Stripes and dimensional crossovers in high-Tc cuprates

PHYSICA ELSEVIER

Physica C 341-348 (2000) 887-890

www.elsevier.nl/locate/physc

Stripes a n d d i m e n s i o n a l c r o s s o v e r s in h i g h - T c cuprates

V.V. Moshchalkov, L. Trappeniers and J. Vanacken Laboratorium voor Vaste-Stoffysica en Magnetisme, Katholieke Universiteit Leuven, Celestijnenlaan 200 D, B-3001 Leuven, Belgium The key problem in high Tc physics is whether the doping is inhomogeneous and holes are expelled into the 1D domains ("stripes"). Direct comparison of the resistivity vs T curves of the ID spin-ladder cuprates and underdoped cuprates below the pseudogap temperature To shows a remarkable similarity which implies that the main scattering mechanism is the same for these two classes of cuprates and therefore in the pseudogap regime the transport in underdoped cuprates is essentially 1D. All the resistivity curves in underdoped cuprates (including high field transport data in fields up to 60 T) can be fitted by the model of the 1D quantum transport with the inelastic length in charge stripes being fully controlled by the magnetic spin correlation length in evenchain spin-ladders formed in the Cu-O plane as a result of inhomogeneous doping. Close to the insulator - metal transition disorder strongly influences the formation of stripes. Knight shift data are also successfully interpreted by taking the same spin-gap value as the one found from resistivity data. The possible dimensional crossovers observed with decreasing temperature at To(x) (2D-1D) and To(x) (1D-2D) are discussed and the main unsolved problems of the evolution of the normal state properties with doping are outlined.

1. INTRODUCTION

2. TRANSPORT AND KNIGHT SHIFT

A rapidly growing experimental evidence [1-8] indicates that the 1D scenario may be relevant for the formation of stripes in underdoped high-To cuprates. Since mobile carriers in this case are expelled from the surrounding Mott insulator phase into the stripes, the latter then provide the lowest resistance paths. This makes the transport properties very sensitive to the formation of the stripes - both static and dynamic. We assume that for the 1D stripes the condition L~ - ~m~D (inelastic length L~ is fully controlled by the 1D magnetic correlation length [9-11]) is enforced by the proximity effect between a metallic stripe and the Mott insulator phase [12]. The conducting stripes could be just doped ladders, spontaneously formed between the Mott-insulator domains in the CuO2 planes under doping. To investigate the possibility of using the 1D quantum scenario for describing transport properties of the 2D CuO planes, we analyze the transport properties and the Knight shift in underdoped cuprates.

Early experiments on twinned high-To samples however, have created an illusion that all planar Cu sites in the CuO2 planes are equivalent. Recent experiments on perfect untwinned single crystals have completely unveiled this illusion. A very large anisotropy in the ab-plane of twin-free samples has been reported for resistivity ( , o a / p b ( Y B a 2 C u 2 0 7 ) = 2.2 [14,15] and p~/pb(YBa2Cu4Oa) = 3.0 [16]), thermal conductivity (tcadtcb(YBa2Cu4Os)= 3-4 [17]), for superfluid density [18,19] and optical conductivity [19,20]. In all these experiments, much better metallic properties have been clearly seen along the direction of the chains (b-axis). And what is truly remarkable, that this large in-plane anisotropy is strongly suppressed by a small (only 0.4 %) amount of Zn [19], which is known to replace copper, at least for Zn concentrations up to 4 %, only in the CuO2 planes [21,22] ! The latter strongly suggests that the ab-anisotropy can not be explained just by assuming the existence of highly conducting CuO-

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chains. Instead, the observation of the anisotropy in the transport properties in the ab-plane for YBa2Cu408 [16] and YBa2Cu307 [14], erroneously interpreted as a large contribution of strongly metallic Cu-O chains p~h~n(T), should be analyzed by taking into account the fact that the in-plane anisotropy is caused by certain processes in the CuO2 planes themselves, where the substitution of Cu by Zn takes place. In this situation we may expect that the chains are actually imposing certain directions in the CuO: planes for the formation of 1D stripes. 0.4 "V-

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T/A Figure 2. Scaled p(T) curves for YBa2Cu30× thin films. The p(T) = Po + C T exp(-A/T) curve perfectly coincides with the data for 0.25 < T/A < 1. Different regimes I-III can be identified as: I - 2D Heisenberg, II - 1D stripe, and III - fragmented and pinned stripes promoting the 2D localization regime.

o. ,, 0 2T(KTv~no ~ 2 0400 0 600 8[800 50

100

150

200

250

300

Temperature (K) Figure 1. Temperature dependence of the resistivity for a YBa2Cu408 single crystal [13] (open circles); the solid line represents the fit using ,o(T) = Po + A T exp(-A/T). The high-temperature data taken on another crystal (after Ref. [23], shown in the inset, illustrate the crossover at T' to the 2D regime.

