This journal is©The Royal Society of Chemistry 2018 J. Mater. Chem. C, 2018, 6, 12689--12697 | 12689

Cite this: J.Mater. Chem. C, 2018,

6, 12689

Hexagonal M2C3 (M = As, Sb, and Bi) monolayers:new functional materials with desirable band gapsand ultrahigh carrier mobility†

Peng-Fei Liu, abcd Tao Bo,bc Zhifeng Liu, ae Olle Eriksson,f Fangwei Wang,cg

Jijun Zhao *ah and Bao-Tian Wang *bcd

Based on first-principles calculations, we propose a new type of two-dimensional (2D) material M2C3

(M = As, Sb, and Bi) showing an infinite hexagonal lattice, in which C atoms adopt sp2 hybridization and

M atoms prefer three-fold coordination with lone pair electrons. Such monolayers are calculated to be

stable verified by their moderate cohesive energies, the absence of imaginary modes in their phonon

spectra, and the high melting points predicted via molecular dynamics simulations. Sb2C3 and Bi2C3

monolayers possess intrinsic band gaps of 1.58 and 1.23 eV (based on HSE06 calculations), values

suitable for photovoltaic applications. The intrinsic acoustic-phonon-limited carrier mobility of the As2C3

sheet can reach up to 4.45 � 105 cm2 V�1 s�1 for electrons at room temperature, higher than that of

(60–200 cm2 V�1 s�1) MoS2 and (B103 cm2 V�1 s�1) few-layer phosphorene, approaching the figure of

merit in graphene (3 � 105 cm2 V�1 s�1). The well-located band edge and visible light absorption make

stretched Sb2C3 a potentially promising optoelectronic material for photocatalytic water splitting.

Besides, Sb2C3/As2C3 excitonic solar cells have been proposed, and their power conversion efficiencies

are estimated to exceed 23%. First-principles calculations have demonstrated that Sb2C3/Bi2C3 hetero-

junctions are indeed 2D node-line semimetals in the absence of spin–orbit coupling.

1 Introduction

Since the mechanical cleavage method for exfoliating graphenefrom graphite in 2004,1 materials with ultrathin two-dimensional(2D) structures have attracted much attention.2–6 With one atomthickness but a very large lateral area, graphene shows novelstructure-related properties, e.g., massless Dirac fermions,1 highelectron/hole mobility,7 superconductivity with proper doping,8

ballistic electronic propagation,9 and the quantum Hall effect.10

However, the trait of zero band gap in pristine graphene greatlyhampers its potential applications in modern electronicsalthough its mobility is ultra-high. Beyond graphene, MoS2 isanother important 2D material with a finite band gap,11,12 butits carrier mobility is relatively low (60–200 cm2 V�1 s�1).13

In this context, phosphorene, atomically thin phosphorusexfoliated from black phosphorus, seems to be an ideal candi-date due to its inherent direct band gap (2.0 eV) and high holemobility (of the order 103 cm2 V�1 s�1).14,15 However, few-layerphosphorene-based devices are environmentally-vulnerablewhen exposed to air and can only be used in an inert atmo-sphere.16,17 Thus, it is necessary to explore new potential 2Dmaterials, which have not only controlled finite band gaps butalso high carrier mobility.

Among the well-known 2D materials, elemental carbon andphosphorus have been studied extensively as both can experi-mentally and theoretically form stable 2D allotropes.1,14,15,18–20

Recently, theoretical studies have beautifully provided evidenceof the existence of monolayer PC allotropes,21–26 which canshow metallic properties with superconductivity,26 semimetallicproperties with an anisotropic Dirac cone, or semiconductingproperties with superior carrier mobility depending on theconfiguration and ratio. Subsequently, few-layer black PC has

a Beijing Computational Science Research Center, Beijing 100094, Chinab Institute of High Energy Physics, Chinese Academy of Sciences (CAS),

Beijing 100049, China. E-mail: wangbt@ihep.ac.cnc Dongguan Neutron Science Center, Dongguan 523803, Chinad Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan,

Shanxi 030006, Chinae School of Physical Science and Technology, Inner Mongolia University,

Hohhot 010021, Chinaf Department of Physics and Astronomy, Division of Materials Theory,

Uppsala University, Box 516, SE-75120 Uppsala, Swedeng Beijing National Laboratory for Condensed Matter Physics, Institute of Physics,

Chinese Academy of Sciences, Beijing 100080, Chinah Key Laboratory of Materials Modification by Laser, Ion and Electron Beams

(Dalian University of Technology), Ministry of Education, Dalian 116024, China.

