7 Energy Bands Bloch Functions Nearly Free Electron

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

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

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

Origin of the Energy Gap

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.

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

→ Delta function potential: Thus so that

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

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

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

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

Solution of the Central Equation 1 -D lattice, only

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

→ with (Kronig-Penney model)

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

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

K << g/2

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 1 st 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 metal semi-metal insulator