This expectation is supported by our observation of a perfect scaling of the ,o(T) curves for 1D spin ladders and (presumably "2D") YBa2Cu408 [11]. Using a simple scaling parameter To, we have found a convincing overlapping of the two sets of data: (P'po)/P(To) versus T / T o (with Po being the residual resistance) for YBa2Cu408 and Sr2.sCallsCU24041. The remarkable scaling behavior (Ref. 11) demonstrates that resistivity vs. temperature dependencies of underdoped cuprates in the pseudogap regime at T< T' and even-chain SL with the spin-gap A are governed by the same underlying mechanism. Since the 1D quantum conductivity model [10,11] works very well for even-chain SL's, then it should be also applicable for the description of the p(T) behavior in the underdoped cuprates.

The fitting of the ,0(7") curve for YBa2Cu40~with the 1D expression [10,11] p(T) = Po + C T exp(-A/T) has resulted in a spin-gap A = (224 + 5) K (fig. 1). A similar approach was applied to the ,o(T) curves of the YBa2Cu3Ox system at various levels of hole doping. These curves were shown to scale onto one universal curve [24] just by linearly scaling both temperature and resistivity. Recently, it was shown [25] that this universal behavior of scaled ,o(T) curves (fig. 2) perfectly obeys the p(T)= po + C Texp(-A/T) law [9-11]. This yields a gap A that increases with lowering hole doping [25]. Therefore, the resistivity p of underdoped cuprates below T* simply reflects the temperature dependence of the magnetic correlation length in the even-chain SL's (,ol(T)- ~miD(T)) and the pseudogap is the spin-gap formed in the 1D stripes. Since in underdoped cuprates the scaling temperature To works equally well for resistivity and Knight-shift data [23, 24], the latter can also be used for fitting with the expressions derived from the 1D SL models [26]: K ( T ) - 7"~%xp(-A/T). Fitting the K(T) data [23] for YBa2Cu4Os with this expression gives an excellent result (figure 3) with the spin-gap A = (222 + 2 0 ) K which is very close to the value A = (224 ± 5) K derived from the p(T) data (fig. 1).

El( Moshchalkovet al.IPhysica C 341-348 (2000) 887-890 30 YBaxCu40s ,.-, 25 0~ 20

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200 300 400 500 T (Kelvin) Figure 3. Knight shift data [23] fitted with K(T)7"U2exp(-~T); a gap of A = (222 ± 20) K was obtained.

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fragmentation and pinning of stripes. This promotes the inter-stripe transport thus effectively recovering the 2D localization regime for transport Ap ~ in(T), seen in high fields [25, 27] (see also the 2D area at T< Tio~)in figure 2). If disorder is removed then the I D area on the T-p phase diagram should be substantially broadened. The rapidly growing experimental evidence of the formation of stripes in the CuO2 planes necessitates the elaboration of the theoretical models taking into account the 1D character of the charge transport in the high-Tc cuprates [28]. A very enlightening recent experimental observation of the dx2y2 character of the A(tp) [29] is perfectly compatible with the stripes formation (fig. 5b): dx2_y2 means two-fold and not four-fold symmetry. Moreover, positive Ax2_y2(~p) values along the stripes could be interpreted as directions with a maximum attraction while negative Ax2_y2(tp) are indicative for the inter-stripe Coulomb repulsion for the direction perpendicular to the stripes.

(a)

(c)

(b)

(d)

Hole doping p Figure 4. cuprates.

Schematic phase diagram of high-To

In any case, depending upon the effective dimensionality, 2D (T> 7") or 1D (stripes at T< T') (figure 2 and also inset of figure 1), conductivity changes from O'2D~ ln(~:m2D) [9] to Cr1D--~mlD [10, l 1] respectively. The ID SL phase of the CuO2 planes becomes unstable at high temperatures, when entropy effects lead to stripe meandering and eventually destroy the 1D stripes [3], and at high doping levels, when the expected distance between the stripes becomes so small that the Mott insulator phase between stripes collapses, thus inducing the 2D regime [10]. At low temperatures T < T~oc, disorder in underdoped cuprates disrupts the 1D stripes and induces the

maximum

stripieity

zero

stripieity

Figure 5. Angular dependence of the stripes in the real space for four different doping levels (a)(d). Note that the dx2_y2 symmetry (b) is perfectly compatible with the presence of the stripes with A> 0 (attraction) and d < 0 (repulsion) for the directions along and perpendicular to the stripes.

