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Understanding atomistic details of diffusion processes in amorphous structures is becoming increasingly important due to the recent advances in various information and energy devices. Atomistic simulations based on the density functional theory (DFT) represent a powerful approach; however, the development of a method characterized by both high reliability and computational efficiency remains a challenge. In this study, a simple neural network (NN) interatomic potential is constructed from the results of DFT simulations to investigate the diffusion of a single Cu atom in amorphous Ta_{2}O_{5}. The proposed technique is as accurate as the DFT in predicting hopping paths and the corresponding barrier energies in a given amorphous structure. Although the developed NN-based approach exhibited some limitations since it was constructed specifically for Cu, the obtained results showed that the NN potential was able to satisfactorily describe the Cu diffusion behavior. Thus, the Cu diffusion activation energy calculated at low temperatures (between 500 and 800 K) using kinetic Monte Carlo simulations and the NN potential matched the experimental data reasonably well.

Amorphous oxides (including *a*-Ta_{2}O_{5}, *a*-HfO_{2}, and *a*-TiO_{2}) are widely used in various energy and electronic devices,^{1}^{)} whose operating mechanism is often related to the atomic diffusion inside the amorphous phase. For example, in the resistance switches^{2}^{–}^{7}^{)} the rate of diffusion in the insulating amorphous oxide layer has a significant effect on the device performance characteristics, such as turn-on voltage, endurance, and switching speed.^{6}^{)} Moreover, several amorphous electrolytes (such as LiPON and Li_{3}PO_{4}) also play key roles in the operation of solid-state thin-film batteries because their charge and discharge rates are determined by the lithium diffusion rate.^{8}^{–}^{10}^{)} Another example is the solid electrolyte interphase in Li batteries, which serves as a protective amorphous layer for the anode and highly affects the battery performance.^{11}^{,}^{12}^{)} Thus, the elucidation of atomic diffusion in such amorphous layers is very important for the development of electronic devices with superior properties.

In recent years, computational simulations have become powerful tools for understanding the mechanism of atomic diffusion in various media. The computational approaches that are most frequently used for this purpose are represented by nudged elastic band (NEB)^{10}^{,}^{13}^{–}^{15}^{)} and molecular dynamics (MD).^{16}^{–}^{21}^{)} Both approaches require the force field provided by either first-principles calculations or classical interatomic potentials. Density functional theory (DFT)-based NEB^{22}^{)} method and ab initio MD calculations can provide reliable descriptions of diffusion mechanisms in solids at an atomic level. However, the application of DFT is limited by its high computational demand, which may be critical for studying atomic diffusion in amorphous solids because modeling realistic amorphous structures typically requires the use of relatively large supercells, and the complexity of local atomic environment significantly exceeds those in the crystal. On the other hand, the simulations using classical interatomic potential with physically motivated simple forms (such as Lennard-Jones, Morse, Tersoff,^{23}^{)} EAM,^{24}^{)} and ReaxFF^{25}^{)} ones) allow relatively fast determination of atomic migration trajectories (as compared to DFT) since it does not take into account electronic states. However, its accuracy remains limited because of the simple formula utilized for describing interatomic interactions.

In order to achieve sufficient accuracy combined with high computational speed during simulations of microscopic atomic movements, various approaches have been proposed. For example, several researchers tried to fit the interatomic potentials used in classical MD simulations with the results of ab initio calculations.^{26}^{,}^{27}^{)} Despite the improvement of the interatomic potential accuracy, its degree was significantly restricted by the simplicity of the potential formula. As a result, the obtained accuracy was not sufficient for many applications, including modeling atomic diffusion in amorphous structures.

Recently, a new method, which involves the construction of interatomic potentials via machine learning techniques to determine the relationship between the studied structure and its corresponding energy, has attracted much attention as a promising way of achieving high computational accuracy and speed. Usually, the results of first-principles calculations are used as the “training” data for the learning process. After sufficient training, the method becomes capable of predicting structural energies that were not included in the training data set. Such machine learning-based interatomic potentials were found to be much more accurate than conventional interatomic potentials with simple formulas and thus were utilized for simulating various materials. The techniques used for this purpose included neural networks (NNs),^{28}^{–}^{37}^{)} kernel regression,^{38}^{,}^{39}^{)} Gaussian process regression,^{40}^{)} and linear regression.^{41}^{,}^{42}^{)}

Among these methods, NNs appear to be especially promising, as has been demonstrated by Behler and coworkers.^{30}^{,}^{33}^{,}^{43}^{,}^{44}^{)} They evaluated the performance of the NN potentials constructed for Si,^{30}^{)} C,^{35}^{)} Cu,^{31}^{)} ZnO,^{33}^{)} TiO_{2},^{43}^{)} H_{2}O dimers,^{45}^{)} Cu clusters supported on Zn oxide,^{32}^{)} and Au/Cu nanoparticles with water molecular.^{44}^{,}^{46}^{)} However, the reported potentials were complicated and contained several thousands of various parameters, which required a relatively large amount of training data, in particular for the systems including more than one atomic species.

