Transcript Diapositiva 1 - e

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
x0
x
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
k
k
TST
A
TST
A
 | v A |  A (r ) A ; in thecanonicalensemble:

 K ( p )  V ( r )
drdp
e
e
)
r
(

)
r
(

|
v
|
 A A A
 K ( p )  V ( r )
drdp
e
e

r  R3N ; p  R3N ;
 A (r )  1 if r  A;
 A (r )  0 if r  A;
Dividingsurface  (3N - 1) - dimensional boundary to stateA;
d
v r
dt
v A  componentof the velocityperpendicular tothedividing surface
1D + harmonic approximation for the potential energy:
A
B
E
n0
k A B  n 0e
E

k BT
Harmonic Transition State Theory
k A B  n 0e
E

k BT
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? The “first-escape” time
distribution (Poisson)
1
  ;
k
f (t )dt  ke
 kt
dt
Valid beyond TST
k2
k1
More than one event:
residence time
f1alone (t )dt  k1e
f1 (t )dt  k1e
 k1t
 k1t
dt;
f 2alone (t )dt  k2e
t
dt  (1   k 2 e
 k 2
 k2t
dt
d ) 
0
 k1e
 k1t  k 2t
e
dt
f 2 (t )dt  k 2 e
 k 2t
t
dt  (1   k1e
 k1
d ) 
0
 k2e
 k 2t  k1t
e
dt
f tot (t )dt  ( f1 (t )  f 2 (t ))dt  (k1  k 2 )e
 ( k1  k 2 ) t
dt
Total rate of escape from a state
kTOT 
k
i
i
First escape-time distribution
f (t )dt  ktot e
k tot t
dt
If I have more ways of exiting from a state, I'll exit sooner
Usually very good approximation at experimental
conditions, i.e. when:
E
 1;  k  n 0e
k BT
E

k BT
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?
MD rate :12.6 10 events/s
9
often ~1012s-1
HTST rate :14.8 109 events/s(EAM;E ~ 0.04 eV;n 0 ~ 9.4  1011 s 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
n j  4 n 0, j exp[0.48*11603/ T (K)]
n 0, j  10 (s );T  300K;
13
1
1
nj
 1 s
ES
Less bonds
More bonds
E2
E1
barrier: ES - E1
barrier: ES - E2
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.
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
….. (here: single-atom moves only)
= empty
= filled
ki  n 0,i e
together with their rates
 Ei / kT
= empty
2
1
= filled
3
6
9
4
5
7
8
14
15
13
16
18
17
10
11
12
For each mechanism, we know that the Poisson
"escape-time" distribution holds:
f i ( )d  ki e
 k i
d ;
a  rand ();  i   ln( a ) / ki
if other events do not bring the system out of the
state before, then mechanism i will do it, after a
time i
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
= filled
0.13ms
0.065ms
0.09ms
0.12ms
201ms
0.13ms
0.1ms
0.1ms 0.09ms
0.07ms
0.17ms
0.11ms
0.15ms
179ms
0.09ms
0.09ms
Mechanism chosen, time advanced by 0.065ms, restart from the new state
= empty
= filled
0.065ms
= empty
= filled
18
ktot   ki
i 1
0
Extract a random number between 0 and ktot ……….
A faster (but totally equivalent) way for doing KMC ....
ktot
18
ktot   ki
i 1
0
Extract a random number between 0 and ktot ……….
high barriers: low probability
ktot
18
ktot   ki
i 1
0
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  where  is
extracted from
f ( )d  ktot e
 ktot 
d ;
a  rand();    ln(a) / ktot
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)
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)
~
Ea  E0  nE
Step-edge atoms:
n=3
Fitting parameters
Isolated adatom:
n=0 (high mobility)
surface atoms: n=4
(almost frozen)
Dimer:
n=1
Adatom attached to a
step with no nbrs: 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)
~
Ea  E0  nE
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:
~
E0  1.3eV; E  0.2eV
Starting point: surface composed of various terraces (more
realistic than flat ones)
Growth mode vs T
F=0.05ML/s
10000 atoms per
layer
T = 300 K
3D growth
7
T = 500 K
Layer-by-Layer
T = 650 K
Step-Flow
(vicinal only)
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”