Artificial Neural Network Model for Atomistic Simulations of \({\rm {Sb/MoS}_{2}}\) van der Waals Heterostructures

  • Yu-Xuan Wang
  • Hsin-An Chen
  • Chun-Wei PaoEmail author
  • Chien-Cheng ChangEmail author
Original Research


van der Waals (vdW) heterostructures have drawn significant amount of attentions because of their potential applications in the future electronic devices as well as quantum computing. Modeling the structural properties has been a challenging task due to the combinatory effects of both chemical complexity and spatial limitations of the first principle calculations. In this work, we trained an artificial neural network (ANN) model for atomistic simulations of \({\rm {Sb/MoS}}_{2}\) vdW heterostructures. The ANN model was trained from thousands of atomistic configurations along with energies computed from density functional theory (DFT) calculations. We demonstrated that the ANN model can successfully predict system energy with high fidelity with respect to DFT calculations with much less consumption of computational resources, manifesting that the ANN model is a powerful tool in atomistic simulations of chemically complex systems.


van der Waals heterostructure Atomistic simulation Neural network Machine learning 


Since the first successful isolation of graphene in 2004, two-dimensional materials (2D materials) have drawn significant amount of attentions from both academia and industries [1, 2, 3, 4, 5]. Recently, the van der Waals (vdW) heterostructures—a new breed of material derived from 2D materials—have emerged because of the span of combinatory space comprised of all 2D materials to date offer significant amount of freedom, allowing material scientists to manipulate these novel composite materials by switching stackings as well as angles of twist for desired material properties [6, 7, 8]. These vdW heterostructure materials have shown great potentials in electronics, optoelectronics, green energy, and even superconductivity applications [8, 9, 10, 11, 12], making them a promising material for the future.

The unusual physical and electronic properties of vdW heterostructures are originated from the interaction, or, in other words, the lattice mismatch between adjacent nanosheets. Hence, characterization of the structure of vdW heterostructures is critical for comprehensive understanding of material properties, and for the fabrication as well as the stability of heterostructure materials. Various microscopy and optical spectroscopy methods have been utilized for characterization of vdW heterostructures. For example, the high-resolution elemental mapping in electron microscopy can be used to characterize vdW heterostructures consisting of two or more types of nanosheets [13, 14], and the scanning tunneling microscopy (STM) and atomic force microscopy (AFM) can be employed to visualize the periodic Moire patterns, which serves as an indicator as formation of vdW heterostructures [15, 16, 17, 18]. The Moire patterns images from STM or AFM experiments must be complemented with extensive modelings to reveal the atomistic details of vdW heterostructures. Hence, atomistic scale simulations can play a critical role in both verification of the stability and analysis of physical/chemical properties of the heterostructures [19, 20, 21, 22].

The atomistic scale calculations of properties of vdW heterostructures are not a trivial task. Ab initio calculations can in principle evaluate the structural stabilities and physical/chemical properties of vdW heterostructures [19, 21]; however, larger simulation supercells relative to 2D materials comprised of single nanosheet component (eg. graphene) must be constructed for calculations to accomodate the lattice mismatch between nanosheets, which critically limits the application of ab initio calculations in vdW heterostructures, in particular, while the angles of twist must be taken into account. In contrast, classical molecular simulations can handle systems with large spatial spans, which seems to be ideal for exploring the structural properties of vdW heterostructures [22, 23, 24]. However, classical molecular simulations suffer from the limitations in available classical interatomic force fields (or, interatomic potentials). Due to the complex combination of chemical species as well as intermolecular interactions, parametrization of interatomic force fields is challenging, thereby limiting applications of classical molecular dynamics (MD) in exploring the large-scale structural properties of vdW heterostructures.

