Electronic structure and the glass transition in pnictide and chalcogenide semiconductor alloys. Part II: The intrinsic electronic midgap states
aa r X i v : . [ c ond - m a t . d i s - nn ] O c t Electronic structure and the glass transition in pnictide and chalcogenidesemiconductor alloys. Part II: The intrinsic electronic midgap states
Andriy Zhugayevych and Vassiliy Lubchenko , ∗ Department of Chemistry, University of Houston, TX 77204-5003 Department of Physics, University of Houston, TX 77204-5005 (Dated: July 4, 2018)We propose a structural model that treats in a unified fashion both the atomic motions andelectronic excitations in quenched melts of pnictide and chalcogenide semiconductors. In Part I(submitted to
J. Chem. Phys. ), we argued these quenched melts represent aperiodic ppσ -networksthat are highly stable and, at the same time, structurally degenerate. These networks are character-ized by a continuous range of coordination. Here we present a systematic way to classify these typesof coordination in terms of discrete coordination defects in a parent structure defined on a simplecubic lattice. We identify the lowest energy coordination defects with the intrinsic midgap electronicstates in semiconductor glasses, which were argued earlier to cause many of the unique optoelectronicanomalies in these materials. In addition, these coordination defects are mobile and correspond tothe transition state configurations during the activated transport above the glass transition. Thepresence of the coordination defects may account for the puzzling discrepancy between the kineticand thermodynamic fragility in chalcogenides. Finally, the proposed model recovers as limitingcases several popular types of bonding patterns proposed earlier, including: valence-alternationpairs, hypervalent configurations, and homopolar bonds in heteropolar compounds.
I. INTRODUCTION
In the preceding article, we presented a chemicalbonding theory for an important class of pnictogen- andchalcogen-containing quenched melts and glasses. Thesematerials exhibit many unique electronic and opticalanomalies not found in crystals, and also are of greatinterest in applications such as information storage andprocessing. We argued these materials can be thoughtof as aperiodic ppσ -networks made of deformed-linearchains that intersect at atomic sites at nearly right an-gles. Extended portions of the chains exhibit a perfect al-ternation of covalent and weaker, secondary bonds, eventhough the lattice as a whole is aperiodic. This bond al-ternation results from a symmetry breaking of a parent,simple cubic structure with octahedral coordination; thisstructure is uniformly covalently bonded. It is the inti-mate relation of the amorphous lattice to its covalentlybonded parent structure that allowed us to rationalize,for the first time to our knowledge, two seemingly contra-dicting features of a bulk glass, i.e., its relative stabilityand structural degeneracy . Yet the argued presence of the degeneracy of the ppσ -network does not, by itself, guarantee that the networkcan be realized as an equilibrated supercooled liquid ora quenched glass: For instance, elemental arsenic can bemade into an amorphous film but does not vitrify readily.To be a liquid, the network should contain a large equilib-rium concentration of structural motifs corresponding tothe transition states for activated transport in quenchedmelts.The goal of the present article is to identify the bondingpatterns of the transition-state structural motifs at themolecular level and describe the rather peculiar midgapelectronic states that are intrinsically associated withsuch motifs. It will turn out that these electronic states are responsible for many of the aforementioned electronicand optical anomalies of amorphous chalcogenide alloys.Identification of structural motifs in vitreous materialsbased on local coordination is difficult because the usualconcept of coordination, which is not fully unambiguouseven in periodic lattices, becomes even less compellingin aperiodic systems. Alternatively, one can try to clas-sify such local motifs in terms of deviation from a puta-tive reference structure, while assigning a correspondingenergy cost. Such deviations could be called “defectedconfigurations.” However, in the absence of long-rangeorder, defining a reference structure in vitreous systemsis, again, ambiguous. To make an informal analogy, isthere a way to identify a typo in a table of random num-bers? The answer, of course, depends on the presenceand specific type of correlation between the random num-bers. The random first order transition (RFOT) theory dictates that glasses do form subject to strict statisticalrules prescribed by the precise degree of structural degen-eracy of the lattice, implying that correlation functionsof sufficiently high order should reveal defects, if any.In fact, already four-point correlation functions in spacecapture the length scale of the dynamic heterogeneity inquenched melts. Yet the only type of order in glassesthat appears to be unambiguously accessible to linear spectroscopy is the very shortest-range order: The veryfirst coordination layer is usually straightforward to iden-tify by diffraction experiments, while the strong covalentbonding between nearest neighbors is identifiable via theindependent knowledge of the covalent radii of the perti-nent elements. Already the next-nearest neighbor bondsappear to exhibit a continuous range of strength and mu-tual angular orientation.We have argued aperiodic ppσ -networks naturally ac-count for this flexibility in bonding in semiconductorglasses, while retaining overall stability. In doing so, weproposed a structural model, by which aperiodic ppσ -networks can be thought of as distorted versions of muchsimpler parent structures defined on the simple cubic lat-tice. There is no ambiguity whatsoever with defining co-ordination on a simple cubic lattice, thus allowing one toclassify unambiguously the parent structures.We will observe that the most important and essen-tially the sole type of defect in ppσ -bonded glasses issingly over- or under-coordinated atoms. These defectsturn out to host peculiar midgap electronic states withthe reversed charge-spin relation, i.e., chemically they re-semble free radicals. We will argue that these defects infact correspond to the electronic states residing on thehigh-strain regions intrinsic to the activated transport insemiconductor glasses proposed earlier by us. On theone hand, these electronic states lie very deep in the for-bidden gap. On the other hand, they are surprisingly ex-tended, calling into question the adequacy of ultralocaldefect models. This large spatial extent reveals itself byredistribution of the malcoordination over a large num-ber of bonds, in a solitonic fashion, and delocalizationof the wave function of the associated electronic state.The extended coordination defects are mobile, consistentwith the conclusions of our earlier semi-phenomenologicalanalysis that the peculiar electronic states are hosted byhigh-strain regions that emerge during activated trans-port in quenched semiconductor melts.The article is organized as follows. Section II reviewsthe conclusions of the RFOT theory on the concentrationand spatial characteristics of the transition state configu-rations for activated transport in supercooled liquids andfrozen glasses, and a general mechanism for the emer-gence of associated electronic states. In Section III, asystematic classification of coordination defects in parentstructures is carried out. In Section IV, we demonstratethat malcoordination defects in parent structures becomedelocalized in the actual, relaxed structure and make aconnection with our earlier, semi-phenomenological anal-ysis of the solitonic states. When making this connec-tion, we perform several independent consistency checksthat the degeneracy of aperiodic ppσ -networks is indeedcompatible with the degeneracy of actual semiconductoralloys. Lastly we argue that the electronic states make atemperature-independent contribution to the activationbarrier for liquid reconfigurations, which helps explainthe apparent disagreement between the thermodynamicand kinetic fragilities in chalcogenides.We have alluded to two vulnerabilities of ad hoc defecttheories, i.e., the difficulty in defining a putative refer-ence structure and the presumption of defects’ being ul-tralocal at the onset. Conversely, such ad hoc theoriesdo not explain self-consistently how the defects combineto form an actual, quite stable lattice. The present ap-proach resolves this potential ambiguity. First, since thedefects are defined on a specific lattice in the first place,the question of their coexistence in a 3D structure is au-tomatically answered. Second, as discussed in Section V,we will see how several popular defect theories proposed much earlier on phenomenological grounds are naturallyrecovered as the ultralocal limit of the present picture.
II. BRIEF REVIEW OF THE RFOT THEORYAND PREVIOUS WORK
Below we outline the minimum set of notions from theRandom First Order Transition (RFOT) theory of theglass transition and other previous work, as necessaryfor the subsequent developments. Detailed reviews ofthe RFOT theory can be found elsewhere.
