Non-equilibrium interface dynamics in crystal growth

Transcription

1 Non-equilibrium interface dynamics in crystal growth Olivier Pierre-Louis ILM-Lyon, France. 24th October 2017 Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

2 1 introduction Models and scales to model surfaces and interfaces 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 Kinetic Roughening Fractal interfaces Continuum models 7 Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

3 introduction Models and scales to model surfaces and interfaces Contents 1 introduction Models and scales to model surfaces and interfaces 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 Kinetic Roughening Fractal interfaces Continuum models 7 Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

4 introduction Models and scales to model surfaces and interfaces Modeling Crystal surfaces at different scales Atomistic Electrons Ab Initio, quantum effects Density Functional Theory (10 3 at., 10 2 CPU, s/day) Atoms Newton equations Molecular Dynamics (10 5 at., 10 CPU, 10 9 s/day) Lattice Effective moves: translation, rotation, etc. Kinetic Monte Carlo (10 6 surface sites, 1 CPU, 10 2 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.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

5 introduction Models and scales to model surfaces and interfaces Direct derivation Vs. Asymptotic analysis Direct derivation of models (phenomenological models) Symmetries Conservation Laws Linear Irreversible Thermodynamics (Onsager)... etc. Asymptotic methods relating different scales Transition state theory Atomistic (DFT or MD) energy landscape rates for KMC on continuum Sharp interface limit Diffuse interface Sharp interface 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 Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

6 introduction Atomic steps and surface dynamics Contents 1 introduction Models and scales to model surfaces and interfaces 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 Kinetic Roughening Fractal interfaces Continuum models 7 Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

7 introduction Atomic steps and surface dynamics The roughening transition Broken bond energy J Length N Free energy F = E TS = NJ k B T ln 2 N = N(J k B T ln 2) F 0 as T T R Roughening transition temperature. T<Tr T>Tr T R = J k B ln 2 γ(θ) hi For usual crystals T r T M NaCl, Métois et al ( o C) Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

8 introduction Atomic steps and surface dynamics Levels of description Solid on Solid Steps Continuum Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

9 introduction Atomic steps and surface dynamics Atomic steps Si(100) Insulin M. Lagally, Univ. Visconsin P. Vekilov, Houston Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

10 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.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

11 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 NH 4 H 2 PO 4 (ADP) (5mm) Chernov 2003 T. Michely, Aachen, Germany Electromigration Si(111) Maroutian, Ernst, Saclay, France SiGe MBE growth Solution Growth SiC E.D. Williams, Maryland, USA Floro et al 1999 Zu et al Cryst. Growth & Des Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

12 introduction Atomic steps and surface dynamics Nano-scale crystal Surface Instabilities out of equilibrium Kinetic instabilities Mass fluxes along the surface Unstable Unstable Stable Stable Mass fluxes to the surface adatom (-) (+) step Energy relaxation driven instabilities: surface energy, elastic energy, electrostatic energy, electronic energy, etc. x z adatom Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

13 introduction Atomic steps and surface dynamics Nano-scale crystal Surface Instabilities out of equilibrium Movie... Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

14 Kinetic Monte Carlo Master Equation Contents 1 introduction Models and scales to model surfaces and interfaces 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 Kinetic Roughening Fractal interfaces Continuum models 7 Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

15 Kinetic Monte Carlo Master Equation Levels of description Solid on Solid Steps Continuum Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

16 Kinetic Monte Carlo Master Equation Master equation Discrete set of configurations, index n = 1,.., N tot Physics: Transition rates R(n m) Markovian dynamics (no memory) Master Equation N tot tp(n, t) = R(m n)p(m, t) R(n m)p(n, t) m=1 N tot m=1 Ising lattice (0,1), L L = L 2 sites, N tot = 2 L2 configurations L = 10 N tot too large for direct numerical solution! Note: number of possible moves from n: N poss(n) N tot R(m n) sparse, i.e. R(m n) = 0 for most values of m, n. Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

