Energy gaps in a 1D crystal possessing a periodic potential with cosine terms

exercise 3_4951

Consider a one-dimensional (1D) solid with lattice constant a and length L with nearly free electrons of effective mass m subject to a weak symmetric lattice potential V ( x ):

V ( x ) = - V 0 [ cos 2 ( π x /a)+2cos 2 (2π x /a)+3cos 2 (3π x /a)].

(a) Show that the energy gap at the n th Brillouin zone (BZ) boundary can be expressed as E gn = 2|< k ג€™ | V ( x ) | k >| where | k > =(1/ Ö L ) exp ( ikx ); k ג€™ and k are values of the wavevectors on opposite ends of the n th BZ boundary. Also, verify that the Bragg condition can be expressed as k ג€™ - k = G = 2 n p / a in 1D. Also, determine the average potential using < V > = < k | V ( x ) | k > and describe its meaning in connection with the Fourier series expansion of the potential. Plot E versus k in the extended zone scheme in the range -7π/2a < k < 7π/2a, showing the magnitude of the energy gaps over the entire range and the energy at k = 0 . Clearly identify regions of the energy dispersion which are best approximated by the free electron dispersion E = 2 k 2 /2 m .

(b) Plot E versus k from -π/a < k < π/a (i.e., in the reduced-zone scheme), showing the magnitude of all energy gaps and the energy at k = 0.