In this study, we have developed a new NN potential to examine the diffusion behavior of a single Cu atom in amorphous Ta_{2}O_{5}. Although the described system contains three different atomic elements, the utilized NN potential is much simpler than those proposed previously for ternary systems.^{32}^{)} We investigate the Cu diffusion in the amorphous Ta_{2}O_{5} because this is a very important process in the operation of Cu/Ta_{2}O_{5}/Pt atomic switch. Considering the facts that Cu filaments grows/shrinks in Ta_{2}O_{5} during the operations and that the network of Ta–O polyhedrons is formed in amorphous Ta_{2}O_{5}, it is believed that the single Cu migration via interstitial sites must be a primary process during the operation.^{1}^{,}^{47}^{,}^{48}^{)} More complicated processes involving other Cu, Ta, and/or O atoms may play roles at certain stages during the operations, but the consideration of such processes is out of scope of the present study. The outline of this paper is as follows. Section 2 describes the details of constructing the NN potential. The potential energy surface of Cu, diffusion paths, and barrier energies obtained using the NN potential (which are compared with the results of DFT calculations) are presented in Sect. 3. In Sect. 4, the diffusion behavior of Cu is further examined by calculating the diffusion network, diffusion coefficients, and effective diffusion activation energy. The obtained conclusions are provided in Sect. 5.

Owing to the high accuracy of high-dimensional NN potentials, they have been extensively used in atomic modeling.^{34}^{,}^{35}^{,}^{49}^{)} However, their application may be restricted by the high computational demand of training data, which usually require many hours of DFT calculations. For this reason, a simple NN potential is proposed in this work to investigate the diffusion of a single atom in amorphous materials.

The main idea of the simplified NN potential utilized in this study is to express the total energy of the structure containing a relaxed amorphous Ta_{2}O_{5} matrix and a single Cu atom, _{2}O_{5} matrix, *before* the relaxation, while that of *after* the relaxation. In this work, however, we implicitly assumed the positional shifts of atoms in the host Ta_{2}O_{5} matrix in structure optimization were not significant. In this case, the magnitudes of both *before* the relaxation. Fortunately, this assumption appears to be valid for NEB calculations since the atomic displacements in the host matrix are usually smaller than 0.5 Å. Then we can forget about the host matrix structure *after* the relaxation and consider only the initial host structure and the Cu insertion position explicitly. It should be noted that although the matrix relaxation was not actually preformed in calculations using the constructed NN potential, the energy after matrix relaxation was directly predicted, because _{2}O_{5} relaxation *after* the Cu insertion were automatically taken into account in the NN potential.

The use of a simplified NN potential considerably reduces the complexity of the computational procedures utilized in previous works because it contains only one NN, while the other high-dimensional NN potentials consist of multiple NNs corresponding to different species. The number of parameters in a simplified NN potential is equal to several hundred, which is much smaller than that in a typical NN potential. The details of the utilized simplified NN potential will be provided in the following sections.

The symmetry functions proposed by Behler et al. were used to describe the atomic environment of Cu.^{50}^{,}^{51}^{)} They provide a unique description of the atomic positions and contain a constant number of functions independent of the number of atoms in the supercell, which make them suitable input coordinates for the atomic NNs. In the present work, the following two kinds of symmetry functions (radial and angular ones) were used: *i*, and *i* and *j*. ^{52}^{)} The detailed description of the utilized symmetry functions can be found in Refs. 50 and 51.

The NN defines a functional relation between the position of a Cu atom in the amorphous matrix (which is described by the symmetry functions in the nodes of the input layer) and the dissolution energy of the Cu atom _{2}O_{5} relaxation after Cu insertion were automatically incorporated into the training process of NN. Similar to the high-dimensional NN potential, the force component acting on the Cu atom *y*, or *z* can be evaluated according to the chain rule:

Vienna ab initio simulation package (VASP)^{53}^{,}^{54}^{)} was utilized to model the amorphous Ta_{2}O_{5} structure and perform all the required DFT calculations for training and validating the NN potential. The projector augmented wave (PAW) method was adopted to treat atomic core electrons, while the generalized gradient approximation (PW91) was adopted to describe the electron–electron interactions.^{55}^{,}^{56}^{)} In addition, a plane-wave basis set with a cutoff energy of 500 eV was used.