Mass transport in liquids slows down with loweringtemperature or increasing density because of increasinglymore frequent molecular collisions. At viscosities of 10Poise and above, however, the transport is no longer dom-inated by collisions but, instead, is in a distinct dynami-cal regime: The equilibrium liquid density profile is nolonger uniform, but instead consists of sharp disparatepeaks, whereby each atom vibrates around a fixed lo-cation in space for an extended period of time. In otherwords, the liquid is essentially an assembly of long-livingstructures that persist for times exceeding the typical re-laxation times of the vibrations by at least three ordersof magnitude. Under these circumstances, mass trans-port becomes activated: Atoms move cooperatively viabarrier-crossing events whereby the current long-lived,low free energy aperiodic configuration transitions lo-cally to another long-lived, low free energy configuration.The multiplicity of alternative aperiodic configurationsis quantified using the so called configurational entropy:A region of size N particles has e s c N/k B distinct struc-tural states, where s c is the configurational entropy perper rigid group of atoms, often called the “bead.” TheRFOT theory predicts that the configurational entropyat the glass transition is ≃ . k B per bead. Individual atomic displacements during the transi-tions are small, i.e., about the vibrational displace-ment at the mechanical stability edge, which is oftencalled the Lindemann length d L . Although the acti-vated transport regime is usually associated with super-cooled liquids, many covalently bonded substances,such as SiO , exhibit activated transport already abovethe melting point. During an activated reconfiguration, two distinct low-energy structures must coexist locally, implying that ahigher free-energy interface region must be present. Be-cause the interface separates aperiodic arrangements, ithas no obvious structural signature; non-linear spec-troscopy is generally required to detect the interface. Thetransitions proceed in a nucleation-like fashion; they aredriven by the multiplicity of the configurations, subjectto the mismatch penalty at the interface γ √ N : F ( N ) = γ √ N − T s c N, (1)see Fig. 1. Here, N gives the number of beads con-tained within the nucleus and the coefficient γ = √ πk B T ln( a /d L πe ). The lengths a and d L ≃ . a N/N ∗ t r an s i t i on f r ee ene r g i e s mismatch energy γ √ N full activation profile ( γ √ N − T s c N ) F ‡ FIG. 1. Typical nucleation profile for structural reconfigu-ration in a supercooled liquid from Eq. (1) and its surfaceenergy component γ √ N , normalized by the typical barrierheight, which reaches (35 − k B T at T = T g . denote the volumetric size of a bead and the Lindemanndisplacement respectively. The reconfiguration timecorresponding to the the activation profile in Eq. (1): τ = τ e F ‡ /k B T = e γ / T s c , (2)grows in a super-Arrhenius fashion with lowering thetemperature because of the rapid decrease of the con-figurational entropy s c ≃ ∆ c p T g (1 /T K − /T ) , (3)where ∆ c p = T ∂s c /∂T | T g is the heat capacity jump at T g , per bead. The quantity T K , often called the Kauz-mann temperature or the “ideal glass” transition tem-perature, corresponds to the temperature at which theconfigurational entropy of an equilibrated liquid wouldpresumably vanish: Hereby the log number of alterna-tive configurations would scale sub-linearly with the sys-tem size. While the temperature T K appears in certainmeanfield models, the ideal-glass state, if any, wouldbe impossible to reach in actual liquids because of thediverging relaxation times, see Eq. (2).The nucleus size N ∗ ≡ ( γ/T s c ) , (4)where F ( N ∗ ) = 0, is special because it correspondsto a region size at which the system is guaranteed tofind at least one typical liquid state. As a result, thesize N ∗ corresponds to the smallest region that can re-configure in equilibrium. Each transition requires work γ √ N to create and grow the interface, at the expenseof relaxing the old interfaces. Thereby, the total num-ber of interfaces remains constant on average. Onethus concludes that the liquid harbors one interfacial re-gion with an associated excess free energy γ √ N ∗ per region of size N ∗ , consistent with the quench being ahigher free energy state than the corresponding crys-tal. In fact, the total mismatch penalty in a sampleof size N is γ √ N ∗ ( N/N ∗ ) = T s c N and thus equalsthe enthalpy that would be released if the fluid crystal-lized at this temperature, save a small ambiguity stem-ming from possible differences in the vibrational en-tropies. The cooperativity size N ∗ grows with the de-creasing configurational entropy (see Eqs. (3) and (4): N ∗ ( T ) ≃ N ∗ ( T g )[( T g − T K ) / ( T − T K )] . Still, it reachesonly a modest value of 200 or so at the glass transition onone hour time scale. N ∗ ( T g ) = 200 corresponds to aphysical size ξ ≡ a ( N ∗ ) / ≃ . a , i.e. almost universallyabout two-three nanometers. This important predictionof the RFOT theory has been confirmed by a numberof distinct experimental techniques and recently, by direct imaging of the cooperative rearrangements on thesurface of a metallic glass. Now, the resulting concen-tration of the domain walls near the glass transition is,approximately, n DW ( T g ) ≃ /ξ ( T g ) ≃ cm − . (5)In the rest of the article we will assume the materialis just above its glass transition temperature T g , so thatit represents a very slow, but equilibrated liquid. Be-low T g , the lattice remains essentially what it was at T g ,save some subtle changes stemming from the decreasedvibrational amplitudes and aging.The present authors have argued that the interfacesmay host special midgap electronic states, subject to anumber of conditions. These conditions are satisfied inmany amorphous chalcogenides and pnictides. The num-ber of the intrinsic midgap states is about 2 per interface,implying a concentration of about 2 /ξ , where ξ is the co-operativity length from Eq. (5). The argument in Ref. isindependent from the present considerations and is basedon the Random First Order Transition (RFOT) theory ofthe glass transition. These intrinsic states are centeredon under- or over-coordinated atoms and are surprisinglyextended for such deep midgap states. In addition, thestates exhibit the reverse charge-spin relation. The ex-istence of the intrinsic states allows one to rationalize anumber of optoelectronic anomalies in chalcogenides in aunified fashion. The interface-based electronic states inglasses are quasi-one dimensional and are relatively ex-tended along that dimension. This quasi-one dimension-ality stems from the structural reconfigurations them-selves being quasi-one dimensional: Activated transitionsbetween typical configurations occur by a nearly uniquesequence of single-atom moves that are nearby in space.In the simplest possible Hamiltonian that couples elec-tronic motions to the atomic moves, by which a transitionbetween two metastable states takes place, only relative positions of the atoms are directly coupled to the elec-tronic density matrix: H el = X n,s h ( − n ǫ n c † n,s c n,s − t n,n +1 ( c † n,s c n +1 ,s + H.C. ) i , (6)where c † n,s ( c n,s ) creates (removes) an electron on site n with spin s . The on-site energy, which reflects localelectronegativity, is denoted with ( − n ǫ n . The electrontransfer integral t n,n +1 between sites n and n +1 dependson the inter-site distance d n,n +1 . The sites are centeredon beads, not atoms. The beads are numbered with index n , 1 ≤ n ≤ N ∗ , in the order they would move during thereconfiguration. III. CLASSIFICATION OF COORDINATIONDEFECTS
The structural model proposed by us in Ref. presentsa systematic way to classify amorphous structures andthe associated electronic-structure peculiarities in vitre-ous ppσ -networks. Without claiming complete general-ity, we will consider the following specific type of par-ent structures, which are largely based on Burdett andMacLarnan’s model of (the crystalline) black phospho-rous and rhombohedral arsenic. In this model, eachvertex on a simple cubic structure is connected to exactlythree nearest neighbors, where only right angles are per-mitted between the links. Two specific periodic ways toplace the links according to this prescription correspondto the crystals of black phosphorous and rhombohedralarsenic. We have seen that this model has a significantlybroader applicability, if one allows for distinct atoms andalso vacancies in the original cubic lattice. For instance,let us take the parent structure for black phosphorus(Fig. 5(b) of Ref. ), place pnictogens and chalcogens inthe rock salt fashion (as in Fig. 6(b) or 7 of Ref. ), andthen omit every third pnictogen in a particular fashion(as in Fig. 7 of Ref. ). This procedure yields both thecrystal structure and the stoichiometry of the archetypalpnictogen-chalcogenide compound Pn Ch (“Pn”= pnic-togen, “Ch”=chalcogen). In the crystal, each pnictogenand chalcogen are three- and two-coordinated, respec-tively, thus conforming to the octet rule. The actualstructure of a crystal or glass is thus viewed as a resultof the following multi-stage procedure: (1) start with thesimple cubic structure with all vertices linked to all sixnearest neighbors; (2) break the links and place