1H-MX_{2}, with M as the transition metal and X as oxygen or dichalcogenide.

## 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-ScO_{2} | 1H-ScS_{2} | 1H-ScSe_{2} | 1H-ScTe_{2} | 1H-TiTe_{2} | 1H-VO_{2} | 1H-VS_{2} | 1H-VSe_{2} | 1H-VTe_{2} |

1H-CrO_{2} | 1H-CrS_{2} | 1H-CrSe_{2} | 1H-CrTe_{2} | 1H-MnO_{2} | 1H-FeO_{2} | 1H-FeS_{2} | 1H-FeSe_{2} | 1H-FeTe_{2} |

1H-CoTe_{2} | 1H-NiS_{2} | 1H-NiSe_{2} | 1H-NiTe_{2} | 1H-NbS_{2} | 1H-NbSe_{2} | 1H-MoO_{2} | 1H-MoS_{2} | 1H-MoSe_{2} |

1H-MoTe_{2} | 1H-TaS_{2} | 1H-TaSe_{2} | 1H-WO_{2} | 1H-WS_{2} | 1H-WSe_{2} | 1H-WTe_{2} |

1T-ScO_{2} | 1T-ScS_{2} | 1T-ScSe_{2} | 1T-ScTe_{2} | 1T-TiS_{2} | 1T-TiSe_{2} | 1T-TiTe_{2} | 1T-VS_{2} | 1T-VSe_{2} |

1T-VTe_{2} | 1T-MnO_{2} | 1T-MnS_{2} | 1T-MnSe_{2} | 1T-MnTe_{2} | 1T-CoTe_{2} | 1T-NiO_{2} | 1T-NiS_{2} | 1T-NiSe_{2} |

1T-NiTe_{2} | 1T-ZrS_{2} | 1T-ZrSe_{2} | 1T-ZrTe_{2} | 1T-NbS_{2} | 1T-NbSe_{2} | 1T-NbTe_{2} | 1T-MoS_{2} | 1T-MoSe_{2} |

1T-MoTe_{2} | 1T-TcS_{2} | 1T-TcSe_{2} | 1T-TcTe_{2} | 1T-RhTe_{2} | 1T-PdS_{2} | 1T-PdSe_{2} | 1T-PdTe_{2} | 1T-SnS_{2} |

1T-SnSe_{2} | 1T-HfS_{2} | 1T-HfSe_{2} | 1T-HfTe_{2} | 1T-TaS_{2} | 1T-TaSe_{2} | 1T-TaTe_{2} | 1T-WS_{2} | 1T-WSe_{2} |

1T-WTe_{2} | 1T-ReS_{2} | 1T-ReSe_{2} | 1T-ReTe_{2} | 1T-IrTe_{2} | 1T-PtS_{2} | 1T-PtSe_{2} | 1T-PtTe_{2} |

Black phosphorus | p-Arsenene | p-Antimonene | p-Bismuthene |

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 |

Silicene | Germanene | Stanene | Indiene |
---|---|---|---|

Blue phosphorus | b-Arsenene | b-Antimonene | b-Bismuthene |

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 |

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

BO | AlO | GaO | InO |

BS | AlS | GaS | InS |

BSe | AlSe | GaSe | InSe |

BTe | AlTe | GaTe | InTe |

Borophene |

## 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 *r* _{0} as the initial bond length. The angle bending gives the energy increment for an angle resulting from an angle variation *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,

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,

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],

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,

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

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

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

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 9.417 | 4.825 | 4.825 | 4.825 |

r _{0} or θ _{0} | 2.090 | 98.222 | 58.398 | 98.222 |

A (eV) | ρ (Å) | B (Å^{4}) | r _{min} (Å) | r _{max}(Å) | |
---|---|---|---|---|---|

Sc-O | 7.506 | 1.380 | 9.540 | 0.0 | 2.939 |

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].

## 3. 1H-SCO_{2}

Most existing theoretical studies on the single-layer 1H-ScO_{2} are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-ScO_{2}.

The structure for the single-layer 1H-ScO_{2} 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

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) | |
---|---|---|---|---|---|---|---|---|---|---|

63.576 | 98.222 | 1.380 | 1.380 | 0.0 | 2.939 | 0.0 | 2.939 | 0.0 | 3.460 | |

85.850 | 58.398 | 1.380 | 1.380 | 0.0 | 2.939 | 0.0 | 2.939 | 0.0 | 3.460 | |

63.576 | 98.222 | 1.380 | 1.380 | 0.0 | 2.939 | 0.0 | 2.939 | 0.0 | 3.460 |

Table 10 shows four VFF terms for the single-layer 1H-ScO_{2}; 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-ScO_{2} 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-ScO_{2}, 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.

ϵ (eV) | σ (Å) | a | λ | γ | cos θ _{0} | A _{L} | B _{L} | p | q | Tol | |
---|---|---|---|---|---|---|---|---|---|---|---|

Sc_{1}─O_{1}─O_{1} | 1.000 | 1.380 | 2.129 | 0.000 | 1.000 | 0.000 | 7.506 | 2.627 | 4 | 0 | 0.0 |

Sc_{1}─O_{1}─O_{3} | 1.000 | 0.000 | 0.000 | 63.576 | 1.000 | −0.143 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Sc_{1}─O_{1}─O_{2} | 1.000 | 0.000 | 0.000 | 85.850 | 1.000 | 0.524 | 0.000 | 0.000 | 4 | 0 | 0.0 |

O_{1}─Sc_{1}─Sc_{3} | 1.000 | 0.000 | 0.000 | 63.576 | 1.000 | −0.143 | 0.000 | 0.000 | 4 | 0 | 0.0 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-ScO_{2} under uniaxial tension at 1 and 300 K. Figure 4 shows the stress-strain curve for the tension of a single-layer 1H-ScO_{2} of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-ScO_{2} 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-ScO_{2}. 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

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 5.192 | 2.027 | 2.027 | 2.027 |

r _{0} or θ _{0} | 2.520 | 94.467 | 64.076 | 94.467 |

A (eV) | ρ (Å) | B (Å^{4}) | r _{min }(Å) | r _{max }(Å) | |
---|---|---|---|---|---|

Sc─S | 5.505 | 1.519 | 20.164 | 0.0 | 3.498 |

There is no available value for nonlinear quantities in the single-layer 1H-ScO_{2}. 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 *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.

## 4. 1H-SCS_{2}

Most existing theoretical studies on the single-layer 1H-ScS_{2} are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-ScS_{2}.

The structure for the single-layer 1H-ScS_{2} 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

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) | |
---|---|---|---|---|---|---|---|---|---|---|

22.768 | 94.467 | 1.519 | 1.519 | 0.0 | 3.498 | 0.0 | 3.498 | 0.0 | 4.132 | |

27.977 | 64.076 | 1.519 | 1.519 | 0.0 | 3.498 | 0.0 | 3.498 | 0.0 | 4.132 | |

22.768 | 94.467 | 1.519 | 1.519 | 0.0 | 3.498 | 0.0 | 3.498 | 0.0 | 4.132 |

Table 14 shows four VFF terms for the single-layer 1H-ScS_{2}; 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-ScS_{2} 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-ScS_{2}, 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 | |
---|---|---|---|---|---|---|---|---|---|---|---|

Sc_{1}─S_{1}─S_{1} | 1.000 | 1.519 | 2.303 | 0.000 | 1.000 | 0.000 | 5.505 | 3.784 | 4 | 0 | 0.0 |

Sc_{1}─S_{1}─S_{3} | 1.000 | 0.000 | 0.000 | 22.768 | 1.000 | −0.078 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Sc_{1}─S_{1}─S_{2} | 1.000 | 0.000 | 0.000 | 27.977 | 1.000 | 0.437 | 0.000 | 0.000 | 4 | 0 | 0.0 |

S_{1}─Sc_{1}─Sc_{3} | 1.000 | 0.000 | 0.000 | 22.768 | 1.000 | −0.078 | 0.000 | 0.000 | 4 | 0 | 0.0 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-ScS_{2} under uniaxial tension at 1 and 300 K. Figure 6 shows the stress-strain curve for the tension of a single-layer 1H-ScS_{2} of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-ScS_{2} 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-ScS_{2}. 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

There is no available value for nonlinear quantities in the single-layer 1H-ScS_{2}. 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 *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 | ||||

Parameter | 5.192 | 2.027 | 2.027 | 2.027 |

2.650 | 92.859 | 66.432 | 92.859 |

A (eV) | ρ (Å) | B (Å^{4}) | r _{min} (Å) | r _{max}(Å) | |
---|---|---|---|---|---|

Sc-Se | 5.853 | 1.533 | 24.658 | 0.0 | 3.658 |

## 5. 1H-SCSE_{2}

Most existing theoretical studies on the single-layer 1H-ScSe_{2} are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-ScSe_{2}.

The structure for the single-layer 1H-ScSe_{2} 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

Table 18 shows four VFF terms for the single-layer 1H-ScSe_{2}; 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.

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) | |
---|---|---|---|---|---|---|---|---|---|---|

21.292 | 92.859 | 1.533 | 1.533 | 0.0 | 3.658 | 0.0 | 3.658 | 0.0 | 4.327 | |

25.280 | 66.432 | 1.533 | 1.533 | 0.0 | 3.658 | 0.0 | 3.658 | 0.0 | 4.327 | |

21.292 | 92.859 | 1.533 | 1.533 | 0.0 | 3.658 | 0.0 | 3.658 | 0.0 | 4.327 |

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-ScSe_{2} 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-ScSe_{2}, 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 | |
---|---|---|---|---|---|---|---|---|---|---|---|

Sc_{1}─Se_{1}-Se_{1} | 1.000 | 1.533 | 2.386 | 0.000 | 1.000 | 0.000 | 5.853 | 4.464 | 4 | 0 | 0.0 |

Sc_{1}─Se_{1}─Se_{3} | 1.000 | 0.000 | 0.000 | 21.292 | 1.000 | −0.050 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Sc_{1}─Se_{1}─Se_{2} | 1.000 | 0.000 | 0.000 | 25.280 | 1.000 | 0.400 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Se_{1}─Sc_{1}─Sc_{3} | 1.000 | 0.000 | 0.000 | 21.292 | 1.000 | −0.050 | 0.000 | 0.000 | 4 | 0 | 0.0 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-ScSe_{2} under uniaxial tension at 1 and 300 K. Figure 8 shows the stress-strain curve for the tension of a single-layer 1H-ScSe_{2} of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-ScSe_{2} 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-ScSe_{2}. 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

There is no available value for nonlinear quantities in the single-layer 1H-ScSe_{2}. 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 *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.

## 6. 1H-SCTE_{2}

Most existing theoretical studies on the single-layer 1H-ScTe_{2} are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-ScTe_{2}.

