Introduction

When two different materials are joined together, the interface is formed. Since the atomic structure of the interface has a significant impact on its electronic properties, understanding and controlling the interfaces at atomistic level has been a challenge of materials science. Particularly, forming high-quality interfaces is important in a wide range of applications. For example, the success of Si-based metal-insulator-semiconductor field-effect transistors (MISFETs) is largely relied on the formation of the atomically-abrupt and low-dangling-bond-density interface between silicon (Si) and silicon dioxide (SiO2)1,2,3,4,5.

Recently, gallium nitride (GaN) has attracted much attention for applications in power electronic devices6. Since the dangling bonds formed at the GaN/insulator interface create trap states which degrade carrier mobility, reducing the interface dangling bonds is a key to improve the performance of GaN-based MISFETs5,7,8,9,10,11.

Given that knowledge of SiO2 has been accumulated in developing Si-based MISFETs, SiO2 is one of choices for the insulator of GaN-based MISFETs5,8,9,10,11. However, since the atomic structure of SiO2 is different from that of GaN, dangling bonds are easily formed at the GaN/SiO2 interface. To overcome this problem, the mechanism which allows for the formation of the high-quality Si/SiO2 interface would be helpful. Although bulk crystalline SiO2 cannot grow epitaxially on crystalline Si due to the lattice mismatch, an atomically-thin ordered layer can form between crystalline Si and amorphous SiO2, serving as a buffer for smoothly connecting the different structures1,2,3,4. Therefore, the formation of a buffer layer between crystalline GaN and amorphous SiO2 would be a key to synthesize a high-quality GaN/SiO2 interface12.

Since GaN has a small lattice mismatch with SiC, and both of GaN and SiC consist of honeycomb layers, knowledge of silicon carbide (SiC)13,14,15,16,17,18,19,20 would also be helpful in searching for an ideal GaN/SiO2 structure. It is known that, although the lattice mismatch prevents the epitaxial growth of bulk crystalline SiO2 on crystalline SiC, it is possible to epitaxially grow atomically-thin silicon-oxide layers on surfaces of crystalline SiC13,16. Specifically, when a SiC(0001) surface is etched in hydrogen, an epitaxial Si2O3 layer grows on the SiC(0001) surface. On the other hand, when a SiC(000 \(\bar{1}\)) surface is etched in hydrogen, an epitaxial Si2O5 layer grows on the SiC(000 \(\bar{1}\)) surface. Although the SiC/Si2O3 and SiC/Si2O5 structures both have atomically-abrupt interfaces, dangling bonds remain at the interfaces. As a result, dangling bond states exist within the band gap of the SiC/Si2O3 and SiC/Si2O5 structures14,15. On the other hand, when the nitrogen treatment is performed after the hydrogen-gas etching, an epitaxial Si4O5N3 layer grows on a SiC(0001) surface16. Interestingly, no dangling-bond states exist within the band gap of the SiC/Si4O5N3 structure because the dangling bonds of Si atoms on the clean SiC surface, as well as those of the Si4O5N3 layer, are all terminated16,17,18,19. The SiC/Si4O5N3 structure is therefore expected to be a seed for a high-quality SiC/SiO2 structure20.

The GaN/Si4O5N3 structure has not been synthesized experimentally yet. However, resting on the experimentally synthesized SiC/Si4O5N3 structure and the small lattice mismatch between GaN and SiC, we argue that the Si4O5N3 layer can epitaxially grow on a GaN(0001) surface. To our knowledge, however, such a possibility has not been recognized so far. Recently, the impact of computer simulations to offer guidelines in the identification of potentially useful new materials has been increasing21,22,23,24. For example, silicene was first predicted by ab initio calculations21 and then later on synthesized experimentally22,23. In this paper, being inspired by the geometrical analogies between SiC and GaN and the experimentally synthesized SiC/Si4O5N3 structure, we investigate the stability of the GaN/Si4O5N3 structure using ab initio calculations, and propose that the GaN/Si4O5N3 structure is experimentally accessible. We also study the band structure of the GaN/Si4O5N3 structure and its derivatives.

