An unsupervised machine learning based approach to identify efficient spin-orbit torque materials

We trained a model using a word embedding model called Word2Vec³⁵ (see Fig. 1) and adopted an unsupervised training approach similar to the report on thermoelectric materials³⁸. Our training dataset contained approximately one million scientific abstracts from various peer-reviewed materials science, physics, and engineering journals published between 1970 and 2020 within the American Physical Society (APS) and the Institute of Electrical and Electronics Engineers (IEEE). We have restricted our data collection until 2020 because one of our key validations of model prediction is based on our experimental demonstration¹⁵ of SOT in iron silicide, and there has not been any discussion of iron silicide in the context of high SOT before 2020 to our knowledge. Additionally, several other studies were published during the peer-review process of this manuscript, which are not included in our training dataset and directly or indirectly supported several of our predictions. We have summarized these studies later in this section. In addition, our trained model inherited a broad range of knowledge beyond the research presented in this manuscript, which will greatly interest different research domains within the scientific community, and a copy of the model is made available online⁵¹. However, in this manuscript, we evaluated the trained model only in the context of magnetism and spintronics. Further details about model training and the training dataset has been discussed later in the Methods section.

Materials and devices knowledge encoded in word embeddings

The relationships among the word embeddings within the trained model captured the knowledge encoded in the text corpus. The related words are closer to each other by the cosine similarity. If two word embeddings are related, the cosine similarity is closer to 1. On the other hand, if the word embeddings are unrelated, the cosine similarity is closer to 0. The cosine similarity among various word embeddings can be used to generate a list of materials, phenomena, and applications related to a target word. For example, the target word “spin Hall effect” is closer to the words “spin-charge conversion”, “nonequilibrium spin polarization”, “spin currents”, and “spin–orbit torque”, among others.

The word embeddings support analogy operations^35,38, which can be useful in connecting the material name to its chemical formula. For example, in the following vector algebraic expression,

Here, “platinum” − “Pt” specifies the context in which we are relating a material name, platinum, to its chemical formula, Pt. Then, the resultant vector, after adding the target word “Ta”, is the corresponding material name, tantalum. We can fix the context and keep changing the target chemical formula to get the corresponding material name, as shown below.

Such vector operations can be useful in populating a list of materials by specifying a material’s class. For example, in the following expression,

Here, “Fe” − “ferromagnet” sets the context to the chemical formula for a material class, “ + semiconductor” specifies the target materials class, and the resultant vector is close to word embeddings for chemical formulas of different semiconductor materials. Changing the target materials class to “topological insulator” also changes the populated materials list that contains materials known to exhibit topological phases. Note that the topological phases of materials are of great interest for large SOT efficiency factors.

We use a similar method to populate a list of SOT materials later in the manuscript.

Our model has been trained with abstracts from both fundamental science and engineering journals; therefore, word embeddings can relate materials and phenomena to relevant devices and applications. For example, the “tunneling magnetoresistance” phenomenon is used to “read” the bit stored in the magnetic tunnel junction devices in the magnetoresistive random access memory (MRAM) application. Within this context, if we ask about the purpose of the “spin-transfer torque” or “spin–orbit torque” phenomenon using the following form, the resultant vector is closest to “write” by the cosine similarity.

Moreover, the addition of word embeddings corresponding to the two phenomena, “tunneling magnetoresistance” and “spin-transfer torque,” yields a vector closest to “magnetic tunnel junctions,” which is the device that functions based on these phenomena⁵². Furthermore, if we add the resultant word embedding “magnetic tunnel junctions” with the word “memory,” it yields a vector closest to ‘MRAM,’ which refers to the application that uses magnetic tunnel junctions⁵².

Note that in some instances, beyond the examples shown here, the closest word embedding to the resultant vector may not be the desirable answer, and the desirable answer may appear as the second or third nearest word embedding. This is mostly due to the nature of the training text corpus, especially how these words have been discussed in the text. However, all the words closer to the resultant vector are relevant and meaningful in the context.

