The rigidity of a material is the property that enables it to preserve its structure when deformed. In a rigid body, no internal motion is possible since the degrees of freedom of the system are limited to translations and rotations only. In the macroscopic scale, the rigidity and response of a material to external load can be studied using continuum elasticity theory. But when it comes to the microscopic scale, a simple yet powerful approach is to model the structure of the material and its interparticle interactions as a ball$-$and$-$spring network. This model allows a full description of rigidity in terms of the vibrational modes and the balance between degrees of freedom and constraints in the system.

In the present work, we aim to establish a microscopic description of rigidity in \emph{disordered} networks. The studied networks can be designed to have a specific number of degrees of freedom and/or elastic properties. We first look into the rigidity transition in three types of networks including randomly diluted triangular networks, stress diluted triangular networks and jammed networks. It appears that the rigidity and linear response of these three types of systems are significantly different. In particular, jammed networks display higher levels of self-organization and a non-zero bulk modulus near the transition point. This is a unique set of properties that have not been observed in any other types of disordered networks. We incorporate these properties into a new definition of jamming that requires a network to hold one extra constraint in excess of isostaticity and have a finite non-zero bulk modulus. We then follow this definition by using a tuning by pruning algorithm to build spring networks that have both these properties and show that they behave exactly like jammed networks. We finally step into designing new disordered materials with desired elastic properties and show how disordered auxetic materials with a fully convex geometry can be produced.

The overall objective of this project is to optimize the development of magnetic ferrite thin films targeted for enabling low-loss broadband communication devices, miniaturized low-microwave inductors and electromagnetic noise suppressors. The focus of this objective is to design and build a reactor and improve the spin-spray process. Each film is then characterized and optimized to have a high permeability and high frequency in the range of 500 MHz - 3 GHz. Films produced by the µ-droplet deposition regime yields a higher Snoek's product than the continuous liquid layer regime. The highest Snoek's product occurs when it is deposited at an oxidant pH of 8.28. The Ni-Zn-Co ferrite magnetic domains were imaged using the Lorentz TEM in which multi-grain domains are experimentally observed for the first time.

The research of this dissertation has involved the nanoscale quantitative characterization of patterned magnetic nanostructures and devices using off-axis electron holography and Lorentz microscopy. The investigation focused on different materials of interest, including monolayer Co nanorings, multilayer Co/Cu/Py (Permalloy, Ni81Fe19) spin-valve nanorings, and notched Py nanowires, which were fabricated via a standard electron-beam lithography (EBL) and lift-off process. Magnetization configurations and reversal processes of Co nanorings, with and without slots, were observed. Vortex-controlled switching behavior with stepped hysteresis loops was identified, with clearly defined onion states, vortex states, flux-closure (FC) states, and Omega states. Two distinct switching mechanisms for the slotted nanorings, depending on applied field directions relative to the slot orientations, were attributed to the vortex chirality and shape anisotropy. Micromagnetic simulations were in good agreement with electron holography observations of the Co nanorings, also confirming the switching field of 700-800 Oe. Co/Cu/Py spin-valve slotted nanorings exhibited different remanent states and switching behavior as a function of the different directions of the applied field relative to the slots. At remanent state, the magnetizations of Co and Py layers were preferentially aligned in antiparallel coupled configuration, with predominant configurations in FC or onion states. Two-step and three-step hysteresis loops were quantitatively determined for nanorings with slots perpendicular, or parallel to the applied field direction, respectively, due to the intrinsic coercivity difference and interlayer magnetic coupling between Co and Py layers. The field to reverse both layers was on the order of ~800 Oe. Domain-wall (DW) motion within Py nanowires (NWs) driven by an in situ magnetic field was visualized and quantified. Different aspects of DW behavior, including nucleation, injection, pinning, depinning, relaxation, and annihilation, occurred depending on applied field strength. A unique asymmetrical DW pinning behavior was recognized, depending on DW chirality relative to the sense of rotation around the notch. The transverse DWs relaxed into vortex DWs, followed by annihilation in a reversed field, which was in agreement with micromagnetic simulations. Overall, the success of these studies demonstrated the capability of off-axis electron holography to provide valuable insights for understanding magnetic behavior on the nanoscale.