Homework Set # 10 - Due April 14

  
   
1) Complex wavevectors in the energy gap: Find an expression for the imaginary part of the wave vector in the energy gap at the boundary of the first Brillouin zone using the same approximation that we used to obtain (8.21). At the boundary of the FBZ the real part of k is K/2. Give a result for H, the imaginary part of the wave vector at the center of the energy gap. Hint: assume that H<<K and give your result in terms of U, K, and m. Assume that ε=ℏ²K²/8m at the center of the gap.

2) Consider an energy band ε(k)=ε₀+c₁k_x²a²+ c₂k_y²a²+ c₃k_z²a² of a cubic crystal with lattice constant a. Assume c_i are positive constants.

a) Obtain the effective mass tensor M of an electron.

b) Assuming that all the c_i=c, consider the motion of an electron confined to a potential U(r)=αr²>0. Write the effective Schrödinger equations for the states of the electron in the representation of the Bloch functions and then in terms of Wannier functions.

3) Derive (19.23).