1 / 20

7. Energy Bands

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:

bud
Télécharger la présentation

7. Energy Bands

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

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

  9. 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

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

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

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

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

  14. (Kronig-Penney model) → with

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

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

  17. K << g/2

  18. 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

More Related