Observation of correlations up to the micrometer scale in sliding charge-density waves

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by  D. Le Bolloc'h

Observation of correlations up to the micrometer scale in sliding charge-density waves
D. Le Bolloc’h,1 V.L.R. Jacques,1 N. Kirova,1 J. Dumas,2 S. Ravy,3 J. Marcus,2 and F. Livet4
1Laboratoire de Physique des Solides (CNRS-UMR 8502),
Bˆat. 510, Universit´e Paris-sud, 91405 Orsay cedex, France
2Institut N´eel, CNRS/UJF, BP166 38042 Grenoble cedex 9, France
3Synchrotron SOLEIL, L’Orme des merisiers, Saint-Aubin BP 48, 91192 Gif-sur-Yvette cedex, France
4LTPCM (CNRS-UMR 5614), ENSEEG-Domaine Universitaire,
BP 75, 38402 Saint Martin d’H`eres cedex, France
High-resolution coherent x-ray diffraction experiment has been performed on the charge density
8 wave (CDW) system K0.3MoO3. The 2kF satellite reflection associated with the CDW has been
0 measured with respect to external dc currents. In the sliding regime, the 2kF satellite reflection
displays secondary satellites along the chain axis which corresponds to correlations up to the mi-
0
crometer scale. This super long range order is 1500 times larger than the CDW period itself. This
2
new type of electronic correlation seems inherent to the collective dynamics of electrons in charge
n density wavesystems. Several scenarios are discussed.
a
J
PACSnumbers:

8

el] anFdecrhroaerlgeectdreicnss,itmyawganveetsicmsayystestmasb,illiizqeuiindcocrmymsteanlss,usrpaitne a) Qs b) (6 0 -3)
-
structures with the underlying lattice[1]. In most cases,
r I=16*I I=16*I
st the stability of incommensurate modulations relies upon I=0 s I=0 s
. acouplingwiththeunderlyinglatticeorintrinsicdefects.
t
a The case of charge density waves is particular because
m the electronic modulation is mainly driven by the elec- b* *** b* ***
2a*-c*
d- tronic structure. The nesting of the Fermi surface stabi- t* *** ~ t*
n lizes the modulation and the band filling fixes the wave c) d)
o vector of the modulation at twice the Fermi wave vec-
c tor 2kF, which may be incommensurate. The K0.3MoO3
[
bluebronzesystem,forexample,stabilizesanincommen-

3 surate Charge Density Wave (CDW) modulation at 2k
F
v =0.748±0.001[2] along the chain axis.
7
2 This incommensurate 2kF wave vector has a well-
2 knownconsequence: the invarianceby translationallows
4 the electronic crystal to slide over the underlying lattice
.
1 for currents greater than a threshold value. The signa- FIG. 1: 3D diffraction patterns of a) the Qs =(6,0.252,3.5)
1 ture of this sliding motion has been mainly observed by satellite and b) the (6,0,3) fundamental Bragg reflection for
7 transport measurements[3, 4]: a large broad band noise twocurrentvaluesatT=75K. Each isosurface hasbeenfixed
0
is observed, as well as periodic voltage oscillations[5, 6]. atImax/18forthe(6,0,3)andImax/7forQs. Forclarity,the
:
v reflections with and without current have been shifted along
i Observingthecollectivedynamicsofelectronsbyclas- b∗. The total size of the boxes is 40*(δθ = 10−3 degree)
X sical diffraction has always been a significant challenge. (vertical axis) by 50*60 pixels (horizontal plane). Each 3D
r The task is difficult inCDWsystems because the CDWs acquisitionlastedlessthan15mn. 2Dpatternscorresponding
a
domainsareusuallylargerthanthemicrometerscaleand tointegrationofthefig1aovertheθaxis,withoutc)andwith
current d). The secondary satellites are indicated by arrows.
any translationalmotion does not change the diffraction
pattern. Only few consequences of the sliding have been
observed by using high-resolution x-ray diffraction, as
for example, the loss of transverse coherence of CDWs using coherent micro diffraction.
