Two-dimensional MSi2N4 (M = Ge, Sn, and Pb) monolayers: promising new materials for optoelectronic applications

The recent growth of two-dimensional (2D) layered crystals of MoSi2N4 and WSi2N4 has sparked significant interest due to their outstanding properties and potential applications. This development has paved the way for a new and large family of 2D materials with a general formula of MA 2 Z 4. In this regard, motivated by this exciting family, we propose two structural phases (1T- and 1H-) of MSi2N4 (M = Ge, Sn, and Pb) monolayers and investigate their structural, vibrational, mechanical, electronic and optical properties by using first-principles methods. The two phases have similar cohesive energies, while the 1T structures are found to be more energetically favorable than their 1H counterparts. The analysis of phonon spectra and ab initio molecular dynamics simulations indicate that all the suggested monolayers, except for 1H-GeSi2N4, are dynamically and thermally stable even at elevated temperatures. The elastic stability and mechanical properties of the proposed crystals are examined by calculating their elastic constants (C ij ), in-plane stiffness ( Y2D ), Poisson’s ratio (ν), and ultimate tensile strain (UTS). Remarkably, the considered systems exhibit prominent mechanical features such as substantial in-plane stiffness and high UTS. The calculated electronic band structures reveal that both the 1T- and 1H-MSi2N4 nanosheets are wide-band-gap semiconductors and their energy band gaps span from visible to ultraviolet region of the optical spectrum, suitable for high-performance nanoelectronic device applications. Lastly, the analysis of optical properties shows that the designed systems have isotropic optical spectra, and depending on the type of the system, robust absorption of ultraviolet and visible light (particularly in 1H-PbSi2N4 monolayer) is predicted. Our study not only introduces new members to the family of 2D MA 2 Z 4 crystals but also unveils their intriguing physical properties and suggests them as promising candidates for diverse nanomechanical and optoelectronic applications.


