1 Introduction
The atomic potential plays a fundamental role in governing both static and dynamic progresses for materials on the microscopic atomic level. With the assistance of the atomic potential $V = V(\pmb {r}_i )$, one can achieve the force for each atom $i$, which can be used to derive the evolution of the whole system.
The first-principles calculations can provide accurate atomic interactions for most materials. However, the first-principles calculations demand rather expensive computational costs for large systems, where empirical or numerical potentials are desirable. According to their procedure, there are two types of potential models: (I) the empirical model with explicit functional form and (II) numerical interpolation model without an explicit functional form.
For potential of type I, many empirical potentials were proposed, including force field (FF) model (Yu 2010), Stillinger-Weber (SW) potential (Stillinger et al. 2000), Tersoff potential (Tersoff 1986), and Brenner potential (Brenner et al. 2002) etc. These empirical potentials are expressed as explicit functions of the cartesian coordinates for all atoms, eg. $V = V(\pmb {r}_i )$ with $\pmb {r}_i $ as the cartesian coordinate for atom $ i$. Their explicit functional forms are constructed based on specific physical considerations. The accuracy of the empirical potential is closely related to the complexity of its functional form. A more complex function can contain more physical effects, so is usually more accurate.
The explicit mathematic form for the empirical potential keeps the same for different materials, but the potential parameters need to be parameterized for different materials. Several approaches have been proposed for the fitting process. In 1994, Ercolessi and Adams proposed the force matching approach for fitting empirical potentials to ab initio atomic forces with the least squares technique (Ercolessi et al. 1994). As an example, they parameterized these 40 parameters for the glue potential for the metal Al, by fitting to lots of ab initio atomic forces. The force matching method has been used to parameterize empirical potentials for many materials, including the EAM potential for magnesium (Liu et al. 1996) or tantalum (Li et al. 2003), the glue-type potential for the Al-Pb system (Landa et al. 2000), a classical interatomic FF for liquid SiO$_{2}$ (Tangney et al. 2002). Izvekov et al. (2004) generalized the force matching approach by solving an overdetermined system of linear equations instead of the direct numerical least-squares fitting used in the original 1994 paper (Ercolessi et al. 1994). Carre et al. (2008) proposed to parameterize a new pair potential for silica by fitting to the partial pair correlations, instead of the atomic forces. The physical quantities used for the fitting process also play an important role in the applicability and accuracy of the empirical potential. The accuracy and applicability of empirical potentials are mainly limited by the physical essence of their mathematic forms, so empirical potentials usually have quite few parameters to be tuned in the fitting process. As a result, some simpler empirical potentials may have difficulty in providing sound descriptions for several physical quantities simultaneously. In this situation, one needs to treat the balance between the accurate description of different quantities.
For potential of type II, some standard interpolation approaches were widely used. In 1994, the moving interpolation technique is proposed for the potential energy surface of polyatomic molecules to find their optimum configurations (Ischtwan et al. 1994). Guo et al. (2007) suggested to improve the efficiency of the interpolating moving least-squares method by local approximations.
The machine learning (ML) is a powerful tool to enhance the performance of the atomic potential model. The ML method is usually used in two different aspects in the development of the atomic potential. Firstly, the ML method can be used as a fitting procedure to parameterize the empirical potential. Secondly, the ML method can be employed to interpolate the energy surface in a numerical manner.
The present review is devoted to the survey of the empirical potential for layered structures, including the 2D atomic layer, the vertical van der Waals (vdW) heterostructure, and the lateral heterostructure as displayed in Fig. 1. There are three types of atomic interactions in these systems shown in Fig. 1, i.e., the intra-layer coupling within the atomic layer, the inter-layer coupling between two neighboring atomic layers, and the intra-layer interaction at the interface between two constitutes. We will discuss these three types of empirical potentials in succession.
Fig. 1
Fig. 1
Atomic layer based structures. Atomic layers (left) can be stacked into the vertical vdW heterostructure (right top), or stitched into the lateral heterostructure (right bottom)
2 Empirical potential for 2D nanomaterials
2.1 FF
In the FF model, the total energy is expanded as a function of the change of bond lengths and angles. There are some usual forms for the FF model
where $\Delta b = b - b_0 $ is the deviation of the bond length from its equilibrium value. Similarly, $\Delta \theta = \theta - \theta _{0} $ is the deviation of the angle. $\Delta \phi = \phi - \phi _{0} $ is the deviation for the twisting angle. $\Delta \gamma = \gamma - \gamma _{0} $ is the deviation of the inversion angle. $k$ is the force constant.
In practice, more terms can be introduced to increase the accuracy of the FF model, but the computation cost will be increased simultaneously. Most interaction terms in Eq. (1) are used by the MM3 (Allinger et al. 1989, Lii et al. 1989a, 1989b) and COMPASS (Sun 1998), so these models are rather accurate and applicable for nonlinear properties. In contrast, the Keating model (Keating 1966) includes only the harmonic terms in $V_b $ and $V_\theta $, so this model is very efficient but can handle only linear properties.
An important task in the FF model is to parameterize its force constant parameters, which are typically determined by numerical fitting to available data. The parameterization can be updated with more accurate data available for the fitting process. Hence, some authors have, as more data becomes available, developed several generations of FF models. As an example, there have been several different versions for the MM model, i.e., MM1 (Allinger 1976), MM2 (Allinger 1977), MM3 (Allinger et al. 1989; Lii et al. 1989a, 1989b), and MM4 (Allinger et al. 1996).
The universal force field (UFF) (Rappe et al. 1992) is a specific model, which includes six terms
where $E_r $ is the bond stretching interaction, $E_\theta $ is the bond angle bending interaction, $E_\phi $ describes the torsional energy for the dihedral angle, $E_\omega $ is the inversion energy and $E_{\rm vdw} $ represents the van der Waals, $E_{\rm el} $ is the electrostatic interaction.
The universality of the UFF is in sense that all parameters needed for the simulation have been provided. More specifically, some fundamental properties (eg. the atomic radius, the effective charge, etc.) for each element have been derived and listed in the UFF paper (Rappe et al. 1992). Based on these atomic properties, all force field parameters can be estimated using general rules.
Let's take the bond stretching term as an example to illustrate the universality of the UFF. The bond stretching interaction is
where $r_0 $ is the initial bond length and $k_r $ is the force constant. The initial bond length $r_0 $ between elements $i$ and $j$ is computed by the following general rule
where $r_i $ and $r_j $ are atom radius, while $r_{\rm BO} $ and $r_{\rm EN} $ are two corrections dependent on the atomic properties of $i$ and $j$. The parameter $(k_r )$ between elements $i$ and $j$, is obtained as follows
where $Z_i^\ast $ and $Z_j^\ast $ are the effective atomic charges. As a result, the bond stretching interaction for any two elements are obtained with the initial bond length given by Eq. (9) and the force constant given by Eq. (10). Similarly, all other terms in Eq. (7) have been determined explicitly in the UFF, i.e., all initial value for geometrical variables (bond length and angle) and force constants are available for all elements.
The distinct feature of the UFF is the universality of this model. It is applicable to describe properties of solids, gases, and liquid. The UFF will be particularly useful as a first simulation tool to study properties for new materials. The UFF has thus been used to simulate a huge number of materials, such as organic molecules (Casewit et al. 1992a), main group compounds (Casewit et al. 1992b), metal complexes (Rappe et al. 1993), and carbon nanotube (Chowdhury et al. 2010), just to name a few of them. As a trade off, the UFF is so universal that it loses some accuracy especially for these materials that has quite limited data to extract the force constant parameters at that time. The data used to determine the force constant parameters for these materials were obtained by interpolating or extrapolating from other elements, so it is necessary to refine the force constants in the UFF for these materials when new data is available.
