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Pert 1-4

Pert 1-4

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  • 1. Pertemuan 1 - 4 FISIKA ZAT PADAT Iwan Sugihartono, M.Si Jurusan Fisika Fakultas Matematika dan Ilmu Pengetahuan Alam
  • 2. Outline
    • Bloch Functions
    • Nearly Free Electron Model
    • Kronig-Penney Model
    • Wave Equation of Electron in a Periodic Potential
    • Number of Orbitals in a Band
    15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 3. 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
    • New concepts:
    • Effective mass
    • Holes
    finite T impurities 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 4. NEARLY FREE ELECTRON MODEL 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | Bragg reflection -> no wave-like solutions -> energy gap Bragg condition: ->
  • 5. ORIGIN OF THE ENERGY GAP 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 6. BLOCH FUNCTIONS 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | Periodic potential -> Translational symmetry -> Abelian group T = { T ( R l )} k -representation of T ( R l ) is Corresponding basis function for the Schrodinger equation must satisfy This can be satisfied by the Bloch function where or -> representative values of k are contained inside the Brillouin zone. Basis =
  • 7. KRONIG-PENNEY MODEL 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | Bloch theorem: ψ  (0) continuous: ψ  ( a ) continuous: ψ (0) continuous: ψ ( a ) continuous:
  • 8. -> Delta function potential: Thus so that 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 9. 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 10. MATRIX MECHANICS 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | Ansatz Secular equation: Matrix equation Orthonormal basis: Eigen-problem
  • 11. FOURIER SERIES OF THE PERIODIC POTENTIAL 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | ->  = Volume of crystal   volume of unit cell For a lattice with atomic basis at positions ρ α in the unit cell is the structural factor ->
  • 12. PLANE WAVE EXPANSION 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | Bloch function  = Volume of crystal Matrix form of the Schrodinger equation: (central equation) n = 0:
  • 13. CRYSTAL MOMENTUM OF AN ELECTRON 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | Properties of k : -> U = 0 -> Selection rules in collision processes -> crystal momentum of electron is  k . Eq., phonon absorption:
  • 14. SOLUTION OF THE CENTRAL EQUATION 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | 1-D lattice, only
  • 15. KRONIG-PENNEY MODEL IN RECIPROCAL SPACE 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | Eigen-equation: -> (only s = 0 term contributes)
  • 16. -> (Kronig-Penney model) with 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 17. EMPTY LATTICE APPROXIMATION 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | Free electron in vacuum: Free electron in empty lattice: Simple cubic
  • 18. APPROXIMATE SOLUTION NEAR A ZONE BOUNDARY 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | k near zone right boundary: Weak U , λ k  2 g >> U -> for E near λ k
  • 19. K << g /2 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 20. 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 21. NUMBER OF ORBITALS IN A BAND 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | 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
  • 22. THANK YOU 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |