From bulk effective mass to 2D carrier mobility accurate prediction via adversarial transfer learning

Hybrid transfer learning framework

Our transfer learning framework consists of two main components. The first part utilizes adversarial transfer learning (ATL) to extract shared features from both bulk materials and 2D materials, as shown in Fig. 1(a). The adversarial transfer learning is composed by three multi-layer perceptron (MLP) models: a feature extractor, an effective mass regressor, and a data source classifier. The feature extractor transforms initial input features into a low-dimensional vector using materials agnostic platform for informatics and exploration (MAGPIE)³². This extractor can be applied to both bulk and 2D materials; initially, without any constraint, the output of the extractor is a random number. Meanwhile, the extracted features are also used to train the bulk material effective mass regressor, and the regression loss is backpropagated to optimize the feature extractor. At this stage, the feature extractor learns the knowledge of effective mass and provides features closely related to it. In contrast to the standard approach, we not only train an effective mass regressor but also an additional data source classifier. This classifier is designed to determine whether the features are extracted from bulk or 2D materials and tell the feature extractor the common features between both types of materials. We achieve this by backpropagating the reversed classification loss to the feature extractor, and iteratively training both extractor and classifier until the classifier can no longer identify the data source. During the training iterations, the feature extractor is trained to fool the classifier about the data source, while the classifier is trained to discriminate between them. Hence, this process is referred to as adversarial training.

The second component of our model involves the embedding of expert knowledge and provides a direct prediction of 2D carrier mobility, as illustrated in Fig. 1(b). The necessity of incorporating expert knowledge lies in the fact that the adversarial approach only ensures the extraction of common knowledge, but it lacks the description of the uniqueness of target materials and their properties. This is particularly critical in cases like 2D materials, where many interesting properties stem from their unique structures. Therefore, in addition to the features extracted by transfer learning, we add features from lattice symmetry, crystal geometry, and electronic properties to describe the unique 2D structures and their electronic behavior. From the perspective of deformation potential theory, we speculate that these features are closely related to carrier mobility. For example, the symmetry can affect phonon vibration mode, while the thickness is related to elastic modulus and reflects the strength of electron-phonon coupling, thus contributing to carrier mobility. The electronic features, such as electronegativity and valence electron distribution, are also regarded as important for carrier transport. The full feature list and description are presented in Supplementary Table 2. It is worth noting that these added features can be directly taken from the structure files without additional density functional theory (DFT) calculation, which is critical for quick-and-direct prediction.

To demonstrate the impact of adversarial training and expert knowledge on the performance of cross-material transfer learning, we conducted comparative tests, as shown in Fig. 2(a). When transfer learning is applied without adversarial training, the extracted features perform worse than MAGPIE (baseline), which is a typical negative transfer. This indicates that although the extracted features work in the source domain, they may not necessarily be helpful in the target domain, especially when the source and target domains are different materials with different properties. However, with the help of adversarial training, common knowledge between bulk and 2D materials is captured, and negative transfer is alleviated. It is also essential to recognize that many appealing properties of 2D materials stem from their unique structure. Therefore, additional features based on expert knowledge describe their special structures, which complements the knowledge acquired from the bulk materials and leads to more accurate predictions. This demonstrates the effectiveness and importance of leveraging adversarial training and expert knowledge to enhance the transfer learning performance.

To further investigate how adversarial transfer learning works, we utilized t-distributed stochastic neighbor embedding (t-SNE) to visualize the output differences between our transfer learning model with and without adversarial training, as shown in Fig. 2(b). The input feature space of 2D and bulk materials is separated, which implies that the overlap is small, making it challenging to transfer knowledge from bulk to 2D materials. We can see that most of the bulk and 2D materials are still separated after the transfer learning without adversarial training. However, after incorporating adversarial training, the extracted features are no longer able to distinguish between bulk and 2D materials, it can be observed that the two types of materials are mixed together in the t-SNE plot. This indicates that the features extracted by adversarial transfer learning are shared by both bulk and 2D materials, therefore, improving the effectiveness of cross-material transfer learning.