E-mail: zhaojj@dlut.edu.cn

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c8tc04165b

Received 21st August 2018,Accepted 22nd October 2018

DOI: 10.1039/c8tc04165b

rsc.li/materials-c

Journal ofMaterials Chemistry C

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been successfully synthesized27–29 and its field-effect transistorexhibits a high hole mobility of 1995 cm2 V�1 s�1 at roomtemperature.27 Black PC could also be a tunable anisotropicplasmonic metasurface which provides a viable platform forhyperbolic metamaterials, bringing widespread application inbiosensors, single-photon sources, nanoantennae, and sub-wavelength resolution imaging.29 These results have shown thatgood semiconducting behavior can be induced in 2D phosphoruscarbide nanomaterials.

Antimonene30 and bismuthene,31 belonging to the samegroup V family as phosphorene, have been successfully synthe-sized experimentally and show tremendous electro-opticalproperties. Besides, several theoretical predictions have suc-cessfully guided the syntheses of some 2D materials, such assilicene,32,33 Janus TMDCs,34,35 Cu2Si sheets,36,37 borophene,38–40

and blue phosphorene.41–43 It is intriguing to find out whetherthe binary compounds of C and M (M = As, Sb, and Bi) also formstable 2D monolayers and possess superior properties. Basedon ab initio calculations, we propose a novel family of 2Dcarbides M2C3 with hexagonal lattices containing C and Matoms. Phonon spectra and ab initio molecular dynamics (AIMD)simulations provide compelling evidence for the thermal anddynamical stabilities of M2C3 monolayers. Remarkably, we findthat M2C3 monolayers are intrinsic semiconductors with bandgaps of 2.27, 1.53, and 1.28 eV for As2C3, Sb2C3, and Bi2C3,respectively, using state-of-the art hybrid density functionalcalculations. The As2C3 monolayer possesses a relatively highelectron mobility of up to 4.45 � 105 cm2 V�1 s�1, which is largerthan that of the MoS2 monolayer and phosphorene. Our resultshave demonstrated that the work function of strained Sb2C3 canwell satisfy the energetic requirements of the reduction andoxidation processes for water splitting. The hexagonal structures,appropriate band gaps, high carrier mobility, desirable bandedge positions, and high optical absorption suggest M2C3 mono-layers could be promising materials for next-generation high-performance devices.

2 Computational methods

First-principles calculations were performed within the frame-work of the Perdew–Burke–Ernzerhof (PBE)44 generalized gradientapproximation,45,46 as implemented in the Vienna ab initiosimulation package.47,48 The plane-wave basis sets of 500 eV wereset with 0.03 Å�1 spacing in the Monkhorst–Pack scheme forstructure optimization, and denser k-point grids with 0.01 Å�1

spacing for property calculation. The length of the unit cell (20 Å)along the z direction was used to get rid of the interaction betweenadjacent images. All geometry structures were fully relaxed untilthe residual forces on each atom were less than 0.01 eV Å�1 andthe total energy variation was less than 1.0 � 10�6 eV. Thescreened exchange hybrid density functional by Heyd–Scuseria–Ernzerhof (HSE06)49,50 was adopted to further correct the electronicstructure. To verify the stability of the system at elevated tem-peratures, AIMD simulations were performed using the Nosealgorithm51 in the NVT ensemble at high temperatures.

The phonon calculations were carried out within theharmonic approximation via the force-constant method52 asimplemented in the PHONOPY code.53 The orbital-resolvedphonon spectra were calculated by postprocessing phononeigenvalues on the basis of the phononic k�p theorem54,55

Xek;s1

�ð jÞ � ekþD;s2ð jÞ��� ��� ¼ ds1;s2 � 0ðDÞ

�� ��; (1)

where ek,s*( j) was the displacement of the atom j in theeigenvector of the (k,s) vibrational mode, and D was a smallwave vector.

