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Microscopic theory of atomic and electronic stretched exponential relaxation in high temperature superconductors
Physica C 340 (2000) 292±298
www.elsevier.nl/locate/physc
Microscopic theory of atomic and electronic stretched exponential relaxation in high temperature superconductors J.C. Phillips * Bell Laboratories, Lucent Technologies, 600 Mountain Avenue, Murray Hill, NJ 07974-0636, USA Received 2 March 2000; received in revised form 24 April 2000; accepted 5 June 2000
Abstract Measured dimensionless room-temperature conductivity relaxation stretching fractions b in YBCO are in excellent agreement with theoretical predictions, which in 1995 identi®ed two magic fractions, b 3=5 and b 3=7. Thus, relaxation studies provide an absolute measure of ``ideality'' in these complex materials, independent not only of composition x but even of crystal structure. The relaxation stretching fractions b associated with Tc itself, reported in 2000, are also explained by the magic fraction b 3=5 predicted by microscopic theory. One can infer that the interactions responsible for high-temperature superconductivity are short range, non-magnetic, and primarily associated with resonant trapping centers in semiconductive layers. Ó 2000 Elsevier Science B.V. All rights reserved. PACS: 74.20.-z; 74.62.Dh; 74.62.Fj Keywords: Relaxation; Stretched exponential; Persistent photoconductivity; Cuprates; Electron±phonon interactions
1. Introduction There is a growing realization of the importance of self-organized defects and nanostructures as a factor in, and possibly even one of the basic elements responsible for, high temperature superconductivity. However, in the complex, multinary pseudoperovskite structures of the cuprates, it is dicult to identify the presence of most defects, and much more dicult to quantify the extent to which the electrically active defects are uniformly distributed. A general method for doing this, independent not only of composition in a given structure but also of the speci®c structure itself, is
*
Tel.: +1-908-582-2528; fax: +1-908-582-4702. E-mail address: [email protected] (J.C. Phillips).
therefore of fundamental interest and may possibly have wide application. 2. Conductivity relaxation at room temperature Two kinds of normal-state relaxation eects have been studied at 300 K in the high temperature superconductor (HTSC) YBa2 Cu3 O6x : the ®rst is associated with relaxation of photoconductivity [1], the second with relaxation of conductivity changes associated with abrupt application or release of hydrostatic pressures [2] of the order of 0.5 GPa. The photoinduced conductivity relaxation eects have been studied, mainly at room temperature, for compositions x somewhat above the metal±insulator transition (MIT) which occurs near x 0:4, where such eects are largest. The
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pressure eects have been studied near x 0:7, where the crossover between the ortho I and ortho II oxygen ordering in the CuO1ÿx plane maximizes dTc =dP . These eects, which can also induce superconductivity near x 0:4, are presumably associated with defects not fundamentally dierent from those which produce HTSC near T 90 K for larger values of x. The observed magnitudes of the conductivity relaxation times are of order 10 h, and exhibit the usual Arrenhius thermal activation with activation energies in good agreement at 0.95(5) eV. The time dependence of the relaxation is given by a stretched exponential, b
Dr t Dr 0 exp ÿ t=s :
1
Such stretched exponential relaxation is ubiquitous in homogeneous glasses; it was ®rst discovered by Kohlrausch about 150 years ago. It is often explained as the result of an arti®cial distribution of relaxation times. This explanation is super®cial; it is actually nothing more than a simple mathematical tautology. This tautology is widely accepted, however, and thus, it is widely believed that the stretching fraction 0 < b < 1 is merely an adjustable parameter which has no microscopic signi®cance, and that the success of the stretched exponential itself is an accidental consequence of this arti®cial distribution. A non-trivial (indeed, mathematically quite profound) microscopic derivation of the peculiar functional form (1) was much discussed by theorists in the mid 1970s, primarily in the context of dispersive transport in amorphous semiconductors. After an initial probe pulse, the relaxation kinetics are represented by diusion of excitations to ®xed traps or sinks which are topologically equivalent to static points and are randomly (i.e., uniformly) distributed. The important result is a rigorous, microscopic derivation of the stretched exponential functional form which also shows that b is not an adjustable parameter. In the high temperature limit, both the traps and the defects diuse, and of course, b ! 1, but in the low temperature (glassy) limit, where the traps are immobile, b is given by the simple relation (inferable by dimensional analysis of the diusion equation),
b d = 2 d ;
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2
where d is the eective fractal dimensionality of the con®guration space in which the relaxation takes place. It would seem that all that this microscopic model has done is to replace one unknown parameter (b) by another (d ), but this is not the case. An extensive survey [3] of a very wide range of experimental data on temporal relaxation (1000 papers), obtained with many dierent probes on many glassy materials, includes discussion of the ®ne points of the microscopic model. This survey showed that in intrinsic materials, where the defects or traps are uniformly distributed, d and b can be speci®ed with high accuracy: (i) The stretched exponential form can be distinguished from other relaxation functions (such as power laws) if the data span more than three decades in time; in the best cases, as many as six decades were observed. (ii) Only in fully disordered samples, such as glasses with no crystallized clusters on any scale, is intrinsic stretched exponential relaxation observed over such a wide range. This is gratifying because the randomness of the static trapping centers is an essential feature of the theoretical model. Moreover, there is a universal pattern of correlation of measured values of b with sample quality and structure on a molecular scale. The most extensive database is polymeric, but the same features are found in a very wide range of other materials. In polymers, extrinsic eects (which can reduce b by more than a factor of 2) are unambiguously associated with bulky or strongly ionic side groups [4] that induce partial crystallization. Similar microscopic signatures are associated with extrinsic behavior in many other materials (for instance, ternary network glass alloys agree much better with the theory than their binary analogues, which are more likely to cluster). (iii) For density relaxation by thermal phonons and local phonons in intrinsic materials, d d 3, and b 3=5. (iv) For charge relaxation by thermal phonons and plasmons, again in intrinsic materials, d d=2 3=2, and b 3=7 0:43. The same value also holds for relaxation in three-dimensional
