Data Mining for New Two- and One-Dimensional Weakly Bonded


Data Mining for New Two- and One-Dimensional Weakly Bonded...

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Letter pubs.acs.org/NanoLett

Data Mining for New Two- and One-Dimensional Weakly Bonded Solids and Lattice-Commensurate Heterostructures Gowoon Cheon,† Karel-Alexander N. Duerloo,‡ Austin D. Sendek,† Chase Porter,§ Yuan Chen,† and Evan J. Reed*,‡ †

Department of Applied Physics, ‡Department of Materials Science and Engineering, and §Department of Mechanical Engineering, Stanford University, Stanford, California 94305, United States S Supporting Information *

ABSTRACT: Layered materials held together by weak interactions including van der Waals forces, such as graphite, have attracted interest for both technological applications and fundamental physics in their layered form and as an isolated single-layer. Only a few dozen single-layer van der Waals solids have been subject to considerable research focus, although there are likely to be many more that could have superior properties. To identify a broad spectrum of layered materials, we present a novel data mining algorithm that determines the dimensionality of weakly bonded subcomponents based on the atomic positions of bulk, three-dimensional crystal structures. By applying this algorithm to the Materials Project database of over 50,000 inorganic crystals, we identify 1173 twodimensional layered materials and 487 materials that consist of weakly bonded onedimensional molecular chains. This is an order of magnitude increase in the number of identified materials with most materials not known as two- or one-dimensional materials. Moreover, we discover 98 weakly bonded heterostructures of two-dimensional and onedimensional subcomponents that are found within bulk materials, opening new possibilities for much-studied assembly of van der Waals heterostructures. Chemical families of materials, band gaps, and point groups for the materials identified in this work are presented. Point group and piezoelectricity in layered materials are also evaluated in single-layer forms. Three hundred and twenty-five of these materials are expected to have piezoelectric monolayers with a variety of forms of the piezoelectric tensor. This work significantly extends the scope of potential low-dimensional weakly bonded solids to be investigated. KEYWORDS: Layered materials, van der Waals solids, van der Waals heterostructures, two-dimensional materials, data mining, piezoelectricity

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of anisotropy, which could contribute to high thermoelectric performance. Moreover, Seebeck coefficients in low-dimensional structures also benefit from having sharp features in the density of states.18 Layered bismuth telluride, bismuth selenide, and their alloys are widely used thermoelectric materials. The thermoelectric research community is also actively studying other layered materials.12,19,20 Layered materials in single- and few-layer forms are being investigated for atomically thin devices21,22In single and fewlayer forms, the most well-known example is graphene, the first two-dimensional layered material to be isolated into monolayers.23 It has been the key platform for a number of groundbreaking physics experiments.24−26 The isolation of graphene has also kindled interest in electronics, optoelectronics, energy storage, biomedical applications, and many more in atomically thin scales. More recently, other layered materials have been studied, including hexagonal boron nitride,

ayered materials have been studied for decades due to the rich physical properties unique to systems with reduced dimensionality. Fractional quantum hall effects1 and Luttinger liquids2−4 are examples of interesting quantum phenomena only observed in low-dimensional systems. Their geometry also provides ample opportunities for applications, both in bulk form and thin layers. Bulk layered materials are widely used in various areas of technology, including solid-state lubricants,5 ion exchange,6 energy storage,7−10 catalysis,11 and thermoelectrics.12,13 Many commercial Li-ion batteries use graphite as anode and layered transition metal oxides as cathode, including LiCoO2. The layered structure of these materials allows for intercalation of compounds in interlayer spaces,14,15 lithium and hydrogen in particular for energy storage applications. Layered transition metal oxides are being investigated for nextgeneration sodium ion batteries as well.16,17 Large surface areas of layered materials are also useful in catalysis. Layered double hydroxides act as catalysts in many useful chemical processes, such as alcohol synthesis, hydrogenation, petroleum reforming, and Fischer−Tropsch process.11 Thermoelectrics are another application where layered materials are widely used. These materials have a high degree © 2017 American Chemical Society

Received: December 16, 2016 Revised: February 11, 2017 Published: February 13, 2017 1915

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Figure 1. Relationship between material dimensionality and covalently bonded cluster sizes used to identify 1D and 2D weakly bonded solids. A pair of atoms is considered to be bonded if they are closer than the sum of their covalent bond radii with a tolerance of 0.45 Å. Max and min in the figure refer to the maximum and minimum number of atoms in the bonded clusters; numbers 1 and 2 refer to the unit cell the clusters are found from 1 for the original unit cell (as in the Materials Project database) and 2 for its 2 × 2 × 2 supercell. Periodic boundary conditions were employed to ensure that the structures of the clusters are preserved regardless of how the unit cell is defined in the crystal. (a−c) In the case of materials without intercalated molecules or atoms, the size of the largest cluster increases by 2d, where d refers to the dimension of the structure. (d) The 1D and 2D structures with intercalated atoms or molecules, which are excluded from our list of 1D and 2D weakly bonded solids. In most cases, the intercalated molecule or atom is smaller than the part of the 1D or 2D structure in the unit cell (panel d, left; LiErS2), so the ratio between max1 and max2 stays the same. The intercalation can be identified with the condition min1 = min2, and we exclude all materials fitting this condition. In the case where the intercalated molecule is bigger than the 1D or 2D structure in the original unit cell (panel d, right; AsPbF7), the ratio between max1 and max2 does not equal 2 (1D structures) or 4 (2D structures). (e) Our algorithm for screening crystal structures to find weakly bonded solids with 1D and 2D substructures from covalently bonded cluster sizes.

There is evidence that there are many more potential layered materials that have yet to receive substantive attention.27,28,36,37 Moreover, there is much activity on assembling layers of different materials to make heterostructures with new material properties,34,35 albeit with practical challenges in fabrication. Materials consisting of weakly bonded units may exhibit fewer interface defects where unwanted charge recombination may occur in photonic applications. While layered materials have received considerable attention, there also exist bulk three-dimensional (3D) crystals that consist of weakly bonded one-dimensional chains, or molecular wires. They have been studied in the context of the physics of one-dimensional systems, such as charge-density waves in NbSe3,38 1D van Hove singularities,39 and Luttinger liquid behavior40 in molybdenum selenide molecular wires. Because of the anisotropic nature of these crystals, they are often observed to grow into needle-shaped structures, which makes them attractive for large-scale synthesis of 1D nanostructures using chemical methods. The synthesis of nanowires from inorganic compounds molybdenum chalcogenides, transition metal chalcohalides,41 chalcogens Se and Te in trigonal phase, SbSI, metal cyanides, and trichalcogenides have been reported.42

transition metal dichalcogenides, black phosphorus, and derivatives of graphene.6,27−29 While single layer forms have received much attention, many of the applications of these materials do not require a single layer. In fact, many devices can benefit from having more than one layer. MoS2 has a thickness-dependent band structure, which could be useful for photonic devices.30 Few-layers are desirable for tuning material properties through intercalation. Intercalation in electronic device application has the advantage of providing higher doping levels than adsorption.31 It can also induce phase changes, as in lithiation of MoS2, which converts semiconducting 2H-MoS2 to metallic 1T-LiMoS2.32 Finally, few-layer devices may be more robust against degradation than single layer devices. Few-layer graphene heat spreaders in integrated circuits are reported to retain mechanical and thermal properties better than than single layers.33 There is value in identifying the full spectrum of layered materials regardless of the potential for exfoliating into a single layer. Despite the fact that the number of researchers worldwide studying layered materials is quite large and these materials are well represented at all major professional meetings, the research is largely focused only on a small number of these materials. 1916

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Each element in the 2D and 1D weakly bonded solids were sorted by groups in the periodic table. Figure 2 shows the

Most molecular wires that have been studied are organic or organometallic.43−45 Inorganic molecular wires have been predicted to possess structural stability and versatile material properties but they have received relatively little attention to date.46 These materials could find applications in a variety of fields, including electronic and thermal applications requiring a high degree of anisotropy or energy applications requiring intercalation of atoms or molecules. In this work, bulk 3D materials in the Materials Project database47 were screened with a data-mining algorithm to identify two- and one-dimensional weakly bonded solids and lattice-commensurate heterostructures. The algorithm finds crystals that are composed of bonded two- or one-dimensional structures held together by weak bonds, that is, those characterized by van der Waals interactions or weakly ionic character. Conventional approaches on finding layered materials have been focused on examining materials of similar chemical composition to those of a few known layered materials. There are numerous review articles on layered materials, each of them listing up to about a hundred materials.6,27−29,48 Information on inorganic molecular wires is much more sparse.42,49 Data mining allows for fast screening of all materials in the database, including but not confined to materials belonging to known families of two- or onedimensional materials. This work reports an order of magnitude increase in the number of identified materials, presenting hundreds of previously unrecognized low-dimensional layered materials. We identify 1173 2D weakly bonded solids and 487 1D weakly bonded solids with unique chemical formulas, consisting of 23 distinct chemical families of 2D weakly bonded solids and 8 chemical families of 1D weakly bonded solids. We found 325 potential two-dimensional piezoelectric monolayers, lacking a center of inversion in the single-layer crystal structure. While piezoelectricity in single layers of specific materials (e.g., MoS2, h-BN) has been reported, a systematic evaluation of monolayer symmetries for a collection of layered materials has not been reported. We also identify weakly bonded lattice-commensurate heterostructures, that is, materials with weakly bonded adjacent units with dissimilar properties. In the following sections, we determine the distribution of physical properties of the twoand one-dimensional weakly bonded solids identified with our algorithm, including chemical families of 1D and 2D weakly bonded solids, band gaps, bulk, and monolayer point groups that determine piezoelectric and nonlinear optical properties. The complete list of materials we identified is provided in the Supporting Information. The majority of weakly bonded solids identified in this work come from experimentally reported crystals composed of stacks of 2D layers or 1D chains of bonded atoms. A total of 943 of 1173 two-dimensional and 401 of the 487 one-dimensional weakly bonded solids come from the Inorganic Crystal Structure Database, which have experimentally reported structures. We exclude all molecular solids, 3D bulk materials with no weakly bonded components, and materials with intercalated atoms or molecules. Examples of each kind of crystal structure is shown in Figure 1. The algorithm used to identify these materials is also summarized in Figure 1 and described in more detail in the last section. To compare our findings with known 2D and 1D weakly bonded solids and to get more information about their distribution, the list of materials identified with our algorithm was categorized into families of similar chemical compositions.

Figure 2. Histogram of families of weakly bonded solids with 2D (a) and 1D (b) subunits with similar chemical composition found in this work. Elements of the same group are represented by one symbol with the exception of N, P, Al, and O, which form families that other members of the same group do not. For example, the family TP4 consists only of transition metal tetraphosphides, and no nitrides or other pnictides. Ac, actinides; As, large pnictogens (As, Sb, Bi); F, halogens; S, chalcogens excluding O; La, lanthanides; T, transition metals. Only the families containing more than five materials are shown in the plot. Each distinct chemical formula is counted only once in the histogram, that is, materials with multiple-layered phases with distinct monolayer phases or stacking sequences are counted only once. Note that the families are grouped only with respect to chemical compositions, and crystal structures may vary within each family.

distribution of each family containing more than five materials. Note that fewer than a quarter of the materials identified in this work belong to the families in Figure 2. This highlights the fact that there are many more 2D and 1D weakly bonded solids to be explored beyond the materials in a few well-known families. Figure 2a shows families of 2D layered materials, including the well-known family of transition metal dichalcogenides indicated by TS2 in the figure. The families of transition metal dihalides and trihalides (TF2, TF3) are known as families of layered materials as well.27,28 They have been known to be layered since the 1970s, but most of them are very reactive to moisture50 and applications exploiting their layered structure have not been widely explored. In addition to transition metal dichalcogenides, other families of transition metal chalcogenides (TS, TS3) are being studied for applications;27,28 for example, TiS3 has been exfoliated into few layers and its electronic and optical properties have been measured experimentally.51 The success with TiS3 has brought transition metal trichalcogenides into attention, and more materials in this family are now being studied.52,53 The pnictogen chalcogenides As2S3 have been studied as topological insulators and for various applications including thermo1917

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Nano Letters electrics54 and battery electrodes.55 Materials in the family TPS3 are reported to exhibit interesting two-dimensional magnetic properties, including experimental reports of quasi-2D antiferromagnetic behavior in TPS3 with T = Mn, Fe, Ni and S = S, Se,56 as well as CoPS3.57,58 These materials have been claimed to be good candidates for 2D Heisenberg magnets on a honeycomb lattice. These materials are also being explored for practical usage. For example, the material MnPS3 has been studied as a hydrogen storage material using its layered structure,9 and its monolayer has been exfoliated along with CdPS3.59 Identification of these known materials verifies that our algorithm effectively finds layered materials. The algorithm also identifies families of materials that have not been studied in a single or few layer form or have not been studied in the bulk form. For example, we are unable to find any references to experimental studies of the seven materials with the formula TF2O, apart from crystal structures.60−62 This is also true for materials in the family La2S5O13, even though some lanthanide oxotellurates have been studied for applications as phosphor hosts. Representative crystal structures for the two-dimensional families in Figure 2a are shown in Table S1. While most materials in each family exist in the representative crystal structures shown in Tables S1 and S2, some individual materials within each family may exist in different crystal structures. The chemical families of one-dimensional weakly bonded solids that the algorithm identifies are shown in Figure 2b. The chain crystal structures of some materials belonging to the families of metal halides63 have been reported, including TiF4, VF4, TiI3, MoCl3, MoBr3, MoI3, PdCl2, PdBr2, and PdI2. Some materials in the family AsSF are actively being investigated for applications in solar cells, such as SbSI, SbSBr, SbSeI,64 BiSI, and BiSeI.65 The materials in this family have band gaps between 0.8 and 2 eV. Some are of lesser toxicity than cadmium- and lead-based compounds, which may be a desirable feature in solar cells.66 Moreover, the 1D structure of these materials reduces dangling bonds when the grain boundaries occur parallel to the 1D chains, which minimizes recombination losses.67 These advantages are not exclusive to the materials in the family AsSF, and other 1D weakly bonded solids identified in this work may have the potential to be useful in optoelectronic applications as well. Representative one-dimensional structures for each onedimensional family of weakly bonded solids are shown in Table S1. The one-dimensional family La(AlF4)3 is a particularly interesting case in that the crystals consist of stacks of not straight but helical chains. The structure is similar to the trigonal chalcogens Se and Te, with larger and chemically complex chains. This also demonstrates the strength of our algorithm; by testing for connectivity, the dimensionality of the material can be identified unrestricted by the geometric details of the crystal elements. While our method of finding families of materials identified most known families of layered materials, some families including the MXenes are absent from our list. The MXenes do not have an entry for a layered phase on the Materials Project database. As there is no single chemical family associated with layered oxides, 20 binary oxides with a spectrum of chemical compositions were identified. For example, Mo9O26 belongs to the family T9O26, PbO belongs to TO, and TeO2 belongs to SO2.

