Question

7. Consider a one-dimensional crystal with N unit cells with one atom per unit cell in the nearly-free electron model. Assume periodic boundary conditions and that each unit cell has Z electrons. (a) Show that you can fill exactly Z/2 energy bands with these electrons. In other words, show that each band can accommodate exactly 2N electrons. (b) For the 1D case above, it is guaranteed that are an even number of electrons per unit cell will result in an insulator (since there will be no partially filled bands). Explain why this is not necessarily true for a 3D crystal.

Answer 1

Pg - 1 Solution 2/2 bands (a) show that fill the energy with with electrous According to Bloch's theorem 14(x) = 5 uk (2) e'kx Here, Q(x) = 4 (2+2) Uk(x) = UK (x+2) Now, substitute 4(x+2) - z Uklxfzje ku 4(x+2) ik (x+4)= UK. (X) 18 KX eikxklk (2+2) eikh x=ė lant where, kh=20th k = 2011 .dk= 217 dk

pg-2 [ Te + "[a] 1. marimum no of electron=an ) (6) For 3D, crystal pike = (kaxta gy+kaz.) conditions an applying boundary kxx + Kx (x+4) elkat = 29nx ka = nx = 1 / kz

pg 3 And anysna = ( 2 ) Akx My Akz Here, considering the 3rd crystal And any ane = ( 2 ) en kak Tontakt = (kart kgh tk22) * K2 = FIT ky=0 kz=0