First-Principles Study on the Microheterostructures of N-GQDs@Si₃N₄ Composite Ceramics

Abstract

In the previous research that aimed to enhance the toughness and tribological properties of silicon nitride ceramics, a lignin precursor was added to the ceramic matrix, which achieved conversion through pyrolysis and sintering, resulting in a silicon nitride-based composite ceramic containing nitrogen-doped graphene quantum dots (N-GQDs). This composite material demonstrated excellent comprehensive mechanical properties and friction-wear performance. Based on the existing experimental results, the first-principles plane wave mode conservation pseudopotential method of density functional theory was adopted in this study to build a microscopic heterostructure model of Si₃N₄-based composite ceramics containing N-GQDs. Meanwhile, the surface energy of Si₃N₄ and the system energy of the N-GQDs@Si₃N₄ heterostructure were calculated. The calculation results showed that when the distance between N-GQDs and Si₃N₄ in the heterostructure was 2.3 Å, the structural energy was the smallest and the structure was the steadiest. This is consistent with the previous experimental results and further validates the coating mechanism of N-GQDs covering the Si₃N₄ column-shaped crystals. Simultaneously, based on the results of the previous experiments, the stress of the heterostructure composed of Si₃N₄ particles coated with different numbers of layers of nitrogen quantum dots was calculated to predict the optimal lignin doping amount. It was found that when the doping amount was between 1% and 2%, the best microstructure and mechanical properties were obtained. This paper provides a new method for studying the graphene quantum dot coating structure.

1. Introduction

As an excellent engineering ceramic material, Si₃N₄ ceramics are widely used in tribological components and the aerospace and automotive industries (e.g., in aviation bearings, tools, etc.) due to their unique thermodynamic, tribological and chemical properties [1]. In practical applications, various sizes of defects and cracks inevitably form in Si₃N₄ ceramic materials, showing the inherent brittleness of Si₃N₄ ceramic materials. On the other hand, the dry tribological properties of single-phase Si₃N₄ against metal pairs are not ideal [2,3,4]. Many researchers have proposed that graphene materials should be added to the silicon nitride matrix in order to enhance the toughness of the ceramic material and reduce friction [5,6]. However, the preparation process of pure graphene is usually very complicated, and graphene is prone to agglomeration in ceramic matrices. Therefore, directly adding graphene to enhance the mechanical properties and tribological properties of ceramic materials still poses some potential problems.

Scholars have found that lignin, as a well-dispersed precursor, can be converted into graphene materials such as graphene quantum dots (GQDs) in situ during the process of pyrolysis conversion [7,8,9]. GQDs are a subset of carbon quantum dots (CDs) derived from graphene or graphene oxide, with a diameter of 2~20 nm. As a new type of carbon nanomaterial, it has excellent properties such as low cytotoxicity, stable photoluminescence and excellent solubility. Some researchers have been focusing on the preparation of GQDs and their application as functional materials in recent years [10,11,12]. In addition, GQDs can be monodispersed well when used as a coating material for Ag. Ag@GQD nanoparticles have higher surface-enhanced Raman scattering (Sers) activity, and its peak intensity is 6–8 times that of Ag nanoparticles [13]. However, the effect of GQDs on the tribological properties and mechanical properties of matrix materials has not been reported by other scholars.

Recently, our research group [14,15,16,17] was surprised to find that, when nano-scale lignin was added to Si₃N₄ ceramics as a precursor, a core–shell structure of N-GQD-coated β-Si₃N₄ columnar crystals appeared after pyrolysis and hot-pressing sintering. The interlayer spacing of the N-GQDs was 2.2~2.3 Å, and the cross-sectional size was 8.99 nm². Meanwhile, the ceramic composite presented excellent mechanical and tribological properties. However, the research on the special core–shell structure of N-GQD-coated β-Si₃N₄ columnar crystals is still in the experiment stage using SEM, TEM and other characterization methods (core–shell phenomenon and core–shell effect) and has failed to explain the formation mechanism.

The formation of the coated structure is closely related to the surface properties of the Si₃N₄ grains. Therefore, in order to study the coating mechanism of N-GQDs to cover Si₃N₄, it is necessary to start with studying its surface properties. Due to the anisotropy of the material, different planes in the bulk material exhibit inconsistent surface characteristics, and the surface energy can be considered as a physical variable to explain these surface characteristics [18]. However, the experimental determination of surface energy is difficult [19], and the surface properties of Si₃N₄ have not been studied. Therefore, many calculation and simulation methods have been proposed to calculate the surface energy of anisotropic materials, including first-principles methods [20,21,22], tight-binding parameterizations [23,24] and semi-empirical methods [25,26,27]. Meanwhile, the first-principles method is considered to be an effective tool for studying the surface properties of materials by accurately calculating the surface energy [28,29,30,31].

