Exploration of the mechanical properties of carbon-incorporated amorphous silica using a universal neural network potential

The progressive miniaturization and integration with stack cells in semiconductor flash memory increase the internal stress in the interlayer dielectric film, resulting in deformation and breakdown. As such, enhancing the mechanical strength of dielectric films is crucial to further realizing their integration for various applications. We particularly focus on amorphous SiO₂ (a-SiO₂) interlayer dielectric films.

C-incorporated amorphous SiO₂ (a-SiOC) containing a significant number of C atoms is particularly referred to as amorphous silicon oxycarbide and has garnered attention as a promising ceramic material.^1–9 In particular, an a-SiOC glass produced by sintering has a higher density and elastic modulus than silica glass.^10,11 Ming et al.² experimentally demonstrated the increase in the elastic modulus of SiOC films with an increase in the fraction of C atoms (F_c). The SiO₂ oxidation film on SiC also exhibited a higher elastic modulus than that on Si.¹² Although the incorporation of C atoms can strengthen SiO₂ films, it can reduce their mechanical strength. SiOCH films,^13–15 which have lower dielectric properties than a-SiO₂ because of the density reduction associated with the presence of hydrocarbons, are used as insulating films in semiconductor devices. However, these materials have low elastic moduli¹⁴ owing to the disruption of the SiO₂ network by the hydrocarbons that form methyl groups.^15–17 As such, incorporating C atoms into SiO₂ can increase or decrease the elastic modulus depending on the deposition process and F_c. Therefore, the relationship between the structure and mechanical strength of a-SiOC films at the atomic scale should be investigated.

Numerical calculations have focused on the atomic structure of a-SiOC.^4,5,18–21 Kroll et al.⁴ fabricated an a-SiOC model by replacing one SiO₄ tetrahedron in a-SiO₂ with a single C atom using the bond switching algorithm by Wooten, Winer, and Waire (WWW method),²² which assumed that the SiO₄ tetrahedron could be replaced by one C atom, that is, the C atom has a four-coordinated structure.²³ They demonstrated the increase in the elastic modulus of the SiOC film as F_c was increased. Ding and Demkowicz²⁴ fabricated an a-SiOC model by replacing the O atom in a-SiO₂ with a single C atom. In this method, the C atom in a-SiOC is assumed to have a two-coordinated structure that bridges two Si atoms. These methods are based on the experimental results, indicating the absence of C–O bonds in a-SiOC.^1,25 As such, they rely heavily on assumptions on the C-atom structure in a-SiOC. As confirmed by the presence of graphene, C nanotubes, and fullerenes, C atoms can have a stable three-coordination structure. Thus, the C atoms in a-SiOC do not necessarily have two- or four-coordinated structures. Matsushita et al.²¹ investigated the formation energies of possible C-related structures in a-SiOC using Density Functional Theory (DFT) calculations, revealing the variations in the stable C-containing structures with the number of O atoms. Therefore, an a-SiOC model should be fabricated without assumptions on the C-atom structure.

Melt quenching is a widely used method for fabricating amorphous structures.^21,26–30 This approach is suitable for fabricating an a-SiOC model without predetermined artificial assumptions on the structure formed by the C atoms. Liao et al.²⁰ and Haseen and Kroll⁵ fabricated a-SiOC models using a melt-quenching method in conjunction with classical molecular dynamics. They investigated the structure and mechanical properties of the precipitates composed of several C atoms, called C clusters or free C, which can be observed with excess C atoms. A combination of the Tersoff potential parameters developed independently for Si,³¹ C,³² SiC,³³ and SiO³⁴ was used in the calculations. However, as these interatomic potential parameter sets have been developed independently, their application to structures with the coexistence of Si, O, and C is uncertain. Furthermore, these studies ignored the role of C–O bonds, which were not experimentally detected in a-SiOC ceramics.^1,25 Although more sophisticated interatomic potentials for SiOC systems have been proposed for the thermal oxidation of SiC,^35,36 their application to the elastic properties of SiOC systems across a wide range of compositions has not been validated.

Ab initio calculations, particularly DFT calculations, can significantly improve the accuracy of structural fabrication. However, a sufficient number of structures should be fabricated for statistical analyses because of the high scattering of amorphous disordered structures. Thus, DFT calculations are unsuitable for such applications because of their high computational costs. Meanwhile, neural network potentials can express a potential energy surface with high accuracy using deep neural networks and a large number of parameters. Takamoto et al. recently proposed a universal graph neural network potential [PreFerred Potential (PFP)³⁷] based on the Tensor-embedded atom Network (TeaNet) architecture,³⁸ which was proposed in collaboration with one of the authors of this study. Graph neural networks are one of the anticipated approaches for highly accurate predictions of material properties, including their use as an interatomic potential.³⁹ Sufficient training with large datasets, including molecules, bulk, cluster, slab, adsorption, and disordered structures, facilitate the arbitrary composition of 45 elements by PFP. Thus, PFP can achieve molecular dynamics calculations with comparable accuracy to that of the DFT calculation with a lower computational cost. Mine et al.⁴⁰ validated its accuracy for bulk formation, surface, and surface O-vacancy formation energies of metal hydrides, carbides, nitrides, oxides, and sulfides in comparison with DFT calculations. Moreover, subsequent research has demonstrated the applicability of PFP to various phenomena.^41–43 We utilize PFP to fabricate a large number of a-SiOC structures.

This study aims to establish the overall relationship between the atomic structure and elastic modulus of a-SiOC to provide a theoretical basis for the potential strengthening of SiO₂ dielectric films through C incorporation. We focused on the atom-network formation and clustering behaviors of C atoms under different O conditions and F_c values. We conducted a comprehensive investigation with different F_c values using molecular dynamics simulations. In particular, we applied PFP, a universal neural network potential, to address the existing problems associated with the accuracy of classical empirical interatomic potentials and the computational cost of DFT calculation. We validated the accuracy of PFP for a-SiOC systems using DFT calculations. The effect of F_c variations in a-SiOC on the elastic modulus was discussed from the viewpoint of the atomic network structure. The remainder of this paper is organized as follows. The fabrication scheme with different atomic compositions is discussed in Sec. II. The validation of the fabricated structures is presented in the  Appendix. The results obtained in this work, including the effect of F_c on the density and bulk modulus and atomic network structures of a-SiOC, are discussed in Sec. III. The relationship between F_c, C-related structures and bulk modulus and the limitations and research direction of the study are discussed in Sec. IV. Finally, Sec. V presents the conclusions of the study.

Similar to previous studies,^4,24 we assumed an a-SiOC composition with some of the O atoms in a-SiO₂ replaced with C atoms. The composition of a-SiOC is expressed as Si_xO_2x−syC_y using the substitution coefficient s (0, 1, or 2), which represents the number of O atoms substituted by a single C atom. When s = 0, the C atoms are simply added to a-SiO₂ without removing the O atoms, as illustrated in Fig. 1(a). As the amount of O atoms is sufficient to form complete SiO₂, we refer to this as the “O-rich condition.” When s = 1, one O atom in a-SiO₂ is replaced by one C atom. In this case, the C atom was assumed to have a two-coordinated structure, as illustrated in Fig. 1(b), which is referred to as the “O–C replacement condition,” which is the same as that used by Ding and Demkowicz.⁴⁴ When s = 2, the C atom has a four-coordinated structure, as illustrated in Fig. 1(c). The composition of the structure with one SiO₄ tetrahedron in a-SiO₂ replaced with a C atom is expressed as Si_x−yO_2x−4yC_y, corresponding to Si_xO_2x′−2yC_y by denoting the number of Si atoms as x′ instead of x−y. This composition was the same as that used in Kroll's WWW method.⁴ As this condition has an insufficient amount of O atoms to form complete SiO₂, we refer to this composition as the “O-poor condition.” These assumptions regarding the coordination number of the C atoms do not directly affect the structural fabrication because the structures were fabricated using the melt-quenching method. Therefore, the fabricated structure may contain CO, CO₂, and C clusters.

