Equations of motion of an electron in the presence of an electric field.

(a) Calculate the velocity of the electron at k = π/a.

(b) If the electric field E is applied in the -x direction, derive the time dependence of k for an electron initially at k = π/a and position x = 0.

(c) Derive the time dependence of the electron velocity, v(t), and the time dependence of the electron position, x(t).

(d) For a = 5 nm, E = 104 V.cm^-1, and m = 0.2 m₀, what are the maximum and minimum values of x that the electron will reach?

(e) What is the period of the oscillation?

(f) For the parameter of part (e), derive an expression for the effective mass as a function of k. Sketch the function.

Transcribed Image Text:

Assuming a dispersion relation : e = €c + - cos (ka)] ma? - 1]