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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:. - PowerPoint PPT Presentation
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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: 3
1cos
4/3ab
Zinc blend structure
Valence orbits
4 bonds
;4/111
;4/111
;4/111
;4/111
4
3
2
1
a
a
a
a
d
d
d
d
sp3 hybridization
]|||[|2
1|
]|||[|2
1|
]|||[|2
1|
]|||[|2
1|
4
3
2
1
zyx
zyx
zyx
zyx
pppsh
pppsh
pppsh
pppsh [111]
[1-1-1]
[-11-1]
[-1-11]
ijjihh | 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 spacelkkl
klVV
rrr
r
)(
2/)(
)(
a
VV
mnklmnklklmn
klmn
RRrr
)( iVV r
rk: position of the k-th atom
Rk: optimized position of the k-th atom
Inner product of two covalent bonds: Keating model (2)
163
1
4
3 22
21
aa
bb
221 16
3ab
b1
b2
a : lattice constant
Keating model potential (3)
l i ijijii
l i ijijii
allBalrA
alla
alra
V
4 4
,
2200
20
4 4
,
22004
22204
])16
1)()(()
4
3)(([
2
1
])16
1)()((
2)
16
3)(([
2
1
rr
rr
1st term: energy of a bond length displacement
2nd term: energy of the bond angle displacement
・ First order term on λklmn vanishes from the optimization condition.
・ Taylor expansion around the optimized structure.
Stillinger Weber potential (1)
),,(
)(
3
2
lji
ij
rrr
r
lji
ljiji
ij rrrrV,,
3,
2 ),,()(
: 2-atom interaction
: 3-atom interaction
Stillinger Weber potential (2)
)/,/,/(),,(
)/()(
33
22
ljilji
ijij
rrrfrrr
rfr
arrf
ararrBrArf qp
:0)(
:])exp[()()(
2
12
211
3
)3
1](cos)()(exp[),,(
),,(),,(),,()/,/,/(
jikikijjikikij
ikjkjkiijkjkjijikikijlji
ararrrh
rrhrrhrrhrrrf
dimensionless 2-atom interaction
dimensionless 3-atom interaction
arar ikij ,
Stillinger Weber potential (3)
)( 02 r
bond length dependence
bond angle dependence
6
1
0 2rrijminimum at minimum at
3
1cos
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 area4
3 2a
2
2
/32
2/)4/3/(3
a
a
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
Wulff’s plot1.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.
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
30 |tan|)(
|tan|)()(),( Tg
hTTfTf p
β: step free energy per unit length
g: interaction energy between steps
Vicinal surfaces (2)
332210
20
)()()()(
sin|tan|)(|sin|
)(cos)(
cos),(),(
TATATATA
Tgh
TTf
TfTf p
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)
substrate
film
Interface energy: σ
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
avsasv
avsasv
σsv
σav
σsa
Wetting angle
Surface free energy: F
Surface tension: σ
Surface free energy is equal to surface tension for isotropic surfaces.
A
dAF
σav
σsv
σsa
θ
av
sasv
cos
Θ: wetting angle
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
s
sa
a
aa 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.