17 Kinetic Monte Carlo Master Equation Equilibrium and Detailed Balance Master Equation N tot tp(n, t) = R(m n)p(m, t) R(n m)p(n, t) m=1 N tot m=1 Equilibrium, steady-state ( tp eq(n, t) = 0), Hamiltonian H(n) P eq(n) = 1 [ Z exp H(n) ] k B T N tot [ Z = exp H(n) ] k n=1 B T leading to P eq(n) P eq(m) Stonger condition:detailed Balance or [ ] H(n) H(m) = exp k B T R(n m)p eq(n) = R(m n)p eq(m) [ ] H(n) H(m) R(n m) = R(m n) exp k B T Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

18 Kinetic Monte Carlo Master Equation Example : Attachment-Detachment model (a) h i i (b) r(hi > hi+1) r(hi > hi 1) State n = {h i ; i = 1.., L} Rates R(h i h i + 1) = F R(h i h i 1) = r 0 exp[ n i J k B T ] (c) n i=1 n i=2 n i=3 i n i number nearest neighbors at site i before detachment (Breaking all bonds to detach / Transition state Theory) i Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

19 Kinetic Monte Carlo Master Equation Example: Attachment-Detachment model Detailed Balance when F = F eq n i=1 R(h i 1 h i )P eq(h i 1) = R(h i h i 1)P eq(h i ) F eqp eq(h i 1) = r 0 exp[ n i J k B T ]Peq(h i ) n i=2 n i=3 Bond energy J Broken Bond energy J/2 Hamiltonian H, surface length L s i Energy change H = L s J 2 Ls > Ls 2 n i=1 Ls > Ls n i=2 Ls > Ls+2 n i=3 L s(h i ) L s(h i 1) = 2(2 n i ) H(h i ) H(h i 1) = (2 n i )J i Thus P eq(h i 1) P eq(h i ) [ H(hi ) H(h i 1) = exp k B T ] = exp [ ] (2 ni )J k B T and [ F eq = r 0 exp 2J k B T Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101 ]

20 Kinetic Monte Carlo Master Equation Example: Attachment-Detachment model Link to the Ising Hamiltonian with field H, S i = ±1 H Ising = J S i S j H S i 4 i,j i Define n i = (S i + 1)/2 J is the bond energy H Ising = J i,j n i n j (J 2H) i n i + const re-writing H H Ising = J i (1 n j ) + n j (1 n i )] 2H 2 i,j [n i n i + const = J 2 Ls µ i n i + const Chemical potential µ = 2H Number of broken bonds L s = i,j [n i (1 n j ) + n j (1 n i )] Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

21 Kinetic Monte Carlo Master Equation Example: Attachment-Detachment model Chemical potential µ = 2H Detailed Balance imposed to the Ising system with field H Peq Ising (h i ) F = R(h i 1 h i ) = R(h i h i 1) Peq Ising (h i 1) [ ] ni J + H Ising (h i ) H Ising (h i 1) = exp k B T [ = exp n i J k B T + [Ls(h i ) L s(h i 1)]J + µ ] 2k B T k B T [ = exp n i J k B T + (n i 2)J 2k B T + µ ] k B T [ ] µ = F eq exp k B T [ F eq = r 0 exp 2J ] k B T Attachement-Detachment model equivalent to Ising in Magnetic field Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

22 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) ] k B T Obeys Detailed Balance [ ] R(n m) R(m n) = exp H(n) H(m) k B T Chain of events conv. to equil. sample configuration space n Thermodynamic averages No time! Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

23 Kinetic Monte Carlo KMC algorithm Contents 1 introduction Models and scales to model surfaces and interfaces 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 Kinetic Roughening Fractal interfaces Continuum models 7 Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

24 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 where R max = max n (R(n n )). P(n m) = 3 implement the time by t 1/(N poss(n)r max ) N poss(n) number of possible moves from state n R(n m) R max Problem: when most R(n m) R max, then most P(n m) 1 most attempts rejected! Questions a rejection-free algorithm? time implementation? Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