The structure for the single-layer 1H-ScTe_{2} 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

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 5.192 | 2.027 | 2.027 | 2.027 |

2.890 | 77.555 | 87.364 | 87.364 |

Table 22 shows four VFF terms for the single-layer 1H-ScTe_{2}; 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-ScSe_{2}, 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 TaSe_{2} and NbSe_{2} 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.

Sc-Te | 4.630 | 1.050 | 34.879 | 0.0 | 3.761 |

11.848 | 77.555 | 1.050 | 1.050 | 0.0 | 3.761 | 0.0 | 3.761 | 0.0 | 4.504 | |

11.322 | 87.364 | 1.050 | 1.050 | 0.0 | 3.761 | 0.0 | 3.761 | 0.0 | 4.504 | |

11.848 | 77.555 | 1.050 | 1.050 | 0.0 | 3.761 | 0.0 | 3.761 | 0.0 | 4.504 |

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Sc_{1}─Te_{1}─Te_{1} | 1.000 | 1.050 | 3.581 | 0.000 | 1.000 | 0.000 | 4.630 | 28.679 | 4 | 0 | 0.0 |

Sc_{1}─Te_{1}─Te_{3} | 1.000 | 0.000 | 0.000 | 11.848 | 1.000 | 0.216 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Sc_{1}─Te_{1}─Te_{2} | 1.000 | 0.000 | 0.000 | 11.322 | 1.000 | 0.046 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Te_{1}─Sc_{1}─Sc_{3} | 1.000 | 0.000 | 0.000 | 11.848 | 1.000 | 0.216 | 0.000 | 0.000 | 4 | 0 | 0.0 |

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-ScTe_{2} 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-ScTe_{2}, 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-ScTe_{2} under uniaxial tension at 1 and 300 K. Figure 10 shows the stress-strain curve for the tension of a single-layer 1H-ScTe_{2} of dimension _{2} 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-ScTe_{2}. 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

There is no available value for nonlinear quantities in the single-layer 1H-ScTe_{2}. We have thus used the nonlinear parameter

## 7. 1H-TITE_{2}

Most existing theoretical studies on the single-layer 1H-TiTe_{2} 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-TiTe_{2}.

The structure for the single-layer 1H-TiTe_{2} 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

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 4.782 | 3.216 | 3.216 | 3.216 |

2.750 | 82.323 | 81.071 | 82.323 |

Table 26 shows the VFF terms for the 1H-TiTe_{2}; 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 *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.

Ti-Te | 4.414 | 1.173 | 28.596 | 0.0 | 3.648 |

22.321 | 82.323 | 1.173 | 1.173 | 0.0 | 3.648 | 0.0 | 3.648 | 0.0 | 4.354 | |

22.463 | 81.071 | 1.173 | 1.173 | 0.0 | 3.648 | 0.0 | 3.648 | 0.0 | 4.354 | |

11.321 | 82.323 | 1.173 | 1.173 | 0.0 | 3.648 | 0.0 | 3.648 | 0.0 | 4.354 |

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-TiTe_{2} 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-TiTe_{2}, 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.

p | q | Tol | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Ti_{1}-Te_{1}-Te_{1} | 1.000 | 1.173 | 3.110 | 0.000 | 1.000 | 0.000 | 4.414 | 15.100 | 4 | 0 | 0.0 |

Ti_{1}-Te_{1}-Te_{3} | 1.000 | 0.000 | 0.000 | 22.321 | 1.000 | 0.134 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Ti_{1}-Te_{1}-Te_{2} | 1.000 | 0.000 | 0.000 | 22.463 | 1.000 | 0.155 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Te_{1}-Ti_{1}-Ti_{3} | 1.000 | 0.000 | 0.000 | 22.321 | 1.000 | 0.134 | 0.000 | 0.000 | 4 | 0 | 0.0 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-TiTe_{2} under uniaxial tension at 1 and 300 K. Figure 12 shows the stress-strain curve for the tension of a single-layer 1H-TiTe_{2} of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-TiTe_{2} 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-TiTe_{2}. 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

There is no available value for the nonlinear quantities in the single-layer 1H-TiTe_{2}. We have thus used the nonlinear parameter *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 *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.

## 8. 1H-VO_{2}

Most existing theoretical studies on the single-layer 1H-VO_{2} are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-VO_{2}.

The structure for the single-layer 1H-VO_{2} 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

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 9.417 | 4.825 | 4.825 | 4.825 |

r _{0} or θ _{0} | 1.920 | 89.356 | 71.436 | 89.356 |

V-O | 5.105 | 1.011 | 6.795 | 0.0 | 2.617 |

43.951 | 89.356 | 1.011 | 1.011 | 0.0 | 2.617 | 0.0 | 2.617 | 0.0 | 3.105 | |

48.902 | 71.436 | 1.011 | 1.011 | 0.0 | 2.617 | 0.0 | 2.617 | 0.0 | 3.105 | |

43.951 | 89.356 | 1.011 | 1.011 | 0.0 | 2.617 | 0.0 | 2.617 | 0.0 | 3.105 |

ϵ (eV) | σ (Å) | a | λ | γ | cos θ _{0} | A _{L} | B _{L} | p | q | Tol | |
---|---|---|---|---|---|---|---|---|---|---|---|

V_{1}-O_{1}-O_{1} | 1.000 | 1.011 | 2.589 | 0.000 | 1.000 | 0.000 | 5.105 | 6.509 | 4 | 0 | 0.0 |

V_{1}-O_{1}-O_{3} | 1.000 | 0.000 | 0.000 | 43.951 | 1.000 | 0.011 | 0.000 | 0.000 | 4 | 0 | 0.0 |

V_{1}-O_{1}-O_{2} | 1.000 | 0.000 | 0.000 | 48.902 | 1.000 | 0.318 | 0.000 | 0.000 | 4 | 0 | 0.0 |

O_{1}-V_{1}-V_{3} | 1.000 | 0.000 | 0.000 | 43.951 | 1.000 | 0.011 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Table 30 shows four VFF terms for the single-layer 1H-VO_{2}; 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-VO_{2} 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-VO_{2}, 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-VO_{2} under uniaxial tension at 1 and 300 K. Figure 14 shows the stress-strain curve for the tension of a single-layer 1H-VO_{2} of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-VO_{2} 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-VO_{2}. 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

There is no available value for nonlinear quantities in the single-layer 1H-VO_{2}. We have thus used the nonlinear parameter *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 *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.

## 9. 1H-VS_{2}

Most existing theoretical studies on the single-layer 1H-VS_{2} 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-VS_{2}.

The structure for the single-layer 1H-VS_{2} 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

Table 34 shows the VFF terms for the 1H-VS_{2}; 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.

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 8.392 | 4.074 | 4.074 | 4.074 |

r _{0} or θ _{0} | 2.310 | 83.954 | 78.878 | 83.954 |

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-VS_{2} 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-VS_{2}, 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) | B (Å^{4}) | r _{min} (Å) | r _{max}(Å) | |
---|---|---|---|---|---|

V─S | 5.714 | 1.037 | 14.237 | 0.0 | 3.084 |

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) | |
---|---|---|---|---|---|---|---|---|---|---|

30.059 | 83.954 | 1.037 | 1.037 | 0.0 | 3.084 | 0.0 | 3.084 | 0.0 | 3.676 | |

30.874 | 78.878 | 1.037 | 1.037 | 0.0 | 3.084 | 0.0 | 3.084 | 0.0 | 3.676 | |

30.059 | 83.954 | 1.037 | 1.037 | 0.0 | 3.084 | 0.0 | 3.084 | 0.0 | 3.676 |

ϵ (eV) | σ (Å) | a | λ | γ | cos θ _{0} | A _{L} | B _{L} | p | q | Tol | |
---|---|---|---|---|---|---|---|---|---|---|---|

V_{1}─S_{1}─S_{1} | 1.000 | 1.037 | 2.973 | 0.000 | 1.000 | 0.000 | 5.714 | 12.294 | 4 | 0 | 0.0 |

V_{1}─S_{1}─S_{3} | 1.000 | 0.000 | 0.000 | 30.059 | 1.000 | 0.105 | 0.000 | 0.000 | 4 | 0 | 0.0 |

V_{1}─S_{1}─S_{2} | 1.000 | 0.000 | 0.000 | 30.874 | 1.000 | 0.193 | 0.000 | 0.000 | 4 | 0 | 0.0 |

S_{1}─V_{1}─V_{3} | 1.000 | 0.000 | 0.000 | 30.059 | 1.000 | 0.105 | 0.000 | 0.000 | 4 | 0 | 0.0 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-VS_{2} under uniaxial tension at 1 and 300 K. Figure 16 shows the stress-strain curve for the tension of a single-layer 1H-VS_{2} of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-VS_{2} 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-VS_{2}. 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

There is no available value for the nonlinear quantities in the single-layer 1H-VS_{2}. 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 *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.

## 10. 1H-VSe_{2}

Most existing theoretical studies on the single-layer 1H-VSe_{2} 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-VSe_{2}.

The structure for the single-layer 1H-VSe_{2} 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

Table 38 shows the VFF terms for the 1H-VSe_{2}; 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-VSe_{2} 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-Vse_{2}, 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.

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 6.492 | 4.716 | 4.716 | 4.716 |

r _{0} or θ _{0} | 2.450 | 82.787 | 80.450 | 82.787 |

A(eV) | ρ(Å) | B(Å^{4}) | r _{min}(Å) | r _{max}(Å) | |
---|---|---|---|---|---|

V─Se | 4.817 | 1.061 | 18.015 | 0.0 | 3.256 |

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) | |
---|---|---|---|---|---|---|---|---|---|---|

33.299 | 82.787 | 1.061 | 1.061 | 0.0 | 3.256 | 0.0 | 3.256 | 0.0 | 3.884 | |

33.702 | 80.450 | 1.061 | 1.061 | 0.0 | 3.256 | 0.0 | 3.256 | 0.0 | 3.884 | |

33.299 | 82.787 | 1.061 | 1.061 | 0.0 | 3.256 | 0.0 | 3.256 | 0.0 | 3.884 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-VSe_{2} under uniaxial tension at 1 and 300 K. Figure 18 shows the stress-strain curve for the tension of a single-layer 1H-VSe_{2} of dimension 100×100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-VSe_{2} 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-VSe_{2}. 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

ϵ (eV) | σ (Å) | α | λ | γ | cos θ _{0} | A _{L} | B _{L} | p | q | Tol | |
---|---|---|---|---|---|---|---|---|---|---|---|

V_{1}─Se_{1}─Se_{1} | 1.000 | 1.061 | 3.070 | 0.000 | 1.000 | 0.000 | 4.817 | 14.236 | 4 | 0 | 0.0 |

V_{1}─Se_{1}─Se_{3} | 1.000 | 0.000 | 0.000 | 33.299 | 1.000 | 0.126 | 0.000 | 0.000 | 4 | 0 | 0.0 |

V_{1}─Se_{1}─Se_{2} | 1.000 | 0.000 | 0.000 | 33.702 | 1.000 | 0.166 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Se_{1}─V_{1}─V_{3} | 1.000 | 0.000 | 0.000 | 33.299 | 1.000 | 0.126 | 0.000 | 0.000 | 4 | 0 | 0.0 |

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 6.371 | 4.384 | 4.384 | 4.384 |

r _{0} or θ _{0} | 2.660 | 81.708 | 81.891 | 81.708 |

There is no available value for the nonlinear quantities in the single-layer 1H-VSe_{2}. We have thus used the nonlinear parameter *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 *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-VTe_{2}

Most existing theoretical studies on the single-layer 1H-VTe_{2} 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-VTe_{2}.

