The conductivity and the magnetization around Tc in optimally-doped YBa2Cu3O7−δ revisited: quantitative analysis in terms of fluctuating superconducting pairs

This is the Accepted Manuscript version of an article accepted for publication in Superconductor Science and Technology. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at https://doi.org/10.1088/1361-6668/AAFE93. The conductivity and the magnetization around Tc in optimally-doped YBa2Cu3O7−δrevisited: quantitative analysis in terms of fluctuating superconducting pairs R I Rey‡, C Carballeira, J M Doval, J Mosqueira, M V Ramallo, A Ramos- ´ Alvarez§, D S´o˜nora, J A Veira, J C Verde, F Vidal Quantum Materials and Photonics Research Group, Particle Physics Department, University of Santiago de Compostela, ES-15782 Santiago de Compostela, Spain December 2018 Abstract. We first present detailed measurements of the rounding behavior around the superconducting transition temperature, Tc, of the in-plane electrical conductivity, magnetoconductivity and magnetization, including the low and moderate magnetic field regimes, in a high-quality single crystal and a thin film of the prototypical optimally-doped YBa2Cu3O7−δ(OPT Y-123), in which the inhomogeneity effects are minimized. Then, it is presented a comparison of these experimental data with the phenomenological Ginzburg-Landau (GL) approach that takes into account the unavoidable contribution of the fluctuating pairs, the only theoretical scenario allowing at present to analyze these roundings at the quantitative level. These analyses demonstrate that the measured rounding effects around Tcmay be explained quantitative and consistently in terms of the GL scenario, even up to the rounding onset temperatures if the quantum localization, associated with the shrinkage of the superconducting wavefunction, is taken into account. The implications of our results on the pseudogap physics of optimally-doped cuprates are also discussed. 1. Introduction The phenomenological descriptions of the superconducting transition of the high temperature cuprate superconductors (HTSC) remain a present not well settled, in spite that their interest is considerably enhanced by the absence of a well-established microscopic pairing mechanism for these superconductors [1, 2]. A central still open issue concerning this phenomenology is the description of the rounding behaviors shown around Tcby different observables. It was early proposed that at least in ‡Present address: Clinical Neurosciences Research Laboratory, Clinical University Hospital, Health Research Institute of Santiago de Compostela (IDIS), Universidade de Santiago de Compostela, Spain §Present address: ICFO-Institut de Ciencies Fot`oniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain The conductivity and the magnetization around Tcin OPT Y-123... 2 optimally-doped HTSC these roundings will be due to the presence of conventional superconducting fluctuations, i.e., to the presence of fluctuating superconducting pairs created by the unavoidable thermal agitation energy. [3–8] In fact, in a wide temperature interval, above but not too close to Tc(above the so-called Ginzburg-Levanyuk temperature), and well below the rounding onset temperature, these rounding effects in optimally doped HTSC were well described in terms of the phenomenological meanfield Gaussian-Ginzburg-Landau (GGL) approach for the superconducting fluctuations of both the amplitude and the phase of the order parameter in multilayered superconductors. [3–8] In relation with the observables studied here, an earlier support to the GGL descriptions was provided by different measurements of the magnetization rounding above Tc(the so-called precursor diamagnetism) [7–10] and of the in-plane electrical resistivity rounding (the so-called paraconductivity). [7,8,11–19] These two observables are not only crucial to characterize the superconducting transition but, in addition, in HTSC they present an excellent ratio between the amplitude of the rounding effects and the corresponding background or bare contributions (orders of magnitude bigger than, for instance, in the heat capacity). [3, 8] As it was also earlier stressed, [20] the GGL approach predicts that the fluctuations effects on both the paraconductivity and the precursor diamagnetism will have the same dependence on the reduced temperature, ε= ln(T/Tc). The confirmation of this prediction in different optimally-doped HTSC (a result that will be extended in our present work to the high reduced-temperature region, ε > 0.1) not only provided further support to the GGL scenario but also was an early indication of the absence of appreciable indirect fluctuation effects on both observables [20]. This conclusion concerns in particular the anomalous Maki-Thompson (MT) contribution to the in-plane paraconductivity, in agreement with the enhanced pair-breaking effects expected for the non-1s0wave pairing in the HTSC [21, 22], and also the density of state (DOS) correction [3]. The GL scenario was further supported by early scaling analyses of the electrical conductivity and the diamagnetism measured around Tc(H) in the so-called critical region [23–27]. This GL scenario was also found compatible with the existence close to the zero-field Tcof a full-critical region where the mean-field-like approaches are no