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The intrinsic heterogeneity of superconductivity in the cuprates

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A Letters Journal Exploring the Frontiers oe Physics

EPL, 109 (2015) 27001 www.epljournal.org

doi: 10.1209/0295-5075/109/27001

Perspective

The intrinsic heterogeneity of superconductivity in the cuprates

A. Shengelaya 1 and K. A. Müller2

1 Department of Physics, Tbilisi State University - Chavchavadze 3, GE-0128 Tbilisi, Georgia

2 Physik-Institut der Universität Zurich - Winterthurerstr. 190, CH-8057 Zürich, Switzerland

received 4 December 2014; accepted in final form 12 December 2014 published online 2 January 2015

pacs 74.20.-z - Theories and models of superconducting state pacs 74.25.-q - Properties of superconductors pacs 74.72.-h - Cuprate superconductors

Abstract - In the hole-doped, high-temperature superconducting cuprates, an intrinsic heterogeneity is found, from the early observations to recent data. Below optimum doping, the heterogeneity consists of dynamic metallic and, at low temperatures, superconducting regions in the form of clusters or stripes, which develop and decay as a function of time and location in the antiferromagnetic lattice. This behaviour is underlined by the interesting linear relation between the oxygen isotope shifts of the magnetic penetration depth and the critical temperature with a slope that is a factor 2 larger than expected for the homogeneous distribution of superfluid density. Allusion is also made to the Bose-Einstein condensation reported in structurally heterogeneous, polycrystalline polymer platelets as well as especially to the heterogeneous distribution of visible and dark matter in the Universe, which point to a change of paradigm in modern physics.

Copyright © EPLA, 2015

In his seminal papers, Fritz London very early assumed that a superconductor is a macroscopic quantum state as is a Bose-Einstein quantum fluid [1,2]. These states are homogeneous in space. This is also presupposed in the Bardeen, Cooper and Schrieffer (BCS) theory [3], which is well confirmed in classical elementary and in-termetallic superconductors [4]. After the discovery of superconductivity in doped copper oxides [5,6], experimental evidence grew that a heterogeneity existed in these compounds, with in part remarkably high transition temperatures. This property was attributed to either the presence of chemical or structural inhomogeneity in the samples or to the existence of intrinsic heterogeneities [7]. With the remarkable advances in sample preparation, the former possibilities could be eliminated [6]. In the following, we will present evidence that in the copper oxides the observed superconductivity is present intrinsically in a dynamic inhomogeneous matrix, contrary to the original assumption of London. This invalidates a substantial part of the theories published on the subject that assume homogeneity at the outset, such as the often used RVB or t-J models. Later we discuss related superconductors, such as granular aluminium, ending with allusions to Bose-Einstein condensates in heterogeneous systems, such as polymer platelets, and the dark matter present in the Universe.

Quite early after the discovery of superconductivity in the doped cuprates, a number of experiments indicated the existence of two intrinsic components. These were reviewed at the conference on "High-Tc Superconductivity 1996: Ten Years after the Discovery" by Mihailovic and Muller [8]. It was pointed out that the different techniques employed supported each other if their characteristic time scales were considered. The results of extended X-ray absorption fine-structure spectroscopy (EXAFS) with a time scale of 10~15 s by Bianconi's group and the somewhat slower neutron pair distribution function (PDF) of Billinge's group supported the presence of an inhomoge-neous structure on the scale of the superconducting coherence length, with two distinctly different Cu-O bond lengths in the CuO2 plane. From NMR investigations on a much slower time scale of ^10~8 s by Hammel and collaborators, a narrow line attributed to tetragonal Cu sites and a broadened signal as resulting from octahedral Cu sites with tilted axis were deduced, in agreement with the EXAFS results [8]. Furthermore, the extensive investigation of 165Tm NMR and Cu NQR in Teplov's group in Kazan also revealed the existence of centers with or-thorhombic and tetragonal symmetry [8]. The authors regarded their finding as a result of an inhomogeneous charge and spin distribution. The slow NQR and NMR observations, as compared to fast EXAFS and PDF's,

Fig. 1: (Colour on-line) Photoinduced signal amplitudes in different cuprate superconductors as a function of temperature. Two distinct components are generally observed. One disappears abruptly at Tc, whereas the other disappears asymptotically at much higher temperatures. From ref. [12].

indicated that the former data could be due to pinned polaronic conformations.

