Computational Synthesis of Substrates by Crystal Cleavage

Joshua T. Paul,1, 2 Alice Galdi,3, 4 Christopher Parzyck,3 Kyle Shen,3 and Richard G. Hennig1, 2, ∗

1Department of Materials Science and Engineering, University of Florida, Gainesville, Florida 32611, USA2Quantum Theory Project, University of Florida, Gainesville, Florida 32611, USA

3Department of Physics, Cornell University, Ithaca, New York 14850, USA4Cornell Laboratory for Accelerator-Based Sciences and Education, Cornell University, Ithaca, New York 14850, USA

(Dated: March 31, 2021)

The discovery of novel substrate materials has been dominated by trial and error, opening the opportunity fora systematic search. To identify stable crystal surfaces, we generate bonding networks for materials from theMaterials Project database with one to five atoms in the primitive unit cell. For three-dimensional crystals in thisset, we systematically break up to three bonds in the bonding network of the primitive cell. Successful cleavagereduces the bonding network to two periodic dimensions, creating a layer of the cleaved crystal. We identify4,693 unique cleavage surfaces across 2,133 bulk crystals, 4,626 of which have a maximum Miller index of 1.To characterize the likelihood of cleavage and the thermodynamic stability of the cleaved surfaces, we createmonolayers of these surfaces and calculate the work of cleavage and the partially-relaxed surface energy usingdensity functional theory to discover 3,991 potential substrates, 2,307 of which do not contain f-valence electronsand 2,183 of which are derived from a bulk precursor with an entry in the Inorganic Crystal Structure Database.Following, we identify distinct trends in the work of cleavage of these layers and relate them to metallic andcovalent/ionic bonding of the three-dimensional precursor. We also assembled a database of commerciallyavailable substrates and show that the database of predicted substrates significantly enhances the diversity andrange of the distribution of electronic properties and lattice parameters, providing opportunities for the epitaxialgrowth of many materials. We illustrate the potential impact of the substrate database by identifying severalnew epitaxial substrates for the transparent conductor BaSnO3, which exhibit low cleavage energies and resultin strains an order of magnitude lower than currently used substrates. The open-source database of predictedand commercial substrates and their properties is available at MaterialsWeb.org.

I. INTRODUCTION

Single-crystal substrates facilitate the epitaxial growth ofcrystalline thin films based on their surface similarity, in termsof lattice parameters and symmetry. Many single-crystal sub-strate materials are commercially available with different facetorientations. However, while exfoliated 2D materials such asgraphene are naturally smooth, cleaving a crystal does notguarantee an atomically smooth surface. Energetic instabil-ities of cleaved facets can result in the formation of variousdefects that limit the quality of a substrate for synthesis ef-forts. Identifying a wider range of substrates with low surfaceenergies would enable the growth of high-quality materials byminimizing the presence of undesirable defects. For example,the cubic perovskite BaSnO3 has diminished electronic prop-erties due to the lattice mismatch of its (100) surface and thesubstrates suitable for its growth.1–3 In addition to lattice mis-match, one must consider the reactivity between a substrateand the thin film being grown, for both the precursor chem-ical reactions and the final thin film composition, which willfurther reduce the number of available options.

The discovery of two-dimensional (2D) materials presentsimilar challenges to substrates, most notably that both re-quire materials with low surface energies. Computational ef-forts to identify novel 2D material have significantly expandedthe list of potential monolayers and helped guide experimentalsynthesis.4–11 One discovery technique is data mining, whichsearches bulk materials databases for yet unidentified mono-layers.6–11 A recent effort in this direction identified and char-acterized the bonding network of a crystal structure using thetopological scaling algorithm (TSA).7 This algorithm identi-

GaO

Cleavageplane

FIG. 1. Example of a cleavable crystal, Ga2O3. The (201) planeexhibits a low density of bonds that, when cleaved, creates a lowenergy surface.

fies bonding clusters within a finite number of unit cells, thenuses the scaling of this network size as a function of super-cell size to define dimensionality. One of the most significantcontributions of this approach is that one does not require oneto define a facet to search along a priori. The algorithm it-self will identify a monolayer with a surface parallel to any

arX

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002.

0090

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at.m

trl-

sci]

29

Mar

202

1

2

facet cut, and thus renders the issue of orientation mute. Thisresulted in the discovery of over 600 low-energy 2D mate-rials7 and the creation of the MaterialsWeb.org 2D materialdatabase.

Previous computational efforts have been made to iden-tify potential substrate materials, e.g., Ding et al.12 searchedthe Materials Project database for a substrate to stabilize ametastable polymorph of VO2. Using lattice mismatch andinduced thin-film strain as criteria, they identify several can-didates. That search considered a set of low-index facets butdid not account for the surface energy of the facets. This il-lustrates the need for an algorithm to identify likely substratesand a database of low-energy work of cleavage facets, whichis the scope of this work.

In this work, we present an approach to identify novelsubstrates, which utilizes the computational cleavage of bulkcrystals. This effort is motivated by our recent 2D struc-ture search using the genetic algorithm software GASP13,14

for the Ga2O3 system. In that search, we identified a mono-layer structure that can be cleaved from the bulk crystal andwas already experimentally synthesized.15,16 Thus, we devel-oped data mining techniques to discover planes of cleavagein fully periodic crystals. Rather than using the TSA to iden-tify van der Waals gaps, we use it first to identify conven-tionally bonded solids; then, we systematically break bondsin the crystal to create a low-dimensional structure. When thebreaking of bonds generates a surface (i.e., gives the materiala 2D structural motif), we extract a unit cell thick monolayerand calculate the work of cleavage (i.e., the energy cost re-quired to cleave the crystal and form two surfaces) using den-sity functional theory (DFT). This approach identifies nearly4,000 potential substrates with surface energies comparable toexperimentally used substrates.

II. RESULTS

Candidate materials. For our study, we select all materialsin the MaterialsProject database17 with five or fewer atoms inthe primitive unit cell and within 50 meV/atom of the con-vex hull, the latter to promote thermodynamic stability of thefinal surface. Next, the topological scaling algorithm (TSA)identifies conventionally networked structures from this sub-set and creates a list of all bonds between neighboring atomsin these crystals. The bonding identification utilizes the em-pirical atomic radii of the elements, as provided in the pymat-gen software package,18 which we increase by 10%.

Bond cleavage. We implement two approaches for cleavingbonds in a crystal structure, which we illustrate in Fig. 2. Thefirst approach breaks all periodic instances of a bond betweentwo atoms in the supercell and is denoted as the “periodicbonds” approach. In this case, the three A-B bonds shown inFig. 2(a) are treated separately. The second approach breaksthe bonds between all periodic instances of the atoms in thesupercell and is referred to as the “periodic atom” approach,illustrated in Fig. 2(b).

We systematically break unique bonds in the primitive cell

(a) Periodic Bond (b) Periodic Atom

A

B

C

A

A A

A A A

AA

B

BBB

B B

B B B

C

C C

CC

A AA

A A

A A A

AA

B B

BBB

B B

B B B

C

C

A

FIG. 2. Bond cleaving algorithm. (a) The periodic bond approachcleaves all periodic images of a bond. (b) The periodic atom ap-proach cleaves the bonds between all periodic images of the atoms.The periodic atom approach requires at least three atoms in the prim-itive cell to generate a cleaved facet.

up to a maximum number of bonds given by

Nbonds = round(

α N2/3atoms

), (1)

where Natoms is the number of atoms in the primitive cell, andα is a scalable parameter based on the desired maximum num-ber of bonds broken. The exponent of 2/3 appropriately scalesthe number of bonds per plane with increasing cell size. Thechoice of α = 1 in this work results in one to three cleavedbonds in primitive cells of one to five atoms. For the peri-odic bond approach, Nbonds scales with the square of the su-percell size. For the periodic atom approach, Nbonds scales atleast with the square of the supercell size, with the potential tobreak a significantly larger number of bonds than the periodicbond approach.