In the present manuscript, we demonstrate that we can successfully construct an interatomic potential for \({\rm {Sb/MoS}}_{2}\) vdW heterostructures by harnessing the power of machine learning (ML). The universal approximation theorem states that a feed-forward neural network can approximate any continuous function [25, 26], allowing representation of the potential energy surface (PES) of molecular systems. Hence, in the present study, an artificial neural (ANN) model was trained by feeding tens of thousands of configurations from DFT calculations as training sets. For identical system configurations, the ANN model we trained can successfully reproduce respective DFT energies. We carried out MD simulations of \({\rm{Sb/MoS}}_{2}\) heterostructures with system size of the order of 1000 atoms—a system size that is difficult for current ab initio molecular dynamics (AIMD) simulations, manifesting the potential of ANN model as an efficient energy/force evaluator for atomistic simulations of chemically complex systems.

Simulation Methodology

Artificial Neural Network for Atomistic Energies

We employed the energy partitioning scheme proposed by Behler and Parrinello to construct the ANN model evaluating system energies [27]. The scheme of the ANN model employed in the present study is depicted in Fig. 1. The target of the ANN model is to replicate the system energy from ab initio calculations for given atomic coordinates. As depicted in the upper panel of Fig. 1, the atomic coordinate of each atoms \(A_i(x_i)\) is firstly transformed into a set of descriptor (or, fingerprint) functions, serving as the input feature vector for the atomic neural network. The output of atomic neural network is the atomic energy \(E_i\) of each atom, and the summation of atomic energies \(E_i\) is the overall system energy E. The architecture of each atomic neural network is displayed schematically in the lower panel of Fig. 1. The atomic neural network is comprised of the input layer, the hidden layers, and the output layer. In our ANN model for atomistic simulations, the input layer is comprised of a series of atomic descriptor functions transformed from coordinates of individual atoms as well as their chemical environments (e.g. bond length or angles with neighboring atoms). The output layer will output atomic energy of each atoms. Each layers in the neural network is comprised of a finite number of nodes (neurons), and each nodes are connected (see the arrows in the lower panel of Fig. 1) via a set of weighting parameters w. The advantage of this energy partitioning scheme is that the energies of systems of arbitrary sizes can be predicted once the atomic neural network function is trained. The expression of the output value of the \(j_{th}\) node in the \(i_{th}\) layer \(o_{i,j}\) can be written as
$$\begin{aligned} o_{i,j}(\{o_{i-1,k}\})=f_a\left( \sum _k w_{kj}^i o_{i-1,k} \right) , \end{aligned}$$
where \(f_a\) is the activation function and \(w_{k,j}^i\) is the weighting parameter connecting \(k_{th}\) node in the \((i-1)_{th}\) layer and the \(j_{th}\) node in the \(i_{th}\) layer. Hence, the vector of all nodes in the \(i_{th}\) layer can equivalently be expressed as
$$\begin{aligned} {\mathbf {o_i}}({\mathbf {o_{i-1}}})=f_a\left( {\mathbf {W_io_{i-1}}} \right) , \end{aligned}$$
where \({\mathbf {W_i}}\) is the weighting matrix connecting the layer i and layer \(i-1\). For an atomic neural network (the lower panel of Fig. 1) with M hidden layers, the atomic energy of atom i can be written as
$$\begin{aligned} E_i = \mathcal {N}({\mathbf {I_i}},\{\mathbf {W}\}) = f_a\left( {\mathbf {W_M}}f_a\left( {\mathbf {W_{M-1}}}f_a(\cdots f_a({\mathbf {W_1I}})\cdots )\right) \right) , \end{aligned}$$
where \(\mathcal {N}({\mathbf {I_i}},\{{\mathbf {W}}\})\) is called the neural network function and \(\mathbf {I_i}\) is the feature vector of atom i. Note that each chemical species in the system (e.g. Sb, Mo, and S in the present study) should have their own atomic neural network function \(\mathcal {N}\) (namely, \(\mathcal {N}_{Sb}\), \(\mathcal {N}_{Mo}\), and \(\mathcal {N}_S\) in the present study). Furthermore, in the present study, we only used the hyperbolic tangent function tanh(o) as the activation function. The essence of the ANN model is the weighting matrix sets \(\{\mathbf {W}\}\) connecting nodes in the neural network, and these parameters needs to be trained to become a valid model that can be utilized for subsequent atomistic simulations. Once the ANN model is trained, we can perform atomistic simulations such as classical MD simulations as well as Monte Carlo simulations to exhaustively sample the configuration space for structural/thermodynamic properties or entropy-related properties such as system free energies. The atomic forces, which are critical for atomistic MD simulations, can be directly obtained by computing the gradient of system energy E, which can be analytically derived by differentiating the atomic neural network function \(\mathcal {N}(\mathbf {I_i},\{\mathbf {W}\})\) with respect to atomic coordinates.
Fig. 1

