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

✉ [email protected]

L-NESS and Department of Materials Science, University of Milano-Bicocca (Italy)

✉ [email protected]

mailto:[email protected]

mailto:[email protected]

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 → classical MD → kinetic Monte-Carlo → → → continuum

Molecular dynamics

Ab-initio MD: 𝑉({𝑹}) is the eigenvalue of the ground-state electronic wave function

Classical MD: 𝑉({𝑹}) 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)

→ time-integration by finite differences methods

Timestep 𝜟𝒕 ≪ 𝝉 “characteristic period” of the system

The elementary processes are atomic vibrations, such that 𝝉𝒗𝒊𝒃 ∼ps.

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 (µs)

𝑉 𝑹 = 𝑉0 +

𝑖

𝑉1 𝑹𝑖 +1

2

𝑖,𝑗

𝑗≠𝑖

𝑉2 𝑹𝑖 , 𝑹𝑗 +1

6

𝑖,𝑗,𝑘

𝑘≠𝑗≠𝑖

𝑉3 𝑹𝑖 , 𝑹𝑗 , 𝑹𝑘

𝑚𝑖

𝜕2𝑹𝑖

𝜕𝑡2= −𝛻𝑖𝑉 𝑹NEWTON LAW

Evolution of the nuclei

according to the classical

hamiltonian:

𝑯 = 𝑲+ 𝑽({𝑹})

e.g. Lennard-Jones,

Morse potential. Ok for

metals (insaturate bonds)

e.g. Tersoff potential

Covalent systems with

strict bond angles

Configuration at

time 𝑡Configuration at

time 𝑡 + Δ𝑡algorithm


Atomistic modeling of deposition


Atom flying in the gas phase.

No perturbation of the surface

Atomic deposition:

• stochastic (random location)

• 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

𝒗

~10Å; t~ps

𝐯

𝜎 → 𝜃

𝑃 ℎ, 𝜃 →𝜃ℎ

ℎ!𝑒−𝜃 POISSON DISTRIBUTION

Stick-Where-You-Hit (SWYH) deposition


• Probability of deposition at site 𝑖: 𝑝 = 1/𝑁

• Probability of ℎ atoms at site 𝑖 after deposition of

𝑡 atoms: BINOMIAL DISTRIBUTION

𝑃 ℎ, 𝑡 =ℎ𝑡

𝑝𝑡 1 − 𝑝 𝑡−ℎ

Thermodynamic limit 𝑁 → ∞:

ℎ =

ℎ=0

𝑡

ℎ𝑃(ℎ, 𝑡) = 𝑡𝑝 = 𝑡/𝑁 = 𝜃

𝜎2 =

ℎ=0

𝑡

ℎ − ℎ 2𝑃(ℎ, 𝑡) = 𝑡𝑝(1 − 𝑝) = 𝜃 1 −1

𝑁

Atoms land randomly on the lattice sites and stick there. T=0 K regime (no diffusion)

Disordered growth:

roughness monotonously increases

roughness

Adatom diffusion


Courtesy by L. Barbisan

All atoms in the crystal vibrate around their equilibrium positions

(𝜈0~1013 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)

[11ത2]

Total time 300ps – 1400K

Adatom diffusion


Courtesy by L. Barbisan

All atoms in the crystal vibrate around their equilibrium positions

(𝜈0~1013 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

[11ത2]

Total time 300ps – 1400K

Adatom diffusion: time-scale separation


𝜏vib = 1/𝜈0 ∼0.1 ps

𝜏dif ≳ 10 ns

𝜏𝛷 ∼ 1/𝐿 ∼ s-min

The system spends most of its time vibrating around an equilibrium position, and only occasionally it

moves to a new site.

→ Time-scale separation (huge at experimental conditions (t~s), negligible at high temperatures, where

the picture does not hold)

𝑘1 = 𝜈0e−EB1𝑘𝑇


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?

