Discussion

As shown in this thesis, the $\mu$SR technique can be used to measure the internal magnetic field distribution in the vortex state of a type-II superconductor. Determination of the fundamental length scale $\lambda _{ab}$ from $\mu$SR measurements, requires that some assumptions be made in the modeling procedure, including the geometry of the vortex lattice. The latter is often determined by neutron scattering, however, due to the unavailability of large samples, the geometry in Pr$_{2-x}$Ce$_x$CuO$_4$ is not known. Nevertheless, there is every reason to expect a hexagonal vortex lattice at the low applied magnetic fields used in this work, where the interactions between vortices is weak.

The spatial field profile of the vortex lattice that was assumed in the analysis of the TF-$\mu$SR data [i.e. Eq. (2.7)], agrees extremely well with the exact numerical solutions of the GL equations at low applied fields. Given that this model has been successful in describing the magnetic field distribution of the vortex lattice in hole-doped HTSCs and conventional superconductors, it is a reasonable starting that approximation for Pr$_{2-x}$Ce$_x$CuO$_4$. However, a rigourous test of the appropriateness of this model awaits future $\mu$SR work on other samples, at low temperatures and higher magnetic fields.

The main objective of this thesis was to study the temperature dependence of the in-plane magnetic penetration depth $\lambda _{ab}$ in Pr$_{2-x}$Ce$_x$CuO$_4$, as the limiting low-temperature behaviour of $\lambda _{ab}$ reflects the nature of the pairing symmetry of the superconducting carriers. Generally speaking, measurements of the magnetic penetration depth by $\mu$SR are best performed at high transverse magnetic field, where the density of vortices in the sample is large. In this case a more uniform vortex lattice is established, because the intervortex repulsion force is able to overcome the flux pinning forces exerted by sample defects. Furthermore, a higher number of muons stop close to the vortex cores, providing a greater sensitivity to the high-field cutoff in fits to the measured internal magnetic field distribution. However, the Pr$_{2-x}$Ce$_x$CuO$_4$ single crystals studied here appear to have a spread in local magnetic susceptibilities (likely due to spatial variations of charge doping) that results in an increased $\mu$SR line width with increasing applied magnetic field. At high applied magnetic field this additional broadening makes it is impossible to isolate the internal magnetic field distribution associated with the vortex lattice. Consequently, the present study of $\lambda _{ab}$ in Pr$_{2-x}$Ce$_x$CuO$_4$ single crystals was restricted to low applied magnetic fields. Despite this limitation a considerable amount of effort was devoted to independently determining the value of the superconducting coherence length $\xi _{ab}$. Attempts were made to fit the $\mu$SR time spectrum at each temperature below $T_c$ assuming a fixed value of $\xi _{ab}$. In particular, $\xi _{ab}$ was determined as the value at which $\chi^2/NDF$ was minimized (where $NDF~\equiv$ number of degrees of freedom). A similar approach was taken, where instead the value of the Ginzburg-Landau parameter $\kappa_{ab}$ (= $\lambda_{ab}/\xi_{ab}$) was fixed in the fitting procedure. However, for both approaches, it was found that $\chi^2/NDF$ did not converge to a minimum at all temperatures, due to the insensitivity of the fits to the high-field cutoff. In the end, a reliable value of $\xi _{ab}$ = $60$ Å was used, which is consistent with reported values of $H_{c2}$ in Pr$_{2-x}$Ce$_x$CuO$_4$. A simple visual inspection of the fits in the frequency domain was also done to verify that this value was reasonable.

The temperature dependence of $\lambda_{ab}^{-2}$ determined in Pr$_{2-x}$Ce$_x$CuO$_4$ above $0.2$ $T_c$ agrees with that determined previously by $\mu$SR in hole-doped HTSCs, although the uncertainty in the measurements is too large for a stringent comparison. A unique identification of the pairing symmetry in Pr$_{2-x}$Ce$_x$CuO$_4$ requires measurements of $\lambda _{ab}$ at lower temperatures. Unfortunately, the small size of the Pr$_{2-x}$Ce$_x$CuO$_4$ single crystals that are currently available requires the use of a specialized low-background experimental setup. This arrangement is incompatible with a dilution refrigerator, so larger Pr$_{2-x}$Ce$_x$CuO$_4$ single crystals are needed to extend the present study to lower temperatures.

A surprising result in our study of superconducting Pr$_{2-x}$Ce$_x$CuO$_4$, was the observation of an enhanced average local magnetic field at the muon site ($B_{0}$) upon cooling the single crystals below $T_c$ in a weak external magnetic field. Since the measurements of the bulk magnetic susceptibility exhibit the usual diamagnetic response characteristic of a superconductor, the increased local magnetic field must arise from the onset of spontaneous magnetic order. Although we calculate that such onset of magnetic ordering will reduce the initial asymmetry by $\sim 19$ % at the lowest temperature, this is not evident from the measurements for two reasons: First, the sample was mounted on a light guide which thermally contracts when it is cooled down. This introduced a systematic uncertainty in the initial asymmetry. Second and more importantly, the unusual situation of having the simultaneous onset of magnetic order and a vortex lattice makes it difficult to extract accurate values of the initial asymmetry across $T_c$, because the functional form of the internal magnetic field distribution dramatically changes.

From the dependence of $B_0$ on the external magnetic field $H$ and dipolar field calculations, it appears that the enhanced local magnetic field detected by $\mu$SR arises from the onset of antiferromagnetic (AF) order of the Cu spins. In particular, the Cu-spin structure required to explain the $\mu$SR results is the noncollinear arrangement previously identified in non-superconducting Pr$_{2-x}$Ce$_x$CuO$_4$ by neutron scattering [30] and $\mu$SR [31]. A slight canting of the Cu spins out of the CuO$_2$ plane produces a dipolar magnetic field at the muon stopping site which is parallel to the basal plane. Since the muon detects the vector sum of local magnetic fields from different sources, the total average internal magnetic field that the muon senses in the presence of AF order exceeds the external field $H$.

Because $\mu$SR is sensitive to magnetic volume fractions, macroscopic phase separation would give rise to more than one distinct $\mu$SR signal. However, essentially all of the implanted muons see an increased local magnetic field below $T_c$, which is visually apparent in the FFTs of the muon-spin precession signals. Thus, one can conclude that the field-induced AF order occurs throughout much of the sample volume. Note the ZF-$\mu$SR measurements indicate that approximately 83% of the sample volume contains static Cu moments.

Recently, neutron scattering measurements of the hole-doped HTSC La$_{1.9}$Sr$_{0.1}$CuO$_4$ showed that a large ($140$ kOe) external field induces AF order in approximately 50% of the sample below $T_c$ [29]. This neutron study followed several other experimental works that detected AF correlations only in the regions near the vortex cores [35]. The emergence of AF correlations in the vortex cores of HTSCs, where the superconducting order parameter is suppressed, is predicted by a number of theoretical models [36]. The results on Pr$_{2-x}$Ce$_x$CuO$_4$ and that of Ref. [29] indicate that the external magnetic field stabilizes AF order well beyond the vortex cores. It is somewhat remarkable that a weak external magnetic field of only $90$ Oe results in long-range AF order in the superconducting state of Pr$_{2-x}$Ce$_x$CuO$_4$. This may be due to the close proximity of the AF and superconducting phases in electron-doped cuprates (see Fig. 1.1), although sample inhomogeneity may also play an important role in extending the AF order into the regions between vortices.