Theoretical study on the structural, electronic, mechanical, vibrational, thermodynamical, and optical properties of the two-dimensional PbC nanomaterials

Two-dimensional structures have attracted attention for application in nanoelectronics and optical devices; then, in this work, we are reporting the predicted physical properties (from first-principles calculations) for the two-dimensional PbC systems. Those physical properties reveal that the PbC monolayers (M-PbCs) in crystallographic planes (111) and (100); moreover, the PbC2 structures (paramagnetic and anisotropic compounds) are thermodynamical, structural, and mechanically stable but energetically and dynamically unstable at T = 0 K. However, the PbC2 non-magnetic (NM) is the most stable system at high temperatures. The M-PbCs exhibit sp 2 hybridization while the PbC2 NM shows sp 3 d 2 hybridization, forming a hexagonal lattice; meanwhile, the strong interaction at the C’s double bond in the PbC2 ferro and antiferromagnetic configurations (MAG) generates a rectangular lattice. These systems are ductile materials; however, the PbC2 (with metallic bonds) is more ductile than the M-PbCs due to the pronounced participation of the Pb 6p-orbitals. The M-PbCs have associated greater values for the hardness (than those for the PbC2 systems), but at high temperatures, the PbC2 MAG exhibits the highest mechanical resistance. The calculated optical data show that the M-PbCs and the PbC2 NM are promising as refractory materials. At the same time, the PbC2 MAG could be helpful in optical and optoelectronic devices capable of operating in the low frequencies of the UV region and in the infrared and visible regions.


Introduction
The study of quantum confinement's effects on many materials' physical and chemical properties has resulted in unique results [1][2][3][4].Since 2004, the most famous method of isolating graphene sheets has been carried out by Novoselov et al [5].It was the first nanomaterial formed by a single layer of carbon atoms.Graphene has high strength, carrier mobility, and thermal conductivity [6], making it the most studied nanomaterial of the last decade.These properties demonstrated that confined atoms in two-dimensional (2D) can be improved compared to their bulk counterpart.Other 2D materials that have been the subject of research in recent years are the monochalcogenides of group III with group IV [7], transition metal dichalcogenides [8], phosphorene [9], MoS 2 [10], MoTe 2 [11], nitrides [12,13], biphenylenes [14] graphynes [15] and carbides [16][17][18] because their physical properties that support the application in nanoelectronics devices.The 2D honeycomb structures made of silicon (silicene) [19], germanium (germanene) [20], and tin (stanene) [21] show the negligible electronic band gap at the K point and a linear band crossing at the Fermi level, similar to graphene, where their electrons behave as a massless Dirac fermion.On the other hand, theoretical calculations of silicon carbide (SiC), germanium carbide (GeC), and tin carbide (SnC) monolayers exhibit a non-zero bandgap [22] and excellent high carrier mobility properties [16,23,24].
Furthermore, bulk binary compounds formed between atoms of group IV A (for example, GeC or SnC), except for SiC, present solubility problems where an element precipitates during crystal growth due to differences in melting points.However, two-dimensional growth can promote regular atomic arrangements, causing group IV A alloys [25]; even the properties of the 2D nanomaterials can change depending on their width and orientation [22].
In 1923, Durand reported on the lead carbide (PbC) for the first time, adding calcium carbide to an aqueous solution of neutral lead acetate to obtain PbC 2 ; however, these results were not reproducible.Later, Mason and Cadiot (1965) reported the preparation of acetylene derivatives from lead without separating the PbC [26].In 2021, a theoretical study reported the structural and electronic properties of the PbC monolayer [27].However, Sengupta et al [28] experimentally found the presence of a PbC 2 layer by isothermal annealing of a lead-bismuth eutectic (LBE) liquid alloy in a graphite crucible.
A problem with graphene is the difficulty in producing high-quality, small-sized graphene, which hinders the creation of electronic devices.One synthesis method for large-scale applications is using (111) and (100) SiC layered epitaxial graphene [29].Another way to obtain high-quality 2D systems is to seek another type of synthesis, ensuring that the layer does not bond strongly with the substrate.The experimental study by Sengupta et al [28] showed that the PbC 2 layer separates from the graphite substrate when it cools down to room temperature.This may facilitate the synthesis of 2D PbC systems on graphite substrates and potentially enable the creation of graphene by separating the PbC 2 layer and C atoms.
For this paper, we systematically studied 2D PbC nanomaterials: the PbC monolayers oriented at planes (111) and (100) and the PbC 2 structures.Different magnetic configurations were analyzed to explore the magnetism in these 2D nanomaterials.Subsequently, the structural, electronic, mechanical, vibrational, thermodynamic, and optical properties were analyzed to determine the stability of the PbC compounds, including their potential applications.Additionally, this work is structured as follows: the computational calculation details and theory are presented in section 2, the results for the different 2D PbC nanomaterials properties are presented in section 3, and the summary of the most relevant results is in section 4.