domains[7], the contraction of the CDW close to electri- We have studied the blue bronze which is a classic
cal contacts[8] or its rotation around a step[9]. By using CDW system. This is a quasi-1D structure, made of
high-resolution and coherent x-ray diffraction, we show clusters of10 MoO6 octahedra,forming chains along the
in this Letter a novel consequence of the sliding CDW [010]direction(b=7.56˚A), andlayersalong the [102]di-
and the incommensurate 2k wave vector: a super long rection. The reciprocal vector b∗ runs along the chains
F
range order, up to the micrometer scale, appears in the and 2a∗−c∗ is perpendicular to the layers. Let u(r) =
sliding regime of the CDW. This observation, as well as u0cos(qc.r+Φ(r)) be the periodic lattice distortion in
an earlier one reported in [10], became possible only by quadrature with the CDW, where q = (1,2k ,0.5) is
c F
2
dfreofinntes.d Aast tehqeuilwibarvieumve,ctthoer CnoDrWmalexthoibtihtse iCntDriWnsicwadvee- D(cid:12) (cid:23)(cid:19) (cid:3) , (cid:3)(cid:19) G(cid:12) (cid:22)(cid:24)(cid:19) (cid:3), (cid:3)(cid:19)
ftaeicocontussmltiikocedmdesios[1dlo2ec,,a1tthi3oe].nssoT[-1hc0ae,ll1de1dis]ppaenhrdasisodoninspcmluaoryvdseei,toswfohtwhicnehvCiisbDrrWae-- E(cid:13)3L[HOV(cid:3)(cid:11)a(cid:3)(cid:12) (cid:20)(cid:21)(cid:22)(cid:19)(cid:19)(cid:19) (cid:3) QVLW\(cid:3)(cid:11)D(cid:17)X(cid:17)(cid:12) (cid:20)(cid:21)(cid:21)(cid:22)(cid:19)(cid:24)(cid:19)(cid:24)(cid:19)(cid:19)(cid:19)(cid:19) (cid:13)(cid:3)(cid:3)(cid:21),, (cid:17)(cid:24)(cid:3)(cid:3)(cid:20)(cid:20)(cid:21)(cid:25)(cid:13)(cid:13),,VV (cid:3)
sponsible for the sliding motion of the CDW as a whole, H (cid:20)(cid:19)(cid:19)
QW
has been measured by inelastic neutron scattering[12]. E(cid:12) (cid:23)(cid:19) , (cid:3)(cid:20)(cid:21)(cid:13),(cid:3) L (cid:24)(cid:19) (cid:18)(cid:21)
V
(cid:19)
obueWernpecreauvrseieofduulsltyehxmepeseaarsimmuereendhti[bg1eh0f]o.qrIeutaaslnietdlyeacstftirneicrgaltelhrecerseyixsspttaeanrlicmaesehnaitns. E(cid:13)3L[HOV(cid:3)(cid:11)a(cid:3)(cid:12) (cid:22)(cid:21)(cid:20)(cid:19)(cid:19)(cid:19) (cid:3) H(cid:12) (cid:20)(cid:24)(cid:19)(cid:19) (cid:3), (cid:3)(cid:19)(cid:20)G(cid:19)(cid:11)(cid:21)D(cid:13)(cid:16)F(cid:13) (cid:12)(cid:3)(cid:21)(cid:11)(cid:19)(cid:20)(cid:19)(cid:16)(cid:23)(cid:3)$(cid:16)(cid:20)(cid:12)(cid:22)(cid:19)
Irnesibsotathncceawseass,oabpserrovneoduantcetdhedtehcrreesahseoldofctuhrerednitff,earcecnotmia-l F(cid:12) (cid:23)(cid:19) (cid:3) , (cid:3)(cid:20)(cid:25)(cid:13),(cid:3) \(cid:3)(cid:11)D(cid:17)X(cid:17)(cid:12) (cid:20)(cid:19)(cid:19) (cid:3)(cid:3),, (cid:3)(cid:3)(cid:20)(cid:20)(cid:21)(cid:25)(cid:13)(cid:13),,VV
piaasttauan7rise5eidKg(nT)bacytr=ueam1rre8aa0opiKnfidet)dhiaennucsdnrliecdathihsaneengogtsfhettdraheteseah.fbtoTrelordhaetdchutberraarenenxnsdpittnei(oroIinisms=ete,e1nmw.t2:hpmiecxArh-- E(cid:13)3L[HOV(cid:3)(cid:11)a(cid:3)(cid:12) (cid:20)(cid:21)(cid:22)(cid:19)(cid:19)(cid:19) V (cid:3) LQWHQVLW (cid:24)(cid:19)(cid:19) (cid:13)(cid:21)(cid:17)(cid:24) (cid:18)(cid:22)
raysdidnotalterthemacroscopicCDW’sproperties. No (cid:20)(cid:19) (cid:21)(cid:19) (cid:22)(cid:19)3L[HOV(cid:3)(cid:11)a(cid:3)(cid:21)D(cid:13)(cid:16)F(cid:13)(cid:12) (cid:20)(cid:19) GE(cid:13)(cid:3)(cid:11)(cid:20)(cid:19)(cid:16)(cid:23)(cid:3)$(cid:16)(cid:21)(cid:20)(cid:19)(cid:12)
heating effect under the x-ray beam has been observed.