Word embeddings capture intrinsic figure-of-merits

Most spintronics phenomena relevant for potential device applications are characterized by multilayered stacks that contain different materials. Thus, the cosine similarities among multiple word embeddings related to different elements of the stack can provide insight into how a given combination of materials is correlated with a particular phenomenon. We define the following function to quantify such a similarity (Γ) among multiple word embeddings (\(\vec{x}\)), as given by

Note that Eq. (1) is not a unique choice for a function to calculate the similarity among multiple word embeddings, and other functions may be more appropriate depending on the nature of the structure and underlying physics. We have defined Eq. (1) such that Γ is highest when all word embeddings under consideration (\(\overrightarrow{x}\)) are highly similar; however, Γ will be zero if one of the word embeddings under consideration is independent or dissimilar to the others.

For three-word embeddings \(\overrightarrow{x}\in \left\{{\mathcal{A}},{\mathcal{B}},{\mathcal{C}}\right\}\), Eq. (1) reduces to the following expression

where \(\cos ({\mathcal{A}},{\mathcal{B}})\) represents cosine similarity between vectors \({\mathcal{A}}\) and \({\mathcal{B}}\). Such a reduced function can be used to understand the likelihood of a phenomenon \({\mathcal{C}}\) being observed in a bilayer consisting of materials \({\mathcal{A}}\) and \({\mathcal{B}}\). The value of Γ will depend on the similarities among \({\mathcal{A}}\), \({\mathcal{B}}\), and \({\mathcal{C}}\) within the trained word embedding model. If both \({\mathcal{A}}\) and \({\mathcal{B}}\) have a high similarity to \({\mathcal{C}}\) but the similarity between materials \({\mathcal{A}}\) and \({\mathcal{B}}\) is low due to growth issues or interfacial dissimilarities, then overall Γ becomes low.

Interestingly, within our trained model, the similarity pattern generated from Γ for a given set of materials in the context of a particular phenomenon often showed a good correlation with the pattern generated by experimental measurements on the intrinsic figure-of-merit for that phenomenon. Here, we show an example by applying Eq. (2) for the phenomenon \({\mathcal{C}}\equiv \,\)“oscillatory exchange coupling” typically observed in magnetic multilayers with transition metal spacers^53,54,55,56 and utilized as a synthetic-antiferromagnet-based reference layer in magnetic tunnel junctions⁵². We have calculated Γ from the trained word embedding model with \({\mathcal{A}}\) being word embeddings for various 3d, 4d, and 5d transition metals and \({\mathcal{B}}\) being “Co” (cobalt). Γ is plotted as a function of the atomic numbers of the transition metals as shown in Fig. 2a. For 3d, 4d, and 5d transition metals, Γ increases with the atomic number, peaks at “Cr”, “Ru”, and “Ir”, respectively, and then decreases with the atomic number. The trend observed from Γ in Fig. 2a agrees well with the experimentally measured first antiferromagnetic oscillatory exchange coupling (OEC) peak energy in various Co/transition metals multilayers^53,54,55,56, as shown in Fig. 2b.

The agreement observed in Fig. 2 is non-trivial since the estimation of Γ did not involve the experimental values of the first OEC energy peak reported in the text corpus. Γ reflects the similarities among word embeddings: “oscillatory exchange coupling”, “Co”, and various transition metals, the vectorial configurations of which were set during the training based on how those words were used to represent knowledge in the text corpus. Thus, Γ can be very useful in ranking materials or materials stacks according to the target functionality. Note that in a given research domain, materials exhibiting larger figure-of-merits are often discussed in more articles than other materials. Hence, such a correlation pattern partly builds up based on how often a particular materials stack has been discussed in different literature in the context of the phenomenon.

On the other hand, the word embedding model can be useful in identifying the correlation between a material and a phenomenon via patterns hidden within the large text corpus, although such a correlation has never been directly discussed in the training dataset. In the next subsection, we will show that such indirect correlation build-up can be very useful in predicting new materials for a target functionality; however, all elements of the new results are part of the training dataset. This method of predictive research differs from those employing advanced generative models^{28,29,30,31,32,33,34}, which can generate results not explicitly part of its training dataset and does not serve solely the purpose of finding hidden correlations among the elements within the dataset.

Materials word embeddings related to large spin–orbit torques

We apply the trained word embedding model to evaluate various material word embeddings in the context of SOT. We first populated about ~2000 word embeddings similar (by the cosine similarity) to the resultant vector of the analogy expression “Fe” − “ferromagnet” + “spin–orbit torque”. We then filtered out the word embeddings that do not correspond to a material to generate the list of materials correlated with SOT (\(\in {\mathcal{A}}\)). Note that the analogy expression used to generate the list of materials is not unique; similar analogy expressions would have provided the same list after thresholding and ranking using Γ as described in the next paragraph.

