the route to the structure determination of amorphous solids: a case study of the ceramic si3b3n7

Amorphous CompoundsDOI: 10.1002/anie.200504193

The Route to the Structure Determination ofAmorphous Solids: ACase Study of the Ceramic Si3B3N7

Martin Jansen,* J. Christian Sch�n, and Leo van W�llen

AngewandteChemie

Keywords:amorphous materials · computerchemistry · structure elucidation ·structure modeling

M. Jansen et al.Reviews

4244 www.angewandte.org 2006 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Angew. Chem. Int. Ed. 2006, 45, 4244 – 4263

http://www.angewandte.org

1. Introduction

1.1. Concepts

Among the states of condensed matter, those classified as“amorphous” are probably the least understood, and dealingwith them remains a scientific challenge.[1–6] Most of theconceptual, experimental, and theoretical intricacies associ-ated with this class of solids originate from their thermody-namic metastability. For every solid chemical compound thereexists a sequence of crystalline polymorphs. Some of these arethermodynamically stable under appropriate conditions,while the remainder are only kinetically stable at sufficientlylow temperatures.[7–9] However, for identical chemical com-positions, an essentially infinite number of “amorphous”atomic configurations, contributing to different metastablestates of the system, can be generated. Consequently, com-pounds for which different amorphous structures are acces-sible—a well-known example is amorphous ice[10, 11]—canexhibit different macroscopic properties, depending on thesynthesis route and the details of their pretreatment. Thediffering properties, thus, reflect the history of the compound.

The nonequilibrium character and history dependence ofamorphous compounds are the major reasons for thedifficulties encountered in defining a comprehensive andunambiguous classification of these compounds. The criteriaof existing classifications, which are typically based onthermodynamics, structure, or chemical composition,[2, 3, 5] forthe most part, apply only to selected aspects of amorphouscompounds. We prefer a standardized definition that, on onehand, covers all phenomena related to noncrystalline solids,and on the other hand, allows a more subtle differentiationamong these phenomena.

The essential features of an amorphous material or a glassare a lack of long-range translational order, as well as aninherent tendency to undergo structural changes over time.Thus, it is possible to differentiate the amorphous state from

other states of matter through the useof a two-dimensional map with thecoordinates “degree of static disorder”and “degree of dynamic disorder”(Figure 1). This diagram is subdividedinto two regions by a line separatingglobally ergodic[12] and nonergodic sys-tems. Of course, the location of this

border cannot be absolutely determined, because whether achemical system appears ergodic or not depends on the timespan that the observer is willing, or able, to follow theevolution of a given material, and on the external conditions,such as temperature and pressure.[8] The diagram covers thefull range of possible dynamic phenomena, including thetransition to ergodic behavior, and differentiates amorphouscompounds from materials with structural periodicity. Inaddition, this description encompasses all aspects of thechemistry and physics of nonperiodic materials, which areotherwise conventionally subsumed under the terms glass,random network, metallic glass, or amorphous solid. Thesecollective terms stem from the different preparation methods,the special structural or physical properties, and the chemicalnatures of the compounds.

From a theoretical point of view, the key to understandingthe static and dynamic properties of an amorphous compoundis the structure of its energy landscape,[13–21] that is, thepotential-energy hypersurface over the space of all atomicconfigurations of the system (the so-called configurationspace). Of course, the full energy landscape of a chemicalsystem contains both amorphous and crystalline regions, eachbeing the union of a large number of configurations. Herein,we focus on the part of the landscape of Si3B3N7 that containsthe amorphous state.

[*] Prof. Dr. M. Jansen, Priv.-Doz. Dr. J. C. Sch-n,Priv.-Doz. Dr. L. van W0llen[+]

Max-Planck-Institut f0r Festk-rperforschungHeisenbergstrasse 1, 70569 Stuttgart (Germany)Fax: (+49)711-689-1502E-mail: [email protected]

[+] Present address:Institut f0r Physikalische ChemieWestf@lische Wilhelms-Universit@t M0nsterCorrensstrasse 30/36, 48149 M0nster (Germany)

Si3B3N7 is the parent compound of a new class of amorphousceramics containing silicon, boron, nitrogen, and carbon that display aunique spectrum of properties. It consists of a random network inwhich the constituent elements are linked by predominantly covalentbonds. Similarly to quartz glass, the composition of amorphousSi3B3N7 is virtually stoichiometric. As all three of its constituentelements can serve as the objects of various structural probes, Si3B3N7

was selected as the basis of a systematic structural investigation, inwhich methods for the structure determination of solids withouttranslational symmetry could be validated and improved. However, asthe complete amorphous structure cannot be deduced from exper-imental data, these results must be complemented by computer simu-lations. Thus, five classes of structure models were generated andcompared to experimental results. Only the models generated byfollowing the actual synthesis route as closely as possible agreed wellwith the experimental data.

From the Contents

1. Introduction 4245

2. Experimental Approaches to theDetermination of CharacteristicStructural Features in Si3B3N7 4247

3. Theory 4255

4. Summary and Conclusion 4260

Amorphous SolidsAngewandte

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4245Angew. Chem. Int. Ed. 2006, 45, 4244 – 4263 2006 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1.2. The Problem of Structure Determination for AmorphousMaterials

As the properties of a solid crucially depend on the spatialarrangement of the atoms, knowledge of the microscopicstructure of a material is an indispensable prerequisite for allattempts to understand and purposefully optimize its proper-ties. The difficulty involved in determining the structures ofamorphous compounds is one of the main reasons that thesecompounds have not always received their due attention.

Although experiments can yield valuable structural infor-mation, this information is diffuse, as a multitude of differentatomic configurations can reproduce the experimental datawithin the error limits. Structure modeling with theoreticalmethods offers a possible solution. However, if we were torely on theory alone, we would need to develop models thatimitate an actual synthesis route, for example, the cooling of amelt, a pressure-induced amorphization, an atom-beam orvapor deposition, or a sol–gel process. Unless these differentroutes all lead to structurally identical amorphous compounds(that is, unless the details of the synthesis route are

unimportant for the structure of the amor-phous state) we face the currently impossibletask of exactly reproducing the synthesisprocess in the simulation.

Clearly, if we want to surmount thisbarrier, it is necessary to combine experimentwith theoretical modeling of the amorphouscompound. In principle, amorphous atomicconfigurations of a given compound can begenerated by employing advanced computa-tional techniques with the incorporation ofexperimental information (without consider-ing the actual synthesis route). However,innumerable plausible solutions will beobtained using this method, and there is nostraightforward way to assign one of thesimulated structure models to a real sample.Furthermore, each of the experimental struc-tural probes provides only partial informa-tion. As the individual analytical tools revealquite different features of the structure, it isdifficult to correlate the results and to gen-

erate a complete and consistent structural model.Therefore, our approach consists of first acquiring as

many structural data as possible for the material underconsideration. These findings are then combined with resultsfrom theoretical model calculations to produce a completestructural picture for the amorphous compound. On thetheoretical side, a large ensemble of structural configurationsmust be generated. The region of this configuration spacecontaining the real amorphous compound is then selected bytaking the experimental data into account. The circle closeswhen the resulting structure reproduces the experimentallydetermined properties, validating its correctness.

1.3. The Case Study: Si3B3N7

Ten years ago, the Sonderforschungsbereich 408 wasfounded in Bonn to develop appropriate theoretical andexperimental methods for the elucidation of the “amorphousstructures” of inorganic nonmetallic solids. One particularlyfascinating compound investigated as part of this long-term

Martin Jansen studied chemistry at the Uni-versit�t Giessen, where he gained his doctor-ate in 1973. After his habilitation in 1978,he accepted a professorial chair for inorganicchemistry at the Universit�t Hannover. In1987, he moved to the Universit�t Bonn.Since 1998, he has been a member of theScientific Council of the Max Planck Societyand a director of the MPI for Solid-StateResearch in Stuttgart. His research interestsinclude preparative solid-state chemistry,crystal chemistry, materials research, andthe structure–property relationships of solids.He is a member of the Advisory Board ofAngewandte Chemie.

J. Christian Sch3n studied physics at theUniversit�t Bonn and at MIT, where hereceived his MSc in mathematical physics in1982, and his PhD in theoretical solid-statephysics in 1988. He then undertook postdoc-toral studies at San Diego State Universityuntil the end of 1991. After being a visitorat the University of Odense, he was apostdoc at Copenhagen University. From1993 to 1999, he worked at the Institute forInorganic Chemistry of the Universit�t Bonn,where he completed his habilitation onenergy landscapes of crystalline and amor-

phous solids in 1997. Since 1999, he has been working as a senior scientistat the MPI for Solid-State Research in Stuttgart.

Figure 1. Placement of amorphous and other states of matter on a map with the coordinates“degree of dynamic disorder” and “degree of static disorder”.




project was the amorphous silicon boron nitride Si3B3N7. Thiscompound consists of a novel prototypical random network ofsilicon and boron atoms, which are interconnected by nitro-gen atoms through chemical bonds of predominantly covalentcharacter.[22–28] The discovery of Si3B3N7 was seminal to theexploration of a new class of amorphous ceramics for high-temperature applications, in the system Si/B/N/C.[22–32] In spiteof their metastable character, these inorganic random net-works exhibit high durability under thermal and mechanicalloads. They surpass the well-known crystalline ceramics SiCand Si3N4 with respect to the combination of all propertiesrelevant to high-temperature applications in air. For example,no weight loss is observed for amorphous Si3B3N7 and SiBN3Cup to approximately 1900 or 2150 K, respectively. The elasticconstants of these materials are quite high (for example, thebulk modulus of SiBN3C is 200–300 GPa), and in spite of theirlow densities of approximately 1.8–1.9 gcm�3, they exhibit ahigh fracture toughness.[23,26,27]

In contrast to many other amorphous materials oftechnological interest, Si3B3N7 cannot be produced by con-ventional glass formation, that is, by quenching from a melt,the reason being that Si3B3N7, and its binary components BNand Si3N4 melt incongruently under standard conditions. Norhas it been possible to produce Si3B3N7 by sintering micro-meter-sized powders of BN and Si3N4. Instead, a sol–gelsynthesis starting from a single-component precursor such as(SiCl3)NH(BCl2) (TADB) can be employed.[22] The use ofmonomers such as TADB, which contain two metalloidcations in the desired ratio linked by a nitrogen bridge,circumvents the concentration fluctuations in the preceramicpolymer that result from the differing aminolysis reactionrates when two independent boron and silicon sources areused.[26, 27] The periphery of the TADB molecule is fullyfunctionalized by chlorine, which allows an easy polymeri-zation of the intermediate amides through aminolysis andsubsequent polycondensation (Figure 2). Thus, in the syn-thesis of amorphous Si3B3N7, the precursor TADB moleculesundergo aminolysis in NH3, are subsequently linked intooligomers and polymers by condensation, and are finallypyrolyzed under an N2 atmosphere to produce theceramic.[22,23,26,33]

The resulting ceramic Si3B3N7 is an air-stable whitepowder for which thermal decomposition, with the evolution

of N2, begins at 1900K and proceeds rapidly at 2000K. Indiffraction experiments with X-rays (synchrotron radiation),neutrons, or electrons, Bragg reflections were not observed.The bulk composition of the product corresponds to Si3B3N7,within the error limits of several analytical methods, and novariation in composition with synthesis charge was noticed.Furthermore, no lateral compositional inhomogeneities wereobserved to a resolution of 1 mm by energy dispersive X-ray(EDX) and wavelength dispersive X-ray (WDX) spectrosco-py, or to a resolution of 1 nm by energy-filtered transmissionelectron microscopy (EFTEM).[34]

While our investigations were motivated in part by thedesire to better understand this new, promising class ofceramic materials, the main driving force was to utilizeSi3B3N7 as a case study, in which the tools available foranalyzing the microstructures of amorphous materials couldbe tested and sharpened. This system has served us as a“Drosophila” for solid-state research. Only through the jointapplication of experimental and theoretical methods was itpossible to develop a consistent structural model for amor-phous Si3B3N7.

2. Experimental Approaches to the Determinationof Characteristic Structural Features in Si3B3N7

2.1. Experimental Methods

The lack of translational periodicity in amorphous solidsrenders it impossible to follow the standard structure-determination protocol of using diffraction techniques toproduce a complete three-dimensional structure in a singleexperiment. As a consequence, one has to resort to as manyalternative structural probes as possible to study variousstructural aspects of the amorphous material. A successfulstrategy for the structure determination of an amorphoussolid commences with a partition of the complex problem intothe three categories of short-range order (1–2 D), medium-range order (2–8 D), and long-range order (> 8 D).[35] Withineach of these categories, as many different experimental toolsas possible should be employed to probe the structuralfeatures, since the immanent distribution of bond lengths andangles often entails broad overlapping signals. These signalsare difficult to disentangle, making an unambiguous inter-pretation problematic. Therefore, only the combined resultsof different experiments within a given category—yieldingoverlapping or complementary information—can pave theway to a successful description of the structure of anamorphous solid.

The microscopic tools available for the characterization ofamorphous solids may be divided into three groups (Table 1):local probes, interference methods, and direct-imaging meth-

Leo van W:llen studied chemistry at theUniversit�t M:nster and completed his doc-torate there with Prof. M:ller-Warmuth in1993. After a two-year postdoctoral staywith Prof. Eckert and Prof. Ford at theUniversity of California in Santa Barbara, hecontinued his research at the Universit�tM:nster (1996) and then at the MPI forSolid-State Research in Stuttgart (1998–2003). In 2002, he completed his habilita-tion in physical chemistry at the Universit�tM:nster, where he has continued hisresearch on new NMR strategies for thecharacterization of the structure and dynam-ics of inorganic materials since 2003.

Figure 2. Synthesis of amorphous Si3B3N7 starting from TADB.


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4247Angew. Chem. Int. Ed. 2006, 45, 4244 – 4263 2006 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org


ods. Local probes that are frequently applied include infrared(IR) spectroscopy, Raman spectroscopy, X-ray photoelectronspectroscopy (XPS), X-ray absorption near-edge fine struc-ture (XANES) spectroscopy, electron energy loss spectros-copy (EELS), and nuclear magnetic resonance (NMR)spectroscopy. Interference methods employing neutrons,electrons, or photons (X-rays) can be used to produceatomic pair correlation functions (PCFs) from which charac-teristic distances can be deduced. In recent years, direct-imaging techniques, such as atomic force microscopy(AFM),[36] scanning electron microscopy (SEM), or trans-mission electron microscopy (TEM), have evolved into verypromising microscopic tools for elucidating structural aspectsof amorphous solids to subnanometer or even atomicresolution.[37] In the following Sections, the body of exper-imental data obtained for Si3B3N7 with the toolbox ofavailable structural probes (Table 1) will be presented.

2.2. Experimental Results and Strategy

According to the approach to the structural character-ization of amorphous solids described in Section 2.1, the firststep is the evaluation of the short-range order (1–2 D). In thecase of Si3B3N7 presented here, a set of complementary andoverlapping approaches, including NMR and XANES spec-troscopy, and scattering techniques, were employed. In asecond step, the medium-range order (2–8 D) of the materialwas addressed using dipolar solid-state NMR spectroscopictechniques and PCFs (from the various scattering techni-ques). In this step, information is harvested about the networkconnectivity—the interconnection of the polyhedral buildingblocks identified in the first step into an extended three-dimensional network. Long-range order is the focus of thefinal step. For Si3B3N7, electron microscopic imaging techni-ques were used to investigate structural features at a lengthscale of > 8 D, including homogeneity and phase separation.

2.2.1. Short-Range Order

The short-range order of Si3B3N7 was evaluated usingsolid-state NMR and XANES spectroscopy, and scatteringtechniques. The B K-edge XANES spectra of Si3B3N7, cubicBN (c-BN), and hexagonal BN (h-BN) are shown inFigure 3a.[38, 39] While the spectrum of h-BN is dominated bya strong 1s!p* resonance at 191.8 eVand a doublet at 198 eV(1s!s* resonances), in the spectrum of c-BN, the 1s!p*

Table 1: Toolbox of probes that can be used to study the structuralaspects of amorphous solids.

Localprobes

Interference methods Direct-imagingmethods

IR PCFs from X-ray, neutron, and electronscattering experiments

SEMRaman AFMNMR TEMEELS EXAFSXANESXPS

Figure 3. a) B K-edge XANES spectra of Si3B3N7, c-BN, and h-BN.[38, 39]

b) Si K-edge XANES spectra of Si3B3N7, a-Si3N4, and b-Si3N4.[40]

c) N K-edge XANES spectra of Si3B3N7, a-Si3N4, and h-BN.




resonance is replaced by a strong 1s!s* resonance at 198 eV.A comparison of these spectra to that of Si3B3N7, in which astrong 1s!p* resonance is observed, suggests that the boronatoms in Si3B3N7 are involved in planar BN3 units comparableto those in h-BN. The Si K-edge XANES spectrum of Si3B3N7

is shown in Figure 3b together with the spectra of a-Si3N4 andb-Si3N4.

