Deep Potential generation scheme and simulation protocol for the Li10GeP2S12-type superionic conductors
Jianxing Huang, Linfeng Zhang, Han Wang, Jinbao Zhao, Jun Cheng, Weinan E
AAutomatic machine-learning potential generationscheme and simulation protocol for theLiGePS-type superionic conductors
Jianxing Huang, † Linfeng Zhang, ¶ Han Wang, § Jinbao Zhao, ∗ , † Jun Cheng, ∗ , † and Weinan E ∗ , ¶ † College of Chemistry and Chemical Engineering, Xiamen University, Xiamen 361005,China ‡ State Key Laboratory of Physical Chemistry of Solid Surfaces, Xiamen University,Xiamen 361005, China ¶ Program in Applied and Computational Mathematics, Princeton University, Princeton,NJ 08544, USA § Laboratory of Computational Physics, Institute of Applied Physics and ComputationalMathematics, Fenghao East Road 2, Beijing 100094, P.R. China (cid:107)
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA
E-mail: [email protected]; [email protected]; [email protected]
Abstract
It has been a challenge to accurately simulate Li-ion diffusion processes in batterymaterials at room temperature using ab initio molecular dynamics (AIMD) due to itshigh computational cost. This situation has changed drastically in recent years dueto the advances in machine learning-based interatomic potentials. Here we implementthe Deep Potential Generator scheme to automatically generate interatomic potentials a r X i v : . [ phy s i c s . c o m p - ph ] J un or LiGePS-type solid-state electrolyte materials. This increases our ability to simulatesuch materials by several orders of magnitude without sacrificing ab initio accuracy.Important technical aspects like the statistical error and size effects are carefully in-vestigated. We further establish a reliable protocol for accurate computation of Li-iondiffusion processes at experimental conditions, by investigating important technicalaspects like the statistical error and size effects. Such a protocol and the automatedworkflow allow us to screen materials for their relevant properties with much-improvedefficiency. By using the protocol and automated workflow developed here, we obtainthe diffusivity data and activation energies of Li-ion diffusion that agree well withthe experiment. Our work paves the way for future investigation of Li-ion diffusionmechanisms and optimization of Li-ion conductivity of solid-state electrolyte materials. All-solid-state Li-ion batteries are amongst the most promising candidates for the next-generation rechargeable batteries.
Desired solid-state electrolyte (SSE) materials shouldhave high Li + conductivity and wide electrochemical windows. Several groups of promisingcandidates, with performance competitive to current commercial liquid electrolytes, e.g.,Li GeP S , Li La Zr O , Li P S , have been reported.Improvement of SSE performance lies in the fundamental understanding of diffusionmechanisms. The ab initio molecule dynamics (AIMD) calculation has been utilized toinvestigate the microscopic details of the diffusion processes. Unfortunately, due to itshigh computational cost, AIMD is typically limited to a system size of hundreds of atomsat the time scale of tens of pico-seconds. This makes it practically impossible to accuratelyestimate the diffusion coefficient of real materials at room temperature. Therefore one oftenresorts to the extrapolation strategy: assuming that a single Arrhenius relationship appliesto a wide temperature range (this implicitly assumes a temperature-independent diffusionmechanism), one can predict the room-temperature ionic conductivity (-40 ℃ - 80 ℃ ) by ex-trapolating from the high-temperature data. For example, a typical dataset for this purposeis ionic conductivity data in the temperature range (600 - 1200K) collected from 100400ps AIMD simulations with 2 × × Even more problematically, this Arrhenius extrapolation approach loses predictive powerwhen the Arrhenius relationship breaks down, and this was already discussed in detail over40 years ago. In particular, as shown in Fig. 1 (a), three different transition behaviors ofthe ionic conductivity with respect to the inverse of the temperature can give rise to threetypes of superionic conductors. Fig. 1 (b) depicts the experimental transition behaviors of 3typical Li-ion ionic conductors. The examples of Li Ge P S and β -Li PS representthe failures of the extrapolation strategy in Type I and II systems. The assumption behindthe extrapolation strategy is only applicable to Type III systems.Figure 1: (a) Schematic illustration of the three kinds of the temperature dependence ofthe conductivity, according to Ref. (b) Temperature dependence of the ionic conductivityof three Li-ion superionic conductors from experiments. Data were taken from the followingliterature: β -Li PS , Li Ge P S
12 19 and Li Ge P S . An obvious solution to the extrapolation problem is to directly simulate the diffusion pro-cess at room temperature, which on the other hand requires significantly longer trajectoriesto ensure convergence of diffusivity data. To accelerate these simulations, there have beenever-increasing efforts to develop empirical potential energy surfaces (PES) or model Hamil-tonians involving simple analytical terms, for systems of interest.
