Data-driven studies of magnetic two-dimensional materials
We use a data-driven approach to study the magnetic and thermodynamic properties of van der Waals (vdW) layered materials. We investigate monolayers of the form ABX, based on the known material CrGeTe, using density functional theory (DFT) calculations and machine learning methods to determine their magnetic properties, such as magnetic order and magnetic moment. We also examine formation energies and use them as a proxy for chemical stability. We show that machine learning tools, combined with DFT calculations, can provide a computationally efficient means to predict properties of such two-dimensional (2D) magnetic materials. Our data analytics approach provides insights into the microscopic origins of magnetic ordering in these systems. For instance, we find that the X site strongly affects the magnetic coupling between neighboring A sites, which drives the magnetic ordering. Our approach opens new ways for rapid discovery of chemically stable vdW materials that exhibit magnetic behavior.
The discovery of graphene ushered in a new era of studies of materials properties in the two-dimensional (2D) limit Bhimanapati et al. (2015). For many years after this discovery only a handful of van der Waals (vdW) materials were extensively studied. Recently, over a thousand new 2D crystals have been proposed Cheon et al. (2017); Mounet et al. (2018). The explosion in the number of known 2D materials increases demands for probing them for exciting new physics and potential applications Chen et al. (2013a); Nourbakhsh et al. (2016). Several 2D materials have already been shown to exhibit a range of exotic properties including superconductivity, topological insulating behavior and half-metallicity Novoselov et al. (2016); Zeng et al. (2016); Choi et al. (2017); Chen et al. (2017). Consequently, there is a need to develop tools to quickly screen a large number of 2D materials for targeted properties. Traditional approaches, based on sequential quantum mechanical calculations or experiments are usually slow and costly. Furthermore, a generic approach to design a crystal structure with the desired properties, although of practical significance, does not exist yet. Research towards building structure-property relationships of crystals is in its infancy Huo and Rupp (2017); Gilmer et al. (2017); Isayev et al. (2017).
Long-range ferromagnetism in 2D crystals has recently been discovered Huang et al. (2017); Gong et al. (2017), sparking a push to understand the properties of these 2D magnetic materials and to discover new ones with improved behavior Cui et al. (2017); Miyazato et al. (2018); Möller et al. (2018); Miao et al. (2018). 2D crystals provide a unique platform for exploring the microscopic origins of magnetic ordering in reduced dimensions. Long-range magnetic order is strongly suppressed in 2D according to the Mermin-Wagner theorem Mermin and Wagner (1966), but magnetocrystalline anisotropy can stabilize magnetic ordering Hope et al. (2000). This magnetic anisotropy is driven by spin-orbit coupling which depends on the relative positions of atoms and their identities. As a result, the magnetic order should be strongly affected by changes in the structural arrangements of atoms and chemical composition of the crystal.
Chemical instability presents a crucial limitation to the fabrication and use of 2D magnetic materials. For instance, black phosphorous degrades upon exposure to air and thus needs to be handled and stored in vacuum or under inert atmosphere. Structural stability is a necessary ingredient for industrial scale application of magnetic vdW materials, such as CrI and CrGeTe Huang et al. (2017); Gong et al. (2017). In addition to designing 2D materials for desirable magnetic properties, it is important to screen for those materials that are chemically stable. In our approach, we employ the calculated formation energy as a proxy for the chemical stability Rasmussen and Thygesen (2015). In particular, we obtain the total energies of systems at zero temperature, and calculate the formation energy as the difference in total energy between the crystal and its constituent elements in their respective crystal phases. This quantity determines whether the structure is thermodynamically stable or would decompose. This formulation ignores the effects of zero-point vibrational energy and entropy on the stability.
Recently, machine learning (ML) has been combined with traditional methods (experiments and ab-initio calculations) to advance rapid materials discovery Rupp et al. (2012); Meredig et al. (2014); Seko et al. (2015); Rasmussen and Thygesen (2015); Ueno et al. (2016); Choudhary et al. (2017); Ju et al. (2017); Cheon et al. (2017); Mounet et al. (2018). ML models trained on a number of structures can predict the properties of a much larger set of materials. In particular, there is presently a growing interest in exploiting ML for discovery of magnetic materials Landrum and Genin (2003); Möller et al. (2018). Data-driven studies of ferromagnetism in transition metal alloys have highlighted the importance of novel data analytics techniques to tackle problems in condensed matter physics Landrum and Genin (2003). It is conceivable that tuning the atomic composition could provide an additional degree of freedom in the search for stable 2D materials with interesting magnetic properties Lu et al. (2017). Even more compelling is the ability of ML tools to assist in uncovering the physics underlying the stability and magnetism of 2D materials Schoenholz et al. (2016); Cubuk et al. (2017). Specifically, ML methods can identify patterns in a high-dimensional space revealing relationships that could be otherwise missed.
In order to develop a path towards discovering 2D magnetic materials, we generate a database of structures based on a monolayer CrGeTe (Fig. 1(a)) using density functional theory (DFT) calculations 111The results of these DFT calculations will be used to build a database of monolayer 2D materials which will be publicly available to the scientific community.. The possible structures amount to a combinatorially large number of type ABX () with different elements occupying the A, B and X sites. We select a subset of 198 structures due to computational constraints. We obtain the total energy, magnetic order, and magnetic moment of each structure. The ground-state properties were determined by examining the energies of the fully optimized structure with several spin configurations, including non-spin-polarized, parallel, and anti-parallel spin orientations at the A sites (Fig. 1(b)).
We then employ a set of materials descriptors which comprise easily attainable atomic properties, and are suitable for describing magnetic phenomena. We employ additional descriptors which are related to the formation energy Heinz and Suter (2004). The performance of descriptors in predicting the magnetic properties or thermodynamic stability sheds some light into the origin of these properties.
To create the database we use DFT calculations 222We used the GGA-PBE for the exchange-correlation functional. The energy cutoff was 300 eV. The vacuum region was thicker than 20 Å. The atoms were fully relaxed until the force on each atom was smaller than 0.01 eV/Å. A -centered 10101 k-point mesh was utilized. with the VASP code Kresse and Furthmüller (1996).
We create the different structures by substituting one of two Cr atoms (A site) in the unit cell with a transition metal atom, from the list: Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Y, Nb, Ru. In the two B sites we place combinations of Ge, Si, and P atoms, namely Ge, GeSi, GeP, Si, SiP, P. The atoms at X sites were either S, Se, or Te, that is, S, Se, Te. Fig 1(c) shows the choice of substitution atoms in the Periodic Table. An example of a structure created through this process is (CrTi)(SiGe)Te.
The careful choice of descriptors is essential for the success of any ML approach Ghiringhelli et al. (2015); Seko et al. (2017). We use atomic properties data from the python mendeleev package 0.4.1 Mentel (2014) to build descriptors for our ML models. We performed supervised learning with atomic properties data as inputs, with target properties the magnetic moment and the formation energy. The choice of the set of descriptors for the magnetic properties was motivated by the Pauli exclusion principle, which gives rise to the exchange and super-exchange interactions. We also consider the magneto-crystalline anisotropy Chikazumi and Graham (2009) by building inter-atomic distances and electronic orbital information into our descriptors. With respect to the formation energy, the choice of descriptors was motivated, in part, by the extended Born-Haber model Heinz and Suter (2004), and include the dipole polarizability, the ionization energy and the atomic radius (see Supplemental Materials for a full list of atomic properties and descriptors used 333See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevX.x.xxxxx for more details and discussions related to this letter. ).
The data were randomly divided into a training set, a cross-validation set and a test set. Training data and cross-validation were typically 60% of the total data while test data comprised 40% of all the data. We employed the following ML models: kernel ridge regression, extra trees regression, and neural networks. Kernel ridge regression with a gaussian kernel has been shown to be successful in several materials informatics studies. Extra trees regression allows us to determine the relative importances of features used in a successful model Tibshirani et al. (2013). An analysis of hidden layers of the deep neural networks could allow us to identify patterns in 2D materials properties data, thereby guiding theoretical studies Cubuk et al. (2017).
Iii Results and discussion
iii.1 Magnetic properties
We find that the non-spin-polarized configuration has the highest energy for all the structures considered. That is, all structures prefer either parallel or anti-parallel ordering in the A plane. Fig. 2(a) shows the energy difference of parallel and anti-parallel spin configurations. Negative (positive) energy difference means the parallel (anti-parallel) is more stable. We note that, because of the supercell size limit, we do not consider more complex spin configurations in this study. For example, the lowest-energy spin configuration of CrSiTe was reported to be zigzag anti-ferromagnetic type Sivadas et al. (2015). Total magnetic moments for the lowest energy spin configuration of each structure are presented in Fig. 2(b). We find that only atoms in the A sites show finite magnetic moments, while the moments in the B and X sites are small. Distinct patterns for regions of high and low magnetic moments are observed for X = Te, Se and S in Fig. 2(b). Structures created by substituting non-magnetic atoms at the A site, such as Cu, have small variations in their relatively small magnetic moments, as seen in the rows of Fig. 2(b). However, substitutions of magnetic atoms, such as Mn, result in a set of structures with a large variation in the magnetic moment, with a much larger upper limit to the range of values observed.
Both the magnetic order and magnetic moment are sensitive to the occupancy of B and X sites, even though the atoms in these sites have negligible contribution to the overall magnetic moment. Atoms in the X sites strongly mediate the magnetic coupling between neighboring A sites Sivadas et al. (2015). Atoms at the B sites can affect the relative positions of A and X sites. Direct exchange between first nearest neighbor A sites competes with super-exchange interactions mediated by the p-orbitals at the X sites. The ground state magnetic order is determined by the interplay between first, second and third nearest neighbor interactions. Changing the identity of one of the A, B or X sites affects the interplay between the direct exchange and super-exchange interactions. Recent work has shown that applying strain to the CrSiTe lattice tunes the first nearest neighbor interaction, resulting in a change in the magnetic ground state from zig-zag antiferromagnetic to ferromagnetic Sivadas et al. (2015). Our work demonstrates that tuning the composition of the ABX lattice can have an equivalent effect. For instance, whereas X=Te structures show more parallel () than anti-parallel (anti-) spin-configurations with lower energy, there is a clear change when X Se or S. As X moves up the periodic table, there are increasingly more regions of anti-parallel spin configuration, as well as regions in which and anti- are degenerate. In particular, we find that the distance between nearest neighbor A and X sites, as well as two adjacent X sites is linked to the magnitude of the magnetic moment (see Supplemental Materials for details).
We use extra trees regression Tibshirani et al. (2013) to approximate the relationship between the total magnetic moment and a set of descriptors designed for magnetic property prediction (see Supplemental Materials). Training and test data are considered for the X = Te, Se, and S structures individually. The model performance for X = Te is shown in Fig. 3(a). We find reasonable prediction performance for X = Te that deteriorates for X = Se and is even worse for X = S. This suggests that our model, along with the set of descriptors used to predict X = Te structures, does not generalize well. This could arise due to the fact that there are more structures that have degenerate and anti- spin configurations if X=Se and S than for X = Te. Nevertheless, subgroup discovery can be exploited to learn more about these systems Goldsmith et al. (2017), implying that the identity of the X site strongly affects the magnetic properties of the structures.