25 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). Choose event with probability Rc(8) Rate that one event occurs Algorithm P(n m) = N tot R(n m) R tot(n) R tot(n) = R(n m) m=1 1 Build cumulative rates R c(m) = m p=1 R(n p), with R c(n tot) = R tot(n). 2 Choose random number R rand, uniformly distributed with 0 < R rand R tot(n). 3 Choose event n such that R c(n 1) < R rand R c(n ). Rrand 0 Rc(7) Rc(6) Rc(5) n*=5 Rc(4) Rc(3) Rc(2) Rc(1) 0 Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

26 Kinetic Monte Carlo KMC algorithm Kinetic Monte Carlo algorithm 3: comments Algorithm 1 Build cumulative rates R c(m) = m p=1 R(n p), with Rc(Ntot) = Rtot(n). 2 Choose random number R rand, uniformly distributed with 0 < R rand R tot(n). 3 Choose event n such that R c(n 1) < R rand R c(n ). Improved algorithm: Number of possible events from one state N poss(n) number of states N tot(n) Groups of events with same rates Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

27 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 R tot(n)dτ) dq(τ) dτ = R tot(n)q(τ) Q(τ) = exp[ R tot(n)τ] probability density ρ, i.e. ρ(τ)dτ first event occuring between τ and τ + dτ ρ(τ)dτ = dq(τ) = R tot(n) exp[ R tot(n)τ]dτ Easy to generate: uniform distribution ρ(u) = 1, and 0 < u 1 Variable change same probability density u = exp[ R tot(n)τ] 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 = log(u) R tot(n) Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

28 Kinetic Monte Carlo KMC Simulations Contents 1 introduction Models and scales to model surfaces and interfaces 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 Kinetic Roughening Fractal interfaces Continuum models 7 Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

29 Kinetic Monte Carlo KMC Simulations Attachment-detachment model Acl (t) / Acl (0) (a) µ /J a 0 ν 0 t / R 0 F = F eqe µ/k B T Saito, Pierre-Louis, PRL 2012 Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

30 Kinetic Monte Carlo KMC Simulations KMC results Thiel and Evans 2007 Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

31 Kinetic Monte Carlo KMC Simulations KMC vs SOI solid-state dewetting h = 3, E S = 1, T = 0.5 E. Bussman, F. Leroy, F. Cheynis, P. Müller, O. Pierre-Louis NJP 2011 Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

32 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.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

33 Kinetic Monte Carlo KMC Simulations Step meandering Instabilities Non-equilibrium meandering of a step Schwoebel effect + terrace diffusion Bales and Zangwill 1990 Y. Saito, M. Uwaha 1994 x adatom z adatom (-) (+) step Solving step dynamics morphology and coupling to a diffusion field Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

34 Phase field models Diffuse interfaces models Contents 1 introduction Models and scales to model surfaces and interfaces 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 Kinetic Roughening Fractal interfaces Continuum models 7 Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

35 Phase field models. Diffuse interfaces models Phase field Model c 11 adatoms Solid f θ φ c+ φ Phase field model Discont. step model Concentration c tc = [M c] + F c τ th/ω Phase field φ τ p tφ = W 2 2 φ f (φ) + λ(c c eq)g (φ) g φ = 0 at min of f Equilibrium at F = F eq = c eq/τ Free energy if f = g/[g] + F = [ W d 2 2 r 2 ( φ)2 + f (φ) + λ ] (c ceq)2 2 Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

36 Phase field models Asympotics Contents 1 introduction Models and scales to model surfaces and interfaces 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 Kinetic Roughening Fractal interfaces Continuum models 7 Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

37 Phase field models Asympotics Asymptotic expansion Two expansions: inner domain and outer domain interface s d (out) c in (η, s, t) = c out (x, y, t) = ɛ n cn in (η, s, t) n=0 ɛ n cn out (x, y, t) n=0 W (in) ɛ = W /l c; η = d/ɛ; d = n Matching conditions c in (η) = c out (ɛη) with η, ɛ 0, ɛη 0 lim η ± cin 0 = lim d 0± cout 0 lim η ± ηcin 0 = 0 lim η ± cin 1 = lim d 0± cout 1 + η lim d 0± n. cout 0 lim η ± ηcin 1 = lim d 0± n. cout 0 lim η ± ηηcin 1 = 0 Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