The positions of the valence-band maximum (VBM) andconduction band minimum (CBM) were defined as

ECBM=VBM ¼ EBGC �1

2EHSE06g ; (2)

where EBGC was the band gap center energy calibrated withrespect to the vacuum level.56

To obtain the optical absorption spectra of M2C3 mono-layers, the imaginary part of the frequency-dependent dielectricfunction was calculated from the interband transition in theindependent-particle picture without including the local fieldeffects using the following equation:57

e2ð�hoÞ ¼2e2pOe0

Xk;n;c

Cck urj jCn

k

� ��� ��2dðEck � Enk � EÞ; (3)

The real part of the dielectric function e1 was obtained by theKramers–Kronig transformation. Then the optical absorptioncoefficients a(o) were evaluated according to the followingequation (where c is the speed of light in a vacuum)58,59

aðoÞ ¼ffiffiffi2p o

c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie12ðoÞ þ e22ðoÞ

q� e1ðoÞ

� �1=2: (4)

The carrier mobility was obtained in the framework ofdeformation potential (DP) theory on the basis of the effectivemass approximation.60 The analytical expression of acoustic-phonon-limited carrier mobility in 2D materials can be definedas follows:61–64

m2D ¼e�h3C2D

kBT m�j j2 Ei1

� 2; (5)

where m* was the effective mass in the transport direction,e was the electron charge, T was the temperature (300 K), E1 wasthe DP constant, and C2D was the 2D elastic modulus.

3 Results and analysis3.1 Structures of M2C3

The geometric configuration of monolayers of M2C3 is shown inFig. 1a. They all share a honeycomb 2D form with a high groupsymmetry of P6/mmm (No. 191). In the structures, the latticeparameter (la) obtained from ab initio calculations was 5.86,6.39, and 6.70 Å for As2C3, Sb2C3, and Bi2C3, respectively(Table 1). There was one unique M (M = As, Sb, and Bi) atom(site symmetry 4h) and one independent C atom (site symmetry6i) in the primitive cell. The 2D structure contained the planar

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sp2 bonding characteristic of graphene and the significantlydifferent nonplanar sp3 bonding feature found in 2D mono-elemental arsenene,65 antimonene,30 and bismuthene.31 TheC–C bond length was 1.33 Å for the three monolayers, which liesclose to the value (1.42 Å) in sp2 bonded graphite (or graphene).1

All M atoms were trigonally coordinated to ambient C atoms viaquasi sp3-hybridization bonding. As shown in Table 1, due to theatom effective radii of M atoms, the M–C bond lengths displayan increasing trend with the following order: As2C3 (2.00 Å) oSb2C3 (2.20 Å) o Bi2C3 (2.31 Å). From the top view of the lattices,the M2C3 sheets can be viewed as an enlarging of arsenene,65

antimonene,30 and bismuthene31 with the inserting of C atomsinto the 2D networks.64 Thus, the M2C3 sheets had periodicallydistributed pores of 5.86, 6.39, and 6.70 Å diameter, for As2C3,Sb2C3, and Bi2C3, respectively, which were equal to their corres-ponding lattice constants, suggesting potential applications ingas separation and storage.

3.2 Stabilities of M2C3

To confirm the stability of M2C3, we have computed thecohesive energy (Ecoh) with respect to the isolated atoms,defined as

Ecoh ¼EM2C3

� nMEM � nCEC

nM þ nC; (6)

where EM2C3, EM, and EC are the total energy of a M2C3 sheet,

single M atom, and single C atom, respectively; nM and nC are thenumbers of M or C atoms per unit cell. The calculated cohesiveenergies of M2C3 sheets are �5.16, �4.89, and �4.66 eV per atomfor As2C3, Sb2C3, and Bi2C3, respectively (Table 1). For comparison,

the cohesive energies we calculated, for graphene,1 silicene,33

black phosphorene,14,15 arsenene,65 antimonene,30 and bis-muthene,31 at the same theoretical level, were �7.85, �3.83,�3.48, �2.97, �2.64, and �2.46 eV per atom, respectively. It isobvious that the cohesive energies of M2C3 are lower than themajority of the well-known 2D materials, which means thatthe experimental synthesis may be achieved with appropriatesubstrates.

