If two periodic potentials have the following unit cells: V(x) = Voi sin (2- nx 2 V2(x) = Voz (+) 1) Derive the necessary equation for the two periodic potentials to have identical dispersion relations Hint: Refer to the delta-barrier approximation. 2) You wish to design the 1st periodic potential so that the width of the 1st allowed energy band is 0.1 eV given that the effective mass is m=0.07* m0 within the structure. Find the values of V01 & L1 that satisfy this condition. Use the values in the equation derived in step #1 to adjust the 2nd periodic potential as well. Hint: Refer to the graphical representation of the condition function in A.J. Levi's Applied Quantum mechanics textbook 3) Using the values found in step #2, Plot the dispersion relation for the energy range from 0 to 5 eV within the first Brillouin zone You can use MATLAB in plotting 4) Indicate on your plot the values of the wavenumber k that would give minimum and maximum group velocity for the electron wave and mention your reasoning behind choosing these values (intuitive, no calculations needed)