Machine Learning Enabled Computational Screening of Inorganic Solid Electrolytes for Suppression of Dendrite Formation in Lithium Metal Anodes.

Next generation batteries based on lithium (Li) metal anodes have been plagued by the dendritic electrodeposition of Li metal on the anode during cycling, resulting in short circuit and capacity loss. Suppression of dendritic growth through the use of solid electrolytes has emerged as one of the most promising strategies for enabling the use of Li metal anodes. We perform a computational screening of over 12 000 inorganic solids based on their ability to suppress dendrite initiation in contact with Li metal anode. Properties for mechanically isotropic and anisotropic interfaces that can be used in stability criteria for determining the propensity of dendrite initiation are usually obtained from computationally expensive first-principles methods. In order to obtain a large data set for screening, we use machine-learning models to predict the mechanical properties of several new solid electrolytes. The machine-learning models are trained on purely structural features of the material, which do not require any first-principles calculations. We train a graph convolutional neural network on the shear and bulk moduli because of the availability of a large training data set with low noise due to low uncertainty in their first-principles-calculated values. We use gradient boosting regressor and kernel ridge regression to train the elastic constants, where the choice of the model depends on the size of the training data and the noise that it can handle. The material stiffness is found to increase with an increase in mass density and ratio of Li and sublattice bond ionicity, and decrease with increase in volume per atom and sublattice electronegativity. Cross-validation/test performance suggests our models generalize well. We predict over 20 mechanically anisotropic interfaces between Li metal and four solid electrolytes which can be used to suppress dendrite growth. Our screened candidates are generally soft and highly anisotropic, and present opportunities for simultaneously obtaining dendrite suppression and high ionic conductivity in solid electrolytes.

Introduction

Increased energy densities of Li-ion batteries are crucial for progress toward complete electrification of transportation.[1−3] Among the many possible routes, Li metal anodes have emerged as one of the most likely near-term commercialization options.[4] Coupled with a conventional intercalation cathode, batteries utilizing Li metal anodes could achieve specific energy of >400 W h/kg, much higher than the current state of the art ∼250 W h/kg.[5,6] Unstable and dendritic electrodeposition on Li metal anode coupled with capacity fade due to consumption of electrolyte has been one of the major hurdles in its commercialization.[5,7−12] For large-scale adoption, a stable, smooth and dendrite-free electrodeposition on Li metal is crucial.

Numerous approaches are being actively pursued for suppressing dendrite growth through the design of novel additives in liquid electrolytes,[13−19] surface nanostructuring,[20,21] modified charging protocols,[22,23] artificial solid electrolyte interphase or protective coatings,[24−26] polymers,[27−29] or inorganic solid electrolytes.[30−33] Among these, dramatic improvements in the ionic conductivity of solid electrolytes[34,35] have made them extremely attractive candidates for enabling Li metal anodes.

A comprehensive and precise criterion for dendrite suppression is still elusive. Interfacial effects[36,37] and spatial inhomogeneities within the solid electrolyte like voids, grain boundaries, and impurities[38] make the problem challenging. Monroe and Newman performed a dendrite initiation analysis and showed that solid polymer electrolytes with shear modulus roughly twice that of Li could achieve stable electrodeposition.[39] In an earlier work, we extended this idea and showed that the criteria for the suppression of dendrite growth gets reversed for inorganic crystalline materials due to the difference in molar volume of Li+. A softer solid electrolyte is required for stability in this case.[40] It is worth highlighting that this requirement applies only for dendrite initiation regime, and other suppression approaches may be possible for the propagation regime. However, once initiated, dendrite growth is extremely hard to mitigate as pointed out by several studies.[41−43] Therefore, it is best to prevent dendrites from initiating to ensure smooth electrodeposition throughout cycling of the battery.

In recent years, high-throughput computational materials design has emerged as a major driver of discovery of novel materials for various applications.[44,45] It typically involves a combination of first-principles quantum-mechanical approaches and database construction and mining techniques. Combined with machine-learning methods that bypass the use of expensive quantum-mechanical calculations through the use of structural descriptors,[46−50] one can accelerate the high-throughput screening by several orders of magnitude.[51−53] Previous high-throughput screening studies of solid electrolytes have used ionic conductivity, stability, and electronic conductivity as screening criteria.[51,52] However, dendrite suppression capability of solid electrolytes is an additional requirement that needs to be assessed.