The number of Si atoms was fixed at x = 30. The number of C atoms varied between y = 0 and 30 with an interval of y = 2. The impact of the initial atom positions was mitigated by fabricating 10 structures for each composition, achieving a total of 480 structures. The average values across these structures are used as representative values. Figure 2 shows the dependence of the numbers of Si, O, and C atoms on the number of C atoms for s = (a) 0, (b) 1, and (c) 2. In all cases, the composition at y = 0 corresponds to SiO₂. Conversely, when y = 30, the compositions of s = 0, 1, and 2 correspond to SiO₂+ C, SiOC, and SiC, respectively. The number of O atoms removed depends on the s value. Meanwhile, F_c, which is the ratio of the number of C atoms to the total number of atoms, was used as an index of the C content.

The melt-quenching method was used to fabricate a-SiOC structures. Initially, the atoms were randomly placed in the simulation cell with the composition and density determined in Sec. II A and heated at 6000 K for 1 ps. The structure was then rapidly quenched to 2000 K in 3 ps and annealed at 2000 K for 10 ps. The structure was quenched again at the same cooling rate to 1600 K, which is slightly below the melting point of SiO₂, and annealed at 1600 K for 10 ps. All the calculations were performed using the NVT ensemble. Subsequently, additional structural relaxation calculations were performed for 10 ps under the NPT ensemble at 0 MPa and 1600 K. During the relaxation process, the densities of the structures were optimized and changed from $d init$⁠. Finally, the structure was optimized. PFP³⁷ was used to determine the interatomic potential. Computing platforms and resources using PFP were provided commercially by Matlantis.⁴⁶ The molecular dynamics calculations were performed using LAMMPS.⁴⁷ Fabricating an a-SiOC structure required approximately 2–4 h. In contrast, the fabrication of the same structure using DFT calculations takes approximately 10 days using a 10-core parallel computer with recent Intel Xeon processors. Some of the structures fabricated using PFP were validated by DFT calculations, as discussed in the  Appendix.

Figure 3(a) shows the dependence of density on F_c with s = 0, 1, and 2. The densities of the fabricated a-SiO₂ and a-SiC structures were approximately 2.2 and 2.8 g/cm³, respectively, depicting the slightly lower density of a-SiC than that of the initial condition (d_SiC = 3.2 g/cm³). Snead and Zinkle⁴⁸ noted that the density of an a-SiC film before densification is approximately 2.8 g/cm³, and recrystallization by sintering at >900 °C for 30 min is necessary to obtain the density of 3.2 g/cm³, which is the typical density of SiC crystals and ceramics. Therefore, the obtained density of 2.8 g/cm³ indicates that the structure of the fabricated a-SiC corresponded to that before densification. As shown in Fig. 3(a), density increases as s increases. In particular, a large difference is noted between the density with s = 0 and s = 1 because C atoms are incorporated into the atomic network of a-SiO₂, instead of the removed O atoms, when s = 1 or 2, whereas C atoms are not easily incorporated into the complete SiO₂ when s = 0. This phenomenon is further discussed in Sec. III B.

The atomic network structure of a-SiOC was analyzed in this study. We focused on the coordination number of each atomic species and structure relevant to the C atom with s = 0, 1, and 2. The bonding and coordination numbers are defined based on the distance between a pair of atoms. The threshold distances were set to 1.99, 2.60, 2.23, 1.82, 1.97, and 1.89 Å for the Si–O, Si–Si, Si–C, O–O, O–C, and C–C bonds, respectively, based on default threshold distances for visualization in the three-dimensional visualization program VESTA.⁵³ 

Figure 4(a) shows the dependence of the average number of bonds in the Si atom on F_c. The numbers of Si–C, Si–O, and Si–Si bonds are also plotted. The result denoted “total” corresponds to the coordination number. Figure 4(b) shows the dependence of the average number of bonds in the O atom. The Si and O atoms maintained four- and two-coordinated structures, respectively.

