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Unformatted text preview: Chapter 7: The Electronic Band Structure of Solids Bloch & Slater April 2, 2001 Contents 1 Symmetry of ( r ) 3 2 The nearly free Electron Approximation. 6 2.1 The Origin of Band Gaps . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Tight Binding Approximation 15 4 PhotoEmission Spectroscopy 24 1 Free electrons FLT Band Structure E E f D(E) V(r) E f E f metal "heavy" metal insulator E D(E) V(r) = V E f Figure 1: The additional effects of the lattice potential can have a profound effect on the electronic density of states (RIGHT) compared to the freeelectron result (LEFT). In the last chapter, we ignored the lattice potential and con sidered the effects of a small electronic potential U . In this chapter we will set U = 0, and consider the effects of the ion po tential V ( r ). As shown in Fig. 1, additional effects of the lattice potential can have a profound effect on the electronic density of 2 states compared to the freeelectron result, and depending on the location of the Fermi energy, the resulting system can be a metal, semimetal, an insulator, or a metal with an enhanced electronic mass. 1 Symmetry of ( r ) From the symmetry of the electronic potential V ( r ) one may infer some of the properties of the electronic wave functions ( r ). Due to the translational symmetry of the lattice V ( r ) is pe riodic V ( r ) = V ( r + r n ) , r n = n 1 a 1 + n 2 a 2 + n 3 a 3 (1) and may then be expanded in a Fourier expansion V ( r ) = X G V G e i G r , G = h g 1 + k g 2 + l g 3 , (2) which, since G r n = 2 m ( m Z ) guarantees V ( r ) = V ( r + r n ). Given this, and letting ( r ) = k C k e i k r the 3 Schroedinger equation becomes H ( r ) =  h 2 2 m 2 + V ( r ) = E (3) X k h 2 k 2 2 m C k e i k r + X k G C k V G e i ( k + G ) r = E X k C k e i k r , k k G (4) or X k e i k r h 2 k 2 2 m E C k + X G V G C k G = 0 r (5) Since this is true for any r , it must be that h 2 k 2 2 m E C k + X G V G C k G = 0 , k (6) Thus the potential acts to couple each C k only with its recip rocal space translations C k + G and the problem decouples in to N independent problems for each k in the first BZ. Ie., each of the N problems has a solution which is a sum over plane waves whos wave vectors differ only by G s. Thus the eigenvalues may be indexed by k . E k = E ( k ) , I.e. k is still a good q.n.! (7) We may now sum over G to get k with the eigenvector sum 4 X X X X X X First B.Z. Figure 2: The potential acts to couple each C k with its reciprocal space translations C k + G (i.e. x x , , and ) and the problem decouples into N indepen dent problems for each k in the first BZ....
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 Fall '11
 Electrodynamics

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