Discovery of 2d materials capable of second harmonic generation

Abstract

In one embodiment, there is provided a method for discovering a two-dimensional (2D) material capable of second harmonic generation (SHG). The method includes training, by a training module, an SHG material discovery module based, at least in part, on one or more training machine learning (ML) descriptors. The method further includes predicting, by the trained SHG material discovery module, an SHG for a candidate material based, at least in part, on a received material descriptor. The received material descriptor is associated with the candidate material.

Description

[0001] DISCOVERY OF 2D MATERIALS CAPABLE OF SECOND HARMONIC GENERATION

[0002] CROSS REFERENCE TO RELATED APPLICATION(S)

[0003] This application claims the benefit of U.S. Provisional Application No. 63 / 548,064, filed November 10, 2023, and U.S. Provisional Application No. 63 / 708,433, filed October 17, 2024, which are incorporated by reference as if disclosed herein in their entireties.

[0004] FIELD

[0005] The present disclosure relates to discovery, in particular to, discovery of two- dimensional (2D) materials capable of second harmonic generation.

[0006] BACKGROUND

[0007] The investigation of two-dimensional (2D) materials have engendered a new era in material science, leading to possibilities for applications that span from flexible electronics to quantum computing. Interest in 2D materials increased following the isolation of graphene in 2004 due to their unique electronic, thermal, and mechanical properties. This interest has since extended to other 2D materials, which exhibit distinctive properties when scaled down to the mono-layer limit. The allure of these materials lies not only in their rich physical phenomena but also in the potential to engineer their properties for tailored functionalities.

[0008] Interesting aspects of 2D materials include, but are not limited to, their nonlinear optical (NLO) properties. Second harmonic generation (SHG) is one aspect of 2D materials that is of particular interest. SHG is an NLO process that may be beneficial for a number of applications including, but not limited to, laser technology, bio-imaging, and optical communication systems.

[0009] Discovering novel materials with desirable properties in a relatively large materials space can be challenging. Existing techniques including, for example, experiments or first- principles quantum calculations such as Density Functional Theory (DFT) can be costly due to high computational costs and the relatively laborious nature of experimental methods. Additionally or alternatively, access to supercomputers has expedited exploration of the material space, thus resulting in an increasing number of novel materials in recent years. SUMMARY

[0010] In some embodiments, there is provided a method for discovering a two-dimensional (2D) material capable of second harmonic generation (SHG). The method includes training, by a training module, an SHG material discovery module based, at least in part, on one or more training machine learning (ML) descriptors. The method further includes predicting, by the trained SHG material discovery module, an SHG for a candidate material based, at least in part, on a received material descriptor. The received material descriptor is associated with the candidate material.

[0011] In some embodiments, the method further includes extracting, by an extraction module, training SHG materials data, the training SHG materials data including second order susceptibility data, and crystal structure data.

[0012] In some embodiments, the method further includes constructing, by a descriptor construction module, the training machine learning descriptors based, at least in part, on one or more elemental properties.

[0013] In some embodiments, the method further includes retrieving, by a discovery management module, materials data from a selected database of 2D materials. The materials data includes one or more of thermodynamic data, elastic data, electronic data, magnetic data and / or optical data.

[0014] In some embodiments of the method, the SHG material discovery module includes a classifier module and a regression module.

[0015] In some embodiments of the method, the extracting includes applying a natural logarithm to the training SHG materials data.

[0016] In some embodiments of the method, the elemental properties are selected from the group including dipole polarizability, ionization energy, electron affinity, atomic radius, van der Waals radius, and a number of valence electrons.

[0017] In some embodiments of the method, the descriptors include at least one elemental property and one or more of a crystal structure, an effective thickness, and / or a Phillips ionicity.

[0018] In some embodiments of the method, the constructing includes performing a mathematical transformation on at least one of the elemental properties.

[0019] In some embodiments, there is provided a second harmonic generation (SHG) material discovery system for discovering a two-dimensional (2D) material capable of SHG. The system includes a computing device including a processor, a memory, an input / output circuitry, and a data store. The system further includes a training module configured to train an SHG material discovery module based, at least in part, on one or more training machine learning (ML) descriptors. The trained SHG material discovery module is configured to predict an SHG for a candidate material based, at least in part, on a received material descriptor. The received material descriptor is associated with the candidate material.

[0020] In some embodiments, the system further includes an extraction module configured to extract training SHG materials data. The training SHG materials data includes second order susceptibility data, and crystal structure data.

[0021] In some embodiments, the system further includes a descriptor construction module configured to construct the training machine learning descriptors based, at least in part, on one or more elemental properties.

[0022] In some embodiments, the system further includes a discovery management module configured to retrieve materials data from a selected database of 2D materials, the materials data including one or more of thermodynamic data, elastic data, electronic data, magnetic data and / or optical data.

[0023] In some embodiments of the system, the SHG material discovery module includes a classifier module and a regression module.

[0024] In some embodiments of the system, the extraction module is configured to apply a natural logarithm to the training SHG materials data.

[0025] In some embodiments of the system, the elemental properties are selected from the group including dipole polarizability, ionization energy, electron affinity, atomic radius, van der Waals radius, and a number of valence electrons.

