Computational Solid State Physics 計算物性学特論 第４回

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 m: electron mass V(r): potential energy h: Planck constant Expansion by base functions Φn : overlap integral

3. :algebraic equation :matrix element of Hamiltonian

4. :expression of algebraic equation by matrixes and vectors

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

6. Solution (2)

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

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

9. Bloch’s theorem for wavefunctions in crystal (1) (2) k is wave vectors in the 1st 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 : potential energy in crystal : secular equation to obtain the energy eigenvalue at k. : Bragg reflection

13. Energy band structure of metals

14. Zincblende structure c b a

15. Brillouin zone for the zincblende lattice

16. Energy band of Si, Ge and Sn Si Ge Sn Empiricalpseudopotential method Empiricalpseudopotential method

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

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

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

20. 1-dimensional lattice (3) Energy dispersion relation ε(k)/-t 1st Brillouin zone ε0=H00: site energy t=H10=H-10: transfer energy ka t < 0

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

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

32. Bond picture (1): sp3 hybridization [111] [-1-1-1] [-11-1] [-1-11]

33. Bond picture (2) Hamiltonian for two hybridized orbits bonding and anti-bonding states : hybridized orbit energy 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. : transfer energy

34. Problems 4 • Calculate the free electron dispersion relation within the 1st 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.