Two-dimensional C6X (X = P2, N2, NP) with ultra-wide bandgap and high carrier mobility

Two-dimensional (2D) materials with ultra-wide bandgap and high carrier mobility are highly promising for electronic applications. We predicted 2D C3P, C3N and C6NP monolayers through density-functional-theory calculations. The phonon spectra and Ab initio molecular dynamics simulation confirm that the three 2D materials exhibit good phase stability. The C3P monolayer shows excellent mechanical flexibility with a critical strain of 27%. The C3P and C6NP monolayers are ultra-wide bandgap semiconductors based on Heyd-Scuseria-Ernzerhof hybrid functional (HSE06) calculation. The C3P monolayer has a direct bandgap of 4.42 eV, and the C6NP and C3N monolayer have indirect bandgaps of 3.94 and 3.35 eV, respectively. The C3P monolayer exhibits a high hole mobility of 9.06 × 104 cm2V−1s−1, and the C3N monolayer shows a high electron mobility of 4.52 × 104 cm2V−1s−1. Hence, the C3P, C3N, and C6NP monolayers are promising materials for various electronic devices.


Introduction
The atomically-layered thin two-dimensional materials have garnered significant attention in recent years due to their unique mechanical and electronic properties [1][2][3][4]. Graphene is one of the most well-known 2D materials with excellent mechanical properties, high carrier mobility and zero bandgap electronic structure [5][6][7][8]. Black phosphorene is another 2D material composed of phosphorus atoms arranged in a puckered, honeycomb-like lattice [9][10][11]. Unlike graphene, black phosphorene has a finite bandgap of 1.51 eV [9]. Although both the graphene and black phosphorene exhibit high carrier mobility, the insufficient bandgaps limit them in the application of high-frequency and high-power electronic devices, such as smart electricity grids, electric vehicles and telecommunications [12]. Ultra-wide bandgaps (>3.5 eV) would allow 2D materials to withstand high electric fields without experiencing dielectric breakdown and high carrier mobility would allow fast device operation, which are essential for the development of high-frequency and high-power electronic devices [12]. However, in the process of developing 2D materials, although few 2D materials, such as monolayer CaFCl [13], hydrogenation of grapheme [14] and h-BN [15], with ultra-wide bandgaps have been found, most of the twodimensional materials discovered by experiments and predicted by theoretical calculations have not simultaneously achieved ultra-wide bandgap and high carrier mobility. Therefore, it is necessary to explore new types of 2D ultra-wide bandgap semiconductors with high carrier mobility for satisfying the needs of electronic devices.
The monolayer carbon phosphides (C x P y ) and carbon nitrides (C x N y ) can be formed by arranging C, N, and P atoms in different bonding configurations, which exhibit a range of electronic properties including metallic, semimetallic and semiconductive properties. For example, the α-CP and β-CP monolayer show semiconductive properties with small bandgaps of 0.63 eV and 0.39 eV respectively [16]. But the Γ-CP monolayer exhibit Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. semimetallic properties with Dirac cones [16]. Wang et al investigated the elastic and electronic properties of honeycombed structure A 3 B (A = C, Si, Ge and B = N, P, As) monolayer and suggested that the honeycombed structure C 3 P and C 3 N have bandgaps of ∼2.8 and ∼2.0 eV, respectively [17]. Recent theoretical research has shown that doping C 3 N monolayers with varying concentrations of B results in an increase in the material's band gap value. Additionally, it was discovered that at a B doping concentration of 12.5%, the C 3 N monolayer transitions from an indirect to a direct gap semiconductor [18]. Most of the previously reported monolayer carbon phosphides are metallic or have bandgaps that are lower than ∼3.5 eV [16,[19][20][21][22][23][24]. In the case of monolayer C x N y , the monolayer g-C 3 N 4 and C 2 N have bandgaps of 2.7 eV [25] and 1.96 eV [26], respectively. While the monolayer C 3 N only has a small band gap of 1.06 eV [27]. In addition, only a few carbon nitrides monolayer, such as monolayer GaSe-like CN [28], exhibit ultra-wide bandgap, and most monolayer C x N y show metallic or semiconductive properties with bandgaps small than 3.5 eV. In addition, carrier mobility is also a crucial parameter for high-frequency and high-power electronic devices. Previous reports indicate that some of the C x P y and C x N y monolayers exhibit high carrier mobility [16,29,30]. For instance, recent theoretical work predicted a high hole mobility of ∼5.19 × 10 4 cm 2 V −1 s −1 for the CP 2 monolayer [29]. Due to the various bonding configuration of C, N and P atoms, there is still possible to discover monolayer carbon phosphide and carbon nitride with ultra-wide bandgaps and high carrier mobility.
