Open access peer-reviewed chapter

Parameterization of Stillinger-Weber Potential for Two- Dimensional Atomic Crystals

Written By

Jin-Wu Jiang and Yu-Ping Zhou

Reviewed: 25 October 2017 Published: 20 December 2017

DOI: 10.5772/intechopen.71929

From the Monograph

Handbook of Stillinger-Weber Potential Parameters for Two-Dimensional Atomic Crystals

Authored by Jin-Wu Jiang and Yu-Ping Zhou

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Abstract

We parametrize the Stillinger-Weber potential for 156 two-dimensional atomic crystals (TDACs). Parameters for the Stillinger-Weber potential are obtained from the valence force field (VFF) model following the analytic approach (Nanotechnology. 2015;26:315706), in which the valence force constants are determined by the phonon spectrum. The Stillinger-Weber potential is an efficient nonlinear interaction and is applicable for numerical simulations of nonlinear physical or mechanical processes. The supplemental resources for all simulations in the present work are available online in http://jiangjinwu.org/sw, including a Fortran code to generate crystals’ structures, files for molecular dynamics simulations using LAMMPS, files for phonon calculations with the Stillinger-Weber potential using GULP, and files for phonon calculations with the valence force field model using GULP.

Keywords

  • layered crystal
  • Stillinger-Weber potential
  • molecular dynamics simulation
  • empirical potential
  • PACS: 78.20.Bh
  • 63.22.-m
  • 62.25.-g

1. Introduction

The atomic interaction is of essential importance in the numerical investigation of most physical or mechanical processes [1]. The present work provides parameters for the Stillinger-Weber (SW) empirical potential for 156 two-dimensional atomic crystals (TDACs). In practical applications, these layered materials are usually played as Lego on atomic scale to construct the van der Waals heterostructures with comprehensive properties [2]. The computational cost of ab initio for the heterostructure will be substantially increased as compared with one individual atomic layer, because the unit cell for the heterostructure is typically very large resulting from the mismatch of the lattice constants of different layered components. The empirical potential will be a competitive alternative to help out this difficult situation, considering their high efficiency.

In the early stage before 1980s, the computation ability of the scientific community was quite limited. At that time, the valence force field (VFF) model was one popular empirical potential for the description of the atomic interaction, since the VFF model is linear and can be applied in the analytic derivation of most elastic quantities [3]. In this model, each VFF term corresponds to a particular motion style in the crystal. Hence, each parameter in the VFF model usually has clear physical essence, which is beneficial for the parameterization of this model. For instance, the bond stretching term in the VFF model is directly related to the frequency of the longitudinal optical phonon modes, so the force constant of the bond stretching term can be determined from the frequencies of the longitudinal optical phonon modes. The VFF model can thus serve as the starting point for developing atomic empirical potentials for different crystals.

While the VFF model is beneficial for the fastest numerical simulation, its strong limitation is the absence of nonlinear effect. Due to this limitation, the VFF model is not applicable to nonlinear phenomena, for which other potential models with nonlinear components are required. Some representative potential models are (in the order of their simulation costs) SW potential [4], Tersoff potential [5], Brenner potential [6], ab initio approaches, etc. The SW potential is one of the simplest potential forms with nonlinear effects included. An advanced feature for the SW potential is that it includes the nonlinear effect, and keeps the numerical simulation at a very fast level.

Considering its distinct advantages, the present article aims at providing the SW potential for 156 TDACs. We will determine parameters for the SW potential from the VFF model, following the analytic approach proposed by one of the present authors (JWJ) [7]. The VFF constants are fitted to the phonon spectrum or the elastic properties in the TDACs.

In this paper, we parametrize the SW potential for 156 TDACs. All structures discussed in the present work are listed in Tables 1 9 . The supplemental materials are freely available online in [1], including a Fortran code to generate crystals’ structures, files for molecular dynamics simulations using LAMMPS, files for phonon calculations with the SW potential using GULP, and files for phonon calculations with the valence force field model using GULP.

1H-ScO2 1H-ScS2 1H-ScSe2 1H-ScTe2 1H-TiTe2 1H-VO2 1H-VS2 1H-VSe2 1H-VTe2
1H-CrO2 1H-CrS2 1H-CrSe2 1H-CrTe2 1H-MnO2 1H-FeO2 1H-FeS2 1H-FeSe2 1H-FeTe2
1H-CoTe2 1H-NiS2 1H-NiSe2 1H-NiTe2 1H-NbS2 1H-NbSe2 1H-MoO2 1H-MoS2 1H-MoSe2
1H-MoTe2 1H-TaS2 1H-TaSe2 1H-WO2 1H-WS2 1H-WSe2 1H-WTe2

Table 1.

1H-MX2, with M as the transition metal and X as oxygen or dichalcogenide.

The structure is shown in Figure 1 .

1T-ScO2 1T-ScS2 1T-ScSe2 1T-ScTe2 1T-TiS2 1T-TiSe2 1T-TiTe2 1T-VS2 1T-VSe2
1T-VTe2 1T-MnO2 1T-MnS2 1T-MnSe2 1T-MnTe2 1T-CoTe2 1T-NiO2 1T-NiS2 1T-NiSe2
1T-NiTe2 1T-ZrS2 1T-ZrSe2 1T-ZrTe2 1T-NbS2 1T-NbSe2 1T-NbTe2 1T-MoS2 1T-MoSe2
1T-MoTe2 1T-TcS2 1T-TcSe2 1T-TcTe2 1T-RhTe2 1T-PdS2 1T-PdSe2 1T-PdTe2 1T-SnS2
1T-SnSe2 1T-HfS2 1T-HfSe2 1T-HfTe2 1T-TaS2 1T-TaSe2 1T-TaTe2 1T-WS2 1T-WSe2
1T-WTe2 1T-ReS2 1T-ReSe2 1T-ReTe2 1T-IrTe2 1T-PtS2 1T-PtSe2 1T-PtTe2

Table 2.

1T-MX2, with M as the transition metal and X as oxygen or dichalcogenide.

The structure is shown in Figure 71 .

Black phosphorus p-Arsenene p-Antimonene p-Bismuthene

Table 3.

Puckered (p-) M, with M from group V.

The structure is shown in Figures 178 or 183 .

p-SiO p-GeO p-SnO
p-CS p-SiS p-GeS p-SnS
p-CSe p-SiSe p-GeSe p-SnSe
p-CTe p-SiTe p-GeTe p-SnTe

Table 4.

Puckered MX, with M from group IV and X from group VI.

The structure is shown in Figure 189 , and particularly Figure 191 for p-MX with X = O.

Silicene Germanene Stanene Indiene
Blue phosphorus b-Arsenene b-Antimonene b-Bismuthene

Table 5.

Buckled (b-) M, with M from group IV or V.

The structure is shown in Figure 222 .

b-CO b-SiO b-GeO b-SnO
b-CS b-SiS b-GeS b-SnS
b-CSe b-SiSe b-GeSe b-SnSe
b-CTe b-SiTe b-GeTe b-SnTe

Table 6.

Buckled MX, with M from group IV and X from group VI.

The structure is shown in Figure 239 .

b-SnGe b-SiGe b-SnSi b-InP b-InAs b-InSb b-GaAs b-GaP b-AlSb

Table 7.

Buckled MX, with both M and X from group IV or M from group III and X from group V.

The structure is shown in Figure 239 .

BO AlO GaO InO
BS AlS GaS InS
BSe AlSe GaSe InSe
BTe AlTe GaTe InTe

Table 8.

Bi-buckled MX, with M from group III and X from group VI.

The structure is shown in Figure 290 .

Borophene

Table 9.

The structure is shown in Figure 323 .

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2. VFF model and SW potential

2.1. VFF model

The VFF model is one of the most widely used linear models for the description of atomic interactions [3]. The bond stretching and the angle bending are two typical motion styles for most covalent bonding materials. The bond stretching describes the energy variation for a bond due to a bond variation Δ r = r r 0 , with r 0 as the initial bond length. The angle bending gives the energy increment for an angle resulting from an angle variation Δ θ = θ θ 0 , with θ 0 as the initial angle. In the VFF model, the energy variations for the bond stretching and the angle bending are described by the following quadratic forms,

V r = 1 2 K r ( Δ r ) 2 , E1
V θ = 1 2 K θ ( Δ θ ) 2 , E2

where K r and K θ are two force constant parameters. These two potential expressions in Eqs. (1) and (2) are directly related to the optical phonon modes in the crystal. Hence, their force constant parameters K r and K θ are usually determined by fitting to the phonon dispersion.

2.2. SW potential

In the SW potential, energy increments for the bond stretching and angle bending are described by the following two-body and three-body forms,

V 2 ( r i j ) = A ( B / r i j 1 ) e [ ρ / ( r i j r m a x ) ] , E3
V 3 ( θ i j k ) = K e [ ρ 1 / ( r i j r m a x 12 ) + ρ 2 / ( r i k r m a x 13 ) ] ( cos θ i j k c o s θ 0 ) 2 E4

where V 2 corresponds to the bond stretching and V 3 associates with the angle bending. The cut-offs r max , r max12, and r max13 are geometrically determined by the material’s structure. There are five unknown geometrical parameters, i.e., ρ and B in the two-body V 2 term and ρ 1, ρ 2, and θ 0 in the three-body V 3 term, and two energy parameters A and K. There is a constraint among these parameters due to the equilibrium condition [7],

ρ = 4 B ( d r m a x ) 2 ( B d d 5 ) , E5

where d is the equilibrium bond length from experiments. Eq. (5) ensures that the bond has an equilibrium length d and the V 2 interaction for this bond is at the energy minimum state at the equilibrium configuration.

The energy parameters A and K in the SW potential can be analytically derived from the VFF model as follows,

A = K r α e [ ρ / ( d r m a x ) ] , E6
K = K θ 2 sin 2 θ 0 e [ ρ 1 / ( d 1 r m a x 12 ) + ρ 2 / ( d 2 r m a x 13 ) ] , E7

where the coefficient α in Eq. (6) is,

α = [ ρ ( d r m a x ) 2 ] 2 ( B / d 4 1 ) + [ 2 ρ ( d r m a x ) 3 ] ( B / d 4 1 ) + [ ρ ( d r m a x ) 2 ] ( 8 B d 5 ) + ( 20 B d 6 ) . E8

In some situations, the SW potential is also written into the following form,

V 2 ( r i j ) = ϵ A L ( B L σ p r i j p σ q r i j q ) e [ σ / ( r i j a σ ) ] , E9
V 3 ( θ i j k ) = ϵ λ e [ γ σ / ( r i j a σ ) + γ σ / ( r j k a σ ) ] ( cos θ i j k cos θ 0 ) 2 . E10

The parameters here can be determined by comparing the SW potential forms in Eqs. (9) and (10) with Eqs. (3) and (4). It is obvious that p = 4 and q = 0. Eqs. (9) and (10) have two more parameters than Eqs. (3) and (4), so we can set ϵ = 1.0 eV and γ = 1.0. The other parameters in Eqs. (9) and (10) are related to these parameters in Eqs. (3) and (4) by the following equations

A L = A , E11
σ = ρ , E12
B L = B / ρ 4 , E13
a = r m a x / ρ , E14
λ = K . E15
VFF type Bond stretching Angle bending
Expression 1 2 K Sc O ( Δ r ) 2 1 2 K Sc O O ( Δ θ ) 2 1 2 K Sc O O ( Δ θ ) 2 1 2 K O Sc Sc ( Δ θ ) 2
Parameter 9.417 4.825 4.825 4.825
r 0 or θ 0 2.090 98.222 58.398 98.222

Table 10.

The VFF model for single-layer 1H-ScO2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Å2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ ijk has atom i as the apex.

A (eV) ρ (Å) B4) r min (Å) r max(Å)
Sc-O 7.506 1.380 9.540 0.0 2.939

Table 11.

Two-body SW potential parameters for single-layer 1H-ScO2 used by GULP [8] as expressed in Eq. (3).

The SW potential is implemented in GULP using Eqs. (3) and (4). The SW potential is implemented in LAMMPS using Eqs. (9) and (10).

In the rest of this article, we will develop the VFF model and the SW potential for layered crystals. The VFF model will be developed by fitting to the phonon dispersion from experiments or first-principles calculations. The SW potential will be developed following the above analytic parameterization approach. In this work, GULP [8] is used for the calculation of phonon dispersion and the fitting process, while LAMMPS [9] is used for molecular dynamics simulations. The OVITO [10] and XCRYSDEN [11] packages are used for visualization. All simulation scripts for GULP and LAMMPS are available online in [1].

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3. 1H-SCO2

Most existing theoretical studies on the single-layer 1H-ScO2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-ScO2.

Figure 1.

Configuration of the 1H-MX2 in the 1H phase. (a) Top view. The unit cell is highlighted by a red parallelogram. (b) Enlarged view of atoms in the blue box in (a). Each M atom is surrounded by six X atoms, which are categorized into the top and bottom groups. Atoms X 1, 3, and 5 are from the top group, while atoms X 2, 4, and 6 are from the bottom group. M atoms are represented by larger gray balls. X atoms are represented by smaller yellow balls.

The structure for the single-layer 1H-ScO2 is shown in Figure 1 (with M = Sc and X = O). Each Sc atom is surrounded by six O atoms. These O atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each O atom is connected to three Sc atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant a = 3.16 Å and the bond length d Sc O = 2.09 Å. The resultant angles are θ ScOO = θ OScSc = 98.222 ° and θ ScO O = 58.398 ° , in which atoms O and O′ are from different (top or bottom) groups.

Figure 2.

Phonon spectrum for single-layer 1H-ScO2. (a) Phonon dispersion along the direction ΓM in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Sc O O 63.576 98.222 1.380 1.380 0.0 2.939 0.0 2.939 0.0 3.460
θ Sc O O 85.850 58.398 1.380 1.380 0.0 2.939 0.0 2.939 0.0 3.460
θ O Sc Sc 63.576 98.222 1.380 1.380 0.0 2.939 0.0 2.939 0.0 3.460

Table 12.

Three-body SW potential parameters for single-layer 1H-ScO2 used by GULP [8] as expressed in Eq. (4).

The angle θ ijk in the first line indicates the bending energy for the angle with atom i as the apex.

Table 10 shows four VFF terms for the single-layer 1H-ScO2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along ГM as shown in Figure 2(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 2(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 11 . The parameters for the three-body SW potential used by GULP are shown in Table 12 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 13 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-ScO2 using LAMMPS, because the angles around atom Sc in Figure 1 (with M = Sc and X = O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 3 (with M = Sc and X = O) shows that, for 1H-ScO2, we can differentiate these angles around the Sc atom by assigning these six neighboring O atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Sc atom.

Figure 3.

Twelve atom types are introduced to distinguish angles around each M atom for the single-layer 1H-MX2. Atoms X1, X3, X5, and X7 are from the top layer. The other four atoms X2, X4, X6, and X8 are from the bottom layer, which are not displayed in the figure.

Figure 4.

Stress-strain for single-layer 1H-ScO2 of dimension 100 × 100 Å along the armchair and zigzag directions.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Sc1─O1─O1 1.000 1.380 2.129 0.000 1.000 0.000 7.506 2.627 4 0 0.0
Sc1─O1─O3 1.000 0.000 0.000 63.576 1.000 −0.143 0.000 0.000 4 0 0.0
Sc1─O1─O2 1.000 0.000 0.000 85.850 1.000 0.524 0.000 0.000 4 0 0.0
O1─Sc1─Sc3 1.000 0.000 0.000 63.576 1.000 −0.143 0.000 0.000 4 0 0.0

Table 13.

