Semiclassical equations of motion of electrons with a “tight-binding” band dispersion – Bloch Oscillations

exercise 3_4959

Suppose that a band dispersion in one dimensions (1D) is described by E ( k ) = -2 b cos( ka ) , where a is the lattice constant.

(a) Plot the E ( k ) versus k band dispersion for just the first Brillouin Zone (1 st BZ).

(b) Using the expression for the group velocity v g = (1/ ħ )( dE / dk ) = dx / dt , show that x ( t ) can be integrated to yield a semi-classical expression for the position of the electron wavepacket :

x ( t ) = x (0) + (2 b / eE ){cos[ k (0) a eaEt / ħ ] – cos[( k (0) a )]}

(c) Setting k ( 0) = 0, show that a Taylor expansion of the motion x ( t ) agrees with the ballistic result to order t 2 , if the ballistic mass is taken to be the effective mass m * at k = 0, where m * is given by the external force F ext such that F ext = m * d 2 x / dt 2 and 1/ m *= (1/ ħ 2 ) ( d 2 E / dk 2 ).

(d) The result in part (b) illustrates a phenomenon called “Bloch Oscillations” which is characterized by electrons oscillating in position and in k - space under the influence of applied electric field E . Let E = 1 V/ cm, a = 3 Å , and the scattering time t c = 2.5 x 10 -14 s at temperature T = 300 K. Determine the period of oscillation in (b) by inserting these numbers and compare the result with t c . How far will an electron traverse in k -space before it will be scattered? How does this compare with the width of the 1 st BZ?