Two-Dimensional Materials from Data Filtering and Ab Initio Calculations

Progress in materials science depends on the ability to discover new materials and to obtain and understand their properties. This has recently become particularly apparent for compounds with reduced dimensionality, which often display unexpected physical and chemical properties, making them very attractive for applications in electronics, graphene being so far the most noteworthy example. Here, we report some previously unknown two-dimensional materials and their electronic structure by data mining among crystal structures listed in the International Crystallographic Structural Database, combined with density-functional-theory calculations. As a result, we propose to explore the synthesis of a large group of two-dimensional materials, with properties suggestive of applications in nanoscale devices, and anticipate further studies of electronic and magnetic phenomena in low-dimensional systems.


I. INTRODUCTION
The recent explosion of scientific activity around graphene [1] has resulted in a large number of works [2][3][4][5] addressing its intriguing physical properties.More recently, efforts have been devoted to study similar, i.e., two-dimensional (2D), systems having a finite band gap, which is a desired feature for many technological applications.Among the directions that have been explored, chemically modified graphene (such as graphane [6,7], graphone [8], fluorographene [9][10][11], or graphXene [12]) and graphene on various substrates [13][14][15][16] and other 2D systems [17][18][19][20][21][22] (e.g., hexagonal boron nitride and transition-metal dichalcogenides) appear to be the most promising.Apart from technological possibilities with new two-dimensional structures, such materials open up new investigations of several unsolved questions, e.g., concerning the coupling between metallic conductivity and dimensionality.Abrahams et al. [23] pointed out that one should not find metallic conductivity as T !0 K in two-dimensional materials.Instead, the resistance is expected to grow logarithmically (in the so-called weak localization regime) or exponentially (for strong localization), becoming infinite for T !0 K.However, experiments primarily on semiconducting heterostructures challenge these theories, pointing to a much more complicated scenario.The possibility to widen the experimental platforms to new classes of two-dimensional materials is expected to shine light on this problem.Also, similar complex couplings between dimensionality and magnetism are well known [24].
However, from an experimental point of view, the study of even a single system requires the ability to synthesize and to characterize it, for example, by probing it with some relevant spectroscopy, such as x-ray surface diffraction, photoemission, low-energy electron diffraction, scanningtunneling spectroscopy, etc.Such a task is unfortunately very time consuming, which makes it difficult to undertake a systematic study of a large number of systems.Computational studies, on the other hand, are somewhat simpler: Once the chemical composition and the crystal structure are known, a large number of properties can be obtained, provided that a reliable theoretical tool is used.This approach has been used, in combination with data-filtering techniques, to identify, e.g., new scintillator materials [25,26], novel superconductors [27], and topological insulators [28].
In the present work, we use the efficiency of densityfunctional theory to reveal the electronic and magnetic structures of candidates of new two-dimensional materials.We report here the parent crystal that is composed of two-dimensional building blocks that can be separated into a two-dimensional unit.We list the International Crystallographic Structural Database (ICSD) number of these crystals and we also provide the electronic structure of the two-dimensional building blocks that these crystals are composed of.Hence, the crystal and electronic structure of a large group of potential new two-dimensional materials is proposed here.These crystals have been identified by means of data-mining algorithms, of compounds listed in the ICSD, and we provide the details of the algorithms in the paragraph below.