Results and Discussions

Atomic structure and stability of the GaN/Si4O5N3 structure

Figure 1a illustrates the optimized structure model of the GaN/Si4O5N3 structure. All the dangling bonds of the Ga atoms at the GaN(0001) surface are terminated by the N atoms of the Si4O5N3 layer. We refer to the Ga atoms bonded to the N atoms of the Si4O5N3 layer as interface Ga atoms. Similarly, we refer to the N atoms of the Si4O5N3 layer bonded to the interface Ga atoms as interface N atoms. As with Ga atoms of GaN, each interface Ga atom of the GaN/Si4O5N3 structure is bonded to four N atoms: one interface N atom of the Si4O5N3 layer and three N atoms of GaN. However, because the interface N atoms of the Si4O5N3 layer are not located right above the interface Ga atoms, the tetrahedra formed by the N atoms surrounding the interface Ga atoms are distorted. Unlike the N atoms of GaN, each interface N atom of the Si4O5N3 layer is bonded to three neighbouring atoms: one interface Ga atom and two Si atoms of the Si4O5N3 layer. Thus, the chemical environments around the interface Ga and N atoms are different from those of Ga and N atoms of GaN, respectively.

Figure 1
figure 1

Structure model. (a) GaN/Si4O5N3 structure. Red, blue, white, green, and orange spheres represent O, Si, N, Ga, and pseudohydrogen atoms, respectively. (b) GaN/Si2O5 structure. Each circled Ga atom has a dangling bond. (c) GaN/Si2O3 structure.

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For comparison, the GaN/Si2O5 structure is illustrated in Fig. 1b. The Si2O5 layer is obtained from the Si4O5N3 layer by removing Si2N3 (all the N atoms and Si atoms bonded to those N atoms). The formation energy per Si2N3 of the GaN/Si4O5N3 structure relative to the GaN/Si2O5 structure is 17.4 eV. The positive value indicates that the dangling bond free GaN/Si4O5N3 structure is energetically more stable than the GaN/Si2O5 structure. Note that the O atoms of the Si2O5 layer are bonded to the Ga atoms of the GaN(0001) surface. However, four surface Ga atoms are not bonded to the Si2O5 layer, in other words, the GaN/Si2O5 structure has four interface dangling bonds per surface supercell.

We also illustrate the GaN/Si2O3 structure in Fig. 1c. This structure is obtained from the GaN/Si2O5 structure by removing O atoms which bridge Ga atoms and Si atoms, and then directly connecting the Si atoms to the Ga atoms. The formation energy per O atom of the GaN/Si2O3 structure relative to the GaN/Si2O5 structure is −3.95 eV. The negative value indicates that the GaN/Si2O5 structure is energetically more stable than the GaN/Si2O3 structure.

The GaN/Si4O5N3 structure is more stable than the GaN/Si2O5 structure which is more stable than the GaN/Si2O3 structure. Thus, the GaN/Si4O5N3 structure is most stable of three. To further examine the stability of the GaN/Si4O5N3 structure, we carried out 2-picosecond molecular dynamics simulations at 1000 K. The GaN/Si4O5N3 structure endured the stability test. Note that the ttt1, tttt1, and SiC-inserted structures, which will be discussed later, also endured the stability test.

Clue to the realization of our proposal

Our results given above suggest that the GaN/Si4O5N3 structure is experimentally accessible. Here, we propose an approach to realize it. For this purpose, we first point out that the SiC/Si4O5N3 structure is formed by the nitrogen treatment followed by the hydrogen-gas etching16. Since the oxygen gas was not used during the process, the O atoms of the SiC/Si4O5N3 structure came from the residual oxygen in the hydrogen or nitrogen gases.

In the case of the GaN/Si4O5N3 structure, we need to supply Si atoms, for GaN has no Si atoms. We therefore propose that the GaN/Si4O5N3 structure can be synthesized by a two-step method as follows:

  1. (1)

    Perform SiCl4-plasma etching of GaN25. As a result, some Si atoms are expected to be adsorbed on the surface of GaN.

  2. (2)

    Perform annealing under nitrogen ambient26, and then N atoms from the nitrogen gas, the adsorbed Si atoms, and residual O atoms are expected to arrange themselves to form the epitaxial Si4O5N3 layer on GaN.

