Multiscale computational understanding and growth of 2D materials: a review

Abstract

The successful discovery and isolation of graphene in 2004, and the subsequent synthesis of layered semiconductors and heterostructures beyond graphene have led to the exploding field of two-dimensional (2D) materials that explore their growth, new atomic-scale physics, and potential device applications. This review aims to provide an overview of theoretical, computational, and machine learning methods and tools at multiple length and time scales, and discuss how they can be utilized to assist/guide the design and synthesis of 2D materials beyond graphene. We focus on three methods at different length and time scales as follows: (i) nanoscale atomistic simulations including density functional theory (DFT) calculations and molecular dynamics simulations employing empirical and reactive interatomic potentials; (ii) mesoscale methods such as phase-field method; and (iii) macroscale continuum approaches by coupling thermal and chemical transport equations. We discuss how machine learning can be combined with computation and experiments to understand the correlations between structures and properties of 2D materials, and to guide the discovery of new 2D materials. We will also provide an outlook for the applications of computational approaches to 2D materials synthesis and growth in general.

Full text

REVIEW ARTICLE OPEN
Multiscale computational understanding and growth of 2D
materials: a review
Kasra Momeni
1,2,3
✉, Yanzhou Ji
4,5
, Yuanxi Wang
6,7
, Shiddartha Paul
1
, Sara Neshani
8
, Dundar E. Yilmaz
9
, Yun Kyung Shin
9
,
Difan Zhang
5
, Jin-Wu Jiang
10
, Harold S. Park
11
, Susan Sinnott
5
, Adri van Duin
9
, Vincent Crespi
6,7
and Long-Qing Chen
3,4,12,13
The successful discovery and isolation of graphene in 2004, and the subsequent synthesis of layered semiconductors and
heterostructures beyond graphene have led to the exploding field of two-dimensional (2D) materials that explore their growth, new
atomic-scale physics, and potential device applications. This review aims to provide an overview of theoretical, computational, and
machine learning methods and tools at multiple length and time scales, and discuss how they can be utilized to assist/guide the
design and synthesis of 2D materials beyond graphene. We focus on three methods at different length and time scales as follows:
(i) nanoscale atomistic simulations including density functional theory (DFT) calculations and molecular dynamics simulations
employing empirical and reactive interatomic potentials; (ii) mesoscale methods such as phase-field method; and (iii) macroscale
continuum approaches by coupling thermal and chemical transport equations. We discuss how machine learning can be combined
with computation and experiments to understand the correlations between structures and properties of 2D materials, and to guide
the discovery of new 2D materials. We will also provide an outlook for the applications of computational approaches to 2D
materials synthesis and growth in general.
npj Computational Materials (2020) 6:22 ; https://doi.org/10.1038/s41524-020-0280-2
INTRODUCTION
The perfection and physical properties of atomically thin two-
dimensional (2D) materials are extremely sensitive to their
synthesis and growth process. Achieving desired characteristics
such as structural uniformity, high carrier mobility
1
, strong
light–matter interactions, tunable bandgap, and flexibility is the
main challenge for the synthesis and growth of next generation,
electronics-grade 2D materials. A reliable and optimized growth
and manufacturing process is essential for the synthesis of 2D
materials with uniform properties at the wafer scale, e.g., for
application in flexible and transparent optoelectronics.
Two main approaches have been employed for the synthesis of
2D materials, i.e., (i) top-down approaches such as mechanical
2
and liquid-phase exfoliation that allows scalability
3
, and
(ii) bottom-up approaches such as chemical vapor deposition
(CVD) and atomic layer deposition techniques
4
. The former
approaches are suitable for mass production of 2D materials but
with typically lower quality, whereas the latter approaches can
produce high-quality 2D materials but in small amounts. For both
types of approaches, the morphology and characteristics of the
synthesized 2D materials are very sensitive to the thermodynamic
or kinetic conditions
5
of the growth processes, e.g., heat transfer
and mass transfer of source chemical species, chemical reaction
kinetics, adsorption of reaction product species on a substrate
surface, and nucleation and growth of the resulting 2D materials.
The goal of this review is to provide an overview of the main
theoretical and computational methods for understanding the
thermodynamics and kinetics of mass transport, reaction, and
growth mechanisms during synthesis of 2D materials. We will
discuss the possibility of synthesis-by-design of new 2D materials
guided by computation to reduce the number of expensive and
time-consuming trial-and-error experimentations.
The critical challenge for developing theoretical and computa-
tional design tools for the synthesis of 2D materials is the broad
range of length and temporal scales involved in their growth
process. For example, it may require quantum mechanical and
atomistic reactive force-field calculations to determine the
activation energies for atomic migration on a surface
6
and
understand the atomistic surface reaction mechanisms
7
, and then
afinite element method (FEM) to model the mesoscale mass
transport phenomena
8
. Other challenges include incorporating
the effects of substrates including the types of substrate defects
9
,
the possible wrinkling of 2D films
10
, the effect of van der Waals
(vdW) interactions at the mesoscale
11
, and the growth kinetics
unique to atomically thin materials
12
. Also reproducing quadratic
dispersion for the flexural acoustic modes of 2D materials using
classical or reactive potentials may not be straightforward, it has
been already formulated
13
. Furthermore, a practically useful
multiscale model should be computationally efficient, numerically
accurate, and, more importantly, able to capture the multi-physical
governing relationships among the growth conditions, growth
morphology, and materials properties. The eventual goal of
developing multiscale computational models is to guide the
design of new growth chambers to produce uniform large-area 2D
materials.
1
Mechanical Engineering Department, Louisiana Tech University, Ruston, LA 71272, USA.
2
Department of Mechanical Engineering, University of Alabama, Tuscaloosa, AL, USA.
3
Institute for Micromanufacturing, Louisiana Tech University, Ruston, LA, USA.
4
Materials Research Institute, The Pennsylvania State University, University Park, PA 16802, USA.
5
Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA.
6
2-Dimensional Crystal Consortium, The Pennsylvania State
University, University Park, PA 16802, USA.
7
Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA.
8
Department of Electrical Engineering, Iowa
State University, Ames, IA 50010, USA.
9
Mechanical Engineering Department, The Pennsylvania State University, University Park, PA 16802, USA.
10
Shanghai Institute of Applied
Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China.
11
Department of Mechanical
Engineering, Boston University, Boston, MA 02215, USA.
12
Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA 16802, USA.
13
