Pert 1-4

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Pert 1-4

  1. 1. Pertemuan 1 - 4 FISIKA ZAT PADAT Iwan Sugihartono, M.Si Jurusan Fisika Fakultas Matematika dan Ilmu Pengetahuan Alam
  2. 2. Outline <ul><li>Bloch Functions </li></ul><ul><li>Nearly Free Electron Model </li></ul><ul><li>Kronig-Penney Model </li></ul><ul><li>Wave Equation of Electron in a Periodic Potential </li></ul><ul><li>Number of Orbitals in a Band </li></ul>15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  3. 3. Some successes of the free electron model: C, κ , σ , χ , … <ul><li>Some failures of the free electron model: </li></ul><ul><ul><li>Distinction between metals, semimetals, semiconductors & insulators. </li></ul></ul><ul><ul><li>Positive values of Hall coefficent. </li></ul></ul><ul><ul><li>Relation between conduction & valence electrons. </li></ul></ul><ul><ul><li>Magnetotransport. </li></ul></ul>Band model <ul><li>New concepts: </li></ul><ul><li>Effective mass </li></ul><ul><li>Holes </li></ul>finite T impurities 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  4. 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. 5. ORIGIN OF THE ENERGY GAP 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  6. 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. 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. 8. -> Delta function potential: Thus so that 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  9. 9. 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  10. 10. MATRIX MECHANICS 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | Ansatz Secular equation: Matrix equation Orthonormal basis: Eigen-problem
  11. 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. 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. 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. 14. SOLUTION OF THE CENTRAL EQUATION 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | 1-D lattice, only
  15. 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. 16. -> (Kronig-Penney model) with 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  17. 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. 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. 19. K << g /2 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  20. 20. 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  21. 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. 22. THANK YOU 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

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