Introduction
In recent years, with the successful isolation of singlelayer graphene [1], an intensive research interest has been attracted towards the field of two-dimensional (2D) materials [2][3][4].The quest for novel ultrathin materials has led to the exploration of a wide range of 2D materials with remarkable physical properties and potential applications in advanced devices [5][6][7][8].Within the 2D dynasty, transition metal dichalcogenides (TMDs), and transition metal nitrides (TMNs) have attracted tremendous interest in many research fields owing to their stability and outstanding electronic properties ranging from semiconducting and metallic to superconducting [9][10][11].According to the coordination configuration of constituent atoms, TMDs, and TMNs monolayers with the general formula of MX 2 can be commonly formed in the centered honeycomb (1T-phase, octahedral lattice), or honeycomb (1H-phase, trigonal prismatic coordination) structures [12,13].The study of the phase engineering and phase transition between various forms of TMDs and TMNs is gaining attention because the possibility of enabling distinct functionalities in the same material offers unique features and opportunities [14,15].Various techniques are available to induce phase transition from the 1T to 1H form (or vice versa), such as applying stresses [16] and changing temperature [17].In addition to TMDs and TMNs, a new family of 2D materials, single-layer group-IV chalcogenides, has gained considerable attention due to their stability and remarkable optoelectronic properties [18][19][20][21].Through the chemical vapor deposition (CVD) technique, Xu et al synthesized ultrathin 2D SnS 2 crystals, which have been utilized as field effect transistors (FETs) [22].Following that, a theoretical study has shown that SnS 2 nanosheet can be dynamically stable in both Tand H-phases [23].Furthermore, a series of group IV dichalcogenide monolayers, MXY (M = Ge, Sn; X, Y = S, Se, and Te) containing T and H phases have been explored using first-principles calculations and proposed for photocatalytic water splitting applications [24].It has also been reported that the PbS 2 monolayer exhibits an unusual negative Poisson's ratio and suitable band edge alignment, which makes it a promising catalyst for photocatalytic water splitting applications [25].Besides the family of group-IV chalcogenides, theoretical and experimental studies also concentrated on the other potentially stable compounds such as group IV-V monolayers [26][27][28].A recent study extensively investigated the structural, vibrational, and electrical properties of two different phases of group IV-V crystals, including GeN, SnN, and PbN monolayers, and demonstrated that they are stable materials with semiconducting nature [29].
In line with the ongoing experimental and theoretical development of 2D materials for the realization of stable nanostructures, Hong et al have successfully synthesized a 2D septuple atomic layer of MoSi 2 N 4 via chemical vapor deposition (CVD) technique [30].They presented a novel approach for synthesizing new materials using an appropriate element (Si) to passivate the surface dangling bonds of MoN 2 during growth.The structure consists of a 1H-MoN 2 monolayer sandwiched between two SiN surface layers.The MoSi 2 N 4 crystal has been reported to be a wide bandgap non-magnetic semiconductor (1.94 eV) with high tensile strength and excellent ambient stability.The growth of MoSi 2 N 4 monolayer led to the synthesis of WSi 2 N 4 monolayer using the same approach [30].Following the experimental progress, many similar 2D materials with a general formula of MA 2 Z 4 (which M: Cr, Mo, W, V, Nb, Ta, Ti, Zr, Hf, Pd, Pt, A: Si or Ge and Z: N, P or As) have been predicted and characterized by ab initio methods [31][32][33][34][35]. Additionally, it has been found that depending on the atomic components, MA 2 Z 4 monolayers exhibit diverse electronic and magnetic properties.For instance, unlike MoSi 2 N 4 and WSi 2 N 4 which are non-magnetic semiconductors [36], VSi 2 N 4 [37] is a ferromagnetic semiconductor with a considerable valley polarization and the TaSi 2 N 4 nanosheet is a metal and a type-I Ising superconductor [38].Recently, it has been reported that MoSi 2 N 4 nanosheets containing single-metal atoms have intriguing properties that make them useful for heterogeneous catalysis applications [39].Moreover, it has been shown that MoSi 2 N 4 and WSi 2 N 4 monolayers in contact to metal surfaces exhibit no Fermi level pinning and highly tunable Schottky barriers, which are promising features for application in microelectronics [40].Using first-principles calculations, Mortazavi et al explored mechanical, electronic, piezoelectric properties, and lattice thermal conductivity of MA 2 Z 4 (M = Cr, Mo, W; A = Si, Ge; Z = N, P) monolayers in two different phases (i.e. the T-phase and the H-phase) [32].In another study, Ding and Wang systematically investigated the structural stability and electronic properties of MSi 2 N 4 monolayers with 4d and 5d TMs and identified 12 stable structures with T-or Hphase configurations [41].So far, the investigations of the MA 2 Z 4 family are mainly focused on the structures in which the core atom is a transition metal and a comprehensive study focusing on the alternative core atoms from post-transition elements has not been performed.With this in mind, in the present work, we study the structural, vibrational, mechanical, electronic, and optical properties of 1T-and 1H-MSi 2 N 4 (M = Ge, Sn, and Pb) monolayers by ab initio methods.The rest of the paper is organized as follows: detailed information about the computational methodology is described in section 2. In section 3, first, the ground state structures of the proposed materials are determined, and the corresponding structural parameters and cohesive energies are reported.Then their dynamical and thermal stabilities are investigated using phonon spectrum analysis and ab initio molecular dynamics (AIMD) simulations, respectively.Next, vibrational, mechanical, electronic, and optical properties of MSi 2 N 4 monolayers are examined.Finally, we draw our conclusions in section 4.

Methodology
In the present study, all first principle calculations were carried out based on the density functional theory [42,43] by using projector augmented wave (PAW) [44] pseudopotentials, as implemented in the Vienna ab initio simulation package (VASP) [45][46][47][48].The Perdew-Burke-Ernzerhof (PBE) [49] functional of the generalized gradient approximation (GGA) was used to parameterize the exchange-correlation interactions.In addition, the Heyd-Scuseria-Ernzerhorf (HSE06) [50,51] functional was utilized to correct the underestimated electronic band gaps.The HSE06 functional was designed by mixing 25% of the exact Hartree-Fock (HF) exchange potential with 75% of PBE exchange and 100% of PBE correlation energy.For the expansion of the electronic wave function, a plane-wave basis set with a kinetic cutoff energy of 520 eV was taken, and a vacuum layer of ∼15 Å was inserted along the non-periodic direction to prevent the unrealistic interactions between the neighboring layers.The convergence value for electronic relaxation between the successive steps in the total energy calculations was less than 10 −5 eV.In addition, during structural optimization (ionic positions and lattice constants), the maximum allowed force on each atom was less than 10 −2 eV Å −1 , and the maximum pressure on the lattice was lowered to 1 kbar.The Gaussian smearing factor was taken to be 0.05 for the total energy calculations.Brillouin zone (BZ) integration was performed by a Γ-centered 16 × 16 × 1 uniform k-point grid.To obtain the net charge transfer between the constituent atoms in the structures, the Bader technique was employed [52].To check the lattice dynamic stability of the examined systems, the phonon band dispersions were calculated for 4 × 4 × 1 supercells using a small-displacement approach as implemented in the PHONOPY [53] code.To analyze the thermal stability, ab initio molecular dynamics (AIMD) simulations were conducted by using a microcanonical ensemble method at constant temperatures (300 and 600 K) with a total simulation time of 10 ps and 1 fs time step.The finite displacement approach was used to calculate the vibrational frequencies and their corresponding offresonant zone-centered Raman activity.For this purpose, at first, each atom in the primitive cell was displaced by ±0.01 Å, and the corresponding dynamical matrix was generated.Subsequently, the vibrational modes were determined by direct diagonalization of the dynamical matrix.Following that, the Raman activity of each vibrational mode was computed by deriving the macroscopic dielectric tensor at the Γ point utilizing the small difference method [54].A suitable Gaussian broadening was used to plot the Raman spectra based on the results acquired from the Raman calculations.In addition, the elastic constants were directly calculated using the density functional perturbation theory method with a sufficiently large k-point grid and cutoff energy of 700 eV.The linear response of a system is characterized by the complex dielectric function, ε(ω) = ε 1 (ω) + iε 2 (ω), where the Kramers-Kronig relation was employed to obtain the real and imaginary components for optical calculations at the level of PBE and HSE functionals [55,56].

