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Ballistic heat transport of quantum spin excitations as seen in SrCuO 2 N. Hlubek,1 P. Ribeiro,1 R. Saint-Martin,2 A. Revcolevschi,2 G. Roth,3 G. Behr,1 B. Büchner,1 and C. Hess1 1IFW-Dresden, Institute for Solid State Research, P.O. Box 270116, D-01171 Dresden, Germany 2Laboratoire de Physico-Chimie de L’Etat Solide, ICMMO, UMR8182, Université Paris-Sud, 91405 Orsay, France 3Institut für Kristallographie der RWTH, D-52056 Aachen, Germany Fundamental conservation laws predict ballistic, i.e., dissipationless transport behaviour in one- dimensional quantummagnets. Experimentalevidence,however, for suchanomalous transport has been lacking ever since. Here we provide experimental evidence for ballistic heat transport in a 0 S = 1/2 Heisenberg chain. In particular, we investigate high purity samples of the chain cuprate 1 SrCuO2 and observe a huge magnetic heat conductivity κmag. An extremely large spinon mean 0 freepathof morethan amicrometer demonstratesthatκmag is onlylimited byextrinsic scattering 2 processes which is a clear signature of ballistic transport in theunderlyingspin model. n PACSnumbers: 75.40.Gb,66.70.-f,68.65.-k,75.10.Pq a J 9 The integrability of the one-dimensional (1D) antifer- increasing temperatures κ is increasingly supressed 1 mag romagnetic S = 1/2 Heisenberg chain implies highly due to spinon-phonon scattering which is the dominant ] anomaloustransportproperties,inparticular,adivergent extrinsic scattering mechanism in this material. el magnetic heat conductivity κmag at all finite tempera- We have grown large single crystals of SrCuO2 by - tures T.1–5 This truly ballistic heat transport suggests the traveling solvent floating zone method,22 where the r t anomalously large life times and mean free paths of the feed rods were prepared using the primary chemicals s . quantumspinexcitations andrenders1D quantummag- CuO and SrCO3 with both 2N (99%) and 4N (99.99%) at nets intriguing candidates for spin transport and quan- purity. Cuboidal samples with typical dimensions of m tuminformationprocessing.6–8 However,despitethe rig- (3×0.5×0.5) mm3 werecut fromthe crystals,with the - orousprediction,experimentalevidenceforballistic heat longest dimension parallel to the principal axes. Four- d transport in quantum magnets is lacking. Nevertheless, probe measurements of the thermal conductivity κ were n promising large κmag has been observed in a number of performed in the 7-300K range23 with the thermal cur- o cuprate compounds which realize 1D S = 1/2 Heisen- rent along the a, b, and c-axes (κ , κ , and κ respec- [c berg antiferromagnets9–18 with the spin chain material tively) for both the 2N and the 4Nasambples. c SrCuO beingaprominentexample10,18althoughaquan- 2 The main structural element in SrCuO is formed 2 titative analysis of κ has always been difficult there 2 v since the phononic anmdagmagnetic heat conductivities are by CuO2 zig-zag ribbons, which run along the crys- 1 tallographic c-axis (see inset Fig. 1). Each rib- of similar magnitude at low temperature. Such exper- 8 bon can be viewed as made of two parallel corner- imental κ is always finite since extrinsic scattering 6 mag sharing CuO chains, where the straight Cu-O-Cu 1 processes due to defects and phonons are inherent to all 2 bonds of each double-chain structure result in a . materials and mask the intrinsic behavior of the chain. 8 very large antiferromagnetic intrachain exchange cou- Formallyit seems reasonableto accountforthe extrinsic 0 pling J/k ≈ 2100 − 2600 K of the S = 1/2 9 scattering via a finite κmag ∼ Dthτ, where τ is a relax- spins at thBe Cu2+sites.19,21 The frustrated and much 0 ationtime,andDth representsthethermalDrudeweight weaker interchain coupling J′ /J ≈ 0.1 − 0.219,24 : which (multiplied by a delta function at zero frequency) v (cid:12) (cid:12) describes the intrinsic heat conductivity. In fact, it was and presumably quantum flu(cid:12)ct(cid:12)uations prevent three- Xi thereby possible to identify the expected low-T linear- dimensional long range magne(cid:12)tic(cid:12)order of the system at r ity of Dth(T) in the case of a “dirty” spin chain material T >TN ≈1.5−2K≈10−3J/kB K.20,25 Hence,atsignif- a where a high density of chain defects generate a large icantly higher T the two chains within one double chain T-independent scattering rate 1/τ.9 structure can be regarded as magnetically independent. In this paper we examine