Computational details
All the first-principles calculations were implemented in the framework of the Density Functional Theory (DFT) [30], employing the Cambridge Sequential Total Energy Package (CASTEP) code [31,32].A cut-off energy of 500 eV was used to truncate the basis set expansion.The Vanderbilt-type ultrasoft pseudo-potentials [33] were utilized to represent the core electrons, while the exchange-correlation interaction was considered by applying the Generalized Gradient Approximation (GGA) by the Perdew-Wang (PW91) functional [34].The Broyden-Fletcher-Goldfarb-Shanno's (BFGS) algorithm was used for the geometry optimization [35] and the integrations in the Brillouin zone (BZ) were performed using 9 × 9 × 1 k-grids on the Monkhorst-Pack scheme [36].The convergence tolerances were also fixed: the forces on atoms were less than 5.0 × 10 -7 eV Å, and the ionic displacement and the highest strain amplitude were reduced to 5 × 10 -3 Å and 0.02 GPa, respectively.
The PbC bulk (B3 structure) was built by the results obtained in [37].The PbC monolayers (M-PbCs) were modeled from the crystallographic planes (111) and (100) considering a layer of C atoms and another of Pb atoms, which are named M-PbC (111) and M-PbC (100).PbC 2 structures were considered a central layer of Pb atoms and two layers (upper and lower) of C atoms in the (111) plane.Furthermore, different magnetic configurations (MCs) were modeled to determine the magnetic properties of the 2D PbC nanomaterials.A nonmagnetic (NM) case and four magnetic (MAG) ones were modeled; these MAG cases were the ferromagnetic (FM) and the I, II, and III type antiferromagnetic (AFM).The formal spin was set to zero for the NM configuration.The FM, AFM I, AFM II, and AFM III cases were modeled as in [38] and a formal spin of 2 unpaired electrons was assigned to each of the Pb atoms in the unit cell.These MCs do not necessarily remain after the geometric relaxation of the 2D models.
Once the most stable structures were found, the mechanical properties were obtained by analyzing the full tensor's second-order elastic constants (SOEC) using the homogeneous deformations method [39].The phonon and optical properties were calculated using a cut-off energy of 820 eV and a BZ sampled by 5 × 5 × 1 k-grids.The phonon dispersion curves were charted under the density functional perturbation theory (DFPT) using the linear response method with norm-conserving pseudo-potentials with a q-vector grid spacing of 0.05 1/Å.The optical properties were calculated with a plasma frequency of 10 eV for the [1 0 0] polarization vector.

Theory
The chemical equilibrium of the 2D PbC nanomaterials is reached when the Gibbs energy is minimized.For our calculations, the temperature (T) is considered at 0 K, so the Gibbs energy equals the enthalpy (H).The normalized H was calculated in terms of the pressure (P), the volume (V ), and the total number of atoms in each cell (n), and it is given by: A system is more stable as the total energy is lesser than the energy of the free atoms set, and the difference between these energies is called the cohesive energy (E COH ) [40].Another indicator of energetic stability is the formation energy (E f ).While the more positive the E c and E f values, the more stable the systems.These energetic stability criteria were defined by:  [42].The Born stability criteria for these 2D nanomaterials were given as [43]: The bulk modulus for the 2D structures is represented by the layer modulus (γ).The g, Young's modulus (Y) for strains in the 'x' and 'y' directions [42], and shear modulus (G) in the 'x' and 'xy' directions [44] were expressed as: The property of a material to be brittle or ductile was estimated using Pugh's ratio (K ).When > K 0.5, the material tends to be brittle, but for < K 0.5, the material is ductile [45].The Poisson's ratio (ν) [34] was evaluated to evaluate the nature of bonds in the structure [42].A covalent compound is assumed for values of n < 0.2, an ionic-covalent compound for the range n   0.2 0.3 0, and a metallic compound if values are greater or equal to 0.33 [46].On the other hand, the higher the Vickers hardness (H V ) [45] values, the higher the hardness associated with the material.The K, n, and H V values were obtained by: The elastic properties correlate with the temperature of a crystal's highest normal vibration mode, called the Debye temperature (q D ), and depend on the averaged sound velocity (v m ).Both parameters were obtained as described in [47].On the other hand, the elastic anisotropy provides information about the plastic relaxation at thin-film metals [48] that was obtained using the elastic anisotropy index (A U ) described in [44].
The phonon dispersion (PD) is the frequency dependence on the wave vector, and the phonon dispersion relationships were achieved by applying the linear response theory to calculate periodic perturbations [49,50].The linear response calculations evaluate the dynamic matrix of k vectors based on evaluating the second-order change in the total energy generated by atomic displacements [51].
The equilibrium structure of a crystal without pressure can be found by minimizing the Helmholtz free energy (F).Considering a perfectly harmonic crystal, the internal energy is the sum of the ground state total energy and the vibrational free energy [52]; moreover, considering the dependence of the phonon frequencies on the structural parameters, the equilibrium structure (at any T ) can be obtained as described in [53].Vibrational parameters such as the entropy (S) and the heat capacity (C v ) can be calculated in [54], and the Debye q D in [55].
The optical properties of a material determine the degree to which an electromagnetic wave is reflected, absorbed, or transmitted, and this can be obtained by evaluating the frequency-dependent complex dielectric function ( w ℇ( )) [56].The information obtained from the real ( w 1 ℇ ( )) and the imaginary ( w 2 ℇ ( )) parts of the dielectric function determine the energy loss function (L), the absorption (I ), the reflectivity (R), the refractive index (N ), and the conductivity (s); these parameters are described in [57].

