Nonlinear conductance of a quantum point contact ($dI/dV=(e^2/h){\sum }_n[T_n({\mu }_S)+T_n({\mu }_D)]$)

At zero temperature, a rigid spin-degenerate transmission spectrum with symmetric source/drain bias has ${\mu }_{S,D}=E_F\pm eV/2$ and the displayed differential conductance. Half-integer plateaus in units $2e^2/h$ occur when the two bias-window edges sample different channel counts. The bias convention and electrostatic response matter.

In a rigid spectrum, source and drain threshold lines in a bias-versus-gate map satisfy $E_n(V_g)={\mu }_S$ or ${\mu }_D$. Where adjacent thresholds align with opposite edges of the bias window, their spacing equals the window width $e|V|$. For symmetric bias and spin degeneracy, neighboring channel counts at those edges give a differential electrical conductance halfway between integer $2e^2/h$ plateaus. The dc ratio $I/V$ is a different observable; barrier motion and heating can modify the interpretation.

A channel supporting ballistic electron transport carries incident populations set by different source and drain chemical potentials, rather than an everywhere-equilibrated local distribution with an Ohmic voltage gradient. Its electrostatic barrier responds self-consistently to contacts, gates and electrostatic screening. A symmetric rigid-barrier model places reservoir energies at $E_F\pm eV/2$ relative to the barrier. Asymmetric partition or bias-dependent transmission changes the weights in the differential electrical conductance, and must be included when extracting an energy scale.

For rigid spin-degenerate channel transmissions and ${\mu }_{S,D}=E_F\pm eV/2$, differentiating $I=(2e/h){\sum }_n{\int }_{{\mu }_D}^{{\mu }_S}{T}_n(E)dE$ gives $G_d=(e^2/h){\sum }_n[T_n({\mu }_S)+T_n({\mu }_D)]$. If the two edges sample neighboring integer channel counts, $G_d$ is a half-integer multiple of $2e^2/h$. Explicit bias dependence of the potential adds terms; the apparent fraction does not imply fractional carriers.