SW potential parameters for single-layer 1H-ScO2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-ScO2 under uniaxial tension at 1 and 300 K. Figure 4 shows the stress-strain curve for the tension of a single-layer 1H-ScO2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-ScO2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-ScO2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 126.3 and 125.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.16 .

VFF type Bond stretching Angle bending
Expression 1 2 K Sc S ( Δ r ) 2 1 2 K Sc S S ( Δ θ ) 2 1 2 K Sc S S ( Δ θ ) 2 1 2 K S Sc Sc ( Δ θ ) 2
Parameter 5.192 2.027 2.027 2.027
r 0 or θ 0 2.520 94.467 64.076 94.467

Table 14.

The VFF model for single-layer 1H-ScS2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of ev/Å2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ ijk has atom i as the apex.

A (eV) ρ (Å) B4) r min (Å) r max (Å)
Sc─S 5.505 1.519 20.164 0.0 3.498

Table 15.

Two-body SW potential parameters for single-layer 1H-ScS2 used by GULP [8] as expressed in Eq. (3).

There is no available value for nonlinear quantities in the single-layer 1H-ScO2. We have thus used the nonlinear parameter B = 0.5d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −652.8 and −683.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 12.2 N/m at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.7 N/m at the ultimate strain of 0.23 in the zigzag direction at the low temperature of 1 K.

Figure 5.

Phonon spectrum for single-layer 1H-ScS2. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

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4. 1H-SCS2

Most existing theoretical studies on the single-layer 1H-ScS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-ScS2.

The structure for the single-layer 1H-ScS2 is shown in Figure 1 (with M = Sc and X = S). Each Sc atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Sc atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant a = 3.70 Å and the bond length d Sc S = 2.52 Å. The resultant angles are θ ScSS = θ SScSc = 94.467 and θ ScS S = 64.076 , in which atoms S and S′ are from different (top or bottom) groups.

Figure 6.

Stress-strain for single-layer 1H-ScS2 of dimension 100 × 100 Å along the armchair and zigzag directions.

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Sc S S 22.768 94.467 1.519 1.519 0.0 3.498 0.0 3.498 0.0 4.132
θ Sc S S 27.977 64.076 1.519 1.519 0.0 3.498 0.0 3.498 0.0 4.132
θ S Sc Sc 22.768 94.467 1.519 1.519 0.0 3.498 0.0 3.498 0.0 4.132

Table 16.

Three-body SW potential parameters for single-layer 1H-ScS2 used by GULP [8] as expressed in Eq. (4).

The angle θ ijk in the first line indicates the bending energy for the angle with atom i as the apex.

Table 14 shows four VFF terms for the single-layer 1H-ScS2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 5(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 5(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 15 . The parameters for the three-body SW potential used by GULP are shown in Table 16 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 17 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-ScS2 using LAMMPS, because the angles around atom Sc in Figure 1 (with M = Sc and X = S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Sc and X = S) shows that, for 1H-ScS2, we can differentiate these angles around the Sc atom by assigning these six neighboring S atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Sc atom.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Sc1─S1─S1 1.000 1.519 2.303 0.000 1.000 0.000 5.505 3.784 4 0 0.0
Sc1─S1─S3 1.000 0.000 0.000 22.768 1.000 −0.078 0.000 0.000 4 0 0.0
Sc1─S1─S2 1.000 0.000 0.000 27.977 1.000 0.437 0.000 0.000 4 0 0.0
S1─Sc1─Sc3 1.000 0.000 0.000 22.768 1.000 −0.078 0.000 0.000 4 0 0.0

Table 17.

SW potential parameters for single-layer 1H-ScS2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-ScS2 under uniaxial tension at 1 and 300 K. Figure 6 shows the stress-strain curve for the tension of a single-layer 1H-ScS2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-ScS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-ScS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 43.8 and 43.2 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.30 .

There is no available value for nonlinear quantities in the single-layer 1H-ScS2. We have thus used the nonlinear parameter B = 0.5d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −146.9 and −159.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.1 N/m at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.9 N/m at the ultimate strain of 0.32 in the zigzag direction at the low temperature of 1 K.

VFF type Bond stretching Angle bending
Expression 1 2 K Sc Se ( Δ r ) 2 1 2 K Sc Se Se ( Δ θ ) 2 1 2 K Sc Se S e ( Δ θ ) 2 1 2 K Se Sc Sc ( Δ θ ) 2
Parameter 5.192 2.027 2.027 2.027
r 0 or θ 0 2.650 92.859 66.432 92.859

Table 18.

The VFF model for single-layer 1H-ScSe2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Å2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ ijk has atom i as the apex.

A (eV) ρ (Å) B4) r min (Å) r max(Å)
Sc-Se 5.853 1.533 24.658 0.0 3.658

Table 19.

Two-body SW potential parameters for single-layer 1H-ScSe2 used by GULP [8] as expressed in Eq. (3).

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5. 1H-SCSE2

Most existing theoretical studies on the single-layer 1H-ScSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-ScSe2.

The structure for the single-layer 1H-ScSe2 is shown in Figure 1 (with M = Sc and X = Se). Each Sc atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Sc atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant a = 3.38 Å and the bond length d S c S e = 2.65 Å. The resultant angles are θ ScSeSe = θ SeScSc = 92.859 and θ ScSeS e = 66.432 , in which atoms Se and Se′ are from different (top or bottom) groups.

Figure 7.

Phonon spectrum for single-layer 1H-ScSe2. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

Table 18 shows four VFF terms for the single-layer 1H-ScSe2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 7(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 7(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 8.

Stress-strain for single-layer 1H-ScSe2 of dimension 100 × 100 Å along the armchair and zigzag directions.

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Sc Se Se 21.292 92.859 1.533 1.533 0.0 3.658 0.0 3.658 0.0 4.327
θ Sc Se S e 25.280 66.432 1.533 1.533 0.0 3.658 0.0 3.658 0.0 4.327
θ Se Sc Sc 21.292 92.859 1.533 1.533 0.0 3.658 0.0 3.658 0.0 4.327

Table 20.

Three-body SW potential parameters for single-layer 1H-ScSe2 used by GULP [8] as expressed in Eq. (4).

The angle θ ijk in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 19 . The parameters for the three-body SW potential used by GULP are shown in Table 20 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 21 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-ScSe2 using LAMMPS, because the angles around atom Sc in Figure 1 (with M = Sc and X = Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Sc and X = Se) shows that, for 1H-ScSe2, we can differentiate these angles around the Sc atom by assigning these six neighboring Se atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Sc atom.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Sc1─Se1-Se1 1.000 1.533 2.386 0.000 1.000 0.000 5.853 4.464 4 0 0.0
Sc1─Se1─Se3 1.000 0.000 0.000 21.292 1.000 −0.050 0.000 0.000 4 0 0.0
Sc1─Se1─Se2 1.000 0.000 0.000 25.280 1.000 0.400 0.000 0.000 4 0 0.0
Se1─Sc1─Sc3 1.000 0.000 0.000 21.292 1.000 −0.050 0.000 0.000 4 0 0.0

Table 21.

SW potential parameters for single-layer 1H-ScSe2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-ScSe2 under uniaxial tension at 1 and 300 K. Figure 8 shows the stress-strain curve for the tension of a single-layer 1H-ScSe2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-ScSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-ScSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 39.4 and 39.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.32 .

There is no available value for nonlinear quantities in the single-layer 1H-ScSe2. We have thus used the nonlinear parameter B = 0.5d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −115.7 and −135.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.1 N/m at the ultimate strain of 0.27 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.9 N/m at the ultimate strain of 0.35 in the zigzag direction at the low temperature of 1 K.

Figure 9.

Phonon spectrum for single-layer 1H-ScTe2. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

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6. 1H-SCTE2

Most existing theoretical studies on the single-layer 1H-ScTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-ScTe2.

The structure for the single-layer 1H-ScTe2 is shown in Figure 1 (with M = Sc and X = Te). Each Sc atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Sc atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant a = 3.62 Å and the bond length d Sc Te = 2.89 Å. The resultant angles are θ ScTeTe = θ TeScSc = 77.555 and θ ScTeT e = 87.364 , in which atoms Te and Te′ are from different (top or bottom) groups.

Figure 10.

Stress-strain for single-layer 1H-ScTe2 of dimension 100 × 100 Å along the armchair and zigzag directions.

VFF type Bond stretching Angle bending
Expression 1 2 K Sc Te ( Δ r ) 2 1 2 K Sc Te Te ( Δ θ ) 2 1 2 K Sc Te T e ( Δ θ ) 2 1 2 K Te Sc Sc ( Δ θ ) 2
Parameter 5.192 2.027 2.027 2.027
r 0 or θ 0 2.890 77.555 87.364 87.364

Table 22.

The VFF model for single-layer 1H-ScTe2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Å2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ ijk has atom i as the apex.

Table 22 shows four VFF terms for the single-layer 1H-ScTe2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 9(a) . The ab initio calculations for the phonon dispersion are from [12]. There is only one (longitudinal) acoustic branch available. We find that the VFF parameters can be chosen to be the same as that of the 1H-ScSe2, from which the longitudinal acoustic branch agrees with the ab initio results as shown in Figure 9(a) . It has also been shown that the VFF parameters can be the same for TaSe2 and NbSe2 of similar structure [15]. Figure 9(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

A ( eV ) ρ ( Å ) B ( Å 4 ) r min ( Å ) r max ( Å )
Sc-Te 4.630 1.050 34.879 0.0 3.761

Table 23.

Two-body SW potential parameters for single-layer 1H-ScTe2 used by GULP [8] as expressed in Eq. (3).

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Sc Te Te 11.848 77.555 1.050 1.050 0.0 3.761 0.0 3.761 0.0 4.504
θ Sc Te T e 11.322 87.364 1.050 1.050 0.0 3.761 0.0 3.761 0.0 4.504
θ Te Sc Sc 11.848 77.555 1.050 1.050 0.0 3.761 0.0 3.761 0.0 4.504

Table 24.

Three-body SW potential parameters for single-layer 1H-ScTe2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Sc1─Te1─Te1 1.000 1.050 3.581 0.000 1.000 0.000 4.630 28.679 4 0 0.0
Sc1─Te1─Te3 1.000 0.000 0.000 11.848 1.000 0.216 0.000 0.000 4 0 0.0
Sc1─Te1─Te2 1.000 0.000 0.000 11.322 1.000 0.046 0.000 0.000 4 0 0.0
Te1─Sc1─Sc3 1.000 0.000 0.000 11.848 1.000 0.216 0.000 0.000 4 0 0.0

Table 25.

SW potential parameters for single-layer 1H-ScTe2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 23 . The parameters for the three-body SW potential used by GULP are shown in Table 24 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 25 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-ScTe2 using LAMMPS, because the angles around atom Sc in Figure 1 (with M = Sc and X = Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Sc and X = Te) shows that, for 1H-ScTe2, we can differentiate these angles around the Sc atom by assigning these six neighboring Te atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Sc atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-ScTe2 under uniaxial tension at 1 and 300 K. Figure 10 shows the stress-strain curve for the tension of a single-layer 1H-ScTe2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-ScTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-ScTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 29.3 and 28.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.38 .

There is no available value for nonlinear quantities in the single-layer 1H-ScTe2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −43.2 and −59.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.7 N/m at the ultimate strain of 0.33 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.7 N/m at the ultimate strain of 0.45 in the zigzag direction at the low temperature of 1 K.

Figure 11.

Phonon dispersion for single-layer 1H-TiTe2. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the Γ M direction. The ab initio results (gray pentagons) are from Ref. [12]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-TiTe2 along Γ MK Γ .

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7. 1H-TITE2

Most existing theoretical studies on the single-layer 1H-TiTe2 are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-TiTe2.

The structure for the single-layer 1H-TiTe2 is shown in Figure 1 (with M = Ti and X = Se). Each Ti atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Ti atoms. The structural parameters are from [12], including the lattice constant a = 3.62 Å and the bond length d Ti Te = 2.75 Å. The resultant angles are θ TiTeTe = θ TeTiTi = 82.323 ° and θ TiTeT e = 81.071 ° , in which atoms Te and Te′ are from different (top or bottom) groups.

Figure 12.

Stress-strain for single-layer 1H-TiTe2 of dimension 100 × 100 Å along the armchair and zigzag directions.

VFF type Bond stretching Angle bending
Expression 1 2 K Ti Te ( Δ r ) 2 1 2 K Ti Te Te ( Δ θ ) 2 1 2 K Ti Te T e ( Δ θ ) 2 1 2 K Te Ti Ti ( Δ θ ) 2
Parameter 4.782 3.216 3.216 3.216
r 0 or θ 0 2.750 82.323 81.071 82.323

Table 26.

The VFF model for single-layer 1H-TiTe2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/ Å 2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

Table 26 shows the VFF terms for the 1H-TiTe2; one of which is the bond stretching interaction shown by Eq. (1), while the other terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the Γ M as shown in Figure 11(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 11(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

A ( eV ) ρ ( Å ) B ( Å 4 ) r min ( Å ) r max ( Å )
Ti-Te 4.414 1.173 28.596 0.0 3.648

Table 27.

Two-body SW potential parameters for single-layer 1H-TiTe2 used by GULP [8] as expressed in Eq. (3).

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Ti Te Te 22.321 82.323 1.173 1.173 0.0 3.648 0.0 3.648 0.0 4.354
θ Ti Te T e 22.463 81.071 1.173 1.173 0.0 3.648 0.0 3.648 0.0 4.354
θ Te Ti Ti 11.321 82.323 1.173 1.173 0.0 3.648 0.0 3.648 0.0 4.354

Table 28.

Three-body SW potential parameters for single-layer 1H-TiTe2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 27 . The parameters for the three-body SW potential used by GULP are shown in Table 28 . Parameters for the SW potential used by LAMMPS are listed in Table 29 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-TiTe2 using LAMMPS, because the angles around atom Ti in Figure 1 (with M = Ti and X = Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Ti and X = Te) shows that, for 1H-TiTe2, we can differentiate these angles around the Ti atom by assigning these six neighboring Te atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one Ti atom.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Ti1-Te1-Te1 1.000 1.173 3.110 0.000 1.000 0.000 4.414 15.100 4 0 0.0
Ti1-Te1-Te3 1.000 0.000 0.000 22.321 1.000 0.134 0.000 0.000 4 0 0.0
Ti1-Te1-Te2 1.000 0.000 0.000 22.463 1.000 0.155 0.000 0.000 4 0 0.0
Te1-Ti1-Ti3 1.000 0.000 0.000 22.321 1.000 0.134 0.000 0.000 4 0 0.0

Table 29.

SW potential parameters for single-layer 1H-TiTe2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = Ti and X = Te).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-TiTe2 under uniaxial tension at 1 and 300 K. Figure 12 shows the stress-strain curve for the tension of a single-layer 1H-TiTe2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-TiTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-TiTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 47.9 and 47.1 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.29 .

There is no available value for the nonlinear quantities in the single-layer 1H-TiTe2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −158.6 and −176.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.6 N/m at the ultimate strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 6.3 N/m at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

Figure 13.

Phonon spectrum for single-layer 1H-VO2. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

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8. 1H-VO2

Most existing theoretical studies on the single-layer 1H-VO2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-VO2.