Thermodynamic stability and structural parameters
The information necessary to determine the most stable MCs of each 2D nanomaterial was obtained through total equilibrium energy calculations.The M-PbCs generally have lower H values than the PbC 2 structures (table 1), indicating that the M-PbCs are the most thermodynamically stable.Similarly, the E COH and E f values in M-PbCs are lower than PbC 2 structures except for the PbC 2 NM structure, whose values are lower than M-PbCs.These values of both energies show that the PbC 2 structures are more energetically than M-PbCs, except for the PbC 2 NM structure.The values for the M-PbC (111) and M-PbC (100) are equal (table 1).Considering the EBS behavior and stability, the NM and MAG configurations in M-PbCs converge at the same paramagnetic relaxation.For the PbC 2 structures, the H, E COH , and E f values for the FM, AFM I, AFM II, and AFM III cases are similar but separate in the NM case.These values and their EBS behavior indicate that the PbC 2 structure converges at two paramagnetic relaxations: PbC 2 NM and PbC 2 MAG.
The 2D PbC systems after the relaxing structure are shown in figure 1.The structure of M-PbCs has a hexagonal lattice (better known as honeycomb type), similar to group IV A carbide monolayers [22].The lattice parameters a and b are slightly more significant for the M-PbC (111) than those for the M-PbC (100) (table 2).The M-PbC (100) has a low buckling angle (q) of 14.403°while the M-PbC (111) presents a Θ value of 33.674°.This difference in buckling angle is mainly due to the Pb-C bond length, where the M-PbC (111) is 4.7% greater than the M-PbC (100).Another different parameter is the internal hexagonal angles: Pb-C-Pb (þ) and C-Pb-C (Φ) angles.The þ and Φ angles in M-PbC (111) are the closest to 120°(an average of 1.43%), while in M-PbC (100), the þ angle is 19.28% larger but Φ angle is 9.64% smaller than 120°.
Furthermore, the PbC 2 structures present two different arrangements.The PbC 2 NM shows a hexagonal lattice (MXene structure [58]), while the PbC 2 MAG has a rectangular lattice (1 T′ phase [58]).The lattice parameters (a and b) for the PbC 2 NM are 2.1% smaller than those for the M-PbCs.The Pb-C bond length at the PbC 2 NM is 2.10% and 7.16% larger than those at the M-PbC (111) and M-PbC (100) systems because the M-PbCs' E COH value indicating a greater force by C atoms, and consequently, the þ angle is 7.9 times greater than Φ angle.Besides, the proximity of C atoms causes a small q angle (10.196°) and a sequential change of its inclination angle between the xy and xz planes to achieve structural stability.

