1. The Quantum Theory of Solids Allowed and forbidden energy bands Pauli Exclusion Principle In any given system, no two electrons can occupy the same state Application Two interacting hydrogen atoms

2. The Quantum Theory of Solids Allowed and forbidden energy bands Application to Silicon nlmspin 3s3001/2 or -1/2 3p31-1,0,11/2 or -1/2 n - principal; l - angular (0..n-1); m- magnetic (-l..+l)

3. The Quantum Theory of Solids Allowed and forbidden energy bands The k-space diagram where

4. The Quantum Theory of Solids Allowed and forbidden energy bands The k-space diagram (cont.)

5. The Quantum Theory of Solids Electrical Conduction in Solids Silicon at T = 0K All valence electrons are in the valence band

6. The Quantum Theory of Solids Electrical Conduction in Solids Silicon at T > 0K Some electrons have moved into the conduction band

7. The Quantum Theory of Solids Effective Mass but, difficult to know Fint, therefore where m* takes into account internal forces Analogy a force acting on ball in air vs. ball in oil Given concept of m*, we can determine acceleration in normal way F= -eE = m*a, therefore a = -eE/m*

8. The Quantum Theory of Solids Concept of a Hole Current flow may be thought of as the movement of valence electrons elevated to the conduction band Alternatively, can think of this process as the flow of (positively-charged) holes

9. The Quantum Theory of Solids Metals, Insulators, and Semiconductors Insulator Semiconductor Conductor non-overlapping overlapping

10. The Quantum Theory of Solids Extension to three dimensions direct bandgap indirect bandgap

11. The Quantum Theory of Solids Density of states E c - lowest energy of the conduction band E v - highest energy of the valence band

12. The Quantum Theory of Solids Statistical mechanics Number of possibilities for N particles in g states For n energy levels Most probable distribution

13. The Quantum Theory of Solids Statistical mechanics(cont.)

14. The Quantum Theory of Solids Statistical mechanics(cont.) Boltzmann approximation