A machine learning-based prediction of crystal orientations for multicrystalline materials

Multicrystalline materials are solids that consist of multiple small crystal grains with different sizes, shapes, and structures. They contain grain boundaries that separate each grain from another, causing a sudden change in crystallographic orientations. Grains normally have random crystallographic orientation, but under particular growth and processing conditions, they may be directed.^1–3 Most inorganic solids, including metals, alloys, and many ceramics, are generally multicrystalline, and the distribution of grain boundaries or crystallographic orientations is one of the important factors that determine material property.^4–8 Lu stated that high grain boundary density increases mechanical strength and hardness in metals, while it is known that crystalline materials tend to crack at their grain boundaries because voids form and link up at the grain boundaries where deformation by sliding is limited.^9,10

The abundance, affordability, and different aspects of multicrystalline materials have made them a widely selected material for various applications. Polycrystalline diamond has a high mechanical strength and is used for polishing pads, while tetragonal zirconia polycrystal (TZP) is used for medical prostheses because of its stiffness, wear resistance, and chemical resistance. Grain refinement that increases grain boundary density is a common technique to enhance strength, making way for a wide range of applications. As for solar cells, the electric properties of the cells have improved over the years, stimulated by the increasing variety of constituent multicrystalline materials, such as silicon, copper indium selenide, cadmium telluride, and perovskites.^11–13 Despite their availability, device performance based on these materials tends to be limited by a high concentration of defects that provoke recombination and deteriorate conversion efficiency. Thus, it is essential to reveal how these defects are generated and to control them. Although crystal structures, such as grain boundary arrangements, grain size distributions, and crystallographic orientations, are closely related to the generation of defects, the variation in the characteristics of the multiple grains has made it a challenge to precisely address the cause.^14,15 Therefore, a systematic and ingot-scale analysis of the crystal structures is required to clarify the mechanisms regarding crystal defects. We particularly focus on the distribution of crystallographic orientations and how they affect the materials’ performance. We aim to clarify the relations between orientation distributions and the generation and behavior of dislocation clusters in multicrystalline silicon. Dislocation clusters have high concentrations of dislocations up to 10⁷ cm⁻² acting as recombination centers for photogenerated carriers. Typically, they form near grain boundaries during the silicon ingot casting due to the relaxation of strain energy, and increasing with crystal growth, they dominate the crystal at the late crystal growth stage. Many crystalline characteristics, including orientation distributions, are considered to affect the presence, but the relationships have not been clarified. Crystallographic orientations are generally measured by electron backscatter diffraction (EBSD) or the Laue scanner method.^16–19 These methods have a high resolution and accuracy while falling behind in measuring speed and area. Other techniques acknowledge the relationship between crystallographic orientations and reflectance, utilizing reflection patterns for determining orientations while facing challenges of developing precise fitting methods.^20,21

Here, we propose a rapid and accurate measurement of crystallographic orientations by optical imaging and machine learning. Machine learning has become widely used for processing images, videos, speech, and texts. Simple convolutional neural networks have been employed for classifying cats apart from dogs or text recognition, and powerful deep nets have allowed for the discovery of new stable materials.^22,23 The combination of effective descriptors and a learning framework can perform a screening on material properties and help create a database relevant for testing candidate materials.²⁴ The ability of machine learning to process enormous amounts of complex data is suitable for addressing the relationships between multiple optical features and crystallographic orientation, as optical features may not be sufficiently reproduced by conventional fitting methods. Our technique is implemented with a home-made apparatus for taking high-resolution images of the sample and a machine learning-based model for estimating crystallographic orientations using the light reflection patterns taken from the optical images of textured multicrystalline wafers.^25,26 Among the various machine learning techniques, a long short-term memory (LSTM) neural network, a type of recurrent neural network (RNN), was employed. We believe that LSTM is effective in preserving the sequential relationships between crystallographic orientations. In this study, we report in detail on the methodology and the results of the attempts to enhance prediction accuracy.

Silicon wafers cut out from a single ingot by a diamond wire saw were used. First, we perform alkaline texturing on the wafer surface. This shaves off extra particles and leaves mainly the {111} facets on the surface, creating a pyramid-like structure. Irradiation on these pyramids will generate crystallographic orientation-specific reflection patterns. A home-made apparatus shown in Fig. 1 is used to generate and capture these reflection patterns. The apparatus mainly consists of an imaging photometer (ProMetric IP-PMY29), a rotating collimated light for illumination, and a stage. As the light rotates around the sample, optical images are taken within every 5 degrees of rotation. We integrate these 72 images (which are obtained after a full rotation around the wafer) into a 72-dimensional signal intensity matrix, which we call the “luminance profile.” The luminance profiles are then normalized to eliminate the background effect from the light source. This is done by dividing the measured luminance with the diffused reflection from a white diffuser plate, which is proportional to the distribution of light intensity. An example of the treated luminance profile is shown in Fig. 2. The elevation angle of the light can be fixed to an arbitrary angle between 15° and 60°. The elevation and the rotation angles determine the incident angle of irradiation on the wafer. The selection of elevation and rotation angles allows for collecting multiple luminance profiles for each grain. All data collection took place in a dark room, minimizing the effects of external lighting.

