Electronic structure and the glass transition in pnictide and chalcogenide semiconductor alloys. Part I: The formation of the ppσ -network
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 I: The formation of the ppσ -network.
Andriy Zhugayevych and Vassiliy Lubchenko , ∗ Department of Chemistry, University of Houston, TX 77204-5003 Department of Physics, University of Houston, TX 77204-5005
Semiconductor glasses exhibit many unique optical and electronic anomalies. We have put fortha semi-phenomenological scenario (
J. Chem. Phys. , 044508 (2010)) in which several of theseanomalies arise from deep midgap electronic states residing on high-strain regions intrinsic to theactivated transport above the glass transition. Here we demonstrate at the molecular level how thisscenario is realized in an important class of semiconductor glasses, namely chalcogen and pnictogencontaining alloys. Both the glass itself and the intrinsic electronic midgap states emerge as a result ofthe formation of a network composed of σ -bonded atomic p -orbitals that are only weakly hybridized.Despite a large number of weak bonds, these ppσ -networks are stable with respect to competing typesof bonding, while exhibiting a high degree of structural degeneracy. The stability is rationalizedwith the help of a hereby proposed structural model, by which ppσ -networks are symmetry-brokenand distorted versions of a high symmetry structure. The latter structure exhibits exact octahedralcoordination and is fully covalently-bonded. The present approach provides a microscopic route toa fully consistent description of the electronic and structural excitations in vitreous semiconductors. I. INTRODUCTION
The electronic and structural excitations in amor-phous semiconductors, and the interplay of these exci-tations, have evaded a self-consistent first-principles de-scription for decades. Amorphous semiconductors areimportant both in applications, e.g., as phase changematerials, and from the basic viewpoint. The elec-tronic structure in a disordered lattice is fundamentallydifferent from the venerable Bloch picture of continuousbands of allowed states separated by strictly forbiddengaps, as would be applicable in periodic solids. Althougha result of multiple electron scattering, the presence ofsuch forbidden gaps in periodic solids is also consistentwith their energetic stability in that such gaps usuallyimply stabilized occupied orbitals. In contrast, in a dis-ordered lattice, strict gaps in the density of states arenot `a priori allowed. Yet the electronic orbitals canstill be subdivided in two relatively distinct classes: (a) extended states, in which electrons move as well-defined wave-packets, and (b) localized states, whosedensity decays nearly exponentially away from mobil-ity bands. Despite the many successes in applying theseideas semi-phenomenologically, developing a first princi-ples description, in which a realistically stable aperiodiclattice and mobility gaps emerge self-consistently, has been difficult. Further compounding this difficulty,several electronic and optical peculiarities of disorderedsemiconductors indicate that there are effects of disor-der beyond those generically expected of a mechanically stable disordered lattice.We have argued that, instead, it is not the aperiodic-ity alone, but the intrinsic metastability of semiconductorglasses that leads to many unique properties of these dis-ordered solids. A glass is made by thermally quenching asupercooled liquid at a rate exceeding the typical liquidrelaxation rate. A supercooled liquid can be thought of as an aperiodic crystal characterized by myriad low freeenergy configurations that are nearly degenerate. Molec-ular motions in the liquid occur via local activated tran-sitions between these low free energy configurations, which are accompanied by the creation of high-strain in-terfacial regions separating the configurations. The in-terfaces are intrinsic to the activated dynamics; theirconcentration, for a given quenched glass, depends (loga-rithmically slow) only on the time scale of the glass tran-sition, i.e., the quench speed. According to the Ran-dom First Order Transition (RFOT) theory of the glasstransition, this concentration just above the glasstransition temperature on one hour scale, T g , is generi-cally about ξ − ≃ cm − , within an order of magni-tude or so, depending on the specific substance. The pa-rameter ξ denotes the cooperativity length for activatedreconfigurations.Our results indicate that the high-strain regionsthat form when the amorphous material is made mayhost deep midgap electronic states of topological ori-gin, which are centered on over- and under-coordinatedatoms. These states share several characteristics with themidgap electronic states in trans-polyacetylene, whichare centered on defects in the perfect alternation pat-tern of the double and single bonds along the polymerchains. Using a semi-phenomenological, coarse-grainedHamiltonian, we have established the spatial and chargecharacteristics of the interface-based midgap states innon-polymeric glasses. We further concluded, based onthe internal consistency of the description, that the statesshould be present only in a limited class of glasses thatsatisfy the following requirements: The bonding shouldexhibit inhomogeneous saturation so that the transfer in-tegrals t in the electronic effective tight-binding Hamil-tonian H = P i ǫ i c † i c i + P h ij i t ij c † i c j should uniformlyexhibit spatial variation. Nevertheless, the magnitude ofthe variation should be modest: | t ′ /t | > ∼ . , (1)where t and t ′ denote the upper and lower limits of thevariation range. More detailed estimates indicate thatthe lower limit on the t ′ /t ratio is probably smaller, i.e.,0.3 or so. Finally, the spatial variation δǫ in electroneg-ativity should not be too large | δǫ | < | t − t ′ | , (2)thus implying the material is a semiconductor, sincethe transfer integral t is at most a few eV. Ofnon-polymeric materials, only certain chalcogen- andpnictogen-containing glasses appear to satisfy all ofthese requirements. On the other hand, these amor-phous arsenic chalcogenides and similar materials doindeed exhibit several electronic and optical anoma-lies that could be accounted for by the interface-basedstates, in a unified fashion. These anomalies includelight-induced electron spin resonance (ESR) and midgapabsorption, two types of photoluminescence, andfield-induced ultrasonic attenuation. Thus general ar-guments, on the one hand, and observation, on the other,seem to converge on the uniqueness of chalcogenide andpnictide glasses with regard to their potential ability tohost topological midgap states. Despite this remark-able convergence, the currently available evidence for theunique interplay between electronic excitations and themetastability in those glasses must be regarded as cir-cumstantial.The purpose of the present effort is to test the conclu-sions of the semi-phenomenological analysis from Ref. directly, based on the local chemistry specific to amor-phous chalcogenide and pnictide alloys. Our basic hy-pothesis for the origin of the topological midgap statesin the semiconductor glasses is that these glasses repre-sent aperiodic networks of σ -bonded p -orbitals that areonly weakly hybridized, as in Fig. 1. These networksexhibit a relatively small number of intrinsic over- andunder-coordinated vertices. The latter are, in fact, re-sponsible for both the midgap states and the transitionstate configurations intrinsic to molecular transport inthe quenched melts. In testing this hypothesis, we facethe deeper question of the actual stability of aperiodic ppσ -bonded networks. Experiment shows (see below),that the enthalpy excess of a glass relative to the cor-responding crystal is less than the typical vibrational en-ergy , i.e., a small fraction - one percent or so - of the totalbonding energy! In other words, despite their aperiod-icity, glasses are nearly defect-free structures, consistentwith their bulk stability, both mechanical and thermo-dynamic. This observation is in conflict with a commonview of glasses as a reconstructed - but otherwise arbi-trary - array of malcoordinated configurations and otherlocal defects, such as vacancies. This common view wouldimply excess enthalpies of the order eV per several atoms,while avoiding to address the mechanism of transport inthe melt. b a d'd FIG. 1. Structure of rhombohedral arsenic as an exampleof a ppσ -bonded network. The solid lines denote regular co-valent bonds that connect the central atom with the near-est neighbors, bond length d . The dashed line denote theweaker, “secondary” bonds connecting the central atom withits next nearest neighbors, bond length d ′ . Angles β < ◦ and α = 90 ◦ reflect the deviation from ideal octahedral co-ordination. These particular crystal fragment and view arebased on Fig. 1 of Shang et al. In the present and companion article, we test thehereby proposed microscopic picture in two relativelyseparate stages. The present article is devoted to thefirst stage, in which we argue that ppσ -networks can rep-resent the quenched liquid and frozen glass forms of thesesubstances in the first place. We will make a case that (a)such networks are stable against other types of bondingin a rather large class of amorphous compounds contain-ing elements from groups 15 and 16; and (b) despite theirrelative stability, aperiodic ppσ networks are multiply de-generate, as are the actual liquids and glasses in question.The present work thus contains, to our knowledge, thefirst chemical bonding theory of a bulk glass.The article is organized as follows. In Section II, wediscuss in detail several key features of ppσ networks, in-cluding their spatial non-uniformity and a hierarchy ofbonding, from strictly covalent to weaker “secondary” toweaker yet van der Waals. Despite the presence of suchweak bonds, the ppσ networks are stable. To trace theorigin of this stability, we formulate a structural modelin Section III, by which both periodic and aperiodic ppσ -networks are symmetry-broken versions of a highly sym-metric, strongly bonded structure. This view is simi-lar but distinct from the common view of many elemen-tal solids as Peierls distorted simple-cubic lattices. Thesymmetry breaking is driven by several competing inter-actions, including in particular sp -mixing; these, never-theless, are only strong enough to perturb but not qual-itatively modify the basic ppσ bonding. We will verifythat the resulting aperiodic ppσ -bonded lattice satisfiesthe three requirements for the existence of the topolog-ical states listed above. Importantly, this lattice will beargued to exhibit multiply degenerate configurations thatdiffer by precise coordination of individual atoms, consis-tent with the possibility of activated transport in the cor-responding quenched melts. The precise degree of degen-eracy and the possibility of activated transport are bothintimately related to the questions of the concentration ofthe corresponding transition-state configurations in themelt and the accompanying electronic excitations. Thelatter questions are analyzed in the companion article. II. ppσ -BONDED SEMICONDUCTORS: THEROLE OF THE SECONDARY ppσ -INTERACTION
The goal of this Section is to provide a detailed descrip-tion of ppσ -networks in pnictides and chalcogenides, asarising from sigma-bonding between p -orbitals that areonly weakly sp –hybridized. Such a description is neces-sitated by the lack of systematic comparative studies ofthe electronic properties of chalcogenide and pnictide al-loys, even though their structure itself has received muchattention. The ppσ -bonding emerges subject to com-petition from other types of local ordering. The presenceof several competing types of local ordering in chalco-genides is evidenced by their broad range of structuraland electronic properties, as could be seen by comparinge.g. As Se and GeSe . While the former exhibits adistorted octahedral coordination and well separated s and p bands, the latter displays tetrahedral ordering andoverlapping sets of s and p orbitals. At the same time,the two substances exhibit opposite trends in terms of in-trinsic and light induced ESR response. Such si-multaneous trends are particularly vivid in the Ge x Se − x series for 1 / < x < /
2, which exhibit coordinationranging from tetrahedral (smaller x ) to distorted octa-hedral (larger x ). In this series, the octahedral orderingseems to correlate with the separation between s and p bands and light induced ESR, and anti-correlatewith the presence of unpaired spins and the glassform-ing ability. Vice versa, the tetrahedral bonding exhibitsthe opposite trend.When sp -hybridization is weak, each atom exhibitsa distorted octahedral coordination, as exemplified inFig. 1: Two or three nearest neighbors are situated at thedistance of the regular covalent bond in an almost right-angled geometry. Opposite to each of these covalentlybonded neighbors, there is an atom at a distance thatexceeds the sum of the corresponding covalent radii, butis closer than the sum of the corresponding van der Waalsradii. Crystalline
As, Se, As Se , GeSe are typical ex-amples of this type of coordination. It is appropriate tothink of crystals that exhibit distorted octahedral coor-dination not as fully covalently bonded, but as networksconsisting partially of fully covalent bonds and weaker,closed-shell interactions. In the physics literature, it iscustomary to call the stronger bonds “front bonds” andto call the weaker bonds “back bonds.” Becausethe back bonds formally correspond to closed-shell in-teractions, chemists call them “secondary” bonds, or donor-acceptor interactions, or, sometimes, hyper-valent or 3-center bonds, where the distinction is onlyquantitative, if any. Importantly, the secondary bondsare stronger than van der Waals interaction and are direc-tional , similarly to their strictly-covalent counterparts. FIG. 2. A fragment of the As Se crystal. The secondarybond between As1 and Se1 is shown with a thin solid line. Thedeviation from the strict octahedral coordination is reflectedin different values of covalent and secondary bond strengths: t ′ σ /t σ = 0 . <
1, a deviation from their strict alignment∆ β = 180 ◦ − β = 21 ◦ , and bond elongation relative to thesum of the covalent radii: δd = +0 .
