Table Of ContentPeierls 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 concen-
tration 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. Stareˇsini´c 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.Stareˇsini´cforgrantingtheexperimentalresultsatour
[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