Magnetic configurations
The material's electronic band structure (EBS) provides information about its electrical behavior (metal, semimetal, semiconductor, or insulator), and the unpaired EBS reveals its magnetic behavior.The different MCs in monolayers show a metallic compound (figures 3(b)-(d)), the NM M-PbC (figure 3(a)) describes a narrow band gap value (direct, 0.003 eV) but it can be considered as a numerical error, so it also describes a metallic material.This behavior described for M-PbCs agrees with the reported in [27].The PbC 2 structures are semiconductor materials: the direct band gap for the NM configuration is 0.384 eV (figure 3(e)), while the indirect band gap value for MAG cases is around 1.000 eV (figures 3(f)-(h)).The band gap value calculated for MAG cases is similar to the calculations carried out with LDA+U (see Supplementary Material), so the use of DFT+U is not necessary in this manuscript.On the other hand, it is observed that for MAG cases in M-PbCs and PbC 2 structures, their contribution of up-and down-spin EBS are the same, indicating that after geometric relaxation, the total magnetic moment in these cases is null.The criteria to obtain the magnetic state is in [62,63] and can be obtained by comparing the calculations for the 2 * Integrated Spin Density

Electronic properties
Due to the quantum confinement in a two-dimensional space, the total (DOS) and partial (PDOS) density of states, the Mulliken population analysis, and the electron density differences were calculated to identify the electrons' behavior in the M-PbCs and PbC 2 structures.The EBS and DOS are directly related because the dispersion relationship depends on the wave vector k of the electronic band structure.The M-PbC (111) and (100) show that the bands close to the fermi energy (figure 3(a) and (C) correspond mostly to the states contributed by C 2p-orbitals followed by Pb 6p-orbitals, these orbitals are those responsible for these systems being metallic materials.Similarly, the gap in PbC 2 NM is formed mainly by the states of the C 2p-orbitals.In the case of PbC 2 MAG, the C 2p-orbitals provide more states near the fermi energy in the valence band (VB); and the Pb 6p-orbitals followed by the C 2p-orbitals provide higher states in the conduction band (CB).

Mechanical properties
For a correct prediction of the mechanical properties and to define a stable structure, the Born stability criteria (equations ( 4)-( 6)) must be fulfilled.As shown in figure 6  MAG.These behaviors indicate that the 2D PbCs have a greater resistance to lengthwise stretching, followed by volumetric compression, but they have a lower resistance to shear stress; the exception is the PbC 2 NM, which has greater resistance to volumetric compression.The PbC 2 MAG has associated a greater resistance to longitudinal stretching and shear stress along the y (xy) axis than the x-axis.Compared with other calculations of 2D binary nanomaterials based on group IV A atoms [43], the moduli g, Y , and G in graphene are greater than M-PbCs, PbC 2 NM, and MAG, but the moduli γ and G of the 2D PbC systems are greater than silicene, germanene, and stenane.In addition, the modulus Y values of silicene are similar to M-PbCs, followed by PbC 2 MAG, germanene, PbC 2 NM, and stanene.Results from the 2D PbC systems calculations show that these are ductile materials with metallic bonds (figure 7(a)).The PbC 2 structures are more ductile than the M-PbCs; this trend could be because the connection between C atoms pairs (PbC 2 MAG) and Pb atoms is mainly by the participation of the Pb 6p-orbitals and that the Pb atoms donate more charge to the C atoms in the PbC 2 structures than those donate in M-PbCs.In addition, the fact that the Pb 5d-orbitals have greater participation in the PbC 2 NM bonds (for sp 3 d 2 hybridization) causes a more metallic bond.On the other hand, figure 7(b) shows that M-PbCs have higher Vickers hardness values, followed by the PbC 2 MAG and NM.The Vickers hardness values for the PbC 2 MAG along the x direction are greater than those at the xy plane.It could be because the parameter a (x direction) is smaller than the parameter b (y direction); also, the bonds of Pb with the pairs of C atoms are closer to the x-axis than to the y-axis projections.Additionally, the 2D PbC systems have lower Vickers hardness values than graphene but greater values than silicene, germanene, and stenane [43].
The v m and q D values for the M-PbC (111) are slightly greater than those for the M-PbC (100) system (figure 8(a)) because the parameters a and b, as well as the buckling angle are lower for the (100); therefore, its geometric dispersion will be less.These factors generate a loss for wave propagation, decreasing the v m and, consequently, the q .
D On the other hand, PbC 2 NM has a higher v m than PbC 2 MAG due to the greater symmetry; however, the q D is lower due to the lower density.From the A U calculations (figure 8(b)), it was found that the PbC 2 MAG is the most anisotropic material, followed by the PbC 2 NM and M-PbC (100), being the most isotropic the M-PbC (111); this indicates that the properties of M-PbC (111) (PbC 2 MAG) vary less (more) at the xy directions than the other systems.