Next, we calculate Γ for each of these materials word embeddings (\(\in {\mathcal{A}}\)) using Eq. (2) with the phenomenon \({\mathcal{C}}\equiv \,\)“spin–orbit torque” and the other layer \({\mathcal{B}}\) as “CoFeB” typically used to form the conductor/magnet (\(\equiv {\mathcal{A}}/{\mathcal{B}}\)) bilayers for SOT characterization. We then select candidates for SOT by taking materials word embeddings that exhibit a Γ higher than a threshold value. The threshold value of Γ for candidate selection was determined by the minimum value at which the model identified a known SOT material, which in this case is PtTe₂⁵⁷ (see Fig. 3). The threshold is ~3% of the calculated Γ for the widely used SOT material platinum⁵⁸. The calculated Γ for different candidate materials are shown in Fig. 3a, b. The rationale for selecting the threshold value is that any new candidate that demonstrates a Γ higher than an established SOT material is likely to be significantly correlated with the SOT phenomenon.

Some of the SOT materials identified by the word embedding model have already been discussed in the literature, e.g., IrMn⁵⁹, Pt⁵⁸, Ta⁴, Ir⁶⁰, CuPt⁶¹ Cu/Bi², Bi/Ag⁶², Bi₂Se₃⁹, (BiSb)₂Te₃⁶³, SnTe⁶³, WTe₂⁶⁴, PtTe₂⁵⁷, NbSe₂⁶⁵, SrIrO₃^66,67, IrO₂⁶⁸, Nb-Ta⁶⁹, and CoGa⁷⁰, which are shown as blue bars in Fig. 3. In addition, the model has identified 97 new materials that have a higher similarity with SOT, which are shown as red bars in Fig. 3. Note that we are referring to the red bars as new materials, as they have not been discussed in the literature dataset in the context of SOT to the best of our knowledge. However, these materials have been discussed in the literature in other contexts, and thus, various knowledge on these materials is included in our training dataset. The model identified the blue bars in Fig. 3 via a direct correlation with SOT; however, the red bars were identified via an indirect correlation with SOT, which was built up within the model according to the information available on these materials within the training dataset. ‘FeSi’ (iron silicide) is one of the new materials identified by the model (green bar in Fig. 3), which we have recently experimentally demonstrated to exhibit a high SOT¹⁵; however, the experimental data were not included in the training text corpus.

Model projection on known spin–orbit torque materials

We first separate the known SOT materials (blue bars in Fig. 3) and compare the relative positions of Γ among these materials (see Fig. 4a) with the experimental reports on the spin Hall conductivities on these materials (see Fig. 4b). Spin Hall conductivity is a widely used figure-of-merit to characterize the SOT materials^1,2,3. Interestingly, the relative position patterns of Γ obtained from the word embeddings reasonably agree with the relative positions of the spin Hall conductivities in these materials. Such an agreement is non-trivial because Γ only calculates the association among the words based on the context in which they appeared in the text corpus. The calculation of Γ did not take into account the numerical values of the spin Hall conductivity reported in the text.

The model projected a larger Γ for IrMn compared to Pt. The relative position of Γ between IrMn and Pt is in agreement with the relative positions obtained from the experimental spin Hall conductivities⁵⁹ (see Fig. 4a, b). Although SOT in IrMn has been discussed in the past⁷¹; the reported strength of the spin Hall conductivity was much lower than in Pt. The experimental report⁵⁹ on a strong spin Hall conductivity in IrMn that is higher than in Pt is not included in our training dataset. Thus, the model is also projecting a high Γ based on an indirect association between the words ‘IrMn’ and ’spin–orbit torque’.

Note that our model did not identify ‘W’ (tungsten) as an SOT material, although it has been reported many times in the literature⁷². One possible explanation is that a majority of the training text-corpus from engineering journals used the symbol “W” to represent watt (unit of power), which in turn reduced the correlation between tungsten and SOT. Another thing to note is that the model identified “Cu/Bi” as an SOT material and the relative position of Γ is in good agreement with the calculated spin Hall conductivity for a Cu/Bi Rashba interface². However, there is an experimental report of a CuBi alloy that exhibited higher spin Hall conductivity at lower temperatures⁷³, which the model has not picked up.