Electronic structure

The band structure of the GaN/Si4O5N3 structure is shown in Fig. 2a. Since its valence band is partially empty, the GaN/Si4O5N3 structure is a degenerate p-type semiconductor. The partially empty valence band is explained by using the electron counting rules as follows27. The dangling bond of each Ga atom at the clean GaN(0001) surface contains 0.75 electrons, while the dangling bond of each N atom of the isolated Si4O5N3 layer contains one electron. When the Si4O5N3 layer is attached to the GaN(0001) surface, each pair of the interface Ga and N atoms forms a covalent bond, which we call an interface Ga-N bond. Unlike a Ga-N bond in GaN, each interface Ga-N bond is occupied by just 1.75 electrons, being short of 0.25 electrons to fulfil itself. Since there are total of twelve interface Ga-N bonds, it needs three more electrons per surface supercell to fulfil the valence band.

Figure 2
figure 2

Band structure. (a) GaN/Si4O5N3 structure. (b) ttt1 structure. (c) ttt2 structure. (d) ttn1 structure. (e) tnn1 structure. (f) nnn1 structure. (g) tttt1 structure. (h) SiC-inserted structure. Each dashed line indicates the Fermi level, and the levels below the dashed line are filled by electrons. Note that the electrostatic potential is set to zero at infinity in the \([000\bar{1}]\) direction.

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According to the rigid band model, since the valency of Si is larger than that of Ga by one, substituting three Ga atoms by three Si atoms makes the GaN/Si4O5N3 structure a semiconductor. We now examine this possibility. As substitution sites, we consider the Ga atoms in the topmost and the next GaN bilayers. According to the location of the doped Si atoms, the substitution patterns can be classified into four groups, which we call “ttt”, “ttn”, “tnn”, and “nnn”. Here, for example, ttn means that two Ga atoms in the topmost GaN bilayer and one Ga atom in the next GaN bilayer are substituted by Si atoms. As for ttt, since all the 12 Ga atoms in the topmost GaN-bilayer are equivalent, we may arbitrarily choose one Ga atom as the first substitution site. Since 11 Ga atoms are left, there are 11 × 10/2 = 55 possible combinations for second and third substitution sites. Note that, although the 55 patterns include all the symmetrically different patterns, but have duplication of symmetrically equivalent patterns. Similarly, we consider 11 × 12 = 134 different patterns for ttn, 12 × 11/2 = 66 for tnn, and 2 × 55 = 110 for nnn. Note that the “2” of “2 × 55” comes from the fact that 12 Ga atoms of the next GaN-bilayer is classified into two groups: eight Ga atoms lie under the Si atoms of the Si4O5N3 layer, while four Ga atoms do not.

To determine the energetically most stable substitution pattern, we first optimized Si-doped structures using a lax convergence criterion of 1 × 10−3 Hartree/Bohr for forces on atoms, and calculated their substitution energies (Fig. 3). We then focused on ttt1, ttt2, ttn1, tnn1, and nnn1 structures (Fig. 4a–e), where the ttt2, for example, indicates a ttt structure whose substitution energy is the second highest in the ttt group. Note that, because of duplication of symmetrically equivalent patterns, three of 55 ttt structures have the highest energy. For reference, the substitution energies of the ttt1, ttt2, ttn1, tnn1, and nnn1 structures are 2.28, 2.18, 2.16, 1.97, and 1.72 eV/atom, respectively. We then optimized further the ttt1, ttt2, ttn1, tnn1, and nnn1 structures using a severe convergence criterion of 1 × 10−4 Hartree/Bohr, and refined their substitution energies. The refined substitution energies of the ttt1, ttt2, ttn1, tnn1, and nnn1 structures are 2.32, 2.19, 2.19, 1.99, and 1.73 eV/atom, respectively. From this result, we find that (1) the substitution doping is energetically favourable, (2) doped Si atoms prefer to locate the interface rather than the interior of GaN, and (3) the ttt1 structure is most stable.