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quasicrystals due to the coexistence of phonons and phasons, while in ideal axial quasicrystals, computer simulations have con®rmed that the phasons carry less weight and b 0:47, in agreement with theory (9/19). We can now apply these very extensively documented and soundly based microscopic principles, which are both substantive and free of arbitrary, arti®cial or exotic assumptions, to the observed values of b for the conductivity in the high pressure (x 0:7) and photoexcited (x 0:4) experiments [1,2] on YBa2 Cu3 O6x . As expected, the high-pressure experiment (x 0:7) studied density relaxation and obtained b 0:60 0:08, in excellent agreement with theory (3/5). Although the relaxation times s and activation energies in the photoexcited experiments (x 0:4) were very close to those in the pressure experiment, the values of b were dierent. The relevant value is the low (room) temperature or ``resting state'' value, which was 0.43, this time in excellent agreement with the theoretical value of 3=7 0:43 predicted for charge transfer, and also observed in amorphous semiconductors [3]. Although these data span only four decades in time, the agreement achieved with theory is almost as good as in the best known (six decades) examples in other materials [3]. By contrast, in a poor material, such as the Tl cuprates, the observed value [5] of b in well-annealed samples undergoing volume relaxation is only 0.25, which is similar to the value found in PVC (polyvinylchloride); because the Cl is so ionic, this is one of the worst polymeric cases [4]. These small values of b appear to be associated with additional nondiusive relaxational channels which are associated with spatial inhomogeneities, for example, partial crystallization in PVC [3]. A comment is in order here. The stretched exponential form (1) was also found in the photoexcited experiments to give a good ®t to the rise of the conductivity under constant illumination to its saturated value, this time with b 0:60(1). During this ``fast'' pumped formation of the prepared state, slow relaxation with b 0:43 is taking place at the same time. The latter involves charge and density ¯uctuations, while the former apparently involves only density ¯uctuations. Note that with
free relaxation, the excitations are being drained at the traps or sinks, but in the pumping process, the excitations are being injected almost uniformly in the thin ®lm. The pumping process still involves a saturation of the regions near the traps, not depleted but ®lled. The presence of a saturation level of free carriers probably screens the charge ¯uctuations, leaving only density ¯uctuations in the excitation ¯uid, and increasing b to 0.60. In view of this success with the dimensionless (topological) stretching factor b, we re-examine the relaxation times s and activation energies EA , which are related by s x s0 x exp ÿEA x=kT :
3
With the common value s0 1:4 10ÿ12 s, the values of EA are 0.935 and 0:98 0:01 eV for photo and pressure relaxation, respectively [1,2]. Although the dierences in EA are small, they are of the order of an oxygen Einstein localized vibrational energy, and the pressure energy is the larger one. This is just as one would expect as volume relaxation by activation of (self-trapped or metastable due to quenched-in stress) oxygen diffusion should require just such an additional Stokes activation energy. By this time, the reader may be asking himself what all the fuss is about. Is it not obvious to anyone knowledgeable in semiconductor physics that photoconductivity, especially persistent photoconductivity, should be associated with charge excitations and trapping? Indeed, it is, but most discussions of HTSC focus on their metallic, not semiconductive layers. However, until now, there was no independent way to decide just how uniform the defect distribution is in samples of a given HTSC or what the chemical trends are from one material to another. It is true that dierent superconductive and normal state transport properties have often been observed on samples with the same nominal composition and indistinguishable diraction patterns. This has led many observers to conclude that ordered defects, such as oxygen vacancies or interstitials, are responsible for HTSC, as suggested very, very early by this author [6] in the same paper which introduced the concept of pseudogaps in these materials. How-
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ever, such dierences do not in themselves provide more than a qualitative guide to the role played by defects and their ordering, whereas the presently identi®ed agreements of b with the predictions of theory connect the properties of ordered defects in HTSC to those in a very wide range of other materials. It might seem at ®rst as if the popular model of HTSC based on charge transfer to the CuO2 planes is consistent with these observations. A detailed study of correlations between photoluminescence and the wavelength dependence of photoinduced conductivity has shown that the traps must be associated with the CuO1ÿx chain segments [7], and the formation of these traps enhances Tc . The authorÕs view of these matters is that they are readily explained by his ®lamentary tunneling model [8,9] according to which HTSC as well as normal-state transport currents are associated with charge ¯ow along ®laments which pass in series through both chains and planes with giant electron±phonon interactions occurring at suitably ordered resonant tunneling centers (defects) in the intervening semiconductive layers [5]. In this model, the trapping centers which are most likely to be eective in ultraslow relaxation processes are just those semiconductive tunneling centers. Such centers are much more likely to be metastable for periods of order 10 h at room temperature because they are the sites most completely isolated from thermally ¯uctuating metallic nanodomain currents of the CuO2 planes and the CuO1ÿx chain segments. In fact, such long relaxation times have been observed previously only in semiconductors (such as a-Si:H) with very high trap densities [3] so that trap impurity bands can form. There are still many papers that discuss HTSC solely in terms of metallic CuO2 planes. Of course, it has long been known that the Ox atoms in the CuOx plane organize into [0 1 0] chain segments [10]. This self-organization, which includes interchain ordering as well is responsible for the abrupt increase [11,12] of Tc (x) from 60 to 90 K with a threshold of x 0:75. (Local oxygen ordering into CuOx [0 1 0] chain segments also explains [13] the onset of the MIT at x 0:35.) Both electron diffraction and neutron scattering have shown that
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this long-range organization is highly sensitive to stress [14] and to whether or not [15] the aging or storage temperature is below a ferroelastic transition at 240 K. Chain reorganization is the only abrupt structural change that occurs for x between 0.6 and 0.9, where nothing changes for the CuO2 planes. Moreover, c-axis magnetoresistance studies [16] have concluded that in overdoped YBa2 Cu3 O6x , (x 0:95) ``the chains are primarily responsible for coherent c-axis transport (q? linear in T), and that those regions of the CuO2 planes not hybridized with the chains remain eectively two-dimensional even in the absence of a normalstate (pseudo) gap''. Thus, the ®lamentary current paths must involve both the CuO2 planes and the CuOx planes, and the fundamental topology must be three-dimensional, as assumed in the ®lamentary model [8]. There is an important point of logic here. All these experiments appeared after the ®lamentary theory was ®rst presented [6,8]. It is generally agreed that one of the most exacting tests for any theory is its ability to predict or to explain in advance new observations. All these new observations associated with the CuOx plane were implicitly predicted by the ®lamentary model at the time when HTSC was almost universally supposed to originate in the CuO2 planes alone. Even today, there is no alternative to the ®lamentary theory that explains all the above chain-related transport phenomena in YBa2 Cu3 O6x . The ®lamentary model also helps us to understand why the ordered defects are so much more uniformly distributed in YBCO than in the Tl cuprates. Linear chain segments are the best structural units for ecient percolation, not only of electrical currents but also of atomic displacements which are needed to relax frozen-in stresses which are an unavoidable part of sample growth. Such linear segments are present not only in YBCO, where they are easily observed by conventional diraction and are responsible for the dierence between the 60 K (ortho I, partially disordered O) and 90 K (ortho II, O completely ordered into chains) plateaus but also in La2ÿx Srx CuO4 , where they produce ``local orthorhombicity''. The latter has been observed by EXAFS, and it remains constant in the superconductive
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®lamentary phase 0:15 < x < 0:21 as the macroscopic orthorhombicity goes to zero [17,18] at x 0:21. Of course, such ``local orthorhombicity'' could also be present in the Tl cuprates (though it has yet to be observed), but one would also expect the latter materials to exhibit a more clustered (non-random) structure because of anti-site disorder (about 10% Cu on Tl sites) due to the similarity of the Pauling ionic radii of these two elements respectively) [19]. Thus, the in(0.96 vs. 0.95 A, homogeneity of the Tl compounds, which gives rise to a very small value of b, is certainly no surprise. 3. Diamagnetic relaxation of Tc and b duality Pressure-(x 0:7) induced relaxation of Tc is also described by stretched exponential relaxation [2], but in this case, experiment gave b 0:60(8). This situation, where dierent pumps (light, pressure) and/or dierent probes, such as electrical conductivity and diamagnetic susceptibility, can give dierent (room-temperature conductivity) or the same (low-temperature diamagnetic susceptibility) values of the magic fractions b 3=5 or 3=7 on the same or dierent samples, has been observed many times [3]. However, the striking fact, which obviously demands a microscopic theory and was highlighted in the abstract of Ref. [3], is that the glassy values observed for b are not arbitrary, but are always close to 3=5 or 3=7, corresponding to d 3, which is not surprising, or 3=2, which is puzzling. Several attempts [3,20] have been made to explain this puzzling b duality, which is not restricted to experiment. It has also been observed in numerical simulations on stretched exponential relaxation in both soft sphere glasses and quasicrystals [3], where it is certainly associated with intrinsic disorder and not with extrinsic sample characteristics (``dirt''). The basic idea is that relaxation takes place in con®guration space. If this con®guration space is simply r space, then of course, d 3. But if it involves new dynamical coordinates and new interactions or additional long-range internal order of some kind, then d* may change. Two examples of this are the long-
range Coulomb interaction, which generates plasmons, which screen interactions between electrons and defects, and phasons, which describe the special order characteristic of quasicrystals. In the latter cases, a diusive ``hop'' can branch or take place in two ways, respectively along the shortrange electron or phonon coordinates, which lead to relaxation or can branch, respectively along the collective plasmon or phason coordinates, which merely rearrange the long-range order, but do not produce short-range relaxation [3]. Because of branching, only half of the ``hops'' are eective, and d d=2. Broadly speaking, if there are p > 1 classes of coordinates that involve, in addition to phonons, other dynamical variables, d d=p. The other dynamical variables could be, for example, phasons, plasmons or magnons. This argument is quite attractive, as far as it goes, but it does not seem to be complete. A different branching argument is needed to explain why polymeric relaxation is often described by d d=2. Thus, it was suggested [20] that two kinds of ``hops'' occur for polymeric chains, one involving longitudinal chain motion, which leads to real relaxation as chains move relative to each other, and the other involving local rotation or twisting around the chain axis, which is ineective for interchain relaxation. How are these con®guration space branching mechanisms to be applied to the diamagnetic susceptibility of HTSC? If electron±magnon interactions were responsible for HTSC, then one would have both electron±phonon and electron±magnon interactions. The latter are similar to phasons or plasmons in the sense that they do not contribute to real-space relaxation. With two classes of interactions, this means that at least p 2, and experimentally, this would mean that b 6 0.43. The fact [2] that b 0:60 therefore excludes electron± magnon interactions and leaves electron±phonon interactions (p 1, d d, b 3=5) as the only possibility. One can simply adopt the ®lamentary or weak link model, and argue that relaxation of these basic interlayer defect structures is short-ranged and takes place in r space unhindered by longrange Coulomb or short-range magnetic interactions. This is the simplest mechanism, and it
J.C. Phillips / Physica C 340 (2000) 292±298
accounts for the fact that Tc relaxation is described by d 3 in the pressure-excited experiments [2]. We notice that in nearly all cases, the eects of additional interactions are either absent or aect b equally with the phonon interactions. One can say that this observation describes a kind of equipartition relaxation kinetics in the glassy state similar to the equipartition principle of equilibrium statistical mechanics. (A few exceptions to this rule are identi®ed in Ref. [3] as being associated with bulky polymeric side groups or clusters (nanoscale phase separation) in binary network glasses. These appear to be absent in YBCO, suggesting that the defects in the semiconductive layers are simple oxygen vacancies or interstitials.) 4. High-resolution NMR echo experiments These branching mechanisms help us to understand the very elegant relaxation processes studied by two-time and four-time stimulated NMR echo experiments [21] on ultrapure OTP (ortho-terphenyl), the best non-alcoholic molecular glass former. The two-time echo experiments measure the average relaxation, which has b 0:42, d 3=2. The four-time echo experiments include two extra pulses which function as a ®lter; as the spacing between these two pulses (the ®lter time) increases, the relaxation time increases, showing that the ®lter progressively selects nuclei with slower relaxation times, and at the same time, b ® 1. In the opposite limit, as the ®lter time goes to zero, the relaxation time shortens, but remains several times longer than the two-time value. In this limit, the stretching exponent b tends to 0.60, not 0.42 (Fig. 4, Ref. [21]). The way in which the ®lter acts to select nuclei with longer relaxation times may depend on shortrange vs. long-range interactions. The ®lter may eectively exclude nuclei with long-range interactions in favor of those with short-range interactions, and in the limit of long ®lter times, the latter become nuclei which are isolated from exchange with other nuclei; for these independent nuclei, the relaxation is simply exponential (b 1). Even in the opposite limit, where local interactions remain, the long-range interactions are excluded by the
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®ltering process, which dephases the latter, thus increasing the relaxation time. It is true that we do not know the details of the glassy state. Nevertheless, we can infer that the same kind of separation may take place in the relaxation interactions available to nuclear spins excited by NMR echoes as occurs for electrons and plasmons or for phonons and phasons in the two-time experiment. When the long-range interaction channel is closed by the ®lter in the four-time experiment, the shortrange interactions alone remain, and b reverts to the simple value of 3=5.
