Common methods of local magnetic imaging display either a high spatial resolution and relatively poor field sensitivity (MFM, Lorentz microscopy), or a relatively high field sensitivity but limited spatial resolution (scanning SQUID microscopy). Since the magnetic field of a nanoparticle or nanostructure decays rapidly with distance from the structure, the achievable spatial resolution is ultimately limited by the probe-sample separation. This thesis presents a novel method for fabricating the smallest superconducting quantum interference device (SQUID) that resides on the apex of a very sharp tip. The nanoSQUID-on-tip displays a characteristic size down to 100 nm and a field sensitivity of 10^-3 Gauss/Hz^(1/2). A scanning SQUID microsope was constructed by gluing the nanoSQUID-on-tip to a quartz tuning-fork. This enabled the nanoSQUID to be scanned within nanometers of the sample surface, providing simultaneous images of sample topography and the magnetic field distribution. This microscope represents a significant improvement over the existing scanning SQUID techniques and is expected to be able to image the spin of a single electron.

On the current status of research activity, providing new information on the applications of SQUIDs, including magnetocardiography, immunoassays, and laser-SQUID microscopes, all of which are close to being commercially available.

Superconductivity is among the most fascinating properties that a material can have. Below the transition temperature $T_c$, electrons condensate into a macroscopic quantum mechanical state and flow without dissipation. The quantum nature of the superconducting state also manifests in its magnetic properties. Superconductors fully expels magnetic field in a weak applied field, referred as Meissner effect. In an intermediate field, superconductors often contain microscopic whirlpools of electrons that carry quantized magnetic flux, called vortices. In this thesis, I present magnetic-force-microscopy (MFM) studies of unconventional superconductors both in the Meissner state and in the mix state. We extend the application of MFM beyond the conventional imaging mode and use it for quantitative analysis. In the mix state, we use MFM manipulating individual vortices with a high level of control and a known force to study the mechanics and dynamics of a single vortex in cuprate superconductors. In the Messiner state, we establish MFM as a novel local technique to measure the magnetic penetration depth $\lambda$ and implement it to study the pairing mechanism of iron-pnictide superconductors. Chapter 1 contains a brief introduction of MFM and its conventional application of imaging. We demonstrate high-spatial resolution images of isolated superconducting vortices. We show that by integrating images of isolated vortices at consecutive heights we are able to reconstruct the force between the MFM tip and vortices. We can also obtain the force by using a tip-vortex model. The two methods agree and both allow us to obtain the force used in vortex manipulation discussed in Chapter 2 and Chapter 3. Chapter 2 discusses the behavior of individual vortices in fully doped YBa$_2$Cu$_3$O$_{7-\delta}$ when subject to a local force. Because the anisotropy of fully doped YBa$_2$Cu$_3$O$_{7-\delta}$ is moderate, the vortex motion can be well described as an elastic string moving through a uniform three dimensional pinning landscape. We find an unexpected and marked enhancement of the response of a vortex to pulling when we wiggle it transversely. In addition, we find enhanced vortex pinning anisotropy that suggests clustering of oxygen vacancies in our sample. We demonstrate manipulation at the nanoscale with a level of control far beyond what has been reported before. We show that a dragged vortex can be used to probe deep into the bulk of the sample and to interact with microscopic structures much smaller than the tip size. Chapter 3 shows the vortex behavior in another limit. In an very underdoped YBa$_2$Cu$_3$O$_{6+x}$ single crystal, a cuprate superconductor with strong anisotropy, a vortex can be regarded as a stack of two-dimensional pancakes with weak interlayer Josephson coupling. We use the MFM tip to split the pancake stacks composing a single vortex and to produce a kinked structure. Our measurements highlight the discrete nature of stacks of pancake vortices in layered superconductors. We also measure the required force in the process, providing the first measurement of the interlayer coupling at the single vortex level. The discovery of iron-pnictide superconductors in 2008 motivates my efforts to locally measure the magnetic penetration depth $\lambda$, one of the two fundamental length scales in superconductors and known to be difficult to measure. Chapter 4 discusses the methodology of measuring $\lambda$ by MFM, which is based on the time-reversed mirror approximation an analytical model of the MFM tip-superconductor interaction in the Meissner state. A calibration run was performed on \YBCO\ single crystals with known $\lambda$. The same time-reversed mirror approximation can be applied to scanning SQUID sysceptometry (SSS) to measure the temperature variation of penetration depth $\Delta\lambda(T)\equiv\lambda(T)-\lambda(0)$. Chapter 5 includes brief introduction of the iron-pnictide superconductors. The multiple conduction bands and the vicinity of the superconducting phase to magnetic phase give additional challenges in $\lambda$ measurements. We demonstrated in this chapter on single crystals of Ba(Fe$_{0.95}$Co$_{0.05}$)$_2$As$_2$ that MFM can measure the absolute value of $\lambda$, as well as obtain its temperature dependence and spatial homogeneity. We observe that $\Delta\lambda(T)$ varies 20 times slower with temperature than previously reported by bulk techniques, and that $\rho_s(T)$ over the full temperature range is well described by a clean two-band fully gapped model, consistent with the proposed $s\pm$ pairing symmetry. Chapter 6 extends the measurements of $\rho_s(T)$ to the family Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ with Co doping level $x$ across the superconducting dome. We observe systematic evolution of $\rho_s(T)$ with $x$ that can be summarized as three main trends. First, $\rho_s(0)$ falls more quickly with $T_c$ on the underdoped side of the dome than on the overdoped. Second, the temperature variation of $\rho_s(T)$ at low temperature increases away from optimal doping. Third, $\rho_s(T)$ increases sharply with cooling through the superconducting transition temperature $T_c$ of both optimally doped and underdoped compounds. These observations hint an interplay between magnetism and superconductivity.

Understanding the nature of vortices in high-Tc superconductors is a crucial subject for research on superconductive electronics, especially for superconducting interference devices (SQUIDs), it is also a fundamental problem in condensed-matter physics. Recent technological progress in methods for both direct and indirect observation of vortices, e.g. scanning SQUID, terahertz imaging, and microwave excitation, has led to new insights into vortex physics, the dynamic behavior of vortices in junctions and related questions of noise. This book presents the current status of research activity and provides new information on the applications of SQUIDs, including magnetocardiography, immunoassays, and laser-SQUID microscopes, all of which are close to being commercially available.

By covering theory, design, and fabrication of nanostructured superconducting materials, this monograph is an invaluable resource for research and development. Examples are energy saving solutions, healthcare, and communication technologies. Key ingredients are nanopatterned materials which help to improve the superconducting critical parameters and performance of superconducting devices, and lead to novel functionalities. Contents Tutorial on nanostructured superconductors Imaging vortices in superconductors: from the atomic scale to macroscopic distances Probing vortex dynamics on a single vortex level by scanning ac-susceptibility microscopy STM studies of vortex cores in strongly confined nanoscale superconductors Type-1.5 superconductivity Direct visualization of vortex patterns in superconductors with competing vortex-vortex interactions Vortex dynamics in nanofabricated chemical solution deposition high-temperature superconducting films Artificial pinning sites and their applications Vortices at microwave frequencies Physics and operation of superconducting single-photon devices Josephson and charging effect in mesoscopic superconducting devices NanoSQUIDs: Basics & recent advances Bi2Sr2CaCu2O8 intrinsic Josephson junction stacks as emitters of terahertz radiation| Interference phenomena in superconductor-ferromagnet hybrids Spin-orbit interactions, spin currents, and magnetization dynamics in superconductor/ferromagnet hybrids Superconductor/ferromagnet hybrids