= 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 $10^8$–$10^9\,\mathrm{cm^{-2}}$, while ordinary dense layers can be much denser. The particular electrically driven device underlying the supplied image had about $5\times10^8\,\mathrm{cm^{-2}}$. These are characteristic scales, not dimensions deducible from the emission image.https://doi.org/10.1126/science.1066790[Device description and figure 1, printed p.103]{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 $XX\to X\to0$: 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>,
$$
E_B=E_g-E_X\simeq1.415-1.3945=0.0205\,\mathrm{eV}.
$$
The three-dimensional screened <Wannier-Mott exciton> has the hydrogenic <exciton binding energy>
$$
E_B=\frac{\mu e^4}{2(4\pi\epsilon_0\varepsilon_r)^2\hbar^2}=13.6057\,\mathrm{eV}\,\frac{\mu/m_0}{\varepsilon_r^2},\qquad \frac1\mu=\frac1{m_e^*}+\frac1{m_h^*}.
$$
Therefore
$$
\boxed{\frac\mu{m_0}=\frac{(0.0205)(15)^2}{13.6057}\simeq0.339,\qquad \mu\simeq3.09\times10^{-31}\,\mathrm{kg}.}
$$
Reading the line as $1.395\,\mathrm{eV}$ instead gives $0.331m_0$; \b[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.
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