Computational Solid State Physics ??????? ?3?

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Computational Solid State Physics ??????? ?3?

• 3. Covalent bond and morphology of crystals,
  surfaces and interfaces

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Covalent bond

• Diamond structure C, Si, Ge
• Zinc blend structure GaAs, InP
• lattice constant a
• number of nearest neighbor atoms4
• bond length
• bond angle

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Zinc blend structure
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Valence orbits
4 bonds
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sp3 hybridization

• 111
• 1-1-1
• -11-1
• -1-11

The four bond orbits are constituted by sp3
hybridization.
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Keating model for covalent bond (1)

• Energy increase by displacement from the
  optimized structure
• Translational symmetry of space
• Rotational symmetry of space

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Inner product of two covalent bonds Keating
model (2)
a lattice constant
b1
b2
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Keating model potential (3)
Taylor expansion around the optimized structure.
First order term on ?klmn vanishes from the
optimization condition.
1st term energy of a bond length
displacement 2nd term energy of the bond angle
displacement
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Stillinger Weber potential (1)
2-atom interaction
3-atom interaction
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Stillinger Weber potential (2)
dimensionless 2-atom interaction
dimensionless 3-atom interaction
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Stillinger Weber potential (3)
bond length dependence
bond angle dependence
minimum at
minimum at
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Stillinger Weber potential (4) crystal structure
most stable for diamond structure.
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Stillinger Weber potential (4) Melting
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Morphology of crystals, surfaces and interfaces

• Surface energy and interface energy

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Surface energy

• Surface energy energy required to fabricate a
  surface from bulk crystal
• fcc crystal lattice constant a
• bond length a /v2
• bond energy e
• (111) surface area of a
  unit cell
• surface energy per unit area

• a/v2

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Close packed surface and crystal morphology
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Equilibrium shape of liquiud

• Sphere
• minimum surface energy, i.e. minimum surface
  area for constant volume

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

• Wulffs 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.

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Wulffs plot
Surface energy has a cusp at the low-index
surface.
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Vicinal surfaces (1)

• Vicinal surfaces constitute of terraces and
  steps.
• Surface energy per unit projected area

ß step free energy per unit length g
interaction energy between steps
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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.
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Equilibrium shape of crystal
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Growth mode of thin film

• Volmer-Weber mode (island mode)
• Frank-van der Merwe mode (layer mode)
• Stranski-Krastanov mode (layerisland mode)

film
substrate
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Interface energy s

• Interface energy energy required to fabricate
  the interface per unit area
• Island mode
• ex. metal on insulator
• Layer mode ex.semiconductor on
• semiconductor
• Layerisland mode
• ex. metal on semiconductor

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Wetting angle

• Surface free energy F
• Surface tension s
• Surface free energy is equal to surface tension
  for isotropic surfaces.

T wetting angle
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Heteroepitaxial growth of thin film

• Pseudomorphic mode (coherent mode)
• growth of strained layer with a lattice
  constant of a substrate
• layer thicknessltcritical thickness
• Misfit dislocation formation mode
• layer thicknessgtcritical thickness

lattice misfit
aa lattice constant of heteroepitaxial
crystal as lattice constant of substarate
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Energy relaxation by misfit dislocation
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Critical thickness of heteroepitaxial growth
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Lattice constant and energy gap of IIIV
semiconductors
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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.