Vibrational properties
The vibrational properties of the 2D PbC systems were analyzed through the PD calculations along the Γ-M-K-Γ trajectory.The PD for the M-PbC (figures 9(a)-(b)) and PbC 2 structures (figures 9(c)-(d)) show six and nine branches, respectively.Three of them are acoustics, and the other branches are optical.Negative frequencies are observed for all the 2D PbC systems, indicating their dynamical instability.The M-PbCs present one branch with negative frequencies along Γ-M-K-Γ directions.The partial phonon density of states (PPDOS) suggests that these branches are mainly associated with the Pb atoms frequencies (figures 10(a)-(b)); also, it is observed that the C atoms frequencies for the M-PbC (111) (figure 10(a)) are greater than those for the M-PbC (100) (figure 10(b)).This trend could indicate that the greater buckling angle and the Pb-C bond length in M-PbC (111) make it a more unstable system.Three branches are in the low-frequency region (LFR), and two branches are located from 0 to 3.5 THz, associated with greater participation of Pb atoms frequencies.The other branch, in the case of the M-PbC (111), has participation from the C atoms (frequencies from 3.9 to 6.2 THz); for the M-PbC (100) case, it is notorious for the greater participation of the C atoms frequencies (from 4.5 to 6.4 THz).The last two branches are in the high-frequency region (HFR).The PPDOS indicates that these branches are associated mainly by C atom frequencies; for M-PbC (111), they are located from 15.2 to 18.6 THz, and a short peak is shown at 19.5 THz, whereas for M-PbC (100), are located from 15.1 to 18.1 THz.
The PbC 2 NM has associated one negative branch along the K-Γ direction (figure 9(c)), which is oriented along the Pb-Pb bond direction.This fact agrees with their PPDOS (figure 10(c)), which indicates that negative frequencies are associated mainly with the Pb atoms.The branches in the LFR have more participation from the Pb atoms (0 to 3.4 THz); meanwhile, the other six branches are in the HFR (7.1 to 4.4 THz), associated with greater participation of the C atoms frequencies.Besides, the PbC 2 MAG has associated two branches with negative frequencies along the Γ-M-K-Γ trajectory (figure 9(d)) and is mostly linked to the C atoms frequencies (figure 10(d)).Three branches in the LFR (0 to 5 THz) have frequencies mainly associated with the Pb atoms.The other three branches in the HFR (from 5 to 8.5 THz) are associated with greater participation of C atom frequencies.It is noticeable that the positive frequency range for the PbC 2 MAG (∼9 THz) is smaller than those for the other systems, but the negative frequency range is greater (∼6 THz).This behavior could indicate that the C double bond in the PbC 2 MAG generates a greater dynamic instability.On the other hand, the PbC 2 NM has the lowest negative frequencies; this system could be the material obtained [28] since these frequencies could change by applying the experimental conditions reported by Sengupta et al.

Thermodynamical properties
The thermodynamic properties describe a system's state and stability in terms of its energy at different temperatures.A system's lowest energy is the zero-point energy, which is correlated with the ground state energy.The calculated zero-point energies are 0.093, 0.093, 0.136, and 0.177 eV for the M-PbC (111), M-PbC (100), PbC 2 NM, and MAG structures.These results agree with the trend of the H atom / calculation of the 2D PbC systems.On the other hand, the calculated values for S and H increase as the T increases (figures 11(a)-(b)), and the values of F decrease (figure 11(c)).The crystalline structure of the M-PbCs is less disordered (figure 11(a)) than those of the PbC 2 systems.The S values for the M-PbC (100) are greater than those for the M-PbC (100), from zero to 1000 K.For the PbC 2 structures, the S values are very similar from zero to 487 K; above this T, the values for the PbC 2 NM are greater than those for the PbC 2 MAG.This trend indicates a greater disorder in the PbC 2 NM structure at higher temperatures than in the PbC 2 MAG case, which could be due to the greater interaction between the C double bonds (MAG) than the C-Pb-C bonds (NM).Similarly, as the temperature increases, the H values are higher for the PbC 2 structures than for the M-PbCs (figure 11(b)).The H values are lower for the M-PbC (111), while that for the PbC 2 NM is higher than the corresponding to the MAG case, above 387 K.Although the zeropoint energy is higher for the PbC 2 MAG, the H values are higher for the PbC 2 NM as temperature increases.This could indicate that the system is more stable at high temperatures because there are fewer negative frequencies and the possibility that these disappear, as observed between 1073 K and 1373 K [28].Furthermore, the F values are smaller for the PbC 2 NM and MAG, while they are greater for M-PbC (111) and M-PbC (100) (figure 11(d)).The F values for the PbC 2 MAG are lower than those for the NM case; however, above 749 K, the F is higher for the PbC 2 NM.This behavior is consistent with the H since, at higher temperatures, a more disordered final state requires less energy transfer to create PbC 2 NM.
The temperature-dependent specific heat approaches 0 cal/cell K as the temperature approaches absolute zero (figure 12(a)).As T increases, the C v values increase proportionally to T 3 until reaching the q D for the case of PbC 2 structures, but this limit is lower for M-PbC (100) and M-PbC (111) (figure 8(a)).Above that temperature, the systems' higher frequencies are excited [64], and C v grows closer to the Dulong-Petit limit.This limit for the M-PbCs is around 11.1 cal/cell K, while for the PbC 2 MAG and NM, the limits are about 15.4 and 17.8 cal/cell K, respectively.The C v values for the PbC 2 NM indicate that it is the most stable system at high temperatures because it absorbs more thermal energy than the other systems.
On the other hand, Q D at = T 0 K is higher (figure 12