Determining which descriptors are most important for making good predictions of a property can be exploited for knowledge discovery, especially when a large number of descriptors are available but their relationships with the target property are not known Reshef et al. (2011). Fig. 3(b) shows the descriptor importances Pedregosa et al. (2011) as derived from extra trees regression. It shows that the ‘the number of valence electrons’ [“nvalence max dif” in Fig. 3(b)], ‘the average covalent radius’ [“covalentrad avg” in Fig. 3(b)] and the ‘average number of spin up electrons’ [“Nup avg” in Fig. 3(b)], linked to the atomic dipole magnetic moment, are among the top six descriptors in the set examined. The magnetic moment per unit cell is a function of the magnetic moments of the individual atoms in the unit cell. We examine the local magnetic moments at the A sites to determine how the magnetic moment per unit cell is constructed. The local magnetic moment at the A sites (A and A) can be different from the atomic dipole magnetic moment of the corresponding element. For instance, while the atomic magnetic moment of Cr is 3 , the local magnetic moment at A fluctuates from 2.7 to 3.2 . Fig. 4 (a) shows the local magnetic moment at A.
iii.2 Formation energy
In addition to identifying structures with specific magnetic properties, the ability to screen for chemical stability is also important. DFT-calculated formation energies (for the lowest energy spin configuration) are shown in Fig. 4 (b). Structures comprising certain elements, such as Y, decrease the formation energy considerably in comparison to those without it. Certain transition metals, such as Cu, tend to destabilize the (CrA)BX structures. The formation energy becomes more negative as the substituted atom at the A site goes from the left to the right of the first and second row of transition metal elements in the Periodic Table. This is linked to the filling of the d-orbital, where elements with a filled d-orbital do not form chemical bonds with other elements. Varying the composition at the B site does not appear to have a strong impact on the formation energy (see Supplemental Materials, Fig. S1). Changing the X site from Te to Se and then S results in the overall trend of decreasing formation energy.
To exploit the trends in the formation energy data, we use statistical models to predict the formation energy and to infer structure-property relationships. We find that some descriptors, such as the atomic dipole polarizability, are strongly correlated with the formation energy, and are therefore important in generating good ML predictions. Since useful descriptors are not always revealed in an analysis of the Pearson correlation coefficient Reshef et al. (2011), we consider other methods to learn descriptor importances such as the extra trees model Pedregosa et al. (2011).
Using the ML models to predict the formation energy of ABX structures permits the quick calculation of the formation energy for a large set of compounds. Whereas DFT calculations of 10 structures could take up to 1 million CPU hours, the ML prediction takes a few seconds. Fig. 5(a) shows the prediction performance for kernel ridge regression using a gaussian kernel. Fig. 5(b) shows the performance of a neural network 444The deep neural network used in this study is implemented by tensorflow. It is comprised of 3 hidden layers with sizes 10, 30 and 10 units while Fig. 5(c) shows the performance of the extra forests regression. Both training set and test set results are displayed, as well as the test scores for kernel ridge regression, extra trees regression, and neural network regression.
Further analysis (see Supplemental Materials) shows that the ‘variance in the ionization energy of atoms’ and the ‘average number of valence electrons’ are the two most important descriptors in the set examined. This demonstrates a link between the formation energy and the atomic ionization energy, emanating from the increased atomic ionizability which produces stronger chemical bonding. In addition, the number of valence electrons is linked to the number of electrons available for bonding. For instance, substitutions by atoms with a filled outer orbital shell will create less stable bonds, leading to chemical instability. The ability of our models to generalize is demonstrated by the high scores on the test data. We further examined how the test set performance varies with the training set size. Fig. 5(d) shows test scores as a function of training set size using extra trees regression. The test score reaches a plateau at about a training set size of 40%, with test score (R) as high as 0.91.
iii.3 High-throughput screening using ML models
We can use our trained ML models to make predictions on a wide range of structures not included in the original DFT data set. Thus far, we have used our ML models to estimate the formation energy for an additional 4,223 ABX structures, constructed as follows: (i) For A site substitutions, we considered transition metals not used in the DFT dataset. (ii) We included Al, Sn and Pb in the set of atomic substitutions for B sites (not shown). (iii) For the X sites, we added O to our previous choice of S, Se and Te. The resulting predictions, partly shown in Fig. 6(a), provide a means to quickly screen a large data set of structures for chemical stability. For instance, our ML predictions suggest that structures based on Er, Ta, Hf, Mo, Zr, and Sc in the A site and Al in the B site are likely to be stable and thus good candidates for further exploration.
