A new paradigm for the low-T glassy-like thermal properties of solids
AA new paradigm for the low- T glassy-like thermal properties of solids Matteo Baggioli ∗ Instituto de Fisica Teorica UAM/CSIC, c/Nicolas Cabrera 13-15,Universidad Autonoma de Madrid, Cantoblanco, 28049 Madrid, Spain.
Alessio Zaccone † Department of Physics ”A. Pontremoli”, University of Milan, via Celoria 16, 20133 Milan, Italy.Department of Chemical Engineering and Biotechnology,University of Cambridge, Philippa Fawcett Drive, CB30AS Cambridge, U.K.Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, CB30HE Cambridge, U.K.
Glasses and disordered materials are known to display anomalous features in the density of states,in the specific heat and in thermal transport. Nevertheless, in recent years, the question whetherthese properties are really anomalous (and peculiar of disordered systems) or rather more universalthan previously thought, has emerged. New experimental and theoretical observations have ques-tioned the origin of the boson peak and the linear in T specific heat exclusively from disorder andTLS. The same properties have been indeed observed in ordered or minimally disordered compoundsand in incommensurate structures for which the standard explanations are not applicable. Usingthe formal analogy between phason modes (e.g. in quasicrystals and incommensurate lattices) anddiffusons, and between amplitude modes and optical phonons, we suggest the existence of a moreuniversal physics behind these properties. In particular, we strengthen the idea that linear in T specific heat is linked to low energy diffusive modes and that a BP excess can be simply induced bygapped optical-like modes. Introduction – Anomalous low-temperature thermalproperties play a big role in the mysterious nature ofthe glassy state – which is considered ”perhaps thedeepest and most interesting unsolved problem in solidstate theory”. On one side, glasses exhibit a linear intemperature contribution to the non-electronic specificheat [1] at low T ; from the other, together with variousdisordered materials, they display a characteristic excessin the vibrational density of states (VDOS) – the bosonpeak (BP) – which violates the standard result fromDebye theory, i.e. C ( T ) ∼ T [1–4].The most (and probably only) accepted theory toexplain the linear in T contribution relies on so-called two-level states (TLS) [5–7], while many theoreticalexplanations for the BP anomaly have been proposed[8–10] over the past decades. Importantly, all thesetheoretical frameworks rely strongly on disorder invarious forms and on the existence of new low energydegrees of freedom, the exact physical nature of whichhas remained somewhat mysterious [11].In addition to this questionable confidence andtheoretical confusion, the “anomalous” nature of glassyfeatures in solids is in deep crisis [12]. This is becausethe same allegedly “anomalous” features have beensystematically observed in non-disordered systems,atomic and molecular cryocrystals [13–15] and in qua-sicrystals/incommensurate structures [16]. ∗ [email protected] † [email protected] Not only these features appear to be quite universalrather than anomalous, and not limited to glasses,but the chase for the universal physics or unifyingprinciples behind these phenomena is completely open.In particular, neither disorder nor Two Level Systems(TLS) can be the ultimate causes of these phenomenasince obviously no disorder and no TLS can be found inordered crystals and in quasicrystals.Recently, the idea that these (not anymore anoma-lous) features can actually be induced by differentphysical mechanisms has gained strength, supported byexperimental observations [13–16] and first principlestheoretical analyses. Robust theoretical frameworks,which fit harmoniously with several experimental data[17] and observations [13–16], have proven that a BPanomaly in crystals can be induced by overdampedlow-energy modes and (Akhiezer-type) anharmonicity[18, 19] and also by the piling up of softly gappedoptical-like modes [20, 21]. Moreover, it has been shown[22] that low-energy diffusive excitations produce a linearin T contribution to the specific heat, in agreement withthe results from random matrix theory [23].In this Letter, we provide more evidence for the caseby considering the low-temperature thermal propertiesof quasicrystals and incommensurate structures. Bothanomalous features (BP and linear in T specific heat)have been experimentally observed in quasicrystals [16].Phenomenological models have been constructed andpinpoint as the origin of these anomalies the dynamicsof the phason