Olivier Pierre-Louis, ILM-Lyon
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
1 / 92
XXIme ”Aux Rencontres de Peyresq”
Dynamiques et Instabilités d’interface
Ecole thématique d’été, 29/05 to 2/06 2017
3 lectures:
Non-equilibrium & nonlinear dynamics of interfaces in growth
Thin films – dewetting – solid-state
Membranes - adhesion and confinement
Leitmotiv:
Modelling non-equilibrium interfaces
Nonlinear phenomena, instabilities, morphology
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
2 / 92
Lecture 1: Non-equilibrium interface dynamics in crystal growth
Olivier Pierre-Louis
ILM-Lyon, France.
27th May 2017
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
3 / 92
1
Introduction
Models and levels of description
Atomic steps and surface dynamics
2
Kinetic Monte Carlo
Master Equation
KMC algorithm
KMC Simulations
3
Phase field models
Diffuse interfaces models
Asympotics
Phase field simualtions
4
BCF Step model
Burton-Cabrera-Frank step model
Multi-scale analysis
5
Nonlinear interface dynamics
Preamble in 0D
Nonlinear dynamics in 1D
6
Conclusion
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
4 / 92
Introduction
Growth shapes
Snowflake
Yoshi Furukawa, Hokkaido, Japan
Growth Cu(1,1,17)
Olivier Pierre-Louis (ILM-Lyon, France.)
Maroutian, Ernst, Saclay, France
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
5 / 92
Introduction
Models and levels of description
Contents
1
Introduction
Models and levels of description
Atomic steps and surface dynamics
2
Kinetic Monte Carlo
Master Equation
KMC algorithm
KMC Simulations
3
Phase field models
Diffuse interfaces models
Asympotics
Phase field simualtions
4
BCF Step model
Burton-Cabrera-Frank step model
Multi-scale analysis
5
Nonlinear interface dynamics
Preamble in 0D
Nonlinear dynamics in 1D
6
Conclusion
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
6 / 92
Introduction
Models and levels of description
Modeling Crystal surfaces at different scales
Atomistic
Electrons
Ab Initio, quantum effects
Density Functional Theory
(103 at., 102 CPU, 10−12 s/day)
Atoms
Newton equations
Molecular Dynamics
(105 at., 10 CPU, 10−9 s/day)
Lattice
Effective moves: translation, rotation, ...
Kinetic Monte Carlo
(106 surface sites, 1 CPU, 102 s/day)
...Intermediate
Lattice Boltzmann
hydrodynamics
Phase field Crystal
continuum but atomic positions
Continuum
Diffusion, hydrodynamics, elasticity, etc.
Diffuse interface
Phase field
sharp interface
Continuum macroscopic
Stochastic Differential Equations
Langevin equations
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
7 / 92
Introduction
Models and levels of description
Modeling Crystal surfaces at different scales
Atomistic
Electrons
Asymptotic methods relating different scales
Ab Initio, quantum effects
Transition state theory
Density Functional Theory
(103 at., 102 CPU, 10−12 s/day)
-Atomistic (DFT or MD)
Atoms
→ energy landscape
Newton equations
→ rates for KMC on continuum
Molecular Dynamics
-Nucleation
(105 at., 10 CPU, 10−9 s/day)
Lattice
Sharp interface limit
Effective moves: translation, rotation, ...
Diffuse interface
Kinetic Monte Carlo
→ Sharp interface
(106 surface sites, 1 CPU, 102 s/day)
...Intermediate
Lattice Boltzmann
hydrodynamics
Phase field Crystal
continuum but atomic positions
Continuum
Diffusion, hydrodynamics, elasticity, etc.
Diffuse interface
Phase field
sharp interface
Continuum macroscopic
Stochastic Differential Equations
Langevin equations
Olivier Pierre-Louis (ILM-Lyon, France.)
Multiple scale expansions
Instability
→ effective nonlinear (amplitude) equations
→ Morphology
Renormalization group
Integrate over small scales
→ large scale behavior
(large distances, large times)
Homogenization
Heterogeneous medium
→ effective medium
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
8 / 92
Introduction
Models and levels of description
Direct derivation Vs. Asymptotic analysis
Atomistic
Electrons
Ab Initio, quantum effects
Density Functional Theory
(103 at., 102 CPU, 10−12 s/day)
Atoms
Newton equations
Molecular Dynamics
(105 at., 10 CPU, 10−9 s/day)
Lattice
Effective moves: translation, rotation, etc.
Kinetic Monte Carlo
(106 surface sites, 1 CPU, 102 s/day)
Direct derivation of models
(phenomenological models)
Symmetries
Conservation Laws
...Intermediate
Lattice Boltzmann
hydrodynamics
Phase field Crystal
continuum but atomic positions
Linear Irreversible Thermodynamics
(Onsager)
... etc.
Continuum
Diffusion, hydrodynamics, elasticity, etc.
Diffuse interface
Phase field
sharp interface
Continuum macroscopic
Stochastic Differential Equations
Langevin equations
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
9 / 92
Introduction
Atomic steps and surface dynamics
Contents
1
Introduction
Models and levels of description
Atomic steps and surface dynamics
2
Kinetic Monte Carlo
Master Equation
KMC algorithm
KMC Simulations
3
Phase field models
Diffuse interfaces models
Asympotics
Phase field simualtions
4
BCF Step model
Burton-Cabrera-Frank step model
Multi-scale analysis
5
Nonlinear interface dynamics
Preamble in 0D
Nonlinear dynamics in 1D
6
Conclusion
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
10 / 92
Introduction
Atomic steps and surface dynamics
The roughening transition
Broken bond energy J
Length N
Free energy
F = E − TS = NJ − kB T ln 2N = N(J − kB T ln 2)
F → 0 as T → TR
Roughening transition temperature
TR =
.
J
kB ln 2
T>Tr
T<Tr
γ(θ)
hi
For usual crystals Tr ∼ TM
Olivier Pierre-Louis (ILM-Lyon, France.)
NaCl, Métois et al (620-710o C)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
11 / 92
Introduction
Atomic steps and surface dynamics
Levels of description
Solid on Solid
Steps
Continuum
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
12 / 92
Introduction
Atomic steps and surface dynamics
Atomic steps
Si(100)
Insulin
M. Lagally, Univ. Visconsin
P. Vekilov, Houston
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
13 / 92
Introduction
Atomic steps and surface dynamics
Nano-scale Relaxation
Ag(111), Sintering
Decay Cu(1,1,1) 10mM HCl at (a) -580mV; and (b) 510 mV
M. Giesen, Jülich Germany
Broekman et al, (1999) J. Electroanal. Chem.
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
14 / 92
Introduction
Atomic steps and surface dynamics
Nano-scale crystal Surface Instabilities out of equilibrium
Growth and Ion Sputtering, Pt(111)
Growth Cu(1,1,17)
Solution Growth
(ADP) (5mm)
NH4 H2 PO4
Chernov
Maroutian, Ernst, Saclay, France
T. Michely, Aachen, Germany
2003
Solution Growth SiC
SiGe MBE growth
Electromigration Si(111)
Zu et al Cryst. Growth & Des. 2013
E.D. Williams, Maryland, USA
Olivier Pierre-Louis (ILM-Lyon, France.)
Floro et al 1999
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
15 / 92
Introduction
Atomic steps and surface dynamics
Nano-scale crystal Surface Instabilities out of equilibrium
Kinetic instabilities
Mass fluxes along the surface
Unstable
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Mass fluxes to the surface
Stable
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Unstable
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Stable
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x
adatom
z
adatom
(-) (+)
step
Energy relaxation driven instabilities:
surface energy, elastic energy, electrostatic energy, electronic energy, etc.
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
16 / 92
Kinetic Monte Carlo
Master Equation
Contents
1
Introduction
Models and levels of description
Atomic steps and surface dynamics
2
Kinetic Monte Carlo
Master Equation
KMC algorithm
KMC Simulations
3
Phase field models
Diffuse interfaces models
Asympotics
Phase field simualtions
4
BCF Step model
Burton-Cabrera-Frank step model
Multi-scale analysis
5
Nonlinear interface dynamics
Preamble in 0D
Nonlinear dynamics in 1D
6
Conclusion
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
17 / 92
Kinetic Monte Carlo
Master Equation
Levels of description
Solid on Solid
Steps
Continuum
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
18 / 92
Kinetic Monte Carlo
Master Equation
Master equation
Discrete set of configurations, index n = 1, .., Ntot
Physics: Transition rates R(n → m)
Markovian dynamics (no memory)
Master Equation
∂t P(n, t) =
Ntot
X
R(m → n)P(m, t) −
m=1
Ntot
X
R(n → m)P(n, t)
m=1
2
Ising lattice (0,1), L × L = L2 sites, Ntot = 2L configurations
L = 10 ⇒ Ntot ∼ 1030 ... too large for direct numerical solution!
Note: number of possible moves from n: Nposs (n) Ntot
⇒ R(m → n) sparse, i.e. ⇒ R(m → n) = 0 for most values of m, n.
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
19 / 92
Kinetic Monte Carlo
Master Equation
Equilibrium and Detailed Balance
Master Equation
∂t P(n, t) =
Ntot
X
R(m → n)P(m, t) −
m=1
Ntot
X
R(n → m)P(n, t)
m=1
Equilibrium, steady-state (∂t Peq (n, t) = 0), Hamiltonian H(n)
1
H(n)
Peq (n) =
exp −
Z
kB T
N
tot
X
H(n)
Z =
exp −
kB T
n=1
leading to
Peq (n)
Peq (m)
=
H(n) − H(m)
exp −
kB T
Stonger condition:Detailed Balance
R(n → m)Peq (n) = R(m → n)Peq (m)
or
R(n → m) = R(m → n) exp
Olivier Pierre-Louis (ILM-Lyon, France.)
H(n) − H(m)
kB T
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
20 / 92
Kinetic Monte Carlo
Master Equation
Example : Attachment-Detachment model
(a)
hi
i
(b)
r(hi −> hi+1)
R(hi → hi + 1)
=
F
R(hi → hi − 1)
=
r0 exp[−
r(hi −> hi−1)
i
(c)
State n = {hi ; i = 1.., L}
Rates
ni number nearest neighbors at site i before detachment
(Breaking all bonds to detach / Transition state Theory)
2J
Feq = r0 exp −
kB T
n i=1
ni =2
ni J
]
kB T
ni =3
i
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
21 / 92
Kinetic Monte Carlo
Master Equation
Example: Attachment-Detachment model
Detailed Balance when F = Feq
R(hi − 1 → hi )Peq (hi − 1)
=
Feq Peq (hi − 1)
=
n i=1
ni =3
ni =2
R(hi → hi − 1)Peq (hi )
ni J
]Peq (hi )
r0 exp[−
kB T
Bond energy J ⇒ Broken Bond energy J/2
Hamiltonian H, surface length Ls
H = Ls
i
J
2
Energy change
Ls −> Ls−2
n i=1
Ls (hi ) − Ls (hi − 1) = 2(2 − ni )
Ls −> Ls+2
Ls −> Ls
ni =2
H(hi ) − H(hi − 1) = (2 − ni )J
ni =3
Thus
i
Peq (hi − 1)
H(hi ) − H(hi − 1)
(2 − ni )J
= exp
= exp
Peq (hi )
kB T
kB T
and
Olivier Pierre-Louis (ILM-Lyon, France.)
2J
Feq = r0 exp −
kB T
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
22 / 92
Kinetic Monte Carlo
Master Equation
Example: Attachment-Detachment model
Link to the Ising Hamiltonian with field H, Si = ±1
HIsing = −
X
J X
Si Sj − H
Si
4
i
hi,ji
Define ni = (Si + 1)/2 ⇒ J is the bond energy
X
X
HIsing = −J
ni nj − (J − 2H)
ni + const
hi,ji
i
re-writing H
HIsing =
X
X
J X
J
[ni (1 − nj ) + nj (1 − ni )] − 2H
ni + const = Ls − ∆µ
ni + const
2
2
i
i
hi,ji
Chemical potential ∆µ = 2H P
Number of broken bonds Ls = hi,ji [ni (1 − nj ) + nj (1 − ni )]
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
23 / 92
Kinetic Monte Carlo
Master Equation
Example: Attachment-Detachment model
Chemical potential ∆µ = 2H
Detailed Balance imposed to the Ising system with field H
F
=
R(hi − 1 → hi ) = R(hi → hi − 1)
Ising
Peq
(hi )
Ising
Peq (hi − 1)
−ni J + HIsing (hi ) − HIsing (hi − 1)
kB T
ni J
[Ls (hi ) − Ls (hi − 1)]J
∆µ
exp −
+
+
kB T
2kB T
kB T
ni J
2(ni − 2)J
∆µ
+
+
exp −
kB T
2kB T
kB T
∆µ
Feq exp
kB T
=
=
=
=
exp
2J
Feq = r0 exp −
kB T
Attachement-Detachment model equivalent to Ising in Magnetic field
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
24 / 92
Kinetic Monte Carlo
Master Equation
Equilibrium Monte Carlo: Metropolis algorithm
Algorithm
1
Choose an event n → m at random
2
implement the event with probability
H(n) ≥ H(m)
⇒
P(n → m) = 1
H(n) ≤ H(m)
⇒
P(n → m) = exp[−
H(m) − H(n)
]
kB T
Obeys Detailed Balance
R(n → m)
H(n) − H(m)
= exp −
R(m → n)
kB T
Chain of events conv. to equil. ⇒ sample configuration space n ⇒ Thermodynamic averages
No time!
