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Vei Wang, Gang Tang, Ya-Chao Liu, Ren-Tao Wang, Hiroshi Mizuseki, Yoshiyuki Kawazoe, Jun Nara, Wen Tong Geng

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This document is the unedited Author’s version of a Submitted Work that was subsequently accepted for publication inJournal of Physical Chemistry Letters, copyright © 2022 American Chemical Societyafter peer review. To access the final edited and published work see https://doi.org/10.1021/acs.jpclett.2c02972[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

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[High-Throughput Computational Screening of Two-Dimensional Semiconductors](https://mdr.nims.go.jp/datasets/13df47f9-c709-4436-9a60-b6db7f82af7a)

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2D_semiconductors_2022_07_23.pdfHigh-Throughput Computational Screening of Two-Dimensional Semiconductors andHeterostructures for Photocatalytic ApplicationsV. Wang,1, ∗ G. Tang,2 R. T. Wang,1 Y. C. Liu,1 Y. Y. Liang,3 Y. Kawazoe,4, 5, 6 J. Nara,7 and W. T. Geng8, †1Department of Applied Physics, Xi’an University of Technology, Xi’an 710054, China2Advanced Research Institute of Multidisciplinary Science,Beijing Institute of Technology, Beijing 100081, China3Department of Physics, Shanghai Normal University, Shanghai 200234, China4New Industry Creation Hatchery Center, Tohoku University, Sendai, Miyagi 980-8579, Japan5Department of Physics and Nanotechnology, SRM Institute ofScience and Technology, Kattankulathur,Tamil Nadu-603203, India6Department of Physics, Suranaree University of Technology, Nakhon, Ratchasima, Thailand7National Institute for Materials Science, Tsukuba 305-0044, Japan8School of Materials Science and Engineering, Hainan University, Haikou 570228, China(Dated: July 23, 2022)By performing high-throughput first-principles calculations combined with a semiempirical vander Waals dispersion correction, we have screened 74 direct- and 184 indirect-gap two dimensional(2D) nonmagnetic semiconductors from near 1000 monolayers according to the criteria for energetic,thermodynamic, mechanical, dynamic and thermal stabilities, and conductivity type. We present thecalculated lattice constants, simulated scanning tunnel microscopy, formation energy, Young’s mod-ulus, Poisson’s ratio, shear modulus, anisotropic effective mass, band structure, band gap, ionizationenergy, and electron affinity for each candidate meeting our criteria. The resulting 2D semiconductordatabase (2DSdb) can be accessed via the website https://materialsdb.cn/2dsdb/index.html orinvoking the VASPKIT program [Comput. Phys. Commun. 267, 108033 (2021)]. We also providethe calculated periodic table of band alignment type for van der Waals heterostructures when pack-ing any two of the 200 screened semiconductor monolayers to form bilayers. Based on the rules ofthumb for photocatalytic water splitting, we have further screened dozens of potential semiconduc-tors and thousands of heterostructures from 2DSdb which are promising for photocatalytic watersplitting. The 2DSdb provides an ideal platform for computational modeling and design of new 2Dsemiconductors and heterostructures in photocatalysis, nanoscale devices, and other applications.I. INTRODUCTIONSince the successful isolation of graphene,1,2 two di-mensional (2D) materials have attracted tremendous at-tentions due to their novel electronic, optical, thermal,and mechanical properties for potential applications ina great variety of fields. Owing to the quantum confine-ment effect along the out-of plane direction, 2D materialsoften exhibit unique features, different from those of theirbulk counterparts.3–15 For examples, an unusual half-integer quantum Hall effect was observed in graphene.7The electronic properties of transition-metal dichalco-genides (TMDs) with MX2 composition (where M = Moor W and X = S, Se or Te) can be tuned from metallicto semiconducting by controlling layer-thickness.6,8,15–18The peculiar puckered honeycomb structure of few-layerblack phosphorene (BP) leads to significant anisotropicelectronic and optical properties on zigzag and arm-chair directions.14,19,20 Remarkably, its band gap is alsothickness-dependent, varying from 0.3 eV in the bulklimit to ∼2.2 eV in a monolayer with a direct bandgap character. Other 2D materials, such as hexago-nal boron nitride (h-BN),21 silicene,22–25 germanene,26,27stanene,28 also exhibit many exotic characteristics thatare absent in their bulk form.A common feature of 2D materials is that they areformed by stacking layers with strong in-plane bondsand weak, van der Waals (vdW)-like interlayer attractionwith typical binding energies of dozens of meV, allow-ing exfoliation into individual and atomically thin layers.This means that 2D materials usually possess in-planestability in the absence of dangling bonds, in contrastto bulk films that are plagued by dangling bonds andsurface state. Inspired by this feature, Inoshita et al.screened the potential 2D binary stoichiometric electridesfrom the layered crystal structures by performing first-principles calculations based on the density functionaltheory (DFT) within the generalized gradient approx-imation (GGA).29 Later, Ahston and co-workers useda topology-scaling algorithm combining high-throughputcalculations to uncover more than 800 monolayers basedon the Materials Project crystal structure databases.30,31Considering the fact that the semi-local density function-als such as GGA functional significantly overestimatesthe lattice constants of crystals having vdW bonds. Arough thumb rule is that if the relative error in lat-tice constant a or b or c (experimental versus GGA-calculated) of one bulk phase is larger than 5%, it mighthave 2D structure. Choudhary et al. identified atleast 1300 monolayers by comparing the experimentallattice constants with those predicted using the GGAfunctional.32 Cheon et al. also identified thousand of 2Dlayered materials based on data mining algorithm.33 An-other important database for 2D materials was buildedby Mounet et al.34 They chose the binding energy ob-2tained by DFT calculations with the vdW correction asthe screening criterion (≤ few tens of meV·Å−1) and iden-tified more than 1800 structures. There are several 2Dcrystals databases publicly available at present, such asMC2D,34 C2DB35, 