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EV. Moshchalkov et al./Physica C 341-348 (2000) 887-890

The presence of stripes in high-Tc's and their effect on the coupling can be modeled by using a simple analytically solvable model with an angular dependent coupling constant 2((0), represented by a square-well form [28]. A remarkable effect of enhancing Tc has been found by introducing the Coulomb repulsion for certain angles (0. The effective coupling A and Tc turn out to be dependent on I tl + I L I, thus providing that the most important factor determining Tc is a 2((0) "contrast" or a 2((0) "modulation depth", which can be maximized by combining attractive ( 2 < 0 ) and repulsive (2 c > 0) interactions along the stripes and perpendicular to them, respectively. We think that the maximum angular 2((0) contrast is realized in high-Tc's in the vicinity of the optimal doping, where addition of 2 c helps a lot in increasing To. In the overdoped regime, due to the doping of the inter-stripe areas, the added charge suppresses the existing stripicity (fig. 5c,d) and strongly reduces the angular modulation of 2((0), which completely disappears around the point T¢_=0 where zero stripicity fully restores the 2D character of the CuO 2 planes and the jellium-like character of doping at the expense of the full suppression of T¢.

ACKNOWLEDGMENTS This work was supported by the GOA and FWOVlaanderen programs. L.T. is a research fellow supported by the Flemish IWT; J.V. is a Research Fellow of the Flemish FWO. The authors are thankful to Y. Bruynseraede for useful discussions.

REFERENCES 1. B. Batlogg, Physica C 282-287 (1997) XXIV. 2. V.J. Emery, S.A. Kivelson and O. Zachar, Physica C 282-287 (1997) 174. 3. J. Zaanen and W. van Saarloos, Physica C 282287 (1997) 178. 4. J.M. Tranquada, Physica C 282-287 (1997) 166.; J.M. Tranquada et al., Nature 375 (1995) 561.;

J.M. Tranquada et al., Phys. Rev. Lett. 73 (1997) 338.; cond-mat/ 9810212 5. E. Dagotto and T.M. Rice, Science 271 (1996) 618. 6. A. Bianconi et al., Phys. Rev. Lett. 76 (1996) 3412. 7. N.L. Saini, Phys. Rev. B 57 (1998) R11101. 8. M.A. Mook et al., Nature 395 (1998) 580. 9. V.V. Moshchalkov, Sol. St. Comm. 86 (1993) 715. 10. V.V. Moshchalkov, cond-mat/9802281. 11. V.V. Moshchaikov, L. Trappeniers, J. Vanacken, Europhys. Lett. 46 (1999) 75. 12. E.W. Carlson et al., Phys. Rev. B 57 (1998) 14704. 13. J. Karpinski et al.,. Nature 336 (1988) 660. 14. R. Gagnon etal., Phys. Rev. B 50 (1994) 3458. 15. T.A. Friedmann et al., Phys. Rev. B 42 (1990) 6217. 16. B. Bucher and P. Wachter, Phys. Rev. B 51 (1995) 3309. 17. J.L. Cohn and J. Karpinski, cond-mat/9810152. 18. R.C. Yu etal, Phys. Rev. Lett. 69 (1992) 1431. 19.N. Wang, S. Tajima, A. Rykov, K. Tomimoto, Phys. Rev. B 57 (1998) R11081. 20. S. Tajima, R. Hauff, W.-J. Jang, A.I. Rykov, Y. Sato and I. Terasaki, dourn. Low Temp, Phys. 105 (1996) 743. 21.J.M. Tarascon et al., Phys. Rev. B 37 (1988) 7458. 22. G. Xiao et al., Nature 332 (1988) 238. 23. B. Bucher et al., Physica B 199-200 (1994) 268. 24. B. Wuyts, V.V. Moshchalkov, Y. Bruynseraede, Phys. Rev. B 53 (1996) 9418. 25.L. Trappeniers, J. Vanacken, P. Wagner, G. Teniers, S. Curras, J. Perret, P. Martinoli, J.-P. Locquet, V.V. Moshchalkov and Y. Bruynseraede d. Low Temp. Phys. 117 (1999) 681-685. 26. M. Troyer, H. Tsunetsugu and D. Wurtz, Phys. Rev. B 50 (1994) 13515. 27. Y. Ando et al., Phys. Rev. Lett. 77 (1996) 2065. 28. V.V. Moshchalkov and V.A. Ivanov, condmat~9912091. 29. J.R. Kirtley et al., Nature 373 (1995) 225.