[40] The broad resonance at 1844 eV is indicative of theSi 1s!3p transition. The similarities (for example, in theposition of the first peak) between the spectrum of Si3B3N7

and those of the reference compounds a-Si3N4 and b-Si3N4

suggest that the silicon atoms in Si3B3N7 are involved in SiN4

groups. The N K-edge XANES spectra of Si3B3N7, a-Si3N4,and h-BN are presented in Figure 3c. The resonances at 402–404 eV in the spectra of a-Si3N4 and h-BN originate from a1s!p* transition, and the resonances at 406 or 408 eV,respectively, originate from a 1s!s* transition. The spectrumof Si3B3N7 can be interpreted as a superposition of the spectraof a-Si3N4 and h-BN and, thus, indicates a coexistence ofvarious approximately trigonal-planar NB3�xSix (0� x� 3)environments. As an important conclusion, XANES spec-troscopy indicates that trigonal-planar BN3 and tetrahedralSiN4 species are the two main constituents of the inorganicnetwork in Si3B3N7.

Further evidence for the existence of these networkpolyhedra can be gained by inspection of the PCFs obtainedfrom neutron,[41, 42] electron,[34] and X-ray (synchrotron radi-ation)[41] scattering experiments. The PCFs determined fromthese experiments are plotted in Figure 4, and the peakmaxima relevant for the characterization of the short-rangeorder are listed in Table 2. Considering all of the possibleelement–element bonds in Si3B3N7 (Si–Si, Si–B, Si–N, B–B,B–N, N–N), the first peak maximum in the PCFs is onlyconsistent with a B–N bond in a trigonal-planar BN3 environ-ment. The B–N bond length in the trigonal-planar BN3 unitsof h-BN is 1.45 D, while that in the tetrahedral BN4 units ofc-BN is 1.56 D. Noting the Si–N bond lengths of 1.71–1.75 Din a-Si3N4, the second peak maximum in the PCFs can beassigned to Si–N bonds. In addition, as the assignment ofbonds is unambiguous, coordination numbers can be calcu-lated from the PCFs. For silicon and boron, the numbers ofnitrogen neighbors determined from this analysis are 3.7–3.8and 2.8–2.9, respectively. The scattering experiments, thus,lend definite support to the suggestion given above that BN3

and SiN4 units are the basic building blocks of the network inSi3B3N7.

To confirm these results and to check for the presence ofadditional local species, high-resolution magic angle spinning(MAS) NMR spectroscopy was performed on Si3B3N7, as anadditional independent probe. In MAS experiments, theinternal interactions whose orientational dependencies scalewith the second Legendre polynomial (P2(cosb)=0.5(3cos2b�1)), that is, the chemical-shift interaction, thehomo- and heteronuclear dipolar couplings, and the first-order quadrupolar interaction, are averaged out by rapidlyrotating the sample about an axis oriented at 54.78 relative tothe magnetic field. The 11B, 15N, and 29Si MASNMR spectra ofSi3B3N7 are compiled in Figure 5. For the spin-1=2 nuclei 15Nand 29Si, only the isotropic part of the chemical-shift tensorsurvives the MAS process. The isotropic chemical shift diso of

�45.5 ppm observed for the 29Si nuclei in Si3B3N7 suggeststhat the silicon atoms occupy tetrahedral coordinationenvironments of nitrogen atoms (for example, for a-Si3N4

diso=�46 and �48 ppm),[43,44] confirming the results men-tioned above. To obtain an acceptable signal-to-noise ratio inthe 15N MAS NMR spectrum of Si3B3N7, a sample enriched to100% with the isotope 15N was necessary. A slightlyasymmetric, broad, and featureless signal at 59 ppm with ahalf width of 25 ppm can be identified in the spectrum. Thechemical-shift range of this signal encompasses the chemical

Figure 4. Total PCFs from a) neutron (DN(R); Qmax=50 M�1),[41,42]

b) electron (DE(R); Qmax=30 M�1),[34] and c) X-ray (DX(R);Qmax=24 M�1)[41] scattering experiments on Si3B3N7. R is theinternuclear distance; Qmax is the maximum momentum transfer; thehorizontal lines in (a) and (c) are to guide the eye.

Table 2: Positions [M] of the first and second peak maxima in the PCFsfrom electron, X-ray, and neutron scattering experiments on Si3B3N7.

Scattering method First peak Second peak

Electron 1.44 1.72X-Ray 1.41 1.71X-Ray 11B[a] 1.40 1.72Neutron 1.45 1.72

[a] On a Si311B3N7 sample.


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shifts for 14,15N nuclei in h-BN at 63 ppm[45,46] and in a-Si3N4 at30 ppm.[47] This evidence points to a mixed NB3�xSix environ-ment; however, a further deconvolution of the signal intocontributions from the different building units was notpossible.

Unlike the spectra for the spin-1=2 nuclei 15N and 29Si,which are determined solely by the isotropic chemical shift,the MAS NMR spectra of the spin-3=2 nucleus

11B in Si3B3N7

are dominated by second-order quadrupolar coupling, theonly interaction that scales with the fourth Legendre poly-nomial and that is, thus, not completely averaged out by theMAS process. A simulation of the line shape observed in theMAS NMR spectrum (lower curve in Figure 5c) yields anisotropic chemical shift diso of 30.4 ppm, a quadrupolarcoupling constant CQ of 2.9 MHz, and an asymmetry param-eter hQ of 0.1. These values are in excellent agreement with

those published for h-BN (diso= 30.4 ppm, CQ= 2.9 MHz,hQ= 0).[46,48] However, the possibility that a minor fraction ofthe boron atoms in Si3B3N7 are involved in tetrahedral BN4

units cannot be excluded on the basis of the MAS NMRspectroscopy. The 11B signal for such a species would resonateat 0–5 ppm and would, therefore, overlap with the quadru-pole-broadened resonance of the boron nuclei in BN3

environments, possibly escaping detection. This problem canbe overcome by employing techniques capable of averagingthe fourth Legendre polynomial. Among the various strat-egies developed—dynamic angle spinning (DAS),[49,50] doublerotation (DOR),[51,52] and multiple quantum MAS(MQMAS)[53–57]—the latter has attracted much attentionbecause of the versatility and ease of use of the approach.

In MQMAS NMR spectroscopy, the evolution of amultiple quantum coherence is correlated with the evolutionof a single quantum coherence in a two-dimensional experi-ment under conditions of fast MAS. This approach results in aseparation of the isotropic chemical shift and the quadrupole-broadened line shape. The 11B MQMAS NMR spectrum ofSi3B3N7 is plotted in Figure 6. The projection onto the

horizontal F2 frequency axis represents the conventionalMAS NMR spectrum of Si3B3N7, whereas the projection ontothe vertical F1 frequency axis produces the isotropic spec-trum; the quadrupole broadening is effectively eliminated inthis projection. The fact that only one isotropic resonance at30.5 ppm is observed confirms that exclusively trigonallycoordinated boron species are present in Si3B3N7.

The combined results from the scattering experiments andMAS NMR spectroscopy allow a deeper insight into thedistribution of local strain within the amorphous network. Asthe various scattering experiments produce rather sharpmaxima in their respective PCFs, indicating a relativelynarrow distribution of bond lengths within the amorphousnetwork, the extremely broad line width in the29Si MAS NMR spectrum necessarily originates in a wide

Figure 5. a) 29Si, b) 15N, and c) 11B MAS NMR spectra of Si3B3N7.[44]

The lower curve in (c) is a simulation of the 11B MAS NMR spectrumwith the parameters given in the text.

Figure 6. Two-dimensional 11B MQMAS NMR spectrum of Si3B3N7, aswell as projections onto the vertical F1 and horizontal F2 frequencyaxes. In the projection onto the F1 axis, only one isotropic signal,assigned to trigonally coordinated boron, is observed.




distribution of N-Si-N bond angles. Together with thecomparatively marginal line broadening in the11B (MQ)MAS NMR spectra, which is a consequence of therigidity of the BN3 units, these findings imply that in Si3N3B7

the majority of the local deformations that are inherentlypresent in an amorphous network are located within the SiN4

tetrahedra.

2.2.2.Medium-Range Order

The combined results from the various local probesunambiguously identified trigonal-planar BN3 and tetrahe-dral SiN4 units as the network-forming polyhedra of amor-phous Si3N3B7. The next logical step in the structureelucidation involves the investigation of the interconnectionof these local building units into an extended three-dimen-sional network. To determine this medium-range order,predominantly two sources of information can be tapped.One strategy involves analyzing the PCFs to longer distances(2–5 D). As shown in Figure 4, the PCFs exhibit distinctmaxima at 2.5–3.0 and 4.3 D. The maxima at 2.49 and 2.9 Ddiscernable in the neutron PCF are typical signatures ofadjacent nitrogen atoms within a BN3 triangle or a SiN4

tetrahedron, respectively. Further information is not readilyavailable from the PCFs, as the inherent shortcoming of thisapproach, the lack of elemental selectivity, entails a severeoverlap of the signatures of partial PCFs for different atompairs. This problem can be at least partially overcome byemploying isotopic contrast techniques that rely on thedifferent scattering lengths bl of two isotopes of the sameelement (for example, bl(

14N)= 9.37 M 10�15 m, bl(15N)=

6.44 M 10�15 m).The results of neutron scattering experiments on Si3B3N7

using isotopic contrast variation for nitrogen are shown inFigure 7. By taking the Fourier transform of the difference of

the intensities measured for two samples with differentisotopic concentrations (in this case, Si3

11B3N7�Si311B3

15N7),the contribution of atom pairs not containing nitrogen can befiltered out of the total PCF.[41,42] The noticeably reducedintensity of the maximum at 2.74 D in the difference function(Figure 7) presents clear evidence that this peak arises froman atom pair not involving nitrogen. This result necessarilyindicates a Si–B correlation and is the first indirect exper-

imental evidence of Si-N-B connectivity within the amor-phous network of Si3B3N7.

Solid-state NMR spectroscopy offers more direct evi-dence of Si-N-B connectivity and allows its quantification. Atthis length scale, the magnetic dipolar interactions, which arecharacterized by the dipolar coupling constant D [Eq. (1)],

D ¼ gI gSm04p �h

2p r3ð1Þ

can be utilized to analyze structural features. In Equation (1),gI and gS denote the magnetogyric ratios for the two differentnuclei involved, r corresponds to their separation, m0 is thepermeability of vacuum, and �h is the Planck constant over 2p.The heteronuclear dipolar interaction, normally averaged outby the process of MAS, can be selectively re-introduced byemploying advanced pulse techniques such as rotational echodouble resonance (REDOR)[58–63] or rotational echo adiabaticpassage double resonance (REAPDOR).[64–66] The commonphysical basis of these approaches is the manipulation of thedipolar interaction through rotor-synchronized radio-fre-quency pulses, as illustrated in Figure 8 for the REDOR

sequence. In the case of an isolated two-spin system, theREDOR evolution curves can be analyzed using Equa-tions (1) and (2) to determine the internuclear distance r.[60] In

DSS0

¼ 1� 14p S0

Zp

0

Z2p

0

cos�2

ffiffiffi2

pN TR D sin 2# sinf

�sin#d# df ð2Þ

Equation (2), DS/S0 denotes the normalized difference signal,N the number of rotor cycles, and TR the rotation period

Figure 7. Fourier transform DDN(R) of the first-order differencefunction DN(Q) determined from neutron scattering experiments onSi3

11B3N7 and Si311B3

15N7. The Fourier transform corresponds to a PCFcontaining only distances R involving nitrogen.[41]

Figure 8. The principle of REDOR. The heteronuclear dipolar interac-tion between nuclei S and I described by the Hamiltonian h

ISD, aver-

aged out over one rotor period TR, is reintroduced by rotor-synchron-ized pulses for the I-spin species. The S-spin signal is detected using arotor-synchronized spin-echo sequence. The dipolar coupling betweenthe I and S spins leads to an attenuation of the intensity of the S-spinsignal (S) relative to that of the reference signal (S0) acquired withoutI-spin pulses. If the evolution time NTR is varied and the normalizeddifference signal (S0-S)/S0=DS/S0 monitored as a function of theevolution time, a REDOR curve is obtained, from which internucleardistances r can be determined.


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(Figure 8); the polar angles # and f relate the principal-axissystem to the laboratory system.

For Si3B3N7, however, such a simplified analysis is notappropriate. A number of complications, for example, thedistribution of bond lengths and angles inherently present inamorphous solids, the possible existence of multiple spininteractions, and the quadrupolar nature of the 11B nucleushave to be taken into account. Consequently, a knowledge ofthe effects of these perturbations is a prerequisite for thesuccessful analysis of the dipolar evolution curves.[67] Fig-ure 9a illustrates the effect of a Gaussian distribution of B–Si

distances on the resulting 11B{29Si} REDOR evolution curve.The important observation is that the initial part of theREDOR curve is virtually unaffected by the bond-lengthdistribution. Only at longer evolution times does a damping ofthe oscillations in the REDOR curve become noticeable.Similarly, a variation in the angle a spanned by the two B–Siinternuclear vectors in a SiB2 three-spin system has littleeffect on the initial part of the evolution curves, as shown inFigure 9b.[67] Thus, when the data analysis is confined to theinitial part of the evolution curves, perturbations such asmultiple spin interactions and distribution effects may beharnessed. The analysis of medium-range order employingdipolar NMR spectroscopic approaches then boils down to

the elucidation of the second coordination sphere around acentral boron or silicon atom. If the internuclear distances(from the analysis of the PCFs; see Section 2.2.1) are used asinput parameters, then the analysis of dipolar NMR spectro-scopic data allows the number of silicon or boron atoms thatare next-nearest neighbors to a central boron or silicon atomto be determined.

For the experiments, we employed a sample of Si3B3N7

that was 100% isotopically enriched with the isotope 29Si. Theresults of the 11B{29Si} REDOR experiment on this sample arecollected in Figure 10.[69, 70] The spectra reveal an attenuation

of the intensity of the REDOR signal relative to that of thereference signal with increasing evolution time. The resultingREDOR evolution curve (filled circles in Figure 10b) cannow be analyzed in terms of the numbers of silicon atoms inthe second coordination sphere around boron. The assump-tion of a single B–Si interaction with an internucleardistance r of 2.74 D produces the dotted line in Figure 10b,and the simulation of a BSi2 three-spin system (r= 2.74 D, a=

608) produces the dashed line. A superposition of these twocontributions according to 0.44(BSi2)+ 0.56(BSi) (solid linein Figure 10b) results in a satisfactory fit to the initial part ofthe REDOR curve. From this fit, an average of 1.4� 0.1

Figure 9. a) Simulated 11B{29Si} REDOR evolution curves for an iso-lated BSi two-spin system calculated assuming a fixed B–Si distance rof 2.74 M (&), or assuming a Gaussian distribution about r=2.74 Mwith half width of 0.5 M (*). b) Simulated REAPDOR evolution curvesfor a SiB2 three-spin system calculated for various B-Si-B angles a of20–1808.[67] The simulation package SIMPSON was used for thecalculations.[68]

Figure 10. a) 11B MAS spin-echo NMR (top), 11B{29Si} REDORNMR (middle), and difference (bottom) spectra of Si3B3N7 as a func-tion of the evolution time NTR (increasing to the right). The increasingattenuation of the REDOR signal with evolution time is clearlyevident.[69] b) Resulting REDOR evolution curve (*), as well as simu-lated REDOR dephasing curves calculated assuming a single B–Sidipolar interaction (g) or a BSi2 three-spin system (a), withr=2.74 M in both cases. The solid line corresponds to a weightedsuperposition of these two contributions according to0.44(BSi2)+0.56(BSi).




silicon atoms per BN3 unit is calculated; that is, boron isconnected (via the three nitrogen atoms) to 1.4 silicon atomson average in Si3N3B7. We note that this number is consid-erably smaller than expected assuming a homogeneouselemental distribution, in which case boron should have3.43 silicon and 2.57 boron next-nearest neighbors per BN3

unit, and silicon should have 4.57 silicon and 3.43 boron next-nearest neighbors per SiN4 unit.

Two further independent experiments were conducted toverify this result. The first was a 29Si{11B} REAPDORexperiment (a variation of the REDOR approach optimizedfor quadrupolar nuclei on the dephasing channel), whichconversely to the 11B{29Si} REDOR experiment, allows thenumber of boron atoms around a SiN4 unit to be determined.The results of the 29Si{11B} REAPDOR experiment arepresented in Figure 11.[67,70] In this case, the simulation leads

to an average value of 1.8� 0.2 boron atoms per SiN4 unit;that is, the silicon atoms are connected (via nitrogen atoms) toan average of 1.8 boron atoms. Again, the result is notcompatible with the assumption of a homogeneous distribu-tion of silicon and boron in the network and, thus, lendssupport to the results of the 11B{29Si} REDOR experiment.