2DMatPedia36 and JARVIS-DFT32.However, one of the major limitations of these databasesis that they mainly focus on the stability analysis andprovide only a small number of the fundamental physi-cal properties such as lattice constants, formation energy,exfoliation energy, and band gap at the GGA level. Al-though the GGA functional can provide sufficiently accu-rate results on forces, structures, and band dispersions,it underestimates band gap of semiconductors, averagelyby 50%.In parallel with the efforts on synthesis of new 2D ma-terials, another strategy has been gaining strength overthe past few years. By stacking together different 2D ma-terials on top of each other, various artificial heterostruc-tures which is known as vdW heterostructures (vdWHs)can be formed.37,38 Compared with the conventional bulksemiconductor-based heterostructures which require sim-ilar lattice structures of the components, vdWHs do notdemand crystal lattice matching. One can build artificialvdWHs with desired functionalities by picking and stack-ing atomic layers of arbitrary compositions. The vdWHsnot only preserve the excellent properties of the origi-nal single layers due to the weak vdW interaction, butalso bear additional features. The invention of vdWHshas enriched greatly the variety of materials and alsoserved as a powerful material platform for exploring newphysics and developing exotic devices in nanoscale. Thereare numerous review articles highlight their potentialapplications in electronics,39 photonics,40 spintronics,41superconductivity,42 energy storage,43 catalysis,44 etc.Despite their widespread applications, a systematic high-throughput investigation of a wide range of vdWHs is stillincomplete, to the best of our knowledge.In this work, combined the high-throughput first-principles calculations with a semiempirical van derWaals dispersion correction, we have chosen the en-ergetic, thermodynamic, mechanical, dynamic, thermalstabilities and conductivity type as the criteria andscreened around 260 2D semiconductors from near 1000structures. We here present the lattice constants, forma-tion energy, scanning tunnel microscopy (STM), Young’smodulus, Poisson’s ratio, phonon dispersions, band struc-ture, effective masses of carriers, band gap, ionization en-ergy and electron affinity for each candidate. In addition,the periodic table of heterostructure types including morethan 20000 possible vdWHs is also presented. Since 2Dsemiconductors and vdWHs have demonstrated unprece-dented performance and ability in the field of photocatal-ysis thanks to their large specific surface area, readilytunable electronic properties, sufficient adsorption andcatalytic sites, high carrier mobility and short carrier mi-gration distance compared to bulk photocatalysts,45–47the potential 2D semiconductors and vdWHs for photo-catalytic water splitting have been further screened. Theremainder of this paper is organized as follows. In Sec.II, methodology and computational details are described.The details of screening criteria are discussed in Sec. III.Sec. IV presents the calculations of structural, mechan-ical and electronic properties. Finally, a short summaryis given in Sec. V.II. METHODOLOGYA. Density functional calculationsOur total energy calculations were performed usingthe Vienna Ab initio Simulation Package (VASP).48,49The electron-ion interaction was described using pro-jector augmented wave (PAW) method50,51 and the ex-change and correlation (XC) were treated with GGAin the Perdew Burke Ernzerhof (PBE) form52. Part ofelectronic structure calculations were also performed us-ing the standard screening parameter of Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional,53–58 upon thePBE-calculated equilibrium geometries. A cutoff energyof 400 eV was adopted for the plane wave basis set, whichyields total energy convergence better than 1 meV/atom.In addition, the non-bonding vdW interaction is incorpo-rated by employing a semi-empirical correction schemeof Grimme’s DFT-D2 method in this study, which hasbeen successful in describing the geometries of variouslayered materials.59,60 In the slab model of 2D systems,periodic slabs were separated by a vacuum layer of 20Å in z direction to avoid mirror interactions. The Bril-louin zone was sampled by the k -point mesh following theMonkhorst-Pack scheme,61 with a reciprocal space reso-lution of 2π×0.03 Å−1. On geometry optimization, boththe shapes and internal structural parameters of pristineunit-cells were fully relaxed until the residual force oneach atom is less than 0.01 eV/Å.B. High-Throughput SettingsThe purpose of this work is to identify the candidatesof 2D semiconductors through large-scale screening ofknown materials, rather than to make the most accu-rate prediction of a specific material. To screen the novel2D semiconductors, we used the VASPKIT package62as a high-throughput interface to pre-process the in-put files and post-process the data obtained by usingVASP code. The overview of the screening process isin Fig. 1. First, VASPKIT generates three input files(POTCAR, KPOINTS, and INCAR) for a given struc-ture file (POSCAR). Then the spin-polarized structure-relaxation was done to determine the magnetic groundstate for each 2D material. If the candidate is non-magnetic, we next calculated the global band structureat the PBE level to determine the accurate positionsof both conduction-band minimum (CBM) and valence-band maximum (VBM) in the reciprocal space. It is well3known that the PBE functional is sufficiently accurateon band dispersion, but underestimates band gaps. TheHSE06 can well describe narrow and middle-sized gapsemiconductors whose valence electrons are not stronglylocalized.57,63 As band structure calculations at HSE06level is rather time-consuming, we have only performedthe static HSE06 calculations on irreducible k-points in-cluding the positions of both CBM and VBM in order toget accurate band gap Eg values at the PBE-calculatedlattice constants. If the candidate meets the energetic,thermodynamic, mechanical, dynamic and thermal sta-bility criteria and bears a non-zero band gap, it could bea potential 2D semiconductor. Finally, we have furtherscreened potential 2D semiconductors and heterostruc-tures for water splitting according to the photocatalyticcriteria which willl be discussed later. This screening al-gorithm is expected to be applicable to other fields, suchas 2D thermoelectricity materials.III. SCREENING CRITERIAA. Thermodynamic StabilityGenerally speaking, a stable material should have ther-modynamic, mechanical, dynamic and thermal stabili-ties simultaneously. Thermodynamic stability measuresthe steadiness of a compound against its decomposition.Three physical quantities are commonly used to evalu-ate the thermodynamic stability of a free-standing 2Dsheet, namely, the exfoliation energy, the energy convexhull and the formation energy. The exfoliation energy isthe energy needed to exfoliate a monolayer from its bulk,an indication of the strength of interlayer bonds hold-ing the layered bulk structure together. However, some2D materials, such as borophene,64 lack any layered bulkstructure from which they can be exfoliated. The energyconvex hull describes the competition between all phaseswith the same composition. Specifically, the phases lyingabove the convex hull have a tendency to decompose intothe ground state compounds on the convex hull. The def-inition of energy convex hull, nevertheless, has the sameproblem as the exfoliation energy.65,66 For example, thesynthesis of 2D sheets by mechanical exfoliation impliesthat it is an endothermic process to break the interlayerbonds. This means that all 2D materials with respectto their corresponding bulk counterparts naturally fallabove the convex hulls. The formation energy which isdefined as the difference between a material and its pureelemental constituents in their ground statesΔEf = Etot −∑nαμα, (1)where Etot is the total energy of pristine 2D monolayer.nα is the number of atoms of species α and μα is theatomic chemical potential of species α which is equal tothe total energy of per atom in its most stable elementalphase. A more negative ΔEf for a material means higherYes2D semiconductor databaseYes2D structure databases YesPerform spin-polarized calculation to optimize 2D structure at PBE levelDetermine formation energy at PBE levelThermodynamic stability?Determine band gap at HSE06 levelSemiconductor?Determine elastic constants at PBE levelMechanical stability?Determine phonon spectrum using DFPT at PBE levelDynamical stability?YesNon-magnetic?YesNo For spintronic applicationsDetermine potential energy using AIMD at PBE levelThermal stability?Yes2D heterostructures for photocatalytic water splitting applications2D semiconductors for photocatalytic water splitting applicationsStructure stabilityMechanical propertiesElectronic propertiesOptical propertiesPBE level HSE06 levelFIG. 1. (Color online) Schematic representation of the funda-mental steps needed to find two-dimensional semiconductors.thermodynamic stability. However, to be thermodynam-ically stable, a material must not only have a negativeformation energy not only with respect to the elementalground states but also have a negative one with respect toall possible competing compound phases. In the presentstudy we mainly focus on the high throughput compu-tational screening of 2D semiconductors, and adopt thePBE-calculated formation energy as the thermodynamicstability criteria. PBE generally underestimate the for-mation energy of solids, especially for the layered ma-terials, with an accuracy of only around 0.2 eV/atomon average.67 We noted that the PBE-calculated forma-4tion energies of Si, Ge and Sn monolayer are higher than0.6 eV/formula-unit (f.u.) but they have recently beensynthesized or isolated by exfoliation.25,26,28 In our high-throughput screening process, we used a threshold of 1.0eV/f.u. as an upper bound on the thermodynamic sta-bility for free-standing monolayers.B. Mechanical StabilityThe mechanical stability of a material describes its re-sistance to deformations or distortions in the presence ofstrain. For a 2D crystal in the linear elastic region, thestress σ = (σ1, σ2, σ6) response to external loading strainε = (ε1, ε2, ε6) follows the generalized Hooke’s law andcan be simplified in the Voigt notation,68,69⎛⎝ σ1σ2σ3⎞⎠ =⎛⎝ C11 C12 C16C21 C22 C26C61 C62 C66⎞⎠ ·⎛⎝ ε1ε2ε6⎞⎠ ,where C ij (i,j=1,2,6) is the in-plane stiffness tensorusing the standard Voigt notation: 1-xx, 2-yy, and 6-xy.The Cij can be obtained using the energy-strain methodimplemented into the VASPKIT code,62 namely,Eelastic (E, {εi}) = E(S, {εi})− E (S0, 0)=S02(C11ε21 +C22ε22 + 2C12ε1ε2+2C16ε1ε6 + 2C26ε2ε6 + C66ε26).(2)In the energy-strain method, the Cij is equal tothe second partial derivative of strain energy Eelasticwith respect to strain ε, and can be written as Cij =(1/S0)(∂2Eelastic/∂εi∂εj), where S0 is the equilibriumarea of the system. Therefore, the unit of elastic stiffnessconstants for 2D materials is force per unit length (N/m).In order to calculate Cij , the Eelastic as a function of εin the strain range -2% � ε � 2% with an increment of0.5% are investigated. The number of independent elas-tic constants is controlled by the symmetry of a 2D crys-tal. For instance, the hexagonal crystals have two butthe oblique ones have six independent elastic constants.This number, together with the necessary and sufficientelastic stability conditions for different 2D lattice typesare summarized in Fig. 2.69,70C. Dynamic StabilityThe dynamic stability reflects the structural toler-ance of a system against small atomic displacements duetothermal motions. It can be determined by calculat-ing the phonon dispersions of a material using either afinite displacement method71 or density functional per-turbation theory72. To be dynamically stable, a materialallows no imaginary phonon spectra in its phonon dis-persions. Shown in Fig. 3 (a) is the phonon spectraof hexagonal MoS2 monolayer. No imaginary modes ap-pear, implying that is dynamically stable. Otherwise, thematerial will undergo reconstructive or martensitic phasetransformations upon a slight lattice distortion. We de-rived phonon dispersions using the finite displacementapproach implemented in the PHONOPY code.73 Theforce constants were calculated using a supercell (20 Å× 20 Å) with atomic displacements of 0.01 Å along thelattice vectors.It is worth mentioning that small negative spectra, i.e.,low imaginary frequency near the Γ point is often ob-served in the phonon spectra of 2D systems, as is thecase for borophene monolayer [Fig. 3 (c)] which has beensynthesized recently.64 Such small imaginary frequenciescould be an artifact of