1

Multiscale Simulation of Thin-Film Epitaxy

Kristen A. FichthornDepartments of Chemical Engineering and Physics

The Pennsylvania State UniversityUniversity Park, PA 16802

USA

What (Generally) Happens in Thin-Film Epitaxy

Deposition

Aggregation

Nucleation

Terrace Diffusion

Edge Diffusion

2

Assembly at Surfaces

Non-Equilibrium Kinetics + Interactions =

.And More!!!

Ag/ 2 ML Ag / Pt(111)H. Brune et al., Nature

394, 451 (1998).

Al / Al(110)F. Bautier de Mongeot et al.,

PRL 91, 016102 (2003).InAs/GaAs(001)M. Xu et al., Surf. Sci.

580, 30 (2005).

Thin-Film Growthe.g. fcc(110) homoepitaxy

Also Crystal Growth,Catalysis at Surfaces and More

Surface Phenomena Involve MultipleLength and Time Scales

K. Fichthorn and M. Scheffler,Nature 429, 617 (2004).

Atoms Hopping (, ps)

Hut Formation(nm, min)

Hut Organization(m, min)

Bautier de Mongeot et al., Phys. Rev. Lett. 91,016102 (2003).

3

Theoretical Techniques that Span theLength and Time Scales in Thin Film Growth:A Challenge is to Link Them!

Continuum Equations

Kinetic Monte Carlo(KMC)

Molecular Dynamics

Time (s)10-15 10-12 10-9 10-6

Leng

th (m

)

10-6

10-8

10-11

ab initio(AIMD) semi-empirical

(MD)

A Practical Goal: Reactor Designfrom First Principles

Continuum Equations forFluid Flow, Heat Transfer,Mass Transfer, Kinetics ina Rotating Disk Reactor (m,h)

Kinetic Monte CarloSimulation of Growthof GaAs(001) (nm, s)

Charge-Density Contoursfor GaAs(001) fromDensity-FunctionalTheory ()

Example: Growth of GaAs Thin Films

Transition-State Theory

Kratzer and Scheffler, Comp. Sci. Eng. 2001.

4

Kinetic Monte Carlo Simulations

Deposition, F

Aggregation

Nucleation

Terrace Diffusion, D

Edge Diffusion

K. Fichthorn et al., Appl. Phys. A 75, 17 (2002).

Kinetic Monte Carlo: Coarse-Graining MD

Rare Events: =

)/)(exp()/)(exp()(

2 TBkVTBkV

TSTvk

RRRR

MD of Co on Cu(001):The Whole Trajectory

KMC: Coarse-GrainedHops

TSTkt 1=

Rotate

5

Kinetic Monte Carlo as an AccurateSolution to the Master Equation

)'(

),(

),'()'(),()'(),(''

xxW

txP

txPxxWtxPxxWdt

txdPrr

rr

r

rrrrrrr

rr

+=

: Probability to be at State at Time txr

: Transition Probability per Unit Timefrom to 'x

rxr

xr 'xr

Kinetic Monte Carlo as a More AccurateSolution to the Master Equation

)(

),(

),()(),()(),(

xxW

txP

txPxxWtxPxxWdt

txdP

xx

+=

rr

r

rrrrrrr

rr

Detailed-Balance Criterion

: Probability to be at State at Time txr

xr: Transition Probability per Unit Time

from to (e.g., TST rate)xr

[ ]TkAxxWxxW

txPxxWtxPxxW

B

eqeq

/exp)()(

),()(),()(

=

=

rr

rr

rrrrrr

KMC Transition Probabilities areAlso Based on a Kinetic Model

6

KMC Transition Probabilities and theDetailed-Balance Criterion

[ ]TkAxxWxxW

B/ exp)()( =

rr

rr

Metropolis MC Satisfies DetailedBalance, but not Kinetics

>

=

if

TBkE

if

EEife

EEiffiW

1)(

/

Ene

rgy

E *

EiEf

Eifb

TST Satisfies DetailedBalance and Kinetics

W(if) = 0 exp( Ebif /kBT )

KMC Simulates a Poisson ProcessEvents Can Happen AnyTime with an Equal Probability per Unit Time r

How Long Must We Wait?

rt

retWnPrrnP

rt

n

/1

)()()1()(

0

=

=

=

Adsorbate Hopping isA Poisson Process

J. Raut and K. Fichthorn, J. Chem. Phys. 103, 8694 (1995).

t = n

31 2

7

Multiple Independent Poisson Processes:One Big Poisson Process

== i

iRt rRtW ;Re)(

t0,A t1,A

t0,B t1,B

+

=t1t0

A

B

AA r

t 1=

BB r

t 1=

BABA rr

t+

=+1

BA

B

BA

Arr

rBP

rrr

AP+

=+

= )(;)(

A Generic KMC Algorithm

Initialize LatticeFinished

?

