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Computational Solid State Physics 計算物性学特論　第３回

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

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Computational Solid State Physics 計算物性学特論　第３回

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

  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 λklmnvanishes 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 tofabricate 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. minimumsurface area for constant volume

  18. Equilibrium shape of crystal • 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. Minimize the surface energy for constant crystal volume.

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

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

  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.

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