Prediction of new phase 2D C 2h group III monochalcogenides with direct bandgaps and highly anisotropic carrier mobilities †

Two-dimensional (2D) group III monochalcogenides (MX, M = Ga, In and X = S, Se, Te) are promising candidates for next-generation ultrathin optoelectronic devices due to their exotic properties. However, the lack of direct band gaps and the low hole mobilities in their conventional single-layer D 3h phase hinder their potential utility for various applications. In this work, new polymorphs of 2D MXs belonging to the space group C 2h are predicted through a global structural search based on artificial swarm intelligence and density functional theory calculations. We demonstrate that such monolayer polymorphs are thermodynamically and kinetically stable through phonon spectrum analysis and ab initio molecular dynamics simulations. Direct band gaps of 2.38 to 2.84 eV are revealed in all C 2h MX monolayers, a property that traditional 2D D 3h MXs do not possess. Calculations based on the Boltzmann Transport Equation method show that electron mobilities in C 2h monolayers are significantly higher in magnitude than those of the conventional D 3h phase. Anisotropic optical properties are predicted and high absorption coeﬃcients covering the UV-visible spectra are also reported. All these features render the new C 2h MX monolayers promising candidates for potential applications in high-eﬃciency solar cells and anisotropic optoelectronic devices.


Introduction
][19][20][21][22] Nevertheless, the zero band gap in graphene, 11 low electron mobility in transition metal dichalcogenides, 13 and the structural instability of black phosphorus under moisture conditions greatly restrict their applications. 16Therefore, novel 2D semiconductors with sizable band gaps, high carrier mobilities, and strong stability are in great request.Fortunately, polymorphic properties in these single-layer materials provide new probability to realize this goal.Examples include the coexistence of metallic octahedral (1T 0 ) and semiconducting trigonal prismatic (2H) monolayer MoS 2 23,24 as well as the presence of blue phosphorus with wider band gaps compared to the black phosphorus polymorphs. 25herefore, the exploration of monolayer polymorphs of known semiconductors leave room for the discovery of unexpected properties and emergent applications.Among these 2D materials, group III monochalcogenides (MX), represented by InSe, have attracted growing research attention because of their idiosyncratic optoelectronic properties and their promising potential applications as field-effect transistors, 17,21 photodetectors, [26][27][28] and photocatalysts. 30Significant efforts have also been spent on investigating the polymorphic nature of MXs.Take InSe as an example.Its most stable 2D structure takes a graphene-like honeycomb lattice, featured by a non-centrosymmetric D 3h point group. 17Bulk InSe consisting of D 3h layers are found to have a direct band gap of about 1.3 eV, but it increases to an indirect band gap of 2 eV as the sample thickness decreases to the atomic scale. 22Moreover, room-temperature hole mobility in D 3h monolayer InSe is only 4.8 cm 2 V À1 s À1 . 29The lack of a direct band gap and the low hole mobility would severely restrict the optoelectronic applications of D 3h monolayer InSe.Recently, a new monolayer polymorph of InSe belonging to the D 3d point group has been theoretically predicted via the Particle Swarm Optimization (PSO) method, followed by the experimental synthesis of single-layer D 3d GaSe sheets. 31Although this new phase is reported to possess a wider band gap and a higher electron mobility, 31,33 there are still some shortcomings that are not fully addressed, such as the absence of a direct band gap and the low hole mobility.
Here, we further explore the polymorphic nature of 2D MX semiconductors through artificial swarm intelligence computational structural searches on CALYPSO (Crystal structure AnaLYsis by Particle Swarm Optimization) 36 and ab initio calculations using the Vienna ab initio simulation package (VASP) software, [37][38][39][40] and identified a monolayer phase belonging to the C 2h point group.It is reported that bulk C 2h InSe with a direct band gap has in fact been experimentally synthesized, 34,35 which hints the possibility of fabricating C 2h monolayers whose properties were largely unknown.We demonstrate the thermodynamic and kinetic stability of the identified C 2h monolayers through phonon spectrum analyses and ab initio molecular dynamics (AIMD).We carried out first-principles computations on the electrical and optical properties of single-layer C 2h MXs and discovered that they all have intrinsic wide direct bandgaps located in the visible spectrum.Furthermore, anisotropic carrier mobilities are reported.It is noteworthy to point out that while C 2h MXs have electron mobilities that are comparable to those of conventional C 2h monolayers, their hole mobilities are significantly higher than those in D 3h sheets.Anisotropic optical properties of 2D C 2h MXs with higher linear absorption coefficients in the UV-visible spectra are also observed.Such superior properties render the C 2h III-VI binary 2D semiconductors worthwhile for further studies to shape them into promising candidates for high-performance electronics and optoelectronics.
Density functional theory (DFT) calculations on the properties of 2D C 2h MXs were carried out based on the generalized gradient approximation (GGA) along with the exchange-correlation functional proposed by Perdew, Burke, and Ernzerhof (PBE) 45,46 as implemented in VASP.The projected augmented-wave (PAW) pseudopotentials were utilized for the description of the electron-ion interactions in the systems, 47,48 and an energy cutoff of 400 eV was adopted for the plane-wave basis set.For all calculations associated with the 2D structures and properties, a vacuum slab of 20 Å was inserted along the c-axis to avoid interactions between the periodic images on the perpendicular directions.For geometry optimizations and electronic structure calculations, 20 Â 20 Â 1 Monkhorst-Pack meshes were applied to sample the Brillouin zone.The lattice structures and atomic positions were relaxed using the conjugated gradient method, and the equilibrium configurations of the systems are determined once the residual energy and force meet their convergence criteria, which are set to be 1 Â 10 À8 eV and 0.001 eV Å À1 respectively.The Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional was adopted to obtain more accurate band structures. 49he carrier mobilities were obtained using the EPW code based on the ab initio Boltzmann transport equation (BTE) 50 which expresses the mobility m e as m e ¼ e j j P n2CB Ð BZ tðn; kÞv 2 ðn; kÞ @f ðn; kÞ @Eðn; kÞ dk where e is the electron charge.The summation is over the band index n, and the integrals are over the electron wavevector k.
For an electronic state denoted as (n, k), its energy is expressed as E(n, k), and its lifetime is denoted as t(n, k); its band velocity, expressed as v(n, k), can be obtained via the gradient of the energy band in the k-space: where k B and T are the Boltzmann constant and the system temperature respectively.Carrier mobility computation using the EPW code also involves the calculation of the electron-phonon coupling (EPC) matrix. 51In our study, the EPC matrix elements were computed through DFPT calculations on a 100 Â 100 Â 2 k-grid and a 100 Â 100 Â 2 q-grid.Moreover, the carrier concentration was set to be 10 13 cm À3 for all systems when calculating the mobilities.
Carrier effective mass is calculated using the following equation: ; where d 2 E dk 2 is the curvature of the electronic band.The optical absorption coefficients of the MXs are derived from the frequency-dependent complex dielectric function calculated with the HSE06 method: where e ð Þ stand for the real and imaginary part of the dielectric function, respectively.With the complex dielectric function at hand, the absorption coefficient (a) can be derived from the following equation: 3. Results and discussion

