Problem Set 6

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 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 k_W, k_W − K₁, k_W − K₂, and k_W − K₃ are degenerate with energies ε_W⁰ = (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_W⁰ U₁ U₁ U₂

U₁ ε_W⁰ U₂ U₁

U₁ U₂ ε_W⁰ U₁

U₂ U₁ U₁ ε_W⁰

where U₁ is the Fourier component of the ionic potential at K₁ and at K₂ (they are equal by cubic symmetry) and U₂ is the Fourier component of the ionic potential at K₃. Show that these eigenvalues are:
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), k_U = (2π/a)(1, 1/4, 1/4), are
ε=ε_U⁰−U₂, ε=ε_U⁰+(1/2)U₂±(1/2)(U₂²+8U₁²)^1/2,
where ε_U⁰=(hbar k_U)²/2m.
Problem 3

Consider a band with an anisotropic dispersion relation,

ε(k) = (hbar)²
2 ( k_x²
m_x + k_y²
m_y + k_z²
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_xm_ym_z)^1/3.