Figure 3 shows the band gaps for the bulk forms of the identified 2D and 1D weakly bonded solids. These are the band

Figure 3. Distribution of semilocal DFT Kohn−Sham band gaps for bulk materials consisting of 2D layers (a) and 1D chains (b) in the Materials Project database. Shown are band gaps reported in the Materials Project database computed for the bulk crystal rather than the isolated 2D and 1D components. If there are multiple layered phases for one material, the value of band gap for the most energetically favorable layered structure (i.e., structure with the lowest energy above hull) was chosen.

gap values that are reported in the Materials Project database for these respective materials. The band gap values are calculated for bulk materials using semilocal Kohn−Sham DFT with computational details described at the Materials Project Web site and summarized in the Supporting Information. Note that standard semilocal DFT functionals frequently underestimate band gaps for many materials. In some cases, including MoS2, Kohn−Sham quasiparticle band gaps have been reported to be similar to optical absorption gaps, although the established presence of strongly bound excitons suggests that this is not generally an appropriate comparison to make.68 To make the distribution clearer, we chose only one value per material for materials with multiple layered phases. For example, CdI2 has over a hundred different 2D layered structures in the Materials Project database, all within the band gap range of 2.3−2.5 eV, which would obscure the trends in the plot. The value of band gap for the lowest energy structure was chosen for such cases. We found that the identified layered materials exhibit a wide and continuous spectrum of intrinsic electronic band gaps, including 144 2D and 22 1D zero band gap weakly bonded solids. The spectrum includes materials in the full visible and telecommunications ranges. The spectrum also includes materials with semilocal DFT band gaps over five electron volts with potential for electronic insulating applications. Some examples of high band gap materials are BHO2 and AlHO2, 1918

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Figure 4. (a,b) Comparison between the inversion symmetries of a bulk crystal and a AA-stacked monolayers of the same material. In a layered material, an inversion center can exist in two locations, at the midpoint between two consecutive layers or in the middle of a monolayer. The two figures illustrate each case: (a) MoS2 in the 2H structure; inversion center is located only between two layers. (b) Bi2Pd, a material found with our algorithm; inversion center is located in each monolayer. In the case of 2H-MoS2, there is no inversion center in a monolayer, and the inversion symmetry arises from how the atomic layers are stacked. The inversion symmetry vanishes when the layers are AA-stacked. However, AA-stacking Bi2Pd monolayers does not break the inversion symmetry of the material. (c) The equivalence of inversion symmetry of a monolayer and an AAstacked material, which was used to identify inversion symmetry of the individual layers of layered materials. The inversion center at the midpoint between two consecutive layers of an AA-stack is denoted by i1, and the inversion center in a monolayer is denoted by i2. From translational symmetry, the black points connected by the red arrows are equivalent. Because all points on the layers have this translational symmetry, the inversion centers i1 between the layers and i2 in a monolayer are equivalent. (d,e) Distribution of point groups for layered materials. (d) The point group distribution of layered materials in the Materials Project database, omitting heterostructures. If there are multiple layered phases for one material, the point group of the most energetically favorable layered structure (i.e., structure with the lowest energy above hull) was chosen. The symmetry of the original bulk 3D crystals plotted in red and the symmetry of their AA-stacked crystals in blue, corresponding to the isolated monolayer symmetry. (e) The distribution of materials in (a) limited to those with nonzero band gaps. Nonzero band gap and noncentrosymmetric point groups are necessary conditions for piezoelectricity, and 325 of the materials found in this work satisfy the conditions.

verified.70,71 Moreover, inversion symmetry breaking through exfoliation can also occur in few layers of MoS2 if the number of layers is odd. We developed a method to calculate monolayer point group symmetries from AA-stacking of the monolayers, as described in methods section. Here we take the monolayers to exhibit the same structure as in the bulk material. This assumption is good for commonly studied two-dimensional materials including graphene and MoS2. This assumption could break down for more strongly bonded materials, largely excluded in the present study. We identify 2D piezoelectric material candidates from point group symmetries. For these calculations, we take a single layer to be isolated on a substrate or freely suspended in vacuum. The potential for isolating a single layer of each of these materials likely depends on a number of factors, including the interlayer binding energy and mechanical stability which require further study. Computed point groups of identified layered materials are presented in Figure 4d,e. Figure 4d depicts inversion symmetries in bulk crystals and AA-stacks and reveals the cases where inversion symmetry is broken upon exfoliation of a single layer as in MoS2. Figure 4d indicates that many materials lose inversion symmetry when exfoliated into monolayers. In

