5. The dispersion relation E-k of an energy band of a one-dimension crystal is E(k)= En cos (ak) (a) What is the effective mass of an electron with wavevector k=0 in this band? (2 points) (b) At t=0, a force F in the positive k direction is applied to this electron. What is the electron's wavevector k as a function of time t? (1 points) (c) What is the electron's velocity vg as a function of time t? (2 points) (d) If this energy band is initially completely filled by electrons, and suddenly an electron with wavevector k, = is removed and thus created a hole, what is the wavevector and the effective mass of the hole? (1 points) 6a 6. What determines the mobility of charge carriers in crystalline materials? (2 points) Equations which may be useful: 1. Relation between lattice vectors and reciprocal lattice vectors: db;= 2m8 where 8 is the Kronecker delta function. 2. Bragg diffraction condition: 2d sin 0 = n where n = 1,2,3,... 3. Velocity of an electron in a crystal with a wavevector k: 1 dE (k) v(k)= dk 4. Equation of motion (in the reciprocal lattice space) of an electron under an applied force-F: dk h == F dt 5. Effective mass of an electron: m = h dE(k) dk