theory of the structures of non-crystalline...

Pure & App!. Chem., Vol. 59, No. 1, pp. 101—144, 1987.Printed in Great Britain.© 1987 UPAC

INTERNATIONAL UNION OF PUREAND APPLIED CHEMISTRY

A state-of-the-art review

THEORY OF THE STRUCTURES OFNON-CRYSTALLINE SOLIDS

Jozef Bicerano' and David Adler2

'Energy Conversion Devices, Inc., Troy, Michigan 48084, USAPresent address: Dow Chemical Company, Midland, Michigan 48674, USA

2Department of Electrical Engineering and Computer Science,Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

commissioned on the occassion of

INTERNATIONAL CONFERENCE ON THE THEORY OF THESTRUCTURES OF NON-CRYSTALLINE SOLIDS

held in Bloomfield Hills, Michigan, USA: 3—6 June 1985

Theory of the structures of non-crystalline solids

Jozef Bicerano* and David Adler

Energy Conversion Devices, Inc., 1675 West Maple Road, Troy, Michigan 48084, USA

Department of Electrical Engineering and Computer Science, Massachusetts Instituteof Technology, Cambridge, Massachusetts 02139, USA

Abstract - The study of the structures and properties of non—crystalline solids and

their interrelationships is a rapidly growing field of research, involvinginterdisciplinary efforts at the frontiers of chemistry, solid state physics, andmaterials science. These materials differ from the better-understood crystallinesolids by the lack of long range three—dimensional translational periodicity intheir atomic arrangements. Consequently, this study cannot make use of thesimplification attendant with the use of repetitions of a small unit cell. Wepresent a review of the state-of-the-art in the theoretical study of the structuresof such non—crystalline solids. The topics covered include structural models, thenature of order and disorder, model systems of low dimensionality, chemical bonding

considerations, thermodynamic and kinetic considerations, dynamic simulations,elastic and vibrational properties, electronic structure and properties, amorphousmetals and alloys, silica glasses and related alloys, liquids and superlattices.It is our hope that this review will generate more interest among chemists in thescientific and technological problems and challenges presented by these materials.

I. INTRODUCTION

The atomic arrangement in non-crystalline (or "amorphous") solids lacks translationalperiodicity; therefore, unlike crystalline solids, these materials do not have a long rangeorder consisting of the repetition in three dimensions of a well-defined unit cell. Theyhave, for this reason, traditionally been in a no man's land between the realms of physicsand chemistry. It has been impossible to describe their electronic wave-functions in termsof the powerful concepts and techniques of crystalline band theory (such as Brillouin zones,Bloch functions, k-space, etc.), discouraging solid state physicists from studying them. Onthe other hand, the fact that they are extended solids rather than molecules with a fixednumber of atoms, has made their study appear to be outside the scope of conventionalchemistry. Their lack of a repeating unit cell has also severely limited the detailedstructural information that can be gained by such standard experimental techniques as x-raydiffraction. The apparent sensitive dependence of the structures of these materials, andhence of their physical, chemical, and electrical properties, on the methods used inpreparation and processing, has further obscured the existing regularities.

During the past two decades, the study of the structures and properties of non-crystallinesolids has gradually emerged as an exciting field of both experimental and theoreticalresearch, involving interdisciplinary efforts at the frontiers of solid state physics,chemistry, and materials science. The barriers to communication between scientists fromthese diverse disciplines have been disappearing, as this field becomes better defined, withits own folklore and mythology, with its own dedicated workers and their characteristictechnical jargon, and with the rapid progress brought about by three important motivatingand enabling factors:

(1) The realization that non-crystalline solids have many important technologicalapplications which go far beyond ordinary window glass. Some of these applicationsare: (i) as thin film photovoltaic materials used in solar cells for theconversion of light into electricity; (ii) in photographic films and xerographicdrums; (iii) in electrical and optical switching; (iv) in electrical and opticaldata storage; (v) in ultra-transparent optical fibers used for telecommunications;(vi) as hard protective coatings for tools and machine parts; (vii) as decorativecoatings; (viii) as metallic glass ribbons used in transformer cores; and (ix) asactive elements in large-area thin film displays such as flat-screen televisionsets. (In addition, polymeric materials, which ordinarily are also a type ofnon-crystalline solid, have many additional applications; however, these will notbe considered in the present review.)

*Current Address: Dow Chemical Company, Central Research, Materials Science and Engineered

Products Laboratory, 1702 Building, Midland, Michigan 48674, USA

102

Theory of the structures of non-crystalline solids 103

(2) The rapid progress in computer technology. The ability to process much largeramounts of data, and the much shorter times required to carry out largecomputations, are the cornerstones of this advance. Consequently, it has becomeincreasingly more efficient to build large—scale structural models fornon-crystalline solids via the use of computer programs, making it less necessaryto go through the tedious process of building "ball-and stick" models; furthermore,it has become feasible to carry out large numerical calculations of electronicstructure by using supercomputers. Finally, realistic simulations of dynamicprocesses such as glass formation via quenching, which require the use oftechniques such as Monte Carlo and Molecular Dynamics, have become possible.

Several current directions in the advance of computer technology insure that computers willplay an increasingly important part in future research on non-crystalline materials. Forexample, advances in parallel processing will probably be especially helpful in dynamicsimulations where the collective behavior of large ensembles of particles is examined. Theability of smaller computers to perform larger calculations, and do this faster than everbefore, will put sophisticated numerical techniques for electronic structure calculationswithin the reach of an increasing number of researchers. In this respect, it is especiallyencouraging to note the new classes of "supermini" and "advanced technology personal"

computers. Finally, advances in three-dimensional color graphics technology, the decliningcost of graphics hardware, and the increasing availability of graphics software, are alldestined to help in computer-aided modeling and visualization of atomic arrangementpatterns, just as they have helped in studying polymers and biological macromolecules.

(3) Recent advances in physics and chemistry. Some new theoretical approachesthat have proved valuable in understanding amorphous solids include localizationtheory, percolation, the utilization of "curved space" techniques which were, inthe past, mainly reserved for problems in general relativity and cosmology, and theuse of higher and fractal dimensions, previously limited primarily to the analysisof unified field theories. On the chemical side, a gradually deepeningunderstanding of the nature of chemical bonding has helped the study ofnon-crystalline solids by contributing to the determination of the local (or shortrange) order around the atoms. In the absence of long range periodic order, thislocal order is of paramount importance in understanding the structures andproperties of these materials.

The theoretical progress has been paralleled and aided by advances both in the array andprecision of experimental techniques available to us and in our understanding of how toapply these methods to non-crystalline solids and then extract structural information fromthe data. The techniques which have been useful in the latter regard include diffraction(x-ray, neutron and electron), EXAFS, various types of spectroscopy (Raman, IR, NMR,Mössbauer and ESR) and measurements of electrical and optical properties such asconductivity, luminescence and reflectivity.

The increased importance of research in non-crystalline solids has been highlighted by theawarding of Nobel Prizes to three scientists (N.F. Mott, P.W. Anderson and P.3. Flory) fortheir contributions to the field. The growth of interest in the area has also beenreflected in a rapid increase in the number of conferences, symposia and workshops held onvarious aspects of the study of these materials. One of the most recent of theseconferences was the International Conference on the Theory of the Structures ofNon-Crystalline Solids, which was held 3-6 June 1985, at the Institute for Amorphous Studiesin Bloomfield Hills, Michigan, U.S.A. The Proceedings of this Conference [1] are stronglyrecommended to readers who want to learn more about these materials. Two other recentbooks, one written by Zallen [2], and the other edited by Adler et al. [3J should also prove

quite useful.

The present review includes a wide range of subjects. Structural models are discussed inSection II, focusing mainly on continuous random network and random close packing models.Several alternative models, and experimental techniques relevant to selecting and evaluatingappropriate models, are also discussed. The subject of "order and disorder" is examined inSection III, in the contexts of short range order, "network momentum", icosahedral order,quasi-crystals and the curved space approach. Model systems of low dimensionality (with oneor two dimensions), which facilitate the performance of certain calculations, are consideredin Section IV. Chemical bonding considerations useful in understanding short range orderare presented in Section V, while fundamental thermodynamic and kinetic considerations arediscussed in Section VI. The use of dynamic (especially molecular dynamics) simulations isexamined in Section VII. Section VIII is devoted to an examination of elastic andvibrational properties, and Section IX discusses the computation of the electronic structureand properties of nonmetallic amorphous solids. This discussion is followed by Section X onamorphous metals and alloys, Section XI on silica glasses and related alloys, Section XII onliquids (which are structurally related to glasses), and Section XIII on superlattices. Ourconclusions are summarized in Section XIV. In order to put the current developments into

104 J. BICERANO AND D. ADLER

perspective, we also provide the general context and background, and present our ownopinions and judgements concerning the work described, in each section, as well as makingrecommendations on how chemists can most easily enter the field and make significantcontributions.

II. STRUCTURAL MODELS

Standard models

Unlike in crystalline solids, where the atomic-scale structure can be precisely determined,at least in principle, by diffraction experiments, in amorphous solids, the lack of aperiodic repeating pattern makes it impossible to determine a unique structure. This mayreflect a real ambiguity, since these materials could have a wide array of alternativestructures with essentially the same total energy or it could represent simply our lack ofknowledge of the precise structure. In any event, it therefore becomes essential to attemptto generate realistic models which will provide insights into the actual structures andproperties of these materials.

The most developed homogeneous models of the atomic—scale structure of non—crystallinesolids are [2,4]:

1. The continuous random network (CRN) model, which has proved valuable in theunderstanding of covalent solids such as amorphous silicon (a-Si), silica (a-5i02)

and a-As2Se3;

2. The random close packing (RCP) model, also sometimes referred to as the denserandom packed (DRP) model, which has been quite useful in studying metallic glasses

such as nickel—phosphorus, gold—silicon and copper-zirconium alloys;

3. The random coil model, which is primarily applicable to polymer-chain organicglasses such as polystyrene.

CRN (continuous random network) models for covalent amorphous solids

Two examples of the similarities and the differences [2,4,5] between a CRN structure and itscrystalline analog are presented in Figs. 1 and 2. The following similarities can be seenbetween the pair of structures in each figure:

(a) (b)

Fig. 1. (a) One layer of a three-fold coordinated crystalline solid, such as arsenic;(b) an analogous continuous random network (CRN) structure.

(a) (b)

Fig. 2. (a) One layer of crystalline As2S3 or As25e3, where the large

open circles represent S or Se; (b) an analogous CRN structure. (Note thatthese two structures can be derived from the corresponding ones in Fig. 1by replacing each bond between the three-fold coordinated atoms by a pair

of bonds to an intervening two-fold coordinated "bridging" atom.This process is called a "decoration" transformation.)

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The algorithm of Lapiccirella et al. [14], on the other hand, starts with a small, initiallyrelaxed cluster, and adds atoms sequentially. Each new atom is added so as to saturate thedangling bond nearest to the center of mass of the cluster. After the addition of eachatom, its position is examined relative to the other atoms in the cluster to prevent theinclusion of unrealistic bonding configurations such as bond lengths not within 10% of thenormal bond length, multiple bonds, or 3—fold and 4-fold rings. After a few atoms are addedin a way which satisfies these constraints, the structure is partially relaxed so as toavoid the generation of highly strained regions frozen in the bulk of the solid. When thesesteps are repeated to generate the full CRN, the structure obtained is both non-defectivend spherical in shape. The spherical shape has the advantages of both minimizing thesurface of the growing cluster and producing a model similar to that derived for an infinitenumber of atoms. In contrast to the algorithm of Wooten et al. [l3J where all the atoms ofthe final CRN are present in the starting large cluster with the diamond structure, thealgorithm of Lapiccirella et al. {14] can be visualized as consisting of the addition ofsuccessive concentric spherical shells to an original nucleus.

Limitations of CRN models

Although the CRN model has occupied a central position in the modeling of the structures ofcovalently bonded non-crystalline solids, it is by no means the only useful model. In fact,there is evidence that it may not be strictly valid for many amorphous alloys.

One obvious practical limitation of type of model for an amorphous material is that,since there is no long range order and no repeating unit cell, any single structure, whetherbuilt by hand or by computer, is non-unique and only represents one example of an entireclass of related conceivable structures. It has therefore been proposed [15] that the word"model" be used for a specific well-defined algorithm or set of rules for generatingexamples of such a related class of structures with any given number of atoms, rather thanbeing used for individual examples. The overall properties of the model can then bedetermined by generating many examples by applying the algorithm and then averaging overthese examples.

Idealized CRN models do not ordinarily admit any defect configurations, which must beintroduced later. It is well-known from studies of crystalline solids, that defects and

impurities, which are often present in concentrations of less than one per 106 atoms, arecrucial in determining important physical properties such as the magnitude of the electricalconductivity. Defects play a similar, very crucial role in non—crystalline solids, in whichit is often much more difficult to pinpoint the specific types of defects involved and the

roles they play [16J. It can be expected that they arise when the energy increase of' aparticular defect is comparable to that arising from the necessary bond distortions in theirabsence. Alternatively, relatively low-energy defects can arise for purely thermodynamicreasons [16]. In addition to dangling bonds, other types of point defects important inamorphous materials include: (1) Valence Alternation Pairs (VAP's) [16,171. Theselow-energy defects are expected to be especially common in alloys which contain normallytwo-fold coordinated chalcogen atoms. The lowest energy electronic state of a chalcogen

atom such as 5, Se or Te is a two-fold coordinated neutral one, denoted by C. In general,

however, it takes very little energy to break a bond and rearrange the resultant bonding

pattern to replace two such atoms by a triply coordinated positively charged (C) and a

singly coordinated negatively charged (Ci) atom, in a reaction which might formally be

represented as 2C-C÷C. Both C and C are low-energy defects since they have a filled

outer octet of electrons, and no electrons in anti-bonding states. (2) Charge TransferDefects (CTD's) [16,18] in tetrahedrally bonded amorphous semiconductors such as a—Si and

a-Ge, in which the lowest energy configuration is denoted T. These CTD's can berepresented as a pair of positively and negatively charged atoms with one or two broken

bonds (T-i-T and T÷T respectively). These are not low—energy defects, but, as is

discussed later, are expected to arise from strains upon processing.

A final limitation of CRN models, relevant to their application to alloys, is theirassumption of chemical ordering, i.e., an inferred strong preference of atoms to bond morefavorably with atoms of different kinds than with the same kind. For example, in a-As2Se3,

it is assumed that only As-Se bonds are present, although one would expect some As-As andSe-Se bonds to also be present as low-energy defect configurations, on simple thermodynamicgrounds. In standard CRN models, homonuclear bonds are only allowed in non-stoichiometricalloys (e.g., in As1 9Se3 2 for which it is clearly impossible to find enough As atoms to

satisfy the valence requirements of all the Se atoms). Because of all the limitationssummarized in this subsection, CRN models should be viewed mainly as useful idealizations ofthe actual structure, and the possibility of significant deviations from them should be keptin mind whenever a new material is studied.


Alternative models for covalent amorphous solids

The earliest structural model for covalent glasses was Lebedev's microcrystallite model [19]which assumed that the structure of glasses consisted of microcrystalline units. This modelwas challenged and eventually replaced by Zacharaisen's CRN model [2,4,5J. Notsurprisingly, however, it is possible to find materials where neither the microcrystallitemodel nor the CRN model is completely adequate to represent the structure and bonding.Borate glasses are examples of such alloys. Extensive nuclear magnetic resonance (NMR) data[20] support the hypothesis that borate glasses such as a-B203 are not composed primarily of

CRN's of individual B04 and 803 units. Instead, these small units form structural groupings

such as boroxyl, diborate and metaborate units similar to the groupings that exist in thecrystalline compounds of the particular borate system. Some of these structural groupingsare illustrated in Fig. 3 [20]. These larger (but still quite small) units are thenconnected randomly to each other to form the glass structure. The situation is thereforeintermediate between the microcrystallite and the CRN models. It has also been pointed out[2lJ that for many types of glass structures it is the intrinsically broken chemical order,i.e., deviations from perfect chemical ordering which are present even in stoichiometricalloys, that generates the defects which control the electronic properties of thematerials. This possible limitation of the Zacharaisen model can be seen from hyperfineeffects, i.e., the proposed clustering into "molecular cluster networks" [22} of largemorphologically and stoichiometrically distinct molecular clusters such as "chain" and"raft" configurations via molecular phase separation with intrinsically broken chemicalorder [21], and exciting recent Mössbauer [23], infrared (IR) [24], and Raman [24]experiments.

Fig. 3. Structural groupings in lithium borate glasses.Solid circles represent boron atoms, and open circlesrepresent oxygen atoms. An open circle with a negativesign indicates a non—bridging oxygen. (a) Boroxol unit;(b) pentaborate unit; (c) triborate unit; (d) diborateunit; (e) metaborate unit; (f) pyroborate unit; (g)orthoborate unit; (h) loose B04 unit. A tetraborate unit

is formed by connecting one oxygen of the B04 unit in the

triborate unit to a B03 unit of the pentaborate unit [20].

These experiments are instructive since they probe the local environments around the atoms,rather than providing statistical averages over the bulk material. Their interpretation,however, is a matter of considerable controversy at the present time. As further Mössbauer,IR and Raman data become available and properly interpreted, a deeper understanding of thenature of the various types of glass structures is likely to emerge. After more than half acentury, the field is still open for significant new conceptual advances!

RCP (random close packed) models for metallic glasses

The RCP (or alternatively ORP) models presently provide the most satisfactoryrepresentations for the structures of amorphous metals [2,4]. A traditional method forgenerating RCP type models is as follows [2]: A large number of hard spheres of uniformsize are quickly poured and packed into a container having irregular surfaces, resulting indisordered configurations. The coordinates of the sphere centers are then determined bydirect measurement. The overall long range interparticle attraction is usually provided bycompression and gravity, while the short range repulsion is provided by the impenetrabilityof the balls used. Just like the CRN models for covalent amorphous solids, the resultingRCP structures are metastable, i.e., local minima on the free energy surface. Such modelsinclude those of Bernal [25], Scott and Kilgour [26], and Finney [27]. Fig. 4 [28] shows anexample of a portion of an RCP structure.

Since atomic orientations in metallic glasses are much less directional than in covalentlybonded solids, it is easier to write satisfactory computer algorithms to generate RCPstructures than CRN structures. The earliest such computer program for the disorderedpacking of hard spheres was written by Bennett [29].

e ¶i'


Fig. 4. Computer-generated portrait of a100-atom portion of a random closepacked (RCP) structure [28].

-- . - - - - I

Fig. 5. (a) A boron atom (small filled circle) surrounded by six Ni atoms at the vertices ofa trigonal prism and three further "capping" atoms forming halfoctahedra (only one of whichis shown). (b) The structure of Ni3B showing a plane through the boron—centered trigonalprisms. Note the edge and vertex linkages. (c) Dissection of the structure into "arrow—like" units consisting of a trigonal prism "tipped" with a tetrahedron and with a half-octahe—dron as the "tail". (d) Dissection emphasizing the arrangement of tetrahedra and octahedra [30].

The RCP model for metallic glasses is clearly a direct analog of the CRN model for covalentamorphous solids. There is, however, evidence that the local environment of certain atomsin glassy metallic alloys is well-defined [30]. For example, the structures of transitionmetal/metalloid glasses such as Ni3B seem to be be best represented by a 9-atom polyhedroncentered on B, and likely to be a tricapped trigonal prism; therefore, the best structuralmodels of such glasses may be generated not by random close packing of individial atoms, butby random close packing of larger structural units such as tricapped trigonal prisms,tetrahedra and octahedra. Fig. 5 [30] provides an illustration of these ideas, which aresimilar, in the context of metallic glasses, to Bray's observations [20] of deviations fromthe simple CRN model for the borate glasses. The lesson to be learned, again, is that thehomogeneous models of the atomic-scale structure of non—crystalline solids should not betaken too literally, but should be viewed more as useful idealizations.

Random coil model for polymeric glasses

This model, sometimes called the spaghetti model, is due to Flory [31]. It can be viewed asan analog of the CRN and the RCP models, subject to the same types of limitations, andespecially designed for organic polymeric glasses such as polystyrene. It is based on therandom-walk configurations adopted by flexible, interpenetrating and intermeshed polymercoils. An example of a random coil model is sketched in Fig. 6 [2,4].

Conversion from RCP to CRN

Raman studies [4] have been carried out on a-As2S3 thin films, prepared by solventdeposition from organic solution, in which the dissolved seed material originated fromcrystalline As2S3, amorphous As2S3, and crystalline As4S4. Films derived from the two As253

sources exhibit Raman spectra very similar to that of the bulk glass, suggesting apredominantly CRN type structure. On the other hand, the Raman spectra of filmssolvent-deposited from the As4S4 source, as well as of vapor-deposited a-As2S3, can be best

visualized as containing predominantly RCP character, with the presence of rather sphericalmolecular units of As4S4 which are in a random close packing configuration. Upon annealing,

these spectra gradually evolve in time to spectra much more similar to those in the bulk

Fig. 6. Schematic sketch of the random coilmodel for polymeric glasses.One polymer chainhas been "labeled" for ease of visualization [2].

d


Fig. 7. 1 possible bond reconstructionmechanism for the digestion of a small moleculeby the extended network of a chalcogenideglass. The bonds represented by the dottedlines replace the ones represented by thedashed lines [2].

Fig. 8. An illustration of ring sizeand bond angle distributions [2].

glass, indicating that the structure is probably undergoing a phase transition from RCP tothe more thermodynamically stable CRN by the "digestion" of the As4S4 molecular units in a

manner that is schematically represented in Fig. 7 [4]. This is a beautiful example of theinterrelatedness of the various types of structural models, as well as of some of thesubtleties involved (such as the significant effect of the method of preparation andprocessing on the type of structure that results).

Comparisons with experimentWe have already mentioned the Mössbauer [23], IR [24], and Raman [4,24] experimentsperformed to probe the local order, and the NMR experiments carried out on borate glasses[20]. The most familiar types of experiments used for analyzing the structures ofnon-crystalline solids, however, are neutron, electron, and x-ray diffraction [32], andcontroversies are quite common between proponents of Mössbauer and Raman experiments andthose of diffraction experiments concerning which type of experiment is more informative.We feel that such controversies are unnecessary, since each type of experiment provides someuseful information not available from any of the others. It seems more fruitful to attempta synthesis of these results, with full awareness of the strengths and limitations of eachmethod. Furthermore, most of these types of experiments are not too different from thetechniques used by structural chemists in examining molecules, crystals, and biologicalmacromolecules. We therefore strongly encourage experimental structural chemists lookingfor new challenges to consider getting involved in such experiments on amorphous solids.

Since amorphous solids lack periodicity, overall structural symmetry and long range order,there is no simple structural formalism (such as a unit cell repeating by translationalsymmetry) that can be used in the calculation of microscopic properties. The results of adiffraction experiment can, in some rare cases, be non—unique even for a crystalline solid,with the calculated set of vector distances corresponding to more than one possible set ofCartesian coordinates. For non-crystalline solids, in which these experiments can produceonly a one-dimensional (scalar) correlation function (whereas the underlying structures are,of course, three—dimensional), the ambiguity is far more severe. Agreement of thepredictions of a structural model with the diffraction data is therefore a necessary but notsufficient condition for the validity of a model. In other words, although any model whichis inconsistent with carefully obtained diffraction data is certainly wrong and must berejected, even a model which gives a perfect fit has no guarantee that it gives the correctstructure, since other models may well exist that fit the data just as well [32].

Diffraction data from covalent non-crystalline solids are often divided into two classes:(1) short range order, involving the order within a single structural unit and itsconnection to its immediate neighbors [32], i.e., characterized by the coordination numbersof the atoms involved, and the bond lengths and bond angles between them [33]; and (2)intermediate (or medium) range order, describing the overall network topology, oftencharacterized in terms of the "ring statistics", i.e., the distribution describing thefrequency of occurrence of n-atom rings as a function of n. An illustration of ringstatistics and bond angle distributions is given in Fig. 8 [2].

Wright remarks that when CRN models are constructed, whether by hand or by computer, and theresults are then compared with carefully obtained diffraction data, as shown in Fig. 9 [32,34] for a—Ge, none of the models fits all of the major experimental features to withinexperimental error, showing that there is probably still room for significant improvement instructural modeling. On the other hand, Zallen [2] also compares diffraction data with

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III. ORDER AND DISORDER

'Amorphous' is not 'formless'

Non—crystalline solids have traditionally been referred to as "amorphous" solids. This isunfortunate, because the word "amorphous", in its direct translation from its Greek roots(a-, without and morphe, form), means "without definite form" or "shapeless" [39. Ittherefore, incorrectly implies that these materials lack any identifiable and well—defined

types of order, making it hopelessly difficult to study them systematically.

In reality, nothing could be further from the truth. A brief review of the structuralmodels presented in the previous section will immediately show that "amorphous" solids arefar from "formless", but in fact, have well—defined types of order, albeit different fromthe monotonous repetitive patterns of crystalline solids. For example, the disappearance ofperiodic order in continuous random networks is associated with a spread in bond angles,while coordination numbers and bond lengths remain at their ideal values. The NMR studieson borate glasses [20] suggest the presence of even more order, since it is whole structuralgroupings, rather than individual atoms or small structural units, which are connectedrandomly. The molecular cluster network models [22] assume the presence of largemorphologically and stoichiometrically distinct molecular clusters in glasses. The randomclose packing models [2] for amorphous metals also have order imposed by the constraints ofthe densest possible packing consistent with the impenetrability of the hard sphere atoms.For many transition metal/metalloid glasses (such as Ni3B), there seems to be even more

ordering, with polyhedral building blocks suct as tetrahedral, octahedral, or tricappedtrigonal prismatic units of atoms forming the solid by random close packing [30]. Finally,in the random coils of a polymeric organic glass [31], the long and flexible coils areintermeshed in random-walk configurations as illustrated in Fig. 6, but each individual coil

is formed by the repetition (up to --lO times) of a monomeric molecular unit, so that thelocal environment of each atom, formed by those atoms (all within the same coil) to which itis bonded, is quite well-defined. An individual styrene molecule, and the bonding patternin a coil of the polystyrene polymer [40] are illustrated in Fig. 10.

CH2=CFI —CH2---CH—CH2—CH—CH2—CH---

1,Jpero,udes ij

Styrene Polystyrene

Fig. 10. (a) A molecule of styrene. (b) One coil of polystyrene,

formed by many (-lO) monomeric units of styrene, three of which are shown.Note the well-defined local environment of each atom in the coil.

Importance of short range order

As we have seen, the types of atomic-scale order observed in solids can be classified asshort range (or local), intermediate (or medium) range, and long range, depending upon howfar a predictable order extends when the arrangement of atoms is considered starting from agiven atom or region in the solid.

Crystalline solids are characterized by long range periodic order. The atoms are located ina lattice generated by the repetition of a unit cell with a finite (usually small) number ofatoms. The constraint of three-dimensional periodicity imposes the severe limitation of the

existence of only seven crystal systems, 32 crystallographic point groups, and 230 spacegroups resulting from the application of the translational symmetry operations of thecrystal [41].

This rigid long range periodic crystalline order, analogous to a regiment of soldiers information marching in a parade, is absent from non-crystalline solids by definition. Acertain amount of intermediate range order may exist, however, as in the large andwell-defined molecular clusters of the molecular cluster network models [221 for covalentglasses, the large structural groupings in borate glasses [20), the tricapped trigonalprismatic building blocks in transition metal/metalloid glasses [30] and the sequence ofdihedral angles in a-Se [38]. Another interesting example of intermediate range order canbe found in the hydrogenated and fluorinated amorphous silicon (a—Si:H:F) alloys [42] usedfor electronic applications (especially as solar cell materials). These alloys, when dopedwith moderate to large amounts of phosphorus to make them n—type semiconductors, manifest anintermediate range order similar to a microcrystalline state with a characteristic length(average particle size) of 20 to 60A, as observed from Raman, electrolyte-electroreflectance(EER) and transmission electron microscopy (TEM) experiments [43].

A very important factor in the determination of the structures and properties ofnon-crystalline solids is the short range order around the atoms in the material [21]. Thislocal order can be considered in the context of a "total interactive environment" dependingon several factors including the nearest—neighbor bonding, the effects of chemical forces,

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promise of providing powerful new theoretical techniques that can be utilized to understanda wide variety of structures within a unified formalism, and to discover hidden order innon—crystalline structures.

An example of a perfect icosahedron occurring in chemistry can be found in the structures of

the boron hydrides [51,52]. Many fascinating polyhedral arrangements are possible with the

general formula BnH where the boron atoms occupy the vertices of a polyhedron, the

hydrogen atoms are directed outwards, and c=—2 in the vast majority of cases. By far, themost stable of these polyhedra occurs for the icosahedral structure with n=12 [52], as shownin Fig. 11.

25.277

25.272

— 2527E, au

25.22

25.257

25.252

25. 247 _________________________________4 + —1 I I #

Fig. 11. Calculated energies per BH unit (E=IEI/n) Fig. 12. Three views of a regularas a function of n for the preferred polyhedral icosahedron [53]: (a) LookingB H2 structures [52]. down a face. There are tenn n three-fold axes of symmetry

connecting the midpoints ofopposite triangular faces. (b)Looking down an edge. There are15 two-fold axes connecting themidpoints of opposite edges. (c)Looking down a vertex. There aresix five-fold axes connectingvertices at opposite ends.

Although the perfect icosahedron is a highly symmetric and very dense arrangement of atoms,full icosahedral order is not possible in a crystal. The reason for this becomes clear whenwe look at the three views of a regular icosahedron, as presented in Fig. 12 [53]. As theview looking down from a vertex shows, an icosahedron possesses six five-fold symmetry axes,each one connecting two vertices at opposite ends. It can easily be shown [41] that onlyone-, two—, three-, four-, or six—fold axes of rotation are consistent with fullthree-dimensional translational periodicity and therefore possible in a crystalline solid.Full icosahedral order is therefore impossible. Any crystalline structure, where the localbonding requirements can be best satisfied by icosahedral arrangements of atoms, has a pointgroup and a space group which are lowered from perfect icosahedral symmetry by distortionsof the icosahedra and/or connections of lower symmetry between the icosahedra. This can beseen from the structure of elemental crystalline boron, where the most stable tetragonal and

rhombohedral allotropic forms contain units of nearly (but not exactly) regular icosahedra,with various types of connections between these distorted icosahedra [53]. Electron

diffraction experiments on vacuum-deposited thin films of amorphous boron produced byevaporation show that icosahedra are also the predominant structural units in a-B [54]. Themajor difference from the crystalline phases is that these icosahedral units form athree-dimensional continuous random network through quite random arrangements of theicosahedra, with a random (disordered) distribution of the possible types of connectionsbetween the icosahedral building blocks [54]. (This is also another example of an amorphousmaterial with intermediate range order, on the scale of the well-defined icosahedral units!)

The Frank—Kasper phases [55] of transition metal alloys are another example of brokenicosahedral symmetry. In these intermetallic crystals, the smaller transition metal elementat the center becomes surrounded by twelve larger atoms arranged like the corners of anicosahedron. Distortions and different orientations of several icosahedra in a unit cellthen reduce the symmetry of the crystal to a point group and a space group consistent withthe existence of three—dimensional translational periodicity.

(b)

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chemists, since they illustrate how the standard computational techniques of molecularorbital theory can be utilized to gain insights into novel structures.

The ideal composition and structure of the icosahedral quasi-crystal phase have beeninferred from the known phase diagrams of aluminum-transition metal alloys, and tilings inthree dimensions by rhombohedral tiles to produce the phases of interest have been discussed

[70].

The mean field theory and density functional theory treatments of quasi-crystals have beenconsidered in the context of the ordering in condensed phases produced by orientationalfreezing in three dimensions [71]. The mean field theory treatment enables connections tobe made with theories of biaxial ordering in liquid crystals. The density functional theorytreatment has the advantage of not having any adjustable or unknown parameters. Thestability of the orientationally ordered phase, its density, and the entropy change upon itsformation, are all seen [711 to be determined, to first order in thermodynamic perturbationtheory, by the structure factor of the corresponding liquid phase. These approachesrepresent the state-of-the-art and are also very valuable in learning more about icosahedralorder; however, they do not constitute easy and obvious entry points into the field fortheoretical chemists more accustomed to the traditional techniques of computationalchemistry.

It has recently been shown [73] that the electron diffraction data [66] can also beexplained in terms of a metastable cubic crystal with a large edge (-26.7A) and a unit cellcontaining approximately 1,120 atoms. This alternative explanation, which is firmlygrounded in traditional crystallography, does not require the postulation of the existenceof "quasi-crystals" as a new kind of matter. More accurate x-ray diffraction experimentsmight help in deciding between these two explanations, one of which has a beautifulconceptual framework, and the other one has the accumulated wisdom of traditionalcrystallography, in its favor.

The curved space approachA unified formal approach to understanding the structures of many classes of materials withdifferent types and amounts of order can be provided by viewing them as arising fromprojections of regular polyhedra (polytopes) in a curved space onto the usual flatthree-dimensional Euclidean space [68,74].

An example of a very simple projection from a two-dimensional space (a square lattice) to aone-dimensional sub-space (a line of incommensurate slope which passes through the originbut through no other lattice point), subject to a well—defined projection rule, isillustrated in Fig. 14 [4]. All lattice points within a fixed distance of the line areprojected on it, defining the atom-center positions on it. Although the number of possibleways one can project between spaces is much larger in higher dimensional spaces (whetherflat or curved), and the projection rules are far more complicated, Fig. 14 illustrates thegeneral idea.

Fig. 14. Projection from a two-dimensionalsquare lattice onto a one—dimensionalsubspace (a line). The solid dots areprojected images of those square latticesites which lie between the dashed lines.Since the line has an incommensurate slopeand passes through the origin but throughno other lattice point of the squarelattice, the site distribution obtained onthe line is only quasi-periodic, andcorresponds to a one-dimensionalquasi-crystal [4].

The curved space approach [74] examines hidden order in non—crystalline structures in thelight of topological and geometrical limitations, arising from the requirements of spacefilling, imposed on the optimal short range order which would completely determine thestructure if the local interactions between atoms were the only important factors.

The failure of certain simple structural patterns, which would normally be preferred andprovide a dense and highly favored packing, to regularly fill (or "tile") the space, isreferred to as "frustration" [68,74]. A very important example of frustration (see Fig. 15)is given by the fact that regular tetrahedra do not tile three-dimensional Euclidean space.

The cause of this frustration is that the dihedral angle of a tetrahedron (70.5°) is not a

sub—multiple of 2ir (as 72° is), causing a misfit when one tries to propagate the tetrahedral

local configuration by completely surrounding a given bond with five tetrahedra, or form anicosahedron by packing tetrahedra.


Fig. 15. Regular tetrahedra do not tile three—dimensional Euclideanspace, resulting in "frustration" [74]: (a) A regular tetrahedron. (b)The failure of five regular tetrahedra to completely surround a givenbond. (c) The resulting failure of the packing of tetrahedra to form aperfect icosahedron. One gets pseudo—icosahedral (dense packing orpolytetrahedral) models out of such a packing.

On the other hand, what is known as "polytope 120" [4,74] or "polytope {3,3,5}", is aregular polyhedron ("Platonic solid") in four dimensions. Its two-dimensional projection isshown in Fig. 16. It has 120 vertices, all lying on the surface of a four-dimensionalsphere, which defines a curved non-Euclidean three—dimensional space, just as the surface of

Fig. 16. Two—dimensional projection of polytope{3,3,5} [68].

a regular sphere defines a curved non—Euclidean two—dimensional space. Within this curvedthree—dimensional space, the 120-vertex polytope forms a configuration in which fivetetrahedra fit perfectly around each bond, so that 600 tetrahedra completely tile thiscurved space. Local dense packing is thus maintained everywhere. Topological frustrationis removed, at the expense of the introduction of curvature [4].

The mapping which is then carried out on the structure in curved space to generate aprojection to Euclidean space, is called "flattening", and can be likened to peeling anorange by spreading out its crust on a plate. The localized distortions (i.e., defects)caused by the mapping from a curved to a flat space, are called "disclinations" [74]. Acomplete set of disclination lines, which forms a network, can cancel the curvature andcompletely define the structure in Euclidean space. Many seemingly unrelated types ofstructures can be generated by applying different types of disclination networks to the{3,3,5} polytope or to other polytopes. This is performed by cutting the structure of thepolytope and adding or removing a wedge of material between the two lips of the cut [74].

The following types of structures are among those that have been treated thus far withinthis unified framework:

1. The Frank-Kasper phases [55], including the Laves phase alloys such as MgZn2,

MgCu2 and MgNi2, are among the types of crystalline solids which can be generated

[68,74].


2. The cubic A15 (or 3-tungsten) structure, which is a type of Frank-Kasper phase

represented by alloys such as W30, W3Si, A1V3, AuTi3, Cr03, etc. [75] can also be

generated [68,74]. These Al5 structures are especially important because thesuperconductors with the highest critical temperatures (such as Nb3Ge and Nb3Sn)

are metastable A15 phases [76].

3. Amorphous metallic alloys can be generated by using non—periodic networks ofdisclination lines [74].

4. Covalently bonded amorphous semiconductor alloys (such as a-Si:H) can also bemodeled [77], although the existence of inhomogeneities in such systems presents a

problem.

5. Icosahedral quasi-crystals can be described in terms of the iterative flatteningof polytopes via hierarchical disclination networks [78].

6. Finally, the defects that introduce octahedral local configurations into RCPmodels of amorphous metals, which can be predominantly regarded as random packingsof tetrahedra, are related to disclinations, and are referred to as "screw" axes or

"dispiration" [79].

The curved space approach is a promising technique for the analysis of amorphous structuresof simple stoichiometric compounds, although it would not be expected to give much insightinto complex multicomponent alloys. It also remains to be decided whether the approach hasreal physical content, or whether it should be viewed more as a mathematical techniqueunifying the treatment of these diverse structures in a mainly formalistic sense.

Iv. MODEL SYSTEMS OF LOW DIM ENSIONALITY

Real materials, of course, have a fully three—dimensional structure, althoughquasi-two-dimensional arrays, e.g., graphite, and quasi—one-dimensional arrays, e.g.,TTF-TCNQ, do exist. In any event, simple model systems with reduced dimensionality can bestudied with more accuracy than three-dimensional networks, and thus can provide usefulinformation about real materials provided that the right questions are asked [59]. For

example, under rather general conditions, concerning the range of the interparticle forces,it can be shown that spontaneous crystalline order should not exist in one or twodimensions. This very important theorem serves as a reminder that the real materials, suchas graphite and TTF—TCNQ, are indeed three — dimensional, even though it might be both aconvenient and a reasonable approximate model, at least for solving certain types ofphysical problems, to view them as consisting of two—dimensional or one-dimensional arraysrespectively, embedded in the three-dimensional space. The application of similarconsiderations to spin correlation functions shows that, in the absence of a finite magnetic

field or magnetic anisotropy, spontaneous ferromagnetic or anti—ferromagnetic order shouldbe thermally unstable, and therefore should not appear, in one or two dimensions [59].Several model systems that have shed some light on specific unsolved problems have beeninvestigated recently, and are discussed in the remainder of this section.

One of the major unanswered questions is whether the non—crystalline atomic arrangement in aglass is indeed a metastable equilibrium configuration that minimizes the interactionpotential. This has been shown to be the case for models with a small number of particles[28], but has not been demonstrated for large systems. Schilling and Reichert [80] recentlyconsidered an anharmonic one-dimensional chain with next—nearest—neighbor interactions thatintroduce structural frustration. By introducing spatial chaos into the interactions via arandom variable in the potential energy, they were able to demonstrate that: (1) nolong-range order exists; (2) the structure factor contains extra peaks at special values ofthe wave number, k, characteristic of short—range order; and (3) the density of states istime—varying, leading to a power-law behavior of the specific heat that has been observed in

some real glasses [81].

A second unsolved problem is that of the dc conductivity of a material with static disorder

[82]. For one—dimensional solids, the exponential localization resulting from elasticscattering leads to conductivity fluctuations even in nominally identical samples. Mello

and Kumar [83] applied the techniques of information theory to this problem, maximizing theentropy subject to flux conservation, time-reversal invariance, and a particular scalinglaw, and were able to calculate the movements of the resistivity in the limit in which thelength of the wire is long compared to the localization length. More work is necessary toextend their solution to smaller wire lengths.

118 J BICERANO AND D. ADLER

Another problem concerns the calculations of the configurational entropy of a CRN, adifficult analysis to carry out in general. Some recent progress has been made by Kleninand Promislow [84], who considered a two—dimensional case in which the four-particlecorrelation function was constrained to take on only discrete values. The structure

analyzed was a two-fold and four-fold coordinated array of rectangular blocks with a 600bond angle at the four-fold—coordinated sites. They found sharp maxima in the entropy per

block at two-fold—coordinated sites with bond angles of 120° and 155°. For the former case,the blocks form a hexagonal grid, while for the latter, a specific ring structure appears.In both cases, the entropy maxima signify a relative lack of interference within theconditions of dense packing. Such models clearly give some insight into the general natureof random networks. It is interesting to speculate that the possibility of doping amorphoussemiconductors arises from the tendency of CRN's to choose the bonding configurations thatavoid interference between atoms.

V. CHEMICAL BONDING CONSIDERATIONS

The importance of chemical bonding considerations

We have already made use of chemical bonding considerations in our discussions of thestructural models used in describing non—crystalline solids (Section II) and of the natureand extent of order in disordered structures (Section III). The short range order in anamorphous solid is primarily determined by the local chemical bonding requirements of itsconstituent atoms. The extension of these local configurations into three—dimensionalsolids generates structures which can be described by\a variety of models such as CRN,molecular cluster network and RCP models. The intermediate range order observed in somematerials, as manifested in the large structural groupings in a-B203 [20], the tricapped

trigonal prismatic building blocks in transition metal/metalloid glasses such as Ni3B [30],

the sequence of dihedral angles in a-Se [38], the microcrystalline—like order in a-Si:H:Falloys doped with moderate to large amounts of phosphorus [43], and the icosahedralstructural units in a—B [54], is also primarily due to the special stability of suchstructural configurations where the chemical bonding requirements of the constituent atomsare well-satisfied. On the other hand, it was seen to be impossible to completely fillspace by packing some simple structural units, such as tetrahedra, which are often thefavored types of short range order for a given set of constituent atoms. This "topologicalfrustration", which often prevents the optimum local environment from being fully propagatedin three dimensions, was seen to possibly form the basis for a whole new and unifiedframework for understanding many types of materials.

The purpose of the present section is to focus in more detail on the chemical bondingconsiderations of importance in understanding non—crystalline solids. There is clearly alot that chemists can contribute to the understanding of non—crystalline solids, along thelines of the work described in this section.

It. is possible to synthesize amorphous materials that have no crystalline analogues, but

posses interesting and even unique physical properties [21]. This extra degree ofcompositional freedom, which introduces a whole new dimension to the science and technologyof non—crystalline solids, is due to the types of chemical bond distributions possible in astructure in which long range order is absent. These considerations have been presented[85] in the form of seven "basic rules of amorphicity", which summarize the basic"anti—crystalline" chemical bonding configurations and their structural and physicalimplications. The most important conclusion of this work [85] is that deviations from localorder, as reflected by the deviant electronic configurations and the total interactiveenvironments, control the fundamental physical properties (such as transport properties) ofnon—crystalline materials just as perturbations of periodicity control those of crystallinematerials.

Glassy alloys

Threshold switching alloys such as Si18Ge7As35Te40, and memory switching alloys such as

Ge15Te81Sb2S2 and Ge24Te72Sb2S2, have a special place among amorphous solids withnocrystalline analogues. The discovery of reversible switching phenomena in such alloys[86], in which the motivating potential can either be electrical (application of a largevoltage, resulting in an increase of electrical conductivity by several orders ofmagnitude), or optical (application of laser or xenon flash lamp pulses, resulting in asharp change in reflectivity), induced a rapid increase in both basic and applied researchon amorphous solids.

The most important short range order parameters in the description of structure and bondingin glassy alloys, such as in the three compositions mentioned above, are the coordinationnumbers C of the elements entering the alloy and the bond energies D between the elements.


w

w

C,)z

Fig. 17. The glass transition temperature Tg as a function of the optical

gap energy E04 for alloy compositions with fixed values of the average

coordination number [87].

An elegant experimental demonstration of the importance of these parameters has beenprovided by deNeufville and Rockstad [87] in their empirical "Tg_Eg_C Correlation":

TgZT + 13(C—2)E04

illustrated in Fig. 17. Here, T is the glass transition temperature of the alloy;

TgZ34O±2O ; C is the average coordination number; E04 is the energy at which the optical

absorption coefficient is lO4cm and is a measure of the optical gap Eg and of the

energies of the_weakest bonds in the alloy; and f3 is a linear proportionality constant foreach value of C. Clearly, Tg increases (i.e., the glass becomes more stable) both as C

(cross—linking) increases, and as E04 (an index of the strengths of the weakest bonds)

increases. The "Tg_Eg_C Correlation" is discussed further in Section VIII, as an example of

the importance of network rigidity in alloy glasses.

A semi-quantitative approach which uses these coordination number and bond energy parametersand some simple rules for obtaining optimal chemical bond distributions in a glassy alloy,has been discussed by Bicerano and Ovshinsky [88]. The only assumptions used in this simplemodel involve: (1) the preferred coordination numbers of the elements; (2) the maximumamount of chemical ordering possible, to determine gross overall structural features; and(3) the formation of bonds in the sequence of decreasing bond energy. This simple approachprovides useful information concerning T, cohesive energies, the numbers and compositions

of phases expected to be present, bond distributions, and electrical properties such as the

conductivity and switching phenomena.

Phillips [89) has presented calculations of the experimental heats of formation ofcrystalline As2X3(X=S,Se,Te) and GeSe2 based on chemical bonding considerations related to

the competition between charge transfer and lone pair resonance screening. Similarconsiderations can also be useful for the amorphous analogues of these and other similartypes of alloys, especially since "metastable" amorphous local minima on a potential surface

are usually only slightly higher in Gibbs free energy than the corresponding crystallineminima for the same material.

Materialswith predominantly tetrahedral coordination

Elemental tetrahedrally bonded non-crystalline solids such as a-Si and a-Ge have extremelyhigh concentrations of point defects (e.g., dangling bonds), making them unsuitable forsemiconductor device applications, including thin film transistors and solar cells. To

OPTICAL GAP, E (eV)

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VI. THERMODYNAMIC AND KINETIC CONSIDERATIONS

It is a basic tenet of modern science that any collection of atoms arrange themselves so asto attain the state of lowest Gibbs Free Energy, G, at the ambient temperature and pressure;however, this turns out to be somewhat naive when applied to the solid state. In fact, itis not clear that any non-crystalline solid attains the state of absolutely lowest C at any

temperature and pressure. By definition, a solid is a material whose shear viscosity

exceeds the order of lO N—s/m2. Such a viscosity implies that a shear stress of lN appliedto a 1cm cube of material would take about a year to produce a displacement of 3Opm, abarely measurable deformation. Although this definition is completely arbitrary, few woulddeny that a system with an atomic relaxation time of the order of a year is indeed solid.

In contrast, typical liquids have viscosities less than 1N—s/m2, so there is ordinarily nodifficulty in differentiating between the two phases; however, it is precisely the lack ofatomic mobility in solids that enables the formation of metastable phases, for which 6 is alocal but not a global minimum. Upon rapid cooling, thermal energy is extracted from thesystem and metastable structures are "frozen in". A typical situation is sketched in Fig.18 [2]. As the temperature of a liquid is reduced, one of two things can happen, dependingon the cooling rate: it can solidify discontinuously via a first—order phase transition to acrystal (curve a in the sketch), or continuously via a higher-order phase transition to aglass (curve b). In the latter case, appropriate to rapid cooling, there is a region inwhich the liquid is supercooled, but at the glass—transition temperature, I , the viscosity

increases by many orders of magnitude to form a solid, identiiable by thecharacteristically low thermal—expansion coefficient (as well as by the large relaxationtimes, often exceeding the age of the universe). Even crystals can be metastable: a diamondmay be forever, but if the universe lasts sufficiently long, it will eventually convert tographite. In popular usage, a non—crystalline solid capable of being quenched from theliquid phase is called a All glasses have a Tg although its value may vary somewhat

with the cooling rate. Some non-crystalline solids (e.g., Si) can be formed only by directcondensation from the gas phase; they are usually referred to as amorphous solids. Suchsystems are not ordinarily describable in terms of thermodynamic equilibria; instead, growthmechanisms and relaxation effects must be handled on a kinetic basis.

The details of the glass transition have remained elusive right up through the present, andremain a fertile subject of research. Gibbs and DiMarzio [99] provided a start byconsidering a one-dimensional polymer and identifying Tg with the temperature at which the

configurational entropy vanishes. An alternative approach is the free-volume theory of

tUi

Fig. 18. The two cooling paths yielding solids:

(a) slow cooling, producing a crystal;

(b) rapid cooling,resulting in a glass [2].

TEMPERATURE


Cohen and Turnbull [100], which focuses on the volume available for the atoms (treated ashard spheres) to move within their environment in the liquid phase. The glass transition isthen identified as the point at which the unoccupied space (the free volume) drops below acritical fraction of the total volume. A further discussion of the glass transition appearsin Section XII.

Clearly, thermodynamic and kinetic considerations are central to an understanding of theformation and structure of non-crystalline solids. One of the prime movers in this fieldhas been Turnbull [101], who has drawn attention to the necessity for suppressingcrystallization. This requires a kinetic resistance either to nucleation or to growth.Recent work [102] has focused on the resistance to nucleation of some structures in thesupercooled-liquid regime between the melting point, Tm and T. Turnbull points out thatmany glasses exhibit a high resistance to nucleation, indicative of a necessaryreconstruction of the short—range order in this process (and also strong evidence againstmicrocrystallite models for the structure of glasses). In many covalently bonded glasses,the activation energy for growth is approximately equal to the primary bond energy. Oneglass, B2O3, may be anomalous because of the unusual chemistry of boron. A modern view of

resistance to nucleation in many systems would involve reconstructions from icosahedralshort-range order to crystalline close-packed structures. Actually, this was firstsuggested over 30 years ago [56]! See Section III for a more detailed discussion ofprogress in this area.

The free-volume model has been refined recently. Cohen and Grest [103) applied percolationtheory to calculate the comunal entropy, the entropy difference between a freely flowingand a confined system of atoms. With this, they were able to analyze the thermodynamics ofthe glass transition. One weakness of their result is the prediction of a first-ordertransition, in disagreement with experiment. Chow [104] was able to derive equations ofstate, non—equilibrium equations, and relaxation functions by analyzing the situation inwhich a distribution of free volumes with different creation energies collapses. Incontrast, Ibar [105] has pioneered a kinetic approach based on the assumption that anon—equilibrium state can be achieved under non—isothermal conditions if the free energyremains at its equilibrium value at the corresponding temperature. He finds it necessary toadd a term to the free energy that has no obvious physical interpretation, but he can thencorrelate a great deal of data on polymeric systems.

