Lecture : Kinetic Monte Carlo Simulations

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1 Lecture : Kinetic Monte Carlo Simulations Olivier Pierre-Louis ILM-Lyon, France. 6th October 2017 Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

2 1 Introduction Models and levels of description Atomic steps and surface dynamics 2 Kinetic Monte Carlo Rates Master Equation Algorithms: Metropolis & KMC 3 KMC Simulations Attachment-detachment model Growth from a surface Solid-state dewetting Step meandering instabilities 3D KMC / Wetting effects Elastic effects 4 Conclusion Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

3 Introduction Growth shapes Snowflake Yoshi Furukawa, Hokkaido, Japan Growth Cu(1,1,17) Maroutian, Ernst, Saclay, France Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

4 Introduction Models and levels of description Contents 1 Introduction Models and levels of description Atomic steps and surface dynamics 2 Kinetic Monte Carlo Rates Master Equation Algorithms: Metropolis & KMC 3 KMC Simulations Attachment-detachment model Growth from a surface Solid-state dewetting Step meandering instabilities 3D KMC / Wetting effects Elastic effects 4 Conclusion Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

5 Introduction Models and levels of description 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,... 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.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

6 Introduction Atomic steps and surface dynamics Contents 1 Introduction Models and levels of description Atomic steps and surface dynamics 2 Kinetic Monte Carlo Rates Master Equation Algorithms: Metropolis & KMC 3 KMC Simulations Attachment-detachment model Growth from a surface Solid-state dewetting Step meandering instabilities 3D KMC / Wetting effects Elastic effects 4 Conclusion Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

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.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

8 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 : Kinetic Monte Carlo Simulations 6th October / 50

9 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 : Kinetic Monte Carlo Simulations 6th October / 50

10 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.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

11 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.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

12 Kinetic Monte Carlo Rates Contents 1 Introduction Models and levels of description Atomic steps and surface dynamics 2 Kinetic Monte Carlo Rates Master Equation Algorithms: Metropolis & KMC 3 KMC Simulations Attachment-detachment model Growth from a surface Solid-state dewetting Step meandering instabilities 3D KMC / Wetting effects Elastic effects 4 Conclusion Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

13 Kinetic Monte Carlo Rates Rates Eb Transition-state theory Exponential activation R = ν 0 e E b/k B T E b barrier energy ν 0 prefactor Microscopic Theory or Simulation Transition rates R(n m) Which event should be considered? Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

14 Kinetic Monte Carlo Master Equation Contents 1 Introduction Models and levels of description Atomic steps and surface dynamics 2 Kinetic Monte Carlo Rates Master Equation Algorithms: Metropolis & KMC 3 KMC Simulations Attachment-detachment model Growth from a surface Solid-state dewetting Step meandering instabilities 3D KMC / Wetting effects Elastic effects 4 Conclusion Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

15 Kinetic Monte Carlo Master Equation Levels of description Solid on Solid Steps Continuum Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

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.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

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.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

18 Kinetic Monte Carlo Master Equation Example : Attachment-Detachment model (a) h i i State n = {h i ; i = 1.., L} Rates (b) r(hi > hi+1) r(hi > hi 1) R(h i h i + 1) = F R(h i h i 1) = r 0 exp[ n i J k B T ] i n i number nearest neighbors at site i before detachment (Breaking all bonds to detach / Transition state Theory) (c) n i=1 n i=2 n i=3 [ F eq = r 0 exp 2J ] k B T i Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

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.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50 ]

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.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

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 + 2(n i 2)J + µ ] 2k B T 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.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

22 Kinetic Monte Carlo Algorithms: Metropolis & KMC Contents 1 Introduction Models and levels of description Atomic steps and surface dynamics 2 Kinetic Monte Carlo Rates Master Equation Algorithms: Metropolis & KMC 3 KMC Simulations Attachment-detachment model Growth from a surface Solid-state dewetting Step meandering instabilities 3D KMC / Wetting effects Elastic effects 4 Conclusion Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

23 Kinetic Monte Carlo Algorithms: Metropolis & KMC 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! A way to evaluate observables in equilibrium statistical mechanics Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