and amplitude modes [24, 25]. Here, weshow that everything can be explained in a universal,simple and elegant way using first principles ideas about a r X i v : . [ c ond - m a t . d i s - nn ] A ug the underlying symmetries.The quasicrystal case provides another direct proofagainst the anomalous nature of glasses and substantialevidence towards the idea that disorder and TLS cannotbe the universal causes behind these phenomena. Thekey of universality simply lies in the formal nature (i.e.mathematical dispersion relations) of the low-energyexcitations from which the (no longer) “anomalous”features can be analytically derived. Quasicrystals – Quasicrystals are crystalline struc-tures possessing long range orientational order butlacking full translational periodicity [26–29]. Differentlyfrom glasses or amorphous systems, they display sharpBragg peaks easily observable with X-ray diffraction.We refer to [30–34] for a more extensive literature.The simplest examples are Penrose tilings and incom-mensurate structures. From a theoretical point ofview, the best way to describe them is by using anextra-dimensional superspace formalism based on themathematical fact that any aperiodic structure in d dimensions is equivalent to a periodic one in a D > d manifold cut at an irrational angle α [35]. In otherwords, to describe the crystalline structure of aperiodiccrystals one needs a number of independent wave-vectors D larger than the real spacetime dimensions d . Thisextra dynamics can be attributed to the so-called phasonmode [36–42]. In the superspace formalism the phasonsimply corresponds to the internal shifts of the cutwithin the extra-dimensional manifold.As a concrete case, let us focus on the dynamics of amodulated structure whose density is described as: ρ ( r ) = ρ + R cos ( k r + φ ( r )) (1)where k is the wave-vector of the substrate periodic lat-tice (e.g. the ionic lattice) and φ the phase of the mod-ulation. Two possible excitations are possible: ρ ( r ) = ρ + ( R + δR ( r )) cos ( k r + φ ( r ) + δφ ( r )) (2)where δR corresponds to the fluctuation of the amplitude– amplitudon – and δφ to that of the phase – phason ,(see Fig.1 for a visual representation). We refer to [43]for a review of the role of these two types of excitationin condensed matter.The crucial point is that, because of the lack of period-icity, there is an extra hydrodynamic ( ≡ massless) mode,the phason. Even more importantly, in the limit of smallwave-vectors, its dispersion relation is diffusive , as con-firmed by several experimental checks [44, 45]. Using astandard hydrodynamic formalism [46], the low energydynamics of the phason is given by the dynamical equa-tion: ∂ t Q φ + γ Q φ = − ∂ F ∂φ , (3) phason amplitudon Figure 1. The two types of excitations in incommensuratestructures. The phason, a shift in the phase of the modulatedstructure and the amplitudon, a variation in its amplitude. where φ is the phason shift, Q φ the conjugate momen-tum and F the free energy that now depends both onthe phonon and phason shifts (see [46] for details). Theparameter γ is the phason damping and the r.h.s. of theequation above is responsible for the propagative con-tribution in terms of the new phason (generalized) elas-tic moduli. Eq.(3) has exactly the same dynamics thatone would find in a simple linear Maxwell model for vis-coelasticity. Not surprisingly, the phason shows a k-gapcrossover, which is postulated also for transverse phononsin liquids [47] . The phason damping is due to the factthat phason shifts are symmetries of the free energy butthey do not commute with the Hamiltonian [42]. Tech-nically, they are symmetries with no associated Noethercurrent [48]; physically, phason shifts cost energy becausethey correspond to atomic jumps and flips of the aperi-odic structure (in analogy to atomic re-arrangements inliquids).All in all, the solution of (3) can be written as ω + i ω γ = v k + . . . , (4)which, for small momenta, predicts a diffusive behaviour ω = − i D k + . . . ; D = v /γ . (5)The same result can be obtained from Frenkel-Kontorovamodels upon introducing a dissipative coefficient for rel-ative motion between the two incommensurate superim-posed lattices [49]. At larger momenta, the phason re-covers the standard propagative behaviour [42], and canbe thought of simply as an additional sound mode. Notice the difference with an attenuated sound mode following ω + i Γ ω k = v k . kIm ( ω ) kRe ( ω ) phason amplitudon Figure 2. The low-energy dispersion relation of the phasonand amplitudon excitations. The phason is a diffusive mode.The amplitudon is a gapped mode.
