1HMX_{2}, with M as the transition metal and X as oxygen or dichalcogenide.
Abstract
We parametrize the StillingerWeber potential for 156 twodimensional atomic crystals (TDACs). Parameters for the StillingerWeber 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 StillingerWeber 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 StillingerWeber potential using GULP, and files for phonon calculations with the valence force field model using GULP.
Keywords
 layered crystal
 StillingerWeber 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 StillingerWeber (SW) empirical potential for 156 twodimensional 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
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],
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.
1HScO_{2}  1HScS_{2}  1HScSe_{2}  1HScTe_{2}  1HTiTe_{2}  1HVO_{2}  1HVS_{2}  1HVSe_{2}  1HVTe_{2} 
1HCrO_{2}  1HCrS_{2}  1HCrSe_{2}  1HCrTe_{2}  1HMnO_{2}  1HFeO_{2}  1HFeS_{2}  1HFeSe_{2}  1HFeTe_{2} 
1HCoTe_{2}  1HNiS_{2}  1HNiSe_{2}  1HNiTe_{2}  1HNbS_{2}  1HNbSe_{2}  1HMoO_{2}  1HMoS_{2}  1HMoSe_{2} 
1HMoTe_{2}  1HTaS_{2}  1HTaSe_{2}  1HWO_{2}  1HWS_{2}  1HWSe_{2}  1HWTe_{2} 
1TScO_{2}  1TScS_{2}  1TScSe_{2}  1TScTe_{2}  1TTiS_{2}  1TTiSe_{2}  1TTiTe_{2}  1TVS_{2}  1TVSe_{2} 
1TVTe_{2}  1TMnO_{2}  1TMnS_{2}  1TMnSe_{2}  1TMnTe_{2}  1TCoTe_{2}  1TNiO_{2}  1TNiS_{2}  1TNiSe_{2} 
1TNiTe_{2}  1TZrS_{2}  1TZrSe_{2}  1TZrTe_{2}  1TNbS_{2}  1TNbSe_{2}  1TNbTe_{2}  1TMoS_{2}  1TMoSe_{2} 
1TMoTe_{2}  1TTcS_{2}  1TTcSe_{2}  1TTcTe_{2}  1TRhTe_{2}  1TPdS_{2}  1TPdSe_{2}  1TPdTe_{2}  1TSnS_{2} 
1TSnSe_{2}  1THfS_{2}  1THfSe_{2}  1THfTe_{2}  1TTaS_{2}  1TTaSe_{2}  1TTaTe_{2}  1TWS_{2}  1TWSe_{2} 
1TWTe_{2}  1TReS_{2}  1TReSe_{2}  1TReTe_{2}  1TIrTe_{2}  1TPtS_{2}  1TPtSe_{2}  1TPtTe_{2} 
Black phosphorus  pArsenene  pAntimonene  pBismuthene 
pSiO  pGeO  pSnO  
pCS  pSiS  pGeS  pSnS 
pCSe  pSiSe  pGeSe  pSnSe 
pCTe  pSiTe  pGeTe  pSnTe 
Silicene  Germanene  Stanene  Indiene 

Blue phosphorus  bArsenene  bAntimonene  bBismuthene 
bCO  bSiO  bGeO  bSnO 
bCS  bSiS  bGeS  bSnS 
bCSe  bSiSe  bGeSe  bSnSe 
bCTe  bSiTe  bGeTe  bSnTe 
bSnGe  bSiGe  bSnSi  bInP  bInAs  bInSb  bGaAs  bGaP  bAlSb 
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
where
2.2. SW potential
In the SW potential, energy increments for the bond stretching and angle bending are described by the following twobody and threebody forms,
where
where
The energy parameters
where the coefficient
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
VFF type  Bond stretching  Angle bending  

Expression 




Parameter  9.417  4.825  4.825  4.825 

2.090  98.222  58.398  98.222 







ScO  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 firstprinciples 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. 1HSCO_{2}
Most existing theoretical studies on the singlelayer 1HScO_{2} are based on the firstprinciples calculations. In this section, we will develop the SW potential for the singlelayer 1HScO_{2}.
The structure for the singlelayer 1HScO_{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 firstprinciples calculations [12], including the lattice constant













