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:

Zinc blend structure

Valence orbits 4 bonds

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

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

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

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

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

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

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

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

Stillinger Weber potential (4): Melting

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

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

Close packed surface and crystal morphology

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

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.

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

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

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.

Equilibrium shape of crystal

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

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

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

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

Energy relaxation by misfit dislocation

Critical thickness of heteroepitaxial growth

Lattice constant and energy gap of IIIV semiconductors

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.