Question

2. Brillouin zone, rectangular lattice. A two-dimensional metal has one atom of valency one in a simple rectangular primitive cell a = 2 Â; b = 4 A. (a) Draw the first Brillouin zone. Give its dimensions, in cm. (b) Calculate the radius of te free electron Fermi sphere, in cm1. (c) Draw this sphere to scale on a drawing d the first Brillouin zone. Make another sketch to show the first few periods of the free electron band in the periodic zone scheme, for both the first and second e ergy bands. Assume there is a small energy gap at the zone boundary

Answer 1

A = 24 degree = 2 time s10^-10 M b = 4A degree = 4 times 10^-10 M Draw the first Brillion Zone K_y pi/b = 0.78 time s10^8 cm sigma pi/a = 1.57 times 10^8 cm N = 2 times pi k^2_f/(2 pi/k)^2 = k^2_F xk^2/2 pi n = N/L^2 = K^2_F/2 pi k_F = squareroot 2 pi n n = 1/8 times 10^16, u s/cm^2 k_F = 0.89 times 10^8 cm^-1