(Solid State) fraction of electrons within kT of fermi level

[wdednam]

Homework Statement

Show that the fraction of electrons within kT of the fermi level is equal to 3kT/2Ef, if D(E) = E^1/2.

Homework Equations

f(E) = 1 / ( exp(E-Ef)/kT + 1 ) fermi distribution

N = integral from 0 to inf of D(E)f(E)dE = total no. of electrons

The Attempt at a Solution

I'm really lost with this problem.

I'm not even sure if I have to solve it at T = 0, in which case I tried to calculate n = integral from Ef - kT to Ef of D(E)f(E)dE over N = integral from 0 to Ef of D(E)f(E)dE with f(E) = 1,

OR at a temperature T which is finite such that now I calculate n = integral from Ef - kT to Ef + kT of D(E)f(E)dE over N as defined in "relevant equations" above. In the first case I get a horrible algebraic expression which does not simplify to what I'm supposed to get and in the second case I get integrals I don't know how to evaluate and can't even calculate using mathcad.

Any hints would be highly appreciated.

Thanks!

 

[wdednam]

Okay, I managed to get the answer.

For those who are interested, the calculation should be done at T = 0. Since the electrons are within kT of Ef, and Ef >> kT, the density of states is just D(Ef) = 3N/2Ef which is constant so it can come out of the integral in the numerator (the integral from Ef - kT to Ef + kT of D(E)f(E)dE). Remember that f(E) = 1 for E < Ef and 0 for E > Ef at T = 0.

To get the final answer just divide the result of this integral by N, the total number of electrons.

Cheers.