07 ˚A, δd ′ = +0 . We will see that, in fact, the distinction between the sec-ondary and covalent bonds in ppσ -networks is not sharp.A common example of the coexistence of covalent andsecondary bonding is crystalline As Se , which consistsof puckered layers of AsSe pyramids, see also Fig. 7-9below, whereby the layers are only loosely bonded. Thepyramids are made of the stronger, covalent bonds, whilethe secondary ppσ -interaction accounts for the rest of theintralayer bonding, see Fig. 2. The interlayer secondarybonds are even weaker; nevertheless they have been ar-gued to be as strong as 0.3 eV in As Se , i.e. signifi-cantly stronger than a typical van der Waals-like bond.Despite thousands of documented instances of sec-ondary bonding and the similarity in its properties acrossa broad spectrum of compounds, both the mech-anism and quantitative description of this type of bond-ing still appear to be a subject of debate. Withoutclaiming full generality, here we will presume the ex-istence and universality of secondary bonds based onthose myriad documented cases. The energetic and spa-tial characteristics of these bonds will be treated empir-ically, using tight-binding (TB) formalism. Such an ap-proach is controllable insofar as the substances in ques-tion are insulators or poor conductors since under thesecircumstances, both localized, Wannier-like and the de-localized Mulliken-Hund orbitals form a complete set ofelectronic wave-functions. Consistent with the closed-shell character of the secondary bonding, the TB expres-sion for the corresponding binding energy contains ex-clusively interaction between occupied and unoccupiedorbitals of the constituent molecules. Indeed, a one-electron Hamiltonian that describes the interaction oftwo molecules A and B can be written as a block ma-trix, where the ket corresponding to the wave function is − t ss t sp t σ t ′ σ − t π ref. structureGe 1.70 2.36 2.56 negli-gible α -AsSe 1.11 2.10 3.37 0.64 0.92 trigonal-SeTABLE I. Transfer integrals (in eV) in particular crys-talline forms of several elements often present in chalcogenidealloys. All integrals are for the nearest neighbors, except t ′ σ . In t ss and t sp , the subscripts indicate the constituent or-bitals. t σ and t ′ σ denote the transfer integrals for the covalentand secondary ppσ bonds, and t π for the ppπ interaction. Agraphical summary of the transfer integral definitions is givenin Fig. 11(a), see also p. 23 of Ref. a vertical stack of the kets pertaining to molecule A andB: H = (cid:18) H A V + V H B (cid:19) , | ψ i = (cid:18) | ψ A i| ψ B i (cid:19) . (3)Matrices H A , H B are the Hamiltonians of the isolatedmolecules A and B, while V contains the correspondingtransfer integrals. According to the standard perturba-tive expression, the binding energy of two closed-shellmolecules A and B, E bind ≡ E ABtot − E Atot − E Btot reads: E bind ≈ ( A occ X n B unocc X m − A unocc X n B occ X m ) | V nm | E A n − E B m , (4)where H A | ψ A n i = E A n | ψ A n i and H B | ψ B n i = E B n | ψ B n i . La-bels “occ” (“unocc”) denote summation over occupied(unoccupied) orbitals of the molecules A and B.Now, the precise geometry of the ppσ -network is sub-ject to several competing interactions, combined with theprecise stoichiometry and other many body effects: the ppσ -interaction, sp -mixing, and ppπ -interaction, such asoften found in conjugated polymers. (See Harrison foran introduction to tight-binding methods.) All these in-teractions have comparable strength as can be inferredfrom the values of the corresponding electron transferintegrals. Table I compiles the values of these transferintegrals for important representatives from groups four-teen, fifteen, and sixteen. Elements of these groups areof particular interest in the context of amorphous semi-conductors, because of comparable electronegativity andsuitable valency, of course. The close magnitude of thelisted competing interactions implies that the local order,which in turn is strongly affected by the stoichiometry,plays the crucial role in determining which interactionwill ultimately dominate.One may list several complementary ways to establishthe presence and significance of ppσ -bonding. A ppσ -network reveals itself structurally in a weak deviationfrom the octahedral coordination. In addition to a nearlyright-angled geometry, the disparity between the lengthsof the secondary and covalent bonds should be modest.In the latter case, the ratio of the corresponding trans-fer integrals, t ′ σ /t σ , is not too small, implying a relatively uniform, stable network. Furthermore, in view of a nearlyuniversal relation − t π /t σ ≃ /
4, Ref. , a large enoughvalue of t ′ σ /t σ automatically guarantees that the effect of ppπ interactions on the geometry is small. On the otherhand, when sp mixing is weak and little ss bonding ispresent, the top of the valence band consists primarilyof p -orbitals, while the s and p subbands are relativelywell separated. Indeed, consider for the sake of argumenttwo identical centers, each having one s and p orbitaland three electrons. The p -orbitals are aligned. Withinthe second order in the sp -mixing, the one-electron ener-gies of the four resulting molecular orbitals are given by ǫ s ∓ t ss − t sp [ ǫ p − ǫ s ± ( t σ + t ss )] − for the ssσ bond and ǫ p ± t σ + t sp [ ǫ p − ǫ s ± ( t σ + t ss )] − for the ppσ bond. If the s and p orbitals are sufficiently separated in energy, it fol-lows automatically that (a) the centers are ppσ -bondedand (b) the effect of the sp -mixing on the ppσ trans-fer integral of the bond is small: t sp / p t σ ( ǫ p − ǫ s ) < S , As Se , and As Te . In Table II, we compile data on the deviation from theideal octahedral coordination and the corresponding ppσ transfer integrals, in several distinct compositions andstoichiometries characteristic of common chalcogenideand pnictide alloys. One observes that a high value of t ′ σ /t σ is indeed characteristic of ppσ -bonded materials,whereby the angular deviation from the ideal coordi-nation does not exceed 10 ◦ . Conversely, a large valueof the t ′ σ /t σ ratio, alone, is often a good predictor of ppσ -networking. Note also a subtle, but neverthelesssignificant trend that in such a network, the strongerbonds are somewhat longer than the sum of the cova-lent radii. Furthermore, this deviation is the more signif-icant, the shorter - and hence stronger - are the secondarybonds. This anti-correlation is a telltale sign of “trans-influence” in which the weaker bonded atom donateselectrons into anti-bonding orbitals of the stronger bond,a common feature with secondary and donor-acceptor in-teractions. For the trans-influence to take place, it isessential that the counterpart covalent and secondarybonds be in a near linear geometry so that the anti-bonding orbital on the stronger bond overlap significantlywith the bonding orbital on the secondary bond. Notethat tight-binding descriptions are consistent with thetrans-influence. Indeed, we show in Appendix B that thebond order for the AB dimer from Eq. (4) is approxi-mately given by the expression: b AB ≈ ( A occ X n B unocc X m + A unocc X n B occ X m ) | V nm | ( E A n − E B m ) . (5)The first (second) double sum is twice the occupationof the antibonding orbitals of molecule B (A), as do-nated by the bonding orbitals of A (B), consistent withthe donor-acceptor nature of the secondary bond. Alter- crystal and Ref. ¯ α, ◦ ǫ ,eV ∆ d ,˚A ∆ d ′ ,˚A ∆ β, ◦ t ′ σ /t σ strong ppσ -secondary bonding/hypervalencyAs Te ppσ -secondary bondingo-As Se Se – 0 0.02 1.0 10 .3Br at 250 K – 0 0.01 1.1 10 .3 α -m-Se > ∼ . < ∼ . O (As) (As) sp -bondingGeSe (Ge) > ∼ . < . ppσ -bonding. ¯ α is the average bond angle (hybridization), ǫ is thehalf-difference in absolute electronegativities, ∆ d and ∆ d ′ arethe deviations of covalent and secondary bond lengths fromthe sum of the covalent radii, ∆ β = 180 − β is the devia-tion from a linear geometry, t ′ σ /t σ is the ratio of ppσ integralsfor covalent and secondary bonds. Crystallographic abbrevi-ations: h – hexagonal, m – monoclinic, o – orthorhombic, r– rhombohedral, t – trigonal. The computational details aregiven in Appendix A. natively, according to Eq. (4), the bond order stronglycorrelates with the binding energy. Since the sum of thebond orders on a given atom equals to the atom’s valency,stronger secondary bonds imply weaker counterpart co-valent bonds.Above said, we should point out that a large valueof the t ′ σ /t σ ratio and the perfect octahedral coordina-tion, separately or together, do not guarantee that the ppσ -bonding is the main contributor to the lattice stabi-lization. An obvious counterexample is provided by ioniccompounds with the rocksalt structure, which exhibit theperfect octahedral coordination. Incidentally, using sto-ichiometry to impose a (distorted) rocksalt structure isnot guaranteed to produce a ppσ -network either. For in-stance, whereas GeSe does indeed exhibit distorted octa-hedral coordination, AsSe forms a molecular crystal com-posed of As Se units, whose symmetry is incompatiblewith uniform octahedral coordination, despite relativelystrong secondary bonding, see Table II and the Supple-mental Material. Finally, other competing types of lo-