Here, we carry out a large-scale data-driven search for solid electrolytes that might be promising candidates for suppressing dendrite growth during the initiation phase with a Li metal anode. We use machine-learning techniques to train and predict the mechanical properties of inorganic solids which play a major role in stabilizing the interface. These properties are fed into the theoretical framework which uses the stability parameter[40,54] to quantify the dendrite initiation with Li metal anode. At a mechanically isotropic interface, the screening results predict the crucial role of surface tension in stabilizing the interface since most solid electrolytes are not intrinsically stabilized by the stresses generated at the interface. Hence, surface nanostructuring may be essential to prevent initiation of dendrites for isotropic interfaces. We rank the materials based on the amount of nanostructuring (surface roughness wavenumber) required for achieving a stable electrodeposition. We then performed a stability analysis of over 15 000 anisotropic interfaces between the Li metal and solid electrolyte using the Stroh formalism. This is essential to account for the highly anisotropic mechanical properties of Li[55] and texturing of electrodeposited Li at the interface.[56] A full anisotropic treatment of the interface reveals over 20 candidate interfaces that are predicted to suppress dendrite initiation. The materials obtained through screening are generally soft and with highly anisotropic mechanical properties. Since softer materials are generally faster ion conductors than stiffer materials due to availability of more volume per atom,[57] the screened candidates present an opportunity to obtain both desirable mechanical properties and fast ion conduction.

Results and Discussion

We describe the procedure for screening solid electrolytes and discuss the material candidates obtained based on isotropic and anisotropic criteria separately. First, we briefly review the criteria for dendrite suppression at a metal–solid electrolyte interface to determine the desirable properties of solid electrolytes. We then discuss our machine-learning models used for predicting these properties. Finally, we perform material screening based on these properties and discuss the implications.

Isotropic Material Screening

In solid electrolytes, the mechanical properties at the interface provide an additional degree of freedom for tuning the stability of electrodeposition. Previously, we developed a generalized stability diagram for assessing the stability of electrodeposition at a metal–solid electrolyte interface for isotropic mechanical response.[40] In these studies, we used the stability parameter first proposed by Monroe and Newman[43] to characterize the growth/decay of dendrites with time. The sign of the stability parameter, denoted hereafter as χ, determines whether the electrodeposition is stable or unstable. A positive χ implies higher current density at the peaks and lower current density at the valley leading to growth of dendrites while a negative χ leads to stabilization or suppression of dendrites. The stability parameter is related to the change in the electrochemical potential of the electron Δμe at a deformed interface z = f(x) between the metal anode and the electrolyte (Figure S1). It is convenient to compute properties of the interface in Fourier space with f(x) = ∫ dk[f1(k) cos(kx) + f2(k) sin(kx)] and then integrate over the surface roughness wavenumber k to obtain the overall behavior. The stability parameter can be calculated in a closed form at a given k. The change in the electrochemical potential at a given k is given by Δμe(k) = χ(k)[f1(k) cos(kx) + f2(k) sin(kx)].[54] This serves to define the stability parameter χ(k) asΔμe can be obtained by including the effect of mechanical stresses and surface tension on the electrochemical potential of the species at a deformed interface as[43]From eq , Δμe depends on k through the surface tension γ, curvature κ, the hydrostatic stress Δp, and deviatoric stress τ generated at the interface. e and s in the subscripts refer to the metal electrode (anode) and solid electrolyte, respectively; VM is the molar volume of the metal atom in the anode, v the ratio of molar volume of the metal ion in electrolyte VM to the metal atom in the anode VM, z the valence of the metal, and the normal to the interface pointing toward the electrolyte. The stability parameter consists of contributions from the surface tension and the stresses developed at the metal–electrolyte interface. For an isotropic metal anode with shear modulus Ge and Poisson’s ratio νe in contact with an isotropic electrolyte with shear modulus Gs and Poisson’s ratio νs, the stability parameter χ(k) can be computed exactly as[40]Using the shear modulus, Poisson’s ratio, and molar volume ratio of a solid electrolyte, it is possible to calculate the stability parameter for its interface with Li metal anode and determine stability of electrodeposition. For a complete understanding of the interface growth and stability, it is necessary to determine the sign of stability parameter at all the Fourier components k. Fortunately, as we will see later, a negative stability parameter at a given k guarantees stability at all higher values.

The molar volume ratio v = VM/VM influences the range of shear moduli over which the electrodeposition is stable. VM was calculated using the coordination number of Li in the crystal structure and mapping them to ionic radius using the values tabulated by Shannon.[58] The coordination number was calculated by generating polyhedra around a species through Voronoi analysis[59,60] as implemented in pymatgen.[61] A linear interpolation was used for computing ionic radius corresponding to coordination numbers not in the Shannon’s tabulated values. Predictions with VM > VM (true for just one candidate in the screening) were ignored since those correspond to very high Li coordination number where Shannon’s tabulated values cannot be used. The partial molar volume of the metal in the electrolyte VM can be measured directly in an experiment on the difference of potentials between a stressed and unstressed electrolyte as done by Pannikkat and Raj[62] and then using the relationship VM = ∂μM/∂p where μM is the electrochemical potential of the metal ion.

