theoretical investigation of the solid state reaction of silicon nitride and silicon dioxide forming...

Theoretical Investigation of the Solid State Reaction of Silicon Nitride andSilicon Dioxide forming Silicon Oxynitride (Si2N2O) under Pressure

Peter Kroll* and Matus Milko

Aachen, Institut für Anorganische Chemie der Rheinisch-Westfälische Technische Hochschule

Received October 1st, 2002.

Abstract. The high-pressure behavior of Si2N2O is studied for pres-sures up to 100 GPa using density functional theory calculations.The investigation of a manifold of hypothetical polymorphs leadsus to propose two dense phases of Si2N2O, succeeding the ortho-rhombic ambient-pressure polymorph at higher pressures:a defectspinel structure at moderate pressures and a corundum-type struc-ture at very high pressures. Taking into account the formation ofsilicon oxynitride from silicon dioxide and silicon nitride and itspressure dependence, we propose five pressure regions of interestfor Si2N2O within the pseudo-binary phase diagram SiO2-Si3N4: (i)stability of the orthorhombic ternary phase of Si2N2O up to 6 GPa,

Theoretische Untersuchung der Festkörperreaktion zwischen Siliciumnitrid undSiliciumdioxid unter Bildung von Siliciumoxidnitrid (Si2N2O) unter Druck

Inhaltsübersicht. Das Verhalten von Si2N2O unter hohem Druck(bis 100 GPa) wird durch Rechnungen auf der Grundlage der Dich-tefunktionaltheorie (DFT) untersucht. Unter der Vielzahl von hy-pothetischen Polymorphen sind zwei dichte Phasen von Si2N2O,die energetisch die orthorhombische Normaldruckphase übertref-fen, eine Defekt-Spinell-Struktur bei moderaten und eine Korund-Typ-Struktur bei sehr hohen Drücken. Die Bildung von Silicium-oxidnitrid aus Siliciumdioxid und Siliciumnitrid und seine Druck-abhängigkeit führt uns zu fünf Druckgebieten, die für Si2N2O in-nerhalb des pseudobinären Phasendiagramms des Systems SiO2-

1 Introduction

The fourfold coordinated silicon atom is the basic structuralmotif of the crystalline phases of silicon dioxide (SiO2), sili-con nitride (Si3N4), and silicon oxynitride (Si2N2O) at am-bient conditions, as well as of non-stoichiometric SiNO-glasses. Such materials are widely used in the ceramic andglass industries and have an enormous application potentialin optical fibers, in microelectronics and in catalysis. Silicondioxide, SiO2 or silica, displays a rich crystal chemistry andseveral different crystalline modifications are known at am-bient conditions [1]: -/β-quartz, -/β-tridymite, -/β-cris-tobalite, all with the structural motif of SiO4-tetrahedra,each two sharing a common corner. At pressures between

* Dr. P. KrollInst. f. Anorg. Chemie der RWTHProf.-Pirlet-Str. 1D-52056 AachenE-mail: [email protected]

Z. Anorg. Allg. Chem. 2003, 629, 17371750 DOI: 10.1002/zaac.200300122 2003 WILEY-VCH Verlag GmbH & Co. KGaA, 69451 Weinheim 1737

(ii) a phase assemblage of coesite, stishovite, and β-Si3N4 between6 and 11 GPa, (iii) a possible defect spinel modification of Si2N2Obetween 11 and 16 GPa, (iv) a phase assemblage of stishovite andγ-Si3N4 above 40 GPa, and (v) a possible ternary Si2N2O phasewith corundum-type structure beyond 80 GPa. The existence ofboth ternary high-pressure phases of Si2N2O, however, depends onthe delicate influence of configurational entropy to the free energyof the solid state reaction.

Keywords: Silicon; Silicon nitride; Silicon oxynitride; Solid state re-actions

Si3N4 von Interesse sind: (i) Die orthorhombische ternäre PhaseSi2N2O ist bis 6 GPa stabil; (ii) ein Phasengemenge aus Coesit, Stis-hovit und β-Si3N4 existiert zwischen 6 und 11 GPa; (iii) zwischen11 und 16 GPa existiert möglicherweise eine Defekt-Spinell-Phase;(iv) oberhalb von 40 GPa existiert ein Phasengemenge aus Stishovitund γ-Si3N4; (v) oberhalb 80 GPa ist eine ternäre Si2N2O-Phasemit Korundstruktur möglich. Die Existenz beider ternärer Hoch-druckphasen von Si2N2O hängt von einer Feinabstimmung derEinflüsse von Konfigurationsentropie und freier Energie der Fest-körperreaktion ab.

2 GPa and 2.5 GPa -quartz transforms into coesite, whichis about 13 % denser than -quartz. Nevertheless, the struc-ture of coesite is still built up from SiO4-tetrahedra. Atpressures above 7 GPa to 12 GPa, the exact phase bound-ary depends on temperature, the only stable polymorph ofSiO2 is stishovite [2]. Its structure is isotypic to the rutilestructure and comprises silicon octahedrally coordinated byoxygen. For a detailed account of the SiO2 phase diagramwe refer to the recent article of Mao et al. [3]. Experimentaland theoretical investigations have given evidence for post-stishovite phases at pressures beyond 50 GPa [47].

The structural complexity of silicon nitride is less diverseand only two polymorphs, - and β-Si3N4, are known atambient conditions. Both structures consist of Si atomstetrahedrally coordinated by four N atoms. Each three ofthese SiN4-tetrahedra share a common corner. Only re-cently, a new polymorph of Si3N4 was synthesized at press-ures of about 15 GPa [8]. γ-Si3N4 adopts the spinel struc-ture and comprises both SiN6-octahedra and SiN4-tetra-hedra in a ratio 2:1. Theoretical calculations suggests that

P. Kroll, M. Milko

post-spinel phases of Si3N4 appear at pressures of 150 GPaand beyond [9, 10]. It is worthy to note that both high-pressure phases, stishovite-SiO2 and γ-Si3N4, can bequenched to zero pressure and are meta-stable under ambi-ent conditions.

Besides these two antipodes, silicon oxynitride Si2N2O isthe only known ternary crystalline modification within thepseudo-binary SiO2-Si3N4-phase diagram [11]. The struc-ture of Si2N2O, its mineral name is sinoite, is built ofSiN3O-tetrahedra: puckered hexagonal Si2N2-layers arelinked together through O atoms, which complete the tetra-hedra around the Si atoms. Si2N2O can be synthesized froma mixture of SiO2 and Si3N4 according to the reaction:

1/2 SiO2 1/2 Si3N4 Si2N2O (1)

Consequently, it is found very often in ceramic materials atthe phase boundary between silicon nitride and silicon di-oxide [12]. A high pressure phase of Si2N2O is yet un-known. A recent study investigated the formation ofSi2N2O from SiO2 and Si3N4 for pressures up to 5 GPa, butdid not detect a new ternary phase [13]. Taking the press-ures of formation of stishovite and γ-Si3N4 and using thecrude assumption of a “mixing rule”, one might estimate ahigh-pressure modification of Si2N2O to appear at approxi-mately 10-15 GPa. A high-coordinated and dense oxyni-tride phase, however, might not be a stable ternary modifi-cation of Si2N2O, since it can in principle at any givenpressure decompose into the binary constituents accord-ing to:

