Introduction

Insulating oxides and nitrides are indispensable materials for diverse applications due to their superior mechanical, thermal, chemical and other outstanding high temperature properties. Furthermore, in the electronic industry these wide band gap materials are being considered for alternative gate oxides [1] and in the field of integrated optics they provide low-loss dielectric waveguides [2]. Recently the subject of wide bandgap oxides and nitrides have gained interest within the context of nanocrystals which offer silicon-based technology for light emitting devices and semiconductor memories [3]. These nanocrystals are embedded in an insulating matrix which is usually chosen to be silica [4–7]. However, other wide bandgap materials are also employed such as germania [8, 9], silicon nitride [10–12], and alumina [13–15]. As a matter of fact, the effect of different host matrices is an active research topic in this field.

Among these insulating oxides and nitrides technologically most important ones are SiO2, Al2O3, Si3N4. The activity around GeO2 is steadily increasing. Another closely-related material, Ge3N4 has attracted far less attention up to now even though it has certain interesting properties [16]. The major obstacle has been the sample growth. However, a very recent study reported an in situ Ge3N4 growth on Ge, demonstrating high thermal stability and large band offsets with respect to the Ge system [17]. In this comprehensive work, we present the ab initio structural and electronic properties of all these materials considering their common polymorphs; these are for SiO2: α-quartz, α- and β-cristobalite and stishovite phases, for GeO2: α-quartz, and rutile phases, for Si3N4 and Ge3N4: α- and β-phases and for Al2O3: α-phase. For amorphous and inherently imperfect matrices, these perfect crystalline phases serve as important reference systems. Moreover, due to their distinct advantages, epitaxial host lattices are preferred over the amorphous ones for specific applications.

With an eye on these technological applications, we focus on several physical properties of these lattices. The elastic constants play an important role on the strain profile of the embedded core semiconductor. Using Eshelby’s continuum elastic consideration [18] the radial and tangential stress fields of the nanocrystal can be determined [19]; these in turn, affect the optical properties [6]. The static and optical dielectric constants of these lattices introduce nontrivial local field effects that modify the absorption spectra of an isolated nanocrystal when embedded inside one of these matrices [20]. Based on the simple effective medium theory which has been tested by ab initio calculations [21], one can assess which host lattice and nanocrystal combination would possess the desired optical properties. Because of the dielectric mismatch between the nanocrystal core and the surrounding lattice, image charges will be produced [22]. These image charges should be taken into account in characterizing nanocrystal excitons [23]. Another promising application is the visible and near infrared electroluminescence from Si and Ge nanocrystals [3]. The electroluminescence is believed to be achieved by the recombination of the electron hole pairs injected to nanocrystals under high bias [3]. In this context the bulk state impact ionization process which can also give rise to electroluminescence is considered to be detrimental leading to dielectric breakdown. For high-field carrier transport, the crucial physical quantity was identified to be the valence and conduction band density of states (DOS) for each of the crystalline polymorph [24]. Based on these technology-driven requirements we compute the elastic constants, band structures, dielectric permittivities and electronic DOS of these aforementioned crystal polymorphs. Our ab initio framework is based on the density functional theory [25, 26], using pseudopotentials and a plane wave basis [27]. With the exception of Ge3N4 which was far less studied, vast amount of theoretical work is already available spread throughout the literature based on a variety of techniques [28–37]. Our first-principles study here enables a uniform comparison of important physical parameters within each material and among different insulating oxides and nitrides.

The plan of the paper is as follows: in Section “Details of ab initio computations" we provide details of our ab initio computations, Section “First-principles results" contains our first-principles results for the structural, electronic properties of the materials considered followed by our conclusions in Section “Conclusions”.

Details of ab initio computations

Structural and electronic properties of the polymorphs under consideration have been calculated within the density functional theory [25, 26], using the plane wave basis pseudopotential method as implemented in the ABINIT code [27]. The results are obtained under the local density approximation (LDA) where for the exchange-correlation interactions we use the Teter Pade parameterization [38], which reproduces Perdew-Zunger [39] (which reproduces the quantum Monte Carlo electron gas data of Ceperley and Alder [40]). We tested the results under two different norm-conserving Troullier and Martins [41] type pseudopotentials, which were generated by A. Khein and D.C. Allan (KA) and Fritz Haber Institute (FHI). For both pseudopotentials, the valence configurations of the constituent atoms were chosen as N(2s 2 p 3), O(2s 2 p 4), Al(3s 2 3p 1), Si(3s 2 3p 2), and Ge(4s 2 4p 2). The number of angular momenta of the KA (FHI) pseudopotentials and the chosen local channel were respectively, for N: 1, p (3, d), for O: 1, p (3, d), for Al: 2, d (3, d), for Si: 2, d (3, d), and for Ge: 1, p (3, s). Our calculated values for these two types of pseudopotentials were very similar, the only exceptional case being the elastic constants for rutile GeO2. Dielectric permitivity and the fourth-order tensor of elastic constants of each crystal are determined by starting from relaxed unit cell under the application of finite deformations within density functional perturbation theory [42] as implemented in ABINIT and ANADDB extension of it. Another technical detail is related with the element and angular momentum-resolved partial density of states (PDOS). To get a representative PDOS behavior we need to specify the spherical regions situated around each relevant atomic site. The radii of these spheres are chosen to partition the bond length in proportion to the covalent radii of the constituent atoms. This resulted in the following radii: for the α-quartz SiO2, r Si = 0.97 Å, r O = 0.65 Å, for the rutile GeO2, r Ge = 1.16 Å, r O = 0.69 Å, for the α-Al2O3, r Al = 1.32 Å, r O = 0.56 Å, and for the β-Si3N4, r Si = 1.03 Å, r N = 0.70 Å. It should be pointed that even though such an approach presents a good relative weight of the elements and angular momentum channels, it inevitably underestimates the total DOS, especially for the conduction bands. Other details of the computations are deferred to the discussion of each crystal polymorph.

First-principles results

First, we address the general organization and the underlying trends of our results. The lattice constants and other structural informations of all crystals are listed in Table 1. Table 2 contains the bond lengths and bond angles of the optimized oxide polymorphs. These results can be used to identify the representation of each polymorph within the amorphous oxides [52]. The elastic constants and dielectric permittivity tensor of each crystal are tabulated in Table 3 and Table 4, respectively. Very close agreement with the existing experimental data and previous calculations can be observed which gives us confidence about the accuracy and convergence of our work. Employing KA pseudopotentials, the band structure for the crystals are displayed along the high-symmetry lines in Figs. 1–4 together with their corresponding total DOS. Such an information is particlulary useful in the context of high-field carrier transport. These results are in good agreement with the previous computations [29, 32, 35, 36]. For all of the considered polymorphs the conduction band minima occur at the Γ point whereas the valence band maxima shift away from this point for some of the phases making them indirect band gap matrices (see Table 5). However, the direct band gap values are only marginally above the indirect band gap values. These LDA band gaps are underestimated which is a renown artifact of LDA for semiconductors and insulators [59]. In this work we do not attempt any correction procedure to adjust the LDA band gap values.

Table 1 Structural information on crystals
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Table 2 Bond lengths and bond angles (in degrees) of SiO2 and GeO2 polymorphs where x represents a Si or a Ge atom
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Table 3 Elastic constants and bulk modulus for each crystal
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Table 4 Dielectric permittivity tensor
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Table 5 Indirect (E g) and direct (E g(Γ)) LDA Band Gaps for each crystal
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Fig. 1
figure 1

LDA band structure and total DOS (electrons/eV cell) of (a) α-cristobalite SiO2, (b) α-quartz SiO2, (c) β-cristobalite SiO2, and (d) stishovite SiO2

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Fig. 2
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LDA band structure and total DOS of (a) α-quartz GeO2, (b) rutile GeO2

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Fig. 3
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LDA band structure of and total DOS of α-Al2O3

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Fig. 4
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LDA band structure and total DOS of (a) α-Si3N4; (b) β-Si3N4; (c) α-Ge3N4 and (d) β-Ge3N4

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We present in Figs. 5–7 the element- and angular momentum-resolved PDOS. A common trend that can be observed in these various lattices is that their valence band maxima are dominated by the p states belonging to O atoms; in the case of Si3N4 and Ge3N4 they are the N atoms. For the conduction band edges, both constituent elements have comparable contribution. This parallels the observation in amorphous SiO2 where due to large electronegativity difference between Si and O, the bonding orbitals have a large weight on O atoms whereas the lowest conduction band states with antibonding character have a significant contribution from the Si atoms [60].