Since bulk and shear moduli are related to second derivatives of energy with respect to lattice constants at equilibrium, their calculation by first-principles requires fitting of the energy–strain relationship or the stress–strain relationship. Calculations on several deformed structures are required in order to get an accurate estimate of the fitting parameters. At each deformed state of the structure, the internal coordinates need to be relaxed to calculate the energy or the stress. The materials project database employed 24 relax calculations for a single material to compute the moduli. To perform a large-scale screening over all Li-containing compounds (over 12 000) for use as solid electrolytes, it is necessary to choose a technique that can predict the properties reasonably accurately and without the high computational cost of multiple first-principles simulations. Hence, we used the crystal graph convolutional neural network (CGCNN) framework[63] to predict the shear and bulk moduli of the crystalline solid electrolyte materials. At the core of the CGCNN is the multigraph representation of the crystal structure which encodes the atomic information and bonding interactions between atoms. The CGCNN builds a graph convolutional neural network directly on top of a multigraph that represents the crystal structure of the electrolytes, and predicts the elastic properties by extracting local structural features from the multigraph representation. Note that this method does not depend on any handcrafted geometric or topological features, and all the features are learned by the neural network automatically. This results in a model that is more general than the usual models relying on descriptors but also requires more data to train.

The training data for the mechanical properties required to compute χ through eq was obtained from the materials project database.[64,65] The calculated values in the database are typically within 15% of the experimental values which is sometimes the uncertainty in experimental data.[65] The moduli have been calculated using density functional theory (DFT) with the Perdew, Burke, and Ernzerhof (PBE) generalized gradient approximation (GGA) for the exchange-correlation functional.[66] GGA-level predictions for 104 systems were within 15% of the experimental value for all but 16 systems for the bulk modulus and 15 systems for the shear modulus.[65] Out of the outliers, many had a discrepancy of less than 10 GPa. Experimentally, the shear and bulk moduli can be calculated using the elastic tensor obtained through inelastic neutron scattering or pulse–echo measurements. The experimental measurements typically have a high degree of variability depending on the experimental technique and conditions. We used 2041 crystal structures with shear and bulk moduli, 60% of the entire data set with elastic properties, to train our CGCNN model. We choose to minimize the mean squared errors between the log values of predicted and calculated elastic properties since we aim to minimize the relative prediction errors instead of absolute errors and avoid overweighing stiffer materials. This also enabled us to always obtain positive values of the shear and bulk moduli. We then performed a hyperparameter optimization on 20% validation data via grid search to select the optimum learning rate, weight decay, and number of convolution layers. The best performing hyperparameters are selected, and the resulting model is evaluated on the rest 20% test data. The CGCNN was implemented in PyTorch,[67] and the details of the architecture and optimized hyperparameters can be found in the Supporting Information and ref (63).

Considering the presence of uncertainty in training data set, we developed a framework for obtaining uncertainty estimates on the results. The uncertainty in the model predictions was obtained by generating an ensemble of 100 CGCNNs using a random 60% of the training data for each model. Using each model, we obtained predictions of the shear and bulk modulus. The ensemble of moduli was then used together with eq to obtain an ensemble of stability parameters for each material. The spread in the distribution of the stability parameter was used to quantify the uncertainty. Figures S4 and S5 show the distribution of the shear modulus and stability parameter for the material Li5Sn2 as an example. Using the ensemble of stability parameters, we calculated the probability of stability Ps as the ratio of number of models that predict a negative stability parameter to the total number of models:Here, N is the total number of models (100 in our calculations), and the indicator function 1{X} is equal to 1 if the condition X is true and 0 if it is false.

The performance of the CGCNN model was evaluated on 680 test data points. In Figure a, we show the comparison between the shear modulus predicted by our model and the DFT-calculated value obtained from the materials project database, and in Table , we show the root mean squared error (RMSE) for the shear and bulk moduli predicted by our model. The RMSE obtained using our model is comparable to previous work by de Jong et al.[49] However, it is worth noting that we evaluated our model on test data while de Jong et al. evaluated on the entire data set, indicating that our model does not overfit and has better generalization capability.

Figure 1

Parity plots comparing the elastic properties: (a) shear modulus G, and elastic constants (b) C11, (c) C12, and (d) C44 predicted by the machine-learning models to the DFT-calculated values. The shear modulus is predicted using CGCNN, and the elastic constants C11 and C44 are predicted using gradient boosting regression while C12 is predicted using kernel ridge regression. The parity plot for shear modulus is on 680 test data points while that for the elastic constants contains all available data (170 points) where each prediction is a cross-validated value.

