University of Virginia

A single contact creates an open boundary condition to Schrodinger's equation through a self-energy or sink (outflow) and a source (inflow). The self-energy depends on the contact Green's function and the contact-device coupling. The device wavefunction is obtained by pre-multiplying the source with the device Green's function. The Green's function shows a delayed, causal response to the contact `source' electrons, and becomes especially large near resonant energy values.
Contacts open up an isolated device by turning sharp eigenstates into well-broadened resonant states. The antiHermitian part of the self-energy, the broadening matrix G, gives a decaying component to the time-dependent part of the wavefunction, making the corresponding time evolution operator non-unitary or irreversible in time. It also broadens the states, as seen from the LDOS at a point which equals the corresponding diagonal element of the spectral function [A] (divided by 2p).
The spectral function A is the imaginary (anti-Hermitian) part of the Green's function, just as the broadening is the anti-Hermitian part of the self-energy.
We now have a way to replace all quantities from our Chapter 1 independent level model by their matrix analogues in a given basis set, calculated within a certain model (e.g. tight binding). Thus, energy level gets replaced by Hamiltonian, electron number by density matrix, DOS by spectral function, etc. This allows us to generalize the equations for N and I derived in chapter 1 to a matrix.
The electron density matrix involves the energy-resolved electron occupancy or correlation function Gⁿ(E), given by the Keldysh equation GSⁱⁿG⁺, where the inscattering function Sⁱⁿ carries information about statistical properties of the contacts. For real metallurgical contact reservoirs each contribution to Gⁿ is equal to the partial spectral function from that contact times its Fermi function. For non-metallurgical contacts such as phonon or photon baths possibly with their own internal dynamics and dissipation, Sⁱⁿ requires a proper microscopic model for these dissipative processes.
For the current, one derives Landauer theory for metallurgical contacts, and the generalized NEGF equations otherwise. In the Landauer limit, the transmission T(E) is given by the Fisher-Lee formula that is the direct matrix analogue of the 1-D case. The real power of the NEGF technique, however, is the ability to incorporate the role of scattering through results that go far beyond Landauer.