Shot noise is the name given to the time-dependent non-equilibrium current (or voltage) fluctuations which persist down to zero temperature and are fundamentally related to the discrete nature of the electron charge. Over the past two decades it has become a major tool for gathering information about microscopic mechanisms of transport and correlations between charges which cannot be extracted from traditional conductance measurements. Recently a handful of theoretical and experimental studies have suggested that shot noise in systems with spin-dependent interactions provides a sensitive probe to differentiate between scattering from magnetic impurities, spin-flip scattering, and continuous spin precession effects on semiclassical or quantum transport of injected spin-polarized currents. This is due to the fact that any spin flip converts spin-↑ subsystem particle into a spin-↓ subsystem particle, where the two subsystems differ when spin degeneracy is lifted. Thus, the nonconservation of the number of particles in each subsystem generates additional source of current fluctuations. Here we generalize the scattering theory of quantum shot noise to include the full spin-density matrix of electrons. This formalism yields the spin-resolved shot noise power applicable for a generic spintronic device where partially polarized charge current or even pure spin current is injected from a spin-filtering or ferromagnetic electrode into a quantum-coherent nanostructure governed by arbitrary spin-dependent interactions.;The developed formalism [2, 5] is applied in Chapter 5 to diffusive multichannel quantum wires with the Rashba spin-orbit (SO) coupling sandwiched between ferromagnetic source and ferromagnetic or normal drain electrodes. The crucial role played by the SO interactions in all-electrical control of spin in semiconductor nanostructures has ignited recent studies of their signatures on the shot noise. We investigate what is the effect of the Rahsba SO coupling on the shot noise and look for a relationship between the degree of quantum coherence of transported spins and the shot noise of charge currents. This allows us to propose electrical shot noise-based scheme to probe spin as a measurable degree of freedom.;Injection of unpolarized charge current through the longitudinal leads of a four-terminal two-dimensional electron gas with the Rashba SO coupling and SO scattering off extrinsic impurities is responsible not only for the pure spin Hall current in the transverse leads, but also for nonequilibrium random time-dependent current fluctuations. We employ the spin-dependent scattering approach in Chapter 6 [3, 5] to analyze the shot noise of transverse pure spin Hall current and zero charge current, or transverse spin current and non-zero charge Hall current, driven by unpolarized or spin-polarized injected longitudinal charge current, respectively. Since any spin-flip acts as an additional source of noise, we argue that these shot noises provide a unique experimental tool to differentiate between intrinsic and extrinsic SO mechanisms underlying the spin Hall effect in paramagnetic devices.;Recently graphene---a one-atom-thick crystal of carbon atoms arranged into a honeycomb lattice---has emerged as one of the most promising materials for future nanoelectronic devices. It combines exceptional sample quality and accessibility with the unique possibility to explore quantum electrodynamics phenomena in a condensed matter system since current is carried by massless Dirac fermions behaving as charged neutrinos. Furthermore, special nanostructures derived from graphene, the so called zigzag nanoribbons, favor ferromagnetic ordering along their edges. Recently shot noise measurements have been used to characterize ballistic transport through evanescent states introduced into clean undoped graphene strips by the attached metallic electrodes. We demonstrate in Chapter 7 [4] that this shot noise can be substantially modified in zigzag nanoribbons due to the topology of their edges inducing localized states that facilitate ferromagnetic ordering along the edge when electron-electron interactions are taken into account.
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