The structure for the single-layer 1H-VO2 is shown in Figure 1 (with M = V and X = O). Each V atom is surrounded by six O atoms. These O atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each O atom is connected to three V atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant a = 2.70 Å and the bond length d V O = 1.92 Å. The resultant angles are θ VOO = θ OVV = 89.356 ° and θ VO O = 71.436 ° , in which atoms O and O′ are from different (top or bottom) groups.

VFF type Bond stretching Angle bending
Expression 1 2 K V O ( Δ r ) 2 1 2 K V O O ( Δ θ ) 2 1 2 K V O O ( Δ θ ) 2 1 2 K O V V ( Δ θ ) 2
Parameter 9.417 4.825 4.825 4.825
r 0 or θ 0 1.920 89.356 71.436 89.356

Table 30.

The VFF model for single-layer 1H-VO2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/ Å 2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ ijk has atom i as the apex.

A ( eV ) ρ ( Å ) B ( Å 4 ) r min ( Å ) r max ( Å )
V-O 5.105 1.011 6.795 0.0 2.617

Table 31.

Two-body SW potential parameters for single-layer 1H-VO2 used by GULP [8] as expressed in Eq. (3).

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ V O O 43.951 89.356 1.011 1.011 0.0 2.617 0.0 2.617 0.0 3.105
θ V O O 48.902 71.436 1.011 1.011 0.0 2.617 0.0 2.617 0.0 3.105
θ O V V 43.951 89.356 1.011 1.011 0.0 2.617 0.0 2.617 0.0 3.105

Table 32.

Three-body SW potential parameters for single-layer 1H-VO2 used by GULP [8] as expressed in Eq. (4).

The angle θ ijk in the first line indicates the bending energy for the angle with atom i as the apex.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
V1-O1-O1 1.000 1.011 2.589 0.000 1.000 0.000 5.105 6.509 4 0 0.0
V1-O1-O3 1.000 0.000 0.000 43.951 1.000 0.011 0.000 0.000 4 0 0.0
V1-O1-O2 1.000 0.000 0.000 48.902 1.000 0.318 0.000 0.000 4 0 0.0
O1-V1-V3 1.000 0.000 0.000 43.951 1.000 0.011 0.000 0.000 4 0 0.0

Table 33.

SW potential parameters for single-layer 1H-VO2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 14.

Stress-strain for single-layer 1H-VO2 of dimension 100 × 100 Å along the armchair and zigzag directions.

Table 30 shows four VFF terms for the single-layer 1H-VO2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 13(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 13(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 31 . The parameters for the three-body SW potential used by GULP are shown in Table 32 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 33 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-VO2 using LAMMPS, because the angles around atom V in Figure 1 (with M = V and X = O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = V and X = O) shows that, for 1H-VO2, we can differentiate these angles around the V atom by assigning these six neighboring O atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one V atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-VO2 under uniaxial tension at 1 and 300 K. Figure 14 shows the stress-strain curve for the tension of a single-layer 1H-VO2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-VO2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-VO2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 133.0 and 132.9 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.17 .

There is no available value for nonlinear quantities in the single-layer 1H-VO2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −652.3 and −705.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 13.3 N/m at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 12.7 N/m at the ultimate strain of 0.22 in the zigzag direction at the low temperature of 1 K.

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9. 1H-VS2

Most existing theoretical studies on the single-layer 1H-VS2 are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-VS2.

The structure for the single-layer 1H-VS2 is shown in Figure 1 (with M = V and X = S). Each V atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three V atoms. The structural parameters are from [12], including the lattice constant a = 3.09 Å and the bond length d V S = 2.31 Å. The resultant angles are θ VSS = θ SVV = 83.954 ° and θ VS S = 78.878 ° , in which atoms S and S′ are from different (top or bottom) groups.

Figure 15.

Phonon dispersion for single-layer 1H-VS2. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from [16]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-VS2 along ΓMKΓ.

Table 34 shows the VFF terms for the 1H-VS2; one of which is the bond stretching interaction shown by Eq. (1), while the other terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 15(a) . The ab initio calculations for the phonon dispersion are from [16]. The phonon dispersion can also be found in other ab initio calculations [12]. Figure 15(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 16.

Stress-strain for single-layer 1H-VS2 of dimension 100 × 100 Å along the armchair and zigzag directions.

VFF type Bond stretching Angle bending
Expression 1 2 K V S ( Δ r ) 2 1 2 K V S S ( Δ θ ) 2 1 2 K V S S ( Δ θ ) 2 1 2 K S V V ( Δ θ ) 2
Parameter 8.392 4.074 4.074 4.074
r 0 or θ 0 2.310 83.954 78.878 83.954

Table 34.

The VFF model for single-layer 1H-VS2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Å2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ ijk has atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 35 . The parameters for the three-body SW potential used by GULP are shown in Table 36 . Parameters for the SW potential used by LAMMPS are listed in Table 37 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-VS2 using LAMMPS, because the angles around atom V in Figure 1 (with M = V and X = S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = V and X = S) shows that, for 1H-VS2, we can differentiate these angles around the V atom by assigning these six neighboring S atoms with different atom types. It can be found that twelve atom types are necessary for the purpose of differentiating all six neighbors around one V atom.

A (eV) ρ (Å) B4) r min (Å) r max(Å)
V─S 5.714 1.037 14.237 0.0 3.084

Table 35.

Two-body SW potential parameters for single-layer 1H-VS2 used by GULP [8] as expressed in Eq. (3).

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ V S S 30.059 83.954 1.037 1.037 0.0 3.084 0.0 3.084 0.0 3.676
θ V S S 30.874 78.878 1.037 1.037 0.0 3.084 0.0 3.084 0.0 3.676
θ S V V 30.059 83.954 1.037 1.037 0.0 3.084 0.0 3.084 0.0 3.676

Table 36.

Three-body SW potential parameters for single-layer 1H-VS2 used by GULP [8] as expressed in Eq. (4).

The angle θ ijk in the first line indicates the bending energy for the angle with atom i as the apex.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
V1─S1─S1 1.000 1.037 2.973 0.000 1.000 0.000 5.714 12.294 4 0 0.0
V1─S1─S3 1.000 0.000 0.000 30.059 1.000 0.105 0.000 0.000 4 0 0.0
V1─S1─S2 1.000 0.000 0.000 30.874 1.000 0.193 0.000 0.000 4 0 0.0
S1─V1─V3 1.000 0.000 0.000 30.059 1.000 0.105 0.000 0.000 4 0 0.0

Table 37.

SW potential parameters for single-layer 1H-VS2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = V and X = S).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-VS2 under uniaxial tension at 1 and 300 K. Figure 16 shows the stress-strain curve for the tension of a single-layer 1H-VS2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-VS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-VS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 86.5 and 85.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.28 .

There is no available value for the nonlinear quantities in the single-layer 1H-VS2. We have thus used the nonlinear parameter B = 0.5d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −302.0 and −334.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.5 N/m at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.9 N/m at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

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10. 1H-VSe2

Most existing theoretical studies on the single-layer 1H-VSe2 are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-VSe2.

The structure for the single-layer 1H-VSe2 is shown in Figure 1 (with M = V and X = Se). Each V atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three V atoms. The structural parameters are from [12], including the lattice constant a = 3.24 Å and the bond length d V Se = 2.45 Å. The resultant angles are θ VSeSe = θ SeVV = 82.787 ° and θ VSeS e = 80.450 ° , in which atoms Se and Se′ are from different (top or bottom) groups.

Figure 17.

Phonon dispersion for single-layer 1H-VSe2. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from [12]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-VSe2 along ΓMKT.

Table 38 shows the VFF terms for the 1H-VSe2; one of which is the bond stretching interaction shown by Eq. (1), while the other terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 17(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 17(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 39 . The parameters for the three-body SW potential used by GULP are shown in Table 40 . Parameters for the SW potential used by LAMMPS are listed in Table 41 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-VSe2 using LAMMPS, because the angles around atom V in Figure 1 (with M = V and X = Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = V and X = Se) shows that, for 1H-Vse2, we can differentiate these angles around the V atom by assigning these six neighboring Se atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 V atom.

Figure 18.

Stress-strain for single-layer 1H-VSe2 of dimension 100×100 Å along the armchair and zigzag directions.

VFF type Bond stretching Angle bending
Expression 1 2 K V Se ( Δ r ) 2 1 2 K V Se Se ( Δ θ ) 2 1 2 K V Se S e ( Δ θ ) 2 1 2 K Se V V ( Δ θ ) 2
Parameter 6.492 4.716 4.716 4.716
r 0 or θ 0 2.450 82.787 80.450 82.787

Table 38.

The VFF model for single-layer 1H-VSe2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Å2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ ijk has atom i as the apex.

A(eV) ρ(Å) B4) r min(Å) r max(Å)
V─Se 4.817 1.061 18.015 0.0 3.256

Table 39.

Two-body SW potential parameters for single-layer 1H-VSe2 used by GULP [8] as expressed in Eq. (3).

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min12 (Å) r max12 (Å) r min13 (Å) r max13 (Å) r min23 (Å) r max23 (Å)
θ V Se Se 33.299 82.787 1.061 1.061 0.0 3.256 0.0 3.256 0.0 3.884
θ V Se S e 33.702 80.450 1.061 1.061 0.0 3.256 0.0 3.256 0.0 3.884
θ Se V V 33.299 82.787 1.061 1.061 0.0 3.256 0.0 3.256 0.0 3.884

Table 40.

Three-body SW potential parameters for single-layer 1H-VSe2 used by GULP [8] as expressed in Eq. (4).

The angle θ ijk in the first line indicates the bending energy for the angle with atom i as the apex.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-VSe2 under uniaxial tension at 1 and 300 K. Figure 18 shows the stress-strain curve for the tension of a single-layer 1H-VSe2 of dimension 100×100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-VSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-VSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 81.7 and 80.6 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.23 .

ϵ (eV) σ (Å) α λ γ cos θ 0 A L B L p q Tol
V1─Se1─Se1 1.000 1.061 3.070 0.000 1.000 0.000 4.817 14.236 4 0 0.0
V1─Se1─Se3 1.000 0.000 0.000 33.299 1.000 0.126 0.000 0.000 4 0 0.0
V1─Se1─Se2 1.000 0.000 0.000 33.702 1.000 0.166 0.000 0.000 4 0 0.0
Se1─V1─V3 1.000 0.000 0.000 33.299 1.000 0.126 0.000 0.000 4 0 0.0

Table 41.

SW potential parameters for single-layer 1H-VSe2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = V and X = Se).

VFF type Bond stretching Angle bending
Expression 1 2 K V Te ( Δ r ) 2 1 2 K V Te Te ( Δ θ ) 2 1 2 K V Te T e ( Δ θ ) 2 1 2 K Te V V ( Δ θ ) 2
Parameter 6.371 4.384 4.384 4.384
r 0 or θ 0 2.660 81.708 81.891 81.708

Table 42.

The VFF model for single-layer 1H-VTe2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Å2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ ijk has atom i as the apex.

There is no available value for the nonlinear quantities in the single-layer 1H-VSe2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −335.2 and −363.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.5 N/m at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 9.0 N/m at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

11. 1H-VTe2

Most existing theoretical studies on the single-layer 1H-VTe2 are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-VTe2.

Figure 19.

Phonon dispersion for single-layer 1H-VTe2. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from [12]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-VTe2 along ΓMKΓ.

A ( eV ) ρ ( Å ) B ( Å 4 ) r min ( Å ) r max ( Å )
V─Te 5.410 1.112 25.032 0.0 3.520

Table 43.

Two-body SW potential parameters for single-layer 1H-VTe2 used by GULP [8] as expressed in Eq. (3).

The structure for the single-layer 1H-VTe2 is shown in Figure 1 (with M = V and X = Te). Each V atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three V atoms. The structural parameters are from [12], including the lattice constant a = 3.48 Å and the bond length d V Te = 2.66 Å. The resultant angles are θ VTeTe = θ TeVV = 81.708 ° and θ VTeT e = 81.891 ° , in which atoms Te and Te’ are from different (top or bottom) groups.

Figure 20.

Stress-strain for single-layer 1H-VTe2 of dimension 100×100 Å along the armchair and zigzag directions.

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ V Te Te 29.743 81.708 1.112 1.112 0.0 3.520 0.0 3.520 0.0 4.203
θ V Te T e 29.716 81.891 1.112 1.112 0.0 3.520 0.0 3.520 0.0 4.203
θ Te V V 29.743 81.708 1.112 1.112 0.0 3.520 0.0 3.520 0.0 4.203

Table 44.

Three-body SW potential parameters for single-layer 1H-VTe2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

Table 42 shows the VFF terms for the 1H-VTe2; one of which is the bond stretching interaction shown by Eq. (1), while the other terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 19(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 19(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 43 . The parameters for the three-body SW potential used by GULP are shown in Table 44 . Parameters for the SW potential used by LAMMPS are listed in Table 45 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-VTe2 using LAMMPS, because the angles around atom V in Figure 1 (with M = V and X = Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = V and X = Te) shows that, for 1H-VTe2, we can differentiate these angles around the V atom by assigning these six neighboring Te atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 V atom.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
V1─Te1─Te1 1.000 1.112 3.164 0.000 1.000 0.000 5.410 16.345 4 0 0.0
V1─Te1─Te3 1.000 0.000 0.000 29.743 1.000 0.144 0.000 0.000 4 0 0.0
V1─Te1─Te2 1.000 0.000 0.000 29.716 1.000 0.141 0.000 0.000 4 0 0.0
Te1─V1─V3 1.000 0.000 0.000 29.743 1.000 0.144 0.000 0.000 4 0 0.0

Table 45.

SW potential parameters for single-layer 1H-VTe2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = V and X = Te).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-VTe2 under uniaxial tension at 1 and 300 K. Figure 20 shows the stress-strain curve for the tension of a single-layer 1H-VTe2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-VTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-VTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 68.1 and 66.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.28 .

VFF type Bond stretching Angle bending
Expression 1 2 K Cr O ( Δ r ) 2 1 2 K Cr O O ( Δ θ ) 2 1 2 K Cr O O ( Δ θ ) 2 1 2 K O Cr Cr ( Δ θ ) 2
Parameter 12.881 8.039 8.039 8.039
r 0 or θ 0 1.880 86.655 75.194 86.655

Table 46.

The VFF model for single-layer 1H-CrO2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/ Å 2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

A ( eV ) ρ ( Å ) B ( Å 4 ) r min ( Å ) r max ( Å )
Cr─O 6.343 0.916 6.246 0.0 2.536

Table 47.

Two-body SW potential parameters for single-layer 1H-CrO2 used by GULP [8] as expressed in Eq. (3).

There is no available value for the nonlinear quantities in the single-layer 1H-VTe2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −237.4 and −260.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.0 N/m at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.6 N/m at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

12. 1H-CrO2

Most existing theoretical studies on the single-layer 1H-CrO2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-CrO2.

Figure 21.

Phonon spectrum for single-layer 1H-CrO2. (a) Phonon dispersion along the Γ M direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The structure for the single-layer 1H-CrO2 is shown in Figure 1 (with M = Cr and X = O). Each Cr atom is surrounded by six O atoms. These O atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each O atom is connected to three Cr atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant a = 2.58 Å and the bond length d Cr O = 1.88 Å. The resultant angles are θ CrOO = θ OCrCr = 86.655 ° and θ CrO O = 75.194 ° , in which atoms O and O′ are from different (top or bottom) groups.

