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Peierls transition as a spatially inhomogeneous gap suppression V. Ya. Pokrovskii,A.V. Golovnya,and S.V. Zaitsev-Zotov Institute of Radioengineering and Electronics, Russian Academy of Sciences, 103907 Moscow, Russia (February 2, 2008) We propose a model of the Peierls transition (PT) taking into account amplitude fluctuations of the charge-density waves and spontaneous thermally activated suppression of the Peierls gap, akin 4 tothephaseslipprocess. Theactivationresultsintheexponentialgrowthofthenormalphasewith 0 increasing temperature. The model fairly describes the behavior of resistance, thermal expansion, 0 Young modulus and specific heat both below and above the PT temperature TP. The PT appears 2 to have a unique nature: it does not comprise TP as a parameter, and at the same time it has n featuresofthe1stordertransition. Thepossiblebasisforthemodelisactivationofnon-interacting a amplitude solitons perturbinglarge volumes around them. J 9 PACS Numbers: 71.45.Lr, 71.30.+h, 68.35.Rh 1 ] Description of the Peierls transition (PT) in quasi 1- intriguing to extend the aproach for the description of l e dimensional(1D)conductorsstillremainsacontroversial the PT. - r problem. The widely used mean-field (MF) approach Inthe presentLetter we proposea model inwhich the st works poorly, firstly because of strong 1D fluctuations. fluctuations of ∆ are phenomenologically introduced as . E.g., it predicts the PT temperature T much above thermally activated local gap suppression (LGS). With t P a the observedvalue. Say, for the typical compound TaS3, increasing T this process gives rise to the activation m the energy gap 2∆ is 1600 K, and the MF value of T growthofconcentrationofnormalphaseasexp(−W/T), P - shouldbe 2∆/3.5=460K,while the experimental value W ≫ ∆. The approach can be extrapolated above the d n is 220 K. The large fluctuations reveal themselves well transition temperature. PT appears to be smeared out, o aboveTP: the valueofthe pseudogapisclosetothelow- butatthe sametime hasfeaturesofthe1stordertransi- c temperaturevalue[1,2],thethresholdnon-linearconduc- tion. The model appears to describe fairly the behavior [ tion [3] indicates the charge-density wave (CDW) state of various parameters near TP. The plausible basics for 1 within the fluctuating volumes. At present only quali- themodelistheexcitationofamplitudesolitonsperturb- v tative attempts to explain these experimental facts are ing considerable volumes around them. 5 undertaken. Though certain success is achieved in fit- To introduce the fluctuations, we shall use the follow- 3 ting the behavior of different values near T [4–6], the ingrelationgivingthefrequencyoftheLGSactsperunit 3 P 1 relations involved are semi-empirical, and their physical volume (see also [14]): 0 sense is not quite clear. Another treatment of the PT is 4 given by the generalized Ehrenfest relation [7] between f =faexp(−W/T), (1) 0 the specific heat, expansivity, Young modulus anoma- / where f is an attempt frequency. Here the essential t lies and stress-induced shift of the transition tempera- a a point is that each LGS act results in a temporal nucle- m ture. This relation works nice for some materials, e.g. for K0.3MoO3 - the blue bronze [4], but fails for others, ation of the normal–state volume v0 having certain life- - time τ. Then the fraction of the normal-state volume d like for TaS3 [8]. due to the spontaneous LGS process is n ThegeneralapproachtotheCDWsistoconsiderthem o as a spatially homogeneous state up to T . However, c P v =v0τf. (2) a recent theoretical study has demonstrated that ther- : v mal fluctuations of the CDW stress may be very large This fraction grows exponentially with increasing T. At Xi at T ≈ TP, so that the r.m.s. shift of the chemical po- highenough temperaturev