FIG. 3. A dimer of Br molecules. The geometry is optimizedat the MP2 level, using the program Firefly with aug-cc-pVTZ basis set and RHF wave-function. The gray shapesshow the lowest energy molecular orbital consisting of thevalence p orbitals of the Br molecules (61st MO, Table V inAppendix B), see also the MO diagram in Fig. 12. The twohues reflect the sign of the wave function. The computationaldetails are provided in Appendix B. cal order are present in solids formed by the elementsfrom groups 14-16. For instance, in GeSe the coordina-tion of Ge atoms is tetrahedral, see Table II. Elemen-tal phosphorus and sulfur at ambient conditions exist asmolecular crystals made of tetraphosphorus P and octa-sulphur S , respectively, that show no signs of octahedralcoordination.Apart from the stoichiometry and peculiar types oflocal ordering, the actual degree of stabilization of the ppσ -network crucially depends on the strength of the sec-ondary bonds since they account for at least a half of thetotal bonds. We are not aware of systematic ab initio studies of the dependence of the strength of these bondson the bond length and the deviation from the preciseoctahedral coordination, in a crystal. Such studies areunderstandably difficult, as the actual chalcogenide crys-tal structures exhibiting this type of bonding are verycomplicated. For instance, the unit cell of As Se has20 atoms. Despite these complications, it is possible toobtain an accurate estimate of the strength of ppσ sec-ondary bonding in semiconductors by analyzing the sim-plest possible micro- and macro-molecular systems thatexhibit this type of bonding, i.e. dimers of diatomic halo-gen molecules and halogen crystals respectively. Specif-ically, bromine is an appropriate example, because theelements of interest are located in periods 3 through 5.The ground state geometry of the bromine dimer is shownin Fig. 3. Here the ppσ molecular orbital of the l.h.smolecule is mixed with the in-plane ppπ molecular or-bitals of the r.h.s. molecule, as can be seen directly inthe electronic density distribution in Fig. 3. Consistentwith this mixing, the strength of the ppσ secondary bondbetween the bromine molecules exceeds 0.3 eV, i.e., sig-nificantly more than expected of a typical van der Waalsbond. According to the estimates on a variety of pseudo-dimer structures by Anderson et al. , the strength of thesecondary bonding, relative to the covalent bonding, de-creases toward the r.h.s. in each period. The above figurefor the binding energy of the bromine dimer thus givesus a secure lower bound on the strength of a secondary bond l eng t h
10 GPa pressure d’d d’ d FIG. 4. Pressure dependence of the lengths of the covalent ( d )and secondary ( d ′ ) bond in rhombohedral arsenic, after Silaset al. At a critical pressure, of about 10 GPa, the latticebecomes simple cubic, while the two types of bonds becomeequivalent. In the inset, we replot the framed region with d as a function of d ′ and the pressure as a dummy parameter.The resulting dependence illustrates the trans-influence of thecovalent and secondary bonds, c.f. Figs. 1 and 2 of Landrumand Hoffmann. bond, consistent with the data in Table II. Furthermore,bromine crystals are comprised of layers in which Br units are arranged in nearly the same geometry as in theground state of an isolated pair of Br molecules, seeSupplementary Material. III. ppσ -NETWORKS AS SYMMETRY BROKENSTATES
Here we propose a specific structural model of ppσ -network formation, both periodic and aperiodic, and ar-gue that aperiodic ppσ -networks are consistent with (a)the structural degeneracy of the corresponding solid and(b) the restrictions on the magnitude of the spatial vari-ation of the electronic transfer integral from Eq. (1) de-rived in Ref. In liquids and glasses, the first coordination shell is de-termined by the stronger bonds and is very similar to thefirst coordination shell in the corresponding crystals.
Much less is known about the precise configurations ofthe weaker-bonded, next-nearest neighbors. A usefulcue is provided by the observation that the crystallinephotoemission spectra of several archetypal ppσ -bondedchalcogenides - As S , As Se , and As Te are verysimilar to their amorphous counterparts. In this and thefollowing Sections, we argue that, indeed, the local in-teractions specific to ppσ -bonded glasses are of the sameorigin as in the corresponding crystals: In both cases, thelocal structures result from a symmetry-lowering transi-tion from the perfect octahedral coordination and thusare comparably stable. The main corollary of this infer-ence is that a supercooled liquid or quenched glass canbe sufficiently stabilized by the ppσ -network alone. Thekey distinction between the crystal and glass is that ow-ing to its aperiodicity, the glass is necessarily structurally (a) (b) FIG. 5. Parent structures of the crystals of (a) elemen-tal arsenic and (b) black phosphorus, after Burdett andMcLarnan. degenerate.Let us begin with crystals. It is longappreciated that the structures of many poly-morphs of elemental pnictogens, chalcogens, andhalogens can be regarded as distorted simple-cubic (sc),with the exception of several of the lightest elements,such as nitrogen or oxygen, which form molecularcrystals. Upon increasing pressure to several tens ofGPa, the structures approach the ideal octahedral coor-dination, while phosphorus and arsenic actually exhibita continuous transition to the simple cubic structure(Refs. and references therein). Specifically in arsenic,which is rhombohedral (A7) at ambient conditions, asthe bond angles approach the right-angled geometry, theratio of bond lengths of the nearest to the next nearestneighbor grows. These two bond types correspond tothe covalent and secondary bonds respectively. In thevicinity of the transition, the covalent bond increases inlength, while the secondary bond continues to shorten, see Fig. 4. Landrum and Hoffmann provide correlationdata on thousands of pnictogen and chalcogen com-pounds that clearly demonstrate similar trans-influencebetween the covalent and secondary bonds, see insetof Fig. 4. The molecular fragments in question are ofthe type X-Q-X, where Q=Sb, Te and X = F, Cl, Br,I. Note the correlation in the inset of Fig. 4 pertains tovalencies 3 (2) for Sb (Te). The combined view of Fig. 4implies that the distinction between the covalent andsecondary bonds is not sharp, but is subject to preciselocal coordination and/or bond tension. In the strict sclimit, both the secondary and covalent bonds becomeequivalent and should be regarded as fully covalent,albeit hypervalent. Similarly in halide crystals, each ppσ -bonded layer transforms into a square lattice ata sufficiently high pressure (80 GPa for bromine ).Isovalent binary compounds A IV B VI transform into thesimple cubic structure not only upon increasing pressure,but also temperature. To rationalize these observations, we hereby pro-pose the following structural model , which draws heav-ily on Burdett and coworkers’ view of the structureof rhombohedral arsenic, black phosphorus and othercompounds.
Place the atoms at the vertices of thecubic lattice, so that each atom is exactly octahedrallycoordinated and the p orbitals are aligned with the prin- (a) (b) FIG. 6. Parent structure for (a) the GeTe crystal and (b)GeSe and GeS crystals, c.f. Fig. 5. Note these structures areby one horizontal layer taller than the repeat unit. cipal axes. All atoms are linked, the links correspondingto bonds. Remove links so that each atom obeys theoctet rule, while making sure the remaining bonds oneach vertex are at 90 degrees, not 180. This procedurecould be interpreted as adding electrons to a rocksalt-like compound while breaking bonds, whereby each filledantibonding orbital transforms into a lone pair of elec-trons pointing away from the remaining bonds. As aresult, each pnictogen and chalcogen, for instance, will bethree- and two-coordinated respectively, whereby all linkspointing from an atom are at right angles. We call theresulting lattice the “parent structure.” Third, estimatethe energy of the resulting parent structure, using a tight-binding Hamiltonian, while assuming that the transferintegrals are significant only for the linked atoms. Now,those bond-breaking patterns that have a particular lowenergy are special. One should expect that to these spe-cial structures, there correspond crystals of actual sub-stances that exhibit a distorted octahedral coordination,in which the covalent bonds will precisely correspond tothe links, while the missing links correspond to secondarybonds or weaker, van der Waals interactions. For in-stance, Burdett and McLarnan have shown there are 36inequivalent ways to arrange three-coordinated atoms onthe cubic lattice with a repeat unit of size 2 × ×
2. Twoof the structures correspond to the lattices of black phos-phorus and rhombohedral arsenic. In actual materials,both of these lattices consist of double layers that arebuckled and mutually shifted, compared with the parentsimple cubic structure. Other specific examples can befound in Refs.