2.2 SW
Stillinger and Weber (1985) proposed the SW model to simulate the bulk silicon in 1985. The SW potential includes the following two terms
The cut-offs $r_{\max } $, $r_{\max 12} $ and $r_{\max 13} $ can be determined according to the atomic geometry. The SW potential has seven parameters, which can be determined by a numerical fitting to some known properties. We have shown that all linear parameters can be derived analytically, while the nonlinear parameter $B$ is the only parameter to be determined (Jiang 2015).
The SW potential in Eq. (11) and Eq. (11) is able to capture the nonlinear process, like the thermal transport (Jiang & Park et al. 2013). However, the SW potential gives zero bending modulus for graphene-like structures without out-of-plane bonds (Jiang & Qi et al. 2013).
The SW potential has been widely used in simulating materials, ranging from atomically layered materials such as $MoS_{2}$ (Jiang & Park et al. 2013, Jiang (2015), Kandemir et al. 2016), to MoSe$_{2 }$ (Kandemir et al. 2016), WSe$_{2}$ (Norouzzadeh et al. 2017), and black phosphorus (Jiang et al. 2015, Jiang 2015, Xu et al. 2015). The SW potential parameters for 156 two-dimensional nanomaterials can be found in a recent work (Jiang et al. 2017).
2.3 Tersoff
The Tersoff potential was initially proposed to describe the coupling within silicon (Tersoff 1986, 1988c). The Tersoff potential takes the following form
The atomic bonding energy is divided into repulsive $(f_{\rm R} (r_{ij} ))$ and attractive $(f_{\rm A} (r_{ij} ))$ terms in Eq. (14). The repulsive pair term is the result of the overlap of electron wave functions, while the attractive pair energy is the bonding energy. The cutoff function is
As a characteristic feature for the Tersoff potential, the strength of the bond is governed by its neighboring atoms. The bond order is represented by the following quantity $b_{ij} $.
The other quantity is $a_{ij} $.
Several versions of the Tersoff potential are available for the coupling of (both 2D and 3D) silicon (Tersoff 1988c, 1988b), carbon (Tersoff 1988a), and germanium (Tersoff 1989). Other parameter sets are also available for the Tersoff potential of graphene (Lindsay et al. 2010, Jiang et al. 2011).
2.4 Brenner
Similar as the Tersoff potential, the Brenner potential energy is divided into the repulsive and attractive terms as follows (Brenner et al. 2002)
$ V_{\rm B} (r) = V_{\rm R} (r) - \bar {B}_{ij} \cdot V_{\rm A} (r) $
where $V_{\rm R} $ and $V_{\rm A} $ are the repulsive and attractive terms
$ V_{\rm R} (r) = \dfrac{D^{(e)}}{S - 1} {\rm e}^{ - \sqrt {2S} \beta (r - R^{(e)})}f_{\rm c} (r) $
$ V_{\rm A} (r) = \dfrac{D^{(e)}S}{S - 1} {\rm e}^{ - \sqrt {2 / S} \beta (r - R^{(e)})}f_{\rm c} (r) $
with the cut-off function
$ f_{\rm c} (r) = \left\{ \begin{array}{ll} 1, &\quad r < R^{(1)} \\ \dfrac{1}{2}\left\{ {1 + \cos \left[ {\dfrac{\pi (r - R^{(1)})}{R^{(2)} - R^{(1)}}} \right]} \right\}, &\quad R^{(1)} < r < R^{(2)} \\ 0, &\quad r > R^{(2)} \\ \end{array} \right. $
The many-body coupling parameter is
$ \bar {B}_{ij} = \dfrac{1}{2}(B_{ij} + B_{ji} ) $
$ B_{ij} = \left[ {1 + \sum\limits_{k \ne ij} {G(\theta _{ijk} )f_{\rm c} (r_{ik} )} } \right]^{ - \delta } $
The angle function $G ( \theta _{ijk} )$ is
$ G(\theta _{ijk} ) = a_0 \left[ {1 + \dfrac{c_0^2 }{d_0^2 } - \dfrac{c_0^2 }{d_0^2 + (1 + \cos \theta _{ijk})^2}} \right] $
The Brenner potential was widely used to simulate mechanical, thermal, and other physical properties for carbon based materials including the 2D graphene. Liang et al. (2009) parameterized the Brenner potential model for MoS$_2$ in 2009.
2.5 Machine learning
2.5.1 Empirical potentials paramerterized by ML methods
Machine learning schemes can be used to parameterize the parameters of some empirical potentials. The genetic algorithm (GA) was introduced to optimize parameters for the Tersoff potential of the silicon (Bukkapatnam et al. 2006). The GA was used to minimize 600 adjustable parameters for the ReaxFF interatomic potential for nitromethane and its decomposition products (Pahari et al. 2012). The GA was also used to optimize 17 independent parameters for a hybrid bond order potential for diverse geometries of gold (Narayanan et al. 2016).
2.5.2 Interpolation with ML methods
Neural networks can be used to fit nonlinear functions of high flexibility. In particular, neural networks can be used as a general interpolation method for the atomic potential. The mapping between the output value and the input value for the neural network is governed by the architecture of the network (Behler et al. 2007). Blank et al. (1995) used the neural networks to interpolate the potential of molecules, where multilayer networks are trained by the feed-forward technique. For crystals, the total energy is represented as a sum of atomic contributions, which can be interpolated by the neural networks with proper descriptors (Behler et al. 2007). Recently, Chen et al. (2018) suggested to interpolate the atomic potential with the convolutional neural network with the descriptors based on interatomic distances.
The Gaussian regression process was implemented to interpolate the atomic potential (Bartok et al. 2010, 2015, 2017; Artrith et al. 2017) which have been applied to tantalum (Thompson et al. 2015), boron carbide (Gao et al. 2015), amorphous carbon (Deringer & Bernstein 2017), graphene (Rowe et al. 2018), amorphous silicon (Deringer et al. 2018), silicon (Bartok et al. 2018), and boron (Deringer & Pickard 2018), the phase-change memory material Ge$_{2}$Sb$_{2}$Te$_{5}$(Mocanu et al. 2018). Several other multidimensional regression or interpolation approaches have also been used to predict the potential energy, including the linear ridge regression (Seko et al. 2014, 2015), the kernel ridge regression (Jacobsen et al. 2018), and the kriging interpolation (Mills et al. 2011).
The ML potential is rather accurate at interpolating atomic potentials for configurations within the `range' of the training configuration. The range of the training set can be measured by recording the minimum and maximum values of the input descriptors (Behler 2011). However, ML potentials based on regressions are not necessarily applicable for extrapolations, i.e., there is no guarantee in providing accurate predictions for those new configurations out of the range of the training configuration. A practical trick to resolve this issue is to call on the quantum-mechanical calculations for these new configurations, and add these new data into the training set to further train the ML model. This is the so-called on-the-fly manner (Li Z et al. 2015, Jacobsen et al. 2018). With this on-the-fly technique, the ML potential can be extended automatically during the application of the ML potential in practical simulations.
2.5.3 Descriptors
For the ML potential, the cartesian coordinate can not be used as direct input variables for the neural network or the kernel. It is because the potential shall be invariant under translational, rotational, and permutational operations on the structure. These invariances are not satisfied for the atomic cartesian coordinates of the structure. Consequently, the ML potential will violate these invariances if the atomic cartesian coordinates are used as input variables. To resolve this issue, the cartesian coordinates are transformed into some intermediate quantities that fulfill these invariances. These intermediate quantities are used as the input for the neural network. These quantities are called descriptors. There are two key requirements for the descriptor. First, descriptors are invariant under the symmetry operations, including the translational, rotational, and the permutational symmetry operations. Second, the descriptor is unique in sense that the descriptors have different values for two different structures.
Descriptors carry the information for the environment around an atom. There have been many types of descriptors to be used to describe the potential energy of crystal structures. Behler and Parrinello (2007) proposed two types of symmetry functions, one of which is for the radial distribution while the other is for the angular distribution. These symmetry functions use the atomic cartesian coordinate as inputs. The values of these symmetry functions are the so-called descriptors, which are used as inputs for the neural network. The radial symmetry function measures the radial distribution of neighboring atoms for the targeted atom. The function of the radial symmetry function is to count the number of neighboring atoms, which is properly weighted by the distance of these neighbors using smooth cutoff functions. In analogous, the angular symmetry function measures the angle distribution around the targeted atom. Behler (2011) suggested several more symmetry functions, so that a more flexible ML potential can be obtained.