𝑘 = 𝜈0 exp −𝐸𝐵𝑘𝑇

Harmonic Transition State Theory


Hypothesis:

• Parabolic approximation of the potential

• No recrossing of the barrier once overtaken

• Adatom thermalization at ech site between the jumps

(no long jumps) → Uncorrelated diffusion events: the

system loses memory of its history by the random

vibrations around the equilibrium site

Arrhenius relation

𝑨 𝑩

𝑝 𝜏 = 𝑘e−𝑘𝜏

From statistics, the escape-time 𝝉 at which the stochastic process 𝐴 → 𝐵, with rate 𝑘, occurs follows the

exponential probability distribution:

𝑘 = 𝑘𝐴→𝐵 = event rate = number of times the event occurs in a unit time

energy of the saddle-point separating state

𝑨 and state 𝑩. It is the activation energy (or

diffusion barrier) for the event that causes

the system to move from 𝑨 to 𝑩frequency prefactor,

i.e. attempt frequency

𝜏 = escape time = time after which the event occurs for the first time

Adatom diffusion: rate


𝑘dif = 𝑁nn 𝜈0 exp −𝐸dif

𝑘𝑇Average diffusion-time = 𝜏 = 1/𝑘dif

Hyp: jump to all equivalent nearest-

neighbour site

1D surface: 𝑁nn = 2 n.n.

2D FCC surface: 𝑁nn = 4 n.n.

2D HEX surface: 𝑁nn = 6 n.n.

All atoms in the crystal are vibrating around their equilibrium positions (𝜈0~1013 Hz).

Diffusion is the (rare) event causing a change in the residence site of one atom.

Typically: 𝐸dif ∼eV, 𝑇 ∼1000K → 𝑘dif ∼ 108 Hz ≪ 𝜈0 (Rare event!)

𝜏dif ≥ 10ns

𝐸𝐵 is strongly dependent on the atom coordination: 𝑘(bulk atom) ≪ 𝑘(complete surface layer) ≪ 𝑘(adatom)

Diffusion rate and random-walk


Einstein theory:

Random-walk between lattice sites

Ԧ𝑟 𝑡 − Ԧ𝑟 0 2 = 𝑘dif𝑎2𝑡 = 𝑁dif𝐷𝑡

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.

𝐷 =𝑘dif𝑎

2

𝑁dif= 𝜈0𝑎

2 exp −𝐸dif𝑘𝑇

𝑘dif𝑎2𝑡

Diffusion coefficient

Diffusion vs. adsorption/desorption

𝑘dif = 𝑁𝑑 𝜈0dif exp −

𝐸dif

𝑘𝑇𝜏ads−1 = 𝜈0

ads exp −Eads

𝑘𝑇

𝜆 = 𝐷𝜏 = 𝜆0 exp𝐸ads − 𝐸dif

2𝑘𝑇

𝜈0𝑎𝑑𝑠 ∼vibrational frequency of adatoms,

normal to the surface


𝜆0 =𝜈0𝑑𝑖𝑓

𝜈0𝑎𝑑𝑠

𝑎

𝑁𝑑

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 𝜏𝑎𝑑𝑠 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


Atoms are stabilized by the interaction with neighbors as they form bonds with them.

Alternative kinetic pathways


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


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, “Introduction to the Kinetic Monte Carlo Method”, 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 𝜏 is:

𝑝1 𝜏 = 𝑘1e−𝑘1𝜏

𝑝2 𝜏 = 𝑘2e−𝑘2𝜏

An escape time for both rare-events can be obtained by extracting a random number for each of them,

following the exponential distribution:

𝜏1 =𝑙𝑛𝜉1𝑘1

; 𝜏2 =𝑙𝑛𝜉2𝑘2

𝜉𝑖 = rand 0,1

The evolution will then follow the path corresponding to the shortest escape time. The selected 𝜏𝑖 is

indeed the correct escape-time of event 𝑖-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




KMC evolution: initial state


= empty

= filled

We take a lattice with some adatoms deposited on the surface.

Hyp:

Only adatoms

can move

KMC evolution: catalogue of possible moves



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


We take a a lattice with some adatoms deposited on the surface.


The corresponding rates are then computed 𝑘𝑖 = 𝜈0,𝑖 exp(−𝐸𝑖/𝑘𝑇).

𝑘1

𝑘4

𝑘3

𝑘2

𝑘11

𝑘12

𝑘13

𝑘15

𝑘17

𝑘18𝑘14

𝑘9𝑘6

𝑘10𝑘5

𝑘8𝑘7

𝑘16

= empty

= filled

Hyp:

Only adatoms

can move

KMC evolution: catalogue of escape-times


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




A random escape-time 𝜏𝑖 = −ln 𝜉𝑖/𝑘𝑖 is computed for each mechanism 𝑖. 𝜉𝑖 = rand 0,1

= empty

= filled

Hyp:

Only adatoms

can move

KMC evolution: selection of the move






The chosen mechanism is the one with shorter 𝜏𝑖….