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 singlelayer 1HScO_{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
The parameters for the twobody SW potential used by GULP are shown in Table 11 . The parameters for the threebody 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 singlelayer 1HScO_{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 1HScO_{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.





cos 




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 singlelayer 1HScO_{2} under uniaxial tension at 1 and 300 K.
Figure 4
shows the stressstrain curve for the tension of a singlelayer 1HScO_{2} of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The singlelayer 1HScO_{2} is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasitwodimensional structure of the singlelayer 1HScO_{2}. The Young’s modulus can be obtained by a linear fitting of the stressstrain 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 

2.520  94.467  64.076  94.467 







Sc─S  5.505  1.519  20.164  0.0  3.498 
There is no available value for nonlinear quantities in the singlelayer 1HScO_{2}. We have thus used the nonlinear parameter
4. 1HSCS_{2}
Most existing theoretical studies on the singlelayer 1HScS_{2} are based on the firstprinciples calculations. In this section, we will develop the SW potential for the singlelayer 1HScS_{2}.
The structure for the singlelayer 1HScS_{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 firstprinciples calculations [12], including the lattice constant













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 singlelayer 1HScS_{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
The parameters for the twobody SW potential used by GULP are shown in Table 15 . The parameters for the threebody 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 singlelayer 1HScS_{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 1HScS_{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.





cos 




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 singlelayer 1HScS_{2} under uniaxial tension at 1 and 300 K.
Figure 6
shows the stressstrain curve for the tension of a singlelayer 1HScS_{2} of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The singlelayer 1HScS_{2} is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasitwodimensional structure of the singlelayer 1HScS_{2}. The Young’s modulus can be obtained by a linear fitting of the stressstrain 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 singlelayer 1HScS_{2}. We have thus used the nonlinear parameter
VFF type  Bond stretching  Angle bending  

Expression 




Parameter  5.192  2.027  2.027  2.027 

2.650  92.859  66.432  92.859 







ScSe  5.853  1.533  24.658  0.0  3.658 
5. 1HSCSE_{2}
Most existing theoretical studies on the singlelayer 1HScSe_{2} are based on the firstprinciples calculations. In this section, we will develop the SW potential for the singlelayer 1HScSe_{2}.
The structure for the singlelayer 1HScSe_{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 firstprinciples calculations [12], including the lattice constant
Table 18
shows four VFF terms for the singlelayer 1HScSe_{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













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 twobody SW potential used by GULP are shown in Table 19 . The parameters for the threebody 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 singlelayer 1HScSe_{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 1HScSe_{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.





cos 




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 singlelayer 1HScSe_{2} under uniaxial tension at 1 and 300 K.
Figure 8
shows the stressstrain curve for the tension of a singlelayer 1HScSe_{2} of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The singlelayer 1HScSe_{2} is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasitwodimensional structure of the singlelayer 1HScSe_{2}. The Young’s modulus can be obtained by a linear fitting of the stressstrain 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 singlelayer 1HScSe_{2}. We have thus used the nonlinear parameter
6. 1HSCTE_{2}
Most existing theoretical studies on the singlelayer 1HScTe_{2} are based on the firstprinciples calculations. In this section, we will develop the SW potential for the singlelayer 1HScTe_{2}.
The structure for the singlelayer 1HScTe_{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 firstprinciples calculations [12], including the lattice constant
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 singlelayer 1HScTe_{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







ScTe  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 twobody SW potential used by GULP are shown in Table 23 . The parameters for the threebody 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 singlelayer 1HScTe_{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 1HScTe_{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 singlelayer 1HScTe_{2} under uniaxial tension at 1 and 300 K.
Figure 10
shows the stressstrain curve for the tension of a singlelayer 1HScTe_{2} of dimension
There is no available value for nonlinear quantities in the singlelayer 1HScTe_{2}. We have thus used the nonlinear parameter
7. 1HTITE_{2}
Most existing theoretical studies on the singlelayer 1HTiTe_{2} are based on the firstprinciples calculations. In this section, we will develop both VFF model and the SW potential for the singlelayer 1HTiTe_{2}.
The structure for the singlelayer 1HTiTe_{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 1HTiTe_{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







TiTe  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 twobody SW potential used by GULP are shown in Table 27 . The parameters for the threebody 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 singlelayer 1HTiTe_{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 1HTiTe_{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.










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 singlelayer 1HTiTe_{2} under uniaxial tension at 1 and 300 K.
Figure 12
shows the stressstrain curve for the tension of a singlelayer 1HTiTe_{2} of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The singlelayer 1HTiTe_{2} is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasitwodimensional structure of the singlelayer 1HTiTe_{2}. The Young’s modulus can be obtained by a linear fitting of the stressstrain 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 singlelayer 1HTiTe_{2}. We have thus used the nonlinear parameter
8. 1HVO_{2}
Most existing theoretical studies on the singlelayer 1HVO_{2} are based on the firstprinciples calculations. In this section, we will develop the SW potential for the singlelayer 1HVO_{2}.
The structure for the singlelayer 1HVO_{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 firstprinciples calculations [12], including the lattice constant
VFF type  Bond stretching  Angle bending  

Expression 




Parameter  9.417  4.825  4.825  4.825 

1.920  89.356  71.436  89.356 







VO  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 





cos 




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 singlelayer 1HVO_{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
The parameters for the twobody SW potential used by GULP are shown in Table 31 . The parameters for the threebody 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 singlelayer 1HVO_{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 1HVO_{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 singlelayer 1HVO_{2} under uniaxial tension at 1 and 300 K.
Figure 14
shows the stressstrain curve for the tension of a singlelayer 1HVO_{2} of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The singlelayer 1HVO_{2} is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasitwodimensional structure of the singlelayer 1HVO_{2}. The Young’s modulus can be obtained by a linear fitting of the stressstrain 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 singlelayer 1HVO_{2}. We have thus used the nonlinear parameter
9. 1HVS_{2}
Most existing theoretical studies on the singlelayer 1HVS_{2} are based on the firstprinciples calculations. In this section, we will develop both VFF model and the SW potential for the singlelayer 1HVS_{2}.
The structure for the singlelayer 1HVS_{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 1HVS_{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
VFF type  Bond stretching  Angle bending  

Expression 




Parameter  8.392  4.074  4.074  4.074 

2.310  83.954  78.878  83.954 
The parameters for the twobody SW potential used by GULP are shown in Table 35 . The parameters for the threebody 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 singlelayer 1HVS_{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 1HVS_{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.







V─S  5.714  1.037  14.237  0.0  3.084 













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 





cos 




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 singlelayer 1HVS_{2} under uniaxial tension at 1 and 300 K.
Figure 16
shows the stressstrain curve for the tension of a singlelayer 1HVS_{2} of dimension 100 × 100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The singlelayer 1HVS_{2} is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasitwodimensional structure of the singlelayer 1HVS_{2}. The Young’s modulus can be obtained by a linear fitting of the stressstrain 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 singlelayer 1HVS_{2}. We have thus used the nonlinear parameter
10. 1HVSe_{2}
Most existing theoretical studies on the singlelayer 1HVSe_{2} are based on the firstprinciples calculations. In this section, we will develop both VFF model and the SW potential for the singlelayer 1HVSe_{2}.
The structure for the singlelayer 1HVSe_{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
Table 38
shows the VFF terms for the 1HVSe_{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
The parameters for the twobody SW potential used by GULP are shown in Table 39 . The parameters for the threebody 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 singlelayer 1HVSe_{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 1HVse_{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 

2.450  82.787  80.450  82.787 







V─Se  4.817  1.061  18.015  0.0  3.256 













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 singlelayer 1HVSe_{2} under uniaxial tension at 1 and 300 K.
Figure 18
shows the stressstrain curve for the tension of a singlelayer 1HVSe_{2} of dimension 100×100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The singlelayer 1HVSe_{2} is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasitwodimensional structure of the singlelayer 1HVSe_{2}. The Young’s modulus can be obtained by a linear fitting of the stressstrain 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





cos 




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 

2.660  81.708  81.891  81.708 
There is no available value for the nonlinear quantities in the singlelayer 1HVSe_{2}. We have thus used the nonlinear parameter
11. 1HVTe_{2}
Most existing theoretical studies on the singlelayer 1HVTe_{2} are based on the firstprinciples calculations. In this section, we will develop both VFF model and the SW potential for the singlelayer 1HVTe_{2}.







V─Te  5.410  1.112  25.032  0.0  3.520 
The structure for the singlelayer 1HVTe_{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













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 1HVTe_{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
The parameters for the twobody SW potential used by GULP are shown in Table 43 . The parameters for the threebody 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 singlelayer 1HVTe_{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 1HVTe_{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 singlelayer 1HVTe_{2} under uniaxial tension at 1 and 300 K.
Figure 20
shows the stressstrain curve for the tension of a singlelayer 1HVTe_{2} of dimension
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 singlelayer 1HVTe_{2}. We have thus used the nonlinear parameter
12. 1HCrO_{2}
Most existing theoretical studies on the singlelayer 1HCrO_{2} are based on the firstprinciples calculations. In this section, we will develop the SW potential for the singlelayer 1HCrO_{2}.
The structure for the singlelayer 1HCrO_{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 firstprinciples calculations [12], including the lattice constant
Table 46
shows four VFF terms for the singlelayer 1HCrO_{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













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 twobody SW potential used by GULP are shown in Table 47 . The parameters for the threebody 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 singlelayer 1HCrO_{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 1HCrO_{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 singlelayer 1HCrO_{2} under uniaxial tension at 1 and 300 K.
Figure 22
shows the stressstrain curve for the tension of a singlelayer 1HCrO_{2} of dimension
There is no available value for nonlinear quantities in the singlelayer 1HCrO_{2}. We have thus used the nonlinear parameter
13. 1HCrS_{2}
Most existing theoretical studies on the singlelayer 1HCrS_{2} are based on the firstprinciples calculations. In this section, we will develop both VFF model and the SW potential for the singlelayer 1HCrS_{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 singlelayer 1HCrS_{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 1HCrS_{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
The parameters for the twobody SW potential used by GULP are shown in Table 51 . The parameters for the threebody 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 singlelayer 1HCrS_{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 1HCrS_{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 singlelayer 1HCrS_{2} under uniaxial tension at 1 and 300 K.
Figure 24
shows the stressstrain curve for the tension of a singlelayer 1HCrS_{2} of dimension
There is no available value for the nonlinear quantities in the singlelayer 1HCrS_{2}. We have thus used the nonlinear parameter
14. 1HCrSe_{2}
Most existing theoretical studies on the singlelayer 1HCrSe_{2} are based on the firstprinciples calculations. In this section, we will develop both VFF model and the SW potential for the singlelayer 1HCrSe_{2}.
The structure for the singlelayer 1HCrSe_{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 1HCrSe_{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







Cr─Se  6.581  1.012  16.043  0.0  3.156 













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 twobody SW potential used by GULP are shown in Table 55 . The parameters for the threebody 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 singlelayer 1HCrSe_{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 1HCrSe_{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 singlelayer 1HCrSe_{2} under uniaxial tension at 1 and 300 K.
Figure 26
shows the stressstrain curve for the tension of a singlelayer 1HCrSe_{2} of dimension
There is no available value for the nonlinear quantities in the singlelayer 1HCrSe_{2}. We have thus used the nonlinear parameter
15. 1HCrTe_{2}
Most existing theoretical studies on the singlelayer 1HCrTe_{2} are based on the firstprinciples calculations. In this section, we will develop both VFF model and the SW potential for the singlelayer 1HCrTe_{2}.
The structure for the singlelayer 1HCrTe_{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 1HCrTe_{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