More descriptors are as follows. Bartok et al. developed the bispectrum descriptor (Bartok & Payne 2010). Bartok, Kondor and Csanyi proposed the SOAP descriptor in 2013, which is constructed based on the expansion of the neighbor density in a basis set of radial functions and spherical harmonics (Bartok et al. 2013). Yang et al. (2014) proposed a descriptor based on the Voronoi polyhedron figure, which reflects the local environment for the atom. Schutt et al. (2014) defined a descriptor using the partial radial distribution function for the distribution of pairwise distances between two atom types. Recently, Himanen et al. (2019) implemented several descriptors in their code.
There are some techniques that can improve the performance of the descriptors. Artrith et al. (2017) proposed descriptors based on the expansion coefficients of the radial and angular distribution function, which can prevent a quadratic increase in the dimensions with increasing number of chemical species. Imbalzano et al. (2018) provided a method to reduce the number of descriptors based on the correlations of the training data, so that the selected descriptors are maximally sparse in the description space.
2.6 Comparison between potentials
Different empirical potentials have been proposed based on the consideration of various physical processes. It is thus reasonable that an empirical potential will be of high accuracy in description these physical properties or physical processes, from which they have been developed. We compare the performance of the above empirical potentials on some typical physical processes.
We first survey the application of different potentials in calculating fundamental elastic properties for graphene and $MoS_{2}$ in Table 1 and Table 2. The thickness for atomic layers is not well defined. The value of the Young's modulus listed in the tables have been rescaled by the thickness of 0.34 nm and 0.61 nm for single-layer graphene and single-layer $MoS_{2}$, respectively. The modulus also depends on the method used in the computation. With the same Brenner potential, the Young's modulus and Poisson's ratio are quite different from the molecular dynamics (MD) or the molecular mechanics (MM) method. In particular, the value of the Poisson's ratio from the MM approach is more than two factors larger than that from the MD simulation. It is closely related to the thermal induced ripples in the graphene at finite temperature in the MD simulation. The flattening of these ripples during the MD simulation of stretching process counteracts the lateral shrinking caused by the Poisson effect, resulting in the reduction of the Poisson's ratio. It should be noted that there are lots of other works on the mechanical properties of graphene. We have listed in the table some representative results to better illustrate the difference between different potentials. The value for the Young's modulus and Poisson's ratio will be close to each other if the same potential is applied and similar simulation methods are used.
Table 1 also lists the thermal conductivity for graphene computed with different potentials or methods. It should be noted that the thermal conductivity is governed by many factors, like the nonlinear properties, various defects, sample size, and etc. The value of the thermal conductivity is thus in a rather diverse range. The comparison between different potentials herein is to illustrate advantages or disadvantages of each potential model. The FF model is linear, so it can be used to compute the thermal conductance in the ballistic regime. In the ballistic thermal transport regime, nonlinear properties and other scattering mechanisms are not considered, so it serves as the upper limit for the thermal conductivity in the experiment. The Tersoff and Brenner potential have accurate nonlinear interactions with reasonable simulation efficiency, so these nonlinear potential models can be used in the MD simulations of the thermal transport. The MD simulations with the Tersoff or Brenner potential are useful in studying the size effect on the thermal conductivity of graphene structures with tens of thousands of atoms. The Tersoff and Brenner potentials can also be used to extract the nonlinear force constants, which are used in the Boltzmann transport equation (BTE) approach in the calculation of the thermal conductivity in the diffusive regime. The DFT calculations are accurate while the computational cost is high. DFT approaches are applied to calculate the linear force constant, which can be used to predict the ballistic thermal conductance. The nonlinear force constants can also be extracted by the DFT methods, which can be used to compute the thermal conductivity in the diffusive regime. Due to its high computational cost, DFT methods are usually not adopted in MD simulations for the thermal transport, where the size effect plays an important role.
Table 1 Graphene's properties calculated from different potentials
 |
Note: The Young's modulus $(E)$ and Poisson's ratio $(\nu )$ are shown in the second and third lines, respectively. The thermal conductivity $(\kappa )$ is listed in the fourth line. MM: molecular mechanics. MD: molecular dynamics. LD: lattice dynamics. BTE: Boltzmann transport equation. OP: optimized version. a:
Table 2 The Young's modulus $(E)$ and the Poisson's ratio $\nu $ for $MoS_{2}$ computed from different potentials
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Note: a:
The formation or breaking of chemical bonds is a typical phenomenon that occurs in the fracture or defect formation, and the crystal growth process. The linear FF potential can describe the strength of the chemical bond. A bond is treated as broken when the atomic distance is outside a manually set cutoff value in the FF model. The FF model can provide a qualitative description for the fracture or defect formation process. However, the FF model does not contain the information for the bond order property, so the FF model is not suitable for the simulation of crystal growth, where the bond order plays an important role. A common characteristic feature for the format of the SW, Tersoff, and Brenner potentials is that these potentials are constructed based on the concept of the bond order. These three models include plenty of information on the chemical bonding properties, so they can be applied to simulate the fracture or defect formation and the crystal growth processes.
The phase transformation is a specific chemical process. The atomic configuration is different in structures before and after the phase transformation. To simulate such process, the empirical potential is required to be rather flexible and can describe both structures simultaneously. The mathematic form for the FF model and the SW potential depends on the structural parameter, so it is rather difficult to implement two structural parameters in a single set of FF or SW parameters. The Tersoff potential and the Brenner potential can be parameterized for several structures in a single set of potential parameters, so they can be applied to simulate the transformation between different structures. For instance, the atomic interaction in graphene and diamond can both be described by the Brenner potential, so the pressure-induced transformation from few-layer graphene into diamond can be simulated by the Brenner potential (Scandolo et al. 1995, Guo et al. 2004). In terms of flexibility, the ML potential has lots of parameters and is able to be parameterized for several structural phases with a set of parameters. In principle, the ML potential can be used to simulate the phase transformation between these structures whose data has been used in the training procedure (Fujikake et al. 2018).
3 Empirical potential for vdW heterostructures
The vdW heterostructure is constructed by stacking atomic layers in the vertical direction (Novoselov et al. 2016, Liu Y et al. 2019). Atomic layers are combined together through the inter-layer vdW interaction, while the intra-layer interaction within each layer can be described by these models discussed in the previous section. We thus discuss several potential models for the inter-layer vdW interaction in this section.
3.1 Lennard-Jones
The Lennard-Jones (LJ) potential was proposed in 1924 to describe the vdw interaction (Jones et al. 1924)
where $r$ is atomic distance. The two potential parameters are $ \in $ and $\sigma $. There are several available forms for the LJ potential, like the AB form (Jones et al. 1924) or the shifted form (Pierro et al. 2015).
The $r^{ - 12}$ term represents the Pauli exclusion principle for the overlap of electronic densities. The $r^{ - 6}$ term represents the weak vdW interaction between two instantaneous polar charges.
The LJ potential has been widely applied to simulate the vdW interaction in the noble gases, liquids, and the coupling between neighboring layers in the vdW heterostructures. A list of the LJ parameters for lots of elements can be found in the work by Rappe et al. (1992).
3.2 Exponential-6
As inspired by the exponential functional form of the wave function for the ground state of the hydrogen atom, the repulsive interaction between two overlap electron orbitals can also be captured by the Buckingham potential (Buckingham 1938). As a result, the vdW interaction can also be described by the following expression
where $ \in $ is the energy parameter, and $r=\sigma$ is the minimum-point for the potential. Figure 2 compares the form of the vdW interaction described by the LJ model and the exponential-6 model. The exponential-6 model has the same behavior as the LJ model for the region $r > \sigma $ with attractive force. For the repulsive force region with $r < \sigma $, the exponential-6 model increases slower than the LJ model with decreasing distance.