longer applicable. In this fullcritical region the superconducting fluctuations have been described in terms of the XY model [11,12,28,29]. Always in relation with the observables studied here, earlier analyses of other aspects of the GL scenario may be seen in Refs. [30] to [37]. These aspects include the interplay with chemical inhomogeneities, particularly relevant in the case of HTSC with doping level different from the optimal [30, 31], and the attempts to extend this scenario to high reduced-temperatures [32–36]. The results of Ref. [36] concern also the so-called regular MT term, whose amplitude at all reduced temperatures was found to be, even in presence of Zn impurities, well below the direct Aslamazov-Larkin (AL) contribution to the in-plane paraconductivity (at least one order of magnitude inferior). This finding of Ref. [36] fully agrees with our earlier conclusions in Ref. [20] and, in The conductivity and the magnetization around Tcin OPT Y-123... 3 addition, it further justifies the conclusions in our present paper on the absence also of an appreciable regular MT contribution, even in the high reduced-temperature region. Mainly in the last ten years, however, different research groups, including some of the most influential, have proposed other physical mechanisms, different from the presence of fluctuating superconducting pairs, to account for the entire or for a part of the measured rounding effects above Tcin all the HTSC, including in those compounds where the rounding onsets are located relatively close to Tc, below twice the measured superconducting critical temperature. [1, 2, 38–60]. In some cases, these proposals were fueled by the observation in different HTSC, including in the optimally doped YBa2Cu3O7−δ(OPT Y-123), of seemingly intrinsic anomalous roundings around Tc, non-describable in terms of the GL approaches summarized above. These anomalous roundings concern not only the in-plane magnetization [38–41] and the dc in-plane electrical conductivity [42, 43], but also other observables [44–60], particularly the Nernst effect. [41,45–60] Although some of these anomalies could in fact be attributed to Tcinhomogeneities, associated with the presence of chemical disorder with long characteristic lengths (larger than the superconducting coherence length amplitude), unavoidable in non-stoichiometric compounds [61–67], most of the works published since then on that issue still attributed the measured rounding effects above Tcto the opening of a pseudogap in the normal state [38–61,68–84]. In this last scenario, the most popular mechanism is based on phase disordering above the measured Tc[1,2,37–58,67–84]. The difficulties to discriminate between the very different theoretical scenarios proposed for the rounding effects around Tcin HTSC are to some extent related to the unavoidable experimental uncertainties that arise when extracting the intrinsic roundings from the measurements. These uncertainties have in turn various and often entangled sources: First at all, those associated with the always somewhat arbitrary approximations needed to determine both the Tcof each theoretical approach and the background or bare contributions, associated with the normal state behavior, to each observable, a difficulty that concerns the descriptions of any phase transition. [86] Secondly, the presence of Tc-inhomogeneities, mainly those associated with chemical disorder [61–67]. In addition, the layered nature of the HTSC and the complexity of their chemistry increases the relevance of these Tc-inhomogeneities, whose influence may be particularly dramatic in compounds with doping level different from the optimal [31, 61, 62, 64]. Indeed, in the case of the theoretical approaches associated with the opening of a pseudogap in the normal state these difficulties are further enhanced by the scarcity until now of quantitative theoretical proposals for the observable effects, in contrast with the GL approaches for the fluctuations of superconducting pairs. [3–10, 23–27] Moreover, in the temperature region where these superconducting fluctuations, thermodynamic-like, are always present, we will see here that up to approximately 1.7Tc, the corresponding rounding must be taken into account when considering other possible effects. The difficulties commented above, some of them unavoidable, affect even the discrimination between the different versions of the GL scenarios proposed until now. The conductivity and the magnetization around Tcin OPT Y-123... 4 In fact, in spite of earlier detailed measurements and analyses [4–37], numerous central aspects of these approaches are are still controversial, as for instance the possible presence of different indirect fluctuation effects (regular and anomalous MT and DOS, in particular), the multilayering influence and the corresponding effective periodicity lengths, the role of the chemical disorder and of the corresponding Tc-inhomogeneities, the behavior at high reduced-temperatures and magnetic fields and the physical origin of the corresponding cutoffs, or the behavior of the phase fluctuations in the so-called full-critical region [43–67,87–95]. In this paper, the experimental uncertainties commented above are strongly mitigated by measuring the rounding behavior of the in-plane electrical conductivity, the magnetoconductivity