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
25 / 92
Kinetic Monte Carlo
KMC algorithm
Contents
1
Introduction
Models and levels of description
Atomic steps and surface dynamics
2
Kinetic Monte Carlo
Master Equation
KMC algorithm
KMC Simulations
3
Phase field models
Diffuse interfaces models
Asympotics
Phase field simualtions
4
BCF Step model
Burton-Cabrera-Frank step model
Multi-scale analysis
5
Nonlinear interface dynamics
Preamble in 0D
Nonlinear dynamics in 1D
6
Conclusion
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
26 / 92
Kinetic Monte Carlo
KMC algorithm
A simple Monte Carlo method with time: random attempts
Using physical rates R(n → m) from a given model
Algorithm
1
Choose an event n → m at random
2
Implement the event with probability
P(n → m) =
R(n → m)
Rmax
where Rmax = maxn0 (R(n → n0 )).
3
implement the time by ∆t ∼ 1/(Nposs (n)Rmax )
Nposs (n) number of possible moves from state n
Problem: when most R(n → m) Rmax , then most P(n → m) 1 ⇒ most attempts rejected!
Questions
a rejection-free algorithm?
time implementation?
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
27 / 92
Kinetic Monte Carlo
KMC algorithm
Kinetic Monte Carlo algorithm 1: Rejection-free algorithm
implement the FIRST event that occurs
Probability first chosen event n → m is ∼ R(n → m)
Rc(10)
Rc(10)
Rc(9)
.
Rc(8)
Choose event with probability
P(n → m) =
R(n → m)
Rtot (n)
Rrand
Rc(7)
Rc(6)
Rc(5)
n*=5
Rc(4)
Rate that one event occurs
Rtot (n) =
Ntot
X
R(n → m)
m=1
Rc(3)
Algorithm
Pm
Build cumulative rates Rc (m) =
Rc (Ntot ) = Rtot (n).
2
Choose random number Rrand , uniformly distributed with
0 < Rrand ≤ Rtot (n).
3
p=1
Rc(2)
Rc(1)
R(n → p), with
1
0
0
Choose event n∗ such that Rc (n∗ − 1) < Rrand ≤ Rc (n∗ ).
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
28 / 92
Kinetic Monte Carlo
KMC algorithm
Kinetic Monte Carlo algorithm 3: comments
Algorithm
Pm
R(n → p), with Rc (Ntot ) = Rtot (n).
1
Build cumulative rates Rc (m) =
2
Choose random number Rrand , uniformly distributed with 0 < Rrand ≤ Rtot (n).
3
Choose event n∗ such that Rc (n∗ − 1) < Rrand ≤ Rc (n∗ ).
p=1
Improved algorithm:
Number of possible events from one state Nposs (n) number of states Ntot (n)
Groups of events with same rates
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
29 / 92
Kinetic Monte Carlo
KMC algorithm
Kinetic Monte Carlo algorithm 2: implementing time
Define τ time with τ = 0 when arriving in state n
Proba Q(τ ) that no event occurs up to time τ , with Q(0) = 1
Q(τ + dτ ) = Q(τ )(1 − Rtot (n)dτ ) ⇒
dQ(τ )
= −Rtot (n)Q(τ ) ⇒ Q(τ ) = exp[−Rtot (n)τ ]
dτ
probability density ρ, i.e. ρ(τ )dτ first event occuring between τ and τ + dτ
ρ(τ )dτ = −dQ(τ ) = Rtot (n) exp[−Rtot (n)τ ]dτ
Easy to generate: uniform distribution ρ(u) = 1, and 0 < u ≤ 1
Variable change
u = exp[−Rtot (n)τ ]
same probability density
ρ(τ )dτ = −ρ(u)du
Algorithm
1 Choose random number u, uniformly distributed with 0 < u ≤ 1.
2 Time increment t → t + τ with
τ =
Olivier Pierre-Louis (ILM-Lyon, France.)
− log(u)
Rtot (n)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
30 / 92
Kinetic Monte Carlo
KMC Simulations
Contents
1
Introduction
Models and levels of description
Atomic steps and surface dynamics
2
Kinetic Monte Carlo
Master Equation
KMC algorithm
KMC Simulations
3
Phase field models
Diffuse interfaces models
Asympotics
Phase field simualtions
4
BCF Step model
Burton-Cabrera-Frank step model
Multi-scale analysis
5
Nonlinear interface dynamics
Preamble in 0D
Nonlinear dynamics in 1D
6
Conclusion
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
31 / 92
Kinetic Monte Carlo
KMC Simulations
Attachment-detachment model
Acl (t) / Acl (0)
10 4
10
(a)
∆µ /J
+0.05
0
−0.05
2
10 0
10−2
10−4
10−2
10 0
10 2
a0 ν0 t / R0
10 4
F = Feq e−∆µ/kB T
Saito, Pierre-Louis, PRL 2012
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
32 / 92
Kinetic Monte Carlo
KMC Simulations
Nano-scale crystal Surface Instabilities out of equilibrium
Movie...