Table 1

Comparison of RMSE in log(GPa) for Shear and Bulk Moduli

method	log(G) RMSE	log(K) RMSE
this work	0.1268	0.1013
de Jong et al.[49]	0.1378	0.0750

The shear modulus, Poisson’s ratio, and molar volume ratio v = VM/VM are the parameters determining the stability of electrodeposition at an interface where both materials are isotropic through eq . The role of surface tension in stabilizing electrodeposition is well established.[20,43,68] Since the contribution of the surface tension to the stability parameter increases as k2 while that of stress increases linearly with k, the surface tension starts dominating the stability parameter as k is increased. This is elucidated in Figure through the contributions of the different terms to the total stability parameter for a material with G = 3.4 GPa and v = 0.1. The red line shows the fraction of contribution of surface tension to the overall stability parameter. All interfaces become stabilized as the value of k is increased beyond the critical surface roughness wavenumber. This motivates a distinction between two types of solid electrolytes—ones that are stabilized by the stress term alone and those that are stabilized by the surface tension beyond the critical value of k. For materials that are stabilized by stresses, the stability parameter remains negative for all values of surface roughness, and therefore, stability is guaranteed. However, for materials that have an overall destabilizing contribution due to stresses (hydrostatic + deviatoric), the stability parameter changes sign at an intermediate value of k since χ → −∞ as k → ∞. For such materials, the electrodeposition becomes stable at the critical surface wavenumber kcrit = 2π /λcrit (Figure ). If the surface roughness wavenumber so obtained is possible to achieve by nanostructuring the interface,[20] the electrodeposition might be stabilized.

Figure 2

Contribution of hydrostatic stress, deviatoric stress, and surface tension to the stability parameter as a function of surface roughness wavenumber. The surface tension term starts dominating at high k and ultimately stabilizes the interface after k = kcrit. The contributions are plotted for a material with shear modulus ratio G/GLi = 1 and Poisson’s ratio ν = 0.33 which is not stable (χ > 0) at k = 108 m–1. The red line shows the fraction of surface tension contribution to the stability parameter obtained by dividing the absolute value of its contribution by the sum of absolute values of all components.

We calculated the stability parameter for 12 950 Li-containing compounds out of which the properties of ∼3400 were in training data, and those of the remaining were predicted using CGCNN. In Figure S3, we visualized the latent space representations of randomly selected 500 training and 500 predicted crystals in a two-dimensional plot using t-distributed stochastic neighbor embedding (t-SNE) algorithm. It is clear that training crystals cover the search space of predicted crystals, indicating the reliability of the prediction. The lower part of Figure S3 includes compounds that do not contain Li atoms, which helps improve the prediction even though they are not directly representative of the search space. The ensemble averaged stability parameter χ and the critical surface roughness λcrit for all materials are shown as a histogram in Figure . We found that none of the materials have a probability of stability over 5% at surface roughness wavenumber k = 108 m–1.[39] The absence of any materials that can suppress dendrites without assistance from surface tension becomes clear from the isotropic stability diagram shown in Figure . All materials have G/GLi ratio higher than the critical value required to stabilize electrodeposition.[40] The highest number of materials are found in the region where G/GLi ∼ 15 and v ∼ 0.1. The critical wavelength of surface roughening was used as the criteria for screening materials since a higher surface roughness is easier to achieve by nanostructuring.

Figure 3

Results of isotropic screening for 12 950 Li-containing compounds. Distribution of ensemble averaged (a) stability parameter for isotropic Li–solid electrolyte interfaces at k = 108 m–1 and (b) critical wavelength of surface roughness required for stability. None of the materials in the database can be stabilized without the aid of surface tension. The required critical surface roughness wavenumber depends on the contribution of the stress term in the stability parameter.

Figure 4

Isotropic stability diagram showing the position of all solid electrolytes involved in the screening. GLi is the shear modulus of Li = 3.4 GPa. The critical G/GLi line separating the stable and unstable regions depends weakly on the Poisson’s ratio, so the lines corresponding to νs = 0.33 and 0.5 are good indicators for assessment of stability. The darker regions indicate more number of materials in the region.

The candidate materials with highest critical wavelength of roughening λcrit are shown in Table along with their stability parameter at different surface roughness wavenumbers and the corresponding probability of stability. While performing screening, we removed all materials which are electronically conducting, i.e., those which have a zero band gap according to the materials project database. However, we retained materials which were thermodynamically metastable (with energy above hull less than 0.1 eV) since many such solid electrolytes like Li10GeP2S12[34] and Li7P3S11[69] have been successfully synthesized. We find several candidate electrolytes with probability of stability Ps over 5% at surface roughness wavelength of 1 nm. It is worth noting that our screening identifies sulfide-, borohydride-, and iodide-based solid electrolytes, classes to which many of the current solid electrolytes belong. The uncertainty in stability parameter is much higher at high surface roughness wavenumber.