Schematics of the artificial neural network-based energy evaluator employed in the present study

Descriptor Functions

The feature vector \(\mathbf {I_i}\) of atom i is comprised of a series of descriptor functions transforming atomic coordinate (usually cartesian) of atom i into translationally/rotationally invariant fingerprints specifying its chemical environment. In this work, the gaussian descriptor functions suggested by Behler [28] were employed as the descriptor functions as the input layer of atomic neural network functions \(\mathcal {N}\). In the present study, the gaussian descriptor functions were divided into two categories, namely, the radial (two-body) descriptor \(G_{i}^{II}\), and the angular (three-body) descriptor \(G_{i}^{III}\). The radial descriptor can be expressed as
$$\begin{aligned} G_{i}^{II} = \sum _{j\ne i}e^{-\eta \frac{(R_{ij}-R_s)^2}{R_c^2}}f_c(R_{ij}), \end{aligned}$$
where \(R_c\) is the cutoff distance for descriptor functions, and \(\eta\), \(R_s\) are predefined parameters for the descriptors, respectively. In the present study, \(R_s\) of all atomic pairs were set to zero. The angular descriptor is expressed as
$$\begin{aligned} G_{i}^{III} = 2^{1-\zeta }\sum _{j,k\ne i}(1+\lambda cos\theta _{ijk})^{\zeta } e^{-\eta \frac{R_{ij}^2+R_{ik}^2+R_{jk}^2}{R_c^2}}f_c(R_{ij})f_c(R_{ik})f_c(R_{jk}), \end{aligned}$$
where \(\zeta\), \(\eta\), and \(\lambda\) are predefined parameters for the angular descriptors. The cutoff function \(f_c(r)\) is expressed as
$$\begin{aligned} f_c(r)={\left\{ \begin{array}{ll} 0.5\left[ 1+cos\left( \frac{\pi r}{R_c} \right) \right] ,r\le R_c\\ 0,r>R_c \\ \end{array}\right. } \end{aligned}$$
The parameter sets defining the descriptor functions are compiled in Tables 2 and 3 in the Appendix.

Training Sets and Training Procedures

The ANN model must be trained to be enabled for evaluating system energies/forces with high fidelity to respective ab initio calculations. During the training processes the weighting parameter set \(\{\mathbf {W_m} \}\) is optimized by minimizing the quadratic error function
$$\begin{aligned} \mathcal {E}(\{{\mathbf {W_m}}\})=\frac{1}{2}\sum _{\sigma }^{structures} \left\{ \left[ \sum _{i}^{atoms} \mathcal {N}_i({\mathbf {I_i}},\{{\mathbf {W_m}}\})\right] -E_{\sigma }^{ref} \right\} ^2, \end{aligned}$$
where \(\sigma\) is a structure within the training set and \(E_{\sigma }^{ref}\) is the respective reference energy of structure \(\sigma\). The weighting parameter set were obtained by minimizing the error function \(\mathcal {E}\). In the present study, we employed the Limited-memory BFGS (L-BFGS) minimizer implemented in the Atomistic Machine-Learning Package (AMP) [29] to train the ANN model of Sb/MoS2 vdW heterostructures.