More recent works have3sed machine learning (ML) tools to represent the many-body and nonlinear depen-dence of the PES on atomic positions for materials modeling.
In particular, applicationsto SSE materials, e.g., Li PO , LiPON, and Li N, have recently been explored.Despite these efforts, two major obstacles have remained. First, when the number ofchemical species is large, a situation often found for SSE materials, one needs a representationthat is capable of fitting multi-element data with sufficient accuracy and efficiency, withoutmuch human intervention. Second, even with the ML tools, a systematic and automaticprocedure to generate uniformly accurate PES models, with a minimal set of training data,is still largely missing. The most straightforward approach is to perform extensive AIMDsimulations at different temperatures and use them as training data for the energies andforces along the AIMD trajectories. However, this procedure is computationally demanding,and the generated snapshots are highly correlated, reducing the quality of the training data.For these reasons, a great amount of trial-and-error process is still involved in most of the ML-based PES models, and consequently, the reliability of these models is very much in doubt.In particular, even though there have been some efforts to build ML potential generationschemes for different materials, there lacks a well-benchmarked, automatic, and efficientpotential generation scheme for superionic conductor materials.In this study, we implement a concurrent learning scheme to generate uniformly accuratePES models for LiGePS-type superionic conductors. With the generated ML potentials,we establish a robust MD protocol to accurately estimate diffusion coefficients at roomtemperature. To represent the multi-element PES, we employ a smooth version of the DeepPotential (DP) model, which is end-to-end, i.e. capable of fitting many-component dataof SSE materials with little human intervention. Using the Deep Potential Generator (DP-GEN) scheme, a minimal set of training data is generated from an efficient and sufficientsampling of the relevant configuration space, thereby guaranteeing a reliable PES modelproduced by training.We notice that a very recent work by Marcolongo et al has used the DP model to study4igure 2: Flowchart of DP-GEN. To start DP-GEN, a dataset with hundreds of initial con-figurations is required. The iteration process is considered converged after a predeterminednumber of loops or only a small number (e.g. 0.1%) of new configurations are found in thelast exploration process.diffusion properties of solidstate electrolytes and proved by numerical results the reliabilityof DP predictions compared with AIMD. Here we shall focus on the data generation schemeand a systematic benchmark of the resulting DP models.A brief description of the DP-GEN scheme is illustrated in Fig. 2. The iteration includes3 stages: labeling , training , and exploration . (a) First, the dataset is labeled by runninghigh precision single-point calculations. (b) Then, an ensemble of DP models with the samearchitecture (i.e. number of neural network layers and nodes) but different random seedsare trained using the whole training set. (c) To explore the configuration space, a fewmolecular dynamics simulations at different thermodynamic conditions are driven the DPmodels from the previous stage. Model deviations are evaluated using all trained models andnew configurations are picked according to the maximum deviation of forces ( σ maxf ), definedas: σ maxf = max i (cid:112) (cid:104)|| f i − (cid:104) f i (cid:105)|| (cid:105) , (1)where f i is the force acting on atom i , and (cid:104)· · · (cid:105) denotes the average of the DP model ensem-5le. Configurations with small force deviations ( σ maxf < σ low , yellow square in Fig. 2(c)) areeffectively covered by the training dataset with high probability. On the contrary, excessiveforce deviation ( σ maxf > σ high , red cross in Fig. 2(c)) implies that the configuration maydiverge from the relevant physical trajectories. Therefore none of them are picked. Onlyconfigurations whose σ maxf fall between a predetermined window are labeled as candidates (blue circles in Fig. 2(c)). According to Ref., a practical rule of thumb is to set σ low slightlylarger than the training error achieved by the model,and set σ high σ low . Therefore, for all the examples in this paper, σ low and σ high are set to 0.12 and 0.25eV/˚A, respectively. In practice, after running several MD trajectories, the selection criterionusually produces hundreds or