chap7 - Chapter 7: The Electronic Band Structure of Solids...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 Photo-Emission 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 free-electron 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 free-electron 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....
View Full Document

Page1 / 29

chap7 - Chapter 7: The Electronic Band Structure of Solids...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online