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
33 / 92
Kinetic Monte Carlo
KMC Simulations
KMC results
Thiel and Evans 2007
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
34 / 92
Kinetic Monte Carlo
KMC Simulations
KMC vs SOI solid-state dewetting
h = 3, ES = 1, T = 0.5
E. Bussman, F. Leroy, F. Cheynis, P. Müller, O. Pierre-Louis NJP 2011
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
35 / 92
Kinetic Monte Carlo
KMC Simulations
KMC conclusion
KMC versatile method to look at crystal shape evolution
Improved methods to obtain physical rates (DFT, MD)
Build-in thermal fluctuations
Problem: difficult to parallelize
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
36 / 92
Kinetic Monte Carlo
KMC Simulations
Step meandering Instabilities
Non-equilibrium meandering of a step
Schwoebel effect + terrace diffusion
Y. Saito, M. Uwaha 1994
Bales and Zangwill 1990
x
adatom
z
adatom
(-) (+)
step
Solving step dynamics
→ morphology and coupling to a diffusion field
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
37 / 92
Phase field models
Diffuse interfaces models
Contents
1
Introduction
Models and levels of description
Atomic steps and surface dynamics
2
Kinetic Monte Carlo
Master Equation
KMC algorithm
KMC Simulations
3
Phase field models
Diffuse interfaces models
Asympotics
Phase field simualtions
4
BCF Step model
Burton-Cabrera-Frank step model
Multi-scale analysis
5
Nonlinear interface dynamics
Preamble in 0D
Nonlinear dynamics in 1D
6
Conclusion
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
38 / 92
Phase field models
Diffuse interfaces models
Phase field Model
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adatoms
.
Solid
θ
Phase
field
model
Concentration c
∂t c = ∇[M∇c] + F −
c
− ∂t h/Ω
τ
Ω atomic volume
Phase field φ
φ
c−
c+
Discont.
step
model
f
τp ∂t φ = W 2 ∇2 φ − f 0 (φ) + λ(c − ceq )g 0 (φ)
Functions
M(φ) Diffusion constant
h(φ) solid height, or concentration
f (φ) double-well potential
g (φ) coupling function
g 0 = 0 at min of f
φ
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
39 / 92
Phase field models
Diffuse interfaces models
Phase field Model
Equilibrium / Free energy?
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adatoms
.
Z
F [c, φ] =
d 2r
λ
W2
(∇φ)2 + f (φ) + (c − ceq )2
2
2
Solid
Phase
field
model
θ
Total mass: U = h + Ωc
If
Equilibrium at F = Feq = ceq /τ
Free energy if h = −g (up to mult const)
φ
c−
Discont.
step
model
c+
Then
λ∂t U
=
∇[M∇
∂t φ
=
−
f
φ
δF
1 δF
[U, φ] ] −
[U, φ]
δU
τ δU
1 δF
[U, φ]
τφ δφ
And
∂t F ≤ 0
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
40 / 92
Phase field models
Asympotics
Contents
1
Introduction
Models and levels of description
Atomic steps and surface dynamics
2
Kinetic Monte Carlo
Master Equation
KMC algorithm
KMC Simulations
3
Phase field models
Diffuse interfaces models
Asympotics
Phase field simualtions
4
BCF Step model
Burton-Cabrera-Frank step model
Multi-scale analysis
5
Nonlinear interface dynamics
Preamble in 0D
Nonlinear dynamics in 1D
6
Conclusion
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
41 / 92
Phase field models
Asympotics
Asymptotic expansion
Two expansions: ”inner domain” and ”outer domain”
interface
c in (η, s, t)
s
d
c out (x, y , t)
(out)
=
∞
X
n=0
∞
X
n cnin (η, s, t)
n cnout (x, y , t)
n=0
Matching conditions c in (η) = c out (η)
with η → ∞, → 0, η → 0
W
c0in
=
∂η c0in
=
c1in
=
∂η c1in
=
∂ηη c1in
=
lim
η→±∞
lim
η→±∞
(in)
lim
η→±∞
lim
η→±∞
∼ W /`c ; η = d/; ∇d = n
lim
η→±∞
Olivier Pierre-Louis (ILM-Lyon, France.)
=
lim c0out
d→0±
0
lim c1out + η lim n.∇c0out
d→0±
d→0±
lim n.∇c0out
d→0±
0
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
42 / 92
Phase field models
Asympotics
Expansion inner domain
Change to inner variables with
∇d = n;
∇ · n = ∆d = κ
Laplacian u
∆u = −2 (∂ηη u + κ∂η u + h.o.t.)
Co-moving frame at velocity V
∂t = −V ∂d
Model equations inner region
o(2 )
=
∂η [M∂η c in ] + (V + Mκ)∂η c in + V ∂η h/Ω
o(2 )
=
∂ηη φin − ∂φ f + λ(c in − ceq )∂φ g + (Va + κ)∂η φin
with
W
= 1,
Olivier Pierre-Louis (ILM-Lyon, France.)
1
W2
=
a
τφ
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
43 / 92
Phase field models
Asympotics
Expansion order by order
λ(c − ceq ) ∼ No coupling to leading order
0th order
∂η [M 0 ∂η c0in ] = 0
→ c0in = cst in step region
∂ηη φ0 − fφ0 = 0
→ frozen step
Diffusion equation in the outer region
∂t c out = D∇2 c out + F −
c out
τ
1st order
Mass conservation
(Ω−1 + c− − c+ )V = −Dn.∇c− + Dn.∇c+
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
44 / 92
Phase field models
Asympotics
Expansion order by order
higher order
Sharp interface asymptotics Caginalp 1989
Weak coupling (permeable steps): λ ∼ c+ = c− = c̃eq
c̃eq = ceq (1 + Γκ) + β̃
V
Ω
with
Olivier Pierre-Louis (ILM-Lyon, France.)