A model for the amorphous Ta_{2}O_{5} matrix^{57}^{)} was constructed using the melt quenching method under the conditions specified in the Ref. 5. The training data utilized for training the NN potential were generated as follows. First, one Cu atom was randomly inserted into the amorphous Ta_{2}O_{5}, and the resulting structure was optimized while keeping the Cu atom position fixed. During optimization, *k*-points were used for all structures, and the force tolerance was equal to 0.05 eV/Å. At last, the Cu dissolution energy _{2}O_{5} matrix from the final total energy.

To construct the simplified NN potential, 2000 training data were obtained via DFT. The resulting data was randomly split into a training set (90%) and an independent testing set (10%). The training set was used to fit the parameters of the NN, which consisted of two hidden layers with 9 nodes each, one input layer with 36 nodes, and one output node. The NN parameters (corresponding to the number of hidden layers and number of nodes in each layer) were determined by performing a special testing procedure.^{58}^{)}

The optimization of NN potential parameters was done iteratively with the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. In each iteration during the optimization, the root-mean-square errors (RMSEs) in both the training set and testing set were monitored. The training process was truncated when testing RMSE stops decreasing. The learning curve of such NN potential was shown in Supplemental Material (Fig. S2).^{59}^{)} The RMSEs for energies after training process were 23 and 39 meV per structure for the reference training and independent testing set, respectively. The mean absolute errors (MAEs) obtained for the training and test structures were 16 and 21 meV/structure, respectively. According to the results presented in Fig. 1, the energies of the training and test structures calculated using the NN potential matched the values obtained via DFT simulations (the corresponding correlation coefficients were equal to 0.998 and 0.997, respectively). In contrast with the high-dimensional NN potentials, whose remaining total energy errors are typically under 5 meV/atom, the accuracy of simplified NN potential appears to be worse. However, it should be pointed out that the RMSEs in the present study are given as values per whole supercell, while they are given as the values per atom in the high-dimensional NN potentials. So the comparison of the potential accuracy between the simplified and high-dimensional NN potentials is not straightforward. On the other hand, the fitting accuracy of NN can be constantly improved by increasing training data at the expense of additional DFT calculations. Considering the fact that the satisfactory accuracy was obtained for diffusion barriers (which will be shown in Sect. 3.2), we decided to perform the following investigation based on the present NN potential.

Figure 1. (Color online) A comparison between the total energies of the training set and testing set structures obtained using the DFT and NN approaches. The inset contains the MAE distributions (

It should be noted that the calculation of the total energy of a single geometric configuration, which was conducted using the current implementation of the NN potential model on a single CPU core (Intel^{®} Xeon^{®} W3565 processor), took only 0.001 s (for comparison, performing the same task using the VASP code and 12 CPU cores of the same type took about 1 h). Hence, the computational speed obtained using the NN potential was

To test the reliability of the NN potential for describing the atomic diffusion process, the former was used to predict a long path of the Cu diffusion in the amorphous Ta_{2}O_{5} matrix and the corresponding energy profile via NEB simulations. During this procedure, the overall motion of the Cu atom was described by several hops between the equilibrium sites. The locations of images after optimization (representing the diffusion path) are shown in Fig. 2(a). The obtained NN potential was in good agreement with the DFT results. The lengths of the diffusion paths defined as the sums of distances between the adjacent Cu transient positions were equal to 28.6 and 28.9 Å for the NN potential and DFT simulations, respectively. The corresponding energy variations along the long-range diffusion path, which are plotted against the diffusion length in Fig. 2(b), also demonstrate a good agreement between the two utilized methods. The largest mismatch between the data obtained via the NN potential and DFT calculations was about 0.15 eV, which corresponded to the saddle point of the highest barrier. Apart from the inherent inaccuracy of the NN potential method, a large discrepancy observed for this point might also result from the NEB images that missed the saddle point of the minimum energy path, which represented a common problem for NEB calculations and could be fixed by using a climbing image NEB method.^{60}^{)} It is worth noting that the range of Cu–O coordination number is wider in the training structures than the transient images and the range of Cu–Ta coordination number is the same between them (for details, see Sect. B of Supplemental Material).^{61}^{)} This suggests that the training dataset includes a variety of local environments sufficient to describe the transient image structures, which may be one of the reasons why the simplified NN potential method can achieve a good accuracy in the NEB calculation.

