Selected Chapters from Semiconductor Physics: Theory and ...
Transcript of Selected Chapters from Semiconductor Physics: Theory and ...
11. Modeling growth at the atomic scale
Dr. Roberto Bergamaschini
Selected Chapters from Semiconductor Physics:
Theory and modelling of epitaxial growth
L-NESS and Department of Materials Science, University of Milano-Bicocca (Italy)
Atomistic processes during crystal growth
Crystal growth is the macroscopic
outcome of infinitely many elementary
events at the atomic scale.
Modeling the growth at the scale of
atomistic processes has the great
advantage of reducing the number of
assumption to describe the system but
poses serious limitations in the size and
time scale that can be explored.
Multi-scale approach: the
characterization of basic mechanisms
and related parameters can be taken as
the input for coarse modeling.
28/05/202111. Modeling growth at the atomic scale
ab-initio โ classical MD โ kinetic Monte-Carlo โ โ โ continuum
Molecular dynamics
Ab-initio MD: ๐({๐น}) is the eigenvalue of the ground-state electronic wave function
Classical MD: ๐({๐น}) is an assigned function of the ionic coordinates. For example:
one-particle potential
(external; usually =0)
three-body potential
(angular)
pair potential
(distance)
Many-particle problem (no analytical solutions)
โ time-integration by finite differences methods
Timestep ๐๐ โช ๐ โcharacteristic periodโ of the system
The elementary processes are atomic vibrations, such that ๐๐๐๐ โผps.
For energy-conservation requirements, typical time steps for simulating realistic systems are of the order of fs.
Ab-initio MD up to 104 steps (10ps) Classical MD up to 109 steps (ยตs)
๐ ๐น = ๐0 +
๐
๐1 ๐น๐ +1
2
๐,๐
๐โ ๐
๐2 ๐น๐ , ๐น๐ +1
6
๐,๐,๐
๐โ ๐โ ๐
๐3 ๐น๐ , ๐น๐ , ๐น๐
๐๐
๐2๐น๐
๐๐ก2= โ๐ป๐๐ ๐นNEWTON LAW
Evolution of the nuclei
according to the classical
hamiltonian:
๐ฏ = ๐ฒ+ ๐ฝ({๐น})
e.g. Lennard-Jones,
Morse potential. Ok for
metals (insaturate bonds)
e.g. Tersoff potential
Covalent systems with
strict bond angles
Configuration at
time ๐กConfiguration at
time ๐ก + ฮ๐กalgorithm
28/05/202111. Modeling growth at the atomic scale
Atomistic modeling of deposition
28/05/202111. Modeling growth at the atomic scale
Atom flying in the gas phase.
No perturbation of the surface
Atomic deposition:
โข stochastic (random location)
โข local (only the impact region is affected)
Atom impacts the surface. Their
kinetic energy is to be dissipated:
THERMALIZATION
Atom starts interacting
with the surface
๐
~10ร ; t~ps
๐ฏ
๐ โ ๐
๐ โ, ๐ โ๐โ
โ!๐โ๐ POISSON DISTRIBUTION
Stick-Where-You-Hit (SWYH) deposition
28/05/202111. Modeling growth at the atomic scale
โข Probability of deposition at site ๐: ๐ = 1/๐
โข Probability of โ atoms at site ๐ after deposition of
๐ก atoms: BINOMIAL DISTRIBUTION
๐ โ, ๐ก =โ๐ก
๐๐ก 1 โ ๐ ๐กโโ
Thermodynamic limit ๐ โ โ:
โ =
โ=0
๐ก
โ๐(โ, ๐ก) = ๐ก๐ = ๐ก/๐ = ๐
๐2 =
โ=0
๐ก
โ โ โ 2๐(โ, ๐ก) = ๐ก๐(1 โ ๐) = ๐ 1 โ1
๐
Atoms land randomly on the lattice sites and stick there. T=0 K regime (no diffusion)
Disordered growth:
roughness monotonously increases
roughness
Adatom diffusion
28/05/202111. Modeling growth at the atomic scale
Courtesy by L. Barbisan
All atoms in the crystal vibrate around their equilibrium positions
(๐0~1013 Hz).