Cr─Te  6.627  1.094  22.154  0.0  3.420 













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 twobody SW potential used by GULP are shown in Table 59 . The parameters for the threebody 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 singlelayer 1HCrTe_{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 1HCrTe_{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 singlelayer 1HCrTe_{2} under uniaxial tension at 1 and 300 K.
Figure 28
shows the stressstrain curve for the tension of a singlelayer 1HCrTe_{2} of dimension
There is no available value for the nonlinear quantities in the singlelayer 1HCrTe_{2}. We have thus used the nonlinear parameter
16. 1HMnO_{2}
Most existing theoretical studies on the singlelayer 1HMnO_{2} are based on the firstprinciples calculations. In this section, we will develop the SW potential for the singlelayer 1HMnO_{2}.
The structure for the singlelayer 1HMnO_{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 firstprinciples 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 singlelayer 1HMnO_{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







Mn─O  4.721  0.961  6.114  0.0  2.540 













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 twobody SW potential used by GULP are shown in Table 63 . The parameters for the threebody 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 singlelayer 1HMnO_{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 1HMnO_{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 singlelayer 1HMnO_{2} under uniaxial tension at 1 and 300 K.
Figure 30
shows the stressstrain curve for the tension of a singlelayer 1HMnO_{2} of dimension
There is no available value for nonlinear quantities in the singlelayer 1HMnO_{2}. We have thus used the nonlinear parameter
17. 1HFeO_{2}
Most existing theoretical studies on the singlelayer 1HFeO_{2} are based on the firstprinciples calculations. In this section, we will develop the SW potential for the singlelayer 1HFeO_{2}.
The structure for the singlelayer 1HFeO_{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 firstprinciples 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 singlelayer 1HFeO_{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







Fe─O  4.242  0.962  6.246  0.0  2.552 













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 twobody SW potential used by GULP are shown in Table 67 . The parameters for the threebody 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 singlelayer 1HFeO_{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 1HFeO_{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.










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 singlelayer 1HFeO_{2} under uniaxial tension at 1 and 300 K.
Figure 32
shows the stressstrain curve for the tension of a singlelayer 1HFeO_{2} of dimension 100×100 Å. Periodic boundary conditions are applied in both armchair and zigzag directions. The singlelayer 1HFeO_{2} is stretched uniaxially along the armchair or zigzag direction. The stress is calculated without involving the actual thickness of the quasitwodimensional structure of the singlelayer 1HFeO_{2}. The Young’s modulus can be obtained by a linear fitting of the stressstrain 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 singlelayer 1HFeO_{2}. We have thus used the nonlinear parameter
18. 1HFES_{2}
Most existing theoretical studies on the singlelayer 1HFeS_{2} are based on the firstprinciples calculations. In this section, we will develop the SW potential for the singlelayer 1HFeS_{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 







Fe─S  4.337  1.097  12.145  0.0  3.000 













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 singlelayer 1HFeS_{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 firstprinciples calculations [12], including the lattice constant
Table 70
shows four VFF terms for the singlelayer 1HFeS_{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
The parameters for the twobody SW potential used by GULP are shown in Table 71 . The parameters for the threebody 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 singlelayer 1HFeS_{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 1HFeS_{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 singlelayer 1HFeS_{2} under uniaxial tension at 1 and 300 K.
Figure 34
shows the stressstrain curve for the tension of a singlelayer 1HFeS_{2} of dimension
There is no available value for nonlinear quantities in the singlelayer 1HFeS_{2}. We have thus used the nonlinear parameter
19. 1HFESE_{2}
Most existing theoretical studies on the singlelayer 1HFeSe_{2} are based on the firstprinciples calculations. In this section, we will develop the SW potential for the singlelayer 1HFeSe_{2}.
The structure for the singlelayer 1HFeSe_{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 firstprinciples calculations [12], including the lattice constant
Table 74
shows four VFF terms for the singlelayer 1HFeSe_{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
The parameters for the twobody SW potential used by GULP are shown in Table 75 . The parameters for the threebody 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 singlelayer 1HFeSe_{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 1HFeSe_{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 







FeSe  4.778  1.139  15.249  0.0  3.168 













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 singlelayer 1HFeSe_{2} under uniaxial tension at 1 and 300 K.
Figure 36
shows the stressstrain curve for the tension of a singlelayer 1HFeSe_{2} of dimension










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 singlelayer 1HFeSe_{2}. We have thus used the nonlinear parameter







Fe─Te  5.599  1.242  20.486  0.0  3.416 
20. 1HFETE_{2}
Most existing theoretical studies on the singlelayer 1HFeTe_{2} are based on the firstprinciples calculations. In this section, we will develop the SW potential for the singlelayer 1HFeTe_{2}.