On the contrary, the amplitudon is not a hydrodynamicgapless mode but rather a massive gapped excitationwith dispersion relation: ω = ω + v k + . . . (6)In a sense, amplitudons can be thought of as additionaloptical-like phonon modes with the subtle differencethat optical phonons do not have a clear interpretationin terms of spontaneous symmetry breaking (SSB) .The amplitude mode is usually labelled as a “Higgsmode”, in analogy to the Higgs particle, and its massivenature comes entirely from the SSB mechanism (e.g.amplitudons in superconducting transitions [52]). Specific Heat –Interestingly, a robust linear in T con-tribution to the low-temperature specific heat has beenobserved in incommensurate structures [16]. Clearly,such feature cannot be attributed to the presence of two-level states, but it was rather connected to the dynamicsof the phason mode. More specifically, a phenomenolog-ical computation `a la Landau [24] found that: C ∼ γ k D v T , (7)where γ is the phason relaxation rate, k D the Debyemomentum and v the asymptotic phason speed.Here, we show that such contribution is the typical anduniversal term appearing because of diffusive low energymodes due to the underlying fundamental symmetries.Using Eq.(5), the result (7) from [24] can be re-writtenas: C ∼ k D D T , (8) Unless the optical-like phonons come from the interaction oflayered structures [50] (in this context they are usually calledpseudo-acoustic modes [51]). k Re [ ω ] T C ( T )/ T Figure 3. The contribution of gapped optical-like modes tothe Debye-normalized specific heat. The orange curve is ob-tained using a single gapped mode; the green one by addingup three gapped modes. The inset shows the dispersion rela-tion of the modes. which is exactly the leading order term in [22] (see Eqs.(10) and (13) therein). In Ref.[22], this result was ob-tained from consideration of “diffusons” [53] as the rele-vant excitations which manifest as a result of structuraldisorder in glasses, but share exactly the same mathe-matical form with (5) for phasons above.The result of [22] is a first-principles analytic, andmuch simpler, computation which goes beyond theresults of [24] and it assumes solely the existence of adiffusive low energy mode (in this case the phason).This suggests that the ”anomalous” linear in T scalingof the specific heat in quasicrystals is due to the diffusivenature of the phason and can be understood in thegeneral picture of [22]. In this sense, phasons in quasy-cristal/incommensurate structures play exactly the samerole as the diffusons in glasses and disordered materials[53]. Given the modest experimental evidence for TLSin glasses, and their non-applicability to quasicrystals,we propose that the presence of low-energy diffusivemode could be the real universal mechanism behind the(not so) anomalous linear in T specific heat.In addition to the linear in T contribution to thespecific heat, a BP excess is experimentally observedin incommensurate structures as well [16]. The originof this anomalous bump has been already attributedto the presence of gapped amplitudon modes [25]. Thesimilarities between this mechanism and those at workin ordered crystals [54] went unnoticed until now. Theresemblances are striking. It has been shown from directmeasurements of the density of states and the low-Tspecific heat that the BP anomaly in ordered crystals[54] is caused by the piling up of optical modes with asoft energy gap.Both the observations, in quasicrystals and in orderedcrystals, point towards the confirmation that gappedoptical-like modes are able to produce a well-definedBP excess without the need of any structural disorder.These experimental facts are backed up by a firstprinciple theoretical computation [20] which is in perfectagreement with the experimental results presented inboth cases.The idea is very simple and again based on symme-tries and the low-energy dispersion relation of the modes.Take an underdamped gapped mode whose dispersion re-lation is well approximated by the solution of: ω − ω − v k + i Γ ω = 0 , (9)where ω is the energy gap of the mode. The parameterΓ is the (Klemens) damping of the optical mode and itis taken to be Γ (cid:28) ω such that the gapped excitationis a well-defined quasiparticle. Eq.(9) corresponds to aquasiparticle Green function of the form: G ( ω, k ) = 1 ω − ω − v k + i Γ ω . (10)Using the spectral representation of the density of states: g ( ω ) = 2 ωπ k D Im (cid:90) k D G ( ω, k ) d k (11)together with the standard formula for the specific heat: C ( T ) = k B (cid:90) ∞ (cid:18) (cid:126) ω k B T (cid:19) sinh (cid:18) (cid:126) ω k B T (cid:19) − g ( ω ) dω , (12)it is straightforward to compute the contribution of thegapped optical-like modes in Eq.