We evaluated the bonding patterns of the C atoms to analyze the atomic structures relevant to them. Figure 5(a) shows the average number of C bonds, and Fig. 5(b) shows the dependence of the average number of bonding patterns relevant to the two-coordinated C atoms on F_c. Here, we defined the bonding pattern as the number of C atoms with a particular coordination and bonding atomic species. CC, OSi, CSi, and CO indicate the C atoms bonded to two C atoms, O and Si atoms, C and Si atoms, and C and O atoms, respectively. Figure 5(c) shows the results of the three-coordinated C atoms. Here, OSiSi indicates that the C atoms bonded with one O atom and two Si atoms similar to that shown in Fig. 5(b). Figure 5(d) shows the results of the four-coordinated C atoms. The average coordination number of the C atoms (total) was maintained at three, regardless of F_c, as shown in Fig. 5(a). As shown in Figs. 5(b) and 5(c), approximately 70% C atoms have three-coordinated structures, whereas approximately 30% have two-coordinated structures. The proportion of four-coordinated C atoms is less than 10%, as shown in Fig. 5(d). When F_c < 0.1, the C atoms tend to bond with Si and O, as shown in Fig. 5(a). Most C atoms exhibit two-coordinated OSi and three-coordinated OOSi and OSiSi structures, as shown in Figs. 5(b) and 5(c).

The structures with F_c= 0.04 (Si₃₀O₆₀C₄), F_c= 0.13 (Si₃₀O₆₀C₁₄), and F_c= 0.24 (Si₃₀O₆₀C₂₈) are shown in Figs. 6(a)–6(c), respectively. OVITO was used for the visualization.⁵⁴ Most C atoms form a terminal structure that bonds with a single-coordinated O atom, as illustrated in Fig. 6(a). As F_c increases, the number of bonds with the C atoms increases, whereas those with the Si and O atoms decrease. The number of bonds with the C atoms exceeds those with the Si and O atoms at F_c = 0.1. The bonding patterns of two-coordinated CC and three-coordinated CSiSi, CCSi, and CCC also increased. This change in the bonding pattern is ascribed to the agglomeration of C atoms, as illustrated in Figs. 6(b) and 6(c). As F_c increases, the structure of C precipitates, referred to as “free C” for SiOC ceramics^5,20 or “C cluster” for thermally oxidized films on SiC substrates,^36,55 transforms from a chainlike structure, as shown in Fig. 6(b), to a graphitelike structure, as shown in Fig. 6(c). For s = 0, the C atoms behave as impurities that are unnecessary for the formation of the a-SiO₂ network. This is likely ascribed to the limitations of C atoms in breaking strong SiO₂ networks.

Figures 7(a) and 7(b) show the dependence of the average number of bonds of the Si and O atoms, respectively, on F_c for s = 1. Similar to s = 0, the Si and O atoms maintained four- and two-coordinated structures, respectively. For the Si atoms, the number of bonds with the C atoms increases as F_c increases. In contrast, almost no bonds are observed between the Si atoms. This indicates that the smaller number of Si–O bonds due to the removal of O atoms was replaced by Si–C bonds without changing the four-coordinated structure of the Si atoms. Most O atoms bonded with two Si atoms but did not bond with C atoms, as shown in Fig. 7(b).

Figure 8(a) shows the average number of bonds of the C atom. Figures 8(b)–8(d) show the dependence of the average number of bonding patterns relevant to the two-, three-, and four-coordinated C atoms, respectively, on F_c for s = 1. As shown in Fig. 8(a), the average coordination number of the C atoms is approximately three. Unlike s = 0, the average coordination number increases as F_c increases. Approximately 60% of C atoms have three-coordinated structures, as shown in Fig. 8(c). The proportion of two-coordinated C atoms decreases, whereas that of the four-coordinated C atoms increases as F_c increases. When F_c < 0.05, most C atoms have a three-coordinated structure and bond with both Si and O atoms, such as OSiSi and OOSi, as shown in Fig. 8(b).