Next, we train the machine learning model to predict orientations for each grain. The model is a supervised neural network that consists of a long short-term memory (LSTM) neural network and two fully connected layers as shown in Fig. 3.^27–29 The LSTM consists of 128 units, and each of the two fully connected layers has 128 nodes. Rectified linear unit (ReLU) was selected as the activation function of the fully connected layers. Dropouts of 0.5 were inserted to prevent overfitting. The pre-obtained luminance profile is used as the input to the model, and the outputs are quaternions that represent orientation.³⁰ Grain orientations measured by the Laue scanner method are used as teacher data (Fig. 4). As the model trains itself, the outputs are compared to the correct values and the weights connecting the nodes are adjusted so that the predictions and the correct values match well. Metrics to evaluate the model’s performance were geodesic distances representing the shortest path between the true and predicted orientations, which were then converted to radians to compare prediction errors. Mean absolute error (MAE) was used as the loss function to be optimized by the Adam optimizer. Batch sizes were 3838, and the number of epochs was 8000. Once the model is trained, it is validated on data from the same wafer and, then, tested on an alternate wafer. Note that the training, validation, and test datasets consist of the luminance profiles of 3838, 960, and 6373 grains, respectively. These were extracted from seven silicon wafers, three for training and validation and four for testing.

Figure 5 shows the [(a), (c), (e), and (g)] validation and [(b), (d), (f), and (h)] test results when the prediction was performed at a variety of elevations. The selected elevations were 30°, 45°, and 60°. The histogram shows the prediction error and the probability density of grains. As the accuracy increases, the red histograms shift to the left and generate a sharp peak. The black line shows the expected shape when predictions were completely random. It is seen that all histograms are inclined to the left, indicating that the prediction was successful. Table I shows the quartiles for each prediction. When trained at 60°, the error median was 8.61°. We can see that there is an optimal elevation angle of 45°. The prediction accuracy varied greatly depending on the elevations. From the results, one would consider that combining all three angles may result in a better-trained model. The results of the combined model are shown in Figs. 5(g) and 5(h). Remarkably, there is a significant decline in the error median of up to 50% in the combined model compared to 60°, and the number of grains with noticeably large errors has also declined. The error median when trained on all three elevation angles was 4.35°.

Furthermore, we improved the predictions by data augmentation. In the previous part, we have attempted to physically increase data size and diversity. In the following part, we practice this mathematically.

Data augmentation is a method to increase data when it is assumed that there are insufficient data to train a network. In many cases, simple transformations, such as rotating or flipping the original image, filtering, or desaturating, are effective.^31,32 The selection of data augmentation techniques is embraced on the condition that they are “safe” as in preserving the labels post-transformation.³¹ Although these minorly altered images are inherently the same data, the neural network treats them as novel data, hence helping the model generalize and improve trustworthiness. Particularly in our case, the original luminance profiles were rotated by a 5° interval, increasing data to 72 times the original size. The quaternions were rotated, respectively, using rotation matrices from SciPy. An example of rotated data is shown in Fig. 6. The model was trained and tested on the augmented data and compared with non-augmented training, as shown in Fig. 7. Table II shows that the prediction accuracy has improved by 25%–35%. It is noted that the time required for training increased from less than 8 min to 7–8 h.

The orientations measured for each grain in a sample wafer are mapped in Fig. 8 on an inverse pole figure (IPF). This mapping was created by training at all elevations with data augmentation. The time required for mapping was 8 min.

Table III shows a comparison of the spatial resolution and the measurement time of our method to two conventional methods, EBSD and the Laue scanner method. Measurement time is calculated for a full-sized wafer of 15 × 15 cm². EBSD has the highest spatial resolution as electron beams can be focused by electromagnetic lenses. Despite their accuracy, the common methods require a great time to measure, and our method excels in this by requiring a total measurement time of 1.5 h. This includes the time required for taking optical images, training, and mapping out the orientation. Training requires a maximum of 7 min and 30 s, and mapping requires up to 8 min. Much of the measurement time is consumed by optical imaging, which can be reduced by redesigning the equipment.

