Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2021 / Paper 2 / Section A(ii) / 15

The map shows Coulomb blockade with discrete single-electron charging: low-current diamonds separate charge-stable regions from finite-bias transport windows. A small conducting island or quantum dot must be tunnel-coupled to source and drain, with a gate shifting its electrochemical potential. Charging prevents sequential transport until an island addition/removal energy enters the bias voltage window.

The tunnel resistances should substantially exceed the resistance quantum $h/e^2\simeq 25.8\mathrm{k\Omega }$, preventing charge fluctuations from washing out integer island occupancy, while still permitting a measurable current. Temperature and tunnel broadening must be small compared with the addition-energy scale. Here the smaller diamonds imply $e^2/C_{\Sigma }\simeq 2.5\mathrm{meV}$, so $k_{B}T\ll 2.5\mathrm{meV}$, or $T\ll 29\mathrm{K}$; resolving the inferred 1 meV orbital structure requires the stricter scale $T\ll 12\mathrm{K}$. Stray capacitance and unintended multiple islands can complicate this simple interpretation.

Assume the usual one-lead-biased measurement, $V_s=V_{sd}$ with drain grounded and the gate referred to that ground; the printed map does not separately specify the wiring. This assumption is needed to assign individual source/drain capacitances. In this Coulomb diamond capacitance extraction, the smaller zero-bias gate intervals are 5 mV wide. In the constant interaction model with no orbital gap between those additions,An upper diamond tip lies at gate 123.75 mV between the zero-bias vertices 120 and 125 mV and at bias 2.50 mV. Its positive and negative edge slopes are therefore $s_{+}=2.50/3.75=2/3$ and $s_{-}=-2.50/1.25=-2$. Reservoir alignment givesThe gate capacitance is not negligible in the second denominator. ConsequentlyThe consistency check is $e^2/C_{\Sigma }=2.50\mathrm{meV}$, matching the smaller tip's bias energy. The conventional electrostatic charging energy $E_C=e^2/(2C_{\Sigma })$ is 1.25 meV; it is half the charging energy contribution to the addition energy. A different applied-bias convention changes the slope formulas, so the convention must accompany the extraction.

The addition energy has both charging and orbital terms,The larger diamonds indicate quantum dot shell closure after 2, 6 and 12 electrons. A two-dimensional isotropic harmonic oscillator confinement, $V(x,y)=m^{\ast }{\omega }_0^2(x^2+y^2)/2$, explains that sequence: shell $q=n_x+n_y$ has $q+1$ orbital states and capacity $2(q+1)$ after spin degeneracy. The cumulative capacities are $(q+1)(q+2)=2,6,12,20,\dots$. Within a degenerate shell, the extra orbital term vanishes to this approximation; crossing a shell boundary costs an extra $\hslash {\omega }_0$.

The gate capacitance lever arm is ${\alpha }_g=C_g/C_{\Sigma }=1/2$. Large and small gate widths differ by $7-5=2\mathrm{mV}$, so $\hslash {\omega }_0\simeq e{\alpha }_g(2\mathrm{mV})=1.0\mathrm{meV}$. Equivalently the tip energies differ by $3.5-2.5=1.0\mathrm{meV}$. Confinement anisotropy, exchange and disorder would split the ideal degeneracies; the diagram supports approximate shell symmetry rather than proving a perfectly parabolic potential.