[0026] In some embodiments of the system, the descriptors include at least one elemental property and one or more of a crystal structure, an effective thickness, and / or a Phillips ionicity.

[0027] In some embodiments of the system, the descriptor construction module is configured to perform a mathematical transformation on at least one of the elemental properties.

[0028] In some embodiments of the system, the descriptor construction module is further configured to perform dimensionality reduction on a representation of a crystal structure.

[0029] In some embodiments, there is provided a computer readable storage device. The device has stored thereon instructions that when executed by one or more processors result in the following operations including any embodiment of the method. BRIEF DESCRIPTION OF DRAWINGS

[0030] The drawings show embodiments of the disclosed subject matter for the purpose of illustrating features and advantages of the disclosed subject matter. However, it should be understood that the present application is not limited to the precise arrangements and instrumentalities shown in the drawings, wherein:

[0031] FIG. 1 illustrates a functional block diagram of a second harmonic generation (SHG) material discovery system, according to several embodiments of the present disclosure;

[0032] FIG. 2 is a flowchart of operations for training an SHG material discovery system, according to various embodiments of the present disclosure; and

[0033] FIG. 3 is a flowchart of operations for predicting a second-order susceptibility (x(2)) using a trained SHG material discovery system, according to various embodiments of the present disclosure.

[0034] Although the following Detailed Description will proceed with reference being made to illustrative embodiments, many alternatives, modifications, and variations thereof will be apparent to those skilled in the art.

[0035] DETAILED DESCRIPTION

[0036] Generally, this disclosure relates to an apparatus, method, and / or system for discovery of two-dimensional (2D) materials capable of second harmonic generation (SHG). The apparatus, method, and / or system are configured to utilize efficient machine learning and training data extracted from existing materials databases to discover materials capable of SHG. The apparatus, method, and / or system are further configured to quantify the SHG of promising discovered materials. The apparatus, method, and / or system may include one or more machine learning modules configured to implement one or more machine learning models.

[0037] In an embodiment according to the present disclosure, a classifier module and a regression module may be implemented. The classifier module may be configured to classify a material according to whether or not the material exhibits a nonzero second order optical susceptibility. As is known, a material with a second order susceptibility of zero may not be capable of SHG. The regression module may be configured to predict a value for second order susceptibility for materials with nonzero second order susceptibility.

[0038] Advantageously, an SHG material discovery system, according to the present disclosure, may identify SHG materials using less computational load than, for example, DFT. Additionally or alternatively, analysis of elements of corresponding machine learning descriptors may provide insight into which elemental properties impact the SHG.

[0039] By way of theoretical background, interesting aspects of 2D materials include, but are not limited to, their nonlinear optical (NLO) properties. Second harmonic generation (SHG) is one aspect of 2D materials that is of particular interest. SHG is an NLO process that may be beneficial for a number of applications including, but not limited to, laser technology, bioimaging, and optical communication systems. In the SHG process, two incident photons at a frequency co generate an emitted photon at a frequency of 2co. As is known, a material capable of SHG has a nonzero 2nd-order susceptibility (%(2)). Whether a 2D material is capable of SHG is related to a lack of inversion symmetry and an atomic thickness. A 2D material that is capable of SHG may have a corresponding enhanced NLO response.

[0040] A 2D material’s capability for SHG may vary with input energy, i.e., pump photon energy. Energy ranges that have practical applications may include, but are not limited to, a relatively low energy range (0.4 to 0.9 eV (electron-volts)), a middle energy range (0.9 to 1.6 eV), and a relatively high energy range (1.6 to 2.1 eV). The low-energy range corresponds to mid- to near-infrared wavelengths, useful for sensing and telecommunications. The middle range corresponds to visible light applications. The high-energy range is useful for relatively deep-UV (ultraviolet) light generation. It may be appreciated that these three energy ranges are merely illustrative and that other energy ranges may be implemented, within the scope of the present disclosure. These energy ranges illustrate a potential of 2D materials to improve nonlinear optical devices across a spectrum of applications.

[0041] The exploration of the material space, and resulting increase in the number of novel materials are related to a corresponding increase in a number of materials databases. It may be appreciated that the increase in number of materials databases and access to relatively efficient machine learning (ML) algorithms has spurred development of materials informatics. Applying ML to materials science may be used to accelerate material discovery and material property prediction. Material property prediction may include, but is not limited to, electronic properties, magnetic properties, thermal properties, optical properties, and / or topological properties. Training ML models on available SHG data may facilitate analyzing a relatively large chemical and structural space of 2D materials to identify candidates likely to exhibit relatively strong SHG responses in a specified energy range. This approach may support a systematic search for novel materials tailored for specific NLO applications.

[0042] As will be described in more detail below, supervised learning, e.g., using an Extra Trees ensemble algorithm may be used to develop a robust predictive model. As is known, the Extra Trees ensemble technique may include constructing a number of decision trees to develop a robust predictive model. In one nonlimiting example, the Extra Trees ensemble technique may be applied to data from the Computational 2D Materials Database (C2DB). The Extra Trees ensemble technique may facilitate a relatively identification of materials with favorable SHG characteristics. The predictive power of these models may facilitate learning patterns and features that enhance understanding the nonlinear optical response in materials. In another nonlimiting example, the Extra Trees ensemble technique may be used to predict the SHG properties of materials listed in another database, e.g., the 2Dmatpedia database, thus broadening the scope of the SHG material discovery system, as described herein, and leveraging it for materials discovery.