poor convergence due to limitedsupercell size, cutoff energy, or k-points; or they mayreflect the actual lattice dynamical instability towardslarge wave undulations of 2D materials. It can possiblybe eliminated by applying a small strain on the film ordepositing the film onto a proper substrate.64,74 Thus, acandidate is still considered to be dynamically stable evenif a tiny imaginary frequency is present near the Γ point.We note that the phonon criterion is still a necessary butnot sufficient condition to evince dynamic stability of amaterial. Since the phonon analysis deals only with smallatomic displacements, it cannot capture phase transitionscoupled with complex lattice reconstructions.66D. Thermal StabilityFinally, the thermal stability of a material reflectsits resistance to decomposition or reconstruction intolower energy structures at high temperatures, and canbe evaluated by performing ab-initio molecular dynam-ics (AIMD) simulations over a long time and wide rangeof temperatures. To verify the dynamic stability of theproposed 2D materials, we employed AIMD simulationsof a 10 Å × 10 Å supercell model at a temperature of400 K. The time step and time duration are set to 1.0fs and 60 ps, respectively. A Nosé-Hoover thermostatwas used to control the temperature.75 To be dynami-cally stable, its potential energy should remain roughlyconstant during the AIMD simulation. For comparativepurpose, we found that the calculated potential energyof BP (Pmma) fluctuates around the equilibrium stateas a function of time [Fig. 4 (a)], indicating a good ther-mal stability. In contrast, the potential energy of MgI2(P3m1) decreases over time, reflecting an irreversiblechange in structure which lowers the formation energy.The snapshot of its atomic configuration at the end ofthe simulation further shows that this material is dras-tically distorted and is unlikely to be fabricated in thefree-standing forms.5(a) Hexagonal (b) Square (c) Rectangular (d) ObliqueC11 > 0,C11 > |C12|C11 > 0,C11 > |C12|C66 > 0, C11 > 0,C66 > 0C11 > 0,det(Cij) > 0,C11C22 > C12C12ZeroNon-zeroEqual1/2(C11-C12)FIG. 2. Classification of crystal systems, independent elastic constants, elastic stability conditions for 2D materials.(a) (b) (c) (d)FIG. 3. (Color online) Calculated (a) and (c) phonon dispersion curves, projected density of states (b) and (d) for H-MoS2 andborophene sheet respectively.E. Semiconductor ScreeningFor nonmagnetic semiconductors, the Kohn-Sham(KS) band gap Eg is defined as the difference betweenthe eigenvalues of CBM and VBM. That is,Eg = εCBM − εVBM, (3)where εCBM and εVBM are the KS eigenvalues of CBMand VBM respectively. It is well known that PBEseverely underestimates the band gap of semiconductorsbecause of the lack of derivative discontinuity of the func-tional with respect to the number of electrons and thelack of clear physical meaning of the unoccupied orbitals.But PBR yields similar band dispersion curves to the hy-brid DFT result. There are five typical 2D Bravais lat-tices, namely, hexagonal, square, rectangular, centeredrectangular, and oblique respectively. The Ball-and-stickmodels, Brillouin zones and suggested k-paths for theBravais lattices adopted in our high-throughput calcula-tions are presented in Fig. 5 and Table I.IV. RESULTS AND DISCUSSIONSBased on the above criteria, we have screened 74 direct-and 184 indirect-gap 2D nonmagnetic semiconductorsfrom near 1000 2D monolayers. By analyzing the oc-currence frequency of each element in the screened 2Dsemiconductors shown in Fig. 6, it is found that themost abundant candidates are oxides, followed by sul-6Initial structureFinal structureFinal structureInitial structure(a) (b)FIG. 4. (Color online) Total potential energy fluctuations of (a) BP (Pmma) and (b) MgI2 (P3m1) during AIMD simulationsat 400 K. The inset shows the snapshots at the begin and end of simulation. The results show that MgI2 tends to reconstructinto lower energy structure and is unlikely to be realized experimentally in the freestanding forms.(a) HexagonalM KΓΓ-M-K-Γ|a1|=|a2|, γ = 120o 2D Brillouin zoneSuggested 2D k-paths2D Bravaislattice(b) SquareΓXMΓ-X-M-Γ|a1|=|a2|, γ = 90o(c) RectangularΓYSXΓ-X-S-Y-Γ|a1|≠|a2|, γ = 90o(d) Centered Rectangular(e) ObliqueΓΓ-X-H1-C-H-Y-ΓYH1CH|a1|≠|a2|, γ ≠ 90o|a2|cosγ=1/2|a1|, γ ≠ 90oΓ-X-H1-C-H-ΓΓX H1Hγ γ γa1a2a2a1a2a1a1a2a1a2γγCXγaγγaγγaaγγaaγaaγ1FIG. 5. (Color online) Overview of the five 2D Bravais lattices and corresponding Brillouin zones. The suggested k-paths forband structure are indicated in blue line. The primitive unit cell is indicated in green box.1/3fides, selenides and halides. Meanwhile, the cations ap-pear to favor heavy metal elements such as Pd, Zr, Hfand Pb. The classifications of these candidates accord-ing to the relative frequencies of lattice type, stoichiom-etry and space group of the crystals are further summa-rized in Figs. 7(a)-(c), respectively. Note that the lat-tice types of 2D semiconductors are dominated by rect-angular (43.4%) and hexagonal (40.3%), and the leastabundant are square (16.3 %). Most of them are bi-nary compounds predominantly bearing by AB2 struc-tures. Moreover, the space groups of these candidates aremainly P21/m and P3m1. It is noteworthy that TMDsare one of the most interesting families in the AB2 lay-ered compounds and display a wide range of importantproperties. The TMD monolayers have three phases,namely, 2H (P6m2), 1T (P3m1) and 1T’ (P21/m), re-spectively. Previous theoretical studies have predictedthat around 50 different transition-metal oxides (TMOs)and TMDs can remain stable as either 2H and/or 1Tfree-standing structures,76,77 even though part of thesepotential MX2 compounds are absent in their bulk coun-terparts. For the sake of completeness, we also revis-7TABLE I. Fractional coordinates of the specific points in reciprocal space for the four nonequivalent two-dimensional Bravaislattices.Bravais Lattice Label and coordinates of specific points Bravais Lattice Label and coordinates of specific pointsΓΓΓ (0, 0) ΓΓΓ (0, 0)Square X (1/2, 0) Oblique X (1/2, 0)M (1/2, 1/2) Y (0, 1/2)ΓΓΓ (0, 0) C (1/2, 1/2)Hexagonal K (1/3, 1/3) Oblique H (η, 1-ν)aM (1/2, 0) H1 (1-η, ν)aRectangular ΓΓΓ (0, 0) Rectangular X (1/2, 0)Y (0, 1/2) S (1/2, 1/2)a η = 1−acosγ/b2sin2γ , ν = 12 − ηbcosγa and γ < 90°.ited the stability and electronic structure of TMOs andTMDs with three