Problems for Tuesday, Jan 23

The point of these problems is to develop intuition related to using density of states and the Fermi function together. Also, to learn about the relationship between \(E_F\) and the number of electrons in a conduction band. Watching the video in the neighboring post, Calculating n, may help you with these problems.

Let’s consider a 1 million atom chain. Suppose you are given that there is a band that extends from 1 eV to 5 eV (bandwidth of 4 eV) and that the density of states within this band is constant with a value of 500,000 states per eV. 

1.a) What is the total number of states in the band?

b) Why does that make sense? What is the correspondence between total number of states in a band and number of atoms?

2. Suppose that kT=.025 eV and that \(E_F = 0.8 eV\).

a) What is the probability that a particular state at the bottom of the band is occupied? Show that you can use an approximate form for the Fermi function, a simple exponential, and that you get pretty good accuracy with that. (What is the difference between the exact answer and the one you get using the approximate form?)

b) What is the probability that a particular state at the top of the band is occupied? This is a much smaller number, right?

3. Using the approximate form of the Fermi function, use integration to calculate:

a) the total number of electrons in the band (i.e., the number of occupied states).

b) the total number of electrons in states with energy between 1 and 2 eV.
c) the total number of electrons in states with energy between 1 and 1.2 eV.
d) the total number of electrons in states with energy between 1 and 1.1 eV.

4. Show that for kT=0.25 eV and \(E_F = 0.8 eV\), as in problem 3, you can extend the upper limit of integration to infinity without much loss of accuracy and thereby get a direct analytic relationship between \(E_F\) and the number of electrons in the conduction band, n.

5. Now suppose that \(E_F = 1.5 eV\).

a) What is the total number of electrons (occupied states) in the band for KT=0?  Sketch a graph of the integrand and show that this can be represented as an area of a rectangle.

b) Roughly, what is the total number of electrons (occupied states) in the band for KT=0.025 eV?  Sketch a graph of the integrand in this case and show how it is not too different, but somewhat different from 4a.

c) Explain why you cannot use the approximate form for the Fermi function (that you could use in problems 1-4) in this case.

6. If you have time, do the same thing for \(E_F = 3 eV\).

a) What is the total number of electrons (occupied states) in the band for KT=0?  Sketch a graph of the integrand and show that this can be represented as an area of a rectangle.

b) Roughly, what is the total number of electrons (occupied states) in the band for KT=0025 eV?  Sketch a graph of the integrand in this case and show how it is not too different, but somewhat different from 4a.

c) Explain why you cannot use the approximate form for the Fermi function (that you could use in problems 1-4) in this case.