[0043] In an embodiment, there is provided a method for discovering a two-dimensional (2D) material capable of second harmonic generation (SHG). The method includes training, by a training module, an SHG material discovery module based, at least in part, on one or more training machine learning (ML) descriptors. The method further includes predicting, by the trained SHG material discovery module, an SHG for a candidate material based, at least in part, on a received material descriptor. The received material descriptor is associated with the candidate material.

[0044] FIG. 1 illustrates a functional block diagram of a second harmonic generation (SHG) material discovery system 100, according to several embodiments of the present disclosure. System 100 includes an SHG material discovery module 102, a discovery management module 104, a computing device 106, and may include a training module 108. SHG material discovery module 102, discovery management module 104, and / or training module 108 may be coupled to or included in computing device 106.

[0045] SHG material discovery module 102 is configured to receive discovery input data 101. Discovery input data 101 may correspond to prediction input data 103 or training input data 107, as will be described in more detail below. Discovery management module 104 is configured to receive the prediction input data 103 and to provide the corresponding discovery input data 101 to the SHG material discovery module 102. During training, training module 108 is configured to receive the training input data 107 and to provide the corresponding discovery input data 101 to the SHG material discovery module 102.

[0046] During operation, SHG material discovery module 102 is configured to provide a classifier binary output 134 and / or a regression output 127 to the discovery management module 104. The discovery management module 104 may then be configured to provide the classifier binary output 134 and / or the regression output 127 as prediction output 105. During training, SHG material discovery module 102 is configured to provide the classifier binary output 134 and / or the regression output 127 to the training module 108.

[0047] SHG material discovery module 102 includes an extraction module 120, a descriptor construction module 122, a classifier module 124 and a regression module 126. The extraction module 120 is configured to receive the discovery input data 101 and to provide extracted data 121 to the descriptor construction module 122. The descriptor construction module 122 is configured to construct a descriptor 123 based, at least in part, on the extracted data 121, and to provide the descriptor 123 to the classifier module 124. The classifier module 124 is configured to provide a classifier binary output 134 to the discovery management module 104 and / or the training module 108. The classifier module 124 may be further configured to provide a classifier output 125 to the regression module 126. The classifier output 125 may include the descriptor 123 and / or the classifier binary output 134. The regression module 126 is configured to provide the regression output 127, e.g., a predicted second order susceptibility, to the discovery management module 104 and / or the training module 108. The discovery management module 104 may then be configured to provide as the prediction output 105, the classifier binary output 134 or the regression output 127. The operations of the SHG material discovery module 102 will be described in more detail below.

[0048] Computing device 106 may include, but is not limited to, a computing system (e.g., a server, a workstation computer, a desktop computer, a laptop computer, a tablet computer, an ultraportable computer, an ultramobile computer, a netbook computer and / or a subnotebook computer, etc.). Computing device 106 includes a processor 110, a memory 112, input / output (I / O) circuitry 114, a user interface (UI) 116, and data store 118.

[0049] Processor 110 is configured to perform operations of SHG material discovery module 102, discovery management module 104, and / or training module 108. Memory 112 may be configured to store data associated with SHG material discovery module 102, discovery management module 104 and / or training module 108. I / O circuitry 114 may be configured to provide wired and / or wireless communication functionality for SHG material discovery system 100. For example, I / O circuitry 114 may be configured to prediction input data 103 and / or training input data 107 and to provide prediction output data 105. UI 116 may include a user input device (e.g., keyboard, mouse, microphone, touch sensitive display, etc.) and / or a user output device, e.g., a display. Data store 118 may be configured to store one or more of prediction input data 103, training input data 107, prediction output data 105, machine learning parameters 109 (i.e., hyperparameters and / or parameters associated with classifier module 124 and / or regression module 126), and / or other data associated with SHG material discovery module 102, discovery management module 104 and / or training module 108. Other data may include, for example, a listing of possible elements of descriptor 123, as well as an indication of the elements included in the descriptor 123. However, this disclosure is not limited in this regard.

[0050] SHG material discovery module 102 may be trained prior to being used to predict a second order susceptibility for a candidate material. Training may be supervised, unsupervised or semi-supervised. Training data for supervised training operations typically include a plurality of data sets that include discovery input data, e.g., discovery input data 101, as well as a corresponding target output for each data set. In an embodiment, discovery input data may include materials data (e.g., thermodynamic, elastic, electronic, magnetic and / or optical properties) and corresponding output data may include an associated second order susceptibility. Training operations may then include setting one or more hyperparameters associated with classifier module 124 and / or regression module 126. Training operations may further include adjusting one or more parameters associated with classifier module 124 and / or regression module 126 to reduce and / or minimize error between a calculated output and a target output included with the training data. After training, the trained SHG material discovery module 102 (e.g., classifier module 124 and regression module 126) may be used to predict a second order susceptibility for a previously unknown material. The predicting may be based, at least in part, on a descriptor constructed by the descriptor construction module using materials data associated with the previously unknown material.