We exploit the remarkable similarity between the Γ pattern among various materials and the corresponding experimental reports of spin Hall conductivities to quantitatively estimate the neural network (NN) projection of the SOT efficiency factor, \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\), using the following expression

where σ is the charge conductivity of the corresponding material \({\mathcal{A}}\), σ_Pt is the spin Hall conductivity of the reference material Pt, and \(\Gamma (`{\text{Pt}}{\mbox{‘}} )\) is the Γ value for the word embedding of ‘Pt’.

The NN projected SOT efficiency factors for the blue bars (i.e., known materials) in Fig. 3 are summarized in Table 1 and compared with corresponding reports in the literature. The NN projected \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\) for various known materials are shown in Fig. 4c as a function of their corresponding experimental values reported in the literature. Fig. 4c is represented on a log-log scale to accurately depict the SOT efficiency factors of these materials, which span several orders of magnitude. We show the ideal line y = x in Fig. 4c, where the y-axis represents the NN projected \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\) and the x-axis represents experimental reports, to show how the prediction deviates from the experimental reports. Due to the functional dependence of y = x, the ideal line also appears linear in the log-log representation. Although a few model projections differed from experimental reports by a sizable factor, the model correctly projected a value of the SOT efficiency factor in the same order of magnitude as expected in these materials, see Table 1. Therefore, Fig. 4c serves as the benchmark for the known material subset out of the total candidates identified by the model (see Fig. 3), and we assert that the predictions on the new material candidates reliably project the order of magnitude of SOT expected in those materials.

Such quantitative projections from the trained word embedding model are highly data-dependent, especially because they are calculated based on the correlations among word embeddings, which may evolve as more data is added to the training text corpus. In order to understand the stability of such a quantitative projection when more knowledge becomes available every year in the form of published literature, we present a timeline analysis in Fig. 4d. For this analysis, we have trained multiple models by cumulatively adding one more year of literature abstracts to the training text corpus. The x-axis of Fig. 4d corresponds to six models that were trained using journal abstracts starting from 1970 until 2015, 2016, 2017, 2018, 2019, and 2020, respectively. For each of these six models, we have evaluated Γ using Eq. (2) for the blue bars in Fig. 3 and then calculated \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\) using Eq. (3), as shown on the y-axis of Fig. 4d.

The magnitudes of the NN projected \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\) in most of these known materials remained relatively stable beyond 2015, with each successive model incorporating additional data. The projection for Cu/Bi became relatively stable around 2017. The model did not pick up PtTe₂ as an SOT material candidate before 2018, and the projected \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\) fluctuates by an order of magnitude onward, which indicates that more data on PtTe₂ is needed for a stable quantitative projection on \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\); however, there is enough data that resulted in a sizable Γ. The model projection on NbSe₂ is reasonably stable beyond 2015; however, the projected value is about three times higher than the experimental report⁶⁵. Note that the quantitative projection of \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\) relies on the relative positions of Γ with respect to the reference word embedding “Pt” and is independent of the absolute value of Γ. Therefore, the stability in the predicted order of magnitude for \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\) corresponds to the stability of the relative position of Γ for new candidates becoming stable with respect to that of “Pt”.

Surprisingly, the NN projected \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\) value for FeSi was relatively stable around 2015, although we first measured this material in the context of SOT in 2020⁷⁴, and subsequently we demonstrated a large SOT efficiency factor closer to the model prediction in 2022¹⁵. Furthermore, the model identified SrIrO₃ as an SOT material and correctly projected the efficiency factor based on abstracts from the literature up to 2015; however, the experimental report appeared four years later^66,67. Remarkably, the word embedding model could have predicted both of these materials (FeSi and SrIrO₃) back in 2015, showing the significant usefulness of such machine learning algorithms in materials and device research.

New materials prediction for large spin–orbit torques

The word embedding model has identified 97 new candidates, shown as red bars in Fig. 3, which, to our knowledge, have not been directly discussed in the context of SOT. Although a few of these candidates have been discussed in the literature in the context of special material phases that are often correlated to SOT, the model correlated most of them to SOT via patterns hidden within the large text corpus. We used Eq. (3) to estimate \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\) for these new candidate materials (red bars in Fig. 3) and distribution of \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\) is shown in Fig. 5a. The estimated values of \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\) and the corresponding conductivities are listed in Table 2.

