Why Do Insulators and Metals Have Different Density of State Curves?

[oddiseas]

Homework Statement

We have been shown the probability curve and coresponding density of state curve for a metal only.


I am trying to figure the probability versus energy, number of energy levels and density of state curves for a insulator.I have found a few on the internet but they seem to contradict what we have been told in lectures.

Namely. we were shown that the "number" of energy levels is proportional to the square root of energy.This was derived by using the energy levels expression for a particle in a 3D box

In wikpedia they say "assuming" that the number of states in the valence band is uniform ... and as a result the number of states is zero at energy=0, and zero at the valence energy> This makes sense because obviously there are no available energy levels in an insulator between the valence band and the conduction band, but this contradicts what we were shown in lectures, that the the total number of states is proportional to the square root of Energy.

Can anyone expain to me what the right way is?

ALso is the probability of being at the fermi level "always half" and if so why is this note reflected in the probability versus energy for a metal?

Homework Equations

The Attempt at a Solution

 

[NateD]

The number of states in an insulator is zero at energy=0 and at the valence band maximum, because there are no available energy levels in the gap between the valence band and the conduction band. This means that the number of states is not proportional to the square root of energy as it is in a metal. The probability of being at the Fermi level is always half in a metal because the Fermi-Dirac distribution predicts that the probability of finding an electron at the Fermi level is 1/2. However, this is not reflected in the probability versus energy curve for a metal, because the Fermi level is not always exactly the same as the energy.