Also, the time-resolved electrodynamic response indicated the presence of two types of carriers [8]. The optical conductivity in the normal state always showed a Drude-like Fermi liquid rising at low frequency plus a characteristic mid-infrared broad response near 400 cm-1 (0.05 eV). The interpretation was in terms of the presence of two dynamic electronic components, with the former more 2D, fermions, and the latter 3D, polarons, thus in line with the structural and magnetic resonance results [8]. The fermion-type carriers had relaxation times of a few picoseconds, the polarons of 10 ns, which implied a slow tunnelling relaxation channel to the ground state. Remarkable at the time were also the data from thermal differential reflectance (TDR) spectroscopy by Little and collaborators at Stanford University [9]. In their experiments, a spectral feature appeared universally in the superconducting to normal reflectivity ratio near 1.5 to 2 eV. Thus, in addition to the usual low-frequency infrared response of the condensate, in the cuprates a high-frequency component exists, giving rise to a two-component spectral response [8]. Already then it was discussed whether a polaronic solid with Fermi liquid-like quasiparticles was present, intertwined with stripes. The high-energy component of the pairing boson obtained from TDR spectroscopy with an energy of 1.5 to 2.0 eV is closely the same as the known Cu2+ Jahn-Teller excitation energy.

More recent experiments confirmed these early findings of a two-component response and inhomogeneous charge and spin distribution in cuprates. Systematic and careful NMR experiments by Haase and collaborators clearly demonstrated that a two-component spin susceptibility is necessary to explain the data in different cuprate families [10]. This is in contrast to the widely held view that the cuprates can be considered as a one-component system [11]. Also, detailed and systematic time-resolved optical reflectivity experiments reconfirmed the presence of two vastly different relaxation times, which appears to be generic in different cuprate superconductors as shown

Fig. 2: (Colour on-line) The mean-square relative displacement of the Cu-O bond distances of Lai.85Sro.15O4 (M = Co, Ni) as a function of temperature. From ref. [13].

in fig. 1 [12]. The two relaxations rates were interpreted as a coexistence of localized and itinerant states.

Using high-resolution EXAFS, in-plane Cu-O bond distances were recently measured in La1.8sSro.15CuO4 single crystals [13]. The temperature dependence of the Cu-O bond mean-square relative displacement ^Cu-o presented in fig. 2 exhibits an anomalous behaviour. Below a certain temperature T* that is much higher than Tc, ^Cu-O deviates from the expected Debye-Waller behaviour and shows a strong upturn, which indicates a local lattice distortion. A sharp drop due to the superconducting coherence at Tc is followed by another upturn upon decreasing the temperature below Tc. These observations were interpreted as formation of an in-plane Cu-O bond splitting below T* that disappears as superconducting coherence is formed. It is clear that the anomalous behaviour of ^Cu-O (T) is due to inhomogeneous local lattice distortions because the average crystal structure studied by diffraction techniques shows no changes at these temperatures. Similar experiments were also performed in La1.85Sr015CuO4 single crystals, in which Cu was partially substituted by Ni and Co impurities. It is well known that such impurities suppress superconductivity. Interestingly, it was found

Fig. 3: Snapshots of the simulated 2D charge distribution patterns for t = 0.04, n = 0.2, and vi(1, 0) = —1 for different vi(1,1). Fromref. [17].

that in such non-superconducting samples, local lattice anomalies are absent and ^cu-o(T) exhibits the conventional behaviour. This result shows that impurities doped in CuO2 planes suppress both local lattice distortions and superconductivity in cuprates. These experimental results have been modelled by Keller and Bussmann-Holder in a two-component vibronic scenario in which the doped holes lead to polaron formation in an antiferromagnetic (AF) background [14]. The local lattice response observed in EXAFS experiments has been shown to originate from the polaronic feedback effect on the lattice degrees of freedom, which results in local mode softening and divergences in the Cu-O mean square displacement.

Inhomogeneous local lattice distortions are directly related to the inhomogeneous hole distribution in CuO2 planes. In fact, while most of the theoretical models assume that the doped holes are uniformly distributed in CuO2 planes, these experiments reveal quite a different picture, pointing towards a highly non-uniform hole distribution that leads to the formation of hole-rich and hole-poor regions. To understand the reason for this electronic phase separation, let us first consider a single hole doped into the insulating AF parent compound.