vacan-cies to satisfy the stoichiometry and the octet rule. Theresulting, lowered-symmetry structure is called the “par-ent” structure. To generate the actual structure from theparent structure, (3) shift the atoms slightly toward thelinked vertices; (4) turn on the interactions by associat-ing an electronic transfer integral to each bond; and (5)geometrically optimize the structure, subject to the re-pulsion between the ionic cores and the variations in theelectronegativity, if any.In the above prescription, it is generally impossible to satisfy the stoichiometry and the octet rule at the sametime except in the case of periodic crystals, as we willsee explicitly in a moment. As a result, the resulting parent structure will generally exhibit a variety of de-fects in the form of under- and over-coordinated atoms,such as three-coordinated chalcogens. Similarly, incor-porating elements of group 14, such as germanium, intothe lattice of strictly three-coordinated vertices wouldintroduce coordination defects. The just noted pres-ence of four-coordinated vertices presents a convenientopportunity to point out that the hereby proposed struc-tural model, though not unique, is special in the con-text of ppσ -bonded solids in distorted octahedral geom-etry. Suppose that, instead, we started from a modelwhere each vertex is four -coordinated. The correspond-ing parent structures, if periodic, could be unstable to-ward a tetrahedrally bonded solid with the β -tin or re-lated structure. As a result, the present discussion islikely limited to compounds where elements from group14 are in the minority, except when these elements comein pairs with chalcogens, such as in GeSe or Ge Sb Te .Such pairs are isoelectronic with a pair of elemental pnic-togens, and so both constituents of the pair are three-coordinated. This is not unlike, for instance, how GaAsforms the diamond (zincblende) structure, in which eachatom is four coordinated.The three-coordinated parent structures on a simplecubic lattice, with right angles between the bonds, haveanother very special property: In any lattice satisfy-ing this constraint, whether periodic or not, each bond-containing line is a strictly alternating bond/gap pat-tern. Informally speaking, one may think of the latticeas made of linear chains, each of which consists of whiteand black segments of equal length in strict alternation;the junctions between adjacent segments correspond tothe lattice vertices, while the white and black segmentscorrespond to no-link and link respectively. This ob-servation allows one to easily estimate the number ofall possible distinct parent structures. Since there areonly two distinct ways to draw a perfect bond/gap al-ternating pattern, the total number of the parent struc-tures in a sample of size L × L × L with no vacanciesis 2 ( L L + L L + L L ) /a , where a is the lattice spacing.Note that in contrast to Ref. , we count as inequivalenttwo structures that can be obtained from each other byrotation, because they will both contribute to the phasespace. We immediately observe that the total number ofdistinct parent structures is sub-thermodynamic in thatit scales exponentially with the surface of the sample, notits volume. As a result, such defect-free structures cannot contribute significantly to the library of bulk aperi-odic states. The only exception to this statement is thehypothetical Kauzmann, or ideal-glass state, in whichthe configurational entropy would strictly vanish. Onethus concludes that an equilibrium liquid must containnot only perfectly bond-alternating configurations, but afinite number of defected configurations per unit lengthin each line, where a “defect” consists of two or more singly perfectalternation undercoordinatedsinglyovercoordinated
FIG. 2. Generation of singly malcoordinated atoms by a “cut-and-shift” procedure, see text. The filled and empty circlesdo not necessarily imply chemically distinct species, but areused to emphasize that it is the bond pattern that is shifted,not the atoms themselves. bonds (gaps) in a row.Furthermore, parent structures of certain common sto-ichiometries, such as Pn Ch , must host a thermody-namic number of vacancies. Adding vacancies to defect-less aperiodic parent structures should generally lead tothe creation of coordination defects. Indeed, to maintainthe stoichiometry, the vacancies should be spaced on av-erage by the same distance. On the other hand, onecan construct a bond-breaking pattern with an arbitrar-ily large period, which can be made arbitrarily differentfrom the average spacing between the vacancies.As a consequence of the deviation from the strict al-ternation pattern along the individual linear chains, athermodynamic number of atoms in a representative par-ent structure must be either under- or over-coordinated.Note that after geometric optimization, defects in theparent structure will be generally mitigated or even re-moved. For example, at least one of the covalent bondsaround a four-coordinated pnictogen will generally elon-gate in the deformed structure, in the pnictogen’s at-tempt to attain the favorable valency 3. As a result,the overcoordination will be spread among a larger num-ber of atoms. An example of defect annihilation is whenan extra bond on an overcoordinated atom annihilateswith a missing bond on a nearby undercoordinate ver-tex. For these reasons, a defect in the deformed lattice isnot well-defined. Here, we analyze several important de-fect configurations in the parent structure, whereby thelocal coordination is entirely unambiguous. Before pro-ceeding, we should note that the discussion is not limitedto atoms of valence three and two. The above discussionapplies even if a fraction of the vertices in the defectlessparent structure shoud be four-coordinated, as would beappropriate for elements of group 14. Estimating thetotal number of distinct parent structures is no longerstraightforward, however it is clear the number could beonly lowered compared to the strict Burdett-McLarnancase. (e) = + += + += + +single malcoordinationdouble malcoordination(a)(b) = + +(d) = + +(c)
FIG. 3. Examples of malcoordination defects of differ-ent orders on a pnictogen site and their relation to single-malcoordination defects along individual directions. A sim-ilar diagram can be drawn for chalcogens, which are doublycoordinated in a defectless parent structure, the bond angleequal to 90 ◦ . Let us begin the discussion of defected parent struc-tures with an ideal parent lattice with one singly over-coordinated vertex. This defect corresponds to two linksin a row in one of the three chains crossing the vertexin question. Formally, one can obtain such a defect froma perfect Burdett-MacLarnan structure, for example, bychoosing a vertex, removing the adjacent gap-segmentfrom one of the chains and shifting the rest of the bondpattern on that side toward the vertex, see Fig. 2. Notethat owing to ppπ -interactions, such shift generally in-vokes an energy cost or gain in addition to the cost ofovercoordinating the chosen vertex, which we will dis-cuss later. One may obtain a singly undercoordinatedvertex analogously, see Fig. 2. It is straightforward to seethat more complicated, multiple-malcoordination config-urations can always be presented as superpositions of thesimple, single-malcoordination configurations describedabove, see Fig. 3. For instance, a doubly-overcoordinatedatom simply corresponds to overcoordination on two in-tersecting chains, at the vertex in question. Anothercommon type of defect could be obtained by simply re-placing a gap in an ideal parent structure by a link. Thisdefect is equivalent to two singly-overcoordinated atomsalong the same line next two each other. Coordinationdefects of higher yet order can be constructed similarly.Note that bond angles other than 90 ◦ also amount to acoordination defect, even if the total number of bondsobeys the octet rule, see Fig. 3(e).Single-malcoordination configurations can move alonglinear chains or make turns by bond switching, see Fig. 4,whereby the defect moves by two bond lengths at a time.Direct inspection shows, however, that an attempt tomake a turn at a pnictogen in Fig. 4 (such as atom 4)will create a double defect, such as in Fig. 3(e), and istherefore likely to be energetically unfavorable. Gener-ally, only turns on atoms of the same parity as the defect
54 6 731 2 54 6 731 2 54 6 731 2 (a)(b)(c)
