Engineering & Science
Urbach Tail Sub-Bandgap Absorption & Urbach Disorder Energy (EU)
Model exponential sub-bandgap optical absorption alpha(E) = alpha_0 * EXP((E - E_0) / (E_U / 1000)) (cm^-1) and extract Urbach disorder energy E_U (meV) from two spectral points in perovskites and amorphous silicon.
Why low Urbach Energy (~15 meV) separates champion perovskite solar cells from defective films!
In an ideal zero-temperature defect-free crystal, optical absorption alpha(E) drops abruptly to zero below bandgap E_0. In real semiconductors, thermal phonons and structural/defect disorder smear the valence and conduction band edges into exponential localized states discovered by Franz Urbach (Physical Review 1953): alpha(E) = alpha_0 * EXP((E - E_0) / E_U) for photon energies E < E_0!
The characteristic decay width E_U—the Urbach Energy (quoted in meV)—is the premier optoelectronic quality metric for photovoltaic absorber layers because steep band edges (E_U ~ 13–15 meV in crystalline GaAs and high-efficiency halide perovskites) minimize open-circuit voltage deficit, whereas disordered hydrogenated amorphous silicon (E_U ~ 45–60 meV) suffers severe tail-state recombination! Watch at photon energy E = 1.56 eV below E_0 = 1.62 eV (alpha_0 = 150,000 cm^-1): a crisp E_U = 15 meV film suppresses sub-gap absorption 55-fold down to 2,747.3 cm^-1, whereas a defective E_U = 45 meV film leaks 39,539.6 cm^-1!
| Semiconductor Band-Edge Disorder EU (E0 = 1.62 eV, E = 1.56 eV) | Exponent (E - E0) / EU | Sub-Bandgap alpha(E) (cm^-1) | Photovoltaic Film Grade |
|---|---|---|---|
| High-Crystallinity Perovskite / GaAs (EU = 15 meV) | -0.060 / 0.015 = -4.00 | 2,747.3 cm^-1 | Minimal Voc Deficit (~kT Sharpness) |
| Disordered / Amorphous Film (EU = 45 meV) | -0.060 / 0.045 = -1.33 | 39,539.6 cm^-1 (14.4x Higher!) | Severe Tail-State Trap Recombination |
| Invalid Energy (E > E0 or EU <= 0) | Sub-Gap Domain Guard | "Invalid Urbach Inputs" | Urbach Law Applies for E <= E0, EU > 0 |
Sub-Bandgap Urbach Tail Absorption Coefficient alpha(E) (cm^-1)
Converts Urbach energy B4 from meV to eV (/ 1000) inside alpha_0 * EXP((E - E_0) / (E_U / 1000)).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 > B3), "Invalid Urbach Inputs", ROUND(B2 * EXP((B5 - B3) / (B4 / 1000)), 1))Inverse e-Folding Spectral Slope S = 1000 / E_U (eV^-1)
Computes the logarithmic d(ln alpha)/dE slope (eV^-1) measured in photothermal deflection spectroscopy (PDS).
=IF(B4 <= 0, "Invalid Urbach Inputs", ROUND(1000 / B4, 2))Try the example
Use the original values or change them to test your understanding.