# Photoelectric absorption in semiconductors

1. Feb 3, 2012

### MicheleC88

Hi everybody. I'm new here and, first of all, sorry for my bad english :-D

I'm studying photoelectric absorption in semiconductors.
The book (and professor too) says that, in the conservation law:

ki + kph = kf

(where ki and kf are wave vectors of initial and final electron state, and kph is the wave vector of incoming photon) we can neglect kph because it is ≈ 2π/λ, whereas ki and kf are ≈ 2π/a, and λ>>a. (a is the length of unitary cell in real space).
But I think that this assumption is good only if electron is at the edge of Brillouin Zone; if the initial and final electrons are near $\Gamma$-point, they should have a very little wave vector, comparable with kph, making the approximation not valid.

what is wrong in my words?

2. Feb 3, 2012

### Cthugha

Just do the math once and sketch the photon dispersion (energy versus k) and the electron dispersion (crystal electron or for simplicity even a free one) into the same graph with the correct dimensions. This is pretty instructive and will give you a good argument for why the approximation your professor gave you is a very good one pretty much everywhere.

3. Feb 3, 2012

### MicheleC88

So, I think having understood from your words, the key is that the conservation of momentum has to be combined with the conservation of energy?

4. Feb 3, 2012

### ZapperZ

Staff Emeritus
Actually, that conservation law is valid only for the in-plane momentum, i.e. parallel to the surface of the material. The out-of-plane momentum is way more complicated than that.

Zz.

5. Feb 3, 2012

### MicheleC88

I tried to make the following math.
from energy conservation: Eph≈Egap
(assuming that initial and final electrons are in proximity of, respectively, the maximum of VB and minimum of CB)

We know that kph = ω/c = Eph/($\hbar$c)

So, from momentum conservation law:

|kf - ki | = |kph|≈Egap/($\hbar$c)

Doing the calc (i assumed 1 eV for Egap):
|kf - ki | ≈ 10-4 angstrom-1, which is about 1 part of thousand of tipical size of Brillouin zone.

Are my reasoning correct to justify the assumpion kph=0 ?

Last edited: Feb 3, 2012
6. Feb 3, 2012

### MicheleC88

However, I think I made the math more complex than necessary:

ki - kf = kph = $\frac{2π}{λ}\widehat{k}_{ph}$

(ki - kf) / (size of Brillouin zone) = $\frac{2π}{λ}\frac{a}{2π}\widehat{k}_{ph}$ << 1 $\cdot$ $\widehat{k}_{ph}$

Thanks to all for the reply!