II. METHODS
Most of the compounds known to exist in two dimensions (such as graphene) have a bulk form where the layers are held together by van der Waals forces.To identify new two-dimensional compounds, we have selected materials by screening compounds in the ICSD on the basis of purely geometrical considerations by gradually refining the selection with a filtering for certain characteristic properties of van der Waals bonded layered compounds.For computational reasons, only high-symmetry compounds are considered that yield square or hexagonal in-plane structures.A study of some known weakly bonded compounds showed that the structures of interest are found in a fairly narrow interval of the packing ratio, defined as the covalent volume (the sum of the volumes of the set of spheres of the covalent radii of each atom in the cell) divided by the total volume of the cell.Higher packing ratios tend to be closepacked structures, and lower packing ratios are typically molecular solids, and so the packing ratio is used as the most coarse selection tool.Packing ratios in the range 0.15-0.50are selected, based on covalent radii compiled from various chemistry textbooks, and databases were obtained from Ref. [29].In the second step of our procedure, we have selected from our preliminary list the compounds that have sufficiently large gaps (> 2:4 # A) between crystallographic planes along the c axis, the direction of stacking.In the last step, to further refine the selection thus obtained, we inspect the gaps found and identify covalent bonds, if any, that reach across the gaps of the crystal structure.This refinement is done by considering a distance between two atoms close to the sum of their covalent radii as being indicative of a covalent bond.If no bonds are found across the gap, we identify the material as likely to be a weakly bonded layered compound and we then extract the crystal structure of a single layer of this material.Notice that nonstoichiometric compounds, which are very common in mineralogy, are de facto excluded from our searching procedure.In the same way, layered compounds not having a sufficiently large gap along the c axis do not appear in our list.Examples of compounds that are layered, but not layered enough for the present search criteria, are, e.g., the high-temperature superconductors.
For each identified single layer, we have performed ab initio calculations within the framework of densityfunctional theory [30] with the Perdew-Burke-Ernzerhof (PBE) [31] version of the generalized-gradient approximation (GGA) [32] as the exchange-correlation potential, as implemented in the Vienna Ab initio Simulation Package (VASP) [33,34], and using the projector augmented-wave method [35] for the basis set.For each structure, we proceed in two steps: Starting from the 2D structure as obtained from the ICSD, we perform a geometry optimization until the force acting on each ion becomes smaller than 0:001 eV= # A. Then, this relaxed geometry is used as the fixed crystal structure for a second calculation that allows us to obtain the corresponding total density of states and band structure.In all our calculations, we have used a plane-wave cutoff of 500 eV, while the k-point mesh is generated automatically to have a distance Ák along each reciprocal lattice vector of Ák ¼ 0:2 # A À1 during the geometry optimization process and then densified to Ák ¼ 0:15 # A À1 to obtain precise total density of states and band structures.For materials containing either V, Cr, Fe, Co, or Ni, we checked for the possibility to obtain a spinpolarized ground state within the crystallographic cell.(To keep the calculations affordable, magnetic orders that can occur in a cell larger than the crystallographic cell are not considered.)Information from the ICSD was processed for geometrical analysis and setup of calculations using python classes from the program CIF2Cell [36].