thousands of candidates. A small fraction of them is typicallyrepresentative enough to improve the model, and therefore a cutoff number ( N maxlabel ) is set torestrict the number of candidates. These candidates are then labeled and added to the origi-nal dataset for the next training. The labeling and training stages are rather standard, whilethere is large flexibility for the sampling strategy on how to explore the relevant phase spacein each iteration. For example, different simulation times, system sizes, thermodynamicconditions, and sampling techniques can be used in different iterations.The structures of LiGePS-type superionic conductors can be grouped into two subsys-tems: MS (M = Ge, Si, Sn, and P) backbone blocks and mobile Li + ions. The solid-likebackbone forms a sublattice, in which the liquid-like Li-ion is free to flow. The dynamicsare characterized by the vibration of the backbone blocks and the diffusion of Li + betweenvacancies. These events can be efficiently captured by running MD at different tempera-tures. In practice, we start the DP-GEN iteration from 590 structures randomly perturbedaround a DFT-relaxed crystalline structure. The exploration strategy of DP-GEN is to runplenty of low-cost DP-based MD and select candidates from the sampled snapshots. Allexplorations are conducted with 10 initial structures and 5 pressures simultaneously. In thefirst 4 iterations, the exploration time is gradually lengthened from 300 fs to 10000 fs (10 ps).Afterward, we use 2 × × All DFT calculations were performed using the projector augmented-wave (PAW) methodapplied in VASP 5.4.4. We use the PBE scheme for the functional approximation, apractice widely adopted in the investigation of SSE materials, and serves as a good bench-mark. LAMMPS was employed to run all MD simulation and DeePMD-kit was used fortraining the DP model. More details of DPT, MD, as well as the DP-GEN setups can befound in the Supplementary Information (SI).Figure 3: For Li GeP S , distribution of maximum deviation of force ( σ maxf ) from iteration1 to 4. Distribution of deviation values at 4 temperatures are plotted and the two verticallines (dashed) correspond to the lower and upper bound of the selection criteria (0.12 and0.25 eV/˚A).To better illustrate the DP-GEN procedure, it is worth taking a thorough look at theexploration results from each iteration. As the first example, we study Li GeP S in-7epth due to its importance and extensive experimental reports. The same protocol isthen extended to Li SiP S and Li SnP S to validate the efficiency of DP-GEN. Fig. 3shows the distribution of σ maxf at different temperatures in the first 4 iterations. Duringthe 1st iteration, it is not surprising that the trajectories given by the preliminary modelssample lots of unreasonable configurations and high-temperature simulations blow up veryquickly. A large fraction of the snapshots sampled in this iteration have a σ maxf larger than0.4 eV/˚A (Fig. 3 (a)). A large portion of the candidates selected for labelling are fromlow-temperature simulations. This situation is drastically improved after just adding 300labeled configurations to the training dataset. In the 2nd iteration, most low-temperaturesnapshots are labeled as “accurate” and the majority of newly selected snapshots comefrom higher-temperature simulations. Going from the 2nd iteration to the 3rd and the 4th,although the time duration of the simulation is extended (i.e. 1000 fs, 5000 fs, and 10000 fs,respectively), most snapshots have their σ maxf value at a satisfactory level, demonstrating aquick convergence of the DP-GEN process. After 4 iterations, the models have converged inthe original cell (50 atoms), i.e. the percentage of candidates is ∼ × × GeP S Li SiP S Li SnP S Energy (meV/atom) 1.65(4) 1.73(3) 1.44(1)Force (meV/˚A) 80.5(5) 79.0(6) 79.0(2)After the DP-GEN procedure, we collect all the training data and use a “fine-training”to generate the production DP models (See SI for details). As shown in Table 1, the lowdeviation of the model ensemble suggests that the potentials have similar accuracy and theDP-GEN scheme gives consistent errors for 3 different systems. The root-mean-square errors8RMSE) of energies and forces are around 2 meV/atom and 80 meV/˚A, respectively. Thevolume of DP-relaxed Li GeP S structure is 998.8 ˚A , which agrees well with the DFTdata (989.8 ˚A ). The resulting models are then used to study the simulation protocol forthe diffusion properties. From now on, the discussion will be focused on the Li GeP S system.Figure 4: Diffusion coefficients of Li