7. Energy Bands

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: 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 ραin 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