A quantum dot (QD) confines electrons and holes to a size R comparable to the exciton Bohr radius. A single electron-hole pair (an exciton) recombines and emits a photon of energy EX; a second pair can be added to the s-shell to form a biexciton (XX), which recombines through a two-step cascade XX → X → 0.
Confinement raises the exciton energy above the bulk gap Eg (simplified Brus model):
E_X(R) = E_g + ħ²π²/(2R²)·(1/m_e* + 1/m_h*) − 1.8e²/(4πεε₀R)
The biexciton binding energy ΔXX is defined by how far the XX line sits from twice the exciton line:
ΔXX = 2·E_X − E_XX
E_XX = 2·E_X − ΔXX (redshifted line if ΔXX > 0, "bound")
ΔXX comes from a real competition inside the dot: direct electron-hole Coulomb attraction pulls the biexciton line down (binding, ΔXX > 0) while exchange/correlation repulsion between the two same-type carriers pushes it up (antibinding, ΔXX < 0). This sim uses the schematic competition
ΔXX(R) ≈ k₁/R − κ·k₂/R²
The direct term (∝ 1/R) wins for larger, weakly-confined dots, giving a small positive (bound, redshifted XX) binding energy that fades toward zero as R grows toward the bulk limit. The correlation term (∝ 1/R², steeper) wins as the dot shrinks, flipping the sign to antibound (blueshifted XX) — the same sign-crossover with size seen in real colloidal quantum dot photoluminescence experiments. There is no single universal formula for ΔXX (it is material- and shape-specific); this is a physically-motivated schematic, not a fit to one compound.
- Dot radius R — sets both the exciton confinement energy and the binding-energy crossover.
- Correlation strength κ — scales the correlation/exchange term, showing how a stronger many-body correlation shifts the antibinding regime to larger dots.
- Pump rate — how often a new electron-hole pair is injected; the dot cycles 0 → X → XX and decays back down, emitting one photon per recombination step.
- Temperature — phonon dephasing broadens each spectral line's linewidth, which can wash out a small ΔXX splitting at high T just as it does in real PL spectroscopy.
The lower panel is a live emission spectrum, plotted relative to E_X, accumulated from the actual simulated recombination events — exactly how ΔXX is measured in the lab: as the separation between the X and XX photoluminescence peaks.