VII. DYNAMIC SIMULATIONS

Dynamic phenomenaIn the previous Section, we discussed the continuous nature of the glass transition, andpointed out that a glass can be regarded as a liquid of extremely high viscosity, in which

crystallization has been bypassed via rapid cooling, preventing the thermodynamically moststable crystalline configurations from being attained. The resulting glassy structures form

thermodynamically metastable states, i.e., local free-energy minima stabilized by the largepotential barriers separating them from the lower-lying crystalline absolute minimum andpossible crystalline local minima for the same composition. A very informative and readable

description of the physical phenomena involved, has been provided by Stillinger and Weber[lO6].These authors show how the classification of potential minima (i.e., of mechanicallystable molecular packings), offers a unifying principle for understanding properties in thesolid state. They show that this approach permits identification of an inherent structurein liquids that is normally obscured by thermal motions. Melting and freezing are shown tooccur through characteristic sequences of molecular packings, with a defect—softeningphenomenon underlying the fact that they are thermodynamically first order phasetransitions. Glass transitions and associated relaxation behavior are then explained by thetopological distribution of feasible transitions between contiguous potential minima.

Types of dynamic simulations

Computer simulations of such dynamic phenomena, are extremely valuable for understandingwhat sequences of events take place, as well as for checking the validity of microscopicmodels, of theoretical assumptions and of empirical approximations. The two major types ofdynamic simulations are the "Molecular Dynamics" and the "Monte Carlo" methods. In thepresent subsection, we will briefly sumarize these two methods, combining the descriptionsgiven by Zallen [2] and by Ziman [59]. The reader interested in more details should referto these two excellent books. Ziman [59] also provides the historical background, and givesreferences to the pioneering contributions of workers such as Alder, Wainwright, Rahman,Verlet, Stillinger and Metropolis.

Consider, first, a collection of "hard—sphere molecules", described by perfectly rigid,non-interpenetrating balls of uniform size. The physical system which most closelyapproaches this description is an ensemble of closed—shell inert gas atoms such as He, Ne,Ar, Kr and Xe. In this idealization, the intermolecular (or pair) potential energy function

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Some recent applicationsA detailed discussion of the vast amount of literature on the applications of dynamicsimulations to physical systems is outside the scope of this review. We will insteadbriefly review some of the most interesting recent results, in order to give the flavor ofthis field, which is likely to gain even more importance with the advent of more powerfulcomputers, as discussed in Section I.

Soules [107] has provided an extensive review of recent Molecular Dynamics and Monte Carlocalculations of glass structure and of diffusion in glasses. He has pointed out that, ingeneral, the structures obtained closely resemble continuous random networks [5]. Forexample: (1) Silica glass has a cage—like structure resembling a randomly distortedhigh—cristobalite. (2) In simulated sodium silicate glasses, Na ions create non—bridgingoxygen atoms and occupy interstitial sites with no well—defined coordination but a mostprobable coordination number of five. (3) Vitreous B203 contains planar regular

triangles linked at apices to form a random network of ribbons and sheets. The MolecularDynamics structure of a-B203 does not contain boroxyl rings. (4) In sodium borosilicate

glasses, the trigonal-to-tetrahedral conversion of B with the addition of Na, agrees withthe NMR results of Vun and Bray [108]. (5) In sodium aluminosilicate glass simulations,aluminum atoms always prefer four-fold coordination, regardless of the concentration of

Na20.

Soules [1071 noted that all of the simulated glasses undergo a kinetic glass transition asthey are cooled. Glass formation occurs when the system reaches a temperature at which thenetwork—forming cations and anions cease to diffuse on the time scale of the MolecularDynamics runs. "Computer experiments" can be meaningfully performed on the thermodynamicand kinetic properties of these dynamically simulated network glasses and liquids,especially at very high temperatures where the atoms are freely diffusing on the time scaleof the "experiment", and at low temperatures where the system is frozen in the atomicconfiguration of a glass. Reasonable agreement is obtained with experimentally determinedvalues of the properties.

Another interesting recent application of Molecular Dynamics simulations has been to theinvestigation of the normal modes of an amorphous solid, to study the onset of localizationof modes in a Lennard—Jones-type frozen fluid, which is a true interacting many—body system[109]. This work has then been extended to the calculation of longitudinal and transverseexcitations in a monoatomic, close-packed glass formed by particles interacting via aLennard-Jones pair potential {llO]. The dispersion relations obtained for the longitudinalmodes look very similar to the results for polycrystalline materials, with the longitudinaldispersion curve showing a remarkably deep minimum at the wave vector where the structurefactor has a peak.

Grabow and Andersen [ill] have used the Molecular Dynamics method to generate modelstructures of amorphous metal alloys such as Fe80P20, Ni80P20, Cu33Zr67 and Cu57Zr43. They

have compared their results with the results obtained by two other methods, namely, theenergy minimization of dense random clusters of atoms and the energy minimization of denserandom periodic structures. They indicate that the energy minimization of dense randomclusters can give structures very different from those of the other two methods even whenexactly the same interatomic potentials are used in the calculations, and that it shouldnot be trusted to generate structures to be compared with experiment. They show thatMolecular Dynamics is the only one of the three methods that provides a valid technique forstudying both the structure of amorphous metals and structural relaxation in thesematerials.

Barnett et al. [112] have studied the glass transition, structure, and dynamics of aCa67Mg33 metallic glass via a new i'lolecular Dynamics approach incorporating the density

dependence of the potentials. They obtain results which are in agreement with neutronscattering data. They find that the accessible configurational energy of the glasspossesses several nearly degenerate potential minima, in agreement with the generalphysical picture presented by Stillinger and Weber [106].

Laakkonen and Nieminen [113] have used Molecular Dynamics to simulate the rapid quenchingof a liquid sample to obtain an amorphous state. The two sample systems they consideredwere a collection of 1372 Ne atoms interacting with a Lennard—Jones pair potential, and anFe85B15 mixture of 1204 atoms interacting with a Morse—type potential. They obtain

reasonable radial distribution functions and distributions of bond angle cosines at atemperature of 4.2K.

These examples should serve to illustrate the wide range of physical systems and importantproblems which can be addressed via dynamic computer simulations, and the increasinglysignificant role such simulations are certain to continue to play in the future. Furtherdiscussion of the application of dynamic simulations will be provided in Sections XI andXII.


VII. ELASTIC AND VIBRATIONAL PROPERTIES

Relation to local structureIn Section II, we mentioned some of the IR and Raman spectroscopic experiments [4,24,38]used to probe the structures of non-crystalline solids. The relationship between thevibrational properties of an amorphous solid and its local atomic structure has beenreviewed by Lucovsky [24] in the contexts of oxide and chalcogenide glasses (such as a—Ge02,

a-As203, a-As2S3 and a-GeSe2), and of amorphous silicon-based alloys. The two major classes

of approaches to the theory of the vibrational properties of amorphous solids are: (1)calculations based on finite clusters either containing a small number of atoms (generallyless than 10), or a large number of atoms (generally more than a few hundred), in eithercase equivalent to calculating the vibrations of a large molecule; and (2) calculationsbased on an infinite aperiodic structure with the same coordination number at each site andwith no closed loops, the Bethe Lattice, which is an artificial but useful model.

Calculations on the small clusters can be carried out with a variety of techniques,including the familiar techniques of ab initio and semi—empirical quantum chemistry. On theother hand, much simpler computational approaches, which make use of the fact that thevibrational properties are largely determined by local structural features, are needed forthe large clusters containing hundreds of atoms. A common example of the use of such alarge cluster is in the calculation of the vibrational properties of a CRN model largeenough to adequately represent the bulk amorphous solid. The simpler computationalapproaches then utilized usually make use of rotationally invariant interatomic potentialswith contributions from bond stretching and bond bending. Their physical meaning istherefore very similar, in the context of modeling solids, to the physical meaning of theMolecular Mechanics technique which also uses a "classical" central force model and is veryfamiliar to chemists who compute the structures and properties of large (but finite) organicmolecules and biological macromolecules.

The Bethe Lattice can easily be defined axiomatically, and used to facilitate thetheoretical solution of many important physical problems, even though it cannot be realizedas a physical system in a space of finite dimensionality since the prohibition againstclosed loops causes an endlessly increasing density as the structure is extended. An

illustration of Bethe Lattices of coordination number 3 [2] and 4 [59] is provided in Fig.20. In order to overcome the divergence problems encountered in the values calculated forphysical observables with the full infinite Bethe Lattice which "overfills" space with aninfinite density, "Cluster Bethe Lattice" calculations are often performed by arbitrarilytruncating the full Bethe Lattice into a cluster with a finite number of atoms, often 20 to25. Once the Bethe Lattice has been truncated into a finite cluster, the rest of thecalculation is clearly no different from a calculation on a finite cluster whose atomicpositions have been determined or assigned by any other method.

For the oxide and chalcogenide glasses, the Bethe Lattice method can be used to calculatelocal vibrational densities of states, the IR dielectric constant, and the polarized anddepolarized Raman response functions [24, 114]. The glasses of interest fall into twoclasses: (1) those displaying strong, sharp polarized features in their Raman spectra,indicative of intermediate range order, usually due to small rings of bonded atoms (examplesof this class include a-As203, a-Ge52 and a—GeSe2); and (2) those that display only broad

features in their Raman spectra, indicative of the absence of small ring configurations in

the intermediate range order (for example, a—6e02, a—5i02, a—As2S3 and a-As2Se3).

(a) (b)

Fig. 20. Bethe Lattice, or "Cayley Tree", of (a) coordination number 3[2., and (b) coordination number 4 [59]. In this type of lattice, notethat the coordination number around each atom is held constant, but noclosed rings are allowed.

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transition or some other more subtle structural change, often preceded by a phaseseparation, so that the material remains in the nonvolatile ON state even after themotivating potential is turned off. To return to the OFF state, a current pulse with asharp trailing edge must be applied, in order to dissolve the crystallites and rapidlyquench the melt in the environment which acts as a heat sink.

The average connectivity of an alloy turns out to be a crucial factor in determining whetherit exhibits threshold or memory switching [86]. The differences between threshold andmemory alloys have been explained in terms of the relative rigidities of the networks formedby different percentages of divalent, trivalent and tetravalent alloying elements[118-120]. To summarize, memory switches have lower energy barriers retardingamorphous-crystalline phase transitions due to weaker bonds and a more flexible networkcontaining large concentrations of divalent elements, thus allowing the setting andresetting of the ON and OFF states. On the other hand, the more strongly bonded andthree-dimensionally rigid threshold switches can only undergo electronic transitions whichare automatically reversed when the applied voltage inducing the transition is turned off.

The other example of the importance of network rigidity, discussed in Section V, is the"T _Eg_t Correlation" of deNeufville and Rockstad [87]. A chain structure would have =2,

an various chains would be held together in the solid by van der Waals interactions and byintertwining. These chains can glide past each other like snakes when the system is heated,melting the solid by overcoming the relatively weak van der Waals forces without thenecessity of breaking covalent bonds. This explains the (-2) factor in the expression forTg which is then essentially independent of bond energies for =2. On the other hand, for

>2, Tg depends linearly both on (-2), which indicates the number of cross-links that have

to be broken to obtain the freely gliding chains, and on E04, which is an index of the

energy that must be supplied to break these weakest bonds. Thus, the final form

Tg T + 13(-2)E04

can be understood in a simple way.

The topological methodA quantitative criterion for predicting which types of materials are likely to be good glassformers has been proposed by Phillips [22, 121], who used a topological viewpoint in whichconnectivity is measured by the average coordination number . He evaluated the number ofconstraints per atom that need to be introduced in order to have fixed bond lengths andfixed bond angles in a covalently bonded alloy, and showed that these can all be satisfiedsimultaneously in three dimensions only if C2.4. One implication of this result is thatalloys such as a—As2Se3, which have C=2.4, have the ideal network rigidity to optimize both

the bond lengths and the bond angles. They should thus be excellent glass formers, i.e.,yield amorphous structures after quenching from the melt at a relatively modest rate. In

contrast, a material such as a-Se, with C=2, will be "under—constrained", with a significantamount of freedom remaining after fixing all the bond lengths and bond angles. It istherefore possible to achieve some amount of dihedral angle ordering. This is the origin ofthe intermediate range order observed in a—Se [38, 122].

At the other extreme from the "under—constrained" a-Se, are the "over — constrained"materials such as a-Si and a-GaAs, both of which have C=4. Such materials possess a great

deal of local strains. In a-Si, much of this strain is relieved by the ±100 spread of bond

angles from the optimum 109.50 value for the tetrahedral angle. Nevertheless, because ofthe extent of the over-constrained nature of the network, some defect centers such asstretched bonds and under-coordinated atoms are also inevitably introduced into a-Si filmsduring preparation, resulting in a material with large defect concentrations. The majorreason why a-Si:H [90] and a-Si:H:F [42] alloys are much better semiconductors than purea—Si is that the monovalent alloying elements H and F reduce the value of C, relieving aconsiderable amount of strain and greatly reducing the defect concentration.

