University of Virginia

At equilibrium the entire system (channel plus contacts) has a common Fermi energy E_F or electrochemical potential m . Current flows under an applied drain bias V_D that moves the contact states relati ve to each other, thereby splitting the contact electrochemical potential by qV_D (q = 1.6 x 10^-19 C).
The difference in Fermi functions f_1,2 creates a non-equilibrium situation. Levels far above (or below) E_F are emptied (or filled) by both contacts, while levels near the Fermi energy (within a few k_BT of m_1,2) suffer a difference in agenda where one contact keeps filling it and the other empties it, leading to a current flow.
The current at each contact depends on the strength of the coupling of the channel to the contacts, described by g_1,2, which also represents the inverse lifetime for the electron to escape into the leads (stronger cou pling creates a smaller lifetime), 1/t_1,2 = g_1,2/ħ.
Current at each contact through an energy level is proportional to the electron charge, the escape rate and the difference between what the contact wants to see, i.e., the Fermi function value at that energy, and the number that it actually supports. Equa ting currents at steady state, the number of electrons (times two for spin) is given by a weighted average of the Fermi functions, with weight factors determined by the couplings g_1,2.
The steady state current depends on the difference in Fermi functions, and a series combination of 1/g_1,2 (since the escape times t_1,2 add). Linearizing at low bias, the maximum conductance seems unbounded, determined only by g_1,2 with no fundamental upper limit.
Experimentally, there is a maximum conductance dI/dV_D that a single level can support. This is given by the quantum of conductance G₀ = 2q²/h which equals 77 mA/V (with a resistance quantum R₀ = 1/G₀ = 12.9 kW). The dependance on fundamental constants of nature indicates that it arises from fundamental physics, in this case, quantum mechanics.
As one couples the level stronger to the contacts to increase the conductance, the contact electron states (which form a continuum of closely spaced levels) spill into the channel and mix with the channel level. This creates a channel density of states (D OS) peaked around the level. Any loss of states from the contact into the channel is precisely compensated by a corresponding loss from the channel into the contact, so that the broadened level still holds one electron. It is this broadening ("spillage") that keeps the conductance of the channel in check even if we keep increasing g_1,2.