It is understood that although thesimple cubic lattice is a convenient parent structure formany compounds, it is by no means unique in this re-gard. For instance, Albright et al. mention two addi-tional formal ways to obtain the arsenic structure, i.e.by adding two electrons per atom to wurzite ZnS or bypuckering graphite sheets. Yet what distinguishes the sc-like parent structure is that, like the actual material, itis ppσ -bonded, whereas the orbitals in the wurzite andgraphite structures are sp and sp hybridized respec-tively. We point out that IV-VI compounds that are iso-electronic with arsenic can be obtained from the parent
11 33 2 22A A AB3 BBB BA A12 33 31BA 22 11 abc pnictogensat positions 1,2 chalcogensat positions 1,2,3vacancy
FIG. 7. A parent structure for a Pn Ch crystal, such ascrystalline As Se and As S , c.f. Fig. 6(b). structures of arsenic or phosphorus, see Fig. 6. Finallynote that the above rules for bond placement, i.e. three-coordinated pnictogens and two-coordinated chalcogenswith right angles between bonds, can be formally re-garded as a subcase of the 64-vertex model, which isthe 3D generalization of the venerable 6-vertex modelof ice and 8-vertex model of anti-ferroelectrics. In thepresent case, 8 configurations on pnictogen vertices and12 configurations on chalcogen vertices have finite en-ergies, while the rest are infinitely costly. This analogyimplies the proposed model is generalizable to more com-plicated coordinations by assigning finite energies to thelatter.The present structural model can be formulated forstoichiometries that can not be arranged on the simplecubic lattice, except if one allows for vacancies. A specificexample of particular relevance for this work is archetypalpnictogen-chalcogenides of stoichiometry Pn Ch , suchas As Se , that can form both a glass and a crystal. (Pn= “pnictogen,” Ch = “chalcogen.”) In Fig. 7 we showthat coordination-wise and symmetry-wise, the struc-ture of this compound consists of double layers similarto those in the black-phosphorus parent structure fromFig. 5(b). Note that by placing the vacancies in a partic-ular fashion, we achieve two things simultaneously: Onthe one hand, the As Se stoichiometry is obeyed, andon the other hand, the octet rule on both pnictogens andchalcogens is satisfied. In drawing individual double lay-ers, we have used the structural model of Vanderbilt andJoannopoulos, see also Ref. Note that based on theactual density of As Se , the bond length in the parentstructure in Fig. 7 would have to be 2 . ppσ networkin the deformed structure. The presence of vacanciesin parent structures should not be too surprising: Thearchetypal phase-change material Ge Sb Te is knownto exhibit a (metastable) distorted cubic structure withvacancies. abc BA BA3 21 BA3 21 BA3 21A1 A31 A31 1B B3 2 22 B A3 1B A32 1 2A3 1 2
FIG. 8. The top view of a portion of the Pn Ch parentstructure from Fig. 7. The labels of the top layer are upside-down, to help distinguish it from the bottom double-layer andindicate that adjacent double layers are related by inversion.The thin dashed lines and adjacent numbers indicate the cor-responding secondary bonds and their length in the deformedstructure, see Fig. 9. The tilted rectangle drawn with dashedlines indicates the unit cell. The parent structure in Fig. 7 is clearly not unique inthat we could have placed the double layers in severaldistinct positions relative to each other. The specific,“homopolar” arrangement in Fig. 7 was chosen becausethe sets of close neighbors in this arrangement and in theactual deformed structure seem to exhibit the greatestoverlap, see Figs. 8 and 9. Nevertheless, several othermutual positions are possible, which exhibit comparablyoverlapping sets of close neighbors with the deformedstructure, and, in addition, minimize the distance be-tween the vacancies in the parent structure better thanthe specific realization in Figs. 7 and 8. For instance, con-sider shifting the top layer in Fig. 8 “north” by one lat-tice spacing. Incidentally, one notices that the vacanciesin the parent structure become “smeared” in the inter-layer space of the distorted structure, see SupplementaryMaterial.
We point out that it would be impossible to“merge” vacancies in the Pn Ch stoichiometry between each two double-layers, if we attempted to use an arsenic-like structure as the parent structure from Fig. 5(a) in-stead of the black-phosphorus structure from Fig. 5(b).This notion is consistent with the stability of the latterstructure in the actual material. At any rate, during thedistortion, the distance between linked atoms decreases,resulting in strong, covalent bonds. Conversely, the sec-ondary bonds will result partially from the cleaved links bcbc bc bc bc bcbc bcbc bcbc bc bcbc bcbcbcbc bC bCbC bC bC bCbCbCbC bCbC bC bC bCbCbCbCbCbC bCbCbC bCbC bc bc bc bc × × × × AAA AA AA AB BBB BB BBBB 1 11 1 1 1112 22 2 2 222333 333 33 ab FIG. 9. A side view of the actual As Se structure. The wavylines indicate inter-layer nearest neighbors in the parent struc-ture, except the A-3 link, which is through a vacancy. Notethat the bonded intra-layer atoms are automatically nearestneighbors in the parent structure. The lengths of the links areonly partially indicative of the actual bond length because thebonds are not parallel to the projection plane. (usually intralayer) or new contacts that formed in thedeformed structure. Figs. 7-9 indicate that because theparent structure is not unique, there is no one-to-one cor-respondence between missing links in the parent struc-ture and the secondary bonds in the distorted structure.It is obvious that in the process of cleaving the bondsin the sc structure, symmetry was lowered resulting in atwice bigger unit cell along the pertinent directions. Forexample, in drawing the structure in Fig. 7, we couldhave split the original sc lattice in double-layers in sixequivalent ways, i.e., two along each coordinate axis.Likewise, there are two equivalent ways to buckle eachdouble layer, in each of the (1,1) and (1,-1) directions.Further symmetry lowering occurs when we place pnic-togens, chalcogens and vacancies at the lattice vertices.While the presence of symmetry breaking itself is herebyobvious, its mechanism appears to be subtler. Severalworkers have argued the symmetry-breaking transitionthat results in the structures of arsenic and phospho-rus is Peierls-like, which is a cooperative analogof the Jahn-Teller (JT) distortion in 1D or quasi-1Dsolids. The precise degree of Fermi surface nesting, requisite for such a structural instability, seems howeverto be subject to a specific approximation employed. The special significance of near- octahedral coordinationand the resulting trans-influence between the correspond-ing covalent and secondary bonds, with regard to thePeierls metal-insulator transition, can be viewed fromyet another angle: Alcock points out that compoundsin which the two bond types display similar length shownoticeable metallic luster.We thus conclude that one should generally regardthe symmetry breaking of periodic sc parent structuresas a second -order (or pseudo) Jahn-Teller distortion, whereby strict electronic degeneracy is not required. Wefurther note that the solid-state analogs of the second-order Jahn-Teller (JT) effect are also well known, suchas the dimerization transition in a hetero polymer, or incoupled homopolar chains, such as polyacene. Here, thepolymer chain is unstable toward dimerization so long as
FIG. 10. The Peierls distorted (AsH ) n chain. The linksand gaps correspond to covalent bond (bond order 0.9) andsecondary bonds (bond order 0.1) respectively. the gap is smaller than the coupling to the symmetrybreaking perturbation, see Eq. (2). Electronic interac-tions, too, can lead to an effective one-particle gap. Al-though second-order JT symmetry breaking, wheneverpresent, is partially hampered by the lack of degeneracy,it is still driven by the very same mechanism as during astrict Jahn-Teller-Peierls distortion.We note that the question of the mechanism of thesymmetry breaking is generally distinct from that of theinteraction that determines the relaxed structure uponthe symmetry breaking. According to Section II, the lat-ter interaction in distorted-octahedral coordinated com-pounds is dominated by sp -mixing. Indeed, Seo and Hoff-mann point out that the distortion of the relaxed struc-tures away from the simple cubic structure is stronger forlighter elements, consistent with stronger sp hybridiza-tion in those elements. For the perturbation caused by sp -mixing to also contribute to the symmetry breakingitself, it is essential that this perturbation can be madeperiodic with the inverse period commensurate with theelectron-filling fraction in the undistorted structure, as isthe case (1 / /
2) for trans-polyacetylene and arsenic.Otherwise, the Peierls-driven destabilization is severelyweakened (Ref. , Chapter 2.6).Specifically three -coordinated lattices with right anglesbetween bonds have a very special property that makethem additionally unstable toward symmetry lowering,even if these lattices are aperiodic . Any such lattice can be thought of as made of linear chains, each of whichconsists of white and black segments of equal length instrict alternation. The junctions between adjacent seg-ments correspond to the lattice vertices, while the whiteand black segments themselves correspond to no-link andlink respectively. In the case of strict octahedral coordi-nation, i.e., no mixing between distinct chains, each suchchain can be thought of as a result of a Peierls distor-tion of a chain of equidistant atoms. As a result, evenaperiodic parent structures are expected to be relatively(meta-)stable, not only the strictly periodic structures ofarsenic or black phosphorus. We illustrate the symme-try breaking in an individual linear ppσ system in thepresence of significant sp -mixing, which will be also ofuse later. Shown in Fig. 10 is a dimerized linear chainof hydrogen-passivated arsenics, (AsH ) n . The details ofthe electronic structure and geometry determination forthe chain are provided in Appendix C. Despite the pres-ence of sp -mixing and other interactions, the chain ex-hibits a clear ppσ -character: (see Fig. 1 for the notations) The As-As bond length are d = 2 .
