Quantum Phenomena in Structural Glasses - American Chemical

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Quantum Phenomena in Structural Glasses: The Intrinsic Origin of Electronic and Cryogenic Anomalies Vassiliy Lubchenko* Departments of Chemistry and Physics, University of Houston, Houston, Texas 77204-5003, United States ABSTRACT: The structural glass transition is often regarded as purely a problem of the classical theory of liquids. The dynamics of electrons enters only implicitly, through the interactions between ionic cores or molecules. Likewise, zero-point effects tied to the atomic masses hardly affect the typical barriers for liquid rearrangements. Yet, glasses do exhibit many quantum phenomenaelectronic, optical, and cryogenic peculiarities that seem to have universal characteristics. These anomalies of the glassy state are uncommon or strongly system dependent in crystals and amorphous solids not made by a quasi-equilibrium quench of a melt. These clearly quantum phenomena include midgap electronic states in amorphous semiconductors, the two-level systems, and the Boson peak. Here, we discuss how these quantum phenomena found in glasses are not merely consequences of any kind of disorder but have universal characteristics stemming from the structural dynamics inherent in the glass transition itself. The quantum dynamics at cryogenic temperatures and electronic dynamics are related to the transition states for relaxational motions above the glass transition temperature, which are partially frozen when the sample is quenched.

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Glassy states, in contrast with most periodic crystals, are degenerate and have finite lifetimes. These times are often astronomically long, as witnessed by the rigidity of common glasses, yet they are also broadly distributed,1,2 so that there is no clear separation between the energetics of reconfigurations between distinct structural states and several intrinsically quantum processes in glassy solids. The typical barrier for structural change is about 35−37kBTg at the glass transition temperature Tg.3 Thus, for a semiconducting material, the peak mechanical perturbation during a structural reconfiguration may exceed the energy of the electronic excitations across the forbidden gap, implying that configurational rearrangements could be accompanied by electronic excitations, if the stress were sufficiently concentrated in space. Further, despite the very high typical barrier, the barrier distribution is finite even at nearly vanishing barriers.2 Reconfigurations corresponding to this low side of the barrier distribution are so facile that they could remain thermally active even at cryogenic temperatures, where they occur by tunneling despite the very large mass of atoms. This Perspective reviews recent work demonstrating that these intriguing quantum mechanical possibilities are indeed realized in structural glasses and the mosaic structure of the glassy energy landscape may explain outstanding questions that have resisted systematic effort since the 1970s. The glass transition is preceded by the emergence of activated transport; the transition states for activated transport events are strained regions separating relatively low free-energy configurations and are similar to interfaces separating two phases during first-order transitions. In amorphous chalcogenides, the transition-state

ittle to no knowledge of quantum mechanics is needed to understand the condensed phases of matter studied by physical chemists, so long as one takes for granted the existence of intermolecular forces. It is the mutual steric hindrance between closed electronic shells that drives the liquid-to-solid transition. The isotropic part of the cohesive interactions between the atoms largely serves as a container that holds the molecules or ionic cores together and establishes the temperature scale for the transitions. The directional part of the bonding interactions controls the details of local coordination and thus, by imposing a specific filling fraction, modifies the effects of the isotropic interactions and steric hindrance. Quantum mechanical effects do not enter explicitly into the analysis of the thermodynamic, long-term stability of the overwhelming majority of crystalline solids. The large mass of the nuclei provides for a f inite time scale in the problem, that is, the vibrational period, while the smaller electronic mass simply sets an energy scale for chemical bonding.

Quantum phenomena in glasses, including midgap electronic states in amorphous semiconductors, the twolevel systems, and the Boson peak, have universal characteristics stemming from the structural dynamics inherent in the glass transition itself.