Carrier mobility prediction and model interpretation

Our hybrid transfer learning framework has been applied to three tasks related to 2D carrier mobility. Figure 3(a) displays the prediction accuracy of carrier mobility under deformation potential theory (DPT), with R² values of 0.88 and 0.90 for the average electron and hole mobility, respectively, and a MAE of 0.19 for both mobilities. Importantly, our trained model only requires the crystal structure file as input when making carrier mobility predictions. This streamlined approach ensures the usability and efficiency. Compared to DFT-based mobility calculation, our approach is five orders of magnitude faster as shown in Supplementary Fig. 10. These results demonstrate that our models can provide accurate and efficient estimation of overall carrier mobility. In addition, feature importance analysis in Fig. 3(d) reveals that the most important predictors for average carrier mobility are ATL features, closely related to effective mass. Furthermore, electronic features such as valence electron distribution, which characterize the electronic distribution of materials, are also crucial predictors. Our models also accurately predict mobility anisotropy, as evidenced by the R² scores of 0.89 and MAE of 0.11 and 0.13 for electron and hole, respectively, as shown in Fig. 3(b). Symmetry features, such as space group and mirror symmetry, play a more prominent role in determining the mobility anisotropy, as shown in Fig. 3(e), compared to that of ATL features. These results indicate that mobility anisotropy is mainly determined by the symmetry of the material.

Recently, a more accurate method to estimate carrier mobility was developed by solving the electron-phonon coupling (EPC) matrix, which provides a valuable re-evaluation of the carrier mobility of common 2D materials and gives more consistent results with experimental measurements³³. We further tested our method of predicting EPC mobilities of transition metal dichalcogenides (TMDs), which show high prediction accuracy as illustrated in Fig. 3(c). The R² score reaches 0.95 for electrons and 0.89 for holes, which is similar to DPT mobility prediction. Differently, the feature importance in Fig. 3(f) suggests that ATL features depicting effective mass are less important than electron features and symmetries in this case. This is in line with the finding that effective mass shows no obvious correlation with carrier mobility for 2D TMDs³⁴. Despite the limited structure types hindering its extrapolation, our method still gives very good accurate predictions. With increasing amounts of high-accuracy data, it can be useful in the future. Nevertheless, this consistency and maintained accuracy prove that our method can provide robust predictions with great generalization ability.

With the well-trained model, we can screen out 2D candidates for high carrier mobility and proper bandgap that is comparable to silicon. As shown in Fig. 4(a), 9115 2D materials were collected from two open-source databases, and a de-duplication process was carried out based on their formula and space group. Then we removed all metals, leaving 4266 semiconductors, of which 3109 are thermodynamically stable. Considering that silicon has a bandgap of around 1 eV, we selected 869 semiconductors with similar bandgaps ranging from 0.5 eV to 1.5 eV. Finally, the trained model was applied to predict their average carrier mobility \(\bar{\mu }\), and 21 materials with electron or hole mobility higher than 10⁴cm²/V·s were screened out. We further validate the accuracy of our ML model by DFT calculations based on effective mass approximation and deformation potential approximation, which provide reliable estimations of carrier mobility at an acceptable computational cost. As shown in Supplementary Fig. 9, for both electron and hole mobility, our ML model gives consistent predictions with the R² score above 0.82, and the MAE values below 0.22, demonstrating the great predictive ability of our ML model. Note that among the screened 21 materials, some have already been synthesized or even been experimentally validated to have high mobility, such as In₄Se₃ and Nb₂SiTe₄^35,36,37.