0.065ms

= empty

= filled

Hyp:

Only adatoms

can move

KMC evolution: move and new state






The chosen mechanism is the one with shorter 𝜏𝑖…. 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)


𝒌𝟏 𝒌𝟐 𝒌𝟑 𝒌𝟒 𝒌𝟏𝟏 𝒌𝟏𝟐 𝒌𝟏𝟑 𝒌𝟏𝟒 𝒌𝟏𝟓 𝒌𝟏𝟔 𝒌𝟏𝟕 𝒌𝟏𝟖𝒌𝟓 𝒌𝟔 𝒌𝟕 𝒌𝟖 𝒌𝟗 𝒌𝟏𝟎

0

2. Extract a random number 𝜌 between 0 and 𝑘𝑡𝑜𝑡 and

identify the corrisponding event 𝑗 in the rate list, such that

3. Evolve the system according to the 𝑗-th mechanism

4. Extract a random number 𝜉 and increment the time by the

exponentially distributed escape-time

𝑘𝑡𝑜𝑡 =

𝑖=1

𝑁

𝑘𝑖

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.

𝑘𝑡𝑜𝑡𝜌 = 𝑟𝑎𝑛𝑑 0, 𝑘𝑡𝑜𝑡

𝑖=1

𝑗−1

𝑘𝑖 < 𝜌 ≤

𝑖=1

𝑗

𝑘𝑖

𝜏 = −ln 𝜉

𝑘𝑡𝑜𝑡

KMC of thin-film growth: the solid-on-solid (SOS) model


Constitutive assumptions:

• No vacancies

• No out-of lattice positions (no interstitials/over-coordinated sites)

• Surface configuration ℎ = ℎ 𝑖, 𝑗• Cubic lattice (extensible/mappable to other lattices)

• Only and all the topmost atom on each column ℎ 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:

• Deposition → stick-where-you-hit or more complex rules

(eventually desorption)

• Surface diffusion → 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)

𝐸𝐵 𝑖, 𝑗 = 𝐸0 + 𝑛1 𝑖, 𝑗 𝐸1 + 𝑛2 𝑖, 𝑗 𝐸2

1st neighbors 2nd neighbors

Hyp: 𝐸𝐵 depends on the

energy of the starting site

and not on the arrival one

Homoepitaxial growth on a flat surface


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

𝐸𝐵 = 𝐸0 + 𝑛1𝐸1 + 𝑛2𝐸2 − Δ𝐸𝑒𝑙 𝑖

KMC for strained layers: Balls-and-Spring SOS


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

𝐾. The adaptation of film atoms to the substrate lattice

stretch/compress the springs thus accumulating an

elastic energy (per atom i-th):

𝐸𝑒𝑙 𝑖, 𝑗 =1

2

𝑛

′

𝐾 𝑑𝑛 − 𝑑𝑛𝑒𝑞 2

When the atom is removed from its original site, the

total elastic energy 𝐸𝑒𝑙 = σ𝑖 𝐸𝑒𝑙 𝑖 decreases by an

amount Δ𝐸𝑒𝑙. Then, the net diffusion barrier, including

strain, must include also such a Δ𝐸𝑒𝑙:

elastic

contribution

Tetragonal relaxation

Toward a comprehensive description: Wetting Layer


Stranski-KrastanovIntermediate misfit 4%

Layer-by-layerLow misfit 2%

𝐸𝐵 = 𝐸𝑠 + 𝑛1𝐸1 + 𝑛2𝐸2 − Δ𝐸𝑒𝑙 𝑖 + 𝐸𝑊𝐿 ℎ 𝑖

chemical bond energyelastic

contribution

Volmer-Weber 3D islandsHigh misfit 6%

Guo et al. Comput. Mater. Sci. 44, 174 (2008)

Thickness dependent interaction 𝐸𝑊𝐿 with the substrate interface, decaying after few

MLs (mimicking 𝛾 = 𝛾 ℎ ) 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

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