Introduction

Thermoelectric generators (TEGs)1,2,3 are capable of directly converting heat into electricity. With low-maintenance and silent operation, TEGs are highly anticipated to reuse waste heat and power sensors and transmitters necessary to gather countless data in Internet of Things (IoT) for future society. The thermoelectric conversion efficiency is mainly determined by the figure of merit zT = S2Tσ/κ, where S, T, σ, and κ are Seebeck coefficient, absolute temperature, electrical and thermal conductivity, respectively. While there are materials such as Bi2Te34, PbTe1, and SnTe5 that exhibit high zTs, the commercialization remains challenging owing to limited abundance and toxicity of these materials. On the other hand, highly abundant and mass-producible silicon6,7,8,9,10 still has a low zT at around room temperature (where most IoT sensors operate) due to its high thermal conductivity10. Improvement in thermoelectric performance of silicon-based materials would help larger spread of the TEG technology.

Nanocrystallization, a process of populating materials with grain boundaries (GBs), has been employed as one of the approaches to improve the zT10,11,12,13,14,15,16,17,18,19,20. The promising effect of GBs on thermoelectrics is generally attributed to phonons, primary heat carriers, being selectively scattered10,11,14,15,16,17,19,20, carrier mobility enhancement13,18, and energy selection of electrons12,17. This shows that GBs can significantly influence the thermoelectric performance of materials, and many experiments have also indicated that the thermoelectric characteristics of a single GB greatly depend on its atomistic structure21,22,23,24. Therefore, knowing the quantitative relationship between GB structure and its thermoelectric properties becomes advantageous when designing thermoelectric nanomaterials.

To investigate the relationship between GB structure and its physical properties, many computational efforts have been made25,26,27,28,29,30,31. For instance, excess volume of MgO at GB has been shown to correlate well with thermal conductivity obtained from molecular dynamics (MD) calculations31. However, the excess volume alone could not fully account for the structure–property relationships in general due to the high-dimensional nature of GB-related problems31. There, structural descriptors containing sufficient information to characterize different atomic structures have recently been shown to be effective in handling the intimidating dimensionality of GBs32,33,34,35,36. Several recent studies have employed these structural descriptors along with machine learning (ML) to gain more insights into the GB structure–property relationship37,38,39,40,41,42,43,44,45. For example, Fujii et al.44 discovered promising descriptors based on suitable smooth overlap of atomic positions (SOAP)46 that were highly correlated with MD-calculated thermal conductivity of MgO GB. They then constructed an accurate prediction model of thermal conductivity using multiple linear regression with predictors based on hierarchical clustering of these descriptors and identified the structure–property relationship from the regression model44.

In this work, we employ the ML-descriptor approach to study the relationship between silicon GB and its thermoelectric properties, obtained by high-throughput calculations based on Green’s function methods47,48. We present a robust ML prediction model of thermal boundary conductance (TBC) covering both crystalline–crystalline and crystalline–amorphous interfaces from input disorder descriptors and atomic density. We also construct high-accuracy ML models for predicting TBC, electrical boundary conductance (EBC), and their ratio (EBC/TBC) of crystalline GBs. Our TBC prediction model exhibits a significantly higher coefficient of determination than earlier studies40,44. In addition, the models reveal insights into the guiding principles to improve the thermoelectric performance that the angular and distance variation at GB are the most predictive descriptors for TBC and EBC, respectively. This suggests that populating materials with GBs that have large angular variation and small distance variation can improve the figure of merit. Moreover, the EBC/TBC prediction model also indicates that priority should be given to angular over distance variation when attempting to decouple TBC from EBC in order to increase the figure of merit.

Results and discussion

Crystalline and amorphous GBs

We obtained 1228 silicon GB structures at various annealing temperatures and pressures. The resulting GB structures depended on the final temperature and pressure of the annealing process. In general, for low final temperatures, the disordered structure in the device region crystallized from both ends and formed a sharp crystalline–crystalline interface as shown in Fig. 1(a, c, d), and for high final temperatures, the structure in the device region remains being disordered resulting in amorphous slab as shown in Fig. 1(b, e, f).