Structural properties
The geometry of the MSi 2 N 4 (M = Ge, Sn, and Pb) monolayers is designed based on the structure of the synthesized 2D MoSi 2 N 4 and WSi 2 N 4 crystals.According to the geometrical structure of the MN 2 layer, two different lattices can be taken into account for each system, namely, 1T and 1H phases.
As illustrated in figure 1, in each form, MN 2 is sandwiched between two buckled honeycomb SiN layers.In both phases, the core atom (M) has six nearest N atoms, and depending on the arrangement of N atoms, either an octahedron (1T) or a triangular prism (1H) can be formed.The primitive cell of the examined systems is constructed from septuple atomic layers of N− Si−N−M−N−Si−N, which are held together with strong covalent bonding.The 1T and the 1H phases are both hexagonal lattices, their symmetries are D 3d and D 3h , respectively, and their space groups belong to P3m1 (No. 164) and P6m2 (No. 187), respectively.The optimized structural parameters including in plane lattice constant (a = b), bond lengths between M-N (d M−N ) and Si-N atoms (d Si−N ), and the thickness (h) of the nanosheets are listed in table 1.The calculated a of 1T-GeSi 2 N 4 , 1T-SnSi 2 N 4 , and 1T-PbSi 2 N 4 are 2.95, 3.05, and 3.12 Å, and a of 1H-GeSi 2 N 4 , 1H-SnSi 2 N 4 , and 1H-PbSi 2 N 4 are 2.93, 3.03, and 3.09, respectively.Notice that a of 1T-and 1H-MSi 2 N 4 monolayers, slightly increases while going down the group IV elements (M atoms), which is correlated with increasing the atomic radius of core atoms.The bond length between M and N atoms (d M−N ) and the thickness of the crystals (h) follows a similar trend as d M−N elongates and h thickens with increasing the atomic radius of M atoms.In both phases, two types of Si-N bonds are formed in the inner and outer layers.The vertical bond between the Si atom and the inner N atom is shorter than the bond with the boundary N atom.Not surprisingly, the variation of bond lengths between Si and N atoms (d Si−N ) remains almost the same for all the systems.The charge partition analysis of the atoms in the framework of the Bader technique reveals that the number of electrons transferred from the M to surrounding N atoms are 2.00, 2.03, and 1.52 | e | in 1T-GeSi 2 N 4 , 1T-SnSi 2 N 4 , and 1T-PbSi 2 N 4 , and 1.87, 1.92, and 1.45 | e |, in 1H-GeSi 2 N 4 , 1H-SnSi 2 N 4 , and 1H-PbSi 2 N 4 monolayers, respectively.In both phases, the charge depletion from the Sn atoms to neighboring N atoms is highest, associated with the larger electronegativity difference between Sn and N atoms.This demonstrates that the bonding between N atoms and Sn possesses more ionic character than Ge and Pb.Furthermore, this reflects the general tendency of Pb to exhibit a more pronounced ionic character in its Pb(II) oxidation state when compared to Pb(IV).Although the variation of ∆ρ (M−N) of 1T-and 1H-MSi 2 N 4 structural phases with the same M atoms is not significant, the ∆ρ (M−N) is slightly larger in 1T than 1H which is consistent with smaller d M−N in 1T structures.The net charge transfer among the constituent atoms in the 1T-and 1H-MSi 2 N 4 structures is illustrated in figure S1 of the supplemental material [57].In addition, the planar averaged electrostatic potentials of the designed structures are computed, and their variations along the z axis are displayed in figure S2 of  Crystal structure the supplemental material [57].Generally, the work function of a material is defined as the energy difference between the vacuum-level potential and the Fermi energy of its surface (Φ = E Vacuum − E Fermi ).The work function is an intrinsic surface property of a material, and understanding and controlling its value is one of the fundamental factors in manipulating electron flow in various electronic applications.The computed results are listed in table 1, and they are found to be larger compared with the reported theoretical values for MoSi 2 N 4 (5.12 eV) [58] and WSi 2 N 4 (4.94)[59] monolayers.Following the structural optimization, to determine the strength of the binding between the constituent atoms in the unit cell, the cohesive energy per atom (E C ) of the examined systems is calculated using the equation below: where E T (M), E T (Si), and E T (N) are the atomic energies of M, Si, and N atoms, respectively; E T (MSi 2 N 4 ) corresponds to the total energy of the MSi 2 N 4 nanosheets; and the denominator (7) is the numbers of the corresponding atoms in a primitive cell.The E C of 1T-and 1H-MSi 2 N 4 monolayers are in the range of 4.72-5.41and 4.69-5.24eV/atom, respectively.These values are within the same range of reported values for PdSi 2 N 4 (5.13 eV/atom) and PtSi 2 N 4 (5.43 eV/atom) [35], implying strong binding energy between the constituent elements in the structures but smaller than MoSi 2 N 4 monolayer (7.34 eV/atom) [60].In accordance with the variation of a and the bond lengths between atoms, in both geometrical configurations, E C gradually decreases when the atomic radii of the M element increase from Ge to Pb.As summarized in table 1, the 1T-phase compounds are energetically more favorable than their 1H counterparts (i.e.1T-phases are more stable than 1H-phases).The E C of GeSi 2 N 4 , SnSi 2 N 4 , and PbSi 2 N 4 structures in 1Tphase are 0.17, 0.09, and 0.03 eV/atom higher than their 1H counterparts.As positive E C solely is not sufficient to determine structural stability, additional analyses, such as phonon band dispersion and AIMD simulations, should be performed.

