More Related Content More from jayamartha (20) Pert 1-41. Pertemuan 1 - 4 FISIKA ZAT PADAT Iwan Sugihartono, M.Si Jurusan Fisika Fakultas Matematika dan Ilmu Pengetahuan Alam 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) 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