Some scholars have studied the stability of the various surfaces of anisotropic materials using first-principles calculations of surface energy [19,32,33]. The lower the surface energy, the more stable the surface is and vice versa. For example, Bao et al. [19] systematically studied the structure and surface energy of Mg₂Pb (001), (110) and (111) surfaces using first-principles calculations. The calculated results showed that the surface energies of the Mg₂Pb (111)-Mg and Mg₂Pb (110)-MgPb surfaces were the lowest, indicating that the Mg₂Pb (111)-Mg and Mg₂Pb (110)-MgPb surfaces were the most stable. Sun et al. [32] studied the surface stability and equilibrium morphology of MoO₃ using first-principles calculations. The results showed that the surface energy of MoO₃ was (010) < (101) < (001) < (100). This is consistent with Tokarz-Sobieraj‘s result [33]: the (010) surface had the lowest surface energy, indicating that the (010) surface should have the lowest reactivity. The higher the surface energy, the smaller the equilibrium morphology area and the higher the chemical activity.

Based on the discussion above, the goal of this study was to select the appropriate crystal plane by calculating the surface energy of β-Si₃N₄, and then build the N-GQDs@β-Si₃N₄ heterogeneous interface structure to calculate the energy to verify the coating mechanism and calculate the stress of the heterostructure of Si₃N₄ grains coated with different numbers of layers of N-GQDs to optimize the lignin doping amount. The surface energies of β-Si₃N₄ along three low Miller index surfaces (100), (110) and (111) and three high-peak crystal surfaces (200), (210) and (101) were studied using first-principles calculations. The two active surfaces (111) and (200) of Si₃N₄ were obtained and used to build heterostructures with N-GQDs. By calculating the energy of N-GQDs@β-Si₃N₄ heterostructures with different spacings, it was verified that the cladding structure produced by the experiment had the most stable structure. Different numbers of layers of N-GQDs were added to the (200) crystal plane of β-Si₃N₄ to obtain heterostructures. The mechanical properties were predicted by calculating the stress of the system, and then the lignin doping amount of was optimized. This study provides theoretical guidance for further study of the surface coating structure of GQDs.

2. Calculation Model and Calculation Method for Core–Shell Structure

2.1. Core–Shell Structure Theoretical Model Construction

It is well known that there are three common crystal structures of Si₃N₄: the α, β and γ phases. The β phase belongs to the hexagonal system, and the space group of β-Si₃N₄ is P6₃/m [34]. The lattice constants are a = 7.66 Å, b = 7.66 Å and c = 2.93 Å. The spatial position of each atom in the unit cell is shown in

Table 1. Atomic coordinates of β-Si₃N₄ crystal.

Wyckoff	Element	x	y	z
2d	N	2/3	1/3	1/4
6h	Si	0.82514	0.59389	1/4
6h	N	0.67006	0.70074	1/4

. According to the experimental results, the Si₃N₄ matrix of the composite material is basically β phase, so based on the above data, this study built the crystal model of β-Si₃N₄ as shown in Figure 1a (blue atoms represent nitrogen and yellow atoms represent silicon).

The lattice constants of graphite are a = 2.46 Å, b = 2.46 Å, c = 6.8 Å, α = β = 90 ° and γ = 120 °. In order to match the actual results, the N-GQD model adopts the approximate structure of graphene doped with nitrogen atoms. The graphite structure shown in Figure 1b is cut by 0.5 layers along the (001) direction, and the obtained structure is graphene.

2.2. Structure Optimization and Surface Energy Calculation Method

The calculation first builds a β-Si₃N₄ crystal model and optimizes the structure, and then calculates the surface energy of β-Si₃N₄ along three low Miller index surfaces ((100), (110) and (111)) and three high-peak crystal surfaces ((200), (210) and (101)) to find the most active surface. Finally, the heterostructure model was built with N-GQDs, and the formation mechanism of the core–shell structure was revealed by calculating the energy of the N-GQDs@β-Si₃N₄ heterostructures with different spacings.

2.2.1. Structural Optimization Method

The reason why the crystal can maintain a stable shape and volume is that there are two types of attraction and repulsion between atoms. In order to construct the surface model of different crystal planes, the structure of Si₃N₄ was optimized first so that the atomic spacing reaches the equilibrium distance, reduces the interaction between atoms, and maintains a stable state. We chose the structure with the smallest volume change rate. At this time, the overall spatial structure and the relative position between atoms did not change much during the optimization process, which ensured the physical stability of the structure. In addition, the structure with a small volume change rate was often in a relatively low energy and stable state. The volume change rate of structural optimization was defined as follows:

$$\mathrm{Deviation} (\%) ={\displaystyle \frac{{\mathrm{V}}_{\mathrm{Final}}-{ \mathrm{V}}_{\mathrm{Initial}}}{{\mathrm{V}}_{\mathrm{Initial}}}}$$

(1)

In the formula, Deviation represents the volume deviation, ${\mathrm{V}}_{\mathrm{I}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}$ represents the volume before structural optimization, and ${\mathrm{V}}_{\mathrm{F}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}$ represents the volume after structural optimization.

In this paper, the calculation was carried out using CASTEP [35] based on the density functional theory (DFT). The exchange–correlation functional was represented by the Perdew–Burke–Ernzerhof (PBE) functional [36] of the generalized gradient approximation (GGA). The valence electrons involved in the calculation for β-Si₃N₄ were 3s²3p² of Si and 2s²2p³ of N. The LBFGS algorithm [37] was used to optimize the structure of the unit cell model, and the pseudopotential was based on the model-conserving pseudopotential. In order to achieve sufficient convergence accuracy, the self-consistent cyclic convergence accuracy was 5.0 × 10⁻⁷ eV/atom, the stress error between atoms was less than 0.02 GPa, the maximum displacement convergence error between atoms was less than 5.0 × 10⁻⁴ Å, and the convergence criterion of total energy was less than 5 × 10⁻⁶ eV/atom.

2.2.2. Surface Energy Calculation Method

Surface energy is an important parameter for studying surface properties. It refers to the energy required per unit area when the crystal is dissociated into two semi-infinite crystals under the action of an external force, which can reflect the surface stability [38]. Due to the anisotropy of the Si₃N₄ material lattice, different surface characteristics would be generated on different surfaces. The anisotropy makes the surface energy of different crystal planes different. The crystal surface with a high surface energy is in a high energy state. In order to reduce the energy, it can react with the external material to form a more stable surface structure. The crystal surface with a low surface energy is relatively stable and the reactivity is low. Since the coating process would preferentially occur on the active surface of the Si₃N₄ crystal and based on the actual test results, three low Miller index surfaces ((100), (110) and (111)) and three high-peak crystal surfaces ((200), (210) and (101)) were selected in this study. The active surface of the Si₃N₄ crystal was found by using surface energy calculations as the basis for the subsequent heterostructure construction. In order to describe the stability of the surface based on the structural optimization and according to the characteristics of different crystal planes of β-Si₃N₄, six crystal cells with different crystal planes were constructed to study the surface energy of the (100), (110), (111), (200), (210) and (101) surfaces.

When the crystal faces in these cells were in contact with other atoms, considering the short-range interaction between the atoms and the surface, we only considered five adjacent atomic layers near the (100), (110), (111), (200), (210) and (101) surfaces. When the atomic layer was equal to or greater than the fifth layer, the interaction energy between the atom and the surface was considered to be approximately zero due to the long-range effect. A 15 Å vacuum layer was applied to separate the model in the Z direction to prevent unnecessary interactions along the model surface.

In order to calculate the surface energy of each crystal face, the variable E_surf was defined as the surface energy of Si₃N₄, which was used to describe the minimum energy in the process of forming two free surfaces. It can be expressed by the following formula:

E_surface = [E_slab − nE_bulk]/2A

(2)

where E_surface represents the surface energy of the unit cell, E_salb is the total energy of the lamellar crystal model, E_bulk is the total energy of the bulk crystal, n is the number of bulk crystals contained in the surface model, and A is the surface area of the surface model.

3. Results and Analysis

3.1. Energy Calculation of β-Si₃N₄ Low-Index Crystal Face Heterostructure

3.1.1. Structural Optimization

The convergence test of the plane wave cutoff energy and k-point density of the β-Si₃N₄ crystal was carried out to complete the structural optimization of β-Si₃N₄. Firstly, the convergence test of plane wave truncation energy was carried out with a Brillouin zone sampling of 2 × 2 × 6. It was found that when the cutoff energy of the plane wave substrate was 780 eV, the structure had the smallest volume change rate of 0.63663%. The specific data are shown in

Table 2. Volume deviation of different plane wave cutoff energies of β-Si₃N₄ crystal.

Cutoff Energy (eV)	760	770	780	790
Deviation (%)	0.64342	0.63978	0.63663	0.63691

. Subsequently, the cut-off energy of the plane wave substrate was set to 780 eV, and the convergence test of the k-point density was performed. It was found that when the Brillouin zone sampling was 2 × 2 × 5, the structure had the smallest volume change rate of 0.63357%. The specific data are shown in

Table 3. Volume deviation of different k-point densities of β-Si₃N₄ crystal.

