Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2018 / Paper 2 / Section B / 19

A common implementation uses a gallium arsenide/aluminium gallium arsenide heterostructure with a two-dimensional electron gas near the interface. electrons supplied by remote donors occupy quantized normal-direction states, while an undoped spacer reduces ionized-impurity scattering. Surface metallic gates, insulated from the conducting layer, deplete selected regions when negatively biased.

Leave a small conducting island bounded by depleted regions and connect it weakly to source and drain reservoirs through two tunable constrictions. A plunger gate controls the island potential/population; barrier gates control tunnel coupling. Ohmic contacts connect the reservoirs to the measurement circuit. Vertical heterostructure confinement plus lateral depletion force defines the dot. A 2DEG without lateral gates would remain extended in its plane.

The original device diagram labels the gates, island, tunnel barriers and reservoirs. Charge stability and spectroscopy versus gate/bias should verify that an isolated weakly coupled island has actually formed; a lithographic outline alone is insufficient.

Let $C$ be the island's total electrostatic capacitance and $Q_{\mathrm{ext}}$ the charge offset induced by gates, reservoir potentials and fixed background charges. In the constant-capacitance model,Taking the difference givesChanging gate voltage changes $Q_{\mathrm{ext}}$ approximately linearly, for example through $C_g{V}_g$.

For fixed $N$, $U_N(Q_{\mathrm{ext}})$ is a parabola. At small temperature/bias, the preferred integer is the one nearest $Q_{\mathrm{ext}}/e$; the equilibrium energy follows the lower envelope of these parabolas. Adjacent charge states become degenerate at $Q_{\mathrm{ext}}=(N+\frac{1}{2})e$, allowing small-bias sequential current when the two barriers are transmissive.

The excess-electron charge $-Ne$ is a staircase. If “net charge” includes the induced compensating charge, define $q_{\mathrm{net}}=Q_{\mathrm{ext}}-Ne$; that residual charge is a sawtooth between $-e/2$ and $e/2$ in equilibrium. The opposite sign convention reverses the sawtooth. State the charge convention: an integer electron-charge staircase and a residual electrostatic-charge sawtooth describe different quantities.

A larger source–drain bias opens an energy window between reservoir electrochemical potentials. When a charge-addition/removal transition enters that window, sequential tunnelling becomes allowed even away from a zero-bias degeneracy. The low-conductance regions in a gate/bias map are Coulomb diamonds.

For an illustrative symmetric device with reservoir potentials $\pm e{V}_{\mathrm{SD}}/2$, let $q=Q_{\mathrm{ext}}/e$ and $E_{\mathrm{ch}}=e^2/C$. In the charge-$N$ interval, the classical blockade domain isIt shrinks to zero at $q=N\pm 1/2$. Unequal capacitances change edge slopes and voltage division; the figure's symmetry is a chosen model, not a universal device property.

Quantum confinement adds discrete single-particle levels and spin structure. In a constant-interaction approximation, successive addition energies contain $e^2/C$ plus the relevant orbital spacing, so diamond sizes need not all match. Excited-state tunnelling can produce extra conductance lines. At stronger coupling, cotunnelling and charge fluctuations can give leakage beyond a simple sequential classical model. Thus temperature and coupling must be small enough to resolve the charging and level structure separately.

An initially localized escaping electron occupies a superposition of lead momentum states. In a uniform free one-dimensional lead,A packet centered at $k_0$ moves with group velocity $\hslash k_0/m^{\ast }$ and generally spreads because its components have different velocities. For an initially unchirped minimum-uncertainty Gaussian wave packet,This illustrates dispersive evolution; the actual emission contact determines $a(k)$ and whether only outward-propagating components are appreciable. It does not instantaneously become a single lead plane wave.

A single-electron transistor used as a sensitive electrometer exploits Coulomb blockade: a nearby charge shifts the offset $Q_{\mathrm{ext}}$ and therefore the conductance peaks. This requires stable calibrated capacitances and low charge noise, rather than merely a small conductor.