The orbital-resolved phonon spectra and projected phonondensity of states (PhDOS) for As2C3 (Sb2C3 and Bi2C3) are shownin Fig. 1b and Fig. S1 (ESI†). For a crystal, its potential energy Fis an analytic function of the displacements (r) of the atoms.66

When the optimized structure locates at the minimum on apotential-energy-surface, the potential energy always increasesagainst any combinations of atomic displacements. More speci-fically, a dynamically stable crystal must satisfy the requirement,@F@r¼ 0. The real frequencies of all modes in the Brillouin Zone

(BZ) indicate that the three 2D crystals are all kineticallystable.67,68 The acoustic branches of As2C3 show a ubiquitousphenomenon in low-dimensional systems with a parabolicdispersion longitudinal acoustic mode and two linear disper-sions transverse modes at the G point.54,55,69 The colours of thephonon spectra are weighted by the contributions of As and Catoms, which clearly show that the vibration modes, in the low-energy region below B10 THz, are mainly from As atoms whilethe C atoms dominate remanent vibration modes. More speci-fically from PhDOS, the phonon mode mainly contains As (z)vibrations especially around 0 THz, while the contributionsfrom hybridizations of As (xy), As (z), and C (xy) vibrationsbecome larger when moving from 0 to 10 THz. In the 10 to25 THz energy region, the main components can be primarilyderived from the C (xy) vibrations. The high-energy region from44 to 49 THz is totally from C (z) vibrations, which is corres-ponding to that of the C (xy) vibrations in pristine graphene.69

The feature of sp2 bonding of C and sp3 bonding of As (Sb and Bi)helps to stabilize the 2D geometry.

To check the mechanical stability of M2C3 sheets, we havecalculated the linear elastic constants based on Hooke’s lawthrough the energy–stress relationship.70 The calculated 2Delastic constants are C11 = 97.12, 73.53, and 58.58 N m�1,C12 = 28.77, 24.03, and 19.92 N m�1, and C66 = 34.15, 24.78,and 19.41 N m�1 for As2C3, Sb2C3, and Bi2C3, respectively.These values meet the requirements of the stability criteriafor 2D structures in the hexagonal class,64,71 e.g. C11 > |C12| andC66 > 0. The in-plane axial Young’s moduli (E = (C11

2 � C122)/C11)

were 88.56, 65.68, and 51.81 N m�1, which are much lower thanthat of graphene (340 N m�1)72 and MoS2 (123 N m�1),73 but inthe same order of magnitude as silicene (62.31 N m�1).74 Thecalculated Poisson’s ratios for M2C3 are 0.296, 0.327, and 0.340for As2C3, Sb2C3, and Bi2C3, respectively, which are similar tothose (0.16–0.59) of monolayer honeycomb structures of group-IVelements and III–V binary compounds.75

The thermal stability of M2C3 sheets has been examined bycarrying out AIMD simulations51 at 1000 and 2000 K for 5 ps.As shown in Fig. S2 (ESI†), the structural integrity of M2C3 is

Fig. 1 (a) Structure of monolayer M2C3 from the top view and side viewwith a unit cell shown by the black solid line. (b) The orbital-resolvedphonon spectra and projected phonon density of states (PhDOS) formonolayer As2C3 calculated with a 2 � 2 � 1 supercell along high-symmetry lines based on the PBE level. xy (green line): in-plane vibrations;z (magenta line): out-of-plane vibrations.

Table 1 Calculated lattice constants (la), bond lengths of C–C (lC–C) andM–C (lM–C) bonds, cohesive energies (Ecoh), band gaps based on PBE andHSE06 methods, Poisson’s ratio (u), and Young’s modulus (E) for As2C3,Sb2C3, and Bi2C3 sheets

Comp.la

(Å)lC–C

(Å)lM–C

(Å)Ecoh (eVper atom)

PBE(eV)

HSE06(eV) u

E(GPa nm)

As2C3 5.86 1.33 2.00 �5.16 1.42 2.27 0.296 88.56Sb2C3 6.39 1.33 2.20 �4.89 0.92 1.53 0.327 65.68Bi2C3 6.70 1.33 2.31 �4.66 0.81 1.28 0.340 51.81

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well retained and can be optimized back to their initial struc-tures via ab initio relaxation calculations after the simulationat 1000 K. The compounds thoroughly degrade and theatomic configurations show serious distortion at 2000 K. Thusthe three monolayers can maintain a stable 2D form at leastup to 1000 K, which suggests good thermal stability at hightemperature.

3.3 Chemical bonding and electronic structure

The electron localization function (ELF)76 can provide a gooddescription of electron localization and chemical bondingcharacter in solids. To highlight the bonds of C–C and M–C,the isosurfaces of ELF maps visualized by different colours werecalculated and are plotted in Fig. 2 using VESTA software.77

For M2C3 sheets, the electrons are well localized around the Matoms and the bonds of C–C, implying that C atoms adopt thefavoured planar sp2-hybridization and M atoms prefer thenonplanar sp3-hybridization with lone pair electrons, which iscommon in graphene and arsenene/antimonene/bismuthenemonolayers.30,31,65

The orbital-resolved band structures and projected densityof states (PDOS) of As2C3, Sb2C3, and Bi2C3 monolayers areshown in Fig. 3. Obviously, As2C3 and Sb2C3 are indirect bandgap semiconductors since the VBM is located at the G pointwhereas the CBM lies at the K point. Monolayer Bi2C3 has adirect PBE band gap of 0.81 eV, which is lower than that ofAs2C3 (1.42 eV) and Sb2C3 (0.92 eV) at the same computing level.Refined calculations based on the HSE06 method49,50 giveenlarged gaps of 2.27, 1.53, and 1.28 eV for As2C3, Sb2C3, andBi2C3 monolayers, respectively. From the orbital-resolved bandstructures, one can see that the bands in the vicinity of theFermi level mainly originate from the hybridization of M andC atomic orbitals. The PDOS analysis further revealed that theedges of the valence bands are dominated by the p orbitals ofM atoms hybridized with visible contributions from the C-porbitals, while the conduction band edges are mainly comingfrom the C-p orbitals and partly from the M-p orbitals. Thecontributions from other orbitals could be almost negligible forM2C3 sheets around the Fermi level. The corresponding real-space charge distributions of the VBM and CBM are shown inFig. 3. The valence bands at the G point are mainly contributed

Fig. 2 ELF maps sliced perpendicular to the (001) direction for (a) As2C3,(b) Sb2C3, and (c) Bi2C3. In the ELF maps, red and blue refer to the highest(1.0) and lowest (0.0) values of ELF, indicating a fully localized electronregion and a low electron density area, respectively.

Fig. 3 Calculated orbital-resolved band structures and projected density of states for (a) As2C3, (b) Sb2C3, and (c) Bi2C3. The colors of band structures areweighted by the contribution from M (blue) and C (red) atoms. The green, magenta, and dark yellow lines in the projected density of states highlight thes, p, and d orbitals. The high-symmetry points of G, K, and M represent (0, 0, 0), (�1/3, 2/3, 0), and (0, 1/2, 0), respectively. Real-space charge distributionof the VBM and CBM for (d) As2C3, (e) Sb2C3, and (f) Bi2C3.

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to by M-pz orbitals with indispensable components from C-px,y

orbitals for As2C3 and Sb2C3 while the VBM of Bi2C3 is mainlyfrom M-px,y and C-px,y orbitals. The distributions of the CBM forthe three monolayers are mainly characterized by the hybridizedbonding and antibonding bands of C-pz and M-px,y orbitals.Meanwhile, the C-pz orbital characteristic of the CBM weakenson going from As2C3 and Sb2C3 to Bi2C3 with the increasing massof the M atoms.

3.4 Band edges and optical properties

The calculations with an HSE06 function revealed that the bandgaps of M2C3 all exceeded the free energy of water splittingof 1.23 eV. Additionally, for a 2D semiconductor to facilitatephotocatalytic water-splitting, the CBM energy must be higherthan the reduction potential of H+/H2 (�4.44 eV) while theVBM energy must be lower than the oxidation potential ofO2/H2O (�5.67 eV).78 In recent years, intensive efforts havebeen devoted to addressing this issue via first-principles frame-works, including in a series of work by the Hybertson group79

and Galli group.80 They promise to be more accurate for theelectronic screening based on quasiparticle correction andexplicit modeling of interface heterostructures. Herein, on thebasis of the HSE06 method,49,50 we have calculated the bandedges of M2C3 with respect to the vacuum level (Fig. 4 andFig. S3, ESI†).56 Comparing the band edge positions of single-layer M2C3 with the redox potentials of hydrogen evolution(H+/H2) and oxygen evolution (O2/H2O) at pH = 0, we find thatthe pristine M2C3 monolayers are not suitable for photocatalyticwater splitting.

Since a 2D film must eventually be deposited or grown on asubstrate for device application, it is quite crucial to check theeffects of strain on the band gaps and band edge positions forM2C3. Using the HSE06 method,49,50 we applied biaxial strainsin the range from �6% to +6% to relax atomic positions andobtain the corresponding CBM and VBM from the relaxed con-figurations (Fig. 4).59 Over large regions of strain, the CBM andVBM levels depend linearly on strain. However, for the threematerials we observed that the slope changes abruptly atcertain strains due to the change of band edge positions, whichinduce a transition between an indirect and a direct band gapphase for M2C3. When stretching M2C3, we can see that only theband edges of Sb2C3 are appropriate for initializing photocata-lytic redox reactions of water.

Besides the above criteria, a photocatalyst must have excel-lent optical absorption mainly in the visible region with wave-lengths of 380–780 nm.56 The calculated optical spectra withpolarization vectors parallel to the layer plane for M2C3 as afunction of photon energy are shown in Fig. 4. It is seen thatAs2C3 shows significant ultraviolet light absorption, whileSb2C3 and Bi2C3 show substantial absorption for both visiblelight and ultraviolet light, suggesting highly efficient utilizationof solar radiation. Moreover, the average absorption intensitiesin the visible light region of Sb2C3 are very strong with values ofmore than 104 cm�1, which are superior to that of the famous2D g-C3N4 photocatalyst.81 For the effects of biaxial strain onthe optical properties of M2C3, the calculated results indicatethat the negative strain will significantly affect the absorptioncoefficients of Sb2C3 and Bi2C3 in the form of red shift while the

Fig. 4 Schematic illustration of band edge positions of M2C3 monolayers relative to the vacuum level, which is set to 0 eV. The reduction potential ofH+/H2 and the oxidation potential of O2/H2O are marked by the red and blue dashed lines, respectively. The CBM and VBM band edge positions aremarked by the solid black lines. (a, b and c) are for the pristine and strained (a) As2C3, (b) Sb2C3, and (c) Bi2C3 within a strain range from �6% to 6% basedon HSE06 level. The energy gaps of semiconductor photocatalysts are marked by the black arrows (solid: direct band gap; dotted: indirect band gap).Strain-dependent optical absorption coefficients a(o) for (d) As2C3, (e) Sb2C3, and (f) Bi2C3 based on HSE06 level. The seven-colour light area betweenthe dashed lines represents the visible light range (380–780 nm).

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positive strain has little influence on these. Within a strainrange of �6% to 6%, monolayer Sb2C3 exhibits strong absorp-tion in the visible-light range.