24 Kinetic Monte Carlo Algorithms: Metropolis & KMC 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.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

25 Kinetic Monte Carlo Algorithms: Metropolis & KMC 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.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

26 Kinetic Monte Carlo Algorithms: Metropolis & KMC Kinetic Monte Carlo algorithm 2: 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: Groups of events with same rates Number of possible events from one state N poss(n) number of states N tot(n) Update list of possible events at each time step Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

27 Kinetic Monte Carlo Algorithms: Metropolis & KMC Kinetic Monte Carlo algorithm 3: 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, with same probability density leading to Algorithm ρ(τ)dτ = ρ(u)du u = exp[ R tot(n)τ] τ = log(u) R tot(n) 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.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

28 Kinetic Monte Carlo Algorithms: Metropolis & KMC Kinetic Monte Carlo algorithm 4: implementing time Complete rejection-free KMC 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 ). 4 Choose random number u, uniformly distributed with 0 < u 1. 5 Time increment t t + τ with τ = log(u) R tot(n) Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

29 KMC Simulations Attachment-detachment model Contents 1 Introduction Models and levels of description Atomic steps and surface dynamics 2 Kinetic Monte Carlo Rates Master Equation Algorithms: Metropolis & KMC 3 KMC Simulations Attachment-detachment model Growth from a surface Solid-state dewetting Step meandering instabilities 3D KMC / Wetting effects Elastic effects 4 Conclusion Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

30 KMC Simulations Attachment-detachment model 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.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

31 KMC Simulations Growth from a surface Contents 1 Introduction Models and levels of description Atomic steps and surface dynamics 2 Kinetic Monte Carlo Rates Master Equation Algorithms: Metropolis & KMC 3 KMC Simulations Attachment-detachment model Growth from a surface Solid-state dewetting Step meandering instabilities 3D KMC / Wetting effects Elastic effects 4 Conclusion Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

32 KMC Simulations Growth from a surface Nano-scale crystal Surface Instabilities out of equilibrium Movie... Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

33 KMC Simulations Growth from a surface KMC results Thiel and Evans 2007 Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

34 KMC Simulations Solid-state dewetting Contents 1 Introduction Models and levels of description Atomic steps and surface dynamics 2 Kinetic Monte Carlo Rates Master Equation Algorithms: Metropolis & KMC 3 KMC Simulations Attachment-detachment model Growth from a surface Solid-state dewetting Step meandering instabilities 3D KMC / Wetting effects Elastic effects 4 Conclusion Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

35 KMC Simulations Solid-state dewetting Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte E. Bussman, Carlo Simulations F. Leroy, F. Cheynis, P. Müller, O. 6th Pierre-Louis October NJP / 50 KMC vs SOI solid-state dewetting ν0 ν 4 ν2 r 2 ν 3 ν1 r 4 r3 r0 KMC simulations SOS Hopping rates A/S: r n = ν 0 e nj/t +E S /T A/A: ν n = ν 0 e nj/t J bong energy h = 3, E S = 1, T = 0.5

36 KMC Simulations Solid-state dewetting Monolayer no dewetting rim OPL, A. Chame, Y. Saito, PRL 2007 Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

37 KMC Simulations Step meandering instabilities Contents 1 Introduction Models and levels of description Atomic steps and surface dynamics 2 Kinetic Monte Carlo Rates Master Equation Algorithms: Metropolis & KMC 3 KMC Simulations Attachment-detachment model Growth from a surface Solid-state dewetting Step meandering instabilities 3D KMC / Wetting effects Elastic effects 4 Conclusion Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

38 KMC Simulations Step meandering instabilities 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.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

39 KMC Simulations 3D KMC / Wetting effects Contents 1 Introduction Models and levels of description Atomic steps and surface dynamics 2 Kinetic Monte Carlo Rates Master Equation Algorithms: Metropolis & KMC 3 KMC Simulations Attachment-detachment model Growth from a surface Solid-state dewetting Step meandering instabilities 3D KMC / Wetting effects Elastic effects 4 Conclusion Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

