Band Gap and Optical Absorption

The band gap E_g is the minimum energy required to excite an electron from the valence band to the conduction band. It determines the optical and electronic properties of semiconductors.

Optical Absorption Edge

A photon is absorbed only if its energy equals or exceeds the band gap:

$E_{\text{photon}} = hf \geq E_g$

This defines the absorption edge wavelength — the longest wavelength (lowest photon energy) that can be absorbed:

$\lambda_{\max} = \frac{hc}{E_g}$

where h = 6.626×10⁻³⁴ J·s and c = 2.998×10⁸ m/s.

Photon Energy from Wavelength

Given a wavelength λ in nanometres, the photon energy in eV is:

$E_{\text{photon}} = \frac{hc}{\lambda \cdot e}$

Direct-Gap Absorption Coefficient

Near the absorption edge in a direct-gap semiconductor, the absorption coefficient scales as:

$\alpha \propto \sqrt{E_{\text{photon}} - E_g}$

A practical model:

$\alpha(E) = \alpha_0 \sqrt{\max(0,\, E_{\text{photon}} - E_g)} \quad [\text{m}^{-1}]$

with α₀ ~ 10⁶ m⁻¹ for typical III-V semiconductors.

Examples

Implement

def absorption_edge_nm(E_g_eV):
    # Returns the absorption edge wavelength in nm
    ...

def photon_energy_eV(lam_nm):
    # Returns the photon energy in eV for wavelength lam_nm (nm)
    ...

def direct_gap_absorption_m(E_photon_eV, E_g_eV, alpha0=1e6):
    # Returns absorption coefficient in m⁻¹
    ...