3.5 Carrier mobilities of M2C3

The carrier mobility of M2C3 has been calculated on the basis ofthe DP theory proposed by Bardeen and Shockley,60 which hasbeen widely used for investigating the electronic transportproperties of 2D atomic layered structures.61–64,82 The E1, C2D,m*, m and relaxation time (t = mm*/e) are summarized inTable 2. The effective mass m* of holes was 1.037, 0.706, and0.549 me for As2C3, Sb2C3, and Bi2C3 monolayers, respectively.On the other hand, the effective mass m* of electrons was 0.581,0.477, and 0.464 me for As2C3, Sb2C3, and Bi2C3 monolayers,respectively. These values are much larger than those of phos-phorene (0.15 and 0.17 me for holes and electrons along theG–X direction),83 2D SnSe (0.13 and 0.16 me for holes andelectrons),84 h-BX (X = P, As, and Sb) (0.04–0.18 me),85 a-PC(0.12 and 0.10 me for holes and electrons along the X direc-tion),22 and b-PC (0.05 me along the X direction),22 but in thesame order of magnitude as MoS2 (0.46–0.60 me)61 and GeP3

(0.55–0.80 me).86 The obtained deformation potentials (E1) forelectrons were 0.096, 0.339, and 0.714 eV for As2C3, Sb2C3, andBi2C3 monolayers (Fig. S4, ESI†), respectively, which are muchsmaller than the corresponding E1 of holes (0.888, 1.708,and 3.850 eV). They are much smaller than those of other 2Dmaterials which have values in the range 3.7–5.0 eV.22,61,83,84,86

These small values can be unraveled by the real-space chargedistribution of the VBM and CBM shown in Fig. 3d–f, where thewavefunctions from p orbitals are extremely robust due to thebonding of sp2-hybridization from the C atoms and the isola-tion of lone pair electrons from M atoms.

The 2D in-plane stiffness (C2D) of As2C3, Sb2C3, and Bi2C3

monolayers was calculated to be 97.04, 73.57, and 58.45 N m�1

(Fig. S5, ESI†), respectively, which was larger than that ofphosphorene (28.94 N m�1 along the G–X direction)83 andsmaller than that of MoS2 (127.44 N m�1)61 and phosphorene(101.60 N m�1 along the G–Y direction).83 The decrease of C2D

for M2C3 shows that the interaction of M and C atoms becomesweak from As to Sb to Bi. The obtained acoustic-phonon-limited carrier mobility for electrons was 4.45 � 105, 4.01 �104, and 7.57 � 103 cm2 V�1 s�1 for As2C3, Sb2C3, and Bi2C3,respectively. The hole mobility was found to be much smallerthan that for electrons with values of 1.63 � 103, 7.20 � 102,

and 1.86� 102 cm2 V�1 s�1. The differences between electron andhole mobility can be mainly ascribed to the different deformationpotentials and effective masses in M2C3. It can be seen that theobtained electron mobility of 4.45 � 105 cm2 V�1 s�1 for As2C3 islarger than that of the MoS2 monolayer (60–200 cm2 V�1 s�1),61

few-layer black phosphorus (B103 cm2 V�1 s�1),83 and graphene(3.38 � 105 cm2 V�1 s�1).87 Meanwhile, the ratio of electron andhole mobility was 273.0, 55.7, and 40.7 for As2C3, Sb2C3,and Bi2C3, respectively. Such large disparity can be used toseparate electrons and holes,88 thus further reducing carrierrecombination rates.

3.6 vdW heterojunctions of M2C3

Deduced from the above analysis, the band gaps of Sb2C3 andBi2C3 were 1.53 and 1.28 eV, respectively, which makes thesemonolayers promising solar-cell absorption materials.89 Forthe Sb2C3 monolayer, we find theoretically that n-As2C3/p-Sb2C3 vdW heterojunctions (Fig. S6 and Table S1, ESI†) aredirect-band-gap semiconductors characterized with typicaltype-II band alignment (Fig. 5, and Fig. S7–S10, ESI†) withhigh thermal stability (Fig. S11, ESI†), which could facili-tate the effective separation of photogenerated electron andhole pairs and therefore become good candidates for next-generation solar cells.90,91 The practical upper limit of thepower conversion efficiency (Z) for studying solar cells can beestimated in the limit of 100% external quantum efficiency asbelow:92,93