possible phases (2H, 1T and 1T’ re-spectively). We find that the band gap of these semicon-ducting candidates is mainly concentrated between 1.0and 3.0 eV. The PBE-calculated lattice constants, forma-tion energy, Young’s modulus, Poisson’s ratio, scanningtunnel microscopy, anisotropic effective mass, as well asthe HSE06-calculated band gap, ionization energy andelectron affinity for each candidate are listed in the Sup-plemental Material.A. Mechanical PropertiesThe mechanical properties of a single-crystal aregenerally anisotropict. The Voigt-Reuss-Hill (VRH)approximation,78,79 is a useful scheme by which one cancalculate isotropic polycrystalline elastic moduli in termsof the anisotropic single-crystal elastic constants. Wepresent the VRH averaged bulk and shear moduli of bi-nary 2D semiconductors as a function of the constituentelements in Fig. 8. One can find that oxides have thelargest bulk modulus, followed by sulfides and then se-lenides. As expected, the shear modulus indicates pos-itive correlations with bulk modulus. Next we com-pare our predicted data with available experimental ortheoretical reports. Up until now, several monolayershave been successfully exfoliated or synthesized, includ-ing graphene (P6/mmm),1 BP (Pmna),14,19,20 borophene(Cmmm),64 BN (P6m2),21,80 MoS2 (P6m2),16 TiS3(P21/m)81. We summarize the calculated in-plane elas-tic stiffness constants, the minimum and maximum ofYoungs’s modulus, shear modulus and Poisson’s ratio forthese systems in Table II. One can find that our pre-dictions are in good agreement with the available pub-lished data. For example, the PBE-calculated Young’smodulus and Poisson’s ratio of graphene are 339 N/mand 0.17, in excellent agreement with the available valuesof 340 N/m and 0.186,82,83 respectively. To investigatethe anisotropic mechanical properties of 2D materials, wealso calculated the orientation-dependent Young’s mod-uli Y (θ), Poisson’s ratio ν(θ) and shear modulus G(θ)using the following formulae,84,851/E(θ) = S11c4 + S22s4 + 2S16c3s+2S26cs3 + (S66 + 2S12) c2s2, (4)ν(θ)/E(θ) = (S66 − S11 − S22) c2s2−S12(c4 + s4)+ (S26 − S16)(cs3 − c3s), (5)and1/4G(θ) = (S11 + S22 − 2S12) c2s2+S66(c2 − s2)2/4− (S16 − S26)(c3s− cs3) , (6)where s = sin(θ), c = cos(θ), and θ ∈ [0, 2π] is theangle with respect to the +x axis. Sij= C−1ij are elas-tic compliance constants. As an example, It is found inFig. 9 that the mechanical properties of a BP mono-layer shows a strong anisotropy. It is expected that allbut hexagonal 2D bravais lattices have the anisotropicmechanical properties.Thermodynamic stability sets limits on the energyand the range of Poisson’s ratio is allowed to be from-1.0 to 0.5. Most materials have a positive Poisson’sratio, shrinking (expanding) longitudinally after beingstretched (compressed) laterally. We do find a few ma-terials with a negative Poisson’s ratio (NPR), also calledauxetic materials. The NPR behavior is mainly at-tributed to some special re-entrant or hinged geomet-ric structures regardless of the chemical composition andelectronic structure of a material. The NPR materials ex-hibit fascinating mechanical properties, such as superiortoughness, higher indentation resistance, larger impactresistance, stronger sound absorption, and better crackpropagation resistance.89 These excellent properties of-fers enormous potential in many important applications,such as automotive, aerospace, marine, and other indus-trial fields.90,91 Recently, the auxetic effect has been re-ported in a number of 2D materials. In addition to mono-layer phosphorus and arsenic allotrope reported in pre-8Occurrence frequencyFIG. 6. (Color online) Heat map of the occurrence frequency of each element in the screened 2D semiconductors.(a) (b) (c)FIG. 7. (Color online) Classification of the screened 2D semiconductors in term of (a) lattice type, (b) stoichiometry and (c)symmetry.vious studies,92–94 we also screened some other 2D semi-conductors with large NPR values, including As2SO6 (-0.392), SiP2 (-0.320), BaIF (-0.256), GeSe (-0.228), SnS(-0.189) and SbSeI (-0.166). Among them, BaIF is theonly one persisting the NPR in all crystal directions.B. Electronic PropertiesBeside the band structure, the projected band struc-ture is also provided illustrate the contributions of dif-ferent atomic orbitals in energy and momentum space,offering a chemist’s perspective of the electronic struc-ture. As examples, the element-resolved and orbital-projected band structures and the corresponding den-sity of states (DOS) of MoS2 and graphene monolay-ers are depicted in Fig. 11. To gain more insight intothe topological characterization of band dispersions nearFermi energy, we calculated the global band structuresof both VBM and CBM for each candidate at the PBElevel. The global band structures of InN (P6m2) andAgI (P4/nmm) are illustrated in Fig. 12. In addition,the orientation-dependent effective mass m∗(θ) of bothholes and electrons can be further obtained from theglobal band structures, with the aim of analyzing theanisotropic band dispersions. The PBE-calculated 2Dpolar representation curves for BP, MoS2 and TiS3 arepresented in Fig. 14 for illustration purpose. One canfind that the effective masses of all representative semi-conductors are highly anisotropic, especially for BP andTiS3. The calculated m∗ along Γ-X and Γ-Y are 0.32(1.52) m0 and 1.06 (0.38) m0 for hole (electron) in TiS3monolayer, in good agreement with previous results, 0.32(1.47) m0 and 0.98 (0.41) m0.95 By comparison, the ef-fective mass of hole (electron) in MoS2 slightly increasesfrom 0.54 (0.44) m0 along K-Γ to 0.61 (0.47) m0 along K-M due to the higher hexagonal symmetry. We define theanisotropy ratios of effective masses, γh= mmaxh /mminhfor hole and γe =mmaxe /mmine for electron carriers. Thecalculated γh (γe) is 1.25 (1.14) for MoS2, 3.18 (3.66) forTiS3 and 128.67 (6.80) for BP.To gain more insights into the band-gap variations ofcompounds, in Fig. 10 we show the HSE06 predictedband gap (Eg) of binary 2D semiconductors as a func-tion of the electronegativity difference between two con-stituent elements. The introduction of electronegativitydifference here is to roughly evaluate the ionic characterof the chemical bond formed between different elements.9FIG. 8. (Color online) Bulk and shear modulus of binary 2D semiconductors as a function of the constituent elements withinVoigt-Reuss-Hill (VRH) approximation. The circle radius represents the magnitude of shear modulus.(a) (b) (c)FIG. 9. (Color online) Calculated orientation-dependent (a) Youngs’s modulus E(θ), (b) Poisson’s ratio ν(θ) and (c) shearmodulus G(θ) for BP respectively.A larger difference in electronegativity implies a strongerionic character. Overall, it is found that a compoundwith a stronger ionic bond tends to own a larger band gapvalue. Nevertheless, there are some exceptional cases inwhich small gaps come along with large electronegativitydifference, such as CrO2, ZrCl2 and HfSe3. In the elec-tronic and optoelectronic devices applications, not onlythe band gap, but also the absolute position of the bandedges relative to vacuum, including ionization energy (I)and electron affinity (A) and work function (φ) are im-portant parameters. I is the minimum energy needed toremove an electron from the highest occupied state tothe vacuum, i.e. at Vvac, I=Vvac − εVBM. A is the nega-tive of the energy change when adding an electron to thelowest unoccupied state, A=Vvac−εCBM. Clearly, the ab-solute positions VBM and CBM with respect to Vvac arethe negatives of I and A, respectively. The work function(φ) is defined as the minimal energy needed to remove anelectron originally at the Fermi level (EF ) deep inside thematerial to just outside its surface, namely, φ=Vvac−EF .In semiconductors, φ varies with the position of the EFbecause EF is strongly sensitive to the preparation condi-tion of the sample in the measurement which determinesto a large extent concentration of various intrinsic andextrinsic defects. Figure 13 provides a schematic illus-tration of different quantities involved. In the KS-DFTscheme, the calculation of Vvac is straightforward as itequals to the asymptotic value of the planar-averagedHartree potential in the vacuum region, as illustratedin Fig. 13. The band edges of several widely studied2D semiconductors, together with available theoreticaldata in literature, are listed in Table 18. One can findthat the HSE06 calculated Eg, I and A of the repre-sentative systems are in good agreement with previousreports.16,21,96,9710TABLE II. PBE-calculated in-plane elastic stiffness constants, Youngs’s modulus Y (θ), shear modulus G(θ) (in units of N/m),and Poisson’s ratio ν(θ). For comparison purposes, the available theoretical or experimental values from the previous literatureare also shown.C11 C22 C12 E(ϕ) G(ϕ) ν(ϕ)Systems Calc. Refs. Calc. Refs. Calc. Refs. Max Min Max Min Max MinGraphene 349 342 [86] 349 342 [86] 60 - 339 339 144 144 0.17 0.17BP 106 105 [87] 34 26 [87] 22 18 [87] 92 29 28 17 0.63 0.08BN 292 289 [86] 292 289 [86] 64 - 277 277 114 114 0.22 0.22MoS2 131 124 [88] 131 124 [88] 33 - 122 122 49 49 0.26 0.26TiS3 88 83 [88] 137 134 [88] 14 - 137 71 47 25 0.42 0.10FIG. 10. (Color online) HSE06 calculated band gap of binary 2D semiconductors as a function of the electronegativity differencebetween two constituent elements. The circle radius indicates the electronegativity difference.(a) (b) (c) (d)FIG. 11. (Color online) Projected band structure (left panel) and density of states (right panel) of (a) MoS2 and (b) graphenemonolayers. The Fermi energy is set to zero eV.11(a) (b) (c) (d)FIG. 12. (Color online) PBE calculated global band structure of (a) InN (P6m2) and (b) AgI (P4/nmm).TABLE III. HSE06-calculated band gap Eg, ionization energy I and electron affinity A. For comparison purposes, the availabletheoretical values from the previous literature are also shown.Band gap (eV) Ionization energy (eV) Electron affinity (eV)Material Our work Literature Our work Literature Our work LiteratureBP 1.57 1.52 [98] 5.46 5.43 [98] 3.89 3.91 [98]BN 5.71 5.68 [99] 6.60 6.56 [99] 0.89 0.88 [99]MoS2 2.18 2.15 [99] 6.38 6.33 [99] 4.20 4.18 [99]WSe2 2.04 1.98 [100] 5.49 5.82 [100] 3.45 3.84 [100]TiS3 1.15 1.06 [101] 5.87 5.34 [101] 4.72 4.28 [101]C. Optical PropertiesThe macroscopic dielectric function of 2D materialscannot be well-defined with the layer thickness d → 0.This is because the calculated dielectric function of anartificial 3D periodic system is affected by the length L ofthe vacuum region in the standard DFT calculations. Toavoid the thickness problem, an L-independent opticalconductivity σ2D(ω) is used to characterize the opticalproperties of 2D sheets,102,103σij(ω) = ε0ωL [εij(ω)− δij ] , (7)where ε(ω) is frequency-dependent complex dielectricfunction calculated in the framework of the independent-quasiparticle approximation104, ε0 is the permittivity ofvacuum, ω is the frequency of incident wave, and L is theslab thickness in the simulation cell. In the present studywe consider only the in-plane component ε(ω) of the di-electric tensor, i.e., only light polarization perpendicularto the sheet normal has been taken into account. Thenormalized reflectance R(ω), the transmittance T (ω),and the absorbance A(ω) can be obtained from the fol-lowing equation:102,103R =∣∣∣∣ σ̃/21 + σ̃/2∣∣∣∣2T =1|1 + σ̃/2|2A =Re σ̃|1 + σ̃/2|2(8)where σ̃(ω) = σ2D(ω)/ε0c is the normalized conductivity(c is the speed of light). We present the linear opticalproperties of graphene in Fig. 15 as an illustrated exam-ple.D. Scanning Tunneling Microscope SimulationsSTM can not only characterize the atomic structureof material surfaces, but also can provide direct local in-sight into the electronic structure.105 Thus, the simulatedSTM image has been obtained for each candidate basedon the Tersoff-Hamann approach.106 In this model thecalculated tunneling current I which depends on the tipposition r and the applied voltage V , is proportional tothe integrated local density of states (LDOS)12CBMElectronenergyVacuum levelAI VvacEgEFVBMFIG. 13. (Color online) Schematic energy diagram of a semi-conductor. The ionization energy I, electron affinity A andwork function φ defined as the energies of VBM, CBM andFermi level EF with respect to the vacuum level Vvac, respec-tively.I(r, V ) ∝∫ εF+eVεF∑knwk |Ψkn(r)|2 δ (ε− εkn) dε, (9)where V is the bias voltage, wk is the k-point weight,Ψkn(r) and εkn are the wave function and eigenvalueat the wave-vector k with band index n, and εF is theFermi-energy. To simulate STM images, we integratedthe LDOS from 0.5 eV below the VBM up to 0.5 eV abovethe CBM. We chose the tunneling tip of 1.0 Å and 2.0 Åabove the upper surface of 2D semiconductors during thesimulations, respectively. Constant current topographsare approximated by constant charge density isosurfaces.In Figs. 16(a)-(c), we give the calculated STM imagesof graphene, BP and h-BN with examples. Clearly, weobserve that the patterns in the computational and ex-perimental STM images are very similar.107–109E. van der Waals HeterojunctionsAccording to the alignments of the CBM and VBM inthe constituent layers, heterojunctions can be classifiedinto three types: type I (straddling gap), type II (stag-gered gap), or type III (broken gap), as illustrated inFig. 