Shrinking a semiconductor crystal to a quantum dot (QD) opens up its bandgap through quantum confinement. A simplified effective-mass (Brus) model gives:
E_g(R) = E_g,bulk + ħ²π²/(2R²)·(1/mₑ* + 1/m_h*)
− 1.8e²/(4πε₀ε_r R)
using PbSe-like parameters (E_g,bulk = 0.28 eV, mₑ*≈m_h*≈0.034m₀, ε_r≈23) — the material most studied for multiple-exciton-generation (MEG) solar cells. Smaller R → stronger confinement → larger E_g.
In a normal solar cell, a photon with energy hν > E_g creates one electron–hole pair (exciton); any energy above E_g is lost as heat (phonons). In a QD, weak carrier–phonon coupling and relaxed momentum conservation let some of that excess energy instead create additional excitons via inverse Auger recombination — this is MEG / carrier multiplication:
N_ideal(hν) = min( floor(hν / E_g),
1 + floor( (hν − E_g) / (ξ·E_g) ) )
The first term is the strict energy-conservation ceiling (each exciton needs at least E_g); the second reflects the empirical MEG onset near ξ·E_g above the gap (ξ ≈ 2–3 in real PbS/PbSe QDs, always ≥ 2 by energy bookkeeping for the *second* exciton). Because internal MEG efficiency is never 100%, each candidate extra exciton beyond the first only survives with ~65% probability here, which is why the histogram's average quantum yield sits below the idealized staircase.
- QD radius — sets Eg via confinement; smaller dots need less absolute excess energy to trigger MEG.
- Photon energy — each incoming photon (yellow, falling) carries this fixed energy when it strikes a dot.
- MEG threshold ξ — how many gap-widths of excess energy are needed per extra exciton (lower ξ ⇒ MEG turns on more easily, as in the best real QD absorbers).
- Generated electrons (blue) drift to the top contact, holes (red) drift to the bottom contact — collected pairs are what raise photocurrent above the single-exciton limit.
Real-world relevance: PbSe and PbS quantum-dot solar cells are the leading experimental platform for MEG, because carrier multiplication can in principle push the single-junction efficiency limit above the ~33% Shockley–Queisser ceiling by harvesting UV/blue photons as more than one electron each.