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA. ✉email: kmomeni@latech.edu
www.nature.com/npjcompumats
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ATOMISTIC COMPUTATIONAL METHODS
First-principles calculations
This section discusses the computational approaches based on the
density functional theory (DFT). The primary application of DFT in
modeling 2D materials is to determine the relative thermody-
namic stabilities of different crystal structures of a 2D material by
computing their chemical potentials or to identify kinetic path-
ways by analyzing the energetics of potential transient structures
from one stable equilibrium structure to another as thermo-
dynamic conditions change. Examples of applications of DFT to
understanding 2D materials and their growth, as well as the
corresponding DFT-based first-principles methodologies and the
corresponding experimental observables for validation are sum-
marized in Table 1.
Thermodynamic stability. The thermodynamic stability of a 2D
material requires its formation energy to be negative. The
formation energies of various 2D materials have been efficiently
calculated by DFT in combination with high-throughput screen-
ing platforms
14
. The success of DFT in predicting energies relies
on the accuracy of well-tempered approximate exchange-
correlation functionals. Standard functionals, e.g., the Perdew-
Burke-Ernzerhof parametrization of the generalized gradient
approximation exchange-correlation functional (GGA-PBE),
accompanied by appropriate corrections
15
, describe well the
formation and atomization energies. However, they are limited
by intrinsic delocalization errors when bonds are stretched,
resulting in underestimated reaction barriers
16
. A more realistic
criterion for a stable 2D material is a low “above-hull”energy
17
.
Exfoliation energies within 0.2 eV/atom are suggested as a
general rule of thumb for the stability in 2D form. A slightly
stricter criterion to have low surface energy (<20 meV/Å) is
suggested to rule out potential high-surface energy but multi-
atom-thick sheets.
Kinetically stabilized 2D Materials. As-grown products may be
metastable structures rather than ground-state polytypes
18
. The
well-known phase-stability competition between the 2H and 1T′
types in transition metal dichalcogenides (TMDs) has been studied
extensively using DFT. Several possible mechanisms for the 1T′-
phase stabilization in the disulfide, including growth-related
factors such as the presence of point defects and residual strain,
have been elucidated through DFT calculations
19
. Phase-stability
analysis for the entire group -IV metal dichalcogenide family
20
revealed that the 1T′phases are generally more stable for
ditellurides. In addition to pure compounds, mixing/segregation
behavior and order–disorder transitions in alloys have also been
discussed
21
. Within the W/Mo +S/Se/Te combinations, mixing
either chalcogens or metals is thermodynamically favorable
22
,
which can be experimentally quantified using the Warren-Cowley
order parameters
23
. For TMDs, theory predicts that at finite
temperatures, the formation of random alloys is favorable.
However, kinetically stabilized atomically thin strips of alternating
W and Mo in a sulfide alloy have been reported
24
.
Interlayer/substrate interactions. Accurate description of vdW
interactions becomes critical when interlayer sheet–sheet or
sheet–substrate interactions are considered, e.g., for identifying
the growth orientation. As a non-local effect, vdW forces are not
correctly described by DFT with semi-local exchange-correlation
functionals and are often corrected by adding pairwise intera-
tomic terms from empirical fittings
25
, based on charge densities
26
,
or by introducing non-local exchange-correlation functionals, i.e.,
van der Waals Density Functional
27
. Thorough testing of these
methods has been examined in ref.
28
. One common technical
problem in this computational approach is that substrates are
modeled in a slab geometry where the one surface not in contact
with the 2D sheet may host surface states, unless it is passivated.
To suppress artificial charge transfer that these may induce,
capping using pseudohydrogen
29
with fractional charge is often
performed. With the advent of new software platforms automat-
ing the generation of solid surfaces in combination with stacked
2D sheets and adsorption geometries (e.g., MPInterfaces
30
), high-
throughput screening of possible substrates may lead to a
systematic approach to substrate engineering.
Precursor chemistry and kinetics. Transient intermediate states are
commonly calculated using transition state theory and the
nudged elastic band (NEB) method
31
by calculating bond
dissociation energies or corresponding activation barriers
32
. One
recently attempted approach to capture precursor reaction
kinetics is constrained molecular dynamics (MD)
33
, where slowly
varying coordination constraints are enforced on reacting species
to map out free energy barriers, as implemented in VASP
34
.Itis
different from NEB calculations in that (1) it calculates free
energies at finite temperatures; (2) the final state of the reaction
need not to be known; and (3) it has a better numerical stability
35
.
Constrained MD was used to study a sulfur precursor, S
2
, reacting
with a MoO
3
surface
36
, concluding that MoO
3
surface vacancies
favor the sulfurization process both kinetically and thermodyna-
mically. The downside of constrained MD is that a reaction
coordinate is chosen a priori, which may bias the system towards
unnatural products with incorrect reaction mechanisms
33
and
higher reaction barriers
34
.
Other methods for sampling of rare events such as nucleation of
a new structure include umbrella sampling, transition interface
sampling, and metadynamics. The umbrella sampling technique
was introduced by Torrie and Valleau
37
, to improve the sampling
of systems with energy landscapes containing high energy
barriers. The weighted histogram analysis method
38
can be used
to analyze a series of umbrella sampling methods. In the transition
interface sampling
39
, the transition region is divided into
subregions of intermediate states. The rate constant of a reaction
in this method is the multiplication of transition probabilities
between different intermediate states. The metadynamics techni-
que was introduced in 2002 and is usually used within an
atomistic modeling framework
40
. Selected collective variables of
the system not only evolve with time but also periodically leave
behind positive Gaussian potentials that are added to the original
Table 1. Examples of applications of DFT to 2D materials, the DFT-based first-principles methodology, and corresponding experimental observables.
Aspect of growth First-principles framework Experimental observable and impact
Thermodynamic stability Thermochemistry, high-throughput screening Successful synthesis
Interlayer interaction Dispersion forces, commensurate supercell construction Orientation control
Precursor chemistry and kinetics Transition state theory/NEB, thermochemistry,
constrained molecular dynamics
Growth rate, residual gas analysis
Growth front advancement Kinetic Monte Carlo, edge energetics Morphology and growth rate, microscopy
image of edge structure
Defects Formation energies Defect population statistics
K. Momeni et al.
2
npj Computational Materials (2020) 22 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences
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potential energy of the system, which effectively push the system
out of a local minimum and into a neighboring energy well.
Informally, metadynamics resembles “filling the free energy wells
with computational sand.”
41
Growth front advancement. Early studies focused on calculating
the surplus energies of edges in the high-symmetry directions and
then applying the thermodynamic Wulff construction
42
.One
difficulty in calculating the edge energies is the polar and non-
centrosymmetric nature of materials, where the usual ribbon
calculation geometry makes the energies of dislike edges insepar-
able. The usual workaround is by constructing triangular flakes
exposing a single type of edge
43
.Another“energy density method”
was introduced in ref.
44
and a recently proposed method
42
aimed
at finding a general method suitable for high-throughput calcula-
tions introduces capping groups to passivate a surface.
One major criticism of further employing thermodynamic Wulff
constructionsisthatgrowthisbydefinition out of equilibrium.
Therefore, edges that dominate over others should not be the
energetically favorable ones, but the slowest growing ones. The
step-flow approach was formulated in ref.
45
for graphene growth
and was further developed in ref.
46
for polar materials, establishing
the use of kinetic Wulff construction. A similar approach involves
DFT calculations and fitting to experimental grain morphologies, to
construct a kinetic Monte Carlo (KMC) model
47
.
Defects. The formation of defects may occur within thermo-
dynamic equilibrium such as thermally generated point defects or
follow from growth imperfections such as dislocations and grain
boundaries (GBs), or reflect the finite size of crystals such as edges
and surfaces. Defects can also be deliberately introduced using
methods such as electron or ion irradiation and chemical
treatment
40
. The calculation of defect formation energies follows
a well-established procedure detailed in ref.
48
. 2D materials also
host lattice-specific defect types, such as the Stone-Wales defect
in graphene
49
. Single-atom vacancies are another type of defects
in 2D materials. A general strategy to ensure correct convergence
behavior for charged defects in 2D systems is presented in ref.
50
.
Defect formation energies in MoS
2
have been comprehensively
studied in ref.
51
; possible strategies of introducing extrinsic
dopants have been examined in ref.
52
. Defect complexes are also
frequently studied to identify likely combinations between simple
intrinsic defects and external contamination
53
, devise possible
defect-pairing strategies to neutralize harmful defects
54
, and
investigate their influence on the growth behavior of 2D materials
on a different 2D sheet
9,55
. Multiple vacancies may exist in 2D
materials such as double vacancies in graphene, resulting in (i)
two pentagons and one octagon—V
2
(5-8-5) defect; or (ii) three
pentagons and three heptagons—V
2
(555-777) defect; or (iii) four
pentagons, a hexagon, and four heptagons—V
2
(5555-6-7777)
defect
56
. The formation of defects with an even number of
vacancies in graphene is energetically favorable due to the lack of
any dangling bonds
57
, whereas a large number of vacancies may
bend and wrap the 2D material
58
. Table 2lists the typical values of
formation and migration energies of these point defects from
atomistic calculations.
Dislocations are line defects in 2D materials that can be of in-
plane edge or out-of-plane screw type. The former consists of
pentagon–heptagon (5–7) pairs
59,60
and the latter makes the
structure to become 3D
61
. In general, dislocations in 2D materials
have high gliding energy barriers and are immobile at room
temperature. They hinder the plastic deformation of 2D materi-
als
62
. The in-plane edge dislocations with the smallest Burgers
vector, b=(1,0), is typically the energetically most favorable
dislocation configuration. The trilayer bi-atomic MoS
2
has a
complex dislocation structure with two types, i.e., Mo-rich that
resembles (5–7) and S-rich that resembles (7–5) pairs. The complex
nature of dislocations in MoS
2
is a source of its rich chemistry. For
example, at specific chemical potentials of S, the (5–7) dislocation
reacts with 2S vacancies to form a (4–6) defect.
GBs in 2D materials can be considered as an array of
dislocations, which can be divided into low-angle and high-
angle GBs depending on the dislocation core density. Atomistic
simulations revealed that the mechanical properties of 2D
materials, similar to the one-dimensional nanostructures, will be
affected by both the density and arrangement of the defects
63
.
Notably, graphene sheets with high-angle tilt GBs can be as strong
as the pristine material and much stronger than those with low-
angle GBs
64
. First‐principles calculations also revealed that sinuous
GB structures are energetically favorable when the straight GB line
cannot bisect the tilt angle
65
. GB energies of different 2D materials
are also reported based on atomistic calculations. For MoS
2
, the
energy of GB with a 20.6° misorientation angle is 0.05 eV/Å
62
. The
GB energy for phosphorene varies between 0.05 and 1.5 eV/Å,
depending on its tilt angle
66
.
Molecular dynamics
The MD approach is one of the main tools that have been used to
study the growth of 2D materials at the atomistic scale. The quality
of potential function governing the atomic interactions is the key
for the accuracy of this technique in predicting the structures and
properties of 2D materials. In general, two classes of interatomic
potentials are used in MD simulations to study the growth of 2D
materials, i.e., empirical and reactive potentials, which we
elaborate on here.
Empirical potentials. Empirical potentials are faster and more
computationally efficient, but less accurate than reactive poten-
tials. Common empirical potentials for modeling 2D materials are
reviewed below.
Lennard-Jones potential: In 1924, Lennard-Jones (LJ) published
the 12–6 pairwise potential to describe the vdW interaction
between two neutral particles (i.e., rare gases)
67
,
VLJ ¼4ϵσ=rðÞ
12σ=rðÞ
6
hi (1)
where ris the distance between two interacting atoms, ϵis the
potential well depth and σis the distance at which the potential is
zero. The r
−12
term captures the strong repulsion occurring when
two atoms get closer. The repulsive effect originates from the Pauli
exclusion principle, which penalizes the overlap of electron
orbitals. The r
−6
term describes the vdW interaction, which is
due to the coupling between the instantaneous polar charges
induced in two neutral molecules and is significantly weaker than
the electron overlap effects.
The popularity of the LJ potential is due to its simplicity and it
has been widely applied to simulate the interlayer interactions in
atomically layered materials. While being used to capture vdW
interactions, a limitation of the pairwise interaction is the inability
to capture frictional processes, such as the sliding of atomic layers.
Table 2. Typical values of formation energies and migration energy
barriers of point defects in 2D materials from atomistic calculations.
Type of 2D
materials
Type of point
defects
Formation
energies
Migration
energy barriers
Graphene Stone-Wales
60
~5 eV
SV
61
~7.5 eV ~1.5 eV
V
2
(5-8-5)
56
~8 eV ~7 eV
V
2
(555-777)
56
~7 eV ~7 eV
V
2
(5555-6-7777)
56
7~8eV ~7eV
Phosphorene SV
195
~1.65 eV ~0.4 eV
MoS
2
V
S59,60
1.22 ~ 2.25 eV ~2.27 eV
K. Momeni et al.
3
Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2020) 22