38 Phase field models Asympotics Expansion inner domain Change to inner variables with d = n; n = d = κ Laplacian u Co-moving frame at velocity V Model equations inner region with u = ɛ 2 ( ηηu + ɛκ ηu + h.o.t.) t = V d o(ɛ 2 ) = η[m ηc in ] + ɛ(v + Mκ) ηc in + ɛv ηh/ω o(ɛ 2 ) = ηηφ in φ f + λ(c in c eq) φ g + ɛ(va + κ) ηφ in 1 a = W 2 τ φ Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

39 Phase field models Asympotics Expansion order by order λ(c c eq) ɛ No coupling to leading order 0th order η[m 0 ηc in 0 ] = 0 c in 0 = cst in step region frozen step Diffusion equation in the outer region ηηφ 0 f 0 φ = 0 1st order Mass conservation tc out = D 2 c out + F cout τ (Ω 1 + c c +)V = Dn. c + Dn. c + Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

40 Phase field models Asympotics Expansion order by order higher order Sharp interface asymptotics Caginalp 1989 Weak coupling (permeable steps): λ ɛ c + = c = c eq c eq = c eq(1 + Γκ) + β V Ω with τ φ β = dη( ηφ 0 ) 2 λw Γc eq = W dη( ηφ 0 ) 2 λ Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

41 Phase field models Asympotics Expansion order by order higher order Thin interface asymptotics Karma Rappel 1996, OPL 2003 Fast kinetics: (c c eq) ɛ Dn. c + = ν +(c + c eq) Dn. c = ν (c c eq) with c eq = c eq(1 + Γκ) + β V Ω 1 = W ν + 1 = W ν ( 1 dη dη M 0 1 D ( 1 M 0 1 D τ φ β = dη( ηφ 0 ) 2 W W λ c eqγ = W dη( ηφ 0 ) 2 λ ) (h 0 h 0 ) ) (h 0 h 0 + ) dη (h0 + h0 )(h 0 h 0 ) M 0 ν ± β ɛ 1 Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

42 Phase field models Phase field simualtions Contents 1 introduction Models and scales to model surfaces and interfaces 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 Kinetic Roughening Fractal interfaces Continuum models 7 Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

43 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 x adatom z time 90 adatom 70 (-) (+) step x Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

44 Phase field models Phase field simualtions Other phase field simulations Dendritic Growth M. Plapp, E.P., France Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

45 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.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

46 BCF Step model Burton-Cabrera-Frank step model Contents 1 introduction Models and scales to model surfaces and interfaces 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 Kinetic Roughening Fractal interfaces Continuum models 7 Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

47 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 + c eq) + ν 0 (c + c ) J = ν (c c eq) ν 0 (c + c ) Gibbs-Thomson µ = Ω γκ, with γ = γ + γ Mass conservation c eq = c 0 eqe µ/k B T V c = J + + J with c = 1/Ω + c c + On terraces: diffusion + incoming flux + evaporation tc = D 2 c + F c/τ D F ν ν 0 ν+ x z ν + > ν Ehrlich-Schwoebel effect ν 0 0 Step transparency or permeability τ Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

48 BCF Step model Burton-Cabrera-Frank step model re-writing Step Boundary Conditions Linear combination of BC Gibbs-Thomson µ = Ω γκ, with γ = γ + γ J + = ν +(c + c eq) J = ν (c c eq) with Identical to thin interface limit of phase field! c eq = c 0 eqe µ/k B T + β V Ω ν + = ν+ν + ν 0 (ν + + ν ) ν ν = ν+ν + ν 0 (ν + + ν ) ν ν 0 β = ν +ν + ν 0 (ν + + ν ) Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

49 BCF Step model Burton-Cabrera-Frank step model Stationary states No permeability (transparency) ν 0 = 0 and quasistatic approx tc = 0 1) Isolated straight step velocity with evaporation diffusion length x s = (Dτ) 1/2 Schwoebel lengths d ± = D/ν ± V = Ω(F c 0 eq /τ)x2 s F τ 2x s + d + + d x 2 s + xsd+ + xsd + d +d V 2) Vicinal surface: Step Flow velocity without evaporation V = ΩF l V F Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