Dynamical stability
To examine the dynamical stability of designed crystals, ab initio phonon calculations at T = 0 are performed, and the corresponding phonon band spectra are represented in figure 2. In the analysis of phonon diagrams, whenever calculated dispersions of vibrational modes have a positive square of frequency (ω 2 > 0) throughout the BZ, the material is regarded as dynamically stable.The existence of an imaginary frequency (ω 2 < 0) corresponds to a non-restorative force, resulting in a reduction in potential energy as the atoms are displaced away from their equilibrium positions along a particular eigenmode.Except 1H-GeSi 2 N 4 monolayer, all phonon spectra of 1T-and 1H-MSi 2 N 4 crystals are found to be free from any imaginary frequencies over the whole BZ, indicating the stability at low temperature.In the 1H-GeSi 2 N 4 system, unstable modes are correlated with a single degenerate acoustic mode and a double degenerate optical mode at the Γ-point of the BZ.The stable structures own similar phonon distributions, and all have twenty-one phonon branches in their phonon dispersions due to the presence of seven atoms in their primitive cells.In 2D ultrathin materials, there are three acoustic vibrational branches; the longitudinal acoustic mode (LA), the transverse acoustic mode (TA), and the out-of-plane flexural acoustic (ZA) branches.For the TA and LA branches, the frequency of acoustic phonon modes (ω) near the Γ point change linearly with phonon wave vector (q) (i.e.ω ∝ q), whereas ZA vibrational mode shows a parabolic dispersion (ω ∝ q 2 ) which is due to the fact that the force constants associated with the transverse vibrations of atoms decay rapidly [61].It is seen that by increasing the weight of the M atom in both configurations, the highest vibrational frequency (ω max ) slightly decreases, and the overall frequency range of phonon modes gradually shrink.Generally, the compression of the phonon bands, particularly for the acoustic modes, brings about lower group velocity and stronger scattering, and, subsequently, lower thermal conductivity.Therefore, in both atomic configurations, due to the narrower dispersion of vibrational modes, PbSi 2 N 4 monolayers possess smallest values of group velocities among the studied systems.To analyze the contribution of atoms to the phonon spectrum of examined systems, we performed the phonon density of states (PhDOS) calculations, and the obtained results are presented in figure S3, supplemental material [57].For both phases, the low-frequency lattice vibrations (up to 200 cm −1 ) are dominated by M atoms due to their heavier atomic masses.The partial DOS of Si ions in the frequency range around 400 cm −1 has the same level as that of N atoms, while above that frequency, the high-frequency phonon modes are attributed to the vibrations of N atoms and, to a lesser extent, Si atoms.
Besides phonon band dispersion analysis, we perform AIMD simulation for 1T-and 1H-MSi 2 N 4 crystals to examine their stability at elevated temperatures.Here, the structures are maintained at temperatures of 300 K and 600 K, and at ambient environment (T = 300 K and aqueous conditions).The simulations are performed for 10 ps.The size constraint of the unit cell is removed by taking a 4 × 4 × 1 supercell for all studied nanosheets.The evolution of energy during the simulation and the final snapshots of the resulting geometries at 300 K, 600 K, and ambient environment are displayed in figures 3, S4, and S5, supplemental material [57], respectively.According to AIMD simulations, all examined systems except 1H-GeSi 2 N 4 monolayer are thermally stable.It is evident from the energy fluctuation graph and the final snapshot of the structure that 1H-GeSi 2 N 4 has a spontaneous phase transition (1H → 1T).Additional investigations involving variation of temperature during AIMD simulations demonstrate that this transition occurs around T = 30-50 K. (see figure S6 supplemental material [57]).For the rest of the crystals, it appears that the gradient values of total energies of the systems fluctuate without any noticeable deviation during the whole simulation process, indicating the dynamical and thermal stability of 1T-and 1H-MSi 2 N 4 monolayers at and above the room temperature.Thermal stability is crucial in various applications, such as thermoelectric devices, which often operate at high temperatures.