A complementary way of acquiring information about themedium-range order in Si3B3N7 is to establish the number ofhomonuclear next-nearest neighbors for a central boron orsilicon atom. This objective can be accomplished with the helpof static spin-echo decay spectroscopy, which producesquantitative results for spin-1=2 nuclei such as 29Si. At shortecho times 2t, the spin-echo decay I/I0 can generally bedescribed by the Gaussian function I/I0= exp(�(2t)2M2/2).

[71]

The second moment M2 is related to the internucleardistances rij by the Van Vleck equation [Eq. (3)], where S is

M2 ¼35

�m04p

�2

gS4 SðS þ 1Þ�h2

Xi 6¼j

rij�6 ð3Þ

the spin quantum number.[72] Because of the r�6 dependence,the contribution of the first two coordination spheres usually

amounts to 90–95% of the total second moment. Therefore,an estimate of M2 allows the number of homonuclear next-nearest neighbors to be calculated, allowing an independentverification of the REDOR and REAPDOR results.

A fit of the 29Si spin-echo data (Figure 12) leads to asecondmoment of 3.1 M 106 rad2 s�2. Assuming a Si–Si distanceof 2.98 D (again, from the PCFs), each Si–Si pair contributes

0.56 M 106 rad2 s�2 to the second moment. Thus, an average of5.6 silicon atoms occupy the second coordination sphere ofeach silicon atom. This value agrees with the results from theREAPDOR experiment: approximately two boron atoms(according to the 29Si{11B} REAPDOR experiment) andapproximately six silicon atoms (according to the 29Si spin-echo decay experiment) complete the second coordinationsphere of each silicon atom.

2.2.3. Long-Range Order

The applicability of all interference-based methods for thestructural characterization of amorphous solids is confined tolength scales of less than 8 D, as the inherent distribution ofbond lengths and angles in these materials entails a rapiddecay of the PCFs at longer distances. Approximately thesame distance cutoff holds for the dipolar NMR spectroscopicapproaches. As a consequence, neither of these methodsholds much promise for the elucidation of structural featuresbeyond 8 D. However, it is in this regime that effects such asthe precipitation of nanocrystals or phase separation inmultinary systems, which severely affect the materialPsproperties, can occur. The relevant length scale is, however,accessible by direct-imaging methods such as SEM, AFM, orTEM (EFTEM).[34] The latter, EFTEM, was used to charac-terize the amorphous ceramic Si3B3N7. Apart from producingthe electron-scattering-based PCFs presented in Sec-tions 2.2.1 and 2.2.2, TEM was also used to provide elementalmaps of silicon, boron, and nitrogen with high lateralresolution. Figure 13 shows the EELS spectrum of Si3B3N7

in the region of the B K edge. With an imaging electron-energy filter attached to the TEM, an energy window of theEELS spectrum can be selected and an image of the spatialenergy distribution of the inelastically scattered electrons inthis window produced. Usually three such images are

Figure 11. 29Si{11B} REAPDOR evolution curve for Si3B3N7 (*;nMAS=5 kHz, nRF(

11B)=89 kHz), as well as simulated REAPDORdephasing curves calculated assuming a single Si–B dipolar interaction(a) or a SiB2 three-spin system (g), with r=2.74 M in bothcases.[67]

Figure 12. Plot of the intensity of the 29Si spin-echo NMR signal ofSi3B3N7 as a function of the echo time 2t (*).[67] A fit of the data(c) leads to a second moment M2 of 3.1 S 106 rad2 s�2.


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acquired: two pre-edge images, which are used to calculate abackground image, and an image beyond the ionization edge(post-edge image). After the background is subtracted fromthe post-edge image, pixel by pixel, the remaining electronintensity stems from the element under investigation. Thus,the distribution of the element can be qualitatively displayedas a map, and the concentration of the element quantified inany pixel of the elemental map. For a Si3B3N7 sample,elemental maps (with a lateral resolution of 0.7 nm) of boron,nitrogen, and silicon, as well as a bright-field image, all of thesame sample area, are shown in Figure 14. The importantresult is that homogeneous distributions are observed forsilicon, boron, and nitrogen in the elemental maps.[34] Inparticular, the images contain no indication of phase separa-

tion, segregation, or clustering. Note, however, that thesample thickness of approximately 5 nm could cause projec-tion effects. At this thickness, the vertical resolution isapproximately a factor of seven less than the lateral reso-lution, and the resulting projection (averaging) effects couldmask a possible phase separation.[73] The macroscopic order-ing of Si3B3N7 was also characterized through a densitymeasurement with a helium pycnometer. The observeddensity of 1.899 gcm�3 can be compared to the densitiescalculated for different structural models.

2.3. The Structure of Si3B3N7 “Seen” by Experiment

In this Section, we summarize the experimental findingsfor Si3B3N7 with respect to short-range and intermediate-range order. Through the use of XANES spectroscopy,scattering techniques, and MAS NMR spectroscopy, tetrahe-dral SiN4, and trigonal planar BN3 and NB3�xSix units wereidentified as the polyhedra that dominate the short-rangeorder of Si3B3N7 (Figure 15). Local deformations seem to

accumulate within the N-Si-N angles. The interconnection ofthe local building blocks in the amorphous network wasprobed with dipolar NMR spectroscopic methods and withthe PCFs obtained from scattering experiments. The Si-N-Blinkages identified in the neutron PCFs by employing isotopiccontrast techniques were confirmed and quantified with thehelp of 11B{29Si} REDOR, 29Si{11B} REAPDOR, and 29Si spin-echo NMR spectroscopy. The results suggest a partialsegregation of the boron and silicon atoms at intermediatelength scales (2–8 D), as indicated in Figure 15d,e. Accordingto the NMR spectroscopic investigations, a central boronatom is surrounded by four to five boron atoms[67] and onlyone or two silicon atoms in the second coordination sphere.Correspondingly, a central silicon atom is connected (via thefour nitrogen atoms in the first coordination sphere) to sixsilicon atoms and two boron atoms, on average, in the secondcoordination sphere. These short- and intermediate-rangenetwork fragments serve as a basis and a test for the modelingwork described in the following Section.

Figure 13. EELS spectrum of Si3B3N7 in the region of the B K edge. Theextrapolated background (calculated from two pre-edge images shownas gray bands) is subtracted from the post-edge image (gray/blackband) to obtain the concentration of boron.[34]

Figure 14. TEM images of Si3B3N7: a) bright-field image, and elementalmaps of b) silicon, c) boron, and d) nitrogen for the same sampleregion (at the same scale; scale bar=5 nm).[34]

Figure 15. Experimentally identified network fragments in amorphousSi3B3N7: local network polyhedra a) SiN4, b) BN3, and c) NB3�xSix ; d)and e) extended fragments consistent with the NMR spectroscopicexperiments. Si red, B blue, N green.




3. Theory

The experimental data alone are not sufficient to derive acomplete structural model for amorphous Si3B3N7. Computa-tional modeling is required to achieve this goal.

3.1. Overview of the Modeling Procedures

As it was not clear at the outset, whether particularfeatures of the actual synthesis would play a role in thestructural and bulk properties of the amorphous compound,five different classes of Si3B3N7 models (A–E) were generatedin different ways:[74]

A: quenching from the melt,B: sintering of subnanometer-sized crystallites,[75]

C: formation of clusters of Si3B3N7 using a molecularmodeling procedure designed by Gladden[76] and adaptedby Wefing,[77] followed by sintering of the clusters,

D: random atom-beam deposition, modeled using a randomclose-packing algorithm[78] ,

E: the actual (rather involved) polymer-precursor route,which is similar to a sol–gel process.[79, 80]

All models of classes A–E involved 700–1300 atoms, andthe simulation/relaxation parts of the modeling process wereperformed using both molecular dynamics and Monte Carlosimulations at constant volume and pressure. In models ofclass A, the starting configurations were all well-equilibratedmelts. In models of class B, the simulations started from amixture of BN and Si3N4 nanocrystallites. The clusters to besintered in models of class C were generated using the citedbuild-up and build-down algorithm. Finally, the startingconfigurations of models of class D were generated by anoptimal placement of silicon and boron atoms in the holes of arandom configuration of nitrogen atoms.

