Absorption, stimulated emission and optical gain

Optical gain is what makes a quantum well amplify light: it occurs when stimulated emission exceeds absorption, which requires a population inversion between the conduction and valence subbands. This page assembles the OghmaNano gain calculation from its physical ingredients — carrier statistics, the two-dimensional density of states, the optical matrix elements and lineshape broadening — and states clearly what the current implementation does and does not include.

1. Occupation and quasi-Fermi levels

Under optical or electrical pumping the electrons and holes are each internally thermalised but not in equilibrium with one another, so each carrier type has its own quasi-Fermi level. The occupation of a state at energy \(E\) is the Fermi–Dirac function

\[ f(E) = \frac{1}{1 + \exp\!\big((E - E_F)/k_B T\big)}, \]

with \(E_F=E_{F,c}\) for the conduction states and \(E_F=E_{F,v}\) for the valence states. The two quasi-Fermi levels move apart as pumping increases; population inversion, and hence gain, requires their separation to exceed the transition energy.

2. Sheet density and the 2D integral

Because motion is free in the plane, each subband is a two-dimensional band and its carrier density is an integral over the in-plane wavevector. The two-dimensional density of states gives the integration weight \(k\,dk/2\pi\) per spin/degeneracy, so the electron sheet density is

\[ n_{2D} = \sum_{n} g\!\int \frac{k\,dk}{2\pi}\; f\!\big(E_n(k)\big), \]

with \(g\) the state degeneracy and the sum running over subbands. OghmaNano evaluates this integral numerically over the computed dispersion. Given a target sheet density it finds the quasi-Fermi level by bisection — adjusting \(E_F\) until the integral matches the target — and does the same for holes with \(1-f\). In the full-spectrum mode, where electron- and hole-like states are mixed, the S-character weight of each state scales its contribution.

3. Bringing in the matrix elements

Each transition contributes in proportion to its optical matrix element \(M^2\), the squared overlap-and-character factor computed from the envelopes. The absolute scale is carried by a prefactor built from the Kane energy and the optical constants,

\[ \text{prefactor} \propto \frac{\pi\,q^2\,p^2}{n_r\,c\,\varepsilon_0\,m_0^2\,\omega\,W}, \qquad p^2 = \tfrac{1}{2} m_0 E_p q, \]

where \(n_r\) is the refractive index, \(W\) the well width, \(\omega\) the optical angular frequency and \(E_p\) the Kane energy. The \(1/W\) converts the sheet response to a bulk-like gain coefficient (units of inverse length).

4. Lineshape broadening

Real transitions are broadened by scattering into a lineshape rather than a sharp delta function. OghmaNano uses a Lorentzian of half-width \(\gamma\),

\[ L(E - \hbar\omega) = \frac{1}{\pi}\,\frac{\gamma}{(E - \hbar\omega)^2 + \gamma^2}, \]

centred on each transition energy \(E = E_e(k) - E_v(k)\). This spreads each transition over a range of photon energies and smooths the spectrum.

5. Stimulated gain, absorption and net gain

For a given photon energy \(\hbar\omega\), each transition contributes to stimulated emission with probability \(f_c(1-f_v)\) (electron present in the upper state, hole in the lower) and to absorption with probability \(f_v(1-f_c)\). Summing over subbands and over the 2D integral,

\[ \begin{aligned} g_{\text{stim}}(\hbar\omega) &= C\!\sum \int \frac{k\,dk}{2\pi}\; M^2\, f_c(1-f_v)\, L, \\ \alpha(\hbar\omega) &= C\!\sum \int \frac{k\,dk}{2\pi}\; M^2\, f_v(1-f_c)\, L, \end{aligned} \]

where \(C\) is the prefactor. The net gain is their difference,

\[ g_{\text{net}}(\hbar\omega) = g_{\text{stim}}(\hbar\omega) - \alpha(\hbar\omega). \]

Net gain is positive only where the population is inverted — where \(f_c > f_v\) — which is the Bernard–Duraffourg condition that the quasi-Fermi separation exceed the photon energy. Below threshold the same expression is negative, correctly describing absorption.

6. TE / TM spectra and density sweep

For the wurtzite model OghmaNano runs the gain integral twice, once with the TE matrix elements (basal, \(P_2\)) and once with the TM matrix elements (c-axis, \(P_1\)), producing separate TE and TM spectra whose ratio reflects the valence orbital character of the uppermost states. The whole calculation is repeated over a sweep of sheet densities, so the output is a family of spectra showing how gain builds up with pumping; a rough volume density \(n_{3D}\approx n_{2D}/W\) is reported for comparison with bulk values.