
The divergence
of Eq. (2.37) along the node directions
means that the extreme nonlocal
limit is obtained near the nodes (i.e.
).
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
will be discussed later in
the thesis.
Next: 2.3 The Magnetic Field
Up: 2 The Characteristic Length
Previous: 2.1.3 Penetration Depth for dx-y-Wave