We will discuss the difference in prediction results between single-angle and triple-angle training. It has been observed that prediction errors vary between training angles. Training at 45° had the best results, while training at 60° brought about a poor accuracy. The orientations of grains with a noticeably large error of over 10° are displayed on an inverse pole figure about the normal direction in Fig. 9, of which the plots indicate the index of the crystallographic plane normal to the sample surface. The figure shows that there is a significant difference in the fatally predicted orientations. For example, at 30°, the model failed at planes near {111} to {101}. The model failed at planes {111} at 45° and at planes {111} to {001} at 60°. This variance is explained by the geometrical configuration of the incident light and surface structures. It is well known that alkaline texturing creates pyramid-like structures consisting of {111} planes on the sample surface. Figure 10(a) shows an octahedron likely to be half-exposed on the surface when the grain is located with plane {100} upward. An example of the luminance profile these grains generate is shown in Fig. 10(d) as well. The measured luminance is generally low with four small peaks corresponding to specular reflection. In addition, low elevations generally create low-intensity profiles compared to higher elevations, but in this case, there is no significant intensity difference between the elevation angles. This is because most of the reflected light travels in the opposite direction of the camera. In grains with orientations near plane {101}, the octahedron generated from {111} planes and their luminance profile is shown in Figs. 10(b) and 10(e). In this case, two {111} planes are facing upward, strongly reflecting light. At low elevations, reflection is weakened, and at high elevations, strong reflection peaks appear. In grains with orientations near {111}, a single {111} plane is facing upward as shown in Fig. 10(c). For grains with exact orientations of {111}, there is seldom a change in reflection intensity responding to rotation. However, when the grain is slightly tilted either way, the profile changes abruptly and results in a significant peak as shown in Fig. 10(f).

These variations in reflection patterns heavily affect the performance of prediction. The clearer the peaks and valleys are, the more capable the model becomes of correctly predicting the orientations. Combining several elevation angles is a good way to take advantage of the different capabilities and get the best results. Figure 11 shows a parity plot displaying the error relevance between each elevation. The correlation coefficients are shown in Table IV. It is seen that the overlap of the predicted results differs depending on elevation. 45° and 60° have more correlation than 30° and 60°. The smaller the correlation is, the more effective it is when these elevations are combined.

Data augmentation resulted in an error median of 2.85°. This is a relatively good score, given that the input images were taken within every 5° of rotation. Combining these two techniques, changing elevations and augmentation, it is possible to adequately measure orientations.

We have established a rapid and accurate measurement of crystallographic orientations by optical images and a series of data science-based techniques. This technique will enable an analysis of a full ingot and its crystal characteristics, which will help us gain a holistic understanding of crystal growth. We believe that this work will contribute to controlling material structure through better fabrication processes and, consequently, enhancing material properties. In future studies, we aim to apply this technique to various multicrystalline materials.

We developed a method to rapidly and accurately measure crystal grain orientations in multicrystalline materials. This was made possible by machine learning techniques and a home-made apparatus for taking high-resolution optical images. By increasing data and their diversity, the prediction accuracy was successfully enhanced to an error median of 2.85°. The results also show that there is an optimal angle for collecting data, associated with the physical configuration of the incident light and surface structures of the grain. Future work seeks an improved prediction for various materials, leading to a clear understanding of crystal structures and how their behaviors affect material properties.

LSTM is known to be sensitive to the selection of hyperparameters and weight initializations, often causing overfitting. In the proposed model, hyperparameters were chosen based on validation accuracy, and no overfitting was observed. The model’s behaviors when initial weights change were examined as well, verifying that the model is unsusceptible to weight initializations. These results are supported by a supplementary material file containing a learning curve and a table showing the changes when trained with different weight initializations.

This work was supported by JST CREST (Grant No. JPMJCR17J1).

The authors have no conflicts to disclose.

Kyoka Hara: Conceptualization (equal); Data curation (lead); Investigation (lead); Methodology (equal); Validation (equal); Writing – original draft (lead). Takuto Kojima: Data curation (supporting); Methodology (equal); Resources (equal); Software (lead); Supervision (supporting); Validation (equal); Writing – review & editing (supporting). Kentaro Kutsukake: Supervision (supporting); Validation (equal); Writing – review & editing (supporting). Hiroaki Kudo: Methodology (equal); Resources (equal); Supervision (supporting); Validation (equal); Writing – review & editing (supporting). Noritaka Usami: Funding acquisition (lead); Project administration (lead); Resources (equal); Supervision (lead); Writing – review & editing (lead).

The data that support the findings of this study are openly available in UsamiCREST/CO-prediction at https://github.com/UsamiCREST/CO-prediction.