50 BCF Step model Multi-scale analysis Contents 1 introduction Models and scales to model surfaces and interfaces 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 Kinetic Roughening Fractal interfaces Continuum models 7 Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

51 BCF Step model Multi-scale analysis Schwoebel effect Meandering instability adatom x z isolated step; one-sided model (ν 0 = 0, ν = 0, ν + ) Model equations tc = zz c + xx c + F c τ (-) (+) step Bales and Zangwill 1990 ( Mullins and Sekerka) adatom c(z = h(x, t)) = c 0 eq exp[ Γκ] 1 th(x, t) ΩD = z c(z = h) ɛ x h x c(z = h) with Γ = Ω γ/k B T, and xx h κ = (1 + ( x h) 2 ) 3/2 Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

52 BCF Step model Multi-scale analysis Meandering/ Linear Analysis: Isolated step isolated step; one-sided model (ν 0 = 0, ν = 0, ν + ) Stationary state at constant velocity V = Ω(F c eq/τ)(d/τ) 1/2 Linear Stability Small perturbations, Fourier Mode An instability occurs if: Re[iω(q)] > 0 Re[iω] = L 2 q 2 + L 4 q 4 Re[i ω] ζ(x, t) Ae iωt+iqx Dispersion relation Long wavelength qx s 1 q iω = x2 s 2 Ω(F Fc)q2 3 4 xsdωc0 eqγq 4 Morphological instability for F > F c = F eq(1 + 2Γ x s ) Bales and Zangwill 1990 L2>0 Weakly unstable ɛ (F F c) q ɛ 1/2 ; iω ɛ 2 x ɛ 1/2 ; t ɛ 2 L2<0 Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

53 BCF Step model Multi-scale analysis Meandering/ Nonlinear behavior: Isolated step, Epavoration Weakly unstable ɛ (F F c) x ɛ 1/2 ; t ɛ 2 ; ζ ɛ x = X ɛ 1/2 ; t = T ɛ 2 ζ(x, t) = ɛh(x, T ) = ɛ[h 0 (X, T ) + ɛh 1 (X, T ) + ɛ 2 H 2 (X, T ) +...] c(x, t) = c 0 (X, T ) + ɛc 1 (X, T ) + ɛc 2 (X, T ) +... Model equations with ɛ 2 T c = zz c + ɛ XX c + F c τ c(z = ɛh(x, T )) = c 0 eq exp[ Γκ] ɛ 3 ΩD T H = z c(z = ɛh) ɛ X H X c(z = ɛh) κ = ɛ 2 XX H (1 + ɛ 3 ( X H) 2 ) 3/2 Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

54 BCF Step model Multi-scale analysis Meandering/ Nonlinear behavior: Isolated step, Epavoration 0th order in ɛ Straight step solution... 3rd order in ɛ Chaotic dynamics Bena-Misbah-Valance 1994 tζ = x2 s 2 (F Fc)Ω xx ζ 3 V 4 xsdωc0 eqγ xxxx ζ + ( x ζ)2 2 Kuramoto-Sivashinsky Equation Out of equilibrium Non-variational Rescaled tζ = xx ζ xxxx ζ + ( x ζ) 2 Geometric origin of nonlin V n = V V n = th [1 + ( x h) 2 ] 1/2 th = V [1 + ( x h) 2 ] 1/2 V [ ( x h)2 ] dh/dt Vn h(x,t) x Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

55 BCF Step model Multi-scale analysis Meandering/ Nonlinear behavior: Isolated step, Epavoration time x Kuramoto-SivashinskyBena-Misbah-Valance 1994 Phase field OPL 2003 KMC Saito, Uwaha 1994 Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

56 BCF Step model Multi-scale analysis Meandering/ Nonlinear behavior: Vicinal surface, no evaporation ɛ F Highly nonlinear equation x ζ ɛ 1/2 [ ( )] ɛ x ζ tζ = x 1 + ( x ζ) + 1 xx ζ ( x ζ) 2 x (1 + ( x ζ) 2 ) 3/2 Step Train (t = -1) Danker, OPL, Misbah 2005 Maroutian et al 2000 Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