Fig. 1: Examples of grain boundaries used in this work.
figure 1

a, c, and d contain crystalline–crystalline boundaries, while b, e, and f contain two crystalline–amorphous boundaries.

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In the case of amorphous slab, there are two crystalline–amorphous interfaces on the left and right side of the slab, therefore the total resistance (Rtotal) consists of the internal resistance of the slab (Rl) and the thermal boundary resistance at the interfaces (R0). To understand their relative contribution, we performed conductance calculation with varying lengths (l) of the amorphous device region as shown in Fig. 2. The result shows that Rtotal grows linearly with l, confirming that the thermal conductivity can be approximated to be constant in the range of variation in l. This then allows us to extract 2R0 by extrapolating the linear profile to l = 0. The analysis finds that, for the range of l, 2R0 is significantly larger than Rl and thus the primary contributor to Rtotal. Therefore, half of the transmission through the amorphous slab case can be seen to represent transmission through a single crystalline–amorphous GB.

Fig. 2: Length dependence of total thermal resistance (Rtotal) of a disorderly structure.
figure 2

Fourier's law gives: Rtotal = 2R0 + Rl = 2R0 + r × l, where R0 is the crystalline–amorphous resistance, and Rl and r is the resistance and resistivity of the amorphous (device) region, respectively.

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Consequently, we now have two types of GBs in this work, crystalline–crystalline GBs and crystalline–amorphous GBs. It becomes necessary, therefore, to classify our structures into crystalline–crystalline and crystalline–amorphous structures based on structural disorder. We employed the k-means algorithm49 using the four disorder descriptors, i.e., σθ, Hϕ, σl, and ARDF to cluster our structures into two groups. As a result, we obtained 426 structures with crystalline–crystalline GBs and 802 structures with crystalline–amorphous GBs.

Combined predictions of TBC

Ten different supervised ML models implemented in the scikit-learn Python library50 were examined: linear regression, k-nearest neighbors (KNN), RandomForest51, ExtraTrees52, GradientBoosting53, CatBoost54, AdaBoost55, XGBoost56, HistGB57, and LightGBM58. The first two are classical models, while the remainings are tree-based models. Five structural descriptors, namely, the standard deviation of bond angles θ and bond lengths l (σθ and σl), the entropy of dihedral angles ϕ (Hϕ), the RDF area (ARDF), and the atomic density (ρ) were used as input predictors, and the calculated conductances of single GBs were used as prediction targets for ML models.

All data were divided into 85% training set and 15% test set. Six-fold cross-validation (CV) on the training set was employed to fine-tune and compare the ML predictive performance. Hyperparameters of each ML model were fine-tuned based on CV results using Tree-structured Parzen Estimator (TPE) algorithm59,60,61 implemented in the Optuna Python library62. Root mean squared error (RMSE) was selected to evaluate the ML models since large errors are undesirable. Moreover, mean absolute percentage error (MAPE) was also calculated to assist the interpretation of results. The model with the lowest RMSE in CV was then refitted to the entire training set and finally evaluated against the test set to assess a generalization capability.

It can be seen from the CV results shown in Table 1 that linear regression had the worst performance of all the ten ML models. This indicates that the relationship between the structural descriptors in this work and thermal boundary conductance (TBC) cannot be explained by a simple linear relationship. Notably, LightGBM also performed poorly, with average error more than 20%. The other ML models performed fairly well, with average errors between 10% and 20%, indicating the robustness of tree-based models in general. On the other hand, KNN, despite being a traditional ML model, showed the second best performance, only inferior to ExtraTrees model.

Table 1 Six-fold cross-validation results for thermal boundary conductance prediction of all training structures, ordered by root mean squared error (RMSE).
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Moreover, by combining the individual predictions from the five best models (ExtraTrees, KNN, GradientBoosting, CatBoost and AdaBoost), we were able to realize a powerful ensemble model with better predictions. This is attributable to the fact that combining a set of equally well performing models could balance out the individual weaknesses of each model63.