Si2N2O 1/2 SiO2 1/2 Si3N4 (2)

Reaction (2) obviously is the reverse of reaction (1).The goal of our study is two-fold: first we investigate

crystal structures of compounds with two cations and threeanions in search of possible high-pressure modifications ofSi2N2O. We calculate a variety of possible polymorphs andfrom the results we extract the phase behavior of a ternarySi2N2O under pressure. Then we access the free enthalpy∆H of reaction (2) as a function of pressure, explicitly look-ing for a possible decomposition of the homogeneous ter-nary phase into a phase assemblage of SiO2 and Si3N4. Thisrequires a calculation of SiO2- and Si3N4-polymorphs underpressure. While the results of our density functional calcu-lations refer to zero temperature only, we access the domi-nant contributions to the free energy ∆G of the reaction byestimating the configurational entropy of the ternary phasein a third step. A discussion in which we propose the opti-mum pressure region for the synthesis of ternary siliconoxynitride and a final summary concludes this study.

2 Computational Method

The total energies, atomic structures and atomic dynamicsof all structures we consider are calculated within densityfunctional theory (DFT) [14]. We used the Vienna ab-initio

2003 WILEY-VCH Verlag GmbH & Co. KGaA, 69451 Weinheim zaac.wiley-vch.de Z. Anorg. Allg. Chem. 2003, 629, 173717501738

Simulation Package (VASP), which combines the total en-ergy pseudopotential method with a plane-wave basis set[1517]. Both approximations to the electron exchange andcorrelation energy, the local density approximation (LDA)[18, 19] as well as the general gradient approximation(GGA) [20] are used. The pseudopotential taken for Si was“hard”, while those taken for N and O were “ultra-soft”.All results presented in this paper are calculated using500 eV as the cut-off energy for the expansion of the wave-function into the plane wave basis set. Brillouin-Zone inte-gration was done via the Monkhorst-Pack scheme [21].Atomic forces were relaxed to values below 1·102 eV/Aand stresses below 1 kbar. The bulk modulus is obtained byvarying the volume around the zero-pressure volume V0 (/1.5 %, in steps of 0.3 %) and subsequently fitting the cal-culated energy E in the E-V graph to Murnaghans equationof state (EOS) [22]. Murnaghans EOS, despite its simplicity,typically describes the compressibility of many compoundsup to surprisingly high pressures [23]. The E-V diagramscan be transformed easily to yield enthalpy vs. pressure dia-grams (H-p): The pressure p can be extracted from the E-Vgraph by a simple numerical differentiation: p E/ V.The enthalpy H is calculated via H E p·V. Havingcalculated p and H it is good practice to plot the enthalpyrelative to a reference phase (∆H-p diagram).

In general, a comparison of LDA and GGA results isvery useful. The LDA typically underestimates the latticevolume by 1-3 % for structures of main-group elementssuch as in this study. GGA on the contrary typically over-estimates volumes by 3-8 %. Elastic properties of such com-pounds are described very good within the LDA, while theGGA underestimates compressibility and elastic constantsby 10-20 %. On the other hand, it has been documentedthat gradient corrections offer significant improvementswhen structures with different environments for the atomsare compared with each other, especially for the estimationof transition pressures [24]. In this study we encounter sucha situation. Consequently, we will provide GGA results forall cases where energy differences between phases are con-sidered especially for the reaction free energy. LDA re-sults are presented for all elastic and vibrational properties.

We did a careful controlling of our results: finer grids forthe k-point sampling were employed as well as larger planewave basis sets. We checked the optimized structures forresidual Pulay-stresses and, furthermore, we re-calculatedthem using “hard” pseudopotentials for N and O and acutoff of 800 eV [25]. All further results confirmed the con-vergence of structural properties at this level of accuracy.The total energy differences are found to be converged tovalues of 0.01 eV per formula unit of Si2N2O (about 2 meVper atom).

The application of such calculations as predictive toolsin solid state physics and solid state chemistry has beendemonstrated in numerous cases. Recent successful ex-amples of related systems are found within prediction andvalidation of SiO2-phases [26, 27], γ-Si3N4 [8], γ-Ge3N4 [28],

Theoretical Investigation of the Solid State Reaction of Silicon Nitride and Silicon Dioxide

Table 1 Energy and volume (per formula unit SiO2), density and bulk modulus of -quartz, coesite, and stishovite calculated within theLDA and GGA in comparison with experimental results.

-quartz-SiO2 coesite-SiO2 stishovite-SiO2

LDA GGA Exp. [35] LDA GGA Exp. [36] LDA GGA Exp. [37]

E/eV 26.16 24.00 26.15 23.91 26.17 23.45V/A3 37.21 40.74 37.66 33.92 35.61 34.18 22.90 24.00 23.3ρ/(g/cm3) 2.68 2.44 2.64 2.93 2.79 2.91 4.34 4.14 4.27B0/GPa 34 29 37 95 92 101 311 265 313

post-spinel phases of Si3N4 [9, 10], and of the P3N5 phasediagram [29].

3 Results

3.1 Binary phases

3.1.1 SiO2

A comprehensive description and theoretical investigationof different phases of SiO2 using the same density func-tional framework as in study is given in Ref. [27]. A theor-etical study of several SiO2 polymorphs which might existat very high pressures is presented in Ref. [26]. We re-calcu-lated these phases of SiO2 and the corresponding phase dia-gram. It turned out that the polymorphs, which are mostimportant for our study of the reaction 1, are those of -quartz, coesite and stishovite. Hereafter, the three relevantphases are briefly described. Phases even denser than sti-shovite, an orthorhombic CaCl2-type form as well as aquenchable -PbO2-type polymorph [30], are known to ap-pear at pressures beyond 50 GPa. The detailed crystallo-graphic data of the calculated phases are given in the ap-pendix and compared with experimental results.

-Quartz. The SiO4-tetrahedra in -quartz are linkedover common corners and form helices. Both a left-handedand a right handed modification of -quartz exists, withspace group P 3221 and P 3121, respectively. The unit cellof -quartz contains three formula units. We did our calcu-lation using 6x6x8 and 3x3x4 k-point meshes.

Coesite. Coesite is a high-pressure polymorph of silicaappearing at 2-2.5 GPa with a rather complex monoclinicstructure [31, 32]. Similar to -quartz the SiO4-tetrahedrain coesite are linked over common corners. We used thespace-group symmetry C 2/c for the crystal structure of co-esite. The calculation we done using a 4x4x4 k-point meshwithin the primitive unit cell containing 8 formula units.

Stishovite. At higher pressures (above 7-12 GPa) SiO2 ad-opts a rutile-type structure with octahedrally coordinatedSi atoms [2]. This modification of SiO2, stishovite, is re-garded the third hardest material after diamond and cubicboron nitride [33]. Although first synthesized in the labora-tory stishovite was also found in meteor craters [34]. Thespace group symmetry of the crystal structure of stishoviteis P 42mnm and we used a 4x4x8 k-point mesh for the Bril-louin zone integration.

Z. Anorg. Allg. Chem. 2003, 629, 17371750 zaac.wiley-vch.de 2003 WILEY-VCH Verlag GmbH & Co. KGaA, 69451 Weinheim 1739

Figure 1 Enthalpy-pressure diagram of -quartz, coesite, and sti-shovite-SiO2. The enthalpy is given per formula unit SiO2 relativeto that of -quartz. Two arrows indicate the transition pressuresfor the -quartz coesite and the coesite stishovite tran-sitions, respectively.

In Table 1 we collect our calculated results of energy, vol-ume, density, and bulk modulus and compare them withexperimental data. The data show the general trend of DFTresults: the experimental volume is confined by the LDAand GGA volumes yielding a lower and an upper bound-ary, respectively. The bulk modulus of a structure is esti-mated more reliable within the LDA. The enthalpy-pressurediagram (at zero temperature) of the three phases of SiO2

is shown in Figure 1. We calculate the transition pressurepqc

t of the -quartz coesite transition to 3.7 GPa withinthe GGA. This agrees fairly well with the experimental dataof 2-2.5 GPa, collected from several experiments as given inthe article of Mao et al.[3]. The transition pressure pcs

t ofthe coesite stishovite transition comes out to 5.9 GPa.This value agrees very good with the compiled experimentalresults extrapolated to low temperatures, at which a pt of6 GPa is expected [3].

With increasing pressure, thus, the phase transitions ofSiO2 can be summarized as follows:

-quartz(4)

14%

coesite(4)

48%

stishovite(6)

where the numbers inside the parentheses are polyhedralcoordination numbers of Si, and the percentages are theincreases in density from one to the other modification cal-culated for zero pressure [38].

P. Kroll, M. Milko

Table 2 Energy and volume (per formula unit Si3N4), density and bulk modulus of β-Si3N4 and γ-Si3N4 calculated within the LDA andGGA and compared with experimental results.

β-Si3N4 γ-Si3N4

LDA GGA Exp. [41, 42] LDA GGA Exp. [43, 44]

E/eV 63.66 57.81 63.15 56.73V/A3 71.34 73.88 72.88 56.58 58.69 57.92ρ/(g/cm3) 3.26 3.15 3.20 4.12 3.97 4.02B0/GPA 251 234 259 319 292 290-302

3.2 Si3N4

The recent successful synthesis of a high-pressure modifi-cation of Si3N4 with spinel structure has initiated a vigorousresearch in high-pressure synthesis of main group elementnitride materials [39]. There are two modifications of Si3N4

under ambient conditions, - and β-Si3N4. γ-Si3N4 isformed between 10 and 13 GPa [8]. Phases even denser thanγ-Si3N4 might appear beyond 150 GPa [9, 10]. The twophases of Si3N4 important for the purpose of this studyare β-Si3N4 and γ-Si3N4. The crystallographic data of bothphases are given in the appendix.

β-Si3N4. β-Si3N4 adopts an ordered phenacite structure.Each three SiN4-tetrahedra are linked through a nitrogenatom at the common corner. The nitrogen atoms are (ap-proximately) trigonal planar coordinated. Si and N atomsare located in layers stacked in an ABAB-sequence. Thestructure of -Si3N4 is very similar to that of β-Si3N4.Consequently the energy difference is calculated to be verysmall and within the error of the method, both within ourcalculations and most recent calorimetric studies [40]. Inour calculations there was no significant enthalpy differencebetween the two polymorphs for up to 30 GPa. Beyond thispressure we find that β-Si3N4 becomes mechanically un-stable, which is rather similar to the behavior proposed forβ-Ge3N4 [28], but in this case with a distortion towards amonoclinic structure. The unit cell of β-Si3N4 contains 14atoms and we used a 3x3x8 mesh for k-point sampling.

γ-Si3N4. γ-Si3N4 adopts the spinel structure with spacegroup Fd3m. Si appears both octahedrally and tetrahedrallycoordinated. The ratio of SiN6-octahedra and SiN4-tetra-hedra is 2:1. Viewed perpendicular to the [111]-direction,the structure is build up from octahedral layers alternatingwith mixed octahedral/tetrahedral layers. For our calcu-lation of γ-Si3N4 we used the primitive unit cell with 14atoms and a 4x4x4 k-point mesh.

Table 2 contains calculated results of energy, volume,density, and bulk modulus of the two Si3N4-phases togetherwith experimental data. Similar trends as observed alreadyfor SiO2 hold also for the two Si3N4-structures: while LDAand GGA results of the volume confine the experimentalresults, the LDA only gives a reliable estimate of the bulkmodulus. Figure 2 shows the calculated enthalpy-pressurediagram of the two Si3N4 phases. For the transition press-ure pt of the β-Si3N4 γ-Si3N4 transition we obtain a valueof 12.1 GPa. This value of pt agrees excellently with theexperimentally observed transition pressures (10-13 GPa)


Figure 2 Enthalpy-pressure diagram of β-Si3N4 and γ-Si3N4. Theenthalpy is given per formula unit Si3N4 relative to that of β-Si3N4.The arrow indicates the transition pressures for the β-Si3N4 γ-Si3N4 transition.

[43]. The LDA result of pt is 5 GPa. We, therefore, summar-ize the phase transitions of Si3N4 with increasing pressureas follows:

β-Si3N4

(4)26%

γ-Si3N4

(4, 6)

Again, the numbers inside the parentheses denote the poly-hedral coordination numbers of Si, and the percentages arethe increase in density calculated for zero pressure.

With regard to the two different approximations to theelectron exchange and correlation energy we conclude thatthe comparison of LDA and GGA results of pt with exper-imental results in both systems SiO2 and Si3N4 demon-strates the superiority of the GGA. The LDA follows thetrend to overrate the bonding energy of high-coordinatedstructures, while the GGA apparently gives more reliableresults for energy differences between structures with differ-ent coordination of atoms.

3.3 Polymorphs of Si2N2O

3.3.1 The low-pressure modification Cmc21-Si2N2O

At ambient conditions Si2N2O adopts an orthorhombicstructure with space group Cmc21 (36) [45, 46]. Its mineralname is sinoite. The structure, which is shown in Figure 3,is built up of SiN3O-tetrahedra linked together to form a 3-dimensional structure. There are at least two different viewsto look at the structure: A first one considers hexagonalSi2N2-layers stacked on top of each other. These layers are


Figure 3 The structure of orthorhombic Cmc21-Si2N2O. All Siatoms are tetrahedrally coordinated. On the left side a ball-and-stick representation (Si black, O dark grey, N light grey), and onthe right side a polyhedral view of the structure. Two unit cellsare indicated.

puckered and connected via O-atoms, which complete thetetrahedra around the Si atoms. One half of the Si atomsare linked to the layer above and the other half to the layerbelow. Nitrogen atoms are approximately trigonal-planarcoordinated, and the Si-O-Si bond angle is bent. The struc-tural formula can be written as Si[4]

2 N[3]2 O[2]; the superscript

denotes the coordination of atoms. The space between theSi2N2-layers is large enough to accommodate additionalatoms; e. g. O can be replaced by the isoelectronic N-Hgroup. The structure of this Si2N2(NH) is indeed isotypicto Si2N2O [47]. Moreover, the O atom can be replaced bythe pseudo-chalcogen -NCN-, the carbodiimide moiety,leading to the very similar and even wider spaced structureof silicon carbodiimide nitride, Si2N2(NCN) [48, 49]. Tak-ing a different perspective, the orthorhombic structure ofSi2N2O is related to a high-pressure modification of B2O3

[50]. This structure can be considered as a defect variant ofwurtzite, in which a third of all cations are discarded [51].This picture becomes even more evident in a projectiononto the x,y-plane of the compressed (at 40 GPa) structureof Si2N2O. We performed all our calculations using the con-ventional unit cell of Cmc21-Si2N2O with 20 atoms and a2x4x4 k-point mesh. Our results of the zero pressure optim-ization of the crystal structure are in agreement with pre-vious calculations [52] and are given in the appendix.