Angle-resolved-photoemission spectroscopy (ARPES) showed that such holes form small polarons in the background of the AF correlations [15]. Now, the hole polarons have a tendency to cluster together because of the vibronic coupling, which induces anisotropic interactions between holes [16]. The size of the hole-rich clusters is limited by Coulomb repulsion. The competition of anisotropic lattice and Coulomb interactions of hole polarons in the CuO2 plane has been modelled theoretically [17]. Figure 3 shows a snapshot of the simulated charge distribution patterns for t = 0.04, n = 0.2, and vl (1, 0) = —1, as a function of vl(1,1), where t is the reduced temperature, n is

\ x 0.5 _,

200 KjîïΙ

170 K^sr:

150IC^

j^f-^ "" ^

120 K^? "•■V----y^ssmv^v 1 v 1 1 1

Fig. 4: EPR spectra of La1.97Sr0.03Cu0.98Mn0.02O4 at different temperatures. The dotted lines show contributions of the broad and narrow EPR signals. From ref. [18].

the density of holes, and vl (1, 0) and vl (1,1) stand for the short-range nearest- and next-nearest-neighbor interactions. One can see the formation of elongated hole clusters/stripes, which, depending on the interaction vl(1,1), are aligned either along {1,0} or {1,1} in the plane [17]. Interestingly, it was found that hole-rich clusters with an even number of particles are more stable than those with an odd number of particles. This has direct implications for the pairing within such clusters.

The existence of hole-rich and hole-poor regions naturally leads to the two-component behaviour observed in cuprates. Coulomb repulsion limits the spatial extent of these regions to the nanoscale. Therefore, it is important to use local microscopic methods to detect the electronic phase separation and its formation as a function of temperature and hole doping. This has been achieved by electron paramagnetic resonance (EPR) measurements in lightly doped La2_KSrKCuO4 (LSCO), using Mn2+ substituted for Cu2+ as an EPR probe [18]. At room temperature, a single EPR line of Lorentzian shape is observed. However, with decreasing temperature, a second line appears and the EPR spectra can be well fitted by a sum of two Lorentzians with different linewidths: a narrow one and a broad one (see fig. 4). Such two-component EPR spectra were observed in samples with different Sr concentrations 0 < x > 0.06. The analysis of the spectra showed that the narrow EPR signal is due to hole-rich metallic regions and the broad signal to hole-poor AF regions. The broad line shows a large oxygen isotope effect, and the narrow one does not [19].

Fig. 5: Temperature dependence of the narrow EPR line intensities for La2-^Sr^Cuo.98Mn0.02O4 and the resistivity anisotropy ratio of Lai.97Sr0.03CuO4. From ref. [18].

Of special interest is the exponential increase of the narrow line intensity upon cooling as shown in fig. 5. Because the narrow line is related to hole-rich regions, an exponential increase of its intensity at low temperatures indicates an energy gap for the existence of these regions. The value of this gap as obtained from EPR experiments is near 500 K, independently of the doping, and gives the formation energy of the hole bipolarons that cluster into metallic domains or stripes [18]. This energy is within experimental error the same as the one deduced for bipolaron formation in cuprates from Raman and inelastic neutron scattering measurements [20].

As EPR is a real-space and local probe, it is not possible to directly determine the shape of the hole-rich regions. Here the comparison with the measured in-plane anisotropy of the resistivity in LSCO single crystals is very instructive. Ando et al. [21] found that at high temperatures the resistivity anisotropy pb/pa is small in LSCO, which is consistent with the weak orthorhombicity present. However, below ^150 K, pb/pa grows rapidly with decreasing temperature. This provides macroscopic evidence that holes self-organize into an anisotropic state.

In fig. 5, the temperature dependence of the resistivity anisotropy and the intensity of the narrow EPR line are plotted together. It is remarkable that such different quantities show practically the same temperature dependence. It means that microscopic EPR measurements and the macroscopic resistivity anisotropy measurements point to the same process: the formation of hole-rich metallic stripes with decreasing temperature below ^150 K in lightly doped LSCO well below x = 0.06. An ARPES study of LSCO also revealed the existence of metallic quasiparticles near the nodal direction below x = 0.06 [22]. Figure 6 shows schematically the AF-ordered insulating matrix in the CuO2 plane with a gas of individual hole polarons at high temperatures, which cluster into holerich stripes upon cooling [23]. This picture emerging from the above-mentioned experiments has also been verified by theoretical calculations [17].

