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Since their discovery by Bednorz and Mueller (1986), high-temperature cuprate superconductors have been the subject of intense experimental research and theoretical work. Despite this large-scale effort, agreement on the mechanism of high-T{sub c} has not been reached. Many theories make their strongest predictions for underdoped superconductors with very low superfluid density n{sub s}/m*. For this dissertation I implemented a scanning Hall probe microscope and used it to study magnetic vortices in newly available single crystals of very underdoped YBa{sub 2}Cu{sub 3}O{sub 6+x} (Liang et al. 1998, 2002). These studies have disproved a promising theory of spin-charge separation, measured the apparent vortex size (an upper bound on the penetration depth {lambda}{sub ab}), and revealed an intriguing phenomenon of ''split'' vortices. Scanning Hall probe microscopy is a non-invasive and direct method for magnetic field imaging. It is one of the few techniques capable of submicron spatial resolution coupled with sub-{Phi}{sub 0} (flux quantum) sensitivity, and it operates over a wide temperature range. Chapter 2 introduces the variable temperature scanning microscope and discusses the scanning Hall probe set-up and scanner characterizations. Chapter 3 details my fabrication of submicron GaAs/AlGaAs Hall probes and discusses noise studies for a range of probe sizes, which suggest that sub-100 nm probes could be made without compromising flux sensitivity. The subsequent chapters detail scanning Hall probe (and SQUID) microscopy studies of very underdoped YBa{sub 2}Cu{sub 3}O{sub 6+x} crystals with T{sub c} {le} 15 K. Chapter 4 describes two experimental tests for visons, essential excitations of a spin-charge separation theory proposed by Senthil and Fisher (2000, 2001b). We searched for predicted hc/e vortices (Wynn et al. 2001) and a vortex memory effect (Bonn et al. 2001) with null results, placing upper bounds on the vison energy inconsistent with the theory. Chapter 5 discusses imaging of isolated vortices as a function of T{sub c}. Vortex images were fit with theoretical magnetic field profiles in order to extract the apparent vortex size. The data for the lowest T{sub c}'s (5 and 6.5 K) show some inhomogeneity and suggest that {lambda}{sub ab} might be larger than predicted by the T{sub c} {proportional_to} n{sub s}(0)/m* relation first suggested by results of Uemura et al. (1989) for underdoped cuprates. Finally, Chapter 6 examines observations of apparent ''partial vortices'' in the crystals. My studies of these features indicate that they are likely split pancake vortex stacks. Qualitatively, these split stacks reveal information about pinning and anisotropy in the samples. Collectively these magnetic imaging studies deepen our knowledge of cuprate superconductivity, especially in the important regime of low superfluid density

Scanning Hall Probe Microscopy of Magnetic Vortices inVery Underdoped yttrium-barium-copper-oxide