Different corrections have been proposed to overcome this
shortcoming
68–70
. The two potential parameters ϵand σcan be
determined by using one energy quantity like the binding energy
and a structural quantity like the density. The LJ parameters for
most elements can be determined systematically, as demon-
strated by Rappe et al.
71
. The LJ parameters for typical 2D
materials and substrates in Table 3are listed in in Table 4(for σ)
and Table 5(for ϵ), respectively.
Stillinger-Weber potential: Stillinger and Weber (SW) proposed
this potential in 1985, to model bulk silicon
72
. The SW potential
includes both pair (two-body) and three-body interactions,
V2¼Ae ρ=rrmax
ðÞ½
B=r41

;V3¼Ke ρ1=r12rmax 12
ðÞþρ2=r13rmax13
ðÞ½
cosθcosθ0
ðÞ
2;
(2)
where V
2
describes the pairwise bond-stretching energy and V
3
captures the bending energy associated with an angle with initial
value θ
0
.
The cut-offs r
max
,r
max12
, and r
max13
can be determined by fitting
to the average of the first- and second-neighboring bond lengths.
The seven unknown potential parameters, including five unknown
geometrical parameters, i.e., ρand Bin V
2
and ρ
1
,ρ
2
, and θ
0
in V
3
,
and two energy parameters Aand K, can be determined either by
a least-squares fit to targeted quantities such as the elastic
constants, Poisson’s ratio, and the phonon spectrum, or by
analytical derivations from the linear valence force-field potential.
It is noteworthy that the nonlinear parameter B could not be
determined by the latter approach, which should be determined
by nonlinear quantities such as the third-order elastic constants
73
.
The SW potential can describe nonlinear processes including
large deformations, as higher-order terms for the variation of
bond length and angle are included in its formulation. Although
the SW potential is efficient and suitable for simulating thermal
transport and other nonlinear phenomena
74
, it is not able to
describe chemical bond formation or breaking. It should also be
noted that the SW potential cannot provide bending energy for
atomically layered materials without out-of-plane bonds such as
graphene
75
. Due to its efficiency, the SW potential has been used
in the simulation of many atomically layered materials such as 2D
nanostructures of Si and their thermal properties
76
, MoS
273,74,76
,
MoSe
276
,WS
277
, and black phosphorus
73,78,79
. The SW potential
has been parametrized for 156 emerging atomic layered materials,
along with other available empirical potentials for these atomically
layered materials
80
.
Force-field potential: The force-field (FF) potential describes the
variation in the potential energy of a deformed structure with
respect to its equilibrium configuration. It is essentially a Taylor
expansion of the total energy in terms of the variation in the bond
length and angle,
V¼VbþVθþVϕþVγþVcþVlj þVel;(3)
Vb¼X
i
kbðΔbiÞ2þX
i
kð3Þ
bðΔbiÞ3þX
i
kð4Þ
bðΔbiÞ4;(4)
Vθ¼X
i
kθðΔθiÞ2þX
i
kð3Þ
θðΔθiÞ3þX
i
kð4Þ
θðΔθiÞ4;(5)
Vϕ¼X
i
kϕðΔϕiÞ2;(6)
Vγ¼X
i
kγðΔγiÞ2;(7)
Vc¼X
ij
kbb0ðΔbiÞðΔb0
jÞþX
ij
kθθ0ðΔθiÞðΔθ0
jÞþX
ij
kbθðΔbiÞðΔθjÞ;
(8)
where Δb,Δθ,Δϕ, and Δγare variations of the bond length, bond
angle, twisting angle, and inversion angle, respectively; k
b
,kð3Þ
b,
kð4Þ
b,k
θ
,kð3Þ
θ,kð4Þ
θ,k
ϕ
,k
γ
,k
bb′
,k
θθ′
, and k
bθ
are force constants. V
b
is
the bond-stretching energy, V
θ
is the bond angle-bending energy,
V
ϕ
is the torsional energy for a bond, V
γ
is the inversion energy
among four atoms, and V
c
is the off-diagonal coupling interaction.
It is noteworthy that the cubic and quartic anharmonic terms in
the bond-stretching and angle-bending interactions enable the FF
model to capture nonlinear phenomena such as thermal
expansion and thermal transport.
The FF potential has two significant features: (i) the number of
interaction terms and (ii) the force-constant parameters and the
equilibrium structural parameters (bond lengths and angles).
MM3
81
and COMPASS
82
use all of the interaction terms and can
simulate large deformations and nonlinear phenomena. In
contrast, the Keating model
83
is the simplest model including
only the harmonic terms. Parameters of the potential are usually
fitted to available data and, thus, the volume and accuracy of the
fitted data set determine the accuracy of the FF model. Several
generations of FF models have been developed as more data
become available, including four versions of the molecular
mechanics FF: MM1
84
, MM2
85
, MM3
81
, and MM4
86
. The UFF
71
provides the FF parameters for all the elements in the
periodic table.
As the energies in the FF model are additive, this potential has
been widely used to calculate physical or mechanical properties
for various materials via analytic expressions, including many 2D
materials. For example, an FF model, with bond stretching and
angle-bending terms, has been used to derive the phonon
dispersion of graphene layers
86
. The bond-stretching and angle-
bending terms were utilized to extract the bending properties of
graphene
87
. A mapping between the FF model and the elastic
beam model has been developed, which is used to derive analytic
expressions for Young’s modulus and Poisson’s ratio of graphene
and carbon nanotubes
88
.
Table 3. List of common substrates that have been used to grow 2D
materials.
TMDs Substrate
MoS
2
Sapphire (Al
2
O
3
)
76
, SiO
276
, Graphene
77
, SiC
78
, hBN
79
MoSe
2
Sapphire (Al
2
O
3
)
80
, SiO
281
, Graphene
77
WS
2
SiO
281
, Sapphire (Al
2
O
3
)
81
, hBN
196
, Graphene
197
WSe
2
SiO
2198
, Sapphire (Al
2
O
3
)
199
, hBN
200
NbS
2
SiO
2
/Si
201
, Sapphire (Al
2
O
3
)
202
NbSe
2
h-BN
203
, Graphene
204
MoTe
2
SiO
2205
h-BN hexagonal boron nitride.
Table 4. List of σ(Å) values of the LJ interactions for different
substrates.
Si O C Al B N
Mo 3.27
206
2.93
206
3.054
207
2.570
208
3.827
209
3.725
209
W 0.094
210
1.75
211
3.9
209
2.865
209
3.927
209
3.825
209
S 3.71
206
3.37
206
4.00
212
2.23
209
3.292
209
3.75
212
Se 2.03
213
3.37
209
3.585
207
2.555
209
3.617
209
3.515
209
Te 2.60
213
3.772
209
3.847
209
2.812
209
3.874
209
3.772
209
Nb 3.5855
209
3.1525
209
3.9
209
2.3375
209
3.399
209
3.2975
209
K. Momeni et al.
4
npj Computational Materials (2020) 22 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences

Reactive interatomic potentials. The reactive-potential approach
has been widely used to explore the synthesis and properties
of different materials. The reactive potentials can capture the
bond breaking and formation during the classical simulations
of 2D materials. They are also computationally less expensive
than first-principles methods such as DFT calculations and ab-
initio MD. Typical reactive potentials that have been frequently
considered are Tersoff Bond Order (Tersoff)
89
,Reactive
Empirical Bond Order (REBO)
90
, Adaptive Intermolecular
Reactive Empirical Bond Order (AIREBO)
91
, Charge Optimized
Many-Body (COMB)
92
, and Reactive Force Field (ReaxFF)
93
.
These different reactive potentials are reviewed in detail in
ref.
94
. Performances of several reactive and non-reactive
potentials in graphene-based materials are compared in ref.
95
.
These potentials allow the realization of the thermodynamic
stability of graphene-based materials under different condi-
tions, environmental chemistry, substrate, interlayer interac-
tions, and structural defects. However, only a few of them have
been employed in the study of computational synthesis of
other 2D materials to reveal thermodynamic stability, simulate
the process of the growth of layers, model the top-down
synthesis of 2D materials such as exfoliation techniques
38
,and
evaluate the properties of 2D nanostructures. This is due to
several challenges including the following: (1) the long time
required to develop a reactive potential, (2) the non-
transferability of potential parameters from one material to
another, and (3) the lack of universal functional forms for the
reactive potentials.
The Tersoff potentials are mainly used to explore various
structures and properties of 2D hexagonal boron nitride (h-BN)
materials
96
. The REBO and AIREBO potentials are used to
simulate the CVD process and growth of 2D amorphous carbon
on substrates, mechanical properties of MoS
2
, and even study
the thermal stability of C
60
2D nanostructures
97
.TheCOMB3
potentials have been used to simulate the experimental CVD
deposition and growth of graphene and metal on substrates
98
.
The ReaxFF potentials have been used to explore the structures
of several 2D materials under intercalation, study the defects
and groups on the surfaces of MXene materials, simulate CVD
growth of MoS
2
layers, and explore the synthesis of 2D
polymeric materials in experiments
7,99
. In general, COMB3 and
ReaxFF can further explore 2D materials, as their functional
forms can describe heterogeneoussystems.Itremainscrucialto
find accelerated methods to develop reactive potentials with
high accuracy and transferability for 2D materials.
Tersoff potential: The Tersoff potential was first published in
1986
100
and further developed in 1988
101
. The second version of
the Tersoff potential has the following form,
E¼X
i
Ei¼1
2X
i;j≠i
Vij;(9)
Vij ¼fcðrijÞ½aij fRðrij Þþbij fAðrij Þ;fRðrÞ¼Aeλ1r;fAðrÞ¼Beλ2r;
(10i-iii)
where fRðrijÞand fAðrij Þare the repulsive and attractive compo-
nents, respectively. The cutoff function is:
fcrðÞ¼
1r<RD
1
21
2sin 1
2πrRðÞ=D