In this paper, we have theoretically designed three novel 2D materials, C 3 P, C 3 N and C 6 NP by using densityfunctional-theory calculations. The calculated cohesive energy, phonon spectrum and AIMD indicate that these three monolayers have high stability. The C 3 P monolayers show high mechanical flexibility with a critical strain of 27%. Moreover, the calculations of their electronic band structures indicate that the C 3 P and C 6 NP monolayers are ultra-wide bandgap semiconductors. The C 3 P monolayer exhibits high hole mobility and the C 3 N monolayer shows high electron mobility. Hence, the C 3 P, C 3 N, and C 6 NP monolayers are promising materials for various electronic devices.

Computational methods
The density functional theory (DFT) calculations were carried out using the VASP code [31]. The electron-ion interaction was described using the projector-augmented wave (PAW) method [32]. We used the generalized gradient approximation (GGA) in Perdew-Burke-Ernzerhof functional (PBE) to describe the exchangecorrelation functional [33,34]. To take the van der Waals interactions into account, the D2-Grimme dispersion correction [35] was used. We used a plane-wave energy cutoff of 500 eV for all the calculations. To reduce the artificial interactions between neighboring layers, we employed a large vacuum slab (∼25 Å) along the z-direction. For geometric optimization, we use 12 × 12 × 1 Γ-centered k-meshes. The cell's volume, shape, and all of its atoms were completely relaxed until the residual force was less than 0.01 eV Å. The electronic structure and optical characteristics of C 3 P, C 3 N and C 6 NP monolayers were acquired using the HSE06 functional [36]. It is worth noting that when using DFT calculations to determine the bandgaps of solids, the results often underestimate the actual values. On the other hand, the Hartree-Fock (HF) method tends to overestimate the bandgaps. However, recent theoretical studies have suggested that by utilizing the B3LYP or B3PW functionals, it is possible to achieve excellent agreement between the calculated band gaps and experimental data for ABO 3 perovskites [37,38]. For the phonon spectra calculation, the density-functional-perturbation theory (DFPT) was used and 5 × 5 × 1 supercell of C 3 P, C 3 N and C 6 NP was adopted [39]. We use a 4 × 4 × 1 supercell of C 3 P, C 3 N and C 6 NP monolayer to perform the Ab initio molecular dynamics (AIMD) simulations. The Nose-Hoover method was used to solve the ionic motion equations. The time step of AIMD simulation is set to be 1.0 fs. We equilibrated the system at 600 K for 10 ps in canonical ensemble (NVT). We used Vaspkit code to derive the mechanical properties of C 3 P, C 3 N and C 6 NP monolayer from the raw VASP calculation results [40]. For chemical bonding analyses, we utilized the COHP method as implemented in the LOBSTER package [41,42].

Results and discussions
Structure and stability of C 3 P, C 3 N and C 6 NP Figures 1(a) and (b) depicted the structure of the proposed C 3 P monolayer and table 1 listed the lattice constants and average bond lengths of C-C, C-P and P-P. The C 3 P monolayer crystallizes in the hexagonal P3m1 space group. The lattice constants of C 3 P monolayer are a = b = 4.180 Å, α = β = 90°, and γ=120°. The C 3 P monolayer is composed of C 3 triangle from the top view and C 4 square from the side view. Each P atom is coordinated with three C atoms. The C -P bond length is 1.83 Å. The C1-C1 and C1-C2 bonds length are 1.54 and 1.58 Å, respectively. The C -P and C -C bond in C 3 P are longer than that in α 1 -PC (1.38 Å for C -C and 1.78 Å for C-P bond) [16]. Recent theoretical studies made a prediction about the structure of a C 3 P monolayer, which consists of a honeycomb arrangement of six carbon atoms and two phosphorus atoms [17]. The phosphorus atoms are uniformly distributed within the structure, and both the carbon and phosphorus atoms exhibit D6h symmetry. In addition, the C 3 P monolayers with a six-membered puckered structure have also been predicted [24]. As illustrated in figures 1(a) and (b), the proposed C 3 P monolayer consists of C3 triangle on the xy plane, with each P atom coordinating with three C atoms, which has a different atomic structure than that described in the earlier work. We have carried out phonon dispersion calculations to investigate the dynamical stability of the predicted C 3 P monolayer. As demonstrated in figure 1(c), there are no imaginary phonon frequencies in the whole Brillouin zone, suggesting that the C 3 P monolayer has good dynamical stability.