GeP S calculated with different (a) simulation timelength (100 ps, 200 ps, 500 ps, 1 ns, 2 ns, 5 ns and 10 ns) and (b) supercell size (50, 200,400 and 900 atoms).To investigate the statistical error of simulation time, we perform 10 ns simulations with3 × × D tr ) at each temperature is estimated by the timederivative of the mean-square displacement (MSD) of Li + . Previous studies based on AIMDhave suggested that a 200 ps MD simulation would be sufficient to ensure the convergence ofdiffusivity at a high temperature ( > − m /s (400 K, 500 K, and 666 K) reach very small variances and convergewithin 1 ns. However, since the diffusivity decreases exponentially with temperature, thestatistics of diffusion processes at low temperatures requires much longer simulation timethan that at high temperatures. At room temperature, extending the simulation time to90 ns ensures convergence of all diffusivity data with the relative uncertainty of 10 − m /s.Thus, we conclude that 10 ns is required for the simulation of room temperature diffusionprocesses.Following the test for the simulation time, we also perform a similar analysis on thesystem size, using 10 ns trajectories. The system-size dependence of the diffusion coefficientand viscosity from MD simulations with periodic boundary conditions is a classic topic andhas been extensively discussed by, e.g., Yeh et. al. Here, as shown in Fig. 4(b), a 2 × × gives decent result. To confirm this, we decide to run simulations with 3 × × In principle, one can simply equilibrate the simulation cell in an N p Tensemble and evaluate relevant quantities. However, considering the non-negligible differencebetween PBE-relaxed cell parameters and the experimental ones (see Table 4) we system-atically evaluate the effect of lattice parameter upon diffusivity at low temperatures (below700 K) by isotropically scaling the lattice volume from -5% to 5% and perform NVT MDsimulations using DP. It is shown in Fig. 5 that such a tiny expansion or contraction issufficient to lead to noticeable differences in the computed diffusion coefficients. This mightbe attributed to the geometrical change of the transport tunnel, which may further changethe diffusion mechanism at some conditions. Volume expansion reduces the strong ionicbonds between lithium and sulfur, which benefits the hopping events. It may also weakenthe repulsion between lithium ions, leading to the suppression of collective motion.Since diffusivity is sensitive to lattice parameters, we also evaluate the temperature de-pendence of the lattice parameters in Figure 6 with 1-ns long N p T simulations. LiGePS-typematerials belong to tetragonal crystal lattice, for which the lattice parameter a is equivalent10igure 5: Diffusivity coefficients of Li GeP S at 400K, 500K and 666K obtained with 1 nsNVT simulations. The lattice volumes are scaled to 95%, 97%, 99%, 100%, 101%, 103% and105%, respectively.Figure 6: Thermal expansion of Li GeP S at seven different temperatures (250 K, 300 K,400 K, 500 K, 600 K, 666 K and 800K), (a) lattice parameters a and c; (b) unit cell volumes.The dashed lines corresponding to the fitting range of thermal expansion coefficients. Theexperimental data is extracted from Weiber et al. to lattice parameter b , and thus the thermal expansion of lattice parameter a and c are pre-sented. Weber et al found that lattice parameters a and c exhibit linear thermal expansionbelow 700 K and anisotropic expansion at higher temperature. Here we focus on the linearregion that is of practical interest. The room temperature thermal expansion coefficient( α L K ) is 3 . × − K − , consistent with the value 3 . × − K − from the literature. Based on the investigation of the time and size convergence as well as the influence of thelattice parameters, in the current study for 3 different systems (Li GeP S , Li SiP S , and11igure 7: Temperature dependence of diffusion coefficients of (a) Li GeP S , (b)Li SiP S and (c) Li SnP S , calculated by Deep Potential from 10 ns trajectories. Exper-imental results of Li GeP S and Li SnP S were extracted from NMR measurements. Li SnP S ), we run NVT simulation with DP models using experimental lattice parametersand with thermal expansion considered. The results are shown in Fig. 7. The activationenergies of the three materials suggested by previous AIMD calculations are 0.21 ± ± ± For Li GeP S , the room-temperaturediffusion coefficient and activation energy calculated by Deep Potential are D = 4 . × − m/s and E a = 0 .
23 eV, in good agreement with the experimental data D = 3 . × − m/sand E a = 0 .