Diffusion is the (rare) event causing a change in the residence site
of one atom.
Let us see for example the dynamics of a Si adatom on top of the
unreconstructed (111) surface, by MD simulation (Tersoff potential)
[11เดค2]
Total time 300ps โ 1400K
Adatom diffusion
28/05/202111. Modeling growth at the atomic scale
Courtesy by L. Barbisan
All atoms in the crystal vibrate around their equilibrium positions
(๐0~1013 Hz).
Diffusion is the (rare) event causing a change in the residence site
of one atom.
Let us see for example the dynamics of a Si adatom on top of the
unreconstructed (111) surface, by MD simulation (Tersoff potential)
The trajectory in the phase-
space is stochastic in nature
and passes through
intermediate and metastable
states which are not easy to
identify a priori. The most-
likely transition path passes
through the lowest activation
barriers, i.e. the saddle points
in the potential, generally not
known a priori.Sketch of the potential curve
[11เดค2]
Total time 300ps โ 1400K
Adatom diffusion: time-scale separation
28/05/202111. Modeling growth at the atomic scale
๐vib = 1/๐0 โผ0.1 ps
๐dif โณ 10 ns
๐๐ท โผ 1/๐ฟ โผ s-min
The system spends most of its time vibrating around an equilibrium position, and only occasionally it
moves to a new site.
โ Time-scale separation (huge at experimental conditions (t~s), negligible at high temperatures, where
the picture does not hold)
๐1 = ๐0eโEB1๐๐
๐2 = ๐0eโEB2๐๐
The MD dynamics gives us more than what we really need to understand the surface processes: it fully
traces the thermal vibrations of the adatom in any basin of minimum energy, even if this is not making any
change to the surface configuration. Can we just focus on the relevant diffusion/deposition dynamics?
๐ = ๐0 exp โ๐ธ๐ต๐๐
Harmonic Transition State Theory
28/05/202111. Modeling growth at the atomic scale
Hypothesis:
โข Parabolic approximation of the potential
โข No recrossing of the barrier once overtaken
โข Adatom thermalization at ech site between the jumps
(no long jumps) โ Uncorrelated diffusion events: the
system loses memory of its history by the random
vibrations around the equilibrium site
Arrhenius relation
๐จ ๐ฉ
๐ ๐ = ๐eโ๐๐
From statistics, the escape-time ๐ at which the stochastic process ๐ด โ ๐ต, with rate ๐, occurs follows the
exponential probability distribution:
๐ = ๐๐ดโ๐ต = event rate = number of times the event occurs in a unit time
energy of the saddle-point separating state
๐จ and state ๐ฉ. It is the activation energy (or
diffusion barrier) for the event that causes
the system to move from ๐จ to ๐ฉfrequency prefactor,
i.e. attempt frequency
๐ = escape time = time after which the event occurs for the first time
Adatom diffusion: rate
28/05/202111. Modeling growth at the atomic scale
๐dif = ๐nn ๐0 exp โ๐ธdif
๐๐Average diffusion-time = ๐ = 1/๐dif
Hyp: jump to all equivalent nearest-
neighbour site
1D surface: ๐nn = 2 n.n.
2D FCC surface: ๐nn = 4 n.n.
2D HEX surface: ๐nn = 6 n.n.
All atoms in the crystal are vibrating around their equilibrium positions (๐0~1013 Hz).
Diffusion is the (rare) event causing a change in the residence site of one atom.
Typically: ๐ธdif โผeV, ๐ โผ1000K โ ๐dif โผ 108 Hz โช ๐0 (Rare event!)