© 2011 American Chemical Society

Received: October 1, 2011 Accepted: December 2, 2011 Published: December 2, 2011 1

dx.doi.org/10.1021/jz201332t | J. Phys. Chem. Lett. 2012, 3, 1−7

The Journal of Physical Chemistry Letters

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configurations can be identified with under- and overcoordinated vertices on a distorted cubic lattice.4−6 In close analogy with the solitonic midgap states in trans-polyacetylene,7 the malcoordination patterns host midgap states, which account for many puzzling anomalies in these semiconducting alloys.8 The low barrier subset of the structural reconfigurations, although relatively small, becomes the main source of dynamics at cryogenic temperatures, accounting for the mysterious twolevel systems (TLS) and the Boson peak.9,10 Although seemingly very different, the high-energy, electronic anomalies and the lowenergy, cryogenic anomalies both result from the structural degeneracy of the glass and are, in fact, characterized by the same length scale; the latter reflects the concentration of the transitionstate configurations2,3 for the activated transport, that is, ∼1020 cm−3. What Is the Structural Glass Transition? By definition, the glass transition occurs when the liquid begins to relax more slowly than the rate of cooling. The liquid relaxation rate decreases dramatically upon lowering the temperature, often in a strongly non-Arrhenius fashion,11 and therefore, even at modest cooling rates, the structure becomes largely arrested within a relatively narrow temperature range. At this point, the heat capacity experiences a relatively sharp drop, justifying the term “transition” in a generic but not the strict thermodynamic sense. (The structure is never fully arrested12 because the relaxation barriers are finite, even if very high.) A theory of the glass transition must rationalize (a) the emergence of the barriers for activated transport in liquids and (b) the apparent temperature dependence of those barriers. The random firstorder transition (RFOT) theory13 is a fully classical theory that accomplishes these goals but, as we shall see, also serves as the microscopic basis to understand the quantum phenomena in glasses. Let me first briefly review several microscopic notions of the RFOT theory. A key nontrivial notion behind the emergence of activated transport in liquids is that at sufficiently high average density, the usual assumption of the uniformity of the particle density profile does not hold on the time scale of vibrational relaxation.14,15 In other words, metastable aperiodic structures begin to form whereby each particle faces the same set of neighbors for extended periods of time. Similarly to the way that the ferromagnetic transition can be formally identified by comparing the Gibbs free energy of a putative polarized state to that of the paramagnetic state, the thermodynamic stability of a given aperiodic structure defined on a lattice {Rp} can be analyzed by employing a suitable particle-density profile

Figure 1. The free energy of an aperiodic LJ crystal, relative to the uniform liquid state, as a function of the spring constant α of the effective Einstein oscillator. (a) Example of pressure dependence at constant temperature (equal, incidentally, to the triple-point temperature). (b) Temperature dependence at constant pressure. In both panels, the thick solid lines correspond to conditions where the metastable minimum just begins to appear. The free energy is per particle, in units of kBT. Pressure, temperature, and length are in units of ϵ/σ3, ϵ/kB, and the effective hard sphere diameter d, respectively. ϵ and σ are the standard LJ parameters.

Still, the lattice is aperiodic and hence degenerate; to compute the free energy of the liquid in the activated regime, one must also account for this degeneracy. The multiplicity of distinct aperiodic states, whose logarithm is often called the configurational entropy Sc, lowers the free energy by an amount TSc. In the meanfield limit, the formation of the aperiodic metastable structures would correspond to a sharp, first-order-like transition,14,18 hence the terminology “random first order”. In finite dimensions, the metastable structures are subject to fluctuations, but at low enough T, they survive for multiple vibrational relaxation times. The emergence of the long-lived structures occurs not as a sharp transition but as a finite-width crossover at higher densities than that in the meanfield limit; the generic temperature of the crossover is denoted with Tcr. Empirically, the crossover occurs at a nearly universal ratio, ∼103, of the liquid to vibrational relaxation rate,19,20 as expected from the RFOT theory.15,21 Once transport becomes predominantly activated, reconfigurations occur by nucleation of a new aperiodic structure within the current aperiodic structure.12,22 The transitions are driven by the multiplicity of the configurations, subject to the mismatch penalty at the interface between the two structures. Individual particle displacements during a transition are small, just above the typical vibrational amplitude.16 The activation profile F(N) is shown as the “most probable path” in Figure 2, where N denotes the number of chemically rigid units contained within the nucleus. This unit is often called the bead;15,23 its volumetric size is denoted with a. The nucleus size N*, where F(N*) = 0, is special because at N > N*, the liquid is guaranteed to find at least one typical aperiodic state. This cooperativity size N* grows with the decreasing configurational entropy22 but reaches only a modest value of 200 or so at the glass transition on a 1 hour time scale,2,3 which corresponds to a physical size ξ = a(N*)1/3 of about two-three nanometers. The interfaces (or “domain walls”) emerge and grow at the expense of relaxing existing interfaces; the total number of interfaces is steady. The resulting concentration of the domain walls near the glass transition is, approximately