becomes of the orderof 1, so tential level from the middle-gap position appears com- r we shouldtake into accountthe shrinking ofthe Peierls– a parable with ∆ [9], and one can expect temporal local statevolume. So,insteadof(2)weobtain: v =v0τf(1− suppression of the gap. So, it is natural to suppose that v), or the Peierls state in the vicinity of T should be consid- P ered as a mixture of the Peierls phase (the CDW) and v0τf v = . (3) the state with suppressed gap. Studies of noise have re- 1+v0τf vealed spontaneous phase-slippage (PS) process [10,11] in the vicinity of T , which also implies local temporal This relation, the principal one for our model, gives P suppression of the Peierl gap. The PS is fairly described growth of v from 0 to 1 with increasing T. The growth as a thermally activated process [12–14], so it would be has the form of a step centered at T =W/ln(v0τfa), at which v = 1/2. The step is smeared out by ≈ T2/W P 1 x 10−5 −1 TaS 3 TaS 102 3 l / Ω 3000 l−2 R, K) ∆ T) (2000 1/ d( R)/1000 g( dlo −3 0 100 150 200 250 4 6 8 10 T(K) 101 103/T (K−1) FIG.2. δl/l vs. T for aTaS3 sample. Thebackgroundap- 3.5 4 4.5 5 5.5 6 6.5 7 proximated with a 2-nd order polynomial is subtracted. The 103/T, K−1 solid line gives a fit with Eq.(3) with W =6500 K. FIG.1. FitsofatypicalR(T)curveforTaS3 (points)with Eq. (4); for the solid line W = 6600 K, v0τfa = 3·1013, ∆ = 710 K. The broken line shows a similar fit with B = 0. note that we have taken B < 0, which is unreasonable The inset shows thecorresponding logarithmic derivatives. for a metal. Even if we take B = 0, i.e. ρ = const, n thefittingaboveT becomesconsiderablyworse(seethe P broken line). Below we shall introduce a modification of (or 1/W in the 1/T-scale). Extrapolation of the LGS the model above T explaining the slower drop of the P description above T gives a way to treat the entire PT. P CDW fluctuations. Within this approachthe transitionconsistsin the LGS- Letus nowprobe the model for the values whichcom- induced destruction of the CDW state. monly characterize thermodynamic transitions, such as Let us compare the resulting relation (3) with the ex- thermal expansion (TE), Young modulus (Y) and spe- periment. WeshallrefermostlytoTaS3,–atypicalCDW cific heat c (see e.g. [5]). p compound, which is among the widely studied quasi 1D To perform TE measurements, we have developed an conductors[15]. The largeratio2∆/TP for TaS3,as well interferometrictechniqueformeasurementsofneedle-like ashighlyanisotropicstructure(theanisotropyofconduc- samples[19,20]. Fig.2givesthetemperaturedependence tivity is about 100 at room temperature) argue for the ofthe relativelengthchangeδl/l for TaS3 in the vicinity strong fluctuations of the CDW order parameter. of the PT. To exclude the contribution of length hys- We are beginning with the temperature dependence teresis [19], we present the half-sum of the results ob- of resistance, the most common curve characterizing the tained upon heating and cooling the sample. A similar PT. Within the present model we should calculate the curve results if we apply electric field exceeding E to T resistance of a mixture of two phases with different re- remove metastability each time before measuring l. Ev- sistivities, ρ and ρ . We shall consider the Peierls-state c n idently, the curve presented is close to equilibrium. For resistivity ρ ∝ exp(∆/T), and the normal state resis- c all our measurements cooling below T results in a drop P tivity ρn = A+BT, where A and B are constants. To of length by about 10−5. This result is quantitatively calculate the resulting resistivity one should consider a similar to that obtained for K0.3MoO3 in the in-chain complexelectricconnectionofthedomainsofeachphase. direction [21]. For simplicity, we take the contribution of each phase to TEatthePThasbeendiscussedin[21]. E.g., itcould