1- 4 a Energy Bands

1. Pertemuan 1 - 4 FISIKA ZAT PADAT Iwan Sugihartono, M.Si Jurusan Fisika Fakultas Matematika dan Ilmu Pengetahuan Alam

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. ORIGIN OF THE ENERGY GAP 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

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. 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. -> Delta function potential: Thus so that 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

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10. MATRIX MECHANICS Ansatz Secular equation: Matrix equation Orthonormal basis: Eigen-problem 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

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. 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. 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. SOLUTION OF THE CENTRAL EQUATION 1-D lattice, only 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

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. -> (Kronig-Penney model) with 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

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. 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. K << g /2 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

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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. THANK YOU 10/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |