{'fraction_non_alphanumeric': 0.031122620246487317, 'fraction_numerical': 0.01529755910420563, 'mean_word_length': 5.063972084908404, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 3, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Efficient Computational Design of 2D van der Waals Heterostructures: Band-Alignment, Lattice-Mismatch, Web-app Generation and Machine-learning

We develop a computational database, web-apps, and machine-learning (ML) models to accelerate the design and discovery of two-dimensional (2D)-heterostructures. Using density functional theory (DFT) based lattice-parameters and electronic band-energies for 674 non-metallic exfoliable 2D-materials, we generate 226,779 possible heterostructures. We classify these heterostructures into type-I, II and III systems according to Andersons rule, which is based on the relative band-alignments of the non-interacting monolayers. We find that type-II is the most common and the type-III the least common heterostructure type. We subsequently analyze the chemical trends for each heterostructure type in terms of the periodic table of constituent elements. The band alignment data can be also used for identifying photocatalysts and high-work function 2D-metals for contacts. We validate our results by comparing them to experimental data as well as hybrid-functional predictions. Additionally, we carry out DFT calculations of a few selected systems (MoS2/WSe2, MoS2/h-BN, MoSe2/CrI3), to compare the band-alignment description with the predictions from Andersons rule. We develop web-apps to enable users to virtually create combinations of 2D materials and predict their properties. Additionally, we use ML tools to predict band-alignment information for 2D materials. The web-apps, tools and associated data will be distributed through JARVIS-Heterostructure website (https://jarvis.nist.gov/jarvish/). Our analysis, results and the developed web-apps can be applied to the screening and design applications, such as finding novel photocatalysts, photodetectors, and high-work function (WF) 2D-metal contacts.


Introduction
Heterostructures are interfaces between dissimilar materials and are of practical importance in a wide variety of optoelectronic devices 1 . Two-dimensional (2D) materials exhibiting van der Waals (vdW) bonding strictly in one of the three crystallographic directions and their heterostructures provide a huge variety of functionality 2, 3 and property-control 4, 5 because their surfaces are generally free from dangling-bonds, extended defects and trap states 6 (which is in stark contrast with semi-coherent and incoherent heterointerfaces found in three-dimensional crystalline solids).
Additionally, they allow integrating various nanoscale materials to create diverse vdW heterostructures because they do not require strict constraints of crystal lattice matching 6 and have a wide variety of bandgaps. Despite their widespread applications, a systematic high-throughput investigation of a wide range of 2D chemistries is still lacking, mainly because of the overwhelmingly large number of underlying combinatorial possibilities. The number of possible 2D materials could be in the thousands as shown by recent computational screenings [7][8][9][10][11] , so there can be millions of potential pair combinations of distinct heterointerfaces. Moreover, since experimental routes for investigating these heterointerfaces usually are labor and resource-intensive, the use of computational methods, such as density functional theory (DFT), to perform screening is a very well justified first step in an elaborate materials design process.
Metallic 2D materials with high work-functions 12 are useful for battery-applications 13 as well as minimizing Schottky barrier 14 , while 2D semiconducting or insulating materials are good candidates for making different types of hetersostructures 15 such as type-I (symmetric), type-II (staggered), and type-III (broken). We determine the types using Anderson's rule, which considers the conduction and valence band information of constituting monolayers only. The information about Conduction Band Minima (CBM) and Valence Band Maxima (VBM) can easily be obtained from DFT. A type-I band alignment is most widely utilized in optical devices, such as lightemitting diodes (LEDs) 16 and lasers. A type-II band alignment is useful for unipolar electronic device applications, including photodetectors, 17-19 as it can allow for more efficient charge separation and rapid transport for valence and conduction band carriers. The type-III heterostructures are used for tunneling field-effect transistors(TFETs) 20 . Magnetic/non-magnetic heterostructures are also very interesting for proximity-effects during phenomena such as bandtuning 21 and valleytronics 22 . Other important applications of 2D heterostructures include selfcleansing 523 , plasmonic devices 24 , electroluminescence 25 , complementary metal-oxidesemiconductor (CMOS) 26 , and DNA biosensors 6,27 .
However, the next generation of 2D materials has gone beyond the known classes and includes many more chemical and structural classes than those investigated in the computational works mentioned above. For a systematic study of heterostructures, a robust algorithm for constructing heterostructures using lattice parameters information and a database of 2D materials' structure and electronic information is essential. Fortunately, recently several computational tools have been developed to identify best-matched configurations including works by Mathew et al. 32  Experimentally, 2D-heterostructures are created either using manual stacking or via direct synthesis using, for instance, chemical vapor deposition (CVD) processes. Unlike direct CVD growth, whereby vdW heterostructures typically adopt a certain fixed orientation, the manual stacking approach offers a new degree of freedom to tailor the relative twist angle (within an error of 1°) 41 between different layers in nearly arbitrary configurations. Moreover, the heterostructures can be created as vertical stacks or as laterally stitched junctions. We consider only vertical junctions in this work. Although 2D heterostructures may not require a strict lattice matching criterion, the presence of sufficient lattice mismatch along with weak vdW bonding between the 2D layers can lead to incoherent lattice matching, and the generation of Moiré patterns 42 .
In this work, we aim to provide a sufficiently accurate and highly efficient model to preselect 2D heterostructures for specific applications among a vast pool of candidates. We develop a database and an open-access web interface to facilitate the generation of 2D heterostructures and their classifications. We use the method of Zur et al. 33,34 to generate interfaces between these 2D materials, and to calculate lattice constants and angle mismatches. The set of 2D materials is obtained from the JARVIS-DFT 2D database, from which we extracted the VBMs, CBMSs, Fermi-energies, equilibrium structures, and wavefunction data. JARVIS-DFT is a database containing about 38000 bulk and 900 low-dimensional materials with their DFT-computed structural, exfoliability 35 , elastic 36 , optoelectronic 37 solar-cell efficiency 39 , and topological 39 properties. It is part of National Institute of Standards and Technology (NIST) effort within the scope of Materials Genome Initiative (MGI) related activities. Next, we develop an easy-to-use web-app for predicting the heterostructure type and lattice mismatches between all the possible 2D materials in our database. As mentioned above, Anderson's rule-based designation of heterostructure type requires only the monolayer CBMs and VBMs of the monolayers forming the heterostructure. Since DFT calculations for a vast number of new 2D materials could still be computationally expensive, we leverage machine-learning (ML) techniques to further accelerate the predictions of CBMs and VBMs of 2D materials given the crystal structure information. We train ML models for CBM, VBM and work-functions of 2D materials using the Atomistic Line Graph Neural Network (ALIGNN) method 43 . We believe that the easily available heterostructure database along with other calculated properties would serve as a useful tool for the community towards novel and application-specific 2D-heterostructure design.

Results and discussion
We start by employing available JARVIS-DFT data for 2D materials to predict the nature of 2Dheterostructure band alignments. A flow-chart associated with this process is shown in Fig. 1. The lattice parameter data is then used to find the best match within the heterointerface twist angle tolerance of 1 o and mismatch of 0.05 % using the algorithm developed by Zur et al. 33,34 , which finds the heterostructure configuration with the minimal mismatch. Note that we avoid highly mismatched heterostructures because they may lead to the generation of dislocation, grainboundaries and internal-stress, interface buckling and other effects that are beyond the scope of this work. These heterostructures can be now classified as type-I, type-II, or type-III structures based on the VBM and CBM data mentioned above based on Anderson's rule 49  For magnetic/magnetic, magnetic/non-magnetic 2D material interfaces, we predict the bandalignments based on net CBMs and VBMs of both spin channels. Note that the electronic structure of heterostructures with internal stress due to mismatch could be different from that of the two constituent materials, and it would be difficult to predict using the Anderson rule. Ozcelik, et al. 15 reported such behavior for MoSe2-WSe2 heterostructure. These results also assume that the two 2D-materials do not interact much. However, charge transfer could be possible especially for lowbandgap materials which would change the band-alignment. Also, note that we discuss intrinsic, defect-free and 0 K results in the current work, while at ambient conditions, equilibrium thermodynamic concentration of defects in systems, and doping levels could also significantly change the heterostructure behavior. We investigate the overall probability that a specific element in the periodic

Type-I heterostructure and optoelectronics applications
Type-I heterostructures are used in transistors, light-emitting diodes (LED) and laser applications 16 . In this case, both the CBM and VBM of the heterointerface are located in the material with narrower bandgap. The quantum confinement of electrons and holes in the same region facilitates their radiative recombination, which is desirable in light-emitting applications.
An example of such experimentally-confirmed type-I materials is MoS2-ReS2 50 , where ReS2 has a narrower gap than MoS2. Using the CBM and VBM datasets, we also find the heterostructure to be type-I, in agreement with the experimental results (shown in the band-alignment diagram in Therefore, more generally, the computational-tools developed in this work can easily predict all possible potential heterostructures for a given 2D-materials and a specific heterojunction type.
Many of these predicted heterostructures are not reported experimentally to the best of our knowledge. Hence, the screening work performed within this study could help guide future experiments towards desired heterostructure types.