V─Te | 5.410 | 1.112 | 25.032 | 0.0 | 3.520 |

The structure for the single-layer 1H-VTe_{2} 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

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

29.743 | 81.708 | 1.112 | 1.112 | 0.0 | 3.520 | 0.0 | 3.520 | 0.0 | 4.203 | |

29.716 | 81.891 | 1.112 | 1.112 | 0.0 | 3.520 | 0.0 | 3.520 | 0.0 | 4.203 | |

29.743 | 81.708 | 1.112 | 1.112 | 0.0 | 3.520 | 0.0 | 3.520 | 0.0 | 4.203 |

Table 42 shows the VFF terms for the 1H-VTe_{2}; 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-VTe_{2} 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-VTe_{2}, 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.

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

V_{1}─Te_{1}─Te_{1} | 1.000 | 1.112 | 3.164 | 0.000 | 1.000 | 0.000 | 5.410 | 16.345 | 4 | 0 | 0.0 |

V_{1}─Te_{1}─Te_{3} | 1.000 | 0.000 | 0.000 | 29.743 | 1.000 | 0.144 | 0.000 | 0.000 | 4 | 0 | 0.0 |

V_{1}─Te_{1}─Te_{2} | 1.000 | 0.000 | 0.000 | 29.716 | 1.000 | 0.141 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Te_{1}─V_{1}─V_{3} | 1.000 | 0.000 | 0.000 | 29.743 | 1.000 | 0.144 | 0.000 | 0.000 | 4 | 0 | 0.0 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-VTe_{2} under uniaxial tension at 1 and 300 K. Figure 20 shows the stress-strain curve for the tension of a single-layer 1H-VTe_{2} of dimension _{2} 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-VTe_{2}. 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

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 12.881 | 8.039 | 8.039 | 8.039 |

1.880 | 86.655 | 75.194 | 86.655 |

Cr─O | 6.343 | 0.916 | 6.246 | 0.0 | 2.536 |

There is no available value for the nonlinear quantities in the single-layer 1H-VTe_{2}. We have thus used the nonlinear parameter

## 12. 1H-CrO_{2}

Most existing theoretical studies on the single-layer 1H-CrO_{2} are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-CrO_{2}.

The structure for the single-layer 1H-CrO_{2} 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

Table 46 shows four VFF terms for the single-layer 1H-CrO_{2}; 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 *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.

65.805 | 86.655 | 0.916 | 0.916 | 0.0 | 2.536 | 0.0 | 2.536 | 0.0 | 3.016 | |

70.163 | 75.194 | 0.916 | 0.916 | 0.0 | 2.536 | 0.0 | 2.536 | 0.0 | 3.016 | |

65.805 | 86.655 | 0.916 | 0.916 | 0.0 | 2.536 | 0.0 | 2.536 | 0.0 | 3.016 |

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-CrO_{2} 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-CrO_{2}, 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.

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Cr_{1}─O_{1}─O_{1} | 1.000 | 0.916 | 2.769 | 0.000 | 1.000 | 0.000 | 6.242 | 8.871 | 4 | 0 | 0.0 |

Cr_{1}─O_{1}─O_{3} | 1.000 | 0.000 | 0.000 | 65.805 | 1.000 | 0.058 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Cr_{1}─O_{1}─O_{2} | 1.000 | 0.000 | 0.000 | 70.163 | 1.000 | 0.256 | 0.000 | 0.000 | 4 | 0 | 0.0 |

O_{1}─Cr_{1}─Cr_{3} | 1.000 | 0.000 | 0.000 | 65.805 | 1.000 | 0.058 | 0.000 | 0.000 | 4 | 0 | 0.0 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-CrO_{2} under uniaxial tension at 1 and 300 K. Figure 22 shows the stress-strain curve for the tension of a single-layer 1H-CrO_{2} of dimension _{2} 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-CrO_{2}. 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

There is no available value for nonlinear quantities in the single-layer 1H-CrO_{2}. We have thus used the nonlinear parameter

## 13. 1H-CrS_{2}

Most existing theoretical studies on the single-layer 1H-CrS_{2} 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-CrS_{2}.

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 8.752 | 4.614 | 4.614 | 4.614 |

2.254 | 83.099 | 80.031 | 83.099 |

Cr─S | 5.544 | 0.985 | 12.906 | 0.0 | 2.999 |

32.963 | 83.099 | 0.985 | 0.985 | 0.0 | 2.999 | 0.0 | 2.999 | 0.0 | 3.577 | |

33.491 | 80.031 | 0.985 | 0.985 | 0.0 | 2.999 | 0.0 | 2.999 | 0.0 | 3.577 | |

32.963 | 83.099 | 0.985 | 0.985 | 0.0 | 2.999 | 0.0 | 2.999 | 0.0 | 3.577 |

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Cr_{1}─S_{1}─S_{1} | 1.000 | 0.985 | 3.043 | 0.000 | 1.000 | 0.000 | 5.544 | 13.683 | 4 | 0 | 0.0 |

Cr_{1}─S_{1}─S_{3} | 1.000 | 0.000 | 0.000 | 32.963 | 1.000 | 0.120 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Cr_{1}─S_{1}─S_{2} | 1.000 | 0.000 | 0.000 | 33.491 | 1.000 | 0.173 | 0.000 | 0.000 | 4 | 0 | 0.0 |

S_{1}─Cr_{1}─Cr_{3} | 1.000 | 0.000 | 0.000 | 32.963 | 1.000 | 0.120 | 0.000 | 0.000 | 4 | 0 | 0.0 |

The structure for the single-layer 1H-CrS_{2} 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

Table 50 shows four VFF terms for the 1H-CrS_{2}; 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 *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-CrS_{2} 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-CrS_{2}, 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-CrS_{2} under uniaxial tension at 1 and 300 K. Figure 24 shows the stress-strain curve for the tension of a single-layer 1H-CrS_{2} of dimension _{2} 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-CrS_{2}. 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 *ab initio* value of 0.27 [18, 19].

There is no available value for the nonlinear quantities in the single-layer 1H-CrS_{2}. We have thus used the nonlinear parameter

## 14. 1H-CrSe_{2}

Most existing theoretical studies on the single-layer 1H-CrSe_{2} 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-CrSe_{2}.

The structure for the single-layer 1H-CrSe_{2} 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

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 9.542 | 4.465 | 4.465 | 4.465 |

2.380 | 82.229 | 81.197 | 82.229 |

Table 54 shows four VFF terms for the 1H-CrSe_{2}; 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) | B (Å^{4}) | |||
---|---|---|---|---|---|

Cr─Se | 6.581 | 1.012 | 16.043 | 0.0 | 3.156 |

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

30.881 | 82.229 | 1.012 | 1.012 | 0.0 | 3.156 | 0.0 | 3.156 | 0.0 | 3.767 | |

31.044 | 81.197 | 1.012 | 1.012 | 0.0 | 3.156 | 0.0 | 3.156 | 0.0 | 3.767 | |

30.881 | 82.229 | 1.012 | 1.012 | 0.0 | 3.156 | 0.0 | 3.156 | 0.0 | 3.767 |

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-CrSe_{2} 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-CrSe_{2}, 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.

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Cr_{1}─Se_{1}─Se_{1} | 1.000 | 1.012 | 3.118 | 0.000 | 1.000 | 0.000 | 6.581 | 15.284 | 4 | 0 | 0.0 |

Cr_{1}─Se_{1}─Se_{3} | 1.000 | 0.000 | 0.000 | 30.881 | 1.000 | 0.135 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Cr_{1}─Se_{1}─Se_{2} | 1.000 | 0.000 | 0.000 | 31.044 | 1.000 | 0.153 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Se_{1}─Cr_{1}─Cr_{3} | 1.000 | 0.000 | 0.000 | 30.881 | 1.000 | 0.135 | 0.000 | 0.000 | 4 | 0 | 0.0 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-CrSe_{2} under uniaxial tension at 1 and 300 K. Figure 26 shows the stress-strain curve for the tension of a single-layer 1H-CrSe_{2} of dimension _{2} 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-CrSe_{2}. 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 *ab initio* value of 0.30 [18, 19].

There is no available value for the nonlinear quantities in the single-layer 1H-CrSe_{2}. We have thus used the nonlinear parameter

## 15. 1H-CrTe_{2}

Most existing theoretical studies on the single-layer 1H-CrTe_{2} 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-CrTe_{2}.

The structure for the single-layer 1H-CrTe_{2} 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

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 8.197 | 4.543 | 4.543 | 4.543 |

2.580 | 82.139 | 81.316 | 82.139 |

Table 58 shows three VFF terms for the 1H-CrTe_{2}; 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) | B (Å^{4}) | |||
---|---|---|---|---|---|

Cr─Te | 6.627 | 1.094 | 22.154 | 0.0 | 3.420 |

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

31.316 | 82.139 | 1.094 | 1.094 | 0.0 | 3.420 | 0.0 | 3.420 | 0.0 | 4.082 | |

31.447 | 81.316 | 1.094 | 1.094 | 0.0 | 3.420 | 0.0 | 3.420 | 0.0 | 4.082 | |

31.316 | 82.139 | 1.094 | 1.094 | 0.0 | 3.420 | 0.0 | 3.420 | 0.0 | 4.082 |

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-CrTe_{2} 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-CrTe_{2}, 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.

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Cr_{1}─Te_{1}─Te_{1} | 1.000 | 1.094 | 3.126 | 0.000 | 1.000 | 0.000 | 6.627 | 15.461 | 4 | 0 | 0.0 |

Cr_{1}─Te_{1}─Te_{3} | 1.000 | 0.000 | 0.000 | 31.316 | 1.000 | 0.137 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Cr_{1}─Te_{1}─Te_{2} | 1.000 | 0.000 | 0.000 | 31.447 | 1.000 | 0.151 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Te_{1}─Cr_{1}─Cr_{3} | 1.000 | 0.000 | 0.000 | 31.316 | 1.000 | 0.137 | 0.000 | 0.000 | 4 | 0 | 0.0 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-CrTe_{2} under uniaxial tension at 1 and 300 K. Figure 28 shows the stress-strain curve for the tension of a single-layer 1H-CrTe_{2} of dimension _{2} 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-CrTe_{2}. 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 *ab initio* value of 0.30 [18, 19].