Fig. 2
Fig. 2
Atomic The vdW interaction described by the LJ model in Eq. (23) and the exponential-6 model in Eq. (24) with parameters $\in = 1.0$ and $\sigma = 3.0$
3.3 Revised models
While it is physically simple, the LJ potential is a pairwise potential without three-body and other many-body interactions. As a result, atoms that interact through the LJ potential favor the formation of close-packed structures, so the LJ potential is not suitable for covalent crystals. However, the LJ potential has been frequently applied to describe the vdW coupling between neighboring 2D atomic layers. It was found that the LJ potential cannot describe the frictional motion of two atoms, because a weak relative frictional, or tangential motion between two atoms does not alter their distance. Therefore, an important shortcoming of the LJ potential is that it underestimates the inter-layer friction in the atomically layered materials. Some corrections have been proposed to overcome this shortcoming (Kolmogorov et al. 2000, 2005; Lebedeva et al. 2011; Jiang & Park 2015).
In 2000, Kolmogorov and Crespi revised the exponential-6 expression for the vdW interaction by including the relative shift between two adjacent atomic layers
where $r$ is the atomic distance, and $z$ is the distance projection to the normal direction while $\rho $ is the projection of the distance onto the atomic layer. Other constants are fitting parameters. This revised form was further improved to accommodate more flexible local geometries (Kolmogorov et al. 2005). Very recently, Wen et al. (2018) further improved this revised form to describe the relative rotation between two adjacent atomic layers with the cross-layer dihedral angle.
Jiang and Park (2015) suggested to improve the LJ potential for the layered structure by including a Gaussian term
where $\rho $ is the relative shift between two adjacent atomic layers, and $g$ is the fitting parameter.
3.4 Comparison between potentials
We list in Table 3 the shift energy $\Delta E^{\max }$ and the rotation force $\Delta F_z^{\max } $ for the bilayer graphene from different inter-layer potentials. The shift energy is the maximum variation of the inter-layer potential energy during the relative shift of two graphene layers, which essentially represents the energy difference between the AA and AB stacking bilayer graphene. The shift energy is related to the frictional property between two graphene layers. The LJ potential underestimates the shift energy by about one order, i.e., the energy difference between the AA and AB stacking is significantly underestimated by the LJ potential. It is due to the intrinsic isotropic property of the LJ potential. The rotation force $\Delta F_z^{\max } $ measures the maximum variation of the inter-layer force along the $z$-direction during the relative rotation of two graphene layers. The rotation motion needs an additional degree of freedom, eg. the dihedral angle. This extra degree of freedom is not included in the previous models like the LJ, RD (Kolmogorov et al. 2005), or the LJ-G (Jiang & Park 2015), so these models provide much smaller value for the rotation force. The DRD model considers this rotational degree of freedom (Wen M et al. 2018), so it is able to give a value close to the DFT result for the rotation force. Overall, there are three typical motion styles between two atomic layers, including the cohesive, shift, and rotation motions. The LJ potential is good enough for the description of the cohesive motion. The RD and LJ-G models provide sufficient degrees of freedom for the description of both cohesive and shift motions between two atomic layers. The latest DRD model can describe all of the cohesive, shift and rotation motions between two atomic layers.
Table 3 Hift and rotation properties for bilayer graphene calculated with different inter-layer potentials
 |
Note: $\Delta E^{\max}$ measures the maximum energy variation during the relative shift between two graphene layers. $\Delta F_z^{\max } $ is the maximum variation in the inter-layer force in the $z$-direction during the relative rotation between two graphene layers. RD: registry dependent. LJ-G: LJ Gaussian. DRD: dihedral corrected registry dependent. a:
4 Empirical potential for lateral heterostructure
The lateral heterostructure is another derived structure to enrich the functionality of 2D nanomaterials. For instance, the growth of transition-metal dichalcogenides (TMDs) in the form of lateral heterostructures seems to be a promising technique to construct lateral p-n diodes with excellent current rectification and other functional devices based on the TMDs (Li M Y et al. 2015, Zhang Z et al. 2017, Sahoo et al. 2018, Xie S et al. 2018).
The interaction of each constitute in the lateral heterostructure can be described by these potential models discussed in the previous section. In contrast to the weak vdW interaction between adjacent atomic layers in the vdW heterostructures, these constitutes in the lateral heterostructure are typically coupled through strong chemical bonds. We discuss some potential models that have been applied to describe the interaction for these strong chemical bonds within the interface area.
4.1 Tersoff
Tersoff (1989) proposed to apply his model to describe the
interaction for multicomponent systems like the Si-Ge compound.
parameters of the Tersoff potential
where parameters like $A_i $ are for a single constitute, while parameters with two subscripts like $A_{ij} $ are for the coupling between two constitutes.
These mixing rules have been applied to simulate the C-Si or Si-Ge multicomponent systems with the Tersoff potential (Tersoff 1989). The Tersoff potential with the mixing rule has also been used to simulate the hexagonal boron nitride (Albe et al. 1997, Jiang & Wang 2011). The mixing rules for the Tersoff potential have been implemented in some codes, eg. GULP (Gale 1997). It is thus quite straightforward to apply this model to simulate multicomponent systems, if the Tersoff potential for each constitute is available.
4.2 SW
In a recent work, we applied the SW potential to describe the interaction between two different TMDs (Jiang 2019). There are two-body and three-body interaction terms in the SW potential model. The energy parameters $({A}$ and $K$ in Eq. (11) and Eq. (12)) in the SW potential for new mixing terms are obtained through the geometric average rule
where $A_i $ and $k_i $ are parameters for the constitute $i$.
5 Future prospects and summary
5.1 Develop new potential model
Physics-inspired model. Most empirical potentials were developed based on some physical mechanisms, eg. each FF term reflects a specific motion. Another example is that the Tersoff potential was developed based on the chemical bond order of each atom. Physics-based models are of concise mathematic forms, so it can greatly speed up the computational simulation while grasp correct physical essences. The development of physics-inspired new potential models is thus a valuable task that can increase both efficiency and accuracy.
Flexible model. However, if no physics-inspired models can be developed to further speed up the simulation, one will inevitably suffer the choose of potential models by considering the trade-off between efficiency and accuracy. An active direction is to develop new potential models that have high flexibility in tuning the trade-off between efficiency and accuracy. The word `flexibility' here means that the efficiency and accuracy of a potential model can be conveniently tuned through some model parameters. The ML potential is one of such models, whose efficiency and accuracy can be tuned by varying the number of parameters in the ML scheme.
5.2 Parameterize existing potential model for new materials
With the development of the chemical synthesis and growth techniques, a huge number of new materials have emerged, like the synthesis of a series of 2D atomic layered materials as inspired by graphene. The mathematic format of the existing potential model does not vary, but its parameters need to be determined for these new materials. Parameterizing existing potential models becomes a key step toward further studying these newly emerged materials. For example, we have parametrized the SW potential for over one hundred 2D materials (Jiang et al. 2017).
5.3 Database for atomic potential
The information has become an important ingredient for scientific research nowadays. There are so many empirical potential models and new materials, that the collection of all available atomic potential will be helpful for researchers to efficiently find the most suitable potential models. One example of such repository project is maintained by Becker et al. (2013) in U.S.
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Conformational analysis. 130. MM2. A hydrocarbon force field utilizing V1 and V2 torsional terms
Molecular Mechanics. The MM3 force field for hydrocarbons. 1
Hyperconjugative effects on carbon-carbon bond lengths in molecular mechanics (MM4)
Efficient and accurate machine-learning interpolation of atomic energies in compositions with many species
Superior thermal conductivity of single-layer graphene
We report the measurement of the thermal conductivity of a suspended single-layer graphene. The room temperature values of the thermal conductivity in the range approximately (4.84+/-0.44)x10(3) to (5.30+/-0.48)x10(3) W/mK were extracted for a single-layer graphene from the dependence of the Raman G peak frequency on the excitation laser power and independently measured G peak temperature coefficient. The extremely high value of the thermal conductivity suggests that graphene can outperform carbon nanotubes in heat conduction. The superb thermal conduction property of graphene is beneficial for the proposed electronic applications and establishes graphene as an excellent material for thermal management.
