PHY 521: Condensed Matter Physics I
Prof. S. Teitel stte@pas.rochester.edu ---- Spring 2022
Due Thursday, November 17, uploaded to Blackboard by 11:59pm
Consider the point W, kW = (2π/a)(1, 1/2, 0), in the first Brillouin Zone of an fcc structure (see A&M Fig. 9.14). Here three Bragg planes meet. These Bragg planes bisect the reciprocal lattice vectors K1 = (2π/a)(1,1,1), K2 = (2π/a)(1,1,-1), and K3 = (2π/a)(2,0,0). Hence the free electron states at wavevectors kW, kW − K1, kW − K2, and kW − K3 are degenerate with energies εW0 = (hbar kW)2/2m.
a) Consider the effect of a weak ion potential in splitting these degeneracies. Show that the new energies, to lowest order in the ion potential, are given by the four eigenvalues ε of the matrix:
εW0 | U1 | U1 | U2 |
U1 | εW0 | U2 | U1 |
U1 | U2 | εW0 | U1 |
U2 | U1 | U1 | εW0 |
ε= ε0−U2 (twice), and ε=ε0+U2±2U1
b) Using a similar method, show that the energies at the point U (see A&M Fig. 9.14), kU = (2π/a)(1, 1/4, 1/4), are
ε=εU0−U2, ε=εU0+(1/2)U2±(1/2)(U22+8U12)1/2,
where εU0=(hbar kU)2/2m.
Consider a band with an anisotropic dispersion relation,
ε(k) = | ℏ2 2 | ( | kx2 mx | + | ky2 my | + | kz2 mz | ) |
b) We learned that, at sufficiently low temperatures, the contribution to the specific heat at constant volume due to the conduction electrons is given by cv = γ T, where γ is a constant independent of temperature. For the anisotropic band structure considered here, explain why γ is proportional to the effective mass m* = (mxmymz)1/3.