Localized absorption of photons and carrier generation

[CB1X]

Perturbation theory predicts rates of transitions between eigenstates of the unperturbed Hamiltonian, which in the independent electron model for a crystal are nonlocal Bloch wave functions or linear combinations of them that extend throughout the crystal. However, photon absorption is localized. With radiation normally incident on a semiconductor, the photon flux N(E;x) decreases with depth according to $$\frac {dN(E;x)}{dx} = -α(E;x)N(E;x)$$ So in a thin layer of thickness dx , $α(E;x)N(E;x)$ photons of energy E (or in a narrow interval dE) will be absorbed per unit volume in unit time. If each photon of energy E absorbed in dx generates a conduction band electron (and/or a valence band hole) in dx, the local rate of generation is then $$G(E;x) = α(E;x)N(E;x)$$
While it can be shown that Fermi’s Golden Rule, provided by perturbation theory, applies to one-electron transitions if the initial one-electron state is a wave packet, e.g. a valence band wave packet, at least under certain assumptions, it appears to me that the possible final state must be an eigenfunction of the unperturbed Hamiltonian, which is nonlocal. Yet I see a number of texts apply perturbation theory, i.e. Fermi’s Golden Rule, to predict local rates of photon absorption and carrier generation (I assume wave packets). How is this justified? How are localized carriers, which I think are wave packets, generated by light absorption? These seem like fundamental questions but I have not been able to extract the answers from books so far.

 

[analogdesign]

Are you asking how it is done physically? Localized carriers are generated via the photoelectric effect.

 

[DrDu]

I think a direct transition from the valence to the conduction band at wavewector k will be excited coherently with a transition at k+\Delta k as long as $(dE_c/dk-dE_v/dk)\Delta k\le \Delta \omega$, here E_c and E_v are the energies of the conduction and valence bands and $\Delta \omega$ is the spectral width of the exciting radiation. So we get a coherent electron-hole pair with width $\Delta k$.

 

[CB1X]

Thank you for your responses. After some effort, I still do not know enough to fully understand what you mean. I think an implication of your statement could be that under certain conditions, for an occupied valence band wave packet with a k-space spread determined by a vector Δk at an initial time, if the condition that you stated is satisfied, then the valence band wave packet will evolve into a conduction band wave packet according to the Schrödinger equation, leaving behind a hole wave packet. I do not know if that is what you are getting at. In any case, how would the direct generation rate G(E; x) for a semiconductor be determined? How does the above condition relate to Fermi’s Golden rule, which I notice is used to predict transition rates? I still have not resolved the issue. Can anyone give some suggestions? Thanys.