Engineering & Science
Varshni Semiconductor Bandgap Temperature Dependence Formula Eg(T)
Calculate semiconductor bandgap shrinkage with temperature E_g(T) = E_g0 - (alpha * T^2) / (T + beta) (eV) and cutoff absorption wavelength lambda_g (nm) for Silicon, GaAs, and LED/laser diodes.
Predict why hot solar cells lose open-circuit voltage and laser diodes red-shift!
As a semiconductor heats up from cryogenic 0 K to operating junction temperatures (300–500 K), thermal lattice dilation and electron-phonon coupling shrink the forbidden energy gap E_g(T)! Y. P. Varshni’s universal 1967 equation captures both the quadratic T^2 Debye regime at low temperatures and linear shrinkage at high temperatures: E_g(T) = E_g(0) - (alpha * T^2) / (T + beta), where E_g(0) is 0 K bandgap (eV), alpha (eV/K) is the high-T limit slope, and beta (K) correlates with the Debye temperature!
Look at crystalline Silicon (E_g0 = 1.1695 eV, alpha = 0.000473 eV/K, beta = 636 K): at room temperature T = 300 K, E_g(300 K) = 1.1240 eV (band-edge cutoff wavelength lambda_g = 1239.842 / E_g = 1,103.0 nm); in a hot power MOSFET or concentrator solar cell at T = 500 K, Silicon’s bandgap shrinks by -58.6 meV down to 1.0654 eV, surging intrinsic leakage carrier concentration exponentially!
| Silicon Junction Temperature T (Eg0 = 1.1695 eV) | Varshni Bandgap Eg(T) (eV) | Band-Edge Cutoff lambda_g (nm) | Optoelectronic Impact |
|---|---|---|---|
| Cryogenic Limit (T = 0 K) | 1.1695 eV | 1,060.1 nm | Zero-Phonon Lattice Limit |
| Room Temperature (T = 300 K) | 1.1240 eV | 1,103.0 nm | Standard Photovoltaic Baseline |
| Hot Power Junction (T = 500 K) | 1.0654 eV (-58.6 meV) | 1,163.7 nm (Red-Shift) | Lower Solar Voc & Higher Leakage |
Varshni Temperature-Dependent Bandgap Energy E_g(T) (eV)
Evaluates E_g0 - (alpha * T^2) / (T + beta) for absolute temperature T >= 0 K.
=IF(OR(B2 <= 0, B3 < 0, B4 <= 0, B5 < 0), "Invalid Bandgap Inputs", ROUND(B2 - ((B3 * (B5 ^ 2)) / (B5 + B4)), 4))Corresponding Optical Band-Edge Cutoff Wavelength lambda_g (nm) = 1239.842 / E_g(T)
Converts temperature-shifted bandgap E_g(T) into nanometer absorption edge wavelength.
=IF(OR(B2 <= 0, B3 < 0, B4 <= 0, B5 < 0), "Invalid Bandgap Inputs", ROUND(1239.842 / (B2 - ((B3 * (B5 ^ 2)) / (B5 + B4))), 1))Try the example
Use the original values or change them to test your understanding.