The coherent x-ray diffraction experiment has been
performed at the ID01 beamline at the ESRF. The FIG. 2: 2D diffraction patterns (30*40 pixels) of the 2kF
0.5×2×0.2mm3 sample was mounted in a top-loading reflection inthe(2a∗−c∗;b∗)planeforI=0mA(a),I=12Is
(b) and I=16 Is (c). The secondary satellites are indicated
Cryostatcooleddownto75KwithHeexchangegas. The by arrows. Corresponding profiles along 2a∗-c∗ in d) and b∗
sample was initially aligned with the b∗ axis vertical, in e). The profiles havebeen rescaled.
and the 2a∗−c∗ axis in the horizontal scattering plane.
ThepatternswererecordedonadirectilluminatedCCD
camera (22µm×22µmpixel size) located 1.20m from the θ axis. The corresponding 2D pattern is displayed with-
sampleposition. Thebeamqualityanditstransverseco- outcurrentinFig.1candwithcurrentinFig.1d. The2D
herencelengthweretestedbyclosingthe entranceslitat patterns shown in Fig.2 correspond to a section of Fig.1
2µm×2µm: the expected cross-like diffraction pattern along the 2a∗−c∗ direction through the 2k reflection.
F
was observed with strong contrast of fringes[14]. First, the host lattice seems to be insensitive to the
Thedegreeofcoherencecanbeestimatedfromtheex- appliedcurrent: the (6,0,3)fundamentalBraggpeakre-
perimentalsetuparound10%[15]. Thepenetrationdepth mains unchanged under the applied current within the
was 18µm−1 and the footprint of the xray beam on the experimental resolution (Fig.1b). On the other hand,
sample’s surface was 45µm (the incident angle equals the CDW modulation displays original features for large
12.6o). In this study, the main interest of using a co- enoughcurrents. Undercurrent,abroadeningofthe2k
F
herent beam is the small beam size and the excellent reflection is observed along the 2a∗-c∗ transverse direc-
Q-resolution. Aresolutionofδq =0.710−4˚A−1 alongb∗ tion which corresponds to decreasing CDW correlation
(i.e. δq =0.8 10−4 in b∗ units.) wasachievedat 7.5 keV lengths from ξ = 1.2µm at I=0mA, to ξ = 0.6µm at
t t
(λ=1.65˚A) by using 10µm×10µm entrance slits. I=12I andto ξ =0.4µm atI=16 I (Fig.2d). The loss
s t s
The experiment consisted in recording the 2D diffrac- ofthetransverseorderunderexternalcurrents,alongthe
tion patterns of the Qs=(5,1,3)+qc satellite reflection, softest directionof blue bronze[17], has alreadybeen ob-
and the (6,0,3) fundamental Bragg peak, far from any served in several studies[7].