The training sets selected for the ANN model of Sb/MoS2 heterostructures are depicted in Fig. 2. The training sets included bulk \(\beta\)-antimonene and MoS2 (Fig. 2a, b), and the Sb/MoS2 heterostructures (Fig. 2c) subjected to hydrostatic strains of \(\pm 5\%\), \(\pm 3\%\), and \(\pm 1\%\) (see orange arrows in Fig. 2a, b, c). Each training sets contains 150 atoms in the system, which is feasible for ab initio calculations. For large molecular systems such as biomolecules, it is not possible to perform ab initio calculations of the whole molecule for training set generation. One potential solution is to partition the whole molecules into pieces that can be handled by first principle calculations and train the ANN models accordingly. Note that in the training set of the Sb/MoS2 heterostructures, the \(\beta\)-antimonene was subjected to a misfit strain of \(-1.773\%\) to accomodate with the lattice of underlying MoS2. To expand range of atomic feature vectors to ensure ANN model transferabilities, the Stone-Wales (SW) defect of single-layer \(\beta -\)antimonene (the lower panel of Fig. 2a) and the sulfur vacancy of single-layer MoS2 (the lower panel of Fig. 2b) were also included in the training set. For each configurations displayed in Fig. 2, atomic coordinates and respective electronic energies from a thousand steps of AIMD simulation subjected to canonical ensemble at T = 300 K were collected as the training data for the training processes. It must be noted that the trained ANN models are likely to fail once the atomic descriptor functions are outside the ranges in which the model was trained; hence, the training sets must be properly chosen to include finger print space (namely, the space spanned by finger print functions) that is relevant for systems of interests.

In the present study, the reference energies of structures in the training sets were computed by density funcitonal theory (DFT) calculations using the Vienna ab initio Simulation Package (VASP) [30, 31] with the project augment wave (PAW) pseudopotential [32], as well as the Perdew, Burke, and Ernzerhof (PBE) exchange-correlation functional [33]. The cutoff energy was 450 eV, and a \(1\times \ 1 \times 1\) Monkhorst Pack k-point mesh was employed. The DFT-D2 method was employed to incorporate van der Waals interactions between atoms [34]. The step size for AIMD simulations were set to 0.5 fs.
Fig. 2

Training sets for ANN model of Sb/MoS2 heterostructures. The orange arrows highlight inclusion of tensile/compressive hydrostatic strains in the training sets

Results and Discussions

Results of Training Processes

Figure 3a displays the correlation between system energies computed from VASP (horizontal axis) and from trained ANN model (vertical axis) for all the structures within the training sets. The red diagonal line in Fig. 3a highlights the equation \(y=x\), or, the perfect fit. It is evident that for all the structures within the training set, the trained ANN model shows good correspondence with respective VASP calculations. Figure 3b displays the absolute errors \(E_{err}=|E_{ANN}-E_{VASP}|\) with respect to VASP energies for all the structures in the training set. We can find that the root mean squared error of the trained ANN model is 0.001 eV/atom, which is in very good agreement with the system energies calculated from VASP. Hence, the ANN model we trained can successfully evaluate the system energies with high accuracy to respective DFT calculations using VASP for all the training set structures. The next step is to feed structures that are NOT in the training set into the trained ANN model to validate our ANN model.
Fig. 3

Training results for the ANN model. a Parity plot between energies from VASP (horizontal axis) and from ANN model trained (vertical axis); b errors in energy per atom with respect to energies from VASP

For validation of the trained ANN model, we fed strained structures of \(\beta\)-antimonene (Fig. 4), MoS2 (Fig. 5), and Antimony/MoS2 heterostructure (Fig. 6) into the ANN model, computed the energies and compared energies from the ANN model with those from respective VASP calculations. Note that each structures in the validation sets were subjected to hydrostatic strains that were not within the training sets, see the applied hydrostatic strains annotated in the upper panels of Figs. 4,  5 and 6a. The structures for validation were obtained from AIMD simulations using VASP subjected to canonical ensemble at T=300K. From Figs. 4,  5 and 6 we can find that the trained ANN model can evaluate the energies with good agreements with those from VASP calculations with a maximal root mean squared error of 0.0014 eV/atom (bulk \(\beta -\)antimonene, \(-2\%\) strain, see Fig. 4b), demonstrating that for given structures of \({\rm {Sb/MoS}}_{2}\) system, this ANN model can evaluate system energies with high fidelity to those from respective ab initio calculations. It must be noted that the simulation system sizes of validation sets were less than 150 atoms, and therefore, allowing direct comparison of energies from trained ANN models and AIMD simulations. Since it has been demonstrated that the trained ANN model can predict system energies with high accuracies to AIMD simulations, in the following subsection, we will demonstrate that the trained ANN model can be employed to carry out MD simulations with system sizes that are too large for AIMD simulations.
Fig. 4