Fig. 6: (Colour on-line) Schematic picture of hole polarons in the AF-ordered insulating matrix in the CuO2 plane at high temperatures, which form bipolarons and cluster into hole-rich metallic domains or stripes upon cooling. From ref. [23].

It is important to note that the hole distribution patterns shown in fig. 6 in general are dynamic, but in some cases they can become static because of pinning by special periodic lattice distortions or impurities. The static charge and spin stripe order stabilized by the so-called low-temperature tetragonal (LTT) phase was detected in the La2_xBaxCuO4 and La1.6_xSrxNd0.4CuO4 cuprates at x =1/8 doping by neutron scattering and hard X-ray diffraction experiments [24]. In the simplest picture of the stripe phase, charge-carrier-poor AF regions are separated by one-dimensional stripes of charge-carrier-rich regions. There is considerable evidence that stripes in the cuprates are intimately related to high-temperature superconductivity. In La2_xBaxCuO4, bulk superconductivity is strongly suppressed at the doping level x = 1/8, for which the static stripe order is stabilized. While the static stripe order appears to suppress superconductivity, some researchers have suggested that dynamic stripe fluctuations may promote superconductivity [25]. Dynamic stripe fluctuations are much more difficult to detect than the static stripe order. However, very recently, fluctuating charge-density waves (CDWs) were observed in a La2_xSrxCuO4 cuprate by ultrafast pump-probe spectroscopy [26]. It was found that in an underdoped La1.gSr01CuO4 film, collective CDW excitations persist up to 100 K. These dynamic CDWs fluctuate with a characteristic lifetime of 2 ps at T = 5 K, which decreases to 0.5 ps at T = 100 K. These experiments provide a direct dynamical measurement of modulated charge in cuprates and establish ultrafast spectroscopies as a valuable probe of fluctuating stripe order.

The intrinsic heterogeneity related to the polaronic character of charge carriers in the AF matrix is reflected

in an essential way in the superconducting properties of cuprates. The nanoscale hole-rich metallic regions formed in underdoped cuprates can host superconducting pairs, but because of the small size and small coupling between these regions, global phase coherence and superconductivity cannot be achieved (see fig. 6). Phase coherence and a macroscopic superconductivity can form if the superconducting regions are coupled via an insulating AF matrix because of the tunnelling of Cooper pairs. In this case, the situation is quite similar to granular superconductors. Interesting similarities between the properties of underdoped high-Tc cuprates and granular superconductors have been discussed in several publications [27-29]. However, there is also an important difference. In cuprates —in contrast to classical granular superconductors— the inhomogeneities are thermally fluctuating. Hole-rich regions may form, disappear, and appear in different spatial locations. At the same time, they can be considered static because their kinetic energy is small compared with their binding energy [12]. Therefore, this situation also differs from the usual Bose-Einstein condensation model of superconductivity, which requires the existence of mobile bipo-larons. Taking this into account, a model was developed in which the kinetic energy of the pairs plays no role and a macroscopic superconducting state is attained by the spreading of phase coherence by percolation (PCP) due to Josephson coupling (pair tunnelling) between hole-rich clusters [30]. This model takes into account temporal fluctuations of the superconducting clusters with a time scale given approximately by t ^ (h/Ep)exp(Ep/kT), where Ep is a formation energy of hole-rich clusters. Using the PCP model with very few underlying assumptions, it is possible to reproduce many features of the cuprate phase diagram. It enables not only the quantitative prediction of the critical concentration of 6% hole doping for the occurrence of superconductivity on the underdoped side, but also an estimate of the Tc from experimentally measured values of the pairing energy Ep [30]. The cluster sizes and therefore the percolation are controlled by Coulomb repulsion (see fig. 3). To obtain larger clusters and, owing to the percolative character of the superconducting transition, therefore also a higher Tc requires a reduction of the electrostatic cluster/stripe repulsion. This can be obtained by having a material with a larger effective dielectric constant present between the clusters. To increase the effective dielectric constant, we recently proposed to create multilayer structures in which an underdoped cuprate thin film is sandwiched between high-dielectric-constant insulator layers, such as ferro- or ferri-electrics, thereby reducing the Coulomb repulsion between the intrinsically present clusters or stripes in the CuO2 layers [31].