Optical properties
The optical properties of the 2D PbC systems were calculated through the w .
ℇ( ) The w 1 ℇ ( ) values in the infrared (IR) and visible regions are negative for all the 2D systems (figure 13 NM has the highest values at low frequencies in the IR region, but at higher frequencies, the values for the PbC 2 MAG are greater than those for the NM case.In the visible region, the PbC 2 MAG exhibits the highest w 2 ℇ ( ) values.For frequencies in the UV region, the w 2 ℇ ( ) values tend to be zero for all systems.The high values of w 2 ℇ ( ) at low frequencies and their decrease as the frequencies increase indicates that the systems are optically anisotropic and transparent to frequencies in the UV region.The L values at the IR and visible regions are close to zero (figure 13(c)).For the M-PbC (111) case, L values are higher from 8.4 to 12.3 eV with a maximum peak at 10.4 eV; this region is known as the bulk plasma frequency (wp).The wp for the different systems are located: from 8.9 to 20.4 eV for the M-PbC (100) case, with a maximum peak at 13.3 eV; from 8.7 to 22.8 eV for the PbC 2 NM, with a maximum peak at 14.3 eV; and from 10.3 to 35.9 eV for the PbC 2 MAG, with a maximum peak at   On the other hand, at low frequencies, the 2D PbC systems show high I values (figure 14(a)).The highest I values are shown in the IR region by the M-PbC (111) and in the visible region by the PbC 2 NM; meanwhile, the PbC 2 MAG shows the lowest I values in these two regions.However, in the UV region (from 4.6 to 34.5 eV), the PbC 2 MAG shows the highest I values, while the M-PbC (111) shows the lowest I values.Above this region, the I values for the 2D PbC systems tend to be zero, except for a peak at 83.4 eV for the PbC 2 NM.This behavior indicates that the PbC 2 NM has the best wave adsorption at IR and visible frequencies, while the PbC 2 MAG absorbs better at low frequencies in the UV region; then, these systems can be promising materials for applications in optical and optoelectronic devices that operate at the mentioned frequencies.The R values are very high in the IR and visible regions (figure 14 The N allows quantifying the wavelength variation through a material and has two components.The real part (n) gives information about the wave phase velocity, and the imaginary part (k) quantifies the absorption loss.At low frequencies (IR region), the n values are very high but decrease rapidly, tending to zero, except for the PbC 2 MAG (figure 14(c)), which reaches a maximum peak at 4.0 eV.In addition, the lowest n values are shown by the M-PbC (111) at frequencies above 11.3 eV.The n values for the 2D PbC systems are around 0.9 (above 63.3 eV).This indicates that the phase velocity is very high at low frequencies, but in the visible region, the phase velocity is close to zero, except for the PbC 2 MAG.At frequencies above 15 eV, the phase velocity improves, and the phase velocities of the systems are very similar.The k values at low frequencies (IR region) are very high but decrease exponentially, reaching values close to zero at frequencies above 22.4 eV (figure 14(d)).The k values for the M-PbC (111) are slightly lower than those for the other systems.The trend for k indicates treater adsorption loss at low frequencies.The behavior of I, R, and N indicate that the M-PbC (111), M-PbC (100), and PbC 2 NM are systems that can be used as refractory materials.At the same time, the PbC 2 MAG could be helpful for applications in optical and optoelectronic devices at low frequencies (IR, visible, and UV regions).
Similar to N, the σ has two components.The real component gives information about the electrical conductivity, while the imaginary part provides information about the phase lag of the charge carriers.The σ real part values are high at low frequencies and decrease exponentially.The real part values (figure 14(e)) at low frequencies (IR and visible region) are higher for the PbC 2 NM and lower for the M-PbC (111).The PbC 2 MAG is associated with the highest values, from 3 to 34.8 eV (UV region), having a maximum peak at 4.5 eV.In addition, the values of the imaginary part also decrease exponentially (figure 14(f)); the lower values are shown by the M-PbC (111).The PbC 2 MAG case has two peaks at frequencies 7.42 and 35.5 eV, decreasing the values as the frequencies increase.The trend of σ reinforces that the PbC 2 MAG is a system that can be used for optoelectronic devices.