22 eV. To the best of our knowledge, no relevant experimental data of thethermal expansion of Li SiP S and Li SnP S have been reported. Here we assumethe two systems have similar thermal expansion coefficients as Li GeP S . The calculatedactivation energies (0.18 eV for Li SiP S and 0.20 eV for Li SnP S are slightly lowerthan the experimental value (0.20 eV for Li SiP S and 0.27 eV for Li SnP S . Whilethe agreement of diffusion coefficients of Li GeP S between calculated and experimentalresults is very good, the same comparison on Li SnP S shows a larger gap.Several factors could contribute to the disagreement between the experimental and simu-lation results. First, the goal of involving machine-learning models is to extend the accessiblesimulation time and cell size without sacrificing accuracy. However, one needs to adopt asuitable functional approximation for DFT to provide training data that are accurate enoughto describe relevant situations. Different functional approximations exhibit different descrip-tions of the bulk properties, the chemical bonding, etc. In Table 4 of SI we also calculated12he relaxed lattice parameters with different functional approximation schemes used in DFT(LDA, PBEsol, PBE with vdw correction and PBE0 ). These results suggest thatPBEsol and PBE with vdw correction offer a better description of the lattice the volume.A more systematic investigation of the influence of the functional approximations is beyondthe scope of this paper but would be facilitated by the accuracy and efficiency of DP andDP-GEN schemes. Secondly, the experimental tetragonal phase Li SiP S is impure and the influence of the disorder seems to be more significant for Li SnP S , whichmight be the reason for the worse agreement between the simulation and experimental re-sults for Li SnP S , compared with that for Li GeP S . We leave the investigation ofthese factors for future studies. Besides, the influence of defects and M-P site (M=Ge, Si,and Sn) disorder in LiGePS-type materials should be considered to further understand thediffusion processes.In conclusion, we developed an efficient and automated workflow to generate ML po-tentials for Li GeP S -type materials. Our focus in this paper is the validation of thesimulation protocol. The effectiveness of the protocol is explained and analyzed in detail.We carefully studied the errors caused by limited simulation time and cell size. We also stud-ied the lattice parameters and thermal expansion. We hope that this work represents a solidstep forward towards the paradigm of generating a multi-component “universal potential”(i.e. Li-Ge-Si-Sn-P-S) to understand and design SSE with higher ionic conductivity. Acknowledgement
J.-X. H. and J.C. are grateful for funding support from the National Natural Science Foun-dation of China (Grants No, 21861132015, 21991151, 91745103). J.-X. H. and J.-B. Z. aresupported by the National Key Research and Development Program of China (Grant No.2017YFB0102000). The work of L. Z. and W. E was supported in part by a gift from iFly-tek to Princeton University, the ONR grant N00014-13-1-0338, and the Center Chemistry13n Solution and at Interfaces (CSI) funded by the DOE Award de-sc0019394. The workof H. W. was supported by the National Science Foundation of China under Grant No.11871110, the National Key Research and Development Program of China under Grants No.2016YFB0201200 and No. 2016YFB0201203, and Beijing Academy of Artificial Intelligence(BAAI).
References (1) Goodenough, J. B.; Park, K.-S. The Li-ion rechargeable battery: a perspective.