Gaussian approximation potentials: The accuracy of quantum mechanics, without the electrons
We introduce a class of interatomic potential models that can be automatically generated from data consisting of the energies and forces experienced by atoms, as derived from quantum mechanical calculations. The models do not have a fixed functional form and hence are capable of modeling complex potential energy landscapes. They are systematically improvable with more data. We apply the method to bulk crystals, and test it by calculating properties at high temperatures. Using the interatomic potential to generate the long molecular dynamics trajectories required for such calculations saves orders of magnitude in computational cost.
On representing chemical environments
Gaussian approximation potentials: A brief tutorial introduction
Machine learning unifies the modeling of materials and molecules
Determining the stability of molecules and condensed phases is the cornerstone of atomistic modeling, underpinning our understanding of chemical and materials properties and transformations. We show that a machine-learning model, based on a local description of chemical environments and Bayesian statistical learning, provides a unified framework to predict atomic-scale properties. It captures the quantum mechanical effects governing the complex surface reconstructions of silicon, predicts the stability of different classes of molecules with chemical accuracy, and distinguishes active and inactive protein ligands with more than 99% reliability. The universality and the systematic nature of our framework provide new insight into the potential energy surface of materials and molecules.
Machine learning a general purpose interatomic potential for silicon
Preprint at http://arxiv.org/abs/1805.01568.
Considerations for choosing and using force fields and interatomic potentials in materials science and engineering
Generalized neural-network representation of high-dimensional potential-energy surfaces
The accurate description of chemical processes often requires the use of computationally demanding methods like density-functional theory (DFT), making long simulations of large systems unfeasible. In this Letter we introduce a new kind of neural-network representation of DFT potential-energy surfaces, which provides the energy and forces as a function of all atomic positions in systems of arbitrary size and is several orders of magnitude faster than DFT. The high accuracy of the method is demonstrated for bulk silicon and compared with empirical potentials and DFT. The method is general and can be applied to all types of periodic and nonperiodic systems.
Atom-centered symmetry functions for constructing high-dimensional neural network potentials
Neural networks offer an unbiased and numerically very accurate approach to represent high-dimensional ab initio potential-energy surfaces. Once constructed, neural network potentials can provide the energies and forces many orders of magnitude faster than electronic structure calculations, and thus enable molecular dynamics simulations of large systems. However, Cartesian coordinates are not a good choice to represent the atomic positions, and a transformation to symmetry functions is required. Using simple benchmark systems, the properties of several types of symmetry functions suitable for the construction of high-dimensional neural network potential-energy surfaces are discussed in detail. The symmetry functions are general and can be applied to all types of systems such as molecules, crystalline and amorphous solids, and liquids.
Stretching and breaking of ultrathin $MoS_{2}$
We report on measurements of the stiffness and breaking strength of monolayer MoS(2), a new semiconducting analogue of graphene. Single and bilayer MoS(2) is exfoliated from bulk and transferred to a substrate containing an array of microfabricated circular holes. The resulting suspended, free-standing membranes are deformed and eventually broken using an atomic force microscope. We find that the in-plane stiffness of monolayer MoS(2) is 180 +/- 60 Nm(-1), corresponding to an effective Young's modulus of 270 +/- 100 GPa, which is comparable to that of steel. Breaking occurs at an effective strain between 6 and 11% with the average breaking strength of 15 +/- 3 Nm(-1) (23 GPa). The strength of strongest monolayer membranes is 11% of its Young's modulus, corresponding to the upper theoretical limit which indicates that the material can be highly crystalline and almost defect-free. Our results show that monolayer MoS(2) could be suitable for a variety of applications such as reinforcing elements in composites and for fabrication of flexible electronic devices.
Neural network models of potential energy surfaces
Elastic constants of compression-annealed pyrolytic graphite
A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons
Atomistic simulations of mechanical properties of graphene nanoribbons
The classical equation of state of gaseous Helium, Neon and Argon
Parametrization of interatomic potential functions using a genetic algorithm accelerated with a neural network
Thermal transport in suspended and supported monolayer graphene grown by chemical vapor deposition
Graphene monolayer has been grown by chemical vapor deposition on copper and then suspended over a hole. By measuring the laser heating and monitoring the Raman G peak, we obtain room-temperature thermal conductivity and interface conductance of (370 + 650/-320) W/m K and (28 + 16/-9.2) MW/m(2) K for the supported graphene. The thermal conductivity of the suspended graphene exceeds (2500 + 1100/-1050) W/m K near 350 K and becomes (1400 + 500/-480) W/m K at about 500 K.
New fitting scheme to obtain effective potential from Car-Parrinello molecular-dynamics simulations: Application to silica
Application of a universal force field to main group compounds
Application of a universal force field to organic molecules
A molecular mechanics approach for the vibration of single-walled carbon nanotubes
Abstract
We investigate the vibrational properties of zigzag and armchair single-wall carbon nanotubes (CNTs) using the molecular mechanics approach. The natural frequencies of vibration and their associated intrinsic vibration modes are obtained. The simulations are carried out for four types of zigzag nanotubes (5, 0), (6, 0), (8, 0), (10, 0) and three types of armchair nanotubes (3, 3), (4, 4), (6, 6). The universal force field potential is used for the molecular mechanics approach. The first five natural frequencies are obtained for aspect ratios ranging from 5 to 20. The results indicate that the natural frequencies decrease as the aspect ratios increase. The results follow similar trends with results of previous studies for CNTs using structural mechanics approach.
Erratum: Nonlinear elastic behavior of two-dimensional molybdenum disulfide (Physical Review B—Condensed matter and materials physics (2013) 87 (035423))
Nonlinear elastic behavior of two-dimensional molybdenum disulfide
Realistic atomistic structure of amorphous silicon from machine-learning-driven molecular dynamics
Preprint at http://arxiv.org/abs/1803.02802.
Machine learning based interatomic potential for amorphous carbon
Data-driven learning of total and local energies in elemental boron
The allotropes of boron continue to challenge structural elucidation and solid-state theory. Here we use machine learning combined with random structure searching (RSS) algorithms to systematically construct an interatomic potential for boron. Starting from ensembles of randomized atomic configurations, we use alternating single-point quantum-mechanical energy and force computations, Gaussian approximation potential (GAP) fitting, and GAP-driven RSS to iteratively generate a representation of the element's potential-energy surface. Beyond the total energies of the very different boron allotropes, our model readily provides atom-resolved, local energies and thus deepened insight into the frustrated beta-rhombohedral boron structure. Our results open the door for the efficient and automated generation of GAPs, and other machine-learning-based interatomic potentials, and suggest their usefulness as a tool for materials discovery.
Interatomic potentials from first-principles calculations: the force-matching method
Modeling the phase-change memory material ($Ge_{2}$Sb_{2}$Te_{5}$) with a machine-learned interatomic potential
Gaussian approximation potential modeling of lithium intercalation in carbon nanostructures
GULP: A computer program for the symmetry-adapted simulation of solids
Machine learning methods for interatomic potentials: Application to boron carbide
Preprint at http://arxiv.org/abs/1512.09110.
Formation of sp3 bonding in nanoindented carbon nanotubes and graphite
Nanoindentation-induced interlayer bond switching and phase transformation in carbon nanotubes (CNTs) and graphite are simulated by molecular dynamics. Both graphite and CNTs experience a soft-to-hard phase transformation at room temperature at compressive stresses of 12 and 16 GPa, respectively. Further penetration leads to the formation of interlayer sp(3) bonds, which are reversible upon unloading if the compressive stress is under about 70 GPa, beyond which permanent interlayer sp(3) bonds form. During nanoindentation, the maximum nanohardness of graphite can reach 109 GPa, and CNTs 120 GPa, which is comparable to that of diamond.