silicic acids SiH2O5 and Si4H2O9, metal halides MgCl2 and AlCl3, metal oxyhalides AlOCl and ErOCl, metal hydrogen sulfates Eu(HSO4)3 and Y(HSO4)3, metal sulfate hydrates Nd2(SO4) 3(H 2O)8 , Er2(SO4) 3(H2O) 8, Tb 2(SO 4)3(H2O) 8, Y2(SO4)3(H2O)8, Dy2(SO4)3(H2O)8, and magnesium phosphate hydrates Mg3P2(HO)16, MgP(HO)7, Mg2P2H12O13. This broad spectrum of band gaps, combined with the lowdimensional geometry of the materials, will find applications in diverse fields. Crystal symmetry affects a wide range of material properties and is a valuable knowledge when seeking materials of some specific properties. For example, the presence of inversion symmetry determines if a material can have nonzero piezoelectric moduli or frequency doubled nonlinear optical response. For a material to be piezoelectric, it needs a nonzero electronic band gap and lack of centrosymmetry. In layered materials, the symmetry of a single layer or few layers may differ from the bulk and lead to different physical properties. An example is piezoelectricity in single layers of MoS2. Bulk MoS2 has an inversion center between the individual layers and hence is nonpiezoelectric but monolayer MoS2 lacks an inversion center. Because monolayer MoS2 also has a nonzero band gap, it was predicted to be piezoelectric69 and was experimentally 1919

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Figure 5. Examples of lattice-commensurate heterostructures identified in this work. The heterostructures come from bulk crystals consisting of stacks of nonidentical layers or chains. Heterostructures were identified from our list of 2D and 1D weakly bonded solids by screening for clusters of distinct chemical compositions or numbers of atoms.

nent layers most of which are experimentally reported. These materials can circumvent all the experimental challenges of fabricating heterostructures from individual layers and have the potential to exfoliate directly into heterostructures for device applications. Because they are derived from 3D crystals, these heterostructures are lattice matched, distinguishing them from the vertical heterostructures that are obtained via mechanical stacking. They have the substantial manufacturing advantage of being amenable to growth using more conventional and scalable methods. The methods section describes how we identified heterostructures by finding materials consisting of clusters with differing number of atoms or chemical compositions. Figure 5 shows some of the heterostructures identified in this work including 2D, 1D, and mixed 1D and 2D heterostructures. All the materials in Figure 5 were reported to be synthesized by heating a stoichiometric mixture of starting materials sealed in a furnace.74−76 Among the heterostructures found in this work is AgPbBrO which consists of alternating layers of PbO (also a layered material) and AgBr. A particulate dispersion of AgBr/ PbO system has been studied for use as photodiodes.77 The bulk structure of AgPbBrO is interesting because it can be considered stacked heterojunctions of these materials. BiCuSeO, another heterostructure identified in this work, is a candidate thermoelectric material with high figure of merit (ZT) that has recently drawn attention in the thermoelectric research community.78,79 We found 69 2D heterostructures with unique chemical composition (89 in total) and 29 1D heterostructures. In the remainder of this article, we describe the method used to obtain the results of this work. We developed an algorithm to determine the dimensionality of van der Waals bonded subunits of bulk materials. Our database of bulk materials comes from the Materials Project database.47 Known layered materials typically have strong covalent bonds within atomic layers and weak van der Waals bonds holding the layers together. Our algorithm distinguishes between these types using bond length criteria, and analyzes how the atoms are connected to identify subunits. Strongly bonded pairs of atoms are identified if the interatomic distance of the pair is less than the sum of their reference atomic bond lengths plus a tolerance of 0.45 Å to account for bond length variability. The reference values of bond radii and tolerance are taken from the values used by Jmol,80 the Java viewer for chemical structures used in the Materials Project Web site. The bond radii used by Jmol are the