Structural search
After unbiased global structural searches based on the PSO algorithm, 3600 possible structures per MX compound are sampled from the free energy landscape, and the ones among them with relatively low energies are mainly composed of three monolayer polymorphs.Structural predictions of InSe in our study are consistent with the results reported in Zhang's study. 31The lowest-energy structure (Fig. 1a) corresponds to the experimentally known D 3h monolayer.Another polymorph belonging to the D 3d space group is found to have an energy comparable to that of the D 3h structure, which has been thoroughly explored in Zhang's study. 31The D 3d phase is beyond the scope of our study, thus not discussed here.The polymorph shown in Fig. 1b is a monoclinic lamellae phase belonging to the C 2h point group.Distinct from the conventional D 3h phase with four atoms per unit cell, the C 2h phase features a larger unit cell in which eight atoms are included.
The lattice constants of the fully relaxed MX monolayers predicted by CALYPSO are tabulated in Table 1.For 2D C 2h InSe, its optimized lattice parameters are a = 11.23 Å, b = 4.10 Å, which show significant similarity to the experimental data of the bulk sample (a = 11.74Å, b = 4.11 Å). 33 Since the C 2h phase of these compounds except for InSe has not been studied before, there are no theoretical or experimental data in previous literature to compare with.Data in Table 1 indicate that the lattice constants b are approximately the same for both of the D 3h and C 2h phases while the lattice parameters a in C 2h MXs are about three times larger than those in D 3h monolayers.As shown in Fig. 1b, the vertically aligned M-M bonds in the D 3h phase adopt a nearly horizontal orientation in the C 2h unit cell, making the latter phase distinct from the graphene-like honeycomb lattice of the former one (Fig. 1a).This structural feature also endows the C 2h structure with potential anisotropic properties along the horizontal axes.Moreover, the presence of inversion symmetry in the C 2h phase differentiates itself from the non-centrosymmetric D 3h phase.