3.3.2 Hypothetical high-pressure modifications ofSi2N2O

In the course of our search for high-pressure modificationsof Si2N2Owe surveyed structure types with two cations andthree anions, namely A2X3, ABX3, ABXY2, A2XY2,A2XYZ and ABXYZ compounds (A,Bcations; X,Y,Z


anions) [5355]. In particular we investigated Si2N2O instructure types of Rh2O3, Sc2O2S, -Al2O3, β-Al2O3, Ga2-S2Te, Ag2O3, Sc2S3, BaTiO3, Bi2O2S, Cu2Cl2O, Mn2O3,Fe2O3, Bi2O3, Pb2F2O, La2O3, La2O2Te Cr2S3, and β-In2S3.Most of these structure types follow the principle of eutaxy.That is, the anions are approximately in a dense packing,either cubic or hexagonal, while cations are distributedamong the tetrahedral and octahedral interstitials. There is,however, no absolute necessity that three types of ions withdifferent radii have to arrange in such a way and many morecrystal structure types, which are still unknown, may bepossible. For some of the structures indicated a mapping ofN and O onto the anion sites is straightforward. In mostcases, however, we had to test several different settings lead-ing to a large amount of possible structures with compo-sition Si2N2O. In some cases the initial coordination ofatoms (e.g. 8-coordination of Si) could not be maintainedand the structure distorted during the optimization pro-cedure. Based on an assessment of energies and volumes wecalculated for more than 50 different structures we concludethat a β-In2S3-type and an -Al2O3-type are the most pro-mising candidates for a high-pressure modification ofSi2N2O.

3.3.3 β-In2S3-type Si2N2O

The indium trisulfide modification β-In2S3 has a spinel-likestructure. S atoms are approximately in a cubic-closedpacking. However, a third of all tetrahedral sites typicallyfilled in the spinel structure are vacant. The ordering ofvacancies causes appreciable atomic displacements and asymmetry reduction from Fd3m, the space group of theideal spinel, to I 41/amd [56]. The unit cell of β-In2S3 con-tains 80 atoms. If the coordination of the ideal spinel isdescribed as A[6]

2 B[4]X4, the β-In2S3-type is described asIn[6]

2 In[4]2/3V[4]

1/3S4 (or, if the composition shall be met, asIn[6]

3/2In[4]1/2V[4]

1/4S3), with V denoting a vacant tetrahedral site.While β-In2S3 is stable under ambient conditions, it un-dergoes a disordering phase transition to the cubic -In2S3

at 420 °C. There is evidence that in -In2S3 both tetrahedraland octahedral sites are partially empty. To derive a poly-morph of Si2N2O from the ordered structure of β-In2S3 wefirst replaced all In atoms by Si. Several different choicesfor the mapping of anions N and O showed that the con-figuration with lowest energy is achieved, if O atoms coor-dinate the tetrahedral vacancies. By doing so, the spacegroup symmetry I 41/amd is preserved. The unit cell of ourβ-In2S3-Si2N2O contains 80 atoms and is shown in Figure4. We performed our calculations using the conventionalunit cell with 80 atoms and a 4x4x2 k-point mesh. Crystal-lographic data are presented in the appendix.

It is possible to construct more such ordered defect-spineltypes of Si2N2O once the derivation of the β-In2S3 structureis understood. For example: by taking the structure of γ-Si3N4 in the hexagonal setting (42 atoms in the unit cell),discarding two of the six tetrahedrally coordinated Si atomsand replacing the N atoms surrounding the vacancies by O

P. Kroll, M. Milko

Figure 4 The structure of β-In2S3-Si2N2O. Si atoms appear bothtetrahedrally and octahedrally coordinated. On the left side a ball-and-stick representation (Si black, O dark grey, N light grey), andon the right side a polyhedral view of the structure. Octahedra areshaded in dark grey, tetrahedra in light grey. Sheets of octahedraare alternating with sheets of mixed octahedra and tetrahedra. Twounit cells are indicated.

atoms, five more spinel-like modification of Si2N2O wereconstructed. We probed different substitution patterns andfound a monoclinic polymorph with space group symmetryC 2/m (the structure is pseudo-orthorhombic and the for-mer hexagonal cell is the primitive unit cell of the base cen-tered structure) with almost identical energy in comparisonto the β-In2S3-type of Si2N2O. The calculations for this ad-ditional spinel-like Si2N2O were done within the primitiveunit cell containing 40 atoms using a 4x4x4 k-point mesh.Detailed crystallographic data of this polymorph is foundin the appendix.

This structural diversity of ordered spinel-like Si2N2O-structures will play a significant role for the discussion ofthe influence of configurational entropy on the stability of adisordered spinel-like modification of Si2N2O lateron. Theenergetics of structures with octahedral vacancies, with N/O exchange, as well as of structures with interstitial anions(no cationic vacancies) will then be discussed as well.

3.3.4 -Al2O3-type Si2N2O

In corundum, -Al2O3, the most stable polymorph ofAl2O3 under ambient conditions, the arrangement of anionsis close to a hexagonal-closed-packing. All cations in thisstructure are octahedrally coordinated. The space group of-Al2O3 is R3c. The mapping of Si2N2O onto the cor-


Figure 5 The structure of -Al2O3-Si2N2O. All Si atoms are octa-hedrally coordinated. On the left side a ball-and-stick represen-tation (Si black, O dark grey, N light grey), and on the right sidea polyhedral view of the structure. Three different kinds of oc-tahedra are shaded.

undum structure leads to a symmetry reduction and re-quires a careful testing of several possible anion arrange-ments. Several of them turned out to be structurally andenergetically quite similar. The polymorph with lowest en-ergy we found adopts space group symmetry C 2/c. We didour calculations within the primitive unit cell containing 30atoms using a 4x4x2 k-point mesh. The crystal structure ofthis -Al2O3-type Si2N2O is shown in Figure 5. The de-tailed crystallographic data are given in the appendix.

Tab. 3 lists energy, volume, and bulk moduli for the or-thorhombic modification of Si2N2O present under ambientconditions, as well as for the possible high-pressure phasesof Si2N2O. We note the increase in density as going fromthe orthorhombic structure of Si2N2O over the spinel-typesto the corundum-like structure of Si2N2O. This trend is re-lated to the increase in connectivity, since the percentage ofoctahedrally coordinated Si atoms increases as well. Thebulk modulus obviously follows the same trend. We remark,however, that for a spinel-like Si2N2O B0 is about 15 % lessthan for γ-Si3N4, and its density is about 5 % less. The -Al2O3-type Si2N2O, in which all Si atoms are now octa-hedrally coordinated, has a bulk modulus comparable to γ-Si3N4 and stishovite, and its density is right between thoseof γ-Si3N4 and stishovite.

