Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2022 / Paper 2 / Section A(ii) / 16

A gallium arsenide/aluminium gallium arsenide heterojunction has a conduction-band offset that can confine electrons on the GaAs side. Donors in the wider-gap AlGaAs supply electrons; band bending produces a narrow interfacial potential well. Motion normal to the interface is quantized into subbands, while motion within the plane remains approximately free with GaAs effective mass. Occupying the lowest confinement subband creates a two-dimensional electron gas.

Modulation doping spatially separates the ionized donors from the mobile electrons. An undoped spacer between the donor layer and the channel reduces Coulomb scattering from donor ions while still allowing charge transfer. This can produce higher mobility than doping the conducting channel itself. The spacer thickness balances lower scattering against weaker carrier supply and electrostatic control; interface roughness, alloy disorder, phonons and background impurities can still limit mobility. The channel is a finite-width quantum wavefunction, not a mathematically zero-thickness sheet.

For an in-plane parabolic band $E(k)=E_0+{\hslash }^2{k}^2/(2m^{\ast })$, one state occupies area $(2\pi {)}^2/A$ in two-dimensional wave-vector space. Including spin degeneracy two and the single GaAs conduction-band valley,Differentiation gives the constant two-dimensional density of statesUsing the supplied $h$ and effective mass givesThis counts available one-electron states, not the number actually occupied at arbitrary temperature. Occupation requires the Fermi-Dirac distribution; each additional subband would add another step above its own threshold.

Interpret the stated Fermi energy as measured above the bottom $E_0$ of the occupied subband, and take a degenerate, nearly zero-temperature gas. ThenFor a $0.100\mathrm{eV}$ filling energy,A Fermi energy referenced instead to an unspecified bulk conduction-band edge would not determine this number without the subband offset. The assumption that only one subband is occupied is imposed by the question; at this comparatively large density, an actual confinement potential should be checked before applying it to a device.

Treat the metal gate and conducting electron sheet as a parallel-plate capacitor separated by a uniform GaAs-like dielectric. Its capacitance per area isA gate-voltage change gives an electron-number change of magnitudeFor the ordinary accumulation convention, a more positive gate voltage attracts more electrons, while a more negative gate depletes them. The magnitude is about $13\%$ of the preceding sheet density.

This neglects depletion, trapped charge, finite well width and quantum capacitance. The occupied-subband quantum capacitance is $C_q/A=e^2{D}_{2D}\simeq 0.0449\mathrm{F}{\mathrm{m}}^{-2}$, much larger than $C_g/A$ here; putting the two in series would reduce the estimate by only about $1.3\%$ within this simple model.