the heat conductivity of Infact,low-T (12K)inelasticneutronscatteringspectra SrCuO which is considered an excellent realization of of the magnetic excitations can be very well described 2 the S = 1/2 Heisenberg chain.19–21 Our samples of ex- within the S = 1/2 Heisenberg antiferromagnetic chain traordinary purity allow an unambiguous separation of model.21 the phononic and magnetic contributions to the thermal Fig.1presentsourresultsforκ andκ ofSrCuO for a c 2 conductivity. This yields the by far highest κ ob- both 2N and 4N purity as a function of T. We first de- mag served10,13 until now. Our analysis reveals a remarkable scribethedatafor2Npuritywhichareingoodagreement lower bound for the low-T limit of the mean free path withearlierresultsbySologubenkoetal.10Apronounced l of more than a micrometer. Thus our data pro- low-T peak at ∼18 K with κ ≈ 215Wm−1K−1 is mag max vide striking evidence that the intrinsic heat transport found for κ , i.e., perpendicular to the chains. This a,2N of the S=1/2 Heisenberg chain is indeed ballistic. With peak and a ∼T−1-decrease at T & 150 K towards a 2 interestingly both curves approach each other and at T &200 K exhibit almost the same T-dependence. Without further analysis some clear-cut conclusions can be drawn. First, the extraordinary enhancement of κ upon the improvement of the material’s purity in c contrast to a concomitantly negligibly small one in κ , a straightforwardly implies that the enhancement primar- ily concerns the magnetic heat conductivity κ which mag is present in κ only. Second, the extreme low-T sensi- c tivity to impurities of κ suggests that spinon-defect mag scattering is the dominating process which relaxes the heat current in this regime. Third, upon rising T, the spinon-defect scattering is increasingly masked by a further scattering process which leads to κ and κ c,2N c,4N being very similar at T & 200 K. The most reasonable candidate for this process is spinon-phonon scattering, since the only thinkable alternative, i.e. spinon-spinon scattering, is negligible3–5,9 in this T-regime. A further analysis of the data requires a reliable separation of the total measured κ into all relevant contri- Figure 1: (Color online) κa and κc of SrCuO2 for different butions which normally add up. Since electronic con- purityvalues. Closed(open)symbolsrepresentc-axis(a-axis) tributions can be excluded in this electrically insulating data,circles(diamonds)correspondto4N(2N)purity. Inset: material, it seems natural to assume that the measured crystal structure of SrCuO2. The symmetry is Cmcm with κ is just the sum of κ and a phononic background lattice constants a=3.56 Å,b=16.32 Å, c=3.92 Å.26 c mag κ ,9–14 where the latter can be approximated by the ph,c purely phononic heat conductivity perpendicular to the chains, κ ≈ κ (see inset of Fig. 2). The thus obtained a b smallvalueatroomtemperature(∼6Wm−1K−1)repre- κ =κ −κ forthe2Nandthe4Nsamplesareshown mag c a sentthecharacteristicT-dependenceofphonon-onlyheat inFig.2. AtT .35K,i.e.,inthe vicinityofthe peakof conductivity κ . The peak originatesfromtwo compet- κ , errors become large and we disregard the data in ph ph,c ing effects:27 at low T, a weakly T-dependent phonon this range for further analysis. For higher T we account mean free path l and a rapidly increasing number of for a possible uncertainty of ±30% in κ . Note that, ph ph,c phonons cause κ to increase strongly. At higher T, in the case of the 4N compound, possible errors in κ ph mag the exponentially risingnumber ofphonon-phononumk- are rendered small because obviously κ ≫κ . mag,4N a,4N lappprocessesincreasinglyshortenslph,whichcausesthe κmag of the 4N sample exhibits a sharp peak at decrease of κph. A similar low-T peak (at ∼ 20 K) ∼37 K with an extraordinary maximum value of about is also present in κc,2N (parallel to the chains). It is 660Wm−1K−1, which is more than a factor of 3 higher however larger (κmax ≈ 335Wm−1K−1) and exhibits a thanthe largestreportedκmag.10,13 The peak isfollowed distinct shoulder at the high-T edge (T & 40 K) of byastrongdecreaseuponraisingT. Similartotheafore the peak. κc,2N decreases at higher T, but remains describedtypicalT-dependenceofa cleanphononicheat much larger than κa,2N and even at room temperature conductor, the overall T-dependence of κmag suggests κc,2N ≈ 40Wm−1K−1. The apparent large anisotropy, that, in a simple picture, two competing effects deter- together with the unusual