2

ρ(r , α) = (α /π)3/2 ∑ e−α(r − R p) p

(1)

This density ansatz has the flexibility to describe both the uniform liquid state, when α = 0, and the formation of the transient cages for finite α, in which case α would yield the spring constant of the effective Einstein oscillator. By the Lindemann criterion,16,17 α ≃ 102 at the transition. As illustrated in Figure 1, the transition is signaled by the appearance of a metastable minimum on the free-energy surface of the liquid at α = 102, which has been originally seen for hard spheres14 and now also for Lennard-Jones (LJ) and, semiphenomenologically, for actual liquids (P. Rabochiy and V. Lubchenko, unpublished results). The latter calculations demonstrate that despite the presence of “soft” interactions, this transient breaking of the translational symmetry is driven primarily by the steric repulsion between the particles, entirely analogously to crystallization of hard spheres.

nDW (Tg) ≃ 1/ξ(Tg)3 ≃ 1020 cm−3

(2)

The RFOT theory, which yields this estimate, has also led to several dozen quantitative predictions confirmed by experiment; see the review in ref 13 and more recent works.4−6,24−32 2



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Figure 2. The black solid line shows the barrier along the most probable path. Thick horizontal lines at low energies and the shaded area at energies above the threshold represent energy levels available at size N. The red and purple lines demonstrate generic paths, and the green line shows the actual (lowest-barrier) path, which would be followed in the thermally activated regime, where ℏω‡ < kBT/2π.

Midgap Electronic States in Semiconductor Glasses. The domain walls separating distinct aperiodic states are a necessary consequence of the symmetry breaking at Tcr in finite dimensions. While analogous to the domain walls in a ferromagnet, in this regard, they are subtler in that an interface between disordered structures may not have an obvious spatial signature, even though it incurs excess strain. The interface surely gives rise to “dynamic heterogeneity”. This heterogeneity has be seen directly by Gruebele et al.33 by using scanning tunneling microscopy. Nonlinear spectroscopy34,35 and monitoring spatially inhomogeneous diffusion of small probes36 also show its presence. In a formal treatment, dynamical heterogeneity manifests itself through multiple-time correlation functions.37−39 Despite this subtlety, the domain walls represent a “lattice defect” that could host midgap electronic states, not unlike the free surface of a solid. The electronic states arise from explicit coupling between the structural transitions in the glassy mosaic and the electrons via the modulation of the bond length, and hence the electron-transfer integral, during the transition.4 A pertinent example of such coupling is the Peierls instability of onedimensional metals, at half-filling, toward doubling-up of the unit cell size and a concomitant metal-to-insulator transition.40 The trans-polyacetylene, with its perfect alternation between single and double bonds, is an example of such a Peierls insulator. Defects in the perfect alternation pattern, represented by either over- or undercoordinated atoms, host midgap electronic states.7 The defects separate two possible dimerization patterns of the chain, both of which are symmetry-broken versions of the original 1D metal. At a defect, the symmetry becomes locally restored, allowing for midgap states; see a specific example in Figure 3. The states are surprisingly delocalized, considering their depth. The defects also show the reverse charge−spin relation, whereby a neutral defect has a spin of 1/2.

Figure 3. (a) Central part of a neutral (AsH2)21 chain, whose ground state contains a coordination defect and the associated midgap level.5 (b) Corresponding electronic energy levels: full MO calculation (crosses) versus a one-orbital model with renormalized ppσ integrals (circles). States below the gap are filled; the midgap state is half-filled. (c) The wave function squared of the midgap state; the circles correspond to the arsenics' pz atomic orbitals (AOs) (total contribution 73%), triangles are As's s-AOs (21%), and crosses are the rest of the AOs (6%).

