List of Figures

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
2. Angular distribution of positrons from $\mu ^{+}$ decay with respect to the initial muon spin direction
3. Schematic diagram for a typical TD-$\mu$SR experiment
4. Schematic diagram of a TF-$\mu$SR experiment
5. Schematic diagram of a ZF-$\mu$SR experiment
6. Crystal structure of La$_{2-x}$Sr$_x$CuO$_4$ and Pr$_{2-x}$Ce$_x$CuO$_4$
7. Bulk magnetic susceptibility measurements taken under FC and ZFC conditions with $H$ = $1$ Oe and resistivity measurements in zero external magnetic field
8. Schematic of the experimental setup
9. Time evolution of the asymmetry in zero external magnetic field at $T$ = $165$ K, $80$ K, $40$ K and $2.3$ K.
10. Time evolution of the asymmetry in longitudinal external magnetic fields of $100$ Oe and $25$ Oe, at $T$ = $25$ K.
11. Temperature dependence of the amplitudes $a_1$ and $a_2$
12. Temperature dependence of the relaxation rates $\Delta$ and $\lambda$.
13. Time evolution of the muon-spin polarization in Pr$_{2-x}$Ce$_x$CuO$_4$ taken under (a) FC and (b) ZFC conditions at $T$ = $2.4$ K
14. Time evolution of the muon-spin polarization in Pr$_{2-x}$Ce$_x$CuO$_4$ taken under (a) FC and (b) ZFC conditions at $H$ = $91$ Oe
15. FFTs of the muon-spin precession signal in Pr$_{2-x}$Ce$_x$CuO$_4$
16. Temperature dependence of $B_{0} - \mu _{0}H$ at $H$ = $91$, $200$, $500$ and $2019$ Oe
    Inset: Magnetic field dependence of $B_{0} - \mu _{0}H$ at $T$ = $5$ and $25$ K
17. $K^{\perp }$ vs. $\chi ^{\perp }$ at $H$ = $3$ kOe.
    Inset: Temperature dependence of $\chi ^{\perp }$ and $\chi^{\Vert}$ at $H$ = $10$ kOe.
18. Temperature dependence $B_{0}$ and $\chi ^{\perp }$ measured under FC conditions at $H$ = $91$ and $500$ Oe
19. Magnetic field dependence of $B_{0}^{S}$ - $B_{0}^{N}$ at $T$ = $5$ K and $10$ K
20. Top: Two proposed ordered Cu-spin structures for Pr$_{2}$CuO$_{4}$
    Bottom: Magnetic field strength at the muon stopping site due to Cu dipole moments plotted as a function of the canting angle $\theta$ between the Cu spins and the CuO$_2$ plane
21. FFTs of the muon-spin precession signal and the theoretical muon-spin polarization functions with $G(t)$ = $e^{-(\Delta t)^\beta }$ where $\beta$ = $1.2$, and $G(t)$ = $e^{-(\sigma ^2 t^2)/2}$
22. Temperature dependence of $\lambda _{ab}$ for a Pr$_{2-x}$Ce$_x$CuO$_4$ single crystal with $\xi _{ab}$ = 30 Å and 60 Å at $H$ = $91$ Oe
23. Temperature dependence of the magnetic penetration depth $\lambda _{ab}^{-2}(T)$ at $H$ = $91$ Oe
24. Temperature dependence of the magnetic penetration depth in a second Pr$_{2-x}$Ce$_x$CuO$_4$ single crystal at $H$ = $200$ Oe
25. Normalized magnetic penetration depth $\lambda ^{2}_{ab}(0)/\lambda ^{2}_{ab}(T)$ as a function of reduced temperature $T/T_{c}$ for Pr$_{2-x}$Ce$_x$CuO$_4$ single crystals at $H$ = $91$ Oe, La$_{2-x}$Sr$_x$CuO$_4$ at $H$ = $2$ kOe and YBa$_2$Cu$_3$O$_{6.95}$ at $H$ = $5$ kOe