The structures with F_c= 0.04 (Si₅₆O₄C₄), F_c= 0.11 (Si₃₀O₅₀C₁₀), and F_c= 0.24 (Si₃₀O₄₂C₁₈) are shown in Figs. 9(a)–9(c), respectively. Similar to s = 0, most C atoms constructed terminal structures in which C atoms bond with single-coordinated O atoms, as shown in Fig. 9(a). The number of bonds with the O atoms decreases, whereas the number of bonds with the C atoms increases as F_c increases. Unlike s = 0, the average number of bonds with Si atoms increases. The proportion of the three- and four-coordinated structures bonded with both C and Si atoms, such as CSiSi, CCSi, and CSiSiSi, increases. In particular, the proportion of C atoms bonded to multiple Si atoms, such as CSiSi and CSiSiSi, is higher than that at s = 0. This difference indicates that the C atoms tend to bond with the Si atoms instead of the reduced O atoms. Small C precipitates consisting of few C atoms were formed, similar to that of s = 0, as illustrated in Figs. 9(b) and 9(c). The size of the cluster is smaller than that at s = 0.

Figures 10(a) and 10(b) show the dependence of the average number of bonds of the Si and O atoms, respectively, on F_c for s = 2. The increase in the Si–C bond is larger than that for s = 1. Unlike s = 0 and s = 1, Si–Si bonds are observed. The Si–Si bond is required to maintain the four-coordinated structure of the Si atoms owing to the lack of O atoms. As shown in Fig. 10(b), most O atoms are bonded with two Si atoms.

Figure 11(a) shows the average number of bonds of a C atom. Figures 11(b)–11(d) show the dependence of the average number of bonding patterns for the two-, three-, and four-coordinated C atoms, respectively, on F_c. When F_c < 0.1, most C atoms bonded with Si atoms and formed SiSiSi structures, where one C atom bonds with three Si atoms, as shown in Figs. 11(a) and 11(c).

The structures with F_c = 0.10 (Si₃₀O₄₄C₈) and F_c = 0.24 (Si₃₀O₂₈C₁₆) are shown in Figs. 12(a) and 12(b), respectively. Most C atoms substitute the reduced O atoms and are incorporated into the SiO₂ network. When F_c > 0.15, the ratio of the four-coordinated C atoms increases notably and exceeds 50%, as shown in Fig. 11(d). Most C atoms have three-coordinated structures of SiSiSi, CSiSi, and CCSi, and four-coordinated structures of SiSiSiSi and CSiSiSi, as shown in Figs. 11(c) and 11(d). In the absence of O atoms, the structure approaches that of a-SiC. The ratio of the structures of CC and CCC is less than those with s = 0 and 1 (approximately 10%–20%). Moreover, large C clusters are not observed.

According to first-principles calculations of the formation energy of C-related defects in SiO₂,²¹ the formation energy of dicarbon (combination of CSiSi structures denoted in Sec. III B) and monocarbon (SiSiSiSi) structures decreases under the O-poor condition. In contrast, under the O-rich condition, the formation energies of structures with C and Si bridged by O, including the CO and CO₂ molecules, decrease. Our calculations indicated that C atoms tend to bond with O atoms, break the SiO₂ network, and form terminated structures for the O-rich condition (s = 0), as illustrated in Fig. 6(a). Meanwhile, the C atoms tend to bond with the Si atoms, instead of with O atoms, for the O-poor condition (s = 2). These results are consistent with the results on the formation energies of C-related defects.

CO and CO₂ molecules were generated and dissociated from the SiOC network during the structural fabrication. Figure 13 shows the dependence of the average numbers of generated CO and CO₂ molecules on F_c for s = 0, 1, and 2. The gas molecules are defined as small clusters of atoms that do not bond to the bulk a-SiOC networks. Less gas molecules were generated for s = 1 and 2 than that for s = 0 because the generation of CO and CO₂ molecules consumes O atoms. For s = 1 and s = 2, C atoms are used to compensate for the lack of O atoms and are incorporated into SiOC networks, suppressing the generation of gas molecules. Meanwhile, for s = 0, C atoms have sufficient capacity to compensate for O atom shortage due to the gas generation, as indicated by the precipitation of C clusters. Furthermore, the formation was observed only with F_c < 0.2. The result is consistent with previous calculations of the formation energies of CO and CO₂ molecules because the low F_c and low s values correspond to the O-rich condition.