In the specific case of chalcogenide glasses, the atomic motifs leading to the midgap states can be identified based on a simple chemical picture. These materials are symmetry-broken and distorted versions of much simpler “parent” structures defined on the simple cubic lattice.5,6 Building on the work of Burdett and co-workers on the structure of solid arsenic and electronically similar compounds,41,42 parent structures for various

The interface-based midgap states in chalcogenides, when neutral, are solid-state analogues of free radicals.

Figure 4. A parent structure for a Pn2Ch3 crystal, such as crystalline As2Se3 and As2S3. The bond placement represents a particular symmetry-breaking pattern. 3



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combinations of chalcogens and pnictogens can be drawn; see Figure 4. Much as the crystal of As2Se3 can be thought of as a distorted version of the parent structure in Figure 4, chalcogenide glasses can also be thought of as being distorted versions of parent structures in which atoms and vacancies are placed aperiodically. In either case, a pnictogen (chalcogen) forms three (two) covalent bonds with its nearest neighbors and a few secondary bonds43 with next-nearest neighbors. The covalent bonds must be approximately orthogonal. The covalent and secondary bonds, depicted by the solid and dashed lines in Figure 4, correspond to the double and single bond in the trans-polyacetylene analogy, while the bonding now is ppσ. Covalent and secondary bonds that are on the opposite sides of an atom and are approximately colinear share electrons in a synergistic fashion; the secondary bond becomes stronger at the expense of its counterpart covalent bond.44 In addition to being aperiodic, a parent structure for glass must also contain under(over)-coordinated atoms. On a pnictogen, for instance, such malcoordination can correspond to two colinear secondary (covalent) bonds, along any of the three principal axes. In actual distorted structures, the malcoordination is difficult to detect because it is distributed over a large region, consistent with the general difficulty in defining coordination in aperiodic lattices. When sp mixing is relatively weak, the distorted cubic lattice can be presented as a collection of long stretches of distorted 1D chains. The rest of the lattice, to a good approximation, renormalizes the parameters of the chain.5,6 The quasi-one dimensionality of the midgap electronic states can be rationalized on general grounds:4 The strain interaction between bead moves decays with distance as 1/r3. Thus, consecutive moves are likely to be nearby in space because the motion of an isolated bead is always subject to the elastic restoring force. The bead moves during a structural reconfiguration can be unambiguously numbered in the order in which they would occur during the transition. In the resulting 1D chain, neighbors are also neighbors in 3D space. Using these notions of quasi-one dimensionality, a concrete example of a malcoordination pattern can be obtained with the help of an actual 1D chain of ppσ-bonded molecules, such as the hydrogen-passivated arsenics in Figure 3. As expected, a neutral chain hosts an unpaired electron in its nonbonding orbital. The latter is quite delocalized indeed. Consequently, such radicals may attract electrons and holes, as borne out by the Lewis octet rules in the ultralocal limit of the theory.6 Such stabilization certainly occurs when a pair of over- and undercoordinated centers are nearby.4 If a midgap state occupied by two electrons is lower in energy, per electron, than a singly occupied state, this implies an effective attraction between like charges, not unlike that in superconductors, but with the difference that the orbitals are localized. Consistent with the mobility of the domain walls, the malcoordination motifs can move and make turns by switching bonds, very much like in the Grotthuss mechanism of bond switching in water; see Figure 5. The presence of the midgap states helps rationalize a number of electronic anomalies in amorphous chalcogenides8 in a unified fashion. Perhaps the most important of these are the observation of a light-induced paramagnetic signal and midgap absorption of light. Two types of photoluminescence and their fatigue also indicate the presence of some sort of “defect states” activated by irradiation at the gap frequency. At the same time, because the Fermi level is apparently pinned close to the middle of the gap while conventional doping of chalcogenides is inefficient,45,46 the defect states must be intrinsic. In the

Figure 5. Illustration of the motion of an overcoordination defect 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) to atom 2 in (c)).

present microscopic picture, the defects are intrinsic because they represent transition-state configurations for activated transport that are inherent in the formation of a glass. Conversely, amorphous materials that cannot be made by quenching a melt, such as amorphous silicon, do not generally exhibit this type of light-induced anomalies, but they can host unpaired spins.8 Detailed analysis4 shows that in a pristine sample of chalcogenide glass, the majority of the states are occupied, as already discussed, and are ESR-inactive. Although the states are not very close to the band edge, there is no subgap absorption because of the Franck−Condon effect: The optical excitation is faster than the atomic movements, implying the energy needed to excite the electron from a filled level also includes the work needed to deform the lattice from the configuration that it would have when the level is half-filled to when it becomes completely filled. When half-filled, the states will absorb light at subgap frequencies.