48 ˚A and d ′ = 3 . α HAsH = 97 ◦ , β AsAsAs = 150 ◦ . The secondarybonding is significant as witnessed by the value of thecorresponding transfer integral: t ′ ≃ . t ≃ . ss , sp -mixing, and ppπ are significantly smaller (see Appendix C), implying thebonding is indeed ppσ . The sp -transfer integral is aslarge as 57% of the ppσ transfer integral, consistent withthe earlier statement that the sp -mixing is the next lead-ing contributor to the geometry of the symmetry brokenstate, after the ppσ interaction itself. As expected of aPeierls insulator, the chain exhibits a perfect bond alter-nation pattern. Lastly note that even though individualchains that make up aperiodic 3D parent structures un-dergo Peierls transitions, the symmetry breaking for the3D structures themselves is generally not quasi-one di-mensional and thus is not Peierls, in contrast with peri-odic systems, such as considered by Burdett and others.The parent structures with three links per atom plusvacancies, if any, see Figs. 5-7, can serve as basic mod-els for compounds with distorted octahedral coordi-nation that consist of two- and three-valent elements.These structures also apply to compounds containingelements from groups 14, such as GeTe, GeSe fromFig. 6 or archetypal phase-change alloys Ge Sb Te andGeSb Te . Hereby each A IV B VI pair is isolectronic witha pair of pnictogens, implying these atoms are three-coordinated. The rest of the atoms are in the Pn Ch stoichiometry, and so the rules leading to the parentstructure in Fig. 7 apply to these atoms. Even in theabsence of A IV B VI pairing, symmetry breaking schemescan be proposed for substances where the number of four-coordinated atoms is large, as in GeP and TlI, or β -tin, and similarly for five-coordinated atoms. Oth-erwise, an atom with coordination 4 or higher can beconsidered as a defect in a lattice of three-coordinatedvertices. We will return to this important point in thecompanion article. The following picture of structure-formation in ppσ -networked materials thus emerges from the above con-siderations: These materials may be thought of assymmetry-broken versions of a simple cubic structure.The symmetry breaking is an interplay of several types ofperturbation: (a) Peierls-instability proper for extendednear-linear chains; (b) cooperative second-order Jahn-Teller distortion that results from mixing of the p -orbitalswith other orbitals, mostly s , and from electronic inter-actions; (c) steric effects due to vacancies, if the lattermust be present owing to stoichiometry, as in Fig. 7; and(d) other coordination variations, as in the case of el-ements from groups 14 and lower. Even though theseperturbations are strong enough to break the symmetry,they are still perturbations , so that the resulting bond-ing is primarily ppσ . This statement can be quantifiedby comparing the strengths of the corresponding transferintegrals in the deformed structure, see Table I and the0(AsH ) n example above.Now, one must recognize that upon geometric opti-mization, the lattice will generally be aperiodic , evenwhen the link-breaking pattern in the parent structureis itself strictly periodic, let alone if we arranged the dis-tinct species or vacancies aperiodically or with a periodincommensurate with the period prescribed by the elec-tron filling fraction. Let us now examine how such ape-riodic lattices maintain the ppσ character, and hence thestability with respect to other types of ordering, while,at the same time, allowing for molecular transport.For concreteness, let us consider a specific prescriptionto deform the parent sc structure. For each atom, con-sider the links in the parent structure as vectors thatstart on the atom. Move each atom by a small distancein the direction which is the sum of its own vectors. Forinstance, if an atom has three links: (1, 0, 0), (0, -1, 0),and (0, 0, -1), move it in the direction (1, -1, -1). Analo-gously for a two coordinated atom, the displacement willbe in the plane containing the two links. Now turn onthe interaction in the form of non-zero transfer integrals,such as listed in Tables I and II. Let the lattice relax,subject of course to the Coulomb repulsion between theionic cores. As already mentioned, there is generally noone-to-one correspondence between the parent and thedistorted structure: this implies there are multiple re-laxed structures and hence multiple metastable minimaon the total energy surface . In discussing Figs. 7-9, wehave mentioned that distinct parent structures for thePn Ch stoichiometry can be obtained by shifting thedouble layers relative to each other. Because such a shiftincurs bond breaking, these alternative parent structuresare separated by barriers. It is understood that since theparent structure is generally aperiodic, Figs. 7-9 applyonly to a small fragment of such an aperiodic structure.Now, since the distinct parent structures are separatedby an energy barrier, at least one of them should be sep-arated by a barrier from the actual deformed structure,implying a presence of additional, metastable minima.When such metastable minima are few, the global po-tential energy minimum is easily accessible, as is proba-bly the case for the periodic parent structure of arsenic,see Fig. 5(a). Elemental arsenic is, in fact, a poor glass-former. (Other distinct three-coordinated parent struc-tures exist, but most of them are energetically costly. )To summarize, the existence of distinct parent structureswith shifted atoms is crucial for the present structuralmodel to be consistent with the presence of moleculartransport in ppσ -networks.When aperiodic, the distorted lattice will exhibit twokey features: First, if the substance forms a periodic crys-tal, it will be lower in energy than any aperiodic coun-terpart. Second, because aperiodic structures are notunique, the lattice will be highly degenerate, as just dis-cussed. We can also understand the emergence of thedegeneracy thermodynamically: Suppose the substancecan crystallize. A mechanically stable aperiodic struc-ture, on the one hand, has a much lower symmetry than the crystal. On the other hand, the aperiodic structurecorresponds to a higher energy and hence higher temper-ature . The only way to reconcile these two conflictingnotions is to recognize that there should be a thermo-dynamically large number of nearly degenerate aperiodicstructures separated by finite barriers. By virtue of bar-rier crossing events, the lattice is able to restore the fulltranslational symmetry at long enough times; the lattersymmetry is higher than that in a mechanically stablecrystal. The lattice therefore corresponds to a liquid inthe activated transport regime, if steady state, or to anaging glass, if below the glass transition. The view of quenched liquids and frozen glasses as bro-ken symmetry phases is supported by an independentargument: According to Fig. 4, the symmetry brokenphase corresponds to a lower pressure. Bevzenko andLubchenko have shown that a covalently-bonded equi-librium melt can be regarded as a high symmetry lat-tice that has been sufficiently dilated and then allowedto relax into one of the many available aperiodic config-urations. Now, are the predicted structural degeneracyof emergent aperiodic ppσ -networks and the barriers foractivated reconfigurations consistent with the configura-tional entropy of actual materials? The contiguity be-tween covalent and secondary bonding, as illustrated inFig. 4, suggests that ppσ -networks support atomic mo-tions that do not involve bond-breaking but only a grad-ual change in the coordination, and as such, may be ther-mally accessible. These special atomic motions will bediscussed in detail in the follow-up article. Available structural data are consistent with the preva-lence of ppσ -bonding in vitreous chalcogenides withthe stoichiometries in question. According to severalstudies, the nearest neighbor bond lengths in amor-phous arsenic chalcogenides are essentially identical onthe average to those in their crystalline counterparts,but have a somewhat broader distribution. In view ofthe argued presence of trans-influence between the cova-lent and secondary bonds in these compounds, we con-clude that the secondary bonds in the vitreous samplesare of strength comparable to those in the correspondingcrystals, again subject to a broader distribution. Conse-quently, based on the applicability of the same generalmechanism of ppσ -network formation and the compara-ble bonding strength, we conclude both periodic and ape-riodic lattices exhibit the same type of bonding. Still, forthe sake of the argument, suppose on the contrary thatthe bonding is dominated not by the ppσ interaction, butby its leading competitor, i.e., sp -mixing, thus resultingin a predominantly tetrahedral coordination. The ra-tio of the filling fractions of the diamond and As Se lattices is approximately 1.18. At the same time, thedensities of the amorphous and crystalline com-pounds differ by less, i.e. is 4.57 vs. 4.81-5.01 g/cm forAs Se and 3.19 vs. 3.46 g/cm for As S , implying thatonly a small fraction of atoms in these glasses, if any,might be regarded as tetrahedrally coordinated. Sucha “defect” is analogous to a small region occupied by an1interface between two distinct lattices, which incurs a sig-nificant free energy cost. The companion article showsthat a mechanism for a reversible formation of such de-fected configurations arises naturally in the present struc-tural model. IV. CONCLUDING REMARKS
The main goal of this article was to establish themechanism of bonding in semiconducting pnictogen- andchalcogen-containing quenched melts and frozen glasses.Representative substances are archetypal glassformers,such as As Se and similar materials whose crystallineforms can be directly argued to exhibit ppσ -bonding,based on their known structures, measured electronicdensity of states, and electronic structure calculations.We have formulated a structural model, by which boththe crystalline and vitreous materials are seen to form bythe same general mechanism, i.e., by symmetry-loweringand distortion of a high-symmetry structure defined onthe simple cubic lattice. By combining this model withthe limited structural data on the vitreous counterpartsof those listed materials and similar compounds, we haveargued the glasses are also ppσ -bonded.Lowering of the symmetry by breaking the bonds inthe fully connected simple cubic structure can be under-stood as lowering of the lattice dimensionality, similarlyto what is seen in the (AsH ) n chain in Fig. 10, whichis a linear array of relatively weakly bonded dimers, orto what one finds in the parent structures from Figs. 5-7, which consist of double-layers, possibly accompaniedby further symmetry lowering. Papoian and Hoffmann have outlined general principles for the interrelation ofdimensionality and deformation of high symmetry pe-riodic structures. These authors argue the bonding inthe parent structures from Fig. 5 can be regarded as aPeierls distortion of a hypervalently bonded lattice builtfrom electron-rich units, whereby the dimensionality ofthe lattice is lowered. In contrast, to preserve the di-mensionality, the electron-rich units comprising the hy-pervalent structure would have to be oxidized instead.In the former case, the deformed structure retains itsoriginal character, while in the latter case, the geometryis expected to change. For instance, a cubic Sb lat-tice exhibits a (distorted) octahedral coordination, whileoxidation would hypothetically result in a tetrahedrallycoordinated Sb + lattice. The semiconducting alloys inquestion do exhibit relatively low variation in electroneg-ativity and, hence, the distorted octahedral coordination.The parent structures for substances at the focus of thepresent study, i.e., pnictogen and chalcogen containing al-loys enjoy a very special property: Each vertex in the par-ent structure is three -coordinated, while the bonds are atnear 90 ◦ angles. Under these special circumstances, thewhole lattice can be thought of as composed of infinitelinear chains in which bond and bond-gaps strictly alter-nate. In view of the weak interaction with the environ- ment, each of these chains can be thought of as a quasi-1D Peierls insulator with renormalized interactions. (Inthe eventual deformed structure, the chains are deformedand, likely, of rather limited length. ) This observationallows one to extend the trends established by Papoianand Hoffmann in periodic crystals to aperiodic lattices.Since the symmetry can be restored partially or fullyby high pressure, the argued view of glasses as lowered-symmetry versions of high-symmetry structures is sup-ported by independent arguments on (negative) pressure-driven glass transition in covalently bonded materials. We have also pointed out that the view of this type ofsymmetry breaking as a second-order, cooperative Jahn-Teller distortion may be equally justified. In addition tothe Peierls instability proper, within each chain, the sym-metry breaking is also driven by local interactions thatcompete with the ppσ -bonding proper, primarily by the sp -mixing.The presence of competing interactions in these glass-forming materials is consistent with their structuraldegeneracy. It appears that the strength of such com-peting interactions should satisfy certain restrictions.Specifically, if the sp -mixing is too strong, it destroysthe ppσ -bonding. Yet, if the mixing is sufficiently weak,the coordination can be made close to perfect octahedral,while decreasing the stability of the glass relative to thecrystal. The latter trend is exploited in making opti-cal drives, using Ge Sb Te or similar compounds. Again consistent with the structural degeneracy of the ppσ -networks is the noted analogy between the bond-placement rules in the proposed structural model and thevertex models known to exhibit rich phase behavior.
