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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

Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences

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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

1234567890():,;

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

12σ=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=rrmax

ðÞ½

B=r41

;V3¼Ke ρ1=r12rmax 12

ðÞþρ2=r13rmax13

ðÞ½

cosθcosθ0

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;fRðrÞ¼Aeλ1r;fAðrÞ¼Beλ2r;

(10i-iii)

where fRðrijÞand fAðrij Þare the repulsive and attractive compo-

nents, respectively. The cutoff function is:

fcrðÞ¼

1r<RD

1

21

2sin 1

2πrRðÞ=D

RD<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 Þ1=2n;ζij ¼X

k≠i;j

fCðrik ÞgðθijkÞeλ3

3ðrijrik Þ3

;

gðθÞ¼1þc2

d2c2

d2þðhcos θÞ2:

(12)

a

ij

is an alternative parameter to improve the accuracy of the

potential, with a similar form

aij ¼ð1þαnηn

ij Þ1=2n;

ηij ¼X

k≠i;j

fCðrikÞeλ3

3ðrijrik 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

Aijeαij 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)

StðÞ¼ΘtðÞþΘtðÞΘ1tðÞ0:51þcos πtðÞ½;(18)

tcrij

¼rij rmin

ij

rmax

ij rmin

ij

:(19)

The attractive interaction is expressed as

VA

ij ¼wij 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

½¼Ues q

fg

;r

fg

½þUshort q

fg

;r

fg

½þUvdW q

fg

;r

fg

½þUcorr q

fg

;frg½

(24)

The total potential energy consists of electrostatic energy

Ues q

fg

;r

fg

½ðÞ, 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

½ðÞ

can be

represented as

Ues qfg;rfg½¼Uself qfg;rfg½þUqq qfg;rfg½þUqZ qfg;rfg½þUpolar 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

½ðÞis associated with the charge

density distribution functions. Nuclear-charge interaction

UqZ q

fg

;r

fg

½depends on the coulombic interaction between

charge density distributions in the system. Polar interaction

energy Upolar q

fg

;frg½depends 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

½¼

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

½¼

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

ðfðϕÞλugðϕÞþ1

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

u¼Ωðcc0

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

δϕ¼f0ðϕÞþλug0ðϕÞþ∇WðθÞ2∇ϕ

∂xWðθÞW0ðθÞ∂yϕ

þ∂yWðθÞW0ðθÞ∂xϕðÞ;

(30)

where τ

ϕ

is the characteristic time. The evolution of the saturation

ﬁeld uis governed by

∂u

∂t¼D∇2uþFdu

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.

8

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.

9

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

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.

11