Figure 2. (Color online) A comparison between the NEB calculation results obtained for the selected diffusion pathway of a Cu atom in amorphous bulk Ta_{2}O_{5} using the DFT and NN potential approaches. (a) Transient images of the Cu atom obtained during the NEB calculations, from which the locations of diffusion pathways can be acquired. The convex polyhedrons represent Ta–O species in the amorphous matrix, while O and Ta atoms are not shown for clarity. (b) Energy profiles computed along the corresponding diffusion paths.

The advantages of the simplified NN potential are as follows: 1) it is based on the multi-layer perceptron NN, which can be easily constructed using various open-source codes; and 2) the number of data sets required for training the NN is about 1–2 orders of magnitude smaller than that utilized for a typical ternary high-dimensional NN potential.

However, the proposed simplified method also has limitations. First, it cannot be extended to MD techniques because of the absence of atomic descriptions (especially forces acting on atoms) in the host matrix. Second, although the NN potential can be trained in a particular amorphous network, it is often desirable to transfer it to another amorphous matrix. However, it was found that the transferability of the simplified potential was not very high (except for the cases with a very similar matrix structure) concerning the energy of the system including a Cu atom inserted at a randomly chosen point.^{62}^{)} For the calculation of metastable Cu positions, migration paths and migration energy barriers, the transferability seems much better, but in some cases the errors are considerably large.^{62}^{)} We expect that the transferability of the proposed method can be significantly improved by training the NN potential with a wide variety of the local environments of Cu atoms (which would require the use of various amorphous host matrices) and/or larger cutoff radii of the symmetry functions (this hypothesis will be tested in future studies).

On the basis of the above, we think that the NN potential method proposed in this paper can be complementary to the high-dimensional NN potential in studying atomic diffusion. Highly reliable description of atomic diffusion behavior is possible by a well-constructed high-dimensional NN potential, while at least rough overview can be obtained by the present NN potential with much lower computational cost.

_{2}O

_{5}

Determining potential equilibrium sites in amorphous bulk materials is a challenging task because of structural disorder. To avoid bias, it is important to locate all interstitial sites without making assumptions about their structure. Owing to the high computation speed of the NN potential, all possible Cu positions in the amorphous Ta_{2}O_{5} matrix structure were investigated in this work. First, Cu atoms were inserted at each of the points defining the ^{63}^{)} (it should be noted that the described technique was used in previous studies to search for equilibrium sites in amorphous bulk materials^{64}^{,}^{65}^{)}). As a result, the calculated interstitial positions were successively merged until all the distances between clusters became longer than the cutoff distance of 0.5 Å (during this procedure, 29 different clusters were obtained). For each cluster, the energy variance of all configurations was below 0.04 eV, and the lowest energy configuration was used to represent the metastable Cu interstitial position.

According to the results obtained in the previous section, 29 metastable interstitial positions of Cu atoms were determined using the NN potential. Assuming that a Cu atom can hop between the two adjacent interstitial positions with a distance between them less than 5 Å, 67 diffusion paths have been obtained (during the search, the paths that can be considered combinations of two or more individual pathways were eliminated). After that, the activation energies of the resulting pathways were determined using the NEB method (the spatial locations of all the obtained diffusion paths are depicted in Fig. 3).

Figure 3. (Color online) Equilibrium sites (blue spheres) and diffusion pathways (blue lines) determined for the Cu atoms in the bulk amorphous Ta_{2}O_{5} structure. The polyhedrons represent Ta–O polyhedrons, while O and Ta atoms have been removed for clarity.

For each individual diffusion path, two different activation energies corresponding to the two opposite hopping directions were computed. In general, for the path connecting sites *i* and *j* with the saddle point energy *i* to *j* is equal to *i*), while the activation energy of the diffusion in the opposite direction equals

Figure 4. (Color online) Distributions of the barrier energies calculated for all diffusion pathways of atomic Cu in bulk amorphous Ta_{2}O_{5}. The black columns correspond to individual hops (single NEB calculations), and the red columns represent the periodical paths, which connect equilibrium sites with their equivalent sites in the adjacent supercells. The frequencies were normalized with respect to the total number of diffusion pathways.