19(a), respectively. In type I heterojunctions, bothVBM and CBM of two independent component semicon-ductors are located at the same side of the heterointer-face. This is beneficial for spatially confining electronsand holes so that efficient recombination can be achiev-able, rendering them potential applications in optoelec-tronic devices such as lightemitting diodes (LEDs).110 Intype II heterojunctions, the CBM and VBM are locatedin different components with electrons accumulating inthe layer with the lower CBM and holes accumulating inthe other layer with the higher VBM. Different from typeI band alignment, the separation of electrons and holesto different layers can increase carrier lifetime, which isdesirable for photocatalysis and unipolar electronic de-vice applications including photovoltaics and photode-tection applications44,111–114 In type III heterojunctions,the VBM of one semiconductor is higher than the CBMof the other, making the whole system overall hetero-junction metallic. Such a property could have great po-tential in tunnel field-effect transistors and wavelengthphotodetectors.115,116The high-throughput design of vdWHs has gained sig-nificant attention because vdWHs have unique physicalproperties and potential applications mentioned above.Rasmussen et al. theoretically predicted the band align-ments of 51 semiconducting TMDs and TMOs mono-layers using GW0 calculations.77 Latterly, äzçelik etal. established a periodic table of band alignmentscover about 900 vdWHs by performing hybrid functionalcalculations.99 Nevertheless, these studies focused onlyon groups IV, III-V and V elemental and/or compoundmonolayers, TMDs and transition-metal trichalcogenides(TMTs). To have a thorough search, we have extendedthe HSE06 calculated periodic table of heterostructuretypes formed by any two of the 200 screened semicon-ductors. The flowchart for the computational design ofvdWHs is illustrated in Fig. 19(b). To determine theband alignment type when A and B monolayers werestacked together, we first compared the absolute posi-tions of VBM and CBM by aligning the vacuum level oftwo composed monolayers to 0 eV based on Anderson’srule117. The band gap of vdWH was then estimated bythe difference between the lower CBM energy and thehigher VBM energy of two monolayers. The resultingvdWH is type I if both the higher VBM and lower CBMare located at the same layer. Otherwise, it belongs totype II or type III. Next the II and III can be separatedby the band gap of vdWH being larger than zero or not.Considering that the dimension table of complete vd-WHs type is very huge (around 20000), we only list theperiodic table of vdWHs for about 2000 hexagonal sys-tems in Fig. 18. Our results show that the hexagonalvdWHs are dominated by type II (44.6%) and type I(43.8%), and the least abundant are type III (11.6 %).In contrast, the complete periodic table provided in theSupplemental Material shows that half of 20000 vdWHsare type I (46.0%), followed by type II (39.1%) and typeII (14.9%). One can find that the vdWHs composedof light elements are dominated by type I and II (nearthe upper left corner); while the vdWHs composed ofheavy elements tends to form type III heterojunctions.This is mainly because the 2D semiconductors composedof heavy elements have higher ionization energy thanthose composed of light ones. Overall, the agreementbetween our calculated results and the corresponding13(a) (b) (c)FIG. 14. (Color online) PBE calculated orientation-dependent effective masses (in units of electron mass m0) of (a) BP, (b)MoS2 and (c) TiS3 monolayers. The red and blue lines indicate the fitted effective mass curves of hole and electron carriers,respectively.(a) (b) (c)FIG. 15. (Color online) Frequency dependence of (a) absorbance, (b) reflectance and (c) transmittance for graphene.(a) (b) (c)FIG. 16. (Color online) Simulated STM image of (a) graphene, (b) BP and (c) h-BN respectively.14+ ++     Type I     Type II  Type III+ + +A B+ ++AABBBuild vdWHs using VASPKIT code2D semiconductor database Import Eg, Vvac,  VBM and CBM of A and B layers, respectivelyFind the global EVBM  and ECBM of A-B vdWH based on Anderson’s rule              YesNo Type I vdWHEVBM  and ECBM in the same material?             ECBM - EVBM > 0? Yes Type II vdWHType III vdWHNoOptional(a) (b)FIG. 17. (Color online) (a) Schematic illustrations of thethree types of semiconductor heterojunctions based on theirenergy band alignments: type I (straddling gap), type II(staggered gap), and type III (broken gap) heterojunctions.(b) flowchart for the computational design of vdWHs.experimental values is very good. For examples, ultra-fast charge transfer was observed in MoS2/WS2 type IIheterojunction,111 and moiré-trapped valley excitons wasobserved in MoSe2/WSe2 type II heterojunction,118 Webelieve the predicted periodic table of vdWHs would pro-vide a useful guidance for experimentalists to design suit-able vdWHs with desired band alignment type.To validate the band-alignment description withthe predictions from Anderson’s rule, three repre-sentative heterojunctions with relatively small lattice,ZrO2/MoSe2, WSe2/MoSe2 and NiS2/ZrCl2 were chosento revisit their band alignments by performing HSE06calculations. Their layer-resolved band structures areshown in Fig. 19. One can find that the band alignmentspredicted using Anderson’s rule is similar to the DFTresults. As expected, no significant interlayer hybridiza-tion between the two components of vdWHs is found. Itneeds to be emphasized that the Anderson’s rule-basedapproach is a rough estimation. It may work only qual-itatively as plays an important role in determining theband structures of 2D materials.38,39,119,120Part of the interlayer coupling arises from the elec-tronegativity difference between the two, which mightcause band shifts