Engineering & Science
Moss’s Rule: Semiconductor Refractive Index From Bandgap (n^4 * Eg = 95 eV)
Estimate infrared semiconductor refractive index n = (95 / E_g)^0.25, high-frequency dielectric constant eps_inf = n^2, and normal-incidence Fresnel air-interface reflectance R (%) from optical bandgap E_g (eV).
Why narrow-bandgap infrared detectors always have high refractive indices!
Compare diamond or GaN (wide bandgap E_g = 3.4–5.5 eV, refractive index n ~ 2.3–2.4) against Silicon (E_g = 1.12 eV, n ~ 3.4) and Germanium or InSb infrared optics (E_g = 0.17–0.66 eV, n ~ 3.5–4.0): narrower bandgaps make valence electrons far easier to polarize, boosting high-frequency permittivity eps_inf = n^2! Trevor S. Moss (1950, updated constant K = 95 eV) derived from Bohr-like photoconductivity scaling the celebrated Moss’s Rule: n^4 * E_g = 95 eV, or n = (95 / E_g)^0.25!
Photonics engineers use Moss’s Rule to estimate refractive index and uncoated Fresnel reflection loss R = ((n - 1) / (n + 1))^2 across novel ternary/quaternary III-V and chalcogenide alloys! At E_g = 1.12 eV (Silicon), Moss predicts n = (95 / 1.12)^0.25 = 3.035 (uncoated single-surface Fresnel reflection = 25.43%); narrowing to E_g = 0.66 eV (Germanium) raises n to 3.464, whereas widening to E_g = 3.40 eV (GaN) drops n to 2.299!
| Semiconductor Bandgap Eg (eV) | Moss Refractive Index n = (95 / Eg)^0.25 | Dielectric eps_inf = n^2 | Uncoated Fresnel Reflection (%) |
|---|---|---|---|
| Germanium IR Optics (Eg = 0.66 eV) | 3.464 | 12.00 | 30.46% |
| Silicon Near-IR (Eg = 1.12 eV) | 3.035 | 9.21 | 25.43% |
| Gallium Nitride Blue LED (Eg = 3.40 eV) | 2.299 | 5.29 | 15.51% |
Moss’s Rule Predicted Refractive Index n = (95 / E_g)^0.25
Evaluates fourth-root scaling (95 / B2)^0.25 for positive semiconductor bandgaps B2 (eV).
=IF(B2 <= 0, "Invalid Bandgap Eg", ROUND((95 / B2) ^ 0.25, 3))Normal-Incidence Air-Semiconductor Fresnel Reflectance R (%)
Computes 100 * ((n - 1) / (n + 1))^2 to size anti-reflection (AR) coating requirements.
=IF(B2 <= 0, "Invalid Bandgap Eg", ROUND(100 * (((((95 / B2) ^ 0.25) - 1) / (((95 / B2) ^ 0.25) + 1)) ^ 2), 2))Try the example
Use the original values or change them to test your understanding.