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For a *d*_{x2-y2}-wave superconductor,
the presence of nodes on the Fermi surface means that the supercurrent
response to a weak applied magnetic field will be nonlinear even at
. This is clearly seen in Fig. 2.4, where,
due to the nodes, quasiparticle excitations will result from even a
small displacement of the Fermi cylinder. For a given shift, the
precise number of quasiparticle excitations will depend on the
slope of the energy gap function at the
nodes and the direction of .
The excited quasiparticles located in a narrow wedge at the nodes
produce a current density which flows in a direction opposite
to that of the superfluid.

For the case in which is directed along
a node, as shown in Fig. 2.4(a), the supercurrent-velocity
relation is

| |
(47) |

where ,
such that ,
is the angular slope of the energy gap at the node, and *v*_{F}^{*} is
the Fermi velocity at the node.
For the case in which is directed along
an antinode, as shown in Fig. 2.4(b), the supercurrent
density is
| |
(48) |

The additional factor of is easy to understand
by comparing the angular size of the wedges at the nodes in
Fig. 2.4(b) to the angular size of the wedge in
Fig. 2.4(a).
Due to the anisotropy of the nonlinear response,
Yip and Sauls [30]
proposed that the field dependence of the penetration depth in the
Meissner state could be used to resolve the structure of the energy
gap in a superconductor.
The magnetic penetration depth can be derived using the expressions
for . The result is that
changes linearly with *H*
at low *T* [30,35,36]
| |
(49) |

where and is a temperature dependent coefficient which
remains finite at due to the nodes in the gap.
The actual value of will of course depend on the
direction of . The definition of in
Eq. (2.50) is the same as that in
Eq. (2.47) (*i.e.* it is related to the initial decay rate
of the field).
As the temperature is increased,
there is eventually a crossover to a situation in which thermal excitation of
quasiparticles also occurs away from the nodes.
Below this crossover temperature *T*^{*}(*H*),
is linear in *H* but quadratic
in *T*, whereas above *T*^{*}(*H*), is quadratic
in *H* and linear in *T* [36].
The first evidence for a linear
*H*-dependence accompanied with a *T*^{2}-dependence
in a high-*T*_{c} material, was obtained by
Maeda *et al.* [37] for measurements of the in-plane
magnetic penetration depth in Bi_{2}Sr_{2}CaCu_{2}O_{y}.
Similar results have since been reported in YBa_{2}Cu_{3}O and
Tl_{2}Ba_{2}CaCu_{2}O_{y} [38]. However, the results of
these experiments are suspect because of the large demagnetization effect
(associated with the shape of the sample) which arises from
applying a magnetic field perpendicular to the flat
- plane.
Early measurements of in a single crystal
of YBa_{2}Cu_{3}O_{6.95} found a large *H*^{2} term [39],
but the sample had a reduced *T*_{c}
indicating there may have been extrinsic effects due to impurities.

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** Up:** 2.3 The Magnetic Field
** Previous:** 2.3.1 Nonlinear Effects in