theresistivitytobejustproportionaltothevolumefrac- beunderstoodwithinthe anharmonicmodel,takinginto tion of the phase, as if for connection in series [16]: accounttheanharmoniceffectofthelatticedistortionas- ρ=vρ +(1−v)ρ . (4) sociatedwiththe CDW.Without deepening intodetails, n c we just assume that the length increase with heating is Note,thatthefluctationsareknowntocontributetothe proportional to the fraction of the normal phase. Thus, conductivity of the Peierls phase due to thermal depin- the l(T) step should be described by Eq. (3). The fit ning of the CDW [14,17,18]. According to the model with W =6500 K is quite nice (Fig. 2, the solid line). [14]this contributionisgovernedbythe LGSaswelland As another example we consider the Young modu- grows as exp(−W/T). Here we shall not distinguish it lus temperature dependence, Y(T). One can expect a from that of the normal phase. drop of Y due to the same anharmonic effects. So, with Fig.1presentsanexampleofafitoftheR(T)curvefor T →TP from above one can expect a decrease of Y pro- TaS3withEq.(4). Thefittingissplendid,butoneshould portional to the fraction of the CDW volume. However, 2 10−2 0.04 TaS TaS 0.035 3 10−3 3 K) 1.005 R/ 0.03 1 Y(130 K)10−4 1.0051 dC/dT (p0.025Y(T)/Y(130 K)00..0099..89995589 lin.fit.−Y(T)/1100−−65Y(T)/Y(130 K)00..0099..89995589 FI0G0.0..01125840.0.09.79T5710h02e00t1e5m0 p2eT22,r 00KTa0 (tKu)re22504d0eriva2t6i0ve of the specific heat 0.975 of TaS3. The fit is given by d2v/dT2 (Eq. (3)) with 10−7 0.97 W =10000 K. Thedata are taken from [8]. 100 150 200 250 10−8 T, K 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 (Y −Y ) = 0.0111 in the normalized units. The fit is 103/T, K−1 r p quitenice,butaboveT thefluctuationsfalldownslower P FIG.3. Arrheniusplot of the deviation of Y from a linear than the fit gives, as it could be expected (recall also dependence Y(T). The slope of the solid line corresponds to Fig. 1). W =5600K.TheinsetshowsthefitofthewholeY(T)curve. It is clear from the examples above that the PT con- The broken line takes into account that for high v (T >TP) 1−v∝exp(−W/2T). The data are taken from [6]. sistsinagradualswitchingtothestateasifhavinghigher free energy, and thus looks as a smeared–out 1st order transition. The common check for the 1st order transi- this is not the whole effect. tion is the latent heat. Because of the smearing out one Let us recall that depinning of the CDW below T can expect a maximum of specific heat c . A cusp-like P p reduces the Young modulus [4,22,23]. (This effect is feature is clearly seen on the c (T) curve for the blue p associated with fast relaxation of the CDW deforma- bronze [5]. Recently a similar feature has been observed tions which in the pinned state contribute to Y [23].) also for TaS3 [8]. Being very faint, it was detected as a As we mentioned above, with increasing T, the fluctu- zigzag pattern on the derivative dc /dT. Fig. 4 presents p ations result in the spontaneous depinning of the CDW the data from [8] together with d2v/dT2, Eq. (3) (the [14,17,18],the concentrationof the depinned state grow- backgroundchangeofc is approximatedwithastraight p ing as exp(−W/T) [14], which results in the drop of Y line). It is clear that the form of the feature is at least withapproachingT frombelow. Thus,adip ofY(T)is approximately described by our model. Other words, P expected at T [24]. The value of the depinning drop of the CDW formation is accompanied by a smeared out P Y depends on the particular compound. Being anoma- step oflatent heatwhose width is of the orderof T2/W. P louslystrongforTaS3 [22],itisnotobservedfortheblue Note that the values of the latent heat Q ≈ 0.25 R· K bronze,δY/Y <5·10−5[25,26]. Thus,thedipinY(T)at =5·104J/m3[8](Ristheuniversalgasconstant)andthe T →TP −0 should be large for TaS3, and much weaker, length change δl/l = 