University of Virginia

The coupling of a channel with contacts gives its electrons a finite lifetime t = ħ/g, with its probability of residing in the device decreasing exponentially with time as e^-t/t. Just as in signal processing, an instantaneous pulse in time corresponds to a wide-band signal in frequency domain (the two connected by a Fourier transform), quantum mechanically a fast escaping electronic probability leads to a broadened level in energy space(energy playing the role of frequency for electrons as it does for photons as well). The Fourier transform of an exponentially decaying time-signal is a Lorentzian in frequency domain.
The applied source-drain bias not only empties/fills levels, but moves the levels as well through a local potential U. One needs to solve Poisson's equation to get the full 3-D potential profile U(r). For starters, we assume a single scalar value of this potential, determined by a combination of the source, drain and gate capacitances. The Laplace part of U is given by the capacitance-weighted average of the imposed source-drain-gate potentials, while the Poisson part is given by the single electron charging energy (i.e., energy to charge a capacitor with one electron) times the charge imbalance in the device.
Both quantum and electrostatic effects are thus crucial for our minimal model. The quantum broadening of the level keeps its conductance in check by siphoning part of the level outside the conducting window. The electrostatic potential can allow the levels to`slip' relative to the source. For instance, ultrasmall transistors suffer from deleterious short-channel effects such as non-saturating currents precisely because their gates are not close enough to hold the channel levels fixed from slippage.
We can augment our minimal model based on rate equations to incorporate these quantum and electrostatic effects. The charge and current expressions need to be integrated over the quantum broadening function (or density of states), which in turn needs to be rigidly shifted by the local electrostatic potential U. The potential, evaluated using a capacitive network, must be calculated self-consistently, since the presence of U shifts the DOS and thus alters the electron number, which in turn affects U by charging the capacitors.