Fig. 5
figure 5

DOS of α-quartz SiO2 (a) Element-resolved; total, PDOS of Si, PDOS of O; (b) Angular momentum-resolved; Si s electrons, Si p electrons, Si d electrons (not visible at the same scale), O s electrons, O p electrons.

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Fig. 6
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DOS of rutile GeO2 (a) Element-resolved; total PDOS of Ge, PDOS of O; (b) Angular momentum-resolved; Ge s electrons, Ge p electrons, Ge d electrons, O s electrons, O p electrons

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Fig. 7
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Element-resolved DOS of (a) β-Si3N4; total, PDOS of Si, PDOS of N, (b) β-Ge3N4; total, PDOS of Ge, PDOS of N

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From another perspective, the band structures and the associated DOS reveal that there is a marked discrepancy between the valence and conduction band edges where for the former there occurs a sharp increase of DOS just below the band edge. As the probabilities of most scattering processes are directly proportional to DOS [61], in the case of high-field carrier transport the electrons should encounter far less scatterings and hence gain much higher energy from the field compared to holes. In this respect Si3N4 and Ge3N4 are further different from the others where for both conduction and valence bands the DOS dramatically increases (cf. Fig. 4) so that the carriers should suffer from excessive scatterings which practically precludes the bulk impact ionization for this material.

Another common trend can be investigated between the density of each polymorph and the corresponding static permittivity, εs. Such a correlation was put forward by Xu and Ching among the SiO2 polymorphs [29]. We extend this comparison to all structures considered in this work and rather use χe =  εs-1 which corresponds to electric susceptibility. It can be observed from Fig. 8 that the trend established by SiO2 polymorphs is also followed by β-Si3N4 and α-Al2O3. On the other hand, Ge-containing structures while possessing a similar trend among themselves, display a significant shift due to much higher mass of the this atom. This dependence on the atomic mass needs to be removed by finding a more suitable physical quantity. We should mention that such a correlation does not exist between the volume per primitive cell of each phase and the static permittivity. After these general comments, now we concentrate on the results of each lattice individually.

Fig. 8
figure 8

Density versus direction-averaged static electric susceptibility

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SiO2

The α-quartz SiO2 is one of the most studied polymorphs as it is the stable phase at the ambient pressure and temperature [30, 34], furthermore its short-range order is essentially the same as the amorphous SiO2 [60]. α-quartz SiO2 has a hexagonal unit cell containing three SiO2 molecules. A plane-wave basis set with an energy cutoff of 60 Ha was used to expand the electronic wave functions at the special k-point mesh generated by 10 × 10 × 8 Monkhorst-Pack scheme [62]. The band structure of α-quartz SiO2 has been calculated by many authors (see, for instance [28, 29]). Our calculated band structure and total DOS shown in Fig. 1a are in agreement with the published studies [29]. The indirect LDA band gap for this crystal is 5.785 eV from the valence band maximum at K to the conduction band minimum at Γ. The direct LDA band gap at Γ is slightly larger than the indirect LDA band gap as seen in Table 5. Calculated values of the elastic constants and bulk modulus listed in Table 3 are in good agreement with the experiments. Apart from C 12, the elastic constants are within 10% of the experimental values. The discrepancy in C 12 can be explained by the fact that C 12 is very soft and this type of deviation also exists among experiments which is also the case for C 14.

α-cristobalite SiO2 has a tetragonal unit cell containing four SiO2 molecules. In the course of calculations an absolute energy convergence of 10−4 Ha was obtained by setting a high plane wave energy cutoff as 60 Ha and 10 × 10 × 8  k-point sampling. Figure 1b shows the band structure of α-cristobalite SiO2 with the 5.525 eV direct band gap at Γ. The bulk modulus of 12 GPa is the smallest among all the host lattice polymorphs considered in this work.