The argued similarity of the bonding mechanisms in ppσ -bonded crystals and glasses explains the puzzlingstability of this important class of glasses. As mentionedin the Introduction, the enthalpy excess ∆ H of the su-percooled liquid, relative to the corresponding crystal, iseasy to estimate. It is directly related to the configu-rational entropy S c of the fluid, i.e. ∆ H = S c T , savea small ambiguity stemming from possible differences inthe vibrational entropies. The liquid configurational en-tropy varies between 0 . k B and about 2 k B per bead, be-tween the glass transition and melting temperatures.The energy 0 . k B T g per bead amounts to less than 0.05eV per atom. The model thus allows one to reconcile twoseemingly conflicting characteristics of quenched meltsand frozen glasses. On the one hand, these materialsexhibit remarkable thermodynamic and mechanical sta-bility, only slightly inferior to the corresponding crystals.On the other hand, these materials are also multiply de-generate thus allowing for molecular transport.Finally, this paper buttresses and clarifies some of thetechnical aspects of the original programme: The ppσ -networks automatically satisfy all the necessary require-ments for the presence of the intrinsic midgap electronicstates, as listed in the Introduction. First, a necessaryrequirement for covalent ppσ bonding is that the elec-tronegativity variation ǫ is not too large, Eq. (2), lest2 bb b b t sp t ss t σ t ′ σ t π b b βp z βp z (a) (b) FIG. 11. (a) A graphical summary of the definitions of thetransfer integrals from Table I. (b) An example of geometryused to compute transfer integrals t and t ′ , see text. the resulting bonding becomes predominantly ionic. Sec-ond, a high-symmetry ppσ -bonded network is unstabletoward Jahn-Teller distortion at each center, implyingthe deformed lattice shows an alternating pattern of bondsaturation in the form of covalent and secondary bonds.Third, based on the stability of the ppσ -network with re-spect to perturbations in the form of competing chemicalinteraction, the secondary bonds are sufficiently strong,i.e. the t ′ /t ratio of the transfer integral of the secondaryand covalent bonds is not too small, see Eq. (1). 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: Detailed explanations of Tables I andII
Table I: A graphical summary of the definitions of thetransfer integrals is given in Fig. 11(a), see also p. 23of Ref. By definition, the ppσ integrals, i.e., t σ and t ′ σ are computed assuming the two p orbitals are alignedwith the bond and hence depend only on the distancebetween the participating atoms. As a result, the transferintegrals t and t ′ from Eq. 1 are equal in value to t σ and t ′ σ respectively only when the three atoms in questionare situated on a straight line. If 180 − β < ◦ , asis the case for almost all compounds from Table I, thedifference between t and t σ is less than a few percent,depending on the specific geometry and basis set. Forinstance, in the geometry from Fig. 11(b), we get t = (cid:10) p z (cid:12)(cid:12) H (cid:12)(cid:12) p z (cid:11) = t σ +( t π − t σ ) cos ( β/ t and t σ for β = 160 ◦ and t π /t σ = − / Table II: For several compounds, the structural datavary somewhat depending on the source, such as forAs Se in Refs. and α -monoclinic Se, Refs. Theresulting ambiguity should be kept in mind. , deviation from the right-angled geome-try. For As O , GeSe , and AsBr only As and Ge wereconsidered as the central atoms. reference structure r cov ,˚AO α -quartz-GeO crystal 0.52Ge crystal 1.225As see text 1.20Se isolated helix and rings 1.17Br diatomic molecule 1.140Te isolated helix and rings . The absolute electronegativity is the av-erage of the electron ionization and affinity energies, asfound in Ref. . The covalent radii used are listed inTable III. These radii were determined using compoundsexhibiting low variations in electronegativity, as perti-nent to the materials in question, except GeO . For As,values found in the literature differ within 0.02 ˚A.We use the value from the middle of this range, whichalso happens to coincide with the result of interpolationacross the sequence of Ge-As-Se-Br. . Only the strongest secondary bonds arecited. To determine these for As Se and Se, we haveused diagrams that show the magnitudes of the atomic p orbitals of the nearby atoms on a sphere centered on achosen atom with a radius equal to the covalent radius ofthat atom, see the Supplementary Material. For thetetrahedrally bonded materials, As O , and AsBr , thenearest neighbor in the direction opposite to the covalentbonded atoms is used. . Accurate values of the tight-binding(TB) parameters are usually determined by obtaining thebest fit to the electronic density of states for each spe-cific systems and are thus system-dependent. Our goalhere is, instead, to highlight the generic trends of tightbinding parameters that apply satisfactorily for bondsranging from the strictly covalent to van der Waals,in as many distinct compounds as possible. Such auniversal parametrization of one-electron transfer (res-onance) integrals is provided, for instance, by the PM6parametrization of the MNDO method, which weuse to estimate the t ′ σ /t σ ratios for the compounds inTable II. Despite the made approximations, we feel thatthe resulting potential ambiguity of the ratio of the ma-trix elements, t ′ σ /t σ , is not large.We can partially judge the reliability of the tight-binding parameters by comparing them to those obtainedby other standard methods. In Table IV below, we com-pare Robertson’s data for transfer integrals in threeelemental solids, as obtained by fitting the spectrumof the one-electron TB Hamiltonian (column 2) and bythe chemical pseudopotential method (column 3) to thepresent TB parametrization (column 4). The Figure onthe r.h.s. compares the result of the parametrization forelemental Ge with accurate calculations of Bernstein et3 Ref. Ref. presentTB fit pseud. workAs 0.54 0.51 0.62Se 0.19 0.26 0.27Te 0.33 0.65 0.74 . . . . . t ′ /t d ,˚ARef. present work TABLE IV. Comparison of the t ′ σ /t σ ratio, as resulting fromthe present tight-binding parametrization with compilationsof Robertson (table) and the calculations of Bernstein etal. (figure), see text. al. Finally, the same TB parametrization was used forcalculation on the dimer of Br molecules, see below. Appendix B: Dimer of bromine molecules
In this appendix we use the electronic structure of Br –Br dimer obtained by ab-initio calculations to estimatethe contribution of the ppσ interaction to the bindingenergy of the dimer and show that the resulting estimateis consistent with the conclusions of a simple molecularorbital theory.The geometry of the dimer (see Fig. 3) is optimized byFirefly program on MP2-level with aug-cc-pVTZ basisset and RHF wave-function. Although more accurate ap-proximations exist, the present method has the advan-tage of simplicity while yielding the geometry consistentwith that of the bromine crystal; it also yields results thatcompare well with accurate calculations for other halogendimers. For clarity, we have not corrected for the basisset superposition error, because here we are interested inthe relative magnitudes of distinct contributions to thebonding, not the absolute value of the binding energy.The resulting ambiguity in the binding energy itself isnot large anyway, about 15%, as can be estimated usingthe standard counterpoise method, Ref. p. 714.The total binding energy of the dimer, 0.13 eV, consistsof several contributions. For instance, the correlation(MP2) contribution is 0.2 eV. To extract the contribu-tion proper of the electrons occupying distinct molecularorbitals, we adhere to the following scheme: First, wesubtract the uniform downshift of all MOs by 20 meVin the dimer, relative to the two isolated Br molecules.Next, we compile the contributions of all molecular or-bitals to the bonding, see Table V. Here, the dimer isin the x-y plane. Because little sp -mixing is present, theMO’s are naturally grouped into classes that consist pri-marily of either p or s orbitals. The p orbitals are furthersubdivided, according to their symmetry, into p x,y and p z orbitals. The former form the ppσ bonds, while the lat-ter are out of the x-y plane and contribute little to thebonding, as is clear from Table V.The magnitude of the binding energy due to the in-plane p -orbitals, 0.284 eV, can be rationalized using a MOs AOs energy range, eV E binding , eV61 , , , , , p x,y ( − . , − . − . , , , s ( − . , − . − . , , , p z ( − . , − . − . ,
72 (unocc.) s, p x,y ( − . , +0 .
2) +0 . -Br dimer from Fig. 3. The quantity E binding is the difference between the total MO energies ofthe dimer and those of the isolated molecules. −15−14−13−12−11−10− 10 E ,eV xy BrBr BrBr Br Br BrBr
FIG. 12. MO diagram of the Br dimer from Fig. 3 that de-picts a subset of the MO’s from Table V, along with pertinentMO’s of the constituent Br molecules. simple molecular orbital theory. Let us first consider onlythe eight valence p -orbitals lying in the plane of the dimershown in Fig. 3. The ab initio diagram of the correspond-ing MO’s is shown in Fig. 12. Let us focus exclusivelyon the intramolecular ppσ - and ppπ -interactions and theintermolecular ppσ -interaction between two atoms con-nected by the dashed line in Fig. 3. In this approxima-tion, the p y orbitals do not interact with the p x orbitals,so that the secondary bonding is exclusively due to the p x orbitals. The specific realization of Hamiltonian (3)for this bonding configuration reads H = ǫ p t σ t σ ǫ p t ′ σ t ′ σ ǫ p t π t π ǫ p . (B1)4By Eq. (4), combined with E Aunocc = ǫ p + t σ , ψ Aunocc = 1 √ ! ,E Bocc = ǫ p ± t π , ψ Bocc = 1 √ ± ! , we obtain the dimer binding energy:( t ′ σ ) /t σ . (B2)Generally, in the perturbative expression (4), the quan-tities V nm correspond to the transfer integrals of the sec-ondary bonds, t ′ , while the denominators to the transferintegrals of the covalent bonds, t . The binding energy isthus second order in the t ′ /t ratio.Note that the result in Eq. (B2) also helps to partiallyassess the reliability of the TB parametrization from Ap-pendix A. Substituting the numerical values for thosetransfer integrals in Eq. (B2) yields the dimer bondingenergy 0.3 eV, in good agreement with the ab-initio anal-ysis leading to Table V. A similar analysis can be usedto show that the s -orbital bonding contribution from Ta-ble V results not from the ss interaction, but primarilyfrom sp -mixing, also consistent with the small overlap ofthe s -orbitals.Finally, we outline derivation of the TB expression(5) for the bond order of a closed-shell interaction. Us-ing Eq. (14.22) of Ref. , for the wave-functions of anAB dimer formed by a closed shell interaction, one canstraightforwardly show that the interaction-induced cor-rection to the density matrix P for the dimer, in thesecond order, is given by A occ X n B unocc X m − A unocc X n B occ X m V nm E A n − E B m | ψ A n ih ψ B m || ψ B m ih ψ A n | ! ., (B3) According to a standard definition of the bond order, (also used in MOPAC), the latter can be written in thechosen basis set as: b AB = A X n B X m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( h ψ A n | , P | ψ B m i !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (B4)leading to Eq. (5).Note that the first (second) double sum inEq. (5) is twice the occupation of the antibond-ing orbitals of the molecule B (A), as donatedby the bonding orbitals of A (B). These occupa-tions are defined as 2 P (AB) occ ν P B unocc m | (cid:10) ψ B m (cid:12)(cid:12) ψ ν (cid:11) | and2 P (AB) occ ν P A unocc m | (cid:10) ψ A m (cid:12)(cid:12) ψ ν (cid:11) | respectively, where ψ ν are MO’s of the AB dimer. Appendix C: Arsenic chain
In our model system, the dimerized (AsH ) n chain(Fig. 10), the shorter and longer As–As bonds correspond to covalent and secondary bonds respectively; the As–H bonds correspond to covalent bonds in chalcogenideglasses. The calculations are performed on semiempiricallevel by MOPAC program with PM6 parametrization.On each arsenic, the axes are labelled as follows: The p x and p y orbitals are oriented toward the hydrogens (butnot strictly along the x and y axis), while p z orbitals aredirected toward the neighboring arsenics, see the geome-try in Fig. 11(b). Note these axes are defined locally oneach individual arsenic; the optimized chain may or maynot be linear on the average.We first check the semiempirical approximationagainst ab-initio calculations for n = 4, see Table VI.The ab initio results clearly indicate a distorted octahe-dral geometry and are consistent with secondary bond-ing between the terminal (AsH ) units: The β (As-As-As) angle is very close to 180 ◦ , while the bond length,3.57 ˚A is close but shorter than the sum of the reportedvan der Waals radii for As, i.e., between 3.7 and 3.9 ˚A,depending on the convention. Now, judging from theangle β between adjacent As–As bonds and H-As-H an-gle, the semiempirical method clearly overestimates thetendency toward sp hybridization. At the same time,the semiempirical method underestimates the secondarybond length, suggesting the two errors in the resultingbond strength will partially cancel out. The discrepancyin the secondary bond lengths and the β angles betweenthe ab initio and PM6 methods should be considereda consequence of the specific parametrization adoptedin MOPAC, which was optimized for covalent, not sec-ondary bonds, and many of which have a significant ioniccomponent and/or coordinations distinct from distortedoctahedral. Thus the TB-based analysis is not expectedto be fully quantitative, unless ad hoc parametrized.Nevertheless, we have seen from the discussion of Eqs. (4)and (5) that TB analysis is qualitatively consistent withthe closed-shell character of the secondary bonding andthe trans-influence between the counterpart covalent andsecondary bonds. Our results for geometric optimizationof rhombohedral arsenic, using the PM6 parametrization,are consistent with these conclusions: Using a 96 atomsupercell, we obtain d = 2 .