With increasing pressure, thus, the phase transitions ofSi2N2O can be summarized as follows:

Cmc21-type(4)

38%

β-In2S3-type(4, 6)

8%

-Al2O3-type(6)

where the numbers inside the parentheses are polyhedralcoordination numbers of Si, and the percentages are theincreases in density from one to the other modification cal-culated for zero pressure.


Table 3 A comparison of energy and volume (per formula unit Si2N2O), density and bulk modulus of several Si2N2O-phases calculatedwithin the LDA and GGA. Experimental results are available for the low-pressure modification only.

Cmc21 β-In2S3 C2/m -Al2O3

LDA GGA Exp. [46] LDA GGA LDA GGA LDA GGA

E/eV 45.02 41.08 44.54 40.08 44.52 40.07 43.79 39.18V/A3 58.78 60.68 58.93 42.23 43.94 42.19 43.90 38.97 40.56ρ/(g/cm3) 2.82 2.74 2.81 3.93 3.78 3.93 3.78 4.26 4.09B0/GPa 133 130 127 258 243 256 240 319 285

Figure 6 Energy-volume (E-V) phase diagram of the orthorhombicCmc21, β-In2S3-, and -Al2O3-type structures of Si2N2O, calculatedwithin the GGA. Each symbol represents a calculation.

3.4 The enthalpy-pressure (∆H-p) diagram ofSi2N2O phases

In a subsequent step we accessed the behavior of the threeSi2N2O phases under pressure. Starting with the zero-pres-sure geometry we reduced the volume of the unit cell byscaling the lattice constants by 0.5 % relative to the initialgeometry. The structures are then optimized under the con-straint of constant volume. Therefore, the cell shape wasoptimized too, since except for cubic structures thelinear compressibility of a crystal is anisotropic in general.After complete optimization for a given volume the scalingprocedure was repeated. All structure-types were tested forpossible distortions of the space group symmetry underpressure (40 GPa and 100 GPa) by displacing atoms andperforming the calculations within P 1. However, no signifi-cant symmetry reduction could be observed. The resultingenergy is plotted against the volume in an E-V diagram.The respective diagram for the three Si2N2O phases isshown in Figure 6.

In the high-pressure experiment, however, pressure p andtemperature T are varied. The thermodynamical variablegoverning structural stability is the free energy GEp·V-T·S, and the difference ∆G between two phases constitutesthe driving force for a structural transformation. Very oftenthe entropy contribution to ∆G is neglected, and ∆G canbe replaced by ∆H, the enthalpy difference between thephases. For each phase we extracted H and p from the re-spective E-V graph, by numerical differentiation using ap-propriate interpolation functions, e.g. Murnaghans EOS ora spline-fit to the data. The resulting data is plotted in Fig-ure 7. Note that we plotted ∆H, therefore, the enthalpy ofthe phases relative to the enthalpy of the orthorhombic


Figure 7 Enthalpy-pressure (∆H-p) phase diagram of Si2N2O calcu-lated within the GGA. The enthalpy is given relative to that of theorthorhombic Cmc21 structure. Two arrows indicate the transitionpressures for the Cmc21 β-In2S3-Si2N2O and the β-In2S3-Si2N2O -Al2O3-Si2N2O transitions, respectively.

Cmc21-structure of Si2N2O. Apparently, a transformationfrom the orthorhombic to the β-In2S3 modification ofSi2N2O should happen at a transition pressure pt of about10.3 GPa. This value of pt fits into the range of the exper-imentally observed transition pressure (10-13 GPa) for theβ-Si3N4 γ-Si3N4 transition [43]. A comparison with thecalculated transition pressure of the β-Si3N4 γ-Si3N4

transition (12.1 GPa), however, indicates a lower value of pt

(by 2 GPa) within the oxynitride system. Obviously, oxygenwithin the spinel phase decreases the transition pressure.Such a result could have been expected following Vegard’srule (“mixing rule”) and inserting the pressures of forma-tion of stishovite and γ-Si3N4. The transition β-In2S3-Si2N2O -Al2O3-Si2N2O occurs at 56.9 GPa. The bend-ing of the ∆H-p curves for both the β-In2S3-type and the -Al2O3-type modification at higher pressures is caused bythe structural deformations of Cmc21 structure. At higherpressures all anions of the defect wurtzite-type are approxi-mately hexagonal-closed-packed and the structure becomesmore resistant against further compression.

3.4.1 The enthalpy of formation of Si2N2O from SiO2and Si3N4 as a function of pressure

The enthalpy-pressure phase diagram of ternary Si2N2Opresented in the last section suggests, that the orthorhombicmodification first transforms into a defect-spinel modifi-cation of Si2N2O at about 10 GPa. Approximately at60 GPa a second phase transformation from the spinelmodification into the corundum-type modification of

P. Kroll, M. Milko

Si2N2O should happen. These results, however, are validonly if the ternary oxynitride is stable against decompo-sition into the binary phases SiO2 and Si3N4. Therefore,we need to investigate the relative enthalpy of the ternaryoxynitride phases with respect to the binary phases at anygiven pressure. This procedure, consequently, yields thepressure dependence of the reaction enthalpy for reaction(1). To calculate it, we take the minimum of enthalpy as afunction of pressure for every composition, Si2N2O, SiO2,and Si3N4, respectively. For example, for Si3N4 we take theenthalpy of β-Si3N4 for pressures up to 12.1 GPa, and thatof γ-Si3N4 for pressures above. For SiO2 we considered ac-cordingly -quartz, coesite, and stishovite. The enthalpyfunctions are then added (SiO2 and Si3N4) and subtractedaccording to reaction (1).

For zero pressure and temperature, a comparison of theenergies of the optimized crystal structures of -quartz, β-Si3N4, and the orthorhombic Cmc21-structure of Si2N2O,given in Tables 1, 2, and 3, respectively already yields thatthe homogeneous phase Si2N2O is more stable than a phaseassemblage of SiO2 (-quartz) and Si3N4 (β-Si3N4). Thetheoretical value of this energy difference is ∆H 0.175 eV/Si2N2O (GGA; the LDA result is 0.11 eV/Si2N2O). A reliable experimental value of ∆H is very diffi-cult to obtain and interpreted carefully, due to the very dif-ferent calorimetric methods which have been employed tomeasure the standard enthalpy of formation of the threephases. If we combine the most recent calorimetric resultsfor -quartz (910.7 kJ/mol [3, 57]), β-Si3N4 (827.8 kJ/mol [40]), and Si2N2O (887.5 kJ/mol [58]), we calculate∆H to 0.19 eV/Si2N2O, however, under standard con-ditions at 298 K. This value is surprisingly close to the cal-culated result, but the should be handled with great care,due to the reasons mentioned above.

