#### 3.1. 2-Dimensional Structures

The structure of W-diamane is relaxed without any passivations or N substitutions. As expected, W-diamane with entire dangling bonds on its surfaces cannot be stabilized after relaxation, but it can be stabilized if the dangling bonds are passivated through hydrogenation (C

${}_{8}$H

${}_{4}$, see

Figure 1a). We find that, instead of hydrogenation, the substitution of N atoms for surficial C atoms can also result in the structural stabilization of W-diamane. Nevertheless, the substitution is needed on both surfaces of W-diamane in order to keep the washboard-like shape stable on two sides after relaxation. For C

${}_{7}$N, an amount of N atoms is not enough to substitute on both of its surfaces, hence it is not considered. Noting that the studied phases have N atoms arranging in highly order by considering the substitution in the unit cell of W-diamane with eight atoms, while the configurational disorder of C and N atoms in W-diamane is beyond the scope and, thus, not considered in this work.

Table 1 reports the relaxed structures of C

${}_{8}$H

${}_{4}$, C

${}_{6}$N

${}_{2}$, C

${}_{5}$N

${}_{3}$, and C

${}_{6}$N

${}_{4}$. Howeve, noting that

c-axes were not fixed during the relaxation, the relaxed

c-axes are long enough to keep all of structures as 2D materials. The lattice parameters,

a and

b, are smaller as the number of N atoms increasing, while their thicknesses from top to bottom layers (

${d}_{0}$) are thicker, as shown in

Table 1. Two parameters presenting the characteristics of the structures are bond length and bond angle. Typically, the C-N and N-N bond lengths are shorter than C-C bond length, so average bond lengths of N-substituted W-diamanes are shorter if the number of N atoms is higher. For an illustration, the average C-C bond length of C

${}_{8}$H

${}_{4}$ is 1.558 Å, while the average bond lengths of C

${}_{6}$N

${}_{2}$, C

${}_{5}$N

${}_{3}$, and C

${}_{4}$N

${}_{4}$ are 1.508, 1.495, and 1.506 Å, respectively. On the contrary, bond angles of atoms between layers of C

${}_{6}$N

${}_{2}$, C

${}_{5}$N

${}_{3}$, and C

${}_{4}$N

${}_{4}$ (see

Figure 1) are 101.85

${}^{\xb0}$, 110.08

${}^{\xb0}$, and 112.37

${}^{\xb0}$, respectively, showing an upward trend that corresponds to the number of N atoms.

Figure 2 shows phonon dispersions of hydrogenated and N-substituted W-diamanes, where the negative value of phonon frequency represents an imaginary mode of the phonon. C

${}_{8}$H

${}_{4}$ and C

${}_{4}$N

${}_{4}$, which are fully passivated and N substituted, respectively, have no imaginary modes, so they are dynamically stable. On the other hand, C

${}_{6}$N

${}_{2}$ and C

${}_{5}$N

${}_{3}$, which are partially N-substituted, have some amounts of the imaginary phonon, so they are not dynamically stable.

Figure 2b–d show the evolution of the imaginary phonon, which is fewer with respect to the number of N substitution in W-diamanes. This is clearly indicating that the stabilization of such a structure can be promoted by suppressing a number of the dangling bonds on the surfaces.

#### 3.2. Electronic Property and Bonding

In the sp

${}^{3}$ 2D carbons, the dangling bonds on their surfaces form energy bands across the Fermi level (E

${}_{F}$) closing the band gap, but the band gap is open if the dangling bonds are all passivated [

11]. For example, the dangling bonds of NCCN and CNCN phases on the surfaces that are substituted by N atoms are replaced by lone pair electrons that open the band gap wildly. C

${}_{8}$H

${}_{4}$ and C

${}_{4}$N

${}_{4}$ are semiconducting with HSE06 band gap, 3.2 and 3.5 eV, respectively, while C

${}_{6}$N

${}_{2}$ and C

${}_{5}$N

${}_{3}$ are metallic. Thus, the band gap of W-diamane is open if no dangling bond is left to be passivated.

The right side of each figure shown in

Figure 3 shows the electronic DOS corresponding to the electronic band dispersion on the left side of each structure. C

${}_{8}$H

${}_{4}$, where its dangling bonds of C

${}_{\mathsf{sf}}$ atoms are fully passivated by H atoms, has the p

${}_{z}$-orbital of C

${}_{\mathsf{sf}}$ atoms hybridizing with the s-orbital of H atoms at energy state −1 eV below the E

${}_{F}$. The electronic density of states (DOS) being occupied near the E

${}_{F}$ are of the p

${}_{x}$-orbitals of C

${}_{\mathsf{sf}}$ atoms in majority and of the p

${}_{x}$-orbitals of C

${}_{\mathsf{in}}$ next below. C

${}_{4}$N

${}_{4}$, instead fully substituted by N atoms on the surfaces, has the electronic DOS, which is C

${}_{\mathsf{in}}$-N

${}_{\mathsf{sf}}$ hybridization near the E

${}_{F}$ in contrast to C

${}_{8}$H

${}_{4}$, where its C-H hybridization is at a lower energy level. The p

${}_{z}$-orbital of N

${}_{\mathsf{sf}}$ atoms obviously dominates the valence states near the E

${}_{F}$.

The electronic band structures of C

${}_{6}$N

${}_{2}$, C

${}_{5}$N

${}_{3}$, and C

${}_{4}$N

${}_{4}$ have similar dispersing feature across the reciprocal space (see

Figure 3b–d). One that makes the electronic property of these three structures different is top two valence bands. C

${}_{6}$N

${}_{2}$, C

${}_{5}$N

${}_{3}$, and C

${}_{4}$N

${}_{4}$ have two, one, and none valence bands, respectively, crossing the E

${}_{F}$. Despite that, their valence states are dominated by p

${}_{z}$-orbital of C

${}_{\mathsf{sf}}$ atoms for C

${}_{6}$N

${}_{2}$ and C

${}_{5}$N

${}_{3}$, and p

${}_{z}$-orbital of N

${}_{\mathsf{sf}}$ atoms for C

${}_{4}$N

${}_{4}$. This is opposite to the diamane that its partial N-substituted phase, CNCN, has an opening band gap, while the p

${}_{z}$-orbital of C

${}_{\mathsf{sf}}$ also dominates the valence states [

21]. Moreover, the lowest conduction state of C

${}_{4}$N

${}_{4}$ is on a

${\Gamma}$-X path, and its highest valence state state is at

${\Gamma}$ point. Thus, C

${}_{4}$N

${}_{4}$ has an indirect band gap, while C

${}_{8}$H

${}_{8}$ has a direct band gap.

It is worth noting that N-substituted W-diamanes have no magnetism, while the CNCN phase has a tiny magnetization [

21]. The magnetism occurs in 2D carbons when the 2D carbons are structurally distorted or defected, for example, hydrogenated graphene, 2D carbon nitrides that are porous structure, diamond surface with Pandey’s reconstruction [

24,

33,

34,

35,

36,

37]. The latter needs HSE06 for calculation in order to obtain magnetism [

24], while magnetism can be acquired using PBE for the former two [

33,

34]. However, the N-substituted W-diamanes investigated while using PBE and HSE06 are non-magnetism.

#### 3.3. Elastic Constants

The hardness is one of the distinguishing properties of diamond and carbon nitrides, where their 2D counterparts are expected to be adopted. However, the hardness of 2D materials, to the best of our knowledge, has not been well-defined. Therefore, the elastic constants that implicitly reflect the hardness are herein considered. Despite the fact that the elastic constants can be calculated by a second derivative of energy with respect to applied strain and devided by a volume of non-strained structure, the 2D materials, such as graphene, are lacking a third dimension. The elastic constants defined for three-dimensional (3D) materials are thus reduced to 2D elastic constants [

38,

39],

where

${A}_{0}$ is an unstrained area of C

${}_{1-x}$N

${}_{x}$, and

${\epsilon}_{i}$ is an applied strain up to

$\pm 2\%$ in order to fit the

${C}_{ij}^{2\mathrm{D}}$. The 2D elastic constants of W-diamanes are reported in

Table 2, by comparing with other 2D sp

${}^{3}$ carbons and carbon nitrides. However, Pakornchote et al. [

21] discussed that the 2D elastic constants cannot be used in order to compare across the 2D materials that have different thickness. Therefore, the 2D elastic constants have to be divided by

${d}_{0}$ of the 2D materials in order to make the values become intrinsic [

21,

40],

where

${d}_{0}$ is reported in

Table 1. The

${C}_{ij}^{2\mathrm{D}}$ and

${C}_{ij}$ are reported in

Table 2 and

Table 3, respectively.

As the number of N atoms in the unit cell of W-dimane increasing, the

${C}_{11}$ of N-substituted W-diamanes are lower, but their

${C}_{22}$ and

${C}_{33}$ are higher.

Figure 1 shows that, in the

x direction, the surficial atoms are bonding with the surficial atoms, which are at the same level, but, in the

y direction, the surficial atoms are bonding with the inner atoms, which are at the lower level. Thus, the N substitution in the surficial layers can enhance (reduce) the stiffness in the direction that the atoms are bonding in the different (same) level. Although, the Voigt bulk modulus (

${K}_{V}$) of C

${}_{6}$N

${}_{2}$, C

${}_{5}$N

${}_{3}$, and C

${}_{4}$N

${}_{4}$, which are 569, 553, and 533 GPa, respectively, are approximately the same if they are compared with the

${K}_{V}$ of NCCN and CNCN phases that are above 600 GPa. The result is in accordance with the simulation in Ref. [

10], showing that diamane has the tensile strength higher than other conformations. Noting that the

${K}_{V}$ is typically valid for polycrystals and might not be valid for 2D materials, so the values of

${K}_{V}$ reported in this work are used for the purpose of comparison.

For C${}_{8}$H${}_{4}$, its ${C}_{ij}$, except C${}_{33}$, are smaller than those of N-substituted W-diamanes. Therefore, ${K}_{V}$ of C${}_{8}$H${}_{4}$ is 310 GPa, which is much smaller that ${K}_{V}$ of C${}_{1-x}$N${}_{x}$. One might argue that ${d}_{0}$ of C${}_{8}$H${}_{4}$ is much larger than ${d}_{0}$ of C${}_{1-x}$N${}_{x}$, so ${C}_{ij}$ is unsurprisingly small. However, even if we consider ${C}_{ij}^{2\mathrm{D}}$, C${}_{8}$H${}_{4}$ has ${C}_{11}^{2\mathrm{D}}$, ${C}_{12}^{2\mathrm{D}}$, ${C}_{13}^{2\mathrm{D}}$, ${C}_{23}^{2\mathrm{D}}$, and ${C}_{66}^{2\mathrm{D}}$ smaller than C${}_{1-x}$N${}_{x}$. The result is in accordance with the result in the H-diamane, which is a fully hydrogenated diamane that its ${C}_{11}^{2\mathrm{D}}$ and ${C}_{ij}$ are smaller than that of NCCN and CNCN phases. Thus, it can be concluded that the hydrogenation seem to soften the 2D diamond, while the N substitution makes the 2D diamond stiffer.