46 ˚A (vs. experimental 2.52˚A), d ′ = 3 .
24 ˚A (3 .
10 ˚A), α = 99 . ◦ (96 . ◦ ). Clearly,the PM6 parametrization is consistent with ppσ bondingin rhombohedral As, and, furthermore, gives reasonablefigures for the lattice parameters.According to the semi-empirical calculation, an infinite AsH chain is dimerized and has a zero net curvature,whereby d = 2 .
48 ˚A, d ′ = 3 . α = 97 ◦ , β = 150 ◦ . De-spite the aforementioned bias toward sp -mixing, analysisof localized MOs in the semiempirical calculations showsthat the bonding contribution of arsenic’s s -orbitals isnegligible: 0.1 (vs. 2 for a regular bond). Significantsecondary bonding and the trans-influence between thecovalent secondary bonds are clearly seen in the valuesof the bond order, which are equal to 0.9 and 0.1 for thecovalent and secondary As-As bond respectively.To independently verify the dominant character of the5 d AsAs d ′ AsAs β d
AsH α HAsH α AsAsH E ,eVPM6 2.463 3.06 148 1.523 95.7 97.4 -0.171.523 95.6 96.9MP2 2.483 3.57 177 1.525 91.7 92.6 -0.131.527 91.6 92.1Ref. (2.441) 3.53 (180) (1.506) (93.9) (93.9) -0.10TABLE VI. The geometry of the dimer of As H molecules:semiempirical PM6 calculations, ab-initio RHF MP2 calcula-tions with acc-pVDZ basis set, and calculations from Ref. The notations are according to Fig. 1, two entries per cell cor-respond to the inner (1st) and the outer (2nd) AsH units, theenergy is the dimer binding energy, the values in parenthesesare not optimized. φ \ ψ p z s p x s H renormalization spp z − . − . . . ǫ − ǫ = +1 . s − . . − . p x − . − . s H − . p cb z . t ) 2 . − . . t − t = +0 . s cb − . . − . p cb x . . p sb z . t ′ ) − . . − . t ′ − t ′ = − . s sb . . − . p sb x . . h φ | H | ψ i of the Fockmatrix of an infinite (AsH ) n chain. The transfer integrals t and t ′ are indicated in brackets next to their numerical values.Column 6: The renormalization of the parameters of the one-particle ppσ -only Hamiltonian, as computed using Eq. (C2).Column 7: The contribution to this renormalization of the s -orbitals. Labels “cb” and “sb” mean covalent and secondarybonds. All AO’s belong to the arsenics except the hydrogen’s s H . ppσ -bonding, we demonstrate that the competing inter-actions are perturbations that mostly amount to a renor-malization of the ppσ -interaction. Indeed, the full one -particle Hamiltonian for a system plus the environmentis a block matrix H = H sys V † V H env ! , (C1)where the matrix H sys contains exclusively the on-siteenergies and transfer integrals for the system, H env forthe environment. The (generally non-square) matrix V and its hermitian conjugate V † contain the system-environment transfer integrals. It is straightforwardto show that, given an energy eigenvalue E for thefull Hamiltonian in Eq. (C1), the portion of the wave-function corresponding to the system is an eigenfunction of the effective Hamiltonian: e H = H sys + V † ( E − H env ) − V (C2)with the same eigenvalue E . Indeed, substituting | ψ i =( h ψ sys | , h ψ env | ) † into H| ψ i = E | ψ i , one gets H sys | ψ sys i + V † | ψ env i = E | ψ sys i and ( H env − E ) | ψ env i = −V| ψ sys i ,from which Eq. (C2) follows.Clearly, in the one-particle picture, the effect of theenvironment can be presented as an (energy-dependent)renormalization of the bare Hamiltonian of the systemproper. One can use this systematic procedure to esti-mate the strength of both the intra-chain and environ-ment’s perturbation to the ppσ bonding within the chain.In doing so, below, we fix the energy E at the gap center.The contribution of s - or d -orbitals to the renormal-ization of the ppσ transfer integrals can be estimated byusing a perturbative expansion of the exact Eq. (C2): e H ij = H sys ,ij + X α t iα t αj E − ǫ α − X αβ t iα t αβ t βj ( E − ǫ α )( E − ǫ β ) + . . . , (C3)where the Latin indices label intra-chain p z orbitals andthe Greek indexes label the rest of the orbitals. The smallparameter t/ ( E − ǫ ) is indeed small, and the more so thefurther the orbital energy ǫ from the gap center.The essential elements of the Fock matrix are listed inTable VII. Note that d -orbitals are also included in PM6parametrization. Despite the large value of the perti-nent transfer integrals, the direct contribution of theseorbitals to the renormalization is at most a few per-cent because the orbitals are almost empty. One infersfrom Table VII that the dominant contribution to As–As bonding stems from ppσ -integrals. The ppπ -integralsare four times smaller than ppσ , while the ss -integralsare essentially negligible. Although s -orbitals do not bythemselves contribute significantly to the bonding in thechain, they provide the main contribution to the renor-malization of the ppσ transfer integrals, specifically by weakening the secondary bonding. This contribution,provided in the last column in Table VII, turns out tobe well approximated by the expressions:∆ ǫ ≈ − t sp /ǫ s , ∆ t ≈ t sp t on-atom sp /ǫ s , (C4)within the error of 20% or less compared with the moreaccurate Eq. (C3). Here t on-atom sp ≈ − g sp P sp / , (C5)where g sp is the Coulomb integral for s and p orbitals ofthe same atom (6 eV for As) and P sp is the correspond-ing entry of the density matrix, which is the measureof actual hybridization. If P sp > chain - then the sp -mixing weakens the secondarybonding, the magnitude of the effect proportional to thehybridization strength.6 ∗ [email protected] K. Morigaki,
Physics of amorphous semiconductors (World Scientific, 1999). R. Zallen,
The physics of amorphous solids (Wiley, 1998). A. Feltz,
Amorphous inorganic materials and glasses (Wi-ley, 1993). S. R. Elliott,
Physics of amorphous materials (Longman,1990). 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). P. W. Anderson, Phys. Rev. , 1492 (1958). N. Mott, Proc. R. Soc. Lond.