Type-II heterostructure and photodetection/absorption applications
In type-II materials, the CBM and VBM of the resultant structure are located in two different materials forming the interface. The excitation of the wide-gap material is followed by the transfer of electrons, but not the holes. The opposite charge transfer occurs when the narrow-gap material is excited. The separation of the electrons and holes to different layers can increase their lifetime and is desirable for photovoltaics and photodetection applications. A well-known example of type-II 2D heterostructure is MoS2/WSe2 interface 51 . Using our methodology, we predicted this system to be type-II as well, which once again validates our approach (Fig. 5b) MoS2. Therefore, compared to the single-layer materials, the heterojunction leads to a better utilization of light and provides more electrons for the reaction, therefore increasing PEC current up to ~6 times compared to single 2D materials 51

Type-III heterostructure and TFETs
Lastly, in type-III heterojunctions, the band-edges do not overlap at all, making the overall heterojunction metallic. The situation for carrier transfer is like type-II, just more pronounced.
Type-III heterojunctions are used in TFETs. TFETs could be used to mitigate the issue of conventional metal-oxide-semiconductor field-effect transistors (MOSFETs) due to scaling down the supply voltage with the increasing power density and the thermionic emission limitation of carrier injection 52,53 . Unlike type-I and II, there are very few type-III heterojunctions known 52 . An experimentally known example of type-III heterojunction is SnSe2/WSe2 52 (JVASP-762, JVASP-652) which we also find as type-III (Fig. 5c). As an example, screening for 2D materials that can form type-III with 2H-MoS2, we find MnO2 (JVASP-6922). Similarly, searches for al l the possible combinations could be performed to guide computations or experimental work.

Water-splitting applications
2D-materials can be used for water-splitting to generate H2 and O2 gases when irradiated with sunlight 54,55 . For a material to be able to split water, its CBM must lie at a more negative potential than the reduction potential (H + /H2: 0.0 eV vs. Normal hydrogen electrode (NHE) at pH 0) and the valence band maximum must exhibit more positive potential than the oxidation potential (O2/H2O: +1.23 eV vs. NHE at pH 0). Out of all the 2D materials in our database, we could find 180 2D monolayer materials that satisfy this condition as shown in Fig. 6. Some of the materials are already known water splitters, including BN 56 , black-P 54 , ZnPS 357 ), but many other materials, including As2O3, InSe, HfBrN, and ZnSiO3 are not known experimentally as photocatalysts for watersplitting. As the band-edges could be tuned using different types of 2D materials, the heterostructures would allow a greater number of materials with water-splitting capabilities. Also, a heterostructure, especially of type-II as discussed above, can prevent electron-hole pairs (EHP) from recombining by separating the electrons and holes, which would increase the photogeneration rate. Fig. 6 Band-alignment of the non-metallic 2D materials with respect to H + /H2 and O2/H2O levels.
VBMs are drawn in red while CBMs are green.

High work-function 2D-materials
Heterostructures, which utilize 2D materials as metallic contacts instead of conventional metallic contacts such as Al, Pt and Cu are of great importance in designing electronic devices. But very few such 2D materials with high WF have been reported [12][13][14] . These materials have been shown to act as an excellent contact for 2D heterostructures and could be advantageous compared to conventional contacts such as Pt because of easier lattice mismatch. In Fig. 7, we show the WF distributions for all the 2D metals. We observe that most of the WFs are centered around the 4 eV