There is no available value for the nonlinear quantities in the single-layer 1H-CrTe_{2}. We have thus used the nonlinear parameter

## 16. 1H-MnO_{2}

Most existing theoretical studies on the single-layer 1H-MnO_{2} are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-MnO_{2}.

The structure for the single-layer 1H-MnO_{2} 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

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 9.382 | 6.253 | 6.253 | 6.253 |

1.870 | 88.511 | 72.621 | 88.511 |

Table 62 shows four VFF terms for the single-layer 1H-MnO_{2}; 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) | B (Å^{4}) | |||
---|---|---|---|---|---|

Mn─O | 4.721 | 0.961 | 6.114 | 0.0 | 2.540 |

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

55.070 | 88.511 | 0.961 | 0.961 | 0.0 | 2.540 | 0.0 | 2.540 | 0.0 | 3.016 | |

60.424 | 72.621 | 0.961 | 0.961 | 0.0 | 2.540 | 0.0 | 2.540 | 0.0 | 3.016 | |

55.070 | 88.511 | 0.961 | 0.961 | 0.0 | 2.540 | 0.0 | 2.540 | 0.0 | 3.016 |

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-MnO_{2} 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-MnO_{2}, 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.

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Mn_{1}─O_{1}─O_{1} | 1.000 | 0.961 | 2.643 | 0.000 | 1.000 | 0.000 | 4.721 | 7.158 | 4 | 0 | 0.0 |

Mn_{1}─O_{1}─O_{3} | 1.000 | 0.000 | 0.000 | 55.070 | 1.000 | 0.026 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Mn_{1}─O_{1}─O_{2} | 1.000 | 0.000 | 0.000 | 60.424 | 1.000 | 0.299 | 0.000 | 0.000 | 4 | 0 | 0.0 |

O_{1}─Mn_{1}─Mn_{3} | 1.000 | 0.000 | 0.000 | 55.070 | 1.000 | 0.026 | 0.000 | 0.000 | 4 | 0 | 0.0 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-MnO_{2} under uniaxial tension at 1 and 300 K. Figure 30 shows the stress-strain curve for the tension of a single-layer 1H-MnO_{2} of dimension _{2} 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-MnO_{2}. 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

There is no available value for nonlinear quantities in the single-layer 1H-MnO_{2}. We have thus used the nonlinear parameter

## 17. 1H-FeO_{2}

Most existing theoretical studies on the single-layer 1H-FeO_{2} are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-FeO_{2}.

The structure for the single-layer 1H-FeO_{2} 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

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 8.377 | 3.213 | 3.213 | 3.213 |

1.880 | 88.343 | 72.856 | 88.343 |

Table 66 shows four VFF terms for the single-layer 1H-FeO_{2}; 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) | B (Å^{4}) | |||
---|---|---|---|---|---|

Fe─O | 4.242 | 0.962 | 6.246 | 0.0 | 2.552 |

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

28.105 | 88.343 | 0.962 | 0.962 | 0.0 | 2.552 | 0.0 | 2.552 | 0.0 | 3.031 | |

30.754 | 72.856 | 0.962 | 0.962 | 0.0 | 2.552 | 0.0 | 2.552 | 0.0 | 3.031 | |

28.105 | 88.343 | 0.962 | 0.962 | 0.0 | 2.552 | 0.0 | 2.552 | 0.0 | 3.031 |

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-FeO_{2} 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-FeO_{2}, 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.

p | q | Tol | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Fe_{1}─O_{1}─O_{1} | 1.000 | 0.962 | 2.654 | 0.000 | 1.000 | 0.000 | 4.242 | 7.298 | 4 | 0 | 0.0 |

Fe_{1}─O_{1}─O_{3} | 1.000 | 0.000 | 0.000 | 28.105 | 1.000 | 0.029 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Fe_{1}─O_{1}─O_{2} | 1.000 | 0.000 | 0.000 | 30.754 | 1.000 | 0.295 | 0.000 | 0.000 | 4 | 0 | 0.0 |

O_{1}─Fe_{1}─Fe_{3} | 1.000 | 0.000 | 0.000 | 28.105 | 1.000 | 0.029 | 0.000 | 0.000 | 4 | 0 | 0.0 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-FeO_{2} under uniaxial tension at 1 and 300 K. Figure 32 shows the stress-strain curve for the tension of a single-layer 1H-FeO_{2} of dimension 100×100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The single-layer 1H-FeO_{2} 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-FeO_{2}. 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

There is no available value for nonlinear quantities in the single-layer 1H-FeO_{2}. We have thus used the nonlinear parameter *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 *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-FES_{2}

Most existing theoretical studies on the single-layer 1H-FeS_{2} are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-FeS_{2}.

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 6.338 | 3.964 | 3.964 | 3.964 |

2.220 | 87.132 | 74.537 | 87.132 |

A (eV) | ρ (Å) | B (Å^{4}) | |||
---|---|---|---|---|---|

Fe─S | 4.337 | 1.097 | 12.145 | 0.0 | 3.000 |

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

33.060 | 87.132 | 1.097 | 1.097 | 0.0 | 3.000 | 0.0 | 3.000 | 0.0 | 3.567 | |

35.501 | 74.537 | 1.097 | 1.097 | 0.0 | 3.000 | 0.0 | 3.000 | 0.0 | 3.567 | |

33.060 | 87.132 | 1.097 | 1.097 | 0.0 | 3.000 | 0.0 | 3.000 | 0.0 | 3.567 |

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Fe_{1}─S_{1}─S_{1} | 1.000 | 1.097 | 2.735 | 0.000 | 1.000 | 0.000 | 4.337 | 8.338 | 4 | 0 | 0.0 |

Fe_{1}─S_{1}─S_{3} | 1.000 | 0.000 | 0.000 | 33.060 | 1.000 | 0.050 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Fe_{1}─S_{1}─S_{2} | 1.000 | 0.000 | 0.000 | 35.501 | 1.000 | 0.267 | 0.000 | 0.000 | 4 | 0 | 0.0 |

S_{1}─Fe_{1}─Fe_{3} | 1.000 | 0.000 | 0.000 | 33.060 | 1.000 | 0.050 | 0.000 | 0.000 | 4 | 0 | 0.0 |

The structure for the single-layer 1H-FeS_{2} 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

Table 70 shows four VFF terms for the single-layer 1H-FeS_{2}; 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.

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-FeS_{2} 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-FeS_{2}, 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-FeS_{2} under uniaxial tension at 1 and 300 K. Figure 34 shows the stress-strain curve for the tension of a single-layer 1H-FeS_{2} of dimension _{2} 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-FeS_{2}. 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

There is no available value for nonlinear quantities in the single-layer 1H-FeS_{2}. We have thus used the nonlinear parameter

## 19. 1H-FESE_{2}

Most existing theoretical studies on the single-layer 1H-FeSe_{2} are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-FeSe_{2}.

The structure for the single-layer 1H-FeSe_{2} 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

Table 74 shows four VFF terms for the single-layer 1H-FeSe_{2}; 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-FeSe_{2} 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-FeSe_{2}, 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.

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 6.338 | 3.964 | 3.964 | 3.964 |

2.350 | 86.488 | 75.424 | 86.488 |

A (eV) | ρ (Å) | B (Å^{4}) | |||
---|---|---|---|---|---|

Fe-Se | 4.778 | 1.139 | 15.249 | 0.0 | 3.168 |

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

32.235 | 86.488 | 1.139 | 1.139 | 0.0 | 3.168 | 0.0 | 3.168 | 0.0 | 3.768 | |

34.286 | 75.424 | 1.139 | 1.139 | 0.0 | 3.168 | 0.0 | 3.168 | 0.0 | 3.768 | |

32.235 | 86.488 | 1.139 | 1.139 | 0.0 | 3.168 | 0.0 | 3.168 | 0.0 | 3.768 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-FeSe_{2} under uniaxial tension at 1 and 300 K. Figure 36 shows the stress-strain curve for the tension of a single-layer 1H-FeSe_{2} of dimension _{2} 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-FeSe_{2}. 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

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Fe_{1}─Se_{1}─Se_{1} | 1.000 | 1.139 | 2.781 | 0.000 | 1.000 | 0.000 | 4.778 | 9.049 | 4 | 0 | 0.0 |

Fe_{1}─Se_{1}─Se_{3} | 1.000 | 0.000 | 0.000 | 32.235 | 1.000 | 0.061 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Fe_{1}─Se_{1}─Se_{2} | 1.000 | 0.000 | 0.000 | 34.286 | 1.000 | 0.252 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Se_{1}─Fe_{1}─Fe_{3} | 1.000 | 0.000 | 0.000 | 32.235 | 1.000 | 0.061 | 0.000 | 0.000 | 4 | 0 | 0.0 |

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 6.338 | 3.964 | 3.964 | 3.964 |

2.530 | 86.904 | 74.851 | 86.904 |

There is no available value for nonlinear quantities in the single-layer 1H-FeSe_{2}. We have thus used the nonlinear parameter

A (eV) | ρ (Å) | B (Å^{4}) | |||
---|---|---|---|---|---|

Fe─Te | 5.599 | 1.242 | 20.486 | 0.0 | 3.416 |

## 20. 1H-FETE_{2}

Most existing theoretical studies on the single-layer 1H-FeTe_{2} are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-FeTe_{2}.

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

32.766 | 86.904 | 1.242 | 1.242 | 0.0 | 3.416 | 0.0 | 3.416 | 0.0 | 4.062 | |

35.065 | 74.851 | 1.242 | 1.242 | 0.0 | 3.416 | 0.0 | 3.416 | 0.0 | 4.062 | |

32.766 | 86.904 | 1.242 | 1.242 | 0.0 | 3.416 | 0.0 | 3.416 | 0.0 | 4.062 |

The structure for the single-layer 1H-FeTe_{2} 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

Table 78 shows four VFF terms for the single-layer 1H-FeTe_{2}; 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.

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Fe_{1}─Te_{1}─Te_{1} | 1.000 | 1.242 | 2.751 | 0.000 | 1.000 | 0.000 | 5.599 | 8.615 | 4 | 0 | 0.0 |

Fe_{1}─Te_{1}─Te_{3} | 1.000 | 0.000 | 0.000 | 32.766 | 1.000 | 0.054 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Fe_{1}─Te_{1}─Te_{2} | 1.000 | 0.000 | 0.000 | 35.065 | 1.000 | 0.261 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Te_{1}─Fe_{1}─Fe_{3} | 1.000 | 0.000 | 0.000 | 32.766 | 1.000 | 0.054 | 0.000 | 0.000 | 4 | 0 | 0.0 |

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-FeTe_{2} 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-FeTe_{2}, 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-FeTe_{2} under uniaxial tension at 1 and 300 K. Figure 38 shows the stress-strain curve for the tension of a single-layer 1H-FeTe_{2} of dimension _{2} 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-FeTe_{2}. 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

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 6.712 | 2.656 | 2.656 | 2.656 |

2.510 | 89.046 | 71.873 | 89.046 |

There is no available value for nonlinear quantities in the single-layer 1H-FeTe_{2}. We have thus used the nonlinear parameter

## 21. 1H-COTE_{2}

Most existing theoretical studies on the single-layer 1H-CoTe_{2} are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-CoTe_{2}.

