Pertemuan 1 - 4 FISIKA ZAT PADAT Iwan Sugihartono, M.Si Jurusan Fisika Fakultas Matematika dan Ilmu Pengetahuan Alam
ENERGY BANDS <ul><li>Bloch Functions  </li></ul><ul><li>Nearly Free Electron Model </li></ul><ul><li>Kronig-Penney Model <...
Some successes of the free electron model: C,  κ ,  σ ,  χ ,  … <ul><li>Some failures of the free electron model:  </li></...
NEARLY FREE ELECTRON MODEL Bragg reflection  -> no wave-like solutions ->  energy gap Bragg condition: -> 10/03/11 ©  2010...
ORIGIN OF THE ENERGY GAP 10/03/11 ©  2010 Universitas Negeri Jakarta  |  www.unj.ac.id  |
BLOCH FUNCTIONS Periodic potential  ->  Translational symmetry  ->  Abelian group  T  = { T ( R l )}  k -representation  o...
KRONIG-PENNEY MODEL Bloch theorem: ψ    (0) continuous: ψ    ( a ) continuous: ψ (0) continuous: ψ ( a ) continuous: 10/...
-> Delta function potential:  Thus so that 10/03/11 ©  2010 Universitas Negeri Jakarta  |  www.unj.ac.id  |
10/03/11 ©  2010 Universitas Negeri Jakarta  |  www.unj.ac.id  |
MATRIX MECHANICS Ansatz  Secular equation: Matrix equation Orthonormal basis: Eigen-problem 10/03/11 ©  2010 Universitas N...
FOURIER SERIES OF THE PERIODIC POTENTIAL ->    = Volume of crystal       volume of unit cell For a lattice with atomic ...
PLANE WAVE EXPANSION Bloch function    = Volume of crystal Matrix form of the Schrodinger equation: (central equation) n ...
CRYSTAL MOMENTUM OF AN ELECTRON Properties of  k : -> U  = 0  -> Selection rules in collision processes -> crystal momentu...
SOLUTION OF THE CENTRAL EQUATION 1-D lattice, only 10/03/11 ©  2010 Universitas Negeri Jakarta  |  www.unj.ac.id  |
KRONIG-PENNEY MODEL IN RECIPROCAL SPACE (only  s  = 0 term contributes) Eigen-equation: -> 10/03/11 ©  2010 Universitas Ne...
-> (Kronig-Penney model) with 10/03/11 ©  2010 Universitas Negeri Jakarta  |  www.unj.ac.id  |
EMPTY LATTICE APPROXIMATION Free electron in vacuum: Free electron in empty lattice: Simple cubic 10/03/11 ©  2010 Univers...
APPROXIMATE SOLUTION NEAR A ZONE BOUNDARY k  near zone right boundary: Weak  U ,  λ k  2 g  >>  U -> for  E  near  λ k 10...
K  <<  g /2 10/03/11 ©  2010 Universitas Negeri Jakarta  |  www.unj.ac.id  |
10/03/11 ©  2010 Universitas Negeri Jakarta  |  www.unj.ac.id  |
NUMBER OF ORBITALS IN A BAND Linear crystal of length  L  composed of of  N  cells of lattice constant  a . Periodic bound...
THANK YOU 10/03/11 ©  2010 Universitas Negeri Jakarta  |  www.unj.ac.id  |
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1- 4 a Energy Bands

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1- 4 a Energy Bands

  1. 1. Pertemuan 1 - 4 FISIKA ZAT PADAT Iwan Sugihartono, M.Si Jurusan Fisika Fakultas Matematika dan Ilmu Pengetahuan Alam
  2. 2. ENERGY BANDS <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>10/03/11 © 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 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  4. 4. NEARLY FREE ELECTRON MODEL Bragg reflection -> no wave-like solutions -> energy gap Bragg condition: -> 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  5. 5. ORIGIN OF THE ENERGY GAP 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  6. 6. BLOCH FUNCTIONS 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 = 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  7. 7. KRONIG-PENNEY MODEL Bloch theorem: ψ  (0) continuous: ψ  ( a ) continuous: ψ (0) continuous: ψ ( a ) continuous: 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  8. 8. -> Delta function potential: Thus so that 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  9. 9. 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  10. 10. MATRIX MECHANICS Ansatz Secular equation: Matrix equation Orthonormal basis: Eigen-problem 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  11. 11. FOURIER SERIES OF THE PERIODIC POTENTIAL ->  = 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 -> 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  12. 12. PLANE WAVE EXPANSION Bloch function  = Volume of crystal Matrix form of the Schrodinger equation: (central equation) n = 0: 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  13. 13. CRYSTAL MOMENTUM OF AN ELECTRON Properties of k : -> U = 0 -> Selection rules in collision processes -> crystal momentum of electron is  k . Eq., phonon absorption: 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  14. 14. SOLUTION OF THE CENTRAL EQUATION 1-D lattice, only 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  15. 15. KRONIG-PENNEY MODEL IN RECIPROCAL SPACE (only s = 0 term contributes) Eigen-equation: -> 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  16. 16. -> (Kronig-Penney model) with 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  17. 17. EMPTY LATTICE APPROXIMATION Free electron in vacuum: Free electron in empty lattice: Simple cubic 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  18. 18. APPROXIMATE SOLUTION NEAR A ZONE BOUNDARY k near zone right boundary: Weak U , λ k  2 g >> U -> for E near λ k 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  19. 19. K << g /2 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  20. 20. 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  21. 21. 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 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  22. 22. THANK YOU 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

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