Two-Dimensional Phase-Field Simulation ofSelf-Assembled Quantum Dot Formation
T. Takaki a, *, T. Hasebe b, Y. Tomita b
a Faculty of Maritime Sciences, Kobe University, Kobe, 658-0022, Japanb Faculty of Engineering, Kobe University, Kobe, 657-8501, Japan* http://www-s.eng.maritime.kobe-u.ac.jp/~takaki/, takaki@maritime.kobe-u.ac.jp

whereφ : phase field

Equilibrium shapeient
ationθ3 = 56.3

0 : vapor phase
1 : solid phase (= substrate + film)φ=which is excluded
from the equilibriumshapeora22
∇φ dVF = ∫  f d (φ ) + f e (φ , ε ij ) +V2θ1 = 11.3
θ2 = 33.8gEarsmissin

k = 16, γ = 0.02

< Free-energy functional >The morphologies of the thin film are assumed to be determined by thecompetition of the surface energy and the elastic strain energy. The
Ginzburg-Landau-type free-energy functional of the system is assumed
to have the form

g orientatio

PHASE-FIELD MODELnWe introduce k = 16 to simulate the multifaceted islands and island
morphological change.

missinPhase-field simulations are performed to investigate the effects of thegrowth parameters on island formation and island morphologicalevolution during deposition for SiGe/Si(001) system. In the phase-fieldmodel, facet morphologies are modeled using the generalized gradientcorrection coefficient for a crystal with a high anisotropy of surface
energy. Furthermore, to enable computation of the island morphologicalchange from faceted pyramids to multifaceted domes with increasing Ge
coverage, we apply the sixteen-fold of the surface energy. The validityof the model developed is confirmed by comparing the numerical resultsdemonstrated here with the reported experimental observations.

< Multifaceted island >SiGe/Si(001) shows the following surface morphological change:
2D wetting layer
SiGe/Si(001)↓morphological change{105} faceted pyramid island↓
Multifaceted dome island with {105}, {113} and {15 3 23} facets↓
Barn island with additional {111} facet (for dilute alloy)

missing orientation

INTRODUCTIONθ

migo
s s intati
rienonθmx-axis

fd : double-well potential

f d (φ ) = Wg (φ )g (φ ) = φ 2 (1 − φ )2fe : elastic strain energy(f e (φ , ε ij ) =)(1Dijkl (φ ) ε ij − ε ij0 ε kl − ε kl0
20: elastic coefficient
Dijkl (φ ) = ρ (φ )Dijkl

The first missing orientation θm iscalculated from

θ1 = 11.3°→ {105} facet planeθ2 = 33.8°→ {15 3 23} facet planea(θ m )sin θ m + aθ (θ m )cosθ m = 0 θ3 = 56.3°→ {111} facet plane)< Generalized gradient coefficient >To provide the island morphology without “ears” corresponding to themissing orientation, we use the regularized gradient coefficient for the
missing orientation.

1
2φ − 1 ρ (φ ) =  tanh+ 1 : density of solid phase22τa : gradient correction coefficienta {1 + γ cos kθ } for (2πi k + θ m ) ≤ θ ≤ (2π (i + 1) k − θ m )a(θ ) =  a(θ m )cosθ
for (2πi k − θ m ) < θ < (2πi k + θ m ) cosθ m(missing orientation)< Faceted island >The surface anisotropy is considered bya(θ ) = a {1 + γ cos kθ }wherea : constant related to the surface energy σ and the surface thickness δγ : strength of anisotropy
k : mode numberθ : angle between the surface normal and the x-axis
Faceted islands are created using a high anisotropy such as γ > 1/(k2-1)

k=4

γ = 0.01

γ = 0.05

γ = 0.10

γ = 0.20

凸凸凹

凹

γ<1k 2 −1γ>high anisotropy
1k 2 −1Ears
Ears

Equilibrium shapes< TDGL (Time-dependent Ginzburg-Landau) Equation >∂φδF  +V n χ= ∇ M∇∂tδφ  d y∂f (φ , ε ij )= M∇ 2 2φ (1 − φ )(1 − 2φ )W + e+∂φ∂  ∂a ∂φ  ∂  ∂a ∂φ 2 2  −  a+  a
 − a ∇ φ  + Vd n y χ∂x  ∂θ ∂y  ∂y  ∂θ ∂x 
σ ij , j = 0 : stress equilibrium condition