T-dependence of κc,2N, is the mine κmag. The low-T increase of κmag is consistent signatureofalargemagneticfractionofκc,2Noveralarge with a regime where the effect of scattering processes is T-range.10,18 weaklyT-dependentsinceD isexpectedtoincreaselin- th We now turn to the new data which have been ob- earlywithT.3–5,9 ThestrongdecreaseathigherT isthen tained for the high-purity compound. The heat trans- the resultofthe increasingimportance ofspinon-phonon port perpendicular to the chains (κa,4N) is slightly en- scattering. κmag of the 2N sample is qualitatively very hanced as compared to κ (κ ≈ 235Wm−1K−1) similar. However,the absolutevalue atthe peakismuch a,2N max which reflects a somewhat reduced phonon-defect scat- lower(∼172W/mK)andthepeak’spositionisshiftedto tering. However, a much more drastic and unexpected a higher T (∼55 K). Similarly to the κc data, at higher large effect of the enhanced purity is observed in the T, the 4N curve approaches that of the 2N sample. The heat transport parallel to the chains, κ . Instead of a latter is consistent with the earlier notion that spinon- c,4N narrow low-T peak and a shoulder as observed in κ , phonon scattering is dominant at high T, while the dif- c,2N a huge and broad peak centered at ∼28 K is present ferences at low T suggest that spinon-phonon scattering in κ (κ ≈ 830Wm−1K−1) which exceeds κ freezes out, upon decreasing T, rendering spinon-defect c,4N max c,2N at T . 70 K by more than a factor of 2. Also at scattering increasingly important. 70K . T ≤ 300 K we observe κ > κ , where We analyze κ quantitatively by extracting the c,4N c,2N mag 3 scattering are practically the same for both samples. In 800 fact,anequallygoodfitisobtainedifthesame T∗isused -1 m)300 κ for both curves. Note that the extracted Tu∗ ∼u 200 K -1 K κa - 2N is indeed of the order of the Debye temperature ΘD W κb - 2N of this material and thus leads to the conjecture that 600 κ (200 κa - 4N mostly acoustic phonons are involved in this scattering b - 4N process.34 Second,the spinon-defectscatteringlength l0, -1 m) 100 which represents a lower bound for the low-T limit of l and which should significantly depend on the sam- -1 K mag W400 ple’s purity turns out to be drastically different for both κ (mag 00 50 100 T (K) 150 cpaosuens.dTaondbeanspeexctirfiaco,rwdeinfianrdyll00≈≈310.60µnmmffoorrtthhee42NNcsaomm-- ple, which correspondto more than 750 and 4100lattice 4N spacings, respectively. These findings provide a further 200 2N confirmation of the above interpretation that, in both cases, κ is determined by the same spinon-phonon mag scattering process and that the difference between the two curves can be described by the different defect den- 0 sityonly. Wementionthatourresultsareconsistentwith 0 100 200 300 T (K) recent data by T. Kawamata et al. 35 Figure2: (Coloronline)κmag ofSrCuO2fordifferentpurities. A major outcome of our study is the unambiguous Open symbols represent low-T κmag which is disregarded in identification of the extrinsic scattering processes as the the further analysis. The shaded areas show the uncertainty onlyrelevantones. Intrinsic spinon-spinonscattering,on of the estimation of κmag due to the phononic background. the other hand, plays no role in our analysis,even in the Inset: κa andκb perpendiculartothechainforbothpurities. case of the very clean sample. The strong enhancement of κ upon reduction of the impurity amountthus ap- mag pears as the manifestation of ballistic heat transport of spinon mean free path l according to9–11 mag the underlying spin model, where κ is rendered finite mag 3~ by extrinsic scattering processes only. One might there- l = κ , (1) forespeculatethatκ ofthismaterialcanbedrivento mag πN k2T mag mag s B much higher values in a perfect crystal. We point out that our analysis relies on a very sim- where N = 4/ab is the number of spin chains per unit s ple theoretical approach which was also successfully area. AscanbeinferredfromFig.3,l ofbothsamples mag show a strongdecrease with increasingT,which directly used in many other low-dimensional S = 1/2 spin sys- tems,9–16,28,29 which is surprising in view of the strong reflects spinon-phonon scattering becoming increasingly quantumnatureofsuchsystems. Moresophisticatedap- important. Bothcurvesareverysimilar,butcleardiffer- proaches might lead to a deeper understanding of the ences are present at low T, where l of the 2N sample mag magnetic heat transport in this system