In addition, two novel structures, C 3 N and C 6 NP monolayer, were discovered by completely or partially replacing the P atoms in C 3 P with N. Figures 1(d)-(e) and 3(h)-(i) show the predicted structure of C 3 N and C 6 NP monolayer and table 1 list the lattice constants and average bond lengths. The C1-C1 and C1-C2 bonds Figure 1. The top (a) and (b) side views, phonon dispersion relation and phonon density of state (c) of the C 3 P sheet; the top (d) and (e) side views, phonon dispersion relation and phonon density of state (f) of the C 3 N sheet; the top (h) and (i) side views, phonon dispersion relation and phonon density of state (j) of the C 3 NP sheet. The brown, blue and pink balls representing C atoms, N atoms and P atoms, respectively. The unit cell is indicated by a black line. Table 1. Calculated structural parameters of C 3 P, C 3 N and C 6 NP monolayers. length of C 3 N are 1.53 and 1.58 Å, respectively, similar to that in C 3 P. The C 3 N monolayer has lattice constants of a = b = 3.76 Å, which is ∼0.4 Å lower than that of C 3 P. This can attribute to the shorter bond length of C-N (1.46 Å) in C 3 N compared to C-P (1.83 Å) in C 3 P. It should be noted that the C 6 NP has a Janus structure with N and P atoms on each side of the surface. The lattice constants of C 6 NP are a = b = 3.93 Å, which is larger than that of C 3 N and smaller than that of C 3 P. As shown in figure 3(b), the C4 square is slightly distorted due to the different bonding on both sides of the C 6 NP surface. The C atoms on the top surface are bonded to N atoms, while the C atoms on the bottom surface are bonded to P atoms. This difference in bonding leads to a slight distortion of the C4 square. The C1-C1, C1-C2 and C2-C2 bonds length are 1.59, 1.58 and 1.48 Å, respectively. The C-N bond length in C 6 NP is 1.50 Å, which is slightly longer than that in C 3 N. The C-P bond length in C 6 NP (1.79 Å) is slightly shorter than that in C 3 P. The phonon spectrum calculations (as shown in figures 1(f) and (j)) demonstrate that there are no imaginary phonon frequencies throughout the Brillouin zone, suggesting the good dynamical stability of the C 3 N and C 3 N monolayer. The AIMD simulations were used to evaluate the thermal stability of the C 3 P, C 3 N and C 6 NP monolayer. The time step of 1 fs was chosen to ensure that the simulations were accurate and reliable. Figure 2 shows the structures after the AIMD simulation as well as the overall energy variation during the simulation. The oscillation of C, N and P atoms around their equilibrium position were observed during the AIMD simulations. The C-C, C-P and C-N bonds are well maintained after 10 ps AIMD simulation. According to the phonon dispersion and AIMD results, the C 3 P, C 3 N and C 6 NP monolayers exhibit good dynamical and thermal stabilities.
We further calculated the cohesive energy of C 3 P, C 3 N and C 6 NP monolayer using the following expression where the E total is the total energy of the unit cell of C x N y P z ; E(C), E(N) and E(P) are the energy of isolated C, N and P atom; the x, y and z represent the number of C, N and P atoms in the unit cell. As summarized in table 2, the three monolayers all have cohesive energies higher than that of graphene but lower than that of phosphorene. The cohesive energy of C 6 NP is comparable to that of C 3 P, while the cohesive energy of C 3 N is ∼0.12 eV lower than that of C 3 P. We also calculated the cohesive energy of the previously reported C 3 P with a six-membered puckered structure [24] for comparison. Although the previously reported C 3 P has a cohesive energy of −6.48 eV, which is 0.35 eV lower than C 3 P in the present work, the phonon spectra and AIMD calculations suggested that the proposed C 3 P in the present work has good phase stability.