The complete reaction enthalpy-pressure (∆H-p) diagramfor reaction 1 is given in Figure 8. According to the phasediagram, the stability of the orthorhombic Si2N2O phaserelative to SiO2 and Si3N4 decreases with increasing press-ure. At a pressure of 5.9 GPa (the LDA result is 2.6 GPa)∆H becomes positive, and consequently a phase assemblageconsisting of SiO2 and Si3N4 is more stable than the homo-geneous Si2N2O. This transition almost coincides with theformation of stishovite within the SiO2 system. The gradi-ent of the ∆H-p curve is positive up to 10.5 GPa, at whichpressure the β-In2S3-modification of Si2N2O gets more fa-vorable in enthalpy in comparison to the orthorhombicSi2N2O phase. The value of ∆H now decreases down to∆H0.18 eV/Si2N2O for p12.1 GPa. This local mini-mum of ∆H is achieved at the transition pressure of the β-Si3N4 γ-Si3N4 transition within the Si3N4-system. Wewill address the value of ∆H at his specific pressure furtherin the course of this paper. Since ∆H is positive, the phaseassemblage is still lower in enthalpy in comparison to ahomogeneous ternary phase. Above 12.1 GPa ∆H increasesagain up to 56.9 GPa, at which pressure the transition β-In2S3-Si2N2O -Al2O3-Si2N2O within the Si2N2O systemhappens. The effect of further volume contraction within


Figure 8 Enthalpy of reaction for the formation of Si2N2O fromSiO2 and Si3N4 as a function of pressure calculated within theGGA; on the top for pressures up to 20 GPa, on the bottom up to100 GPa. The letters label certain relevant points: A) the -quartz coesite transition; B) the decomposition of a homogenuousSi2N2O into the binary phases of SiO2 and Si3N4; C) the coesite stishovite transition; D) the Cmc21 β-In2S3-Si2N2O transition;E) the β-Si3N4 γ-Si3N4 transition; F) the β-In2S3-Si2N2O -Al2O3-Si2N2O transition.

the Si2N2O-system all cations become octahedrally coor-dinated has a drastic effect of the phase diagram. Theslope of the reaction enthalpy ∆H turns towards negativevalues. However, it does not become negative for pressuresbelow 100 GPa. At 100 GPa ∆H is still 0.44 eV/Si2N2O.Pushing the calculations further (and neglecting the influ-ence of post-stishovite phases of SiO2 as well as post-spinelphases of Si3N4), we found that ∆H may become negativeonly at pressures beyond 150 GPA.

3.5 Contributions of entropy to the free energy of aspinel-like or corundum-like Si2N2O

The results of the previous subsection indicate that a homo-geneous ternary Si2N2O phase can not be synthesized atpressures exceeding 6 GPa. A spinel-like phase of Si2N2Ohas a positive enthalpy of formation ∆H of at least0.18 eV/Si2N2O (at 12.1 GPa). A corundum-like Si2N2Ohas ∆H of 0.44 eV/Si2N2O at 100 GPa. Therefore, if webase our argument on enthalpy, which is correct for zerotemperature, it is impossible to synthesize either a spinel-like or corundum-like modification of Si2N2O. The experi-ment, however, proceeds at about 2000 °C [43]. At this tem-perature, which is typical for laser-heating within the dia-mond-anvil cell, the calculated enthalpy difference ∆H of0.18 eV corresponds to an entropy difference of justabout 9 J·K1mol1. This is a very typical value for entropydifferences between solid state structures. The likelihood of


a ternary Si2N2O with spinel structure will thus depend onsome favorable thermodynamic properties.

There are several experimental indications for a spinelsilicon oxynitride. Chemical compositions of γ-Si3N4-samples presented by Schwarz et al. show an oxygen con-tent of up to 13.9 wt% (see Ref. [43], Tab. 2, sample 6). Thecomposition Si2N2O corresponds to an oxygen content of16 wt%. According to the authors, the O bulk content wasgreater than 5 wt% even after a careful and thorough treat-ment with fluoric acid [59]. Another indication for a spinelsilicon oxynitride is found in the report of Jiang et al. [60].Their Rietfeld-refinement shows a cation deficiency in thetetrahedral and octahedral sites of approximately 15 % and9 %, respectively. Moreover, the density of their sample was3.75 g/cm3, more than 5 % less than the expected 4.0 g/cm3

of a pure γ-Si3N4 measured by other groups, and close toour calculated value for a spinel-type Si2N2O, see Table 3.Both the cation deficiency and the reduced density can beinterpreted in terms of a significant oxygen contamination,leading to a composition close to Si2N2O. More argumentsfor the possibility of silicon oxynitrides with spinel modifi-cation stem from spinel-like γ-AlONs and γ-Al2O3, bothwhich obviously can be synthesized. Recent calculations ofFang et al. found the difference in enthalpy between thebinary constituents -Al2O3 and AlN and a defect modelof γ-AlON (Al23O27N5) to be 3 kJ/atom [61]. This matchesour value of ∆H very closely. Moreover, we calculated asimilar value of ∆H between the zero pressure structures of-Al2O3 and γ-Al2O3, using a spinel-type model with octa-hedral defects for the latter. Therefore, it appears likely thateven a spinel silicon oxynitride does exist, accessible by bal-ancing the free energy of formation at a high temperature.It eventually can be quenched to zero pressure, since likethe other spinel phases it apparently is kinetically andmechanically stable towards phase decomposition.

To evaluate the free energy ∆G∆H-T·∆S of reaction 1at elevated temperatures, it becomes necessary to estimatethe entropy difference ∆S between products and the educt.Unfortunately, a rigorous calculation of the entropy of eachcrystal phase (or likewise its partition function) is imposs-ible. Our approach for an estimation of ∆S is based on thefact that the bonding situation of the atoms in the struc-tures of stishovite-SiO2 and γ-Si3N4 on one side, and of β-In2S3-Si2N2O on the other side, is very much alike. This isthe same assumption, which usually justifies the neglect of∆S, with appreciable success as we have seen for the calcu-lated transition pressures within the SiO2 and Si3N4 phasediagram. The leading term contributing to ∆S now stemsfrom two sources: i) the inherently defective structure of aspinel-like Si2N2O, in which vacancies can order in differentpatterns, resulting in different, but energetically degenerate,structures, and ii) the presence of two kinds of anions whichcan mix within the anionic sublattice. Therefore, we are toestimate the entropy difference between the well-ordered bi-nary phases and a homogeneous but disordered spinel-like Si2N2O.


We present two approaches for the estimation of ∆S, amaximum entropy and a minimum entropy ansatz. The firstone is based on the entropy of mixing and gives an upperboundary ∆S. In a second approach we simply count thenumber of energetically degenerate states we found and esti-mate ∆S using the standard formula of statistical thermo-dynamics. This procedure yields a lower boundary for ∆S.

3.5.1 Maximum configurational entropy

In spinel-like Si2N2O we have two sources which can con-tribute to the configurational entropy: within the cationsublattice of the structure Si atoms and vacancies can mixon tetrahedral sites, and within the anion sublattice O andN can mix. The mixing entropy per formula unit Si2N2O isthus given by two independent contributions:

Smix Scatmix San

mix R · [Mcat (xSi[4] lnxSi[4] xV[4] lnxV[4]) (3)

Man (xN · lnxN xO lnxO)] .