57 BCF Step model Multi-scale analysis Step bunching Inverted ES effect, or electromigration Electromigration Si(111) E.D. Williams, Maryland, USA Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

58 BCF Step model Multi-scale analysis t Step bunching with evaporation distance to instability threshold ɛ Generic equation Benney: th = ɛ xx h + b xxx h xxxx h + ( x h) 2 Ordered/chaotic bunches ρ = x h without evaporation: Highly nonlinear dynamics singular facets appear ρ = x h migration force ɛ V Ω = a y [ ɛceq + aρ y c eq 1 + (d + + d )ρ Separation of the bunches 9813 ] 9812 step density ρ y Electromigration on vicinal surfaces M. Sato, M. Uwaha, 1995 O. Pierre-Louis, C. Misbah Step position Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

59 BCF Step model Multi-scale analysis Non-equilibrium mass fluxes and morphological stability Growth with Ehrlich-Schwoebel effect mass flux Mass fluxes along the surface Unstable Stable Unstable Stable Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

60 BCF Step model Multi-scale analysis Ehrlich-Schwoebel effect: Mound formation Growth with Ehrlich-Schwoebel effect mass flux J = F l 2 = F 2 h Continuum limit: th = F x J Separation of varliables: h = Ft + A(t)g(x) leading to 2AȦ = g g(g ) 2 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.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

61 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.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

62 Nonlinear interface dynamics Preamble in 0D Contents 1 introduction Models and scales to model surfaces and interfaces 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 Kinetic Roughening Fractal interfaces Continuum models 7 Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

63 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.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

64 Nonlinear interface dynamics Preamble in 0D One dimensional dynamics with one parameter: ta = F (A, λ) Linear stability around threshold ɛ = λ λ c da/dt = ɛa A ε Steady state loosing its stability Time scale: t ɛ 1 critical slowing down Unstable large amplitude nonlinear dynamics Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

65 Nonlinear interface dynamics Preamble in 0D Symmetry A A da/dt = ɛa + c 3 A 3 Super-critical or Sub-critical bifurcation A ε A ε Scales t ɛ 1, A ɛ 1/2 next order A 5 ɛ 5/2 negligible other terms ɛ 3/2 Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

66 Nonlinear interface dynamics Preamble in 0D Two springs in a plane x x A A x x Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

67 Nonlinear interface dynamics Preamble in 0D Constraint on ɛ ta = F (A, ɛ) If F (A, ɛ) 0 when ɛ 0 da/dt = ɛq(a) + o(ɛ 2 ) A ε with Q(A) = ɛf (A, ɛ) ɛ=0 Scales t ɛ 1, A 1 Highly nonlinear dynamics Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

68 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] ɛ = sin(θ) Dynamics are highly nonlinear around θ = 0 f (A) ta = νmg sin(θ) 1 + f (A) 2 Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

69 Nonlinear interface dynamics Nonlinear dynamics in 1D Contents 1 introduction Models and scales to model surfaces and interfaces 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 Kinetic Roughening Fractal interfaces Continuum models 7 Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

70 Nonlinear interface dynamics Nonlinear dynamics in 1D Front dynamics h(x,t) 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 x Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

71 Nonlinear interface dynamics Nonlinear dynamics in 1D Systematic analysis Linear analysis, Fourier modes: h(x, t) = h ωq exp[iωt + iqx] Dispersion relation D[iω, iq] = 0 Local dynamics Expansion of the dispersion relation at large scales: iω = L 0 + il 1 q L 2 q 2 il 3 q 3 + L 4 q th = L 0 h + L 1 x h + L 2 xx h + L 3 xxx h + L 4 xxxx h Translational invariance L 0 = 0 Gallilean transform x x + L 1 t ( t t + L 1 x ) th = L 2 xx h + L 3 xxx h + L 4 xxxx h +... Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