(9).The result is shown in Fig.(3) for an arbitrary choiceof parameters. Importantly, the contribution of thegapped modes gives a BP excess whose location is mostlygoverned by the energy gap of the mode. Moreover, it isclear that piling up a large number of close-by gappedmodes increases the amplitude of the BP and amplifiesthe effect. This simple computation provides a cleartheoretical confirmation for the experimental resultsfound in ordered crystals [54] and quasicrystals [16].Once again, the similar physical behaviour obtained intwo completely different systems calls for a universalmechanism, which can be found in the gapped nature ofthe low-energy optical-like modes. Discussion –In this Letter, we demystify the natureof anomalous glassy features in the low- T properties ofsolids. We collected experimental and theoretical evi-dence which supports the following emerging paradigm:(I) the anomalous thermal properties of glasses are notpeculiar but can be found in different structures suchas ordered crystals and incommensurate systems. Inthis sense, the label “anomalous” becomes meaningless.(II) Disorder (and all the theoretical models whichincarnates it) and the presence of “randomly distributedtwo-level systems” cannot be the universal causes behind d i ff u s i ve m od es D e b ye O p t i ca l li ke g a pp e d m od es T0C ( T )/ T Figure 4. The typical structure of the Debye-normalized spe-cific heat in glasses, ordered crystals and incommensuratestructures. At low temperature, a ∼ /T fall-off is visiblebefore the Debye regime ∼ cost . Finally, a BP bump is ob-served at higher temperatures. In our interpretation, all thesefeatures can be universally explained by the presence of quasi-localized diffusive modes and soft optical-like gapped modes. this much more general physics. They certainly cannotaccount for these features in ordered systems and inincommensurate systems.We propose that a universal mechanism behind thelinear in T specific heat can be identified with theexistence of low-energy quasi-localized diffusive modes.The existence of these modes can be caused by diversephysical features such as disorder-induced scattering(e.g. diffusons in glasses) or new low-energy dynamicsdue to incommensuration and aperiodicity (e.g. phasonsin quasicrystals). We are able to prove analyticallythat low-energy diffusive modes give a linear in T contribution to the specific heat, whose leading term isin perfect agreement with the phenomenological analysisof [24].In the same spirit, we argue that the presence ofa boson peak anomaly can be generically induced bystrong Akhiezer damping effects (due to high anhar-monicities) and/or by the piling up of soft optical-likemodes. The first mechanism could be responsible forthe BP observed in the low-T specific heat of molecularcrystals and cryocrystals [13], where anharmonicity isstrong due to the shallow attractive part of the (van derWaals-type) interaction potential, thus leading to strongAkhiezer damping, which implies a diffusive mode, andhence to a BP [18]. The second mechanism is the oneat work in certain ordered crystals, where the BP hasbeen successfully linked to low-lying optical phonons[15] and quasicrystals, where the gapped modes are theamplitude fluctuations of the incommensurate structure[25].Given these interesting formal analogies betweendiffusive excitations in glasses and aperiodic crystals, itis likely that the same low-temperature anomalies couldbe found in the thermal conductivity κ . Not surprisingly,a ∼ T scaling and a glass-like plateau have alreadybeen observed in the thermal conductivity of aperiodiccrystals [55, 56]. It is tempting to attribute them to theexistence of diffusive phason modes playing the samerole of diffusons [53] in glasses. This would provide afinal confirmation of the universal picture presented inthis work.Moreover, phasons appear also in twisted bilayergraphene (TBG), where they might play a key rolefor transport and thermodynamics [57]. Followingthe same logic, it is tempting to predict the pres-ence of a linear in T term in the vibrational specificheat of TBG. To the best of our knowledge, measure-ments of the specific heat in TBG have not appeared yet. In summary, we suggest that the universal (andnot anymore anomalous) glassy-like features have to begenerically understood from the nature of the low-energyexcitations (see Fig.4 for a visual summary). The latteris the key behind these, more ubiquitous than originallythought, features, which importantly do not dependexclusively on the presence of disorder or any amorphousstructure. 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