K-Point Density	2 × 2 × 4	2 × 2 × 5	2 × 2 × 6	3 × 3 × 6
Deviation (%)	0.63406	0.63357	0.63663	0.63927

.

In summary, when the cutoff energy of the plane wave substrate was 780 eV and the Brillouin zone sampling was 2 × 2 × 5, the structure had the smallest volume change rate, which was closest to the experimental value. Thus, the structural optimization of the β-Si₃N₄ crystal model was completed and the most stable structure was obtained. It has a plane wave energy close to the cutoff of 770 eV and the k-point density of 2 × 2 × 6 adopted by Jiang et al. [34].

In the theoretical study, the surface energy is the energy consumed in order to destroy the chemical bonds between molecules on the surface of the material. Therefore, the surface energy is an index to measure the surface stability. The smaller the surface energy, the better the surface stability. Because the surface formed by the core–shell structure must be the most active surface of the β-Si₃N₄ crystal, the surface with the largest surface energy was obtained by calculating and comparing the surface energy of different crystal planes of the β-Si₃N₄ crystal to determine the most active surface of the β-Si₃N₄ crystal. Combined with the results of the actual test (shown in Figure 2), three low Miller index surfaces (100), (110) and (111) were selected in this study. The active surface of the β-Si₃N₄ crystal was found by calculating the surface energy of the above surfaces.

Due to the long-range effect, when the number of atomic layers was equal to or greater than five layers, the interaction energy between the internal atoms and the surface was considered to be approximately zero so the maximum thickness of this study was only calculated to five adjacent atomic layers near the surface. The unit cell model of the different crystal planes of the four-layer β-Si₃N₄ crystal is shown in Figure 3.

3.1.2. Surface Energy Calculation of Low Miller Index Surfaces

In order to select the crystal plane with the highest surface energy, the surface energy convergence test with one to five layers was carried out on the (100), (110) and (111) surfaces of the β-Si₃N₄ crystal. The surface energy data of the (100), (110) and (111) surfaces are shown in

Table 4. Surface energy of (100), (110) and (111) β-Si₃N₄ surfaces with different thicknesses.

	N (Layer Number)	S1	S2	S3	S4	S5
ESurf (J/M2)	(100)	2.218	1.646	0.923	0.274	−0.343
	(110)	2.357	2.570	2.089	1.718	1.347
	(111)	2.179	1.543	3.241	3.229	3.162

.

The results show that the surface energy of β-Si₃N₄ was (100) < (110) < (111). The surface energy of the (111) surface was the highest, which indicates that the (111) surface is more active, the surface interaction is stronger, and it is easier to bind to graphene quantum dots. It can be seen from Table 4 that the surface energy of the (111) surface reached convergence in four layers, so the four-layer structure of the (111) surface was selected for the construction of the heterogeneous interface structure. In particular, when the number of layers of the β-Si3N4 (100) crystal plane was five, the surface energy was negative. Our analysis suggests that the core reason for the negative surface energy of the (100) crystal plane is that the model thickness does not meet the convergence requirements, resulting in a deviation between the volumetric energy simulation and the actual situation. The strong relaxation effect and low intrinsic surface energy characteristics that may exist in the β-Si3N4 (100) crystal plane itself eventually lead to the calculated value deviating from the physical reality. Negative values are not “negative surface energy” in the physical sense, but rather a manifestation of the limitations of the model.

3.1.3. Surface Relaxation of Low Miller Index Surfaces

During the surface cutting process, the symmetry of the surface layer is destroyed relative to the bulk interior, and the CN (coordination number) of the surface atoms is reduced, so the surface atoms will move to a new position [19]. Surface relaxation occurs because the force on the surface atom is different from the bulk structure so the surface atom layer moves relative to the inner atom to reduce the energy of the system. The relaxation degree of the crystal surface plays an important role in the properties of the surface [38]. Therefore, it is necessary to perform surface relaxation on the surface structure.

The surface relaxation was performed on the four-layer cell of the (111) crystal plane of β-Si₃N₄, the surface relaxation was performed on the surface of the five-layer atoms of the cell, and the remaining part was fixed as a bulk structure.

The convergence test of the plane wave cutoff energy and k-point density was carried out on the unit cell with four layers on the (111) surface of the β-Si₃N₄ crystal to complete the surface relaxation of the unit cell model with four layers on the (111) surface of the β-Si₃N₄ crystal. Firstly, using a Brillouin zone sampling of 2 × 2 × 5, the convergence test of plane wave truncation energy was carried out. It was found that when the cutoff energy of the plane wave substrate was 930 eV, the structure had the smallest volume change rate of 0.01520%. The specific data are shown in

Table 5. Volume deviation of different plane wave cutoff energies of 4-layer β-Si₃N₄ crystal ((111) crystal plane).