FIG. 4. Illustration of the motion of an overcoordination de-fect by bond-switching along a linear chain (from atom 6 in(a) to atom 4 in (b)) or making a turn (from atom 4 in (b) toatom 2 in (c)). center are allowed. Note that the distance between thenearest singly over- and under-coordinated atoms, alonga chain, should be always an even number of bonds, ifthe malcoordination is along the very same chain. These“opposite” defects can be brought together and mutuallyannihilated. Conversely, one observes that two oppositedefects separated by an odd number of bonds can notannihilate. A pair of such defects can be said to be topo-logically stable, see Fig. 5. In this Figure, we also illus-trate that this pair of defects is mobile, subject to the“turning” rule above, of course. In addition, the specificdefect-pair configuration in Fig. 5 is special because it al-lows for defect motion without the creation of Π-shapedbond motifs, in which three bonds constitute three sidesof a rectangle. These motifs are somewhat energeticallyunfavorable, see the Appendix. Finally note that oneof the nearest neighbors of the four-coordinated atom inFig. 5(c) is not covalently bonded to any other atoms, im-plying the orientation of its bond with the central atomis relatively flexible. We will see below that owing tothis extra flexibility and the said topological stabilityof the defect pair, the central atom is particularly un-stable toward sp hybridization when positively charged.Nevertheless, since each of the individual defects can beremoved, such sp hybridized units do not represent adistinct type of defect. Conversely, the formation of sp hybridized atoms in ppσ -networks is reversible .We thus conclude that singly-malcoordinated atomscomprise the only distinct type of defect in the par-ent structure. The number of the motifs in the ac-tual deformed structure that originate from the coordi-nation defects in the parent structure is determined bythe corresponding free energy cost. This cost consistsof the energy cost proper of single-malcoordination de-fects, the energy of their interaction, and a favorable en-tropic contribution reflecting the multiplicity of distinctdefect configurations. With regard to the interaction,it is easy to see that like defects should repel and de- (a)(c)(b) FIG. 5. A specific example of the motion of a topologicallystable pair of an over- and under-coordinated defect, wherebythe two defects move to the right while switching chains. fects of the opposite kind should attract. Indeed, thehigher the over (under) coordination is, the larger theenergy cost; conversely, mutual annihilation of a singlyover- and under-coordinate defects is energetically favor-able. One may thus conclude that for both energeticand entropic reasons, there are no bound states of single-malcoordination defects to speak of that could be clas-sified as a distinct type of defect. The only such boundstate is a topologically-stable pair of a singly over- andunder-coordinated vertex, as in Fig. 5.We have pointed out earlier that defectless parentstructures can be formally represented as a subset ofconfigurations of the 64-vertex model, which is the 3Dgeneralization of the 6-vertex model of ice and 8-vertexmodel of anti-ferroelectrics. The presence of malcoordi-nation can still be implemented in the 64-vertex model,by assigning an appropriate energy cost to such coordi-nation. However, the presence of ppπ -interaction impliesnon-adjacent bonds interact directly. One concludes thatparent structures of pertinence to ppσ -networks are sub-ject to a more general class of models, such as the Isingmodel with next nearest interactions. In concluding the discussion of defect classification, weemphasize that stoichiometry-based vacancies in parentstructures should be regarded not as defects, but as anintrinsic part of the parent structure. As in the exam-ple of the parent Pn Ch structure, such stoichiometry-based vacancies contribute to driving the distortion ofthe original simple cubic lattice and are minimized in thedeformed structure, subject to competing interactions.Formally, this means the energy cost of local negativedensity fluctuations in the deformed structure that stemfrom the vacancies in the parent structure is a pertur-bation to the ppσ bonding, similarly to the sp -mixing.Elementary estimates of the energy cost of a vacancy ina glass show the cost is prohibitively high, dictatingthat an equilibrium melt above the glass transition willcontain negligibly few voids of atomic size or larger. Sim-ilarly, homopolar bonds are not regarded as defects fromthe viewpoint of the present structural model. In fact, FIG. 6. Central part of neutral (AsH ) chain, whoseground state contain a coordination defect and the associatedmidgap level. The calculations are performed on semiempir-ical level by MOPAC with PM6 parametrization and theRHF/ROHF method, see Ref. for full computational detailand Supplementary Material for comparison between theresults of optimization for several chain lengths. we have argued a thermodynamic quantity of homopolarbonds in parent structures are required for the presenceof transport in supercooled melts. These notions are con-sistent with the relatively low electronegativity variationsin ppσ -bonded materials and hence a weak applicabilityof Pauling’s rules. To avoid possible confusion, we re-iterate that the present argument strictly applies only toequilibrium melts or to glass obtained by quenching suchmelts. Amorphous films made by deposition generally donot correspond to an equilibrated structure at any tem-perature and, presumably, could host a greater variety ofstructural motifs.
IV. THE INTRINSIC MIDGAP ELECTRONICSTATES
To characterize the motifs arising in the deformedstructure as a result of one singly undercoordinated atomin a parent structure, we consider first an isolated chainof ppσ -bonded arsenic atoms passivated by hydrogens:(AsH ) n . For concreteness, the chain is oriented alongthe z axis. An infinite undistorted chain is Peierls-unstable. The two ground state configurations of an in-finite (AsH ) n chain consist each of a perfect alterna-tion pattern of covalent and secondary bonds. A goodapproximation for these ground states can be obtainedusing a finite linear chain containing an even number ofmonomers, see the preceding article. Even if the chain is embedded in a lattice, each of thesetwo ground states can still be regarded as the result of aPeierls transition, so long as the deviation from the strictoctahedral coordination is weak. It is shown in Ref. that the ppσ -chain portion of the full one-particle wave-function is an eigenfunction of the effective Hamiltonian: e H = H ppσ + V † ( E − H env ) − V (7)with the same eigenvalue E , where H ppσ contains ex-clusively the p orbitals in question, H env the rest of theorbitals, and V the transfer integrals between the two setsof orbitals. Using Eq. (7), the effect of the environmentcan be presented as an (energy-dependent) renormaliza-tion of the on-site energies and transfer integrals of the ppσ -network proper, see Appendix C of Ref. The most − − − − − − − E , eVeigenstate bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bCbCXX X X X X XXXX XX X X X X X X X X X X X X X X X X X X X X XXXXXXXXXXX bCX FIG. 7. Electronic energy levels of the defected chain(AsH ) chain from Fig. 6: full MO calculation (crosses) vs.one-orbital model with renormalized ppσ integrals (circles).States below the gap are filled; the midgap state is half-filled. significant contributor to this perturbation is the com-peting sp interactions. The perturbation resulting fromthese competing interactions turns out to be sufficientlyweak, as could be inferred from (a) its magnitude beingclose to the corresponding estimate using a perturbationseries in the lowest non-trivial order; and (b) the bondmorphology in the chain conforming to what is expectedfrom a ppσ -network. Now, in contrast to an even-numbered chain, theground state of a (4 n + 1)-membered open chain can bethought of as having one singly undercoordinated atom,see Fig. 6, while the ground state of a (4 n + 3)-memberedopen chain contains one singly overcoordinated atom. Ei-ther of these “defected” chains is a neutral radical withan unpaired spin. The spectrum of the defected chain,shown with crosses in Fig. 7, exhibits a singly-populatedstate exactly in the middle of the forbidden gap. Theelectronic wave function of this midgap state, shown inFig. 8, is centered on the undercoordinated atom and ex-hibits a significant degree of delocalization. Likewise, thedeviation from the perfect bond alternation, although thestrongest in the middle of the chain, is delocalized overseveral atoms. The partitioning of the electronic den-sity between the p z orbitals proper (73%), the arsenics’ s -orbitals (21%), and the rest of the orbitals (6%) in-dicate that the interactions that compete with the ppσ -bonding proper are significant but, nevertheless, may beregarded as a perturbation to the latter bonding, simi-larly to the perfectly dimerized chain. Note that the po-sition of the nodes of the wave-function is easy to under-stand using elementary H¨uckel considerations: Consideran odd-numbered chain of identical atoms, each hostingan odd number of orbitals. A moment thought showsthat the resulting set of orbitals (in one electron approx-imation) has a half-filled non-bonding orbital, wherebythe wave-function has nodes on every second atom.The delocalization of the malcoordination is manifestin the broad sigmoidal profile of the ppσ transfer integralas a function of the coordinate, see Fig. 9. Because themalcoordination is distributed among a large number ofbonds in the deformed structure, the resulting structuralsignature of such a defect is far from obvious. In fact,the magnitude of the transfer integrals for the bonds ad- . . . site | ψ | bC bC bC bC bC bC bC bC bCuT uT uT uT uT uT uT uTX X X X X X X X FIG. 8. The wave function squared of the midgap state ofthe (AsH ) chain: the circles correspond to the arsenics’ p z atomic orbitals (AO) (total contribution 73%), triangles As s -AO’s (21%), crosses – the rest of AO’s (6%). All contributionsare nearly zero on every other atom; these very small valuesare omitted for clarity. The solid line corresponds to