Sketch the energy band diagram for a 1-D crystal for the approximate case we discussed in class, where the only 2 terms we considered important were and . Derive the expressions for the dispersion relations near k=0 and for this approximate case. Assume the periodic potential in a 1-D crystal with atoms a distance apart is given by: eV Set up the plane wave expansion matrix for this potential. Using an eigenvalue solver (IMSL3 or MATLAB), calculate the first 5 bands for this potential. Give a graph of E vs k of these bands in the first Brillouin zone. Use the effective mass tensor to show that overall average electron acceleration in silicon is in the direction of the electric field. (As long as the carrier concentration the respective k-space minima are the same.) Use your results from number 3 to show that mobility and thus current density would also be in the direction of the electric field in silicon. Derive the expression for the density of states for ellipsoidal, nonparabolic energy bands. Under what conditions do you expect the instantaneous velocity of an electron to be parallel to its wave vector?