Percolation

Percolation theory has been rapidly gaining popularity as a technique for the analysis ofdisordered systems. Percolation refers to the propagation of random interconnectionsthrough a network. When the available fraction of sites, or of bonds connecting thesesites, exceeds a critical "percolation threshold", it becomes possible to find a continuouspathway going through the network by only traveling through the available sites or availablebonds. Although the results of percolation theory ordinarily refer to the mathematicallimit of an infinitely extended network, they are applicable to solids since even a tiny

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We have already seen that the computational techniques most commonly used for thecalculation of the vibrational properties of amorphous solids are finite clustercalculations and Cluster Bethe Lattice calculations [24, ll4J. Since these methods areequally useful in the calculation of electronic structure and properties, they will bediscussed further in Section IX.

Another versatile technique is the Recursion Method, which was originally developed forcalculations of the electronic structure of disordered materials [128], and later extendedto calculations of their vibrational properties [129]. In this very general, butalgebraically quite complex approach, the required density of states (whether electronic orvibrational) is obtained without recourse to the direct calculation of the eigenvectors oreigenvalues of the appropriate dynamical matrix. In other words, matrix diagonalization isDQt needed. The technique can therefore be used for very large calculations, with lessexpenditure of computer time than even the familiar Extended Hückel approach which requiresa matrix diagonalization step. In the Recursion Method, the basis set of states {Is>} isgenerated from some starting state Ii> by successive applications of the appropriatedynamical matrix, via an algorithm that ensures the mutual orthogonality of all the statesin the set {Is>J. The method is best applied in a local site representation, where theinitial function Ii> is a site orbital selected by judicious consideration of the shortrange order, and new neighbors are introduced at each successive stage of the calculation.The technique gets its name from the recursion relations which arise from its application.

Another technique which enables the calculation of smooth spectral functions withoutrequiring the diagonalization of a large matrix, is the Equation-of—Motion Method [130],which produces results quite similar to those produced by the Recursion Method, in spite ofthe seemingly quite different algorithm used. In the Equation-of-Motion Method, theequation of motion describing the lattice vibrations is integrated forward in time (t), fromt = 0 to t = T, adopting reasonable values for the initial displacements and assuming thatthe velocities vanish at t = 0.

A third technique which enables the calculation of the density of vibrational states without

diagonalizing a large matrix, is the Negative Eigenvalue Method [131, 132]. This methoduses a theorem which allows the efficient computation of the distribution of eigenvalues ofa real symmetric matrix. The frequency spectra are obtained directly, while further work isrequired if information on a specific eigenvalue or eigenvector is needed.

Theoretical chemists can clearly make significant contributions to the development andapplication of computational methods for the vibrational properties of non-crystallinesolids. The most popular of the computational techniques mentioned in this section, namelythe finite cluster and Cluster Bethe Lattice calculations, are natural extensions to solidsof methods routinely used to treat finite molecules. The Recursion Method, which appears tobe less directly related to the familiar techniques of theoretical chemistry, in realityrequires just as careful a consideration of the short range order in the solid (i.e., theuse of chemical considerations) as the Hückel—type methods require of the bondingenvironment of the atoms in a molecule.

IX. ELECTRONIC STRUCTURE AND PROPERTIES OF NONMETALLIC MATERIALS

The subject of the electronic structure of non—crystalline solids has been replete withmisunderstanding, confusion, and even erroneous conclusions. This evolved from the factthat the band theory of solids developed by Bloch [133] in 1928 was really only the bandtheory of crystalline solids. By making use of the mathematical simplifications arisingfrom long-range periodicity, Bloch was able to derive some general properties of theelectronic states in crystals, from which Wilson [134] developed a theory of electronictransport. Bloch's work followed the discovery of the periodicity of crystals by less than20 years and the development of quantum theory by only three years. His demonstration thatall the electronic states of a crystal were extended throughout the solid cleared up the"mystery of the unscattered electrons". The first theory of electronic transport proposedthat the negatively charged electrons would necessarily be scattered by the positive ioncores, resulting in a mean free path of the order of an interatomic spacing; however, inmany materials, the actual electronic mean free path is hundreds of interatomic spacings.After the development of Bloch-Wilson theory, it became clear that only deviations fromperiodicity scatter electrons in extended states, and the mystery was solved.

To this day, the standard treatments of solid—state theory are based on crystallineperiodicity. To avoid handling the problem of non—crystalline solids, many texts define asolid as a periodic array of atoms, relegating non—crystalline materials to the realm of

supercooled liquids. Of course, all this is nonsense. Liquids ordinarily exhibitelectronic mean free paths of the order of the corresponding solids [135], so thatperiodicity is clearly not the answer to the mystery of the unscattered electrons. As acorollary, the Bloch—Wilson theory should be abandoned as the general approach to theelectronic structure of solids.


The modern theory of solids is actually based on chemistry rather than crystallography[16]. We can begin with a gedanken theory. In principle, a reasonable starting point is acluster that contains at least the nearest and next-nearest neighbors of the atoms whoseouter electrons are near the Fermi energy of the solid. If periodic boundary conditions areapplied, the band structure can usually be calculated. The deviations from periodicity canthen be handled by perturbation theory. The sharp band edges of a periodic system thendisappear, and band tails result. Experimentally, these band tails presently appear todecay exponentially as the energy changes. The most important question to answer, however,is the order of magnitude of the mean free path of electrons in these bands. Very fewrigorous results exist in this regard. Mott and Twose [136] showed that all electronicstates in disordered one-dimensional systems are localized rather than extended, butone-dimensional systems have special properties that do not generalize (e.g., thepercolation threshold is 100%). Recent numerical calculations [137] indicate that somestates in a disordered two-dimensional system are, in fact, extended. Mott [138] suggestedthat the bands in a real non-crystalline solid contain both localized and extended states,separated by a sharp mobility edge. Simple models based on either classical percolation[139] or quantum—mechanical tunneling [140] tend to confirm that the electronic mobilitysharply increases from very small values to much larger values over a relatively narrowrange of energy in the band tails of disordered systems. If these results hold up toincreased rigor, the mystery of the unscattered electrons will once again appear to beresolved.

Cohen et al. [141] pointed out that the extent of the band tails should be a measure of thedisorder in the system, and noted that highly disordered solids such as multicomponentglasses should always have a finite density of states at the Fermi energy, g(E).Nevertheless, if EF falls in a region of localized states, the material would benonmetallic. They proposed the density of states sketched in Fig. 23 for highly disorderedamorphous semiconductors. Except for the band tails, this density of states is essentiallyfeatureless, in sharp contrast to that of a crystalline solid. The structure in the gap ofa crystalline semiconductor arises from the presence of so—called defects, e.g., vacancies,interstitials, and impurities forced into the wrong chemical environment by the constraintsof the periodic lattice. Cohen et al. [141] proposed that such defects are not present inglasses, since in the liquid state from which they are quenched, there is no reason why amobile atom should not be in its chemical ground state.

1018 020 Jo2'

Density of states (cm3eV')

Fig. 23. CFO model for the density of states of a highly disorderedamorphous semiconductor. and E are the respective valence andconduction band mobility edges, Eg is the mobility gap, and EF is the

Fermi energy [16].

Now, many years later, the last assumption is known to be quite naive. If it were the case,substitutional doping of amorphous semiconductors would have proved impossible and theelectronic properties of glasses would have been totally incomprehensible. Well-definedchemical defects can appear in glasses either to minimize the free energy at the glasstransition temperature or because of internal strains upon solidification [142]. Sincethese defects can give rise to electronic states located near EF, they can, in fact, control

the electronic properties of amorphous semiconductors, just as in crystalline semiconductors.

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recently carried out by Bar—Yam and Joannopoulos [162]. The sign of this energy is stillone of the majorunresolved problems in the study of a—Si:H [163].

Hayes and Beeby [164] suggested a new multiple—scattering approach to estimate the densityof states of disordered solids. Although few calculations have actually been carried outwithin their scheme, the technique has the advantage of a great degree of flexibility. In avirtually orthogonal approach, Blaisdell et al. [165] have developed a computer program forthe ab initio self—consistent calculations of molecular orbitals in polymeric and molecularcrystals and have applied it to a linear chain of nine H2 molecules.

As an example of the current state—of—the—art, we consider the technologically importantmaterial, a—Si:H. Experimentally, pure a—Si is a poor semiconductor, with EF pinned by a

large density of localized states in the gap. As hydrogen is introduced during the growthprocess, the energy gap increases and the density of states in the gap decreases. Thestarting point in an understanding of the electronic structure of a—Si:H is the realizationthat a CRN with an average coordination number of 4 is very overconstrained (recall thediscussion in Section II). Consequently, a—Si is always characterized by a largeconcentration of strain—induced defects, especially dangling bonds. The incorporation ofhydrogen lowers the average coordination number and relieves a large part of the overallstrain. In addition, the Si—H bond strength is about 40% greater than the strength of theSi—Si single—bond. Since H is more electronegative than Si, tight—binding calculationsindicate that the Si—H bonding states are deep in the a—Si:H valence band, while the Si—Hanti—bonding orbitals are not too far from the conduction—band mobility edge. Sketches ofthe tight—binding estimates [166] for the configurations Si5, Si4H, and the dangling bond

Si4 are shown in Fig. 24. Note that even in these crude estimates, it appears that the band

gap of a—Si:H should increase with hydrogen concentration, by shifting states from near thetop of the valence band to considerably lower energy.

Fig. 24. Tight—bindingapproach to the bandstructure of a—Si:H: (a) Acentral Si atomtetrahedrally coordinated byform neighboring Si atoms;

the eight sp3 hybridizedorbitals are split into four

bonding and four_______ anti—bonding orbitals, which

_______ respectively spread into thevalence and conduction bandsof the solid. (b) A centralSi atom tetrahedrallycoordinated by threeneighboring Si atoms and oneH atom; the Si—H bondingorbital falls deep withinthe valence band, while theSi—H anti—bonding orbitallies somewhat above thelower edge of the conductionband. (c) A central Si atomsurrounded by only three

_______ neighboring Si atoms; thefourth electron on thecentral atom occupies a

non—bonding orbital (a

dangling bond), separatedfrom its unoccupied partnerby the correlation energy, V[166].

TfflONDIN6EF

g(E)

(0)

(4)Si-Si

(6) —x-x—- (8) //p

/ Sp3 \//'—x--.x--x—x— (4)/

Si-Si/(2). —x—x—"

(b)_________ (3)/ Si-Si

Si-H(8)p

/ Sp3 '\/ 'ii/\ '—X—X—X— (3)/ Si-Si

(2) —x—x—Si-H

(c)______ (3)I,

//(6)—X-X----——\ (8) / _______p ________

-f-S/ Sp3 \I

1—X—X—X-— (3)

(2) —x——x——'

- EF

BONDING

g(E)

ANTIBONDING

DANGLING-ROND STATES

EF

DING

— g(E)


The results of more accurate SCF—Xcz—SW calculations [153] are shown in Fig. 25. Theseresults confirm the increase in band gap via recession of the valence band edge as itshydrogen concentration increases, and shows several hydrogen—induced peaks in the density of

states (A—D). Cluster Bethe Lattice calculations [145] are presented in Fig. 26 forhydrogenated surfaces of c—Si and a—Si and a model for a—Si:H and compares the results to

experimental photoemission data [167]. It is clear that such calculations allow anunderstanding of the origin of the hydrogen—related structure in the data (C,D,E). OLCAOresults [168] are shown in Fig. 27. Although the details of the band structure differ fromthe other calculations, the gap and the hydrogen—induced structure are consistent.

Fig. 25. SCF—X—SW molecular—orbital calculations for several atomic clusters

expected to be important in a—Si:H. The density—of—states curves are broadened

according to a program appropriate for comparison with photoemissionexperiments [153].

Coa.

(I)wI-4I-U)

U-0>-I-U)zwci

-12 -8 -4 0 4ENERGY (eV)

• S o H

H monolayer on Si (Ill) surface

Isolated H in Si BL bulk

Isolated H on Si BL surface

Fig. 26. Cluster Bethe Lattice

calculations for c—Si, a—Si,and a—Si:H.

(a) Photoemission results;(b) density—of—states (dashed

curve) and local densityon H atoms (solid curve)for a hydrogenated c—Si

surface;(c) same as part (b), but

for an isolated Hembedded in a Si BetheLattice;

(d) same as part (b), butfor an isolated H on aSi Bethe Lattice surface[145].

St5 (sat)12a-Si

0

0>.

