Past exam of the quantum condensed matter course of the University of Cambridge / 2017 / 1 / b / Solution

A self-assembled InAs quantum-dot single-photon source uses small indium arsenide islands inside a larger-gap gallium arsenide host. Strain during epitaxial growth favors islands after a thin wetting layer has formed. Their characteristic diameter is of order $10$–$30\mathrm{nm}$ and their height a few nanometres, for example $2$–$5\mathrm{nm}$. Growth conditions determine the areal density: low-density samples suitable for isolating dots can be around $10^8$–$10^9\mathrm{c}{\mathrm{m}}^{-2}$, while ordinary dense layers can be much denser. The particular electrically driven device underlying the supplied image had about $5\times 10^8\mathrm{c}{\mathrm{m}}^{-2}$. These are characteristic scales, not dimensions deducible from the emission image.^[ref]

Quantum confinement in the dot traps both Electrons and electron holes in three directions, giving discrete energy levels. In a junction device, injected carriers relax into these levels. Electron-hole recombination then emits a photon, and the dot must be repopulated before emitting again on that same transition. Spatial selection of one dot, spectral selection of one line, and well-separated injection pulses make this a single-photon source. Photon antibunching, rather than low average brightness alone, establishes suppression of simultaneous photons. In particular, collecting both photons of a cascade without selecting a line would not produce exactly one collected photon per excitation.

The lower-current feature $X$ is the transition from one confined exciton to the empty dot. The higher-current feature $X_2$ is the first step of the biexciton cascade $XX\to X\to 0$: it becomes strong when the dot can contain two electron-hole pairs. In the weak-pumping limit the probabilities of one and two pairs scale approximately as $I$ and $I^2$, respectively. Coulomb and confinement energies shift the two transitions relative to one another; $X_2$ is not a photon of twice the $X$ energy. In this particular image the $X_2$ photon is about $4$–$5\mathrm{meV}$ higher in energy. With $B_{XX}=2E_X-E_{XX}$, this is a negative biexciton binding energy, since $E_{XX}-E_X=E_X-B_{XX}$.

For the mass estimate, the original plot places the $X$ line at approximately $E_X=1.3945\mathrm{eV}$, with graph-reading precision of order $0.5\mathrm{meV}$. Under the prescribed identification of photon energy with the full lowest exciton excitation energy,The three-dimensional screened Wannier-Mott exciton has the hydrogenic exciton binding energyThereforeReading the line as $1.395\mathrm{eV}$ instead gives $0.331m_0$; about $0.33$–$0.34m_0$ is an appropriate precision. This inference uses the supplied pair-creation energy and a three-dimensional screened hydrogenic model. In a real dot, strain and confinement also alter transition energies, so it is an effective model estimate and does not independently determine either individual carrier mass.