University of Virginia

We thus have three kinds of expression for the charge and currents, an independent level model that works with a set of energy levels (perhaps catalogued as a vector) and giving a common sense expression for these, a matrix model that works for a device Hamiltonian between two contacts obtained by matricizing the input parameters of the independent level model, and a most general model that defines generic in-scattering and self-energy functions for any contact (real contact leads or virtual scattering `contacts' due to vibrations, spins, etc).
In deriving the most general expressions, we start with the partitioned Schrodinger equation with open boundary conditions at the contacts. We then impose certain external restrictions on the electrons in the contacts, stipulating that they are uncorrelated with each other like photons from a light bulb (as opposed to device electrons that can interfere like photons from a laser). This puts a condition on the contact electron density matrix, dictating that off-diagonal components lose their phase coherence through scattering that is essential for thermalizing the electrons, and diagonal terms simply pick up this equilibrium Fermi-Dirac distribution multiplied by the corresponding spectral functions.
The current flow problem thus amounts to a process dictated by two reservoirs that are each separately in equilibrium but out of equilibrium with each other (due to the applied voltage). The current is determined not by the source wavefunctions themselves (which are fluctuating rapidly in the source), but by the electron `noise' in the contacts (akin to the power in a noisy transmission line) that determine statistical averaged properties of bilinear products of device electronic wavefunctions, such as their averaged charge density or time-averaged steady-state current.
The real power of the NEGF equations is the relative readiness with which one can start modeling electronic properties of various systems at different levels of detail -- from atomic scale transport in thin films, nanotubes or molecular wires to more compact level circuit theories for silicon devices. As an example, one could set up a calculation of the DOS and transmission of a 1-D wire, which requires us to first calculate the device Hamiltonian, the lead self-energies, thence the Green's functions, and finally the DOS through the spectral function and the transmission.
It is easy to check interesting physics through this approach -- how a dopant defect creates a sharp level outside a metallic band, how a multimoded conductor has integer transmission blocks that set in as each new subband gets energetically accessed, and how the transmission changes from monotonic increasing to oscillatory as we impose a potential on this wire that varies from a tunneling barrier to a double barrier structure. The current-voltage characteristics for these systems (ohmic, exponentially increasing, or showing negative differential resistance) also follow by integrating the transmission over the relevant voltage window set by the bias-dependent contact electrochemical potentials.