Eq. (10). jacent to the overcoordinated atom in a (4 n + 3) chainis very similar to that for the undercoordinated atomin a (4 n + 1) chain. This magnitude is approximatelythe average of the magnitude of the transfer integral forthe covalent and secondary bond in a perfectly dimerizedchain, see Fig. 9. The same comment applies to the cor-responding bond lengths in the first place, see Fig. 11, inview of the direct relationship between d and t . As a re-sult, the malcoordination is essentially fully “absorbed”by the chain. For instance, the distances between atoms1 and 19 in 19- and 21-member chains are 48.71 and48.70 ˚A respectively. These notions emphasize yet an-other time the ambiguity of defining coordination or coor-dination defects in actual, deformed structures. Despitebeing relatively extended, the undercoordination defectcould be thought of as separating two alternative stateswith perfect dimerization. This defect exhibits the re-verse charge-spin relation, because it is neutral when itsspin is equal 1/2. Adding or removing an electron wouldyield a singly-charged defect with spin 0. These char-acteristics of the midgap state in the (AsH ) n chain areentirely analogous to those of the midgap state in trans-polyacetylene, which is a classic example of a Peierls insu-lator. Until further notice, we will consider only neutralchains. Unlike the perfectly dimerized chain, a geometryoptimized defected chain is somewhat curved on average;this effect is small, however.The surprisingly large spatial extent of the midgapstate in the arsenic chain results from an interplay ofseveral factors, similarly to how the spatial character-istics of the solitonic state in polyacetylene depends onthe lattice stiffness, electronic interactions etc. Never-theless, it turns out that in both systems the extent ofdelocalization is determined essentially by only two pa-rameters, namely the renormalized transfer integrals ofthe covalent and secondary bond in the perfectly dimer-ized chain, denoted with t and t ′ respectively. (In thecase of polyacetylene, the transfer integrals are ppπ , ofcourse.) This simplification takes place because, similarlyto the case of a perfectly dimerized chain, the effects of the competing intra-chain and external interactions onthe midgap state largely reduce to the renormalizationof the ppσ interaction proper. To demonstrate this no-tion, we take the geometrically-optimized chain, extractonly the ppσ integrals, and renormalize them accordingto Eq. (7) while setting E equal exactly to the center ofthe forbidden gap. The spectrum of the resulting Hamil-tonian is shown with circles in Fig. 7. Setting E at thecenter of the gap should result in an error the greater,the further the state in question from the gap center.Yet one infers from Fig. 7 that the error in the spectrumis relatively small, consistent with the smallness of theperturbation.The spatial dependence of the renormalized ppσ in-tegrals is shown in Fig. 9 with circles. In the con-tinuum limit of the Su-Schreiffer-Heeger (SSH) modelfor midgap states in trans-polyacetylene, the trans-fer integral depends on the coordinate according to theexpression : t n = t + t ′ t − t ′ − n tanh (cid:18) n − n ξ s /d (cid:19) , (8)where the soliton is centered on bond n and its half-width ξ s is given by: ξ s = t + t ′ t − t ′ d, (9)where d is the length of the bond in the parent structure.Using the renormalized values of the parameters t and t ′ from Ref. , we obtain the solitonic profile shown by thesolid line in Fig. 9. Note the difference between the twosets of transfer integrals is small. Finally, the electronicwavefunction in the same continuum limit is given by thefunction: ψ n = (1 / d/ξ s ) cosh − [( n − n ) d/ξ s ] , (10)as shown with the solid lines in Fig. 8. Again, the agree-ment with the result of an electronic structure calculationusing renormalized transfer integrals is good. The agree-ment is not expected to be perfect, however, because ofthe approximations inherent in the continuum limit ofthe SSH model.We thus observe that to describe both the ground stateand defected configurations of linear chains from a ppσ network, we can apply a Hamiltonian that contains ex-plicitly only the ppσ interactions whose parameters arerenormalized by other interactions. This resulting tight-binding Hamiltonian, which corresponds with the renor-malized Hamiltonian (7), reads as: H el = X n,s h ( − n ǫ n c † n,s c n,s − t n,n +1 ( c † n,s c n +1 ,s + H.C. ) i , (11)where c † n,s ( c n,s ) creates (annihilates) an electron withspin s in the p z -orbital on atom n . The on-site en-ergy, which reflects the local electronegativity, is denoted site t , eV bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bCX X X X X X X X X X X X X X X X X X X X FIG. 9. Spatial dependence of the ppσ transfer integral in the(AsH ) chain. The actual transfer integrals from H ppσ inEq. 7 are shown with crosses. The renormalized integrals from e H in Eq. 7 are shown with circles. The solid line correspondsto Eq. (8). with ( − n ǫ n . The renormalized electron transfer inte-gral t n,n +1 between sites n and n +1 depends on the bondlength d n,n +1 . The latter is determined by the restoringforce of the lattice, in addition to the Peierls symmetrybreaking force stemming from the electronic degree offreedom in Eq. (11). The restoring force of the latticeincludes both the intra- and extra-chain perturbationsto the ppσ interaction. We have seen in Ref. that aportion of the intra-chain perturbation, namely the sp -mixing also contributes to the symmetry-breaking thatresults in bond-dimerization of the chain. Yet accord-ing to Eqs. (9) and (10), the participating competinginteractions determine the spatial characteristics of thecoordination defect largely through the values of only twoparameters: t and t ′ , in addition to the lattice spacing d .We are now ready to argue that the defect states cen-tered on singly malcoordinated atoms in ppσ -chains, as inFigs. 7-9 should be identified with the intrinsic states pro-posed in Ref. On the one hand, ppσ -bonded glasses meetthe sufficient conditions for the presence of the intrinsicstates, as argued in the preceding article. On the otherhand, we have seen here a singly malcoordinated atomis the only type of defect present in ppσ -bonded glasses,while all of its characteristics are identical to those estab-lished independently for the intrinsic states, including thereverse charge-spin relation and the relative delocaliza-tion. In addition, both types of states emerge at the co-existence region of two distinct lowered-symmetry states,which originated from a higher symmetry state. In therest of this Section, we discuss the microscopic insightsarising from both the common aspects and complemen-tarity of the semi-phenomenological approach from Ref. (case 1) and the present, molecular model for the midgapstates in ppσ -bonded glasses (case 2).Let us begin with the electronic Hamiltonian gov-erning the formation of the midgap states. In bothcases, the electronic Hamiltonians are quasi-one dimen-sional, whereby the sites are situated along a deformedline in space. Although the Hamiltonians are identicalnotation-wise, they are distinct microscopically: In case1, Eq. (6), the sites refer to rigid molecular units, often called “beads.” By construction, the beads are not signif-icantly disturbed by liquid motions and, conversely, inter-act only weakly with each other, comparably to the vander Waals coupling. Beads usually contain several atomseach; they are essentially chemically equivalent and fillout the space uniformly. The volumetric size of the beadis denoted with a . In contrast, in case 2, Eq. (11) thesites are actual atomic orbitals which are generally notchemically equivalent. To check whether the two Hamil-tonians are consistent, we recall the RFOT theory’s pre-diction that the smallest cooperativity size ξ is just above2 a , implying a supercooled melt can host at most onemalcoordinated atom per ∼ T cr . The volumetricsize a of the bead can be estimated based on the RFOT’sprediction that the configurational entropy at the glasstransition on one hour time scale is universally 0.8 perbead. For As Se , this estimate yields about a half ofthe As Se unit per bead, i.e. roughly one bead per ar-senic atom, and a ≃ . An inspection of the parentstructure of As Se in Fig. 8 of Ref. demonstrates thatit is possible to accommodate one malcoordination de-fect per 4 arsenics so that the defects are separated by atleast two atoms. Since the defects are not immediatelyadjacent in space, the geometric optimization of the par-ent structure will not necessarily lead to their mutualannihilation.Next, we address the width of the interface and therelated restriction on the spatial variation of the elec-tronic transfer integral. In Ref. , we used the simplestspatial profile of bead displacement during a cooperativerearrangement to suggest that the soliton half-width ξ s is bounded from below by ξ/