Further crystal orbital Hamilton population (COHP) investigations were conducted to investigate the bonding strength of C-P, C-C, and C-N in monolayer C 3 P, C 3 N, and C 6 NP. The COHP and integrated COHP are shown in figure S1. Table 3 also displays the ICOHP value, which is the energy value after integrating COHP Figure 2. The AIMD simulation at 600 K for C 3 P, C 6 NP and C 3 N. The insert figures present the top view of the atomic structure of C 3 P, C 6 NP and C 3 N after AIMD simulation. up to the Fermi energy. As shown in figure S1, for the C-P and C-N interaction, a few antibonding states near the fermi level are observed. The calculated -ICOHP of C-C is in the range of 7.8 to 8.6 eV for all the three monolayers. The -ICOHP of C-C in C 3 P, C 6 NP and C 3 N are significantly lower than that in graphene, indicating relatively weaker bonding strength in those three sheets. This can be attributed to the unique structural arrangement of C 3 P, C 6 NP and C 3 N monolayer, in which the six C atoms are composed of a triangular prism. The calculated -ICOHP of C-P is 7.22 and 7.31 eV in C 3 P and C 6 NP, respectively, which are lower than that of C-C, indicating that C-P bonds are relatively weak compared to that of C-C bonds. The -ICOHP of C-N in C 3 N and C 6 NP is 10.86 and 10.35 eV, respectively, which are significantly higher than that of C-C and C-P bonds. In general, the bond strength is following the trend: C-N > C-C > C-P in C 3 P, C 3 N and C 6 NP monolayer. The high bond strength of C-N compared to C-C and C-P may contribute to the stability and durability of the C 3 N and C 6 NP monolayers under different conditions. On the other hand, the weaker bonding interaction of C-P may make the C 3 P monolayer more flexible and suitable for applications requiring high stretchability.

Mechanical properties
To investigate the mechanical stability of the C 3 P, C 3 N and C 6 NP monolayer, the elastic constants were calculated. The elastic modulus C D 2 can be calculated by where l , 0 S 0 and E 0 are the lattice constant, lattice volume and total energy of the unstrained monolayer, respectively; E and Dl are the total energy and deformation of the strain monolayer.
As shown in table 4, the calculated elastic constants of C 3 P, C 3 N, C 6 NP satisfactorily meet mechanical stability criteria: C 11 > 0 and C 11 > | | C 12 [40], demonstrating the mechanically stability of the proposed monolayer. In general, the elastic constants of C 11 follows the trend: C 3 N > C 6 NP > C 3 P, which may be ascribed to the fact that C-N bonds are stronger than C-P bonds. The following expressions can be used to directly calculate the in-plane Young's modulus (Y) and Poisson's ratio (ν): n = / C C 12 11 According to table 4, the computed Young's moduli for C 3 P, C 6 NP, and C 3 N are 163.92, 204.365, and 264.679 N m −1 , respectively. Mechanically, all three monolayers are more stable than black phosphorene (Young's modulus: 104.4 N m −1 ). The Young's modulus of C 3 N is comparable to that of h-BN monolayer (270 N m −1 ) [43] and lower than that of graphene (340 N m −1 ). The trend in Young's modulus of C 3 P, C 6 NP, and C 3 N monolayers can be attributed to the strength of the bonds between the constituent atoms. As shown in table 3, the C-N bonds are the strongest followed by C-C and then C-P bonds, leading to an increase in stiffness from C 3 P to C 6 NP and then to C 3 N. Although the Young's modulus of C 3 P, C 6 NP, and C 3 N are considerably different, their Poisson's ratios are the same at a value of 0.25, indicating that the three materials exhibit similar deformation behavior under stress. The comparable Poisson's ratios observed in C 3 P, C 6 NP, and C 3 N could be attributed to the unique structural features of these materials. As illustrated in figure 1, all three materials consist of C3 triangle on the xy plane, with each P or N atom coordinating with three C atoms. The in-plane Poisson's ratio is influenced by the proportion of C 12 and C 11 . Table 4 indicates that the elastic constants for both C 11 and Table 3. ICOHP of C-N, C-P and C-C.  