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PHY 521/321: Condensed Matter Physics I
Prof. S. Teitel stte@pas.rochester.edu  Spring 2010
Problem Set 6
Due Wednesday, April 21, in lecture
 Problem 1
Consider a two dimensional square Bravais lattice with lattice constant a. Suppose the periodic ionic potential is given by,
U(x, y) = 4Ucos(2πx/a)cos(2πy/a)
a) Find the Fourier components U_{K} for reciprocal lattice vectors K.
b) For k = (k, 0) along the k_{x} axis, how does the single electron energy ε(k) behave at the boundary of the first Brillouin Zone, (π/a, 0)?
c) For k = (k, k) along the diagonal of the first Brillouin Zone, how does ε(k) behave at the corner of the first Brillouin Zone, (π/a, π/a)?
 Problem 2 (Ashcroft and Mermin problem 9.3)
Consider the point W, k_{W} = (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 K_{1} = (2π/a)(1,1,1), K_{2} = (2π/a)(1,1,1), and K_{3} = (2π/a)(2,0,0). Hence the free electron states at wavevectors k_{W}, k_{W} − K_{1}, k_{W} − K_{2}, and k_{W} − K_{3} are degenerate with energies ε_{W}^{0} = (hbar k_{W})^{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:
ε_{W}^{0}  U_{1}  U_{1}  U_{2} 
U_{1}  ε_{W}^{0}  U_{2}  U_{1} 
U_{1}  U_{2}  ε_{W}^{0}  U_{1} 
U_{2}  U_{1}  U_{1}  ε_{W}^{0} 
where U_{1} is the Fourier component of the ionic potential at K_{1} and at K_{2} (they are equal by cubic symmetry) and U_{2} is the Fourier component of the ionic potential at K_{3}. Show that these eigenvalues are:
ε= ε^{0}−U_{2} (twice), and ε=ε^{0}+U_{2}±2U_{1}
b) Using a similar method, show that the energies at the point U (see A&M Fig. 9.14), k_{U} = (2π/a)(1, 1/4, 1/4), are
ε=ε_{U}^{0}−U_{2}, ε=ε_{U}^{0}+(1/2)U_{2}±(1/2)(U_{2}^{2}+8U_{1}^{2})^{1/2},
where ε_{U}^{0}=(hbar k_{U})^{2}/2m.
 Problem 3
Consider a band with an anisotropic dispersion relation,
ε(k) =  (hbar)^{2} 2  (  k_{x}^{2} m_{x}  +  k_{y}^{2} m_{y}  +  k_{z}^{2} m_{z}  ) 
a) Using the fact that an ellipsoid given by the equation, (x/a)^{2} + (y/b)^{2} + (z/c)^{2} = 1, encloses a volume (4π/3)abc, calculate the density of states g(ε). Hint: One can write for the density of states, g(ε)=dG(ε)/dε, where G(ε) is the number of single electron states per unit volume with energy less than ε.
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 c_{v} = 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^{*} = (m_{x}m_{y}m_{z})^{1/3}.

