In this chapter, recent SR measurements of the
- plane magnetic penetration depth and the vortex core radius *r _{0}* in the conventional
type-II superconductor NbSe

Figures 6.1 and 6.2 show
the Fourier transforms of the muon precession
signal in NbSe_{2} as functions of temperature and applied magnetic
field, respectively.
The horizontal axes are in terms of the internal
magnetic field *B* relative to the average field of the background
signal , which by definition is centered at 0 G. As the temperature
or magnetic field is lowered, the line shape broadens and the high-field
tail becomes longer due mainly to a decrease
in . The high-field cutoff is clearly visible in all of the
measured line shapes for NbSe_{2}. This implies that the vortex cores
occupy a significant volume of the sample. This fractional volume
depends on both the size of the vortex cores and the areal density
of vortices.

In order to test the effects of the analysis procedure on the determined behaviour of and , three different models for the theoretical internal field distribution corresponding to the vortex lattice were considered:

- 1.
- the modified London (ML) model given in Eq. (4.10) with a Gaussian cutoff factor .
- 2.
- the ML model with a Lorentzian cutoff factor .
- 3.
- the analytical GL model given in Eq. (4.13).

Figure 6.3 shows the magnetic field dependence of
at (*i.e.* K)
obtained by fitting the SR time spectra with a polarization function
which assumes one of the three models for the field distribution due
to the vortex lattice.
From magnetization measurements, T, so that
the results extend over the field range ,where . A clear linear *H*-dependence for
is obtained for
all three types of analysis, although there is some difference in the
absolute value of and the strength of the linear
term. The solid lines in Fig. 6.3 are fits to the linear relation

(1) |

Model for | [Å] | |
---|---|---|

ML: Gaussian Cutoff |
1659(1) | 1.85(4) |

ML: Lorentzian Cutoff |
1398(2) | 0.81(3) |

Analytical GL Model |
1323(2) | 1.62(3) |

Figure 6.4 shows the magnetic field dependence of the quality of the fits at , obtained for the three different models. The ratio of to the number of degrees of freedom (NDF) is significantly greater than 1.0 in most cases due to the high statistics of the measured magnetic field distribution. For a non-perfect fit, higher statistics magnify the value of .Fits to the ML model with a Gaussian cutoff generally yield the worst value. On the other hand, fits assuming a Lorentzian cutoff are only slightly better than fits to the analytical GL model.

Figure 6.5 shows, in the frequency domain, how the quality of the
fits obtained (in the time domain) from the ML model using a
Gaussian cutoff factor and from the analytical GL model are virtually
indistinguishable. One would expect the results from these two models
to converge at higher magnetic fields. However, as shown
in Fig. 6.3,
determined for the two different models appear
to diverge slowly at high *H*.
The reason is that the analytical GL model deviates significantly
from the exact numerical GL solutions at high reduced
fields [121], and also, according to Brandt [123],
the ML model is really only applicable when .

Despite the quantitative differences between the three
phenomenological models, which is related to their validity in
different field ranges, the finding of a linear-*H* dependence for
is common to all.
Since the analytical GL model properly accounts for the finite size of the
vortex cores and our measurements are taken essentially at low
reduced fields (especially in the case of YBa_{2}Cu_{3}O which
we consider later) the results obtained using this model should
most faithfully reflect the behaviour of the fundamental length scales.
Unless otherwise stated, results presented in the remainder of
this thesis were obtained assuming this model.

Figure 6.6 shows a comparison between in NbSe_{2}
at two different temperatures. A linear-*H* dependence is observed
between and 0.6 *T*_{c}.
The field dependence at lower *T* was not investigated because
the ^{4}He gas flow cryostat limited us to temperatures
above K.
In the Meissner state of a conventional *s*-wave superconductor,
is expected to increase
quadratically as a function of magnetic field, due to nonlinear effects.
The nonlinear corrections to the supercurrent response
are the same in both the Meissner and vortex states.
However, the average supercurrent density
scales quite differently in the
Meissner and vortex states, as shown in Fig. 6.7. The
curve in the top panel of Fig. 6.7 (*i.e.* the Meissner state)
was generated
assuming that the magnetic field decays exponentially
[see Eq. (2.15)] and that is field independent.
Thus in the Meissner state, .It follows that if , then
.

The solid curve in the bottom panel of Fig. 6.7
(*i.e.* the vortex state)
was generated with the field profile from the analytical
GL model. The dashed curve in this figure shows that the average
supercurrent density in the vortex state is approximately proportional to
. Thus, if
and
(as measured here), then as in the Meissner
state .This suggests that the field dependence of in the
vortex state of NbSe_{2} is due to nonlinear effects.
However, measured in our SR experiment is by definition
not the same as the penetration depth which appears in the nonlinear theory
or which is measured in the Meissner state.
Relating from the
nonlinear theory to the effective
measured by SR is nontrivial and
requires a proper account of the vortex source term.

