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Homework07_solution

# Homework07_solution - Homework 7 Solution Problem 1...

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Homework # 7 - Solution Problem # 1 Consider the free electron energy bands of an fcc crystal lattice in the reduced zone scheme in which all k 's are transformed to lie in the first Brillouin zone. Plot roughly in the [111] direction the energies of all bands up to six times the lowest band energy at the zone boundary at k = (2 π /a)(½,½,½). Explain what happens with these bands in the presence of a weak crystal potential. The reciprocal lattice is a bcc lattice with primitive translation vectors: 1 2 ˆˆˆ () a π =− + + bx y z , 2 2 a + y , z 3 2 a = +− y z All the reciprocal lattice vectors are given by 11 2 2 33 2 ( nn n h k l a ) =++ = + + Gbbb x y z The energy within the first Brillouin zone: 2 22 2 xx yy zz Ek G k G k G m ⎡⎤ =+ + + + + ⎣⎦ = 2 z Along [111] direction: and therefore xy kk k k === 2 2 G k G k G m 2 z = + ++ = ; k varies from a to a Low energy bands are given in the table and plotted in the figure below. bands n 1 n 2 n 3 h k l E / 2 2 m = E / 2 2 m = at k =0 E / 2 2 m = at k = a 1 000 000 2 3 k 0 2 3( ) a 2,3,4 100, 010, 001 111, 111 2( ) ( ) kk aa 2 12( ) a 2

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Homework07_solution - Homework 7 Solution Problem 1...

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