III. RESULTS AND DISCUSSION
Applying our computational procedure, we have been able to identify 92 two-dimensional compounds, in total.Their chemical composition, the ICSD number of the corresponding bulk (3D) material, the value of the minimum band gap (if present), as well as the magnetic order of the ground state (when relevant), are presented in Tables I, II, and III.The band structure of these 92 materials is provided in the Supplemental Material [37].As usual with a functional such as the GGA, the true value of the band gap is larger than the calculated one, which should therefore be seen as the lower bound, with a correction of between about 50% and 100% [38].However, the GGA functional very often makes an accurate distinction between metallic and semiconducting materials and describes the structural parameters well.
During our search for new two-dimensional compounds, several already known materials emerge out of the datafiltering procedure.Hence, we successfully identify graphene (from graphite, ICSD number 617290) and 2D hexagonal boron nitride (from three-dimensional hexagonal boron nitride, ICSD number 035538) as relevant materials.At the same time, 2D Bi 2 Se 3 (from bulk Bi 2 Se 3 ,  I, we briefly summarize some of their properties but we do not discuss here in detail their electronic structure, since it has already been reported [18,39,40]. An important family of two-dimensional compounds is the layered dichalcogenides, which appear as possible competitors to modified graphene, as it has been shown that they can be obtained in the form of two-dimensional sheets [17,[19][20][21]41,42].So far, only MoS 2 , NbSe 2 , WS 2 , MoSe 2 , MoTe 2 , TaSe 2 , NiTe 2 , and VS 2 have been synthesized as nanosheets, but the family of dichalcogenides is known from bulk studies to be larger than that.The full list of dichalcogenides found from our data filtering of the ICSD database is presented in Table II [43], and our computed band structures, which are essentially equivalent to earlier studies (see the Supplemental Material [37]), will not be commented upon any further here.From our calculations, the vanadium dichalcogenides VX 2 are all found to be ferromagnetic, a fact that was noticed before [44] for VS 2 and VSe 2 .(VTe 2 was not studied in Ref. [44].)In the case of TaS 2 and TaSe 2 , two polymorphs are found, depending on how the S and Se atoms are located on each side of Ta.In Table II, we use the symbol ÀAA for the polymorph when the S (or Se) are on top of each other and ÀAB when they are not.To our knowledge, the electronic structure of the other dichalcogenides in their two-dimensional form has not been presented so far.While a majority of them are metals (see Table II), several present a sizable band gap, and in particular ZrSe 2 and HfSe 2 have a GGA band gap very close to the one obtained for bulk silicon (ZrSe 2 and HfSe 2 have calculated band gaps of 0.4 and 0.6 eV, respectively, while bulk silicon has a GGA band gap of 0.46 eV).Because of the similarity in calculated band gap to that of Si, we suggest that ZrSe 2 and HfSe 2 are potentially interesting for applications.In Fig. 1, we present their band structures along the high-symmetry directions À-M-K-À in the 2D Brillouin zone.The two band structures are very similar, with a maximum of the valence band at the À point and a minimum of the conduction bands at the M point.Furthermore, the GGA band gap of the tellurides PdTe 2 and PtTe 2 is less than 1 eV, suggesting that the true band gap is probably between 0.5 and 1.5 eV.It can be modulated to a suitable value by alloying the telluride with the corresponding sulfide or selenide, which are insulating.Also, the series of chromium dichalcogenides appears to be particular, since it presents magnetic ordering: antiferromagnetic for CrS 2 and ferromagnetic for CrSe 2 and CrTe 2 .
In Table III, we present a list of materials that have not been previously reported as possible two-dimensional compounds and that are outside the group of dichalcogenides.This list includes materials with a relatively simple chemical composition and structure, such as GaS, FeS, FeTe, and PbO, and more complex compounds, such as P 2 AgSe 6 Bi, P 2 CuSe 6 Bi, ScP 2 AgSe 6 , and K 2 C 9 FeO 3 N 6 .Therefore, it is not surprising that the different electronic structures span everything from metals to wide-band-gap insulators, also including materials showing a magnetic order, such as FeTe, FeBr 3 , and CrSiTe 3 .These last two compounds present a moderate band gap (0.5 eV for FeBr 3 and 0.6 eV for CrSiTe 3 ) that could be of interest for electronics applications.The details of their electronic  structures are presented in Figs. 2 and 3.As for FeBr 3 , which we find to be an antiferromagnet, the top of the valence band is composed of Fe and Br derived states, while the bottom of the conduction band arises mainly from Fe states.The band gap is found to be indirect, with a maximum of the valence band along the K-À direction, while the minimum of the conduction band is at À. CrSiTe 3 is found to be ferromagnetic, with the top of the valence bands and the bottom of the conduction bands made from Cr and Te states.Because of the magnetic moment of the Cr atoms, the Te and Si atoms become slightly spin polarized as well, as shown by the exchange splitting between spin-up and spin-down states.For this compound, the top of the valence band is at the À point with a majority spin polarization, while the bottom of the conduction band is at the K high-symmetry point with minority spin polarization.Also, Bi 14 Te 13 S 8 , P 2 CuSe 6 Bi, PbSb 2 Te 4 , and Sb 2 Ge 2 Te 5 have a GGA band gap lower than 1.0 eV and are of interest as well.Our calculated band structures for the compounds presented in Table III are given in the Supplemental Material [37].