๐dif โฅ 10ns
๐ธ๐ต is strongly dependent on the atom coordination: ๐(bulk atom) โช ๐(complete surface layer) โช ๐(adatom)
Diffusion rate and random-walk
28/05/202111. Modeling growth at the atomic scale
Einstein theory:
Random-walk between lattice sites
ิฆ๐ ๐ก โ ิฆ๐ 0 2 = ๐dif๐2๐ก = ๐dif๐ท๐ก
100000 diffusion steps on a square lattice
It holds true as an
ensemble average
over a large
number of random
walk trajectories
(here 100 or 1000)!
Let us consider the erratic motion of a single adatom on a planar surface, hopping between nearest-neighoring
sites on a square lattice.
๐ท =๐dif๐
2
๐dif= ๐0๐
2 exp โ๐ธdif๐๐
๐dif๐2๐ก
Diffusion coefficient
Diffusion vs. adsorption/desorption
๐dif = ๐๐ ๐0dif exp โ
๐ธdif
๐๐๐adsโ1 = ๐0
ads exp โEads
๐๐
๐ = ๐ท๐ = ๐0 exp๐ธads โ ๐ธdif
2๐๐
๐0๐๐๐ โผvibrational frequency of adatoms,
normal to the surface
28/05/202111. Modeling growth at the atomic scale
๐0 =๐0๐๐๐
๐0๐๐๐
๐
๐๐
Adatoms on the surface are not expected to stay there and randomly move forever. During their erratic
motion they can interact with each other, eventually returning nucleation of critical clusters, they can
aggregate to existing islands or get captured at steps. Even if none of these happen, an adatom after a
certain time ๐๐๐๐ will be lost, because of adsorption in the crystal bulk or desorption back in the gas phase.
Diffusion
regime
Kinetic
regime
Adatom diffusion at island edges or terrace steps
28/05/202111. Modeling growth at the atomic scale
Atoms are stabilized by the interaction with neighbors as they form bonds with them.
Alternative kinetic pathways
28/05/202111. Modeling growth at the atomic scale
Fast diffusion of
the adatom across
the surface while
the island is
virtually frozen
(on the typical
adatom time scale)
2D
3D
?Interaction with other adatoms
on the surface, e.g. added by
deposition (especially at high
supersaturation).
possible nucleation
center for a new island
From MD to KMC
28/05/202111. Modeling growth at the atomic scale
MD trajectories are set on the continuum
potential energy surface. Most of the
time is spent around the minima and
only at certain times the system escape
from one basin and fall into the next one.
Representative states corresponds
to regions of the potential energy
surface pertaining to a single basin.
Each of them is distinguished as an
element in a lattice of configurations
The continuum MD trajectory
connecting the different configurations
in the lattice is replaced by a Markov
chain of discrete hops between each
state in the lattice.
Kinetic Monte Carlo is a method for visiting the phase space by following a Markov chain of hops between the
distinct configurations of the system, in a statistically consistent way. The temporal sequence of events is
tracked by knowing the full catalog of mechanisms and the corresponding rates of occurence.
Andersen, Panosetti and Reuter. Front. Chem. 7, 202 (2009) ; Voter Phys. Rev. B 34, 6819 (1986); Fichthorn & Weinberg J. Chem. Phys. 95, 1090 (1991)
Voter, โIntroduction to the Kinetic Monte Carlo Methodโ, in Radiation Effects in Solids,(Springer, NATO Publishing Unit, Dordrecht, The Netherlands, 2005)
Let us consider two events with different activation barriers, i.e. different rates.
For each event alone, the probability that it
occurs after a time ๐ is:
๐1 ๐ = ๐1eโ๐1๐
๐2 ๐ = ๐2eโ๐2๐
An escape time for both rare-events can be obtained by extracting a random number for each of them,
following the exponential distribution:
๐1 =๐๐๐1๐1
; ๐2 =๐๐๐2๐2
๐๐ = rand 0,1
The evolution will then follow the path corresponding to the shortest escape time. The selected ๐๐ is
indeed the correct escape-time of event ๐-th since all other mechanisms has not occurred yet.
This suggests a route for implementing an algorithm to evolve the system in time.
