= Solution
Transverse confinement in a short clean channel gives subband dispersions
$$
E_n(k)=E_n^\perp+\frac{\hbar^2k^2}{2m^*}.
$$
A gate varies the subband thresholds and channel width. If elastic backscattering is negligible and <spin> is degenerate, the two-terminal linear conductance is
$$
\boxed{G=\frac{2e^2}{h}\sum_n T_n(E_F).}
$$
A fully open subband has $T_n\simeq1$, 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 $\mu_S-\mu_D=eV_{\rm sd}$ and take positive current from source to drain. For spin-degenerate channels,
$$
\boxed{I=\frac{2e}{h}\sum_n\int T_n(E;V_{\rm sd})[f_S(E)-f_D(E)]\,dE.}
$$
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 $\mu_{S,D}=E_F\pm eV_{\rm sd}/2$ and neglecting an explicit bias response of $T_n$,
$$
\boxed{\frac{dI}{dV_{\rm sd}}=\frac{e^2}{h}\sum_n
[T_n(\mu_S)+T_n(\mu_D)].}
$$
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.
\Image[/past-exam-of-the-physics-of-nanoelectronic-systems-course-of-the-university-of-cambridge/2015/paper-1-channel-bias.png]
{title=Original ideal one-dimensional subbands sampled by a symmetric bias window, with integer and half-integer differential-conductance plateaus}
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 $k_BT$; 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.
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