Validation of trained ANN model. Upper panels: comparison of ANN energies with DFT energies from VASP MD trajectories of \(\beta -\)antimonene subjected to hydrostatic strains of (a) \(2\%\) (b) and \(-2\%\). The lower panels display respective ANN energy deviations from VASP calculations

Fig. 5

Validation of trained ANN model. Upper panels: comparison of ANN energies with DFT energies from VASP MD trajectories of \({\rm {MoS}}_{2}\) subjected to hydrostatic strains of (a) \(4\%\) (b) and \(-4\%\). The lower panels display respective ANN energy deviations from VASP calculations

Fig. 6

Validation of trained ANN model. Upper panels: comparison of ANN energies with DFT energies from VASP MD trajectories of \({\rm{Sb/MoS}}_{2}\) heterostructure subjected to hydrostatic strains of (a) \(2\%\) (b) and \(-2\%\). The lower panels display respective ANN energy deviations from VASP calculations

Molecular Dynamics Simulations of \({\rm {Sb/MoS}}_{2}\) Heterostructures

The most important objective of training ANN model in the present study is to perform atomistic simulations with system size larger than those can be achieved by typical ab initio simulations. Hence, to test if the trained ANN model can be utilized for actual atomistic simulations, we carried out classical MD simulations of two different \({\rm {Sb/MoS}}_{2}\) heterostructures under canonical ensemble using the ANN model trained in the present study. We expanded the system size to the order of one thousand atoms, which is extremely expensive simulation for ab initio calculations and therefore infeasible for AIMD simulations as well as structural analysis from AIMD trajectories. The Langevin thermostat was employed to control the system temperature at 300 K with friction constant 0.8 and time step size of 0.5 fs. The structures of \({\rm {Sb/MoS}}_{2}\) heterostructures for MD simulations are displayed in the insets in the upper panels of Fig. 7. The first system contains 900 atoms and is comprised of three layers of \(\beta -\)antimonene and two layers of \({\rm {MoS}}_{2}\) (namely, the tri-layer heterostructure). The second system contains 1056 atoms and was comprised of monolayer \(\beta\)-antimonene and \({\rm {MoS}}_{2}\) (namely, the monolayer heterostructure), such that a lateral dimension of 51.14 Å × 51.14 Å can be achieved. The evolution of system temperatures and total/potential energies of the two heterostructures during classical MD simulations are displayed in the upper panels and lower panels of Fig. 7a, b, respectively. By examining the temperature and total/potential energies evolution from MD simulations (Fig. 7a, b), it is evident that the system is stable against finite-temperature MD simulations. It is noteworthy that these MD simulations were performed using only one computational core. For AIMD simulations using VASP for systems of such size, usually hundreds of cores are required. In Table 1, we compiled the computational resources as well as speed (time required per MD step) of the ANN model and respective AIMD simulations. It is evident that the ANN model trained in the present study is much more efficient in evaluating system energies/forces comparing with ab initio calculations. Since the current code for MD simulations using the ANN model is neither optimized nor parallelized, there are still more spaces to further boost the computational efficiencies for MD simulations based on the ANN model trained in the present study, and currently we are actively working on this subject.
Fig. 7

Temperature (upper panels) and total energy/potential energy evolution of classical MD simulation using interatomic potential from the trained ANN model

Table 1

Computational costs of tri-layer antimonene (900 atoms) and monolayer antimonene (1056 atoms) using the ANN model and ab initio calculations (VASP)