Magnetic moment predictions are shown in Fig. 6(b). From the results of the ML predictions we select structures with formation energies below -1 eV and magnetic moments above 5 (for X=Te only). From the 4,223 predictions, we obtained 40 that satisfied our constraints. 15 of these were randomly selected for verification with DFT. 5 of these 15 structures were confirmed to have the expected properties within uncertainty. These 15 structures were then added to the training data to build an improved model for predicting magnetic moment. A second iteration of prediction and verification by DFT generated three structures, all of which satisfied the constraints within uncertainty: (CrTc)(SiSn)Te, (CrTc)SnTe, Cr(SiP)Te.
We presented evidence that the magnetic properties of ABX monolayer structures can be tuned by making atomic substitutions at A, B, and X sites. This provides a novel framework for investigating the microscopic origin of magnetic order of 2D layered materials and could lead to insights into magnetism in systems of reduced dimension Huang et al. (2017); Gong et al. (2017). Our work represents a path toward tailoring magnetic properties of materials for applications in spintronics and data storage Han (2016). We showed that ML methods are promising tools for predicting the magnetic properties of 2D magnetic materials. In particular, our data-driven approach highlights the importance of the X site in determining the magnetic order of the structure. Changing the composition of the ABX structure alters the inter-atomic distances and the identity of electronic orbitals. This impacts the interplay between first, second and third nearest neighbor exchange interactions, which determines the magnetic order.
One goal of this work was to find magnetic 2D materials that are also thermodynamically stable. ML models were trained to predict chemical stability that allow the rapid screening of a large number of possible structures. We showed that the chemical stability of ABX structures based on CrGeTe can be tuned by making atomic substitutions. Examples of structures that satisfy both magnetic moment and formation energy requirements include the following: (CrTc)(SiSn)Te and (CrTc)SnTe, not included in our original DFT database. In addition, we found structures in our set of DFT calculations that also satisfied our requirements: Cr(SiP)Te, (TiCr)(SiP)Te, (YCr)GeS and (NbCr)SiTe.
This work provides the impetus for further exploration of structures with other architectures not considered here, that is, with more complex atomic substitutions beyond 1 in 2 replacement of Cr atoms at the A site. We estimate a total number of at least 310 structures of the ABX type described in Fig. 1. A computationally efficient estimation of the magnetic properties and formation energy is required to quickly explore this vast chemical space. We also expect the ML methods explored here, with proper modification, to allow an efficient exploration of other families of 2D magnets, such as CrI, CrOCl and FeGeTe Chen et al. (2013b); Huang et al. (2017); Miao et al. (2018).
Acknowledgements.We thank Marios Mattheakis, Daniel Larson, Robert Hoyt, Matthew Montemore, Sadas Shankar, Ekin Dogus Cubuk, Pavlos Protopapas and Vinothan Manoharan for helpful discussions. For the calculations we used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation (grant number ACI-1548562) and the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University. T.D.R. is supported by the Harvard Future Faculty Leaders Postdoctoral Fellowship. We acknowledge support from ARO MURI Award W911NF-14-0247.
- Bhimanapati et al. (2015) G. R. Bhimanapati, Z. Lin, V. Meunier, Y. Jung, J. Cha, S. Das, D. Xiao, Y. Son, M. S. Strano, V. R. Cooper, et al., ACS Nano 9, 11509 (2015).
- Cheon et al. (2017) G. Cheon, K.-A. N. Duerloo, A. D. Sendek, C. Porter, Y. Chen, and E. J. Reed, Nano Lett. 17, 1915 (2017).
- Mounet et al. (2018) N. Mounet, M. Gibertini, P. Schwaller, D. Campi, A. Merkys, A. Marrazzo, T. Sohier, I. E. Castelli, A. Cepellotti, G. Pizzi, et al., Nat. Nanotechnology 13, 246 (2018).
- Chen et al. (2013a) W. Chen, E. Santos, W. Zhu, E. Kaxiras, and Z. Zhang, Nano Lett. 13, 509 (2013a).
