Energy Bands in Solids

1. Energy Bands in Solids Physics 355

2. Conductors, Insulators, and Semiconductors Consider the available energies for electrons in the materials. As two atoms are brought close together, electrons must occupy different energies due to Pauli Exclusion principle. Instead of having discrete energies as in the case of free atoms, the available energy states form bands.

3. Free Electron Fermi Gas Monovalent Crystal

5. Fermi-Dirac Function

6. For free electrons, the wavefunctions are plane waves:

7. Felix Bloch 1905-1983

8. Band Gap zone boundary

9. “thermally excited” “doped”

10. +  Origin of the Band Gap To get a standing wave at the boundaries, you can take a linear combination of two plane waves:

11. Origin of the Band Gap Electron Density

12. Origin of the Band Gap

13. Bloch Functions Felix Bloch showed that the actual solutions to the Schrödinger equation for electrons in a periodic potential must have the special form: where u has the period of the lattice, that is Felix Bloch 1905-1983

14. U(x) U0 (a+b) b 0 a a+b x Kronig-Penney Model The wave equation can be solved when the potential is simple... such as a periodic square well.

15. Kronig-Penney Model Wave Equation Region I - where 0 < x < a and U = 0 The eigenfunction is a linear combination of plane waves traveling both left and right: The energy eignevalue is:

16. Kronig-Penney Model Wave Equation Region II - where b < x < 0 and U = U0 Within the barrier, the eigenfunction looks like this and

17. U(x) U0 (a+b) b 0 a a+b x Kronig-Penney Model II I III To satisfy Mr. Bloch, the solution in region III must also be related to the solution in region II.

18. Kronig-Penney Model A,B,C, and D are chosen so that both the wavefunction and its derivative with respect to x are continuous at the x = 0 and a. At x = 0... At x = a...

19. Kronig-Penney Model Result for E < U0: To obtain a more convenient form Kronig and Penney considered the case where the potential barrier becomes a delta function, that is, the case where U0 is infinitely large, over an infinitesimal distance b, but the product U0b remains finite and constant. and also goes to infinity as U0. Therefore:

20. Kronig-Penney Model What happens to the product Qb as U0 goes to infinity? • b becomes infinitesimal as U0 becomes infinite. • However, since Q is only proportional to the square root of U0, it does not go to infinity as fast as b goes to zero. • So, the product Qb goes to zero as U0 becomes infinite. • As a results of all of this...

21. Kronig-Penney Model

22.   ka ka 0  2 3  0  Kronig-Penney Model Plot of energy versus wavenumber for the Kronig-Penney Potential, with P = 3/2. Extended Zone Scheme Reduced Zone Scheme

23. Conductors, Insulators, and Semiconductors Free Electron Model Crucial to the conduction process is whether or not there are electrons available for conduction.

24. Conductors, Insulators, and Semiconductors

25. Conductors, Insulators, and Semiconductors

26. Conductors, Insulators, and Semiconductors “thermally excited” “doped”