Harnessing light: optical absorption in semiconductors
The phenomenon of optical absorption in semiconductors is a fundamental aspect of solid-state physics, particularly significant for understanding electronic transitions in direct gap semiconductors. Through the application of Fermi’s golden rule, one can predict the transition rates between electronic states in a semiconductor when subjected to electromagnetic waves.
Theoretical framework
The transition rate for absorption from an initial state |\psi_i \rangle with energy E_i to a final state |\psi_f \rangle with energy E_f is given by:
w_{if} = \frac{2\pi}{\hbar}\left|\langle \psi_f | \mathbf{H}_{p0} | \psi_i \rangle\right|^2 \delta(E_f - E_i - \hbar\omega)
where \hbar is the reduced Planck’s constant, and \omega is the frequency of the electromagnetic wave.
Perturbing Hamiltonian
The interaction between the electromagnetic field and the electronic states in a semiconductor is encapsulated by the perturbing Hamiltonian \mathbf{H}_p(t), which varies with both time and the spatial dependence of the electromagnetic wave amplitude. For practical purposes, we approximate the perturbing Hamiltonian as follows:
\mathbf{H}_{p0}(\mathbf{r}) \approx -\frac{e}{m_0}\mathbf{A} \cdot \mathbf{p}
Here, e is the electron charge, m_0 is the electron mass at rest, \mathbf{A} is the vector potential, and \mathbf{p} = -i\hbar\nabla is the momentum operator.
Transition matrix element
To advance, we assume the electronic wavefunctions are in Bloch state, and we apply the normalization condition over a unit cell, leading to the simplification:
\langle \psi_f | \mathbf H_{p0} | \psi_i \rangle = -\frac{e A_0}{2m_0N} \sum_m e^{i(\mathbf k_c - \mathbf k_v + \mathbf k_{op}) \cdot \mathbf R_m} \int u_c(\mathbf r) p_x ; u_v(\mathbf r) \mathrm d^3 \mathbf r
Conditions for effective absorption
For the summation above to not average to zero, the phase condition must be satisfied:
\mathbf k_v - \mathbf k_c + \mathbf k_{op} = 0
When this condition is met, the transition matrix element simplifies significantly, and using Fermi’s golden rule, the transition rate becomes:
w_{if} = \frac{2\pi}{\hbar}\frac{e^2 A_0^2}{4m_0^2} |p_{cv}|^2 \delta(E_f - E_i - \hbar\omega)
Conclusion
The intricate interplay between electronic states and electromagnetic fields in semiconductors reveals the delicate balance of physical forces at the quantum level. For more insights into this topic, you can find the details here.