Among the 97 new materials, 16 of the candidate materials are likely to exhibit an SOT efficiency factor much larger than or closer to unity (see Fig. 5a). These 16 candidates include EuSe, CrSe, DyBa₂Cu₃O₇, MnGe, CrVTiAl, EuTe, EuMnBi₂, CrSi, BiSbTeSe₂, CrTe, EuFe₂As₂, NaCo₂O₄, FeSi, FeGa, TaSe₃, and ZrZn₂, where EuSe, CrSe, DyBa₂Cu₃O₇, and MnGe are projected to exhibit a SOT efficiency factor >10. The estimated \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\) for these 16 candidates are shown in Fig. 5b as a function of their corresponding Γ values. Γ values for these candidates range from 10 to 80% of Pt and FeSi, EuTe, EuSe, MnGe, CrSi, CrTe, and FeGa have Γ > 50% of Pt while exhibiting \({\xi }_{\,\text{SOT}}^{\text{NN}\,} \,> \,1\). We recently measured a high SOT efficiency factor on the order of ~2 in one of these 16 candidates, FeSi¹⁵, which is in good agreement with the quantitative projection of the word embedding model (see Table 1). Note that the measured data on FeSi or related information was not included in the training text corpus. Thus, such an agreement between model prediction and experimental observation demonstrates the quantitative prediction capability of the machine learning approach, which will be useful for application-specific materials search.

In addition to FeSi, the word embedding model predicted SOT in various other silicides, including CrSi, MoSi, CoSi, PdSi, PtSi, NiSi, FeSiAl, LaPt₂Si₂, Co₂FeSi, Co₂MnSi, CePt₃Si, CaIrSi₃, and YbRh₂Si₂. The model predicts that CrSi will exhibit an efficiency factor ~3 × higher than FeSi, close to a value of ~5.3. Other silicides identified by the model are expected to exhibit SOT efficiencies close to what we observe for transition metals. SOT in silicides could be interesting for SOT device applications, as silicides are integrable within standard CMOS technologies.

In order to understand the stability of such a quantitative projection on the high SOT candidates, we have performed a timeline analysis similar to that done for the known materials in the previous subsection. We used the six models trained with journal abstracts starting from 1970 until 2015, 2016, 2017, 2018, 2019, and 2020, respectively, to calculate the NN projected \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\) using Eq. (3) for the 16 high SOT candidate materials identified by the model. Fig. 5c shows \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\) for these materials as a function of accumulated training data over each consecutive year. The predictions for EuSe, CrSe, DyBa₂Cu₂O₇, MnGe, EuTe, and EuFe₂As₂ were already reasonably stable since 2015. The predictions for CrSi, CrTe, TaSe, and ZrZn₂ became stable between 2016 and 2018. CrVTiAl, EuMnBi, and BiSbTeSe₂ were not identified as SOT candidate material before 2019, and the projected \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\) is still fluctuating as more data is added to the training text corpus.

Among the other candidate materials identified by the model, 49 of them are predicted to exhibit a sufficiently large SOT efficiency factor within the range of \(0.1\le {\xi }_{\,\text{SOT}}^{\text{NN}\,} \,<\, 1\) (see Fig. 5a), where Co₂TiSn, SrMnBi₂, NiMn, MnAl, Co₂MnGe, MnPt, CeRh₃B₂, CeRu₂Al₁₀, NbSe₃, and MoSi are expected to exhibit \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\ge 0.5\). Furthermore, 26 materials are expected to have SOT efficiency factors ranging within \(0.01\le {\xi }_{\,\text{SOT}}^{\text{NN}\,} \,< \,0.1\), and 6 materials are expected to show very weak SOT efficiency factors \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\, <\, 0.01\). For example, the model identified highly conductive delafossite metals, specifically PdCrO₂ and PdCoO₂, as SOT materials, although their expected SOT efficiencies are roughly an order of magnitude lower than that of platinum. A detailed list of the candidate materials and their expected \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\) are given in Table 2.