Multiple concurrent events
28/05/202111. Modeling growth at the atomic scale
๐1 = ๐0eโEB1๐๐
๐2 = ๐0eโEB2๐๐
KMC evolution: initial state
28/05/202111. Modeling growth at the atomic scale
= empty
= filled
We take a lattice with some adatoms deposited on the surface.
Hyp:
Only adatoms
can move
KMC evolution: catalogue of possible moves
28/05/202111. Modeling growth at the atomic scale
We take a lattice with some adatoms deposited on the surface.
Possible moves must be known in advance (here: single-atom moves only).
1
4
3
2
11
12
13
15
17
1814
96
105
87
16
= empty
= filled
Hyp:
Only adatoms
can move
KMC evolution: catalogue of rates
28/05/202111. Modeling growth at the atomic scale
We take a a lattice with some adatoms deposited on the surface.
Possible moves must be known in advance (here: single-atom moves only).
The corresponding rates are then computed ๐๐ = ๐0,๐ exp(โ๐ธ๐/๐๐).
๐1
๐4
๐3
๐2
๐11
๐12
๐13
๐15
๐17
๐18๐14
๐9๐6
๐10๐5
๐8๐7
๐16
= empty
= filled
Hyp:
Only adatoms
can move
KMC evolution: catalogue of escape-times
28/05/202111. Modeling growth at the atomic scale
0.1ms
0.13ms
0.09ms
0.14ms
0.065ms
0.07ms
0.12ms
0.11ms
0.09ms
0.1ms0.1ms
179ms201ms
0.15ms0.17ms
0.09ms0.13ms
0.09ms
We take a a lattice with some adatoms deposited on the surface.
Possible moves must be known in advance (here: single-atom moves only).
The corresponding rates are then computed ๐๐ = ๐0,๐ exp(โ๐ธ๐/๐๐).
A random escape-time ๐๐ = โln ๐๐/๐๐ is computed for each mechanism ๐. ๐๐ = rand 0,1
= empty
= filled
Hyp:
Only adatoms
can move
KMC evolution: selection of the move
28/05/202111. Modeling growth at the atomic scale
We take a lattice with some adatoms deposited on the surface.
Possible moves must be known in advance (here: single-atom moves only).
The corresponding rates are then computed ๐๐ = ๐0,๐ exp(โ๐ธ๐/๐๐).
A random escape-time ๐๐ = โln ๐๐/๐๐ is computed for each mechanism ๐. ๐๐ = rand 0,1
The chosen mechanism is the one with shorter ๐๐โฆ.
0.065ms
= empty
= filled
Hyp:
Only adatoms
can move
KMC evolution: move and new state
28/05/202111. Modeling growth at the atomic scale
We take a a lattice with some adatoms deposited on the surface.
Possible moves must be known in advance (here: single-atom moves only).
The corresponding rates are then computed ๐๐ = ๐0,๐ exp(โ๐ธ๐/๐๐).
A random escape-time ๐๐ = โln ๐๐/๐๐ is computed for each mechanism ๐. ๐๐ = rand 0,1
The chosen mechanism is the one with shorter ๐๐โฆ. and the system is evolved in a new state.
= empty
= filled
Hyp:
Only adatoms
can move
Bortz-Kalos-Lebowitz algorithm: J. Comp. Phys. 17, 10 (1975)
28/05/202111. Modeling growth at the atomic scale
๐๐ ๐๐ ๐๐ ๐๐ ๐๐๐ ๐๐๐ ๐๐๐ ๐๐๐ ๐๐๐ ๐๐๐ ๐๐๐ ๐๐๐๐๐ ๐๐ ๐๐ ๐๐ ๐๐ ๐๐๐
0
2. Extract a random number ๐ between 0 and ๐๐ก๐๐ก and
identify the corrisponding event ๐ in the rate list, such that
3. Evolve the system according to the ๐-th mechanism
4. Extract a random number ๐ and increment the time by the
exponentially distributed escape-time
๐๐ก๐๐ก =
๐=1
๐
๐๐
1. Lets list the escape-rates in an array.
Each mechanism is identified by the index of the cell in
the array, matching the catalogue labeling.