5. Other possible perovskite applications Pseudoperovskite structures are favorable not only to HTSC, but to giant magnetoresistance as well. In spite of obvious chemical similarities between manganite perovskites and HTSC pseudoperovskites, it is dicult to ®nd similar unambiguous anomalies in physical properties, although these are seen in a few notable cases, such as ion channeling [22±24]. The generality of the present analysis suggests that relaxation studies could provide quantitative microscopic insight into these similarities, and might again lead to useful insights into the roles played by spatial and spin coordinates.
6. Conclusions The pattern of relaxation bs obtained in the photo (x 0:4) and pressure (x 0:7) excited room-temperature conductivity in YBa2 Cu3 O6x , together with that obtained for the diamagnetic susceptibility at Tc , is remarkably similar to that obtained in other atomic and molecular materials, with the same magic fractions 3=5 and 3=7. At ®rst, it seems that the new experiments merely con®rm the universality of the trap model and b duality [3]. However, close examination of the temperature-dependent conductivity data shows that the strong similarity to relaxation in amorphous semiconductors, for example, is totally inconsistent with popular models of HTSC which
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ascribe both the normal state transport anomalies and HTSC to metallic currents con®ned to the CuO2 planes. In the charge-transfer model, the interactions of the dopants with carriers or traps in the semiconductive barrier layers are neglected. This is analogous to the d-function doping geometry used to observe quantum Hall eects, where the carrier± dopant interaction is designed to be as weak as possible. Instead, it is clear that the currents must be forced to pass through dopants in the semiconductive layers, thus following zig-zag paths from one metallic plane to adjacent metallic planes. This has the eect of maximizing the dopant±carrier interactions. The only microscopic model in which this can happen is the ®lamentary model, where the CuO2 ``metallic'' planes are in fact not completely metallic, but instead consist of metallic domains separated by non-metallic domain walls. The data on the relaxation of superconductive diamagnetism furthermore are interpreted within the branching mechanism for b duality, which in most cases emphasizes the way in which long-range interactions become ineective for producing real-space relaxation. Thus, the observation of b 3=5 (single channel) in Tc relaxation experiments [2] automatically excludes branching, and therefore is taken as an indication that neither electron±electron interactions nor spin modes aect Tc . All that remains is shortrange electron±phonon interactions at the defects in the semiconductive layers, as discussed above. These must be simple interactions, which involve diusion of oxygen alone, and not more complex interactions, such as might arise from weak links formed from small clusters associated with antisite defects.
