1. 7. Energy Bands Bloch Functions Nearly Free Electron Model
Kronig-Penney Model
Wave Equation of Electron in a Periodic Potential
Number of Orbitals in a Band

2. Some successes of the free electron model: C, κ, σ, χ, …
Some failures of the free electron model:
Distinction between metals, semimetals, semiconductors & insulators.
Positive values of Hall coefficent.
Relation between conduction & valence electrons.
Magnetotransport.
Band model
finite T
impurities
New concepts:
Effective mass
Holes

3. Nearly Free Electron Model
Bragg reflection → no wave-like solutions
→ energy gap
Bragg condition: →

4. Origin of the Energy Gap

5. Bloch Functions
Periodic potential → Translational symmetry → Abelian group T = {T(Rl)}
k-representation of T(Rl) is
Basis =
Corresponding basis function for the Schrodinger equation must satisfy or
This can be satisfied by the Bloch function
where
→ representative values of k are contained inside the Brillouin zone.

6. Kronig-Penney Model Bloch theorem: ψ(0) continuous: ψ(a) continuous:

7. →
Delta function potential:
Thus
so that

9. Matrix Mechanics Ansatz Matrix equation Eigen-problem
Secular equation:
Orthonormal basis:

10. Fourier Series of the Periodic Potential→
V = Volume of crystal
  volume of unit cell →
For a lattice with atomic basis at positions ρα in the unit cell
is the structural factor

11. Plane Wave Expansion Bloch function
V = Volume of crystal
Matrix form of the Schrodinger equation:
n = 0:
(central equation)

12. Crystal Momentum of an Electron
Properties of k: →
U = 0 →
Selection rules in collision processes → crystal momentum of electron is  k.
Eq., phonon absorption:

13. Solution of the Central Equation
1-D lattice, only

14. Kronig-Penney Model in Reciprocal Space
(only s = 0 term contributes)
Eigen-equation: →

15. (Kronig-Penney model)→
with

16. Empty Lattice Approximation
Free electron in vacuum:
Free electron in empty lattice:
Simple cubic

17. Approximate Solution Near a Zone Boundary
Weak U, λk2g >> U
k near zone right boundary: →
for E near λk

18. K << g/2

20. Number of Orbitals in a Band
Linear crystal of length L composed of of N cells of lattice constant a.
Periodic boundary condition:
→ N inequivalent values of k →
Generalization to 3-D crystals:
Number of k points in 1st BZ = Number of primitive cells
→ Each primitive cell contributes one k point to each band.
Crystals with odd numbers of electrons in primitive cell must be metals,
e.g., alkali & noble metals
insulator
metal
semi-metal