Figure 3
figure 3

Substitution pattern dependence of substitution energy per Si atom. Each point indicates the substitution energy of a Si-doped GaN/Si4O5N3 structure.

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Figure 4
figure 4

Structure model. (a) ttt1 structure. The arrow indicates the top view of the topmost and next GaN bilayers. (b) The topmost and next GaN bilayers of the ttt2 structure. (c) ttn1 structure. (d) tnn1 structure. (e) nnn1 structure. (f) tttt1 structure. (g) SiC-inserted structure. Brown spheres represent C atoms.

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To further study the stability of the ttt1 structure, we examine structures where four Si atoms are doped. Since doped Si atoms prefer to locate the interface, we consider only the tttt patterns. The tttt1 structure (energetically most stable structure in the tttt group) is illustrated in Fig. 4f. Its substitution energy is 1.89 eV/atom, and is smaller than that of the ttt1 structure. This suggests that Si atoms of the ttt1 structure do not aggregate to form the tttt1 structure.

The band structures of the ttt1, ttt2, ttn1, tnn1, nnn1 and tttt1 structures are shown in Fig. 2b–g. When three Si atoms are doped, the most stable ttt1 structure is a semiconductor. On the other hand, the tttt1 structure doped with four Si atoms is a degenerate n-type semiconductor (Fig. 2g), for the electrons are supplied to the conduction band. These results suggest that the type of the band structure can be tuned by the number of doped Si atoms which can be controlled by the amount of the adsorbed Si atoms.

The substitution pattern affects the band structure (Fig. 2b–f). Specifically, the band gap energies of the ttt1, ttt2, ttn1 and tnn1 structures are 0.90, 0.94, 0.58 and 0.11 eV, respectively. Even a metallic band structure is obtained when the Si atoms are doped in the nnn1 pattern (Fig. 2f). The substitution pattern dependence of the band gap energies can be explained in terms of the electrical double layer formed at the interface28. To show this, we first explain why the band gap energy of the ttt1 structure (0.90 eV) is smaller than that of the GaN/Si4O5N3 structure (1.50 eV). For this purpose, we first study the wave functions (WFs) of the GaN/Si4O5N3 structure. The conduction-band-minimum wave function (CBM-WF for short) consists of the s orbitals of the N atoms, resembling the CBM-WF of bulk GaN(Fig. 5a). On the other hand, the valence-band-maximum wave function (VBM-WF) of the GaN/Si4O5N3 structure consists of the \({p}_{\parallel }\) orbitals of the interface N atoms (Fig. 5b). Here, the \({p}_{\parallel }\) orbitals are the p orbitals pointing in directions parallel to the (0001) plane. We therefore refer to the VBM-WF as the interface-nitrogen band maximum wave function (INBM-WF for short). The CBM- and INBM-WFs of the ttt1 structure are essentially the same as those of the GaN/Si4O5N3 structure, respectively (Fig. 5c and d). Note that the INBM of the ttt1 structure is occupied, while that of the GaN/Si4O5N3 structure is unoccupied. Since electrons are transferred from the doped Si atoms to the interface N atoms to fulfil the interface-nitrogen band, an electrical double layer is formed at the interface of the ttt1 structure. As a result, an electrostatic potential difference is created between the Si4O5N3 layer and the GaN region. Because of the potential difference, the energy of the INBM of the ttt1 structure (−2.82 eV) is larger than that of the GaN/Si4O5N3 structure (−3.43 eV). On the other hand, since the CBM-WF is bulklike and consists of the atomic orbitals of the GaN region, the formation of the electrical double layer only minimally affects the energy of the CBM. In fact, the energy of the CBM of the ttt1 structure (−1.92 eV) is comparable to that of the GaN/Si4O5N3 structure (−1.93 eV). As a result, \({E}_{{\rm{g}}}^{{\rm{ttt}}1}={\varepsilon }_{{\rm{C}}}^{{\rm{ttt}}1}-{\varepsilon }_{{\rm{I}}}^{{\rm{ttt}}1}=0.90\,{\rm{eV}}\) is smaller than \({E}_{{\rm{g}}}^{{\rm{GaN}}/{{\rm{Si}}}_{4}{{\rm{O}}}_{5}{{\rm{N}}}_{3}}={\varepsilon }_{{\rm{C}}}^{{\rm{GaN}}/{{\rm{Si}}}_{4}{{\rm{O}}}_{5}{{\rm{N}}}_{3}}-{\varepsilon }_{{\rm{I}}}^{{\rm{ttt}}1}=1.50\,{\rm{eV}}\). Here, \({E}_{{\rm{g}}}^{{\rm{X}}}\), \({\varepsilon }_{{\rm{C}}}^{{\rm{X}}}\), and \({\varepsilon }_{{\rm{I}}}^{{\rm{X}}}\) are the band gap energy, the energy of the CBM, and the energy of the INBM of the structure X, respectively.