๐๐ก๐๐ก๐ = ๐๐๐๐ 0, ๐๐ก๐๐ก
๐=1
๐โ1
๐๐ < ๐ โค
๐=1
๐
๐๐
๐ = โln ๐
๐๐ก๐๐ก
KMC of thin-film growth: the solid-on-solid (SOS) model
28/05/202111. Modeling growth at the atomic scale
Constitutive assumptions:
โข No vacancies
โข No out-of lattice positions (no interstitials/over-coordinated sites)
โข Surface configuration โ = โ ๐, ๐โข Cubic lattice (extensible/mappable to other lattices)
โข Only and all the topmost atom on each column โ are active
It is sufficient to keep track of the top exposed layers and of the
nearest neighbour list and one immediately builds on the fly the list
of mechanisms to be used in KMC. The key processes to be
modelled are:
โข Deposition โ stick-where-you-hit or more complex rules
(eventually desorption)
โข Surface diffusion โ energy barrier
SOS KMC is extremely fast and allows to match typical experimental time scales at typical temperatures,
also for rather large systems!
Young & Schubert JCP 1965; Gordon JCP 1968; Abraham and White JAP 1970; Gilmer & Bennema
JAP 1972; Vvedensky et al. 1987 and later (important modification for treating semiconductors)
๐ธ๐ต ๐, ๐ = ๐ธ0 + ๐1 ๐, ๐ ๐ธ1 + ๐2 ๐, ๐ ๐ธ2
1st neighbors 2nd neighbors
Hyp: ๐ธ๐ต depends on the
energy of the starting site
and not on the arrival one
Homoepitaxial growth on a flat surface
28/05/202111. Modeling growth at the atomic scale
Layer-by-layer: adatoms have sufficient time
to migrate and attach at the borders of existing
2D islands before nucleating a new layer
Multi-layer growth: adatom diffusion is limited so
that new nuclei form on top of incomplete layers
T=600KT=800K
๐ธ๐ต = ๐ธ0 + ๐1๐ธ1 + ๐2๐ธ2 โ ฮ๐ธ๐๐ ๐
KMC for strained layers: Balls-and-Spring SOS
28/05/202111. Modeling growth at the atomic scale
chemical bond energy
MD can naturally describe the elastic relaxation of a
strained structure (e.g. the tetragonal relaxation) but it
is not adequate to model the islanding process as size
and time scales exceed the capabilities.
KMC algorithms have been developed such to include
strain effects.
Atoms are connected by springs with elastic constant
๐พ. The adaptation of film atoms to the substrate lattice
stretch/compress the springs thus accumulating an
elastic energy (per atom i-th):
๐ธ๐๐ ๐, ๐ =1
2
๐
โฒ
๐พ ๐๐ โ ๐๐๐๐ 2
When the atom is removed from its original site, the
total elastic energy ๐ธ๐๐ = ฯ๐ ๐ธ๐๐ ๐ decreases by an
amount ฮ๐ธ๐๐. Then, the net diffusion barrier, including
strain, must include also such a ฮ๐ธ๐๐:
elastic
contribution
Tetragonal relaxation
Toward a comprehensive description: Wetting Layer
28/05/202111. Modeling growth at the atomic scale
Stranski-KrastanovIntermediate misfit 4%
Layer-by-layerLow misfit 2%
๐ธ๐ต = ๐ธ๐ + ๐1๐ธ1 + ๐2๐ธ2 โ ฮ๐ธ๐๐ ๐ + ๐ธ๐๐ฟ โ ๐
chemical bond energyelastic
contribution
Volmer-Weber 3D islandsHigh misfit 6%
Guo et al. Comput. Mater. Sci. 44, 174 (2008)
Thickness dependent interaction ๐ธ๐๐ฟ with the substrate interface, decaying after few
MLs (mimicking ๐พ = ๐พ โ ) is to be included in the diffusion barrier to account for the
possible formation of a pseudomorphic WL: the stronger the interaction the higher the
barrier:
Wetting energy