The document describes different algorithms for performing kinetic Monte Carlo simulations of epitaxial crystal growth processes. It introduces linear search, standard inverted list, minimal allocation, and flexible allocation algorithms. The flexible allocation algorithm synthesizes aspects of the standard and minimal allocation approaches to provide memory efficient simulations while maintaining low computational cost per iteration. Performance comparisons on test systems show the flexible approach performs well even for large system sizes.

1. Flexible Memory Allocation in
Kinetic Monte Carlo Simulations
Aaron Craig

2. Context & Motivation: Epitaxy
• Kinetic Monte Carlo Monte Carlo Markov Processes Stochastic Processes⇢ ⇢ ⇢
Source: physics.lancs.ac.uk/lamsa/pages/page3.htm
“Surface of a GaAs film grown by liquid phase epitaxy”
Lancaster Materials and Surface Analysis Service
(Lancaster University (UK))

3. Context & Motivation: Epitaxy
• Monte Carlo Simulations use a random number to select state-
transitions in Markov processes.
• Kinetic Monte Carlo simulations simulate the time-evolution of
systems in nature.
• Only a small fraction of transitions are accessible: sparse
transition matrix
• K.M.C. for crystal growth: Markov process modeling the
behavior of a system of atoms
• Stochastic simulation: discrete or continuous time-scales

4. Simple Cubic, Solid-on-solid Epitaxial Growth
• Cubic Lattice of lateral dimension
• Permitted transitions in system state: changes in
height array
• Moves or ‘Hops’: Atoms move from one lattice site to another
• Deposition: Stochastic ‘rain’
• Determination of rates: local bond-coordination
• Simplest case: Only nearest neighbors determine rate
• Single species
M2
H

5. Simple Cubic, Solid-on-solid Epitaxial Growth
• Our model allows five possible rates: one for each
coordination:
, .
• The parameter reflects bond-strength.
• The pre-factor is adjusted to reflect the model.
• Periodic boundary conditions
• Exploit the finite number of rates, categories
E
n = 0, ..., 4rn = Ke En
K

6. Tour of Methods
• Linear Search
• Inverted List Methods
• (1) Standard Inverted List Algorithm
• (2) Minimal Allocation Algorithm
• Cascade procedure
• (3) Flexible Allocation Algorithm
• Synthesis of methods (1) and (2)

7. Linear Search Algorithm
1. Compute the partial sums by summing over the
event index:
!
!
2. Select a random number .
!
3. Find the integer for which .
!
4. Carry out event . Update height array , rates.
!
5. Return to step 1.
Ri =
iX
j=1
rj
u 2 [0, R]
i
i
Ri 1 < u  Ri
H

8. Standard Inverted List Algorithm
1. Sum the rates by category:
!
!
!
!
2. Select a random number .
!
3. Find the integer for which .
!
4. Carry out a randomly-selected event with rate .
!
5. Update the list , inverse list , and occupation array .
!
6. Return to step 1.
Ri =
iX
j=1
Cj ˆrj
u 2 [0, R]
i Ri 1 < u  Ri
ˆri
L A C

9. Standard Inverted List Algorithm
• Two types of list updates:
• (1) Updates to the list of events : categorization by rate
• (2) Updates to the inverse list : -‘addresses’
• Advantages of Inverted List Methods:
• (1) Fixed cost per iteration: computational costs do not scale
with , the size of the system.
• (2) Both and can be maintained with a small number of
updates per iteration.
L
LA
M2
L A

10. Standard Inverted List Algorithm
• Inverted list methods require more memory:
• The standard list requires a total allocation of
• Our model:
L nM2
n = 5

11. Standard Inverted List Algorithm
kf
(A(e, 1), ki)
Updates:
1) Replacement
2) Event moved to kf
Standard List Update

12. Minimal Allocation Algorithm
• Total Memory Allocation:
• Each iteration necessitates a cascade procedure.
• Cases requiring special treatment
• Problem: as increases, performance suffers.
• Fixed cost per iteration (scales with )
• Slower than standard inverted list method, but uses less
memory
M2
n
n

13. Cascade Procedure:
1) Replacement
2) Cascade
3) Event moved to kf
Minimal Allocation Algorithm: Cascade
Case: kf < ki

14. Cascade Procedure:
1) Replacement
2) Cascade
3) Event moved to kf
Minimal Allocation Algorithm: Cascade
Case: kf > ki

15. Flexible Allocation Algorithm
• Idea: synthesis of previous two methods
• Total memory allocation , .
• Size of is model-dependent.
• Memory Considerations for large systems
• Performance outlook for more complex models
 (1 + ↵)M2
↵ << n 1
↵

16. Overview of the Algorithm
• Two cases:
• Case 1: Destination sub-list contains unoccupied space
• Two updates required
• Case 2: No unoccupied space
• Cascade procedure required
• Cases requiring special treatment

17. Performance Considerations
• Cascades & Allocation Considerations
• Periodicity and System-Size
• Deposition and its complications
• Cache

18. Updates:
1) Replacement
2) Event moved to kf
Flexible Allocation Algorithm: Non-cascade

19. Updates:
1. Replacement
2. Cascade
3. Event moved to kf
Flexible Allocation Algorithm: Cascade

20. Observations
• Total allocation always grows.
• Saturation eventually occurs.
• Cascade procedures are increasingly rare

21. Further Application
• Many models: larger number of possible rates
• Face-centered cubic lattice
• System size: lattice sites
• Rates determined by neighbors and ‘next-nearest’
neighbors
• 91 distinct rates
M3
 5003

22. Parameters!
• System-size:
• Rates: ,
!
M2
= 22p , p = 5, ..., 12
108
rij = e Enij
E = 3
• iterations
ˆr1 = 1
ˆr2 = e 3
⇡ 4.98 ⇥ 10 2
ˆr3 = e 6
⇡ 2.48 ⇥ 10 3
ˆr4 = e 9
⇡ 1.23 ⇥ 10 4
ˆr5 = e 12
⇡ 6.14 ⇥ 10 6
• Deposition rate: d = 10 6 monolayers
M2 · s

23. CPU-time vs. P

24. CPU-time vs. P (Inverted List Methods)

25. Allocation Ratio vs. Number of Iterations

26. Cascade Count vs. Number of Iterations

27. Conclusions
• Effects of System-size
• Intersection of inverted list algorithms’ curves
results from this model’s small number of rates.
• For models with more rates, we expect a larger
difference in the performances of the inverted
list algorithms.
• Other Models

28. References
[1] T.P. Schulze, "Efficient Kinetic Monte Carlo Simulation," Journal of
Computational Physics 227 (2008) 2455 - 2462
[2] T.P. Schulze, "Kinetic Monte-Carlo Simulations with Minimal Searching," Phys.
Rev. E 65 (2002) Art. no. 036704