In this chapter, recent SR measurements of the - plane magnetic penetration depth and the vortex core radius r0 in the conventional type-II superconductor NbSe2 are presented.
Figures 6.1 and 6.2 show the Fourier transforms of the muon precession signal in NbSe2 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 NbSe2. 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:
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
|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 , and also, according to Brandt , 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 YBa2Cu3O 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 NbSe2 at two different temperatures. A linear-H dependence is observed between and 0.6 Tc. The field dependence at lower T was not investigated because the 4He 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 NbSe2 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
the value of Hc2 (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
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
H-dependence of in the vortex state.
Furthermore, according to the calculations of Amin et al. ,
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.
|8.4 (2)||7.4 (2)||5.7(2)||8.2(3)|
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 93Nb nuclear moments () in NbSe2 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 93Nb nuclear moments to the muon depolarization rate can be subtracted in quadrature from the fitted value of
For a perfect triangular vortex lattice the intervortex spacing is given by
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 r0 to be the distance from the vortex center for which Js (r) reaches its maximum value. As shown in Fig. 6.10, Js (r) rises steeply from zero at the vortex center to its maximum value at r0.
The magnetic field dependence of r0 in NbSe2 is shown in Fig. 6.11, where r0 is obtained from Js (r) profiles created from the fitted field profiles B (r). The deduced values of r0 are less sensitive (than ) to the choice of the theoretical model for B (r). This is because a good approximation of Js (r) can be obtained by taking the curl of any function B (r) which fits the measured field distribution well. This includes an insensitivity to the assumed vortex-lattice geometry, provided a good fit is obtained. Since r0 is fairly robust to the validity of the theoretical field distribution used to fit the data, then provided a good fit is obtained, the vortex-core radius can be determined from SR with few theoretical assumptions. We note that discrepancies between the three models considered here do appear at low fields (see Fig. 6.11) because of the reduced statistics at the high-field tail of the measured internal field distribution. Since there are far fewer vortices in the sample at these low fields, there is a reduction in the number of events originating from muons which stop in the vicinity of the vortex cores. As a result, the high-field tail shows more ``statistical wiggles'', which in turn allows for a greater variation in the tail of the fitted B (r). Increasing the number of recorded muon decay events in the SR spectra at low H would rectify this problem and should lead to better agreement between the three models in Fig. 6.11 at all magnetic fields.
Golubov and Hartmann  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 r0 (H) in NbSe2 . The magnetic field dependence of r0 in NbSe2 determined by STM at is shown in Fig. 6.12 along with that determined by SR at and 0.33 Tc. The definition of r0 in the STM experiment was arbitarily chosen to be the radius at which the measured I(r) had diminished to of its maximum value at the vortex-core center. It was shown in Ref.  that this gives a value of r0 which is somewhat larger than the commonly used theoretical definition, i.e. the radius at which rises from zero at the core center to of its maximum value well away from the core. The different definitions of r0 are the main reason for the difference in magnitude of r0 between the STM and SR experiments at . Also, we found from the microscopic theory (see Ref. ) that Js (r) does not reach its maximum value at exactly the radius where reaches of its maximum value, at all temperatures and magnetic fields. For this reason, our definition of r0 is robust to changes in T and H and should better reflect the actual H-dependence of the vortex-core radius.
The authors of Ref.  reported good agreement between the STM measurements of r0 (H) and the dirty-limit microscopic theory. Although the magnitude of r0 is reasonably predicted from their calculations, the error bars in the STM measurements are too large to conclude whether there is precise agreement with the theory. In a SR experiment, typically muons sample the local magnetic field of approximately 1011 vortices in the bulk of a few mm2 sample, as opposed to an STM experiment which averages r0 from a few vortices at the surface. As a result of this statistical improvement, the SR data shown in Fig. 6.12 have smaller error bars and less scatter. This improvement allowed us to show in Ref.  that r0 decreases more strongly with magnetic field than predicted by the dirty-limit microscopic theory. This finding was not surprising since NbSe2 is in the clean limit. To our knowledge, there have yet to be any calculations of the H-dependence of r0 from the microscopic theory in the clean limit. However, in Ref.  we showed that the SR results fit well to the simple phenomenological equation
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 r0(H,T), which was shown earlier to be model independent. The curves through the data points were generated from the fitted relations for and given in Table 6.3. It should be kept in mind that must be considered an ``effective'' coherence length. For instance, according to Eq. (2.73) of the GL theory, the coherence length near Hc2 at should be 106.5 Å, where T. However, at 2.9 T the fitted curve in Fig. 6.13 gives Å. Similarly, at , Eq. (2.73) yields Å, whereas the fitted curve in Fig. 6.13 gives 78.7 Å. A reduced value of may be obtained if the fitted theoretical field distribution overestimates the length of the high-field tail in the SR line shape. However, according to Fig. 6.5, it is highly unlikely that the fits are substantially overestimating the length of the high-field tail. The discrepancy between the measured and that predicted in Eq. (2.73) is most likely due to the theoretical difference between in Eq. (4.13) and the ``true'' GL coherence length. Given that GL theory is really only valid near the phase boundary, it is reasonable that deviations occur at low T and low H.
Assuming that the shrinking of the cores is associated with the strength of the vortex-vortex interactions, the increase in r0 and with decreasing magnetic field should saturate when the vortices are sufficiently far apart (i.e. when ). From Eq. (6.4) and the fitted expressions for in Table 6.3, the field at which this saturation occurs can be estimated. In particular, at there should be no change in the size of the vortex cores below T, whereas this crossover field is T at .
A few remarks are now necessary with regard to the behaviour of and in Fig. 6.14. The behaviour of implies that NbSe2 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 NbSe2 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 r0 and with increasing temperature which is shown in Fig. 6.12 and Fig. 6.13, respectively, is expected from theoretical predictions for a s-wave vortex [140,148,150]. However, as shown in Fig. 6.18(b), the vortex core radius does not decrease as steeply with temperature as predicted by theory. The dashed line in Fig. 6.18(b) is a fit to the theory of Kramer and Pesch  [see Eq. (4.11)] where ,with Å. Part of the problem is that these theoretical calculations pertain to a single isolated vortex. Given the apparent strong influence of vortex-vortex interactions, the vortex-lattice effect should not be ignored in theoretical calculations for r0 (T). For a given magnetic field, will grow with increasing temperature [see Fig. 6.18(a)], whereas the intervortex spacing L remains constant. The strength of the vortex-vortex interactions will increase at higher field as the ratio increases, leading to additional changes in the electronic structure of the vortex cores. Since these interactions become stronger with increasing T, the difference between the measured value of r0 and that predicted for an isolated vortex core will increase monotonically with temperature.
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  (which shows a T-dependence of which is close to (1-t2)), 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-T dependence for ,the temperature dependence of the vortex-core radius is given by