The heterogeneous character of superconductivity in cuprates is further substantiated by several experimental facts. Ando et al. were able to suppress superconductivity in underdoped LSCO and Bi2Sr2-KLaKCuO6+5 by applying very high magnetic fields of up to 61 T [32]. The resulting normal state showed insulating behavior with an

Fig. 7: (Colour on-line) Plot of the OIE shift AAab(0) /Xab (0) vs. the OIE shift -ATc/Tc for different cuprate superconductors obtained by using various experimental techniques [36]. Only results for underdoped cuprates are shown. The solid line corresponds to the linear relation AAab(0)/Aab(0) = |ATc/Tc|. The dashed line corresponds to the differential Uemura relation with AAab(0)/Aab(0) = 1/2|ATc/Tc |.

unusual logarithmic increase of the resistivity at low temperatures. Traditional mechanisms, such as localization or the Kondo effect, could not explain these results. Theoretical calculations revealed that the logarithmic temperature dependence of the resistivity is expected in granular metals because of the Coulomb interaction [33,34]. Furthermore, analysis of the temperature dependence of the magnetic-field penetration depth A in different cuprate superconductors shows a finite-size effect on A, revealing the existence of superconducting nanoscale domains [35]. The observed oxygen isotope effect (OIE) on the size of the superconducting domains clearly indicates the existence and relevance of the coupling between the superfluid and local lattice distortions [35]. Finally, we would like to discuss an interesting correlation between the OIE shifts of the in-plane penetration depth AAab(0)/Aab(0) and the critical temperature ATc/Tc in underdoped cuprate superconductors shown in fig. 7 [36]. First of all, it is remarkable that the results obtained in different cuprate families with different types of samples (single crystals, powders) and different experimental techniques (magnetization, torque, muon-spin rotation) yield consistent results. This indicates that the observed isotope effects are intrinsic and do not depend on the particular experimental method or the sample used. The solid line shows that the linear relation AAab(0)/Aab(0) = |ATc/Tc\ describes quite well results obtained in underdoped cuprate superconductors. The generic trend presented in fig. 7 is qualitatively consistent with the Uemura relation Tc ^ C/A2ab(0), where C is a constant for all cuprate families [37]. However, there is an important quantitative difference: from the Uemura relation, it follows that AAab(0)/Aab(0) = 1/2|ATc/Tc|,

which differs by factor 2 from experimental results (solid line in fig. 7). While the physical meaning of this difference still is open at present, one possibility is that it is related to the microscopically heterogeneous superconductivity in underdoped cuprates. Indeed, it is known that the London equation for the penetration depth is valid only for a homogeneous distribution of the superfluid. When the super-fluid density varies within the sample, the London formula should be modified, and the penetration depth starts to depend on the spatial variation of superconducting charge carriers [38]. This points to a different character of the superconductivity in cuprates with respect to that observed in classical superconductors, where a homogeneous distribution of charge and Cooper pairs are present.

The dynamic intrinsic heterogeneity described above is present not only in the field of superconductivity as that of charge, but in a different way also in Bose-Einstein condensation (BEC): There the first realisation was at low temperatures in alkali-metal vapours [39,40], then of po-laritons in quantum wells of semiconductor single crystals, again with homogenous quasiparticle distribution. Most recently, BEC was reported at room temperature for polaritons in inhomogeneous polymer platelets [41]. The occurrence of inhomogeneity is, however, not limited to condensed-matter physics: Pietronero was the first to point towards the fractal nature of the visible cosmos [42]. This has since been confirmed up to distances of 900 Mps [43]. Moreover, there is increasing evidence that the mass distribution of the invisible matter in the cosmos, the so-called dark matter, is fractal [43]. Here we note that Einstein [44] and later Friedmann [45] have assumed a homogeneous distribution of mass in the Universe in their theories of general relativity, as London assumed a homogeneous charge distribution in the superconducting quantum fluid [1,2]. This indicates that inhomogeneous distributions of physical quantities, aside from homogenous ones, play an important role in physics.

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