and band gap variations in vdWHs.The magnitude of interlayer coupling dependent onlayer-thickness, interlayer distance, stacking order, in-terlayer twist angle, constituent elements, the symmetryof two components, etc. Furthermore, for the lattice-mismatched vdWHs, the internal stress between the con-stituent layers is more likely to modify their band struc-tures in addition to direct interlayer coupling. Thus, itis expected that the vdWH type can vary due to the in-terlayer coupling and internal stress if the difference ofabsolute band extrema of two components is not signif-icant. Nevertheless, the effect of interlayer coupling onthe electronic structures of vdWHs is rather complicatedand beyond the scope of the present work.F. 2D Semiconductors and Heterojunctions forWater Splitting PhotocatalystsPhotocatalytic water splitting has been attractingtremendous attention as an energy-efficient and envi-ronmental protective route to produce hydrogen. 2Dmaterials possess inherent advantages to improve pho-tocatalytic performance for water splitting. To be apromising candidate for water splitting, a semiconduc-tor should meet three basic requirements:45,121 (i) be-ing chemically stable and insoluble in water; (ii) bandgap being larger than the free energy of water splittingof 1.23 eV and smaller than 3 eV to enhance solar ab-sorption; and (iii) band edge position crossing the re-dox potentials of water,122 i.e.,, the CBM being higherthan the reduction potential of H+/H2 (-4.44 eV at pH= 0) and the VBM being lower than the oxidation po-tential (O2/H2) (-5.67 eV at pH = 0). Moreover, theredox potentials is influenced by the pH value in thewater splitting reaction. Specifically, the pH-dependentreduction potential for H+/H2 and oxidation potentialfor O2/H2 are EredH+/H2 = −4.44 + pH × 0.059 eV andEoxO2/H2O= −5.67 + pH× 0.059 eV respectively.On the basis of above-mentioned criteria (ii) and (iii),we have extended the photocatalysis screening proce-dure to our 2D semiconductor database. We obtained53 kinds of monolayers possessing band edge positionswhich meet the requirement for photocatalytic watersplitting at a certain pH value within 0-7, as shown inFig. 20. These results include the extensively studied2D semiconductors. For examples, recent experimentalor computational studies have revealed that graphene-like g-C3N4,123 C2N,124 MoS2,125 BP,126 PdSeO3,127SiP2,128 Bi2Te2S and Bi2Te2Se129 are potential candi-dates for photocatalytic water splitting. Besides thepreviously reported monolayer materials, we have alsoscreened around 40 novel monolayer semiconductors withgood structural stability and proper band edge positions.Among them, there are 17 potential candidates being ap-plicable to a wide range of pH value from 0 to 7. However,it should be emphasized here that the presence of appro-priate band edge positions cannot guarantee an effectivephotocatalyst for overall water-splitting. The chemicalstability of these candidates in exposure to water andair were not considered in our high-throughput screen-ing calculations. Furthermore, the fast recombination ofphotoinduced carriers is the main limitations for photo-catalytic water-splitting since the photo-excited carriersin such thin layer can quickly recombine. Zhang and co-workers reported that the semiconductors with indirectband gap character can reduce the possibility of recom-bination of photogenerated electrons and holes.130Aside from monolayer 2D semiconductors, vdWHs15Type I Type IIIType IIABAB2AB3FIG. 18. (Color online) HSE06 calculated periodic table of hexagonal vdWHs. Type I (44.5%), II (44.9%), and III (10.6%)band alignments are represented by blue, red, and green boxes, respectively.have also been developed as a new avenue to de-sign high-performance photocatalysts for water split-ting as discussed in several reviews.45,46,131–133 Espe-cially, in the type-II heterostructures, the lowest energystates of electrons and holes are on different layers, en-suring efficient separation of photogenerated electronsand holes. This makes them excellent candidates forphotocatalysts.134,135 Zhang et al. screened 44 kinds ofpotential type-II heterojunctions for water splitting un-der the constraint of suitable band edge positions andsimilar lattice parameters.46 Here from more than 20000possible vdWHs aforementioned, we have screened 927(pH = 0), 1410 (pH = 7) and 846 (pH = 0∼7) type-IIvdWHs which have potential for water splitting photo-catalysts. Detailed information are given in the Supple-mental Material, respectively. Previous experimental andtheoretical studies focus on a limited number of vdWHs,such as C2N/MoS2,136,137C2N/WS2,137 g-C3N4/C2N,13816(a) (c)(b)FIG. 19. (Color online) Layer-resolved band structures of (a) ZrO2/MoSe2 with type I junction, (b) WSe2/MoSe2 with type IIjunction and (c) NiS2/ZrCl2 with type III junction, respectively. The Fermi level is shown with black dashed line.g-C3N4/MoS2,139 g-C3N4/BP,140 and InSe/g-C3N4.141It is expected that out results will serve as a guide toimprove the photocatalytic performance of 2D materials.V. SUMMARYIn conclusion, we have identified 258 2D nonmag-netic semiconductors from near 1000 2D monolayersby performing first-principles high-throughput calcula-tions. The calculated properties include lattice con-stants, formation energy, Young’s modulus, Poisson’s ra-tio, scanning tunnel microscopy, band gap, band struc-ture, anisotropic effective mass, ionization energy andelectron affinity. The types of band alignment for morethan 20000 van der Waals heterostructures are also pro-vided. Based on the rules of thumb for photocatalytic wa-ter splitting, we have screened dozens of monolayer semi-conductors and thousands of heterostructures promisingfor photocatalytic water splitting. We hope that ourcomputational screening database cout stimulate furtherexploration of 2D semiconductors and heterostructuresin photocatalysis, nanoscale devices, and other impor-tant applications.ACKNOWLEDGMENTSV.W. acknowledges the support of National NaturalScience Foundation of China (Grant No. 62174136),Natural Science Basic Research Program of Shaanxi(Program Nos. 2022JQ-063 and 2021JQ-464) and Cen-ter for Computational Materials Science, Institute forMaterials Research, Tohoku University for the use ofMASAMUNE-IMR (Project No.2112SC0503). 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