Conclusions
The physical properties of the 2D PbC systems were estimated using ab initio calculations based on the DFT.These systems are thermodynamical, structural, and mechanically stable but energetically and dynamically unstable.However, the PbC 2 NM is the most stable at high temperatures, indicating that Sengupta et al could develop this system experimentally [28].The 2D PbC systems are paramagnetic and anisotropic compounds, where the M-PbCs and the PbC 2 NM have a hexagonal lattice that mainly exhibit sp 2 and sp 3 d 2 hybridizations, respectively; the PbC 2 MAG has associated a rectangular lattice, generated by a strong interaction at the C double bond.The x-ray diffraction patterns predicted for the 2D PbC systems are similar to previous experimental results [60,61].The 2D PbC systems are ductile materials with metallic bonds where the PbC 2 structures are more ductile than the M-PbCs due to the Pb 6p-orbitals.Compared with other 2D materials calculations [43], the moduli γ and G of the 2D PbC systems are greater than those for silicene, germanene, and stanene but lower than those for graphene.Also, the modulus Y values are similar between 2D PbC systems, silicene, germanene, and stenane.The H V values are greater for the M-PbCs than for the PbC 2 systems, but at high temperatures, the PbC 2 MAG shows the highest mechanical resistance.At low frequencies (IR, visible, and UV regions), the M-PbC (111), M-PbC (100), and PbC 2 NM are systems that can be used as refractory materials, while the PbC 2 MAG could be helpful for applications in optical and optoelectronic devices.These systems are optically transparent for the higher frequencies in the UV region.

Figure 1 .
Figure 1.Top and side views for 2D PbC systems.A honeycomb type is formed after relaxing structure for (a) M-PbC (111) and M-PbC (100); an MXene type for (b) PbC 2 NM; and a 1 T′ type for (c) PbC 2 MAG.The internal hexagonal angles Pb-C-Pb (þ) and C-Pb-C (Φ), as well as the buckling angle (q), are shown.The structures were built with VESTA Software [59].

Table 2 .
Structural parameters, Pb-C, Pb-Pb, and C-C bond lengths for the 2D PbC systems.In addition, the values of þ, Φ, and q angles are shown. Structure

( 2 *
ISD) and the 2 * Integrated |Spin Density| (2 * I| SD |).The 2 * ISD and 2 * I| SD | values for all the MCs for the M-PbCs and PbC 2 structures are zero, indicating that these 2D nanomaterials are paramagnetic compounds.The paramagnetism of these compounds agrees with the (same behavior) results obtained for the contribution and the values of both spins (up and down) channels, including the MAG cases, after the relaxation of the structure models.

Figure 2 .
Figure 2. The x-ray diffraction patterns predicted for the M-PbC (a) (111) and (b) (100); also, the corresponding diffraction patterns expected for the PbC 2 (c) NM and (d) MAG are shown.The intensity is in arbitrary units, and the reflections are the predicted Bragg positions.

Figure 3 .
Figure 3. Electronic bands structure for (a) NM, (b) FM, AFM I, II, and III for the M-PbC (111); (c) NM, (d) FM, AFM I, II, and III; and the PbC 2 structure (e) NM, (f) FM, AFM I, (g) II and (h) III.The Fermi energy was set at zero.

Figure 4 (
Figure 4(d) shows a higher state concentration in the BV than CB for PbC 2 MAG.At the energy intervals from −13.71 to −12.43, from −9.20 to −6.80, and from −5.34 to −0.01 eV, there is a significant contribution from the C 2s-, the Pb 6p-and the C 2p-orbitals, respectively.The Mulliken population analysis indicates a C atoms' charge gain of 0.48 |e| and a large electron cloud density between pairs of C atoms.This behavior could indicate the formation of a C double bond.Besides, the Pb atoms transfer their charge (0.96 |e|) to the neighboring carbon atoms (figure 5(d)), forming ionic bonds, which would explain why the Pb-Pb bond lengths are larger than the Pb-C and C-C bond lengths.

