= Solution
For a clean, adiabatically connected GaAs channel at low <temperature> and small bias, opening successive spin-degenerate subbands gives
$$
\boxed{G=N\frac{2e^2}{h}.}
$$
Conductance therefore shows a \b[staircase of plateaux], rather than the continuously geometry-dependent classical Ohmic result. Each fully transmitted mode contributes approximately $77.5\,\mu\mathrm S$. More positive, or less negative, depletion-gate <voltage> opens a conventional GaAs constriction; more negative <voltage> closes it.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2017/paper-2-conductance.png]
{title=Conductance rises in quantum steps as a depletion-gate constriction opens}
{source=paper-2-conductance.py}
Require subband spacings larger than thermal and bias broadening and a <mean free path> exceeding the constriction length. The general <Landauer formula> is $G=(e^2/h)\sum_\alpha T_\alpha$ with each spin/valley channel counted separately. A material with additional valley degeneracy need not have the same step sequence as GaAs, and imperfect transmission or contact resistance blurs the ideal plateaux.
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