Figure 14(a) shows the dependence of the bulk modulus on the bond density, which is defined as the number of bonds normalized by the volume. The bulk modulus strongly depends on bond density. Figures 14(b) and 14(c) show the dependence of the bulk modulus on the Si–C and Si–O bond densities, respectively. Opposite trends were observed for the Si–C and Si–O bonds. In particular, the bulk modulus increases as the number of Si–C bonds increases, whereas it decreases as the number of Si–O bonds increases. Therefore, it is crucial for increasing the elastic modulus to increase the bond density by effectively substituting the two-coordinated O atoms with three- or four-coordinated C atoms. Although concrete evidence linking bond density to the elastic properties of amorphous materials is not available, a denser crosslinking network, represented by bond density, is expected to increase material strength.⁵⁶ Kumagai et al.³⁰ provided insights into this relationship, in the case of hydrogenated Si-incorporated amorphous C, by explaining that a higher atom coordination, indicating denser crosslinking networks, decreases the potential for internal displacement of atoms. The small increase in the bulk modulus for s = 0, as shown in Fig. 3(b), can also be attributed to the bond density. In particular, the bond density of Si–C with s = 0 is lower than those with s = 1 and s = 2 because C atoms are difficult to incorporate into the SiOC networks under the O-rich condition, as discussed in Sec. III B 1. These results suggest that an O-poor condition during deposition is crucial for fabricating stronger dielectric films.

This work successfully clarified the relationship between the composition, atomic structure, and elastic properties of a-SiOC, providing a significant contribution to the understanding of the fabrication of stronger dielectric films and revealing the mechanisms behind C cluster formation. Nonetheless, some limitations remain owing to the computational costs of current MD calculations with neural network potentials. A specific concern is the spatial limitations in this study. The number of atoms in the a-SiOC model in this work was limited to approximately 100 because we focused on elucidating the overall tendencies among different compositions with a low calculation cost for structural fabrications. This limitation is particularly notable when F_c< 0.1, resulting in a restricted number of C atoms and hindering a comprehensive discussion on the potential agglomeration of C atoms. Although a-SiOC structures with thousands of atoms can be fabricated, this would require several days, resulting in high computational costs for fabricating sufficient structures to reveal overall tendencies. The second concern is related to the timescale limitations. As discussed in Sec. III A, our fabricated structures are not densified because of the limited time for annealing and structural relaxation during the fabrication. Although studies have fabricated well-relaxed structures through lengthy MD calculations,^27,57 such would incur high computational costs for fabricating sufficient structures. We anticipate that these potential limitations will be addressed in the future with the ongoing rapid advancements in computational capabilities.

There are several directions for future research. First, although this study indicated that a-SiOC films with increased elastic properties can be fabricated by controlling the deposition conditions to reduce the O content, the direct correspondence between such conditions and actual manufacturing conditions remains unclear. Further investigations to bridge this gap are needed to realize practical applications. Moreover, the other mechanical properties of a-SiOC, such as fracture toughness, need to be investigated for long-term stability and reliability as dielectric films for semiconductor devices. The electrical properties should also be investigated because C incorporation can degrade the insulation abilities of a-SiO₂, as discussed in the thermal oxidation of SiC.^58–60 Exploring other materials is also an interesting challenge. In particular, N incorporation into a-SiO₂, i.e., a-SiON, provides an attractive research direction. However, as N atoms tend to become N₂ molecules, whereas C atoms can solely form solid deposits, the structural characteristics of a-SiOC and a-SiON are expected to differ significantly. Additionally, the application of the method presented in this paper is not limited to a-SiO₂ considering the ability of PFP to handle an arbitrary composition of atoms. For example, the structural characteristics and mechanical properties of phosphate glass with different cations^61–63 are still under discussion.