on a microscopic is somewhat lower, in accordance with a higher spinon- level. Inthisregarditisinterestingtonotethat,inclean defect scattering. We evoke Matthiesen’s rule to model samples (i.e. with large l ), l = (A T)−1exp(T∗/T) itnhgepTr-odceepsesnesdevnizc.elo−f1lm=agla−n1d+tol−a1c.cHouenret,folrdbeostchrisbceastttehre- leads to κc ≈ κmag ∝ ex0p(Tsu∗p/T) wisth Tu∗ ∼ 20u0 K mag 0 sp 0 at high T, in agreement with the theory proposed T-independent spinon-defect scattering whereas l (T) sp by Shimshoni et al.30 However, we do not observe accounts for the T-dependent spinon-phonon scattering. κ =κ ∝exp(2T∗/T) as expected in the same model. Forthelatter,weassumeageneralumklappprocesswith a ph u ∗ It seems worthwhile mentioning in this regard that the acharacteristicenergyscalek T oftheorderoftheDe- byeenergy,whichiscommonlyBusuedinliterature.10,15 We only slight enhancement of κa observed upon increasing purityisquiteunexpected. Onemightspeculatethatthis thus have is an indication of phonon scattering off the spin chains exp(T∗/T) −1 which in principle should be relevant.31,32 l−1 =l−1+ u , (2) mag 0 (cid:18) A T (cid:19) To sum up, we have investigated the spinon heat con- s ductivity κ of the antiferromagnetic S=1/2 Heisen- mag ∗ which can be used to fit the data with l , A and T berg chaincuprate SrCuO for standard(99%) andhigh 0 s u 2 (A describes the coupling strength) as free parameters. (99.99%)purity. The higher purity leads to a drasticen- s We find an excellent agreement between such fits and hancementofκ atlowT andwefindtheup-to-present mag the experimental l , see Fig. 3. Inspection of the fit by far highest reported κ in the high-purity sample. mag mag parameters33 yields two remarkable aspects which cor- For higher T, we provide clear-cut evidence that spinon- roborateour previous qualitative findings. First, the pa- phonon scattering is the most relevant scattering which ∗ rameters A and T which determine the spinon-phonon leads to a very efficient reduction of κ . An extreme s u mag 4 sensitivityofκmagtoimpuritiesispresentatlowT,which impliesthatthespinon-defectscatteringisdominatingin 1.6 m) 1 thisregime. Asimpleanalysisrevealsaremarkablelower µ 1.4 (mag bound for the low-temperature limit of the spinon mean l free path lmag of more than a micrometer. Our results 1.2 0.1 therefore suggest that κmag is only limited by extrinsic scattering processes which appears as the manifestation m) µ 1.0 of the ballistic nature of heat transport in the S = 1/2 l (mag0.8 0.0110 100 antiferromagnetic Heisenberg chain. T (K) 0.6 0.4 4N Acknowledgments 2N 0.2 We thank W. Brenig, A. L. Chernyshev, S.-L. Drech- 0.0 sler, F. Heidrich-Meisner, P. Prelovšek, X. Zotos and 0 100 200 300 T (K) A. A. Zvyagin for stimulating discussions. This work Figure3: (coloronline)MagneticmeanfreepathsofSrCuO2 was supported by the Deutsche Forschungsgemeinschaft fordifferentpurities. Thesolidlineswerecalculatedaccording through grant HE3439/7, through the Forschergruppe toEq.2. Theshadedareaillustratestheuncertaintyfromthe FOR912 (grant HE3439/8) and by the European Com- estimation of the phononicbackground. mission through the NOVMAG project (FP6-032980). 1 X. Zotos, F. Naef, and P. Prelovsek, Phys. Rev. B 55, 70, 437 (2001). 11029 (1997). 18 P.Ribeiro, C. Hess, P. Reutler,G. Roth,andB. Büchner, 2 X. Zotos, Phys. Rev.Lett. 82, 1764 (1999). J. Mag. Mag. Mater. 290-291, 334 (2005). 3 A.KlümperandK.Sakai,J.Phys.A:Math.Gen.35,2173 19 N. Motoyama, H. Eisaki, and S. Uchida, Phys. Rev. Lett. (2002). 76, 3212 (1996). 4 F. Heidrich-Meisner, A. Honecker, D. C. Cabra, and 20 M. Matsuda, K. Katsumata, T. Osafune, N. Motoyama, W. Brenig, Phys.Rev.B 68, 134436 (2003). H.Eisaki,S.Uchida,T.Yokoo,S.M.Shapiro,G.Shirane, 5 F. Heidrich-Meisner, A. 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(58.6±5.4) 10−16m/K and l0,2N = (305±5)nm, 34 Modeling lmag with lsp ∝ exp(T∗/T) which accounts for sTpu∗e,2cNtive=ly.(2S1e7tt±in3g)KT,∗AS,2=N T=∗(78±=1)20140−K16maf/tKer, firet-- tphheonaoltnesr2n9arteisvueltssciennaarfiiotooffsismpiilnaornqsuascliattytewriitnhgcoomffpoaprtaibcalel ting the 4N data gui,v2Nes l0,2Nu,4N= (320±13)nm and l0 and T∗ ∼300 K. As,2N =(72±5)10−16m/Kfor2N.Theerrorsaccountfor 35 T. Kawamata, N. Kaneko, M. Uesaka, M. Sato, and Y. theaccuracy of thefit.Itis also possible to obtain agood Koike, unpublisheddata. fitwiththesameAs forthe4Nand2Ncases.However,in- dividualAs account for errorsin theabsolutevalue,while