Table 46 shows four VFF terms for the single-layer 1H-CrO2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the Γ M as shown in Figure 21(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 21(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 22.

Stress-strain for single-layer 1H-CrO2 of dimension 100 × 100 Å along the armchair and zigzag directions.

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Cr O O 65.805 86.655 0.916 0.916 0.0 2.536 0.0 2.536 0.0 3.016
θ Cr O O 70.163 75.194 0.916 0.916 0.0 2.536 0.0 2.536 0.0 3.016
θ O Cr Cr 65.805 86.655 0.916 0.916 0.0 2.536 0.0 2.536 0.0 3.016

Table 48.

Three-body SW potential parameters for single-layer 1H-CrO2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 47 . The parameters for the three-body SW potential used by GULP are shown in Table 48 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 49 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-CrO2 using LAMMPS, because the angles around atom Cr in Figure 1 (with M = Cr and X = O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Cr and X = O) shows that, for 1H-CrO2, we can differentiate these angles around the Cr atom by assigning these six neighboring O atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Cr atom.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Cr1─O1─O1 1.000 0.916 2.769 0.000 1.000 0.000 6.242 8.871 4 0 0.0
Cr1─O1─O3 1.000 0.000 0.000 65.805 1.000 0.058 0.000 0.000 4 0 0.0
Cr1─O1─O2 1.000 0.000 0.000 70.163 1.000 0.256 0.000 0.000 4 0 0.0
O1─Cr1─Cr3 1.000 0.000 0.000 65.805 1.000 0.058 0.000 0.000 4 0 0.0

Table 49.

SW potential parameters for single-layer 1H-CrO2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-CrO2 under uniaxial tension at 1 and 300 K. Figure 22 shows the stress-strain curve for the tension of a single-layer 1H-CrO2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-CrO2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-CrO2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 210.6 and 209.0 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.13 .

There is no available value for nonlinear quantities in the single-layer 1H-CrO2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −1127.7 and −1185.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 19.4 N/m at the ultimate strain of 0.18 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 18.7 N/m at the ultimate strain of 0.20 in the zigzag direction at the low temperature of 1 K.

Figure 23.

Phonon dispersion for single-layer 1H-CrS2. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the Γ M direction. The ab initio results (gray pentagons) are from [17]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-CrS2 along Γ MK Γ .

13. 1H-CrS2

Most existing theoretical studies on the single-layer 1H-CrS2 are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-CrS2.

VFF type Bond stretching Angle bending
Expression 1 2 K Cr S ( Δ r ) 2 1 2 K Cr S S ( Δ θ ) 2 1 2 K Cr S S ( Δ θ ) 2 1 2 K S Cr Cr ( Δ θ ) 2
Parameter 8.752 4.614 4.614 4.614
r 0 or θ 0 2.254 83.099 80.031 83.099

Table 50.

The VFF model for single-layer 1H-CrS2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/ Å 2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

A ( eV ) ρ ( Å ) B ( Å 4 ) r min ( Å ) r max ( Å )
Cr─S 5.544 0.985 12.906 0.0 2.999

Table 51.

Two-body SW potential parameters for single-layer 1HCrS2 used by GULP [8] as expressed in Eq. (3).

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Cr S S 32.963 83.099 0.985 0.985 0.0 2.999 0.0 2.999 0.0 3.577
θ Cr S S 33.491 80.031 0.985 0.985 0.0 2.999 0.0 2.999 0.0 3.577
θ S Cr Cr 32.963 83.099 0.985 0.985 0.0 2.999 0.0 2.999 0.0 3.577

Table 52.

Three-body SW potential parameters for single-layer 1H-CrS2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Cr1─S1─S1 1.000 0.985 3.043 0.000 1.000 0.000 5.544 13.683 4 0 0.0
Cr1─S1─S3 1.000 0.000 0.000 32.963 1.000 0.120 0.000 0.000 4 0 0.0
Cr1─S1─S2 1.000 0.000 0.000 33.491 1.000 0.173 0.000 0.000 4 0 0.0
S1─Cr1─Cr3 1.000 0.000 0.000 32.963 1.000 0.120 0.000 0.000 4 0 0.0

Table 53.

SW potential parameters for single-layer 1H-CrS2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = Cr and X = S).

Figure 24.

Stress-strain for single-layer 1H-CrS2 of dimension 100 × 100 Å along the armchair and zigzag directions.

The structure for the single-layer 1H-CrS2 is shown in Figure 1 (with M = Cr and X = S). Each Cr atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Cr atoms. The structural parameters are from [17], including the lattice constant a = 2.99 Å and the bond length d Cr S = 2.254 Å. The resultant angles are θ CrSS = θ SCrCr = 83.099 ° and θ CrS S = 80.031 ° , in which atoms S and S’ are from different (top or bottom) groups.

Table 50 shows four VFF terms for the 1H-CrS2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the Γ M as shown in Figure 23(a) . The ab initio calculations for the phonon dispersion are from [17]. Similar phonon dispersion can also be found in other ab initio calculations [12]. Figure 23(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 51 . The parameters for the three-body SW potential used by GULP are shown in Table 52 . Parameters for the SW potential used by LAMMPS are listed in Table 53 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-CrS2 using LAMMPS, because the angles around atom Cr in Figure 1 (with M = Cr and X = S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14] According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Cr and X = S) shows that, for 1H-CrS2, we can differentiate these angles around the Cr atom by assigning these six neighboring S atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Cr atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-CrS2 under uniaxial tension at 1 and 300 K. Figure 24 shows the stress-strain curve for the tension of a single-layer 1H-CrS2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-CrS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-CrS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 98.4 and 97.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are in reasonably agreement with the ab initio results, e.g., 112.0 N/m from [18], or 111.9 N/m from [19]. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.26 , which agrees with the ab initio value of 0.27 [18, 19].

There is no available value for the nonlinear quantities in the single-layer 1H-CrS2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −364.8 and −409.3 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 12.4 N/m at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.8 N/m at the ultimate strain of 0.26 in the zigzag direction at the low temperature of 1 K.

Figure 25.

Phonon dispersion for single-layer 1H-CrSe2. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from [21]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-CrSe2 along ΓMKΓ.

14. 1H-CrSe2

Most existing theoretical studies on the single-layer 1H-CrSe2 are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-CrSe2.

The structure for the single-layer 1H-CrSe2 is shown in Figure 1 (with M = Cr and X = Se). Each Cr atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Cr atoms. The structural parameters are from [12], including the lattice constant a = 3.13 Å and the bond length d Cr Se = 2.38 Å. The resultant angles are θ CrSeSe = θ SeCrCr = 82.229 ° and θ CrSeS e = 81.197 ° , in which atoms Se and Se’ are from different (top or bottom) groups.

Figure 26.

Stress-strain for single-layer 1H-CrSe2 of dimension 100 × 100 Å along the armchair and zigzag directions.

VFF type Bond stretching Angle bending
Expression 1 2 K Cr Se ( Δ r ) 2 1 2 K Cr Se Se ( Δ θ ) 2 1 2 K Cr Se S e ( Δ θ ) 2 1 2 K Se Cr Cr ( Δ θ ) 2
Parameter 9.542 4.465 4.465 4.465
r 0 or θ 0 2.380 82.229 81.197 82.229

Table 54.

The VFF model for single-layer 1H-CrSe2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/ Å 2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

Table 54 shows four VFF terms for the 1H-CrSe2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 25(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 25(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

A (eV) ρ (Å) B4) r min (Å) r max (Å)
Cr─Se 6.581 1.012 16.043 0.0 3.156

Table 55.

Two-body SW potential parameters for single-layer 1H-CrSe2 used by GULP [8] as expressed in Eq. (3).

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Cr Se Se 30.881 82.229 1.012 1.012 0.0 3.156 0.0 3.156 0.0 3.767
θ Cr Se S e 31.044 81.197 1.012 1.012 0.0 3.156 0.0 3.156 0.0 3.767
θ Se Cr Cr 30.881 82.229 1.012 1.012 0.0 3.156 0.0 3.156 0.0 3.767

Table 56.

Three-body SW potential parameters for single-layer 1H-CrSe2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 55 . The parameters for the three-body SW potential used by GULP are shown in Table 56 . Parameters for the SW potential used by LAMMPS are listed in Table 57 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-CrSe2 using LAMMPS, because the angles around atom Cr in Figure 1 (with M = Cr and X = Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Cr and X = Se) shows that, for 1H-CrSe2, we can differentiate these angles around the Cr atom by assigning these six neighboring Se atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Cr atom.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Cr1─Se1─Se1 1.000 1.012 3.118 0.000 1.000 0.000 6.581 15.284 4 0 0.0
Cr1─Se1─Se3 1.000 0.000 0.000 30.881 1.000 0.135 0.000 0.000 4 0 0.0
Cr1─Se1─Se2 1.000 0.000 0.000 31.044 1.000 0.153 0.000 0.000 4 0 0.0
Se1─Cr1─Cr3 1.000 0.000 0.000 30.881 1.000 0.135 0.000 0.000 4 0 0.0

Table 57.

SW potential parameters for single-layer 1H-CrSe2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = Cr and X = Se).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-CrSe2 under uniaxial tension at 1 and 300 K. Figure 26 shows the stress-strain curve for the tension of a single-layer 1H-CrSe2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-CrSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-CrSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 90.0 and 89.0 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are in reasonably agreement with the ab initio results, e.g., 88.0 N/m from [18], or 87.9 N/m from [19]. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.30 , which agrees with the ab initio value of 0.30 [18, 19].

There is no available value for the nonlinear quantities in the single-layer 1H-CrSe2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most two-dimensional atomic layered materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −279.6 and −318.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 13.0 N/m at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 12.4 N/m at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Figure 27.

Phonon dispersion for single-layer 1H-CrTe2. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from [12]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-CrTe2 along ΓMKΓ.

15. 1H-CrTe2

Most existing theoretical studies on the single-layer 1H-CrTe2 are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-CrTe2.

The structure for the single-layer 1H-CrTe2 is shown in Figure 1 (with M = Cr and X = Te). Each Cr atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Cr atoms. The structural parameters are from [12], including the lattice constant a = 3.39 Å and the bond length d Cr Te = 2.58 Å. The resultant angles are θ CrTeTe = θ TeCrCr = 82.139 ° and θ CrTeT e = 81.316 ° , in which atoms Te and Te’ are from different (top or bottom) groups.

Figure 28.

Stress-strain for single-layer 1H-CrTe2 of dimension 100 × 100 Å along the armchair and zigzag directions.

VFF type Bond stretching Angle bending
Expression 1 2 K Cr Te ( Δ r ) 2 1 2 K Cr Te Te ( Δ θ ) 2 1 2 K Cr Te T e ( Δ θ ) 2 1 2 K Te Cr Cr ( Δ θ ) 2
Parameter 8.197 4.543 4.543 4.543
r 0 or θ 0 2.580 82.139 81.316 82.139

Table 58.

The VFF model for single-layer 1H-CrTe2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/ Å 2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

Table 58 shows three VFF terms for the 1H-CrTe2; one of which is the bond stretching interaction shown by Eq. (1), while the other two terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 27(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 27(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

A (eV) ρ (Å) B4) r min (Å) r max (Å)
Cr─Te 6.627 1.094 22.154 0.0 3.420

Table 59.

Two-body SW potential parameters for single-layer 1H-CrTe2 used by GULP [8] as expressed in Eq. (3).

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Cr Te Te 31.316 82.139 1.094 1.094 0.0 3.420 0.0 3.420 0.0 4.082
θ Cr Te T e 31.447 81.316 1.094 1.094 0.0 3.420 0.0 3.420 0.0 4.082
θ Te Cr Cr 31.316 82.139 1.094 1.094 0.0 3.420 0.0 3.420 0.0 4.082

Table 60.

Three-body SW potential parameters for single-layer 1H-CrTe2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 59 . The parameters for the three-body SW potential used by GULP are shown in Table 60 . Parameters for the SW potential used by LAMMPS are listed in Table 61 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-CrTe2 using LAMMPS, because the angles around atom Cr in Figure 1 (with M = Cr and X = Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Cr and X = Te) shows that, for 1H-CrTe2, we can differentiate these angles around the Cr atom by assigning these six neighboring Te atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Cr atom.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Cr1─Te1─Te1 1.000 1.094 3.126 0.000 1.000 0.000 6.627 15.461 4 0 0.0
Cr1─Te1─Te3 1.000 0.000 0.000 31.316 1.000 0.137 0.000 0.000 4 0 0.0
Cr1─Te1─Te2 1.000 0.000 0.000 31.447 1.000 0.151 0.000 0.000 4 0 0.0
Te1─Cr1─Cr3 1.000 0.000 0.000 31.316 1.000 0.137 0.000 0.000 4 0 0.0

Table 61.

SW potential parameters for single-layer 1H-CrTe2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = Cr and X = Te).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-CrTe2 under uniaxial tension at 1 and 300 K. Figure 28 shows the stress-strain curve for the tension of a single-layer 1H-CrTe2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-CrTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-CrTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 77.2 and 76.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are in reasonably agreement with the ab initio results, e.g., 63.9 N/m from [18, 19]. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.30 , which agrees with the ab initio value of 0.30 [18, 19].

There is no available value for the nonlinear quantities in the single-layer 1H-CrTe2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most two-dimensional atomic layered materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −237.1 and −280.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.2 N/m at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.7 N/m at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

Figure 29.

Phonon spectrum for single-layer 1H-MnO2. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

16. 1H-MnO2

Most existing theoretical studies on the single-layer 1H-MnO2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-MnO2.

The structure for the single-layer 1H-MnO2 is shown in Figure 1 (with M = Mn and X = O). Each Mn atom is surrounded by six O atoms. These O atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each O atom is connected to three Mn atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant a = 2.61 Å and the bond length d Mn O = 1.87 Å. The resultant angles are θ MnOO = θ OMnMn = 88.511 ° and θ MnO O = 72.621 ° , in which atoms O and O′ are from different (top or bottom) groups.

Figure 30.

Stress-strain for single-layer 1H-MnO2 of dimension 100 × 100 Å along the armchair and zigzag directions.

VFF type Bond stretching Angle bending
Expression 1 2 K Mn O ( Δ r ) 2 1 2 K Mn O O ( Δ θ ) 2 1 2 K Mn O O ( Δ θ ) 2 1 2 K O Mn Mn ( Δ θ ) 2
Parameter 9.382 6.253 6.253 6.253
r 0 or θ 0 1.870 88.511 72.621 88.511

Table 62.

The VFF model for single-layer 1H-MnO2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/ Å 2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

Table 62 shows four VFF terms for the single-layer 1H-MnO2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 29(a) . The ab initio calculations for the phonon dispersion are from [12]. Typically, the transverse acoustic branch has a linear dispersion, so is higher than the flexural branch. However, it can be seen that the transverse acoustic branch is close to the flexural branch, which should be due to the underestimation from the ab initio calculations. Figure 29(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

A (eV) ρ (Å) B4) r min (Å) r max (Å)
Mn─O 4.721 0.961 6.114 0.0 2.540

Table 63.

Two-body SW potential parameters for single-layer 1H-MnO2 used by GULP [8] as expressed in Eq. (3).

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Mn O O 55.070 88.511 0.961 0.961 0.0 2.540 0.0 2.540 0.0 3.016
θ Mn O O 60.424 72.621 0.961 0.961 0.0 2.540 0.0 2.540 0.0 3.016
θ O Mn Mn 55.070 88.511 0.961 0.961 0.0 2.540 0.0 2.540 0.0 3.016

Table 64.