In this equation, R is the universal gas constant(8.314 JK1mol1). xN and xO are the fraction of N and Oatoms (xN2/3, xO1/3) and Si[4] and V[4] are the fractionof Si atoms and vacancies at tetrahedral sites (Si[4]2/3,V[4]1/3). Mcat and Man are multipliers (Mt3/4, Man3)to account for the number (mol) of available sites (cationsamong the tetrahedral sites only, anions at all anion sites)per formula unit (mol) Si2N2O. The effect of mixing withinthe anion sublattice obviously is four times larger than mix-ing within the cation sublattice. Inserting the values in theequation above yields for the mixing entropy Smix19.8 JK1mol1. At T2000 °C the entropic contributionto the free energy T·∆S0.46 eV/Si2N2O, which is about 2.5times larger than the minimum value of ∆H. The free en-ergy difference of reaction (1) calculated at 12.1 GPa nowcomes out to ∆G0.28 eV/Si2N2O, and clearly indicatesa “stable” spinel-like Si2N2O. Under the given assumptions,therefore, a spinel modification of Si2N2O with vacanciesdisordered among the tetrahedral sites and a complete mix-ing of N and O within the anionic sublattice should existwithin the p-T phase diagram of Si2N2O. A very similarresult of ∆G is calculated, if we focus on a single unit cellof the spinel structure only and describe the disorderamong cations and anions with fractional occupancies.Such an approach was used for an estimation of entropytowards the stability of aluminum oxynitride phasesAlON’s [61].

The contribution of mixing entropy to the free energy ofa corundum-type polymorph of Si2N2O is calculated in asimilar way. While there is no mixing within the cationicsublattice, the contribution from the anionic sublattice again assuming a complete mixing of O and N among theanionic sites is 15.9 JK1mol1. At 2000 °C, therefore, acorundum-type Si2N2O with a complete mixing of anionsis stabilized by an additional 0.37 eVSi2N2O. The valuewe obtained for ∆H at 100 GPa (0.44 eVSi2N2O) wouldrequire a temperature of greater than 2500 °C.

P. Kroll, M. Milko

3.5.2 Minimum configurational entropy

Using the Boltzmann-equation S R · lnΩ it is in prin-ciple possible to calculate the amount of entropy S rigor-ously, if the complete partition function Ω is known (R isthe the universal gas constant). The partition function Ωitself separates into electronical, vibrational, and configura-tional part and each part adds to the entropy. The configu-rational part ΩC (we will briefly discuss the contribution ofthe vibrational part later and will not consider the elec-tronic part) for a canonical ensemble of i independent statesat a given temperature T can be written as

ΩC i

gi exp(∆Ei/kT) , (4)

with gi counting the number of degenerate states with en-ergy Ei and k the Boltzmann factor. At zero temperaturethe leading term in this expression is given by the degener-acy of the ground state. At higher temperatures, however,configurations with higher energy (“exited” states) play amore and more important role.

We can roughly estimate the degeneracy of the groundstate and, therefore, access the leading term of the partitionfunction for low temperatures. We start from the cubic unitcell of γ-Si3N4 (56 atoms), choose one direction of the axes,and transform the structure into the tetragonal setting (28atoms). Then we triple the unit c-axis (84 atoms). The newunit cell has 12 occupied tetrahedral sites. Among the ( 4

12) 495 possible mappings of the four vacancies onto the 12sites there are three equivalent ones (this stems from thetripling of c) corresponding to the lowest energy structure.Proceeding in a similar way along the two other directionsof space yields 9 different space-patterns of distributing vac-ancies in the (infinite) spinel structure. Each structure is en-ergetically degenerate to the β-In2S3-Si2N2O modification.The second spinel-like polymorph we found with C 2/msymmetry yields the same amount of degenerate configura-tions. As a result, we have a degeneracy of 18 for a groundstate with 80 atoms or 16 f.u. Si2N2O, and the contributionto ∆S is 1.5 JK1/Si2N2O. At T2000 °C this yields an en-tropic contribution to the free energy T·∆S of 0.04 eV.While this apparently is not enough (just 25 %) to yield astable spinel-like modification of Si2N2O, this is certainly alower boundary of the entropy and thus a minimum valueof ∆S. Once the temperature (kT) is comparable to the en-ergy difference ∆E between the ground state and further“exited” states, however, more and more configurations willcontribute substantially to the partition function in equa-tion (4). For the corundum-type polymorph of Si2N2O weestimate a degeneracy of 6 the ground state with 6 f.u.Si2N2O, and the contribution to ∆S is 2.4 JK1/Si2N2O.

3.5.3 The formation free energy ∆G at 2000 °C as afunction of pressure

Combining the results for the enthalpy of reaction given inFigure 8 and the estimated entropy differences for the spi-


Figure 9 Free energy of reaction at 2000 °C for the formation ofSi2N2O from SiO2 and Si3N4 as a function of pressure (∆G-p) cal-culated within the GGA. The upper curve represents the assump-tion of minimum entropic contributions, the lower curve is for themaximum entropy assumption. The curve in-between (labelled50 %) uses the half of the maximum entropy.

nel-like and corundum-like phases of Si2N2O, it becomes in principle possible to calculate the complete p-T phasediagram of Si2N2O. In Figure 9 we have plotted the press-ure dependence of the free energy of reaction 1 (∆G-p) forT2000 °C using both the maximum and minimum esti-mates of ∆S. Apparently, using the maximum entropy esti-mate a spinel-like silicon oxynitride now has a range of sta-bility, approximately from 6 GPa to 38 GPa. We expect,however, that the “true” pressure dependence of ∆G will beconfined by the two curves, which give an upper and lowerboundary for ∆G. If we assume an intermediate value of∆S, say just 50 % of the maximum estimate, this has theinteresting consequence that a multiple decomposition andreconstruction of a ternary phase will take place. At 6 GPathe orthorhombic ternary phase of Si2N2O decomposesinto coesite and β-Si3N4. At 10.3 GPa the defect spinelmodification of Si2N2O will reconstruct from a phase as-semblage of coesite, stishovite, and β-Si3N4. We point outthat the maximum driving force for the formation of ahomogeneous ternary spinel-type Si2N2O is still at a press-ure of 12.1 GPa, the pressure of the phase transformationβ-Si3N4 γ-Si3N4. Interestingly, as the ∆G-p curve furtherpoints out, a spinel-like Si2N2O, if it exists, will decomposeagain into a phase assemblage of stishovite and γ-Si3N4,at 16.5 GPa within 50 %-assumption, but certainly below40 GPa, even using the maximum estimate. A corundum-type phase of Si2N2O appears unlikely at 2000 °C, evenwithin the maximum entropy assumption. For higher tem-peratures, however, the ∆G-p curves will be shifted towardsmore negative values of ∆G, increasing the likelihood offormation and stability of ternary Si2N2O-phases.

4 Discussion

The results presented in the preceding sections now suggestthat high-pressure polymorphs of Si2N2O can be synthe-sized, because at elevated temperature they have a range ofstability within the pressure-temperature phase diagram ofSi2N2O. Their existence obviously depends on a delicatebalance between reaction enthalpy and entropy difference


between a homogeneously disordered ternary phase and aphase assemblage of the binary constituents. If, for ex-ample, a spinel-like Si2N2O compound exists, the maximumdriving force for its synthesis will be at 12.1 GPa, the press-ure of the transition β-Si3N4 γ-Si3N4. A corundum-typeSi2N2O will be accessible only at very high pressures andtemperatures, and care has to be taken on the formation ofpost-stishovite phases.