Finally, the ensemble model was refitted to the entire training set and evaluated against the test set. As shown in Fig. 3 and Table 2, the ensemble model was able to predict TBC of both GB types with exceptional accuracy on both training and test set. Moreover, the close errors on the training and test set indicate that the ensemble model did not overfit, which commonly occurs when a ML model captures unnecessary noise and detail in the training set. The prediction results on the test set also show robustness of the ensemble model when predicting data it has never encountered before.

Fig. 3: Parity plot between calculated and predicted thermal boundary conductance (TBC) from the ensemble model.
figure 3

Parity plots of a training set and b test set. For the test set, the model performed better on crystalline–crystalline GBs than on crystalline–amorphous GBs.

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Table 2 Summary of thermal boundary conductance (TBC) prediction results on training and test set.
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Furthermore, it becomes apparent that the model performed slightly better on CV than on the test set. This is expected since we fine-tuned the hyperparameters of the models based on CV results, thus the ensemble model was adjusted to give an optimal performance on CV. Consequently, the test set in fact evaluates the generalization ability of the ensemble model. Moreover, as the performance on the test set was comparable to that in cross-validation, which is the average result from different splits, it can be said that it is not by chance that the randomly selected test set in this work yielded remarkable prediction results.

In addition, it can be observed that, while the ensemble model approximately had an equivalent performance on both GB types for the training set, it performed better on crystalline–crystalline GBs than on crystalline–amorphous GBs for the test set. As shown in Table 2, the ensemble model had smaller RMSE on crystalline–crystalline GBs than crystalline–amorphous GBs despite the former type having higher overall TBCs for both training and test set.

The somewhat superior performance of the ensemble model on crystalline–crystalline GBs may be attributed to the fact that the calculated inverse of Rtotal in the crystalline–amorphous case is not exactly the TBC at the interface with minor contribution from the thin amorphous slab (Fig. 2). Another possible explanation is the origin of the disorder features (σθ, Hϕ, σl, and ARDF). While the disorder features were obtained from the entire structure in the simulation domain, for crystalline–crystalline GBs, they reflect the disorder of the atoms near the GB, i.e., the local interface disorder since there is virtually no disorder in the internal crystal region away from the interface. On the other hand, in the case of crystalline–amorphous GB, the disorder features reflect that of the amorphous slab in addition to the interface. Therefore, the act of approximating the resistance and features at the crystalline–amorphous interface could potentially have confused the prediction model, hence the inferior performance. Nevertheless, the interesting point here is that despite the approximation, the prediction of conductance was quite successful even for crystalline–amorphous interfaces.

Figure 5 shows the feature importances of the ensemble model, which are the average relative importances of features over the five best models (except KNN) fitted on the entire training set. In general, tree-based models attempt to minimize loss, which is the variance of a target variable within each leaf node in case of regression, as illustrated in Fig. 4. Feature importance of tree-based models is the total reduction in loss resulting from splitting each node within each tree by that feature, averaged over all trees. The values of feature importances are normalized so that the total importance of all features is unity, i.e., the relative importances are computed. In essence, the importance of a feature represents the contribution of that feature to accurately predicting the target variable, which is the calculated conductance in this work.

Fig. 4: Demonstration of computing loss reduction resulting from one split in a decision tree.
figure 4

This reduction value for a given feature is summed all over a single tree, then gets averaged over all trees. y, \(\hat{y}\), and x1 denote the target variable, predicted value, and selected feature for splitting, respectively.

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It is obvious from Fig. 5 that all of the disorder features (descriptors), i.e., σθ, Hϕ, σl, and ARDF are important to predict TBC. Particularly, the angles-related features (σθ and Hϕ) were the strongest predictors of TBC, i.e., they yielded the highest reduction in TBC variance in a tree node. These two features accounted for more than 50% of the total importance for TBC prediction. The distance-related features (σl and ARDF) are also crucial for the prediction, with 36% total importance. While atomic density (ρ) was the weakest predictor, it is still a relevant feature since excluding it from the input features greatly exacerbated the predictive performance of the ensemble model. The fact that this feature is still important is attributed to its accounting for varying pressure during the annealing process of GB structure.