Three-body SW potential parameters for single-layer 1H-MnO2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 63 . The parameters for the three-body SW potential used by GULP are shown in Table 64 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 65 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-MnO2 using LAMMPS, because the angles around atom Mn in Figure 1 (with M = Mn and X = O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Mn and X = O) shows that, for 1H-MnO2, we can differentiate these angles around the Mn atom by assigning these six neighboring O atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Mn atom.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Mn1─O1─O1 1.000 0.961 2.643 0.000 1.000 0.000 4.721 7.158 4 0 0.0
Mn1─O1─O3 1.000 0.000 0.000 55.070 1.000 0.026 0.000 0.000 4 0 0.0
Mn1─O1─O2 1.000 0.000 0.000 60.424 1.000 0.299 0.000 0.000 4 0 0.0
O1─Mn1─Mn3 1.000 0.000 0.000 55.070 1.000 0.026 0.000 0.000 4 0 0.0

Table 65.

SW potential parameters for single-layer 1H-MnO2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-MnO2 under uniaxial tension at 1 and 300 K. Figure 30 shows the stress-strain curve for the tension of a single-layer 1H-MnO2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-MnO2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-MnO2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 161.1 and 160.2 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.10 .

There is no available value for nonlinear quantities in the single-layer 1H-MnO2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −915.9 and −957.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 14.1 N/m at the ultimate strain of 0.17 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 13.7 N/m at the ultimate strain of 0.20 in the zigzag direction at the low temperature of 1 K.

Figure 31.

Phonon spectrum for single-layer 1H-FeO2. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

17. 1H-FeO2

Most existing theoretical studies on the single-layer 1H-FeO2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-FeO2.

The structure for the single-layer 1H-FeO2 is shown in Figure 1 (with M = Fe and X = O). Each Fe atom is surrounded by six O atoms. These O atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each O atom is connected to three Fe atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant a = 2.62 Å and the bond length d Fe O = 1.88 Å. The resultant angles are θ FeOO = θ OFeFe = 88.343 ° and θ FeO O = 72.856 ° , in which atoms O and O′ are from different (top or bottom) groups.

Figure 32.

Stress-strain for single-layer 1H-FeO2 of dimension 100 × 100 Å along the armchair and zigzag directions.

VFF type Bond stretching Angle bending
Expression 1 2 K Fe O ( Δ r ) 2 1 2 K Fe O O ( Δ θ ) 2 1 2 K Fe O O ( Δ θ ) 2 1 2 K O Fe Fe ( Δ θ ) 2
Parameter 8.377 3.213 3.213 3.213
r 0 or θ 0 1.880 88.343 72.856 88.343

Table 66.

The VFF model for single-layer 1H-FeO2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/ Å 2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

Table 66 shows four VFF terms for the single-layer 1H-FeO2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 31(a) . The ab initio calculations for the phonon dispersion are from [12]. Figure 31(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

A (eV) ρ (Å) B4) r min (Å) r max (Å)
Fe─O 4.242 0.962 6.246 0.0 2.552

Table 67.

Two-body SW potential parameters for single-layer 1H-FeO2 used by GULP [8] as expressed in Eq. (3).

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Fe O O 28.105 88.343 0.962 0.962 0.0 2.552 0.0 2.552 0.0 3.031
θ Fe O O 30.754 72.856 0.962 0.962 0.0 2.552 0.0 2.552 0.0 3.031
θ O Fe Fe 28.105 88.343 0.962 0.962 0.0 2.552 0.0 2.552 0.0 3.031

Table 68.

Three-body SW potential parameters for single-layer 1H-FeO2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 67 . The parameters for the three-body SW potential used by GULP are shown in Table 68 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 69 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-FeO2 using LAMMPS, because the angles around atom Fe in Figure 1 (with M = Fe and X = O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Fe and X = O) shows that, for 1H-FeO2, we can differentiate these angles around the Fe atom by assigning these six neighboring O atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Fe atom.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Fe1─O1─O1 1.000 0.962 2.654 0.000 1.000 0.000 4.242 7.298 4 0 0.0
Fe1─O1─O3 1.000 0.000 0.000 28.105 1.000 0.029 0.000 0.000 4 0 0.0
Fe1─O1─O2 1.000 0.000 0.000 30.754 1.000 0.295 0.000 0.000 4 0 0.0
O1─Fe1─Fe3 1.000 0.000 0.000 28.105 1.000 0.029 0.000 0.000 4 0 0.0

Table 69.

SW potential parameters for single-layer 1H-FeO2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-FeO2 under uniaxial tension at 1 and 300 K. Figure 32 shows the stress-strain curve for the tension of a single-layer 1H-FeO2 of dimension 100×100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-FeO2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-FeO2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 100.2 and 99.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.23 .

There is no available value for nonlinear quantities in the single-layer 1H-FeO2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −423.4 and −460.2 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.4 N/m at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.9 N/m at the ultimate strain of 0.25 in the zigzag direction at the low temperature of 1 K.

18. 1H-FES2

Most existing theoretical studies on the single-layer 1H-FeS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-FeS2.

VFF type Bond stretching Angle bending
Expression 1 2 K Fe S ( Δ r ) 2 1 2 K Fe S S ( Δ θ ) 2 1 2 K Fe S S ( Δ θ ) 2 1 2 K S Fe Fe ( Δ θ ) 2
Parameter 6.338 3.964 3.964 3.964
r 0 or θ 0 2.220 87.132 74.537 87.132

Table 70.

The VFF model for single-layer 1H-FeS2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Å2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

A (eV) ρ (Å) B4) r min (Å) r max (Å)
Fe─S 4.337 1.097 12.145 0.0 3.000

Table 71.

Two-body SW potential parameters for single-layer 1H-FeS2 used by GULP [8] as expressed in Eq. (3).

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Fe S S 33.060 87.132 1.097 1.097 0.0 3.000 0.0 3.000 0.0 3.567
θ Fe S S 35.501 74.537 1.097 1.097 0.0 3.000 0.0 3.000 0.0 3.567
θ S Fe Fe 33.060 87.132 1.097 1.097 0.0 3.000 0.0 3.000 0.0 3.567

Table 72.

Three-body SW potential parameters for single-layer 1H-FeS2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Fe1─S1─S1 1.000 1.097 2.735 0.000 1.000 0.000 4.337 8.338 4 0 0.0
Fe1─S1─S3 1.000 0.000 0.000 33.060 1.000 0.050 0.000 0.000 4 0 0.0
Fe1─S1─S2 1.000 0.000 0.000 35.501 1.000 0.267 0.000 0.000 4 0 0.0
S1─Fe1─Fe3 1.000 0.000 0.000 33.060 1.000 0.050 0.000 0.000 4 0 0.0

Table 73.

SW potential parameters for single-layer 1H-FeS2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 33.

Phonon spectrum for single-layer 1H-FeS2. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from Ref. [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The structure for the single-layer 1H-FeS2 is shown in Figure 1 (with M=Fe and X=S). Each Fe atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Fe atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant a = 3.06 Å and the bond length d Fe S = 2.22 Å. The resultant angles are θ FeSS = θ SFeFe = 87.132 ° and θ FeS S = 74.537 ° , in which atoms S and S’ are from different (top or bottom) groups.

Table 70 shows four VFF terms for the single-layer 1H-FeS2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 33(a) . The ab initio calculations for the phonon dispersion are from Ref. [12]. Figure 33(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 34.

Stress-strain for single-layer 1H-FeS2 of dimension 100 × 100 Å along the armchair and zigzag directions.

The parameters for the two-body SW potential used by GULP are shown in Table 71 . The parameters for the three-body SW potential used by GULP are shown in Table 72 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 73 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-FeS2 using LAMMPS, because the angles around atom Fe in Figure 1 (with M=Fe and X=S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M=Fe and X=S) shows that, for 1H-FeS2, we can differentiate these angles around the Fe atom by assigning these six neighboring S atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Fe atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-FeS2 under uniaxial tension at 1 and 300 K. Figure 34 shows the stress-strain curve for the tension of a single-layer 1H-FeS2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-FeS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-FeS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 83.6 and 83.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.20 .

There is no available value for nonlinear quantities in the single-layer 1H-FeS2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −377.5 and −412.7 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 9.0 N/m at the ultimate strain of 0.20 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.6 N/m at the ultimate strain of 0.23 in the zigzag direction at the low temperature of 1 K.

19. 1H-FESE2

Most existing theoretical studies on the single-layer 1H-FeSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-FeSe2.

The structure for the single-layer 1H-FeSe2 is shown in Figure 1 (with M=Fe and X=Se). Each Fe atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Fe atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant a = 3.22 Å and the bond length d Fe Se = 2.35 Å. The resultant angles are θ FeSeSe = θ SeFeFe = 86.488 ° and θ FeSeS e = 75.424 ° , in which atoms Se and Se’ are from different (top or bottom) groups.

Figure 35.

Phonon spectrum for single-layer 1H-FeSe2. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from Ref. [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

Table 74 shows four VFF terms for the single-layer 1H-FeSe2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 35(a) . The ab initio calculations for the phonon dispersion are from Ref. [12]. Figure 35(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 75 . The parameters for the three-body SW potential used by GULP are shown in Table 76 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 77 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-FeSe2 using LAMMPS, because the angles around atom Fe in Figure 1 (with M=Fe and X=Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M=Fe and X=Se) shows that, for 1H-FeSe2, we can differentiate these angles around the Fe atom by assigning these six neighboring Se atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Fe atom.

Figure 36.

Stress-strain for single-layer 1H-FeSe2 of dimension 100 × 100 Å along the armchair and zigzag directions.

VFF type Bond stretching Angle bending
Expression 1 2 K Fe Se ( Δ r ) 2 1 2 K Fe Se Se ( Δ θ ) 2 1 2 K Fe Se S e ( Δ θ ) 2 1 2 K Se Fe Fe ( Δ θ ) 2
Parameter 6.338 3.964 3.964 3.964
r 0 or θ 0 2.350 86.488 75.424 86.488

Table 74.

The VFF model for single-layer 1H-FeSe2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Å2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

A (eV) ρ (Å) B4) r min (Å) r max (Å)
Fe-Se 4.778 1.139 15.249 0.0 3.168

Table 75.

Two-body SW potential parameters for single-layer 1H-FeSe2 used by GULP [8] as expressed in Eq. (3).

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Fe Se Se 32.235 86.488 1.139 1.139 0.0 3.168 0.0 3.168 0.0 3.768
θ Fe Se S e 34.286 75.424 1.139 1.139 0.0 3.168 0.0 3.168 0.0 3.768
θ Se Fe Fe 32.235 86.488 1.139 1.139 0.0 3.168 0.0 3.168 0.0 3.768

Table 76.

Three-body SW potential parameters for single-layer 1H-FeSe2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-FeSe2 under uniaxial tension at 1 and 300 K. Figure 36 shows the stress-strain curve for the tension of a single-layer 1H-FeSe2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-FeSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-FeSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 77.3 and 77.4 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.23 .

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Fe1─Se1─Se1 1.000 1.139 2.781 0.000 1.000 0.000 4.778 9.049 4 0 0.0
Fe1─Se1─Se3 1.000 0.000 0.000 32.235 1.000 0.061 0.000 0.000 4 0 0.0
Fe1─Se1─Se2 1.000 0.000 0.000 34.286 1.000 0.252 0.000 0.000 4 0 0.0
Se1─Fe1─Fe3 1.000 0.000 0.000 32.235 1.000 0.061 0.000 0.000 4 0 0.0

Table 77.

SW potential parameters for single-layer 1H-FeSe2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

VFF type Bond stretching Angle bending
Expression 1 2 K Fe Te ( Δ r ) 2 1 2 K Fe Te Te ( Δ θ ) 2 1 2 K Fe Te T e ( Δ θ ) 2 1 2 K Te Fe Fe ( Δ θ ) 2
Parameter 6.338 3.964 3.964 3.964
r 0 or θ 0 2.530 86.904 74.851 86.904

Table 78.

The VFF model for single-layer 1H-FeTe2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Å2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

There is no available value for nonlinear quantities in the single-layer 1H-FeSe2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −323.8 and −360.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.8 N/m at the ultimate strain of 0.21 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.4 N/m at the ultimate strain of 0.25 in the zigzag direction at the low temperature of 1 K.

Figure 37.

Phonon spectrum for single-layer 1H-FeTe2. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from Ref. [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

A (eV) ρ (Å) B4) r min (Å) r max (Å)
Fe─Te 5.599 1.242 20.486 0.0 3.416

Table 79.

Two-body SW potential parameters for single-layer 1H-FeTe2 used by GULP [8] as expressed in Eq. (3).

20. 1H-FETE2

Most existing theoretical studies on the single-layer 1H-FeTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-FeTe2.

Figure 38.

Stress-strain for single-layer 1H-FeTe2 of dimension 100 × 100 Å along the armchair and zigzag directions.

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Fe Te Te 32.766 86.904 1.242 1.242 0.0 3.416 0.0 3.416 0.0 4.062
θ Fe Te T e 35.065 74.851 1.242 1.242 0.0 3.416 0.0 3.416 0.0 4.062
θ Te Fe Fe 32.766 86.904 1.242 1.242 0.0 3.416 0.0 3.416 0.0 4.062

Table 80.

Three-body SW potential parameters for single-layer 1H-FeTe2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

The structure for the single-layer 1H-FeTe2 is shown in Figure 1 (with M=Fe and X=Te). Each Fe atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Fe atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant a = 3.48 Å and the bond length d Fe Te = 2.53 Å. The resultant angles are θ FeTeTe = θ TeFeFe = 86.904 ° and θ FeTeT e = 74.851 ° , in which atoms Te and Te’ are from different (top or bottom) groups.

Table 78 shows four VFF terms for the single-layer 1H-FeTe2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the two in-plane acoustic branches in the phonon dispersion along the ΓM as shown in Figure 37(a) . The ab initio calculations for the phonon dispersion are from Ref. [12]. Figure 37(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Fe1─Te1─Te1 1.000 1.242 2.751 0.000 1.000 0.000 5.599 8.615 4 0 0.0
Fe1─Te1─Te3 1.000 0.000 0.000 32.766 1.000 0.054 0.000 0.000 4 0 0.0
Fe1─Te1─Te2 1.000 0.000 0.000 35.065 1.000 0.261 0.000 0.000 4 0 0.0
Te1─Fe1─Fe3 1.000 0.000 0.000 32.766 1.000 0.054 0.000 0.000 4 0 0.0

Table 81.

SW potential parameters for single-layer 1H-FeTe2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

The parameters for the two-body SW potential used by GULP are shown in Table 79 . The parameters for the three-body SW potential used by GULP are shown in Table 80 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 81 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-FeTe2 using LAMMPS, because the angles around atom Fe in Figure 1 (with M=Fe and X=Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M=Fe and X=Te) shows that, for 1H-FeTe2, we can differentiate these angles around the Fe atom by assigning these six neighboring Te atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Fe atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-FeTe2 under uniaxial tension at 1 and 300 K. Figure 38 shows the stress-strain curve for the tension of a single-layer 1H-FeTe2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-FeTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-FeTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 69.6 and 69.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.25 .