Cutoff Energy (eV)	830	880	930	980
Deviation (%)	0.02445	0.02607	0.01520	0.01521

.

Subsequently, the cutoff energy of the plane wave substrate was set to 930 eV, and the convergence test of the k-point density was performed. It was found that when the Brillouin zone sampling was 2 × 2 × 5, the structure had the smallest volume change rate of 0.01520%. The specific data are shown in

Table 6. Volume deviation of different k-point densities of 4-layer β-Si₃N₄ crystal ((111) crystal plane).

K-Points	3 × 3 × 1	2 × 2 × 5	6 × 6 × 2
Deviation (%)	0.02566	0.01520	0.01551

.

Combined with the above calculation results, when the cutoff energy was 930 eV and the k-point density was 2 × 2 × 5, the unit cell structure of the (111) surface of the β-Si₃N₄ crystal with four layers had the smallest volume change rate and the system energy converged well.

3.1.4. Energy Calculation of Heterostructures

In terms of crystal structure, N-GQDs are graphene fragments with a size of less than 100 nm and have obvious graphene lattice properties. For the modeling of N-GQDs@β-Si₃N₄ heterostructures, the four-layer structure model of the (111) plane of the β-Si₃N₄ crystal and the N-GQD structure model were combined in this study.

In this study, the optimized lattice constant of graphene was a = b = 2.46 Å, and the optimized lattice constant of the β-Si₃N₄ crystal (111) crystal plane was a =b = 8.22 Å. The modeling method of the N-GQDs@β-Si₃N₄ heterostructure is as follows. The 3 × 3 supercells on the (111) surface of the β-Si₃N₄ crystal are used as the bottom layer, and the GQDs form 10 × 10 supercells. Following the method of a related study [39], three C atoms are substituted with N atoms on each 5 × 5 supercell model to construct the N-GQD model (as shown in Figure 4a). Finally, the double-layer N-GQDs are matched with the (111) surface of the β-Si₃N₄ crystal to form the N-GQDs@β-Si₃N₄ heterostructure (as shown in Figure 4b). The interlayer spacing between N-GQDs was 2.3 Å.

During the formation of the heterostructures, the N-GQDs were subjected to 0.077% and 0.008% strain on the a and b lattice parameters when adapting to the lattice of the heterostructures, and 0.150% and 0.016% strain was exerted on the (111) surface of the β-Si₃N₄ crystal. The model is periodic in two dimensions (a and b), and the third dimension is associated with the thickness of the system. When the double-layer N-GQDs were matched with the (111) plane of the β-Si₃N₄ crystal, the resulting lattice mismatch was tolerable for the system. The optimized lattice parameters of the N-GQDs (001) and β-Si₃N₄ (111) surfaces are reported in

Table 7. The calculated structural properties of the N-GQDs (001) and β-Si₃N₄ (111) parts compared with the calculated structural properties of the N-GQDs@β-Si₃N₄ model. We report the cell vector (a, b), angle (γ) and surface area (S).

System	a (Å)	b (Å)	γ (°)	S (Å2)
N-GQDs (001)	24.600	24.600	120.0	492.710
β-Si3N4 (111)	24.656	24.594	107.2	561.807
N-GQDs@β-Si3N4	24.619	24.598	115.7	519.743

along with the lattice parameters of the heterojunctions.

In this study, the energy of different distances between the N-GQDs and β-Si₃N₄ heterostructure interface was calculated as the starting point. The distances between the N-GQDs and β-Si₃N₄ were 1.0, 1.5, 2.3, 3.0, 4.0 and 10.0 Å. The energy calculation results for the heterostructures with different spacings between the N-GQDs and β-Si₃N₄ are shown in

Table 8. Energy of N-GQDs@β-Si₃N₄ (111) heterostructures with different spacings.

Spacing (Å)	1.0	1.5	2.3	3.0	4.0	10.0
E (eV)	−165,086.85	−165,086.92	−165,087.23	−165,086.43	−165,086.93	−165,085.86

.

The results show that, as expected, the calculated data from the heterostructure are consistent with the experimental data. Compared with other spacings, when the spacing between the N-GQDs and β-Si₃N₄ was 2.3 Å, the minimum energy of the system was −165,087.23 eV, indicating that this heterostructure system was the most stable at this time. Further research shows that the energy of the N-GQDs@β-Si₃N₄ heterostructure basically showed an increasing trend as the distance between the N-GQDs and β-Si₃N₄ increased, which means that the microstructure of the N-GQDs and β-Si₃N₄ dispersed with each other was unstable. Therefore, in the pyrolysis–hot pressing sintering coupling process, from an energy point of view, in order to obtain a stable crystal structure, the composite material is bound to spontaneously form a core–shell structure.