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 singlelayer 1HFeTe_{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 firstprinciples calculations [12], including the lattice constant
Table 78
shows four VFF terms for the singlelayer 1HFeTe_{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 inplane acoustic branches in the phonon dispersion along the ΓM as shown in
Figure 37(a)
. The










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 twobody SW potential used by GULP are shown in Table 79 . The parameters for the threebody 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 singlelayer 1HFeTe_{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 1HFeTe_{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 singlelayer 1HFeTe_{2} under uniaxial tension at 1 and 300 K.
Figure 38
shows the stressstrain curve for the tension of a singlelayer 1HFeTe_{2} of dimension
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 singlelayer 1HFeTe_{2}. We have thus used the nonlinear parameter
21. 1HCOTE_{2}
Most existing theoretical studies on the singlelayer 1HCoTe_{2} are based on the firstprinciples calculations. In this section, we will develop the SW potential for the singlelayer 1HCoTe_{2}.







Co─Te  6.169  1.310  19.846  0.0  3.417 
The structure for the singlelayer 1HCoTe_{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 firstprinciples calculations [12], including the lattice constant













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 singlelayer 1HCoTe_{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
The parameters for the twobody SW potential used by GULP are shown in Table 83 . The parameters for the threebody 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 singlelayer 1HCoTe_{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 1HCoTe_{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 singlelayer 1HCoTe_{2} under uniaxial tension at 1 and 300 K.
Figure 40
shows the stressstrain curve for the tension of a singlelayer 1HCoTe_{2} of dimension
VFF type  Bond stretching  Angle bending  

Expression 




Parameter  6.933  3.418  3.418  3.418 

2.240  98.740  57.593  98.740 







NiS  6.425  1.498  12.588  0.0  3.156 
There is no available value for nonlinear quantities in the singlelayer 1HCoTe_{2}. We have thus used the nonlinear parameter
22. 1HNIS_{2}
Most existing theoretical studies on the singlelayer 1HNiS_{2} are based on the firstprinciples calculations. In this section, we will develop the SW potential for the singlelayer 1HNiS_{2}.
The structure for the singlelayer 1HNiS_{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 firstprinciples calculations [12], including the lattice constant
Table 86
shows four VFF terms for the singlelayer 1HNiS_{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













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 twobody SW potential used by GULP are shown in Table 87 . The parameters for the threebody 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 singlelayer 1HNiS_{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 1HNiS_{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 singlelayer 1HNiS_{2} under uniaxial tension at 1 and 300 K.
Figure 42
shows the stressstrain curve for the tension of a singlelayer 1HNiS_{2} of dimension
There is no available value for nonlinear quantities in the singlelayer 1HNiS_{2}. We have thus used the nonlinear parameter
23. 1HNISE_{2}
Most existing theoretical studies on the singlelayer 1HNiSe_{2} are based on the firstprinciples calculations. In this section, we will develop the SW potential for the singlelayer 1HNiSe_{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 







Ni─Se  4.004  1.267  15.249  0.0  3.213 













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 singlelayer 1HNiSe_{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 firstprinciples calculations [12], including the lattice constant
Table 90
shows four VFF terms for the singlelayer 1HNiSe_{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
The parameters for the twobody SW potential used by GULP are shown in Table 91 . The parameters for the threebody 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 singlelayer 1HNiSe_{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 1HNiSe_{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 singlelayer 1HNiSe_{2} under uniaxial tension at 1 and 300 K.
Figure 44
shows the stressstrain curve for the tension of a singlelayer 1HNiSe_{2} of dimension
There is no available value for nonlinear quantities in the singlelayer 1HNiSe_{2}. We have thus used the nonlinear parameter
24. 1HNITE_{2}
Most existing theoretical studies on the singlelayer 1HNiTe_{2} are based on the firstprinciples calculations. In this section, we will develop the SW potential for the singlelayer 1HNiTe_{2}.
The structure for the singlelayer 1HNiTe_{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 firstprinciples 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 singlelayer 1HNiTe_{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