Regarding β-cristobalite, its actual structure is somewhat controversial, as a number of different symmetries have been proposed corresponding to space groups Fd3m, \({I\overline{4}2d},\) and P213 [34]. Recently, incorporating the quasiparticle corrections the tetragonal \({I\overline{4}2d}\) phase was identified to be energetically most stable [63]. However, we work with the structure having the space group of Fd3m that was originally proposed by Wyckoff [64] and which is widely studied primarily due to its simplicity [28, 30]. This phase has a cubic conventional cell with two molecules. We used 60 Ha plane wave energy cutoff and 10 × 10 × 10  k-point sampling. Figure 1c shows the band structure of β-cristobalite SiO2 with the 5.317 eV direct band gap at Γ. Unlike their band structures, total DOS of α- and and β-cristobalite SiO2 are very similar (cf. Fig. 1c). This similarity can be explained by the fact that their local structure are very close. On the other hand there is a considerable difference between the DOS spectra of the α-quartz SiO2 and the β-cristobalite SiO2. In Table 3, we present elastic constants of the β-cristobalite SiO2 calculated by two types of pseudopotentials, FHI and KA. There is no considerable difference between them. Dielectric constants of β-cristobalite SiO2 are the smallest among the five polymorphs of SiO2 studied here (see Table 4).

Stishovite is a dense polymorph of SiO2 with octahedrally coordinated silicon, unlike the previous phases [34]. It has a tetragonal cell with two molecules. Calculations were done by using 60 Ha plane wave energy cutoff and 8 × 8 × 10 k-point sampling. The band structure of stishovite with a wide single valence band is markedly different from that of the previous three crystalline phases of SiO2 having two narrow upper valence bands. The cause of this increased valence bandwidth is the lack of separation between bonding and nonbonding states [36]. Hence, the total DOS for stishovite shows no gap at the middle of the valence band (see Fig. 1d). Our calculations yield a direct LDA band gap of 5.606 eV at Γ. As seen in Table 3, the differences between our computed elastic constants and the experimental values are less than 3%; this is an excellent agreement for LDA. Its bulk modulus is the largest among all the host lattice polymorphs considered in this work. Moreover, dielectric constants of stishovite is the largest of the five polymorphs of SiO2 considered in this work (see Table 4).

GeO2

For α-quartz GeO2 we used the same energy cutoff and k-point sampling as with α-quartz SiO2 which yields excellent convergence. The band structure of the α-quartz GeO2 is displayed in Fig. 2a. The similarity of the band structures of the α-quartz GeO2 and the α-quartz SiO2 is not surprising as they are isostructural. Similarly their total DOS resemble each other (cf. Fig. 2a). The indirect LDA band gap for this phase is 4.335 eV from the valence band maximum at K to the conduction band minimum at Γ. The direct band gap at Γ is slightly different from indirect band gap as seen in Table 5. This gap is smaller than that of the α-quartz SiO2. The perfect agreement between calculated elastic constants of the α-quartz GeO2 and experimental values [45, 54] can be observed in Table 3.

The rutile structure of GeO2, also known as argutite [65] is isostructural with the stishovite phase of SiO2. The same energy cutoff and k-point sampling values as for stishovite yield excellent convergence. The direct LDA band gap at Γ for rutile-GeO2 is less than that of stishovite with a value of 3.126 eV. The two upper valence bands are merged in the total DOS (see Fig. 2b) as in the case of stishovite. The increased valence bandwidth in the band structure can be explained by the same reason as in the case of stishovite. The results of the elastic constants calculated with KA type pseudopotential shown in Table 3 deviate substantially from the experiment whereas the agreement with the FHI pseudopotentials is highly satisfactory. The similarity of the dielectric constants of rutile GeO2 and stishovite can be observed in Table 4.