Thermodynamic and kinetic stability
Cohesive energy (E c ) calculations are carried out based on the following expression, and the corresponding data are recorded in Table 1.
where N M and N X are the number of group III (M) and VI (X) atoms per unit cell respectively, while E M and E X stand for the single atom energies of their corresponding elements, and E MX is the system's total energy.It is indicated in Table 1 that there is an energy difference of about 40 to 50 meV per atom between the energetically less favored C 2h polymorph and the common D 3h phase for all III-VI compounds.This difference in cohesive energy for GaX compounds (49 meV per atom for both GaSe and GaS) is larger than that in InX compounds (39 meV per atom for InSe and 37 meV per atom for InS).More importantly, these energy differences can be considered tolerable compared to that between the 2H and 1T phases of MoS 2 (E70 meV per atom). 52This suggests that the monolayer C 2h phase could possibly be fabricated experimentally given their favorable energetics.
Thermodynamic and kinetic stability of C 2h MX monolayers under ambient conditions is demonstrated via AIMD computations and phonon dispersion spectra.For all four MX compounds of interest, AIMD simulation results of monolayer D 3h and C 2h phases are demonstrated respectively in Fig. S1 (ESI †) and Fig. 2. For the entire simulation processes, all atoms vibrate around their equilibrium lattice sites without causing structural disorders.As shown in Fig. 2, the fluctuations in potential energies of the    ), 34 a significant gap broadening of 0.73 eV is observed due to the quantum-confinement effect when the thickness of the material is down to the atomic level. 22This conclusion could be applied to other MXs as well.
As shown in Fig. 4 and Table S2 (ESI †), while the conduction band edge of D 3h InSe monolayer has approximately equal contribution from In and Se (51.8%In and 48.2% Se), that of 2D C 2h InSe is dominated by In (71.0%In and 29.0%Se).This finding can also be applied to the other three MX monolayers.Such different band-edge states in C 2h MXs might give rise to exceptional electronic properties.It is also critical to emphasize that among all four single-layer MXs, C 2h polymorph is the only phase that features a direct band gap, a property that the conventional D 3h monolayers never possess.The origin of this extraordinary electronic property could be found in the band structures and the projected density of states (PDOS) diagrams.To begin with, just like what plenty of previous studies indicated, the single-layer D 3h InSe features a non-parabolic ''Mexican hat'' shaped structure at the top of the valence band as shown in Fig. 4a. 21,53This unique band structure occurs when the material is thinned down from its bulk form to a single layer.The same ''Mexican-hat'' band edge occurs in other D 3h MX monolayers as well, as shown in Fig. S5 (ESI †).
The band structures of the bulk D 3h MXs feature a direct band gap in which both the VBM and CBM locate at the G point, but during the process of thinning down, the top of the valence band at the G point shifts downward until it forms a doublepeak shape. 21,53As a result, the VBM migrates away from the G point and converts the direct band into an indirect one.Such ''Mexican hat'' band structure of InSe results in a 2D van Hove singularity which is visible as a sudden peak of density of states right below the Fermi energy (Fig. 4a). 53Similar phenomena are observed in our calculated band structures for all monolayer D 3h MXs besides InSe, and sudden increases in the density of states are found near the top of the valence bands, contributed mainly by the p z orbitals (shown in orange), according to our calculated PDOS diagrams (Fig. S5, ESI †).However, in comparison, the top of valence bands at the G points of the 2D C 2h MXs remain parabolic in nature (Fig. 4): there is no ''Mexican hat'' structure at the top of the valence bands.Thus, the VBMs remain at the G points, and the direct band features are maintained.At the VBMs of the C 2h monolayers, the PDOS's in Fig. 4 and Fig. S5 (ESI †) also reveal a lack of the sudden peak of density of states that is associated with the van Hove singularity in D 3h monolayers.In fact, the PDOS diagrams unveil that there are still sudden increases in the density of states contributed by the p z orbitals in C 2h MXs between about À1.3 to À1.8 eV as shown in Fig. 4 and Fig. S5 (ESI †).However, the p z orbitals are no longer dominating the top valence bands in C 2h monolayers; instead, they are rather lower in energy compared to the p x/y orbitals which now compose the topmost valence bands.The unique appearance of the latter in the C 2h phase overshadows the influence of the ''Mexican-hat'' shaped band responsible for the direct-indirect transition in the 2D D 3h polymorphs, and thus retains the direct band gap feature in the monolayer C 2h MXs.We also consider the effect of spin-orbit coupling (SOC) on the electronic band structures (Fig. S6, ESI †).Small differences between the band structures with and without SOC indicate its weak and negligible influence.
We also explore the transport properties of the 2D C 2h MXs based on the ab initio Boltzmann transportation equation (BTE) method.Carrier mobilities of all four MX monolayers belonging to both D 3h and C 2h space groups are calculated and tabulated in Table 2; their corresponding effective masses are reported in Table S3 (ESI †).The validity of our computation is tested since our calculated room-temperature electron mobility of 2D D 3h InSe (136 cm 2 V À1 s À1 ) is similar to that reported in the study by