References [1] V.I. Kudinov, I.L. Chaplygin, A.I. Kirilyuk, N.M. Kreines, R. Lahio, E. Lahderanta, C. Ayache, Phys. Rev. B 47 (1993) 9017. [2] S. Sadewasser, J.S. Schilling, A.P. Paulikas, B.W. Veal, Phys. Rev. B 61 (2000) 741. [3] J.C. Phillips, Rep. Prog. Phys. 59 (1996) 1133. [4] J.C. Phillips, Rep. Prog. Phys. 59 (1996) 1173. [5] A.-K. Klehe, C. Looney, J.S. Schilling, H. Takahashi, N. Mori, Y. Shimakawa, Y. Kubo, T. Manako, S. Doyle, A.M. Herman, Physica C 257 (1996) 105. [6] J.C. Phillips, Phys. Rev. Lett. 59 (1987) 1856. [7] J.F. Federici, D. Chew, B. Welker, W. Savin, J. GutierrezSolana, T. Fink, W. Wilber, Phys. Rev. B 52 (1995) 15592. [8] J.C. Phillips, Phys. Rev. B 41 (1990) 8968. [9] J.C. Phillips, Phil. Mag. B 79 (1999) 1477. [10] J.D. Jorgensen, B.W. Veal, A.P. Paulikas, L.J. Nowicki, G.W. Crabtree, H. Claus, W.K. Kwok, Phys. Rev. B 41 (1990) 1863. [11] M. Akoshima, Y. Koike, J. Phys. Soc. Jpn. 67 (2000) 3653. [12] K. Hatada, H. Shimizu, Physica C 304 (1998) 89. [13] T. Zeiske, R. Sonntag, D. Hohlwein, N.H. Andersen, T. Wolf, Nature 353 (1991) 542. [14] O. Kraut, C. Meingast, G. Brauchle, H. Claus, A. Erb, G. Muller-Vogt, H. Wuhl, Physica C 205 (1993) 139. [15] W.H. Fietz, R. Quenzel, H.A. Ludwig, K. Grube, S.I. Schlachter, F.W. Hornung, T. Wolf, A. Erb, M. Klaser, G. Muller-Vogt, Physica C 270 (1996) 258. [16] N.E. Hussey, H. Takagi, Y. Iye, S. Tajima, A.I. Rykov, K. Yoshida, Phys. Rev. B 61 (2000) R6475. [17] D. Haskel, E.A. Stern, D.G. Hinks, A.W. Mitchell, J.D. Jorgensen, J.I. Budnick, Phys. Rev. Lett. 76 (1996) 439. [18] J.C. Phillips, Phil. Mag. B 79 (1999) 1481. [19] S. Sasaki, K. Kawaguchi, M. Nakao, Jpn. J. Appl. Phys. 31 (1992) L467. [20] J.C. Phillips, J.M. Vandenberg, J. Phys.: Condens. Matter 9 (1997) L251. [21] R. Bohmer, G. Hinze, G. Diezemann, B. Geil, H. Sillescu, Europhys. Lett. 36 (1996) 55. [22] R.P. Sharma, T. Venkatesen, Z.H. Zhang, J.R. Liu, R. Chu, W.K. Chu, Phys. Rev. Lett. 77 (1996) 4624. [23] R.P. Sharma, G.C. Xiong, C. Kwon, R. Ramesh, R.L. Greene, T. Venkatesen, Phys. Rev. B 54 (1996) 10014. [24] J.C. Phillips, Physica C 252 (1995) 188.
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Microscopic theory of atomic and electronic stretched exponential relaxation in high temperature superconductors J.C. Phillips * Bell Laboratories, Lucent Technologies, 600 Mountain Avenue, Murray Hill, NJ 07974-0636, USA Received 2 March 2000; received in revised form 24 April 2000; accepted 5 June 2000
Abstract Measured dimensionless room-temperature conductivity relaxation stretching fractions b in YBCO are in excellent agreement with theoretical predictions, which in 1995 identi®ed two magic fractions, b 3=5 and b 3=7. Thus, relaxation studies provide an absolute measure of ``ideality'' in these complex materials, independent not only of composition x but even of crystal structure. The relaxation stretching fractions b associated with Tc itself, reported in 2000, are also explained by the magic fraction b 3=5 predicted by microscopic theory. One can infer that the interactions responsible for high-temperature superconductivity are short range, non-magnetic, and primarily associated with resonant trapping centers in semiconductive layers. Ó 2000 Elsevier Science B.V. All rights reserved. PACS: 74.20.-z; 74.62.Dh; 74.62.Fj Keywords: Relaxation; Stretched exponential; Persistent photoconductivity; Cuprates; Electron±phonon interactions
1. Introduction There is a growing realization of the importance of self-organized defects and nanostructures as a factor in, and possibly even one of the basic elements responsible for, high temperature superconductivity. However, in the complex, multinary pseudoperovskite structures of the cuprates, it is dicult to identify the presence of most defects, and much more dicult to quantify the extent to which the electrically active defects are uniformly distributed. A general method for doing this, independent not only of composition in a given structure but also of the speci®c structure itself, is
*
Tel.: +1-908-582-2528; fax: +1-908-582-4702. E-mail address: [email protected] (J.C. Phillips).
therefore of fundamental interest and may possibly have wide application. 2. Conductivity relaxation at room temperature Two kinds of normal-state relaxation eects have been studied at 300 K in the high temperature superconductor (HTSC) YBa2 Cu3 O6x : the ®rst is associated with relaxation of photoconductivity [1], the second with relaxation of conductivity changes associated with abrupt application or release of hydrostatic pressures [2] of the order of 0.5 GPa. The photoinduced conductivity relaxation eects have been studied, mainly at room temperature, for compositions x somewhat above the metal±insulator transition (MIT) which occurs near x 0:4, where such eects are largest. The
0921-4534/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 0 ) 0 1 5 0 6 - 9
J.C. Phillips / Physica C 340 (2000) 292±298
pressure eects have been studied near x 0:7, where the crossover between the ortho I and ortho II oxygen ordering in the CuO1ÿx plane maximizes dTc =dP . These eects, which can also induce superconductivity near x 0:4, are presumably associated with defects not fundamentally dierent from those which produce HTSC near T 90 K for larger values of x. The observed magnitudes of the conductivity relaxation times are of order 10 h, and exhibit the usual Arrenhius thermal activation with activation energies in good agreement at 0.95(5) eV. The time dependence of the relaxation is given by a stretched exponential, b
Dr t Dr 0 exp ÿ t=s :
1
Such stretched exponential relaxation is ubiquitous in homogeneous glasses; it was ®rst discovered by Kohlrausch about 150 years ago. It is often explained as the result of an arti®cial distribution of relaxation times. This explanation is super®cial; it is actually nothing more than a simple mathematical tautology. This tautology is widely accepted, however, and thus, it is widely believed that the stretching fraction 0 < b < 1 is merely an adjustable parameter which has no microscopic signi®cance, and that the success of the stretched exponential itself is an accidental consequence of this arti®cial distribution. A non-trivial (indeed, mathematically quite profound) microscopic derivation of the peculiar functional form (1) was much discussed by theorists in the mid 1970s, primarily in the context of dispersive transport in amorphous semiconductors. After an initial probe pulse, the relaxation kinetics are represented by diusion of excitations to ®xed traps or sinks which are topologically equivalent to static points and are randomly (i.e., uniformly) distributed. The important result is a rigorous, microscopic derivation of the stretched exponential functional form which also shows that b is not an adjustable parameter. In the high temperature limit, both the traps and the defects diuse, and of course, b ! 1, but in the low temperature (glassy) limit, where the traps are immobile, b is given by the simple relation (inferable by dimensional analysis of the diusion equation),
b d = 2 d ;
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2
