A. Calculate the magnetic susceptibility, \(\chi\), and specific heat of a metal in the normal state for a given bandwidth and several values of e-e interaction strength. Make them both intensive quantities. Figure out the units of susceptibility (this is an E&M problem. I think that perhaps magnetization (m) and the thing called H in E&M have the same units? Is that wrong?). Graph them as a function of T from T=0 to 100 K. Show how you can use specific heat measurement to infer both the bandwidth (or density of states at the Fermi level) and the strength of the e-e interaction in a non-magnetic (aka paramagnetic, aka normal) metal. Make up a problem that has about 20 simulated data points for \(\chi\) and \( C_v\) and where the student os asked to fit the data points and infer bandwidth and e-e interaction strength.
Suggestion: Start with \(e^2/a =0\). Then work up to 0.5 eV, 1 eV, 1.5 eV. Use a bandwidth of 2 or 3 eV. What is the essential nature of \(\chi\) vs T in a reasonable range of T?
B. Consider a degenerate "Fermi liquid" at low temperature. Specifically, consider a half-filled band with upward curvature. (You can do downward curvature as well as you like.) Explain how an applied electric field shifts the occupation boundaries and leads to a modest number of electrons (much less than N) carrying current. Explain what fraction of electrons are responsible for the net current and what their typical speed is. Get an estimate of the conductivity. You can assume that the size of the boundary shift is very small compared to \(\pi/a\). That assumption can be used to greatly simplify the integral over q and to simplify your final result for the conductivity. (There is no benefit, and significant cost, to not making that assumption!)
Then redo the whole thing for a 1/8 or a 1/4 filled band using the approximation that E is quadratic in k and with an effective mass in the denominator (i.e., \( E_q = \hbar^2 q^2/(2 m_o m^*)/) . Try to get simple expression for the conductivity. Use n = N/V where appropriate. Conductivity should be defined to be intensive.
C. Use energy eigenstate formulation to calculate the time dependent spreading of a wave function expressed as a wave packet. Start with a free electron in a gaussian initial state. Then do this also for an electron in a crystal using the energy eigenstates associated with a single band of crystal energy eigenstates. What range of crystal eigenstates does it take to localize an electron to one lattice site? What range of energy eigenstates does it take to localize an electron to a finite, but larger region, say 100 lattice sites. For the latter problem you could use a Gaussian envelope function to define your initial t=0 wave function.
(For this project I think you will need an atomic wave-function to work with. My suggestion would be a gaussian, due to the absence of a singularity at the origin. The delta function ground state might also be a possibility. Actual square well states will be difficult to work with because of the piece-wise nature. I think that the results are not much effected by this choice, as long as the atomic state is exponentially localized.
D. Use a matrix formulation to calculate the time dependence (spreading) of a wave function representing one electron in a spatially periodic perfect lattice.
(Do this numerically, say for a 1000 site lattice.) After doing this
for a perfect lattice, try doing it for a model in which the
off-diagonal elements are all the same (as before), but the diagonal
elements have some degree of random disorder with respect to a central
mean value (a mean value the same as you used in the previous part).
E. Consider a lattice with only two sites and two electrons. Each electron can be localized on either site in single-electron eigenstate in which it has an energy of -8 eV (or call it \(E_o\). If the electrons are on the same site then there is a coulomb energy of +2 eV. If they are not on the same site the coulomb energy is zero. Including spin, enumerate all possible states of this system and what the energy of each state is. Draw simple pictures to illustrate.
Do the same thing for 3 electrons on 3 sites, and then perhaps for 4 electrons on 4 sites.
F. Investigate and discuss the nature and origin of anti-ferromagnetism. What is a half-filled Hubbard band? What does it have to do with magnetism? ... .... (perhaps we can flesh this out more. Also, I think this is related to E.)
G. Understanding superconductivity: The key thing in superconductivity is not so much the specific nature of the interaction that leads to paring (which can vary), but rather the tendency of electrons at a Fermi surface to be highly susceptible to pairing. The Fermion nature of electrons this plays a critical role. In 3-dimensions a Fermi surface could be a sphere in k-space inside which all states are occupied (by electrons) and outside which all states are empty. Cooper showed that electrons at the Fermi surface could be unstable with respect to pairing to form pairs of Fermions which then act as bosons. These boson pairs then develop a phase coherence that leads to a superfluidity know as superconductivity. As a project one could study Cooper pair formation, phase coherence and superfluidity or some other aspect of superconductivity.
H. Calculations of basic quantities in the effective mass approximation. ... more to follow
Feel free to ask questions and suggest other projects. Allow yourself enough time to understand the problem etc… There may be flaws in these problems that will take time and discussion to work out. (or maybe not?)
No comments:
Post a Comment