[0051] Generally, training operations may include selecting a database of 2D materials data and then retrieving and / or acquiring materials data. The materials data may include materials properties (e.g., thermodynamic, elastic, electronic, magnetic, and / or optical properties), chemical composition as well as crystal structure information. Second order susceptibility data and crystal structure data may then be extracted. Machine learning descriptors (i.e., materials descriptors) may then be constructed based, at least in part, on elemental properties and based, at least in part, on crystal structure. The machine learning models, e.g., classification module 124, and regression module 126, may then be trained using second order susceptibility data across a plurality of ranges of photon pump energy as the target. In one nonlimiting example, the machine learning models may correspond to extra tree ensemble techniques. Training may begin with selecting a materials database that contains materials data for a plurality of 2D materials. In one nonlimiting example, the materials database may correspond to C2DB (Computational 2D Materials Database, maintained by the Computational Materials Repository, https: / / cmr.fysik.dtu.dk / index.htm). However, this disclosure is not limited in this regard. Other databases with materials data, as described herein, may be selected. The selecting may be performed by and / or using training module 108.

[0052] Materials data and ground-state crystal structures may then be retrieved and / or acquired from the selected database, i.e., may be mined from the selected database, e.g., C2DB. The retrieving / acquiring may be performed by and / or using training module 108.

[0053] 2nd-order susceptibility (%(2)) data and crystal structure data may then be extracted from the selected database of materials data. Machine-learning descriptors may then be constructed from the extracted materials data. The descriptors may be constructed based, at least in part, on elemental properties as well as crystal structure. A machine learning (ML) module may then be trained using a supervised machine learning technique based, at least in part, on training materials data. The materials data may include target second order susceptibility, %(2). Materials data from a number of materials may be used. In one nonlimiting example, the number of materials may be on the order of hundreds of materials (e.g., 700). In some embodiments, the training may be performed across one or more (e.g., three) ranges of photon pump energy, with each range corresponding to a respective application. After training, the SHG materials discovery module 102 may then be used to predict the NLO (nonlinear optical) properties of 2D materials in other databases.

[0054] In the following, reference is made to the C2DB database and characteristics specific to this materials database. It should be noted that the training operations described herein may be implemented with other materials databases and such operations are within the scope of this disclosure.

[0055] C2DB is an open-source database that contains on the order of thousands of 2D material structures and their thermodynamic, elastic, electronic, magnetic, and optical properties. As is known, these material properties are determined based, at least in part, on DFT (density functional theory) and based, at least in part on, many-body perturbation theory. It may be appreciated that the C2DB materials data is calculated without spin-orbit coupling. Additionally or alternatively, the C2DB data does not consider the scissor shift in its original calculations. It may be appreciated that DFT may underestimate the electronic band gaps of materials due to self-interaction error and a lack of a proper exchange-correlation potential. This limitation may be significant in 2D materials where quantum confinement and body effects alter electronic properties. A scissor shift is a common correction applied in post-DFT calculations to adjust the underestimated band gaps obtained from standard DFT calculations. The scissor shift may include shifting the conduction band uniformly in energy, without altering the wave functions, to better match experimental data or more accurate calculations like those from GW approximation.

[0056] It may be appreciated that bulk susceptibility (units of nm V1) is not well defined for monolayer 2D materials, thus the C2DB uses sheet susceptibility (units of nm2V1). Continuing with the C2DB, retrieving materials data may include selecting materials whose materials data includes2)data. For C2DB, the materials data for approximately 2% of the total materials (345 materials out of 15,733) includes %(2)data. For model performance evaluation purposes, an equal amount of materials without %(2)data may be selected, yielding a dataset of 690 materials total. For the materials with %(2)data, a tensor component, and its respective spectra, may be selected that exhibit a relatively highest nonlinear optical response for that material. The peak %(2)of the SHG spectra in each of the three specified photon pump energy ranges may then be extracted. For example, for a monolayer of material (MoTe?) and its corresponding spectrum, an absolute value of Xyyycomponent for monolayer MoTe? provided the peak2), for each of three energy ranges. Continuing with this example, the energy ranges may include 0.4-0.9eV range, 0.9-1.6eV range and 1.6-2. leV range. However, this disclosure is not limited in this regard.