Vibrational properties
Raman spectroscopy is a strong tool for investigating and characterizing various nanomaterials, including 2D ultrathin crystals.The analysis and interpretation of experimental Raman data can be considerably simplified by comparison with the theoretically computed spectra.Vibrational properties and the corresponding Raman spectra are closely correlated with the geometry of the structures.Thus, their analysis can provide an insight into the physical properties associated with lattice symmetries.As mentioned above, the 1T-MSi 2 N 4 monolayers are characterized by the space group P3m1 (D 3d ) in the Hermann-Mauguin notation and the corresponding irreducible representation for vibrational modes in their spectrum can be represented by Γ D 3d = 3E g + 3E u + 3A 1g + 3A 2u , where E g and E u phonon modes are attributed to the doubly degenerate symmetric and asymmetric in-plane vibrations, respectively, and the A 1g and A 2u modes represent the symmetric and asymmetric vibrations along the z axis, respectively.The E g and A 1g are Raman active (RA) as they correspond to quadratic functions in the character table of D 3d , whereas E u and A 2u are infrared active (IRA) modes.Thus, the investigated monolayers in 1T phase should exhibit six peaks at their Raman spectrum.The calculated Raman activity of the 1T-MSi 2 N 4 nanosheets is depicted in figure 4(a), and the atomic displacements of each peak in the Raman spectra are labeled to identify the origins of Raman peaks.By going down the group IV elements in the nanosheets, the Raman spectra slightly shift toward lower frequencies, primarily due to the increased mass of M atoms.The most prominent RA peaks in the spectral profile of 1T-MSi 2 N 4 structures are the first and  (1) and A 1g (1)] phonon modes arise from the in-plane and out-ofplane vibration of buckled Si-N sublayers, respectively, while the inner N atoms have small contributions to the vibration.The E g (2) mode originates from the strong opposite in-plane vibration of inner N atoms in the unit cell, and the A 1g (2) mode corresponds to symmetric out-of-plane vibrations of N-Si-N sub-layers against each other with respect to the M atoms.The E g (3) vibrational mode is attributed to the in-plane shear vibration of buckled Si-N sublayers.The E g (3) is followed by A 1g (3) phonon modes which represent the symmetric out-of-plane vibration of Si and inner N pairs against each other, while the vibration of N atoms is more dominant.In the case of 1H structures (which possess D 3h symmetry), the group theory analysis shows that the irreducible representation for the optical phonon modes at the Γ point can be assigned by in figure 4(c).Analysis of the most prominent peaks in the 1H structures shows that the A ′ 1 (1) phonon mode originates mainly from the symmetric out-ofplane vibration of the two N-Si-N sublayers with respect to the M atoms and the A ′ 1 (3) vibrational mode arises from the symmetric vibration of Si and inner N atoms along z axis, while the vibration of N atoms is more dominant.

