Question

Consider a quantum well (QW) with electron and hole effective masses both equal to 0.5 m0 (i.e., the conduction and valence band structures are symmetric). The width of the QW is z = 10 nm. The interband transition energy from the n = 1 to m = 1 subbands is ℎm = 1 eV. Assume = 3.1 and p = 25.7 eV.

At T = 0K, what is the maximum optical gain of the QW when the separation between the quasi-Fermi levels is 52 meV higher than the transition energy? Consider only the n = 1 to m = 1 transition. Assume the perturbing photons have TE polarization and that the transitions occur between the conduction band and heavy-hole band and that an infinite potential well model can be used. Note that the quasi-Fermi levels are higher (lower) in energy than c (), so that at T = 0K c = 1 below c in the conduction band and = 0 above in the valence band.

Answer 1

The maximum optical gain in a quantum well occurs when there is a population inversion due to the separation between quasi-Fermi levels being greater than the interband transition energy. At T = 0K and with a 52 meV separation above the transition energy, the condition for maximum optical gain is met in the given quantum well.

Step-by-step explanation:

For a quantum well (QW) with electron and hole effective masses equal to 0.5 m0 and a well width of 10 nm, determining the maximum optical gain involves understanding the interaction between photons and the electron-hole pairs within the well. Given that the interband transition energy (from the n=1 to m=1 subbands) is 1 eV and that the separation between the quasi-Fermi levels is 52 meV higher than this transition energy, the QW can exhibit optical gain when the photon absorption leads to stimulated emission.

At T = 0K, the distribution of electrons and holes at the quasi-Fermi levels is such that all states below the electron quasi-Fermi level are filled and all states above the hole quasi-Fermi level are empty. The condition for maximum optical gain is achieved when the difference in quasi-Fermi levels exceeds the interband transition energy, which is the case in this scenario by 52 meV. This will result in a population inversion necessary for stimulated emission.

To calculate the maximum optical gain, one would typically use the semiconductor gain equation, which involves factors like the density of states, the transition matrix element, and the photon density. Since specifics of these are not provided in the question, a quantitative answer cannot be given without more information. However, the gain will be proportional to the number of electrons available for the transition, which is directly influenced by the excess energy of 52 meV above the transition energy.