Interpolating moving least-squares methods for fitting potential energy surfaces: Improving efficiency via local approximants
Dscribe: Library of descriptors for machine learning in materials science
Preprint at http://arxiv.org/abs/-1904.08875.
Automatic selection of atomic fingerprints and reference configurations for machine-learning potentials
Machine learning of atomic-scale properties is revolutionizing molecular modeling, making it possible to evaluate inter-atomic potentials with first-principles accuracy, at a fraction of the costs. The accuracy, speed, and reliability of machine learning potentials, however, depend strongly on the way atomic configurations are represented, i.e., the choice of descriptors used as input for the machine learning method. The raw Cartesian coordinates are typically transformed in
Molecular potential energy surfaces by interpolation
Effective force fields for condensed phase systems from ab initio molecular dynamics simulation: A new method for force-matching
On-the-fly machine learning of atomic potential in density functional theory structure optimization
Machine learning (ML) is used to derive local stability information for density functional theory calculations of systems in relation to the recently discovered SnO_{2}(110)-(4x1) reconstruction. The ML model is trained on (structure, total energy) relations collected during global minimum energy search runs with an evolutionary algorithm (EA). While being built, the ML model is used to guide the EA, thereby speeding up the overall rate by which the EA succeeds. Inspection of the local atomic potentials emerging from the model further shows chemically intuitive patterns.
Thermal conductance of graphene and dimerite
Manipulation of heat current by the interface between graphene and white graphene
Theoretical study of thermal conductivity in single-walled boron nitride nanotubes
Elastic bending modulus of single-layer molybdenum disulphide ($MoS_{2}$): Finite thickness effect
We derive, from an empirical interaction potential, an analytic formula for the elastic bending modulus of single-layer MoS2 (SLMoS2). By using this approach, we do not need to define or estimate a thickness value for SLMoS2, which is important due to the substantial controversy in defining this value for two-dimensional or ultrathin nanostructures such as graphene and nanotubes. The obtained elastic bending modulus of 9.61 eV in SLMoS2 is significantly higher than the bending modulus of 1.4 eV in graphene, and is found to be within the range of values that are obtained using thin shell theory with experimentally obtained values for the elastic constants of SLMoS2. This increase in bending modulus as compared to monolayer graphene is attributed, through our analytic expression, to the finite thickness of SLMoS2. Specifically, while each monolayer of S atoms contributes 1.75 eV to the bending modulus, which is similar to the 1.4 eV bending modulus of monolayer graphene, the additional pairwise and angular interactions between out of plane Mo and S atoms contribute 5.84 eV to the bending modulus of SLMoS2.
Molecular dynamics simulations of single-layer molybdenum disulphide ($MoS_{2}$): stillinger-weber parametrization, mechanical properties, and thermal conductivity
A Gaussian treatment for the friction issue of Lennard-Jones potential in layered materials: Application to friction between graphene, MoS2, and black phosphorus
A stillinger-weber potential for single-layer black phosphorus, and the importance of cross-pucker interactions for negative Poisson's ratio and edge stress-induced bending
The distinguishing structural feature of single-layered black phosphorus is its puckered structure, which leads to many novel physical properties. In this work, we first present a new parameterization of the Stillinger-Weber potential for single-layered black phosphorus. In doing so, we reveal the importance of a cross-pucker interaction term in capturing its unique mechanical properties, such as a negative Poisson's ratio. In particular, we show that the cross-pucker interaction enables the pucker to act as a re-entrant hinge, which expands in the lateral direction when it is stretched in the longitudinal direction. As a consequence, single-layered black phosphorus has a negative Poisson's ratio in the direction perpendicular to the atomic plane. As an additional demonstration of the impact of the cross-pucker interaction, we show that it is also the key factor that enables capturing the edge stress-induced bending of single-layered black phosphorus that has been reported in ab initio calculations.
Parametrization of stillinger-weber potential based on valence force field model: Application to single-layer $MoS_{2}$ and black phosphorus
We propose parametrizing the Stillinger-Weber potential for covalent materials starting from the valence force-field model. All geometrical parameters in the Stillinger-Weber potential are determined analytically according to the equilibrium condition for each individual potential term, while the energy parameters are derived from the valence force-field model. This parametrization approach transfers the accuracy of the valence force field model to the Stillinger-Weber potential. Furthermore, the resulting Stilliinger-Weber potential supports stable molecular dynamics simulations, as each potential term is at an energy-minimum state separately at the equilibrium configuration. We employ this procedure to parametrize Stillinger-Weber potentials for single-layer MoS2 and black phosphorous. The obtained Stillinger-Weber potentials predict an accurate phonon spectrum and mechanical behaviors. We also provide input scripts of these Stillinger-Weber potentials used by publicly available simulation packages including GULP and LAMMPS.
Parameterization of stillinger-weber potential for two-dimensional atomic crystals
Preprint at http://arxiv.org/abs/1704.03147.
Misfit strain induced buckling in MX2 (with $M=Mo$, $W$ and $X=S$, $Se$, $Te)$ lateral heterostructures: A molecular dynamics study
On the determination of molecular fields. II. From the equation of state of a gas
Thermal transport properties of $MoS_{2}$ and $MoSe_{2}$ monolayers
The isolation of single- to few-layer transition metal dichalcogenides opens new directions in the application of two-dimensional materials to nanoelectronics. The characterization of thermal transport in these new low-dimensional materials is needed for their efficient implementation, either for general overheating issues or specific applications in thermoelectric devices. In this study, the lattice thermal conductivities of single-layer MoS2 and MoSe2 are evaluated using classical molecular dynamics methods. The interactions between atoms are defined by Stillinger-Weber-type empirical potentials that are developed to represent the structural, mechanical, and vibrational properties of the given materials. In the parameterization of the potentials, a stochastic optimization algorithm, namely particle swarm optimization, is utilized. The final parameter sets produce quite consistent results with density functional theory in terms of lattice parameters, bond distances, elastic constants, and vibrational properties of both single-layer MoS2 and MoSe2. The predicted thermal properties of both materials are in very good agreement with earlier first-principles calculations. The discrepancies between the calculations and experimental measurements are most probably caused by the pristine nature of the structures in our simulations.
Effect of invariance requirements on the elastic strain energy of crystals with application to the diamond structure
Smoothest bearings: Interlayer sliding in multiwalled carbon nanotubes
The well-defined geometry and extreme structural anisotropy of a multiwalled carbon nanotube can bring qualitatively new features to its nanometer-scale tribology. Efficient cancellation of registration-dependent interactions in incommensurate tubes (and also, surprisingly, certain axial commensurate tubes) can induce extremely small and nonextensive shear strengths.
Registry-dependent interlayer potential for graphitic systems
First-principles analysis of lattice thermal conductivity in monolayer and bilayer graphene
$C_2$F, BN, and C nanoshell elasticity from ab initio computations
Development of glue-type potentials for the Al-Pb system: Phase diagram calculation
Interlayer interaction and relative vibrations of bilayer graphene
The van der Waals corrected first-principles approach (DFT-D) is for the first time applied for investigation of interlayer interaction and relative motion of graphene layers. A methodological study of the influence of parameters of calculations with the dispersion corrected and original PBE functionals on characteristics of the potential relief of the interlayer interaction energy is performed. Based on the DFT-D calculations, a new classical potential for interaction between graphene layers is developed. Molecular dynamics simulations of relative translational vibrations of graphene layers demonstrate that the choice of the classical potential considerably affects dynamic characteristics of graphene-based systems. The calculated low values of the Q-factor for these vibrations Q approximately 10-100 show that graphene should be perfect for the use in fast-responding nanorelays and nanoelectromechanical memory cells.