Identify All Processesand Rates Ri

Do Process , Increment Time

( )10

)ln(R1

i

K

=

u

ut

i

YesNo

Choose a Process

=

i

RP

iR)(

0 1=

1)(

iiP

=

1

1)(

iiP

K. Fichthorn and W. Weinberg, J. Chem. Phys. 95, 1090 (1991).

8

KMC of Langmuir Adsorption / Desorption

Initialize Lattice,N Sites

Finished?

Count of VacantSites, V

YesNo

Choose Adsorption with

DA

AA rVNVr

VrP)( +

=

AD PP =1

Choose Desorption with

PA PD

Find a Site, Do Process, Increment Time

( )10

)ln()(

1

K+

=

u

urVNVr

tDA

NVN )(

=0 1

rDrA

Application of KMC to LangmuirAdsorption / Desorption

( )( )[ ]

1-1- s 2;s 1

exp1)(

0)0(;)1(

==

++

=

==

DA

DADA

A

DA

rr

trrrr

rt

rrdtd

rDrA

K. Fichthorn and W.H. Weinberg,J. Chem. Phys. 95, 1090 (1991).

9

Rates from Transition-State Theory (TST)

)/exp(

0

*

0

)/)(exp(

)/)(exp(*)(,

TkEqq

k

BA

TkV

TkV

BATSTA B

A B

=

=

R

RRRR*

A

B

Nudged Elastic Band Method: Henkelman and JonssonRidge Method: Ionova and CarterDimer Method: Henkelman and JonssonStep and Slide Method: Miron and FichthornString Method: E, Ren, Vanden-Eijnden

TST Search Methods

MD Naturally Gives TST Rates, but its SLOW

)/exp( 073

1

*

63

1 TkEv

vk BN

jj

N

jj

TST

=

=

=Potential-Energy Minima

Saddle Point

Harmonic Transition-State TheoryG. Henkelman, G. Jhannesson, and H. Jnsson,in Progress on Theoretical Chemistry and Physics,(Kluwer Academic Publishers, 2000).

10

[ ] [ ]

02

021

1)(;0

MEP AlongGradient

)(

Gradient

)(

RRRRu

uuRRuuRR

=

== iiii U

dtd

iiiUiU 44 344 2143421

The Minimum-Energy Path(AKA The Reaction Coordinate)

ui = unit vector pointingalong the path at i

Elber and Co-Workers

R3

R2R1

R0

The Nudged Elastic Band (NEB) Method

[ ]{

( ) i1iii1isi

si

k

0iiiUiU

uRRRRF

FuuRR

=

=+

+

==Images)t Equidistanfor 0(Path Along Force Spring0) (Path toOrthogonal Force

)()(4444 34444 21

Springs Keep ImagesDistributed On the Path

( ) siiiii Udtd FuuR += 1

H. Jnsson, G. Mills, K. W. Jacobsen, in Classical and Quantum Dynamics inCondensed Phase Simulations, Ed.B. J. Berne, G. Ciccotti and D. F. Coker (World Scientific, 1998)

11

Hopping of Al on Al(110): The NEB Method in First-Principles DFT (VASP)

In-Channel Cross-Channel

E = 0.47 eV E = 0.71 eV

In-Channel Cross-Channel

Exchange of Al on Al(110): The NEB Method in First-Principles DFT (VASP)

E = 0.39 E = 0.38

12

Rates from Transition-State Theory (TST)

)/exp(

0

*

0

)/)(exp(

)/)(exp(*)(,

TkEqq

k

BA

TkV

TkV

BATSTA B

A B

=

=

R

RRRR*

A

B

Accelerate MD to Find andSimulate Rare Events!!

MD Naturally Gives TST Rates, but its SLOW

HyperdynamicsParallel Replica DynamicsTemperature-Accelerated Dynamics

Art Voter

=A

TBkVA

TBkV

BATSTkB

)/)(exp(

)/)(exp(*)(

, R

RRR

Accelerated Molecular Dynamics(Hyperdynamics)

A. Voter, J. Chem.Phys. 106, 11 (1997).

Detailed Balance!

