A 2D solid material has primitive lattice translation vector ae and be with b>a. There is one divalent atom per unit cell. Electrons in the solid experience a weak periodic potential V(x, y), with 2 2 Vo + Y [EXIS: (21) + CIX (2777) cos a b V(x, y) = V + y cos Problem: (1) Draw the first and second Brillouin zones. (2) Calculate the energy gap at (0,7) on the first Brillouin zone boundary from first principles. (3) Show how an increase in the parameter y can change the solid from a metal into an insulator and estimate the value of y at which this occurs.