40 KMC Simulations 3D KMC / Wetting effects 3D KMC Model 3D KMC Hopping along the surface ν = ν 0 e (n 1J 1 +n 2 J 2 +n s1 J s1 +n s2 J s2 )/T J bond energy, n i nb neighbors i = 1, 2 NN, NNN adsrobate i = s1, s2 NN, NNN substrate Moves to NN Allowed when there is NN or NNN Shape controlled by ζ = J 2 J 1 = J s2 J s1 Wetting controlled by χ = J s1 J 1 Link T O: 1 χ = S 2γ(0) χ 0: Complete de-wetting χ 1: Complete Wetting Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

41 KMC Simulations 3D KMC / Wetting effects Wetting on a flat substrate Wulff-Kaishiew Contact angle not a good parameter for facetted crystals! ψ = S AV S AS Wetting control parameter Cube ζ = 0 χ = J s1 J 1 ψ = 5 4χ KMC: N = 11025, ζ = 0.2, χ = 0.4, T /J 1 = 0.5 Error: Energy 1%; ψ 3%. Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

42 KMC Simulations 3D KMC / Wetting effects Nanocrystals in Cassie-Baxter state Growth of GaN on Si nano-pillars Hersee et al J.A.P Avoiding dislocations? Growing without collapse? Stability? Zang et al, APL 2006 χ = 0.390, χ = Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

43 KMC Simulations 3D KMC / Wetting effects Migration-induced switching Nanoswitch controlled by an electron beam KMC wih imposed migration M. Ignacio, OPL, PRE (2015) Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

44 KMC Simulations 3D KMC / Wetting effects Nucleation-limited vs diffusion-limited imbibition front motion χ = 0.8, `x = 6, h = 3, `p = 2, 4 (a) (b) (c) (d) (e) (f) P. Gaillard, Y. Saito, OPL, Phys Rev Lett 2011 Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

45 KMC Simulations Elastic effects Contents 1 Introduction Models and levels of description Atomic steps and surface dynamics 2 Kinetic Monte Carlo Rates Master Equation Algorithms: Metropolis & KMC 3 KMC Simulations Attachment-detachment model Growth from a surface Solid-state dewetting Step meandering instabilities 3D KMC / Wetting effects Elastic effects 4 Conclusion Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

46 KMC Simulations Elastic effects Elastic islands on nano-pillars lp/l0 3D KMC with elastic effects Extended stability Asymmetric CB state χc(ns=2) χc(ns=0) χel(ns=2) χel(ns=0) χ*(ns=2) (a) 0.2 Partially collapsed state χ M. Ignacio, Y. Saito, P. Smereka, OPL, PRL 2014 (b) Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

47 KMC Simulations Elastic effects Elastic islands on flat substrate 2D KMC with elastic effects Green function method P. Gaillard, T. Frisch, J.N. Aqua, PRB (2013) Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

48 Conclusion Conclusion Contents 1 Introduction Models and levels of description Atomic steps and surface dynamics 2 Kinetic Monte Carlo Rates Master Equation Algorithms: Metropolis & KMC 3 KMC Simulations Attachment-detachment model Growth from a surface Solid-state dewetting Step meandering instabilities 3D KMC / Wetting effects Elastic effects 4 Conclusion Conclusion Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

49 Conclusion Conclusion KMC conclusion KMC versatile method to look at crystal shape evolution Improved methods to obtain physical rates (DFT, MD) Built-in thermal fluctuations Difficulties parallelization finding the list of events Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

50 Conclusion Conclusion Conclusion References Books Reviews 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) A.L. Barabási, H.E. Stanley, Fractal Concepts in Surface growth, Cambridge, (1995) M. Kotrla, Computer Physics Communications (1996) A. Voter, Introduction to the KMC method, Radiation effects in solids, Springer Nato Series 235 C. Misbah, O. Pierre-Louis, Y. Saito, Rev Mod Phys (2010). Burton Carbrera Frank, The Growth of Crystals and the Equilibrium Structure of their Surfaces, Phil. Trans. A, (1951) Olivier Pierre-Louis (ILM-Lyon, France.) Lecture : Kinetic Monte Carlo Simulations 6th October / 50

Non-equilibrium interface dynamics in crystal growth

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