>-0

Atoms0.2 —

.0.0 - Si3d—

-0.2

-0.4- THis

0)

S

3s

514H (sat)9Si Monohydride

(I. H

(4) — 1

AE 1.7ev

(20) —

S15H3) sat)9Si Trihydride

/ H

(8)— 's

toE = 2.5ev

(60) —A

(30) —8

(20) —

(13)

a) — a-Si:Ha-Si

H

c-Si

St-BL

d)-HSi-BL

134 J. BICERANO AND D. ADLER_. !iTiIJlJ1J1> b Energy[eV]

SiHX:O.20j::c; :' -12-10-8-6-4-202468z

C Energy[eV]

ENERGY (eV)

Fig. 28. (a) Total density—of—statesFig. 27. Local density—of—states for an for c—Si containing 5% vacancies, asisolated H atom embedded in a Si CRN, as calculated by the CPA method. (b)calculated by the OLCAO technique [168]. Same as part (a), but with theThe dashed line shows the photoemission vacancies each saturated by four Hresults. atoms. (c) Local density—of—states of

the model of part (b) on Si. (d)Local density—of—states of the modelof part (b) on H [156].

Finally, the CPA results [156] of Fig. 28 show clearly how the defect states are snowplowedout of the gap by hydrogenation, as expected.

In summary, we have come a long way since people questioned whether amorphous semiconductorswere theoretically possible [169] or whether disordered materials even have a bandstructure. Some realistic calculations have been carried out and many new and promisingtechniques have been developed. Nevertheless, there are a multitude of unsolved problemsthat cry out for more accurate theoretical calculations to resolve [163]. The electronicstructure of non—crystalline solids is certain to be an active area of research for manyyears to come.

X. AMORPHOUS METALS AND ALLOYS

Metallic glasses are technologically exciting because of their remarkable mechanical andmagnetic properties. Many of them are both extremely strong and ductile, as well as beingvirtually non—corrosive. In addition, they can be magnetically soft (i.e., easilymagnetized) while mechanically hard, so they have found applications as transformer coresand magnetic disks, among many others. The mechanical strength arises from the lack ofmobile dislocations, the origin of fracture in crystalline metals and alloys. We note thisis one more example of an advantage of the amorphous compared to the crystalline state.Another is the integrity of surface bonding relative to that of cleaved crystals, believedto be the origin of the low corrosion rates in amorphous solids.

There are many classes of amorphous metals. We have already mentioned the transitionmetal/metalloid (T—M) glasses in Section II, typically consisting of 80% from one or morelate transition metal (e.g., Pd, Ni, Fe) and 20% from one of the metalloids (e.g., P, Si, B,Sb). Another class (T—R glasses) is composed of alloys of late transition metals (e.g.,


Ni, Co, Fe) with rare—earth atoms (e.g., Gd, La). There are also alloys of early (e.g., Nb,Zn) and late transition metals (T—T glasses) and the alkaline—earth alloys, (AE glasses)which contain Ca, Mg, Zn, or Cd combined with one or more of a large variety of other metals.

In the absence of directional covalent bonds, the question of glass—forming ability is moresubtle in metallic glasses than in amorphous semiconductors, for which the continuous randomnetwork model is appropriate. In order to achieve the dense random packing structurebelieved applicable to a glass, other criteria have been suggested. Clearly, a deepeutectic composition helps glass forming by allowing for less—rapid cooling rates in orderto reach Tg from Tm without crystallization, and this is borne out by experiment. Nagel and

Tauc [170] suggested that glass forming is further aided by tailoring the composition sothat EF lies near a minimum in the electronic density of states, so that small relaxations

can lead to local stabilization of somewhat distorted structures; however, experimental[171] and theoretical [172] confirmation of this idea are thus far dubious at best, and muchremains to be done before this area can be considered to be basically understood.

Recent work on glassy metals has tended to concentrate on deviations from dense randompacking. We have already mentioned Gaskell's analysis [3O of the short and intermediaterange order in T—M glasses, in which a 9—atom polyhedron centered on the metalloid is amajor structural unit. The philosophy behind the dense packing of moderate—sized low—energyclusters is that the existence of more than one (but not too many) can act as a suppressantto crystallization. Some support for the general idea of dense random packing of structuralunits 4.5—6A in size comes from a recent understanding of the first sharp diffraction peakin the structure factor of a large number of diverse glasses [173]. Other work has involvedcomputer simulations of DRP models with two different atomic radii and the introduction ofatomic relaxations via Lennard—Jones or Keating potentials. Saw and Faber [174] haveinvestigated the DRP model of a binary alloy in which either the atomic size ratio or thecomposition can vary. They found only small effects arising from short—range order afterallowing for atomic relaxations with a Keating potential. Saw and Schwartz [175] appliedthe DRP model to a T—T glass, Ni35Zr65, and found essentially no short—range order unless

different strengths for the Ni—Ni, Zr—Zr, and Ni—Zr bonds were explicitly introduced. Usinga computer simulation, Suzuki and Egami [176] were able to show that DRP structures areunstable against shear deformations, at least for T=O, consistent with some experimentaldata.

Electronic density—of—states calculations for glassy metals have not been performed with anydegree of frequency. Kelly and Bullett [172] investigated a T—M glass, Pd80Si20, usingclusters of about 200 atoms and a Green's function approach. Frota—Pessoa [177] and Jaswal[178] have both investigated Ni—Zr glasses, the former using an empirical method, and thelatter a self—consistent cluster approach with periodic boundary conditions and therecursion technique (see Section VIII). For metallic systems, the calculated density ofstates can thus far be tested only by the results of photoemission experiments, which arenot ordinarily sufficiently sensitive to make firm conclusions about the accuracy of thetheoretical approach. In any event, more work in this area is essential.

XI. SILICA GLASSES AND RELATED ALLOYS

Historically, fused silica (a—SiO2) and its related alloys have received a great deal of

attention for thousands of years, since window glass is one member of the class. Theincreasing technological importance of ceramics, composites, and optical fibers is certainto ensure continued interest in these materials. It is now generally (but not universally)believed that a—Si02 forms a classic CRN (an example of which is the model of Bell and Dean

[10]), with each Si atom surrounded by four 0 atoms with tetrahedral bond angles (109.5°),

and each 0 atom surrounded by two Si atoms with a spread in bond angles (l5O°±l5). Whenthe CRN is modified to allow for atomic relaxations, agreement between theory and experiment

becomes quite good [179]. Even Molecular Dynamics approaches have given good results forthe structure of silica glasses [107,180] with the tetrahedral structure being predicteddespite use of a purely ionic potential [180]. There has also been a great deal of effortexpended on the study of defects in a—SiO2 [181], and microscopic models have been studied

theoretically by tight—binding approaches [182] as well as by more sophisticated methods[183,184]. In Section VIII, we also discussed some of the work on the vibrationalproperties of these glasses [24,114].

Studies of vibrational densities of states from CRN models have recently been reported.When only short range interactions with atomic relaxations are included [185] in a modifiedKeating potential, the resulting structure factor shows very good agreement with elasticneutron scattering experiments, while the vibrational density of states also shows fairlygood agreement with the experiments. The fact that the double maximum observed at highfrequencies of the spectrum is also found in the simulation using only short range forces[185] is encouraging. Alternatively, calculations have also been reported where, for the


first time, long range Coulomb effects were included [186] in a network with periodicboundary conditions. These calculations show that optical excitations with wavelengths muchlonger than the nearest neighbor separation can propagate in amorphous media, and that alongitudinal—transverse optical mode splitting is observed as in crystalline materials.

An analytical model has been built for the distribution of dihedral angles in AX2—type

tetrahedral glasses, and its predictions compared with available experimental data [187].The comparison is favorable for a—Si02, but becomes less impressive for glasses such as

GeSe2, which have much smaller intertetrahedral angles and more regular rings than a—Si02.

Considering the basic simplicity of the model, whose only assumptions are a local order withtetravalent A and divalent X, and a simple type of non—bonded steric hindrance which doesnot include any information concerning the existence or effect of completed rings of bonds,these results are quite encouraging. Since predicting and understanding the distribution ofdihedral angles in a covalently bonded non—crystalline solid can provide valuable insightson the intermediate range order (see, for example, Sections II and III, and the discussionon amorphous selenium [38]), further work along these lines can be very useful.

Another example of promising work on intermediate range order in a—5i02 is the examination,

by Tadros et al. [188], of ring statistics in very large computer—generated models with upto 30,000 atoms, including constraints imposed by chemical bonding on the dihedral angledistribution. These authors find that ring formation is dictated primarily by the bondangle centered at the twofold coordinated atomic site, thus explaining the absence of strongspectroscopic features [24] in the oxides of Si and Ge. The calculations also suggest (nottoo surprisingly in our view!) that small ring formation is more probable at smaller bondangles.

The Molecular Dynamics simulation of Ochoa and Simmons [189] examining the response ofa—Si02 to longitudinal strain, is an example of the use of dynamic simulations to predict

the bulk properties of a solid. For example, the effect of various atomic motions on thebrittle strength, and the effect of internal cracks on fracture, are among properties whichcan be studied.

High resolution solid state 29Si NMR studies of the local environment of Si in an 5i02—A1203

glass ceramic [190] using magic angle spinning, is an example of state—of—the—artexperimental work of the type that could be very useful in studying short and long rangeorder in these complex materials. A valuable theoretical contribution to this field wouldbe to attempt to systematize the understanding of solid state NMR results, so that the local

environment of a Si atom can be inferred reliably from a properly taken solid state 29NMRspectrum, just as the local environment of a hydrogen atom in an organic molecule can beinferred from the proton NMR spectrum of that molecule.

Finally, it is interesting to note that according to IR, X—ray photoelectron spectroscopy,and Auger electron spectroscopy experiments, suboxides of silicon (a_SiOx, x<2) produced by

plasma enhanced chemical vapor deposition are also describable by a CRN model involvingtetrahedral building blocks [191]. since there are not enough oxygen atoms to obtain afully chemically ordered CRN (see Sections II and III), each central Si atom in atetrahedral building block contains four near neighbors, one or more of which might be otherSi atoms.

XII. LIQUIDS

Relations between liquids and glassesWe have already encountered many of the fundamental structural, thermodynamic and kineticconsiderations which relate liquids and glasses. For example, in Section VI, we discussedthe continuous nature of the glass transition, and pointed out that a glass can be regardedas a liquid of extremely high viscosity, in which crystallization has been bypassed viarapid cooling, thus preventing the thermodynamically most stable crystalline configurationsfrom being attained. The resulting glassy structures were seen to be thermodynamicallymetastable states, i.e., local free—energy minima, stabilized by the large kinetic barriersseparating them from the lower—lying crystalline absolute minimum and possible othercrystalline local minima for the same composition. In Section VII, we discussed the dynamicsimulation methods which can be used to model these rapid quenching phenomena, and gaveseveral examples of recent applications of these techniques. In Section III, we discussedliquid—to—icosahedral phase transitions, pointed out how the possibility of an icosahedralcondensed phase with long range orientational order was computationally predicted on thisbasis, and went on to mention the relationship between the structure factor of the liquidphase and the properties of the orientationally ordered phase.

In the present section, we carry the discussion of the relationship between liquids andnon—crystalline solids a little further, and then describe some of the most exciting workgoing on at the frontiers of this field.


Mechanisms of phase separations

Most of the standard structural models treat non—crystalline solids as homogeneousmaterials. This type of approach, in addition to its important limitations at themicroscopic atomic scale as discussed in Section II, also neglects the presence ofstructural inhomogeneities on a larger scale. One common type of structural inhomogeneityin glassy alloys is compositional phase separation caused by the immiscibility of componentsof the melt which is rapidly cooled to form the glass. This type of phase separation occurswhen the Gibbs free energy of the phase—separated mixture in the liquid is lower than thatof the single homogeneous phase. A simple example of this situation is a binary system AB,which can exist in the liquid either as a single phase or as a phase—separated mixture,depending on which alternative minimizes the Gibbs free energy. A nice description of thephenomena involved appears in Section 3.6 of the book by Elliott [126]. To summarize, thereare two distinct mechanisms which can lead to phase separation. These differ in thedependence of free energy on composition. If the compositional fluctuations must be greaterthan a certain size to be stable and lead to phase separation, the phase separation is saidto occur by "nucleation and growth". Conversely, the single phase might be unstable to allfluctuations, of any size. In this case, phase separation does not require a nucleationstep, but rather is diffusion—limited. A theory for the initial stages of this so—called"spinodal decomposition" process has been provided by Cahn [192], and by Cahn and Charles[193J. A specific physical process that can be described by the spinodal decomposition

mechanism is the initial stages of the heat treatment at 450°C of a B203—PbO—A12O3 glass

quenched from 1150°C [194J.

RecentworkThe study of liquids and their relation to glasses is a vast field of research, and we willonly attempt to provide the fundamental underlying physical ideas without going into thedetails. Several recent papers [195—199], examine some of the problems at the frontiers ofthe study of liquids. The most promising techniques used include dynamic simulations,density functional calculations, hydrodynamic techniques in conjunction with considerationof phenomena which occur on a molecular length scale, and electronic structure calculations.

