5. Assume the the bandstructuref a metal is well-described in the tightbinding approximationsing a square latticewith a dispersion relation E(q) = E, - 2y cos(q, a) - 27 cos(q, a). (2) For simplicity choose E, =0, y =0 5 eV and a =1 A. (a)In the q, - q,-plane draw the boundariesof the Brillouin zone and label the axes with the proper units (b)Sketch the Fermi surfaces at Fermi-energies of E = -0.95, E =0.0, and E, =0.95. For each Fermi-surface indicateif it would be easier to use an electron or hole descriptionof the electrical carrier conduction (c) For each Fermi-surface mark the points where the line 4 - q, intersects the Fermi-surface and indicate the direction of the Fermi-velocity at these points. (d) Using thesame parametersexpandE( q) around the two points in the first Brillouin zone where an effective- mass approximationin boththe q, and q, directionssimultaneously is useful. Thereare actually four points but I only want you to consider the points where the effective mass matrix is proportional to the identity matrix so that the diagonal clements have the same sign. HINT: One expansion point will have a positive effective mass, and one will have a negative effective mass. Do these look similar or different to the same Fermienergies in part (b)? (c) On a separatedrawing of the first Brillouin zone in the 4, - q-plane draw the Fermi-surfaces for (i) Ep = -0.95 for the positiveeffective mass case from part(d) and (ii) E, =0.95 for the negative effective mass case from part(d). HINT. It will be helpful to use Mathematica'scontour plot function to determinethe shapes of the Fermi surfaces in part (b) and for comparisonin part (c) (f) Writedown a suitablymodified analog of Eq 2 if we slightly stretchthe latticein the r-directionso that it becomes rectangular. (g) Do you expect the tunnelingrate in the r-direction to increase decrease or remain constantduring this deformation