Lattice gas models and Kinetic Monte Carlo simulations of epitaxial crystal growth Theoretische...
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Transcript of Lattice gas models and Kinetic Monte Carlo simulations of epitaxial crystal growth Theoretische...
Lattice gas models and Kinetic Monte Carlo simulations of epitaxial crystal growth
Theoretische Physik und Astrophysik
& Sonderforschungsbereich 410
Julius-Maximilians-Universität Würzburg
Am Hubland, D-97074 Würzburg, Germany
http://theorie.physik.uni-wuerzburg.de/~biehl
Mathematics and Computing Science
Intelligent Systems
Rijksuniversiteit Groningen, Postbus 800,
NL-9718 DD Groningen, The Netherlands
Michael Biehl Michael Biehl
Outline
• Motivation Non-equilibrium growth - Molecular Beam Epitaxy (MBE)
• Theory / modeling / simulation several levels of theoretical description
• Summary
• Lattice gas and Solid-On-Solid (SOS) models
• Kinetic Monte Carlo simulations deposition and transient kinetics thermally activated processes, Arrhenius dynamics problems and limitations
• Example applications I) unstable growth, mound formation and coarsening dynamics II) Atomic Layer Epitaxy (ALE) growth of II-VI(001) systems
Molecular Beam Epitaxy ( MBE )
control parameters: substrate/adsorbate materialsdeposition rate substrate temperature T
ultra high vacuumdirected deposition of adsorbatematerial onto a substrate crystal
production of, for instance, high quality · layered semiconductor devices · magnetic thin films · nano-structures: quantum dots, wires
theoretical challenge · clear-cut non-equilibrium situation · interplay: microscopic processes macroscopic properties · self-organized phenomena, e.g. dot formation
Mikrostrukturlabor, Würzburg
oven
UHV
T
Theory / modelling of (growing) surfaces
Quantum Mechanics
faithful material specific descriptionincluding electronic propertiesoften: configuration of a few atoms/molecules, unit cells of periodic structures, zero temperature treatment
important tool: Density Functional Theory (DFT)
description in terms of electron densities
typical problem:
energy/stability of surface reconstructions,
preferred arrangement of surface atoms
CdTe (001) surface reconstructions
Molecular Dynamics
numerical integration of equations of motion + thermal fluctuations
effective interactions, e.g. classical short range pair potentials
(QM treatment: e.g. Car Parinello method )
microscopic dynamics of particles
limited system size and real time ( 10-6 s )
example: diffusion on a surfaceatomic vibrations ( ~10-12 s )with occasional hops to the next local minimum
typical problem:
dissociation of deposited
dimers at the surface,
transient mobility of arriving atoms
Kinetic Monte Carlo (KMC) simulations
stochastic dynamics, consider only significant changes of the configuration
simplifying lattice gas models: pre-defined lattice of empty / occupied siteshops from site to site
Solid-On-Solid (SOS) models:exclude bulk vacancies, overhangs,defects, stacking faults, etc.
d+1 dim. crystal represented by integer array above d-dim. substrate lattice
Deposition of particles, e.g. with flux F = 1 atom / site / s (incoming momentum, attraction to the surface...)
incorporation processes, examples:
Transient effects upon deposition
knockout-processes
at terrace edges
downhill funnelling
steering
weakly bound, highly mobile intermediate states
regular lattice sitepote
nti
al en
erg
ydistance from the surface
vac.
Arrhenius law: waiting time TBk
Δ
o eτ τ
rate TBk
Δ
o e R
attempt frequency , e.g. o
energy barrier , e.g. for hopping diffusion
thermally activated processes, simplifying representation:
112o 10~ s
after incorporation: mobile adatoms at surface sites
R (ab) = 0 exp[ / (kBT) ]
R (ba) = 0 exp[ ( Ea-Eb+ ) /
(kBT) ]a
b Eb
Ea
more general:
transition states and energy barriers affect „only“
the non-equilibrium dynamics of the system
Et
t
R (ab) exp[ - Ea / (kBT) ] = R (ba) exp[ - Eb / (kBT) ]
detailed balance condition stationary P(s) exp[- Es / (kBT) ]
for states of type a,b,...
in absence of deposition and desorption:
system approaches thermal equilibrium
an example: Ehrlich-Schwoebel instability
ES
E
E
diffusion bias: adatoms attach to
upper terraces preferentially
uphill current favors mound formation
additional Schwoebel barrier
hinders inter-layer diffusion
non-equilibrium, kinetic effect:
additional barrier ES is irrelevant for equilibrium properties of the system
implicitly used simplifications and assumptions
deep (local) minima, infrequent eventsexclude, e.g., double or multiple jumps:
transition state theory: correct treatment takes into account entropies / free energies
constant prefactor (attempt frequency) - temperature independent - state independent disregard actual shape of the energy landscape a
b
t
o(ab) = o(ba) ?
consistent with discretized state spaceand concept of detailed balance
frozen crystal : e.g. single, mobile particle in a static environment, neglectconcerted rearrangements of the entire crystal / neighborhood
desorption
islands
diffusion
nucleation
deposition
downward diffusion
edge diffusion
some microscopic processes on the growing surface
+more: incorporation, knockout attachment to edges / islands detachment processes, ...
Kinetic Monte Carlo Simulation (rejection free)
· perform the selected event
(evaluate physical real time step)
· perform the selected event
(evaluate physical real time step)
· initial configuration of the (SOS) system
· catalogue of all relevant processes i=1,2,...n
and corresponding Arrhenius rates
· initial configuration of the (SOS) system
· catalogue of all relevant processes i=1,2,...n
and corresponding Arrhenius rates
R1
R2
R3
Rn
... rate
s ...