β
=
Γceq
=
Z
τφ
dη(∂η φ0 )2
λW
Z
W
dη(∂η φ0 )2
λ
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
45 / 92
Phase field models
Asympotics
Expansion order by order
higher order
Thin interface asymptotics Karma Rappel 1996, OPL 2003
Fast kinetics: (c − ceq ) ∼ Dn.∇c+ = ν̃+ (c+ − c̃eq )
−Dn.∇c− = ν̃− (c− − c̃eq )
with
c̃eq = ceq (1 + Γκ) + β̃
1
ν̃+
1
ν̃−
1
1
0
−
(h−
− h0 )
0
M
D
Z
1
1
0
W
dη
−
(h0 − h+
)
M0
D
Z
Z
0 − h0 )(h0 − h0 )
(h+
τφ
−
dη(∂η φ0 )2 − W
dη
Wλ
M0
Z
W
dη(∂η φ0 )2
λ
Z
=
=
β̃
=
ceq Γ
=
V
Ω
W
dη
ν̃± ∼ β̃ ∼ −1
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
46 / 92
Phase field models
Phase field simualtions
Contents
1
Introduction
Models and levels of description
Atomic steps and surface dynamics
2
Kinetic Monte Carlo
Master Equation
KMC algorithm
KMC Simulations
3
Phase field models
Diffuse interfaces models
Asympotics
Phase field simualtions
4
BCF Step model
Burton-Cabrera-Frank step model
Multi-scale analysis
5
Nonlinear interface dynamics
Preamble in 0D
Nonlinear dynamics in 1D
6
Conclusion
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
47 / 92
Phase field models
Phase field simualtions
Simulations of step dynamics with phase field
Phase field models → no interface tracking
Phase field model with Schwoebel effect
OPL 2003
130
110
z
time
x
adatom
90
70
adatom
50
(-) (+)
−10
step
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
0
x
10
27th May 2017
48 / 92
Phase field models
Phase field simualtions
Other phase field simulations
Dendritic Growth
M. Plapp, E.P., France
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
49 / 92
Phase field models
Phase field simualtions
Phase field with liquids
Liquid-Liquid phase separation
Granasy et al 2005
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
50 / 92
Phase field models
Phase field simualtions
Conclusion
Interface tracking vs Interface Capturing → very effective numerical method
Can add, elastic strain, hydrodynamics, temperature fields, etc...
Can include fluctuations
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
51 / 92
BCF Step model
Burton-Cabrera-Frank step model
Contents
1
Introduction
Models and levels of description
Atomic steps and surface dynamics
2
Kinetic Monte Carlo
Master Equation
KMC algorithm
KMC Simulations
3
Phase field models
Diffuse interfaces models
Asympotics
Phase field simualtions
4
BCF Step model
Burton-Cabrera-Frank step model
Multi-scale analysis
5
Nonlinear interface dynamics
Preamble in 0D
Nonlinear dynamics in 1D
6
Conclusion
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
52 / 92
BCF Step model
Burton-Cabrera-Frank step model
Burton-Cabrera-Frank step model
At the steps
Thermodynamic Fluxes
J± = ±Dn.∇c±
Fluxes prop. to forces + Onsager reciprocity
J+
=
ν+ (c+ − ceq ) + ν0 (c+ − c− )
J−
=
ν− (c− − ceq ) − ν0 (c+ − c− )
Gibbs-Thomson µ = Ωγ̃κ, with γ̃ = γ + γ 00
On terraces:
diffusion + incoming flux + evaporation
∂t c = D∇2 c + F − c/τ
τ
F
ν0
ν−
D
0 µ/kB T
ceq = ceq
e
ν+
x
z
Mass conservation
V ∆c = J+ + J−
ν+ > ν− Ehrlich-Schwoebel effect
ν0 6= 0 Step transparency or permeability
with ∆c = 1/Ω + c− − c+
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
53 / 92
BCF Step model
Burton-Cabrera-Frank step model
re-Writing Step Boundary Conditions
Linear combination of BC
J+
J−
Gibbs-Thomson µ = Ωγ̃κ, with γ̃ = γ +
=
ν̃+ (c+ − c̃eq )
=
ν̃− (c− − c̃eq )
γ 00
0 µ/kB T
c̃eq = ceq
e
+ β̃
V
Ω
with
ν̃+
=
ν̃−
=
β̃
=
ν+ ν− + ν0 (ν+ + ν−)
ν−
ν+ ν− + ν0 (ν+ + ν−)
ν−
ν0
ν+ ν− + ν0 (ν+ + ν−)
Identical to thin interface limit of phase field!
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
54 / 92
BCF Step model
Burton-Cabrera-Frank step model
Stationary states
No permeability (transparency) ν0 = 0 and quasistatic approx ∂t c = 0
1) Isolated straight step velocity with evaporation
0
V̄ = Ω(F − ceq
/τ )xs2
2xs + d+ + d−
xs2 + xs d+ + xs d− + d+ d−
diffusion length xs = (Dτ )1/2
Schwoebel lengths d± = D/ν±
F
τ
V
2) Vicinal surface: Step Flow velocity without evaporation
V̄ = ΩF `
F
V
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
55 / 92
BCF Step model
Multi-scale analysis
Contents
1
Introduction
Models and levels of description
Atomic steps and surface dynamics
2
Kinetic Monte Carlo
Master Equation
KMC algorithm
KMC Simulations
3
Phase field models
Diffuse interfaces models
Asympotics
Phase field simualtions
4
BCF Step model
Burton-Cabrera-Frank step model
Multi-scale analysis
5
Nonlinear interface dynamics
Preamble in 0D
Nonlinear dynamics in 1D
6
Conclusion
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
56 / 92
BCF Step model
Multi-scale analysis
Schwoebel effect → Meandering instability
isolated step; one-sided model
(ν0 = 0, ν− = 0, ν+ → ∞)
Model equations
x
adatom
z
adatom
(-) (+)
=
∂zz c + ∂xx c + F −
c(z = h(x, t))
=
0
ceq
exp[−Γκ]
1
∂t h(x, t)
ΩD
=
∂z c(z = h) − ∂x h∂x c(z = h)
with Γ = Ωγ̃/kB T , and
step
Bales and Zangwill 1990
(∼ Mullins and Sekerka)
Olivier Pierre-Louis (ILM-Lyon, France.)
c
τ
∂t c
κ=
−∂xx h
(1 + (∂x h)2 )3/2
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
57 / 92
BCF Step model
Multi-scale analysis
Meandering/ Linear Analysis: Isolated step
isolated step; one-sided model (ν0 = 0, ν− = 0, ν+ → ∞)
An instability occurs if: <e[iω(q)] > 0
Stationary state at constant velocity
V̄ = Ω(F − ceq /τ )(D/τ )1/2
<e[iω]
Linear Stability
Small perturbations, Fourier Mode
Re[i ω]
=
−L2 q 2 + L4 q 4
ζ(x, t) ∼ Ae iωt+iqx
q
Dispersion relation
Long wavelength qxs 1
iω =
xs2
3
0
Ω(F − Fc )q 2 − xs DΩceq
Γq 4
2
4
→ Morphological instability
for F > Fc = Feq (1 + 2Γ
)
x
s
Bales and Zangwill 1990
Olivier Pierre-Louis (ILM-Lyon, France.)