Measurement of the elastic properties and intrinsic strength of monolayer graphene
We measured the elastic properties and intrinsic breaking strength of free-standing monolayer graphene membranes by nanoindentation in an atomic force microscope. The force-displacement behavior is interpreted within a framework of nonlinear elastic stress-strain response, and yields second- and third-order elastic stiffnesses of 340 newtons per meter (N m(-1)) and -690 Nm(-1), respectively. The breaking strength is 42 N m(-1) and represents the intrinsic strength of a defect-free sheet. These quantities correspond to a Young's modulus of E = 1.0 terapascals, third-order elastic stiffness of D = -2.0 terapascals, and intrinsic strength of sigma(int) = 130 gigapascals for bulk graphite. These experiments establish graphene as the strongest material ever measured, and show that atomically perfect nanoscale materials can be mechanically tested to deformations well beyond the linear regime.
The effect of $VMoS_3$ point defect on the elastic properties of monolayer $MoS_2$ with REBO potentials
Structural defects in monolayer molybdenum disulfide (MoS2) have significant influence on the electric, optical, thermal, chemical, and mechanical properties of the material. Among all the types of structural defects of the chemical vapor phase-grown monolayer MoS2, the VMoS3 point defect (a vacancy complex of Mo and three nearby S atoms) is another type of defect preferentially generated by the extended electron irradiation. Here, using the classical molecular dynamics simulation with reactive empirical bond-order (REBO) potential, we first investigate the effect of VMoS3 point defects on the elastic properties of monolayer MoS2 sheets. Under the constrained uniaxial tensile test, the elastic properties of monolayer MoS2 sheets containing VMoS3 vacancies with defect fraction varying from 0.01 to 0.1 are obtained based on the plane anisotropic constitutive relations of the material. It is found that the increase of VMoS3 vacancy concentration leads to the noticeable decrease in the elastic modulus but has a slight effect on Poisson's ratio. The maximum decrease of the elastic modulus is up to 25 %. Increasing the ambient temperature from 10 K to 500 K has trivial influences on the elastic modulus and Poisson's ratio for the monolayer MoS2 without defect and with 5 % VMoS3 vacancies. However, an anomalous parabolic relationship between the elastic modulus and the temperature is found in the monolayer MoS2 containing 0.1 % VMoS3 vacancy, bringing a crucial and fundamental issue to the application of monolayer MoS2 with defects.
Epitaxial growth of a monolayer $WSe_{2}$-$MoS_{2}$ lateral $p$-$n$ junction with an atomically sharp interface
Two-dimensional transition metal dichalcogenides (TMDCs) such as molybdenum sulfide MoS2 and tungsten sulfide WSe2 have potential applications in electronics because they exhibit high on-off current ratios and distinctive electro-optical properties. Spatially connected TMDC lateral heterojunctions are key components for constructing monolayer p-n rectifying diodes, light-emitting diodes, photovoltaic devices, and bipolar junction transistors. However, such structures are not readily prepared via the layer-stacking techniques, and direct growth favors the thermodynamically preferred TMDC alloys. We report the two-step epitaxial growth of lateral WSe2-MoS2 heterojunction, where the edge of WSe2 induces the epitaxial MoS2 growth despite a large lattice mismatch. The epitaxial growth process offers a controllable method to obtain lateral heterojunction with an atomically sharp interface.
Embedded-atom-method tantalum potential developed by the force-matching method
Molecular dynamics with on-the-fly machine learning of quantum-mechanical forces
We present a molecular dynamics scheme which combines first-principles and machine-learning (ML) techniques in a single information-efficient approach. Forces on atoms are either predicted by Bayesian inference or, if necessary, computed by on-the-fly quantum-mechanical (QM) calculations and added to a growing ML database, whose completeness is, thus, never required. As a result, the scheme is accurate and general, while progressively fewer QM calls are needed when a new chemical process is encountered for the second and subsequent times, as demonstrated by tests on crystalline and molten silicon.
Parametrization of a reactive many-body potential for Mo-S systems
Molecular mechanics. The MM3 force field for hydrocarbons. 2. Vibrational frequencies and thermodynamics
Molecular mechanics. The MM3 force field for hydrocarbons. 3. The van der Waals' potentials and crystal data for aliphatic and aromatic hydrocarbons
Optimized Tersoff and Brenner empirical potential parameters for lattice dynamics and phonon thermal transport in carbon nanotubes and graphene
Ab initio calculation of ideal strength and phonon instability of graphene under tension
EAM potential for magnesium from quantum mechanical forces
Van der Waals integration before and beyond two-dimensional materials
Material integration strategies, such as epitaxial growth, usually involve strong chemical bonds and are typically limited to materials with strict structure matching and processing compatibility. Van der Waals integration, in which pre-fabricated building blocks are physically assembled together through weak van der Waals interactions, offers an alternative bond-free integration strategy without lattice and processing limitations, as exemplified by two-dimensional van der Waals heterostructures. Here we review the development, challenges and opportunities of this emerging approach, generalizing it for flexible integration of diverse material systems beyond two dimensions, and discuss its potential for creating artificial heterostructures or superlattices beyond the reach of existing materials.
Elastic properties of carbon nanotubes and nanoropes
Intramolecular polarisable multipolar electrostatics from the machine learning method Kriging
We describe an intramolecularly polarisable multipolar electrostatic potential model for ethanol, which acts as a pilot molecule for this proof-of-concept study. We define atoms via the partitioning prescribed by quantum chemical topology (QCT). A machine learning method called Kriging is employed to capture the way atomic multipole moments vary upon conformational change. The multipole moments predicted by the Kriging models are used in the calculation of atom-atom electrostatic interaction energies. Charge transfer is treated in the same way as dipolar polarisation and the polarisation of higher rank multipole moments. This method enables the development of a new and more accurate force field. (C) 2011 Elsevier B.V.
Nitrogen doping and vacancy effects on the mechanical properties of graphene: A molecular dynamics study
In this Letter, we used classical Molecular Dynamics (MD) simulations to investigate the tensile behavior of graphene. The validity of the proposed MD architecture is verified by comparing the simulation results with the available experimental results. By performing uniaxial tension simulations, we studied the effects of strain rate, chirality, nanoribbons width and number of atomic planes on the mechanical properties of graphene. We particularly investigated the effects of doped nitrogen atoms and point vacancies concentrations on the Young's modulus and tensile strength of graphene. By plotting the deformation process of graphene at various strain levels, the failure behavior is discussed. (C) 2012 Elsevier B.V.
Describing the diverse geometries of gold from nanoclusters to bulk—a first-principles based hybrid bond order potential
Thermal conductivity of single-layer $WSe_2$ by a Stillinger-Weber potential
In this study, we present the parameters of the Stillinger-Weber (SW) potential for single-layer WSe2 and calculate its in-plane thermal conductivity using a reverse non-equilibrium molecular dynamics simulation. The parameters are fitted to the phonon dispersion curves from literature density functional perturbation theory and experimental structural properties. The set reproduces the phonon dispersion well. The in-plane thermal conductivity of single-layer WSe2 and its dependency on sample length and temperature are calculated and the results are in good agreement with experimental values. Our developed SW-type potential facilitates further investigations on thermal, mechanical and other properties of WSe2.
2D materials and van der Waals heterostructures
ReaxFF reactive force-field study of molybdenum disulfide $MoS_{2}$
Two-dimensional layers of molybdenum disulfide, MoS2, have been recognized as promising materials for nanoelectronics due to their exceptional electronic and optical properties. Here we develop a new ReaxFF reactive potential that can accurately describe the thermodynamic and structural properties of MoS2 sheets, guided by extensive density functional theory simulations. This potential is then applied to the formation energies of five different types of vacancies, various vacancy migration barriers, and the transition barrier between the semiconducting 2H and metallic 1T phases. The energetics of ripplocations, a recently observed defect in van der Waals layers, is examined, and the interplay between these defects and sulfur vacancies is studied. As strain engineering of MoS2 sheets is an effective way to manipulate the sheets' electronic and optical properties, the new ReaxFF description can provide valuable insights into morphological changes that occur under various loading conditions and defect distributions, thus allowing one to tailor the electronic properties of these 2D crystals.
