21
Phase field models for crystal steps Olivier Pierre-Louis PRE 68 012604 (2003) http://consoude.ujf-grenoble.fr Spectro, Grenoble, France

Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

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Page 1: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

Phase field models

for crystal steps

Olivier Pierre-Louis

PRE 68 012604 (2003)

http://consoude.ujf-grenoble.frSpectro,

Grenoble, France

Page 2: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

Solid on Solid

Steps

Continuum

Page 3: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

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adatoms

Solid

c+c−

θ

φ

Phasefieldmodel

stepmodel

.

Discont.

Page 4: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

Phase field model for crystal steps:

• Lui and Metiu (1994): surface relaxation

• Plapp and Karma (1998): spiral growth

Step kinetics?

• Schwoebel effect

• Step transparency

Phase field model with 1 global concentration field→ Fast step kinetics

P.f. model with 1 concentration field per terrace→ Arbitrary step kinetics

Page 5: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

Discontinuous interface modelOn terraces

∂tc = D∇2c+ F − c/τ

At the step (Standard form)

D∂nc+ = ν+(c+ − c∗eq) + ν0(c+ − c−)

−D∂nc− = ν−(c− − c∗eq) + ν0(c− − c+)

c∗eq = ceq(1 + Γκ)

1

ΩV = (−D∂nc− − V c−) + (D∂nc+ + V c+)

νν+

ν0

zx

S

D

E

Page 6: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

Re-formulation of boudary conditions:

Form I: Modified equilibrium concentration

D∂nc+ = ν+(c+ − ceq)

−D∂nc− = ν−(c− − ceq)

ceq = ceq(1 + Γκ) + βV

Ω

(i) Other physical meaning of transparency:modification of ceq on a moving step.

(ii) Link to solidificationlimit of strong transparency ν0 → ∞:

c+ = c− = ceq = ceq(1 + Γκ) + βV

Ω

Page 7: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

Re-formulation of boudary conditions:

Form II: Global exchange

D∂nc+ = αV

Ω+ ν(c+ − c−)

−D∂nc− = (1 − α)V

Ω+ ν(c− − c+)

V

Ω=

1

β

[

α(c+ − c∗eq) + (1 − α)(c− − c∗eq)]

Page 8: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

2 fields model

∂tc = ∇[M∇θ] + F −θ

τ− ∂th

τp∂tφ = W 2∇

2φ− fφ + λgφ(θ − θeq)

variational if g ∝ h.

F =

∫[

W 2

2(∇φ)2 + f +

λ

2(θ − θeq)

2

]

f

φ

Page 9: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

Asymptotics narrow interface:W → 0, τp ∼ D/W 2.

Sharp interface limit: λ ∼Wweak coupling limit→ strong transparency

ν+ ∼ 1, ν− ∼ 1, ν0 ∼1

W

Thin interface limit: (θ − θeq) ∼Wsmall departure from equil at step→ Fast attachement kinetics

ν+ ∼1

W, ν− ∼

1

W, ν0 ∼

1

W

Page 10: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

How many concentration fields?

φ

φ−φ+

φ

θ+

θ−

θ

Page 11: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

1 conc field per terrace

Diffusion flux

Jd ∼ ∇θ

ψ∼ [ψ∇θ − θ∇ψ]

Model equations:

∂tθ+ + ∂th+ = D[φ+∇2θ+ − θ+∇

2φ+]

−1

τ(θ+ − θ∞+ )

−(B++θ+ − B+−θ−)

∂tθ− + ∂th− = D[φ−∇2θ− − θ−∇

2φ−]

−1

τ(θ− − θ∞− )

−(B−−θ− − B−+θ+)

τφ∂tφ = W 2∇

2φ− fφ

+ λ[g+φ(θ+ − θeqφ+) + g−φ(θ− − θeqφ−)]

Page 12: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

variational if B±± ∼ φ±and g±φφ± ∼ h±φ + θeqφ±φ

Sharp interface limit: λ ∼W→ Arbitrary kinetics

ν± ∼ 1 ν0 ∼ 1

Page 13: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

A numerical trick: the 3 fields model

n n+1n−1n−2 n+2

1 2 1 2 1θ

θ

x

x

Page 14: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

10−1 100 101 102

d+ / W−0.15

−0.10

−0.05

0.00

0.05

V/V nu

m−1

10−1 100 101 102

d+ / W

0.01

0.02

0.03

0.04

V

Figure 1: Step veloc, 1 global conc field

100 101 102 103 104

d+/W

10−4

10−3

10−2

V

100 101 102 103 104

d+/W

−0.05

−0.03

−0.01

0.01

0.03

0.05

Figure 2: Step velocity, 1 conc field per terrace

Page 15: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

−1 −0.5 0 0.5 1 α

0.006

0.007

0.008

0.009

0.01

0.011

0.012

V

Figure 3: Step velocity as a function of Schwoebeleffect (ν+ − ν−)/(ν+ + ν−)

10−2 10−1 100 101

ν 0

0.01

0.011

0.012

0.013

0.014

0.015

0.016

V

Figure 4: Velocity as a function of transparency ν0

Page 16: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

adatom z

x

(+)(-)step

adatom

Figure 5: Bales and Zangwill instability

0 0.2 0.4 0.6q

−0.005

−0.003

−0.001

0.001

ω

Figure 6: Dispersion relation for step meandering

Page 17: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

Nonlinear dynamics of an isolated step

Discontinuous Model→ multi-scale analysis (Bena et al 1994)→ Kuramoto-Sivashinsky Equation

∂th = −∂xxh− ∂xxxxh+ (∂xh)2

→ Spatio-temporal chaos

Page 18: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

−100 −50 030

80

130

Figure 7: KS dynamics

−10 0 10x

50

70

90

110

130

time

Figure 8: Phase field model

Page 19: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):
Page 20: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

−1 −0.5 0 0.5 1−10−4

0

10−4

Figure 9: Pairing rate during growth with invertedShwoebel effect

Page 21: Phase eld models for crystal stepsilm-perso.univ-lyon1.fr/~opl/docs/phaseOberw.pdf · Phase eld model for crystal steps: Lui and Metiu (1994): surface relaxation Plapp and Karma (1998):

Conclusion

• Model with 1 global concentration field→ fast step kinetics

• Model with 1 concentration per terrace→ arbitrary step kinetics

Perspectives

• Link to microscopic dynamics

• Numerical simulations in complex geometry

• Nucleation, fluctuations

• Elasticity