3.2. Energy Calculation of β-Si₃N₄ High Peak Crystal Plane Heterostructure

In order to further study the actual crystal plane of β-Si₃N₄ combined with N-GQDs, the high peak surface energy of the crystal plane was calculated. The peak values of the β-Si₃N₄ (200), (210) and (101) crystal planes were the highest (as shown in Figure 2), indicating that they were the most unstable crystal planes. Among them, the surface energy of the crystal plane with a high crystal plane index was high, and its particle density was small, the spacing was large, and it was unstable, making it easier to combine with N-GQDs. Therefore, the surface energy of the three β-Si₃N₄ crystal planes (200), (210) and (101) was calculated, and the heterostructure was further built to calculate the energy of the system

3.2.1. Surface Energy Calculation of High-peak Crystal Surfaces

The unit cell models of the β-Si3N4 (101), (200) and (210) crystal planes are shown in Figure 5. The surface energies of the β-Si3N4 (101), (200) and (210) crystal planes with different thicknesses are shown in

Table 9. Surface energy of (101), (200) and (210) crystal planes of β-Si₃N₄ crystal with different thicknesses.

	N (Layer Number)	S1	S2	S3	S4	S5
ESurf (J/M2)	(101)	0.1985	0.326	0.329	0.327	0.326
	(200)	31.342	62.588	93.798	125.013	156.224
	(210)	0.058	0.137	0.159	0.126	0.110

.

The results show that the surface energy of β-Si3N4 was (210) < (101) < (200). The surface energy of the (200) surface was the highest, which indicates that the (200) surface is more active, the surface interaction is stronger, and it is easier for it to bind to N-GQDs.

A sufficiently thick Slab supercell has very similar internal properties and bulk material properties so the surface energy will tend to be stable with an increased Slab thickness. If the accuracy of E_slab and E_bulk is exactly the same, Equation (2) can be used to accurately calculate the surface energy. However, there are some inevitable differences in the calculation (e.g., the k-point grid cannot be exactly the same), and the accuracy of the bulk and surface calculations cannot be completely consistent, which will cause the surface energy to deviate linearly with increasing n. It can be seen from Table 9 that the surface energy of the (200) surface tended to diverge as the number of layers increased.

In order to calculate the surface energy more accurately, the surface energy of the (200) surface was calculated using the following method [40]: The bulk energy E_bulk is not calculated using a separate bulk phase and the energy E_bulk (n) corresponding to the gradually increasing number of layers Slab is calculated first, and the relationship between E_bulk (n) and the number of layers n is fitted. When n is large enough, the bulk energy E_bulk is obtained using the fitted linear slope, which avoids the error caused by inconsistent parameters. For the (200) surface, by calculating the surface energy corresponding to the Slab supercell with 1–5 atomic layers, the change in the surface energy with the thickness of Slab is calculated. The fluctuation in the surface energy was also very small and it converged easily (as shown in

Table 10. The change in surface energy of the β-Si₃N₄ (200) crystal plane with the number of layers.

Tier Number (n)	nEbulk (eV)	Eslab (eV)	A (Å2)	Esurf (J/M2)
1	−1404.948	−1402.172	22.527	0.062
2	−2809.896	−2808.617	22.527	0.028
3	−4214.844	−4216.712	22.527	−0.041
4	−5619.792	−5624.620	22.527	−0.107
5	−7024.739	−7032.667	22.527	−0.176

). Therefore, the β-Si3N4 (200) crystal plane with four atomic layers was selected for subsequent heterostructure construction and calculation.

3.2.2. Surface Relaxation of High-peak Crystal Surfaces

The plane wave truncation energy and the convergence of the k-point density of the unit cell with four layers of thickness on the (200) surface of the β-Si3N4 crystal were tested, and the surface relaxation of the unit cell model with four layers on the (200) surface of the β-Si3N4 crystal was completed. Firstly, the Brillouin zone sampling was set to 2 × 2 × 5 and the convergence test of the plane wave truncation energy was carried out. It was found that when the cutoff energy of the plane wave substrate was 830 eV, the structure had the smallest volume change rate of 0.61589%. The specific data are shown in

Table 11. Volume deviations of different plane wave cutoff energies of 4-layer β-Si₃N₄ crystal ((200) crystal plane).

Cutoff Energy (eV)	730	780	830	880
Deviation (%)	0.64801	0.62742	0.61589	0.61641

.

Subsequently, the cutoff energy of the plane wave substrate was set to 830 eV, and the convergence test of the k-point density was performed. It was found that when the Brillouin zone sampling was 2 × 2 × 5, the structure had the smallest volume change rate of 0.61589%. The specific data are shown in

Table 12. Volume deviations of different k-point densities of 4-layer β-Si₃N₄ crystal ((200) crystal plane).