10−5 appear to be consistant with if any, for the blue bronze. This expectation agrees with the Clausius-Clapeyron equation dT /dσ = −T δl/Q if p p l theexperiment: insettoFig.3showsY(T)forTaS3from onetakesdTp/dσ∼1K/kbar[15,6](σisthestressalong [6]. LargedropofY is seenwith T approachingT both the chains) [27]. P from above and from below, whereas only a small dip of Above T one should bear in mind the small sizes of P Y(T) at T → T from below is observed for the blue the remnant CDW volumes v . As soon as they shrink P c bronze in the in-chain direction [4,5]. down below v0, Eq. (2) is no longer valid, because the Fig. 3 shows the dependence δY(T) below T in the new normal volume due to the LGS cannot exceed v . P c Arrheniusaxes(alineardependenceY(T)issubtracted). With simple assumptions at high enough T one can ob- The slope of the solid line gives the activation energy tain(1−v)∝exp(−W/2T);theappropriatefitforY(T) 5600 K, giving a good fit nearly up to T . To fit the isgivenwithabrokenlineintheinsettoFig.3. Thiscon- P whole Y(T) curve we present Y as vY +v Y +v Y , sideration also explains the behavior of the R(T) curve n p p r r where the indices n, p and r refer to the normal, pinned above T (Fig. 1). P and relaxed states respectively (v +v +v = 1). The Thus, the LGS model fairly describes the temperature p r drop of Y in comparison with the normal state can be evolution of the principle parameters in the vicinity of presented as δY = (Y − Y )v + (Y − Y )v , where the PT. All the fits proposed have transparent physical r n r p n p v =(1−v)min(f exp(−W/T),1),andv =(1−v−v ). sense. The main parameter of the model, - the energy r r p r The inset to Fig. 3 shows the fit with W = 6000 K, W, is close to the values for the barrier characterizing fr = 7.8·1011, v0τfa = 6·1011, (Yr −Yn) = 0.0443, thermally initiated PS [12–14]. Evidently, LGS is gov- 3 erned by the same process as PS. (Some indications of S.N. Artemenko for helpful discussions. This work has the connection between PS and the PT have been given beensupportedbytheRFBR(grantsNo01-02-17771,02- in the early works [10,11,14,28].) So, it would be nat- 02-17301),Jumelage (CNRS and RFBR), INTAS (grant ural to consider excitation of dislocation loops [29] as a No 01-0474) and by the State programme “Physics of precursor effect below T . Excitation of the dislocation Solid-State nanostructures”. P loopsis being consideredas apossibleoriginofsoftening of solids [30] or similar transitions in liquid helium and HTSCs [31]. This approach gives critical expansion and proliferation of the loops due to their mutual screening. The apparentabsenceofthe criticalbehaviorinourcase couldmeanthat the CDW excitationspracticallydo not [1] M.E.Itkis,F.Ya.Nad’Pis’maZh.Eksp.Teor.Fiz.39,373 interact up to TP. Note that while for a conventional (1984) [JETP Lett 39, 448 (1984)]. crystal the smallest possible radius of a dislocation loop [2] B.P.Gorshunovet al.,Phys.Rev.Lett., 73,308(1994). is of the order of the lattice constant, for the CDW it is [3] V. Ya. Pokrovskii, S. V. Zaitsev-Zotov, P. Monceau, of the order of ξ⊥. Such an object (i.e. an amplitude Phys. Rev.B 51, R13377 (1997) soliton) covers a volume ∼ ξ3 ≡ ξ ξ2, where ξ and ξ [4] J. W. Brill, in Handbook of Elastic Properties of Solids, k ⊥ k ⊥ arethein-chainandthetransverseamplitudecorrelation Liquids, and Gases, edited by M. Levy, H. E. Bass, and lengthsrespectively;thesolitoncanperturbastillhigher R. R. Stern (Academic Press, New York, 2001), Vol. II, volume, where, say, the conductivity is increased. Thus, pp. 143-162. [5] J.W. Brill et al., Phys.Rev.Lett 74, 1182 (1995). the condition v =1/2 can be achieved when the concentration of the excitations is still ≪ 1/λs, where λ is the [6] G. Mozurkewich, R.L. Jacobsen, Synth. Met. 60, 137 (1993). CDW wavelength, and s is the area per chain. [7] P. Ehrenfest, Leiden Comm. Suppl. 