where d is the eective fractal dimensionality of the con®guration space in which the relaxation takes place. It would seem that all that this microscopic model has done is to replace one unknown parameter (b) by another (d ), but this is not the case. An extensive survey [3] of a very wide range of experimental data on temporal relaxation (1000 papers), obtained with many dierent probes on many glassy materials, includes discussion of the ®ne points of the microscopic model. This survey showed that in intrinsic materials, where the defects or traps are uniformly distributed, d and b can be speci®ed with high accuracy: (i) The stretched exponential form can be distinguished from other relaxation functions (such as power laws) if the data span more than three decades in time; in the best cases, as many as six decades were observed. (ii) Only in fully disordered samples, such as glasses with no crystallized clusters on any scale, is intrinsic stretched exponential relaxation observed over such a wide range. This is gratifying because the randomness of the static trapping centers is an essential feature of the theoretical model. Moreover, there is a universal pattern of correlation of measured values of b with sample quality and structure on a molecular scale. The most extensive database is polymeric, but the same features are found in a very wide range of other materials. In polymers, extrinsic eects (which can reduce b by more than a factor of 2) are unambiguously associated with bulky or strongly ionic side groups [4] that induce partial crystallization. Similar microscopic signatures are associated with extrinsic behavior in many other materials (for instance, ternary network glass alloys agree much better with the theory than their binary analogues, which are more likely to cluster). (iii) For density relaxation by thermal phonons and local phonons in intrinsic materials, d d 3, and b 3=5. (iv) For charge relaxation by thermal phonons and plasmons, again in intrinsic materials, d d=2 3=2, and b 3=7 0:43. The same value also holds for relaxation in three-dimensional
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quasicrystals due to the coexistence of phonons and phasons, while in ideal axial quasicrystals, computer simulations have con®rmed that the phasons carry less weight and b 0:47, in agreement with theory (9/19). We can now apply these very extensively documented and soundly based microscopic principles, which are both substantive and free of arbitrary, arti®cial or exotic assumptions, to the observed values of b for the conductivity in the high pressure (x 0:7) and photoexcited (x 0:4) experiments [1,2] on YBa2 Cu3 O6x . As expected, the high-pressure experiment (x 0:7) studied density relaxation and obtained b 0:60 0:08, in excellent agreement with theory (3/5). Although the relaxation times s and activation energies in the photoexcited experiments (x 0:4) were very close to those in the pressure experiment, the values of b were dierent. The relevant value is the low (room) temperature or ``resting state'' value, which was 0.43, this time in excellent agreement with the theoretical value of 3=7 0:43 predicted for charge transfer, and also observed in amorphous semiconductors [3]. Although these data span only four decades in time, the agreement achieved with theory is almost as good as in the best known (six decades) examples in other materials [3]. By contrast, in a poor material, such as the Tl cuprates, the observed value [5] of b in well-annealed samples undergoing volume relaxation is only 0.25, which is similar to the value found in PVC (polyvinylchloride); because the Cl is so ionic, this is one of the worst polymeric cases [4]. These small values of b appear to be associated with additional nondiusive relaxational channels which are associated with spatial inhomogeneities, for example, partial crystallization in PVC [3]. A comment is in order here. The stretched exponential form (1) was also found in the photoexcited experiments to give a good ®t to the rise of the conductivity under constant illumination to its saturated value, this time with b 0:60(1). During this ``fast'' pumped formation of the prepared state, slow relaxation with b 0:43 is taking place at the same time. The latter involves charge and density ¯uctuations, while the former apparently involves only density ¯uctuations. Note that with
free relaxation, the excitations are being drained at the traps or sinks, but in the pumping process, the excitations are being injected almost uniformly in the thin ®lm. The pumping process still involves a saturation of the regions near the traps, not depleted but ®lled. The presence of a saturation level of free carriers probably screens the charge ¯uctuations, leaving only density ¯uctuations in the excitation ¯uid, and increasing b to 0.60. In view of this success with the dimensionless (topological) stretching factor b, we re-examine the relaxation times s and activation energies EA , which are related by s x s0 x exp ÿEA x=kT :
3
With the common value s0 1:4 10ÿ12 s, the values of EA are 0.935 and 0:98 0:01 eV for photo and pressure relaxation, respectively [1,2]. Although the dierences in EA are small, they are of the order of an oxygen Einstein localized vibrational energy, and the pressure energy is the larger one. This is just as one would expect as volume relaxation by activation of (self-trapped or metastable due to quenched-in stress) oxygen diffusion should require just such an additional Stokes activation energy. By this time, the reader may be asking himself what all the fuss is about. Is it not obvious to anyone knowledgeable in semiconductor physics that photoconductivity, especially persistent photoconductivity, should be associated with charge excitations and trapping? Indeed, it is, but most discussions of HTSC focus on their metallic, not semiconductive layers. However, until now, there was no independent way to decide just how uniform the defect distribution is in samples of a given HTSC or what the chemical trends are from one material to another. It is true that dierent superconductive and normal state transport properties have often been observed on samples with the same nominal composition and indistinguishable diraction patterns. This has led many observers to conclude that ordered defects, such as oxygen vacancies or interstitials, are responsible for HTSC, as suggested very, very early by this author [6] in the same paper which introduced the concept of pseudogaps in these materials. How-