72 Nonlinear interface dynamics Nonlinear dynamics in 1D An instability occurs if: Re[iω(q)] > 0 Re[iω] = L 2 q 2 + L 4 q 4 Re[i ω] q Long wavelength instab. L 4 < 0, L 2 = ɛ Instability scales t ɛ 2, x ɛ 1/2 Fast propagative time scale t ɛ 3/2 L2>0 L2<0 Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

73 Nonlinear interface dynamics Nonlinear dynamics in 1D Weakly nonlinear expansion Generic formal expansion Power counting scaling of h ɛ α Linear terms ɛ 2+α Nonlinear term ɛ γ+n/2+l+mα th = ɛ xx h + L 3 xxx h + L 4 xxxx h + ɛ γ [ x ] n [ t] l [h] m α = 2 γ n/2 l m 1 Nonlinearity which saturates the amplitude the soonest wins! selection of the nonlinearity having the biggest α Condition for WNL th, x h 1 α > 1/2 Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

74 Nonlinear interface dynamics Nonlinear dynamics in 1D Generic equation Benney: b = 0 in the presence of x x symmetry Kuramoto-Sivashinsky equation th = ɛ xx h + b xxx h xxxx h + ( x h) 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.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

75 Nonlinear interface dynamics Nonlinear dynamics in 1D Conservation law th = x j h(x,t) Vicinity to thermodynamic equilibrium (or variational steady-state) driving force ɛ [ ] δf th = x M x δh ɛj + o(ɛ 2 ) Highly nonlinear dynamics where A, B, C functions of x h 1 x x symmetry; non-variational Examples in Molecular Beam Epitaxy x th = x [ɛa + B xx C] j Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

76 Nonlinear interface dynamics Nonlinear dynamics in 1D Mound formation in MBE Experiments: T. Michely, Aachen, Germany. Deposition of 300 atomic layers of Pt at 440K; 660 nm x 660 nm. Theory: J. Villain, CEA Grenoble, France Non-equilibrium mass flux due to ES effect used Highly nonlinear equations ɛ = F th = x [ɛa + B xx C] Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

77 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 ɛ = F th = x [ɛa + B xx C] Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

78 Nonlinear interface dynamics Nonlinear dynamics in 1D In phase meander Free energy F = dsγ Chemical potential µ = γκ κ step curvature Relaxation via terrace diffusion [ Dl c eq V n = s k B T sµ [ ( )] α x ζ tζ = x 1 + ( x ζ) + β xx ζ ( x ζ) 2 x (1 + ( x ζ) 2 ) 3/2 ] λ c λ m λ a) 0 m 1 λ/λ m m with α = F l 2 /2 and β = Dc eqlγ/k B T. obtained from step model m = max[ x ζ/(1 + ( x ζ) 2 ) 1/2 ] t x Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

79 Nonlinear interface dynamics Nonlinear dynamics in 1D Can steps be pinned by anisotropy?? Energetic: γ(θ) or Kinetic: Ehrlich-Schwoebel effect, Edge diffusion. Same qualitative change: 1) θ c = 0 no effect 2) θ c = π/4 interrupted coarsening. θ c =0 θ c =π/4 λ < λ m λ λ > λ m λ c λ m 0 0 m 1 Si(111) growth Hiroo Omi, Japan x Anisotropy Interrupted coarsening Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

80 Nonlinear interface dynamics Nonlinear dynamics in 1D λ Elastic interactions in homo-epitaxy: λ m λ c σ f σ m 0 0 m 1 Force dipoles at step: interaction energy 1/l 2 between straight steps. f x Elastic interactions endless coarsening Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

81 Nonlinear interface dynamics Nonlinear dynamics in 1D From the study of step meandering 3 scenarios: frozen wavelength (no coarsening) growing amplitude OPL et al PRL 1998 interrupted coarsening (nonlin wavelength selection) Danker et al PRL 2004 endless coarsening 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.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

82 Nonlinear interface dynamics Nonlinear dynamics in 1D Facetting+Electromigration φ = x h tφ = xxxx [ xx φ + φ φ 3] j E xx φ. 4th order equation for steady-states 3 regimes: j E > 0 reinforced instability; 0 > j E > 1/4 weak instability; 1/4 > j E stability (a) j E > 0 (b) -1/4 < j E < 0 h(x,t) time (c) j E < -1/4 minima maxima x/l x/l Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