75b, 8 (1933); The concentration of the solitons could be estimated L.R. Testardi, Phys. Rev. B 3, 95 (1971); ibid 12, 3849 as 1 exp(−W/T), where W is the energy of such an λs (1975). excitation. Then, v ∼ ξ3 exp(−W/T). At T = T we [8] D. Stareˇsinić et al.,Eur.Phys.J.B 29, 71 (2002). have ξ3 exp(−W/T)≈1λ,sand come to the estimateP: [9] S.N. Artemenko, J.Phys.IV 12, Pr9-77 (2002); private λ3 communication. W [10] V.Ya. Pokrovskii and S.V. Zaitsev-Zotov, Euro- TP ≃ ln(ξ3/λs) (5) phys.Lett., 13, 361 (1990). [11] V.Ya. Pokrovskii, S.V. Zaitsev-Zotov, Synth. Met, 41- With ξ3/λs = 103 we obtain T = W/7, which can give 43, 3899 (1991) P [12] J.C. Gill, J.Phys.C: Solid StatePhys., 19, 6589 (1986). an idea of the low value of T in comparison with W. P [13] D.V. Borodin, S.V. Zaitsev-Zotov and F.Ya. Nad’, Zh. A higher ratio W/T might be obtained if we take into P Eksp. Teor. Fiz. 90, 618 (1986); [Sov Phys. JETP 63, considerationthelargewavelengthsofthefluctuationsof 194 (1986)]. the CDW stress along the chains. According to [9] they [14] V.Ya.PokrovskiiandS.V.Zaitsev-Zotov,Phys.Rev.B can considerably exceed ξk and, consequently, the LGS 61, 13261 (2000) 3 volumes could appear much larger than ξ . [15] P.Monceau in: Electronic Properties of Inorganic Quasi- According to our estimates, the excitations would be- one-dimensionalConductors,Part2.Ed.byP.Monceau. gin to turn into dislocation loops, only above T . So, Dortrecht: D.Reidel Publ. Comp., 1985. P within the model, the metallic state develops at lower [16] If instead of resistivities we sum up conductivities (con- temperaturethanthe criticalbehaviorisexpectedtobe- nection in parallel), the result is similar. Actually, one can expectaweakcriticality associated with thecurrent gin. As far as we comprehend, the approach proposes a percolation through the normal-phase cluster, but this new type of phase transition, which does not comprise concerns only thetransport properties. T as a parameter. At the same time, the PT in a sense P [17] J.C. Gill, Synth.Met. 43, 3917 (1991). resembles a 1st order transition. The model successfully [18] J. McCarten et al.,Phys. Rev.B 46, 4456 (1992). works both below and above T , though further extrap- P [19] A. V. Golovnya, V. Ya. Pokrovskii, and P. M. Shadrin, olation of the approach to higher temperatures requires Phys. Rev.Lett. 88, 246401 (2002). furtherdevelopmentofthemodel. Theunderlyingmicro- [20] A.V.Golovnya,V.Ya.Pokrovskii,Rev.Sci.Instrum.74, scopic mechanisms of the LGS also needs deeper under- 4418 (2003). standing. Though the model requires further grounds it [21] M.R. Hauser, B.B. Plapp, and G. Mozurkevich, Phys. gives a limpid insight into the processesinside the CDW Rev. B 43, 8105 (1991). near T . [22] R.L. Jacobsen and G. Mozurkewich, Phys. Rev. B 42, P 2778 (1990). We are thankful to J.W. Brill, G. Mozurkewich and [23] G. Mozurkewich, Phys. Rev.B 42, 11183 (1990). D.Stareˇsinićforgrantingtheexperimentalresultsatour [24] Note, that thetreatment of theY(T)anomaly proposed disposal, to V.V. Frolov and R.E. Thorne for furnish- in [K. Maki, Phys.Rev. B. 41, 2657 (1990).] also distin- ing the samples, to V.I. Anisimkin, L. Burakovsky and 4 guishes pinned and depinned CDW. [25] L.C.Bourne,andA.Zettl,SolidStateCommun.60,789 (1986). [26] Toourunderstanding[19],theeffectisdueto3Dfeatures of theelectronic structure, which are individual for each compound. [27] J.W. Brill, 2003, privatecommunication. [28] S.V. Zaitsev-Zotov, V.Ya. Pokrovskii, Abstracts of the XXV All-Union Session on the Low-Temperature Physics, Leningrad, 1988, part 3, p.112 (in Russian). [29] K. Maki, Physica 143B, 59 (1986). S. Ramakrishna et al.,Phys.Rev. Lett.68, 2066 (1991). [30] S. Panyukov and Y. Rabin, Phys. Rev. B 59, 13657 (1999). [31] G.A. Williams, Phys.Rev.Lett. 82, 1202 (1999). 5