A Cooper-pair (CP) splitter is a device capable of spatially separating a pair of entangled electrons by sending a weak current from a superconductor (SC) to a pair of quantum dots or quantum wires. In this thesis, CP splitters based on quantum spin Hall insulators (QSHI), also known as two-dimensional topological insulator, and quantum wires are theoretically studied. Spin-entangled electrons can be extracted from the CP's in the SC, and transmitted by the helical electronic states hosted by these quantum heterostructures. In the introduction, the background information on the integer quantum Hall effect, QSHI, and CP splitters is provided.

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This thesis advances the theory of spintronics by exploring ways in which collective spin dynamics can be manipulated in magnetic materials, with an emphasis on sytems which can exhibit low dissipation spin currents and nontrivial response to external driving.

In Chapter 1, we provide an introductory review of mesoscopic ferromagnetic dynamics. The Landau-Lifshitz equation is motivated and derived, with a brief discussion on its application in the presence of dissipative forces and external driving. Then we discuss the notion of spin superfluidity in easy plane ferromagnets and compare this to traditional Ginzberg-Landau superconductivity.

In Chapter 2, we further extend the analogy between ferromagnetism and superconductivity, and apply this anology to the study of the ferromagnet with strong coherent easy-plane anisotropy and weak in-plane coherent anisotropy. The low bias non-equilibrium phase diagram is mapped, and the potential for producing applications with superconductor-based circuit functionality at elevated temperatures is discussed.

In Chapter 3, the spin superfluid is studied in strong driving regimes in which the anaology to superconductivity begins to break down. Multiple non-equilibrium phases are discovered which are associated with the onset of non-linear effects, such as choatic oscillations and stationary soliton formation, near the spin torque injection region. Using numerical simulations and analytical modeling, we observe a robustness in spin superfluid transport and its high bias phase diagram despite the presence of symmetry breaking dipole-dipole interactions, finite size effects, and dissipation.

In Chapter 4, we turn to microscopic magnetic lattices, in particular weakly coupled arrays of quantum spin chains with fermionizable, via a Jordan-Wigner transformation, Hamiltonians. Using mean field theory, we find models which may exhibit dissipationless spin currents closely analogous to that of superconductivity and the quantum Hall effect.

In Chapter 5, we study the manipulation of magnetic domain walls on a wire with mechanical waves. We show how ferromagnetic and anitferromagnetic domain walls can be driven by circularly and linearly polarized waves, respectively. We note the potential for applications using mechanical waves as a means for manipulating magnetic solitons in insulators.

For Chapter 6, in collaboration with an experimental study on a cleaved edge overgrowth sample, we analyze the chiral edge states of the quantum Hall effect and their tunneling properties. Our modeling provides a theoretical fooundation from which the experimental method of momentum resolved spectroscopy can provide insight into how the quantum Hall effect evolves with changing magnetic field. Spin splitting of the chiral edge states is observed in the presence of a strong in-plane magnetic field.

We close with Chapter 7 giving an outlook for future work which could build off of the research presented in this thesis.

This thesis advances the theory of quantum and semiclassical transport in magnetic heterostructures. In the solid state, angular momentum can be carried by individual electrons and collective modes. The flow of angular momentum (a spin current), central to the operation of spintronic devices, is generated by the application of electric and magnetic fields and temperature gradients. In what follows, we explore the physics of such nonequilibrium spin currents in magnetic structures, involving an interplay of charge and magnetic dynamics and thermoelectric effects.

Chapter 1 provides an introduction to the transport of spin in magnets, carried by electrons and collective excitations of the magnetic order. Chapters 2-6 study the role of thermal fluctuations in transport and magnetic dynamics. In Chapter 2, we describe how incoherent thermal fluctuations of the spin density (magnons), which open inelastic scattering channels, contribute to spin and energy transport between a normal metal and a magnet. Such (temperature-dependent) transport may arise from a thermal gradient applied across the metal/magnet interface or a spin accumulation inside the normal metal and may alter or even drive magnetic dynamics.

Chapter 3, is dedicated to the realization of Bose-Einstein condensed magnons (previously observed by microwave pumping) in a normal metal/insulating ferromagnet heterostructure. As is described in Chapter 2, the combination of a temperature gradient and normal metal spin accumulation can drive spin into the insulating ferromagnet, accumulating as magnons; upon reaching a critical density, the magnons, which are bosonic, spontaneously form a quasi-equilibrium condensate.

Chapter 4 focuses on thermally driven spin-torques in electrically insulating structures, wherein direct electric control of magnetic dynamics is prohibited. In contrast to the interfacial transport described in Chapter 2, where a spin accumulation is to necessary to observe magnetic dynamics, here we demonstrate how spin-torques can arise from a pure thermal gradient in a heterostructure. These spin-torques can be measured by ferromagnetic resonance and can, under a sufficiently strong bias, actuate magnetic switching.

Chapter 5 concerns charge transport in a single-electron transistor, consisting of a magnetic quantum dot in contact with magnetic and normal metal leads. Microwave-driven precession by the dot induces a pumped electric current, which can be enhanced and made highly nonlinear by electron interactions (Coulomb blockade). The dependence of the resulting electrical response on the power and spectrum of microwave irradiation may be utilized to develop nanoscale microwave detectors analogous to single-electron transistor-based electrostatic sensors and nanoelectromechanical devices.

In Chapter 6 we study bilayers, composed of a nonmagnetic conducting and a magnetic layer. We develop a general phenomenology for the magnetic and charge dynamics, which are coupled by spin-orbit interactions. In contrast to Chapters 2-4, we focus on the long-wavelength magnetic dynamics, which is subject to current-induced torques and produces fictitious electromotive forces that drive charge dynamics.}

The main text is presented in four parts. In chapter 2, we develop a phenomenological theory of voltage induced torques in magnetic tunnel junctions. The reciprocal of this effect and spin-transfer torque can pump charge into an attached circuit when the magnet precesses. We calculate the resulting change in impedances due to this pumping as a function of applied magnetic field and thickness of tunneling spacer. Because the impedances due to voltage and current pumping are qualitatively distinct under variation of the magnetic field, we suggest that this measurement could be used as experimental differentiation between these effects.

In chapter 3 we study magnetic Josephson junctions wherein spin polarized Ohmic and supercurrent exert a torque on the magnetic layer. As a result, there is a nonlinear dynamic interplay between the magnetic order parameter and the phase of the superconducting parameter. This results in a modified stability diagram for both the magnet and superconductor. In particular, we find a nonmonotonic dependence of the critical current on the applied magnetic field and current. When the temperature is raised above the superconducting critical temperature, the leads become metallic and the equations of motion coincide with those of chapter 3.

In contrast to the monodomain models studied in other chapters, chapter 4 examines the effects of micromagnetics on the thermal stability of a typical MRAM bit. In addition to noting that a finite stiffness parasitically effects bit stability, we find that domain-wall nucleation and propagation is the dominant mode of thermal bit flipping.

Finally, in chapter 5 we derive a nonequilibrium expression for spin current between two magnetic leads biased by voltage, temperature or spin. The interaction on the dot is left general and the equation for current can be written as a function of the full retarded Green's function on the dot. We apply this methodology to a dot with large on-site Coulomb interaction.