RD<r<RþD
0r>RþD
:
8
>
<
>
:
(11)
A distinct feature of the Tersoff potential is the absence of an
explicit three-body interaction term. Instead, the concept of bond
order is introduced, in which the strength of a given bond
depends strongly on its local environment. Bond order is
accounted for via the function b
ij
, which has the following form
bij ¼ð1þβnζn
ij Þ1=2n;ζij ¼X
k≠i;j
fCðrik ÞgðθijkÞeλ3
3ðrijrik Þ3
;
gðθÞ¼1þc2
d2c2
d2þðhcos θÞ2:
(12)
a
ij
is an alternative parameter to improve the accuracy of the
potential, with a similar form
aij ¼ð1þαnηn
ij Þ1=2n;
ηij ¼X
k≠i;j
fCðrikÞeλ3
3ðrijrik Þ3(13)
The 13 unknown parameters are typically determined by fitting
to data from first-principles calculations or experiments.
The Tersoff potential was used for modeling of graphene, which
gives a better description of the phonon spectrum
102
. It was also
used to model germanene
103
and silicene
103
. By introducing one
scaling parameter for the quantity b
ij
with atoms iand jof
different elements
104
, the Tersoff potential was generalized to
describe the interaction for multicomponent systems including
C-Si and Si-Ge
104
, and boron nitride
105
. Using the Tersoff potential,
the thermal conductivity of silicene was calculated to be 61.7 and
68.5 W mK
−1
for the armchair and zigzag configurations, which is
reduced when it is doped with heavier silicon isotopes
106
. This
potential has also been used for modeling the misfit dislocations
in 2D materials, where a critical thickness was found for the
formation of an interface misfit dislocation
107
.
ReaxFF potential: The ReaxFF
93
is a bond-order-based force
field, which allows formation and breaking of bonds. It has a
general form of
Esystem ¼Ebond þEover þEunder þElp þEval þEtor þEvdW þECoulomb;
(14)
where the energy terms on the right-hand side in the order of
appearance are bond, over-coordination penalty, under-
coordination stability, lone pair, valence, torsion, non-bonded
vdW interactions, and Coulomb energies, respectively. The bond-
order parameter determines the bonded interactions between the
Table 5. List of ɛ(eV) values of the LJ interactions for different substrates.
Si O C Al B N
Mo 0.00562
206
0.004
206
0.003325
207
0.00585
208
0.00155
209
0.00208
209
W 0.00835
209
0.006035
209
0.003862
209
0.00478
209
0.00406
209
0.00543
209
S 0.00562
206
0.00884
206
0.00750
212
0.00889
209
0.00755
209
0.007762
212
Se 0.86
213
0.0114
209,214,215
0.00758
207
0.00905
209,214,216
0.00767
209
0.01033
209
Te 1.01
213
0.0114
209,215,217
0.03104
209
0.0327
209
0.00766
209
0.0334
209
Nb 0.05115
209
0.037
209
0.023651
209
0.0293
209
0.0248
209
0.03325
209
K. Momeni et al.
5
Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2020) 22

atoms in the system,
BO0
ij ¼BOσ
ij þBOπ
ij þBOππ
ij
¼exp pbo1rij
rσ
o

pbo2
hi
þexp pbo3rij
rπ
o

pbo4
hi
þexp pbo5rij
rππ
o

pbo6
hi
;(15)
where BOσ
ij ;BOπ
ij ;and BOππ
ij are the partial contributions of
associated σ,π, and ππ bonds between the iand jatoms; r
ij
is
the distance between the iand jatoms; rσ
o,rπ
o, and rππ
oare the
associated bond radii; and p
bo
are empirical constants that will be
determined from ab-initio simulations or experiments. The energy
terms in Eq. (14) are then formulated as a function of the bond-
order parameter (see ref.
93
for details).
ReaxFF studies on 2D materials were initiated from studies of
various aspects of graphene
108
and inspired investigations for a
number of other 2D materials, which are elaborated below.
Graphene—Effect of the environment’s chemistry on properties
and responses of graphene has been investigated using ReaxFF.
For example, effect of oxygen on the mechanical response of
graphene
109
, effect of the addition of fluorine and hydrogen
110
on
the structural quality of graphene, as well as the Ni-catalyzed
growth process of single-layer graphene
111
, are investigated. The
growth mechanism involves atomic deposition and segregation of
C atoms on the Ni substrate, forming dome-like carbon nanotube
cap and polygonal ring patterns via atoms→chains→graphene-
like layer formation (Fig. 1a). The ReaxFF has also been applied to
study reactive events at the graphene–water interface (Fig. 1b)
112
and selective desalination via the bare and hydrogenated
graphene nanopore
113
.
Silicene and phosphorene—Silicene and black phosphorene
(BP) are the silicon and phosphorus analogs of graphene,
respectively. A favorable route for silicene formation has been
proposed by employing the graphene bilayer as a template (Fig.
1c)
114
. It was found that vacancy defects reduce the thermal
stability of silicene, where the critical temperature reduces by
more than 30%
115
. A P/H parameter set has been developed for
monolayer BP
114
to examine the mechanical, thermal, and
chemical stability of both pristine and defective BP.
Molybdenum disulfide—The ReaxFF potential for 2D MoS
2
systems was introduced in ref. 6. This potential could precisely
calculate the formation energy of ripplocations in MoS
2
(Fig.
1d)
116
, which result from the slippage of the upper layer against
the lower layer to relax the strain without breaking covalent
bonds. This potential was also trained over the vacancy formation
energies, diffusion barriers, bending rigidity, and kinetics. It has
Fig. 1 Application of the ReaxFF method to various 2D materials. a Ni-catalyzed growth of single-layer graphene on a Ni(100) surface using
MD/UFMC simulations. [Adapted from ref.
111
with permission from The Royal Society of Chemistry]. bWater-mediated proton transfer
through O/OH-terminated vacancy defects on a graphene layer. [Adapted from ref.
112
under Creative Commons Attribution 4.0 International
License]. cFormation of silicene between graphene bilayers. [Reprinted figure with permission from ref.
114
. Copyright (2019) American
Chemical Society]. dAtomic structures of ripplocations with different numbers of extra units added on top MoS
2
layer. [Adapted with
permission from ref.
116
. Copyright (2019) American Chemical Society]. eHistograms corresponding to averaged interlayer distance without
and with K
+
ions [Reprinted from ref.
121
. Copyright (2019), with permission from Elsevier].
K. Momeni et al.
6
npj Computational Materials (2020) 22 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences

also been applied to design S vacancy defects on MoS
2
surface
117
.
Reactive MD simulations have also been used to model the
synthesis of MoS
2
layers via sulfidation of MoO
3
surfaces
7
. These
simulations identified a three-step reaction pathway as follows: (i)
evolution of O
2
and reduction of MoO
3
surface; (ii) S
2
-assisted
reduction and formation of SO/SO
2
; and (iii) sulfidation of the
surface via Mo-S bond formation. The ReaxFF MD simulations have
also been employed within a multiscale framework to capture
thermal properties of MoS
2118
. The predicted thermal conductivity
of MoS
2
structures with infinite length (37 ± 3 W/mK) falls within
the range of experimental results (34.5–110 W/mK)
119
.
MXenes—MD simulations with ReaxFF of MXenes revealed that
intercalating with potassium cations drastically improves water
stability and homogeneity of MXenes, and reduce the self-
diffusion coefficient of water by two orders of magnitude
120
.
The dynamical response of MXenes with different surface
terminations to intercalating ions was studied with DFT calcula-
tions and ReaxFF MD simulations
121
. Figure 1e compares the
interlayer distance of Ti
3
C
2
(OH)
2
MXene bilayer with and without
K
+
ions. Influence of metal ions intercalation on vibrational
properties of water molecules trapped between MXene layers was
studied using ReaxFF MD simulations
122
. Widening of the
interlayer gap may enable the penetration of molecular reactants
such as urea, which decomposes readily and leads to the
intercalation of ammonium cations
123
. ReaxFF simulations were
performed for defect formation and homoepitaxial growth of TiC
layer on Ti3C2 MXene structures
124
.
REBO/AIREBO potentials: The REBO potential ðEREBO
ij Þis a
combination of attractive ðVA
ij Þand repulsive ðVR
ij Þinteractions
with certain ratio (b
ij
), EREBO
ij ¼VR
ij þbijVA
ij . The bond-order interac-
tion ratio, b
ij
, is dependent on the local coordination in the atomic
environment. The REBO potential can define the conjugation of
bonds using b
ij
, and therefore the potential has been widely used
for modeling hydrocarbon systems
125
. The repulsive term is
expressed by the Brenner equation
90
,
VR
ij ¼wij rij
1þQij
rij

Aijeαij rij :(16)
Here, the Q
ij
,r
ij
, and α
ij
parameters depend on iand j. The bond
weighting parameter w
ij
(r
ij
) depends on switching function S(t) as
wij rij

¼St
crij

;(17)
StðÞ¼ΘtðÞþΘtðÞΘ1tðÞ0:51þcos πtðÞ½;(18)
tcrij

¼rij rmin
ij
rmax
ij rmin
ij
:(19)
The attractive interaction is expressed as
VA
ij ¼wij rij

X
3
n¼1
BðnÞ
ij eBðnÞ
ij rij ;(20)
the bond-order interaction ratio b
ij
is
bij ¼1
2pσπ
ij þpσπ
ji
hi
þπrc
ij þπdh
ij :(21)
Here, pσπ
ij and pσπ
ji are not necessarily equal as they depend on
the penalty function (g
i
) of bond angle θ
jik
between the vector r
ij
,
and vector r
ki
, i.e.,
pσπ
ij ¼1þX
k≠i;j
wik ðrik Þgiðcosθjik Þeλjik þPij
"#
1=2
:(22)
The AIREBO potential is the modification over the REBO
potential, which also considers the torsional ðETORSION
kijl Þand LJ
ðElj
ij Þinteractions. The integration of pairwise interactions in
AIREBO potential can be represented by the following equation
E¼1
2X
iX
j≠i
EREBO
ij þElj
ij þX
k≠i;jX
l≠i;j;k
ETORSION
kijl
"#
:(23)
The REBO potential can describe the breaking, formation, and
hybridization of covalent bonds. The REBO and AIREBO potentials
have been used to study the growth of 2D amorphous carbon,
mechanical properties of MoS
2
, and even thermal stabilities of C
60
2D nanostructures using atomic coordination, internal stress, and
mass density analysis
126
. The subplantation phenomenon during
CVD, i.e., penetration of the substrate bulk carbon by carbon
atoms, can be explained by this REBO potential. REBO potential is
also suitable for modeling and analysis of TMD materials such as
MoS
2
. The potential results in stabilized structure and good
agreement of the structural and mechanical properties
97,127,128
.It
was also used for modeling the synthesis of diamane from
graphene and its stability under different conditions
129
.
COMB3 potential: The general formation of the COMB3
potential can be expressed as
92
Utot q
fg
;r
fg
½¼Ues q
fg
;r
fg
½þUshort q
fg
;r
fg
½þUvdW q
fg
;r
fg
½þUcorr q
fg
;frg½
(24)
The total potential energy consists of electrostatic energy
Ues q
fg
;r
fg
½ðÞ, short-range interactions Ushort q
fg
;r
fg
½

, long-
range vdW interaction UvdW q
fg
;r
fg
½

, and the correction terms.
Here, qand rrepresent the charge and coordinate array of the
elements. The electrostatic energies Ues qfg;rfg
½ðÞ
can be
represented as
Ues qfg;rfg½¼Uself qfg;rfg½þUqq qfg;rfg½þUqZ qfg;rfg½þUpolar qfg;frg½
(25)
The self-interaction energy Uself q
fg
;r
fg
½

consists of field-
effect and ionization energy, or the affinity energy. Charge-to-
charge interaction Uqq q1s
fg
;r
fg
½ðÞis associated with the charge
density distribution functions. Nuclear-charge interaction
UqZ q
fg
;r
fg
½depends on the coulombic interaction between
charge density distributions in the system. Polar interaction
energy Upolar q
fg
;frg½depends on the charge interactions and
dipole distributions in response to the external electric field.
The charge-dependent short-range interaction in Eq. (24)is
associated with bond distance functions of the atom and the
associated charge functions. This energy term can be expressed
by
Ushort½fqg;frg ¼ X
iX
j>i
Vbond
ij ¼X
iX
j>i
fFcðrijÞ½VRðrij ;qi;qjÞ  bij VAðrij;qi;qjÞg
(26)
Here, Vbond
ij is the bond energy that is associated with pairwise
attraction VAðrij;qi;qjÞ

and pairwise repulsion energy
ðVRðrij;qi;qjÞÞ. They exponentially decrease with the interatomic
distance r
ij
.F
c
(r
ij
) is the cutoff function.
The long-range interaction energy is defined by the vdW
interaction, which can be expressed by the LJ interaction
formula
130
Usq
fg
;r
fg
½¼
X
iX
NN
j>i
4εvdW
ij
σvdW
ij
rij
!
12
σvdW
ij
rij
!
6
"#
(27)
K. Momeni et al.
7
Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2020) 22

Here, σvdW
ij and εvdW
ij are the equilibrium distance and interaction
energy, respectively; r
ij
is the cutoff radii where the interaction
assumes zero.
The correction factor in Eq. (24) adjusts the energy penalties
associated with specified angles. It can be calculated by a sixth-
order Legendre polynomial and bond bending term.
Usq
fg
;r
fg
½¼
X
iX
N
j>i X
N
k≠iX
6
n¼1
KLPn
ijk cos θijk