IV. CONCLUSION
Beside our proposed list of compounds, further materials are anticipated: The transition-metal dichalcogenides NiS 2 , NiSe 2 , ReTe 2 , TcSe 2 , and TcTe 2 have not been reported experimentally (and are therefore not present in the ICSD as bulk materials), but they are likely to exist.Another direction to explore is the synthesis of hybrid structures: This synthesis was achieved recently [45] with h-BN or graphene.The concept can be extended to other two-dimensional structures provided that a suitable match between the different subsystems is found.Also, these new materials can be put on substrates, as is currently done with graphene.Note that the two-dimensional materials that we find to be metallic should not be disregarded, since their electronic structure may be modified by functionalizing the material in the same way that graphene is nowadays modified in graphane or flurographene.Transparent metals also are sought after, e.g., as replacements to indium tin oxides [46].In addition, as discussed above, metallic conductivity in two dimensions is far from understood, and the identification of new two-dimensional materials enables further studies of this complex phenomenon.Similar conclusions can be made about the complex coupling between magnetic properties and dimensionality.Notice that our proposed list certainly does not cover all possible 2D materials, since, as explained in Sec.II, we have only searched through a subset of the ICSD.
From our results, it appears that the number of 2D materials and the variety of their electronic structures is quite large: The list of possible 2D dichalcogenides has only begun to be explored by experimentalists [17,19,41], while the compounds presented in Table III have not been thought to exist as 2D crystals.Since the compounds that we propose have already been synthesized in bulk form, their 2D counterparts can probably be obtained using the same techniques [1,17,19] that are now routinely used to obtain graphene, 2D h-BN, and 2D transition-metal dichalcogenides.Therefore, the path now seems to be open to explore and engineer these new materials.

FIG. 3 .
FIG.3.Density of states (upper panel) and band structure along the high-symmetry directions À-M-K-À (lower panel) of ferromagnetic CrSiTe 3 .Note that the density of states is also shown as an atomic projected property.The Fermi level is at zero energy.In the inset of the upper panel, the two-dimensional crystal structure is also shown.

TABLE I .
List of known two-dimensional materials also found by our search algorithm.The ICSD number of the corresponding bulk material, the calculated band gap of the two-dimensional material, and its magnetic state (FM refers to a ferromagnetic ordering) are given as well.(The cells of the table are left blank if the material is not magnetically ordered.)Bi 2 Te 3 (from bulk Bi 2 Te 3 , ICSD number 158366), which are the building blocks of the corresponding bulk materials presenting electronic properties of topological insulators, as well as 2D FeSe and 2D LiFeAs, which belong to the group of iron-based superconductors, are found as well.In Table . We have identified the following transition-metal dichalcogenides (with X ¼ S, Se, or Te): TiX 2 , ZrX 2 , HfX 2 , VX 2 , NbX 2 , TaX 2 , MoX 2 , WX 2 , PdX 2 , and PtX 2 from their 3D counterparts in the ICSD database.CrX 2 compounds are also present in our list: While CrS 2 and CrSe 2 exist as bulk materials, we have obtained 2D CrTe 2 as a layered building block of bulk thallium chromium telluride (TlCrTe 2 , ICSD number 152836; bulk CrTe 2 is not referenced in the ICSD database).We have also identified the sulfides YS 2 , TcS 2 , and ReS 2 , the selenide ReSe 2 , and the tellurides CoTe 2 , RhTe 2 , IrTe 2 , and NiTe 2 .Also, SiTe 2 , SnS 2 , and SnSe 2 , which are layered dichalcogenides but not composed of a transitionmetal ion, are reported.The electronic structure of MoX 2 , NbX 2 , WX 2 , and TaX 2 has already been investigated in the past

TABLE II .
List of two-dimensional dichalcogenides found by our search algorithm with the ICSD number of the corresponding bulk material and with our calculated band gap (in eV).The corresponding magnetic state (AFM refers to an antiferromagnetic ordering) is given in the last column to the right.(The cells of the table are left blank if the material is not magnetically ordered.) table are left blank if the material is not magnetically ordered.)