C ; 11 12
(a), the C 11 values for all the 2D PbC systems are greater than zero, and > C in addition, the C 66 values comply with equation (6).Besides, C 11 and C 22 are greater than C 12 and C 66 for all 2D PbC systems indicating that the resistance to mechanical stress is very similar along the x and y axes but has less resistance to transverse expansion and shear stress.The C , 11 C , 22 C , 12 and C 66 are very similar for M-PbCs; the lowest values for C , 11 C , 22 and C 66 are associated with the PbC 2 NM, while the highest values for C 22 and C 12 are linked to PbC 2 MAG.The difference between C 11 and C 22 for the PbC 2 MAG could be due to their atomic interactions and the difference between their lattice parameters a and b.The observed C ij behaviors are reflected on the calculated moduli (figure 6(b)).The Y values are generally greater than the g values, but the G moduli values are the smallest.The g, Y , and G values are very similar between M-PbCs; the smallest g, Y , and G values are those for the PbC 2 NM, while the highest g and Y y values are linked to PbC 2

Figure 4 .
Figure 4. Total DOS (dark blue) and atomic contributions from the Pb s (green), p (light green), and d (orange) orbitals, as well as from the C s (purple) and p (red) orbitals, are shown for M-PbC (a) (111) and (b) (100) and for the PbC 2 (c) NM and (d) MAG.

Figure 5 .
Figure 5. Mulliken population analysis for the C (gray spheres) and Pb (purple spheres) atoms in the 2D PbC systems.The electron density difference between atoms for the M-PbC (a) (111) and (b) (100), and for the PbC 2 (c) NM and (d) MAG.

Figure 6 .
Figure 6.(a) Elastic constants values (C ij ) and (b) layer (g ), Young in x (Y x ) and y (Y y ) directions, and shear (in x G x and y G y ) directions moduli for the 2D PbC systems.

Figure 7 .
Figure 7. (a) Pugh's ratio in x (K x ) and y (K y ) directions, Poisson's ratio in x (v x ) and y (v y ) directions, and (b) Vickers hardness (H V ) for the modeled 2D PbC systems.

Figure 8 .
Figure 8.(a) Average sound velocity (v m ), Debye temperature (q D ), and (b) universal anisotropy index (A U ) for the modeled 2D PbC systems.

Figure 10 .
Figure 10.Partial and total phonon density of states for the 2D PbC systems (green) and the Pb (purple) and C (red) atoms.
(b)) for the M-PbC (100) and M-PbC (111), followed by the PbC 2 NM and MAG.The Q D values for the PbC 2 NM are higher at = T 110 K and change to a linear trend until they reach 692 K at the maximum calculated temperature.The trend modification for the M-PbCs appears around = T 190 K and reaches a maximum Q : D 1320 K and 1490 K for (100) and (111), respectively.The Q D grows linearly for the PbC 2 MAG case until 1870 K.The highest Q D for the PbC 2 MAG indicates that, among the studied systems, this one has the highest mechanical resistance at high temperatures, which is linked to the C double bond.

23. 3
eV.The wp with the smaller and larger regions are linked to M-PbC (111) and PbC 2 MAG, respectively.Considering the wp region, the 2D PbC systems are refractive materials for frequencies between 8.4 and 10.3 eV.At the same time, they are transparent for frequencies greater than 12.3, 20.4,22.8, and 35.9 eV for the M-PbC (111), M-PbC (100), PbC 2 NM, and MAG, respectively.
(b)); the highest R values are linked to the M-PbC (111), while the lowest correspond to the PbC 2 MAG.In the UV region, the R values decrease; from 2 to 10.4 eV, the highest R values are for M-PbC (111), from 10.4 to 13.1 for M-PbC (111), and from 10.4 to 37 eV for PbC 2 MAG.Above these frequencies, the R values are very close to zero.The behavior of I and R are in accordance with L because the 2D PbC systems are transparent materials at frequencies above 35.9eV (UV region).
E Pb and E C are the energies for isolated Pb and C atoms; n Pb and m C are the numbers of Pb and C atoms, respectively, in the system's cells; E ,

Table 1 .
Enthalpy per atom (H/atom), cohesive (E COH ), and formation energy (E f ) calculated for 2D PbC nanomaterials considering their magnetic configurations.