[0057] It may be appreciated that, for some materials databases, the distribution of SHG data may be skewed towards zero, presenting challenges for machine learning models due to the limited variability and high kurtosis. High kurtosis in the distribution can lead to model overfitting, particularly as the tails of the distribution dominate the learning process. A natural logarithmic transformation may be applied to the %(2)values to address the high kurtosis, resulting in a more Gaussian distribution. It may be appreciated that Gaussian distributions facilitate the inference of conditional distributions, useful for probabilistic predictions. This transformation is advantageous because Gaussian distributions are fully parameterized by their mean and variance, simplifying the statistical handling of data. This approach may thus mitigate skewness and tail dominance in the data. The machine learning (i.e., materials) descriptors may then be constructed. It may be appreciated that the choice of descriptors affects a predictive capability of machine learning models in materials science. In an embodiment, descriptors may be constructed based, at least in part, on elemental properties and based, at least in part, on crystal structure information of 2D materials. Elemental properties may include, but are not limited to, dipole polarizability, ionization energy, electron affinity, atomic radius, van der Waals radius, and a number of valence electrons. In one nonlimiting example, one or more elemental properties may be extracted using a mendeleev Python package (Python Software Foundation https: / / www.python.org / about / and / or https: / / pypi.org / project / mendeleev / . The elemental properties may be further processed, e.g., via mathematical transformations including, but not limited to, calculating the mean, standard deviation, maximum difference, and sum of these properties for each material. One or more of the processed elemental property values may then be included in the machine learning descriptor.

[0058] Crystal structure may affect determination of the properties of 2D materials including, but not limited to, electronic properties, optical properties and mechanical properties. In one nonlimiting example, a Sine Coulomb Matrix (SCM) descriptor from the DScribe package may be utilized to represent the crystal structure. DScribe is a Python package for transforming atomic structures into fixed-size numerical fingerprints, i.e., “descriptors” available at https: / / singroup.github.io / dscribe / latest / . The SCM may be determined as: where Z, and are the atomic number and the position of the ithatom. B is a matrix formed by the lattice vectors and ekare the cartesian unit vectors. The SCM can capture properties of the Coulomb interaction, such as the periodicity of the crystal lattice. In some embodiments, due to a relatively high dimensionality of the SCM, principal component analysis (PCA) may be used to reduce its dimensions. This reduction is configured to make the SCM effective for use in machine learning.

[0059] The structural descriptors may further include effective thickness (Zeff) of the material. The effective thickness may be determined by considering maximum and minimum z-coordinates (zmax and zmin) of the atoms in the material, as well as the van der Waals radii (vdwmax and vdwmin) of the atoms at these positions in angstroms. The effective thickness is given by:

[0060] Along with the elemental properties and crystal structure descriptors, a Phillips Ionicity of the atomic bonds may be determined. Phillips Ionicity may provide insight into the electronic properties of the material, as it measures the ionic character of a bond. The Phillips Ionicity for a bond between atoms A and B is given by: where XA and XB are the electronegativities of atoms A and B in the material. The average Phillips Ionicity for a material may then be calculated as the mean ionicity of all bonds in the material.

[0061] The materials descriptors, including elemental properties, crystal structure information, and Phillips Ionicity, thus form the basis of the machine learning models.

[0062] Each descriptor Di for the zthstructure may be defined based on a range of atomic properties, including, but not limited to, dipole polarizability, ionization energy, atomic radius, number of valence electrons, electronegativity, atomic volume, van der Waals radius, covalent radius, lattice constant, electron affinity, C& dispersion coefficient, and chemical hardness. In one nonlimiting example, the atomic properties may be sourced from the mendeleev Python package, as described herein.

[0063] The function / is configured to aggregate these properties through a variety of statistical measures including, but not limited to, mean, variance, and maximum differences. The statistical measures may enhance the descriptor set with comprehensive data on each material’s intrinsic characteristics. In some embodiments, machine learning descriptors may further include a Space group number. In some embodiments, the Sine Coulomb Matrix may be constructed using the Dscribe Python package, as described herein, to enhance the materials (including, but not limited to, molecules, crystal structure and / or corresponding composites) representations. In some embodiments, a principal component analysis dimensionality reduction technique may then be applied to the Sine Coulomb Matrix configured to reduce their dimensionality to facilitate machine learning. In some embodiments, an initial materials descriptor set that included 108 descriptors may be reduced using backward elimination techniques. In one nonlimiting example, a final descriptor set for the classification model was 9 descriptors. For the regression model, the final descriptor set was 15 descriptors.

[0064] In an embodiment, the classifier module 124 and the regression module 126 may be implemented as respective machine learning models using the decision-tree-based techniques. The machine learning models are configured to predict the SHG properties of 2D materials. The classifier module corresponds to a classification model configured to predict whether or not a material will exhibit SHG. In other words, the classification model was trained to distinguish between materials that exhibit SHG and those that do not. The regression module corresponds to a regression model configured to predict the2)of the material. In other words, the regression model was developed to predict the %(2)of materials that exhibit SHG. In some embodiments, the models may be optimized using grid search with 5-fold cross- validation to identify appropriate hyperparameters.

[0065] In some embodiments, a regressor chain model may be employed configured to enhance the prediction of multi-target properties. This approach involves sequentially modeling each target variable while using the predictions of previous target variables as additional inputs. By capturing the dependencies among the multiple target variables, the regressor chain model aims to improve the prediction accuracy for each %(2)energy range, reflecting the interconnected nature of these properties in 2D materials.

[0066] In some embodiments, to optimize the predictive models, descriptors may be selected using recursive feature elimination (RFE) with cross-validation. This method iteratively removes the least important features, based on model performance, to enhance the overall predictive accuracy. A 5-fold cross-validation scheme with implemented within RFE to enhance robustness. The final models may then be evaluated on a separate test dataset to confirm their ability to generalize to new, unseen data.