Z ¼ JscJocbFFPsolar

¼0:65 Ed

g � DEc � 0:3Ð1EdgPð�h$Þdð�h$Þ

�Ð10 Pð�h$Þdð�h$Þ

; (7)

where 0.65 is the fill factor (bFF), P(h�$) is the AM1.5 solarenergy flux at the photo energy (h�$), DEc is the conductionband offset, Joc is the maximum open circuit voltage, and Jsc isan integration in the limit external quantum efficiency of 100%.The efficient n-As2C3/p-Sb2C3 heterojunctions could have apower conversion efficiency (Z) exceeding B23%, which is com-parable with g-SiC2 based systems,94 AlxC based heterobilayers,95

and phosphorene/MoS2 systems.96

For the Sb2C3/Bi2C3 heterojunctions (Fig. S6 and Table S1,ESI†), first-principles calculations show that they are indeed 2Dnode-line semimetals97 in the absence of spin–orbit coupling(Fig. 5, Fig. S7 and S12, ESI†). It can be clearly seen that theSb2C3 and Bi2C3 orbitals overlap around the K point and theorbital characters exchange after passing through the bandcrossing points, indicating an intrinsic band inversion from theformation of a heterojunction. To clearly show the node-linenatures of the Sb2C3/Bi2C3 heterojunctions, we have calculated3D band structures in the whole BZ using a two-dimensionwave vector mesh. It is obvious that the conduction band andvalence band meet at a circle centered at the K point rather thanisolated points, exhibiting features of node-line semimetalssimilar to other 2D monolayers, such as MX (M = Pd and Pt;X = S, Se, and Te) monolayers,97 honeycomb-kagome Hg3As2,98

and line-centered square Be2C.99

Table 2 Calculated deformation potential constant (E1), 2D in-planestiffness (C2D), effective mass (m*), mobility (m), and relaxation time (t)for electrons and holes of As2C3, Sb2C3, and Bi2C3 sheets at 300 K

Comp.Carriertype

E1

(eV)C2D

(N m�1)m*(me)

m (cm2

V�1 s�1) t (fs)

As2C3 Electrons 0.096 97.04 0.581 4.45 � 105 1.47 � 105

Holes 0.888 97.04 1.037 1.63 � 103 9.62 � 102

Sb2C3 Electrons 0.339 73.57 0.477 4.01 � 104 1.09 � 104

Holes 1.708 73.57 0.706 7.20 � 102 2.14 � 101

Bi2C3 Electrons 0.714 58.45 0.464 7.57 � 103 2.00 � 103

Holes 3.850 58.45 0.549 1.86 � 102 5.82 � 101

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4 Conclusions

In summary, we theoretically identify the existence of two-dimensional (2D) materials M2C3 (M = As, Sb, and Bi) with aninfinite hexagonal lattice. In the structure, each C atom adoptssp2 hybridization with one C atom and two M atoms, while Matoms retain the sp3 hybridization character with lone pairelectrons. Calculated results indicate that As2C3, Sb2C3 andBi2C3 are intrinsic semiconductors with band gaps of 2.27, 1.58and 1.23 eV. The room-temperature electron mobility of theAs2C3 sheet is calculated to be 4.45 � 105 cm2 V�1 s�1, which ishigher that of graphene (3 � 105 cm2 V�1 s�1). The well-locatedband edge and visible light absorption make strained Sb2C3

monolayer a potentially promising optoelectronic material forphotocatalytic water splitting. Besides, Sb2C3/As2C3 hetero-junction solar cells have power conversion efficiencies exceedingB23%, while Bi2C3/Sb2C3 heterojunctions possess features of 2Dnode-line band structures in momentum space in the absence of

SOC. Our results highlight the unique electronic and structuralproperties of a new family of 2D materials and are expected toguide future studies toward experimental realization of efficientphotoelectric applications based on these binary materials.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

Zhifeng Liu acknowledges financial support from the NationalNatural Science Foundation of China under Grant No. 11604165.We acknowledge the computational support from the Super-computing Center of Dalian University of Technology (DUT) andthe China Spallation Neutron Source (CSNS).

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