  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 $4\times4$ matrices. Starting from the
Non-Equilibrium Green's Function (NEGF) formalism in the elastic,
phase-coherent transport regime, we develop spin generalized
Landauer-B\"uttiker formulas involving such $4\times 4$ conductances, for
multi-terminal devices in the presence of Normal-Metal (NM) 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 (FM) 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 2D. Our results allows 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.