A (eV) | ρ (Å) | B (Å^{4}) | |||
---|---|---|---|---|---|

Co─Te | 6.169 | 1.310 | 19.846 | 0.0 | 3.417 |

The structure for the single-layer 1H-CoTe_{2} 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

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

23.895 | 89.046 | 1.310 | 1.310 | 0.0 | 3.417 | 0.0 | 3.417 | 0.0 | 4.055 | |

26.449 | 71.873 | 1.310 | 1.310 | 0.0 | 3.417 | 0.0 | 3.417 | 0.0 | 4.055 | |

23.895 | 89.046 | 1.310 | 1.310 | 0.0 | 3.417 | 0.0 | 3.417 | 0.0 | 4.055 |

Table 82 shows four VFF terms for the single-layer 1H-CoTe_{2}; 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-CoTe_{2} 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-CoTe_{2}, 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.

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Co_{1}─Te_{1}─Te_{1} | 1.000 | 1.310 | 2.608 | 0.000 | 1.000 | 0.000 | 6.169 | 6.739 | 4 | 0 | 0.0 |

Co_{1}─Te_{1}─Te_{3} | 1.000 | 0.000 | 0.000 | 23.895 | 1.000 | 0.017 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Co_{1}─Te_{1}─Te_{2} | 1.000 | 0.000 | 0.000 | 26.449 | 1.000 | 0.311 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Te_{1}─Co_{1}─Co_{3} | 1.000 | 0.000 | 0.000 | 23.895 | 1.000 | 0.017 | 0.000 | 0.000 | 4 | 0 | 0.0 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-CoTe_{2} under uniaxial tension at 1 and 300 K. Figure 40 shows the stress-strain curve for the tension of a single-layer 1H-CoTe_{2} of dimension _{2} 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-CoTe_{2}. 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

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 6.933 | 3.418 | 3.418 | 3.418 |

2.240 | 98.740 | 57.593 | 98.740 |

A (eV) | ρ (Å) | B (Å^{4}) | |||
---|---|---|---|---|---|

Ni-S | 6.425 | 1.498 | 12.588 | 0.0 | 3.156 |

There is no available value for nonlinear quantities in the single-layer 1H-CoTe_{2}. We have thus used the nonlinear parameter

## 22. 1H-NIS_{2}

Most existing theoretical studies on the single-layer 1H-NiS_{2} are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-NiS_{2}.

The structure for the single-layer 1H-NiS_{2} 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

Table 86 shows four VFF terms for the single-layer 1H-NiS_{2}; 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.

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

46.062 | 98.740 | 1.498 | 1.498 | 0.0 | 3.156 | 0.0 | 3.156 | 0.0 | 3.713 | |

63.130 | 57.593 | 1.498 | 1.498 | 0.0 | 3.156 | 0.0 | 3.156 | 0.0 | 3.713 | |

46.062 | 98.740 | 1.498 | 1.498 | 0.0 | 3.156 | 0.0 | 3.156 | 0.0 | 3.713 |

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-NiS_{2} 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-NiS_{2}, 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.

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Ni_{1}─S_{1}─S_{1} | 1.000 | 1.498 | 2.107 | 0.000 | 1.000 | 0.000 | 6.425 | 2.502 | 4 | 0 | 0.0 |

Ni_{1}─S_{1}─S_{3} | 1.000 | 0.000 | 0.000 | 46.062 | 1.000 | −0.152 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Ni_{1}─S_{1}─S_{2} | 1.000 | 0.000 | 0.000 | 63.130 | 1.000 | 0.536 | 0.000 | 0.000 | 4 | 0 | 0.0 |

S_{1}─Ni_{1}─Ni_{3} | 1.000 | 0.000 | 0.000 | 46.062 | 1.000 | −0.152 | 0.000 | 0.000 | 4 | 0 | 0.0 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-NiS_{2} under uniaxial tension at 1 and 300 K. Figure 42 shows the stress-strain curve for the tension of a single-layer 1H-NiS_{2} of dimension _{2} 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-NiS_{2}. 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

There is no available value for nonlinear quantities in the single-layer 1H-NiS_{2}. We have thus used the nonlinear parameter

## 23. 1H-NISE_{2}

Most existing theoretical studies on the single-layer 1H-NiSe_{2} are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-NiSe_{2}.

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 4.823 | 2.171 | 2.171 | 2.171 |

2.350 | 90.228 | 70.206 | 90.228 |

A (eV) | ρ (Å) | B (Å^{4}) | |||
---|---|---|---|---|---|

Ni─Se | 4.004 | 1.267 | 15.249 | 0.0 | 3.213 |

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

20.479 | 90.228 | 1.267 | 1.267 | 0.0 | 3.213 | 0.0 | 3.213 | 0.0 | 3.809 | |

23.132 | 70.206 | 1.267 | 1.267 | 0.0 | 3.213 | 0.0 | 3.213 | 0.0 | 3.809 | |

20.479 | 90.228 | 1.267 | 1.267 | 0.0 | 3.213 | 0.0 | 3.213 | 0.0 | 3.809 |

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Ni_{1}─Se_{1}─Se_{1} | 1.000 | 1.267 | 2.535 | 0.000 | 1.000 | 0.000 | 4.004 | 5.913 | 4 | 0 | 0.0 |

Ni_{1}─Se_{1}─Se_{3} | 1.000 | 0.000 | 0.000 | 20.479 | 1.000 | −0.004 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Ni_{1}─Se_{1}─Se_{2} | 1.000 | 0.000 | 0.000 | 23.132 | 1.000 | 0.339 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Se_{1}─Ni_{1}─Ni_{3} | 1.000 | 0.000 | 0.000 | 20.479 | 1.000 | −0.004 | 0.000 | 0.000 | 4 | 0 | 0.0 |

The structure for the single-layer 1H-NiSe_{2} 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

Table 90 shows four VFF terms for the single-layer 1H-NiSe_{2}; 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-NiSe_{2} 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-NiSe_{2}, 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-NiSe_{2} under uniaxial tension at 1 and 300 K. Figure 44 shows the stress-strain curve for the tension of a single-layer 1H-NiSe_{2} of dimension _{2} 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-NiSe_{2}. 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

There is no available value for nonlinear quantities in the single-layer 1H-NiSe_{2}. We have thus used the nonlinear parameter

## 24. 1H-NITE_{2}

Most existing theoretical studies on the single-layer 1H-NiTe_{2} are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-NiTe_{2}.

The structure for the single-layer 1H-NiTe_{2} 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

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 6.712 | 2.656 | 2.656 | 2.656 |

2.540 | 89.933 | 70.624 | 89.933 |

Table 94 shows four VFF terms for the single-layer 1H-NiTe_{2}; 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) | B (Å^{4}) | |||
---|---|---|---|---|---|

Ni─Te | 6.461 | 1.359 | 20.812 | 0.0 | 3.469 |

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

24.759 | 89.933 | 1.359 | 1.359 | 0.0 | 3.469 | 0.0 | 3.469 | 0.0 | 4.114 | |

27.821 | 70.624 | 1.359 | 1.359 | 0.0 | 3.469 | 0.0 | 3.469 | 0.0 | 4.114 | |

24.759 | 89.933 | 1.359 | 1.359 | 0.0 | 3.469 | 0.0 | 3.469 | 0.0 | 4.114 |

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-NiTe_{2} 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-NiTe_{2}, 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.

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Ni_{1}─Te_{1}─Te_{1} | 1.000 | 1.359 | 2.553 | 0.000 | 1.000 | 0.000 | 6.461 | 6.107 | 4 | 0 | 0.0 |

Ni_{1}─Te_{1}─Te_{3} | 1.000 | 0.000 | 0.000 | 24.759 | 1.000 | 0.001 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Ni_{1}─Te_{1}─Te_{2} | 1.000 | 0.000 | 0.000 | 27.821 | 1.000 | 0.332 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Te_{1}─Ni_{1}─Ni_{3} | 1.000 | 0.000 | 0.000 | 24.759 | 1.000 | 0.001 | 0.000 | 0.000 | 4 | 0 | 0.0 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-NiTe_{2} under uniaxial tension at 1 and 300 K. Figure 46 shows the stress-strain curve for the tension of a single-layer 1H-NiTe_{2} of dimension _{2} 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-NiTe_{2}. 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

There is no available value for nonlinear quantities in the single-layer 1H-NiTe_{2}. We have thus used the nonlinear parameter

## 25. 1H-NBS_{2}

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

The structure for the single-layer 1H-NbS_{2} 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

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 8.230 | 4.811 | 4.811 | 4.811 |

2.470 | 84.140 | 78.626 | 84.140 |

Table 98 shows four VFF terms for the 1H-NbS_{2}; 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-NbS_{2}. We have used these phonon frequencies as the phonon dispersion of the single-layer 1H-NbS_{2}, as the interlayer interaction in the bulk 2H-NbS_{2} 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-NbS_{2} (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) | B (Å^{4}) | |||
---|---|---|---|---|---|

Nb─S | 6.439 | 1.116 | 18.610 | 0.0 | 3.300 |

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

35.748 | 84.140 | 1.116 | 1.116 | 0.0 | 3.300 | 0.0 | 3.300 | 0.0 | 3.933 | |

36.807 | 78.626 | 1.116 | 1.116 | 0.0 | 3.300 | 0.0 | 3.300 | 0.0 | 3.933 | |

35.748 | 84.140 | 1.116 | 1.116 | 0.0 | 3.300 | 0.0 | 3.300 | 0.0 | 3.933 |

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-NbS_{2} 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-NbS_{2}, 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.