As the polymer-precursor route was the one used tosynthesize the experimentally investigated samples ofSi3B3N7, we give a short description of the stepping-stoneprocess that we used to model (E) this route (details can befound in references [79,80]). Initially, the precursor moleculesare in solution with an excess of NH3; the rate of reaction ofthe molecules depends on the number of available reactionsites per molecule and on the concentration of molecules inthe solution. Once several precursor molecules have joinedinto a larger aggregate, this oligomer will move much moreslowly than the remaining precursor and NH3 molecules.Thus, the oligomers become essentially stationary and serveas condensation centers for the still-mobile reactants. We can,therefore, model this latter phase of the polymerization as amultiple condensation process of individual oligomers andmonomers. At the end of this stage, we are left with manyisolated oligomers, which begin to cross-link to form thepolymer. This development is modeled by placing theoligomers randomly on a lattice and shrinking the averagedistance between them until the oligomers can interact andform bonds. The final pyrolysis stage, which transforms thepolymer into a ceramic, is simulated by a Monte Carlosimulation at 1200 K, well below the (calculated) melting

temperature Tmelt of the system (Tmelt(Si3B3N7cryst)= 2500 K

and Tmelt(Si3B3N7amorph)= 2000 K) of the system. Nevertheless,

under these conditions, most of the dangling bonds stillpresent in the polymer are eliminated.

In each of the cases A–E, raw models were generated,which were subsequently (locally) minimized by energy, usingMonte Carlo and conjugate-gradient methods at constantvolume and pressure. We employed an empirical potentialfrom the literature,[81] which was derived from ab initiocalculations on molecular fragments and from structural andelastic data for the binary compounds Si3N4 and BN. Variousstructural properties of the models generated were analyzed:the radial PCFs and angle distribution functions (ADFs), themean coordination numbers within the first and secondneighbor spheres of the atoms, and the ring statistics. Directcomparisons can then be made with the experimentallydetermined PCFs and the observed first and second coordi-nation spheres. The ADFs are only indirectly accessible fromthe experimental data; the same holds true for informationabout the third coordination sphere, and the size anddistribution of rings within the network. In Section 3.2, thestructural properties of the models are briefly described, andtheir similarities and differences, both with respect to eachother and with respect to the experimental data, arediscussed.

3.2. Structural Properties of Models Belonging to the Five Classes3.2.1. Pair Correlation Functions

In Figure 16, the experimental and calculated X-ray andneutron PCFs are compared. The functions calculated fromall of the models have the same general shape as theexperimental functions. As is apparent from Table 3, thevarious peaks in the experimental PCFs can be associatedwith specific interatomic distances within the first and secondneighbor spheres (see also Figure 17). The best quantitativeagreement is found for the crystal-fragment (class B) and thesol–gel (class E) models.

A great advantage of the availability of structure modelsgenerated by computer is that they allow the analysis offeatures of the amorphous structure that are not directlyaccessible by experiment. Thus, we can, for example, extendthe peak assignments given in Section 2. In Figure 17, thepartial PCFs for neutron and X-ray scattering experimentscalculated for models of class A and B are presented. Theassignment of the first peaks and shoulders up to 3.1 D agreeswith that in Table 3, while the shallow peaks between 3.8 and4.2 D in the experimental PCFs (Figures 16 and 17) can nowbe associated with third-neighbor Si–N and B–N distances,and the oscillations at 5.0–5.5 D with N–N and Si–N distances(Table 3). In addition, small peaks occur at approximately2.05, 2.2, and 2.6 D in the B–B, Si–B, Si–Si, and N–N partialPCFs (Figure 17), indicating the presence of edge-sharingpolyhedra in the structural models. However, these featuresare not clearly visible in the experimental PCFs.


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3.2.2. Local Coordination and Angle Distribution Functions

Regarding the distribution of coordination numbers in theoptimized models, the silicon atoms are mainly fourfoldcoordinated, whereas the boron and nitrogen atoms aremainly threefold coordinated. Comparing the silicon andboron atoms, we find that the boron atoms achieve their idealcoordination number more often than the silicon atoms (thatis, the variation about the mean coordination number is largerfor silicon), and that coordination defects are more frequently

encountered at the silicon centers than at the boron centers.These results hold for all model classes. Differences betweenthe models of different classes are found in the distribution ofthe nitrogen coordination spheres NB3�xSix. For modelsderived from classes A, C, and D, the distribution ofNB3�xSix units reveals a preference for NBSi2 and NB2Siunits, while the models of classes B and E contain a higherpercentage of NSi3 and NB3 units.

Similarly to the ring-size distributions discussed in Sec-tion 3.2.3, the ADFs are easily calculated for computer-generated models, but are not directly accessible by experi-ment. Nevertheless, knowledge of these distribution functionscan assist us in interpreting the more subtle aspects of theexperimental data. As is apparent in Figure 18, the maxima inthe ADFs for the N-Si-N, N-B-N, and X-N-Y (X,Y= Si,B)angles in all models correspond to those expected fortetrahedral or trigonal coordination. Furthermore, the N-Si-N and, in particular, the X-N-Y distributions for all modelsexhibit a small peak near 90o. This result agrees with theobservation that a small number of edge-sharing SiN4

tetrahedra and BN3 triangles are present in all of themodels. The interatomic distances d90(X-Y) correspondingto these 908 angles are d90(B-B)� 2.02 D, d90(Si-B)� 2.23 D,and d90(Si-Si)� 2.44 D. As these values are similar to otherrelevant interatomic distances, they will at most result in smallshoulders on larger peaks in the total experimental PCFs.

3.2.3. Second Coordination Spheres and Ring Statistics

Whereas all models yielded similar distributions for thedistances and angles in the first coordination spheres, thesecond (cation–cation) coordination spheres reveal the differ-ences between the model classes. Models of classes B and Econtain more Si–Si and B–B contacts between next-nearestneighbors than all other models and are, thus, in bestagreement with experiment. A comparison of the ring-sizedistributions for the different models (Table 4) shows thatmodels of classes A–C contain approximately 5–7% four-membered rings (Si2N2, B2N2, and SiBN2), and models ofclasses D and E approximately 8–10%. The four-memberedrings are associated with small peaks near 908 in the ADFsand with small shoulders in the partial PCFs. No preference

Figure 16. Comparison of PCFs from a) X-ray (DX(R)) and b) neutron(DN(R)) experiments (c) on Si3B3N7 with those calculated (c) formodels of the five classes A (melt), B (nanocrystallite), C (molecularmodeling), D (random close packing), and E (sol–gel).[74]

Table 3: Positions [M] of the peak maxima in the experimental PCFs ofSi3B3N7 and the corresponding interatomic distances [M] in the models.

Peak(experiment)[a]

Pairs/distances (model)[b]

1.43 B–N(I) 1.431.72 Si–N(I) 1.722.5 N–N(I) 2.50, B–B(I) 2.502.8 N–N(II) 2.71, Si–B(I) 2.753.0 B–N(II) 2.90, Si–Si (I) 3.03.8 B–N(III) 3.804.2 Si–N(II) 4.20, B–B(II) 4.255.0–5.5 N–N(III) 5.2, B–B(III) 5.0, B–N(IV) 5.2, Si–Si (II) 5.5

[a] Average of the peak maxima in the PCFs from X-ray, neutron, andelectron scattering experiments. [b] Average of the interatomic distancesfrom models of classes A–E; roman numerals distinguish differentdistances for the same atom pairs.




for edge sharing between two SiN4 tetrahedra over edgesharing between a SiN4 tetrahedron and a BN3 triangle is

observed in the compositional distribution of the four-membered rings. The highest fraction of six-membered ringsis found in the crystal-fragment models of class B, whichinitially contain a considerable number of borazene (B3N3)and Si3N3 rings, approximately 40% of the six-memberedrings each. A significant part of the initial structure remainsintact after the sintering. In contrast, in the models ofclasses A, C, and D, mainly mixed-cation six-membered ringsare found. Models of class E exhibit a larger number ofborazene rings.

3.2.4.Main Structural Differences Among the Model Classes

The differences in the structures of the models of differentclasses are most apparent in the second coordination spheres(in particular, the B–B, Si–B, and Si–Si partial PCFs;Figure 17). The melt models of class A (blue curves) containmore Si–B, and less B–B and Si–Si next-nearest-neighborpairs than the crystal-fragment models of class B (red curves).An accumulation of homoatomic B–B and Si–Si contactsbetween next-nearest neighbors corresponds to concentrationfluctuations at a subnanometer scale, as is clearly visible in thecomplete structure models of classes A and E depicted inFigure 19. With respect to the second coordination sphere,models of class B resemble those of class E, while models ofclasses C and D resemble those of class A. This division of themodels into two groups is also apparent in the ring-sizedistributions, as models of classes B and E show an increasedpresence of borazene rings. The alternating arrangement ofthe cations in the structural models generated by quenchingfrom the melt (class A) is reminiscent of the empiricalLRwenstein rule.[82] This observation suggests that the actuala-Si3B3N7 ceramic synthesized by the sol–gel route wouldprobably be less stable than a hypothetical ceramic synthe-sized by quenching from the melt, a hypothesis that issupported by the fact that the average energy of models ofclass A is lower than that of models of class E (see Sec-tion 3.3).