Energy model


No. atoms

No. cores



Tri-layer antimonene



12.5 min/step


Tri-layer antimonene



50.0 min/step


monolayer antimonene



13.3 min/step


monolayer antimonene



55.6 min/step

Figure 8 displays the radial distribution functions (RDFs) of antimony atoms in the upper layer of the tri-layer heterostructures (left panel in the inset of Fig. 8) and in the monolayer heterostructures (the right panel of Fig. 8) after 1000 classical MD steps using the trained ANN model. Note that the RDF of bulk pristine \(\beta\)-antimonene is also displayed for comparison. The locations of the first peak of RDFs in both the upper layer Sb and monolayer Sb are smaller than those in the bulk pristine antimonene, which can be attributed to the compressive misfit strains imposed on the \(\beta\)-antimonene. Furthermore, the peaks from the monolayer Sb (the right panel in the inset of Fig. 8) is noticeably broader than those in the upper layer Sb, which can be attributed to the distortion of the antimonene bonds in the monolayer antimonene. Figure 9 displays the structures of monolayer antimonene/\({\rm {MoS}}_{2}\) heterostructures after 1000 step of MD simulations and subsequent static relaxation. It is evident that some of the hexagons in antimonene underwent substantial distortions, see the polygons outlined in red in Fig. 9. The distortions originated from the misfit strain between antimonene and underlying \({\rm {MoS}}_{2}\). The antimony atoms tend to occupy the hollow sites surrounded by three sulfur atoms, see the shaded regions in Fig. 9. As a result, the hexagons are distorted once antimony atoms belonging to the same hexagons tries to fit themselves into neighboring hollow sites. Hence, we demonstrate that the ANN model can be utilized to study the misfit effects in vdW heterostructures, and can be easily extended to twisted vdW heterostructures in the near future.
Fig. 8

Radial distribution function of \(\beta\)-antimonene layers of bulk pristine \(\beta\)-antimonene, the upper layer of the tri-layer heterostructure (layer enclosed in red rectangle), and the Sb monolayer in the monolayer heterostructure (layer enclosed in blue rectangle) after 1000 step of MD simulations using the trained ANN model

Fig. 9

Distortion of hexagons (outlined in red) in antimonene. Shaded areas highlight Sb atoms occupying the hollow sites surrounded by three sulfur atoms


In conclusion, the present study successfully constructed an ANN-based model for efficient energy/force evaluation of \({\rm {Sb/MoS}}_{2}\) van der Waals heterostructures with high fidelity to ab initio calculations. The ANN model was trained by feeding a large number of structures of \(\beta\)-antimonene, \({\rm {MoS}}_{2}\), and \({\rm {Sb/MoS}}_{2}\) heterostructures subjected to hydrostatic strains up to \(\pm 5\%\) along with their respective energies from DFT calculations into the training set. The ANN model can successfully evaluate system energies of given structures in the validation set—the set of structures NOT within training sets—with good agreements with respective DFT energies. We performed classical MD simulations of \({\rm {Sb/MoS}}_{2}\) heterostructures using the trained ANN model, and demonstrated that the heterostructures are stable against finite temperature MD simulations with much lower computational expense relative to DFT calculations. Hence, the present study demonstrates that the ANN model can be utilized as an efficient energy/force evaluator for atomistic simulations, allowing researchers to explore structures of vdW heterostructures with system size beyond the reach of conventional ab initio calculations.



We thank the Academia Sinica Career Development Award, Grant no. 2317-1050100, and Ministry of Science and Technology, Taiwan, Grant no. MOST 105-2112-M-001-009-MY3 for financial support, and the National Center for High-performance Computing for computational support.


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Copyright information

© Korean Multi-Scale Mechanics (KMSM) 2019

Authors and Affiliations

  1. 1.Institute of Applied MechanicsNational Taiwan UniversityTaipeiTaiwan
  2. 2.Research Center for Applied Sciences, Academia SinicaTaipeiTaiwan

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