electric contact. Both reflections have been measured The most striking feature appears along the b∗ direc-
successively after each current variation. Due to the ex- tion. Withoutcurrent,the2k reflectiondisplaysasingle
F
perimental geometry, the 2D reciprocal plane probed by peak,thewidthofwhichcorrespondstotheentranceslit
theCCDatthesatelliteanglecorrespondstothe(b∗,t∗) aperture: the CDWs domain is larger than 10µm along
plane, where t∗ is the direction tilted by 19.5o from the b∗. Aboveapproximately12timesthethresholdcurrent,
2a∗-c∗ direction[16]. SeveralCCDacquisitionshavebeen theQ satellitestartstobroadenalongb∗ (Fig.2b). This
s
recorded for different incident θ angles, with δθ = 10−3 broadeningincreasescontinuouslyforlargercurrents. At
degreesteps. This allowsus togetthe threedimensional acurrent16timeslargerthanthethreshold,severalmax-
intensity distribution of the reflections (see Fig.1). In ima located at regular positions along b∗ are clearly
thisway,thebehaviorofthe2k reflectionandthemain distinguished from the 2k reflection (see Fig.1d). All
F F
Bragg reflection was observed with respect to external of them display approximately the same width, slightly
current, especially along the chain direction b∗ and the larger than the 2k width without current. As shown
F
transverse direction 2a∗-c∗. In order to get more inten- in Fig.1, these new intensity maxima correspond to well
sity, we have integrated the volume of the Fig.1 over the defined 3D peaks and will be called secondary satellite
3
reflection in the following. These secondary satellite re- decayingintensity. Itsignifiestheappearanceundercur-
flections are located at (6,0.252±nδq ,3.5) with δq = rent of non-harmonic periodic structure, modulating the
s s
4.910−4b∗units(seeFig.1a,1d,2cand2e). Thereduced CDW state along b∗. An amplitude modulation of the
wave vector δq leads to a period of L=2π/δq =1.5µm. CDWwouldyieldsymmetricalsecondarysatellite reflec-
s s
We did not increase the current further for fear of tions, that only structure factor or Debye-Waller effects
destroying the electrical contacts. The appearance of could make asymmetric in intensity. On the other hand,
secondary satellite reflections is reversible: after heat- a phase modulation would directly result in asymmet-
ing up the sample above T under zero field and cooling ric secondary satellite reflections [1, 21]. This remark,
c
downto 75Kagain,the samediffrationpatternhasbeen along with the well-known fact that phase excitations
observed[18]. These secondary satellite reflections were cost much less energy that amplitude ones, lead us to
also observed in a second high quality crystal. Note the consider this latter case in the following.
strong intensity of higher-order satellites and the asym- Consider first a long range modulation provided by a
metry ofthe profile inFig.2e[19]. Fig.1dalsodisplaysan static soliton lattice. In blue bronzes, the CDW wave
asymmetric diffuse intensity along the transverse direc- vector (Q = 0.748b∗) is close to 0.750b∗ corresponding
tion. This is surely due to inhomogeneities in the CDW to the quarterization along the chain. The actual crys-
current density within the probed volume. Those inho- tallographic space group requires an 8-order commensu-
mogeneities have been observed by transport measure- rability (8qc is a fundamental Bragg reflection) which
mentsintheblue bronze[6]. We alsonoticedtheabsence locks the CDW phase ϕ at multiples of π/4 via the en-
of any speckle neither on the 2k reflection nor on sec- ergy ∼−αcos(8ϕ)[22]. Thus, the proximity to the com-
F
ondary satellites, despite the coherence properties of our mensurability point results in a lattice of discommensu-
x-ray beam. This last point will be brought up later. rations, or solitons with the phase increment ∆ϕ=π/4.