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ever, such dierences do not in themselves provide more than a qualitative guide to the role played by defects and their ordering, whereas the presently identi®ed agreements of b with the predictions of theory connect the properties of ordered defects in HTSC to those in a very wide range of other materials. It might seem at ®rst as if the popular model of HTSC based on charge transfer to the CuO2 planes is consistent with these observations. A detailed study of correlations between photoluminescence and the wavelength dependence of photoinduced conductivity has shown that the traps must be associated with the CuO1ÿx chain segments [7], and the formation of these traps enhances Tc . The authorÕs view of these matters is that they are readily explained by his ®lamentary tunneling model [8,9] according to which HTSC as well as normal-state transport currents are associated with charge ¯ow along ®laments which pass in series through both chains and planes with giant electron±phonon interactions occurring at suitably ordered resonant tunneling centers (defects) in the intervening semiconductive layers [5]. In this model, the trapping centers which are most likely to be eective in ultraslow relaxation processes are just those semiconductive tunneling centers. Such centers are much more likely to be metastable for periods of order 10 h at room temperature because they are the sites most completely isolated from thermally ¯uctuating metallic nanodomain currents of the CuO2 planes and the CuO1ÿx chain segments. In fact, such long relaxation times have been observed previously only in semiconductors (such as a-Si:H) with very high trap densities [3] so that trap impurity bands can form. There are still many papers that discuss HTSC solely in terms of metallic CuO2 planes. Of course, it has long been known that the Ox atoms in the CuOx plane organize into [0 1 0] chain segments [10]. This self-organization, which includes interchain ordering as well is responsible for the abrupt increase [11,12] of Tc (x) from 60 to 90 K with a threshold of x 0:75. (Local oxygen ordering into CuOx [0 1 0] chain segments also explains [13] the onset of the MIT at x 0:35.) Both electron diffraction and neutron scattering have shown that
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this long-range organization is highly sensitive to stress [14] and to whether or not [15] the aging or storage temperature is below a ferroelastic transition at 240 K. Chain reorganization is the only abrupt structural change that occurs for x between 0.6 and 0.9, where nothing changes for the CuO2 planes. Moreover, c-axis magnetoresistance studies [16] have concluded that in overdoped YBa2 Cu3 O6x , (x 0:95) ``the chains are primarily responsible for coherent c-axis transport (q? linear in T), and that those regions of the CuO2 planes not hybridized with the chains remain eectively two-dimensional even in the absence of a normalstate (pseudo) gap''. Thus, the ®lamentary current paths must involve both the CuO2 planes and the CuOx planes, and the fundamental topology must be three-dimensional, as assumed in the ®lamentary model [8]. There is an important point of logic here. All these experiments appeared after the ®lamentary theory was ®rst presented [6,8]. It is generally agreed that one of the most exacting tests for any theory is its ability to predict or to explain in advance new observations. All these new observations associated with the CuOx plane were implicitly predicted by the ®lamentary model at the time when HTSC was almost universally supposed to originate in the CuO2 planes alone. Even today, there is no alternative to the ®lamentary theory that explains all the above chain-related transport phenomena in YBa2 Cu3 O6x . The ®lamentary model also helps us to understand why the ordered defects are so much more uniformly distributed in YBCO than in the Tl cuprates. Linear chain segments are the best structural units for ecient percolation, not only of electrical currents but also of atomic displacements which are needed to relax frozen-in stresses which are an unavoidable part of sample growth. Such linear segments are present not only in YBCO, where they are easily observed by conventional diraction and are responsible for the dierence between the 60 K (ortho I, partially disordered O) and 90 K (ortho II, O completely ordered into chains) plateaus but also in La2ÿx Srx CuO4 , where they produce ``local orthorhombicity''. The latter has been observed by EXAFS, and it remains constant in the superconductive
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®lamentary phase 0:15 < x < 0:21 as the macroscopic orthorhombicity goes to zero [17,18] at x 0:21. Of course, such ``local orthorhombicity'' could also be present in the Tl cuprates (though it has yet to be observed), but one would also expect the latter materials to exhibit a more clustered (non-random) structure because of anti-site disorder (about 10% Cu on Tl sites) due to the similarity of the Pauling ionic radii of these two elements respectively) [19]. Thus, the in(0.96 vs. 0.95 A, homogeneity of the Tl compounds, which gives rise to a very small value of b, is certainly no surprise. 3. Diamagnetic relaxation of Tc and b duality Pressure-(x 0:7) induced relaxation of Tc is also described by stretched exponential relaxation [2], but in this case, experiment gave b 0:60(8). This situation, where dierent pumps (light, pressure) and/or dierent probes, such as electrical conductivity and diamagnetic susceptibility, can give dierent (room-temperature conductivity) or the same (low-temperature diamagnetic susceptibility) values of the magic fractions b 3=5 or 3=7 on the same or dierent samples, has been observed many times [3]. However, the striking fact, which obviously demands a microscopic theory and was highlighted in the abstract of Ref. [3], is that the glassy values observed for b are not arbitrary, but are always close to 3=5 or 3=7, corresponding to d 3, which is not surprising, or 3=2, which is puzzling. Several attempts [3,20] have been made to explain this puzzling b duality, which is not restricted to experiment. It has also been observed in numerical simulations on stretched exponential relaxation in both soft sphere glasses and quasicrystals [3], where it is certainly associated with intrinsic disorder and not with extrinsic sample characteristics (``dirt''). The basic idea is that relaxation takes place in con®guration space. If this con®guration space is simply r space, then of course, d 3. But if it involves new dynamical coordinates and new interactions or additional long-range internal order of some kind, then d* may change. Two examples of this are the long-