Table 2

Solid Electrolyte Screening Resultsa for Stable Electrodeposition with Li Metal Anode Together with Their materials project id (MP id) Ranked by Critical Wavelength of Surface Roughening λcrit Required to Stabilize Electrodeposition

			low k		high k
formula	space group	MP id	χ	Ps	χ	Ps	λcrit (nm)
Li2WS4	P4̅2m	mp-867695	0.62		–109.26		3.64
Li2WS4	I4̅2m	mp-753195	1.75		–38.54		1.34
LiBH4	P1̅	mp-675926	1.98		–40.13		1.32
LiAuI4	P21/c	mp-29520	2.7 ± 0.9	0	16.1 ± 55.2	0.43	1.02 ± 0.40
LiGaI4	P21/c	mp-567967	3.2 ± 1.1	0	48.6 ± 67.0	0.28	0.85 ± 0.29
LiWCl6	R3	mp-570512	3.2 ± 0.9	0	51.3 ± 56.6	0.17	0.82 ± 0.27
Cs3LiI4	P21/m	mp-569238	3.1 ± 0.7	0	46.9 ± 43.4	0.15	0.80 ± 0.17
LiInI4	P21/c	mp-541001	3.5 ± 1.0	0	68.5 ± 62.8	0.12	0.74 ± 0.20
Cs2Li3I5	C2	mp-608311	3.6 ± 0.9	0	77.2 ± 59.0	0.05	0.71 ± 0.17
Ba19Na29Li35	F4̅3m	mp-569025	4.2 ± 1.3	0	101.9 ± 81.3	0.08	0.68 ± 0.19
Ba38Na58Li26N	F4̅3m	mp-570185	4.2 ± 1.3	0	104.5 ± 82.3	0.08	0.67 ± 0.20
Li2UI6	P3̅1c	mp-570813	4.2 ± 1.4	0	111.5 ± 86.8	0.11	0.66 ± 0.29

χ is the stability parameter in kJ/(mol nm) which needs to be negative for stability, and k = 2π/λ is the surface roughness wavenumber. Low k corresponds to k = 108 m–1 while high k corresponds to a wavelength λ = 2π/k = 1 nm. Only materials with probability of stability Ps > 0.05 at high k are shown. Uncertainties in χ and λcrit (standard deviation of their distributions) and Ps are only shown for materials whose properties were predicted using CGCNN and not for those whose properties were available in training data.

Anisotropic Material Screening

The isotropic assumption provides a good starting point for determining the stability of electrodeposition in terms of the mechanical and chemical properties of the solid electrolyte. However, this assumption does not take into account the huge anisotropy of Li metal resulting in an anisotropy index of 8.52.[55] This may also arise when the surface of the solids in contact are single crystalline or dominated by a particular crystallographic orientation as has been seen for Li metal.[56] Therefore, a full anisotropic treatment of the mechanics taking into account the elastic tensor and orientation of Li metal at the interface is essential. Previously, we also observed a large variation in the stability parameter, with the [010] orientation of Li and solid electrolyte at the interface being the most compliant.[54] Here, we analyze the stability of electrodeposition at an anisotropic interface involving crystalline solid electrolyte in contact with crystalline Li metal.

For this analysis, we calculate the stability parameter using the full elastic tensor of the electrode and the electrolyte. A knowledge of surface orientations at the interface is necessary to determine the elastic tensor. Specifically, we considered ⟨1 0 0⟩, ⟨1 1 0⟩, ⟨1 1 1⟩, and ⟨2 1 1⟩ crystallographic directions of Li normal to the interface, in contact with seven low-index facets of the solid electrolyte for the screening procedure. The last three crystallographic directions have been observed through cryo-electron microscopy as the dominant directions for Li dendrite growth in carbonate-based electrolytes.[70] Additionally, X-ray diffraction pole figure analysis has revealed that Li gets electrodeposited along certain preferential crystallographic orientations.[56] The crystallographic orientation provides one more parameter for tuning the mechanical properties at the interface. In an earlier work, we used the Stroh formalism[71,72] to calculate the stresses and deformations at the interface which were used to obtain the stability parameter χ.[54]

We developed a model different from CGCNN for predicting the full elastic tensor of each material since the size of training data is much smaller for the full elastic tensor. Considering the success of atomistic descriptors such as Li neighborhood, anionic framework,[57] bond ionicity, etc. in predicting Li-ion conduction,[52] we developed a regression model for predicting the elastic tensor of all Li-containing compounds using several of these relevant descriptors. A complete list of descriptors can be found in the Supporting Information. All descriptors used in the model are geometric or topological and can be obtained from the structure of the material in CIF format.[73] We further employed the crystal symmetries to reduce the number of parameters in the elastic tensor. This procedure reduces the number of constants in the elastic tensor to train from 21 to 3 in the cubic crystal class, for example, and, by construction, ensures the predicted elastic tensor obeys its crystal symmetry.[74] We trained separate models for each elastic constant, which were used to build the full elastic tensor of the material. We believe the differences in the uncertainty of the different elastic constants[75] result in different amounts of noise in the training data, thereby, justifying the need for separate models.