Surface energies

σ (θ ) = σ {1 + γ cos kθ }low anisotropy

where i denotes the integers from 0 to k−1.

whereM : mobility representing the surface diffusion
Vd : deposition rateny : y-direction component of the surface normal
χ : random numberThe phase field φ is assumed to be a conserved parameter. We assume
the stress equilibrium conditions σij = 0, since the elastic relaxation
occurs much faster than the surface diffusion.

NUMERICAL SIMULATIONRESULTS

Phase field equation is descretized spatially using second-order finite
difference method and explicit time differencing is employed for φ.
Stress field is solved by finite element method as a plane strain problem.

The effects of Ge composition, mobility and deposition rate on island
formation and the evolution of surface morphology are investigated, andfollowing results are obtained:● For the 3D pyramid formation, it is observed that, at a low mismatchstrain, 3D pyramid islands are formed from surface ripples as a result ofstrain-driven instability, while at high mismatch strain, islands are formedby nucleation.
● Si0.5Ge0.5 islands tend to show improved spatial ordering and a narrowsize distribution compared to Ge islands.
● In the growth of a Si0.5Ge0.5 layer at low temperatures, a rippled surface
morphology continues to grow without the formation of individual islands.

εx0 = 0Vapor

Computational
domain forStress field

Filmtsεx0 = -ε misyFE mesh

Substrate(0,0)

x-dXε misdX< Parameters >Surface energy σ = 1.23 J/m2
Young’s modulus E = 104 GPa
Poisson’s ratio ν = 0.27
Strength of anisotropy γ = 0.1
Lattice mismatch strain εmis = 0.04x(where x is Ge composition)
Mesh size dx = 5 nm
Time increment dt = dx4/(62.0a2M)

0.625 µm

Computational domains and boundary conditions

900 s

Vd = 0.005 nm/s

εx0 = -ε misdY

Computational
domain forPhase-field

1200 s

2.025 µm (405 mesh)
Finite element mesh and computational domain size

1500 s1800 s

855 s

885 s

900 s

945 s

1140 s1200 s1035 s1395 s1500 s1125 s1650 s1800 sVd = 0.01 nm/s

M = 1×10-25 m5/Js

M = 5×10-25 m5/Js

Effects of
750 s

Growth temperature

1219 sGe composition

1688 s

Deposition rate

900 s

1200 s1500 son the surface morphological change

1800 s

938 s

938 s

938 s

1875 s1875 s1875 s2813 s2813 s2813 s3750 s3750 s3750 s2156 sVd = 0.05 nm/s

x = 0.75 (Si0.25Ge0.75/Si)

x = 1.0 (Ge/Si)

M = 1×10-24 m5/Js

x = 0.5 (Si0.5Ge0.5/Si)

Close-up views

M = 1×10-24 m5/Js

M = 5×10-25 m5/Js

M = 1×10-25 m5/Js

CONCLUSIONS
2D wetting layer

island nucleation

pyramidpyramid

layer to island

525 s

750 s

975 s

1219 s

dome

dome

dome

dome

2156 s1913 s1688 s1444 sClose-up views (Si0.25Ge0.75/Si)−3.5 −3.0

−2.0

−1.0

0.0

1.0stress σx [MPa]pyramid to dome

The heteroepitaxial growth of Si1-xGex/Si(001) alloys during deposition
are simulated by the phase-field method as a two-dimensional problem,
which is developed so as to simulate a typical sequential growth of a
wetting layer, faceted pyramid islands and multifaceted dome islands.
The effects of Ge composition, mobility and deposition rate on island
formation and the evolution of surface morphology are investigated. It is
confirmed that the phase-field model developed can simulate sometypical and important phenomena for the formation and evolution of
SiGe islands grown on Si, as observed in previous experiments. We
believe that this model can be applied to the analysis of otherheteroepitaxial systems for the formation of faceted 3D islands and canbe expanded to solve more realistic three-dimensional problems.