Consistent with the mobility of the domain walls, the malcoordination motifs can move and make turns by switching bonds, very much like in the Grotthuss mechanism of bond switching in water. The midgap states are a direct consequence of the local metastability in glasses. Above the glass transition temperature, transitions between the distinct states occur on the time scale of the experiment but are largely frozen at temperatures significantly below Tg. A small subset of these motions remains mobile, however, even at cryogenic temperature. Two-Level Systems. The puzzling low-temperature anomalies in glassy solids, often associated with the so-called two-level systems (TLS),9 have enjoyed a reputation not unlike that of the dark matter in astrophysics. There are clearly low-energy 4



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structural excitations in glasses active at cryogenic temperatures T < 10 K. These account for the majority of the specific heat, phonon scattering, and spectral diffusion in single-molecule experiments, 47 among many other things. A puzzling observation is the universality of phonon scattering by the TLS. The ratio of the phonon mean-free path to the wavelength is about 150 for all substances, independent of chemistry!48 Identification of these excitations in terms of atomic motions has resisted direct experiment for 4 decades.49 In a recent breakthrough, TLS-like cooperative rearrangements have been observed on the surface of a metallic glass.33 We have argued that these mysterious structural excitations are a necessary consequence of the activated dynamics, which precedes the glass transition.2,10 At fixed spatial density of the domain walls, from eq 2, the spectral density of states, near the glass transition, must follow an exponential distribution with the energy scale given by Tg itself2,10

n(E) ≃

1 Tg ξ3

e

Figure 6. Tunneling to the alternative state at energy ε can be accompanied by a distortion of the domain boundary, thus populating the vibrational states of the domain walls. All transitions exemplified by solid lines involve tunneling between the intrinsic states and are coupled linearly to the lattice distortion and contribute the strongest to the phonon scattering. The “vertical” transitions, denoted by the dashed line, are coupled to higher-order strain and contribute only to Rayleigh-type scattering, which is much lower in strength than that due to the resonant transitions.

E / kBTg

(3)

Aside from a decreased magnitude of thermal vibrations and some aging, the structure is expected to persist to all temperatures below the glass transition, no matter how low. Thus, in the classical limit, the spectrum from eq 3 should apply even at cryogenic temperature. The majority of transitions between these states are subject to very high barriers, however, and would be inactive. Nevertheless, a simple argument, which has been called the library construction,12 demonstrates that the spectrum of the transition states for the transport is the same as that for the typical states, except that it is shifted up in energy owing to the mismatch penalty between distinct aperiodic structures; see Figure 2. The resulting distribution of the transition-state energies is exponential,2 similarly to eq 3; it is small but nonzero even significantly below the typical trajectory. This is expected because the glass is well above the ground state of the solid, roughly by TgSc(Tg) energy-wise. Some of the states intermediate between the glass and the ground state (which is often a periodic crystal) are accessible by tunneling. A region already of size 10% larger than the cooperative region size at Tg has at least one tunneling trajectory that possesses a nearly zero barrier. The resulting density of the quantum excitations, 1/Tgξ3, is thus determined by the characteristics of the lattice set at the temperatures of preparation. This preparation temperature scale is 2 orders of magnitude larger than the ambient cryogenic one! The tunneling events encompass several hundred atoms and correspond to transitions between the two lowest states in Figure 6. The tunneling transitions are coupled to the phonons; the coupling is set by the crossover temperature, where the transitions are marginally stable against vibrations.2,10 Because of the flat density of states, the phonon mean-free path is inversely proportional to the temperature.50 As a result, the ratio of the mean-free path to the thermal phonon wavelength is a constant. This turns out to depend only on the size of the cooperative region by the present argument:2,10 lmfp/λ = 1/N*(a/ξ)3, thus explaining the aforementioned universality48 in phonon scattering. At somewhat higher temperatures, the domain walls are expected to be vibrationally excited.10,51 In structural terms, these vibrations correspond to the uncertainty of the domain wall position. The density of these excitations is similar to the spectrum of a hollow vibrating sphere of diameter ξ (see Figure 6)