· pick one of the possible events randomly
with probability pi Ri
· pick one of the possible events randomly
with probability pi Ri
0
1
random
num
ber
· update the catalogue of possible processes
and associated energy barriers and rates
· update the catalogue of possible processes
and associated energy barriers and rates
R3
exchange processes / concerted moves
e.g. exchange vs. hopping diffusion
dimer and island mobility
material specific input ?
direct / indirect experimental measurement
calculations/estimates: first principles semi-empirical potentials
quantitative match of simulations and experiments
complete catalogue of events ?
potentially relevant processes:
Problems and limitations
lattice gas / SOS description:
defects, dislocations ?
hetero-epitaxial growth ?
strain and other mismatch effects ?
material specific simulations
realistic lattices or off-lattice simulations
interaction potentials, realistic energy barriers
particularities of materials / material classes
Applications:abstract models, further simplifications
basic questions
example: (universal?) dynamical scaling behavior
I) Unstable growth: slope selection and coarsening
model features / simplifications
SOS lattice (e.g. simple cubic) neglect overhangs, defects
knock-out process upon deposition momentum of incoming particles
irreversible attachment
immobile islands
forbidden downward diffusion
high barriers (large enough flux)
limited diffusion length for
terrace / step edge diffusion
effective representation of
nucleation events
single particle picture
lsed : characteristic length of step edge diffusion
• initial mound formation
due to Schwoebel effect
• coarsening process
merging of mounds driven by
- deposition noise
and/or - step edge diffusion
• saturation state
finite system size single mound
example: slow step edge diffusion (associated length lsed=1 lattice const. )
16 ML 256 ML 4096 ML
• selection of a stable slope:
compensating particle currents
upward (Schwoebel)
downward (knockout)
dynamic scaling behavior time t <h> (film thickness)
surface width ~mound height
w =t for t< tx
wsat L for t> tx
growth exponent
roughness exponent
saturation time tx Lz dynamic exponent z= /
systemsizesL = 80 100 125 140 256 512
w /
Lscaling plot, data collapse
=1 (slope selection)
z=4=1/4
relatively slow coarsening
The role of step edge diffusion (sed)
for the morphology and coarsening dynamics
64ML
fast sed
(lsed L)
1.00
0.45
sed driven
coarsening 128ML
slow sed
(lsed 1)
1.00
0.25
noise assisted
coarsening
128ML
absent sed
0.70 < 1
0.20
absence of
slope selection,
rough surface
additional
corner barrier
hindered sed,
noise assisted
coarsening
128ML
significant step edge diffusion
characteristic exponents:
= 1, 1/3, z 3
for 1 << lsed << L
lsed
universality: observed in various types of lattices
simple cubic (001), body centered cubic (001)
simple hexagonal (001), hcp (001)
contradicts continuum model predictions:
0.24 for cubic lattices
1/3 for all other lattices
Siegert, 1998Moldovan, Golubovic, 2000
Ahr, Kinne, Schinzer
anisotropicbinding structure:
][ 011
][110
x
y
example system: II-VI (001) semiconductor surfaces
· zincblende lattice, (001) orientation:
alternating layers (square lattices) of, e.g., Cd / Te
SOS representation, four sub-lattices
· surface reconstructions observed:
- c(2x2), (2x1) vacancy structures Cd-terminated
- (2x1) dimerization Te-terminated
II) Competing surface reconstructions in non-equilibrium
CdTe (001)
properties of Cd-terminated surfaces
maximum coverage 50 % two competing vacancy structures: checkerboard or missing rows
simultaneous occupation
of NN sites in y-direction,
i.e. [1-10], is forbidden
(extremely unfavorable)
TeCd
x empty
electron counting rule, DFT
[Neureiter et al., 2000]
small difference in surface energies
favors checkerboard c(2x2)-order at low temperatures
e.g. DFT: E 0.008 eV per site in CdTe [Gundel, private comm.] 0.03 eV ZnSe
Te at the surface
isotropic N.N. interaction
additional Te
dimerization
motivation: coverage 3/2 observed under flux of excess Te
allows for Te deposition on a perfect c(2x2) Cd surface
beyond SOS
weakly bound, highly mobile Te-atoms ( Te* ) on the surface, e.g.
at a Cd-site (Te-trimers)
bound to a single Cd (neutralizes repulsion)temporary position
time consuming explicit treatment / mean field like Te* reservoir
Kinetic Monte Carlo simulations
Arrhenius rates for elementary processes = o e –/ (kT) o = 1012/s
choice of parameters: qualitative features, plausibility arguments semi-quantitative comparison,prospective first principle results
Atomic Layer Epitaxy (ALE)
alternating pulses (1s) of Cd and Te flux: 5ML/s dead time: 0.1s
Cd Te Cd Te
reconstructions self-limitation of the growth rate at high temperature
experiment [Faschinger, Sitter] simulation [M. Ahr, T. Volkmann]
overcome at lower T due to presence of highly mobile, weakly bound Te* :
Summary
• Motivation Non-equilibrium growth - Molecular Beam Epitaxy (MBE)
• Theory / modeling / simulation several levels of theoretical description following talks: continuum descriptions, multi-scale approach,...
• Lattice gas and Solid-On-Solid (SOS) models
• Kinetic Monte Carlo simulations deposition and transient kinetics thermally activated processes, Arrhenius dynamics problems and limitations
• Example applications I) unstable growth, mound formation and coarsening dynamics II) Atomic Layer Epitaxy (ALE) growth of II-VI(001) systems
Outlook (Wednesday)
application of KMC method in off-lattice modelstreatment of
- hetero-epitaxy, mismatched lattices
- formation of dislocations
- strain-induced island growth
- surface alloys of immiscible materials