Timothy, Us-china 2009

Enhanced Kinetic Monte Carlo

T. P. Schulze Department of Mathematics University of Tennessee

Research supported by: NSF-DMS-0707443 & DE-FG02-03ER25586

Kinetic Monte-Carlo Simulations

• Configuration space is typically discrete, e.g. occupation arrays for a fixed lattice. • Dynamics are imposed via a Markov Chain model • Transition rates are based on configuration changes • Example: qi = ν exp



− ∆φ kb T



,

∆φ = ES + mEN

Early Stages of Growth

≈ 104 and 105 atoms colored by coordination Nijk

Effect of Surface Diffusion

K = 106

K = 105

K = 104

KMC with Elastic Interactions Collaborator: P. Smereka (Michigan) Reference: Lam, Lee & Sander Phys. Rev. Lett. 2002 • 1+1 SOS with height hi • Nearest & next nearest neighbor linear springs: long springs if short springs if • Hopping rate:

j>0 j < 0.

ri = r0 exp [(−γn + ∆W + E) /kB T ] ,

• Elastic energy barrier: ∆W = W (with atom i)−W (without atom i), 1 X W = wi,j , 2 (i,j)∈Ω

• Equilibrium configuration:

∂W ∂ui,j

=0

and

∂W ∂vi,j

= 0.

Energy Localization Method

• The exact elastic energy barrier is ∆W = W (u; Ω) − W (ua ; Ωa ). • The energy localization is approximation is ∆WL = W (u; Ωρ ) − W (uaρ ; Ωaρ ), where the displacement of the atom-off solution uaρ is constrained to agree with the atom-on solution u on the boundary.

Energy Barrier and Film Profile

Continuum Analog—Displacement Field Within this framework, the displacement field is give by ∂i2 uj + 2∂j ∂k uk Teij nj ui

=

x∈Ω

0,

= σ1 n1 δi1 , x ∈ ∂Ω, → 0, |x| → ∞.

The energy density can be expressed as w = 1/2Eij Tij , —consistent with the continuum limit of our discrete model. For a finite region Ωρ ⊂ Ω is the total energy is Z W (u; Ωρ ) = w dx. Ωρ

The elastic energy barrier is then   ∆W = lim W (u; Ωρ ) − W (ua ; Ωaρ ) . ρ→∞

Energy Localization Approximation

For the local solution, we define the atom-off solution on a domain Ωaρ with lower boundary constrained by

uaρ = u,

x ∈ Γρ .

Our approximation for atom-off displacement field is then  a uρ if |x| < ρ w= u if |x| ≥ ρ. and our approximation for the elastic energy barrier is ∆WL = W (u; Ωρ ) − W (uaρ ; Ωaρ ).

(1)

Energy Localization Approximation Theorem 1 - Residual Estimate. Suppose that h(x) is a compactly supported function whose support includes x = 0. Further, suppose that h(x) is modified by a localized change centered at x = 0 then the following is true RL = O(ρ−2 )

as

ρ → ∞.

Theorem 2 - Principle of Energy Localization. Under the same hypotheses of Theorem 1 the following is true ∆W = ∆WL(1 + O(ρ−2 ))

as

ρ → ∞.

This explains the high accuracy of our method.

Energy Truncation Approximation Naively, one might expect an approximation based on a simple truncation of the energy integrals, ∆WT = W (u; Ωρ ) − W (ua ; Ωaρ ), would be better. However, we also have Theorem 3 - Nonlocality of the Energy Density. Under the same hypotheses of Theorem 1 the following is true ∆W = ∆WT (1 + O(Hρ−1 ))

as

ρ → ∞.