There are, however, arguments that reduce our expec-tations on ∆S caused by configurational disorder. We haveindicated an energetical preference for oxygen atoms tocluster around the vacant tetrahedral Si sites. Furthermore,an inspection of the low-energy structures yields that theVO4-clusters seek a homogenous distribution among thelattice. Both effects lead to substantial order within the cat-ion and anion sublattice, and consequently the entropy con-tribution caused by disorder of anions will not gain itsmaximum value. We have, however, shown that just 40 % ofthe entropy contributions are necessary to “stabilize” a spi-nel-like Si2N2O, and less at even higher temperatures.

There are, on the other hand, arguments for a larger en-tropy difference between the defect-spinel type of Si2N2Oand stishovite and γ-Si3N4 as well. In the course ofmodeling spinel-like structures we considered many possibleconfigurations, some derived from a hexagonal setting,others derived from a tetragonal setting of the spinel struc-ture. Many of them are within 0.1-1 eV/Si2N2O above ourminimum energy configuration of β-In2S3-Si2N2O. All thesestructures will give a significant contribution to the par-tition function in equation 4 at elevated temperatures. Thesite exchange of oxygen and nitrogen as well as the forma-tion of octahedral vacancies will also lead to an increase of∆S. We calculated the energy associated with one single O/N site exchange within an unit cell with 80 atoms to 1.4 eV.The energy difference between our β-In2S3-structure andone with an octahedral vacancy comes out to 2.2 eV, ren-dering a spinel-type Si2N2O with octahedral vacancies onlyabout 0.5 eV/Si2N2O higher in energy. We also calculatedseveral models which have all cation sites occupied, buthave anions in interstitial sites. Such “constant-cation-models” turn out to have an energy much higher (> 1.2 eV/Si2N2O) than the cation deficient models we considered.The small energy differences towards a manifold of similarconfigurations they all differ only by the ordering of theanions and vacancy sites shows that there is certainlya contribution to the configurational entropy by this kindof disorder.

Besides the configurational disorder, we have indicationthat the free energy of a spinel-like Si2N2O benefits fromdifferences in the vibrational entropy too. Presently neg-lected, this contribution certainly is larger for the defectphase than it is for Si3N4 and SiO2. We calculated the vi-brational spectra of β-In2S3-Si2N2O and summarize thetrends as follows: Most obvious the principal frequency ofthe pure nitride spinel γ-Si3N4 is shifted towards higher wavenumbers in β-In2S3-Si2N2O, from 965 cm1 to 1040 cm1.This optical mode originates from atomic displacement


within the anion sublattice. And while the electrostaticcharge within the anion sublattice of a spinel-like Si2N2O islarger than for γ-Si3N4, the associated potential and forceconstants are larger too. The observed shift of the principalfrequency in general will be useful to determine the oxygencontent in a γ-Si3N4. For modes with low frequencies in γ-Si3N4 we observe a shift towards even lower wave numbersin β-In2S3-Si2N2O. The reason for this is grounded in thedefect nature of the spinel-like Si2N2O. Anions coordinat-ing a defect have more accessible space for vibrations intothe vacant tetrahedral site and encounter less repulsive for-ces connected with such movements. This effect can be dis-tinguished from the consequences that arise by extendingthe primitive unit cell of the simple spinel structure from 14atoms in γ-Si3N4 to 40 atoms in β-In2S3-Si2N2O. By doingso, we implicitly calculate vibrations at finite q-values (withregard to the smaller unit cell) and consequently have theeffect of “band-folding”, folding explicitly the acousticbranch of the phonon dispersion curve, which leads to anincrease in the number of modes of the vibrational spectra.Another consequence of this procedure is found for the se-cond spinel-like polymorph of Si2N2O with space groupsymmetry C2/m. Within the reduced point group symmetryparts of the acoustic branch, which were Raman-inactivefor the ideal γ-Si3N4 with its higher symmetrical pointgroup, now become Raman-active. The intensity of suchmodes is expected to increase with increasing oxygen con-tent. Experimental indication of both phenomena, whichcan be summarized as an increase of low-wave numbermodes in the Raman-spectra of a spinel oxynitride in com-parison to a pure nitride spinel, is given in the recordedRaman-spectra of γ-Ge3N4 [62, 63]. Interestingly, at thattime the Arizona group already speculated on a possible“defective spinel”, spinelloid, or other structure to interpretthe appearance of several “unexpected” low-frequency lines[63]. However, these effects can be re-interpreted as beingdue to significant oxygen contamination within the bulk ofthe samples.

Last in our discussion we note that the enthalpy differ-ences are rather small, and comparable to the error of thecomputational method we used. For once, we can not ex-clude that using a much larger supercell it might be possiblefind yet another spinel-like modification of Si2N2O, whichwill be lower in energy in comparison to the β-In2S3

modification. Enlarging the unit cell will allow for a morehomogenous distribution of the VO4-cluster and might helpto lower the energy. On the other side, although we workwithin well-converged basis sets and k-points schemes, themethod overall is affected with an uncertainty of about0.1 eV/Si2N2O. This value stems mainly from experience,but also from a detailed comparison of calculated andmeasured transition pressures in various cases, e.g. SiO2 andSi3N4. Therefore, the question as to whether high pressurephases of Si2N2O exist needs careful synthetical work, andprecisely conducted experiments. It can be expected that thekinetics of the phase transformations will pose further chal-lenges to phase construction and identification.

P. Kroll, M. Milko

5 Summary

We investigated a manifold of possible structures of high-pressure phases of crystalline silicon oxynitride, Si2N2O, toelucidate the pressure-dependent phase diagram of Si2N2O.Two candidates, a spinel-type β-In2S3-Si2N2O and a cor-undum-type -Al2O3-Si2N2O, are found to succeed the or-thorhombic low-pressure phase of Si2N2O. Taking into ac-count the formation of Si2N2O from the binary phases SiO2

and Si3N4, we propose that a spinel-type Si2N2O will bemost easily accessible at 12.1 GPa, the pressure of the β-Si3N4 γ-Si3N4 transition. The existence of a homo-geneous ternary spinel silicon oxynitride, however, dependson entropic effects. While configurational entropy of the de-fective spinel structure certainly decreases the free energy ofthe Si2N2O phase, a definite answer can not be given. Asimilar conclusion is made for the corundum-type Si2N2O,that might be accessible at 60-100 GPa and 2500-3000 °C.

Acknowledgments. The authors thank Professor Dr. Richard Drons-kowski for his continuous support and encouragement to pursuethis work. Useful discussions with Professor Dr. Manfred Martin,Dr. Bernhard Eck, and Dr. Marcus Schwarz are gratefully acknow-ledged. This work was sponsored by the Deutsche Forschungsge-meinschaft (Kr1805/3-1).

Appendix

Crystal Structures



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theoretical investigation of the solid state reaction of silicon nitride and silicon dioxide forming...

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