Non-Equilibrium Green's Function Based Circuit Models for Coherent Spin Devices

With recent developments in spintronics, it is now possible to envision “spin-driven” devices with magnets and interconnects that require a new class of transport models using generalized Fermi functions and currents, each with four components: one for charge and three for spin. The corresponding impedance elements are not pure numbers but
$\text{4}\times \text{4}$
matrices. Starting from the Non-Equilibrium Green's Function formalism in the elastic, phase-coherent transport regime, we develop spin generalized Landauer-Büttiker formulas involving such
$\text{4}\times \text{4}$
conductances, for multi-terminal devices in the presence of normal-metal leads. In addition to usual “terminal” conductances describing currents at the contacts, we provide “spin-transfer torque” conductances describing the spin currents absorbed by ferromagnetic regions inside the conductor, specifying both of these currents in terms of Fermi functions at the terminals. We derive universal sum rules and reciprocity relations that would be obeyed by such matrix conductances. Finally, we apply our formulation to two example Hamiltonians describing the Rashba and the Hanle effect in two-dimensional. Our results allow the use of pure quantum transport models as building blocks in constructing circuit models for complex spintronic and nano-magnetic structures and devices for simulation in SPICE-like simulators.