Journalof the American Chemical Society , , 1167–1176.(2) Tarascon, J.-M.; Armand, M. Materials for Sustainable Energy: A Collection of Peer-Reviewed Research and Review Articles from Nature Publishing Group ; World Scientific,2011; pp 171–179.(3) Chu, S.; Majumdar, A. Opportunities and challenges for a sustainable energy future. nature , , 294.(4) Hu, Y.-S. Batteries: getting solid. Nature Energy , , 16042.(5) Zhang, Z.; Shao, Y.; Lotsch, B.; Hu, Y.-S.; Li, H.; Janek, J.; Nazar, L. F.; Nan, C.-W.;Maier, J.; Armand, M., et al. New horizons for inorganic solid state ion conductors. Energy & Environmental Science , , 1945–1976.(6) Kamaya, N.; Homma, K.; Yamakawa, Y.; Hirayama, M.; Kanno, R.; Yonemura, M.;Kamiyama, T.; Kato, Y.; Hama, S.; Kawamoto, K., et al. A lithium superionic conduc-tor. Nature materials , , 682.(7) Murugan, R.; Thangadurai, V.; Weppner, W. Fast lithium ion conduction in garnet-type Li7La3Zr2O12. Angewandte Chemie International Edition , , 7778–7781.148) Seino, Y.; Ota, T.; Takada, K.; Hayashi, A.; Tatsumisago, M. A sulphide lithium superion conductor is superior to liquid ion conductors for use in rechargeable batteries. Energy & Environmental Science , , 627–631.(9) Car, R.; Parrinello, M. Unified approach for molecular dynamics and density-functionaltheory. Physical Review Letters , , 2471.(10) Ong, S. P.; Mo, Y.; Richards, W. D.; Miara, L.; Lee, H. S.; Ceder, G. Phase stabil-ity, electrochemical stability and ionic conductivity of the Li 10 ± Energy &Environmental Science , , 148–156.(11) Ceder, G.; Ong, S. P.; Wang, Y. Predictive modeling and design rules for solid elec-trolytes. MRS Bulletin , , 746751.(12) He, X.; Zhu, Y.; Mo, Y. Origin of fast ion diffusion in super-ionic conductors. Naturecommunications , , 15893.(13) Nolan, A. M.; Zhu, Y.; He, X.; Bai, Q.; Mo, Y. Computation-accelerated design ofmaterials and interfaces for all-solid-state lithium-ion batteries. Joule ,(14) Van der Ven, A.; Ceder, G. Lithium diffusion in layered Li x CoO2.
Electrochemicaland Solid-State Letters , , 301–304.(15) Mo, Y.; Ong, S. P.; Ceder, G. First Principles Study of the Li 10 GeP 2 S 12 LithiumSuper Ionic Conductor Material. Chemistry of Materials , , 15–17.(16) He, X.; Zhu, Y.; Epstein, A.; Mo, Y. Statistical variances of diffusional properties fromab initio molecular dynamics simulations. npj Computational Materials , , 18.(17) Boyce, J. B.; Huberman, B. A. Superionic conductors: Transitions, structures, dynam-ics. Physics Reports , , 189–265.1518) Tachez, M.; Malugani, J.-P.; Mercier, R.; Robert, G. Ionic conductivity of and phasetransition in lithium thiophosphate Li3PS4. Solid State Ionics , , 181 – 185.(19) Kwon, O.; Hirayama, M.; Suzuki, K.; Kato, Y.; Saito, T.; Yonemura, M.;Kamiyama, T.; Kanno, R. Synthesis, structure, and conduction mechanism of thelithium superionic conductor Li10+ δ Ge1+ δ P2- δ S12.
Journal of Materials ChemistryA , , 438–446.(20) Kanno, R.; Murayama, M. Lithium Ionic Conductor Thio-LISICON: The Li 2S-GeS2-P 2S 5 System. Journal of the Electrochemical Society , , 742–746.(21) Xiao, R.; Li, H.; Chen, L. Candidate structures for inorganic lithium solid-state elec-trolytes identified by high-throughput bond-valence calculations. Journal of Materi-omics , , 325–332.(22) Kahle, L.; Marcolongo, A.; Marzari, N. Modeling lithium-ion solid-state electrolyteswith a pinball model. Physical Review Materials , , 065405.(23) Behler, J.; Parrinello, M. Generalized neural-network representation of high-dimensional potential-energy surfaces. Physical review letters , , 146401.(24) Bart´ok, A. P.; Payne, M. C.; Kondor, R.; Cs´anyi, G. Gaussian approximation potentials:The accuracy of quantum mechanics, without the electrons. Physical review letters , , 136403.(25) Artrith, N. The Atomic Energy Network (´cnet)(release 2.0. 0). ,(26) Zhang, L.; Han, J.; Wang, H.; Saidi, W.; Car, R.; E, W. Advances in Neural InformationProcessing Systems 31 ; Curran Associates, Inc., 2018; pp 4436–4446.(27) Artrith, N.; Urban, A.; Ceder, G. Constructing first-principles phase diagrams of amor-phous Li x Si using machine-learning-assisted sampling with an evolutionary algorithm.