Let us discuss the interpretation. First, the secondary The soliton lattice is characterized by the soliton size l
satellites cannot be due to the fragmentation of CDW andthe distance betweenthe solitonsL. The single soli-
in isolated CDWs domains, 1.5 µm wide along b∗. The ton size l∼ (C/α)1/2 is then determined jointly by the
truncation effect due to finite CDWs domain would lead energy α and the CDW longitudinal elastic modulus C.
to very weak satellites with respect to the main 2k re- Atlowappliedcurrentthepreexistingsolitonlatticecor-
F
flection. It can not be explained either by the presence responds to an almost sinusoidal phase, and manifests
of a single CDW dislocation as it has been observed in itself as the pure shift of the CDW peak without sec-
blue bronzewithoutcurrentin[10]. Indeed, the distance ondary satellites. At higher applied currents however,
betweeneachfringewouldbeconstantwhateverthecur- a strong decrease of elastic modulus C (corresponding
rent and equal to the beam size, i.e. δq = 2π/10µm = to a decreasing l) leads to non-harmonic modulation of
0.610−4˚A−1. Inaddition,themain2k reflectionshould the phase ϕ and to higher-order satellites at 2π/L as in
F
disappearbecauseofthepresenceofthedislocation. The Fig.2e. ThestrongvariationofC necessarytogetanon-
last two points are in contradiction with our measure- harmonic modulation could be due to a screening effect
ment. Finally, phase shifts randomly distributed in the by free carriers. In any case, the existence of the soli-
volumewouldleadtotypicalspecklepatterns[20]andnot ton lattice allows the electronic crystal to approach the
toregularsatellitereflectionsasobservedinFig.1d. Note commensuratemodulation. Inthisworkhowever,wedid
also that purely dynamical phenomena, like phase slips, not observedany clear shift of the 2k reflection toward
F
would only have a negligible effect on the 2k reflection. 0.75b∗ undercurrent. Notethatthismechanismisbased
F
We interpret the experimental data as due to a long onthe interactionbetweenthe CDW andthe underlying
rangeperiodicstructurewhichmodulatestheCDWstate lattice, without taking explicitly into account the pres-
along b∗. This super long range order is surprising since ence of extrinsic defects[25].
the period L is 1500 times larger than the CDW period Another explanation of the secondary satellites could
along the chain direction (λCDW ≈4/3 b=10.08 ˚A). To bethepresenceofanarrayofCDWedge-dislocations. A
our knowledge, long range electronic correlations up to singleCDWdislocationhasalreadybeenobservedinthe
the micrometer scale have never been observed in elec- bulkin[10]. Theycanbeformedbelowthesurfacedueto
tronic systems. the charge transfer or other types of stress near the sur-
The existenceofthis superlongrangeorderisa direct face. In the sliding state but for moderate electric field,
consequence of the incommensurability of the 2k wave therandomlydistributeddislocationscanbestillpinned,
F
vector and of the sliding motion of the electronic crys- leadingto amere changeof the 2k profiles[23]. Athigh
F
tal. Several scenarios are briefly discussed here, based electricfields,theCDWdislocationscanbedepinnedand
on static or dynamic phenomena: a soliton lattice, an form a periodic structure. Correlated dislocations of the
ordering of CDW dislocations or a new vibration mode host lattice are knownin crystalsstressedby the epitax-
induced by sliding. ial layers (see for example [24]). The above model corre-
The observed periodic sequence of satellites flanking sponds also to a non-harmonic modulation and leads to
the 2k reflectiondisplays a remarkablyhigh andslowly higher-order satellites and to an asymmetric profile.
F
4
As already mentioned in the introduction, the sec- M. Renard, P. Gressier, A. Meeschaut, L. Guemas and
ondary satellites have been observed several times. We J. Rouxel,Solid State Comm. 46, 325 (1983);
never observed speckles but smooth profiles, despite the [6] P. Beauchˆene, J. Dumas, A. Janossy, J.Marcus, C.
Schlenker,PhysicaB143,126(1986);M.H.Hundleyand
coherence properties of our x-ray beam. The absence of
A. Zettl, Phys. Rev.B 39, 3026 (1989).
any speckle is a strong indication that this phenomenon
[7] T.Tamegai et al., Solid State Commun. 51, 585 (1984);
could also have a dynamic origin. The secondary satel-
R.M.Fleming,R.G.Dunn,L.F.Schneemeyer,Phys.Rev.
lites could be the consequence of the presence of a prop- B 31, 4099 (1985).
agating mode, associatedwith the sliding of the conden- [8] S. Brazovskii, N. Kirova, H. Requardt, F. Ya. Nad, P.