Kirkpatrick [195] attempted to explain the results of computer experiments using MolecularDynamics, and considered anomalous mode coupling effects in dense classical liquids. He wasable to explain the discrepancies between the results of conventional mode coupling theoryand computer "experiments" in terms of a generalized mode coupling theory. This generalizedmode coupling theory takes into account phenomena which occur on a molecular length scalewhere liquid structure is important, in addition to the phenomena usually treated withhydrodynamic mode coupling theories.

Wolynes [196] considered application of techniques from liquid state theory, such as freeenergy functionals for inhomogeneous systems, along with self—consistent phonon theories, inorder to relate the stability of glasses to their underlying structure and to theintermolecular forces. He also reviewed recent work along these lines for the hard sphereglass. He recommends the use of more flexible trial densities, the development of moresophisticated density functionals, and the inclusion of nonequilibrium effects andfluctuations.

Chen et al. [197] discussed transverse atomic correlations in supercooled liquids describedin terms of the orientational ordering of the local shear stresses. They used MolecularDynamics studies of liquid iron to show that the onset of the correlation as a function oftemperature is rather sharp, and is accompanied by changes in diffusivity, viscosity, andthe phonon density of states. They also suggested that the glass transition may be definedby this transition in a dynamical sense, by the elastic percolation of the correlatedregions.

Alder et al. [198] have discussed the use of Quantum Monte Carlo calculations to simulateliquids. The Quantum Monte Carlo simulations are, in principle, quite similar to theclassical simulations, although there are several additional serious technical difficultiescaused by the use of quantum mechanical wave functions and probability densities instead ofclassical particles, and the corresponding replacement of the classical equations of motionby the Schrödinger equation. They have applied the Quantum Monte Carlo simulations to thecalculation of the properties of the electron gas (the simplest model system for thecondensed phase), of hydrogen, and of liquid helium. The study of the bulk properties ofincreasingly complex systems by Quantum Monte Carlo simulations is a promising research areawhich will progress as algorithms are improved and faster computers become available;however, it will probably take many years for these techniques to reach the level ofmaturity of the classical simulations and become as easily usable for the same types of verycomplex systems.

Finally, St. Peters and Roth [199] have presented an analysis of a muffin—tin modelcalculation of the electronic density of states of liquid copper. An ionic pairdistribution function was used in this work to account for the fact that due to the short


range liquid order, the ionic potential of a liquid is neither periodic nor random.Multiple scattering theory was used to handle the strong interactions between the localizedd states and delocalized s states. The results of these calculations were seen to comparewell with optical density of states measurements on molten Cu.

In general, the electronic structure of liquids has been understood in terms of a nearlyfree electron approach [59), which appears to work because the temperature is large comparedto the Debye temperature of the corresponding solid. Unfortunately, this is not the case inamorphous solids, so that a more complex analysis is required. The work reported by St.Peters and Roth [l99J is an example, in the context of a liquid system, of how the metallicmodel can be modified to take some of the more complex effects into account.

An alternative to the metallic model is the so—called "distorted crystal" model [200], wherethe crystalline phase of a semiconductor is used as a starting point for the treatment ofthe electronic structure of the corresponding liquid semiconductor. This provides azeroth—order approximation where a band structure with a band gap is already present. Twotypes of disorder can then be applied to this initial band structure. One of these is thetopological disorder arising from the loss of long range order while some kind of shortrange order persists. The second type of disorder is due to fluctuations in atomic densityand potential. There is clearly some vagueness and arbitrariness involved in the nature ofthe disorder to be applied.

Another very chemically oriented alternative to the metallic model is the so—called"molecular bond" model [200], where the idealized starting point consists of discretemolecules which are far enough that their electronic levels are discrete. When thesediscrete molecules are brought closer together, the discrete electronic energies broadeninto bands which may be described by tight binding theory. The approximation that theelectronic states of different bonds and non—bonding electrons are independent for thediscrete molecules, is an additional simplification that can be used in the context of thismodel. This model has the advantages of being in accord with chemical intuition; of beingequally applicable whether the bonding leads to network structures as in As25e3, or to small

molecules as in Tl2Te; and of being easy to apply, since the semi—empirical methods of

structural chemistry can be used for making estimates of the energy and electronic characterof the bond orbitals, and these estimates can then be used to infer the electronic bands via

tight binding calculations [200].

Research on the structure, dynamics and properties of liquids is certain to continue tocontribute to a better understanding of glasses as well as to making new types of glassymaterials for technological applications. Another (somewhat unrelated but equallyimportant) reason for chemists to become more interested in liquids is the fact that a great

deal of interesting chemistry, especially in biological systems, takes place in complexliquid media. The book by Cutler [200] is highly recommended to the reader interested inlearning more about liquids.

XIII. SUPERLATTICES

There has been a great deal of recent interest in artificial structures, in which complexmultilayered geometries employing two or more different materials are fabricated. A simpleexample is an inversion layer [201], which consists of a narrow (30A) region near thesurface of a doped semiconductor in which minority carriers move freely. For example, ifsufficient negative charge is induced across the insulator of a metal/insulator/p—typesemiconductor (MIS) structure by applying a positive potential to the metal, conduction—bandstates begin to become filled. Since the electric field acts like a potential well for theconduction—band electrons, quantum effects can be important. According to the HeisenbergUncertainty Principle, confining electrons to a region of width L introduces an additionalkinetic energy of the order of:

E = p2/2m>h2/2mL2

Confining an electron (m—l030kg) to a distance L30A yields E—0.leV. This means that inorder to fill the lowest quantum level in the well, an applied field sufficient to populatelevels about 0.1eV above the conduction—band edge is required. Note that the quantizationoccurs in only one dimension, the direction of the applied field; the electrons move freelyin the plane of the layer. When a current is passed within the layer and a magnetic fieldis applied, normal to the layer, the Lorentz force causes the electrons to move within thelayer, perpendicular to the current. When they reach the surface, they induce a transverseelectric field called the Hall field. Because of the quantum—well structure, however, theHall voltage is also quantized. The discovery of this so—called quantum Hall effect led tothe 1985 Nobel Prize in Physics [202].

The most novel artificial structures to date have involved periodic alternating thin layersof different materials, e.g., ABABAB... When the materials are crystalline semiconductors,the structures are called superlattices [203]. If the two materials in the ABAB...structure are semiconductors with different energy gaps, the different potentials in the two


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0(b)

U)CU)

regions act like periodic potential wells for both electrons and holes in the directionperpendicular to the layers. This type of problem, a one—dimensional periodic potential,was first investigated over 50 years ago [204], and is called the Kronig—Penney model. Themodel suggests that in superlattices, quantum effects occur for both electrons and holes,the level separations becoming larger as the thickness of each layer decreases. Since theoptical gap is the minimum energy necessary to excite an electron from a filled to an emptystate, the gap in superlattices should increase with decreasing layer thickness.

One problem with superlattices is the mismatch in lattice constants when a crystalline layeris deposited on a different one, resulting in large concentrations of interface states whichbecome important when many layers are present, often to the extent that they mask the bulkproperties. This problem can be overcome by using alternating amorphous semiconductors

[205]. In addition to the elimination of lattice mismatches, amorphous multilayerstructures have the flexibility of an essentially infinite number of possible alloys for anyof the layers as well as the possibility of controlled compositional gradients within the

layers. Furthermore, there is the interesting theoretical problem of analyzing afundamentally disordered system containing a periodic structure on the lO—100A scale.

Two of the most investigated amorphous multilayered structures are a_Si:H/a_Si1_xNx:H and

a—Si:H:P/a—Si:H:B arrays. In both cases, the potential energy creates quantum wells forboth electrons and holes; the band structure of the former is sketched in Fig. 29. Assuggested by the Kronig—Penney model, the quantum wells should yield an increasingsemiconductor band gap as the thickness of the layers decreases. This is borne out by theexperimental data [205], shown in Fig. 30. An interesting conclusion from these results is

(a)Si:H

ioAa.SiNx:H

Fig. 29. (a) Potential well model for the a_Si:H/a_SiixNx:H

superlattice; the energy levels, Ec and for the lowest quantum

states for electrons and holes, the discontinuities in conduction andvalence band edges, Ec and E and the Fermi energy, EF. are indicated.

(b) Density—of—states of a free—electron gas in the presence of thepotential of part (a) for a—Si:H layer thicknesses, L. of 20A and lOOA

[205].

0

Energy (ev)

2

•

:Av'

I

103.i , 0: /0 /

jV1 11111111 I L,I I1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4

E(eV)Fig. 30. Optical—absorption coefficient as a function of photon energy,E, for a series of a—Si:H/a—Si1_xNx:H superlattices with LN=3SA and

several different values of L:L5=40A (0), L=20A (a), L=l2A(V),L5=8A(o).

The dotted line represents the data for bulk a—Si:H, the

dashed line for bulk a_Si1_xNx:H [205].


the observation of quantum size effects in a—Si:H layers as thick as 40A, implying awave—function coherence length greater than about 20 interatomic spacings. Tsu [206] hascalculated that such a value is consistent with an electronic mean free path of about 5A.In alternating layers of a—Si:H doped n—type and p—type, a persistent photoconductivity hasbeen observed, lasting for many hours after removal of the incident light [207]. Althoughthis could result from the reduced recombination attendant with electrons and holes beingtrapped in different regions of space (i.e., in the p—type and n—type layers, respectively),the effect now appears to be due to deep trapping in the interface regions [208].Structural studies [209] have indicated that the difference between the length of the Si—Siand Si—N bonds leads to interface distortions in a_Si:H/a—Si1N:H multilayer structures.

xlv. CONCLUSIONS

Any study of a condensed collection of atoms must begin with an understanding of thestructure. For a solid, which tetains its shape under at least moderate shear stresses, weare aided by the conclusion that each atom is located near a local potential minimum. Theempirical investigations of the structure of non—crystalline solids reviewed in this paperhave, above all, focused attention on the importance of the chemical nature of theconstituent atoms in the formation of such materials. Great strides have been made bothexperimentally and theoretically over the past few years, and the results finally appear tobe converging, even if slowly [210]. Theoretical approaches have progressed far beyond theball—and—stick models of 15 years ago, and now not only computer simulations with atomicrelaxations but Monte Carlo and Molecular Dynamics techniques have produced results thatportend a useful future for such calculations.

Given the structure of the material, it becomes possible to analyze both the vibrationalproperties (i.e., phonon modes) and the electronic behavior (i.e., electronic density of

states). The former often controls the thermal properties of the solid, such as specificheat, thermal expansion, and thermal conductivity. But the electronic structure determinesthe response of the material to applied electric fields, visible and ultraviolet light, anda host of coupled effects such as thermoelectric, magnetoelectric, and optoelectronicbehavior. The modern theory of crystalline solids has achieved its major successes from thedetailed quantitative understanding of such behavior as functions of temperature and eventime (under transient conditions). This is an area in which a great deal of work is stillnecessary in the case of non—crystalline solids.

The recent insights afforded by curved space approaches and three—dimensional projections ofhigher—dimensional structures has opened up new avenues for theoretical investigation. In

addition, the recognition of the importance of well—defined defects in non—crystallinesolids has suggested additional uses for such points of view. The fact that an infinitenumber of dimensions are available for modeling suggests that this area will remain fruitfulfor quite some time to come.

Since all solids are formed at finite temperatures and pressures, thermodynamic and kineticconsiderations are vital. Difficulties in calculating such quantities as entropy and Gibbsfree energy for systems with no long—range order have retarded progress, but makes thesequantities no less important. Just because we are not yet smart enough to evaluate thestates of local free—energy minima does not retard the atom's ability to find them rather

quickly upon solidification.

A classic problem in the study of solids has been the rapid progress in experimentaltechniques, which often has led to the feeling that theoretical understanding will nevercatch up. Recent work in picosecond spectroscopy, superlattice structures, and ultra—rapidsolidification, among many other developments, has intensified such feelings. But sooner orlater, the theoretical insight must come or experimental progress will decelerate. Thepresent gaps between theory and experiment stand as an exciting challenge and opportunity.The fact that closing this gap necessarily requires the cooperation between chemists,physicists, and materials scientists should only add to the appeal of working in this field.

Acknowledgements

This review article was stimulated by the International Conference on the Theory of theStructures of Non—Crystalline Solids, held at the Institute for Amorphous Studies,Bloomfield Hills, Michigan, U.S.A. We would therefore like to especially thank Stanford R.

Ovshinsky, President of Energy Conversion Devices, Inc., whose support and encouragementmade this Conference possible; Martin C. Steele and Brian B. Schwartz, respectively theformer and present Directors of the Institute; and Ghazaleh Koefod, Executive Administratorof the Institute. We are also indebted to the International Union for Pure and AppliedChemistry for their invitation to prepare this review. In addition, we owe a greatintellectual debt to our many colleagues who have helped us to understand various aspects ofthis vast field of research, and especially to those who participated in the Conference andits active discussion periods. Finally, we wish to thank Ella M. Norman, who patientlytyped the entire manuscript.


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