Past exam of the physics of Nanoelectronic Systems course of the University of Cambridge / 2015 / Paper 1 / 3 / a

Transverse confinement in a short clean channel gives subband dispersionsA gate varies the subband thresholds and channel width. If elastic backscattering is negligible and spin is degenerate, the two-terminal linear conductance isA fully open subband has $T_n\simeq 1$, producing conductance steps in units $2e^2/h$. Magnetic spin splitting can resolve $e^2/h$ steps. A finite two-terminal resistance remains in a perfect ballistic channel because reservoirs supply and absorb a finite number of transport modes; it does not prove momentum-randomizing scattering inside the constriction.

At finite source-drain bias the two reservoir distributions differ over an energy window rather than at one Fermi energy. Choose ${\mu }_S-{\mu }_D=e{V}_{\mathrm{sd}}$ and take positive current from source to drain. For spin-degenerate channels,At zero temperature with a rigid transmission spectrum, $I=(2e/h){\sum }_n{\int }_{{\mu }_D}^{{\mu }_S}{T}_n(E)dE$. For symmetric bias ${\mu }_{S,D}=E_F\pm e{V}_{\mathrm{sd}}/2$ and neglecting an explicit bias response of $T_n$,If the two window edges lie above different numbers of subbands, the differential conductance can lie at half-integer values in units $2e^2/h$. For example, two open subbands at one edge and one at the other give $3e^2/h$. This is a bias-window effect, not fractional charge or a fractional Hall state.

Mapping transconductance versus gate voltage and source-drain bias reveals boundaries where a reservoir chemical potential crosses a subband bottom, enabling energy-spacing spectroscopy under a specified lever arm and bias division. Finite temperature rounds thresholds over $k_{B}T$; reflection, disorder, series resistance and a saddle-shaped barrier reduce or smooth plateaus. Large bias can alter the electrostatic potential, heat carriers or produce inelastic scattering, so the rigid-transmission differential formula is not generally valid without its assumptions. One-dimensional interaction effects can add anomalies beyond this independent-channel model.