Engineering & Science
Burstein-Moss Shift: Degenerate Doping Optical Bandgap Widening
Calculate the conduction-band Pauli-blocking blue-shift ΔE_BM = 3.64645E-15 * n^(2/3) / m_eh^* (eV) and apparent optical bandgap E_g,opt = E_g0 + ΔE_BM in ITO and AZO transparent conducting oxides.
How heavy tin doping makes Indium Tin Oxide (ITO) BOTH metallic and UV-transparent!
How can smartphone touchscreens and solar-cell top electrodes made of Indium Tin Oxide (ITO) conduct electricity like a metal while staying crystal-clear across visible light? Elias Burstein (1954) and Trevor S. Moss (1954) explained the quantum Pauli-exclusion effect: when donor concentration n exceeds ~10^19–10^21 cm^-3, degenerate electrons flood the bottom of the conduction band up to Fermi wavevector k_F = (3π^2 n)^(1/3)—blocking vertical valence-to-conduction photon transitions below k_F and blue-shifting the optical absorption edge by ΔE_BM = (ħ^2 / (2 * m_eh^*)) * (3π^2 n)^(2/3)!
Evaluating exact CODATA constants for carrier density n in cm^-3 and reduced electron-hole effective mass m_eh^* = (1/m_e^* + 1/m_h^*)^-1 (in units of m_0) yields ΔE_BM (eV) = 3.64645 × 10^-15 * n^(2/3) / m_eh^*! Watch ITO (intrinsic E_g0 = 3.550 eV, m_eh^* = 0.30): at moderate n = 1.0 × 10^20 cm^-3, ΔE_BM = +0.262 eV (E_g,opt = 3.812 eV); doping to n = 1.0 × 10^21 cm^-3 widens the optical gap by +1.215 eV up to E_g,opt = 4.765 eV—pushing absorption deep into the ultraviolet!
| ITO Degenerate Carrier Density n (Eg0 = 3.55 eV, m_eh* = 0.30) | Burstein-Moss ΔE_BM (eV) | Apparent Optical Eg,opt (eV) | UV Cutoff Wavelength |
|---|---|---|---|
| Undoped In2O3 Limit (n -> 0) | 0.000 eV | 3.550 eV | 349 nm (Near-UV Edge) |
| Moderate TCO Film (n = 1.0 × 10^20 cm^-3) | +0.262 eV | 3.812 eV | 325 nm |
| Heavy Degenerate ITO (n = 1.0 × 10^21 cm^-3) | +1.215 eV | 4.765 eV | 260 nm (Deep UV Window!) |
Burstein-Moss Conduction-Band Filling Blue-Shift ΔE_BM (eV)
Evaluates 3.64645E-15 * (n_cm3 ^ (2 / 3)) / m_eh^* in electron-volts.
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0), "Invalid TCO Inputs", ROUND(3.64645E-15 * (B4 ^ (2 / 3)) / B3, 3))Apparent Widened Optical Bandgap E_g,opt = E_g0 + ΔE_BM (eV)
Adds the Burstein-Moss Pauli-blocking shift to fundamental bandgap B2 (eV).
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0), "Invalid TCO Inputs", ROUND(B2 + 3.64645E-15 * (B4 ^ (2 / 3)) / B3, 3))Try the example
Use the original values or change them to test your understanding.