Fig. 5: Feature importances for TBC prediction.
figure 5

Angles-related features are the most predictive. Atomic density is the weakest predictor but still relevant.

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As above, the ML-descriptor approach employed in this work is capable of capturing the complex relationship between the atomistic details of GB structure and TBC. The obtained insight cannot be discovered by the traditional methods such as acoustic mismatch model (AMM)64 and diffusive mismatch model (DMM)65 as these models do not account for the atomic structure at GB.

Thermoelectric properties predictions of crystalline–crystalline GBs

Regarding electrical boundary conductance (EBC), we discovered that a large number of GBs had zero or extremely small values of maximum EBCs within a realistic range of chemical potential (−1 eV ≤ μ ≤ 1 eV). Excluding these GBs left us with 75 crystalline–crystalline and 211 crystalline–amorphous GBs that are electronically realistic.

We repeated the ML-training process on all 286 electronically realistic GBs and the prediction results were mediocre (30.9% and 37.7% average error for TBC and EBC cross-validation, respectively). These inferior results indicate that a large amount of data is necessary in order to predict the conductances of both GB types simultaneously. However, using only 75 crystalline–crystalline GBs (with 20% randomly selected test data) yielded extremely accurate prediction results, as shown in Fig. 6 and Table 3. Six-fold cross-validation (CV) on the training set was also employed to fine-tune ten ML models based on RMSE. An ensemble model, however, was constructed by averaging the individual predictions from the three best models of each property instead of five since this delivered better results.

Fig. 6: Prediction results of 75 crystalline–crystalline grain boundaries that are electroincally realistic.
figure 6

Panels a, c, and e demonstrate parity plots between calculated and ensemble-predicted values with error bars indicating standard deviations of the ensemble predictions, while b, d, and f show feature importance of thermal boundary conductance (TBC), electrical boundary conductance (EBC), and EBC/TBC respectively.

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Table 3 Summary of prediction results for thermal and electrical boundary conductance (TBC and EBC) of 75 electronically realistic crystalline–crystalline grain boundaries.
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In addition to TBC and EBC, we also trained the models to directly predict EBC/TBC, which is proportional to the thermoelectric figure of merit (zT). By directly predicting EBC/TBC, we can identify the important descriptors that need to be prioritized when attempting to decouple one conductance from the other in order to increase zT of silicon materials. The Seebeck coefficient was excluded in this work since it is a bulk property in general, in contrast to the boundary conductance which is always an interfacial property (see Supplementary Discussion).

It can be observed from Table 3 that TBC was predicted with merely around 2% average error. Moreover, the training and test results were extremely close, indicating that the ensemble model barely overfitted the training data, thus should have correctly captured the underlying structure–property relationship. In addition, our model also has a significantly higher coefficient of determination (R2 = 0.99) on the test set than previous works predicting the thermal transport property of grain boundary40,44 despite using less data.

As is the case with combined predictions of both GB types, the angles-related structural descriptors, particularly the entropy of dihedral angles (Hϕ), were the strongest predictors for TBCs of crystalline–crystalline GBs, as illustrated in Fig. 6b. This result further reinforces the previous insight that the angles variation at GB is highly predictive of TBC.

For EBC, although the errors were higher than those of TBC, the ensemble model still enabled good predictions with approximately 9% average error on the test set. In contrast to TBC, distance-related features, i.e., σl and ARDF were the most predictive for EBC, as shown in Fig. 6d. This shows potential decoupling of EBC from TBC by decoupling distance-related features from angles-related features. Notably, atomic density was also a strong predictor of EBC.

Finally, predicting EBC/TBC directly was the most difficult task, with ~10% average error on the test set. This result is somewhat intuitive since the ML models should be able to predict each individual property more easily. However, by directly predicting EBC/TBC, we were able to discover the most important features to prioritize when attempting to decouple TBC from EBC, which are the angles-related features, as shown in Fig. 6f.

Furthermore, more data than 75 crystalline–crystalline GBs did not significantly improve the ML predictive performance (see Supplementary Fig. 1). This indicates that, given predictive input descriptors and high-quality data, even a small amount of data can be sufficient for a ML model to capture the structure–property relationship and thus accurately predict the property. On the other hand, a large amount of data was necessary for predicting the conductances of crystalline–amorphous GBs. This suggests that while the quantity of data is important for a large search space, the quality of data is more important for a small search space.