L2<0
L2>0
Weakly unstable ∼ (F − Fc )
q ∼ 1/2 ; iω ∼ 2
x ∼ −1/2 ; t ∼ −2
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
58 / 92
BCF Step model
Multi-scale analysis
Meandering/ Nonlinear behavior: Isolated step, Epavoration
Weakly unstable ∼ (F − Fc )
x ∼ −1/2 ; t ∼ −2 ; ζ ∼ x = X −1/2 ;
t = T −2
ζ(x, t)
=
H(X , T ) = [H0 (X , T ) + H1 (X , T ) + 2 H2 (X , T ) + ...]
c(x, t)
=
c0 (X , T ) + c1 (X , T ) + c2 (X , T ) + ...
Model equations
c
τ
2 ∂T c
=
∂zz c + ∂XX c + F −
c(z = H(X , T ))
=
0
ceq
exp[−Γκ]
=
∂z c(z = H) − ∂X H∂X c(z = H)
3
ΩD
∂T H
with
κ = 2
Olivier Pierre-Louis (ILM-Lyon, France.)
−∂XX H
(1 + 3 (∂X H)2 )3/2
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
59 / 92
BCF Step model
Multi-scale analysis
Meandering/ Nonlinear behavior: Isolated step, Epavoration
0th order in → Straight step solution
...
3rd order in ∂t ζ = −
Chaotic dynamics
Bena-Misbah-Valance 1994
3
V̄
xs2
0
(F − Fc )Ω∂xx ζ − xs DΩceq
Γ∂xxxx ζ + (∂x ζ)2
2
4
2
130
Kuramoto-Sivashinsky Equation
Out of equilibrium → Non-variational
Rescaled
80
2
∂t ζ = −∂xx ζ − ∂xxxx ζ + (∂x ζ)
Geometric origin of nonlin Vn = V̄
30
.
Vn
∂t h
=
=
∂t h
[1 + (∂x h)2 ]1/2
2 1/2
V̄ [1 + (∂x h) ]
Olivier Pierre-Louis (ILM-Lyon, France.)
−100
−50
0
dh/dt
Vn
h(x,t)
1
≈ V̄ [1 + (∂x h)2 ]
2
Lecture 1: Non-equilibrium interface dynamics in crystal growth
x
27th May 2017
60 / 92
BCF Step model
Multi-scale analysis
Meandering/ Nonlinear behavior: Isolated step, Epavoration
130
130
time
110
90
80
70
50
−10
30
−100
−50
0
Kuramoto-SivashinskyBena-Misbah-Valance 1994
Olivier Pierre-Louis (ILM-Lyon, France.)
0
x
Phase field
10
OPL 2003
Lecture 1: Non-equilibrium interface dynamics in crystal growth
KMC
Saito, Uwaha 1994
27th May 2017
61 / 92
BCF Step model
Multi-scale analysis
Conclusion
Steps dynamics → nanoscale morphology
Growth, or evaporation/dissolution ...
but also: electromigration, sputtering, dewetting dynamics, strain-induced instabilities, etc.
Non-equilibrium steady-states and morphological instabilities
Universality in pattern formation: Kuramoto Sivashinsky example
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
62 / 92
Nonlinear interface dynamics
Preamble in 0D
Contents
1
Introduction
Models and levels of description
Atomic steps and surface dynamics
2
Kinetic Monte Carlo
Master Equation
KMC algorithm
KMC Simulations
3
Phase field models
Diffuse interfaces models
Asympotics
Phase field simualtions
4
BCF Step model
Burton-Cabrera-Frank step model
Multi-scale analysis
5
Nonlinear interface dynamics
Preamble in 0D
Nonlinear dynamics in 1D
6
Conclusion
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
63 / 92
Nonlinear interface dynamics
Preamble in 0D
What happens when the surface is unstable? Nonlinear Dynamics?
Preamble: simpler case with 1 degree of freedom A(t)
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
64 / 92
Nonlinear interface dynamics
Preamble in 0D
One dimensional dynamics with one parameter:
∂t A = F (A, λ)
Linear stability around threshold = λ − λc
dA/dt = A
A
ε
Steady state loosing its stability
Time scale: t ∼ −1
critical slowing down
Unstable → large amplitude → nonlinear dynamics
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
65 / 92
Nonlinear interface dynamics
Preamble in 0D
Symmetry A → −A
dA/dt = A + c3 A3
Super-critical or Sub-critical bifurcation
A
ε
A
ε
Scales t ∼ −1 , A ∼ 1/2
next order A5 ∼ 5/2 negligible other terms ∼ 3/2
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
66 / 92
Nonlinear interface dynamics
Preamble in 0D
Two springs in a plane
x
x
A
x
A
x
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
67 / 92
Nonlinear interface dynamics
Preamble in 0D
Constraint on ∂t A = F (A, )
If F (A, ) → 0 when → 0
dA/dt = Q(A) + o(2 )
A
ε
with Q(A) = ∂ F (A, )|=0
Scales t ∼ −1 , A ∼ 1
Highly nonlinear dynamics
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
68 / 92
Nonlinear interface dynamics
Preamble in 0D
A tube on a plane
f(A)
θ
A
h local height
f (A) in plane meander of the tube
m particle mass
g gravity
ν friction
Particle velocity
V
=
−ν∂s [mgh]
∂t A
=
−νmg sin(θ)
f 0 (A)
1 + f 0 (A)2
= sin(θ)
Dynamics are highly nonlinear around θ = 0
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
69 / 92
Nonlinear interface dynamics
Nonlinear dynamics in 1D
Contents
1
Introduction
Models and levels of description
Atomic steps and surface dynamics
2
Kinetic Monte Carlo
Master Equation
KMC algorithm
KMC Simulations
3
Phase field models
Diffuse interfaces models
Asympotics
Phase field simualtions
4
BCF Step model
Burton-Cabrera-Frank step model
Multi-scale analysis
5
Nonlinear interface dynamics
Preamble in 0D
Nonlinear dynamics in 1D
6
Conclusion
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
70 / 92
Nonlinear interface dynamics
Nonlinear dynamics in 1D
Front dynamics
h(x,t)
x
front h(x, t)
Translational invariance in x, and in h
(→ dynamics depends on derivs of h only)
local dynamics
Instability at long wavelength
Examples: crystal growth, flame fronts, sand ripple formation, etc....