The Journal of chemical physics , , 241711.1628) Deringer, V. L.; Merlet, C.; Hu, Y.; Lee, T. H.; Kattirtzi, J. A.; Pecher, O.; Cs´anyi, G.;Elliott, S. R.; Grey, C. P. Towards an atomistic understanding of disordered carbonelectrode materials. Chemical communications , , 5988–5991.(29) Fujikake, S.; Deringer, V. L.; Lee, T. H.; Krynski, M.; Elliott, S. R.; Cs´anyi, G. Gaussianapproximation potential modeling of lithium intercalation in carbon nanostructures. The Journal of chemical physics , , 241714.(30) Lacivita, V.; Artrith, N.; Ceder, G. Structural and Compositional Factors That Controlthe Li-Ion Conductivity in LiPON Electrolytes. Chemistry of Materials , , 7077–7090.(31) Li, W.; Ando, Y.; Minamitani, E.; Watanabe, S. Study of Li atom diffusion in amor-phous Li3PO4 with neural network potential. The Journal of chemical physics , , 214106.(32) Deng, Z.; Chen, C.; Li, X.-G.; Ong, S. P. An electrostatic spectral neighbor analysispotential for lithium nitride. npj Computational Materials , , 75.(33) Bernstein, N.; Cs´anyi, G.; Deringer, V. L. De novo exploration and self-guided learningof potential-energy surfaces. arXiv preprint arXiv:1905.10407 ,(34) Nyshadham, C.; Rupp, M.; Bekker, B.; Shapeev, A. V.; Mueller, T.; Rosenbrock, C. W.;Cs´anyi, G.; Wingate, D. W.; Hart, G. L. Machine-learned multi-system surrogate mod-els for materials prediction. npj Computational Materials , , 1–6.(35) Marcolongo, A.; Binninger, T.; Zipoli, F.; Laino, T. Simulating Diffusion Propertiesof Solid-State Electrolytes via a Neural Network Potential: Performance and TrainingScheme. ChemSystemsChem n/a .(36) Deringer, V. L.; Proserpio, D. M.; Csnyi, G.; Pickard, C. J. Data-driven learning andprediction of inorganic crystal structures.
Faraday Discuss. , , 45–59.1737) Mortazavi, B.; Podryabinkin, E.; Novikov, I. S.; Roche, S.; Rabczuk, T.; Zhuang, X.;Shapeev, A. Efficient machine-learning based interatomic potentials for exploring ther-mal conductivity in two-dimensional materials. Journal of Physics: Materials ,(38) Podryabinkin, E. V.; Tikhonov, E. V.; Shapeev, A. V.; Oganov, A. R. Accelerating crys-tal structure prediction by machine-learning interatomic potentials with active learning.
Physical Review B , , 064114.(39) Gubaev, K.; Podryabinkin, E. V.; Shapeev, A. V. Machine learning of molecular proper-ties: Locality and active learning. The Journal of Chemical Physics , , 241727.(40) Zhang, L.; Lin, D.-Y.; Wang, H.; Car, R.; E, W. Active learning of uniformly accurateinteratomic potentials for materials simulation. Physical Review Materials , ,023804.(41) Zhang, Y.; Wang, H.; Chen, W.; Zeng, J.; Zhang, L.; Wang, H.; E, W. DP-GEN: Aconcurrent learning platform for the generation of reliable deep learning based potentialenergy models. Computer Physics Communications , 107206.(42) Bl¨ochl, P. E. Projector augmented-wave method.