Materials Advances Paper
Li et al. (120 cm 2 V À1 s À1 ). 54According to Table 2, except for GaS, all four C 2h MX monolayers exhibit higher electron mobilities along the y-direction compared to their corresponding D 3h counterparts at 300 K.However, electron mobilities along the x-direction are comparatively lower in C 2h MXs than in D 3h monolayers, which can be attributed to the larger effective masses (m Ã e;x ) in the former ones, as shown in Table S3 (ESI †).For example, compared to the electron mobility of 2D D 3h InSe (136 cm 2 V À1 s À1 ) monolayer C 2h InSe exhibits lower mobilities in the x-direction (23 cm 2 V À1 s À1 ).Correspondingly, the electron effective mass along the x-direction in monolayer C 2h InSe (0.743 m 0 ) is higher than that in single-layer D 3h InSe (0.190 m 0 ), suggesting that it might be the heavier electron effective mass that results in a lower mobility.On the other hand, hole mobilities along both in-plane directions in 2D C 2h MXs are prominently higher than those in the D 3h phase.While hole mobilities along the y-direction (m h,y ) in the C 2h phase is about two to three times greater than those in the D 3h monolayers (e.g., m h,y = 9.7 cm 2 V À1 s À1 and 4.8 cm 2 V À1 s À1 in C 2h and D 3h InSe, respectively), m h,x in the former phase could be up to one order of magnitude higher than those in the latter (e.g., m h,x = 67 cm 2 V À1 s À1 and 4.8 cm 2 V À1 s À1 in C 2h and D 3h InSe, respectively).This phenomenon could be explained by the much smaller densities of states near the valence band edges of the C 2h monolayers, a parameter identified by Li et al. 54 as the key factor influencing the hole mobility in 2D materials.As mentioned before, the ''Mexican-hat'' shaped valence band edge in a D 3h monolayer MX gives rise to a sharp peak (van Hove singularity) in the density of states which increases the hole scattering rate, decreasing the carrier lifetime and the mobility.However, the lack of a similar peak in density of states associated with the van Hove singularity at the VBM of a C 2h monolayer spares the holes from a high probability of scattering, thus  extending the carrier lifetime and maintaining a high hole mobility.But what really makes C 2h monolayers stand out is their strong anisotropic characteristic, featuring higher electron mobilities along the y-direction and higher hole mobilities along the x-direction.In other words, the dominant transport direction in monolayered C 2h MXs exhibits a strong dependency on the carrier type.This property renders single-layer C 2h MXs suitable materials in nanoelectronics devices in which the superior transport direction can be readily altered by a change of carriers induced by switching the sign of the applied gate voltage.The anisotropic transport property might be contributed by different effective masses of the carriers along x-and y-directions, as shown in Table S3 (ESI †).Take C 2h InSe as an example.On one hand, while its hole effective mass along the y-direction is as high as 4.02 m 0 , that in the x-direction is approximately an order smaller, having a value of 0.304 m 0 .Correspondingly, the hole mobility in the x-direction (67 cm 2 V À1 s À1 ) is almost one magnitude greater than that in the y-direction (9.7 cm 2 V À1 s À1 ).
On the other hand, a lower electron mobility along the x-direction (23 cm 2 V À1 s À1 and 149 cm 2 V À1 s À1 in the x-and y-direction respectively) corresponds to a larger effective mass (0.743 m 0 and 0.216 m 0 in the x-and y-direction respectively).