Table 4. Elastic constants, Young's modulus (Y) and Poisson's ratio (ν) of C 3 P, C 3 N and C 6 NP. We further calculated the stress-strain relation to investigate the ideal strength of the C 3 P, C 3 N and C 6 NP monolayer. Figure 3 shows that all three sheets display a linear stress-strain relationship when small uniaxial strains along the x-direction are applied. And the stress-strain relationship becomes increasingly nonlinear as the strain rises. The C 3 P can withstand a maximal uniaxial strain of 27%, corresponding to a maximal stress of 22.14 N m −1 . This high critical strain demonstrates the excellent mechanical flexibility of C 3 P monolayer and its potential for use in flexible electronic applications. In contrast, the recently predicted honeycombed structure C 3 P monolayer can withstand a maximal uniaxial strain of 15% along the zigzag direction [17]. The C 6 NP and C 3 N can withstand the maximal uniaxial strain of 19% and 18%, corresponding to a maximal stress of 22.63 and 27.75 N m −1 , respectively. In contrast, graphene can withstand tensile loads of up to 19% in the armchair directions [44]. The peak stress of C 6 NP is secondary only to graphene (34 N m −1 ) and similar to that of B 4 N monolayer (25 N m −1 ) [44]. After reaching the critical tensile point, the stress values for loading suddenly decrease. According to the maximal strain and stress value, the C 3 P monolayer shows higher stretchability than C 3 N and C 6 NP under external loading.

Electronic structure
To comprehend the electronic structure of C 3 P, C 3 N, and C 6 NP, we calculated the band structure, electron localization function (ELF), and density of states (DOS). Due to the fact that GGA often underestimates the bandgap, the hybrid Heyd, Scuseria, Ernzerhof (HSE06) functional form was utilized to get electronic characteristics. The C 3 P is a semiconductor with a direct bandgap of 4.42 eV (figure 4(a)), which is larger than the previously reported honeycombed structure C 3 P (2.81 eV) [17]. Both the valence band maximum (VBM) and the conduction band minimum (CBM) are situated at K-point in reciprocal space. The DOS (figures 4(b) and S2) demonstrates that the valence band near the fermi level is mostly formed from the p orbitals of P and C, whereas the density state around CBM is primarily obtained from the s and p orbitals of C and p orbitals of P. As illustrated in figures 4(c) and 4(d), the ELF reveal that the C-C bonds are in the sp 3 configuration. The P atoms adopt the sp 3 hybridization configuration, where one of the hybrid orbitals is occupied by a lone electron pair while the other three hybrid orbitals establish covalent connections with the surrounding C atoms.
In contrast to C 3 P, as shown in figures 5(a) and 6(a), C 3 N and C 6 NP are indirect bandgap semiconductors. C 3 N and C 6 NP have bandgaps of 3.35 and 3.94 eV, respectively. The VBM and CBM of C 3 N are situated at K-point and Γ-point in reciprocal space, respectively. For C 3 N, the DOS (See figure 5(b) and S3) indicates that the valence band near the fermi level is mostly formed of p orbitals of C and N atoms, whereas the bands around CBM are primarily composed of p orbitals of C. Each N atom is surrounded by a pair of nonbonding lone electrons and three hybrid orbitals that participate in covalent bonding with the surrounding C atoms (figures 5(c)-(d)), indicating that the N atoms adopt sp 3 hybridization configuration, similar to that of P in C 3 P. In the case of C 6 NP, the VBM and CBM are situated at K-point and M-point in reciprocal space, respectively. As seen in figure 6(b) and S4, the bands near VBM consist mostly of p orbitals of C, N, and P atoms, whereas the bands near CBM consist primarily of s and p orbitals of C and P. Figures 6(c)-(e) demonstrates that there is a pair of nonbonding lone electrons surrounding each P and N atom. In addition, each P and N atom has three hybrid orbitals that are involved in covalent bonding with surrounding C atoms. Hence, P and N atoms both exhibit sp 3 hybridization configurations.