þKBB
ijk cos θijk

cos Kθ
ijk
hi
2
()
(28)
Here, Kθ
ijk indicates the specificbondanglesandθ
ijk
is the
bond angle.
The COMB3 potential has been introduced to study the CVD
process for various 2D materials. The charge-optimized COMB3
potentials have been used to simulate the experimental CVD
deposition and growth of graphene on metal substrates.
Especially, the introduction of dihedral interaction terms in the
COMB3 potential can capture the delocalized bonding and bond
bending during the CVD process on a particular substrate. The
wrinkle formation due to the size mismatch of graphene and
Cu-substrate has been studied using COMB3 potential, which was
comparable to experimental results
98
. This potential was also used
to understand the role of growth parameters such as absorption
energy, migration barrier, and temperature in the Cu deposition
on ZnO substrate
131
. COMB3 potential has also been used to
investigate the effect of surface hydroxylation of amorphous SiO
2
substrate on heat conduction of supported graphene
132
.
MESOSCALE SIMULATIONS
Phase-field model
At mesoscale, a simple and effective model for 2D materials
growth is the classical Burton-Cabrera-Frank (BCF) theory
133
,
where the island growth is realized by the deposition of adatoms
from the supersaturated gas atmosphere and the incorporation of
diffusing atoms at the step edge. The direct numerical imple-
mentation of the BCF model requires tracking the evolving step
edges and applying the boundary conditions due to the sharp-
interface nature of the model; in contrast, phase-field diffuse-
interface models
134
avoid the explicit tracking of the interfaces.
The BCF-based phase-field models have been developed since
the 1990s for a series of problems related to crystal growth,
including collective step motions in a 1-D step train
135
, spiral
surface growth during thin-film epitaxy
136
, and step flow under
different kinetic regimes
137
. More recently, the combined effects
of edge diffusion, the Ehrlich-Schwoebel barrier, deposition, and
desorption for epitaxial growth have been investigated in 2-D
138
.
Based on these studies, Meca et al.
139,140
applied the model to the
growth of 2D materials with anisotropic diffusion on substrates,
using the epitaxial graphene growth on copper foil during CVD as
an example, with experimental validations. The typical BCF-based
phase-field model for 2D materials growth uses a smoothly
varying order parameter ϕto distinguish the phases, e.g., ϕ=1in
the 2D island, ϕ=−1 in the substrate, and −1<ϕ< 1 at step
edges. The free energy functional Fof the system has the general
form of:
F¼ZV
ðfðϕÞλugðϕÞþ1
2WðθÞ2j∇ϕj2ÞdV (29)
where the first term f(ϕ) is a double-well (or multi-well, e.g., see
refs.
136,141
) function with global minima at ϕ=1 and ϕ=−1; the
second term describes the coupling with the reduced saturation
field u, with g(ϕ) an interpolation function and λthe coupling
coefficient; the third term is the gradient energy due to the
inhomogeneous distribution of ϕat interfaces, where W(θ) is the
angle-dependent interface thickness with θ¼arctanðϕy=ϕxÞ.u
can be directly related to the adatom concentration cthrough
u¼Ωðcc0
eqÞ, where c0
eq is the equilibrium concentration for a
flat interface and Ωis the atomic area of the solid phase. The
evolution of ϕis governed by
τϕ
∂ϕ
∂t¼
δF
δϕ¼f0ðϕÞþλug0ðϕÞþ∇WðθÞ2∇ϕ

∂xWðθÞW0ðθÞ∂yϕ

þ∂yWðθÞW0ðθÞ∂xϕðÞ;
(30)
where τ
ϕ
is the characteristic time. The evolution of the saturation
field uis governed by
∂u
∂t¼D∇2uþFdu
τs
1
2
∂ϕ
∂t(31)
where Dis the diffusion coefficient, F
d
is the effective deposition
rate, and τ
s
is the characteristic time scale for atom desorption.
The phase-field equations (Eqs. (30) and (31)) can recover the
sharp-interface BCF model in the limit of W(θ)→0
140
.
The applications of the phase-field models in 2D materials
growth can be categorized into the following three aspects. (1)
Reproducing and explaining experimentally observed growth
morphologies. Examples include the reconstruction of the spiral
growth of SnSe
2141
, the verification of impurity-induced bilayered
graphene growth
142
, and the explanation to island shape change
from quasi-hexagons to triangles and dendrite-like morpholo-
gies
143
. (2) Investigating the effect of experimental parameters on
growth morphologies. Taking the CVD growth of graphene as an
example, the effects of flux of carbon species
144
, deposition
rates
145
, and equilibrium saturation
145
, and substrate orienta-
tion
139
on graphene island morphologies have been investigated
using phase-field simulations, which could provide useful
guidance to the experimental control of the growth qualities. (3)
Further development of the model to include the effect of
additional physical/chemical processes during growth. In addition
to the deposition, edge diffusion, and edge anisotropies, phase-
field simulations have been extended to model multi-island
interactions
146
, tilt GB topology of 2D materials on substrates with
topological cones
147
, and to include the chemical reaction kinetics
in the gas phase using a microkinetic model
148
. These works
improve the fidelity and applicable range of the phase-field model
for 2D materials growth.
Nevertheless, a primary obstacle to applying phase-field models
for computational synthesis of 2D materials is the lack of
connection between the model parameters and the experimen-
tally controllable CVD processing parameters, which is largely due
to the difference in size scales of interest. Most of the existing
phase-field simulations only focus on the growth of a few 2D
islands (up to micrometer scale), which can hardly represent the
effect of the macroscopic CVD parameters. One possible solution
is to enable the “multiscale”modeling approach
8
, integrating the
FEM (see the section on “Macroscale models”) to account for the
effect of macroscopic CVD parameters on transport phenomena
and the mesoscale phase-field model for 2D crystal growth within
selected zones of the whole substrate. In particular, the steady-
state velocity, concentration, and temperature profiles from FEM
can be incorporated in the phase-field model, as it has been
proposed and applied by Momeni et al.
8
(see Fig. 2). The
incorporation of chemical reaction kinetics in the phase-field
model can improve the accuracy of the predictions and expand its
applicability. Phase-field models have also been applied to phase
changes in 2D materials. Notable examples include the multi-
domain microstructure during H-T’structure transformation in
MoTe
2
and the domain switching behaviors under external
stimuli
149
. A fast Fourier transform algorithm has been imple-
mented to study the domain switching during bending of MoTe
2
(see Fig. 3
149
).
K. Momeni et al.
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npj Computational Materials (2020) 22 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences

Phase-field-crystal model
Different from the continuous mesoscale phase-field approach,
the phase-field-crystal (PFC) approach
150
describes the thermo-
dynamics and dynamics of phase transformations through an
atomically varying order parameter related to the atomic density
field. With the application of the classical dynamic DFT, PFC can
capture the atomistic-scale morphology and evolution dynamics,
within diffusional time scales, which is usually challenging for
conventional atomistic methods. Recently, PFC has been actively
applied to investigate the defect formation and the atomic
structure of interfaces, GBs, and triple junctions during the growth
of 2D materials
151
. The unique GB structure and collective domain
dynamics in binary 2D materials, e.g., h-BN, was investigated
(see Fig. 4
152
).
Compared with the continuous phase-field approach, the
advantage of PFC lies in resolving atomistic-scale interface
structures and growth dynamics of 2D islands; however, due to
the limitation of simulation size, interface dynamics under the
effect of realistic CVD parameters are difficult to realize. Future
directions of PFC simulations for 2D materials growth include
the consideration of atomistic-scale island-substrate interactions
and the integration with the larger-scale phase-field or FEM
simulations.
Kinetic Monte Carlo
KMC is another mesoscale technique for predicting the growth
morphology and kinetic mechanisms of 2D materials, where all
possible kinetic events are listed in an event catalog, and their
stochastic sequence are randomly selected based on their
activation energies
153
. Activation energies of the related kinetic
processes are usually obtained from atomistic-scale calculations.
KMC identifies the governing kinetic pathways and simulates the
kinetic process. Details of its implementation can be found in
ref.
154
.
KMC has been extensively applied to simulate the growth of
graphene, e.g., the evolution of vacancy complexes and formation
of vacancy chains
155
, the formation and kinetic effect of multi-
member carbon ring complexes
156
, etching of graphene GBs due to
oxygen migration and reaction
157
, and GB evolution following the
Stone-Wales mechanism
158
. Growth of nanoscale graphene islands
Fig. 2 A coupled macroscale/phase-field simulation and experimental comparison for MoS
2
.aInitial precursor concentration distribution
from FEM. bPhase-field simulation results for MoS
2
morphology on substrate at t=0.25 h, the enlarged images of i–iii boxes are shown in d–f.
cExperimental results for deposition of MoS
2
after deposition for t=0.25 h, the enlarged images of 1–3 boxes are shown in g,h. Reprinted
from ref. 8under Creative Commons Attribution 4.0 International License. The scale bar in a,bshows 400 μm and the one in d–fshows
400 μm.
K. Momeni et al.
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Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2020) 22