It is important to note that the differences among thePCFs of the various model classes are greater than thosewithin a given model class (Table 5). Thus, it should bepossible to distinguish among various amorphous modifica-tions of Si3B3N7, once their syntheses by the various routeshave been realized.

3.3. Density and Stability of Voids

The macroscopic density is another quantity that can beused to compare the theoretical models with experiment.However, the density is one of the most difficult quantities tomodel in amorphous compounds, as it depends crucially onthe conformation of the network on a length scale of 0.5–2 nm, which is nearly inaccessible by experiment (exceptpossibly by small-angle X-ray or neutron scattering underfavorable conditions). Therefore, information important forthe modeling, for example, whether small voids of 0.5–2 nmdiameter are contained in the real compound, is not available.The densities calculated for the various models are shown in

Table 4: Fraction of rings with four, six, or eight atoms in models ofSi3B3N7 of classes A–E.

4-rings 6-rings[a] 8-rings

A 0.05 0.28 [0.01] 0.39B 0.06 0.41 [0.35] 0.32C 0.07 0.27 [0.02] 0.37D 0.10 0.24 [0.06] 0.36E 0.08 0.28 [0.11] 0.31

[a] Relative fraction of borazene rings given in brackets.

Figure 17. Comparison of the a) X-ray (DABX (R)) and b) neutron (DAB

N (R))partial PCFs calculated for models of Si3B3N7 of classes A (c) andB (c).[74]


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Figure 20.[74] Again, the closest agreement with experiment isobserved for the crystal-fragment models of class B and thesol–gel models of class E. However, if the void volume issubtracted from the total volume of each model, the remain-ing (void-free) regions of the models exhibit densities of 2.5–2.8 gcm�3. These values compare well with the density ofhypothetical crystalline Si3N3B7 polymorphs (2.9 gcm�3),[7,83]

and the appropriately weighted average of the experimentaldensities of crystalline h-BN and b-Si3N4 (2.75 gcm

�3). Thisobservation suggests that the reason for the surprisingly lowexperimental of amorphous Si3B3N7 might be the presence ofstable voids (cages) with radii of 3–10 D, a hypothesis that isdifficult to verify experimentally.

The polymer-precursor route is expected to lead to manydefects and, hence, to the formation of cavities of all sizes inthe actual structure of amorphous Si3B3N7. The fact that the

final densities of the sol–gel models (class E) agree best withthe experimental density is self-consistent, and justifies faithin the model-generating procedure. According to the plots ofdensity versus potential energy for models chosen from allclasses (Figure 20), the higher the density, the lower theenergy. This finding underlines the importance of taking thedetails of the synthesis route into account in the models ofamorphous Si3B3N7. Attempts to simply find the amorphousconfiguration with the lowest energy would clearly miss themark and instead generate structures with densities that aremuch too high.

The distribution of void sizes calculated for models ofSi3B3N7 of classes A and E is shown in Figure 21.[84] Bothstructures exhibit tiny voids with radii of less than 2 D, whichreflect the general packing constraints of the SiN4 polyhedraand BN3 triangles within the covalent network. In this respect,

Figure 18. ADFs for a) N-Si-N, b) N-B-N, and c) X-N-Y (X,Y=Si,B)calculated for models of Si3B3N7 of the five classes A (melt), B (nano-crystallite), C (molecular modeling), D (random close packing), andE (sol–gel). The dotted line indicates the expected maximum in eachcase.




both classes of model resemble the (hypothetical) crystallinecompound,[7] which by its nature, exhibits a very narrow sizedistribution. However, the model configurations generated by

the sol–gel route (class E) contain many comparatively largevoids with radii up to 7 D. Annealing at low temperaturesover a long period of time leads to redistributive coarsening inboth the quenched-melt (class A) and the sol–gel (class E)models, but the effects are not very large, and no significantchanges in the overall densities of the models are observed. Incontrast, when the crystal-fragment (class B) or the sol–gel(class E) models are heated beyond the melting temperatureat constant pressure, the densities increase from 2.2 or1.9 gcm�3, respectively, to 2.6 gcm�3, owing to the eliminationof larger voids. This observation again shows that thequenched-melt models (class A) are thermodynamicallymore stable than, for example, the sol–gel models (class E).However, during the polymer-precursor synthesis, the systemremains far below the melting temperature. Thus, the largevoids in amorphous Si3B3N7 should be relatively stable (toapproximately 1500 K).

An investigation of the stability of voids in a 5200-atomsystem with an initial density of 1.9 gcm�3 over a wide rangeof temperatures below the melting point produced a slowoverall densification.[84] The density increased approximatelylogarithmically with time. In this process, larger voids weremore stable than smaller ones. Therefore, the large voidscontained in amorphous Si3B3N7 should remain stable to atemperature of approximately 1500 K. The same logarithmic

Figure 19. Structure models of amorphous Si3B3N7 of a) class A (melt)and b) class E (sol–gel). The cation distribution is homogeneous in (a)and heterogeneous in (b). The structures are depicted as ball modelswith an edge length of ca. 4.5 nm; Si red, B blue, N green.

Table 5: Spread in the calculated neutron PCFs within the models(DYYDN(R)) and between the models (DYZDN(R)) of Si3B3N7 of classesY, Z=A–E.[a]

A B C D E

A 0.15 0.38 0.32 0.17 0.93B 0.38 0.09 0.44 0.30 0.60C 0.32 0.44 0.20 0.25 0.81D 0.17 0.30 0.25 0.18 0.71E 0.93 0.60 0.81 0.71 0.13

[a] The internal spread in the PCFs DDY(R) within a class Y containing

nY models is DYYDY(R)= (1/NR)

PNR

i¼1

PnY

m¼1[Dm(Ri)�hDY(Ri)i]2, where

hDY(Ri)i= (1/nY)PnY

m¼1Dm(Ri) is the average value of the PCF at a distance

Ri, Dm(Ri) is the value of the PCF at Ri in model m of class Y, and NR is the

number of distances Ri included. To compare the PCFs of models of

different classes Y and Z, we calculate DYZD(R)= (1/NR)PNR

i¼1-

[hDY(Ri)i�hDZ(Ri)i]2.

Figure 20. Plot of the density 1 versus the energy difference DE(between the energy of the model of the amorphous compound andthat of the hypothetical crystalline compound[7]) for models of Si3B3N7

of classes A (melt), B (nanocrystallite), C (molecular modeling),D (random close packing), and E (sol–gel).[74] The horizontal lineindicates the experimentally determined density of Si3B3N7.

Figure 21. Distribution of voids of radii Rvoid for models of Si3B3N7 ofclasses A (melt) and E (sol–gel), expressed in terms of the voidvolume Vvoid normalized to the total volume V total

void of all voids in themodel.


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time dependence is found for the slow improvement in energyduring the aging process, as observed in long-time simulationsfor Si3B3N7 models at temperatures below the meltingpoint.[85] The effect of high pressures on the nanoporousmaterial could have great ramifications for potential applica-tions.[84] Constant-pressure (1–50 GPa) simulations at 300 Kshow that the system can be compacted significantly, and thatmuch of the densification is irreversible. For pressuresexceeding 15 GPa, a saturation density value of approxi-mately 2.8–3.0 gcm�3 is observed after pressure release. Incomparison, at a temperature of 1500 K, a much fasterdensification to approximately 2.8 gcm�3 is observed for allpressures above 1 GPa.[86]

3.4. Bulk Properties

Once a credible structural model has been constructed,the physical properties of the compound can be calculated. Inthis Section, a short overview of various bulk properties ofamorphous Si3B3N7 is given.