Figure 5
figure 5

Isosurface of WF. (a) CBM-WFs of the GaN/Si4O5N3 structure and bulk GaN. (b) INBM-WF of the GaN/Si4O5N3 structure. (c) CBM-WF of the ttt1 structure. (d) INBM-WF of the ttt1 structure. (e) CBM-WF of the SiC-inserted structure. (f) INBM-WF of the SiC-inserted structure. The isovalue is ±0.015 \(({{\rm{electron}}}^{1/2}/{{a}_{0}}^{3/2})\) except for bulk GaN where ±0.030 is used. Here, a0 is the Bohr radius.

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The potential difference increases as the distance between the electrical layers increases. Therefore, the energy of the INBM of the Si-doped GaN/Si4O5N3 structure increases when the Si atoms are doped farther from the interface N atoms. Accordingly, the band gap energy decreases as the number of Si atoms doped in the next bilayer increases: \({E}_{{\rm{g}}}^{{\rm{ttn}}1}=0.58\,{\rm{eV}}\) and \({E}_{{\rm{g}}}^{{\rm{tnn}}1}=0.11\,{\rm{eV}}\). The nnn1 structure is a metal, for \({\varepsilon }_{{\rm{I}}}^{{\rm{nnn}}1}=-1.84\,{\rm{eV}}\) becomes larger than \({\varepsilon }_{{\rm{C}}}^{{\rm{nnn}}1}=-1.95\,{\rm{eV}}\). Since the potential difference in the ttt2 structure is comparable to that in the ttt1 structure, \({E}_{{\rm{g}}}^{{\rm{ttt}}2}=0.94\,{\rm{eV}}\) is comparable to \({E}_{{\rm{g}}}^{{\rm{ttt}}1}=0.90\,{\rm{eV}}\). For reference, Eg, εC, and εI of the GaN/Si4O5N3, ttt1, ttt2, ttn1 and tnn1 structures are summarized in Table 1.

Table 1 Band gap energy (Eg), energy of the bulklike CBM-MO band (εC), and energy of the INBM-MO (εI).
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The calculated band gap energy of the ttt1 structure (0.90 eV) is smaller than that of bulk GaN(1.84 eV). Note that the density functional theory underestimates the band gap energy and that the experimental value of bulk GaN is 3.42 eV6. Since a larger band gap energy is desirable for the application of GaN to power electronic devices, we attempt to increase the band gap energy. Given that the interface-nitrogen band is the origin of the narrow band gap, modifying the chemical components around the interface N atoms is a possible approach to enlarge the band gap. To examine this possibility, we point out that the interface-nitrogen band of the SiC/Si4O5N3 structure lies below its VBM17. In addition, since GaN can grow on a SiC substrate experimentally6, it would be experimentally possible to deposit SiC on a GaN substrate as well. We therefore examine a structure where two SiC bilayers (Si24C24) are inserted between the Si4O5N3 layer and the Si-doped GaN of the ttt1 structure (Fig. 4g).