Optical properties
The optical properties of C 2h MXs monolayers are demonstrated and compared to those of their corresponding conventional D 3h polymorphs in their optical absorption spectra along x-and y-directions as shown in Fig. 5. Prominent optical absorption coefficients (on the order of 10 5 cm À1 ) are observed in all C 2h MXs monolayers in the range of visible-UV lights.Especially when comparing the x-direction absorption spectra of C 2h (orange curves in Fig. 5) and D 3h (green curves in Fig. 5) MXs, the former exhibits significantly stronger absorption of visible and near-ultraviolet light than the latter does (approximately two times the latter in the range of 2 to 3 eV), implying a higher utilization efficiency of solar energy in C 2h MXs.The absorption spectra (Fig. 5) also indicate the conspicuously anisotropic optical characteristics of 2D C 2h MXs materials which are never found in their traditional D 3h phases.More specifically, along the x-direction of 2D C 2h MXs (orange curves in Fig. 5), absorption is much stronger in the visible and the nearultraviolet but is significantly weaker as the energy further increases in the ultraviolet range when compared to that in the y-direction (black curves in Fig. 5).This intrinsic optical anisotropy of C 2h MXs monolayers endows themselves with promising potential for anisotropic optoelectronic applications.

Conclusions
To summarize, properties of single-layer C 2h group III monochalcogenides are predicted.We identified the stable 2D C 2h polymorph via global structural searches based on the particle swarm optimization algorithm.Notably, first-principles investigations on the electronic structures indicate that all C 2h monolayer MXs feature a wide direct bandgap ranging from 2.38 to 2.84 eV, a property that the conventional D 3h phase does not possess.In addition, carrier mobilities in C 2h monolayers are found to be anisotropic and higher in magnitude compared to their D 3h counterparts.Moreover, anisotropic and strong optical absorption ability in the order of 10 5 cm À1 is observed in the visible to ultraviolet region, indicating a high utilization efficiency of solar energy.All these excellent properties render the C 2h MXs monolayers potential candidates for future high efficiency solar cells and optoelectronics.

Fig. 1
Fig. 1 The monolayer structures of (a) D 3h and (b) C 2h MXs.The top and side views are presented on the left and the right side of each panel respectively.The structural coordinates of C 2h monolayers can be found in the ESI.†

Fig. 2
Fig. 2 Fluctuations of the potential energies of monolayer C 2h (a) InSe, (b) InS, (c) GaSe, and (d) GaS during the process of molecular dynamics under a temperature of 300 K.

Fig. 3
Fig. 3 Computed electronic band structures of monolayer C 2h (a) InSe, (b) InS, (c) GaSe, and (d) GaS at the HSE06 level.The red arrows point from VBM to CBM (direct band gaps at the G points).The values of the band gaps can be found in Table S1 (ESI †).

Fig. 4 A
Fig. 4 A close comparison between the projected band structures (computed at the HSE06 level) of monolayer (a) D 3h and (b) C 2h InSe.

Fig. 5
Fig. 5 Calculated absorption coefficients (a a ) in monolayer (a) InSe, (b) InS, (c) GaSe, and (d) GaS.The isotropic absorption coefficients of the D 3h phases are shown in green.For the C 2h monolayers, the absorption coefficients along the x-direction are shown in orange and those along the y-direction are shown in black.The insets show the two in-plane directions with respect to the top view of the C 2h structure.

Table 1
The lattice constants (a and b) of D 3h and C 2h MXs and their corresponding cohesive energies (E c )

Table 2
Computed carrier mobilities (m e for electrons and m h for holes) of both single-layer D 3h and C 2h MXs along x-and y-direction at 100 K and 300 K m e (cm 2 V À1 s À1 ) m h (cm 2 V À1 s À1 )