Web-app development
We now describe the details of the developed website application (web-app), which is specifically designed to facilitate easy predictions and further analysis of 2D-heterostructure. A snapshot of the JARVIS-Heterostructure web-app is shown in Fig. 8. In this app, the electronic bands data and lattice-matching algorithm are used to predict the possible 2D heterostructure and its type.
Currently, there are two sub-apps: a) a user can select options of two 2D materials (for which computed data is already available for 2D materials) to predict the lattice-mismatch and bandalignments of the heterostructures, b) computationally design a heterostructure for arbitrary 2D materials. The possible options are all the semiconducting 2D materials in the database. In addition to the mismatch and band-alignment information, the app shows the band-diagram to help visually analyze the results. In case some material is not in the database, the user can put two 2D materials crystallographic information in the second app, which then makes the 2D heterostructure under the pre-defined mismatch criteria. The user can then run the calculations on the three structures to predict the heterostructure properties. The code to generate band-alignments and the heterostructure is freely available at the jarvis-tools github page (https://github.com/usnistgov/jarvis). The web-app will be made available at: https://jarvis.nist.gov/jarvish/ .

Machine-learning applications
To treat possible 2D materials that are not yet in our database, we develop machine-learning (ML) models for the band diagrams of 2D materials, using the ALIGNN method 43 . The data are taken from the JARVIS 2D-dft dataset, for a total of 1,062 2D materials. We use 80% of the data for training, 10% for hyperparameter optimization, and 10% for test scoring; hyperparameters and other training details are provided in the Methods section. We consider CBMs, VBMs, WFss separately from the JARVIS-DFT calculation results as the target data for ML. In the Fig. 9a-c we visualize the target data distribution. This helps us understand the boundaries in which the ML algorithm would work well, as ML algorithms are good for interpolation purposes only and generally fail at extrapolation. The results of this test are shown in Table. 1. As it is possible that a 80/10/10 split could be biased; therefore, we perform five-fold cross-validation on each model with complete data. We report the corresponding mean absolute error and standard deviations as shown in the Table. 1 as well. In Fig. 9d-f we show the DFT vs ML predictions after training the ALIGNN model on 80 % training data and testing on 10 % test set data. The results for the trained models are given in the Table. 1 and clearly suggest that for most of the models we achieve reasonable-accuracy compared to the average-predicting baseline model and the models could be used on new materials which are not currently available in our database for a first line of screening.
On the second app discussed above, we plan to integrate the ML models so that users can input crystal structure and find the best-matched interface as well as get ML predictions for the heterostructure design.

Limitations
Although the above tools and datasets provide a unique resource for predicting 2D heterostructures, there are various limitations that we mention here. First, the conventional semilocal DFT functionals we use in this work are known to underestimate the bandgaps of materials 58 .
Higher levels methods such as hybrid-functionals 59   In summary, we present a computational database, tools, web-apps and machine-learning models to accelerate the design and discovery of 2D-heterostructures. Using lattice-parameter and DFT based electronic level information of 674 non-metallic 2D-materials, we generate 226779 heterostructures and classify them into type-I, II and III systems according to Anderson's rule. We find that type-II is the most common and the type-III the least common heterostructure type. We analyze the chemical trends of all the generated heterostructures in terms of the periodic the band-alignment data their applications and types of heterostructures are determined. We require lattice-mismatch less than/equal to 0.05 % and 1-degree angle tolerance to build the heterostructures. However, we provide all the data and tools to enable users to construct heterostructures with other tighter or looser tolerances. For metallic systems, we determine the workfunction as the difference between the Fermi-energy and the VAC. The web-app is developed using Flask-python 47 and jarvis-tools packages. The Flask-app allows integration of python basedprograms and web using very light coding requirements. Finally, we train three neural network models for CBMs, VBMs and work-functions, according to the ALIGNN system previously described elsewhere. 43 To optimize hyperparameters we train on 80% of the data and optimize the performance on the 10% validation set. The ALIGNN is trained for 200 epochs and appears to be fully converged at that point, as shown in the convergence plots of Figure 12. We begin with a default choice of 0.01 for both learning rate and decay rate, with a batch size of 32, and test varying these parameters separately by a factor of two in either direction. The validation accuracy proved to be relatively insensitive to this variation, so we adopt the highest-performing settings discovered this way but do not optimize any further. For CBM, accuracy is maximized with the defaults for each parameter, while VBM is slightly more accurate with a batch size of 64, and work-function is most accurate with a learning rate of 0.02. We also perform a five-fold cross-validation study on the whole data to analyze the sensitivity of the selection of dataset-splits.