1. Computational Solid State Physics 計算物性学特論 第３回
3. Covalent bond and morphology of crystals, surfaces and interfaces

2. Covalent bond Diamond structure: C, Si, Ge
Zinc blend structure: GaAs, InP
lattice constant : a
number of nearest neighbor atoms=4
bond length:
bond angle:

3. Zinc blend structure

4. Valence orbits
4 bonds

5. sp3 hybridization [111] [1-1-1] [-11-1] [-1-11]
The four bond orbits are constituted by sp3 hybridization.

6. Keating model for covalent bond (1)
Energy increase by displacement from the optimized structure
Translational symmetry of space
Rotational symmetry of space
rk: position of the k-th atom
Rk: optimized position of the k-th atom

7. Inner product of two covalent bonds: Keating model (2)
a : lattice constant b1 b2

8. Keating model potential (3)
・Taylor expansion around the optimized structure.
・First order term on λklmn vanishes from the 　 　optimization condition.
1st term: energy of a bond length displacement
2nd term: energy of the bond angle displacement

9. Stillinger Weber potential (1)
: 2-atom interaction
: 3-atom interaction

10. Stillinger Weber potential (2)
dimensionless 2-atom interaction
dimensionless 3-atom interaction

11. Stillinger Weber potential (3)
bond length dependence
bond angle dependence
minimum at
minimum at

12. Stillinger Weber potential (4): crystal structure
most stable for diamond structure.

13. Stillinger Weber potential (4): Melting

14. Morphology of crystals, surfaces and interfaces
Surface energy and interface energy

15. Surface energy
Surface energy: energy required to fabricate a surface from bulk crystal
fcc crystal: lattice constant: a
bond length: a /√2
bond energy: ε
(111) surface: area of a unit cell
・ surface energy per unit area
a/√2

16. Close packed surface and crystal morphology

17. Equilibrium shape of liquiud
Sphere
　minimum surface energy, i.e. minimum　surface area for constant volume

18. Equilibrium shape of crystal
Minimize the surface energy for constant crystal volume.
Wulff’s plot
1.Plot surface energies on lines starting from the center of the crystal.
2.Draw a polyhedron enclosed by inscribed planes at the cusp of the calculated surface energy.

19. Wulff’s plot
Surface energy has a cusp at the low-index surface.

20. Vicinal surfaces (1)
Vicinal surfaces constitute of terraces and steps.
・Surface energy per unit projected area
β: step free energy per unit length
g: interaction energy between steps

21. Vicinal surfaces (2) Surface energy per unit area of a vicinal surface
Surface energy of the vicinal surface is higher than that of the low index surface.
Orientation dependence of surface energy has a cusp at the low-index surface.

22. Equilibrium shape of crystal

23. Growth mode of thin film
Volmer-Weber mode (island mode)
Frank-van der Merwe mode (layer mode)
Stranski-Krastanov mode (layer+island mode)
film
substrate

24. Interface energy: σ
σsv
σav
σsa
Interface energy: energy required to fabricate the interface per unit area
Island mode
ex. metal on insulator
Layer mode　ex.semiconductor on
　semiconductor
Layer+island mode
ex. metal on semiconductor

25. Wetting angle Surface free energy: F Surface tension: σ
Surface free energy is equal to surface tension for isotropic surfaces.
Θ: wetting angle
σav
σsv
σsa θ

26. Heteroepitaxial growth of thin film
Pseudomorphic mode (coherent mode)
growth of strained layer with a lattice constant of a substrate
layer thickness<critical thickness
Misfit dislocation formation mode
layer thickness>critical thickness
lattice misfit:
aa: lattice constant of heteroepitaxial crystal
as: lattice constant of substarate

27. Energy relaxation by misfit dislocation

28. Critical thickness of heteroepitaxial growth

29. Lattice constant and energy gap of IIIV semiconductors

30. Problems 3
Calculate the most stable structure for (Si)n clusters using the Stillinger-Weber potential.
Calculate the surface energy for (111), (100) and (110) surface of fcc crystals using the simple bond model.
Calculate the equilibrium crystal shape for fcc crystal using the simple bond model.
Calculate the equilibrium crystal shape for diamond crystal using the simple bond model.