2, which is about 3 a at T g .Inspection of Fig. 9 indicates that the soliton, thoughstill extended, is somewhat more narrow, i.e. ξ s ≃ a .(Recall also that d < a .) Using the condition ξ s < t + t ′ t − t ′ a, (12)we then should revise the lower limit on the variation ofthe transfer matrix element to: t ′ /t > ∼ . . (13)We have seen earlier that the ppσ bonded materials inquestion do indeed meet this revised condition very well.Finally we note that the prediction of Ref. that the ex-tent of the wave function somewhat exceeds 2 ξ s is indeedborne out by the result in Fig. 8.The RFOT theory makes specific predictions as to thefree energy cost of a structural reconfiguration. For in-stance, the total mismatch penalty near the glass transi-tion, according to the discussion of Fig. 1, is four timesthe typical barrier for the reconfiguration. At the glasstransition, this corresponds to about 140 k B T g , or gener-ically about 3 to 4 eV. For the present microscopic pic-ture to be valid, this energy should exceed the energycost of the structural defect, which includes the energy0 FIG. 10. Central portion of a geometrically optimized doublechain of hydrogen-passivated arsenics. The top and bottomchains, containing 19 and 17 arsenics, host an over- and under-coordinated arsenic respectively, c.f. Fig. 6. The numbersdenote the bond lengths in Angstroms. of the associated lattice deformation and the electronicenergy proper. We have established earlier that the elec-tronic energy associated with the intrinsic state can beestimated via the optical gap E gap and the effective at-traction U eff between like particles occupying the state: ∼ ( E gap −| U eff | ) ≃ . E gap . This figure is about one quar-ter of the full interface energy predicted by the RFOTtheory. Alternatively, according to the solution of thecontinuum limit of the SSH model, the cost of the intrin-sic state on an isolated defect can be expressed throughits width ξ s : 4 π t ξ s /a , (14)yielding a similar figure to the above estimate. One ob-serves that, indeed, the present microscopic picture isinternally consistent. There yet is another potential elec-tronic contribution to the defect energy. We have alreadymentioned that the creation or motion of a defect may byaccompanied by a creation of Π-like patterns of bonds inthe parent structure; the latter are analyzed in the Ap-pendix.A chain hosting a singly under- and over-coordinateddefect can always lower its energy via mutual annihi-lation of the defects, consistent with conclusions of thecontinuum approach of Ref. Two such defects on differ-ent chains, however, may be topologically stable againstsuch annihilation, as in Fig. 5(c). Such a configurationpresents a felicitous opportunity to examine the interac-tion of the defects in the deformed structure, for instance,by geometrically optimizing a double (4 n + 1) − (4 n + 3)chain of hydrogen-passivated arsenics, see Fig. 10. Oneobserves that in the ground state of a resulting defectpair, the under- (over-) coordinated atom is negatively(positively) charged, so as to obey the octet rule, see alsoSection V. The resulting dipole moment of the chain is3 . The As-As angles at the overcoordinated ar-senic are 110, 110, and 128 degrees. Note in perfect tetra-hedral coordination, these angles would be 109.5 degrees.In addition, the defects become more localized when inclose proximity, as shown in Fig. 11, because of the par-tial cancellation of the malcoordination. Alternatively, − − − − − − − − − d ,˚Asite b b b b b b b bb b b b b b b b bbC bC bC bC bC bC bC bCbC bC bC bC bC bC bC bC bCbC bC bC bC b b b b FIG. 11. Site dependence of the As-As bond length in sin-gle and double chains of hydrogen-passivated arsenics, aslabeled by “1-chain” and “2-chain” respectively. The “1-chain, under” corresponds to a (AsH ) chain similar to the(AsH ) chain from Fig. 6 and “1-chain, over” correspondsto a (AsH ) chain. The double-chain entries pertain to theindividual chains in Fig. 10, “over” and “under” correspondto the top and bottom chain respectively. one may say the energy cost of this localization partiallyoffsets the stabilization resulting from the binding of thedefects.The extended nature of the coordination defects is con-sistent with their mobility: In the continuum limit, thepinning of a soliton is in fact strictly zero. Consistentwith the latter notions, the energy gradients during ge-ometrical optimization of the (AsH ) n chain above havevery small numerical values. Notwithstanding this com-plication, an estimate for the barrier for soliton hoppingevent (by two bond lengths) can be made within theSSH model: Using the effective transfer integrals fromEqs. (8) and (9) we estimate the energy of the transi-tion state configuration (corresponding to a move by onebond length) at approximately 0.1 meV, which is about10 − E gap . ppπ interactions may also contribute to pin-ning, see the Appendix for quantitative estimates. Whilethe above notions may apply relatively directly to neutraldefects even in actual 3D lattices, the situation appearsto be more complicated for charged solitons (which in factgreatly outnumber the neutral ones ) because of latticepolarization. This is work in progress. Finally, note thehigh mobility of the soliton seems particularly importantin the context of cryogenic anomalies of glasses, whichhave been argued to arise from the low-barrier subset ofthe structural reconfigurations. As an additional dividend of the present discussion,let us see that the presence of an electronic - and hence T -independent - contribution to the RFOT’s mismatchpenalty between distinct low free energy aperiodic con-figurations allows one to partially explain the deviationof chalcogenides from the RFOT-predicted correlationbetween the thermodynamics and kinetics of the glasstransition. Indeed, because of this T -independent con-tribution we expect corrections to the detailed tempera-ture dependence of the reconfiguration barrier F ‡ fromEq. (2) and the value of the so called fragility coeffi-cient m = (1 /T ) ∂ log( τ ) /∂ (1 /T ) | T = T g . The RFOT1theory predicts that this coefficient should be equal towhat one may call a “thermodynamically” determinedfragility m thermo = 34 . × ∆ c p ( T g ) where ∆ c p ( T g ) is theheat capacity jump at the glass transition per bead . Using the measured ∆ c p ( T g ) per mole , and the beadsize determination from the fusion enthalpy produces ex-cellent agreement for dozens of substances, howeveramong the outliers are the chalcogenides. For instance,for As Se , the measured m ≃
40, while the theoret-ically computed m thermo ≃ .
5. Determination of thebead size is ambiguous in As Se , because the corre-sponding crystal is highly anisotropic. In fact, in thefusion-entropy based estimate, the bead comes out tobe a single atom, in conflict with the aforementionedrequirement of chemical homogeneity and with Ref. ,where a bead was argued to contain at least two atoms.Using the RFOT’s prediction that s c ( T g ) ≃ . k B perbead, gives a better value m thermo ≃ still not enoughto account for the disagreement. Simple algebra showsthat in the presence of a T -independent contribution tothe surface tension, the estimate of the thermodynamicfragility from Ref. should be increased by a factor of[1 + 2( δ Σ)( T g − T K ) /T K ], where δ Σ is the relative contri-bution of the electronic states to the total surface energyat T g . Assuming the mentioned δ Σ ≃ .
25 and using pa-rameters for As Se from Ref. , we get an increase by afactor of 1 .
45, which is a considerable correction in theright direction. Besides the experimental uncertainty indetermination of T K , we note also that more than onepair of solitonic states might accompany the transition.This would significantly increase the corresponding con-tribution to the surface tension and the fragility.We note that since ( T g − T K ) /T K ∝ / ∆ c p , this T -independent correction to the surface tension would beless significant for fragile substances. (( T g − T K ) /T K < ∼ s c is the lead-ing contribution to the viscous slowdown in a super-cooled melt, except in strong liquids whose slowdown isnearly Arrhenius-like. Incidentally, the strongest liquidsfrom the survey by Stevenson and Wolynes, , namelyGeO , and BeF , and ZnCl , do conform to the relation m thermo = 34 . × ∆ c p ( T g ). This is consistent with thepresent results, since we do not expect solitonic states inthese wide-gap materials. V. THE CHARGE ON THE COORDINATIONDEFECTS AND THE RELATION TO EARLIERDEFECT THEORIES
We have argued previously that the intrinsic midgapstates should be typically charged: A half-filled defect isessentially a neutral molecule embedded in a dielectricmedium and should be generally stabilized by addinga charge, because of ensuing polarization. To analyzedistinct charge states of the coordination defects in thepresent approach, we will use the same methodology Lewis octet −4 hypervalenthypervalent pnictogen−like chalcogen−liketetrahedral halogen−like Lewis octet Pn Ch Ch Pn Pn +4 Ch +3 coord’ncharge positive negativeoverunder Ch −3 Pn FIG. 12. A compilation of the possible charged statesof singly malcoordinated atoms. Here, “Ch”=chalcogen,“Pn”=pnictogen. Neutral states, not shown, imply danglingbonds and would be energetically costly. as in Sections III and IV above: We will consider dis-tinct charge states on the defects in the parent struc-ture, with the usual understanding that the correspond-ing motif in the actual, deformed structure will be signif-icantly delocalized. It is during the analysis of chargeddefects in the parent structure that we will be able tomake a connection between the present approach andthe much earlier specific proposals on the defect statesin chalcogenides.
