Introduction

In 1986, Bednorz and Müller discovered superconductivity in the La$_{2-x}$Ba$_x$CuO$_{4+y}$ system, below a critical transition temperature $T_c \approx 30$ K [1]. This finding was subsequently followed by the discovery of high-$T_c$ superconductivity in a number of related layered compounds consisting of CuO$_2$ planes. The CuO$_2$ planes in these materials are separated by ``charge reservoir layers'', which control the oxidation state of the copper ions. High-$T_c$ superconductivity is achieved when the undoped parent compound (e.g. La$_2$CuO$_4$), which is an antiferromagnetic insulator, is doped with holes (Fig. 1.1) [2]. An exception is the R$_{2-x}$Ce$_x$CuO$_4$ system, where R = {La, Nd, Pr, Sm or Eu}. In this case, high-$T_c$ superconductivity occurs upon partial substitution of Ce$^{4+}$ for R$^{3+}$, which dopes electrons into the CuO$_2$ layers. The most studied electron-doped high-$T_c$ superconductor (HTSC) is Nd$_{2-x}$Ce$_x$CuO$_4$, which was synthesized by Tokura and co-workers in 1989 [3]. Compared to hole-doped HTSC's, less is known about the electron-doped cuprates. One reason is that it is difficult to synthesize high-quality single crystals. In contrast to the hole-doped compounds, as-grown crystals are made superconducting by subjecting them to an oxygen reduction process [4], which often degrades the crystal surface. In addition, single crystals of electron-doped cuprates are generally plagued by spatial inhomogeneity of the cerium and oxygen concentration [5,6].

Fig. 1.1: Generic Phase diagram of the hole-doped HTSC La$_{2-x}$Sr$_x$CuO$_4$ and the electron-doped HTSC Pr$_{2-x}$Ce$_x$CuO$_4$ as a function of charge doping concentration, showing the anitferromagnetic (AF), spin glass (SG) and superconducting (SC) phases.

One of the fundamental properties of superconductors is that they exhibit a diamagnetic response called the ``Meissner effect.'' When a superconducting material is cooled below $T_c$ in the presence of a weak applied magnetic field $H$, the field is expelled from the bulk of the sample. When the applied magnetic field exceeds a critical value $H_c$, superconductivity is destroyed. For type-II superconductors, like the high-$T_c$ cuprates, the response to an applied magnetic field can be one of the following:

1. Meissner State: At $H$ < $H_{c1}$, (where $H_{c1} \equiv$ the lower critical magnetic field), supercurrents circulate near the surface of the sample, screening the magnetic field from the bulk. However, the external magnetic field partially penetrates the sample at the surface. The magnitude of the penetrating field-component parallel to the surface decays exponentially [7] as a function of distance from the surface. The characteristic length scale of the exponential field decay is the ``magnetic penetration depth'' $\lambda$. This quantity is of fundamental importance, as $\lambda^{-2}$ is proportional to the density of superconducting carriers $n_s$.

2. Vortex State: At $H_{c1}$ < $H$ < $H_{c2}$ (where $H_{c2} \equiv$ the upper critical magnetic field), magnetic flux penetrates the bulk of the superconductor in the form of a periodic arrangement of quantized flux lines, called a ``vortex lattice''. Each vortex in the lattice is comprised of one flux quantum $\Phi$ = $hc/2e$. The vortex core is a region where the ``superconducting order parameter'' $\psi(r)$ is suppressed and the local magnetic field $B(r)$ is maximum ($r$ is the radial distance from the centre of the vortex). The length scale which governs spatial variations of $\psi(r)$ is called the ``superconducting coherence length'' $\xi$. Supercurrents circulating around the individual vortices screen the magnetic field within the vortex core from the surrounding material, in the same way that supercurrents near the sample surface screen the external magnetic field in the Meissner state. Consequently, the magnetic field decays outside the vortex core region over the length scale $\lambda$.

3. Normal State: $H$ > $H_{c2}$, the external magnetic field fully penetrates the sample and superconductivity is destroyed.

One of the outstanding issues concerning the electron-doped HTSCs is the pairing symmetry of the superconducting carriers in these compounds. While there are phase sensitive [8], angle resolved photoemission spectroscopy [9] and microwave [10] measurements that suggest the pairing symmetry is $d_{x^2-y^2}$-wave (like in hole-doped HTSCs), other experiments [11,12] favour $s$-wave symmetry. Measurements of the temperature dependence of the magnetic penetration depth are one way of distinguishing between $s$-wave and $d_{x^2-y^2}$-wave symmetry. Thus far, most measurements of $\lambda$ in electron-doped HTSCs have been performed in the Meissner state. In this thesis, the results of measurements of the in-plane magnetic penetration depth $\lambda _{ab}$ in the vortex state of the electron-doped HTSC Pr$_{2-x}$Ce$_x$CuO$_4$, by muon spin rotation ($\mu$SR) spectroscopy are presented. As a bulk local probe, $\mu$SR has the advantage that it is insensitive to inhomogeneities at the sample surface. The samples measured are the smallest single crystals studied so far by the $\mu$SR techniques.A complication in studying electron-doped HTSCs, is the electronic magnetic moments of the rare-earth ions. For example, their presence has prevented an accurate determination of the values of the magnetic penetration depth in the Meissner state [13,14]. The Pr$_{2-x}$Ce$_x$CuO$_4$ compound is appealing for study, because the crystal-electric-field ground state of the Pr ion is non-magnetic.

In the next chapter an introduction to the $\mu$SR method is given. Chapter 3 provides a brief description of the crystal growth process, sample characteristics and the experimental setup. In Chapters 4 and 5, the experimental results are presented. A discussion of the results and conclusions are given in Chapter 6.