University of Virginia

A minimal set of transport equations thus involves solving rate equations through levels broadened by quantum mechanics (`spillage') and shifted by their local electrostatic potential (`slippage').
To simulate the conductance in any system we need to know the equilibrium electronic structure of the channel material (electron number N₀ and density of states D(E), measured through spectral techniques or calculated through its bandstructure). Next we need the electrostatic parameters, namely, contact capacitances which depend on their geometries and dielectric constants. We need to know the contact properties such as their Fermi energy and coupling strengths with the channel. Finally, we need to know the operating conditions of the device (temperature, drain and gate voltages etc).
We thus have a universal transport formalism that works for various materials, devices and dimensions. This provides a microscopic, `bottom-up' derivation of all well-known classical equations for resistors, MOSFETs, thermionic and tunneling currents. In addition, the formalism works where classical theories do not, at nanoscale dimensions where concepts such as mobility, diffusion constant or effective mass are plain meaningless.
If the levels act independently as we have assumed so far, one can simply add contributions from each level contributing to the DOS and get a quick-and-dirty answer. In general, however, levels hardly act independently but tend to show quantum interference. Under these conditions, we cannot just add terms for each level, but need to deal with matrices instead. We will develop this matricized generalization of the rate equations, known as the Non-Equilibrium Green's Function (NEGF) formalism.
Coherent scattering, such as due to impurities, can be readily included in our matricized band-structure. But the NEGF formalism also allows us to capture incoherent dephasing due to processes that actually alter the environment, such as due to vibrations or photons that also communicate with the external environment.
The charging term has been oversimplified with an effective capacitor that is being charged up, ignoring the simple fact that an electron cannot feel a potential due to itself. In reality, electrons tend to exclude each other and correlate cooperatively to cut down the charging cost, which cannot be captured with a simple potential U. Weak correlations can be included in the NEGF formalism, but stronger ones may need a completely different transport approach involving many-electron physics.