V (R)

**

ABC

**

BA

C

-V (R) -V (R)

-V (R) -V (R)

=

=

=

)/exp(/)(/)/exp()/exp(/)/exp()(

)(exp)(

)(/)/)(exp()()(/)/)(exp()()(

*

*

TkWTkTkTkk

TkVW

WTkVWWTkVWk

BB

BBBTST

B

B

BBTST

RRR

RR

RRRRRRRR

13

Accelerated Molecular Dynamics(Hyperdynamics) A. Voter, J. Chem. Phys. 106, 11 (1997).

Detailed Balance!

ABC

**

kTST,AC / 1/W(R)A

=

=

=

CATST

BATST

CATST

BATST

kk

k

k

,

,

,

, kTST,AB / 1/W(R)A

kTST,ABkTST,AC

MD Time: tMD= Nt

AMD Time: ( )==

=

=N

ii

N

i i

kTVtRWtt

11

/exp)(

=

kTVexpBoost

Accelerated Molecular DynamicsThe Bond Boost Method

R. Miron & K. Fichthorn, J. Chem. Phys. 119, 6210 (2003)

EmpiricalThreshold

14

Accelerated Molecular DynamicsDetails of the Bond Boost Method

Channels the Boost intothe Bond thats Readyto Break

Overview of the Bond Boost MethodR. Miron & K. Fichthorn, J. Chem. Phys. 119, 6210 (2003)

15

Diffusion on Cu(100): Elementary Processes

Adatom Hop

Vacancy HopDimer Hop

Dimer ExchangeAdatom Exchange

R. Miron & K. Fichthorn,J. Chem. Phys. 119, 6210(2003)

The Bond-Boost Method: Diffusion on Cu(100)R. Miron & K. Fichthorn,J. Chem. Phys. 119, 6210(2003)

16

Boost = Physical Time / Simulation Time

exp

=

TkVBoost

B

The Bond-Boost Method: Diffusion on Cu(100)

log 10(Boost)

(kBT)-1 (eV-1)

V(R)

R. Miron & K. Fichthorn,J. Chem. Phys. 119, 6210 (2003)

Accelerated AIMD (VASP): Diffusion on Al/Al(110)

Climbing-ImageNudged ElasticBand Method

AcceleratedAIMD

vs.

The Winner!!

EB = 0.38 eV EB = 0.33 eV

17

The Boost in ab initio MD

73 ns

Co/Cu Heteroepitaxy

Promising for spintronic recording media

1 ML of Co on Cu(001) Pentcheva and Scheffler, Phys. Rev. B 60 (2000).

Interesting heteroepitaxial growth modes

18

ab initio KMC of Submonolayer Co/Cu(001)Heteroepitaxy

R. Pentcheva, K. Fichthorn, M. Scheffler,et al., PRL 90, 076101 (2003).

Experiment

Co Grows on Top of Cu

Co Trapped at Exchanged Co

Co, Cu Escape from Exchanged Co

Spin-Polarized, FP-LAPW DFTFor Energy Barriers..

Hopping / Exchangeof Co & Cu Adatoms

Cu Hopping Awayfrom Exchanged Co

Co Hopping Awayfrom Exchanged Co

Tight Binding Potential

R. Miron and K. Fichthorn, Phys. Rev. B. 72, 115433 (2005).

Based on potential by Levanov et al.,Phys. Rev. B 61 (2000).

Levanov Miron Levanov Miron

19

Thin Film Growth at 250 K, F = 0.1 ML/s

Note ClusterMobility

R. Miron and K. Fichthorn, PRL 93, 138201 (2004).PRB 72, 115433 (2005).

R. Miron, K. Fichthorn,Phys. Rev. Lett. 93, 2004.

Accelerated MD SimulationOf Cluster Diffusion

Static Barriers Agreewith MD Values

20

State-Bridging Accelerated MD of Co/Cu(001)Heteroepitaxy: T = 250 K, F = 0.1 ML/s, = 0.54 ML

Mechanism of BilayerIsland Formation

MD Simulations were run for 5.4 s

When an atom is pulled up, it stays there!

Co/Cu(001): Bilayer FormationMechanisms

21

ConclusionsSurface Phenomena are Complex, Interesting, and Multiscale

KMC Coarse-Grains MD, SimulatesExperiments

TST Searches Can Characterize RateProcesses (e.g., NEB method)

Accelerated MD Finds Rate Processes,Simulates Experiments

Theres Room for New Developments!