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Since the high-*T*_{c} compounds are extreme type-II superconductors,
they should be well described by the London theory.
The underlying assumption in the linear relation between
the supercurrent density and the vector potential
that appears in the London model is that
the applied field is both weak and slowly varying over a distance .
This in turn implies that the wave functions of the
superconducting carriers are independent of magnetic field *H* and
that scales exactly with the velocity of the superfluid [see Eq. (2.7)].

Let us first consider an *s*-wave superconductor in the
Meissner state for the case where , so that there
is no contribution to from thermally excited quasiparticles
[*i.e.* the second term in Eq. (2.22) is zero].
If a static magnetic field is then applied to the BCS ground state,
each electron with initial momentum will have a new
momentum given by

| |
(37) |

The other electron in a pair has momentum , so
that the new momentum of these electrons is simply
| |
(38) |

The net effect is to shift the entire Fermi circle in -space
as shown in Fig. 2.3.
The change in kinetic energy of an electron pair is

| |
(39) |

Using Eq. (2.7) and Eq. (2.18) and summing
up the contribution from all Cooper pairs gives the total increase
in kinetic energy due to the applied magnetic field as
| |
(40) |

This is just the classical equation for *n*_{s} electrons per unit volume
moving with a velocity .
Thus, the supercurrents
are determined solely by the applied field, and only exist in its
presence. Provided the displacement of the Fermi circle does not
exceed the width of the energy gap [as in Fig. 2.3(a)],
the supercurrent response will
remain linear. However, if the applied field is large enough, the shift
will exceed this limit and quasiparticle excitations will occur. This latter
case is illustrated in Fig. 2.3(b).
Naturally, this will lead to a reduction in *n*_{s} and a corresponding
increase in . Near the Fermi surface, a quasiparticle
has kinetic energy . The change in kinetic energy
of the quasiparticle due
to the superflow velocity is
| |
(41) |

so that there is a shift in the quasiparticle excitation spectrum such that
| |
(42) |

The shift in quasiparticle energy relative to the energy gap ,can be used to define a critical velocity for *v*_{s}, below which
the supercurrent response is linear in the
Meissner state of an *s*-wave superconductor
| |
(43) |

Thus, if then . For
velocities greater than *v*_{c}, there is a quasiparticle contribution
such that the supercurrent density is
| |
(44) |

where is the reduced value of the superfluid density
(relative to its maximum value *n*_{s} in the linear regime).
Thus drop rapidly when so that the supercurrent response becomes nonlinear.
If the energy shift is larger than the condensation energy (which will
be derived later), the sample will become normal.
For , the size of the energy gap is reduced, so
that the value of *v*_{c} decreases.
The precise form of the supercurrent-velocity relation
of Eq. (2.45) is [30]

| |
(45) |

where the prefactor is
a *T*-dependent coefficient that
decreases rapidly with
*T*, with as .At low temperatures, .Using Eq. (2.46), the nonlinear London equations can be
constructed to determine the field dependence of .It is found that for a conventional *s*-wave superconductor at
low *T*, changes quadratically with *H*
^{}
| |
(46) |

where is
a characteristic field on the order of the
thermodynamic critical field *H*_{c}(*T*), which is of course related to *v*_{c}.
Above *H*_{0} (*T*) the nonlinear effects become important.
The *H*^{2} behaviour in Eq. (2.47) has been
confirmed experimentally in the
type-I superconductors Sn and In [31,32,33]
and the type-II superconductor V_{3}Si [34].
The coefficient in Eq. (2.47)
is typically small in conventional superconductors.

** Next:** 2.3.2 Nonlinear Effects in d_{x-y}-Wave
** Up:** 2.3 The Magnetic Field
** Previous:** 2.3 The Magnetic Field