with a characteristic frequency scale

ωBP ≃ (a /ξ)ωD

(4)

where ωD is the Debye frequency. The resulting excess heat capacity and phonon scattering were shown to match observation well. Summary and Outlook. The degeneracy of an equilibrated liquid above the glass transition leads to the emergence of a nontrivial length scale and a distribution of time scales for atomic motions. These scales are not typically present in periodic crystals or materials that cannot be quenched into an aperiodic lattice in a quasi-equilibrium fashion. The length scale, ξ, characterizes the dynamical heterogeneity of the glass in the form of a “mosaic”3 of transition states for activated transport, while the time scales correspond to the (broadly distributed) activation times. The dynamical heterogeneity serves as an efficient electron scatterer in chalcogenide alloys. The corresponding resonance gives rise to midgap states in these materials that are in excess of the allowed states,52 at a concentration of 1/ξ3. On the other hand, a small subset of the structural reconfigurations are low-barrier and remain thermally active down to very low temperatures. These degrees of freedom reveal themselves in excess phonon scattering and specific heat in excess of the Debye density of states, again at the characteristic length ξ.

The tunneling events encompass several hundred atoms. In the above discussion, the quantum degrees of freedom play a perturbative role on a still classically determined glassy landscape. There may be systems in which the glass transition and quantum fluctuations are more intimately related, such as the stripe glass in electronic systems.53 Current research 5