Mechanical properties
Next, we study the mechanical features of the stable structures in terms of in-plane stiffness (or 2D Young's modulus, Y 2D ), Poisson's ratio (ν), and ultimate tensile strain (UTS).Mechanical properties, together with elastic stability, can be obtained by analyzing the elastic strain tensors (C ij ).Our calculated C ij 's are listed in table 1.The elastic stability for the designed 2D hexagonal systems is ascertained when all C ij 's are positive and satisfy the Born and Huang criteria [62,63] 1, it can be found that the dynamically stable structures are also mechanically stable.In the 2D limit, Y 2D is defined as the measure of rigidity (or flexibility) of a crystal under the applied external load and can be calculated by the formula; The calculated Y 2D 's are 452, 394, and 328 N m −1 for 1T-GeSi 2 N 4 , 1T-SnSi 2 N 4 , and 1T-PbSi 2 N 4 monolayers, respectively, and the Y 2D of 1H-SnSi 2 N 4 and 1H-PbSi 2 N 4 are computed to be 406 and 341 N m −1 , respectively.Owing to the symmetric and isotropic crystal structure of the examined systems, the values of Y 2D along the x direction are identical to those along the y direction.When compared with the other members of MA 2 Z 4 family, Y 2D is smaller than that of MoSi 2 N 4 (491 N m −1 ) [58] but larger than 1T-and 1H-YSi 2 N 4 monolayers with 301, 236 N m −1 [41], respectively, indicating the rigidity of the proposed monolayers.In both phases, Y 2D decreases down the group of M atoms correlated to the reduction of E C .Thus the smallest value is obtained for PbSi 2 N 4 .To clarify the role of applied stress on the mechanical features of the systems, we calculated the Poisson's ratio (ν) of the materials.ν is defined as the negative ratio of transverse contraction strain to longitudinal extension strain in the direction of the stretching force.Based on C ij , ν is calculated by utilizing the relation; ν = C 12 /C 11 and the acquired results are in the range of 0.25-0.33 and are given in table 1.For comparison, the reported values of ν for MoSi 2 N 4 and WSi 2 N 4 crystals are found to be 0.28 and 0.27 [41], respectively.Different from Y 2D , ν increases with increasing the atomic mass of M atoms in the structures.According to the Christensen criterion, ν ≈ 2/7 proposed as a limit between ductile and brittle transition in 2D ultra-thin materials [64].Thus, our calculated results indicate that 1T-GeSi 2 N 4 and 1T-SnSi 2 N 4 monolayers are brittle with ν < 2/7 which is associated with the more ionic nature of M-N bonding in the structures, and 1T-PbSi 2 N 4 and both stable 1H crystals with higher values of 2/7 possess a ductile character.We further examine the UTS, which is another important intrinsic mechanical feature of 2D materials.UTS is defined as the maximum value of tensile strains that a material can withstand without breaking.Generally, strain is present in experiments, whether intrinsically or intentionally, and during the process of strain engineering, extreme strains can be applied to the 2D materials.Therefore, a suitable material for strain-coupled applications of 2D materials should retain its integrity, even at high levels of induced strain.To estimate UTS of the systems, we employed a 3 × 3 × 1 supercell to restrict the artificial effects of small unit cell size and then measured the amount of stress imposed into the system upon various biaxial tensile strains.The stress-strain curve of stable 1T-and 1H-MSi 2 N 4 structures is depicted in figures 5(a) and (b), respectively.The figure shows that the stress-strain curve appears linear for all monolayers up to ∼ 8%, specifying the systems' elastic regime.Beyond the elastic limit, the examined systems undergo plastic deformation, and stress gradually increases until reaching the maximum point.Our results indicate that the UTS of the studied systems is in the range of 16-18 %, which is comparable with that of ultimate strain for MoSi 2 N 4 (19.5%)[65] monolayer.It can be seen that 1T-and 1H-MSi 2 N 4 nanosheets exhibit prominent mechanical features such as sizeable in-plane stiffness and high UTS that make them promising candidates for elastic energy storage applications.In addition, the piezoelectric properties of the proposed crystals have been calculated, and the acquired results are presented in the supplemental materials [57].These findings are subsequently compared with other members of the MSi 2 N 4 family [66].