The calculation of the vibrational (phonon) density ofstates (VDOS)[74,89] and the bulk modulus B[74] (70–180 GPa)of the different models was performed with the GULPprogram.[88] The overall shapes of the VDOS curves are thesame for models of all classes (Figure 22). All of the curvesexhibit peaks at approximately 1500 and 1000 cm�1, which aremost likely associated with B–N and Si–N stretching modes,respectively.[90] The speed of sound vS can be estimated bystudying the time evolution of a heat pulse;[92] in the case of

amorphous Si3B3N7, a value of vS� 3–4 M 103 ms�1 isobtained.[93] This value falls in the range of speeds of soundfound experimentally for crystalline b-Si3N4 (1.1 M 103 ms�1)and h-BN (7M 103 ms�1). Similarly, we can simulate andmeasure the heat conductivity k by generating a steady-stateheat flow through the system;[92] for amorphous Si3B3N7, wefind a heat conductivity of approximately 4 Wm�1K�1, whichvaries only slightly with temperature.[93] As experimentalconductivity values typically exceed the simulated values by afactor 3–5,[92,93] the true value for amorphous Si3B3N7 shouldbe k� 15 Wm�1K�1.

We have also calculated a metastable phase diagram forthe Si3B3N7 system by performing long-time Monte Carlosimulations over a large range of temperatures and volumesfor a system containing 702 atoms.[89, 94] The term “metastablephase diagram” seems appropriate here, as in the solid state,the phase with the lowest energy corresponds to a weightedmixture of b-Si3N4 and h-BN. The phases with the next-lowestenergies are the crystalline modifications of Si3B3N7.

[7, 83] Ofcourse, for very small crystallites, the polycrystalline mixtureof b-Si3N4 and h-BN begins to lose stability compared to theamorphous phase; the critical particle diameter at which thisoccurs is approximately 30 D.[89, 94] This result explains thefailure of attempts to synthesize amorphous Si3B3N7 bysintering a mixture of BN and Si3N4 microcrystallites (diam-eter of approximately 500 nm).[23] A comparison of thecorresponding free energies yields a very rough estimate ofthe pressure needed to suppress the evolution of N2 from thecondensed phase of Si3N3B7 near its melting temperature; at2000 K, this limiting pressure is approximately 4 GPa.

Finally, it is of great practical interest to gain some insightinto possible aging phenomena in the amorphous ceramicSi3B3N7. Information about the likelihood of structuralchanges, such as the separation of crystalline phases, attypical application temperatures is especially important, asthese changes would probably weaken the amorphous net-work. Our simulations for Si3B3N7 indicate a slow aging with alogarithmic time dependence at temperatures between 1200and 1700 K.[85] Furthermore, Si3B3N7 quenched from the meltshould exhibit a glass transition in the temperature range1700–2500 K, assuming the application of a N2 partialpressure that is high enough to prevent decomposition ofthe ternary melt.

4. Summary and Conclusion

Thus far, Si3B3N7 has been only accessible as a noncrystal-line random network. Since no significant deviations from theideal composition are observed in a-Si3B3N7, the compoundparallels the prototypical glassy system, quartz glass, in thisrespect, and may be regarded as a ternary nitride analogue ofa-SiO2. Of course, the more complicated composition ofSi3B3N7 implies that structure analyses are more difficult. Onthe other hand, the broader choice of atoms that are sensitiveto various structural probes is expected to yield a largerspectrum of information.

The use of as many different structure-sensitive probes aspossible has provided quite consistent results. In particular,

Figure 22. Calculated VDOS for models of Si3B3N7 of classesA (bottom) to E (top).[74] The red and blue vertical lines correspond tovibrations dominated by contributions from SiN4 and BN3 buildingunits, respectively; the dashed vertical lines indicate the positions ofN–Si and B–N vibrations in experimental IR spectra of Si3B3N7.

[91]




the application of overlapping probes that monitor the samestructural property, as well as combinations of complemen-tary probes has helped to generate less ambiguous and morereliable experimental results. It is now well established thatthe first coordination sphere of silicon is tetrahedral inSi3B3N7, and that those of boron and nitrogen are trigonalplanar. Interestingly, the line shapes observed in the NMRspectra of Si3B3N7 vary significantly in their inhomogeneousline broadening. While the quadrupolar transition for boronyields a line shape that fully matches the boron signalrecorded for h-BN, the resonance for silicon exhibits apronounced broadening, compared to that of a-Si3N4. Thus,in amorphous Si3B3N7, boron occupies a virtually idealtrigonal-planar coordination environment, while most of thedeformations, which are a consequence of the inevitablemechanical strain in the amorphous network, occur in thesilicon coordination spheres. Similarly, the nitrogen resonancein the NMR spectrum of Si3B3N7 is also clearly broadened,which is understandable given that four different approxi-mately trigonal-planar NB3�xSix units contribute to the signal.Surprisingly, in the next-nearest neighbor shells, a kind ofsegregation is observed: instead of a random distribution ofcations, a preference for cations of the same type as thecentral cation in the second coordination sphere (for example,silicon atoms as the next-nearest neighbors of silicon) isfound.

These observations from probes sensitive to the localenvironment were complemented by measurements ofmedium- and long-range spatial correlations in the Si3B3N7

system. The PCFs determined by X-ray (synchrotron radia-tion) and neutron scattering revealed all of the expectedstructural features; however, it is not possible to generate acomplete structure from these pieces of the mosaic. Onereason for this difficulty, is simply the lack of solid informa-tion concerning structural features at length scales between afew SngstrRms and a few nanometers. A second reason is thatall of the structural probes yield only statistical information atbest. As a consequence, a multitude of different structuralmodels that reproduce the measurement data can be con-structed, and no further criteria are available to distinguishamong them.

Thus, it is necessary to complement the experiments withcomputer simulations. Structural models were generated bycomputational techniques and validated through comparisonto the experimental results. As it was not obvious at the outsetto what extent the synthesis route would determine thestructural features of the amorphous ceramic, five differentclasses of models were generated for Si3B3N7, each corre-sponding to a different (hypothetical) synthesis route(quenching from the melt, sintering of nanocrystals, growthand sintering of clusters, film deposition, and the actualpolymer-precursor route). A comparison of these models tothe experimental findings reveals that the short-range orderin models of all classes agrees with the experimental data.However, beyond the nearest-neighbor peaks, the PCFs of thedifferent model classes exhibit statistically significant differ-ences. Satisfactory agreement with the experimental data isonly found for two models, the nanocrystallite and thepolymer-precursor models. These two models also show the

best agreement with experiment regarding the inhomoge-neous distribution of next-nearest neighbors. Note that theatomic configurations of these two classes of structuralmodels have systematically higher energies than those ofthe other three model classes. This result implies that thepartial separation of the cations at a subnanometer scalecannot be understood on a purely thermodynamic basis withenergy minimization as the only criterion.

The best simulated models even agree with details of theexperimental results. For example, the largest deviations froman ideal local environment in the models are found for thesilicon atoms. These deviations are visible in the variation ofthe local coordination number and in the broad distribution ofN-Si-N angles found by experiment. Even the edge sharing ofpolyhedra found in the calculated models can be identified asshoulders in the experimental PCFs, which were not pre-viously assigned. The good agreement observed for bothglobal and local structural properties makes us confident thata structural model coming close to reality has been achieved.

The rather low density found experimentally for Si3B3N7

(only two thirds of the value expected for a binary mixture ofh-BN and b-Si3N4) indicates that the amorphous compoundhas a rather open structure. Comparably low densities werealso found for models simulated according to the polymer-precursor route (as the only one of the five model classes).The simulated relaxation of configurations with subnanom-eter-sized pores at constant temperature reveals a highstructural stability to relatively high temperatures. With aknowledge of the structure, additional bulk properties ofSi3B3N7, such as the specific heat, the bulk modulus, or thethermal conductivity, can now be calculated; however, thecorresponding experiments have not yet been performed.

To conclude, the investigations presented here stronglyemphasize the fact that the structure of an amorphousmaterial can only be successfully modeled by considering itssynthesis route. Thus, in the case of amorphous Si3B3N7, themodeling process must include the steps of polycondensationin solution, the aging of the polymer, and finally, the pyrolysisthat forms the amorphous ceramic.

The work presented here was generously funded by the Fondsder Chemischen Industrie, the Max Planck Society, and theDeutsche Forschungsgemeinschaft (SFB 408 at the Rheinische-Friedrich-Wilhelms-Universit9t Bonn). The authors would liketo thank all students and colleagues for their support inachieving the results. Their contributions are acknowledged byciting the respective publications. It is our particular pleasure toacknowledge the always pleasant and efficient cooperationwith all of our colleagues in the SFB 408. Finally, we would liketo thank Dr. Alexander Hannemann for his competentassistance in preparing the manuscript.

Received: November 24, 2005

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the route to the structure determination of amorphous solids: a case study of the ceramic si3b3n7

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