These two approaches are complementary, as illustrated inFig. 2. For example, applying the periodic atom approach tothe structure in Fig. 2(a) or the periodic bond approach to thestructure in Fig. 2(b) would both fail to cleave a surface withα = 1 in Eq. 1. As the unit cell size increases, the periodicbond and periodic atom approaches become equivalent. Witha high enough α value, the periodic bond approach will alsoidentify the cleaved surfaces of the periodic atom approach.However, this would be significantly more computationallyexpensive due to the increase in possible combinations of bro-ken bonds, which need to be considered.

Topology of resulting structure. To determine if a cleav-age surface has been generated, the TSA is run on the three-dimensional primitive cell with Nbonds or fewer bonds bro-ken (using either approach). If the TSA identifies a two-dimensional structural motif with the same stoichiometry asthe overall cell, we have created a cleavable surface. Thissurface is isolated as a unit cell thick monolayer and orientedsuch that the ~a and ~b lattice vectors span the 2D lattice ofthe monolayer structure and the ~c lattice parameter is chosen

3

perpendicular to the (~a,~b) plane. Due to the crystal symme-try, our algorithm can identify multiple instances of some sur-faces. We remove these duplicates and identify the unique sur-faces extracted from each crystal using the pymatgen structurematching algorithm.17

Predicted substrates. From a starting set of 120,612 mate-rials in the Materials Project database,17 9,089 crystals meetthe criteria of exhibiting (i) a formation energy within 50meV/atom of the thermodynamic hull (72,316), (ii) a prim-itive cell of 5 or fewer atoms (9,980), (iii) all lattice anglesgreater than 16◦ and less than 164◦ (9,973), and (iv) a bond-ing network of three-dimensional topology. Criteria (iii) isintroduced as a result of the exfoliation algorithm, which isfound to return incorrect cleaved surfaces if the lattice has anextreme lattice angle. Though applying the algorithm to theconventional cell of these crystals resolves the issue, this re-sults in a unit cell with greater than 5 atoms and thus are notconsidered. Applying the periodic bond and periodic atomcleavage approaches to the 9,089 materials generates 1,925and 4,006 unique surfaces, respectively. We apply the pymat-gen structure tool to the combined list of 5,931 surfaces toidentify a total of 4,693 unique cleaved surfaces across 2,133bulk crystals. respectively. We apply the pymatgen structuretool to the combined list of 5,845 surfaces to identify a totalof 4,693 unique cleaved surfaces across 2,133 bulk crystals.It is worth noting here that Ga2O3 is not identified as cleav-able by this search, as (i) the number of atoms in the primitivecell of the crystal is greater than five and (ii) the TSA identi-fies the crystal as layered when using our choice of 1.1 timesthe atomic radii of Ga and O, rather than as a fully networkedstructure.

To determine the stability of the surfaces, we perform DFTcalculations with VASP.19,20 We extract a single unit cell thick

Å

(0001) AlN

(0001)ZnO

(0001)CdS

WurtziteStructure

Work of Cleavage (meV/Å2)

Surf

ace

Ener

gy (m

eV/Å

2 )

FIG. 3. The work of cleavage and the surface energy after ten relax-ation steps of the cleaved surfaces. Red indicates a higher density ofpoints. The solid, dashed, and dotted black lines represent the workof cleavage of the (0001) CdS, ZnO, and AlN substrates, respectively.The grey region indicates materials have a work of cleavage greaterthan common substrates.

slab for each of the identified 4,693 unique cleavage sur-faces and perform geometric optimization of atomic positionswithin the slabs and a single-point calculation for the bulkprecursor to maintain consistency across our energy compar-isons. Of the 4,693 monolayers, 4,614 converged or reachedour maximum number of iterations with a final surface energyof at most half the work of cleavage. A sizable number of2,015 of these surfaces contain f-valence elements, for whichsemilocal exchange-correlation functionals exhibit larger for-mation energy errors.21 Therefore, care should be taken whenconsidering these materials in the MaterialsWeb.org substratedatabase.

We use the work of cleavage of a monolayer, Ecleave, tomeasures the energy required to cleave a unit area of the bulkprecursor material and the surface energy, Esurf, to describethe thermodynamic stability of a material’s facet:

Ecleave =(Esub,as−cut −Ebulk

NsubNbulk

)

Asub(2)

and

Esurf =(Esub,opt −Ebulk

NsubNbulk

)

2Asub, (3)

where Esub,as−cut and Esub,opt are the energies of the as-cut sub-strate immediately after cleaving and of the optimized sub-strate after relaxing the atomic positions; Ebulk is the energyof the bulk precursor, Nbulk and Nsub represent the number ofatoms in the bulk and substrate, respectively, and Asub denotesthe substrate area. The factor of two difference is due to thework of cleavage being a measure of the energy needed tocreate two surfaces, while the surface energy is the thermody-namic stability of a single surface.

To validate that a monolayer accurately approximates thework of cleavage for cleaving a crystal, we calculate thechange in work of cleavage with slab thickness for a subsetof 21 (001) surfaces, from one to four unit cells. We findchanges in the work of cleavage of less than 3 meV/A2 inthese systems, confirming that one unit cell thick slabs suf-ficiently describe the work of cleavage for a cleaved surfacein this high-throughput effort. To verify that the work ofcleavage indicates surface stability, we compare the calcu-lated work of cleavage for 4,614 cleaved, free-standing slabsto their partially-optimized surface energy in Fig. 3. We ob-serve that 2,557 materials display surface energies within 10%of half their work of cleavage and 1,615 within 5%.

We note that there is the possibility of surface reconstruc-tions for some of the predicted substrates. However, due tothe high computational cost, we do not perform a search forpossible reconstructions. The thermodynamic driving forcefor surface reconstructions is the surface energy. Hence, theprobability for reconstructions decreases for lower energy sur-faces. However, before pursuing synthesis of a substrate inthis work, we recommend determining at least the dynamicstability of a surface structure using phonon calculations and,if possible, searching for reconstructions using, e.g., geneticalgorithm searches.14 Note that as this is a global optimiza-tion problem,22 there is no guarantee that any reconstruction,

4

ÅÅÅ

Bond Density (bonds/Å2)Bond Density (bonds/Å2) Bond Density (bonds/Å2)

Wor

k of

Cle

avag

e (m

eV//Å

2 )

Wor

k of

Cle

avag

e (m

eV//Å

2 )

Wor

k of

Cle

avag

e (m

eV//Å

2 )

(a) Any precursor (4,614) (b) Metallic precursor (2,795) 1.3 eV/bond (c) Insulating precursor (1,819) 1.8 eV/bond

FIG. 4. Distribution of the work of cleavage vs. bond density of cleaved surfaces, represented by kernel density estimation: (a) for 4,614cleavage surfaces, (b) for surfaces with metallic precursors, and (c) for surfaces with precursors exhibiting an electronic bandgap. We usethe bandgap reported by Materials Project for the bulk precursors of the surfaces. The dashed lines indicate the mean energy per bond. Onaverage, the metallic precursors exhibit a higher coordination number, and hence bond density for the cleavage plane in conjunction with alower energy per bond than the more covalent and ionically bonded precursors. This trend is reflected in the average energy of the cleavedbonds for metals being lower at 1.3 eV/bond compared to 1.8 eV/bond for the cleaved covalent and ionic precursors.

let alone the most stable reconstruction, will be identified withcomputational techniques. Reconstructions will also be sig-nificantly altered by the adsorption of species in the first stagesof the film growth on the substrate, making this problem moreclosely related to the specific film-substrate interface.

III. DISCUSSION

Our search identifies three surfaces currently used as sub-strates: hexagonal CdS, ZnO, and AlN. These crystals sharethe same structure and are all commercially available as(0001) substrates. In our search, we identify four substratesfor each material: one (110), one (101), and two (001) facets.These facets are equivalent to the Miller-Bravais conventionof (1100), (1011), and (0001). The most stable facet for thesecrystals is (110), with a work of cleavage of 56, 123, and 275meV/A2 for CdS, ZnO, and AlN, respectively. The most sta-ble (001) facet of each crystal has a work of cleavage of 92,196, and 407 meV/A2, respectively. Fig. 3 highlights 3,991surfaces (2,307 f-valence free and 2,183 with an entry in theInorganic Crystal Structure Database (ICSD) entry) with awork of cleavage below that of (001) AlN, and 935 (718 f-valence free and 418 with an ICSD entry) surfaces below thatof (001) CdS.

We consider any slab with a work of cleavage less than(001) AlN to be a suitable substrate and included in oursubstrate database. Some facet cuts could have differentterminations for the same cleaved unit. For example, thecleaved (001) AlN can has both an aluminum-terminated andnitrogen-terminated facet. For these cases, the reported workof cleavage and surface energy are an average of these termi-nations. In addition to the energy benchmarks, we identify theconventional cell hkl index of the facets we cut. Of the 4,693

facets, 4,626 have Miller indices no greater than 1, showingthat the vast majority of planes we cleave are low-index facets.Of the remaining facets, 63 have a maximum Miller index of 2and 4 have a maximum Miller index of 3. While our choice ofprimitive cells biases the indices found, these results demon-strate how our approach is agnostic to the orientation of thefacet cuts and the potential for future searches to expand thissubstrate database.

Fig. 4 shows the trend between the number of cleaved bondsand the work of cleavage across the 4,614 converged cleavedsurfaces. The joint distribution in Fig. 4(a) indicates that ourcleavage criterion of Eq. (1) results in both a low density ofcleaved bonds and a moderate spread of work of cleavage.Furthermore, the distribution indicates two clustering trendsin the plot, which correspond to different average bond ener-gies. We attribute these trends to different types of chemicalbonds being broken. Metallic systems typically display highcoordination numbers and somewhat lower bond energies.Covalent and ionic bonds share localized electrons, which re-sults in lower coordination numbers and higher energies perbond.

To test this hypothesis, Figures 4(b) and (c) show the workof cleavage and bond density distribution for monolayers de-rived from metallic and insulating precursors, respectively,based on the precursor bandgap reported in the MaterialsPro-ject database.17 We observe that the two clusters in the dis-tribution indeed correspond predominantly to metallic andcovalent/ionic bonding. Metallic bonds typically occur inmaterials with higher coordination numbers and have loweraverage energy than covalent or ionic bonds, resulting in abroad spread of bond densities in Fig. 4(b). In contrast, co-valent/ionic bonds typically correspond to lower coordinationnumbers with higher energies for each bond, which is re-flected in the distribution in Fig. 4(c). The average bond en-

5

426

464

385 (tetragonal)

Lattice Parameter Ratio (b/a)

Lattice Parameter a (Å)

Metals

(d)Lattice Parameter a (Å)

Ban

d G

ap (e

V)

327 Insulatorsand Semiconductors

Lattice Parameter a (Å)

Ban

d G

ap (e

V)

97 Insulatorsand Semiconductors

Lattice Parameter a (Å)

Ban

d G

ap (e

V)

585 Insulatorsand Semiconductors

(a) (b) (c)

FIG. 5. Lattice vectors and bandgaps for (a) hexagonal, (b) square, and (c) tetragonal surfaces of semiconducting or insulating substratesidentified in this work. For the tetragonal substrates in (c), the a and b lattice vectors are both plotted and connected with a dashed line, andthe b kernel density is plotted in grey. 93 of the 106 commercially available insulating substrates identified are shown in red filled markers.For metallic substrates, (d) shows boxplots for the hexagonal and square surface structures and the b/a ratio of the tetragonal ones. The23 monoclinic surfaces are not represented in these figures. Surfaces are classified with a maximum lattice vector difference of 1% and amaximum in-plane angle difference of 0.5◦. Substrates containing f -valence elements are not shown.

ergies of the metallic and insulating precursors are 1.3 and1.8 eV/bond, respectively, further demonstrating the differ-ence of bonding in these systems. As bonds exist across acontinuous spectrum of metallic, covalent, and ionic charac-ter, the average bond energy is not a precise description of thebonding in these systems. However, the observed bond ener-gies in the metallic and insulating precursor materials revealan overall consistent trend.