The large spatial extent of the malcoordination-basedstates is a result of the lattice’s attempt to mitigate theenergy cost of what would have been a local defect in theparent structure. The converse is also true: Accordingto Eq. (14), the energy cost of localizing malcoordina-tion would well exceed the energy density typical of anequilibrium melt. Thus in the analogies below betweenthe present and earlier approaches, one should keep inmind that on the one hand, the earlier proposals ascrib-ing midgap states to specific local defects can be thoughtof as an ultralocal limit of the present theory. On theother hand, we see that this ultralocal limit is somewhatartificial in that it greatly overestimates the energy costof the defects.Because there is only one basic type of the defect,namely a singly under- or over-coordinated atom, andthree distinct charges: -1, 0, 1, the possible combina-tions of these variables are only few, see Fig. 12. Alreadyan elementary analysis yields that charged states are ex-pected to be stabilized, consistent with independent ar-guments from Ref. Indeed, four of the resulting configu-rations, namely those listed in the “Lewis octet” sectors,satisfy the octet rules and thus are expected to be partic-ularly stable. Specifically, a positively charged overcoor-dinated chalcogen is chemically equivalent to a pnictogenand a negatively charged undercoordinated pnictogen is2equivalent to a chalcogen. These two configurations areexpected to stabilize the distorted octahedral arrange-ment of the ambient lattice. On the other hand, the Pn +4 configuration is unstable toward tetrahedral order, c.f.Fig. 5(c) and Fig. 10, and as such, would tend to frus-trate the ppσ -network. The situation with the entries inthe “hypervalent” and “hypovalent” sectors appears lesscertain. First off, we have used the label “hypervalent” inthe right bottom sector in reference to the formal electroncount around the atom exceeding 8 and the low magni-tude of electronegativity variation in the compounds inquestion. The T-shaped bond geometry in the Ch − casewas chosen by analogy with small molecules with thesame electron count, such as ClF . If the T-shaped bondgeometry is indeed favored for this type of defect, therewill be an additional penalty for a negatively chargeddefect to make a turn at a chalcogen. Now, the left up-per section represents a hypovalent configuration. In theworst case, these hypovalent configurations would favora triangular bipyramidal arrangement, in which there islinear alignment for at least two bonds. We have notlisted neutral defects in the table. These would amountto having dangling bonds and thus are deemed energeti-cally costly. Finally note that pairs of oppositely chargeddefects with complementary coordinations will either an-nihilate or be particularly stable, if their annihilation istopologically forbidden. This attraction is indicated bythe double-ended arrows in Fig. 12. A specific realizationof an attractive, topologically stable pair of defects in a deformed structure is shown in Fig. 10. This configura-tion formally corresponds to a bound Pn − -Pn +4 pair.Several specific configurations from Fig. 12, includ-ing Ch +3 , Ch − , Ch +1 , and Pn +4 , have been proposedas relevant defect species earlier, based on the puta-tive presence of dangling bonds vector-type charge-density waves, and molecular dynamics studies usingenergies determined by tight-binding methods. Specif-ically in the venerable approach of Kastner et al. ,pairs of defects, called valence-alternation pairs (VAP),could spontaneously arise in glass because of a pre-sumed instability of a pair of dangling bonds towardcreation of two charged defects: 2Ch → Ch +3 +Ch − or2Pn → Pn +4 +Ch − . Despite common features with theVAP theory, the present picture is distinct in severalkey aspects, in addition to the aforementioned ones. Wehave seen in Section III the coordination defects are anecessary prerequisite for molecular transport and arethemselves mobile. In other words, (the delocalized ver-sions of) the specific configurations from Fig. 12 are sim-ply transient configurations arising during motions of thestructural interfaces in a quenched melt or frozen glass. VI. CONCLUDING REMARKS
We have proposed a structural model, by which thestructure and electronic anomalies of an important classof vitreous alloys emerge in a self-consistent fashion. The model allows one to reconcile two seemingly conflictingcharacteristics of quenched melts and frozen glasses. Onthe one hand, these materials exhibit remarkable thermo-dynamic and mechanical stability, only slightly inferiorto the corresponding crystals. On the other hand, thesematerials are also multiply degenerate thus allowing formolecular transport. The stability of the lattice stemsfrom its close relationship with a highly symmetric, fullybonded structure. The lattice’s aperiodicity and the nec-essary presence of a thermodynamic number of transitionstates for structural reconfigurations dictate that the lat-tice exhibit a thermodynamic number of special struc-tural motifs, roughly one per several hundred atoms at T g . These motifs host midgap electronic states that aresimilar to solitonic states in trans-polyacetylene. Themotifs correspond to coordination defects in the parent structure, which can be defined entirely unambigously. Incontrast, in the actual materials, coordination is poorlydefined because the interatomic distances and bond an-gles are continuously distributed. The presence and thecharacteristics of these defects have been established us-ing an exhaustive, systematic protocol, implying the clas-sification of the defects is complete.The actual molecular mechanism of both the stabil-ity of semiconductor glasses and the mobility of the de-fected configurations relies on the very special propertyof chalcogen- and pnictogen- containing semiconductors,namely their fully developed ppσ -bond network. Hereby,most atoms exhibit a distorted octahedral coordination.In each pair of collinear bonds on such an atom, oneis fully covalent and the other is weaker, but still di-rectional. The two bonds are intrinsically related be-cause they exchange electron density, similarly to thedonor-acceptor interactions. During defect transport, thestronger and weaker bond exchange identities, resultingin bond switching not unlike the Grotthuss mechanism ofbond switching in water. The electronic states residingon these mobile coordination defects are thus an intrinsicfeature of transport in ppσ -bonded melts and glasses. Incontrast, bond switching in known tetrahedrally bondedsemiconductors does not occur because the bonds in thesematerials are homogeneously saturated throughout, ex-cluding the possibility of intrinsic states in these mate-rials, consistent with our earlier conclusions. Note thatall types of glassformers host the high strain interfacialregions that separate low free energy aperiodic config-urations and are intrinsic to activated transport. Yetglassformers exhibiting distorted octahedral coordinationappear to be unique in their ability to host topological electronic states. The corresponding glasses are thus ex-pected to exhibit anomalies that we have associated withthese electronic states, including light-induced electronspin resonance. This expectation is consistent with thevariation of the magnitude of light-induced anomaliesacross the Ge x Se − x series for 1 / < x < /
2, whichdisplay coordination ranging from tetrahedral (smaller x )to distorted octahedral (larger x ). In this series, the octa-hedral ordering seems to correlate with the light induced3ESR and vice versa for the tetrahedral bonding. The coordination defects exhibit topological features,such as stability against annihilation, if there is a mis-alignment in the motion of two defects. This peculiarityis related to another topological feature of these states,which is especially transparent in the continuum limit ofEq. (6): H = − ivσ ∂ x + ∆( x ) σ + ǫ ( x ) σ . Here, σ i arethe Pauli matrices, while − iv , ǫ ( x ), and ∆( x ) correspond,respectively, to the kinetic energy, local one-particle gapand variation in electronegativity. The local gap ∆( x ) isspace dependent and, in fact, switches sign at the defect,thus corresponding to a rotation of a vector (∆ , ǫ ). Theorientation angle of this vector is the topological phaseassociated with the defect. The phases of defects trav-eling along different paths can not cancel, resulting in astability against annihilation.Finally, what are the implications of the present re-sults for direct molecular modeling of amorphous chalco-genides and pnictides? Parent structures differing onlyby defect locations can be used as the initial configu-rations for MD simulations and may, conceivably, helpfind transition state configurations corresponding to re-alistic quenching rates. Yet even assuming the inter-actions can be estimated accurately and efficiently, theproblem is still very computationally expensive, as high-lighted by the present work: The number of defectlessparent structures, although sub-thermodynamic, is stillvery very large. We have seen it is plausible that thesestructures comprise the elusive Kauzmann state for thesesystems. Acknowledgments : The authors thank David M. Hoff-man, Thomas A. Albright, Peter G. Wolynes, and theanonymous Reviewer for helpful suggestions. We grate-fully acknowledge the Arnold and Mabel Beckman Foun-dation Beckman Young Investigator Award, the Donorsof the American Chemical Society Petroleum ResearchFund, and NSF grant CHE-0956127 for partial supportof this research.