Al2O3

Al2O3 is regarded as a technologically important oxide due to its high dielectric constant and being reasonably a good glass former after SiO2 [1]. The α-Al2O3 (sapphire) has the rhombohedral cell with two molecules. Computations about Al2O3 were done by using 60 Ha plane wave energy cutoff and a total of 60 k-points within the Brillouin zone. Figure 3 shows the computed band structure and total DOS of the α-Al2O3. These are in excellent agreement with the previous calculation [31, 33]. For Al2O3, minimum of the conduction band is at Γ and maximum of the valence band is at a point along Γ−X close to the Γ point. The corresponding LDA band gap is 6.242 eV. Because of the very small difference between the direct and indirect band gaps, Al2O3 is considered as a direct band gap insulator. Measured band gap of this crystal is 8.7 eV. However the precise value of the gap of Al2O3 is still elusive because of the existence of an excitonic peak near the absorbtions edge [66]. As seen in Table 3, computed values of the elastic constant and bulk modulus of Al2O3 are in excellent agreement with the experiments. As a further remark, the α-Al2O3 unit cell can be described as hexagonal or rhombohedral depending on the crystallographical definition of the space group \({\hbox{R}\overline{3}\hbox{C}}.\) During our first-principles calculations it has been defined as rhombohedral in which case C 14 vanishes. Although the sign of C 14 is experimentally determined to be negative for the hexagonal-Al2O3, previous calculations reported a positive value [67]. To check this disagreement we have calculated the elastic constant of the hexagonal-Al2O3 and found it to be around −3.0.

Si3N4 and Ge3N4

The research on silicon nitride has largely been driven by its use in microelectronics technology to utilize it as an effective insulating material and also as diffusion mask for impurities. Recently it started to attract attention both as a host embedding material for nanocrystals [10–12] and also for optical waveguide applications [2]. The α- and β-Si3N4 have hexagonal conventional cells with four and two molecules, respectively. We used 60 Ha plane wave energy cutoff and 6 × 6 × 8  k-point sampling. The computed band structures of these two phases shown in Fig. 4a and b are identical to those reported by Xu and Ching [32]. The top of the valence band for β-Si3N4 is along the Γ-A direction, and for α-Si3N4 it is at the M point. The bottom of the conduction band for two phases are at the Γ point. The direct and indirect LDA band gaps of these two phases are respectively, 4.559, 4.621 eV for α-Si3N4 and 4.146, 4.365 eV for the β-Si3N4. The general band structure of two phases are very similar, except that the α-Si3N4 has twice as many bands because the unit cell is twice as large. The total DOS of these two phases shown in Fig. 4a and b are only marginally different. Calculated values of the elastic constants and bulk modulus of β-Si3N4 listed in Table 3 are in excellent agreement with the quoted experiments. Those for the α-Si3N4 which is thermodynamically less stable with respect to β-phase [68] were left out due to excessive memory requirements for the desired accuracy.

Ge3N4 is the least studied material among the oxides and nitrides considered in this work. Recently its high-pressure γ-phase has attracted some theoretical interest [69]. However, the available Ge3N4 samples contain a mixture of α and β-phases as in the case of Si3N4 and these are the polymorphs that we discuss in this work. The band structures of both of these phases of Ge3N4 (cf. Fig. 4) are very similar to those of Si3N4. Regarding the elastic constants of β-Ge3N4, our theoretical results listed in Table 3 await experimental verification. In terms of density, the β phases of Si3N4 and Ge3N4 fill the gap between the α-quartz and stishovite/rutile phases of their oxides. As can be observed from Fig. 8 their electric susceptibility versus density behavior strengthens the correlation established by the remaining polymorphs. Finally it should be pointed that β-Ge3N4 has the largest high-frequency dielectric constant (ε∞) among all the materials considered in this work.

Conclusions

A comprehensive first-principles study is presented which is unique in analyzing common polymorphs of the technologically-important insulating oxides and nitrides: SiO2, GeO2, Al2O3, Si3N4, and Ge3N4. The structural parameters, elastic constants, static and optical dielectric constants are obtained in close agreement with the available results. The computed dielectric constants are observed to display a strong correlation with their mass densities. For all of the considered polymorphs the conduction band minima occur at the Γ point whereas the valence band maxima shift away from this point for some of the phases making them indirect band gap matrices. However, the direct band gap values are only marginally above the indirect band gap values. The investigation of band structure and DOS data reveal that the holes in all polymorphs considered and the electrons for the case of Si3N4 and Ge3N4 should suffer excessive scatterings under high applied field which will preclude bulk impact ionization for these carrier types and polymorphs. This can be especially important for applications vulnerable to dielectric breakdown.