Fig. 4 PFC simulations of the grain growth and coalescence in h-BN. a–fGrain coalescence and inversion domain dynamics. cAtomic site
number Nvs. t, showing two regimes N=−0.065t+3.045 × 10
4
and N=−0.085t+3.612 × 10
4
via fitting. d,eDomain shrinking in the white
boxed region of b.fTransient of three merging heart-shaped defects before their annihilation. g–iTime evolution of collective atomic
displacements in the yellow boxed corner of d. Adapted figure with permission from ref.
152
. Copyright (2017) by the American Physical Society.
Fig. 3 Variant selection and transformation morphologies in MoTe
2
under an applied in-plane strain εii .aPhase diagram in εxx ;εyy

space.
Colored (hatched) regions correspond to one (two) phase regions. Solid lines (analytic) and filled data points (simulations) indicate the strain at
which H and a given T′variant have equal energies. Dashed lines and open symbols similarly represent two-phase coexistence bounds. (▲,△)
and (●,○) correspond to simulations with and without bending, respectively, and blue circles (blue solid circle and blue open circle) represent
DFT results at 0. b–dImages of the microstructures with increasing time/strain from left to right for strain paths b–din phase diagram a. Strain
values εxx;εyy

are given below each panel. eCorresponding maximum principal stress (GPa) maps for microstructures in d. All simulations are
for 50 μm×50μm simulations. Adapted with permission from ref.
149
. Copyright (2017) American Chemical Society.
K. Momeni et al.
10
npj Computational Materials (2020) 22 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences

from carbon monomer nuclei or pre-existing growth fronts has
been captured, showing catalytic growth behaviors
159
, anisotropic
morphological patterns
160
, temperature- and deposition-flux-rate-
dependent size and shapes
161
, inhomogeneous and nonlinear
growth kinetics due to lattice mismatch
162
, and geometry-
determined growth mechanisms
163
.The“coastline”graphene
morphology during sublimation
164
and the step-flow growth of
epitaxialgraphenehavealsobeenreported
165
. Based on KMC
simulations for graphene growth on Cu (111) with and without
hydrogen, the growth protocol was designed for bilayer growth
and N-doped graphene growth
166
.
Regarding TMDs, the KMC simulations for the growth of WSe
2
monolayer on graphene have been used to develop the phase
diagram of domain morphologies as a function of flux and
precursor stoichiometry (Fig. 5)
154,167
. It was found that the fast
kink nucleation and propagation, rather than edge attachment and
diffusion, could lead to ultrafast growth of monolayer WSe
2168
.KMC
simulations also guided CVD growth of large-scale WSe
2
grains by
controlling the three-stage adsorption–diffusion–attachment
mechanisms
169
. KMC model is also a key component in a more
generalized mechanistic model for growth morphology predictions
of 2D materials
47
.
In summary, the KMC model can be a useful tool for
investigating the kinetic pathways and morphologies during the
growth of 2D materials. However, the probability-based nature of
KMC makes it most suitable for cases where atomic fluctuations
are high, i.e., the atomistic and nanoscale morphology and kinetics
of 2D materials. For larger-scale simulations, due to the
significantly increased system size and the disparate rates of
different KMC events, a full-KMC is computationally expensive. As
a compromise, multiscale KMC has been developed
163
; simplifica-
tions should also be made to account for the key events that are
most relevant to the large-scale kinetics
163
. Under such situations,
the phase-field approach could be a more efficient option.
Meanwhile, the KMC model can provide the governing kinetic
mechanisms for phase-field simulations to improve the validity
and accuracy of the simulation.
MACROSCALE MODELS
Although the growth of 2D materials occurs at the nano- and
mesoscale, it is controlled by physics and parameters that have a
macroscopic nature, e.g., heat and mass transfer, furnace
configuration, and gas-phase reactions. Thus, having a thorough
understanding of the macroscale physics and processes is
essential for controlling the growth of 2D materials and their
synthesis by design. We can classify the macroscale models of the
growth chamber into four groups: (i) experiment-based models,
where rate equations and their constants used to describe the
growth
170
are determined from experiments; (ii) analytical models,
where the governing equations are simplified and solved
analytically
171
; (iii) adaptive models, where a set of experiments
are used to train the model
172
; and (iv) multiphysics models,
where the coupled system of governing equations at different
length and temporal scales are solved numerically
173
. Among
these methods, the last group of models have key advantages,
providing a profound understanding to the growth process,
flexibility to apply to different growth conditions, and the ability to
optimize the process.
A practical macroscale model of the growth chamber should
capture the critical governing physics, e.g., the heat and mass
transports and chemical reactions. Setting up these models
requires several key information that can be obtained from lower
scale simulations
7
or experiments
174
. Identifying the gas-phase
reactions, we may approximate some of the reaction parameters
using classical theories, e.g., the collision rate to estimate the
chemisorption rate of species
175
or the group contribution
methods to determine the diffusion coefficient
176
. The other
approximation is for low concentrations of reactive species, where
the change in pressure and heat of reaction can be neglected. In
the latter case, we may decouple the fluid and heat transfer of
gaseous materials from the mass transfer and kinetics. In contrast,
for high concentrations of reactive species in the gas phase, such
as in metalorganic chemical vapor deposition (MOCVD), the
coupled system of equations must be solved
176
. A summary of
main equations in macroscale models are presented below.
Gas flow
The Navier-Stokes equation governs the flow rates in the growth
chamber,
ρ∂u
∂tþρu∇ðÞu¼∇pIþμ∇uþ∇uðÞ
T
hi
þF;ρ∇u¼0;
(32)
where uis the velocity field, pis pressure, Iis the unit matrix, μis
the dynamic viscosity, and Fis the volumetric applied force, i.e.,
Fig. 5 KMC simulations of WSe
2
growth. a Reaction energy diagram of the growth process based on DFT calculations
167
. The simulation
starts with adatoms. They react with each other to form the TMD domains. After bonded into the domain, the in-flake atoms are mobile
through edge diffusion and vacancy diffusion. Each state is defined by its state energy, and each process is defined by its transition activation
energy. Adapted from ref.
154
under Creative Commons Attribution 4.0 International License. bDomain morphology diagram on the metal
flux-C/M ratio plane at 973 K from KMC simulations. Five regions are identified: I: no growth, II: quasi-equilibrium compact domain, III: fractal,
IV: dendrite, and V: semi-compact domain. The arrows show a proposed scenario to obtain high-quality domains with high growth rate.
Adapted from ref.
167
.
K. Momeni et al.
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weight. Furthermore, the density ρis a function of temperature
and precursor concentration. In general, the concentration of
precursor in the gas phase is low and ρcan be assumed to be only
a function of temperature.
Heat transfer
The two main heat transfer mechanisms in the growth chamber
are convection and conduction that respectively are