Appendix A: The ppπ -interaction
When the coordination is exactly octahedral, Π-shapedbonds patterns, such as those formed by atoms 2-1-6-7and 4-3-8-9 in Fig. 13(b), give rise to an additional energycost stemming from the ppπ -interaction, as explained inthe following. It is partially owing to this cost, for in-stance, that the layers of AsSe pyramids in the actualstructure of As Se are so greatly puckered, see Figs. 7-9of Ref. To elucidate the origin and estimate the strengthof the ppπ -interaction, we consider for concreteness theconfigurations shown in Fig. 13, periodically continuedalong the double chain. Only atomic p -orbitals lyingin the plane of the figure are considered; these orbitalshost two electrons per atom. The nonzero entries of theHamiltonian matrix are indicated in Fig. 13(a); the nota- tions are from Ref. The total energy of the configurationin Fig. 13(a) can be expanded in a series in terms of theratios t ′ σ /t σ and t π /t σ . In the second order in the expan-sion, the total energy is the sum of three contributions(assuming ǫ = 0): The covalent bond energy is − t σ perbond, the secondary bond energy equals − ( t ′ σ ) / t σ perbond, and the contribution due to the ppπ -interaction isgiven by − t π t σ (A1)per each vertical pair of atoms. One can similarly showthat the energy of the configuration in Fig. 13(b) has thesame contributions as in (a) except the stabilizing con-tribution from Eq. (A1), owing to the Π shaped motifs.As a result, the configuration in Fig. 13(b) is less en-ergetically favorable. According to Eq. (A1), such a Πmotif generically costs about 0 . This fact can be interpreted equally well either as stabi-lization due to the staircase motifs in configuration (a) ordestabilization due to the Π motifs in configuration (b).The presence of the ppπ -interaction generally affectsthe energetics of defect movement, as could be seen inFig. 13. The top chains in panels (a) and (b) can beobtained from each other by switching all the bonds, ascould be achieved, for instance, by moving a malcoor-dination defect along the chain. As a consequence, themotion of such a defect is subject to a uniform potential.Now, each structural transition in a supercooled melt isaccompanied by the creation of two defects of oppositecharge and malcoordination. It follows from the abovediscussion that in the worst case scenario, the cost ofseparating two defects scales linearly with the distance,i.e. as N / , where N is the size of the rearranged re-gion. The scaling of this cost is clearly subdominant tothat of the mismatch penalty term γN / from Eq. (1).Furthermore, at large enough N , it will be energeticallypreferable to remove the Π patterns separating the origi-nal pair of defects by inserting another pair of malcoordi-nation defects between them. By this mechanism, whichis not unlike the mechanism of quark confinement, thecost of the Π patterns is limited by a fixed number whichis comparable, but smaller than the typical energy of theinterface, i.e., γN / . We suggest the actual effect ofthe Π-configurations is yet smaller. Indeed, distinct liq-uid configurations should have, on the average, the samenumber of Π-configurations, implying the defect move-ments are subject to a zero bias, on average. As a result,the magnitude of the overall bias for defect separationscales at most with a square root of the distance, leadingto a correction to the energy cost of the reconfigurationthat is proportional to ( N / ) / = N / . This correc-tion is clearly inferior to the γN / term.4 (b)(a) t s ’ t p t s -e+e t p FIG. 13. Double chain interacting via ppπ -overlaps: theconfiguration (a) is lower in energy than (b) due to ppπ -interaction. ∗ [email protected] A. Zhugayevych and V. Lubchenko, “Electronic struc-ture and the glass transition in pnictide and chalco-genide semiconductor alloys. Part I: The formation ofthe ppσ -network,” (2010), submitted to
J. Chem. Phys. ,cond-mat/1006.0771. K. Shimakawa, A. Kolobov, and S. R. Elliott, Adv. Phys. , 475 (1995). M. Wuttig and N. Yamada, Nature Mat. , 824 (2007). D. Lencer, M. Salinga, B. Grabowski, T. Hickel, J. Neuge-bauer, and M. Wuttig, Nature Mat. , 972 (2008). V. Lubchenko and P. G. Wolynes, Annu. Rev. Phys. Chem. , 235 (2007). L. Berthier, G. Biroli, J.-P. Bouchaud, L. Cipelletti, D. ElMasri, D. L’Hˆote, F. Ladieu, and M. Perino, Science ,1797 (2005). C. Dalle-Ferrier, C. Thibierge, C. Alba-Simonesco,L. Berthier, G. Biroli, J.-P. Bouchaud, F. Ladieu,D. L’Hˆote, and M. Perino, Phys. Rev. E , 041510 (2007). S. Capaccioli, G. Ruocco, and F. Zamponi, J. Phys. Chem.B , 10652 (2008). A. Zhugayevych and V. Lubchenko, J. Chem. Phys. ,044508 (2010). V. Lubchenko and P. G. Wolynes, Adv. Chem. Phys. ,95 (2007). R. Evans, Adv. Phys. , 143 (1979). Y. Singh, J. P. Stoessel, and P. G. Wolynes, Phys. Rev.Lett. , 1059 (1985). V. Lubchenko and P. G. Wolynes, J. Chem. Phys. ,9088 (2003). X. Xia and P. G. Wolynes, Proc. Natl. Acad. Sci. , 2990(2000). F. A. Lindemann, Phys. Z. , 609 (1910). C. A. Angell, K. L. Ngai, G. B. McKenna, P. F. Millan,and S. W. Martin, Appl. Phys. , 3113 (2000). C. A. Angell, J. Phys.: Condens. Matter , 6463 (2000). T. R. Kirkpatrick, D. Thirumalai, and P. G. Wolynes,Phys. Rev. A , 1045 (1989). V. Lubchenko and P. G. Wolynes, J. Chem. Phys. ,2852 (2004). V. Lubchenko and P. G. Wolynes, Phys. Rev. Lett. ,195901 (2001), cond-mat/0105307. V. Lubchenko, J. Phys. Chem. B , 18779 (2006). M. Tatsumisago, B. L. Halfpap, J. L. Green, S. M. Lindsay, and C. A. Angell, Phys. Rev. Lett. , 1549 (1990). R. Richert and C. A. Angell, J. Chem. Phys. , 9016(1998). T. R. Kirkpatrick and P. G. Wolynes, Phys. Rev. B ,8552 (1987). J. Stevenson and P. G. Wolynes(2010), unpublished,arXiv:1006.5840v1. U. Tracht, M. Wilhelm, A. Heuer, H. Feng, K. Schmidt-Rohr, and H. W. Spiess, Phys. Rev. Lett. , 2727 (1998). E. V. Russel and N. E. Israeloff, Nature , 695 (2000). M. T. Cicerone and M. D. Ediger, J. Chem. Phys. ,7210 (1996). S. Ashtekar, G. Scott, J. Lyding, and M. Gruebele, J. Phys.Chem. Lett. , 1941 (2010). J. K. Burdett and T. J. McLarnan, J. Chem. Phys. ,5764 (1981). J. K. Burdett,
Chemical Bonding in Solids (Oxford Uni-versity Press, 1995). J. K. Burdett and S. Lee, J. Amer. Chem. Soc. , 1079(1983). W. Kauzmann, Chem. Rev. , 219 (1948). R. J. Baxter,
Exactly Solved Models in Statistical Mechan-ics (Academic Press, 1982). V. Lubchenko and P. G. Wolynes, “Comment on the “novelisotope effects observed in polarization echo experiments inglasses”,” (2004), cond-mat/0407581. MOPAC2009, J. J. P. Stewart, Stewart Compu-tational Chemistry, Colorado Springs, CO, USA,http://OpenMOPAC.net (2008). H. Takayama, Y. R. Lin-Liu, and K. Maki, Phys. Rev. B , 2388 (1980). J. Stevenson and P. G. Wolynes, J. Phys. Chem. B ,15093 (2005). D. Bevzenko and V. Lubchenko, J. Phys. Chem. B ,16337 (2009). M. Kastner, D. Adler, and H. Fritzsche, Phys. Rev. Lett. , 1504 (1976). R. A. Street and N. F. Mott, Phys. Rev. Lett. , 1293(1975). Y. Shimoi and H. Fukutome, J. Phys. Soc. Jap. , 2790(1990). S. I. Simdyankin, S. R. Elliott, Z. Hajnal, T. A. Niehaus,and T. Frauenheim, Phys. Rev. B , 144202 (2004). A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W. P. Su, Rev. Mod. Phys. , 781 (1988). P. S. Salmon, J. Non-Crys. Solids , 2959 (2007). F. Mollot, J. Cernogora, and C. Benoit `a la Guillaume,Phil. Mag. B , 643 (1980). See supplementary material at URL for an illustration ofthe HOMO and LUMO, the spatial variation of the elec-trostatic potential in a double chain of passivated arsenics,and comparison between the results of geometry optimiza-tion for several chain lengths of the (AsH ) n chain. Supplementary Material
FIG. 14. Local charge distribution at a topologically stable pair of an over- and under-coordinated arsenic in a double chainof hydrogen-passivated arsenics from Fig. 10 of the main text. The LUMO and HOMO are shown in the top and middlepanels respectively. The electrostatic potential is shown in the bottom panel, red and blue colors corresponding to positive andnegative values respectively. − − − − − − − − − −
10 2.52.62.72.82.93.0 d ,˚A site bC bC bC bC bC bC bC bC bC bCbC bC bC bC bC bC bC bC bC bCbC bC bC bC bC bC bC bC bC + + + + + + + + ++ + + + + + + ++ + bC bC bC FIG. 15. Site dependence of As-As bond lengths in a (AsH ) n chain for several chain lengths is shown, to partially assess thesensitivity of the calculation to the chain length, n . (See the main text for calculational details.) For the reader’s reference wenote that the energy gradients in the geometry-optimized chains were about 0.01 kcal/mol/˚A for even n , and 0.001 kcal/mol/˚Afor odd n . ewis octet −4 hypervalenthypervalent pnictogen−like chalcogen−liketetrahedral halogen−like Lewis octet Pn Ch Ch Pn Pn +4 Ch +3 coord’ncharge positive negativeoverunder Ch −3−3