We investigated the influence of F_c in a-SiOC on the elastic modulus and discussed the relationship between the elastic modulus and atomic structure. Up to 480 structures with different SiOC compositions were randomly fabricated by melt-quenching. The universal graph neural network potential PFP was employed to realize high-accuracy and high-speed simulations of multicomponent systems.

The bulk modulus increased with F_c. Specifically, a higher bulk modulus was obtained under the O-poor condition, increasing from 45 to 60 GPa with 10% C. The density and bulk modulus under the O-poor condition could be expressed through linear interpolation between a-SiO and a-SiC using F_c. Meanwhile, a higher O content decreased both the bulk modulus and density. In particular, 20% C incorporation under the O-rich condition is required to increase the bulk modulus from 45 to 60 GPa.

The atomic network structure was also analyzed. Under the O-rich condition, where C atoms were added to complete SiO₂, C atoms tend to bond with O atoms and form terminal structures, in which a C atom bonded with a single-coordinated O atom. As F_c increased, C atoms formed C clusters under the O-rich condition. Under the O–C replacement condition, in which an O atom was substituted with a C atom, most C atoms exhibited a three-coordinated structure and bonded with both Si and O atoms, which was also observed for the O-rich condition. C atoms tended to bond with the Si atoms as F_c increased, unlike the O-rich condition. Under the O-poor condition, where two O atoms were replaced by a C atom, most of the C atoms substituted the reduced O atoms and were incorporated into the SiO₂ network. The C atoms tended to bond with Si atoms, and Si–Si bonds were observed because of insufficient O. The structure approached that of a-SiC. Our results were consistent with the DFT calculations for the formation energy of C-related defects.

The relationship between the bulk modulus and atomic network structures is attributed to the formation of denser crosslinking atomic network structures, as represented by the bond density. The bulk modulus increased as the bond density increased by effectively substituting the two-coordinated O atoms with the three- or four-coordinated C atoms. Such structures can be fabricated under O-poor conditions. Therefore, stronger dielectric films should be fabricated to realize O-poor conditions during deposition.

This work was supported by JSPS KAKENHI Grant No. JP23K13216.

The authors have no conflicts to disclose.

Hiroki Sakakima: Conceptualization (lead); Investigation (equal); Methodology (lead); Visualization (equal); Writing – original draft (lead). Keigo Ogawa: Data curation (lead); Formal analysis (lead); Investigation (equal); Methodology (supporting); Validation (equal); Visualization (equal). Sakurako Miyazaki: Data curation (supporting); Methodology (supporting); Validation (equal). Satoshi Izumi: Supervision (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The structures fabricated by PFP were validated by comparing the Radial Distribution Functions (RDFs) and cohesive energies of parts of the fabricated structures with experimental values and ab initio calculations. Ab initio calculations were performed using DFT with generalized gradient approximation with Perdew–Burke–Ernzerhof functionals using the projector-augmented wave method with the PHASE/0 code.⁶⁴ The energy cut-off was set at 16 hartree. A single Γ point was used for the k-point sampling.

Figure 15 shows a comparison of the calculated partial RDFs of a-SiO₂ obtained using the PFP and DFT methods.⁶⁵ The calculated RDFs using PFP were averaged over 10 structures. The calculated values for all cases correspond well with the DFT results.

The cohesive energy and bulk modulus of parts of the SiOC structures fabricated by PFP were calculated using DFT. The cohesive energy in the DFT calculations was determined by subtracting the sum of the energies of the isolated atoms from the total energy of the fabricated structures. Figure 16 shows a comparison of the normalized cohesive energy per atom and bulk modulus. For the cohesive energy, the values calculated using PFP and DFT are similar, indicating that PFP provides highly accurate predictions of the energy of a-SiOC structures. The accuracy of the bulk modulus was slightly lower than that of the energy. PFP reproduced the DFT calculation results with an average accuracy of approximately 7 GPa and a maximum deviation of 26 GPa.