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(10) Lubchenko, V.; Wolynes, P. G. The Microscopic Quantum Theory of Low Temperature Amorphous Solids. Adv. Chem. Phys. 2007, 136, 95−206. (11) Martinez, L.-M.; Angell, C. A. A Thermodynamic Connection to the Fragility of Glass-Forming Liquids. Nature 2001, 410, 663. (12) Lubchenko, V.; Wolynes, P. G. Theory of Aging in Structural Glasses. J. Chem. Phys. 2004, 121, 2852. (13) Lubchenko, V.; Wolynes, P. G. Theory of Structural Glasses and Supercooled Liquids. Annu. Rev. Phys. Chem. 2007, 58, 235−266. (14) Singh, Y.; Stoessel, J. P.; Wolynes, P. G. The Hard Sphere Glass and the Density Functional Theory of Aperiodic Crystals. Phys. Rev. Lett. 1985, 54, 1059−1062. (15) Lubchenko, V.; Wolynes, P. G. Barrier Softening near the Onset of Nonactivated Transport in Supercooled Liquids: Implications for Establishing Detailed Connection between Thermodynamic and Kinetic Anomalies in Supercooled Liquids. J. Chem. Phys. 2003, 119, 9088−9105. (16) Lubchenko, V. A Universal Criterion of Melting. J. Phys. Chem. B 2006, 110, 18779−18786. (17) Lindemann, F. A. Ü ber die Berechnung Molekularer Eigenfrequenzen. Phys. Z. 1910, 11, 609. (18) Kirkpatrick, T. R.; Wolynes, P. G. Connections Between some Kinetic and Equilibrium Theories of the Glass Transition. Phys. Rev. A 1987, 35, 3072−3080. (19) Casalini, R.; Roland, C. M. Viscosity at the Dynamic Crossover in o-Terphenyl and Salol under High Pressure. Phys. Rev. Lett. 2004, 92, 245702. (20) Novikov, V. N.; Sokolov, A. P. Universality of the Dynamic Crossover in Glass-Forming Liquids: A “Magic” Relaxation Time. Phys. Rev. E 2003, 67, 031507. (21) Stevenson, J. D.; Schmalian, J.; Wolynes, P. G. The Shapes of Cooperatively Rearranging Regions in Glass-Forming Liquids. Nat. Phys. 2006, 2, 268−274. (22) Kirkpatrick, T. R.; Thirumalai, D.; Wolynes, P. G. Scaling Concepts for the Dynamics of Viscous Liquids Near an Ideal Glassy State. Phys. Rev. A 1989, 40, 1045−1054. (23) Stevenson, J.; Wolynes, P. G. Thermodynamic-Kinetic Correlations in Supercooled Liquids: A Critical Survey of Experimental Data and Predictions of the Random First-Order Transition Theory of Glasses. J. Phys. Chem. B 2005, 109, 15093− 15097. (24) Lubchenko, V. Shear Thinning in Deeply Supercooled Liquids. Proc. Natl. Acad. Sci. U.S.A. 2009, 106, 11506−11510. (25) Lubchenko, V. Charge and Momentum Transfer in Supercooled Melts: Why Should Their Relaxation Times Differ? J. Chem. Phys. 2007, 126, 174503. (26) Bevzenko, D.; Lubchenko, V. Stress Distribution and the Fragility of Supercooled Melts. J. Phys. Chem. B 2009, 113, 16337− 16345. (27) Hall, R. W.; Wolynes, P. G. Intermolecular Forces and the Glass Transition. J. Phys. Chem. B 2008, 112, 301−312. (28) Stevenson, J. D.; Wolynes, P. G. On the Surface of Glasses. J. Chem. Phys. 2008, 129, 234514. (29) Wolynes, P. G. Spatiotemporal Structures in Aging and Rejuvenating Glasses. Proc. Natl. Acad. Sci. U.S.A. 2009, 106, 1353− 1358. (30) Stevenson, J. D.; Wolynes, P. G. A Universal Origin for Secondary Relaxations in Supercooled Liquids and Structural Glasses. Nat. Phys. 2010, 6, 62−68. (31) Bhattacharyya, S. M.; Bagchi, B.; Wolynes, P. G. Subquadratic Wavenumber Dependence of the Structural Relaxation of Supercooled Liquid in the Crossover Regime. J. Chem. Phys. 2010, 132, 104503. (32) Stevenson, J. D.; Wolynes, P. G. The Ultimate Fate of Supercooled Liquids. J. Phys. Chem. A 2011, 115, 3713−3719. (33) Ashtekar, S.; Scott, G.; Lyding, J.; Gruebele, M. Direct Visualization of Two-State Dynamics on Metallic Glass Surfaces Well Below Tg. J. Phys. Chem. Lett. 2010, 1, 1941−1945. (34) Tracht, U.; Wilhelm, M.; Heuer, A.; Feng, H.; Schmidt-Rohr, K.; Spiess, H. W. Length Scale of Dynamic Heterogeneities at the Glass

suggests that quantum fluctuations may directly influence the glass transition depending on their strength. When weak, the fluctuations seem to augment the glass transition temperature, but when sufficiently strong, they suppress the transition. The magnitude of the effect seems to depend on the specific model and approximation.54−56 Finally, the extent of the midgap states in chalcogenides scales inversely proportionally with the gap. In view of the effective attraction between electrons, one expects that at high enough pressure, the midgap orbitals will overlap, leading to superconductivity.4 Chalcogenides are known to become superconducting at high pressure.57 It will be interesting to see whether this very quantum phenomenon originates in things glassy.

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AUTHOR INFORMATION

Corresponding Author *E-mail: [email protected]. Biography Vassiliy Lubchenko is an Associate Professor of Chemistry at the University of Houston. His research involves strongly nonequilibrium phenomena with applications to materials science and biology, including the structural glass transition, electronic structure of vitreous materials, and protein aggregation. http://www.chem.uh.edu/people/ faculty/lubchenko/

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ACKNOWLEDGMENTS The author gratefully acknowledges Peter G. Wolynes, Andriy Zhugayevych, and Pyotr Rabochiy, with whom much of the described work was performed. His work has been supported by the Arnold and Mabel Beckman Foundation Beckman Young Investigator Award, the National Science Foundation (Grants MCB-0843726 and CHE-0956127), the Alfred P. Sloan Research Fellowship, and the Welch Foundation.

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REFERENCES

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Quantum Phenomena in Structural Glasses - American Chemical

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