Homework 5. Metals. Due Monday Feb 19.

1. Consider a 1D metal with dispersion relationship
\(E_k = E_{atom} - 2 \gamma cos(ak)\)
where the hopping integral, \(\gamma\), is 2 eV and E_atom won't really matter.
Values of k range from \(-\pi/a\) to +\(\pi/a\). Let's assume that there is a total of \(10^8 states/cm\) in this 1D band.
a) Show that in the presence of an electric field, the current, J, can be written as \(e (2 \Delta k) v_F\) where \(\Delta k\) refers to the shift in the occupation boundaries due to the applied electric field.
b) Calulate \(v_F\) for the following cases:
i) a 1/8 filled band; ii) a 1/4 filled band; iii) a 1/2 filled band; iv) a 3/4 filled band*(see below for hint)
c) Graph \(v_F\) as a function of \(k_F\).  Graph also n, the density of electrons in the conduction band, as a function of \(k_F\). To what extent does \(v_F\) scale with n? Discuss.
d) Calculate \(\Delta k\) for the half-filled band case. Does \(\Delta k\) depend on filling? For the half filled case, what is the total number of electrons in the conduction band and what is the number of electrons the are "unbalanced" and thus participate directly in the net current, J. (email me this number when you get something. or you can post it here for discussion)

2. (Fermi surfaces in 2 dimensions) Consider a 2D metal with dispersion relationship
\(E_k = E_{atom} - 2 \gamma cos(ak_x) - 2 \gamma cos(ak_y)\)
where the hopping integral, \(\gamma\), is 2 eV and E_atom..., well, lets make it zero.
a) At what energy is the bottom of the band? At what value of k does that occur?
b) What is the shape of the Fermi boundary, in k-space, for \(E_F = 0\)? What filling of the band does that correspond to?
c) What is the shape of the Fermi boundary, in k-space, for \(E_F = -2 \gamma \)? What filling of the band does that correspond to? (see hint #2 below if you like)

3. extra credit. With reference to the proceeding  problem, what is the shape of the Fermi boundary for the following fillings:
1/8, 1/4, 3/8, 1/2, 3/4, 7/8?  What is the Fermi energy for each case?

4. extra credit:  For the artificial case in which one uses: \(E_k = \hbar^2 k^2/(2m)\) instead of a valid band dispersion relationship,
a) show that v_f is \(\hbar k_F/m\) and that therefore the current is proportional to
\(e^2 \tau k_F/m\).
b) Write \(k_F\) in terms of n and see what you get for current vs electric field.*(see below)
c) Does the n is this expression come from the number of electrons carrying current or from the Fermi velocity?
* hint#1. \(k_F = \pi/(8a)\) corresponds to a 1/8 filled band. \(k_F = \pi/(4a)\) corresponds to a 1/4 filled band. This reflects the fact that the density of states is uniform along the k axis.

hint #2. The relationship between filling and Fermi energy is not a simple closed-form relationship in 2 or 3D due to the non-trivial nature of the Fermi boundary in k space. You may have to estimate that numerically. Feel free to be creative.

* hint #3. This problem is kind of vague so  I made a video so you wouldn't get stuck on it.