The training data for the elastic tensor was obtained from the materials project database[64,65] retrieved using pymatgen,[61] with the same level of theory as the isotropic case. We performed a data cleaning step in which all materials which were mechanically unstable (116 in number), i.e., with a nonpositive definite elastic tensor, were removed from training data. Since sufficient data was available only for the cubic crystal class (see Table S2 of the Supporting Information), we developed a regression model for predicting the elastic tensor of only cubic Li-containing compounds. However, our framework is very general and can accommodate more data as it is generated in training set as well as add models for other symmetries once the size of training data is sufficient in the materials project database. For other crystal symmetries besides cubic, we screened over materials with elastic tensor available from first-principles. Our screening process consisting of DFT-calculated data for all symmetries and predicted data for cubic symmetries that covers >15 000 Li metal–solid electrolyte interfaces.

The regression model was chosen on the basis of cross-validation performance (coefficient of determination) on the values of the elastic constants (in GPa). The models tested for performance were Lasso regression, elastic net, kernel ridge regression, Gaussian process regression, gradient boosting regression[76,77] used in an earlier work on predicting mechanical properties of zeolite frameworks,[53] AdaBoost regression,[78,79] support vector regression,[80] random forest regression, Bayesian ridge regression,[81] least angle regression, and automatic relevance determination. All hyperparameters were optimized using grid search with nested 3-fold cross-validation. The optimized hyperparameter values can be found in the Supporting Information. The scikit-learn package,[82] which has implementations of all models listed above, was used for training, hyperparameter optimization, and prediction.

Figure b–d shows the comparison of predicted and calculated values of elastic constants C11, C12, and C44. We obtain an overall R2 of 0.93, 0.92, and 0.92 and a cross-validation R2 of 0.60, 0.79, and 0.6 for the elastic constants C11, C12, and C44, respectively. A general idea of the effect of different atomistic features on the elastic constants can be obtained through the coefficients of a linear model fitted to the elastic constants. The dominant atomistic features affecting the elastic tensor based on the coefficients were mass density, volume per atom, sublattice electronegativity, and ratio of bond ionicities. The material stiffness increases with mass density and decreases with volume per atom. The stiffness also increases with ratio of bond ionicity of Li and sublattice and decreases with sublattice electronegativity.

While evaluating the models, the inherent uncertainty in DFT-calculated values due to exchange-correlation functional should be taken into account.[75,83] The uncertainty in DFT-calculated values generates noise in the training data. For example, DFT-calculated C12 generally has high uncertainty[75] which degrades the performance of boosting techniques. We believe the lower uncertainty in the training data for C11 and C44 makes boosting algorithms perform better. These algorithms are sensitive to the presence of outliers since a higher weight will be assigned to them. Further investigation is needed to quantify the effect of DFT uncertainty in training data on the machine-learning model to be used. Overall, our models have satisfactory performance to be used for the next step of prediction and screening. A two-dimensional t-SNE plot of all training and predicted feature data (Figure S6) also shows that out training data covers the feature space of the predicted crystals.

Our anisotropic screening procedure involves the determination of electrodeposition stability for 482 electrolyte materials with DFT-computed elastic tensor from the materials project database and 548 materials with cubic crystal structure whose elastic tensor was predicted using the regression model developed earlier. The total number of Li–solid electrolyte interfaces involved in the screening is over 15 000 (see also Table S2 in the Supporting Information). The best candidate interfaces obtained from the screening procedure are shown in Table . It is worth pointing out that all the screened candidates are stabilized by the stress term in the stability parameter and do not depend on surface tension for stability. We observe two major features in the materials obtained through screening: either the material is mechanically soft, i.e., with small eigenvalues of the elastic tensor, or the material has highly anisotropic mechanical properties. The mechanical softness of these materials is particularly attractive since it means the mechanical properties for dendrite suppression and ionic conductivity can be optimized simultaneously through methods like increasing volume per atom.[57] The high anisotropy results in certain weak directions along which the electrolyte may comply with the Li metal anode. This degree of freedom in crystallographic orientation is probably the reason why there are several screened candidates in the anisotropic case compared to the isotropic case. All screened candidates in Table have a universal anisotropy index greater than 10, which is zero for an isotropic material.[84] The highest anisotropy index of 113.29 and 31.30 is found for LiOH and Li2WS4. The mechanically soft nature of these materials is reminiscent of density-driven stability[40] in the isotropic case for materials with v < 1.