sate. An estimate of the frequency of the mode with- Monceau, R. Currat, J. E. Lorenzo, G. Gru¨bel and Ch.
out current can be obtained by extrapolating, at δq , Vettier, Phys. Rev. B 61, 10640 (2000); H. Requardt,
s
the phason dispersion curve measured by inelastic neu- F.Ya.Nad,P.Monceau,R.Currat,J.E.Lorenzo,S.Bra-
tron scattering[12]. For δq = 4.9 10−4 in b∗ units, one zovskii, N. Kirova, G. Gru¨bel, Ch. Vettier, Phys. Rev.
s Lett. 80, 5631 (1998).
obtains frequencies in the GigaHertz range (15 GHz at
[9] A.F.Isakovic,P.G.Evans,J.Kmetko,K.Cicak,Z.Cai,
175K and approximately 4 times larger at 75K), i.e. a B. Lai, and R. E. Thorne, Phys. Rev. Lett. 96, 046401
particulary low frequency mode. In the sliding state, a (2006).
softening of this phason mode at δqs could explain the [10] D. Le Bolloc’h, S. Ravy, J. Dumas, J. Marcus, F. Livet,
observed data since the diffracted intensity is inversely C.Detlefs,F.Yakhou,andL.Paolasini,Phys.Rev.Lett.
proportional to the frequency squared of the vibration 95, 116401 (2005).
[11] P. A. Lee, T. M. Rice, Phys.Rev.B, 19, 3970 (1979).
mode. The origin of the very soft mode remains to be
[12] B.Hennion,J.-P.Pouget andM.Sato,Phys.Rev.Lett.
clarified.
68 2374 (1992); J.-P, Pouget, B. Hennion, C. Escribe-
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terscaleareobserved,alongthechainaxis,inthesliding [13] S. Ravy, H. Requardt, D. Le Bolloc’h, P. Foury-
regimeofthebluebronzebyusingcoherentx-raydiffrac- Leylekian, J.-P. Pouget, R.Currat, P. Monceau, and M.
tion. However,theappearanceofsuperlong-rangecorre- Krisch, Phys. Rev.B 69, 115113 (2004).
lationsinducedbyCDWmotionisstilldifficulttounder- [14] D. Le Bolloc’h, F. Livet, F. Bley, T. Schulli, M. Veron
and T.H. Metzger, J. SynchrotronRad. 9, 258 (2002).
stand. Several scenarios are presented, based on the or-
[15] F. Livet,Acta Cryst. A 63, 87 (2007).
dering of topologicaldefects (solitons, edge-dislocations)
[16] The scheme of thereciprocal lattice can be found in our
or the softening of a phason mode. Additional exper-
previous article[10].
iments are necessary to clarify the origin of this phe- [17] In blue bronze, the constant force along the 2a∗-c∗ di-
nomenon. rection is 100 times weaker than along b∗ (see [12]).
TheauthorswouldliketoacknowledgetheID01beam- [18] Memory effects are well-known in blue bronze. A virgin
linestaffattheESRF,especiallyC.MocutaandT.Met- stateisobtainedbyincreasingthetemperatureaboveTc
and bycooling down without current.
zger, and P. Van den Linden for technical support; J.P.
[19] We have also observed that the whole 2kF profile was
Pouget, M. Marsi and S. Brazovskii for fruitful discus- shifted along b∗ for some current values. We consider
sions. N. Kirova acknowledges the financial support of
that this global shift is mainly due to the the finite re-
INTASNo05-10000008-7972. Partofthisworkwassup- producibility of the diffractometer since the (6,0,3) and
ported by ANR grant LoMaCoCup. the2kF reflectionshavebeenmeasuredsuccessivelyafter
eachcurrentvariation.Wecanestimatethattheshiftsof
the2kF profilewithrespecttothecurrent,ifany,remain
less than δq=10−3˚A−1∼2.5×δqs.
[20] To see typical specklepattern at wide angles, see for ex-
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[3] For a review on the blue bronze see: Low Dimensional Commun. 14, 703 (1974).
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