The previous works demonstrated that applying strain on 2D materials is an effective approach to tune their electronic properties. Therefore, we further investigate the change of bandgaps under the uniaxial strain along  Figure 7 shows that as the uniaxial tensile strain increases from 0 to 5%, the bandgap of C 3 P decreases at a nearly linear rate and reaches a value of 4.12 eV. The applied uniaxial compressive strain of −1% lead to an increased bandgap of C 3 P monolayer to 4.55 eV. Further increase in uniaxial compressive strain from −1% to −5% leads to reduce bandgap of C 3 P monolayer to ∼4.32 eV. In contrast, the bandgap of C 6 NP increases as the uniaxial tensile strain increases from 0 to 4%. As the uniaxial compressive strain increases from 0% to −5%, the bandgap of C 6 NP decreases to a value of 3.86 eV. In the case of C 3 N, the bandgap decreases from 3.57 to 3.22 eV as the strain increase from −5% to 5%. The changes in bandgap with strain may be attributed to changes in the bond length and bond angle in the monolayers, leading to the change in the energy difference between the valence and conduction bands.
Since carrier mobility is also a crucial parameter for high-frequency and high-power electronic devices, we estimated the electron and hole mobility of C 3 P, C 6 NP and C 3 N monolayer using the deformation potential approach. The carrier mobility m D 2 was obtained using the following expression [9,45,46].  where  is the Planck's constant, e is the electron charge, and T is the temperature; * m represents the carrier effective mass (electron or hole) in the transport direction. The calculated carrier mobility of C 3 P, C 6 NP and C 3 N monolayer along the x-direction at 300K were summarized in table 5. The predicted hole mobility of C 3 P is 9.06 × 10 4 cm 2 V −1 s −1 , which is significantly higher than that of electron mobility (237 cm 2 V −1 s −1 ). The extremely high hole mobility of C 3 P can be attributed to the small value of deformation-potential constant. In comparison, recent theoretical work predicted electron and hole mobility of 2804 and 483 cm 2 V −1 s −1 along the zigzag direction, and 807 and 621 cm 2 V −1 s −1 along the armchair direction for honeycombed structure C 3 P monolayer [17]. The proposed C 3 P monolayer has higher hole mobility and lower electron mobility compared to the previously reported honeycomb-structured C 3 P monolayer. The C 6 NP shows electron mobility of 3.59 × 10 3 and hole mobility of 7.74 × 10 3 . In the case of C 3 N, it exhibits a high electron mobility of 4.52 × 10 4 cm 2 V −1 s −1 and hole mobility of 7.33 × 10 3 cm 2 V −1 s −1 . The exceptionally high electron mobility of C 3 N arises from the small value of the effective mass of electron and deformation-potential constant. The proposed C 3 P and C 3 N monolayer have significantly high carrier mobility, which are higher than the previously reported carrier mobility of black phosphorus (1 ∼ 2.1 × 10 4 cm 2 V −1 s −1 ) [9]. The carrier mobility in the C 3 P and C 3 N monolayer are also much higher than that of many other 2D

Conclusion
In summary, two-dimensional C 3 P, C 3 N and C 6 NP were predicted density-functional-theory calculations. The phonon spectra and Ab initio molecular dynamics simulation confirm that the three compositions have good phase stability. The C 3 P monolayer exhibits excellent mechanical flexibility with a critical strain of 27%. The C, P and N atoms are in sp 3 hybridization configuration. The COHP analyses indicate that bond strength is following the trend: C-N > C-C > C-P. The C 3 P and C 6 NP monolayers are ultra-wide bandgap semiconductors. The C 3 P monolayer has a direct bandgap of 4.42 eV, and the C 6 NP and C 3 N monolayer have indirect bandgaps of 3.94 and 3.35 eV, respectively. The C 3 P monolayer exhibits a high hole mobility of 9.06 × 10 4 cm 2 V −1 s −1 , and the C 3 N monolayer shows a high electron mobility of 4.52 × 10 4 cm 2 V −1 s −1 . These results demonstrate the potential of C 3 P, C 3 N and C 6 NP monolayers as new materials for various electronic devices. Table 5. The calculated carrier effective mass(m), deformation-potential constant(E 1 ) and carrier mobility (μ 2D ) for C 3 P, C 6 NP and C 3 N monolayer along the x-direction at 300 K. m/m 0 E 1 (eV) μ 2D (cm 2 V −1 s −1 )