1. 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.)

2. 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.

3. TST: general form

4. 1D + harmonic approximation for the potential energy:B A E n0

5. 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.

6. (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

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

8. More than one event: residence timek2 k1
More than one event: residence time

10. 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

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

12. 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 ...)

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

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

15. Adatom jump (saddle point)
E=0.48 eV

16. Adatom jump (final point)
E=0 eV

17. Adatom jump: rate

18. 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

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

20. Almost a potential-energy minimum

21. 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

22. possible nucleation
center for a new
island

23. Other critical example: 2D vs 3D

24. Other critical example: 2D vs 3D

25. Other critical example: 2D vs 3D

26. 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).

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

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

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

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

31. 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

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

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

34. = empty
= filled

35. 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 ....

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

37. 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.

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

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

40. 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").

41. 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)

42. 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

43. 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:

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

45. 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

46. 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).

47. Exchange mechanism

48. Exchange mechanism

49. Exchange mechanism

50. Exchange mechanism

51. Exchange mechanism
Feibelman (1990)

52. 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”