Surface diffusion as a sequence of rare, uncorrelated events Oversimplified picture (1 dimension, 1 particle, “rigid” potential, etc.) tvib~1ps tD The system spends most of its time vibrating around equilibrium position, and it occasionally moves to a new position  Time-scale separation (huge at experimental conditions (tD~1s), negligible at high temperatures, where the picture does not hold.)

Transition State Theory -x x0 x State A System at equilibrium -x x x0 The rate of escape from A is given by the (equilibrium ensemble average of the) flux exiting through the boundary to state A.

TST: general form

1D + harmonic approximation for the potential energy: B A E n0

Harmonic Transition State Theory Arrhenius relation n0 frequency prefactor; E = energy of the saddle-point separating state A and state B E is the “activation energy” (or “diffusion barrier”) for the event that causes the system to move from A to B.

(H)TST is not always valid: anharmonic effects, recrossings, long jumps (dynamically correlated events) ….. All these problems get serious at high T E n0 Recrossings Long jumps

When does the event take place When does the event take place? The “first-escape” time distribution (Poisson) Valid beyond TST

More than one event: residence time k2 k1 More than one event: residence time

Total rate of escape from a state First escape-time distribution If I have more ways of exiting from a state, I'll exit sooner

Usually very good approximation at experimental conditions, i.e. when:

Want to estimate a rate? Fast but not always easy: compute E and n0, and use the Arrhenius relation to estimate their rates Slow but very accurate (beyond hTST): run MD and count the number of events! (should be done at different temperatures ...)

Example: Ag on Ag(111) at T=100K how do we compute E? often ~1012s-1

Adatom jump (initial minimum) E=0 eV Ag/Ag(100); EAM (Voter-Chen) potentials

Adatom jump (saddle point) E=0.48 eV

Adatom jump (final point) E=0 eV

Adatom jump: rate

ES Simple (often useful) way of thinking: energy is proportional to bonds  barriers are proportional to the difference between bonds at the saddle and bonds at the initial minimum. Less bonds More bonds E2 E1 barrier: ES - E1 barrier: ES - E2

Knowing the real moves and their typical time scale is fundamental ...

Almost a potential-energy minimum

The isolated adatom will reach the island if: the adatom can fast diffuse across the surface the island is virtually frozen (on the typical adatom time scale) if under deposition, I also need to consider the role played by "new" adatoms

possible nucleation center for a new island

Other critical example: 2D vs 3D

Other critical example: 2D vs 3D

Other critical example: 2D vs 3D

Want to simulate both kinetics and thermodynamics? You can try to use KMC A.B. Bortz, M.H. Kalos, and J.L. Lebowitz, J. Comp. Phys. 17, 10 (1975). A.F. Voter, Phys. Rev. B 34, 6819 (1986). K.A. Fichthorn and W.H. Weinberg, J. Chem. Phys. 95, 1090 (1991).

KMC: let us consider a lattice with some atoms deposited on (state 1) = empty = filled

Goal: simulate the evolution out of state 1 = empty = filled

Possible moves must be known in advance … Possible moves must be known in advance ….. (here: single-atom moves only) = empty = filled

together with their rates = empty 2 = filled 1 3 6 9 4 5 7 10 11 8 12 14 15 13 16 18 17

For each mechanism, we know that the Poisson "escape-time" distribution holds: if other events do not bring the system out of the state before, then mechanism i will do it, after a time ti the KMC recipe is very simple: extract an escape time for each possible mechanism and choose the event with the fastest one

= empty 0.1ms 0.14ms 0.13ms = filled 0.065ms 0.11ms 0.09ms 0.07ms

Mechanism chosen, time advanced by 0.065ms, restart from the new state = empty = filled 0.065ms

= empty = filled

A faster (but totally equivalent) way for doing KMC .... ktot Extract a random number between 0 and ktot ………. A faster (but totally equivalent) way for doing KMC ....

high barriers: low probability ktot Extract a random number between 0 and ktot ……….

ktot Choose the corresponding mechanism (6), and evolve the system Rates can be estimated from experiments, previous MD simulations, hTST, etc. Growth simulations: Deposition can be added in a very natural way.

Mechanism 6 is chosen: move to state 2 accordingly = empty = filled 6

Mechanism 6 is chosen: move to state 2 accordingly, and restart = empty = filled

Time is advanced by t where t is extracted from If all of the rates are known exactly and hTST holds, KMC gives the exact dynamics !!! It is much faster than MD and it allows to reach very long time scales (experimental) and to consider large system sizes. Notice that vibrations are filtered out (only "useful dynamics").

The simplest KMC of thin-film growth: the solid on solid model Historical bibliography: Young & Schubert JCP 1965; Gordon JCP 1968; Abraham and White JAP 1970; Gilmer & Bennema JAP 1972; Vvedensky & C 1987 and later (important modification for treating semiconductors)

surface atoms: n=4 (almost frozen) Cubic lattice, atoms adsorbing on-top of each other. No vacancies or out-of lattice positions allowed. Barriers proportional to the number of nearest neighbors (excluding the supporting one) Step-edge atoms: n=3 Fitting parameters Isolated adatom: n=0 (high mobility) surface atoms: n=4 (almost frozen) Adatom attached to a step with no nbrs: n=1 Dimer: n=1

It is sufficent to keep track of the top exposed layers and of the nearest neighbor list and one immediately builds on the fly the list of mechanisms to be used in KMC. Deposition can be easily trated (random + stick where you hit or more complex rules) SOS KMC is extremely fast and allows to match typical experimental time scales at typical temperatures, also for rather large systems! Let us see an application for parameters typical of Si:

Starting point: surface composed of various terraces (more realistic than flat ones)

Step-Flow (vicinal only) Growth mode vs T F=0.05ML/s 10000 atoms per layer 7 T = 300 K 3D growth T = 650 K Step-Flow (vicinal only) T = 500 K Layer-by-Layer

MD vs KMC MD simulations allow the system to evolve “exactly” (for a given potential ...). Problem: short time-scales KMC simulations allow one to simulate over long time scales. Problem: reliability of the catalogue Mechanisms are not always easy to guess!!! Present-day research is aimed at making it possible for MD to reach longer time scales OR finding a way to make reliable KMC’s (complete catalogue).

Exchange mechanism

Exchange mechanism

Exchange mechanism

Exchange mechanism

Exchange mechanism Feibelman (1990)

What do we learn from the exchange mechanism ? It is dangerous to rely on pure intuition The reaction coordinate is not trivial More than one atom can be involved in the process Finding the saddle-point energy requires some care Since 1990, several “unexpected diffusion mechanisms were found”