The band gap energy of the SiC-inserted structure is 1.14 eV (Fig. 2h and Table 1). Although the band gap energy is still smaller than that of bulk GaN, tuning chemical environments around the interface N atoms is found to be effective for increasing the band gap energy. To explain why the band gap energy of the SiC-inserted structure is larger than that of the ttt1 structure, we study the profiles of the averaged difference Hartree potential29 \((\overline{{\rm{\delta }}{V}_{{\rm{H}}}}(z))\), which represents the electrostatic potential induced by charge transfer (See Methods for details). The averaged difference Hartree potential significantly drops in the SiC bilayers (Fig. 6a). On the other hand, such a drop is not found near the interface of the ttt1 structure (Fig. 6b). As with the ttt1 structure, the CBM-WF of the SiC-inserted structure mainly consists of the s orbitals of the N atoms. Therefore, the potential drop in the SiC bilayers does not affect the energy of the CBM. Therefore, \({\varepsilon }_{{\rm{C}}}^{{\rm{SiC}}-{\rm{inserted}}}=-1.87\,{\rm{eV}}\) is comparable to \({\varepsilon }_{{\rm{C}}}^{{\rm{ttt}}1}=-1.92\,{\rm{eV}}\) (Table 1). On the other hand, although the INBM-WF of the SiC-inserted structure resembles that of the ttt1 structure, the atomic orbitals of the SiC bilayers are mixed in it (Fig. 5f). Since the electrons in the SiC bilayers contribute to decrease the electrostatic potential energy of the INBM, \({\varepsilon }_{{\rm{I}}}^{{\rm{SiC}}-{\rm{inserted}}}=-3.01\,{\rm{eV}}\) is smaller than \({\varepsilon }_{{\rm{I}}}^{{\rm{ttt}}1}=-2.82\,{\rm{eV}}\) (Table 1). As a result, the band gap energy of the SiC-inserted structure is larger than that of the ttt1 structure.

Figure 6
figure 6

Profile of the average difference Hartree potential \(\overline{\delta {V}_{H}}(z)\). (a) SiC-inserted structure. (b) ttt1 structure. The z axis is set to be along the [0001] direction. The average z coordinate of the interface N atoms is set to zero. Note that \(\overline{{\rm{\delta }}{V}_{{\rm{H}}}}(-\infty )=0\).

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Finally, we note that there are pros and cons to both the ttt1 and SiC-inserted structures. The latter is more advantageous for power electric devices than the former in terms of the band gap energy. On the other hand, because of its structural complexity, synthesizing the SiC-inserted structure may be more difficult than the ttt1 structure.

Conclusion

We have proposed that the Si4O5N3 layer can epitaxially grow on a GaN(0001) surface without creating interface dangling bonds. Our ab initio calculations have shown that the dangling-bond-free GaN/Si4O5N3 structure is energetically more stable than the GaN/Si2O5 and GaN/Si2O3 structures. The electronic properties of the GaN/Si4O5N3 structure can be tuned by modifying the chemical components near the interface N atoms. We have also proposed a possible approach to experimentally synthesize the GaN/Si4O5N3 structure. Since it can be used as a seed for a high-quality GaN/SiO2 interface, the demonstration of the GaN/Si4O5N3 structure would be a milestone for improving the performance of GaN-based MISFETs.

Methods

Ab initio calculation

The ab initio calculations were performed using the OpenMX code30 which is based on the density functional theory with the generalized gradient approximation31 and the norm-conserving pseudopotentials32. The wave functions were expressed by the linear combination of pseudo-atomic orbitals (LCAO). The basis sets, Ga7.0-s2p2d2, N5.0s2p2d1, Si7.0-s2p2d1, O5.0-s2p2d1, PH5.0-s2p1, and C5.0-s2p2d1, were used for Ga, N, Si, O, pseudohydrogen with a fractional valence of 0.7533,34,35 and C atoms, respectively. Here, the abbreviation, for example Ga7.0-s2p2d2, indicates that two s, two p, and two d orbitals of a Ga atom with the cutoff radius of 7.0 Bohr were employed. Note that it was reported that the poor minimal basis set (Si7.0-s1p1 and C5.0-s1p1) did not reproduce the conduction band edge state of SiC36. However, we have confirmed that the proper basis set (Si7.0-s2p2d1 and C5.0-s2p2d1) reproduces it (Supplementary Fig. S1).