Determination of best-fit potential parameters for a reactive force field using a genetic algorithm
The ReaxFF interatomic potential, used for organic materials, involves more than 600 adjustable parameters, the best-fit values of which must be determined for different materials. A new method of determining the set of best-fit parameters for specific molecules containing carbon, hydrogen, nitrogen and oxygen is presented, based on a parameter reduction technique followed by genetic algorithm (GA) minimization. This work has two novel features. The first is the use of a parameter reduction technique to determine which subset of parameters plays a significant role for the species of interest; this is necessary to reduce the optimization space to manageable levels. The second is the application of the GA technique to a complex potential (ReaxFF) with a very large number of adjustable parameters, which implies a large parameter space for optimization. In this work, GA has been used to optimize the parameter set to determine best-fit parameters that can reproduce molecular properties to within a given accuracy. As a test problem, the use of the algorithm has been demonstrated for nitromethane and its decomposition products.
A stochastic algorithm for the isobaric-isothermal ensemble with Ewald summations for all long rang forces
We present an algorithm termed COMPEL (COnstant Molecular Pressure with Ewald sum for Long range forces) to conduct simulations in the NPT ensemble. The algorithm combines novel features recently proposed in the literature to obtain a highly efficient and accurate numerical integrator. COMPEL exploits the concepts of molecular pressure, rapid stochastic relaxation to equilibrium, exact calculation of the contribution to the pressure of long-range nonbonded forces with Ewald summation, and the use of Trotter expansion to generate a robust, highly stable, symmetric, and accurate algorithm. Explicit implementation in the MOIL program and illustrative numerical examples are discussed.
Elastic properties of single-walled carbon nanotubes
UFF, a full periodic table force field for molecular mechanics and molecular dynamics simulations
Application of a universal force field to metal complexes
Equilibrium configuration and continuum elastic properties of finite sized graphene
Development of a machine learning potential for graphene
One-pot growth of two-dimensional lateral heterostructures via sequential edge-epitaxy
Two-dimensional heterojunctions of transition-metal dichalcogenides have great potential for application in low-power, high-performance and flexible electro-optical devices, such as tunnelling transistors, light-emitting diodes, photodetectors and photovoltaic cells. Although complex heterostructures have been fabricated via the van der Waals stacking of different two-dimensional materials, the in situ fabrication of high-quality lateral heterostructures with multiple junctions remains a challenge. Transition-metal-dichalcogenide lateral heterostructures have been synthesized via single-step, two-step or multi-step growth processes. However, these methods lack the flexibility to control, in situ, the growth of individual domains. In situ synthesis of multi-junction lateral heterostructures does not require multiple exchanges of sources or reactors, a limitation in previous approaches as it exposes the edges to ambient contamination, compromises the homogeneity of domain size in periodic structures, and results in long processing times. Here we report a one-pot synthetic approach, using a single heterogeneous solid source, for the continuous fabrication of lateral multi-junction heterostructures consisting of monolayers of transition-metal dichalcogenides. The sequential formation of heterojunctions is achieved solely by changing the composition of the reactive gas environment in the presence of water vapour. This enables selective control of the water-induced oxidation and volatilization of each transition-metal precursor, as well as its nucleation on the substrate, leading to sequential edge-epitaxy of distinct transition-metal dichalcogenides. Photoluminescence maps confirm the sequential spatial modulation of the bandgap, and atomic-resolution images reveal defect-free lateral connectivity between the different transition-metal-dichalcogenide domains within a single crystal structure. Electrical transport measurements revealed diode-like responses across the junctions. Our new approach offers greater flexibility and control than previous methods for continuous growth of transition-metal-dichalcogenide-based multi-junction lateral heterostructures. These findings could be extended to other families of two-dimensional materials, and establish a foundation for the development of complex and atomically thin in-plane superlattices, devices and integrated circuits.
Pressure-induced transformation path of graphite to diamond
How to represent crystal structures for machine learning: Towards fast prediction of electronic properties
Sparse representation for a potential energy surface
First-principles interatomic potentials for ten elemental metals via compressed sensing
Computer simulation of local order in condensed phases of silicon
COMPASS: An ab initio force-field optimized for condensed-phase applicationssoverview with details on alkane and benzene compounds
An ab initio parametrized interatomic force field for silica
New empirical model for the structural properties of silicon
Empirical interatomic potential for carbon, with applications to amorphous carbon
Empirical interatomic potential for silicon with improved elastic properties
New empirical approach for the structure and energy of covalent systems
Modeling solid-state chemistry: Interatomic potentials for multicomponent systems
Spectral neighbor analysis method for automated generation of quantum-accurate interatomic potentials
Dihedral-angle-corrected registry-dependent interlayer potential for multilayer graphene structures
Coherent, atomically thin transition-metal dichalcogenide superlattices with engineered strain
Epitaxy forms the basis of modern electronics and optoelectronics. We report coherent atomically thin superlattices in which different transition metal dichalcogenide monolayers-despite large lattice mismatches-are repeated and laterally integrated without dislocations within the monolayer plane. Grown by an omnidirectional epitaxy, these superlattices display fully matched lattice constants across heterointerfaces while maintaining an isotropic lattice structure and triangular symmetry. This strong epitaxial strain is precisely engineered via the nanoscale supercell dimensions, thereby enabling broad tuning of the optical properties and producing photoluminescence peak shifts as large as 250 millielectron volts. We present theoretical models to explain this coherent growth and the energetic interplay governing the ripple formation in these strained monolayers. Such coherent superlattices provide building blocks with targeted functionalities at the atomically thin limit.
Atomic energies from a convolutional neural network
Understanding interactions and structural properties at the atomic level is often a prerequisite to the design of novel materials. Theoretical studies based on quantum-mechanical first-principles calculations can provide this knowledge but at an immense computational cost. In recent years, machine learning has been successful in predicting structural properties at a much lower cost. Here we propose a simplified structure descriptor with no empirical parameters,
Direction dependent thermal conductivity of monolayer phosphorene: Parameterization of Stillinger-Weber potential and molecular dynamics study
Proposed definition of crystal substructure and substructural similarity
Robust epitaxial growth of two-dimensional heterostructures, multiheterostructures, and superlattices
We report a general synthetic strategy for highly robust growth of diverse lateral heterostructures, multiheterostructures, and superlattices from two-dimensional (2D) atomic crystals. A reverse flow during the temperature-swing stage in the sequential vapor deposition growth process allowed us to cool the existing 2D crystals to prevent undesired thermal degradation and uncontrolled homogeneous nucleation, thus enabling highly robust block-by-block epitaxial growth. Raman and photoluminescence mapping studies showed that a wide range of 2D heterostructures (such as WS2-WSe2 and WS2-MoSe2), multiheterostructures (such as WS2-WSe2-MoS2 and WS2-MoSe2-WSe2), and superlattices (such as WS2-WSe2-WS2-WSe2-WS2) were readily prepared with precisely controlled spatial modulation. Transmission electron microscope studies showed clear chemical modulation with atomically sharp interfaces. Electrical transport studies of WSe2-WS2 lateral junctions showed well-defined diode characteristics with a rectification ratio up to 10(5).
Size and chirality dependent elastic properties of graphene nanoribbons under uniaxial tension
We investigate the mechanical strength and properties of graphene under uniaxial tensile test as a function of size and chirality using the orthogonal tight-binding method and molecular dynamics simulations with the AIREBO potential. Our results on Young's modulus, fracture strain, and fracture strength of bulk graphene are in reasonable agreement with the recently published experimental data. Our results indicate that fracture strain and fracture strength of bulk graphene under uniaxial tension can have a significant dependence on the chirality. Mechanical properties such as Young's modulus and Poisson's ratio can depend strongly on the size and chirality of the graphene nanoribbon.