Ni─Te  6.461  1.359  20.812  0.0  3.469 













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 twobody SW potential used by GULP are shown in Table 95 . The parameters for the threebody 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 singlelayer 1HNiTe_{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 1HNiTe_{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 singlelayer 1HNiTe_{2} under uniaxial tension at 1 and 300 K.
Figure 46
shows the stressstrain curve for the tension of a singlelayer 1HNiTe_{2} of dimension
There is no available value for nonlinear quantities in the singlelayer 1HNiTe_{2}. We have thus used the nonlinear parameter
25. 1HNBS_{2}
In 1983, the VFF model was developed to investigate the lattice dynamical properties in the bulk 2HNbS_{2} [21]. In this section, we will develop the SW potential for the singlelayer 1HNbS_{2}.
The structure for the singlelayer 1HNbS_{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 1HNbS_{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 2HNbS_{2}. We have used these phonon frequencies as the phonon dispersion of the singlelayer 1HNbS_{2}, as the interlayer interaction in the bulk 2HNbS_{2} only induces weak effects on the two inplane acoustic branches. The interlayer coupling will strengthen the outofplane acoustic branch (flexural branch), so the flexural branch from the present VFF model (blue line) is lower than the theoretical results for bulk 2HNbS_{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.







Nb─S  6.439  1.116  18.610  0.0  3.300 













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 twobody SW potential used by GULP are shown in Table 99 . The parameters for the threebody 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 singlelayer 1HNbS_{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 1HNbS_{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 singlelayer 1HNbS_{2} under uniaxial tension at 1 and 300 K.
Figure 48
shows the stressstrain curve for the tension of a singlelayer 1HNbS_{2} of dimension
There is no available value for the nonlinear quantities in the singlelayer 1HNbS_{2}. We have thus used the nonlinear parameter
26. 1HNBSE_{2}
In 1983, the VFF model was developed to investigate the lattice dynamical properties in the bulk 2HNbSe_{2} [15, 21]. In this section, we will develop the SW potential for the singlelayer 1HNbSe_{2}.
The structure for the singlelayer 1HNbSe_{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 1HNbSe_{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 2HNbSe_{2}. We have used these phonon frequencies as the phonon dispersion of the singlelayer 1HNbSe_{2}, as the interlayer interaction in the bulk 2HNbSe_{2} only induces weak effects on the two inplane acoustic branches. The interlayer coupling will strengthen the outofplane acoustic branch (flexural branch), so the flexural branch from the present VFF model (blue line) is lower than the theoretical results for bulk 2HNbSe_{2} (gray pentagons). It turns out that the VFF parameters for the singlelayer 1HNbSe_{2} are the same as the singlelayer NbS_{2}. The phonon dispersion for singlelayer 1HNbSe_{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.







NbSe  6.942  1.138  22.849  0.0  3.460 













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 twobody SW potential used by GULP are shown in Table 103 . The parameters for the threebody 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 singlelayer 1HNbSe_{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 1HNbSe_{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 singlelayer 1HNbSe_{2} under uniaxial tension at 1 and 300 K.
Figure 50
shows the stressstrain curve for the tension of a singlelayer 1HNbSe_{2} of dimension
There is no available value for the nonlinear quantities in the singlelayer 1HNbSe_{2}. We have thus used the nonlinear parameter
27. 1HMoO_{2}
Most existing theoretical studies on the singlelayer 1HMoO_{2} are based on the firstprinciples calculations. In this section, we will develop the SW potential for the singlelayer 1HMoO_{2}.
The structure for the singlelayer 1HMoO_{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 firstprinciples 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 