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Nb_{1}─S_{1}─S_{1} | 1.000 | 1.116 | 2.958 | 0.000 | 1.000 | 0.000 | 6.439 | 12.014 | 4 | 0 | 0.0 |

Nb_{1}─S_{1}─S_{3} | 1.000 | 0.000 | 0.000 | 35.748 | 1.000 | 0.102 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Nb_{1}─S_{1}─S_{2} | 1.000 | 0.000 | 0.000 | 36.807 | 1.000 | 0.197 | 0.000 | 0.000 | 4 | 0 | 0.0 |

S_{1}─Nb_{1}─Nb_{3} | 1.000 | 0.000 | 0.000 | 35.748 | 1.000 | 0.102 | 0.000 | 0.000 | 4 | 0 | 0.0 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-NbS_{2} under uniaxial tension at 1 and 300 K. Figure 48 shows the stress-strain curve for the tension of a single-layer 1H-NbS_{2} of dimension _{2} 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-NbS_{2}. 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

There is no available value for the nonlinear quantities in the single-layer 1H-NbS_{2}. We have thus used the nonlinear parameter

## 26. 1H-NBSE_{2}

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

The structure for the single-layer 1H-NbSe_{2} 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

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 8.230 | 4.811 | 4.811 | 4.811 |

2.600 | 83.129 | 79.990 | 83.129 |

Table 102 shows four VFF terms for the 1H-NbSe_{2}; 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-NbSe_{2}. We have used these phonon frequencies as the phonon dispersion of the single-layer 1H-NbSe_{2}, as the interlayer interaction in the bulk 2H-NbSe_{2} 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-NbSe_{2} (gray pentagons). It turns out that the VFF parameters for the single-layer 1H-NbSe_{2} are the same as the single-layer NbS_{2}. The phonon dispersion for single-layer 1H-NbSe_{2} 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) | B (Å^{4}) | |||
---|---|---|---|---|---|

Nb-Se | 6.942 | 1.138 | 22.849 | 0.0 | 3.460 |

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

34.409 | 83.129 | 1.138 | 1.138 | 0.0 | 3.460 | 0.0 | 3.460 | 0.0 | 4.127 | |

34.973 | 79.990 | 1.138 | 1.138 | 0.0 | 3.460 | 0.0 | 3.460 | 0.0 | 4.127 | |

34.409 | 83.129 | 1.138 | 1.138 | 0.0 | 3.460 | 0.0 | 3.460 | 0.0 | 4.127 |

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Nb_{1}─Se_{1}─Se_{1} | 1.000 | 1.138 | 3.041 | 0.000 | 1.000 | 0.000 | 6.942 | 13.631 | 4 | 0 | 0.0 |

Nb_{1}─Se_{1}─Se_{3} | 1.000 | 0.000 | 0.000 | 34.409 | 1.000 | 0.120 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Nb_{1}─Se_{1}─Se_{2} | 1.000 | 0.000 | 0.000 | 34.973 | 1.000 | 0.174 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Se_{1}─Nb_{1}─Nb_{3} | 1.000 | 0.000 | 0.000 | 34.409 | 1.000 | 0.120 | 0.000 | 0.000 | 4 | 0 | 0.0 |

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-NbSe_{2} 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-NbSe_{2}, 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-NbSe_{2} under uniaxial tension at 1 and 300 K. Figure 50 shows the stress-strain curve for the tension of a single-layer 1H-NbSe_{2} of dimension _{2} 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-NbSe_{2}. 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

There is no available value for the nonlinear quantities in the single-layer 1H-NbSe_{2}. We have thus used the nonlinear parameter

## 27. 1H-MoO_{2}

Most existing theoretical studies on the single-layer 1H-MoO_{2} are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-MoO_{2}.

The structure for the single-layer 1H-MoO_{2} 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

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 14.622 | 8.410 | 8.410 | 8.410 |

2.000 | 88.054 | 73.258 | 88.054 |

A (eV) | ρ (Å) | B (Å^{4}) | |||
---|---|---|---|---|---|

Mo-O | 8.317 | 1.015 | 8.000 | 0.0 | 2.712 |

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

72.735 | 88.054 | 1.015 | 1.015 | 0.0 | 2.712 | 0.0 | 2.712 | 0.0 | 3.222 | |

79.226 | 73.258 | 1.015 | 1.015 | 0.0 | 2.712 | 0.0 | 2.712 | 0.0 | 3.222 | |

72.735 | 88.054 | 1.015 | 1.015 | 0.0 | 2.712 | 0.0 | 2.712 | 0.0 | 3.222 |

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Mo_{1}─O_{1}─O_{1} | 1.000 | 1.015 | 2.673 | 0.000 | 1.000 | 0.000 | 8.317 | 7.541 | 4 | 0 | 0.0 |

Mo_{1}─O_{1}─O_{3} | 1.000 | 0.000 | 0.000 | 72.735 | 1.000 | 0.034 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Mo_{1}─O_{1}─O_{2} | 1.000 | 0.000 | 0.000 | 79.226 | 1.000 | 0.288 | 0.000 | 0.000 | 4 | 0 | 0.0 |

O_{1}─Mo_{1}─Mo_{3} | 1.000 | 0.000 | 0.000 | 72.735 | 1.000 | 0.034 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Table 106 shows four VFF terms for the single-layer 1H-MoO_{2}; 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-MoO_{2} 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-MoO_{2}, 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-MoO_{2} under uniaxial tension at 1 and 300 K. Figure 52 shows the stress-strain curve for the tension of a single-layer 1H-MoO_{2} of dimension _{2} 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-MoO_{2}. 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

There is no available value for nonlinear quantities in the single-layer 1H-MoO_{2}. We have thus used the nonlinear parameter

## 28. 1H-MoS_{2}

Several potentials have been proposed to describe the interaction for the single-layer 1H-MoS_{2}. In 1975, Wakabayashi et al. developed a VFF model to calculate the phonon spectrum of the bulk 2H-MoS_{2} [22]. In 2009, Liang et al. parameterized a bond-order potential for 1H-MoS_{2} [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-MoS_{2} [24]. The present author (J.W.J.) and his collaborators parameterized the SW potential for 1H-MoS_{2} 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-MoS_{2} [25].

VFF type | Bond stretching | Angle bending | |
---|---|---|---|

Expression | |||

Parameter | 8.640 | 5.316 | 4.891 |

2.382 | 80.581 | 80.581 |

A (eV) | ρ (Å) | B (Å^{4}) | |||
---|---|---|---|---|---|

6.918 | 1.252 | 17.771 | 0.0 | 3.16 |

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

67.883 | 81.788 | 1.252 | 1.252 | 0.0 | 3.16 | 0.0 | 3.16 | 0.0 | 3.78 | |

62.449 | 81.788 | 1.252 | 1.252 | 0.0 | 3.16 | 0.0 | 3.16 | 0.0 | 4.27 |

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Mo_{1}─S_{1}─S_{1} | 1.000 | 1.252 | 2.523 | 0.000 | 1.000 | 0.000 | 6.918 | 7.223 | 4 | 0 | 0.0 |

Mo_{1}─S_{1}─S_{3} | 1.000 | 0.000 | 0.000 | 67.883 | 1.000 | 0.143 | 0.000 | 0.000 | 4 | 0 | 0.0 |

S_{1}─Mo_{1}─Mo_{3} | 1.000 | 0.000 | 0.000 | 62.449 | 1.000 | 0.143 | 0.000 | 0.000 | 4 | 0 | 0.0 |

We show the VFF model and the SW potential for single-layer 1H-MoS_{2} 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-MoS_{2} 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-MoS_{2} 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

The VFF model for single-layer 1H-MoS_{2} 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-MoS_{2} in Table 110 , neglecting other weak interaction terms. The SW potential parameters for single-layer 1H-MoS_{2} used by GULP are listed in Tables 111 and 112 . The SW potential parameters for single-layer 1H-MoS_{2} 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-MoS_{2} 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-MoS_{2}, 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-MoS_{2} 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 | ||||

Parameter | 7.928 | 6.945 | 6.945 | 5.782 |

2.528 | 82.119 | 81.343 | 82.119 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-MoS_{2} under uniaxial tension at 1 and 300 K. Figure 54 shows the stress-strain curve during the tension of a single-layer 1H-MoS_{2} of dimension _{2} 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-MoS_{2}. 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.,

## 29. 1H-MoSe_{2}

There is a recent parameter set for the SW potential in the single-layer 1H-MoSe_{2} [25]. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-MoSe_{2}.

A (eV) | ρ (Å) | B (Å^{4}) | |||
---|---|---|---|---|---|

Mo-Se | 5.737 | 0.913 | 18.787 | 0.0 | 3.351 |

The structure for the single-layer 1H-MoSe_{2} 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

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

32.526 | 82.119 | 0.913 | 0.913 | 0.0 | 3.351 | 0.0 | 3.351 | 0.0 | 4.000 | |

32.654 | 81.343 | 0.913 | 0.913 | 0.0 | 3.351 | 0.0 | 3.351 | 0.0 | 4.000 | |

27.079 | 82.119 | 0.913 | 0.913 | 0.0 | 3.351 | 0.0 | 3.351 | 0.0 | 4.000 |

Table 114 shows four VFF terms for the 1H-MoSe_{2}; 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, 31–34]. 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-MoSe_{2} 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-MoSe_{2}, 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.

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Mo_{1}─Se_{1}─Se_{1} | 1.000 | 0.913 | 3.672 | 0.000 | 1.000 | 0.000 | 5.737 | 27.084 | 4 | 0 | 0.0 |

Mo_{1}─Se_{1}─Se_{3} | 1.000 | 0.000 | 0.000 | 32.526 | 1.000 | 0.137 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Mo_{1}─Se_{1}─Se_{2} | 1.000 | 0.000 | 0.000 | 32.654 | 1.000 | 0.151 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Se_{1}─Mo_{1}─Mo_{3} | 1.000 | 0.000 | 0.000 | 27.079 | 1.000 | 0.137 | 0.000 | 0.000 | 4 | 0 | 0.0 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-MoSe_{2} under uniaxial tension at 1 and 300 K. Figure 56 shows the stress-strain curve during the tension of a single-layer 1H-MoSe_{2} of dimension _{2} 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-MoSe_{2}. 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 *ab initio* value of 0.23 [18].

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 6.317 | 6.184 | 6.184 | 5.225 |

2.730 | 81.111 | 82.686 | 81.111 |

We have determined the nonlinear parameter to be *ab initio* calculations [35]. We have extracted the value of *ab initio* calculations to the function

## 30. 1H-MoTe_{2}

Most existing theoretical studies on the single-layer 1H-MoTe_{2} 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-MoTe_{2}.

The structure for the single-layer 1H-MoTe_{2} 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 (eV) | ρ (Å) | B (Å^{4}) | |||
---|---|---|---|---|---|

Mo-Te | 5.086 | 0.880 | 24.440 | 0.0 | 3.604 |

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

23.705 | 81.111 | 0.880 | 0.880 | 0.0 | 3.604 | 0.0 | 3.604 | 0.0 | 4.305 | |

23.520 | 82.686 | 0.880 | 0.880 | 0.0 | 3.604 | 0.0 | 3.604 | 0.0 | 4.305 | |

20.029 | 81.111 | 0.880 | 0.880 | 0.0 | 3.604 | 0.0 | 3.604 | 0.0 | 4.305 |

Table 118 shows four VFF terms for the 1H-MoTe_{2}; 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-MoTe_{2} 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-MoTe_{2}, 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.

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Mo_{1}─Te_{1}─Te_{1} | 1.000 | 0.900 | 4.016 | 0.000 | 1.000 | 0.000 | 5.169 | 37.250 | 4 | 0 | 0.0 |

Mo_{1}─Te_{1}─Te_{3} | 1.000 | 0.000 | 0.000 | 24.163 | 1.000 | 0.143 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Te_{1}─Mo_{1}─Mo_{3} | 1.000 | 0.000 | 0.000 | 20.416 | 1.000 | 0.143 | 0.000 | 0.000 | 4 | 0 | 0.0 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-MoTe_{2} under uniaxial tension at 1 and 300 K. Figure 58 shows the stress-strain curve for the tension of a single-layer 1H-MoTe_{2} of dimension _{2} 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-MoTe_{2}. 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 *ab initio* value of 0.24 [18].