K-Points	4 × 10 × 1	2 × 2 × 5	7 × 19 × 2
Deviation (%)	0.71097	0.61589	0.70273

.

Combined with the above calculation results, when the cutoff energy was 830 eV and the k-point density was 2 × 2 × 5, the unit cell structure of the (200) surface of the β-Si3N4 crystal with four layers had the smallest volume change rate, and the system energy converged well.

3.2.3. Energy Calculation of Heterostructures

According to the modeling method for the N-GQDs and β-Si3N4 heterostructures, the β-Si3N4 (111) surface carries a 1 × 3 supercell as the bottom layer, and the GQD (001) surface carries a 3 × 2 supercell. One C atom on each 3 × 2 supercell model is substituted with a N atom, and the doping ratio is 4.17%. As shown in Figure 6a, 3 × 2 double-layer N-GQDs were selected to match the surface of β-Si3N4, resulting in a lattice mismatch that is tolerable for the system. The optimized lattice parameters of the N-GQDs (001) and β-Si3N4 (200) surfaces are reported in

Table 13. The calculated structural properties of the N-GQDs (001) and β-Si₃N₄ (200) surface compared with the calculated structural properties of the N-GQDs@β-Si₃N₄ model. We report the cell vector (a), (b), angle (γ) and surface area (S).

System	a (Å)	b (Å)	γ (°)	S (Å2)
N-GQDs (001)	7.379	8.520	90	62.865
β-Si3N4 (200)	7.799	8.652	89.9997	67.478
N-GQDs@β-Si3N4	7.519	8.564	89.9999	64.390

and compared with the lattice parameters of the heterojunctions. In the formation of the heterostructures, the N-GQDs (001) were subjected to 1.9% and 0.52% strain on the a and b lattice parameters when adapting to the lattice of the heterostructures, and the β-Si3N4 (200) surface was subjected to 3.6% and 1% strain in turn.

Figure 6 b,c show heterostructure models composed of two layers of N-GQDs and β-Si3N4 (200) crystal planes. The interlayer spacing between the N-GQDs is 2.3 Å, and the spacings between the N-GQDs and β-Si3N4 are 2.3 and 10 Å, respectively. When the distance between the N-GQDs and β-Si3N4 is 2.3 Å, it represents the experimental coating structure, and when the distance is 10 Å, it is not coated. The energy calculation results for the N-GQDs and β-Si3N4 heterostructures with different spacings are shown in

Table 14. Energy of N-GQDs@β-Si₃N₄ (200) crystal plane heterostructures with different spacings.

Spacing (Å)	2.3	10
E (eV)	−24,433.42	−24,433.29

.

The results show that, as expected, the calculated data from the heterostructure are consistent with the experimental phenomenon data. Compared with the distance of 10 Å, when the distance between the N-GQDs and β-Si3N4 was 2.3 Å, the system energy was smaller than −24,433.42 eV, indicating that the heterostructure system was more stable at this time.

In short, Si3N4 has a high surface energy, and the surface is in an unstable state with high energy. When N-GQDs are in contact with Si3N4, N-GQDs will cover the surface of Si3N4, reducing the contact area between the surface of Si3N4 and the external environment, thereby reducing the energy of the whole system and making the system reach a more stable state.

3.2.4. Stress Calculation of Graphene and β-Si₃N₄ Heterostructure with Different Numbers of Layers

Stress describes the internal forces per unit area generated within a crystalline material due to external forces (e.g., lattice distortion) or internal states (such as thermal vibration or defects). Through the calculation of stress, the changes in the macroscopic mechanical properties of the material can be reflected to a certain extent [41]. The changes in the mechanical properties of pure Si₃N₄ with a (200) crystal plane and graphene with different numbers of layers are shown in

Table 15. The stress of heterostructures of N-GQDs with different numbers of layers and β-Si₃N₄ (200) crystal plane.

GQD Content (Number of Layers)	0	1	2	3	4	5	7
Stress (GPa)	−0.012 2	−0.227	0.367 1	0.059	0.006	−0.034	0.027

¹ Positive values indicate compressive stress. ² Negative values indicate compressive stress.

. The crystal structure model of the pure Si₃N₄ (200) crystal plane and Si₃N₄ with different numbers of layers of graphene is shown in Figure 7.

It can be seen from Table 15 that the maximum stress was observed with two layers and the minimum was observed with four layers. The smaller the stress, the lower the hardness, the greater the toughness, the greater the bending strength, and the greater the density while the bulk density is constant. The change in the apparent porosity depends on the properties of the material and the specific application conditions. Therefore, when the number of layers of N-GQDs was two, the hardness of the composite was the highest. When the number of layers was four, the toughness, bending strength and density of the composite were the highest.