VFF type Bond stretching Angle bending
Expression 1 2 K Co Te ( Δ r ) 2 1 2 K Co Te Te ( Δ θ ) 2 1 2 K Co Te T e ( Δ θ ) 2 1 2 K Te Co Co ( Δ θ ) 2
Parameter 6.712 2.656 2.656 2.656
r 0 or θ 0 2.510 89.046 71.873 89.046

Table 82.

The VFF model for single-layer 1H-CoTe2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Å2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å2) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

There is no available value for nonlinear quantities in the single-layer 1H-FeTe2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −267.5 and −302.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.6 N/m at the ultimate strain of 0.22 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.1 N/m at the ultimate strain of 0.26 in the zigzag direction at the low temperature of 1 K.

21. 1H-COTE2

Most existing theoretical studies on the single-layer 1H-CoTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-CoTe2.

Figure 39.

Phonon spectrum for single-layer 1H-CoTe2. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from Ref. [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

A (eV) ρ (Å) B4) r min (Å) r max (Å)
Co─Te 6.169 1.310 19.846 0.0 3.417

Table 83.

Two-body SW potential parameters for single-layer 1H-CoTe2 used by GULP [8] as expressed in Eq. (3).

The structure for the single-layer 1H-CoTe2 is shown in Figure 1 (with M=Co and X=Te). Each Co atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Co atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant a = 3.52 Å and the bond length d Co Te = 2.51 Å. The resultant angles are θ CoTeTe = θ TeCoCo = 89.046 ° and θ CoTeT e = 71.873 ° , in which atoms Te and Te’ are from different (top or bottom) groups.

Figure 40.

Stress-strain for single-layer 1H-CoTe2 of dimension 100 × 100 Å along the armchair and zigzag directions.

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Co Te Te 23.895 89.046 1.310 1.310 0.0 3.417 0.0 3.417 0.0 4.055
θ Co Te T e 26.449 71.873 1.310 1.310 0.0 3.417 0.0 3.417 0.0 4.055
θ Te Co Co 23.895 89.046 1.310 1.310 0.0 3.417 0.0 3.417 0.0 4.055

Table 84.

Three-body SW potential parameters for single-layer 1H-CoTe2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

Table 82 shows four VFF terms for the single-layer 1H-CoTe2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 39(a) . The ab initio calculations for the phonon dispersion are from Ref. [12]. Figure 39(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 83 . The parameters for the three-body SW potential used by GULP are shown in Table 84 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 85 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-CoTe2 using LAMMPS, because the angles around atom Co in Figure 1 (with M=Co and X=Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M=Co and X=Te) shows that, for 1H-CoTe2, we can differentiate these angles around the Co atom by assigning these six neighboring Te atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Co atom.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Co1─Te1─Te1 1.000 1.310 2.608 0.000 1.000 0.000 6.169 6.739 4 0 0.0
Co1─Te1─Te3 1.000 0.000 0.000 23.895 1.000 0.017 0.000 0.000 4 0 0.0
Co1─Te1─Te2 1.000 0.000 0.000 26.449 1.000 0.311 0.000 0.000 4 0 0.0
Te1─Co1─Co3 1.000 0.000 0.000 23.895 1.000 0.017 0.000 0.000 4 0 0.0

Table 85.

SW potential parameters for single-layer 1H-CoTe2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-CoTe2 under uniaxial tension at 1 and 300 K. Figure 40 shows the stress-strain curve for the tension of a single-layer 1H-CoTe2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-CoTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-CoTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 53.7 and 54.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.32 .

VFF type Bond stretching Angle bending
Expression 1 2 K Ni S ( Δ r ) 2 1 2 K Ni S S ( Δ θ ) 2 1 2 K Ni S S ( Δ θ ) 2 1 2 K S Ni Ni ( Δ θ ) 2
Parameter 6.933 3.418 3.418 3.418
r 0 or θ 0 2.240 98.740 57.593 98.740

Table 86.

The VFF model for single-layer 1H-NiS2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Å2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

A (eV) ρ (Å) B4) r min (Å) r max (Å)
Ni-S 6.425 1.498 12.588 0.0 3.156

Table 87.

Two-body SW potential parameters for single-layer 1H-NiS2 used by GULP [8] as expressed in Eq. (3).

There is no available value for nonlinear quantities in the single-layer 1H-CoTe2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −157.2 and −187.9 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.2 N/m at the ultimate strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.9 N/m at the ultimate strain of 0.33 in the zigzag direction at the low temperature of 1 K.

22. 1H-NIS2

Most existing theoretical studies on the single-layer 1H-NiS2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-NiS2.

Figure 41.

Phonon spectrum for single-layer 1H-NiS2. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from Ref. [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

The structure for the single-layer 1H-NiS2 is shown in Figure 1 (with M=Ni and X=S). Each Ni atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Ni atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant a = 3.40 Å and the bond length d Ni S = 2.24 Å. The resultant angles are θ NiSS = θ SNiNi = 98.740 ° and θ NiS S = 57.593 ° , in which atoms S and S’ are from different (top or bottom) groups.

Table 86 shows four VFF terms for the single-layer 1H-NiS2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 41(a) . The ab initio calculations for the phonon dispersion are from Ref. [12]. Figure 41(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

Figure 42.

Stress-strain for single-layer 1H-NiS2 of dimension 100 × 100 Å along the armchair and zigzag directions.

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Ni S S 46.062 98.740 1.498 1.498 0.0 3.156 0.0 3.156 0.0 3.713
θ Ni S S 63.130 57.593 1.498 1.498 0.0 3.156 0.0 3.156 0.0 3.713
θ S Ni Ni 46.062 98.740 1.498 1.498 0.0 3.156 0.0 3.156 0.0 3.713

Table 88.

Three-body SW potential parameters for single-layer 1H-NiS2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 87 . The parameters for the three-body SW potential used by GULP are shown in Table 88 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 89 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-NiS2 using LAMMPS, because the angles around atom Ni in Figure 1 (with M=Ni and X=S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M=Ni and X=S) shows that, for 1H-NiS2, we can differentiate these angles around the Ni atom by assigning these six neighboring S atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Ni atom.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Ni1─S1─S1 1.000 1.498 2.107 0.000 1.000 0.000 6.425 2.502 4 0 0.0
Ni1─S1─S3 1.000 0.000 0.000 46.062 1.000 −0.152 0.000 0.000 4 0 0.0
Ni1─S1─S2 1.000 0.000 0.000 63.130 1.000 0.536 0.000 0.000 4 0 0.0
S1─Ni1─Ni3 1.000 0.000 0.000 46.062 1.000 −0.152 0.000 0.000 4 0 0.0

Table 89.

SW potential parameters for single-layer 1H-NiS2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-NiS2 under uniaxial tension at 1 and 300 K. Figure 42 shows the stress-strain curve for the tension of a single-layer 1H-NiS2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-NiS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-NiS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 84.0 and 82.5 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.19 .

There is no available value for nonlinear quantities in the single-layer 1H-NiS2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −403.2 and −414.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.7 N/m at the ultimate strain of 0.20 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 8.3 N/m at the ultimate strain of 0.24 in the zigzag direction at the low temperature of 1 K.

Figure 43.

Phonon spectrum for single-layer 1H-NiSe2. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from Ref. [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

23. 1H-NISE2

Most existing theoretical studies on the single-layer 1H-NiSe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-NiSe2.

VFF type Bond stretching Angle bending
Expression 1 2 K Ni Se ( Δ r ) 2 1 2 K Ni Se Se ( Δ θ ) 2 1 2 K Ni Se S e ( Δ θ ) 2 1 2 K Se Ni Ni ( Δ θ ) 2
Parameter 4.823 2.171 2.171 2.171
r 0 or θ 0 2.350 90.228 70.206 90.228

Table 90.

The VFF model for single-layer 1H-NiSe2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/c2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of eV/Å2) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

A (eV) ρ (Å) B4) r min (Å) r max (Å)
Ni─Se 4.004 1.267 15.249 0.0 3.213

Table 91.

Two-body SW potential parameters for single-layer 1H-NiSe2 used by GULP [8] as expressed in Eq. (3).

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Ni Se Se 20.479 90.228 1.267 1.267 0.0 3.213 0.0 3.213 0.0 3.809
θ Ni Se S e 23.132 70.206 1.267 1.267 0.0 3.213 0.0 3.213 0.0 3.809
θ Se Ni Ni 20.479 90.228 1.267 1.267 0.0 3.213 0.0 3.213 0.0 3.809

Table 92.

Three-body SW potential parameters for single-layer 1H-NiSe2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Ni1─Se1─Se1 1.000 1.267 2.535 0.000 1.000 0.000 4.004 5.913 4 0 0.0
Ni1─Se1─Se3 1.000 0.000 0.000 20.479 1.000 −0.004 0.000 0.000 4 0 0.0
Ni1─Se1─Se2 1.000 0.000 0.000 23.132 1.000 0.339 0.000 0.000 4 0 0.0
Se1─Ni1─Ni3 1.000 0.000 0.000 20.479 1.000 −0.004 0.000 0.000 4 0 0.0

Table 93.

SW potential parameters for single-layer 1H-NiSe2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 44.

Stress-strain for single-layer 1H-NiSe2 of dimension 100 × 100 Å along the armchair and zigzag directions.

The structure for the single-layer 1H-NiSe2 is shown in Figure 1 (with M=Ni and X=Se). Each Ni atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Ni atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant a = 3.33 Å and the bond length d Ni Se = 2.35 Å. The resultant angles are θ NiSeSe = θ SeNiNi = 90.228 ° and θ NiSeS e = 70.206 ° , in which atoms Se and Se’ are from different (top or bottom) groups.

Table 90 shows four VFF terms for the single-layer 1H-NiSe2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 43(a) . The ab initio calculations for the phonon dispersion are from Ref. [12]. The lowest acoustic branch (flexural mode) is almost linear in the ab initio calculations, which may be due to the violation of the rigid rotational invariance [20]. Figure 43(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 91 . The parameters for the three-body SW potential used by GULP are shown in Table 92 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 93 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-NiSe2 using LAMMPS, because the angles around atom Ni in Figure 1 (with M=Ni and X=Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M=Ni and X=Se) shows that, for 1H-NiSe2, we can differentiate these angles around the Ni atom by assigning these six neighboring Se atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Ni atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-NiSe2 under uniaxial tension at 1 and 300 K. Figure 44 shows the stress-strain curve for the tension of a single-layer 1H-NiSe2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-NiSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-NiSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 47.6 and 47.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.27 .

There is no available value for nonlinear quantities in the single-layer 1H-NiSe2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −173.9 and −197.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 6.1 N/m at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 5.9 N/m at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

Figure 45.

Phonon spectrum for single-layer 1H-NiTe2. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from Ref. [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

24. 1H-NITE2

Most existing theoretical studies on the single-layer 1H-NiTe2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-NiTe2.

The structure for the single-layer 1H-NiTe2 is shown in Figure 1 (with M=Ni and X=Te). Each Ni atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Ni atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant a = 3.59 Å and the bond length d Ni Te = 2.54 Å. The resultant angles are θ NiTeTe = θ TeNiNi = 89.933 ° and θ NiTeT e = 70.624 ° , in which atoms Te and Te’ are from different (top or bottom) groups.

Figure 46.

Stress-strain for single-layer 1H-NiTe2 of dimension 100 × 100 Å along the armchair and zigzag directions.

VFF type Bond stretching Angle bending
Expression 1 2 K Ni Te ( Δ r ) 2 1 2 K Ni Te Te ( Δ θ ) 2 1 2 K Ni Te T e ( Δ θ ) 2 1 2 K Te Ni Ni ( Δ θ ) 2
Parameter 6.712 2.656 2.656 2.656
r 0 or θ 0 2.540 89.933 70.624 89.933

Table 94.

The VFF model for single-layer 1H-NiTe2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Å2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

Table 94 shows four VFF terms for the single-layer 1H-NiTe2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 45(a) . The ab initio calculations for the phonon dispersion are from Ref. [12]. The lowest acoustic branch (flexural mode) is almost linear in the ab initio calculations, which may be due to the violation of the rigid rotational invariance [20]. The transverse acoustic branch is very close to the longitudinal acoustic branch in the ab initio calculations. Figure 45(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

A (eV) ρ (Å) B4) r min (Å) r max (Å)
Ni─Te 6.461 1.359 20.812 0.0 3.469

Table 95.

Two-body SW potential parameters for single-layer 1H-NiTe2 used by GULP [8] as expressed in Eq. (3).

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Ni Te Te 24.759 89.933 1.359 1.359 0.0 3.469 0.0 3.469 0.0 4.114
θ Ni Te T e 27.821 70.624 1.359 1.359 0.0 3.469 0.0 3.469 0.0 4.114
θ Te Ni Ni 24.759 89.933 1.359 1.359 0.0 3.469 0.0 3.469 0.0 4.114

Table 96.

Three-body SW potential parameters for single-layer 1H-NiTe2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 95 . The parameters for the three-body SW potential used by GULP are shown in Table 96 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 97 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-NiTe2 using LAMMPS, because the angles around atom Ni in Figure 1 (with M=Ni and X=Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M=Ni and X=Te) shows that, for 1H-NiTe2, we can differentiate these angles around the Ni atom by assigning these six neighboring Te atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Ni atom.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Ni1─Te1─Te1 1.000 1.359 2.553 0.000 1.000 0.000 6.461 6.107 4 0 0.0
Ni1─Te1─Te3 1.000 0.000 0.000 24.759 1.000 0.001 0.000 0.000 4 0 0.0
Ni1─Te1─Te2 1.000 0.000 0.000 27.821 1.000 0.332 0.000 0.000 4 0 0.0
Te1─Ni1─Ni3 1.000 0.000 0.000 24.759 1.000 0.001 0.000 0.000 4 0 0.0

Table 97.

SW potential parameters for single-layer 1H-NiTe2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-NiTe2 under uniaxial tension at 1 and 300 K. Figure 46 shows the stress-strain curve for the tension of a single-layer 1H-NiTe2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-NiTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-NiTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 53.2 and 53.6 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.32 .

There is no available value for nonlinear quantities in the single-layer 1H-NiTe2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −156.6 and −184.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 8.1 N/m at the ultimate strain of 0.26 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 7.8 N/m at the ultimate strain of 0.33 in the zigzag direction at the low temperature of 1 K.

Figure 47.

Phonon dispersion for single-layer 1H-NbS2. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The theoretical results (gray pentagons) are from Ref. [21]. The blue lines are from the present VFF model. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-NbS2 along ΓMKΓ.

25. 1H-NBS2

In 1983, the VFF model was developed to investigate the lattice dynamical properties in the bulk 2H-NbS2 [21]. In this section, we will develop the SW potential for the single-layer 1H-NbS2.

The structure for the single-layer 1H-NbS2 is shown in Figure 1 (with M=Nb and X=S). Each Nb atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Nb atoms. The structural parameters are from Ref. [21], including the lattice constant a = 3.31 Å and the bond length d Nb S = 2.47 Å. The resultant angles are θ NbSS = θ SNbNb = 84.140 ° and θ NbS S = 78.626 ° , in which atoms S and S’ are from different (top or bottom) groups.

Figure 48.

Stress-strain for single-layer 1H-NbS2 of dimension 100 × 100 Å along the armchair and zigzag directions.

VFF type Bond stretching Angle bending
Expression 1 2 K N b S ( Δ r ) 2 1 2 K N b S S ( Δ θ ) 2 1 2 K N b S S ( Δ θ ) 2 1 2 K S N b N b ( Δ θ ) 2
Parameter 8.230 4.811 4.811 4.811
r 0 or θ 0 2.470 84.140 78.626 84.140

Table 98.