Moreover, since both TBC and EBC had negative correlations with the degree of structural disorder represented by four descriptors in this work, one possible solution for thermoelectrics is to increase the population of GBs with large angular variation and small distance variation.

In summary, we employed the ML-descriptor approach identify the relationship between silicon GB structure and its thermoelectric properties. Using structural descriptors representing angular and radial variation as well as atomic density, together with high-throughput conductance calculations based on Green’s function method, we were able to build a robust ML prediction model of thermal conductance covering both crystalline and amorphous GBs. Moreover, we presented high-accuracy ML models for predicting EBC and TBC and their ratio which is directly proportionate to thermoelectric figure of merit, using only small data of 75 crystalline GBs. We also discovered that the variations of interatomic angles and distance at GB are the most predictive descriptors of TBC and EBC respectively, and that angular variation is more important to EBC/TBC prediction than distance variation. This suggests that constructing GBs with large angular and small distance variation in materials can improve thermoelectric performance of silicon. The insights from this work also open the way to reveal the underlying mechanisms governing GB physical properties. In addition, the extreme robustness of the ML model predicting a large range of interfaces expands the applicability of ML methodology to explore wider search space of material structures and phenomena.

Methods

Grain boundary (GB) structure construction

Firstly, fifteen Si GBs were generated by tilting or twisting two crystals about their [001], [110], [310], [212] and [100] axes for various misorientation angles, implemented in a Python library called Aimsgb66. These structures were relaxed by an annealing method using molecular dynamics (MD) simulation performed in Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) software67. The two tilted or twisted Si crystals were placed as the contact regions in Fig. 7 and a melt slab of Si, obtained by heating a single-crystal Si up to 7000 K, was inserted into the device region in between to form an initial structure for further annealing.

Fig. 7: The model set up in Green’s function calculations.
figure 7

The contact regions are the unit layers fixed during the annealing process.

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Annealing of the initial structure started at 4000 K with a stepwise temperature decrease to 25 K over 1275 ps. The pressure during the process was set to 1 bar. The GB structure was finally optimized with a conjugate gradient algorithm. The optimized Stillinger-Weber interatomic potential68 was used and one unit layer on each side was fixed throughout the whole process. We repeated the process 10 times for each GB using different initial velocities and selected the most energetically stable structure.

To obtain more structures for training ML models, we varied the values of the final temperature and pressure of the annealing process. Particularly, in addition to the standard condition of 25 K and 1 bar, the annealing calculations with the final temperature of 3000, 2000, and 1000 K, and the constant pressure from 0.1 to 0.6 Mbar with an increment of 0.1 Mbar were performed69. All structures were also energetically optimized after the annealing. Furthermore, we took four additional structures with different initial velocity distributions for each amorphous condition (2000 and 3000 K), giving us the total of 84 structures for each of 15 base GBs obtained initially. After excluding 32 unstable structures, we overall obtained 1228 stable or metastable GB structures.

Green’s function methods

Green’s function methods47,48 were selected as a means to calculate boundary conductance since they account for the quantum distribution. Moreover, these deterministic methods are also significantly faster than other common alternatives such as non-equilibrium classical molecular dynamics (NEMD) which requires the ensemble averaging of statistical computation. Thus, Green’s function methods are exceptional for the ML tasks in this work where a large amount of data is advantageous.

