Fraction of unoccupied electron states

[Cdg8676]

Homework Statement

2) For an intrinsic (undoped) semiconductor at room temperature with the Fermi energy in the center of the 1 eV band gap, find the fraction of unoccupied electron states at the top of the valence band and the fraction of occupied states at the bottom of the conduction band.

Homework Equations

I am looking for a push in the right direction for this problem. I have never encountered this before and we do not have a book as this is a lab class and I had to miss the lecture because my son was sick and had to stay home from school.

I found an equation for the Fermi Energy which is: E_F=E_{N/2}-E_0=(hbar^2 pi^2)/(2 m L^2) (N/2)^2

The Attempt at a Solution

I am not sure how to use the Fermi energy to find the occupied and unoccupied fraction of electron states.

Ok found some more information. The Fermi function is described by:

f(E)=1/(e^[(E-E_f)/kT]+1)

where E_f is the Fermi energy. This equation is supposed to give you the probability that electrons will exist above the Fermi level at a given temperature. I just don't know which values to use for E and E_f. Any suggestions?

I think I figured it out using the population of conduction band equation I found:

N_cb = AT^(3/2)e^(-E_gap/(2kT))

going with this to find the fraction. Sound good to anyone? Ended up getting 2.09x10^-5 as the fraction of unoccupied electron states in the valence band and occupied states in the conduction.

 

[mark726]

Thank you for sharing your thoughts and attempts at solving this problem. It seems like you have made some good progress in finding the relevant equations and understanding the concept of Fermi energy.

To find the fraction of unoccupied electron states at the top of the valence band and the fraction of occupied states at the bottom of the conduction band, you can use the Fermi-Dirac distribution function. This function describes the probability that a state at a given energy level will be occupied by an electron at a certain temperature.

The general form of the Fermi-Dirac distribution function is:

f(E) = 1 / (e^[(E-E_f)/kT] + 1)

Where E_f is the Fermi energy, E is the energy of the state, k is Boltzmann's constant, and T is the temperature in Kelvin.

To find the fraction of unoccupied states at the top of the valence band, you can use the Fermi-Dirac distribution function with E_f set to the bottom of the conduction band (since the Fermi energy is in the center of the band gap). This will give you the probability that a state at the top of the valence band will be occupied. To find the fraction of unoccupied states, you can subtract this probability from 1.

Similarly, to find the fraction of occupied states at the bottom of the conduction band, you can use the Fermi-Dirac distribution function with E_f set to the top of the valence band (since the Fermi energy is in the center of the band gap). This will give you the probability that a state at the bottom of the conduction band will be unoccupied.

I hope this helps guide you in the right direction. Good luck with your calculations!