83 Nonlinear interface dynamics Nonlinear dynamics in 1D Facetting+Electromigration j E > 0 reinforced instability endless coarsening (a) (b) j E > A/j E A j E = 0.45 j E = 0.35 j E = 0.25 j E = 0.15 λ/j E 0 0 λ- λ m λ t > j E > 1/4 weak instability frozen wavelength -1/4 < j 0,6 E < 0 0,4 A 0,2 L1-0,02 F. Barakat, K. Martens, O. Pierre-Louis, Phys Rev Lett /2 (a) λ λ π 0 λ- λ m 10 λ+ λ (c) 0,4 0 φ 0-0,4 λ +, λ - λ λ m λ L2 (b) -0,2 j E -0,1 (d) 0 1 x/λ Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

84 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.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

85 Kinetic Roughening Fractal interfaces Contents 1 introduction Models and scales to model surfaces and interfaces 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 Kinetic Roughening Fractal interfaces Continuum models 7 Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

86 Kinetic Roughening Fractal interfaces Example: Ballistic Deposition (from Barabasi and Stanley Fractal Concepts in Surf. Growth) Zinc Electrodeposition Sawada 1985 Irreversible attachment Ballistic Deposition Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

87 Kinetic Roughening Fractal interfaces Roughness [ 1 L ] 1/2 W (t, L) = dx ζ 2 (x, t) L 0 ζ(x, t) = h(x, t) 1 L dx h(x, t) L 0 Short times t t, roughening, growth exponent β W (t, L) t β Long times t t, Steady-state, roughness exponent α W (t, L) L α Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

88 Kinetic Roughening Fractal interfaces Scaling Growth of the correlation length, dynamic exponent z ξ t 1/z Correlations reach the system size at ξ L t L z Crossover condition W (t, L) t β Lα β = α z Family-Viscek ansatz f (u) u β for u 0 f (u) cst for u. W (t) L α f ( ) t L z. Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

89 Kinetic Roughening Continuum models Contents 1 introduction Models and scales to model surfaces and interfaces 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 Kinetic Roughening Fractal interfaces Continuum models 7 Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

90 Kinetic Roughening Continuum models Continuum Models 1: Edwards-Wilkinson model Linear Equation: η white Gaussian, zero average η(x, t) = 0 th(x, t) = V + ν xx h(x, t) + η(x, t) η(x, t)η(x, t ) = 2Dδ(x x )δ(t t ) For d = 1, α = 1/2, β = 1/4, z = 2 W (t) 2 1/2 = f (u) = ( ) D 1/2 ( ) νt L 1/2 f ν L 2 [ ] 1 e 8π2 n 2 1/2 u 2 4π n=1 2 n 2 f (u) 1/12 1/2 for u, f (u) (2u/π) 1/4 for u 0. For arbitrary d α = 2 d 2, β = 2 d, z = 2 4 Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

91 Kinetic Roughening Continuum models Continuum Models 2.1: Kardar Parisi Zhang model Nonlinear Equation: th(x, t) = V + ν xx h(x, t) + λ 2 ( x h)2 + η(x, t) For d = 1, many exact results α = 1/2, β = 1/3, z = 3/2 h(t) = v t + (Γt) β χ +... χ obeys the Tracy-Widom distribution Spohn et al 2000, 2010 T. J. Oliveira, S. C. Ferreira, and S. G. Alves et al 2012 For d > 1 no exact expression of the exponents But for arbitrary d: α + z = 2 (Galilean invariance) Numerical results, e.g. for d=2: β , and α Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101

92 Kinetic Roughening Continuum models Continuum Models 2.2: Kardar Parisi Zhang model Roughening is a universal phenomena also valid for other systems Topological-defect turbulence in electroconvection of nematic liquid crystals Takeuchi, Sano PRL 2010 Paper burning M. Myllys et al Phys rev E 2001 Olivier Pierre-Louis (ILM-Lyon, France.) Non-equilibrium interface dynamics in crystal growth 24th October / 101