The vortex lattice in a type-II superconductor results in a spatially
varying magnetic field
inside the superconductor. Transverse
-field muon spin rotation (TF-SR) can be utilized to investigate
the local magnetic field distribution function [72,77]:

The function

The field distribution pertaining to a perfect vortex lattice free of distortions resembles that of Fig. 3.1. The magnetic lineshape exhibits van Hove singularities corresponding to:

A: the minimum value of
which occurs at the center of the
vortex lattice.

B: the saddle point value of
which occurs midway between
two vortices.

C: the maximum value of
which occurs at the vortex cores.

The average local magnetic field
strength inside the superconductor is denoted
.
The spatial average of *B* is just the *first moment* of
*n*(*B*) [73]:

where

If the applied field
then the London model is valid. At
such fields the spacing *L* between adjacent vortices is large compared
to the coherence length. In the
London theory, the local magnetic field
in an isotropic superconductor is given as [68]:

where the sum extends over the reciprocal lattice vectors of the vortex lattice. Combining (3.4) and (3.5), the second moment of

It is clear from Eq. (3.6) that one can estimate the penetration depth from the second moment , so that is inherently related to the line width of the measured field distribution. Eq. (3.6) is valid at intermediate fields where the second moment and hence is independent of the applied field [34,74]. At such fields the inter-vortex spacing [66]. Rammer argues that in the case of , Eq. (3.6) is useful only for fields significantly smaller than 50kG (

This result was derived for a triangular flux-line lattice. According to Eq. (3.7), increasing the applied field in this region leads to a corresponding increase in the second moment. This statement of course assumes that the penetration depth is a field independent quantity.

At high fields the second moment decreases with increasing applied field *H*as the vortex cores start to overlap. The second moment at such fields can
be estimated by using Abrikosov's numerical solution of the GL-equations
for
:

where and , with the lower limit given by [34] and the upper limit given by [74]. London theory is not valid in this high-field regime, where the inter-vortex spacing of the lattice is comparable to [

In London theory, where
is assumed to be zero,
the local magnetic field in the center of a vortex is
not finite. To get around this divergence, Brandt [34,72]
approximates the vortex core by a gaussian function of width ,
which provides a dome-shaped peak for the field. An
additional term (1-*b*) can be incorporated to roughly account for the
field dependence of the superconducting order parameter [76].
As in Eq. (3.8) this term is significant for applied
fields close to the upper critical field *H*_{c2}, leading to a
noticeable reduction in the width of the magnetic lineshape. Including
this effect, the vortex core can be approximated by a gaussian function
of width
.
Thus, according to Brandt
[34,72,76],
one can modify Eq. (3.5) so that the local magnetic field
of an isotropic superconductor for applied fields
*H* < *H*_{c2}/4 is

The second exponential in Eq. (3.9) introduces an upper
cutoff for the reciprocal lattice vectors at
[77]. This value emerged from Brandt's numerical solution
of the GL-equations [34,72]. The upper cutoff for leads to a finite value of the local field *B* at
the vortex core. Fig. 3.2
illustrates the field distribution
as determined
by Eq. (3.9) for the average internal magnetic fields
=5kG and 15kG. As the field is increased, the distance
between vortex cores decreases. Also the difference between the
maximum and minimum field in the distribution decreases with increasing
applied field. Fig. 3.3 shows the effects on the field
distribution of increasing the magnetic penetration depth ,
while Fig. 3.4 illustrates the dependence on the
GL-parameter .

The variation of the supercurrent density
with
position
from the vortex centre can be determined by
substitution of Eq. (3.9) into Eq. (2.3).
Fig. 3.5 illustrates the dependence of the absolute value
of the supercurrent density distribution on magnetic field.
The supercurrent density is zero at the centre of a vortex core, corresponding
to the maximum in the field distribution; *J*_{s} rises steeply to its
maximum value at the edge of the vortex core and then falls off
exponentially. The singularity observed halfway between two vortices
denotes a change in the flow direction of the supercurrents.

Fig. 3.8 shows the dependence of the average value of the supercurrent density on the average field for . As the applied field ( ) is increased above the lower critical field, the average supercurrent density rises steeply. This is because the supercurrents which screen the vortex cores have less room to spread out at higher fields when the cores are closer together. However, as shown in Fig. 3.8, the relationship is not linear. This is due to the overlapping of vortices, which tends to reduce . As the field is increased, the inter-vortex separation decreases. Eventually the overlapping of vortices dominates the contribution to the average supercurrent density and the curve turns over, so that decreases with further increase in the applied magnetic field. Interestingly, the turnover in Fig. 3.8 occurs around 50kG; the maximum field at which Rammer claims Eq. (3.6) is valid [32]. The dependence of the supercurrent distribution on and can be seen in Fig. 3.6 and Fig. 3.7, respectively.

As mentioned in the previous chapter, the orientation of the applied field
with respect to the crystallographic axes
of the superconductor is important. For an applied field
parallel to the c-axis, the second moment of *n*(*B*) valid at intermediate
fields can be written as [66]

and for an applied field parallel to the

In the latter case, the supercurrents flow across the

The peak in the measured SR field distribution

2001-09-28