O. Pierre-Louis EPL 2005
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
71 / 92
Nonlinear interface dynamics
Nonlinear dynamics in 1D
Systematic analysis
Linear analysis, Fourier modes:
h(x, t) = hωq exp[iωt + iqx]
Dispersion relation D[iω, iq] = 0
Local dynamics
Expansion of the dispersion relation at large scales:
iω
=
L0 + iL1 q − L2 q 2 − iL3 q 3 + L4 q 4 + ...
∂t h
=
L0 h + L1 ∂x h + L2 ∂xx h + L3 ∂xxx h + L4 ∂xxxx h
Translational invariance L0 = 0
Gallilean transform x → x + L1 t (∂t → ∂t + L1 ∂x )
∂t h = L2 ∂xx h + L3 ∂xxx h + L4 ∂xxxx h + ...
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
72 / 92
Nonlinear interface dynamics
Nonlinear dynamics in 1D
An instability occurs if: <e[iω(q)] > 0
<e[iω]
=
−L2 q 2 + L4 q 4
Re[i ω]
q
L2<0
L2>0
Long wavelength instab. L4 < 0, −L2 = Instability scales t ∼ −2 , x ∼ −1/2
Fast propagative time scale t ∼ −3/2
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
73 / 92
Nonlinear interface dynamics
Nonlinear dynamics in 1D
Weakly nonlinear expansion
Generic formal expansion
∂t h = −∂xx h + L3 ∂xxx h + L4 ∂xxxx h + γ [∂x ]n [∂t ]` [h]m
Power counting
scaling of h ∼ α
Linear terms ∼ 2+α
Nonlinear term ∼ γ+n/2+2`+mα
α=
2 − γ − n/2 − 2`
m−1
Nonlinearity which saturates the amplitude the soonest wins!
→ selection of the nonlinearity having the biggest α
Condition for WNL
∂t h, ∂x h 1 → α > −1/2
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
74 / 92
Nonlinear interface dynamics
Nonlinear dynamics in 1D
Largest possible α:
m = 2, γ = 0, n = 2, ` = 0 → γ [∂x ]n [∂t ]` [h]m = (∂x h)2
Generic equation Benney:
∂t h
=
−∂xx h + b∂xxx h − ∂xxxx h + (∂x h)2
b = 0 in the presence of x → −x symmetry
→ Kuramoto-Sivashinsky equation
130
80
30
−100
−50
0
Step growth with Schwoebel effect
I Bena, A. Valance, C. Misbah, 1993
Electromigration on vicinal surfaces
M. Sato, M. Uwaha, 1995 O. Pierre-Louis, C. Misbah 1995
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
75 / 92
Nonlinear interface dynamics
Nonlinear dynamics in 1D
Kuramoto-Sivashinsky equation
Flame fronts
Anders, Dittmann, Weinberg 2012
Reaction-Diffusion
etc.
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
76 / 92
Nonlinear interface dynamics
Nonlinear dynamics in 1D
Conservation law
∂t h = −∂x j
j
h(x,t)
x
Vicinity to thermodynamic equilibrium
(or variational steady-state)
driving force δF
− J + o(2 )
∂t h = ∂x M∂x
δh
Highly nonlinear dynamics h ∼ −1/2
∂t h = ∂x [A + B∂xx C ]
where A, B, C functions of ∂x h ∼ 1
x → −x symmetry; non-variational
→ Examples in Molecular Beam Epitaxy
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
77 / 92
Nonlinear interface dynamics
Nonlinear dynamics in 1D
Step meandering in MBE
Experiments:
on Cu(100) Maroutian, Néel, Douillard, Ernst, CEA-Saclay, France.
Si(111) H. Hibino, NTT, Japan.
Theory: O. Pierre-Louis, C. Misbah,LSP Grenoble.
G. Danker, K. Kassner, Univ. Otto von Guerricke Magdeburg, Germany.
Multi-scale expansion from step model
→ Highly nonlinear dynamics
∂t h = ∂x [A + B∂xx C ]
=F
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
78 / 92
Nonlinear interface dynamics
Nonlinear dynamics in 1D
In phase meander
λ
Free energy F = dsγ
Chemical potential µ = γκ
κ step curvature
Relaxation via terrace diffusion
D`⊥ ceq
∂s µ
V n = ∂s
kB T
"
∂t ζ = −∂x
α∂x ζ
β
+
∂x
1 + (∂x ζ)2
1 + (∂x ζ)2
a)
λm
R
1
λ/λm
λc
m
0
∂xx ζ
!#
00
1
m
1
1400
(1 + (∂x ζ)2 )3/2
1200
1000
with α = F `2 /2 and β = Dceq `γ/kB T .
obtained from step model
m = max[∂x ζ/(1 + (∂x ζ)2 )1/2 ]
800
t
600
400
200
0
0
20
40
60
80
x
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
79 / 92
Nonlinear interface dynamics
Can steps be pinned by anisotropy??
Energetic: γ(θ)
or
Kinetic: Ehrlich-Schwoebel effect, Edge diffusion.
Same qualitative change:
1) θc = 0 → no effect
2) θc = π/4 → interrupted coarsening.
θc =0
θc =π/4
Nonlinear dynamics in 1D
λ
∼
λ < λm
∼
λ
∼
λ > λm
λc
m
0
∼
0
1
m
3000
Si(111) growth
2500
2000
1500
1000
500
0
0
25
50
75
100
125
x
Anisotropy → Interrupted coarsening
Hiroo Omi, Japan.
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
80 / 92
Nonlinear interface dynamics
Nonlinear dynamics in 1D
λ
λm
Elastic interactions in homo-epitaxy:
λc
f
σ
m
0
σ
f
Force dipoles at step:
→ interaction energy ∼ 1/`2
between straight steps.
∼
0
1
m
1400
1200
1000
800
600
400
200
0
0
20
40
60
80
x
Elastic interactions → endless coarsening
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
81 / 92
Nonlinear interface dynamics
Nonlinear dynamics in 1D
From the study of step meandering
3 scenarios:
frozen wavelength (no coarsening) growing amplitude
interrupted coarsening (nonlin wavelength selection)
endless coarsening
OPL et al PRL 1998
Danker et al PRL 2004
Paulin et al PRL 2001
Can coarsening dynamics can generically be guessed from the branch of steady states only?