[0067] FIG. 2 is a flowchart 200 of operations for training an SHG material discovery system, according to an embodiment of the present disclosure. In particular, the flowchart 200 illustrates training a SHG material discovery module. The operations may be performed, for example, by the SHG material discovery system 100 (e.g., SHG material discovery module 102, discovery management module 104, and / or training module 108) of FIG. 1.

[0068] Operations of this embodiment may begin with selecting a database of materials data at operation 202. Materials data may be retrieved and / or acquired at operation 204. Operation 206 includes extracting SHG materials data. SHG materials data may include second order susceptibility data and crystal structure data. Machine learning (e.g., materials) descriptors may be constructed at operation 208. At least one machine learning model may be trained at operation 210. The machine learning model may be selected from the group including a classifier model and a regression model. In some embodiments, the trained SHG material discovery system may be verified at operation 212. Program flow may then end at operation 214.

[0069] Thus, a SHG material discovery module may be trained.

[0070] FIG. 3 is a flowchart 300 of operations for predicting a second-order susceptibility (X(2)) using a trained SHG material discovery system, according to various embodiments of the present disclosure. In particular, the flowchart 300 illustrates predicting a second-order susceptibility (x(2)) using a trained SHG material discovery system. The operations may be performed, for example, by the SHG material discovery system 100 (e.g., SHG material discovery module 102, discovery management module 104, and / or training module 108) of FIG. 1.

[0071] Operations of this embodiment may begin with selecting a database of materials data at operation 302. Materials data may be retrieved and / or acquired at operation 304. Operation 306 includes extracting SHG materials data. SHG materials data may include crystal structure data. Machine learning (e.g., materials) descriptors may be constructed at operation 308. At least one descriptor may be provided to the trained machine learning models at operation 310. The machine learning model may be selected from the group including a classifier model and a regression model. A second order susceptibility may be predicted at operation 312. Program flow may then end at operation 314.

[0072] Thus, a second-order susceptibility (x(2)) of a 2D material may be predicted using a trained SHG material discovery system.

[0073] Generally, this disclosure relates to an apparatus, method, and / or system for discovery of two-dimensional (2D) materials capable of second harmonic generation (SHG). The apparatus, method, and / or system are configured to utilize efficient machine learning and training data extracted from existing materials databases to discover materials capable of SHG. The apparatus, method, and / or system are further configured to quantify the SHG of promising discovered materials. The apparatus, method, and / or system may include one or more machine learning modules configured to implement one or more machine learning models. Advantageously, an SHG material discovery system, according to the present disclosure, may identify SHG materials using less computational load than, for example, DFT. Additionally or alternatively, analysis of elements of corresponding machine learning descriptors may provide insight into which elemental properties impact the SHG and / or insight into one or more characteristics of the crystal structure that have previously unknown impact on the SHG.

[0074] Experimental data

[0075] Density functional theory calculations were performed using the Abinit package. The projector augmented wave (PAW) type of pseudopotential along with the local density approximation (LDA) exchange-correlation functional was used. A plane wave basis set of energy cutoff 408 eV was used for the expansion of electronic wavefunction. The Gammacentered k-point grids of 12 x 12 x 1 and 24 x 24 x 1 were used for the geometrical optimizations and second-order susceptibility (%(2)) calculations, respectively. To avoid the vdW interactions between any periodic replicas, a vacuum space of at least 20 A was used along the z-axis of all promising candidate structures. The electronic and force convergence criteria were set to 108eV and 103eV / A respectively. To obtain smooth SHG spectra, a smearing of 0.0272 eV was applied. The %(2)components were calculated within the independent particle approximation. The expressions for %(2)were derived within the dipole approximation.

[0076] The Extra Trees classification model was evaluated on a test set, yielding an accuracy of 0.92 and an Fl score of 0.93. A confusion matrix illustrated a relatively high number of true positives and true negatives.

[0077] Experimental results indicated that a material’s intrinsic atomic characteristics and crystal symmetry help determine its SHG properties.

[0078] In some embodiments, the regression model may be implemented using a regressor chain approach with Extra Trees. This approach allows each model in the chain to use all available features along with the predictions from previous models in the chain, enhancing predictive accuracy by incorporating interdependencies among targets.

[0079] It may be appreciated that the physical significance of each descriptors may offer insights into the mechanisms that govern SHG. The ‘ionization standard deviation’ is indicative of the energy needed to displace an electron from an isolated atom or molecule, which affects how the electron cloud within the material responds to an external electric field — a relatively important factor in SHG. Materials with a higher threshold for ionization are likely to have a strong electron binding, which can enhance their nonlinear optical behavior due to increased polarization upon interaction with light. The ‘space group number’ conveys symmetry information about the crystal structure. Certain symmetries or the lack thereof are known to either enable or suppress SHG. In particular, non-centrosymmetric materials are useful for SHG as they allow for asymmetric electronic distributions that can lead to a net dipole moment under an electric field. The ‘lattice constant average’ may reflect the periodicity within the material’s crystal lattice, which plays a role in the dispersion of electronic bands. Variations in the crystal lattice can modify the band gap, as well as a material’s SHG response. ‘Philips ionicity’ provides insights into the ionic versus covalent nature of bonding within the material. This influences the material’s dielectric response and contributes to the effective polarization — this response has been shown to correlate with the strength of the SHG signal. A higher ionicity implies a greater charge separation, which could facilitate stronger SHG responses.