A 382 , 1 (1982). M. H. Cohen, H. Fritzsche, and S. R. Ovshinsky, Phys.Rev. Lett. , 1065 (1969). N. F. Mott,
Metal-Insulator Transitions (Taylor and Fran-cis, London, 1990). P. W. Anderson, Nature - Phys. Sci. , 163 (1972). P. W. Anderson, Phys. Rev. Lett. , 953 (1975). N. F. Mott, Rev. Mod. Phys. , 203 (1978). A. Zhugayevych and V. Lubchenko, J. Chem. Phys. ,044508 (2010). T. R. Kirkpatrick, D. Thirumalai, and P. G. Wolynes,Phys. Rev. A , 1045 (1989). X. Xia and P. G. Wolynes, Proc. Natl. Acad. Sci. , 2990(2000). V. Lubchenko and P. G. Wolynes, J. Chem. Phys. ,2852 (2004). V. Lubchenko and P. G. Wolynes, Annu. Rev. Phys.Chem. , 235 (2007). A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W. P. Su,Rev. Mod. Phys. , 781 (1988). A. Zhugayevych and V. Lubchenko, “Electronic struc-ture and the glass transition in pnictide and chalcogenidesemiconductor alloys. Part II: The intrinsic electronicmidgap states,” (2010), submitted to
J. Chem. Phys. ,cond-mat/1006.0776. D. K. Biegelsen and R. A. Street, Phys. Rev. Lett. ,803 (1980). J. Hautala, W. D. Ohlsen, and P. C. Taylor, Phys. Rev. B , 11048 (1988). T. Tada and T. Ninomiya, Sol. St. Comm. , 247 (1989). T. N. Claytor and R. J. Sladek, Phys. Rev. Lett. , 1482(1979). S. Shang, Y. Wang, H. Zhang, and Z.-K. Liu, Phys. Rev.B , 052301 (2007). J. K. Burdett,
Chemical Bonding in Solids (Oxford Uni-versity Press, 1995). M. A. Popescu,
Non-crystalline chalcogenides (Kluwer,2000). J. Li and D. A. Drabold, Phys. Rev. B , 11998 (2000). M. Cobb, D. A. Drabold, and R. L. Cappelletti, Phys.Rev. B , 12162 (1996). S. G. Bishop, U. Strom, and P. C. Taylor, Phys. Rev. B , 2278 (1977). F. Mollot, J. Cernogora, and C. Benoit `a la Guillaume,Phil. Mag. B , 643 (1980). P. S. Salmon, J. Non-Crys. Solids , 2959 (2007). K. Hachiya, Journal of Non-Crystalline Solids , 160(2001). L. Makinistian and E. A. Albanesi, J. Phys. Cond. Mat. , 186211 (2007). R. Azoulay, H. Thibierge, and A. Brenac, J. Non-Crys.Solids , 33 (1975). W. B. Pearson,
The crystal chemistry and physics of met-als and alloys (Wiley, 1972). W. A. Harrison,
Electronic Structure and the Propertiesof Solids (Freeman, San Francisco, 1980). G. N. Greaves, S. R. Elliott, and E. A. Davis, Adv. Phys. , 49 (1979). J. Robertson, Adv. Phys. , 361 (1983). N. Alcock, Adv. Inorg. Chem. Radiochem. , 1 (1972). P. Pyykk¨o, Chem. Rev. , 597 (1997). G. A. Landrum and R. Hoffmann, Angew. Chem. Int. Ed. , 1887 (1998). G. A. Papoian and R. Hoffmann, Angew. Chem. Int. Ed. , 2408 (2000). H. A. Bent, Chem. Rev , 587 (1968). A. Antonelli, E. Tarnow, and J. D. Joannopoulos, Phys.Rev. B , 2968 (1986). J. K. Burdett, Chem. Soc. Rev. , 299 (1994). J. N. Murrell, S. F. A. Kettle, and J. M. Tedder,
Thechemical bond , 2nd ed. (Wiley, 1985). W. Harrison,
Elementary electronic structure (WSPC,2004). S. G. Bishop and N. J. Shevchik, Phys. Rev. B , 1567(1975). A. C. Stergiou and P. J. Rentzeperis, Z. Kristallogr. ,139 (1985). D. Schiferl and C. S. Barrett, J. Appl. Cryst. , 30 (1969). P. M. Smith, A. J. Leadbetter, and A. J. Apling, Philos.Mag. , 57 (1974). H. Wiedemeier and H. G. von Schnering, Z. Kristallogr. , 295 (1978). R. Keller, W. B. Holzapfel, and H. Schulz, Phys. Rev. B , 4404 (1977). A. C. Stergiou and P. J. Rentzeperis, Z. Kristallogr. ,185 (1985). A. L. Renninger and B. L. Averbach, Acta Crystallogr. B , 1583 (1973). Y. Miyamoto, Jpn. J. Appl. Phys. , 1813 (1980). B. M. Powell, K. M. Heal, and B. H. Torrie, Mol. Phys. , 929 (1984). P. Cherin and P. Unger, Acta Cryst. B , 313 (1972). F. Pertlik, Monatshefte fur Chemie , 277 (1978). H. Braekken, Kongelige Norske Videnskapers Selskab,Forhandlinger , 3 (1935). G. von Dittmar and H. Schafer, Acta Crystallogr. B ,2726 (1976). Y. Zhang, Z. Iqbal, S. Vijayalakshmi, S. Qadri, andH. Grebel, Sol. St. Comm. , 657 (2000). R. W. G. Wyckoff,
Crystal structures (Interscience Pub-lishers, New York, 1963). D. R. Armstrong, P. G. Perkins, and J. J. P. Stewart, J.Chem. Soc. Dalton Trans. , 1973 (1973). A. A. Granovsky, Firefly version 7.1.G,http://classic.chem.msu.su/gran/firefly/index.html. W. P. Anderson, J. K. Burdett, and P. T. Czech, J. Amer.Chem. Soc. , 8808 (1994). P. Silas, J. R. Yates, and P. D. Haynes, Phys. Rev. B ,174101 (2008). R. Bellissent, C. Bergman, R. Ceolin, and J. P. Gaspard, Phys. Rev. Lett. , 661 (1987). R. Bellissent and G. Tourand, J. Non-Cryst. Solids ,1221 (1980). S. Hosokawa, A. Goldbach, M. Boll, and F. Hensel, Phys.Stat. Sol. (b) , 785 (1999). R. Zallen, R. E. Drews, R. L. Emerald, and M. L. Slade,Phys. Rev. Lett. , 1564 (1971). L. M. Falicov and S. Golin, Phys. Rev. , A871 (1965). M. Takumi and K. Nagata, J. Phys. Soc. Jpn. Suppl. A , 17 (2007). H. Fujihisa, Y. Fujii, K. Takemura, and O. Shimomura,J. Phys. Chem. Solids , 1439 (1995). H. Katzke and P. Tol´edano, Phys. Rev. B , 024109(2008). P. B. Littlewood, J. Phys. C , 4875 (1980). J. K. Burdett and T. J. McLarnan, J. Chem. Phys. ,5764 (1981). J. K. Burdett, P. Haaland, and T. J. McLarnan, J. Chem.Phys. , 5774 (1981). J. H. Bularzik, J. K. Burdett, and T. J. McLarnan, Inorg.Chem. , 1434 (1982). T. A. Albright, J. K. Burdett, and M.-H. Whangbo,
Or-bital Interactions in Chemistry (Wiley, 2011). X. N. Wu and F. Y. Wu, J. Phys. A , L55 (1989). R. J. Baxter,
Exactly Solved Models in Statistical Mechan-ics (Academic Press, 1982). D. Vanderbilt and J. D. Joannopoulos, Phys. Rev. B ,2596 (1981). Y. Shimoi and H. Fukutome, J. Phys. Soc. Jap. , 1264(1990). A. V. Kolobov, P. Fons, A. I. Frenkel, A. L. Ankudinov,J. Tominaga, and T. Uruga, Nature Mat. , 703 (2004). C. Steimer, V. Coulet, W. Welnic, H. Dieker, R. Detem-ple, C. Bichara, B. Beuneu, J. Gaspard, and M. Wuttig,Adv. Mat. , 4535 (2008). J. K. Burdett and S. Lee, J. Amer. Chem. Soc. , 1079(1983). J. B. Bersuker,
The Jahn-Teller Effect (Cambridge, 2006). E. Canadell and M.-H. Whangbo, Chem. Rev , 965(1991). D. Seo and R. Hoffmann, J. Sol. State Chem. , 26(1999). M. J. Rice and E. J. Mele, Phys. Rev. Lett. , 1455(1982). P. J´ov´ari, I. Kaban, J. Steiner, B. Beuneu, A. Sch¨ops, andM. A. Webb, Phys. Rev. B , 035202 (2008). V. Lubchenko and P. G. Wolynes, J. Chem. Phys. , 9088 (2003). D. Bevzenko and V. Lubchenko, J. Phys. Chem. B ,16337 (2009). Y. Iwadate, T. Hattori, S. Nishiyama, K. Fukushima,Y. Mochizuki, M. Misawa, and T. Fukunaga, J. Phys.Chem. Sol. , 1447 (1999). Semiconductors other than group IV elements and III-V compounds , edited by O. Madelung (Springer-Verlag,Berlin, 1992). V. Lubchenko, Proc. Natl. Acad. Sci. , 10635 (2008). R. A. Stern and G. F. Tuthill, Int. J. Mod. Phys. B ,3331 (2001). P. Goldstein and A. Paton, Acta Crystallogr. B , 915(1974). B. Cordero, V. Gomez, A. E. Platero-Prats, M. Reves,J. Echeverria, E. Cremades, F. Barragan, and S. Alvarez,Dalton. Trans. , 2832 (2008).
P. Pyykk¨o and M. Atsumi, Chem. Eur. J. , 186 (2009). M. Springborg and Y. Dong, Int. J. Quant. Chem. ,837 (2009).
P. Ghosh, J. Bhattacharjee, and U. V. Waghmare, J.Phys. Chem. C , 983 (2008).
J. J. P. Stewart, J. Mol. Model. , 1173 (2007). M. J. S. Dewar and W. Thiel, J Am Chem Soc , 4899(1977). N. Bernstein, M. J. Mehl, and D. A. Papaconstantopoulos,Phys. Rev. B , 075212 (2002). J. Moilanen, C. Ganesamoorthy, M. S. Balakrishna, andH. M. Tuononen, Inorg Chem , 6740 (2009). M. H. Karimi-Jafari, M. Ashouria, and A. Yeganeh-Jabrib, Phys. Chem. Chem. Phys. , 5561 (2009). I. N. Levine,
Quantum chemistry (Prentice Hall, 2009)6th edition, page 714.
MOPAC2009, J. J. P. Stewart, Stewart Compu-tational Chemistry, Colorado Springs, CO, USA,http://OpenMOPAC.net (2008).
A. Bondi, J. Phys. Chem. , 441 (1964). K. W. Klinkhammer and P. Pyykk¨o, Inorg. Chem. ,4134 (1995). See supplementary material at URL for an illustrationof voids in crystalline As Se , image of Br crystal, andangular maps of the strength of the ppσ -interaction inselected substances. Supplementary Material
FIG. 13. Voids in As Se crystal. The graph is generated as follows: First, we compute the distance r min to the nearest atomin each point ( ξ, η, ζ ) in the lattice coordinates. (The lattice is monoclinic.) Call the resulting function r min ( ξ, η, ζ ). Next, wemake a projection of this function onto the plane of Fig. 9 of the main text, which is perpendicular to the ζ axis, according tothe following recipe. For each point ( ξ, η ), we vary ζ to find the largest value of r min ( ξ, η, ζ ), call it r ( ξ, η ), and assign a colorto the point ( ξ, η ) so that redder hues correspond to larger values of r ( ξ, η ), more violet hues to smaller values of r ( ξ, η ). Thefunction r ( ξ, η ) varies in the [1 . , .
52] ˚A range, while the sum of the covalent radii is 2.4 ˚A. As a result, intensely red regionsmark voids in As Se crystal. Atoms are marked by white balls. The largest void is distorted octahedral; it is centered at point(1 / , / , /
2) with Se atoms as vertices. bb bbbb b bb b b bb b ◦ ◦ FIG. 14. Fragment of a slice of bromine crystal. Compare the bond angles with those in Fig. 3 of the main text. FIG. 15. Illustration of distorted octahedral coordination of the arsenic atom in the position As1 of As Se crystal. Draw asphere around the As1 atom at its covalent radius (1.2 ˚A). For each point, compute the logarithm of the sum of the radialparts of the p orbitals on the atoms within 4.5 ˚A. Assign a color to the value of this function so that redder hues correspondto larger values and more violet hues to smaller values. The resulting map is Merkator-projected on a rectangular area, wherethe top edge corresponds to the north pole. The resulting map is shown in panel (a). In perfect octahedral coordination, redareas would be centered at the black dots. (The line connecting the poles coincides with a C axis of the octahedron.) In panel(b), the nearest neighbors are excluded, to specifically highlight back-bonding. The pertinent neighbors are indicated, togetherwith their distance to the As1 atom, in Angstroms. FIG. 16. Distorted tetrahedral coordination of the selenium atom in the position Se1 of As Se crystal. See Fig. 15 forexplanation. Black and white dots indicate octahedral and tetrahedral coordination respectively.2