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 8.230 | 4.811 | 4.811 | 4.811 |

2.480 | 83.879 | 78.979 | 83.879 |

A (eV) | ρ (Å) | B (Å^{4}) | |||
---|---|---|---|---|---|

Ta-S | 6.446 | 1.111 | 18.914 | 0.0 | 3.310 |

We have determined the nonlinear parameter to be *ab initio* calculations [35]. We have extracted the value of *ab initio* calculations to the function

## 31. 1H-TaS_{2}

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

The structure for the single-layer 1H-TaS_{2} 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

35.396 | 83.879 | 1.111 | 1.111 | 0.0 | 3.310 | 0.0 | 3.310 | 0.0 | 3.945 | |

36.321 | 78.979 | 1.111 | 1.111 | 0.0 | 3.310 | 0.0 | 3.310 | 0.0 | 3.945 | |

35.396 | 83.879 | 1.111 | 1.111 | 0.0 | 3.310 | 0.0 | 3.310 | 0.0 | 3.945 |

Table 122 shows the VFF terms for the 1H-TaS_{2}; 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 _{2}. We have used these phonon frequencies as the phonon dispersion of the single-layer 1H-TaS_{2}, as the interlayer interaction in the bulk 2H-TaS_{2} 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-TaS_{2} (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.

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Ta_{1}─S_{1}─S_{1} | 1.000 | 1.111 | 2.979 | 0.000 | 1.000 | 0.000 | 6.446 | 12.408 | 4 | 0 | 0.0 |

Ta_{1}─S_{1}─S_{3} | 1.000 | 0.000 | 0.000 | 35.396 | 1.000 | 0.107 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Ta_{1}─S_{1}─S_{2} | 1.000 | 0.000 | 0.000 | 36.321 | 1.000 | 0.191 | 0.000 | 0.000 | 4 | 0 | 0.0 |

S_{1}─Ta_{1}─Ta_{3} | 1.000 | 0.000 | 0.000 | 35.396 | 1.000 | 0.107 | 0.000 | 0.000 | 4 | 0 | 0.0 |

The parameters for the two-body SW potential used by GULP are shown in Table 123 . The parameters for the three-body SW potential used by GULP are shown in Table 124 . Parameters for the SW potential used by LAMMPS are listed in Table 125 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-TaS_{2} using LAMMPS, because the angles around atom Ta in Figure 1 (with M = Ta 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 = Ta and X = S) shows that, for 1H-TaS_{2}, we can differentiate these angles around the Ta 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 Ta atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-TaS_{2} under uniaxial tension at 1 and 300 K. Figure 60 shows the stress-strain curve for the tension of a single-layer 1H-TaS_{2} of dimension _{2} 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-TaS_{2}. 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 87.4 and 86.6 N/m along the armchair and zigzag directions, respectively. The Poisson’s ratio from the VFF model and the SW potential is

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 8.230 | 4.811 | 4.811 | 4.811 |

2.590 | 83.107 | 80.019 | 83.107 |

^{4}) | |||||
---|---|---|---|---|---|

Ta─Se | 6.885 | 1.133 | 22.499 | 0.0 | 3.446 |

There is no available value for the nonlinear quantities in the single-layer 1H-TaS_{2}. We have thus used the nonlinear parameter

## 32. 1H-TaSe_{2}

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

The structure for the single-layer 1H-TaSe_{2} is shown in Figure 1 (with M = Ta and X = Se). Each Ta 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 Ta atoms. The structural parameters are from Ref. [21], including the lattice constant

34.381 | 83.107 | 1.133 | 1.133 | 0.0 | 3.446 | 0.0 | 3.446 | 0.0 | 4.111 | |

34.936 | 80.019 | 1.133 | 1.133 | 0.0 | 3.446 | 0.0 | 3.446 | 0.0 | 4.111 | |

34.381 | 83.107 | 1.133 | 1.133 | 0.0 | 3.446 | 0.0 | 3.446 | 0.0 | 4.111 |

Table 126 shows the VFF terms for the 1H-TaSe_{2}; 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 _{2}. We have used these phonon frequencies as the phonon dispersion of the single-layer 1H-TaSe_{2}, as the interlayer interaction in the bulk 2H-TaSe_{2} 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-TaSe_{2} (gray pentagons). Figure 61(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.

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Ta_{1}─Se_{1}─Se_{1} | 1.000 | 1.133 | 3.043 | 0.000 | 1.000 | 0.000 | 6.885 | 13.668 | 4 | 0 | 0.0 |

Ta_{1}─Se_{1}─Se_{3} | 1.000 | 0.000 | 0.000 | 34.381 | 1.000 | 0.120 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Ta_{1}─Se_{1}─Se_{2} | 1.000 | 0.000 | 0.000 | 34.936 | 1.000 | 0.173 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Se_{1}─Ta_{1}─Ta_{3} | 1.000 | 0.000 | 0.000 | 34.381 | 1.000 | 0.120 | 0.000 | 0.000 | 4 | 0 | 0.0 |

The parameters for the two-body SW potential used by GULP are shown in Table 127 . The parameters for the three-body SW potential used by GULP are shown in Table 128 . Parameters for the SW potential used by LAMMPS are listed in Table 129 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-TaSe_{2} using LAMMPS, because the angles around atom Ta in Figure 1 (with M = Ta 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 = Ta and X = Se) shows that, for 1H-TaSe_{2}, we can differentiate these angles around the Ta 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 Ta atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-TaSe_{2} under uniaxial tension at 1 and 300 K. Figure 62 shows the stress-strain curve for the tension of a single-layer 1H-TaSe_{2} of dimension _{2} 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-TaSe_{2}. 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 80.8 and 81.1 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

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 15.318 | 10.276 | 10.276 | 10.276 |

2.030 | 87.206 | 74.435 | 87.206 |

^{4}) | |||||
---|---|---|---|---|---|

W─O | 8.781 | 1.005 | 8.491 | 0.0 | 2.744 |

There is no available value for the nonlinear quantities in the single-layer 1H-TaSe_{2}. We have thus used the nonlinear parameter

## 33. 1H-WO_{2}

Most existing theoretical studies on the single-layer 1H-WO_{2} are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1H-WO_{2}.

The structure for the single-layer 1H-WO_{2} is shown in Figure 1 (with M = W and X = O). Each W 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 W atoms. The structural parameters are from the first-principles calculations [12], including the lattice constant

Table 130 shows four VFF terms for the single-layer 1H-WO_{2}; 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 *ab initio* calculations for the phonon dispersion are from Ref. [12]. Figure 63(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.

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

85.955 | 87.206 | 1.005 | 1.005 | 0.0 | 2.744 | 0.0 | 2.744 | 0.0 | 3.262 | |

92.404 | 74.435 | 1.005 | 1.005 | 0.0 | 2.744 | 0.0 | 2.744 | 0.0 | 3.262 | |

85.955 | 87.206 | 1.005 | 1.005 | 0.0 | 2.744 | 0.0 | 2.744 | 0.0 | 3.262 |

The parameters for the two-body SW potential used by GULP are shown in Table 131 . The parameters for the three-body SW potential used by GULP are shown in Table 132 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 133 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-WO_{2} using LAMMPS, because the angles around atom W in Figure 1 (with M = W 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 = W and X = O) shows that, for 1H-WO_{2}, we can differentiate these angles around the W 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 W atom.

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

W_{1}─O_{1}─O_{1} | 1.000 | 1.005 | 2.730 | 0.000 | 1.000 | 0.000 | 8.781 | 8.316 | 4 | 0 | 0.0 |

W_{1}─O_{1}─O_{3} | 1.000 | 0.000 | 0.000 | 85.955 | 1.000 | 0.049 | 0.000 | 0.000 | 4 | 0 | 0.0 |

W_{1}─O_{1}─O_{2} | 1.000 | 0.000 | 0.000 | 92.404 | 1.000 | 0.268 | 0.000 | 0.000 | 4 | 0 | 0.0 |

O_{1}─W_{1}─W_{3} | 1.000 | 0.000 | 0.000 | 85.955 | 1.000 | 0.049 | 0.000 | 0.000 | 4 | 0 | 0.0 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-WO_{2} under uniaxial tension at 1 and 300 K. Figure 64 shows the stress-strain curve for the tension of a single-layer 1H-WO_{2} of dimension _{2} 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-WO_{2}. 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 237.1 and 237.2 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

There is no available value for nonlinear quantities in the single-layer 1H-WO_{2}. We have thus used the nonlinear parameter

## 34. 1H-WS_{2}

Most existing theoretical studies on the single-layer 1H-WS_{2} 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-WS_{2}.

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 8.701 | 7.421 | 7.421 | 6.607 |

2.390 | 81.811 | 81.755 | 81.811 |

Table 134 shows the VFF terms for the 1H-WS_{2}; 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 *ab initio* calculations for the phonon dispersion are from Ref. [31]. Similar phonon dispersion can also be found in other *ab initio* calculations [12, 26, 34, 38, 39]. Figure 65(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.

^{4}) | |||||
---|---|---|---|---|---|

W─S | 5.664 | 0.889 | 15.335 | 0.0 | 3.164 |

37.687 | 81.811 | 0.889 | 0.889 | 0.0 | 3.164 | 0.0 | 3.164 | 0.0 | 3.778 | |

37.697 | 81.755 | 0.889 | 0.889 | 0.0 | 3.164 | 0.0 | 3.164 | 0.0 | 3.778 | |

33.553 | 81.811 | 0.889 | 0.889 | 0.0 | 3.164 | 0.0 | 3.164 | 0.0 | 3.778 |

The parameters for the two-body SW potential used by GULP are shown in Table 135 . The parameters for the three-body SW potential used by GULP are shown in Table 136 . Parameters for the SW potential used by LAMMPS are listed in Table 137 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-WS_{2} using LAMMPS, because the angles around atom W in Figure 1 (with M = W 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 = W and X = S) shows that, for 1H-WS_{2}, we can differentiate these angles around the W 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 W atom.