fact, the percentage of noncentrosymmetric structures increased from 18% in bulk crystals to 43% in monolayers. Furthermore, we find the spectrum of noncentrosymmetric point groups to be diverse. Figure 4e shows the symmetries of only the materials that exhibit a nonzero electronic band gap (from semilocal DFT), a requirement for the piezoelectric effect. Because different point group symmetries allow for different forms of nonzero elements of the piezoelectric tensors, there is a wide range of possibilities for piezoelectric responses. In MEMS applications, this data makes it possible to find a material with the desired piezoelectric coupling for specific applications. For example, the 2H structure of MoS2 exhibits a nonzero e11 coefficient, coupling in-plane strain to in-plane electrical polarization. For some MEMS applications, one might prefer other types of coupling including an e13 coefficient that couples in-plane strain to out-of-plane electrical polarization. For example, we find that candidates for nonzero e11 include materials of the form MT(PS3)2 (M = metal, T = transition metal, S = S or Se), which are ScAg(PS3)2, TmAg(PSe3)2, ErAg(PSe3)2, ScAg(PSe3)2, InAg(PS3)2, InAg(PSe3)2, and InCu(PSe3)2, all with point group 32. Four of the materials in the family TAsS2F (CdSbS2Cl, MnSbS2Cl, MnBiS2Cl, MnSbSe2Br, all with point group mm2) exhibit nonzero e13 coefficient and zero e11 coefficient. Similar forms of the piezoelectric tensor can arise when adatoms exist on one surface of a 2D material.72 There has been significant attention focused on van der Waals vertical heterostructures including vertically stacked dissimilar layered materials. van der Waals heterostructures provide an atomically sharp interface between two different materials without dangling bonds, opening new possibilities for applications in photonics and electronics. There have already been applications in electronics and optoelectronics, including vertical diodes and transistors, photodetectors, solar cells, and LEDs.34,73 To date, methods of creating vertical heterostructures have largely been of a labor-intensive nature that is not amenable to scaling for large-scale manufacturing. These methods involve the isolation of single layers through mechanical exfoliation or chemical vapor deposition, and stacking them on top of each other. The preparation of quality monolayers and aligning the monolayer flakes together pose a significant experimental and manufacturing challenge. In the process of seeking weakly bonded solids, we have also discovered that there are a number of bulk materials that occur naturally as heterostructures of chemically distinct subcompo1920

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only one of the coexisting layers or chains. To find heterostructures, we screened our list of 1D and 2D weakly bonded solids to find materials with clusters of different sizes or chemical compositions. For all clusters found in a material, if the number of atoms or chemical compositions are not identical, the material is identified as a heterostructure. The choice of bond lengths used in this algorithm is expected to have some impact on the classification of the materials. To examine the effects of changing the bond lengths on the set of materials we identify as 2D and 1D weakly bonded solids, we applied the same algorithm with a different set of bond lengths based on a statistical analysis of crystallographic data from the Cambridge Structural Database.81 Excluding the 20 elements that have the greatest difference between those two data sets, the values of the radii match within 10%. The number of 2D weakly bonded solids identified with these new bond lengths are 2083 (counting materials with the same chemical formula), compared with 1664 identified using the original bond length set. The increase mostly comes from materials with alkali metals and alkaline earth metals for which the new bond lengths are significantly larger than the original values due to consideration of primarily ionic bonds with oxygen in this database. The number of newly identified materials containing group I or II elements are 484. Out of the materials originally classified as 2D, only 13% were not included in the new classification with these alternative bond length definitions.