In the vortex state, the strength of the term which is linear in *H* is
almost the same at both temperatures considered, when normalized with
respect to
the value of *H*_{c2} (*T*) (see parameter
in Table 6.3).
As the temperature is increased, the energy gap in the quasiparticle
excitation spectrum shrinks, leading to the thermal
excitation of quasiparticles.
The reduction in the size of the energy gap also means that quasiparticles
can be excited by relatively smaller magnetic fields.
For this reason, in the Meissner state, the strength of the term
quadratic in *H*
is found to increase with increasing *T*.
Since the strength of the coefficient for the term linear in *H* in
Eq. (6.1)
does not appear to change over a large range of temperature
in the vortex state, it seems unlikely that the mechanism responsible
for the nonlinear Meissner effect can be solely responsible for the
observed
*H*-dependence of in the vortex state.
Furthermore, according to the calculations of Amin *et al.* [40],
it seems unlikely that nonlinear corrections to the supercurrent response
in the vortex state can result in a field dependence for the effective
penetration depth measured by SR which is as strong
as that found here. However, as just mentioned, the calculation of the
effective is rather sensitive to the vortex source term,
so that the size of the vortex cores should be included in such calculations.

Equation | 2c| | 2c| | ||

[Å] | [Å] | |||

1323(2) | 1.62(3) | 1436(3) | 1.56(2) | |

8.4 (2) | 7.4 (2) | 5.7(2) | 8.2(3) | |

6.9(2) | 9.5(3) | 5.1(2) | 10.2(4) | |

Since is not the only parameter which contributes
to the fitted SR line width, it is necessary to monitor the behaviour
of the additional broadening parameter . Besides
disorder in the vortex lattice, the large ^{93}Nb nuclear moments
() in NbSe_{2}
also slightly broaden the SR line shape.
For instance, in the normal state the muon depolarization rate at K is found to increase linearly
from at T to at
T. To determine the degree of disorder
in the vortex lattice, the contribution of the ^{93}Nb
nuclear moments to the muon depolarization rate
can be subtracted in quadrature from the fitted value of

(2) |

(3) |

For a perfect triangular vortex lattice the intervortex spacing is given by

(4) |

The radius of a vortex core is not a uniquely defined quantity, since
there exists no sharp discontinuity between a normal vortex core and the
superconducting material. Nevertheless, a useful definition can be made
taking into account the dramatic spatial changes observed in quantities
such as the order parameter , the local density of
states ,
the supercurrent
density and the local magnetic field strength near the center of a vortex line.
Since the supercurrent density can be easily obtained
from the fitted field profile through the Maxwell relation
, we define an effective core radius *r _{0}* to be
the distance from the vortex center for which

The magnetic field dependence of *r _{0}* in NbSe

Golubov and Hartmann [205] have shown that the shrinking
of the vortex core radius with increasing magnetic field can be
attributed to increased vortex-vortex interactions. They solved
the microscopic equations in the dirty limit (*i.e.* the Usadel
equations) self-consistently and showed that the order parameter
and the LDOS reach their maximum
values closer to the vortex
center when *H* is increased. From the LDOS, these authors calculated
tunneling current *I*(*r*) profiles from the vortex center as a
function of *H* in order to model STM measurements of *r _{0}* (

The authors of Ref. [205] reported good agreement between
the STM measurements of *r _{0}* (

(5) |

The effective coherence length in
Eq. (4.13) which best fits the data is
plotted in Fig. 6.13 as a function of magnetic field.
The variation of is similar to that of *r _{0}*(

Assuming that the shrinking of the cores is associated with the strength of
the vortex-vortex interactions,
the increase in *r _{0}* and with decreasing magnetic field
should saturate when the vortices are sufficiently far apart (

A few remarks are now necessary with regard to
the behaviour of and
in Fig. 6.14.
The behaviour of implies that NbSe_{2} becomes more type-II like
with increasing magnetic field.
In GL theory, is independent
of both *H* and *T*. However, this definition
is strictly valid only near the superconducting-to-normal phase
transition. Our results imply that the conventional GL equations
with field-independent and are not applicable
deep in the superconducting state.
Even if were field independent, an
increase of and with *H* would still
arise from the decrease of which has been independently
observed in NbSe_{2} by STM. Furthermore, attempts to fix
and to constant values
in the fitting procedure yield higher values of
and unphysical results--such as a
residual background signal which is of the total signal
amplitude.

Figure 6.15 shows a typical muon precession signal displayed for convenience in a reference frame rotating at about 1.5 MHz below the Larmor precession frequency of a free muon. The curves through the data points are examples of fits to the theoretical polarization function for fixed values of , where and all other parameters were free to vary. Only the first 3 s of data are shown in Fig. 6.15 since the signal from the vortex lattice essentially decays over this time range--although the fits were actually performed over the first 6 s. Figure 6.16 shows the difference between the data points and the fitted curve for the fits in Fig. 6.15. There is a clear oscillation for the fits corresponding to Å and Å, indicating a missed frequency or frequencies. The ratio of to the number of degrees of freedom (NDF) is shown in Fig. 6.17(a) as a function of for two different applied magnetic fields. Note that the value of for which /NDF reaches its minimum value is quite different for the two fields. Figure 6.17(b) shows the behaviour of the free parameter for these same fits. The best fits indicate that is dependent on magnetic field.

The reduction in *r _{0}* and with increasing temperature
which is shown in Fig. 6.12 and Fig. 6.13, respectively,
is expected from theoretical predictions for a

The solid line through the data in Fig. 6.18(a) is a fit to
the empirical relation with m^{-2},
and .Although this is considered to be consistent with a
weak-coupling BCS superconductor [1]
(which shows a *T*-dependence of which is close
to (1-*t ^{2}*)), there is no real low temperature data to obtain
a proper fit to the BCS expression of Eq. (2.27).
From the empirical fit and an observed weak
linear-

(6) |