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

W_{1}─S_{1}─S_{1} | 1.000 | 0.889 | 3.558 | 0.000 | 1.000 | 0.000 | 5.664 | 24.525 | 4 | 0 | 0.0 |

W_{1}─S_{1}─S_{3} | 1.000 | 0.000 | 0.000 | 37.687 | 1.000 | 0.142 | 0.000 | 0.000 | 4 | 0 | 0.0 |

W_{1}─S_{1}─S_{2} | 1.000 | 0.000 | 0.000 | 37.697 | 1.000 | 0.143 | 0.000 | 0.000 | 4 | 0 | 0.0 |

S_{1}─W_{1}─W_{3} | 1.000 | 0.000 | 0.000 | 33.553 | 1.000 | 0.142 | 0.000 | 0.000 | 4 | 0 | 0.0 |

The parameters for the two-body SW potential used by GULP are shown in Table 131 . The parameters for the three-body SW potential used by GULP are shown in Table 132 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 133 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-WO_{2} using LAMMPS, because the angles around atom W in Figure 1 (with M = W 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 = W and X = O) shows that, for 1H-WO_{2}, we can differentiate these angles around the W 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 W atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-WS_{2} under uniaxial tension at 1 and 300 K. Figure 66 shows the stress-strain curve for the tension of a single-layer 1H-WS_{2} of dimension _{2} 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-WS_{2}. 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 121.5 N/m along both armchair and zigzag directions. These values are in reasonable agreement with the *ab initio* results, e.g., 139.6 N/m from Ref. [18], or 148.5 N/m from Ref. [35]. The Poisson’s ratio from the VFF model and the SW potential is *ab initio* value of 0.22 [18].

We have determined the nonlinear parameter to be *ab initio* calculations [35]. We have extracted the value of *ab initio* calculations to the function

## 35. 1H-WSe_{2}

Most existing theoretical studies on the single-layer 1H-WSe_{2} are based on the first-principles calculations. Norouzzadeh and Singh provided one set of parameters for the SW potential for the single-layer 1H-WSe_{2} [40]. In this section, we will develop both VFF model and the SW potential for the single-layer 1H-WSe_{2}.

The structure for the single-layer 1H-WSe_{2} is shown in Figure 1 (with M = W and X = Se). Each W 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 W atoms. The structural parameters are from [12], including the lattice constant

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 8.286 | 8.513 | 8.513 | 7.719 |

2.510 | 80.693 | 83.140 | 80.693 |

Table 138 shows three VFF terms for the 1H-WSe_{2}; 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 *ab initio* calculations for the phonon dispersion are from [31]. Similar phonon dispersion can also be found in other *ab initio* calculations [12, 33, 34, 39, 41]. Figure 67(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.

^{4}) | |||||
---|---|---|---|---|---|

W─Se | 5.476 | 0.706 | 16.273 | 0.0 | 3.308 |

25.607 | 80.693 | 0.706 | 0.706 | 0.0 | 3.308 | 0.0 | 3.308 | 0.0 | 3.953 | |

25.287 | 83.240 | 0.706 | 0.706 | 0.0 | 3.308 | 0.0 | 3.308 | 0.0 | 3.953 | |

23.218 | 80.693 | 0.706 | 0.706 | 0.0 | 3.308 | 0.0 | 3.308 | 0.0 | 3.953 |

The parameters for the two-body SW potential used by GULP are shown in Table 139 . The parameters for the three-body SW potential used by GULP are shown in Table 140 . Parameters for the SW potential used by LAMMPS are listed in Table 141 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-WSe_{2} using LAMMPS, because the angles around atom W in Figure 1 (with M = W 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 = W and X = Se) shows that, for 1H-WSe_{2}, we can differentiate these angles around the W 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 W atom.

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

W_{1}─Se_{1}─Se_{1} | 1.000 | 0.706 | 4.689 | 0.000 | 1.000 | 0.000 | 5.476 | 65.662 | 4 | 0 | 0.0 |

W_{1}─Se_{1}─Se_{3} | 1.000 | 0.000 | 0.000 | 25.607 | 1.000 | 0.162 | 0.000 | 0.000 | 4 | 0 | 0.0 |

W_{1}─Se_{1}─Se_{2} | 1.000 | 0.000 | 0.000 | 25.287 | 1.000 | 0.118 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Se_{1}─W_{1}─W_{3} | 1.000 | 0.000 | 0.000 | 23.218 | 1.000 | 0.162 | 0.000 | 0.000 | 4 | 0 | 0.0 |

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-WSe_{2} under uniaxial tension at 1 and 300 K. Figure 68 shows the stress-strain curve for the tension of a single-layer 1H-WSe_{2} of dimension _{2} 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-WSe_{2}. 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 124.1 and 123 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., 116 N/m from [18], or 126.2 N/m from [35]. The Poisson’s ratio from the VFF model and the SW potential is *ab initio* value of 0.19 [18].

We have determined the nonlinear parameter to be *ab initio* calculations [17]. We have extracted the value of *ab initio* calculations to the function

## 36. 1H-WTe_{2}

Most existing theoretical studies on the single-layer 1H-WTe_{2} 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-WTe_{2}.

The bulk WTe_{2} has the trigonally coordinated H phase structure [43]. However, it has been predicted that the structure of the single-layer WTe_{2} can be either the trigonally coordinated H phase [12] or the octahedrally coordinated _{2} (1H-WTe_{2}), while the SW potential for the _{2} (1T-WTe_{2}) is presented in another section.

VFF type | Bond stretching | Angle bending | ||
---|---|---|---|---|

Expression | ||||

Parameter | 5.483 | 7.016 | 7.016 | 5.718 |

2.730 | 81.111 | 82.686 | 81.111 |

The structure for the single-layer 1H-WTe_{2} is shown in Figure 1 (with M = W and X = Te). Each W 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 W atoms. The structural parameters are from [42], including the lattice constant

^{4}) | |||||
---|---|---|---|---|---|

W─Te | 4.326 | 0.778 | 22.774 | 0.0 | 3.604 |

21.313 | 81.111 | 0.778 | 0.778 | 0.0 | 3.604 | 0.0 | 3.604 | 0.0 | 4.305 | |

21.147 | 82.686 | 0.778 | 0.778 | 0.0 | 3.604 | 0.0 | 3.604 | 0.0 | 4.305 | |

17.370 | 81.111 | 0.778 | 0.778 | 0.0 | 3.604 | 0.0 | 3.604 | 0.0 | 4.305 |

Table 142 shows the VFF terms for the 1H-WTe_{2}; 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 *ab initio* calculations for the phonon dispersion are from [42]. Similar phonon dispersion can also be found in other *ab initio* calculations [12]. Figure 69(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.

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

W_{1}─Te_{1}─Te_{1} | 1.000 | 0.778 | 4.632 | 0.000 | 1.000 | 0.000 | 4.326 | 62.148 | 4 | 0 | 0.0 |

W_{1}─Te_{1}─Te_{3} | 1.000 | 0.000 | 0.000 | 21.313 | 1.000 | 0.155 | 0.000 | 0.000 | 4 | 0 | 0.0 |

W_{1}─Te_{1}─Te_{2} | 1.000 | 0.000 | 0.000 | 21.147 | 1.000 | 0.127 | 0.000 | 0.000 | 4 | 0 | 0.0 |

Te_{1}─W_{1}─W_{3} | 1.000 | 0.000 | 0.000 | 17.370 | 1.000 | 0.155 | 0.000 | 0.000 | 4 | 0 | 0.0 |

The parameters for the two-body SW potential used by GULP are shown in Table 143 . The parameters for the three-body SW potential used by GULP are shown in Table 144 . Parameters for the SW potential used by LAMMPS are listed in Table 145 . We note that 12 atom types have been introduced for the simulation of the single-layer 1H-WTe_{2} using LAMMPS, because the angles around atom W in Figure 1 (with M = W 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 = W and X = Te) shows that, for 1H-WTe_{2}, we can differentiate these angles around the W 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 W atom.

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1H-WTe_{2} under uniaxial tension at 1 and 300 K. Figure 70 shows the stress-strain curve for the tension of a single-layer 1H-WTe_{2} of dimension _{2} 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-WTe_{2}. 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 82.7 and 81.9 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., 86.4 N/m from [18] or 93.9 N/m from [35]. The Poisson’s ratio from the VFF model and the SW potential is *ab initio* value of 0.18 [18].

We have determined the nonlinear parameter to be *ab initio* calculations [35]. We have extracted the value of *ab initio* calculations to the function

## 37. 1T-ScO_{2}

Most existing theoretical studies on the single-layer 1T-ScO_{2} are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ScO_{2}.

The structure for the single-layer 1T-ScO_{2} is shown in Figure 71 (with M = Sc and X = O). Each Ni 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

VFF type | Bond stretching | Angle bending | |
---|---|---|---|

Expression | |||

Parameter | 11.926 | 3.258 | 3.258 |

2.07 | 102.115 | 102.115 |

Table 146 shows three VFF terms for the single-layer 1T-ScO_{2}; 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). We note that the angle bending term *ab initio* calculations for the phonon dispersion are from [12]. Figure 72(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}) | |||
---|---|---|---|---|---|

Sc─O | 10.187 | 1.493 | 9.180 | 0.0 | 2.949 |

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

50.913 | 102.115 | 1.493 | 1.493 | 0.0 | 2.949 | 0.0 | 2.949 | 0.0 | 4.399 | |

50.913 | 102.115 | 1.493 | 1.493 | 0.0 | 2.949 | 0.0 | 2.949 | 0.0 | 4.399 |

The parameters for the two-body SW potential used by GULP are shown in Table 147 . The parameters for the three-body SW potential used by GULP are shown in Table 148 . Some representative parameters for the SW potential used by LAMMPS are listed in Table 149 .

We use LAMMPS to perform MD simulations for the mechanical behavior of the single-layer 1T-ScO_{2} under uniaxial tension at 1 and 300 K. Figure 73 shows the stress-strain curve for the tension of a single-layer 1T-ScO_{2} of dimension _{2} 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 1T-ScO_{2}. 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 100.9 and 100.4 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

Tol | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Sc─O_{1}─O_{1} | 1.000 | 1.493 | 1.975 | 50.913 | 1.000 | −0.210 | 10.187 | 1.847 | 4 | 0 | 0.0 |

VFF type | Bond stretching | Angle bending | |
---|---|---|---|

Expression | |||

Parameter | 3.512 | 1.593 | 1.593 |

2.50 | 92.771 | 92.771 |

A (eV) | ρ (Å) | B (Å^{4}) | |||
---|---|---|---|---|---|

Sc─S | 3.516 | 1.443 | 19.531 | 0.0 | 3.450 |

There is no available value for nonlinear quantities in the single-layer 1T-ScO_{2}. We have thus used the nonlinear parameter

## 38. 1T-ScS_{2}

Most existing theoretical studies on the single-layer 1T-ScS_{2} are based on the first-principles calculations. In this section, we will develop the SW potential for the single-layer 1T-ScS_{2}.

The structure for the single-layer 1T-ScS_{2} is shown in Figure 71 (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

K (eV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

16.674 | 92.771 | 1.443 | 1.443 | 0.0 | 3.450 | 0.0 | 3.450 | 0.0 | 4.945 | |

16.674 | 92.771 | 1.443 | 1.443 | 0.0 | 3.450 | 0.0 | 3.450 | 0.0 | 4.945 |

Table 150 shows three VFF terms for the single-layer 1T-ScS_{2}; 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). We note that the angle bending term