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

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.