Moreover, the elastic constants can be calculated from group velocities of phonons for cross-validation [

41]. In the orthorhombic system, the sound waves can be expressed as [

42]

where

$\mathbf{q}={q}_{x}\widehat{i}+{q}_{y}\widehat{j}+{q}_{z}\widehat{k}$ is a wave vector,

$\mathbf{u}={u}_{x}\widehat{i}+{u}_{y}\widehat{j}+{u}_{z}\widehat{k}$ is a phonon eigenvector,

$\omega \left(\mathbf{q}\right)=\left|\mathbf{q}\right|{v}_{\left[{q}_{x}{q}_{y}{q}_{z}\right]}$ is a phonon eigenvalue that is limited to acoustic modes, and

$\rho $ is a 3D density of W-diamanes. By solving Equations (

3)–(

5), each

$\mathbf{q}$ yields three values of sound velocities, where

$\left[q00\right]$ and

$\left[0q0\right]$ are herein considered,

The above two equations lead to a condition that

Superscripts of the velocities are just numbering, but have not yet been assigned to any phonon modes. The sound velocities can be computed from three acoustic phonons while using Phonopy package that takes the derivative on a dynamical matrix with respect to

**q** divided by

$2\omega \left(\mathbf{q}\right)$ at

$q=0.05$. Noting that only C

${}_{4}$N

${}_{4}$ and C

${}_{8}$H

${}_{4}$ are discussed, since the dispersion of acoustic phonons of C

${}_{6}$N

${}_{2}$ and C

${}_{5}$N

${}_{3}$ are dropping, causing the instability of the structures (see

Figure 2).

In

Table 4, the

${C}_{ij}$ were calculated from the group velocities of three acoustic phonons, ZA, TA, and LA modes while using Equations (

6) and (

7). If we assigned

${v}^{3}$ to be

${v}^{\left(TA\right)}$, this satisfies an unrestricted condition of Equation (

8) that

${v}_{\left[q00\right]}^{TA}\approx {v}_{\left[0q0\right]}^{TA}$. Therefore, from Equations (

6) and (

7),

${C}_{66}$ is either 725 or 716 GPa, which is similar to the 719 GPa of

${C}_{66}$ that was reported in

Table 3. Thus, if we assigned

${v}^{\left(1\right)}$ to be

${v}^{LA}$,

${C}_{11}=\rho {\left({v}_{\left[q00\right]}^{LA}\right)}^{2}=1666$ GPa and

${C}_{22}=\rho {\left({v}_{\left[0q0\right]}^{LA}\right)}^{2}=1097$ GPa. These two values are similar to

${C}_{11}$ and

${C}_{22}$ reported in

Table 3. By these assignments,

${C}_{ij}$ calculated using Equations (

2), (

6) and (

7) are in correspondence. Noting that, because of a convex dispersion of ZA mode around

${\Gamma}$-point,

${v}^{ZA}$ abruptly changes along

**q**, yielding inconsistent values of

${C}_{44}$ and

${C}_{55}$, so it needs to be further investigated in the future work.

#### 3.4. Formation Energy

An equation that is used to calculate a formation energy is

where

$E\left({\mathrm{C}}_{x}{\mathrm{N}}_{y}\right)$ is the energy of 2D carbon nitrides,

$E\left(\mathrm{C}\right)$ is an energy of graphene, and

$E\left(\mathrm{N}\right)$ is an energy of N

${}_{2}$ molecule. The formation energies of N-substituted W-diamanes plotted in circle symbols in

Figure 4 are 612, 626, and 630 meV for C

${}_{6}$N

${}_{2}$, C

${}_{5}$N

${}_{3}$, and C

${}_{4}$N

${}_{4}$, respectively. They are relatively high by comparing with the formation energy of synthesizable phases, triazine and polyheptazine [

43], which are 171 and 126 meV, respectively.

On the one hand, the NCCN phase has the lowest formation energy among 2D diamond-like carbon nitrides. Its formation energy is positive at 0 GPa, but the formation energy is on a convex hull at 10 GPa if only layered phases of carbon nitrides are considered. Hence, it is possible to be synthesized if the precursor is limited to be 2D materials [

21]. On the other hand, the N-subsituted W-diamanes have the formation energy as triple the NCCN phase (see

Figure 4), which is too high for synthesizing such materials. Nw starting materials other than graphene and N

${}_{2}$ molecule must be examined in order to find the possible pathway to synthesize W-diamane, but this is beyond the scope of the present work.