Compute conductance from transmission channels

A transverse orbital mode supplies right-moving and left-moving states. The excess incident populations from reservoirs at slightly different electrochemical potentials generate net current. For an elastic scattering region, each transmission eigenvalue $T_n$ lies between zero and one. A unit incoming state flux times the one-dimensional density of states cancels the group velocity, giving an electrical conductance quantum per spin-resolved channel. The transmission eigenvalues are probabilities, not wave amplitudes that should be squared a second time.

If each orbital mode has two identical unresolved spin channels, it is convenient to use $G_0=2e^2/h$ and sum only orbital transmissions. If a field or spin-dependent scattering removes that duplication, use $e^2/h$ and enumerate spin-resolved channels explicitly. Valley multiplicity likewise must be either enumerated or included once as a prefactor under a justified degeneracy. The two propagation directions are already part of the reservoir imbalance argument; they do not provide an additional conductance factor two.

At finite temperature, a small voltage changes occupations over an energy window of order $k_BT$ around the chemical potential. Conductance is a weighted average of transmission over that window, with weight minus the energy derivative of the Fermi function. Evaluating transmission only at the Fermi energy is reliable when it varies little across the window. Near a subband threshold, a resonance, or a narrow tunneling feature, thermal averaging can substantially change the result even in linear response.

For an ideal step transmission above a threshold, the weighted integral reduces to the Fermi occupation evaluated at that threshold. The resulting conductance rises smoothly rather than jumping sharply at finite temperature. Larger bias requires an integral over the difference between two reservoir Fermi functions, and transmission can itself depend on the bias-dependent potential. Substituting a large terminal voltage into a small-signal conductance law without checking that potential response is not a general nonlinear transport calculation.

A perfectly transmitting single spin-degenerate orbital channel has two-terminal resistance $h/(2e^2)$, rather than zero. The finite count of available incident modes limits conductance even without internal backscattering. In the ideal reservoir picture, equilibration and dissipation occur in the reservoirs; the scattering region need not possess a local Ohmic voltage gradient. A four-terminal resistance depends on how voltage probes couple and cannot automatically be equated with the inverse of this two-terminal conductance.

Phase coherence allows amplitudes along alternative paths to interfere and makes resonant transmission strongly phase sensitive. Ballistic motion refers to the lack of momentum-randomizing scattering over the relevant length; an elastically disordered conductor may remain phase coherent while being diffusive. The simple coherent scattering description is therefore not restricted to perfectly clean wires. Strong inelastic processes or interactions may require additional reservoirs, kinetic equations, or a many-body treatment, rather than merely replacing every channel by an arbitrary classical transmission number.