Table 3

Screening Results for Anisotropic Interfaces (Top 20) for Stable Electrodeposition with Li Metal Anode with Their materials project id, Interface Normal, and Stability Parameter (Last Column Shows the Universal Anisotropy Index AU Which Is Zero for a Completely Isotropic Material)[84]

			interface normal
formula	space group	MP id	Li	electrolyte	χ at k = 108 m–1 (kJ/(mol nm))	AU
Li2WS4	P4̅2m	mp-867695	[1 1 1]	[0 0 1]	–1.92	31.30
Li2WS4	P4̅2m	mp-867695	[2 1 1]	[0 0 1]	–1.87	31.30
Li2WS4	P4̅2m	mp-867695	[0 1 0]	[0 0 1]	–1.68	31.30
Li2WS4	I4̅2m	mp-753195	[0 1 0]	[0 0 1]	–1.23	12.84
LiBH4	P1̅	mp-675926	[0 1 0]	[0 1 0]	–1.12	13.65
Li2WS4	I4̅2m	mp-753195	[1 1 1]	[1 0 1]	–1.00	12.84
LiOH	P4/nmm	mp-23856	[0 1 0]	[0 0 1]	–1.00	113.29
Li2WS4	I4̅2m	mp-753195	[1 1 1]	[0 0 1]	–1.00	12.84
LiOH	P4/nmm	mp-23856	[2 1 1]	[0 0 1]	–0.99	113.29
LiOH	P4/nmm	mp-23856	[1 1 1]	[0 0 1]	–0.98	113.29
Li2WS4	I4̅2m	mp-753195	[2 1 1]	[0 0 1]	–0.89	12.84
Li2WS4	I4̅2m	mp-753195	[0 1 0]	[1 0 1]	–0.79	12.84
LiBH4	P1̅	mp-675926	[1 1 1]	[1 1 0]	–0.77	13.65
LiBH4	P1̅	mp-675926	[1 1 1]	[0 1 0]	–0.75	13.65
Li2WS4	I4̅2m	mp-753195	[0 1 0]	[0 1 1]	–0.49	13.84
LiBH4	P1̅	mp-675926	[0 1 0]	[1 1 0]	–0.47	13.65
Li2WS4	P4̅2m	mp-867695	[1 1 0]	[0 0 1]	–0.40	31.30
Li2WS4	P4̅2m	mp-867695	[1 1 1]	[1 0 1]	–0.28	31.30
Li2WS4	P4̅2m	mp-867695	[0 1 0]	[1 0 1]	–0.17	31.30
Li2WS4	I4̅2m	mp-753195	[2 1 1]	[1 0 1]	–0.07	13.84

Other Considerations for Screened Electrolytes

Although the literature on the candidates we obtained through screening is rather limited, we do find some similarities with solid electrolyte materials currently being explored. For example, Li borohydrides have been recently explored as fast ion conductors for solid state batteries.[85,86] Their anion substituted analogues also exhibit high conductivities and provide opportunities for tuning other desirable properties such as electrochemical and thermodynamic stability.[86−88] Li2WS4 phases are sulfides with structures similar to the tetragonal phase of LiFePO4. One of the candidates, LiOH, is often found as an electrochemically stable product in the solid electrolyte interphase formed at the anode.[89,90] The candidates we obtained through screening will only be useful if they satisfy several other requirements imposed by solid electrolytes. Here, we focus on the most important ones: ionic conductivity, electronic conductivity, and thermodynamic stability. We develop quantitative metrics for each of them and show them for the screened candidates in Table .

Table 4

Quantitative Metricsa for Other Important Requirements of Screened Solid Electrolytes for Their Use in Li-Ion Batteries: Ionic Conductivity, Electronic Conductivity, and Thermodynamic Stability

formula	space group	MP id	Pion	band gap (eV)	energy above hull per atom (eV)
LiOH	P4/nmm	mp-23856	0.05	6.34	0.000
LiAuI4	P21/c	mp-29520	0.94	1.92	0.000
LiGaI4	P21/c	mp-567967	0.18	4.33	0.000
LiBH4	P1̅	mp-675926	0.27	8.57	0.071
Li2WS4	I4̅2m	mp-753195	0.15	3.61	0.032
Li2WS4	P4̅2m	mp-867695	0.23	3.52	0.037
Cs3LiI4	P21/m	mp-569238	0.01	6.07	0.018
LiInI4	P21/c	mp-541001	0.13	3.96	0.000
Cs2Li3I5	C2	mp-608311	0.33	6.58	0.000
Ba19Na29Li35	F4̅3m	mp-569025	0.00	0.94	0.000
Ba38Na58Li26N	F4̅3m	mp-570185	1.00	0.96	0.009
Li2UI6	P3̅1c	mp-570813	0.26	1.17	0.000

Ionic conductivity is quantified through Pion, the probability of superionic conduction; electronic conductivity through the band gap; and thermodynamic stability through energy per atom above the convex hull.