Figure 4(b) shows the element and crystal system distribution of selected materials with high carrier mobility. The most frequent elements belong to p-block, and the most common crystal syngony are parallelogram, orthorhombic, and rhombic syngony. To gain a deeper understanding about why these materials possess high carrier mobility, we conducted partial dependence analysis based on Shapley additive explanation (SHAP) values, as shown in Fig. 4(c–f). Regarding the elemental features, an increase in the p-valence electrons fraction is positively correlated with carrier mobility (Fig. 4(c)), consistent with the element distribution results in Fig. 4(b). Moreover, Fig. 4(d) shows that the smaller the difference in electronegativity, the more positive the contribution to carrier mobility. This may be because smaller electronegativity difference tends to facilitate the formation of covalent bonds, in which the electrons are more free and easier to move, resulting in higher mobility. Structural features such as mirror symmetries are also found to correlate to carrier mobility. As shown in Fig. 4(e–f), materials with three in-plane mirror operations, such as TMDs, generally exhibit lower electron mobility, while materials with out-of-plane mirror operations, such as hexagonal boron nitride, have higher electron mobility. This may be because the mirror symmetry restricts some specific phonon vibration modes, which weakens the electron-phonon coupling and results in higher carrier mobility³⁸. It is worth noting that although the trend regarding symmetry is obvious, the overall impact of symmetry on the model is not as significant as elemental features according to SHAP values. Therefore, in materials with different compositions, the effect of symmetry can be easily masked. Moreover, research on the correlation between symmetry and mobility is still limited, and further study based on phonon vibration modes is needed to provide deeper insights.

Figure 4(g, h) presents the distribution of carrier mobility in 2D semiconductors using t-SNE. Space groups including P6/mcc, P4/nmm, P4₁2₂, and some low symmetry systems are observed to have high carrier mobility. A significant trend is that the majority of 2D semiconductors exhibit higher electron mobility than hole mobility, except for some low symmetry structures where the hole mobility is higher, the t-SNE plot of hole mobility is shown in Supplementary Fig. 7. Further DFT calculations show that the effective mass of electrons and holes in these systems are similar, while the deformation potential exhibits significant differences, as listed in Table 1 and Supplementary Table 3. Therefore, the high hole mobility in these low symmetry structures may be influenced by different phonon scattering mechanisms for electrons and holes. Although the materials selected in this study exhibit higher charge carrier mobility than traditional 2D semiconductors such as MoS₂ and black phosphorus, a considerable fraction of these materials have bandgaps that are either too large or too small for application in the semiconductor industry. However, these materials may still have potential applications in fields such as catalysis and photovoltaic applications. Figure 4(i) presents the electron and hole carrier mobility of the screened materials with proper bandgap around 1 eV, most materials have high mobility of only one carrier type, which are promising candidates for p-type or n-type semiconductors. Notably, BiAs and BiSb possess both high electron and hole mobility, serving as the compelling choice for complementary logic devices.

2D semiconductors with high carrier mobility

To gain insight into the mechanisms underlying the high mobility of these 2D semiconductors, we performed an electronic structure analysis of two representative structures (group V AB and group IV-V AB₂), as shown in Fig. 5. We also summarized their carrier mobility, bandgap, effective mass and deformation potential in Table 1 and Supplementary Table 3. Group V AB, they have a structure similar to black phosphorus, as shown in Fig. 5(a)³⁹. However, with different elements substituted, their band edges shift from the Γ point to a point within the Y-Γ high-symmetry path. Moreover, the shape of the shifted band edges is sharper, indicating smaller effective masses and higher carrier mobilities as shown in Fig. 5(b, c). These findings suggest that bandgap engineering through elemental substitution can be an effective approach to enhance carrier transport, and the elements within the p-block may be good choices according to the above partial dependence analysis. For the group IV-V AB₂ semiconductors, as shown in Fig. 5(d), they exhibit strong structural anisotropy, which leads to remarkable electronic anisotropy, as shown in Fig. 5(e, f). Local magnifications of conduction band minimum and valence band maximum reveal obvious differences in effective mass along orthogonal directions. Specifically, the effective mass is smaller along a lattice direction, interestingly, the deformation potential is far smaller along b direction than a direction (see Table 1). Consequently, the mobility is higher along the b direction, which indicates that the electron-phonon interaction plays a decisive role in this system. In addition, the a, b plane also show a symmetry difference. Specifically, the a direction exhibits mirror symmetry, while the b direction does not, further implying a potential correlation between symmetry and mobility. This highlights the importance of further investigation on how symmetry affects carrier mobility, as well as the potential for modulating carrier mobility through symmetry-protection or symmetry-broken structural engineering.