Introduction

In recent years, ``high-T_c'' superconductors have generated a frenzy of experimental and theoretical endeavours, motivated by the likelihood of spawning revolutionary technologies. Their mere existence challenges our present-day understanding of many facets of condensed matter physics. The particular features which distinguish this class of materials from conventional BCS superconductors have been well documented. Besides the extraordinarily high transition temperatures in excess of 120K, some of the more noteable differences include: anomalous normal-state properties, strong anisotropy, a very small coherence length and unconventional behaviour in the superconducting state.

Early experiments intended to answer some of the broader questions concerning the nature of the mechanism forming the basis for superconductivity in the high-T_c materials, yielded conflicting and questionable results. More recently, the experiments have become more reliable, mainly because of vast improvements in sample quality. Researchers now recognize that the inherent differences between high-T_c powders, thin films and single crystals, are significant enough that they must be accounted for when interpreting and comparing results. Furthermore, it is now understood that impurities play a major role in the outcome of many of these experiments, so that often only experiments done with the same sample are comparable.

Theories regarding the high-T_c compounds are divided into two major categories: ``Fermi liquid'' and ``non-Fermi liquid'' theories. The normal state of a conventional BCS superconductor is well described by a Fermi liquid. However, the anomalous normal-state properties of the high-T_c compounds hint at the possible absence of a Fermi surface in this class of materials. The interpretation of experimental measurements is dependent upon which class of theories one assumes. For now at least, there is an unwillingness to stray too far from conventional thinking and the successes of BCS theory, so that experimental results are usually interpreted in the context of Fermi liquid theories.

With the general assumption that the electronic ground state of the high-T_csuperconductors is composed of paired charge carriers, determining the symmetry of the paring state has been the subject of much controversy. Despite many of the earlier reports, it is now generally agreed that conventional s-wave pairing of carriers is unlikely. The pairing state is almost certainly anisotropic, but beyond this, little is known for certain. A leading candidate in the high-T_c materials is the d-wave form of pairing called d_x²-y² [1,2]. For d_x²-y² symmetry, the wave-vector dependence of the energy gap which develops at the Fermi surface in the superconducting state is such that line nodes appear in the gap.

Measurements of the temperature dependence of the magnetic penetration depth $\lambda$ are one way to probe the nature of the low-energy excitations and the symmetry of the pairing state. For an isotropic s-wave superconductor, the magnetic penetration depth $\lambda (T)$ varies exponentially at low temperatures. Even if the superconducting gap is anisotropic, as long as it does not vanish anywhere on the Fermi surface, the penetration depth $\lambda (T)$ will still exhibit exponential behaviour at temperatures such that k_B Tis less than the minimum value of the energy gap for excitations [3].

However, if the symmetry of the superconducting state is d_x²-y², with the associated line nodes running along the cylindrical Fermi surface, Cooper pairs can be broken more easily and $\lambda (T)$ is expected to change linearly with temperature at low temperatures [4,5]. Recent microwave cavity perturbation measurements of $\Delta \lambda$ (i.e. $\lambda (T) - \lambda (0)$) on high quality $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{6.95}$ }$ single crystals show a strong linear term below 30K [6]. Studies of Tl₂CaBa₂Cu₂O $_{8- \delta}$ single crystals also show a linear dependence on T for $\lambda (T)$ at low temperatures [7]. However, previous experiments in Bi₂Sr₂CaCu₂O₈ and $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7- \delta}$ }$ thin films showed a T² dependence on $\lambda (T)$ [8]. It has been pointed out that impurity scattering in a d_x²-y²-wave superconductor can change the low-temperature behaviour from $\lambda(T) \sim T$ to $\lambda(T) \sim T^{2}$[5,6,7].

Muon spin rotation is the most direct way to measure $\lambda_{ab}$ (the penetration depth in the ab-plane) in the bulk of the sample. Unlike other techniques, which are performed in zero-static magnetic field, $\mu ^{+}$SR directly measures the magnetic field distribution associated with the vortex lattice of a type II superconductor. The penetration depth $\lambda_{ab}$ is the length scale over which magnetic flux leaks into the superconducting regions around the vortex cores. The $\mu ^{+}$SR technique provides a means by which one can investigate not only the temperature dependence of $\lambda_{ab}$, but also its field dependence. The difficulty in the $\mu ^{+}$SR technique lies in the extraction of $\lambda_{ab}$ from the measured data. Here, one relies heavily on a phenomenological model of the magnetic field distribution and the ability of the fitting program to extract the relevant parameters.

Previous $\mu ^{+}$SR studies on sintered powders and crystals of $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7- \delta}$ }$, which were of lower quality than the ones used in this study, provide evidence in support of an isotropic s-wave pairing state [9]. However, impurities or other crystalline imperfections could lead to a T² dependence. Furthermore, conclusions regarding the pairing mechanism were based on the overall behaviour of $\lambda (T)$ rather than that in the low-temperature regime. It has been argued that the temperature dependence of $\lambda (T)$at higher temperatures can be drastically altered by strong-coupling corrections, impurity scattering, the precise shape of the Fermi surface and gap anisotropy [4].

A universal equation for $\lambda (T)$ at all T < T_c, based on the d-wave formalism, has yet to be determined. One must rely solely on the low-temperature deviations from s-wave theory, anticipated from purely qualitative arguments. Consequently, conclusions with regard to the pairing mechanism in $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7- \delta}$ }$ requires precise low temperature data. Previous $\mu ^{+}$SR measurements of $\lambda (T)$ either lacked good low-temperature data or were done on sintered powders. Low-temperature measurements of $\lambda (T)$ should also be able to reconcile the nature of the nodes in the gap, if they do indeed exist. That is to say, point nodes on the Fermi surface should be distinguishable from the line nodes associated with d_x²-y²pairing [4].

This present study is concerned with recent $\mu ^{+}$SR measurements of $\lambda_{ab}$ in single crystals of $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{6.95}$ }$. Some words of caution must be provided to the reader. Recent discovery of a significant $\vec{a}$-$\vec{b}$anisotropy in the magnetic penetration depth [10] may have a dramatic influence on the interpretation of measurements of $\lambda _{ab} (T)$. Although the existence of $\vec{a}$-$\vec{b}$anisotropy is acknowledged in this study, it is not included in the final analysis, as the matter is presently under investigation. Also, during the final stages of this thesis there have been recent suggestions that $\lambda (T)$ is controlled by critical fluctuations down to very low temperatures [11]. In this case, the measured $\lambda (T)$ has little to do with the pairing mechanism, but is determined by the 3D XY and 2D XY universality classes. This possibility is not discussed any further in this thesis.