Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2017 / Paper 2 / Section A(ii) / 15

Let $x$ run along the wire and $y$ across it. In a separable approximation,Transverse confinement gives discrete levels $E_n^{\perp }$, but the long unconfined direction has a nearly continuous longitudinal wave vector and a plane-wave factor. The result is a family of one-dimensional subbands, not a completely discrete spectrum in both in-plane directions. For a hard-wall width $W$, $E_n^{\perp }={\hslash }^2{\pi }^2{n}^2/(2m^{\ast }{W}^2)$; an electrostatic confinement potential gives different thresholds. The suppressed out-of-plane degree of freedom is already frozen into the 2DEG subband.

One suitable platform is a monolayer molybdenum disulfide semiconductor. An electrostatic accumulation gate supplies conduction electrons whose motion is confined to the atomic sheet; source and drain contacts act as reservoirs. A clean insulating substrate and dielectric or encapsulation reduce disorder and prevent gate leakage. Two split gates deplete the adjacent sheet and leave a narrow conducting gap, thereby confining the in-plane transverse direction while retaining transport along the gap.

Apply a small source-drain voltage, measure the current and determine linear-response conductance while tuning the split-gate voltage and carrier density. A graphene sheet is another two-dimensional carrier system, but its lack of an ordinary gap complicates purely electrostatic confinement; etched boundaries or a suitable gapped architecture may be needed. A two-dimensional material does not by itself guarantee ballistic transport or resolved subbands.^[ref]

For a clean, adiabatically connected GaAs channel at low temperature and small bias, opening successive spin-degenerate subbands givesConductance therefore shows a 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.

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.