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Algorithms, Math, and Physics

Understanding optical transitions in semiconductors

Continuing from my previous post on optical absorption in semiconductors here, I explore deeper into the mechanics of optical transitions. In this discussion, I primarily focus on the quantum mechanical aspect of these transitions and their impact on the semiconductor’s optical absorption coefficients.

Theoretical foundations

The total transition rate, W_{tot}, is crucial for understanding optical absorption, represented by the formula:

W_{tot} = \frac{2\pi}{\hbar}\frac{e^2 A_0^2}{4m_0^2} |p_{cv}|^2 \sum_{\mathbf k, \text{spin}} \delta(E_c(\mathbf k) - E_v(\mathbf k) - \hbar\omega)

Here, p_{cv} represents the momentum matrix element, assumed constant across the Brillouin zone. The delta function, \delta, ensures energy conservation during transitions, reflecting the ‘vertical’ nature of optical transitions in the energy-momentum diagram.

Simplifications and assumptions

For simplicity, I assume parabolic energy bands, which allows us to express the energy difference E_J(\mathbf k) between the valence and the conduction band as:

E_J(\mathbf k) = \frac{\hbar^2 k^2}{2}\left(\frac{1}{m_{eff_{e}}} + \frac{1}{m_{eff_{v}}}\right) + E_g = \frac{\hbar^2 k^2}{2\mu_{eff}} + E_g

This simplification leads to the formulation of the joint density of states g_J(E_J) for transitions, which can be expressed as:

g_J(E_J) = \frac{1}{2\pi^2}\left(\frac{2\mu_{eff}}{\hbar^2}\right)^{\frac{3}{2}} \sqrt{E_J - E_g}

Calculating the absorption coefficient

Integrating these concepts into the calculation of the absorption coefficient \alpha yields:

\alpha = \frac{\pi \hbar e^2}{2m_0 c \varepsilon_0}\frac{1}{n_r} \frac{E_p}{\hbar \omega} g_J(\hbar\omega)

where E_p = \frac{2}{m_0}|p_{cv}|^2 encapsulates the influence of the momentum matrix element squared. The formula highlights how the absorption coefficient depends on the photon energy \hbar\omega, directly reflecting the energy states involved in the transition.

Practical implications and further insights

The formulation and results discussed are fundamental in understanding how semiconductors interact with light. This not only aids in designing better optoelectronic devices but also enhances our theoretical knowledge base, making it possible to predict new phenomena in semiconductor physics.

For more insights into this topic, you can find the details here.