standard elemental radii in the Cambridge Structural Database from October 1997 release, which are listed in Table S3. After identification of bonded pairs of atoms, the algorithm finds all clusters of connected atoms, taking into account periodic boundary conditions. These clusters are molecules for molecular solids, parts of chains for 1D weakly bonded solids, parts of layers for 2D weakly bonded solids, and a part of the bulk structure for 3D bulk solids. This cluster analysis is subsequently repeated for a 2 × 2 × 2 supercell, that is, one that was obtained by doubling the length of each of the lattice vectors. The change in the number of clusters when going from a 1 × 1 × 1 to a 2 × 2 × 2 super cell reveals the dimensionality of the subunits. The size of the largest cluster remains invariant in molecular solids, doubles in one-dimensional weakly bonded solids, and increases 4-fold in 2D weakly bonded solids. Examples of the dependence of cluster size on supercell size are described in Figure 1. This algorithm is similar to that proposed in ref 37. To construct our list of 2D and 1D weakly bonded solids, we exclude all molecular solids, 3D bulk materials, and materials with intercalated atoms or molecules. The materials identified by the algorithm are energy-stable 1D and 2D weakly bonded solids without intercalated atoms or molecules. Materials with energy above the convex hull greater than 0.1 eV are excluded for being potentially metastable rather than stable. While this gives information on the energy stability of bulk crystals, it does not guarantee that the material will be energy or mechanically stable as a monolayer, for example, in a freely suspended form. Most materials identified by our algorithm are largely covalently bonded solids, which are likely to be more stable as monolayers than the ionic counterparts. The decision process for materials’ dimensional structure is summarized in Figure 1e. For the 1D and 2D weakly bonded solids identified with the algorithm, the largest cluster is saved into a separate crystal structure file for crystal symmetry analysis. This is accomplished by removing all atoms from the primitive cell except those contained in the largest cluster. We developed a method to calculate monolayer point group symmetries by using AA-stacked crystals, that is, a bulk crystal consisting of monolayers stacked directly on top of each other. The two-dimensional point symmetry elements of an AAstacked crystal are clearly the same as its monolayer in the plane of the layer, and we only need to consider an inversion center and a mirror plane in the plane of the layers as additional symmetry elements. An AA-stacked crystal consists of monolayers periodically translated in the direction perpendicular to the mirror plane; mirror and glide symmetry in the plane of the layers are preserved in AA-stacked crystals. The equivalence of inversion symmetry in monolayers and AAstacked crystals is demonstrated in Figure 4. Therefore, the point group of an AA-stacked bulk crystal reflects the symmetries of the monolayer. To create AA-stacked crystals, we first isolated monolayer unit cells (consisting of the largest cluster in the unit cell) from bulk material unit cells as described above. Then we modified the unit cell so that the a- and b-axes are in the plane of the layer, and c-axis is perpendicular to the plane of the layer. We used Python Materials Genomics (pymatgen), an open-source Python library for materials analysis, to analyze symmetries of the AA-stacked materials. In the case of weakly bonded heterostructures, layers or chains of different chemical composition or structure coexist in the crystal. In this case, selecting the largest cluster would select



ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.6b05229. Representative crystal structures for families of 2D weakly bonded solids, representative crystal structures for families of 1D weakly bonded solids, table of covalent bond radii used in this work, computational details for material properties (PDF) Spreadsheet of all materials found in this work (XLSX)



AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: (650) 723-2971. Fax: (650) 725-4034. ORCID

Gowoon Cheon: 0000-0002-3026-6796 Notes

The authors declare no competing financial interest.



ACKNOWLEDGMENTS This work was partially supported by Army Research Office Grant W911NF-15-1-0570, by Office of Naval Research Grant N00014-15-1-2697, by the U.S. Army Research Laboratory through the Army High Performance Computing Research Center, Cooperative Agreement W911NF-07-0027, by NSF Grant DMR-1455050 and EECS-1436626, and by the Stanford Graduate Fellowship program.



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