To study the Si4O5N3 layer on a GaN(0001) surface, a slab model composed of six GaN bilayers was used (Fig. 1a). Note that GDIS37 was used to construct the structure models and VESTA38 was used to illustrate the structure models. The effective screening medium method was employed to simulate an isolated slab39,40. The electrostatic potential is set zero at infinity in the \([000\bar{1}]\) direction. The dangling bonds of N atoms at the artificial GaN(\(000\bar{1}\)) surface are terminated by pseudohydrogen atoms with a fractional valence of 0.7533,34,35. The two GaN bilayers adjoining to the pseudohydrogen atoms were fixed in optimizing the geometry of the slab model. The \((\sqrt{3}\times \sqrt{3})R{30}^{^\circ }\) surface supercell is the primitive one for the GaN/Si4O5N3 structure. However, to study the substitutional doping effects on the band structure, we used the \((2\sqrt{3}\times 2\sqrt{3})R{30}^{^\circ }\) surface supercell, except for molecular dynamics simulations where the \((4\sqrt{3}\times 4\sqrt{3})R{30}^{^\circ }\) surface supercell was used. The in-plane unit-cell vectors were constrained to match the bulk GaN(0001) surface. In the geometry optimizations, the convergence criterion of 1 × 10−4 Hartree/Bohr for forces on atoms was used unless noted otherwise. The 2 × 2 × 1 mesh of k points was used in the geometry optimizations. The 1 × 1 × 1 mesh of k points was used in the molecular dynamics simulations. The 11 × 11 × 1 mesh of k points was used in the band structure calculations.

Formation energy

The formation energy of the GaN/Si4O5N3 structure per Si2N3 relative to the GaN/Si2O5 structure is defined as

$${f}_{{\rm{GaN}}/{{\rm{Si}}}_{4}{{\rm{O}}}_{5}{{\rm{N}}}_{3}}=-({E}_{{\rm{GaN}}/{{\rm{Si}}}_{4}{{\rm{O}}}_{5}{{\rm{N}}}_{3}}-12{\mu }_{{\rm{N}}}-8{\mu }_{{\rm{Si}}}-{E}_{{\rm{GaN}}/{{\rm{Si}}}_{2}{{\rm{O}}}_{5}})/4,$$
(1)

where \({E}_{{\rm{GaN}}/{{\rm{Si}}}_{4}{{\rm{O}}}_{5}{{\rm{N}}}_{3}}\) is the total energy of the GaN/Si4O5N3 per supercell, μN and μSi correspond to the energies per atom calculated for an N2 molecule and bulk Si, respectively, and \({E}_{{\rm{GaN}}/{{\rm{Si}}}_{2}{{\rm{O}}}_{5}}\) is the total energy of the GaN/Si2O5 structure per supercell. Note that the GaN/Si2O5 supercell is obtained by removing four Si2N3 units from the GaN/Si4O5N3 supercell, and the number of N (Si) atoms in the GaN/Si4O5N3 supercell is larger than that in the GaN/Si2O5 supercell by 12 (8).

We also define the formation energy of the GaN/Si4O5N3 structure per N atom relative to the GaN/Si2O5 structure as

$${f}_{{\rm{GaN}}/{{\rm{Si}}}_{4}{{\rm{O}}}_{5}{{\rm{N}}}_{3}}^{\ast }=-({E}_{{\rm{GaN}}/{{\rm{Si}}}_{4}{{\rm{O}}}_{5}{{\rm{N}}}_{3}}-12{\mu }_{{\rm{N}}}-8{\mu }_{{\rm{Si}}}-{E}_{{\rm{GaN}}/{{\rm{Si}}}_{2}{{\rm{O}}}_{5}})/12.$$
(2)

Note that \({f}_{{\rm{GaN}}/{{\rm{Si}}}_{4}{{\rm{O}}}_{5}{{\rm{N}}}_{3}}^{\ast }={f}_{{\rm{GaN}}/{{\rm{Si}}}_{4}{{\rm{O}}}_{5}{{\rm{N}}}_{3}}/3\).