To determine the potential of our data mining approachto expand the set of known substrates, we characterize thecleavage surfaces’ symmetry and lattice parameters. Fig. 5illustrates the lattice parameter distribution of the hexagonal,square, and tetragonal substrates identified in this work whichare free of f -valence elements, as well as the electronic prop-erties of their bulk precursors. The broad range of electronicbehavior and lattice parameters for each symmetry indicatesthat these cleaved crystals could provide suitable substratesfor a variety of thin film systems.

We follow by creating a database of commercially availablesubstrates totaling 190 substrates and include the data in Fig. 5using crystal structures and band gaps sourced from the Mate-rials Project database. Comparing the database of predictedsubstrates to the one of commercially available substratesshows two significant advancements. First, we greatly expandthe list of substrates exhibiting band gaps within the visiblespectrum and beyond, creating more opportunities for opticalexcitation of thin films from beneath the substrate rather thanfrom above. Second, we identify substrates with a broaderrange of lattice parameters, which can help synthesis efforts.For example, currently, 23 commercially available hexagonalsubstrates exhibit lattice parameters between 2.5 and 5.0 A.The computational database expands this list to 984 hexago-nal substrates within the same range. This increase providesmore opportunity to identify a substrate that both epitaxiallymatches the growth crystal and is chemically compatible.

To demonstrate the application of this substrate database,we epitaxially match cubic perovskite (100) BaSnO3, using

the crystal structure and stiffness tensor provided by Mate-rialsProject17,23 to substrates with a work of cleavage belowthat of (0001) AlN. We use the pymatgen18 lattice match-ing algorithm to epitaxially match the perovskite to the lay-ers extracted in our search. We identify 42 cleavage sur-faces as potential substrates when using the screening crite-ria of (i) a work of cleavage less than that of (0001) AlN,(ii) an induced strain energy below 2 meV/atom, (iii) no f-valence species, and (iv) epitaxial matches to a single unitcell of BaSnO3. Substrate matching also requires consider-ing the chemical compatibility between the substrate and thethin film, and thus we focus further discussion on substrateswith chemically inert surfaces.

We highlight here three potential substrates: (001)Rb2NiO2,24 (001) NiO,25 and (001) CaSe.26 The work ofcleavage for each substrate is small, being 19, 72, and56 meV/A2, respectively. The resulting epitaxial strain ofBaSnO3 for Rb2NiO2 and NiO are also small, with only+0.3% and +0.4%, which correspond to strain energies of 0.2and 0.4 meV/atom, respectively. These are an order of mag-nitude smaller than currently used substrates such as (001)SrTiO3 (−5.4%) and (001) MgO (+2.2%) indicating the op-portunity for defect-free growth of BaSnO3 on these newsubstrates. The strain energy when using CaSe is larger at1.3 meV/atom, though still with a low lattice mismatch of+0.7%. All three precursor materials – Rb2NiO2, NiO, andCaSe – have been experimentally synthesized,24–26 providingnew opportunities for the growth of BaSnO3. In addition, theformer two being oxides indicates that the surfaces will befairly inert and unlikely to form strong chemical bonds withBaSnO3.

To facilitate the use of these substrates for further stud-ies and future synthesis efforts of epitaxial single-crystal thinfilms, we provide the substrate structures and the data on theirstability under an open-source license at MaterialsWeb.org.Furthermore, we make the software for cleaving crystal struc-tures available as part of the open-source MPInterfaces pack-

6

age.27,28

IV. SUMMARY

In conclusion, we developed a data mining approach thatsystematically breaks bonds in three-dimensional crystals toidentify cleavage planes for substrate synthesis. We identify4,693 unique cleavage surfaces across 2,133 periodic crys-tals and determine their structure and stability. We show that3,991 surfaces display a work of cleavage comparable to thatof the known substrate material (0001) AlN. These cleavagesurfaces show a broad distribution of electronic properties andlattice parameters. We illustrate their utility by identifying 42substrates for BaSnO3 with epitaxial matches that are an orderof magnitude better than currently used substrates, explicitlydiscussing two oxide and one chalcogenide substrate. Thoughwe limited our search to small primitive cells, the low sur-face energy of Ga2O3 indicates that there are many more low-energy substrates which can be cleaved from periodic crystals.