range Coulomb interaction, which generates plasmons, which screen interactions between electrons and defects, and phasons, which describe the special order characteristic of quasicrystals. In the latter cases, a diusive ``hop'' can branch or take place in two ways, respectively along the shortrange electron or phonon coordinates, which lead to relaxation or can branch, respectively along the collective plasmon or phason coordinates, which merely rearrange the long-range order, but do not produce short-range relaxation [3]. Because of branching, only half of the ``hops'' are eective, and d d=2. Broadly speaking, if there are p > 1 classes of coordinates that involve, in addition to phonons, other dynamical variables, d d=p. The other dynamical variables could be, for example, phasons, plasmons or magnons. This argument is quite attractive, as far as it goes, but it does not seem to be complete. A different branching argument is needed to explain why polymeric relaxation is often described by d d=2. Thus, it was suggested [20] that two kinds of ``hops'' occur for polymeric chains, one involving longitudinal chain motion, which leads to real relaxation as chains move relative to each other, and the other involving local rotation or twisting around the chain axis, which is ineective for interchain relaxation. How are these con®guration space branching mechanisms to be applied to the diamagnetic susceptibility of HTSC? If electron±magnon interactions were responsible for HTSC, then one would have both electron±phonon and electron±magnon interactions. The latter are similar to phasons or plasmons in the sense that they do not contribute to real-space relaxation. With two classes of interactions, this means that at least p 2, and experimentally, this would mean that b 6 0.43. The fact [2] that b 0:60 therefore excludes electron± magnon interactions and leaves electron±phonon interactions (p 1, d d, b 3=5) as the only possibility. One can simply adopt the ®lamentary or weak link model, and argue that relaxation of these basic interlayer defect structures is short-ranged and takes place in r space unhindered by longrange Coulomb or short-range magnetic interactions. This is the simplest mechanism, and it
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accounts for the fact that Tc relaxation is described by d 3 in the pressure-excited experiments [2]. We notice that in nearly all cases, the eects of additional interactions are either absent or aect b equally with the phonon interactions. One can say that this observation describes a kind of equipartition relaxation kinetics in the glassy state similar to the equipartition principle of equilibrium statistical mechanics. (A few exceptions to this rule are identi®ed in Ref. [3] as being associated with bulky polymeric side groups or clusters (nanoscale phase separation) in binary network glasses. These appear to be absent in YBCO, suggesting that the defects in the semiconductive layers are simple oxygen vacancies or interstitials.) 4. High-resolution NMR echo experiments These branching mechanisms help us to understand the very elegant relaxation processes studied by two-time and four-time stimulated NMR echo experiments [21] on ultrapure OTP (ortho-terphenyl), the best non-alcoholic molecular glass former. The two-time echo experiments measure the average relaxation, which has b 0:42, d 3=2. The four-time echo experiments include two extra pulses which function as a ®lter; as the spacing between these two pulses (the ®lter time) increases, the relaxation time increases, showing that the ®lter progressively selects nuclei with slower relaxation times, and at the same time, b ® 1. In the opposite limit, as the ®lter time goes to zero, the relaxation time shortens, but remains several times longer than the two-time value. In this limit, the stretching exponent b tends to 0.60, not 0.42 (Fig. 4, Ref. [21]). The way in which the ®lter acts to select nuclei with longer relaxation times may depend on shortrange vs. long-range interactions. The ®lter may eectively exclude nuclei with long-range interactions in favor of those with short-range interactions, and in the limit of long ®lter times, the latter become nuclei which are isolated from exchange with other nuclei; for these independent nuclei, the relaxation is simply exponential (b 1). Even in the opposite limit, where local interactions remain, the long-range interactions are excluded by the
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®ltering process, which dephases the latter, thus increasing the relaxation time. It is true that we do not know the details of the glassy state. Nevertheless, we can infer that the same kind of separation may take place in the relaxation interactions available to nuclear spins excited by NMR echoes as occurs for electrons and plasmons or for phonons and phasons in the two-time experiment. When the long-range interaction channel is closed by the ®lter in the four-time experiment, the shortrange interactions alone remain, and b reverts to the simple value of 3=5.
5. Other possible perovskite applications Pseudoperovskite structures are favorable not only to HTSC, but to giant magnetoresistance as well. In spite of obvious chemical similarities between manganite perovskites and HTSC pseudoperovskites, it is dicult to ®nd similar unambiguous anomalies in physical properties, although these are seen in a few notable cases, such as ion channeling [22±24]. The generality of the present analysis suggests that relaxation studies could provide quantitative microscopic insight into these similarities, and might again lead to useful insights into the roles played by spatial and spin coordinates.
6. Conclusions The pattern of relaxation bs obtained in the photo (x 0:4) and pressure (x 0:7) excited room-temperature conductivity in YBa2 Cu3 O6x , together with that obtained for the diamagnetic susceptibility at Tc , is remarkably similar to that obtained in other atomic and molecular materials, with the same magic fractions 3=5 and 3=7. At ®rst, it seems that the new experiments merely con®rm the universality of the trap model and b duality [3]. However, close examination of the temperature-dependent conductivity data shows that the strong similarity to relaxation in amorphous semiconductors, for example, is totally inconsistent with popular models of HTSC which
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ascribe both the normal state transport anomalies and HTSC to metallic currents con®ned to the CuO2 planes. In the charge-transfer model, the interactions of the dopants with carriers or traps in the semiconductive barrier layers are neglected. This is analogous to the d-function doping geometry used to observe quantum Hall eects, where the carrier± dopant interaction is designed to be as weak as possible. Instead, it is clear that the currents must be forced to pass through dopants in the semiconductive layers, thus following zig-zag paths from one metallic plane to adjacent metallic planes. This has the eect of maximizing the dopant±carrier interactions. The only microscopic model in which this can happen is the ®lamentary model, where the CuO2 ``metallic'' planes are in fact not completely metallic, but instead consist of metallic domains separated by non-metallic domain walls. The data on the relaxation of superconductive diamagnetism furthermore are interpreted within the branching mechanism for b duality, which in most cases emphasizes the way in which long-range interactions become ineective for producing real-space relaxation. Thus, the observation of b 3=5 (single channel) in Tc relaxation experiments [2] automatically excludes branching, and therefore is taken as an indication that neither electron±electron interactions nor spin modes aect Tc . All that remains is shortrange electron±phonon interactions at the defects in the semiconductive layers, as discussed above. These must be simple interactions, which involve diusion of oxygen alone, and not more complex interactions, such as might arise from weak links formed from small clusters associated with antisite defects.
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