[0080] For the regression model, the descriptor ‘Zeff’, representing the effective thickness, reflects how the thickness of a material influences its optical properties. This parameter could be relevant to phase-matching conditions necessary for efficient SHG. The principal components of the Sine Coulomb matrix signify the influence of the arrangement and charge of atoms in a material on its SHG. The interactions captured by this descriptor are useful in determining how a material’s internal electrostatic environment can affect its susceptibility to second-order non-linear effects.

[0081] It may be appreciated that these descriptors suggest which characteristics affect SHG. By encoding both the atomic properties and the crystal structure of materials, these features guide the material selection and design process for optimizing non-linear optical responses.

[0082] An SHG material discovery system, trained as described herein, was used to predict the SHG properties of materials within the 2Dmatpedia database. 2Dmatpedia is a comprehensive open-source repository, hosting over 6,000 monolayer structures identified using both top-down and bottom-up discovery methods. These approaches not only cover the exfoliation of layers from existing 3D materials but also explore the relatively vast chemical space through systematic elemental substitutions, expanding the scope of potential 2D materials beyond those found in traditional databases. To identify potential candidates for SHG applications, materials were filtered to exclude those with inversion symmetry and those without a bandgap. The predictions were then transformed from peak sheet susceptibility (units of nmV2) to peak bulk susceptibility (units of nm V ') for comparison with DFT results. The predicted SHG properties identified materials with promising non- linear optical properties. The top 10 materials with the highest predicted SHG susceptibilities are listed in Table 1, laying the groundwork for future Density Functional Theory (DFT) validations and experimental verifications. Two of the materials (VS2, In2Se3) highlighted have been either experimentally or computationally proven to have a second harmonic response. Advantageously, an SHG material discovery system, according to the present disclosure may be used to search for 2D materials that exhibit SHG and may thus accelerate the discovery of new materials suited for non-linear optical applications.

[0083] Table 1

[0084] It may be appreciated that the promising candidate monolayers have space group of P- 62m and the group number 189 with D311 symmetry. The materials in this space group have the following non-zero2)components: yyy = yxx = xxy = xyx. The calculated SHG response with the highest susceptibility was xy2yy(2w, w, w) |for the candidate structures. Before calculating the SHG spectra for promising candidates, the SHG spectrum for monolayer 2H-M0S2 was reproduced. Experimental data illustrates that monolayers of SnC, B2S3, B2Se3, and Ga2Te3 have larger second harmonic responses than monolayer 2H-M0S2. It may be appreciated that the effective bulk susceptibility of the promising candidates in nmV1may be determined by multiplying the intensity of the SHG spectra (i.e. y-axis) by a dimensionless quantity Lz / L , where Lzis the length of the simulation cell along the z-axis and L is the effective thickness of the cell. The scaling factors thus obtained are 4.93, 5.75, 5,45, 5.55, 5.15 for SnC, B2S3, 82863, AhSes, and Ga2Te3 respectively. It should be noted that the DFPT calculations were performed only to evaluate the accuracy of the SHG material discovery system, as described herein, in predicting the SHG spectra.

[0085] It may be appreciated that the Extra Trees model, implemented as described herein has facilitated identifying descriptors that correlate with SHG. For example, descriptors including, but not limited to, total ionization energy, space group number, average lattice constant, and Philips ionicity were useful in identifying materials that exhibit SHG. The regression models illuminated how variations in valence electron configuration and the material’s effective thickness impact SHG intensity. It is contemplated that these insights may offer a pathway to accelerate the search for materials with optimized non-linear optical properties.

[0086] It is contemplated that a predictive framework corresponding to the SHG material discovery system, as described herein, can be expanded to include a broader range of materials classes including, but not limited to, organic and inorganic materials, types of crystal structures, heterostructures, and / or stacks of existing materials. The methodology is adaptable and can be tailored to explore materials with diverse crystal structures and composite systems. As such, it holds promise for accelerating the design and discovery of new materials with bespoke optical properties, particularly those suited for advanced photonic applications.

[0087] As used in any embodiment herein, the terms “logic” and / or “module” may refer to an app, software, firmware and / or circuitry configured to perform any of the aforementioned operations. Software may be embodied as a software package, code, instructions, instruction sets and / or data recorded on non-transitory computer readable storage medium. Firmware may be embodied as code, instructions or instruction sets and / or data that are hard-coded (e.g., nonvolatile) in memory devices.

[0088] “Circuitry”, as used in any embodiment herein, may include, for example, singly or in any combination, hardwired circuitry, programmable circuitry such as computer processors comprising one or more individual instruction processing cores, state machine circuitry, and / or firmware that stores instructions executed by programmable circuitry. The logic and / or module may, collectively or individually, be embodied as circuitry that forms part of a larger system, for example, an integrated circuit (IC), an application-specific integrated circuit (ASIC), a system on-chip (SoC), desktop computers, laptop computers, tablet computers, servers, smart phones, etc. Memory 112 may include one or more of the following types of memory: semiconductor firmware memory, programmable memory, non-volatile memory, read only memory, electrically programmable memory, random access memory, flash memory, magnetic disk memory, and / or optical disk memory. Either additionally or alternatively system memory may include other and / or later-developed types of computer-readable memory.