V. METHODS

Characterization of cleaved materials. To calculate the sta-bility of the cleaved surfaces, we perform density functionaltheory (DFT) calculations with the projector-augmented wave(PAW) method as implemented in the VASP package.19,20 Thechoice of PAW potentials follows the recommendation of py-matgen.17 We employ the Perdew-Burke-Ernzerhof (PBE)29

approximation for the exchange-correlation functionals. Toobtain convergence of the energy to 1 meV/atom, we use a Γ-centered k-point mesh with a density of 60 k-points per A−1

and a cutoff energy for the plane-wave basis set of 600 eV.For the cleaved slabs, we employ a vacuum spacing of at least14 A and reduce the number of k-points in the out-of-planedirection to 1. The Brillouin zone integration uses Gaussiansmearing with a width of 0.03 eV. The DFT calculations areperformed spin-polarized with an initial ferromagnetic config-uration. Transition metal and f -valence atoms are initializedwith a magnetic moment of 6 µbohr and all others with a mo-ment of 0.5 µBohr. We perform single-point calculations on thebulk precursors sourced from the Materials Project database toensure accurate energy and consistency of calculation settingsacross all systems. We allow a maximum of ten iterations(ionic steps) during the optimization of the atomic positionsin the slabs, and as not all surfaces converged within that win-dow, we refer to this energy as “partially-relaxed.” For thesystems that reached ten iterations, the average and mediansurface energy gained are 1.6 and 0.2 meV/A2, respectively,from the relaxations. We keep lattice constants of the slabsfixed at the cleaved values for these calculations.

Workflow. We use the software packages pymatgen18 andMPInterfaces27 to prepare the input files, organize the resultsfor the cleavage algorithm, and analyze the results of the DFT

calculations. We use the pymatgen structure matching algo-rithm with an atomic position tolerance of 10−4 and not per-mitting any primitive cell reduction, lattice scaling, or super-cell transformations.17 Decreasing the tolerance to 10−5 onlymarginally increases the number of unique surfaces by 27,indicating that the choice of 10−4 provides high confidencein the uniqueness of the identified substrates. Any surfaceswhich diverged in energy or displayed a final surface energygreater than half the work of cleavage are excluded from theanalysis.

VI. DATA AVAILABILITY

The database of substrates identified in this work is freelyavailable at MaterialsWeb.org. The software we developedand used to search for novel substrates is freely available aspart of the MPInterfaces software package.28

ACKNOWLEDGMENTS

We thank S. Xie, A. M. Z. Tan, W. DeBenedetti, I. Bazarov,and S. Karkare for helpful discussions. This work was sup-ported by the National Science Foundation under grants Nos.PHY-1549132, the Center for Bright Beams, DMR-1542776and OAC-1740251, and the UF Informatics Institute. Thisresearch used computational resources of the University ofFlorida Research Computing Center. Part of this research wasperformed while the author was visiting the Institute for Pureand Applied Mathematics (IPAM), which is supported by theNational Science Foundation under grant No. DMS-1440415.

COMPETING INTERESTS

The Authors declare no Competing Financial or Non-Financial Interests.

AUTHOR CONTRIBUTION

All authors contributed extensively to the work presented inthis paper. J.T.P. and R.G.H. conceived the cleavage algorithmand structure search strategy. J.T.P. implemented the algo-rithm and performed the structure search and analysis. A.G.devised the application of the substrate database to materi-als and compiled the list of commercially available substrates.A.G., C.P., and K.S contributed the list of commercially avail-able substrates. J.T.P. created the structure files for these com-mercial substrates. J.T.P., A.G., and R.G.H. contributed to thewriting of the manuscript.

7

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