6.4 ELECTRONIC BAND STRUCTURES

1. 6.4 ELECTRONIC BAND STRUCTURES Dongwoo, Shin

2. Contents 6.4.1. Reciprocal Lattices and the First Brillouin Zone 6.4.2. Bloch’s Theorem 6.4.3. Band Structures of Metals and Semiconductors

3. FT for time Reciprocal lattice FT for space 6.4.1. Reciprocal Lattices and the First BrillouinZone Reciprocal Lattice • Crystal is a periodic array of lattices Performing a spatial Fourier transform • Reciprocal Lattice • Expression of crystal lattice in fourier space

4. Reciprocal Lattice Primitive Vector for a simple orthorhombic lattice Reciprocal primitive vectors

5. Reciprocal Lattice Next lattice plane b-c plane

6. First Brillouin Zone The smallest of a Wigner-Seitz cell in the reciprocal lattice The reciprocal lattices (dots) and corresponding first Brillouin zones of (a) square lattice and (b) hexagonal lattice.

7. The first Brillouin zone of a FCC structure

8. The first Brillouin zone of a FCC structure - Primitive vectors : -Primitive basis vectors of the face-centered cubic lattice

9. The first Brillouin zone of a FCC structure • Reciprocal primitive vectors : General reciprocal lattice vector:

10. The first Brillouin zone of a FCC structure 1st Brillouin zone : the shortest

11. 6.4.2 Bloch’s Theorem Hamiltonian Operator (for the one-electron model) From (3.68), “The one-electron Schrödinger equation”

12. - can be expanded as a fourier series : Bloch’s Theorem - The periodicity of the lattice structure : - The solution of the Schrödinger equation for a periodic potential must be a special form : Where is a periodic function with the periodicity of the lattices

13. Central Equation The wavefunction can be expressed as a Fourier series: From the one-electron Schrödinger equation, the coefficients of each Fourier component must be equal on both sides of the equation. : Central equation

14. Central Equation for 1-D From the first Brillouin zone where

15. Central Equation for 1-D - Near the zone boundary • Nontrivial solutions for the two coefficient

16. Standing Waves - Wave function at the zone edge Forming 2 standing waves

17. Electron band structure - Representation of the electronic band structure (a) The extended-zone scheme (b)The reduced –zone scheme

18. - Calculated energy band structure of copper Band Structures of Metals • Copper outermost configuration : • Electron in the s band can be easily excited from below the to above the “CONDUCTOR” • Interband transition The absorption of photons will cause the electrons in the s band to reach a higher level within the same band. 6.4.3 Band Structures of Metals and Semiconductors

19. Band structure of semiconductor Calculated energy band structure of Silicon Calculated energy band structure of GaAs - Interband transitions : The excitation or relaxation of electrons between subbands - Indirect gap :The bottom of the conduction band and the top of the valence band do not occur at the same k - Direct gap :The bottom of the conduction band and the top of the valence band occur at the same k

20. Band structure of semiconductor - Energy versus wavevector relations for the carriers - Effective mass