The experimental results show that the lignin doping amount was proportional to the number of layers of N-GQDs. When the nano-lignin doping amount was 2 wt.%, the best comprehensive performance was obtained. At this time, the number of layers of N-GQDs was 6–8 layers. Figure 7 shows the crystal structure model of Si₃N₄-based composites with different numbers of layers of graphene. It is predicted that in order to improve the mechanical properties of the composites, the doping amount should be between 1 wt.% and 2 wt.%.

4. Analysis of Theoretical Calculation Results and Experimental Results

In previous studies [14,15,16,17] using transmission electron microscopy, the composite material was shown to be composed of N-GQDs coated on the surface of β-Si₃N₄ columnar crystals. The size of the N-GQDs was 8.99 nm², the number of layers was 6~8, and the interlayer spacing was 2.2~2.3 Å (as shown in Figure 8b).

Previous research [42] showed that the lower the energy of the system, the more stable the system is. Therefore, in order to make the structure of the system more stable, the material will spontaneously form a microstructure with the lowest energy. Therefore, the stability can be verified by calculating the structural energy, and the formation mechanism of the core–shell structure can be further revealed. In this chapter, the energy of N-GQDs@β-Si₃N₄ heterostructure is calculated. It was found that the system had the smallest energy and the most stable microstructure when the distance between the two was 2.3 Å (as shown in Figure 4b and Figure 6b).

The theoretical calculation results show that the N-GQDs@β-Si₃N₄ heterostructure had the smallest overall energy and the most stable microstructure when the spacing was 2.3 Å. The actual test results show that the microstructure of the composite material formed a core–shell structure with a layer spacing of 2.2~2.3 Å after the pyrolysis–hot pressing sintering coupling process. The theoretical calculation results are consistent with the actual test results. Therefore, the formation mechanism of the core–shell structure can be obtained. In order to obtain a more stable microstructure, during hot-pressing sintering, N-GQDs spontaneously coat the surface of β-Si₃N₄ columnar crystals, thus forming a lower energy and more stable core–shell structure.

Previous experimental results [17] showed that the core–shell structure formed is also different when different amounts of nano-lignin are doped. When the doping amount is too much, the thickness of the N-GQD aggregation also increases. From the calculation results, it can be seen that when the number of N-GQDs layers is too high, the adsorption capacity of the active surface of the β-Si₃N₄ columnar crystal for external N-GQDs is reduced so it cannot be subjected to uniform stress, resulting in an uneven coating.

5. Conclusions

In this study, a β-Si₃N₄ crystal model was built and the structure was optimized. Then, the surface energy of the low-plane index crystal plane and the high-peak crystal plane of the β-Si₃N₄ crystal was calculated and studied. The (111) crystal plane in the low-plane index crystal plane and the (200) crystal plane in the high-peak crystal plane were determined to be the most active surfaces. Finally, the heterostructure model was built by combining the crystal model with N-GQDs. The formation mechanism of the core–shell structure was revealed by calculating the energy of the heterostructures with different spacings of N-GQDs@β-Si₃N₄. The results showed the following:

(1)

Si₃N₄ had a high surface energy, and surface was in an unstable state with high energy. When N-GQDs were in contact with Si₃N₄, the N-GQDs would cover the surface of Si₃N₄, reducing the contact area between the surface of Si₃N₄ and the external environment, thereby reducing the energy of the whole system and making the system reach a more stable state.

(2)

Comparing the energy of N-GQDs@β-Si₃N₄ heterostructures with different spacings, the lowest system energy was obtained and the heterostructure system was the most stable when the spacing between the N-GQDs and β-Si₃N₄ was 2.3 Å. This spacing was consistent with the previous experimental data. As the distance between the N-GQDs and β-Si₃N₄ increased, the energy of the N-GQDs@β-Si₃N₄ heterostructure basically increased, indicating that the microstructure of the N-GQDs and β-Si₃N₄ dispersed with each other was unstable. Because the material always spontaneously changed into a stable structure, the N-GQDs would spontaneously coat on the surface of the β-Si₃N₄ crystal during the pyrolysis–hot pressing sintering coupling process so that the system had the minimum energy, and the composite material also formed the most stable microstructure-core–shell structure.

(3)

N-GQDs formed a core–shell structure by adsorbing onto the surface of the β-Si₃N₄ crystal, and the content of nano-lignin had a certain influence on the performance of the core–shell structure. It was predicted that the best mechanical properties of the composites would be obtained with a lignin doping amount between 1 wt.% and 2 wt.%.