[0089] Embodiments of the operations described herein may be implemented in a computer- readable storage device having stored thereon instructions that when executed by one or more processors perform the methods. The processor may include, for example, a processing unit and / or programmable circuitry. The storage device may include a machine readable storage device including any type of tangible, non-transitory storage device, for example, any type of disk including floppy disks, optical disks, compact disk read-only memories (CD-ROMs), compact disk rewritables (CD-RWs), and magneto-optical disks, semiconductor devices such as read-only memories (ROMs), random access memories (RAMs) such as dynamic and static RAMs, erasable programmable read-only memories (EPROMs), electrically erasable programmable read-only memories (EEPROMs), flash memories, magnetic or optical cards, or any type of storage devices suitable for storing electronic instructions.

[0090] The terms and expressions which have been employed herein are used as terms of description and not of limitation, and there is no intention, in the use of such terms and expressions, of excluding any equivalents of the features shown and described (or portions thereof), and it is recognized that various modifications are possible within the scope of the claims. Accordingly, the claims are intended to cover all such equivalents.

[0091] Various features, aspects, and embodiments have been described herein. The features, aspects, and embodiments are susceptible to combination with one another as well as to variation and modification, as will be understood by those having skill in the art. The present disclosure should, therefore, be considered to encompass such combinations, variations, and modifications.

Claims

CLAIMSWhat is claimed is:

1. A method for discovering a two-dimensional (2D) material capable of second harmonic generation (SHG), the method comprising: training, by a training module, an SHG material discovery module based, at least in part, on one or more training machine learning (ML) descriptors; and predicting, by the trained SHG material discovery module, an SHG for a candidate material based, at least in part, on a received material descriptor, the received material descriptor associated with the candidate material.

2. The method of claim 1, further comprising extracting, by an extraction module, training SHG materials data, the training SHG materials data comprising second order susceptibility data, and crystal structure data.

3. The method of claim 1, further comprising constructing, by a descriptor construction module, the training machine learning descriptors based, at least in part, on one or more elemental properties.

4. The method of claim 1, further comprising retrieving, by a discovery management module, materials data from a selected database of 2D materials, the materials data comprising one or more of thermodynamic data, elastic data, electronic data, magnetic data and / or optical data.

5. The method of claim 1, wherein the SHG material discovery module comprises a classifier module and a regression module.

6. The method of claim 2, wherein the extracting comprises applying a natural logarithm to the training SHG materials data.

7. The method of claim 3, wherein the elemental properties are selected from the group comprising dipole polarizability, ionization energy, electron affinity, atomic radius, van der Waals radius, and a number of valence electrons.

8. The method of claim 3, wherein the descriptors comprise at least one elemental property and one or more of a crystal structure, an effective thickness, and / or a Phillips ionicity.

9. The method of claim 3, wherein the constructing comprises performing a mathematical transformation on at least one of the elemental properties.

10. A second harmonic generation (SHG) material discovery system for discovering a two-dimensional (2D) material capable of SHG, the system comprising: a computing device comprising a processor, a memory, an input / output circuitry, and a data store; and a training module configured to train an SHG material discovery module based, at least in part, on one or more training machine learning (ML) descriptors, the trained SHG material discovery module configured to predict an SHG for a candidate material based, at least in part, on a received material descriptor, the received material descriptor associated with the candidate material.

11. The system of claim 10, further comprising an extraction module configured to extract training SHG materials data, the training SHG materials data comprising second order susceptibility data, and crystal structure data.

12. The system of claim 10, further comprising a descriptor construction module configured to construct the training machine learning descriptors based, at least in part, on one or more elemental properties.

13. The system of any one of claims 10 to 12, further comprising a discovery management module configured to retrieve materials data from a selected database of 2D materials, the materials data comprising one or more of thermodynamic data, elastic data, electronic data, magnetic data and / or optical data.

14. The system of any one of claims 10 to 12, wherein the SHG material discovery module comprises a classifier module and a regression module.

15. The system of claim 11, wherein the extraction module is configured to apply a natural logarithm to the training SHG materials data.

16. The system of claim 12, wherein the elemental properties are selected from the group comprising dipole polarizability, ionization energy, electron affinity, atomic radius, van der Waals radius, and a number of valence electrons.

17. The system of claim 12, wherein the descriptors comprise at least one elemental property and one or more of a crystal structure, an effective thickness, and / or a Phillips ionicity.

18. The system of any one of claims 12, 16 or 17, wherein the descriptor construction module is configured to perform a mathematical transformation on at least one of the elemental properties.

19. The system of any one of claims 12, 16 or 17, wherein the descriptor construction module is further configured to perform dimensionality reduction on a representation of a crystal structure.

20. A computer readable storage device having stored thereon instructions that when executed by one or more processors result in the following operations comprising the method according to any one of claims 1 to 9.

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