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ID: past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2021/2/a/solution

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Past exam of the laser science and quantum information processing course of the University of Oxford / 2021 / 2 / a / Solution by

Codex  0 2026-10-25

In a direct band gap semiconductor, the small photon momentum makes optical transitions approximately vertical in wave vector. Their energy is

E (k) = E_{c} (k) - E_{v} (k) = E_{g} + \frac{ℏ ^{2} k ^{2}}{2 m _{r}}, \frac{1}{m _{r}} = \frac{1}{m _{c}^{*}} + \frac{1}{m _{v}^{*}} .

The joint density of states per unit volume per transition energy, including the supplied spin counting, is

D_{J} (E) = \frac{k ^{2}}{π ^{2}} \frac{d k}{d E} = \frac{m _{r} k}{π ^{2} ℏ ^{2}} = \frac{2 m _{r}^{3/2}}{π ^{2} ℏ ^{3}} E - E_{g}, E > E_{g} .

For an upper electron state of occupation

f_{c}

and a lower electron state of occupation

f_{v}

, stimulated emission has occupation factor

f_{c} (1 - f_{v})

; absorption has factor

f_{v} (1 - f_{c})

. Their difference is exactly

f_{c} - f_{v}

. Multiplication by the joint density of states therefore gives

g (ω) \propto m_{r}^{3/2} ℏ ω - E_{g} [f_{c} (E_{2}) - f_{v} (E_{1})] .

The proportionality suppresses the optical transition matrix element and other smoothly varying optical prefactors. It assumes parabolic isotropic energy bands, an allowed direct transition, and negligible linewidth and many-body corrections. Here

f_{v}

counts electrons, not electron holes.

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