MoO  8.317  1.015  8.000  0.0  2.712 













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 singlelayer 1HMoO_{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
The parameters for the twobody SW potential used by GULP are shown in Table 107 . The parameters for the threebody 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 singlelayer 1HMoO_{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 1HMoO_{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 singlelayer 1HMoO_{2} under uniaxial tension at 1 and 300 K.
Figure 52
shows the stressstrain curve for the tension of a singlelayer 1HMoO_{2} of dimension
There is no available value for nonlinear quantities in the singlelayer 1HMoO_{2}. We have thus used the nonlinear parameter
28. 1HMoS_{2}
Several potentials have been proposed to describe the interaction for the singlelayer 1HMoS_{2}. In 1975, Wakabayashi et al. developed a VFF model to calculate the phonon spectrum of the bulk 2HMoS_{2} [22]. In 2009, Liang et al. parameterized a bondorder potential for 1HMoS_{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 2HMoS_{2} [24]. The present author (J.W.J.) and his collaborators parameterized the SW potential for 1HMoS_{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 singlelayer 1HMoS_{2} [25].
VFF type  Bond stretching  Angle bending  

Expression 



Parameter  8.640  5.316  4.891 

2.382  80.581  80.581 








6.918  1.252  17.771  0.0  3.16 













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 singlelayer 1HMoS_{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 singlelayer 1HMoS_{2} are from the firstprinciples calculations as shown in
Figure 1
(with M = Mo and X = S) [26]. The Mo atom layer in the singlelayer 1HMoS_{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 singlelayer 1HMoS_{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 singlelayer 1HMoS_{2} in Table 110 , neglecting other weak interaction terms. The SW potential parameters for singlelayer 1HMoS_{2} used by GULP are listed in Tables 111 and 112 . The SW potential parameters for singlelayer 1HMoS_{2} used by LAMMPS [9] are listed in Table 113 . We note that 12 atom types have been introduced for the simulation of the singlelayer 1HMoS_{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 1HMoS_{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 singlelayer 1HMoS_{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 singlelayer 1HMoS_{2} under uniaxial tension at 1 and 300 K.
Figure 54
shows the stressstrain curve during the tension of a singlelayer 1HMoS_{2} of dimension
29. 1HMoSe_{2}
There is a recent parameter set for the SW potential in the singlelayer 1HMoSe_{2} [25]. In this section, we will develop both VFF model and the SW potential for the singlelayer 1HMoSe_{2}.







MoSe  5.737  0.913  18.787  0.0  3.351 
The structure for the singlelayer 1HMoSe_{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













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 1HMoSe_{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
The parameters for the twobody SW potential used by GULP are shown in Table 115 . The parameters for the threebody 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 singlelayer 1HMoSe_{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 1HMoSe_{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 singlelayer 1HMoSe_{2} under uniaxial tension at 1 and 300 K.
Figure 56
shows the stressstrain curve during the tension of a singlelayer 1HMoSe_{2} of dimension
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
30. 1HMoTe_{2}
Most existing theoretical studies on the singlelayer 1HMoTe_{2} are based on the firstprinciples calculations. In this section, we will develop both VFF model and the SW potential for the singlelayer 1HMoTe_{2}.
The structure for the singlelayer 1HMoTe_{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







MoTe  5.086  0.880  24.440  0.0  3.604 













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 1HMoTe_{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
The parameters for the twobody SW potential used by GULP are shown in Table 119 . The parameters for the threebody 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 singlelayer 1HMoTe_{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 1HMoTe_{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 singlelayer 1HMoTe_{2} under uniaxial tension at 1 and 300 K.
Figure 58
shows the stressstrain curve for the tension of a singlelayer 1HMoTe_{2} of dimension
VFF type  Bond stretching  Angle bending  

Expression 




Parameter  8.230  4.811  4.811  4.811 

2.480  83.879  78.979  83.879 







TaS  6.446  1.111  18.914  0.0  3.310 
We have determined the nonlinear parameter to be
31. 1HTaS_{2}
In 1983, the VFF model was developed to investigate the lattice dynamical properties in the bulk 2HTaS_{2} [21]. In this section, we will develop the SW potential for the singlelayer 1HTaS_{2}.
The structure for the singlelayer 1HTaS_{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 1HTaS_{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










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 