1. Computational Solid State Physics 計算物性学特論 第４回 4. Electronic structure of crystals

2. Single electron Schroedinger equation Expansion by base functions Φ n : overlap integral m: electron mass V(r): potential energy h: Planck constant

3. : matrix element of Hamiltonian :algebraic equation

4. :expression of algebraic equation by matrixes and vectors

5. : ortho-normalized bases : unit matrix eigenvalue equation condition of existence of inverse matrix of secular equation Solution (1)

6. Solution (2)

7. Potential energy in crystals a,b,c: primitive vectors of the crystal n.l.m: integers G: reciprocal lattice vectors :periodic potential Fourier transform of the periodic potential energy

8. Primitive reciprocal lattice vectors : volume of a unit cell Volume of 1st Brilloluin zone Properties of primitive reciprocal lattice vectors

9. Bloch ’ s theorem for wavefunctions in crystal (1) (2) k is wave vectors in the 1 st Brillouin zone. Equations (1) and (2) are equivalent.

10. Plane wave expansion of Bloch functions G : reciprocal lattice vectors

11. Normalized plane wave basis set :satisfies the Bloch’s theorem V : volume of crystal

12. Schroedinger equation for single electron in crystals : Bragg reflection : potential energy in crystal : secular equation to obtain the energy eigenvalue at k.

13. Energy band structure of metals

14. Zincblende structure a b c

15. Brillouin zone for the zincblende lattice

16. Empirical pseudopotential method Energy band of Si, Ge and Sn Empirical pseudopotential method Si Ge Sn

17. Tight-binding approximation i-th atomic wavefunction at (n,l,m)-lattice sites Linear Combination of Atomic Orbits (LCAO) satisfies the Bloch theorem.

18. 1-dimensional lattice (1) a S(n-m)

19. :Schroedinger equation 1-dimensional lattice (2)

20. 1-dimensional lattice (3) ε 0 =H 00 : site energy t=H 10 =H -10 : transfer energy ka ε(k)/-t t < 0 Energy dispersion relation 1 st Brillouin zone

21. Valence orbits for III-V compounds 4 bonds

22. Matrix elements of Hamiltonian between atomic orbits

23. Matrix element of Hamiltonian between atomic orbit Bloch functions

24. Calculation of Hamiltonian matrix element

25. Matrix element between atomic orbits

26. Hamiltonian matrix for the zincblende structure

27. 1-fold Bottom of conduction band: s-orbit Top of valence band: p-orbit Energy at Gamma point (k=0) 3-fold

28. Energy band of Germanium

29. Energy band of GaAs, ZnSe, InSb, CdTe

30. Spin-orbit splitting at band edge

31. Efficiency and color of LED Periodic table B C N Al Si P Ga Ge As In Sn Sb PL energy is determined by the energy gap of direct gap semiconductors.

32. Bond picture (1): sp 3 hybridization [111] [-1-1-1] [-11-1] [-1-11]

33. Bond picture (2) Hamiltonian for two hybridized orbits : hybridized orbit energy : transfer energy bonding and anti-bonding states Successive transformations of linear Combinations of atomic orbitals, beginning with atomic s and p orbitals and proceeding to Sp3 hybrids, to bond orbitals, and finally to band states. The band states represent exact solution of the LCAO problem.

34. Problems 4 Calculate the free electron dispersion relation within the 1 st Brillouin zone for diamond structure. Calculate the energy dispersion relation for a graphen sheet, using a tight-binding approximation. Calculate the dispersion relation for a graphen sheet, using pane wave bases.