The VFF model for single-layer 1H-NbS2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Å2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

Table 98 shows four VFF terms for the 1H-NbS2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 47(a) . The theoretical phonon frequencies (gray pentagons) are from Ref. [21], which are the phonon dispersion of bulk 2H-NbS2. We have used these phonon frequencies as the phonon dispersion of the single-layer 1H-NbS2, as the interlayer interaction in the bulk 2H-NbS2 only induces weak effects on the two in-plane acoustic branches. The interlayer coupling will strengthen the out-of-plane acoustic branch (flexural branch), so the flexural branch from the present VFF model (blue line) is lower than the theoretical results for bulk 2H-NbS2 (gray pentagons). Figure 47(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

A (eV) ρ (Å) B4) r min (Å) r max (Å)
Nb─S 6.439 1.116 18.610 0.0 3.300

Table 99.

Two-body SW potential parameters for single-layer 1H-NbS2 used by GULP [8] as expressed in Eq. (3).

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Nb S S 35.748 84.140 1.116 1.116 0.0 3.300 0.0 3.300 0.0 3.933
θ Nb S S 36.807 78.626 1.116 1.116 0.0 3.300 0.0 3.300 0.0 3.933
θ S Nb Nb 35.748 84.140 1.116 1.116 0.0 3.300 0.0 3.300 0.0 3.933

Table 100.

Three-body SW potential parameters for single-layer 1H-NbS2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

The parameters for the two-body SW potential used by GULP are shown in Table 99 . The parameters for the three-body SW potential used by GULP are shown in Table 100 . Parameters for the SW potential used by LAMMPS are listed in Table 101 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-NbS2 using LAMMPS, because the angles around atom Nb in Figure 1 (with M=Nb and X=S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M=Nb and X=S) shows that, for 1H-NbS2, we can differentiate these angles around the Nb atom by assigning these six neighboring S atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Nb atom.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Nb1─S1─S1 1.000 1.116 2.958 0.000 1.000 0.000 6.439 12.014 4 0 0.0
Nb1─S1─S3 1.000 0.000 0.000 35.748 1.000 0.102 0.000 0.000 4 0 0.0
Nb1─S1─S2 1.000 0.000 0.000 36.807 1.000 0.197 0.000 0.000 4 0 0.0
S1─Nb1─Nb3 1.000 0.000 0.000 35.748 1.000 0.102 0.000 0.000 4 0 0.0

Table 101.

SW potential parameters for single-layer 1H-NbS2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M=Nb and X=S).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-NbS2 under uniaxial tension at 1 and 300 K. Figure 48 shows the stress-strain curve for the tension of a single-layer 1H-NbS2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-NbS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-NbS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 87.7 and 87.2 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.27 .

There is no available value for the nonlinear quantities in the single-layer 1H-NbS2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −315.3 and −355.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.4 N/m at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.8 N/m at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

Figure 49.

Phonon dispersion for single-layer 1H-NbSe2. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The theoretical results (gray pentagons) are from Ref. [15]. The blue lines are from the present VFF model. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-NbSe2 along ΓMKΓ.

26. 1H-NBSE2

In 1983, the VFF model was developed to investigate the lattice dynamical properties in the bulk 2H-NbSe2 [15, 21]. In this section, we will develop the SW potential for the single-layer 1H-NbSe2.

The structure for the single-layer 1H-NbSe2 is shown in Figure 1 (with M=Nb and X=Se). Each Nb atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Nb atoms. The structural parameters are from Ref. [21], including the lattice constant a = 3.45 Å and the bond length d Nb Se = 2.60 Å. The resultant angles are θ NbSeSe = θ SNbNb = 83.129 ° and θ NbSeS e = 79.990 ° , in which atoms Se and Se’ are from different (top or bottom) groups.

Figure 50.

Stress-strain for single-layer 1H-NbSe2 of dimension 100 × 100 Å along the armchair and zigzag directions.

VFF type Bond stretching Angle bending
Expression 1 2 K Nb Se ( Δ r ) 2 1 2 K Nb Se Se ( Δ θ ) 2 1 2 K Nb Se S e ( Δ θ ) 2 1 2 K Se Nb Nb ( Δ θ ) 2
Parameter 8.230 4.811 4.811 4.811
r 0 or θ 0 2.600 83.129 79.990 83.129

Table 102.

The VFF model for single-layer 1H-NbSe2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV/Å2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

Table 102 shows four VFF terms for the 1H-NbSe2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 49(a) . The theoretical phonon frequencies (gray pentagons) are from Ref. [21], which are the phonon dispersion of bulk 2H-NbSe2. We have used these phonon frequencies as the phonon dispersion of the single-layer 1H-NbSe2, as the interlayer interaction in the bulk 2H-NbSe2 only induces weak effects on the two in-plane acoustic branches. The interlayer coupling will strengthen the out-of-plane acoustic branch (flexural branch), so the flexural branch from the present VFF model (blue line) is lower than the theoretical results for bulk 2H-NbSe2 (gray pentagons). It turns out that the VFF parameters for the single-layer 1H-NbSe2 are the same as the single-layer NbS2. The phonon dispersion for single-layer 1H-NbSe2 was also shown in Ref. [12]. Figure 49(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

A (eV) ρ (Å) B4) r min (Å) r max (Å)
Nb-Se 6.942 1.138 22.849 0.0 3.460

Table 103.

Two-body SW potential parameters for single-layer 1H-NbSe2 used by GULP [8] as expressed in Eq. (3).

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Nb Se Se 34.409 83.129 1.138 1.138 0.0 3.460 0.0 3.460 0.0 4.127
θ Nb Se S e 34.973 79.990 1.138 1.138 0.0 3.460 0.0 3.460 0.0 4.127
θ Se Nb Nb 34.409 83.129 1.138 1.138 0.0 3.460 0.0 3.460 0.0 4.127

Table 104.

Three-body SW potential parameters for single-layer 1H-NbSe2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Nb1─Se1─Se1 1.000 1.138 3.041 0.000 1.000 0.000 6.942 13.631 4 0 0.0
Nb1─Se1─Se3 1.000 0.000 0.000 34.409 1.000 0.120 0.000 0.000 4 0 0.0
Nb1─Se1─Se2 1.000 0.000 0.000 34.973 1.000 0.174 0.000 0.000 4 0 0.0
Se1─Nb1─Nb3 1.000 0.000 0.000 34.409 1.000 0.120 0.000 0.000 4 0 0.0

Table 105.

SW potential parameters for single-layer 1H-NbSe2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M=Nb and X=Se).

The parameters for the two-body SW potential used by GULP are shown in Table 103 . The parameters for the three-body SW potential used by GULP are shown in Table 104 . Parameters for the SW potential used by LAMMPS are listed in Table 105 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-NbSe2 using LAMMPS, because the angles around atom Nb in Figure 1 (with M=Nb and X=Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M=Nb and X=Se) shows that, for 1H-NbSe2, we can differentiate these angles around the Nb atom by assigning these six neighboring Se atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Nb atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-NbSe2 under uniaxial tension at 1 and 300 K. Figure 50 shows the stress-strain curve for the tension of a single-layer 1H-NbSe2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-NbSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-NbSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 80.2 and 80.7 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.29 .

There is no available value for the nonlinear quantities in the single-layer 1H-NbSe2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −258.8 and −306.1 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.2 N/m at the ultimate strain of 0.24 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 10.7 N/m at the ultimate strain of 0.28 in the zigzag direction at the low temperature of 1 K.

Figure 51.

Phonon spectrum for single-layer 1H-MoO2. (a) Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the ab initio results (pentagons) from Ref. [12]. (b) The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

27. 1H-MoO2

Most existing theoretical studies on the single-layer 1H-MoO2 are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-MoO2.

The structure for the single-layer 1H-MoO2 is shown in Figure 1 (with M = Mo and X = O). Each Mo atom is surrounded by six O atoms. These O atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each O atom is connected to three Mo atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant a = 2.78 Å and the bond length d Mo O = 2.00 Å. The resultant angles are θ MoOO = θ OMoMo = 88.054 ° and θ MoO O = 73.258 ° , in which atoms O and O′ are from different (top or bottom) groups.

VFF type Bond stretching Angle bending
Expression 1 2 K Mo O ( Δ r ) 2 1 2 K Mo O O ( Δ θ ) 2 1 2 K Mo O O ( Δ θ ) 2 1 2 K O Mo Mo ( Δ θ ) 2
Parameter 14.622 8.410 8.410 8.410
r 0 or θ 0 2.000 88.054 73.258 88.054

Table 106.

The VFF model for single-layer 1H-MoO2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV / Å 2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

A (eV) ρ (Å) B4) r min (Å) r max (Å)
Mo-O 8.317 1.015 8.000 0.0 2.712

Table 107.

Two-body SW potential parameters for single-layer 1H-MoO2 used by GULP [8] as expressed in Eq. (3).

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Mo O O 72.735 88.054 1.015 1.015 0.0 2.712 0.0 2.712 0.0 3.222
θ Mo O O 79.226 73.258 1.015 1.015 0.0 2.712 0.0 2.712 0.0 3.222
θ O Mo Mo 72.735 88.054 1.015 1.015 0.0 2.712 0.0 2.712 0.0 3.222

Table 108.

Three-body SW potential parameters for single-layer 1H-MoO2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Mo1─O1─O1 1.000 1.015 2.673 0.000 1.000 0.000 8.317 7.541 4 0 0.0
Mo1─O1─O3 1.000 0.000 0.000 72.735 1.000 0.034 0.000 0.000 4 0 0.0
Mo1─O1─O2 1.000 0.000 0.000 79.226 1.000 0.288 0.000 0.000 4 0 0.0
O1─Mo1─Mo3 1.000 0.000 0.000 72.735 1.000 0.034 0.000 0.000 4 0 0.0

Table 109.

SW potential parameters for single-layer 1H-MoO2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Figure 52.

Stress-strain for single-layer 1H-MoO2 of dimension 100 × 100 Å along the armchair and zigzag directions.

Table 106 shows four VFF terms for the single-layer 1H-MoO2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the acoustic branches in the phonon dispersion along the ΓM as shown in Figure 51(a) . The ab initio calculations for the phonon dispersion are from Ref. [12]. Figure 51(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 107 . The parameters for the three-body SW potential used by GULP are shown in Table 108 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 109 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-MoO2 using LAMMPS, because the angles around atom Mo in Figure 1 (with M = Mo and X = O) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Mo and X = O) shows that, for 1H-MoO2, we can differentiate these angles around the Mo atom by assigning these six neighboring O atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Mo atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-MoO2 under uniaxial tension at 1 and 300 K. Figure 52 shows the stress-strain curve for the tension of a single-layer 1H-MoO2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-MoO2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-MoO2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 210.0 and 209.3 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.17 .

There is no available value for nonlinear quantities in the single-layer 1H-MoO2. We have thus used the nonlinear parameter B = 0.5 d 4 in Eq. (5), which is close to the value of B in most materials. The value of the third-order nonlinear elasticity D can be extracted by fitting the stress-strain relation to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −1027.8 and −1106.8 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 21.0 N/m at the ultimate strain of 0.19 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 20.1 N/m at the ultimate strain of 0.22 in the zigzag direction at the low temperature of 1 K.

28. 1H-MoS2

Several potentials have been proposed to describe the interaction for the single-layer 1H-MoS2. In 1975, Wakabayashi et al. developed a VFF model to calculate the phonon spectrum of the bulk 2H-MoS2 [22]. In 2009, Liang et al. parameterized a bond-order potential for 1H-MoS2 [23], which is based on the bond order concept underlying the Brenner potential [6]. A separate force field model was parameterized in 2010 for MD simulations of bulk 2H-MoS2 [24]. The present author (J.W.J.) and his collaborators parameterized the SW potential for 1H-MoS2 in 2013 [13], which was improved by one of the present authors (J.W.J.) in 2015 [7]. Recently, another set of parameters for the SW potential were proposed for the single-layer 1H-MoS2 [25].

VFF type Bond stretching Angle bending
Expression 1 2 K M o S ( Δ r M o S ) 2 1 2 K M o S S ( Δ θ M o S S ) 2 1 2 K S M o M o ( Δ θ S M o M o ) 2
Parameter 8.640 5.316 4.891
r 0 or θ 0 2.382 80.581 80.581

Table 110.

The VFF model parameters for single-layer 1H-MoS2 from Ref. [22].

The second line gives the expression for each VFF term. Parameters are in the unit of eV / Å 2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

A (eV) ρ (Å) B4) r min (Å) r max (Å)
r Mo S 6.918 1.252 17.771 0.0 3.16

Table 111.

Two-body SW potential parameters for single-layer 1H-MoS2 used by GULP [8] as expressed in Eq. (3).

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ MoSS 67.883 81.788 1.252 1.252 0.0 3.16 0.0 3.16 0.0 3.78
θ SMoMo 62.449 81.788 1.252 1.252 0.0 3.16 0.0 3.16 0.0 4.27

Table 112.

Three-body SW potential parameters for single-layer 1H-MoS2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Mo1─S1─S1 1.000 1.252 2.523 0.000 1.000 0.000 6.918 7.223 4 0 0.0
Mo1─S1─S3 1.000 0.000 0.000 67.883 1.000 0.143 0.000 0.000 4 0 0.0
S1─Mo1─Mo3 1.000 0.000 0.000 62.449 1.000 0.143 0.000 0.000 4 0 0.0

Table 113.

SW potential parameters for single-layer 1H-MoS2 used by LAMMPS 9 as expressed in Eqs. (9) and 10.

Atom types in the first column are displayed in Figure 2 (with M = Mo and X = S).

Figure 53.

Phonon spectrum for single-layer 1H-MoS2. Phonon dispersion along the ΓM direction in the Brillouin zone. The results from the VFF model (lines) are comparable with the experiment data (pentagons) from Ref. [22]. The phonon dispersion from the SW potential is exactly the same as that from the VFF model.

Figure 54.

Stress-strain for single-layer 1H-MoS2 of dimension 100 × 100 Å along the armchair and zigzag directions.

We show the VFF model and the SW potential for single-layer 1H-MoS2 in this section. These potentials have been developed in previous works. The VFF model presented here is from Ref. [22], while the SW potential presented in this section is from Ref. [7].

The structural parameters for the single-layer 1H-MoS2 are from the first-principles calculations as shown in Figure 1 (with M = Mo and X = S) [26]. The Mo atom layer in the single-layer 1H-MoS2 is sandwiched by two S atom layers. Accordingly, each Mo atom is surrounded by six S atoms, while each S atom is connected to three Mo atoms. The bond length between neighboring Mo and S atoms is d = 2.382 Å, and the angles are θ MoSS = 80.581 ° and θ SMoMo = 80.581 ° .

The VFF model for single-layer 1H-MoS2 is from Ref. [22], which is able to describe the phonon spectrum and the sound velocity accurately. We have listed the first three leading force constants for single-layer 1H-MoS2 in Table 110 , neglecting other weak interaction terms. The SW potential parameters for single-layer 1H-MoS2 used by GULP are listed in Tables 111 and 112 . The SW potential parameters for single-layer 1H-MoS2 used by LAMMPS [9] are listed in Table 113 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-MoS2 using LAMMPS, because the angles around atom Mo in Figure 1 (with M = Mo and X = S) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Mo and X = S) shows that, for 1H-MoS2, we can differentiate these angles around the Mo atom by assigning these six neighboring S atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Mo atom.

