Fermi Energy, number of electrons and holes

[bcjack]

Fermi Energy, Quantum mechanics

Electron levels and degeneracies thereof in a system are:
W1 = 0 eV, 10^23 /cm3 Valence band
W2 = 0.9 eV, 5x10^20 /cm3 Donor level
W3 = 1 eV, 2x10^23 /cm3 Conduction level

Total number, n, of electrons in the system is (10^23 + 5 x 10^23) /cm3.

Determine Fermi energies, number of electrons in level W3, and number of vacancies (holes) in levels W1 and W2 at i) 290 K and ii) 174 K.

My first thoughts of doing this problem is by adding the number of electrons in the 3 states together and solve for Ef with the fermi distribution.

f(e)=1 / [1 +e^ (E-Ef)/kT ]
n = g(e) * f(e) density of states times the fermi distribution

n_total = n1 + n2 + n3 = 10^23 / [1 +e^ (0-Ef)/kT ] + 5x10^20/ [1 +e^ (0.9-Ef)/kT ] + 2x10^23 / [1 +e^ (1-Ef)/kT ] = (10^23 + 5 x 10^23) /cm3

and solve for Ef.

Is this a reasonable approach?

To find the number of electrons and holes, i would then plug the Ef back into the equation for each energy level.

any thoughts would be appreciated. thanks!

 

[NateD]

Yes, that is a reasonable approach. You can use the Fermi-Dirac distribution to calculate the Fermi energy. To find the number of electrons and holes, you can then plug the Fermi energy back into the equation for each energy level to find the number of electrons and vacancies.