Partition function for electrons/holes

  • Thread starter Welshy
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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 [tex]\lambda[/tex] for a particle is [tex]\lambda[/tex] = [tex]\frac{h}{\sqrt{2 \pi mkT}}[/tex]


Homework Equations




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?
 

Answers and Replies

  • #2
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.
 
  • #3
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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.
 

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