Electronic properties
We further investigated the electronic properties of the 1T-and 1H-MSi 2 N 4 nanosheets.The electronic band structure, the atomic contributions to the electronic band dispersions in 1T and 1H forms, and partial density of states (PDOS) of dynamically stable systems are shown in figures 6 and S7-S9, supplemental material [57], respectively.At first, electronic band structures are computed at the level of GGA-PBE (E PBE g ).Then, the SOC effect is included with the GGA functional (E PBE-SOC g ) to analyze the band splitting near the energy band gap.The obtained results are summarized in table 2. It can be seen that all structures are wide band gap semiconductors, and except for 1T-GeSi 2 N 4 and 1T-SnSi 2 N 4 nanosheets, they have direct band gaps.In both phases, by increasing the atomic radius of M atoms, E PBE-SOC g narrows, which is caused by the reduced orbital overlap in the suggested systems.In addition, according to the acquired results, the 1T structures possess wider E PBE-SOC g than the corresponding 1H counterparts.As seen, there are no significant effects of   are found to be 3.16 and 1.87 eV, respectively.Accordingly, the proposed structures can be considered wide-bandgap semiconductors whose energy band gap covers a range from ultraviolet to the visible light parts of the optical spectrum.Recent studies have shown that 2D wide-band-gap materials are desirable for a variety of applications such as solar cells, photoelectrochemical catalysts, and field effect transistors (FET) [67,68].In addition, a flat valence band can be observed in the structures, which indicates that the holes as charge carriers in the systems have low mobility.This flatness is more noticeable in 1H-SnSi 2 N 4 and 1H-PbSi 2 N 4 monolayers.Analysis of atom-resolved projection of electronic band structure shows that in all monolayers, N atoms are the main contributors to the highest valence band (HVB).In the lowest conduction band (LCB), the dominance of N states diminishes, and a mixture of states from all atomic species can be found, especially around the conduction band minimum (CBM).Therefore, due to the dominance of N atoms in both HVB and LCB of all examined nanosheets, we further explored the orbital decomposed band structures projected onto p-orbitals of N atoms.There is a consistent pattern among all crystals that the HVB consists primarily of p y and p z orbitals, and p x orbitals are only present around the degenerate Γ-point of the HVB, except for 1H-PbSi 2 N 4 where the degeneracy at the Γ-point is lifted, and no significant contribution for p x orbitals can be found.On the other hand, the LCBs of the examined systems are mainly attributed to the moderate contribution of the N-p orbitals, and the main contribution to the CBM arises from the p x of the N atoms.Moreover, the analysis of PDOS results demonstrated that in 1T-and 1H-MSi 2 N 4 monolayers, the valence band maximum (VBM) are mainly derived from p orbitals of N atoms.

Optical properties
Finally, the in-plane optical response of the stable 1Tand 1H-MSi 2 N 4 structures is analyzed.The invest-igation of a material's optical properties is a useful tool in envisioning its potential industrial applications.The optical properties can be determined from the frequency-dependent complex dielectric function, ε(ω), which can be formulated as [ε(ω) = ε 1 (ω) + iε 2 (ω)].Calculating ε 2 (ω) of the dielectric function enables us to determine various optical properties, such as absorbance and absorption coefficient.The variation of ε 2 (ω) as a function of photon energy for in-plane light polarization of the designed systems calculated within PBE and HSE06 functionals is presented in figure 7.According to the figure, HSE06 optical band gaps for both 1T and 1H nanosheets are notably larger than those obtained from PBE calculations.Due to the geometrical symmetry of the hexagonal lattice along the x-and y-directions, the optical spectra in the considered systems are found to be isotropic.Analysis of the HSE06 results reveals that for 1T-GeSi 2 N 4 , 1T-SnSi 2 N 4 , and 1T-PbSi 2 N 4 monolayers, the first prominent optical peaks are noticed at ≈5.7, 5 and 3.6 eV (UV region), respectively, while for 1H-SnSi 2 N 4 and 1H-PbSi 2 N 4 crystal structures, the onset of optical absorption emerges in the near ultraviolet and visible light (3.79 and 2.22 eV) region of the optical spectrum, respectively, in accordance with their direct electronic band gap.The first absorption peak of MoSi 2 N 4 monolayer is calculated at 2.31 eV, which is in the visible range of the optical spectrum [58].According to the calculated imaginary part of the dielectric function, the first prominent peak point corresponds to the transition from the valence band to the conduction band states.Therefore, the aforementioned values are attributed to the fundamental bandgap of proposed nanostructures.In both phases, the absorption peak positions for the suggested crystals move toward lower energy (redshifts) by increasing the atomic mass of the core element.The main peak region of absorption for 1T-and 1H-PbSi 2 N 4 monolayers appears around 4-5 eV.Our results show that the suggested 2D systems are wide-bandgap semiconductors with intense optical peaks within the ultraviolet and visible spectral region, making them promising materials for various optoelectronic applications such as solar energy utilization and UV detectors.