The transmission function of phonons Ξ(ω) across a GB plane can be calculated as follows48:

$${\Xi }_{{{{\rm{ph}}}}}(\omega )\,=\,{{{\rm{Tr}}}}[{\Gamma }_{{{{\rm{L}}}}}(\omega ){G}_{{{{\rm{ph}}}}}(\omega ){\Gamma }_{{{{\rm{R}}}}}(\omega ){G}_{{{{\rm{ph}}}}}^{{\dagger} }(\omega )]$$
(1)

where ω is the phonon frequency, Gph is the retarded Green’s function of the device region, \({\Gamma }_{{{{\rm{L}}}}}\,=\,{{{\rm{i}}}}({{{\Sigma }}}_{{{{\rm{L}}}}}\,-\,{{{\Sigma }}}_{{{{\rm{L}}}}}^{{\dagger} })\) and \({\Gamma }_{{{{\rm{R}}}}}\,=\,{{{\rm{i}}}}({{{\Sigma }}}_{{{{\rm{R}}}}}\,-\,{{{\Sigma }}}_{{{{\rm{R}}}}}^{{\dagger} })\) represent the flow rate from the left contact into the device and from the device into the right contact, and ΣL and ΣR are the self-energy matrices of the left and right contact. Using the Landauer formula, we can obtain the following thermal boundary conductance (TBC)70:

$${{{\rm{TBC}}}}\,=\,\frac{1}{2\pi A}\int\nolimits_{0}^{\infty }{\Xi }_{{{{\rm{ph}}}}}(\omega )\hslash \omega \left(\frac{\partial {f}_{{{{\rm{BE}}}}}}{\partial T}\right)\,d\omega$$
(2)

where ℏ is the reduced Planck constant, fBE is the Bose-Einstein distribution, T is the temperature, and A is the GB cross-sectional area.

The transmission function of el ectrons Ξ(E) at energy E can be calculated similarly47:

$${\Xi }_{{{{\rm{el}}}}}(E)\,=\,{{{\rm{Tr}}}}[{\Gamma }_{{{{\rm{L}}}}}(E){G}_{{{{\rm{el}}}}}(E){\Gamma }_{{{{\rm{R}}}}}(E){G}_{{{{\rm{el}}}}}^{{\dagger} }(E)]$$
(3)

Electrical boundary conductance (EBC) can also be derived in the same manner as TBC:

$${{{\rm{EBC}}}}\,=\,\frac{2{{{{\rm{e}}}}}^{2}}{hA}\int\nolimits_{-\infty }^{\infty }{\Xi }_{{{{\rm{el}}}}}(E)\left(-\frac{\partial {f}_{{{{\rm{FD}}}}}}{\partial E}\right)\,dE$$
(4)

where h is the Planck constant and fFD is the Fermi-Dirac distribution. Since the value of EBC varies with the chemical potential (μ), we calculated the maximum EBC over a realistic range of chemical potential, −1 eV ≤ μ ≤ 1 eV. All conductance calculations in this work were implemented in an Atomistix ToolKit package71 with the optimized Stillinger-Weber interatomic potential68 and the pbc-0-3 Slater Koster file72 for phonon and electron calculation, respectively. The temperature was set to 300 K in all cases.

Structural descriptors

To represent GB quantitatively, we computed five global structural descriptors. Here, the descriptors were calculated for the entire region of a structure. The first two are the standard deviation of bond angles θ and bond lengths l, i.e., σθ and σl. Next is the entropy of dihedral angles ϕ, which is calculated as follows:

$${H}_{\phi }\,=\,\mathop{\sum}\limits_{\phi }{p}_{\phi }\log \left(\frac{1}{{p}_{\phi }}\right)$$
(5)

We also computed a descriptor based on radial distribution function (RDF) g(r), which is a measure of the probability of finding an atom at a distance r away from a given reference atom. Motivated by the fact that the value of g(r) converges to unity at smaller r as a structure becomes more disordered73, we calculated the area between g(r) and g(r) = 1 as a descriptor:

$${A}_{{{{\rm{RDF}}}}}\,=\,\int\nolimits_{0}^{{r}_{{{{\rm{cutoff}}}}}}| g(r)\,-\,1| \,dr$$
(6)

where the cutoff radius rcutoff was set to 10 Å in this work. These four descriptors (σθ, Hϕ, σl, and ARDF) quantify the degree of global disorder in a structure. σθ and Hϕ represent angular variation, while σl and ARDF represent radial variation. Finally, the global atomic density or number density ρ, which is total number of atoms divided by total volume, was computed as the last structural descriptor. All of the descriptors were calculated using the Atomic Simulation Environment (ASE) python library74.