- Nourbakhsh et al. (2016) A. Nourbakhsh, A. Zubair, R. Sajjad, A. Tavakkoli KG, W. Chen, S. Fang, X. Ling, J. Kong, M. Dresselhaus, E. Kaxiras, K. Berggren, D. Antoniadia, and P. T., Nano Lett. 16, 7798 (2016).
- Novoselov et al. (2016) K. Novoselov, A. Mishchenko, A. Carvalho, and A. C. Neto, Science 353, aac9439 (2016).
- Zeng et al. (2016) J. Zeng, W. Chen, P. Cui, D. Zhang, and Z. Zhang, Phys. Rev. B 94, 235425 (2016).
- Choi et al. (2017) J. Choi, P. Cui, W. Chen, J. Cho, and Z. Zhang, Wiley Interdiscip. Rev. Comput. Mol. Sci. 7 (2017).
- Chen et al. (2017) W. Chen, Y. Yang, Z. Zhang, and E. Kaxiras, 2D Mater. 4, 045001 (2017).
- Huo and Rupp (2017) H. Huo and M. Rupp, arXiv:1704.06439 (2017).
- Gilmer et al. (2017) J. Gilmer, S. S. Schoenholz, P. F. Riley, O. Vinyals, and G. E. Dahl, in Proceedings of the 34th International Conference on Machine Learning, Proceedings of Machine Learning Research, Vol. 70, edited by D. Precup and Y. W. Teh (PMLR, International Convention Centre, Sydney, Australia, 2017) pp. 1263–1272.
- Isayev et al. (2017) O. Isayev, C. Oses, C. Toher, E. Gossett, S. Curtarolo, and A. Tropsha, Nat. Commun. 8, 15679 (2017).
- Huang et al. (2017) B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu, Nature 546, 270 (2017).
- Gong et al. (2017) C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G. Louie, J. Xia, and X. Zhang, Nature 546, 265 (2017).
- Cui et al. (2017) P. Cui, J. Choi, W. Chen, J. Zeng, C. Shih, Z. Li, and Z. Zhang, Nano Lett. 17, 1097 (2017).
- Miyazato et al. (2018) I. Miyazato, Y. Tanaka, and K. Takahashi, J. Phys. Condens. Matter 30, 06LT01 (2018).
- Möller et al. (2018) J. J. Möller, W. Körner, G. Krugel, D. F. Urban, and C. Elsässer, Acta Materialia 153, 53 (2018).
- Miao et al. (2018) N. Miao, B. Xu, L. Zhu, J. Zhou, and Z. Sun, Journal of the American Chemical Society 140, 2417 (2018), pMID: 29400056, https://doi.org/10.1021/jacs.7b12976 .
- Mermin and Wagner (1966) N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).
- Hope et al. (2000) S. Hope, B.-C. Choi, P. Bode, and J. Bland, Phys. Rev. B 61, 5876 (2000).
- Rasmussen and Thygesen (2015) F. A. Rasmussen and K. S. Thygesen, J. Phys. Chem. C 119, 13169 (2015).
- Rupp et al. (2012) M. Rupp, A. Tkatchenko, K.-R. Müller, and O. A. von Lilienfeld, Phys. Rev. Lett. 108, 058301 (2012).
- Meredig et al. (2014) B. Meredig, A. Agrawal, S. Kirklin, J. E. Saal, J. W. Doak, A. Thompson, K. Zhang, A. Choudhary, and C. Wolverton, Phys. Rev. B 89, 094104 (2014).
- Seko et al. (2015) A. Seko, A. Togo, H. Hayashi, K. Tsuda, L. Chaput, and I. Tanaka, Phys. Rev. Lett. 115, 205901 (2015).
- Ueno et al. (2016) T. Ueno, T. D. Rhone, Z. Hou, T. Mizoguchi, and K. Tsuda, Materials Discovery 4, 18 (2016).