The diffusivity and diffusion activation energy of Cu atoms were examined by performing KMC simulations. During the utilized procedure, the probability for the Cu atom hopping from interstitial site *i* to site *j* was determined using the harmonic transition state theory: *k* represents the jump frequency (equal to about ^{−1}), and *i* and *j*. At the beginning of the simulations, Cu atoms were distributed randomly among the interstitial sites according to the Boltzmann-weighted energy relationship. Afterwards, the initial state was equilibrated by running 5000 MC events per atom. After reaching equilibrium, additional 10 million MC events were run for 1000 atoms, and the self-diffusivity of Cu atoms was evaluated from their respective trajectories using the Einstein expression: *a*-Ta_{2}O_{5} estimated at temperatures ranging from 500 to 1400 K. After fitting the obtained temperature dependence of the Cu self-diffusivity with the Arrhenius expression _{0} and

Figure 5. (Color online) (a) Cu self-diffusivity in the amorphous bulk Ta_{2}O_{5} structure calculated from the results of KMC simulations. The two black lines represent the linear fits of the simulation data obtained at high temperatures (900–1400 K) and low temperatures (500–800 K). (b) The comparison of Cu diffusivity between experimental observations and KMC simulation. The data of experiments 1 and 2 are taken from Refs. 6 and 7, respectively.

Lastly, we compared the diffusion behaviors extracted from KMC simulation with the experimental observations. Two independent experiments about the Cu diffusion in amorphous Ta_{2}O_{5} thin film have been reported.^{6}^{,}^{7}^{)} The diffusivities evaluated from the experiments are shown in Fig. 5(b), as well as the KMC simulation results in the same temperature range. In the first experiment, the Cu diffusivity at 200–500 °C was measured with secondary ion mass spectroscopy.^{6}^{)} By fitting the measured data to the Arrhenius relation, the activation energy for Cu diffusion is estimated to be 0.64 eV, while the room temperature diffusivity is estimated to be ^{2}/s. In the other experiment, the Cu diffusivity in amorphous Ta_{2}O_{5} film deposited by electron-beam was evaluated to be ^{2}/s from cyclic voltammogram measurements.^{7}^{)} The discrepancy of about eight orders of magnitude in the room temperature diffusivity may be ascribed to the difference in the sample thickness and the structure of the deposited Ta_{2}O_{5} film.^{7}^{)} Though the large discrepancy between the two experiments makes the quantitative comparison with our simulation results difficult, we may be able to say that our KMC results are reasonable. Firstly, the low temperature activation energy obtained from KMC simulation, 0.67 eV, is close to the one estimated from the first experiment, 0.64 eV. Secondly, the room temperature diffusivity extrapolated from the KMC results, ^{2}/s, is in between the two experimental estimates.

In this work, a simplified NN potential was developed to investigate the diffusion behavior of a single Cu atom in amorphous Ta_{2}O_{5} structure. In contrast to the NN potentials that were used for describing binary and ternary systems in previous studies, the proposed technique was found to be much simpler, easier to use, and required much less training data. The reliability of the utilized NN potential was validated via energy predictions and NEB calculations. Thus, its RMSE for energy predictions was equal to 23 meV/structure for the training set and 39 meV/structure for the testing set, and the NEB calculation results obtained for selected diffusion pathways were in good agreement with the DFT simulation data.

Using the developed NN potential, the equilibrium positions and diffusion paths of Cu atoms in the amorphous Ta_{2}O_{5} structure were located and characterized. The obtained energy barriers of single-hop paths ranged between 0.01 and 1.79 eV. The lowest energy barrier determined for the periodical pathways connecting equilibrium interstitial sites and their equivalent sites in the adjacent supercells was 0.68 eV. This magnitude matched the Cu diffusion activation energy calculated from the Arrhenius plot of the low-temperature self-diffusivity values generated via KMC simulations (0.67 eV) as well as the number obtained from experimental observations (0.64 eV). The conducted KMC simulations also revealed that the Cu diffusion activation energy obtained at high temperatures (0.87 eV) was larger than the magnitude calculated at low temperatures. Though the proposed simplified NN potential has disadvantages such as the inability to perform molecular dynamics simulations, it can be used as an important complementary technique, especially for studying atomic diffusion processes.

## Acknowledgements

This work is partially supported by JSPS KAKENHI Grant Number JP16H01535, the Support Program for Starting up Innovation Hub funded by the Japan Science and Technology Agency (JST), and the Core Research for Evolutional Science and Technology program of the Japan Science and Technology Agency (CREST–JST). Some computations were performed using the facilities of the Supercomputer Center of the Institute of Solid State Physics, University of Tokyo.

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