9 Conclusions

The linear temperature dependence of $\lambda_{ab}^{-2}$ at low T found in the vortex state of both YBa₂Cu₃O_6.95 and YBa₂Cu₃O_6.60 provides strong support for a superconducting energy gap with lines nodes. The strength of the coefficient linear in T was shown to agree extremely well with other measurements of $\lambda_{ab}$ in the Meissner state. This agreement implies that the change in the superfluid fraction as a function of temperature is identical in both phases.

The absolute value of $\lambda_{ab}(T \! = \! 0)$ extrapolated to zero magnetic field from the $\mu$SR measurements is approximately 10-15$\%$smaller than that obtained from far infrared measurements of $\lambda_{ab}(T \! = \! 0)$ in zero field. The difference is perhaps reasonable given the very different nature of the $\mu$SR and far infrared methods. Although the absolute magnitude of $\lambda_{ab}$ is sensitive to the assumed theoretical model for the field distribution of the vortex lattice, the temperature dependence is relatively independent of the choice of model. It is possible that the presence of vortices also influences the absolute value of $\lambda_{ab}$ in the vortex state. The ratio of the penetration depths for the underdoped and optimally doped samples determined by $\mu$SR is comparable to that found from far infrared measurements on similar crystals.

The magnetic field dependence of $\lambda_{ab}$ measured here in the vortex state of both NbSe₂ and YBa₂Cu₃O$_{7-\delta}$ is not completely understood. The field dependence of $\lambda_{ab}$ in NbSe₂ may be due to nonlinear effects, since it is found that $\langle J_s \rangle \! \propto \! \sqrt{\lambda_{ab}}$--which is the same relation for nonlinear effects in the Meissner state. However, according to the calculations in Ref. [40], the nonlinear effects are only a small correction to the supercurrent response and therefore cannot account for the size of the field dependence (measured here) in the vortex state of NbSe₂. It is possible that the assumption of a small vortex-core radius by the authors in Ref. [40] affected their calculations of the effective penetration depth measured by $\mu$SR. The field dependence of $\lambda_{ab}$ in YBa₂Cu₃O$_{7-\delta}$ is likely predominantly due to the nonlocal effects associated with nodes on the Fermi surface, as outlined in Ref. [40]. However, more detailed measurements of the field dependence on untwinned samples of YBa₂Cu₃O$_{7-\delta}$ may be required to confirm this. The stronger field dependence measured in YBa₂Cu₃O$_{7-\delta}$ (relative to that in NbSe₂) could be explained by this additional effect alone--although, the effects of a nonlinear supercurrent response are also expected to be stronger in a superconductor with nodes on the Fermi surface than in a conventional superconductor.

In NbSe₂, $\lambda_{ab}$ was found to increase linearly with increasing field. The precise form of the field dependence of $\lambda_{ab} (H)$ in the vortex state of YBa₂Cu₃O$_{7-\delta}$could not be determined, due to the narrow range of reduced field which the measurements cover. The strength of the field dependence in the vortex state is found to be weaker than that reported from microwave cavity perturbation measurements in the Meissner state, which find $\Delta \lambda_{ab} \! \propto \! H$in YBa₂Cu₃O_6.95 [38]. However, very recent AC susceptibility measurements [208] suggest that the field dependence is much weaker in the Meissner state than that reported in Ref. [38]. In the vortex state, nonlocal effects associated with nodes on the Fermi surface likely dominate the behaviour of $\lambda_{ab} (H)$, whereas nonlinear effects are believed to be the primary source of the H-dependence in the Meissner state. One must be careful in making comparisons between measurements of the penetration depth in the Meissner and vortex states. Differences may be solely due to the way in which the penetration depth is defined in the techniques used.

The measurements of r₀ as a function of temperature and magnetic field are an important contribution to the general understanding of the characteristic length scale $\xi_{ab}$. It has been shown here that in the conventional theory of the vortex state, $\xi_{ab}$ behaves essentially in the same manner as the vortex-core size. The sharp decrease in the vortex-core radius r₀ (and hence $\xi_{ab}$) with increasing magnetic field is attributed to increased vortex-vortex interactions. In YBa₂Cu₃O$_{7-\delta}$, $\xi_{ab}$ is generally assumed to be a small quantity (e.g. typical values being 12-14 Å). However, the results herein indicate that at least in the vortex state, this is really only the case in moderate magnetic fields. The extrapolated zero-field value of $\xi_{ab}$ in YBa₂Cu₃O_6.95 is $\approx \! 80$ Å, which ``may'' imply that $\xi_{ab}$ is larger in the Meissner state than what is generally assumed. Deoxygenation is found to increase the magnitude of $\xi_{ab}$, which in the vortex state implies that the cores will overlap at a reduced value of H_c2. It is important to note that while r₀ is rather insensitive to the choice of the fitted model, the precise relationship between r₀ and $\xi_{ab}$does depend on the model.

The shrinking of the vortex-core radius with decreasing temperature in NbSe₂ is consistent with the traditional picture of discrete bound quasiparticle states in the core. As r₀ shrinks, the energy level spacing increases. The change in the size of the vortex core should saturate when the thermal energy is less than the energy level spacing. The substantially weaker temperature dependence of r₀ found in YBa₂Cu₃O$_{7-\delta}$ suggests that this occurs at much higher temperatures in this compound. The smaller core size and the reduction of the T-dependence in YBa₂Cu₃O$_{7-\delta}$ both imply that there are fewer bound quasiparticle states in the vortex cores than in NbSe₂.

It should be noted that since the temperature and field dependence of $\xi_{ab}$ found here originates from changes in the electronic structure of the vortex cores, there is no reason to expect that $\xi_{ab}$ should exhibit similar behaviour in the Meissner state. Furthermore, it is really the vortex core size which has been measured in this thesis. Although this is generally considered to be an indirect measurement of the coherence length, it is not entirely clear whether this coherence length is fundamentally the same as the coherence length in the Meissner state.

Finally, the results of these measurements indicate that the London and GL models with field independent $\lambda$ and $\xi$ are not applicable deep in the superconducting state. The fact that the data were analyzed with models in which $\lambda$ and $\xi$ were not defined as functions of magnetic field does not invalidate this conclusion. The field dependence of both $\lambda$ and $\xi$ appears to be associated with the unique properties of the vortex lattice.