Physical review B , , 17953.(43) Kresse, G.; Furthm¨uller, J. Efficient iterative schemes for ab initio total-energy calcu-lations using a plane-wave basis set. Physical review B , , 11169.(44) Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Physical Review B , , 1758.(45) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation MadeSimple. Phys. Rev. Lett. , , 3865–3868.(46) Plimpton, S. Fast parallel algorithms for short-range molecular dynamics. Journal ofcomputational physics , , 1–19.1847) Wang, H.; Zhang, L.; Han, J.; E, W. DeePMD-kit: A deep learning package for many-body potential energy representation and molecular dynamics. Computer Physics Com-munications , , 178–184.(48) Yeh, I.-C.; Hummer, G. System-Size Dependence of Diffusion Coefficients and Viscosi-ties from Molecular Dynamics Simulations with Periodic Boundary Conditions. TheJournal of Physical Chemistry B , , 15873–15879.(49) Ong, S. P.; Mo, Y.; Richards, W. D.; Miara, L.; Lee, H. S.; Ceder, G. Phase stability,electrochemical stability and ionic conductivity of the Li10+-1MP2X12 (M = Ge, Si, Sn,Al or P, and X = O, S or Se) family of superionic conductors. Energy and EnvironmentalScience , , 148–156.(50) Bachman, J. C.; Muy, S.; Grimaud, A.; Chang, H.-H.; Pour, N.; Lux, S. F.; Paschos, O.;Maglia, F.; Lupart, S.; Lamp, P.; Giordano, L.; Shao-Horn, Y. Inorganic Solid-StateElectrolytes for Lithium Batteries: Mechanisms and Properties Governing Ion Conduc-tion. Chemical Reviews , , 140–162, PMID: 26713396.(51) Fitzhugh, W.; Ye, L.; Li, X. The effects of mechanical constriction on the operation ofsulfide based solid-state batteries. J. Mater. Chem. A , , 23604–23627.(52) Weber, D. A.; Senyshyn, A.; Weldert, K. S.; Wenzel, S.; Zhang, W.; Kaiser, R.;Berendts, S.; Janek, J.; Zeier, W. G. Structural Insights and 3D Diffusion Pathwayswithin the Lithium Superionic Conductor Li 10 GeP 2 S 12. Chemistry of Materials , , 5905–5915.(53) Kuhn, A.; Duppel, V.; Lotsch, B. V. Tetragonal Li10GeP2S12 and Li7GePS8 exploringthe Li ion dynamics in LGPS Li electrolytes. Energy Environ. Sci. , , 3548–3552.(54) Kuhn, A.; Gerbig, O.; Zhu, C.; Falkenberg, F.; Maier, J.; Lotsch, B. V. A new ultra-fast superionic Li-conductor: ion dynamics in Li11Si2PS12 and comparison with othertetragonal LGPS-type electrolytes. Phys. Chem. Chem. Phys. , , 14669–14674.1955) Bron, P.; Johansson, S.; Zick, K.; Schmedt auf der Gnne, J.; Dehnen, S.; Roling, B.Li10SnP2S12: an affordable lithium superionic conductor. Journal of the AmericanChemical Society , , 15694–15697.(56) Whiteley, J. M.; Woo, J. H.; Hu, E.; Nam, K.-W.; Lee, S.-H. Empowering the lithiummetal battery through a silicon-based superionic conductor. Journal of the Electrochem-ical Society , , A1812–A1817.(57) Wang, Z. Q.; Wu, M. S.; Liu, G.; Lei, X. L.; Xu, B.; Ouyang, C. Y. Elastic proper-ties of new solid state electrolyte material Li10GeP2S12: A study from first-principlescalculations. International Journal of Electrochemical Science , , 562–568.(58) Hu, C. H.; Wang, Z. Q.; Sun, Z. Y.; Ouyang, C. Y. Insights into structural stabilityand Li superionic conductivity of Li 10 GeP 2 S 12 from first-principles calculations. Chemical Physics Letters , , 16–20.(59) Kohn, W.; Sham, L. J. Self-Consistent Equations Including Exchange and CorrelationEffects. Phys. Rev. , , A1133–A1138.(60) Perdew, J. P.; Ruzsinszky, A.; Csonka, G. I.; Vydrov, O. A.; Scuseria, G. E.; Con-stantin, L. A.; Zhou, X.; Burke, K. Restoring the Density-Gradient Expansion forExchange in Solids and Surfaces. Phys. Rev. Lett. , , 136406.(61) Klimeˇs, J.; Bowler, D. R.; Michaelides, A. Chemical accuracy for the van der Waalsdensity functional. Journal of Physics: Condensed Matter , , 022201.(62) Perdew, J. P.; Ernzerhof, M.; Burke, K. Rationale for mixing exact exchange withdensity functional approximations. The Journal of Chemical Physics , , 9982–9985.(63) Adamo, C.; Barone, V. Toward reliable density functional methods without adjustableparameters: The PBE0 model. The Journal of Chemical Physics , , 6158–6170.2064) Kuhn, A.; K¨ohler, J.; Lotsch, B. V. Single-crystal X-ray structure analysis of the supe-rionic conductor Li10GeP2S12. Physical Chemistry Chemical Physics ,15