The formation energy of the GaN/Si2O3 structure per O atom relative to the GaN/Si2O5 structure is defined as

$${f}_{{\rm{GaN}}/{{\rm{Si}}}_{2}{{\rm{O}}}_{3}}=-({E}_{{\rm{GaN}}/{{\rm{Si}}}_{2}{{\rm{O}}}_{3}}+8{\mu }_{{\rm{O}}}-{E}_{{\rm{GaN}}/{{\rm{Si}}}_{2}{{\rm{O}}}_{5}})/8,$$
(3)

where \({E}_{{\rm{GaN}}/{{\rm{Si}}}_{2}{{\rm{O}}}_{3}}\) is the total energy of the GaN/Si2O3 per supercell, μo corresponds to the energy per atom calculated for an O molecule. Note that the number of O atoms in the GaN/Si2O3 supercell is smaller than that in the GaN/Si2O5 supercell by 8.

One may claim that LCAO methods cannot properly calculate the energies of molecules, hence the formation energies. To check the accuracy, we compared the formation energies of β-Si3N4 and α-SiO2 obtained using the OpenMX code with those obtained using the QMAS code24,41 which is based on the projector-augmented-wave method. Here, the formation energy of β-Si3N4 per N atom is defined as

$${f}_{{\rm{\beta }}-{{\rm{Si}}}_{3}{{\rm{N}}}_{4}}=-({E}_{{\rm{\beta }}-{{\rm{Si}}}_{3}{{\rm{N}}}_{4}}-3{\mu }_{{\rm{Si}}}-4{\mu }_{{\rm{N}}})/4,$$
(4)

where \({E}_{{\rm{\beta }}-{{\rm{Si}}}_{3}{{\rm{N}}}_{4}}\) is the total energy of β-Si3N4 per Si3N4. The formation energy of α-SiO2 per O atom is defined as

$${f}_{{\rm{\alpha }}-{{\rm{SiO}}}_{2}}=-({E}_{{\rm{\alpha }}-{{\rm{SiO}}}_{2}}-{\mu }_{{\rm{Si}}}-2{\mu }_{{\rm{O}}})/2,$$
(5)

where \({E}_{{\rm{\alpha }}-{{\rm{SiO}}}_{2}}\) is the total energy of α-SiO2 per SiO2. We found that the OpenMX code underestimates \({f}_{{\rm{\beta }}-{{\rm{Si}}}_{3}{{\rm{N}}}_{4}}\) by 0.01 eV and overestimates \({f}_{{\rm{\alpha }}-{{\rm{SiO}}}_{2}}\) by 0.06 eV. The errors are considerably small compared to \({f}_{{\rm{GaN}}/{{\rm{Si}}}_{4}{{\rm{O}}}_{5}{{\rm{N}}}_{3}}^{\ast }\)= 5.80 eV and \({f}_{{\rm{GaN}}/{{\rm{Si}}}_{2}{{\rm{O}}}_{3}}\) = −3.95 eV.

Substitution energy

The substitution energy per Si atom is defined as

$$s=-({E}_{\mathrm{Si}-\mathrm{doped}}-n{\mu }_{{\rm{Si}}}+n{\mu }_{{\rm{Ga}}}-{E}_{{\rm{GaN}}/{{\rm{Si}}}_{4}{{\rm{O}}}_{5}{{\rm{N}}}_{3}})/n,$$
(6)

where ESi-doped is the total energy of the Si-doped GaN/Si4O5N3 structure per supercell, n is the number of doped Si atoms, and μGa is the energy per atom calculated for bulk α-Ga.

Average difference Hartree potential

The averaged difference Hartree potential is defined as

$$\overline{{\rm{\delta }}{V}_{{\rm{H}}}}(z)=\iint {\rm{\delta }}{V}_{H}(x,y,z)dxdy/\iint dxdy$$
(7)

where the z axis is set to be along the [0001] direction, δVH(x, y, z) is the difference Hartree potential29 at (x, y, z) associated with the difference electron density \(({\rm{\delta }}n({\boldsymbol{r}})=n({\boldsymbol{r}})-\sum _{i}{n}_{i}^{({\rm{a}})}({\boldsymbol{r}}))\), where n(r) is the electron density and \({n}_{i}^{({\rm{a}})}({\boldsymbol{r}})\) is the electron density of the atom i. Note that \(\overline{{\rm{\delta }}{V}_{{\rm{H}}}}(-\infty )=0\).