- Choudhary et al. (2017) K. Choudhary, I. Kalish, R. Beams, and F. Tavazza, Sci. Rep. 7, 5179 (2017).
- Ju et al. (2017) S. Ju, T. Shiga, L. Feng, Z. Hou, K. Tsuda, and J. Shiomi, Phys. Rev. X 7, 021024 (2017).
- Landrum and Genin (2003) G. A. Landrum and H. Genin, J. Solid State Chem. 176, 587 (2003).
- Lu et al. (2017) A.-Y. Lu, H. Zhu, J. Xiao, C.-P. Chuu, Y. Han, M.-H. Chiu, C.-C. Cheng, C.-W. Yang, K.-H. Wei, Y. Yang, Y. Wang, D. Sokaras, D. Nordlund, P. Yang, D. A. Muller, M.-Y. Chou, X. Zhang, and L.-J. Li, Nat. Nanotechnol. 12, 744 (2017).
- Schoenholz et al. (2016) S. S. Schoenholz, E. D. Cubuk, D. M. Sussman, E. Kaxiras, and A. J. Liu, Nat. Phys. 12, 469 (2016).
- Cubuk et al. (2017) E. D. Cubuk, B. D. Malone, B. Onat, A. Waterland, and E. Kaxiras, J. Chem. Phys. 147, 024104 (2017).
- (32) The results of these DFT calculations will be used to build a database of monolayer 2D materials which will be publicly available to the scientific community.
- Heinz and Suter (2004) H. Heinz and U. W. Suter, J. Phys. Chem. B 108, 18341 (2004).
- (34) We used the GGA-PBE for the exchange-correlation functional. The energy cutoff was 300 eV. The vacuum region was thicker than 20 Å. The atoms were fully relaxed until the force on each atom was smaller than 0.01 eV/Å. A -centered 10101 k-point mesh was utilized.
- Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
- Ghiringhelli et al. (2015) L. M. Ghiringhelli, J. Vybiral, S. V. Levchenko, C. Draxl, and M. Scheffler, Phys. Rev. Lett. 114, 105503 (2015).
- Seko et al. (2017) A. Seko, H. Hayashi, K. Nakayama, A. Takahashi, and I. Tanaka, Phys. Rev. B 95, 144110 (2017).
- Mentel (2014) L. Mentel, “Mendeleev – a python resource for properties of chemical elements, ions and isotopes,” (2014), 0.4.1.
- Chikazumi and Graham (2009) S. Chikazumi and C. D. Graham, Physics of Ferromagnetism, Vol. 94 (Oxford University Press on Demand, 2009).
- (40) See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevX.x.xxxxx for more details and discussions related to this letter. .
- Tibshirani et al. (2013) R. Tibshirani, G. James, D. Witten, and T. Hastie, “An introduction to statistical learning with applications in R,” (2013).
- Sivadas et al. (2015) N. Sivadas, M. W. Daniels, R. H. Swendsen, S. Okamoto, and D. Xiao, Phys. Rev. B 91, 235425 (2015).
- Goldsmith et al. (2017) B. R. Goldsmith, M. Boley, J. Vreeken, M. Scheffler, and L. M. Ghiringhelli, New J. Phys. 19, 013031 (2017).
- Reshef et al. (2011) D. N. Reshef, Y. A. Reshef, H. K. Finucane, S. R. Grossman, G. McVean, P. J. Turnbaugh, E. S. Lander, M. Mitzenmacher, and P. C. Sabeti, Science 334, 1518 (2011).
- Pedregosa et al. (2011) F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, et al., J. Mach. Learn. Res. 12, 2825 (2011).
- (46) The deep neural network used in this study is implemented by tensorflow. It is comprised of 3 hidden layers with sizes 10, 30 and 10 units.
- Han (2016) W. Han, APL Mater. 4, 032401 (2016).
- Chen et al. (2013b) B. Chen, J. Yang, H. Wang, M. Imai, H. Ohta, C. Michioka, K. Yoshimura, and M. Fang, J. Phys. Soc. Jpn 82, 124711 (2013b).