Quantum Dots — Exciton & Size Quantization

Quantum Dot Parameters

Material: Dot radius R (nm): 2.5 nm Temperature T (K): 300 K Quantum levels shown: 3 Dielectric constant ε_r: 6.2

Brus equation: E_QD = E_gap(bulk) + ℏ²π²/(2μR²) − 1.786e²/(4πε₀ε_r·R), where μ = m_e·m_h/(m_e+m_h) is reduced exciton mass. Three terms: bulk gap + confinement energy (∝1/R²) − Coulomb binding (∝1/R).
Confinement regimes: Strong (R≪a_B), intermediate, weak (R≫a_B), a_B = exciton Bohr radius. CdSe: a_B≈5.5 nm, E_gap=1.74 eV, m_e=0.13m₀, m_h=0.45m₀.
Size quantization: Particle-in-sphere eigenstates E_{n,l} = ℏ²α_{n,l}²/(2m*R²), α_{n,l} = nth zero of jₗ(x). Photoluminescence wavelength blueshifts as R decreases.