Conclusion
In conclusion, inspired by the theoretical and experimental studies on 2D MA 2 Z 4 the stability and fundamental properties of 1T-and 1H-MSi 2 N 4 (M = Ge, Sn, and Pb) monolayers are investigated via ab initio calculations.Our results reveal that both phases have similar cohesive energies, however, it has been found that the 1T structures are more energetically favorable than their 1H counterparts.The stability analyses performed by phonon dispersion calculations and AIMD simulations, reveal that except 1H-GeSi 2 N 4 monolayer, which is dynamically and thermally unstable, the other designed nanosheets are stable at and above ambient temperature.Our vibrational analysis reveals that the 1T-MSi 2 N 4 monolayers possess six Raman peaks in their spectrum whereas 1H-MSi 2 N 4 structures reveals nine characteristic RA mode.It is also found that the distinctive peaks of the same structural phases have similar frequencies and intensities.The elastic stability and mechanical features of the designed nanosheets are explored by calculating their elastic constants (C ij ), in-plane stiffness (Y 2D ), Poisson's ratio (ν), and UTS and they are found to be isotropic.The calculated Y 2D values for stable monolayers range from 328 to 452 N m −1 , indicating their rigidity, and the estimated values of UTS lie between 16%-18% which are comparable to the reported values of MoSi 2 N 4 and WSi 2 N 4 monolayers.The electronic structure investigation shows all suggested crystals are wide band gap semiconductors with energy band gaps ranging from 1.87 to 4.52 eV at the level of HSE06+SOC which spans a range from visible light to the UV region of the optical spectrum.Lastly, the analysis of optical properties shows that the designed systems have isotropic optical spectra, and depending on the type of the system, high absorption of ultraviolet and visible light (particularly in 1H-PbSi 2 N 4 monolayer) is predicted.Our results provide a comprehensive overview concerning the fundamental physical properties of 1T-and 1H-MSi 2 N 4 (M = Ge, Sn, and Pb) semiconductors and highlight their intriguing properties and suggests them as promising candidates for diverse nanomechanical and optoelectronics applications.

Figure 1 .
Figure 1.For 1T-MSi2N4 and 1H-MSi2N4 monolayers, top (above) and side (below) views of the crystal structures, respectively.The relevant geometrical parameters are displayed.

Figure 3 .
Figure 3.The total energy profiles from ab initio molecular dynamics simulations of the 1T-MSi2N4 and 1H-MSi2N4 structures with respect to the simulation time at 300 K.The insets represent the corresponding atomic configurations at the end of simulation.

Figure 4 .
Figure 4. (a) The calculated Raman activity spectra of the 1T-and 1H-MSi2N4 monolayers.The corresponding atomic vibrations of the optical phonon modes in the (b) 1T-and (c) 1H-MSi2N4 structures.
in which the E ′ and E ′ ′ phonon modes are attributed to the doubly degenerate inplane vibrations whereas A ′ 1 and A ′ ′ 2 modes correspond to the non-degenerate out-of-plane vibrations.Among the vibrational modes, A ′ ′ 2 is IRA and the E ′ , E ′ ′ and A ′ 1 phonon modes are RA modes.This analysis reveals that the Raman spectrum of 1H-MSi 2 N 4 systems contains nine peaks.The calculated Raman spectrum of 1H-SnSi 2 N 4 and 1H-PbSi 2 N 4 monolayers are illustrated in figure 4(a).The most significant RA peaks for 1H-SnSi 2 N 4 monolayer are A ′ 1 (1) and A ′ 1 (3) vibrational modes which appear at 275 and 1003 cm −1 , respectively.Similar to 1H-SnSi 2 N 4 , in 1H-PbSi 2 N 4 monolayer, the most intense peaks belong to the A ′ 1 (1) and A ′ 1 (3) phonon modes which are observed at 253 and 962 cm −1 , respectively.The next prominent peaks correspond to the E ′ (2), A ′ 1 (2) vibrational modes, which appear at 451 and 612 cm −1 , respectively.To better explain the vibrational character of the 1H structures, the atomic displacements (eigenvectors) of RA modes are displayed 2D Mater.11 (2024) 015016 M Jahangirzadeh Varjovi et al

Figure 5 .
Figure 5.The variation of stress as a function of applied biaxial strain for 1T-and 1H-MSi2N4 monolayers.

Figure 6 .
Figure 6.Calculated electronic band structures for the 1T-MSi2N4 and 1H-MSi2N4 crystals within the level of GGA-PBE + SOC and HSE06 + SOC are shown by red solid and dashed blue lines, respectively.The Fermi level is set to zero.

Figure 7 .
Figure 7.The variation of the imaginary part [ε2(ω)] of the dielectric function with respect to the photon energy for in-plane polarization in 1T-MSi2N4 and 1H-MSi2N4 monolayers.The red and blue lines represent the calculation within the level of GGA-PBE and HSE06 functionals, respectively.

Table 2 .
For 1T-and 1H-MSi2N4 (M = Ge, Sn, and Pb) monolayers, energy band gaps at the level of GGA-PBE, E PBE g ; GGA-PBE + SOC, E PBE-SOC g ; HSE06 + SOC, E HSE-SOC g ; and the locations of VBM and CBM edges in the BZ.