Some of the materials identified by the model have a relatively straightforward correlation with SOT. The correlation built up between SOT and BiSbTeSe₂ is straightforward, as this material has been discussed as a topological insulator⁷⁵ in the literature. In particular, many other bismuth-based compounds^9,63 were discussed as a host for the topological insulator phase, and some of them have experimental reports of a strong SOT. Another candidate material, MnGe, has been discussed in the context of a topological texture⁷⁶, which is a possible correlation to a strong SOT. Another candidate, CaMnBi₂, has been discussed to exhibit a nonzero Berry phase⁷⁷, which is associated with the physical origin of SOT in many materials^1,3.

Interestingly, a magnetism-induced topological phase has been discussed for EuAs₃ in the literature⁷⁸; however, the article appeared beyond the time frame of the dataset used to train the model. Similarly, GdPtBi⁷⁹ and TaSe₃⁸⁰ have been discussed to host a topological semi-metallic phase, and such a phase is known to exhibit a strong SOT⁶⁴; however, these articles were published beyond the dataset collected to train the model. Moreover, there has been a discussion of a Rashba channel formation in GeTe⁸¹, which is another intrinsic mechanism for SOT³. An article also discussed a strong field-like SOT in a GeTe/NiFe bilayer⁸²; however, the article also appeared beyond the time frame of the collected dataset used to train the model.

Note that the calculation of the Γ value depends only on the similarities among various word embeddings within the trained model, but the calculation of the NN projection of the SOT efficiency factor also takes into account the corresponding material conductivity (see Eq. (3)). Here, we use the nominal conductivity reported in the literature for these materials (see Table 2 and supplementary information); however, some of the NN projections of SOT efficiency may need to be re-calibrated as conductivity varies in different growth and crystallinity conditions.

t-SNE projection of the possible origin of spin–orbit torque

SOT in nonmagnetic materials can arise from various intrinsic mechanisms^1,2,3, e.g., interfacial mechanisms such as a Rashba interface or a topological phase of material^9,63,64 or bulk mechanisms such as the spin Hall effect. Current-induced SOT has been observed in ferromagnetic materials^83,84 that arise from the spin anomalous Hall effect⁸⁵, the planar spin Hall effect⁸⁶, or a magnetization-dependent spin Hall effect⁸⁷. In addition, large SOT has also been discussed in antiferromagnetic phases of materials^59,88.

The word embedding model can provide insight into a possible intrinsic mechanism of SOT for the new candidate materials based on the correlation among related word embeddings within the high-dimensional space. To understand such a correlation, we use a dimensionality reduction methodology called t-distributed stochastic neighbor embedding (t-SNE)⁸⁹, which is a widely used method for visualizing high-dimensional data on a two-dimensional (2D) plane. In a t-SNE plot, the data points corresponding to similar word embeddings tend to cluster together, and dissimilar data points are located farther apart.

To generate a cluster of word embeddings related to a target word, we have collected all word embeddings closer to the target word by a cosine similarity ≥0.8 and used the t-SNE method to reduce these high-dimensional word embeddings onto a 2D plane. The t-SNE plots were generated using scikit-learn with perplexity = 50, n − components = 2, init = ‘pca’, n − iterations = 5000, and random states = 10. For visual clarity, we represent the cluster for a given target word with a semi-transparent colored region constructed by the boundary data points of the cluster. We have plotted such clusters for the target words “topological insulator”, “topological semimetal”, “spin Hall effect”, “Rashba”, “ferromagnet”, and “antiferromagnet” in Fig. 6 with the x and y axes having arbitrary units. Note that the choice of target words is based on the known intrinsic mechanisms that can give rise to SOT.

We also projected the word embeddings for the new candidate materials using t-SNE to understand which candidates overlap with various clusters for different intrinsic mechanisms, as shown in Fig. 6. Such an overlap insinuates a possible origin of SOT in these materials, which are summarized in Table 2. For a benchmark of this methodology, we have also projected the word embeddings for known materials “Pt” and “Ta”, and they overlap with the ‘spin Hall effect’ cluster, which is the known origin of SOT in these transition metals^4,58. We have also projected the word embeddings for known topological insulator materials “Bi₂Se₃” and “Bi₂Te₃” on the 2D t-SNE plot, and they overlap with the “topological insulator” cluster, confirming the intrinsic mechanism.

