# Partition function for electrons/holes

## Homework Statement

By shining and intense laser beam on to a semiconductor, one can create a collection of electrons (charge -e, and effective mass me) and holes (charge +e, and effective mass mh) in the bulk. The oppositely charged particle may pair up (as in a hydrogen atom) to form a gas of excitons, or they may dissociate into an electron hole plasma.

a) Write down the single particle partition functions Ze(1) and Zh(1) at temperature T in a volume V for electrons and holes respectively. (The thermal wavelength $$\lambda$$ for a particle is $$\lambda$$ = $$\frac{h}{\sqrt{2 \pi mkT}}$$

## The Attempt at a Solution

I know that the partition function is exp[-E/kT] summed over all energies or integrated with a density of states over all energies. But how do I go about it in this case?

In our notes somewhere (section 11/12/13), you can see that he uses the equation:

Z(1) = (V/λ^3), where V is the volume.

I'm not sure where it comes from, but I think we just need to learn it.

You can obtain that formula by summing over all the energy states by approximating the summation by an integral using the fact that the number of quantum states within a volume V_p of momentum space is

V V_p/h^3

So, if you integrate over all momenta, the energy is E = p^2/(2m) and you can write:

Z1 = Integral of V d^3p/h^3 exp[-beta p^2/(2m)] =

V/h^3 Integral from |p| = 0 to infinity of 4 pi p^2
exp[-beta p^2/(2m)] d|p|

For electrons you must multiply this by 2 because there are two spin states for each energy eigenstate.