Ionic Conductivity

We obtained the probability of superionic conduction by using the logistic regression framework proposed by Sendek et al.[52] which relates the atomistic features to conductivity. The probability can be obtained using the features 6, 7, 9, 12, and 14 from the feature list in the Supporting Information. These probabilities are based on atomic structure of the bulk. Of the screened candidates, LiAuI4 and Ba38Na58Li26N are predicted to be superionic conductors. Further enhancement in conductivities may be achieved through doping, defects, and disorder. The high-temperature hexagonal phase of LiBH4 has a Li-ion conductivity as high as 10 mS/cm.[91] The screened triclinic P1̅ phase of LiBH4 is disordered and is expected to have higher Li ionic conductivity than that obtained for bulk atomic structure. It will be worth investigating the possibility of tuning its ionic conduction and mechanical properties through anion substitution similar to its hexagonal analogue which undergoes order of magnitude improvements in ionic conductivity.[87,88] LiOH is known to undergo a solid state transition at ∼415 °C accompanied by a large increase in ionic conductivity.[92] We also expect the high anisotropy in mechanical properties of some electrolytes to affect the activation energy landscape for Li-ion conduction as well, possibly leading to anisotropic conductivities.

Thermodynamic Stability

While screening, we only removed solids with energy greater than 0.1 eV/atom above hull. Besides LiBH4 and Li2WS4, all screened candidates are thermodynamically stable; i.e., energy above the convex hull is 0 eV, or the energy above hull per atom is within thermal fluctuations at room temperature. The two phases of Li2WS4 have energies 31 and 36 meV per atom above hull. The hexagonal phase of LiBH4 has been explored previously as a solid electrolyte with promising performance.[91,93,94] This phase is thermodynamically unstable at room temperature. However, doping with Li halides improved the stability of the hexagonal phase even below room temperature.[93] It is possible that a similar technique might help stabilize the screened triclinic P1̅ phase obtained through our screening as well.

Electronic Conductivity

All the screened solid electrolytes in the list are electronic insulators since we removed those with a zero GGA band gap. Since GGA DFT generally underestimates band gaps, we used an empirical scheme to correct for the band gap[95] (see the Supporting Information). All candidates apart from Ba19Na29Li35 and Ba38Na58Li26N have corrected band gaps greater than 1 eV.

We found several candidates that could enable dendrite suppression through the anisotropic criteria. It is worth highlighting that grain misorientation needs to be avoided for these materials. Textured growth of Li film on Li metal anode has been observed for liquid electrolytes.[56] A similar opportunity for textured growth of Li metal is present with solid electrolytes; however, it would require precise engineering of film growth at the interphase which requires further investigation.

While the candidate list identified is small, we strongly believe that there is a lot of room for further search for candidates. In particular, we find that anisotropy plays a crucial role in determining the stability, and thus, there may be additional noncubic Li-containing compounds that could suppress dendrites at certain crystallographic orientations. Further, the degree of disorder is another important factor, and many glassy or amorphous materials are known to be good solid electrolytes. These avenues will be the subject of our future investigations.

Conclusions

We applied criteria for stable electrodeposition together with machine-learning techniques to computationally screen solid electrolytes for suppressing Li dendrite growth. The machine-learning techniques accelerate the process of screening by predicting the properties of solid electrolytes through the identification of structure–property relationships. We train a graph convolutional neural network on shear and bulk moduli. We employ gradient boosting regression and kernel ridge regression for training the elastic constants of cubic materials. The choice of machine-learning models used was rationalized by the model’s ability to handle noise in the training data. Our approach is readily applicable for screening materials for other properties of interest and can easily accommodate more data as it is generated.

Our predictive models enabled us to screen 12 950 solids using isotropic stability criteria and over 15 000 interfaces using anisotropic stability criteria of electrodeposition on the Li metal anode. In the isotropic case, we found no materials that could be stabilized solely by the stresses generated at the interface; however, a surface tension-mediated stabilization was found to be possible at high surface roughness wavenumbers. In the anisotropic case, the additional degree of freedom related to crystallographic orientation of the solid electrolyte at the interface enabled us to find over 20 interfaces with six solid electrolytes that are predicted to be stable to dendrite initiation. We identify some common features like anisotropy and mechanical softness present in the screened candidates based on which one can simultaneously optimize properties required for dendrite suppression as well as fast ion conduction. We believe that the use of techniques like doping and defect generation will be crucial to ensure simultaneous satisfaction of other solid electrolyte requirements for screened candidates.