P. Politi - C. Misbah PRL 2004, Pierre-Louis 2014 → phase stability
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
82 / 92
Nonlinear interface dynamics
Nonlinear dynamics in 1D
Meandering/ Nonlinear behavior: Vicinal surface, no evaporation
∼F
Highly nonlinear equation x ∼ ζ ∼ −1/2
1
∂x ζ
+
∂x
1 + (∂x ζ)2
1 + (∂x ζ)2
∂xx ζ
!#
(1 + (∂x ζ)2 )3/2
0
10
20
"
∂t ζ = −∂x
30
40
50
-10
0
10
20
30
40
50
60
Step Train (t = -1)
Danker, OPL, Misbah 2005
Olivier Pierre-Louis (ILM-Lyon, France.)
Maroutian et al 2000
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
83 / 92
Nonlinear interface dynamics
Nonlinear dynamics in 1D
Step bunching
Inverted ES effect, or electromigration
Electromigration Si(111)
E.D. Williams, Maryland, USA
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
84 / 92
Nonlinear interface dynamics
Nonlinear dynamics in 1D
Step bunching
with evaporation
distance to instability threshold Generic equation Benney:
∂t h
=
−∂xx h + b∂xxx h − ∂xxxx h + (∂x h)2
without evaporation: Highly nonlinear dynamics
→ singular facets appear
ρ = ∂x h
migration force ∼ V
ceq + aρ∂y ceq
= a∂y
Ω
1 + (d+ + d− )ρ
Separation of the bunches
Ordered/chaotic bunches ρ = ∂x h
9813
step density ρ
9812
9811
9810
0
50
100
10000
t
20000
y
Electromigration on vicinal surfaces
−20
80
180
0
M. Sato, M. Uwaha, 1995 O. Pierre-Louis, C. Misbah 1995
Step position
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
85 / 92
Nonlinear interface dynamics
Nonlinear dynamics in 1D
Non-equilibrium mass fluxes and morphological stability
Growth with Ehrlich-Schwoebel effect → mass flux
Mass fluxes along the surface
Unstable
Stable
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Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
86 / 92
Nonlinear interface dynamics
Nonlinear dynamics in 1D
Ehrlich-Schwoebel effect: Mound formation
Growth with Ehrlich-Schwoebel effect
→ mass flux
J=F
`
F
=
2
2|∇h|
Continuum limit:
∂t h = F − ∂x J
Separation of varliables: h = Ft + A(t)g (x)
2AȦ =
g 00
g (g 0 )2
leading to
A ∼ t 1/2 and g ∼ erf −1 (x)
Singular shape → mounds separated by sharp
trenches
2D nucleation controls the shape of mound tops
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
87 / 92
Nonlinear interface dynamics
Nonlinear dynamics in 1D
Facetting+Electromigration
∂t φ = ∂xxxx ∂xx φ + φ − φ3 − jE0 ∂xx φ .
φ = ∂x h
4th order equation for steady-states
3 regimes: jE0 > 0 reinforced instability; 0 > jE0 > −1/4 weak instability; −1/4 > jE0 stability
j’E > 0
(b)
-1/4 < j’E < 0
(c)
j’E < -1/4
time
h(x,t)
(a)
0
Olivier Pierre-Louis (ILM-Lyon, France.)
x/L
0.2 0
minima
maxima
x/L
0.2
Lecture 1: Non-equilibrium interface dynamics in crystal growth
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Nonlinear interface dynamics
Nonlinear dynamics in 1D
Facetting+Electromigration
jE0 > 0 reinforced instability → endless coarsening
(a) j’ > 0
(b)
E
10
4
j’E= 0.45
j’E= 0.35
j’E= 0.25
j’E= 0.15
1
A/j’E
10
A
0
10
2
2
1/2
λ/j’E
λ- λm
0
0
20
10
1
40
λ
10
0
10
1
10
t
2
10
3
0 > jE0 > −1/4 weak instability → frozen wavelength
0,6
(a)
-1/4 < j’E < 0
λ+, λ-
0,4
λ
A
0,2
(b)
∗
λ
λm
λL2
20
10
√8 π
0 λ- λ
m 10
λ
λ+
-0,2
(c)
0
j’E
-0,1
(d)
0,4
φ0
L1
-0,02
-0,4
10
λ
20
0
x/λ
1
F. Barakat, K. Martens, O. Pierre-Louis, Phys Rev Lett 2012
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
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Nonlinear interface dynamics
Nonlinear dynamics in 1D
Conclusion
weakly vs highly nonlinear dynamics
Fronts close to thermodyn equil, or variational steady-state (Lyapunov functional)
Higher order problems?
Other examples of Highly nonlinear dynamics
Step bunching
J. Chang, O. Pierre-Louis, C. Misbah, PRL 2006
Oscillatory driving of crystal surfaces
O Pierre-Louis, M.I. Haftel, PRL 2001
Similar results without translational invariance
O. Pierre-Louis EPL 2005
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
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Conclusion
Conclusion
Modelling Crystal Surface Dynamics
Tools
KMC
Phase field
Sharp interface models (such as BCF Step models)
Phenomena
Growth-dissolution rate
Kinetic Roughening
Morphology: relaxation or instabilities → patterns
Global remarks
Surface Diffusion vs Bulk diffusion
Faithful microscopic vs Effective dynamics
Multi-scale modeling
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
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Conclusion
Conclusion
References
Books
A.L. Barabási, H.E. Stanley, Fractal Concepts in Surface growth, Cambridge, (1995)
Y. Saito, Statistical Physics of Crystal Growth, Worl Scientific (1996)
A. Pimpinelli and J. Villain, Physics of crystal growth, Cambridge, (1998)
J. Krug, and T. Michely, Islands, Mounds, and Atoms, Springer (2004)
C. Misbah, Complex Dynamics and Morphogenesis: An Introduction to Nonlinear Science
(2017)
Reviews
Burton Carbrera Frank, The Growth of Crystals and the Equilibrium Structure of their
Surfaces, Phil. Trans. A, (1951)
M. Kotrla, Computer Physics Communications 97 82-100 (1996)
C. Misbah, O. Pierre-Louis, Y. Saito, Rev Mod Phys 82 981 (2010).
Olivier Pierre-Louis (ILM-Lyon, France.)
Lecture 1: Non-equilibrium interface dynamics in crystal growth
27th May 2017
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