Note that the t-SNE method effectively preserves the local structure of the data, but may not accurately represent the overall data distribution. As a result, while any overlap of data points and spatial separation of data clusters may indicate a possible qualitative relation, the quantitative distance between two data points may not be meaningful.

Among the 16 candidate materials with \({\xi }_{\,\text{SOT}}^{\text{NN}\,}\ge 1\), EuSe, DyBa₂Cu₃O₇, EuTe, CrTe, FeGa, and FeSi overlap with the “spin Hall effect” cluster (see Fig. 6a), indicating that the spin Hall effect is a possible mechanism expected in these materials. On the other hand, CrSe, CrSi, BiSbTeSe₂, and TaSe₃ are expected to exhibit a topological phase based on the overlap with ‘topological insulator’ and ‘topological semimetal’ clusters, as shown in Fig. 6a. Note that BiSbTeSe₂ has been established as a topological insulator⁷⁵ in the literature. Moreover, TaSe₃ has been discussed as a topological semimetal in ref. ⁸⁰, in agreement with the overlap with the ‘topological semimetal’ cluster (see Fig. 6a); however, the article was published beyond the dataset collected to train the model. MnGe and EuFe₂As₂ overlap with both “antiferromagnet” or “ferromagnet” clusters, indicating a possible mechanism similar to that observed in ferromagnetic or antiferromagnetic conductors. Other high SOT candidates such as CrVTiAl, EuMnBi₂, NaCo₂O₄, and ZrZn₂ did not overlap with any clusters for known mechanisms, and we leave the further evaluations of these candidates for the future.

Other candidate materials likely to exhibit a spin Hall effect-induced SOT include Co₂MnGe, MnPt, MoSi, Co-doped ZnO, GeTe, CoZrNb, CuMnAs, Co₂FeAl, CoS₂, Co₂CrAl, GdAl₂, FeZr, PdSi, GdPtBi, CeNiSn, NiMnSb, CrGe, RuO₂, AuFe, PdNi, NiSi, PtSi, NiGe, FeCu, CuCr, and PdCoO₂, as shown in Fig. 6a–d. Note that the Rashba mechanism has been discussed in GeTe⁸¹ with a strong field-like SOT in a GeTe/NiFe bilayer⁸²; however, a direct discussion of the spin Hall effect in GeTe is not present in the literature. In addition, a topological Weyl semimetal state in GdPtBi has been discussed⁷⁹ under hydrostatic pressure. In a recent article⁹⁰, RuO₂ has been discussed as a powerful spin current generator to achieve field-free SOT switching. However, these articles appeared beyond the timeframe of the dataset used to train the model.

Candidate materials that are likely to exhibit a topological semi-metallic phase include SrMnBi₂, SrPtAs, LaPt₂Si₂, CuTe, and EuS/Al. Two of the candidate materials, CaMnBi₂ and MnBi, overlapped with both “topological insulator” and “topological semimetal” clusters; indicating a possible topological phase in these materials. Note that two-dimensional Dirac fermions and a nonzero Berry phase have been discussed for CaMnBi₂⁷⁷. Candidates such as CoSi, Cu₂MnAl, NbZr, and PtIr overlapped with clusters for both ‘topological insulator’ and ‘spin Hall effect’; while other candidates such as Co₂TiSn, YCo₅, and PtMnSb overlapped with clusters for both ‘topological semimetal’ and ‘spin Hall effect’.

Some candidate SOT materials only overlapped with the “antiferromagnet” or “ferromagnet” clusters. Candidate materials such as MnAl, CeRh₃B₂, CeRu₂Al₁₀, FeGe, SmAl₂, CoS₂, EuAs₃, YbPtIn, GdPtBi, NbFe₂, YbRh₂Si₂, RhSn, PtGa, TiBe₂ overlap with the “antiferromagnet” cluster and may exhibit SOT using a similar mechanism as seen in a few known antiferromagnets^59,88. Note that a magnetism-induced topological transition has been discussed⁷⁸ for EuAs₃ where a topological massive Dirac metal phase has been studied in the antiferromagnetic ground state at low temperatures; however, the article appeared beyond the time frame of the dataset used to train our model. On the other hand, TaS₂ and FeAs overlap with the “ferromagnet” cluster only. The proper magnetic phase in these materials and associated mechanisms can be further evaluated using conventional theoretical and experimental methodologies.