Engineering & Science
Wannier-Mott Exciton Binding Energy (meV) & Exciton Bohr Radius (nm)
Calculate semiconductor electron-hole exciton binding energy E_b = 13605.7 * (μ / m_0) / ε_r^2 (meV) and exciton Bohr radius a_B^* = 0.052918 * ε_r / (μ / m_0) (nm) to determine room-temperature exciton stability and quantum-dot confinement size.
Why GaAs excitons ionize instantly at 300 K—while ZnO excitons survive room temperature!
Gregory Wannier (1937) and Nevill Mott (1938) showed that a photoexcited conduction electron and valence hole form a hydrogen-like bound state inside a crystal whose Coulomb attraction is screened by dielectric constant ε_r and scaled by reduced mass μ / m_0 = (m_0/m_e^* + m_0/m_h^*)^-1! Scaling the 13,605.7 meV hydrogen Rydberg and 0.052918 nm Bohr radius immediately gives E_b = 13605.7 * (μ / m_0) / ε_r^2 (meV) and a_B^* = 0.052918 * ε_r / (μ / m_0) (nm)!
Compare thermal energy k_B T = 25.85 meV at 300 K across two classic optoelectronics semiconductors: in GaAs (ε_r = 12.9, μ / m_0 = 0.058), dielectric screening slashes E_b to 4.74 meV (< 25.85 meV, so excitons dissociate into free carriers at 300 K!) while ballooning exciton Bohr radius to a_B^* = 11.77 nm (setting the < 23 nm quantum-dot strong-confinement regime!). In wide-gap ZnO (ε_r = 8.5, μ / m_0 = 0.318), E_b leaps to 59.88 meV (> 2.3 × k_B T, enabling room-temperature UV exciton lasing!) with compact a_B^* = 1.41 nm!
| Semiconductor Crystal | εr & Reduced Mass μ/m0 | Exciton Binding Eb (meV) | Exciton Bohr Radius aB* (nm) |
|---|---|---|---|
| GaAs Infrared / Quantum Dot | εr = 12.9, μ = 0.058 | 4.74 meV (< 25.9 meV kT) | 11.77 nm (Large QD Threshold) |
| CdSe Colloidal Quantum Dot | εr = 9.5, μ = 0.102 | 15.38 meV | 4.93 nm (Tunable Visible QDs) |
| ZnO UV Exciton Laser | εr = 8.5, μ = 0.318 | 59.88 meV (> 2.3× Room kT!) | 1.41 nm (Tightly Bound) |
Wannier-Mott Exciton Binding Energy E_b = 13605.7 * μ / ε_r^2 (meV)
Scales the 13,605.7 meV hydrogen Rydberg by reduced effective mass ratio B3 and inverse squared dielectric constant B2^2.
=IF(OR(B2 < 1, B3 <= 0), "Invalid Exciton Inputs", ROUND(13605.7 * B3 / (B2 ^ 2), 2))Exciton Bohr Radius a_B^* = 0.052918 * ε_r / μ (nm)
Computes the characteristic electron-hole orbital radius in nanometers—quantum dots with radius R < a_B^* enter strong 3D quantum confinement.
=IF(OR(B2 < 1, B3 <= 0), "Invalid Exciton Inputs", ROUND(0.052918 * B2 / B3, 2))Try the example
Use the original values or change them to test your understanding.