We use GULP to compute the phonon dispersion for the single-layer 1H-MoS2 as shown in Figure 53 . The results from the VFF model are quite comparable with the experiment data. The phonon dispersion from the SW potential is the same as that from the VFF model, which indicates that the SW potential has fully inherited the linear portion of the interaction from the VFF model.

VFF type Bond stretching Angle bending
Expression 1 2 K M o S e ( Δ r ) 2 1 2 K M o S e S e ( Δ θ ) 2 1 2 K M o S e S e ( Δ θ ) 2 1 2 K S e M o M o ( Δ θ ) 2
Parameter 7.928 6.945 6.945 5.782
r 0 or θ 0 2.528 82.119 81.343 82.119

Table 114.

The VFF model for single-layer 1H-MoSe2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV / Å 2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-MoS2 under uniaxial tension at 1 and 300 K. Figure 54 shows the stress-strain curve during the tension of a single-layer 1H-MoS2 of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-MoS2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-MoS2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 97 and 96 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is isotropic in the armchair and zigzag directions. These values are in considerable agreement with the experimental results, e.g., 120 ± 30 N/m from Refs [27, 28], or 180 ± 60 N/m from Ref. [29]. The third-order nonlinear elastic constant D can be obtained by fitting the stress-strain relation to σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D are −418 and −461 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.27 .

29. 1H-MoSe2

There is a recent parameter set for the SW potential in the single-layer 1H-MoSe2 [25]. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-MoSe2.

Figure 55.

Phonon dispersion for single-layer 1H-MoSe2. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from Ref. [30]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-MoSe2 along ΓMKΓ.

A (eV) ρ (Å) B4) r min (Å) r max (Å)
Mo-Se 5.737 0.913 18.787 0.0 3.351

Table 115.

Two-body SW potential parameters for single-layer 1H-MoSe2 used by GULP [8] as expressed in Eq. (3).

The structure for the single-layer 1H-MoSe2 is shown in Figure 1 (with M = Mo and X = Se). Each Mo atom is surrounded by six Se atoms. These Se atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Se atom is connected to three Mo atoms. The structural parameters are from Ref. [30], including the lattice constant a = 3.321 Å and the bond length d Mo Se = 2.528 Å. The resultant angles are θ MoSeSe = θ SeMoMo = 82.119 ° and θ MoSeS e = 81.343 ° , in which atoms Se and Se′ are from different (top or bottom) groups.

Figure 56.

Stress-strain for single-layer 1H-MoSe2 of dimension 100 × 100 Å along the armchair and zigzag directions.

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Mo Se Se 32.526 82.119 0.913 0.913 0.0 3.351 0.0 3.351 0.0 4.000
θ Mo Se S e 32.654 81.343 0.913 0.913 0.0 3.351 0.0 3.351 0.0 4.000
θ Se Mo Mo 27.079 82.119 0.913 0.913 0.0 3.351 0.0 3.351 0.0 4.000

Table 116.

Three-body SW potential parameters for single-layer 1H-MoSe2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

Table 114 shows four VFF terms for the 1H-MoSe2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 55(a) . The ab initio calculations for the phonon dispersion are from Ref. [30]. Similar phonon dispersion can also be found in other ab initio calculations [12, 3134]. Figure 55(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 115 . The parameters for the three-body SW potential used by GULP are shown in Table 116 . Parameters for the SW potential used by LAMMPS are listed in Table 117 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-MoSe2 using LAMMPS, because the angles around atom Mo in Figure 1 (with M = Mo and X = Se) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Mo and X = Se) shows that, for 1H-MoSe2, we can differentiate these angles around the Mo atom by assigning these six neighboring Se atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Mo atom.

σ (eV) a (Å) λ γ cos θ 0 A L B L p q 100 × 100 Tol
Mo1─Se1─Se1 1.000 0.913 3.672 0.000 1.000 0.000 5.737 27.084 4 0 0.0
Mo1─Se1─Se3 1.000 0.000 0.000 32.526 1.000 0.137 0.000 0.000 4 0 0.0
Mo1─Se1─Se2 1.000 0.000 0.000 32.654 1.000 0.151 0.000 0.000 4 0 0.0
Se1─Mo1─Mo3 1.000 0.000 0.000 27.079 1.000 0.137 0.000 0.000 4 0 0.0

Table 117.

SW potential parameters for single-layer 1H-MoSe2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = Mo and X = Se).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-MoSe2 under uniaxial tension at 1 and 300 K. Figure 56 shows the stress-strain curve during the tension of a single-layer 1H-MoSe2 of dimension 100 × 100 Å . Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-MoSe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-MoSe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 103.0 and 101.8 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are in considerable agreement with the experimental results, e.g., 103.9 N/m from Ref. [18], or 113.9 N/m from Ref. [35]. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.24 , which agrees quite well with the ab initio value of 0.23 [18].

VFF type Bond stretching Angle bending
Expression 1 2 K Mo Te ( Δ r ) 2 1 2 K Mo Te Te ( Δ θ ) 2 1 2 K Mo Te T e ( Δ θ ) 2 1 2 K Te Mo Mo ( Δ θ ) 2
Parameter 6.317 6.184 6.184 5.225
r 0 or θ 0 2.730 81.111 82.686 81.111

Table 118.

The VFF model for single-layer 1H-MoTe2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV / Å 2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

We have determined the nonlinear parameter to be B = 0.46 d 4 in Eq. (5) by fitting to the third-order nonlinear elastic constant D from the ab initio calculations [35]. We have extracted the value of D = 383.7 N/m by fitting the stress-strain relation along the armchair direction in the ab initio calculations to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −365.4 and −402.4 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 13.6 N/m at the ultimate strain of 0.23 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 13.0 N/m at the ultimate strain of 0.27 in the zigzag direction at the low temperature of 1 K.

30. 1H-MoTe2

Most existing theoretical studies on the single-layer 1H-MoTe2 are based on the first-principles calculations. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-MoTe2.

The structure for the single-layer 1H-MoTe2 is shown in Figure 1 (with M = Mo and X = Te). Each Mo atom is surrounded by six Te atoms. These Te atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each Te atom is connected to three Mo atoms. The structural parameters are from Ref. [36], including the lattice constant a = 3.55 Å and the bond length d Mo Te = 2.73 Å . The resultant angles are θ MoTeTe = θ TeMoMo = 81.111 ° and θ MoTeT e = 82.686 ° , in which atoms Te and Te′ are from different (top or bottom) groups.

Figure 57.

Phonon dispersion for single-layer 1H-MoTe2. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the ΓM direction. The ab initio results (gray pentagons) are from Ref. [36]. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-MoTe2 along ΓMKΓ.

A (eV) ρ (Å) B4) r min (Å) r max (Å)
Mo-Te 5.086 0.880 24.440 0.0 3.604

Table 119.

Two-body SW potential parameters for single-layer 1H-MoTe2 used by GULP [8] as expressed in Eq. (3).

Figure 58.

Stress-strain for single-layer 1H-MoTe2 of dimension θ 0 Å along the armchair and zigzag directions.

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Mo Te Te 23.705 81.111 0.880 0.880 0.0 3.604 0.0 3.604 0.0 4.305
θ Mo Te T e 23.520 82.686 0.880 0.880 0.0 3.604 0.0 3.604 0.0 4.305
θ Te Mo Mo 20.029 81.111 0.880 0.880 0.0 3.604 0.0 3.604 0.0 4.305

Table 120.

Three-body SW potential parameters for single-layer 1H-MoTe2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

Table 118 shows four VFF terms for the 1H-MoTe2; one of which is the bond stretching interaction shown by Eq. (1), while the other three terms are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ΓM as shown in Figure 57(a) . The ab initio calculations for the phonon dispersion are from Ref. [36]. Similar phonon dispersion can also be found in other ab initio calculations [12, 34, 37]. Figure 57(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

The parameters for the two-body SW potential used by GULP are shown in Table 119 . The parameters for the three-body SW potential used by GULP are shown in Table 120 . Parameters for the SW potential used by LAMMPS are listed in Table 121 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-MoTe2 using LAMMPS, because the angles around atom Mo in Figure 1 (with M = Mo and X = Te) are not distinguishable in LAMMPS. We have suggested two options to differentiate these angles by implementing some additional constraints in LAMMPS, which can be accomplished by modifying the source file of LAMMPS [13, 14]. According to our experience, it is not so convenient for some users to implement these constraints and recompile the LAMMPS package. Hence, in the present work, we differentiate the angles by introducing more atom types, so it is not necessary to modify the LAMMPS package. Figure 2 (with M = Mo and X = Te) shows that, for 1H-MoTe2, we can differentiate these angles around the Mo atom by assigning these six neighboring Te atoms with different atom types. It can be found that 12 atom types are necessary for the purpose of differentiating all 6 neighbors around 1 Mo atom.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Mo1─Te1─Te1 1.000 0.900 4.016 0.000 1.000 0.000 5.169 37.250 4 0 0.0
Mo1─Te1─Te3 1.000 0.000 0.000 24.163 1.000 0.143 0.000 0.000 4 0 0.0
Te1─Mo1─Mo3 1.000 0.000 0.000 20.416 1.000 0.143 0.000 0.000 4 0 0.0

Table 121.

SW potential parameters for single-layer 1H-MoTe2 used by LAMMPS [9] as expressed in Eqs. (9) and (10).

Atom types in the first column are displayed in Figure 2 (with M = Mo and X = Te).

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-MoTe2 under uniaxial tension at 1 and 300 K. Figure 58 shows the stress-strain curve for the tension of a single-layer 1H-MoTe2 of dimension 100 × 100 Å . Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-MoTe2 is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasi-two-dimensional structure of the single-layer 1H-MoTe2. The Young’s modulus can be obtained by a linear fitting of the stress-strain relation in the small strain range of [0, 0.01]. The Young’s modulus is 79.8 and 78.5 N/m along the armchair and zigzag directions, respectively. The Young’s modulus is essentially isotropic in the armchair and zigzag directions. These values are in considerable agreement with the experimental results, e.g., 79.4 N/m from Ref. [18], or 87.0 N/m from Ref. [35]. The Poisson’s ratio from the VFF model and the SW potential is ν x y = ν y x = 0.25 , which agrees with the ab initio value of 0.24 [18].

VFF type Bond stretching Angle bending
Expression 1 2 K Ta S ( Δ r ) 2 1 2 K Ta S S ( Δ θ ) 2 1 2 K Ta S S ( Δ θ ) 2 1 2 K S Ta Ta ( Δ θ ) 2
Parameter 8.230 4.811 4.811 4.811
r 0 or θ 0 2.480 83.879 78.979 83.879

Table 122.

The VFF model for single-layer 1H-TaS2.

The second line gives an explicit expression for each VFF term. The third line is the force constant parameters. Parameters are in the unit of eV / Å 2 for the bond stretching interaction and in the unit of eV for the angle bending interaction. The fourth line gives the initial bond length (in the unit of Å ) for the bond stretching interaction and the initial angle (in the unit of degrees) for the angle bending interaction. The angle θ i j k has atom i as the apex.

A (eV) ρ (Å) B4) r min (Å) r max (Å)
Ta-S 6.446 1.111 18.914 0.0 3.310

Table 123.

Two-body SW potential parameters for single-layer 1H-TaS2 used by GULP [8] as expressed in Eq. (3).

We have determined the nonlinear parameter to be B = 0.44 d 4 in Eq. (5) by fitting to the third-order nonlinear elastic constant D from the ab initio calculations [35]. We have extracted the value of D = 278.2 N/m by fitting the stress-strain relation along the armchair direction in the ab initio calculations to the function σ = E ϵ + 1 2 D ϵ 2 with E as the Young’s modulus. The values of D from the present SW potential are −250.5 and −276.6 N/m along the armchair and zigzag directions, respectively. The ultimate stress is about 11.7 N/m at the ultimate strain of 0.25 in the armchair direction at the low temperature of 1 K. The ultimate stress is about 11.1 N/m at the ultimate strain of 0.29 in the zigzag direction at the low temperature of 1 K.

31. 1H-TaS2

In 1983, the VFF model was developed to investigate the lattice dynamical properties in the bulk 2H-TaS2 [21]. In this section, we will develop the SW potential for the single-layer 1H-TaS2.

Figure 59.

Phonon dispersion for single-layer 1H-TaS2. (a) The VFF model is fitted to the three acoustic branches in the long wave limit along the Γ M direction. The theoretical results (gray pentagons) are from Ref. [21]. The blue lines are from the present VFF model. (b) The VFF model (blue lines) and the SW potential (red lines) give the same phonon dispersion for single-layer 1H-TaS2 along Γ MK Γ .

The structure for the single-layer 1H-TaS2 is shown in Figure 1 (with M = Ta and X = S). Each Ta atom is surrounded by six S atoms. These S atoms are categorized into the top group (e.g., atoms 1, 3, and 5) and bottom group (e.g., atoms 2, 4, and 6). Each S atom is connected to three Ta atoms. The structural parameters are from Ref. [21], including the lattice constant a = 3.315 Å and the bond length d Ta S = 2.48 Å . The resultant angles are θ TaSS = θ STaTa = 83.879 ° and θ TaS S = 78.979 ° , in which atoms S and S′ are from different (top or bottom) groups.

Figure 60.

Stress-strain for single-layer 1H-TaS2 of dimension 100 × 100 Å along the armchair and zigzag directions.

K (eV) θ 0 (°) ρ 1 (Å) ρ 2 (Å) r min 12 (Å) r max 12 (Å) r min 13 (Å) r max 13 (Å) r min 23 (Å) r max 23 (Å)
θ Ta S S 35.396 83.879 1.111 1.111 0.0 3.310 0.0 3.310 0.0 3.945
θ Ta S S 36.321 78.979 1.111 1.111 0.0 3.310 0.0 3.310 0.0 3.945
θ S Ta Ta 35.396 83.879 1.111 1.111 0.0 3.310 0.0 3.310 0.0 3.945

Table 124.

Three-body SW potential parameters for single-layer 1H-TaS2 used by GULP [8] as expressed in Eq. (4).

The angle θ i j k in the first line indicates the bending energy for the angle with atom i as the apex.

Table 122 shows the VFF terms for the 1H-TaS2; one of which is the bond stretching interaction shown by Eq. (1), while the others are the angle bending interaction shown by Eq. (2). These force constant parameters are determined by fitting to the three acoustic branches in the phonon dispersion along the ϵ as shown in Figure 59(a) . The theoretical phonon frequencies (gray pentagons) are from Ref. [21], which are the phonon dispersion of bulk 2H-TaS2. We have used these phonon frequencies as the phonon dispersion of the single-layer 1H-TaS2, as the interlayer interaction in the bulk 2H-TaS2 only induces weak effects on the two in-plane acoustic branches. The interlayer coupling will strengthen the out-of-plane acoustic (flexural) branch, so the flexural branch from the present VFF model (blue line) is lower than the theoretical results for bulk 2H-TaS2 (gray pentagons). Figure 59(b) shows that the VFF model and the SW potential give exactly the same phonon dispersion, as the SW potential is derived from the VFF model.

ϵ (eV) σ (Å) a λ γ cos θ 0 A L B L p q Tol
Ta1─S1─S1 1.000 1.111 2.979 0.000 1.000 0.000 6.446 12.408 4 0 0.0