and the magnetization around Tcin high quality OPT Y-123 samples. These measurements extend in temperature up to 300 K and up to 9 T in applied magnetic field, which when analyzed on the ground of the GL approaches cover both the low magnetic field limit (Schmidt regime) and the moderate field (or Prange) regime. In addition to their relevance to probe the superconducting transition, these observables were chosen because, as noted before, in HTSC they present an excellent ratio between the amplitudes of the rounding effects and the corresponding background or bare contributions. In turn, the choice of the OPT Y-123 superconductor was first motivated by the high chemical and structural quality of the samples that may be prepared, what minimizes the Tc-inhomogeneities. In addition, the normal-state resistivity and the magnetic susceptibility are linear in an extended temperature region, allowing a simple and reliable background extraction. In fact, these characteristics make the OPT Y-123 the prototypical compound to analyze the rounding effects in HTSC, and in the last twenty years these effects were extensively measured on it and analyzed in terms of superconducting fluctuation effects [7–61, 88–90]. However, our present experimental results covering the rounding effects on three central observables and under different magnetic field regimes are, therefore, particularly useful to discriminate between different scenarios and theoretical approaches proposed until now. The importance of the results presented here, as well the unique role played by the prototypical OPT Y-123, are further enhanced by the fact that they provide, as we will demonstrate in the present paper, some compelling thermodynamic constraints for the different scenarios being proposed for the physical origin of the pseudogap of the HTSC, at present another highly debated issue which implications concern some of the proposals for the origin of the superconducting pairing [1,2,38–60,96–99]. 2. Fabrication of the samples and details of the measurements 2.1. Thin film and measurements of the electrical conductivity and magnetoconductivity To determine the electrical conductivity and the magnetoconductivity roundings around Tcwe have grown OPT Y-123 thin films over (100) SrTiO3substrates by high-pressure dc sputtering. Details of the growth and characterization of these films are similar The conductivity and the magnetization around Tcin OPT Y-123... 5 100 200 300 1 2 3 120 140 160 180 0.6 0.8 1.0 85 90 95 100 0 20 40 60 80 0 H=0 T ab (10 -8 m) 0 H=0 T (b) T =0 T c T c d ab /dT 0 100 200 300 0 50 100 150 200 T onset T low onset ab (10 -8 m) (a) fitting region 90 100 110 0 50 100 T(K) T onset d ab /dT (10 -8 m/K) (c) T low onset T low onset T onset Figure 1. (a) The in-plane resistivity versus temperature under zero applied magnetic field measured in the microbridge described in the main text. The solid line is the background contribution. Inset: Detail around Tc. (b) ρab(T) curve around Tctogether with its temperature derivative (in arbitrary units). (c) Overview of dρab(T)/dT showing the onset temperature of the resistivity rounding (Tonset) and its lower limit (Tlow onset). Inset: Detail around Tonset. to those described elsewhere. [101] Microbridges were patterned on those films by photolithography and wet chemical etching into a four-probe configuration suitable for the electrical measurements. Au pads were then deposited onto the current and voltage terminals. The final contact resistance achieved was less than 0.1 Ω/contact. The in-plane resistivity (ρab) was measured in presence of magnetic fields up to 9 T perpendicular to the ab-layers with a Quantum Design’s physical property measurement system (PPMS) by using an excitation current of ∼50 µA at 23 Hz. The uncertainties in the geometry and dimensions of the microbridges lead to an uncertainty in the ρab(T) amplitude below 20%. The conductivity and the magnetization around Tcin OPT Y-123... 6 An example of the ρab(T) curves obtained in the microbridge studied in our present work is shown in Fig. 1(a). This microbridge was 2.1 mm long, 90 nm thick, and 40 µm wide. A detail of the resistive transition is presented in Fig. 1(b), where the solid line is dρab(T)/dT. The superconducting transition temperature Tc, and the resistive transition width, ∆Tc, are obtained as the temperature location of the maximum of the derivative peak and, respectively, its width at half maximum. For this microbridge, Tc= 91.10 K and ∆Tc= 0.55 K. Complementarily, the temperature at which the resistance becomes unobservable is denoted Tρ=0, and is of the order of 90.5 K. These results are summarized in Table I. The linear temperature dependence of the normal-state resistivity well above Tc, between 150 K and 300 K, further confirms the high quality of our Y-123 film. This is illustrated in Figs. 1(a) and (c). Such a linear behaviour was already observed in high-quality OPT Y-123 single-crystals and epitaxial thin films by different groups. [7, 8, 11–20, 23–30, 102, 103] In addition, our microbridges present a normal-state resistivity ratio ρab(300K)/ρab(100K) around 3.1, as expected for OPT Y-123 [104]. Also, the extrapolation to T= 0 K of the normal state resistivity leads to very low residual values, as shown in Fig. 1(a). These results already suggest the absence of appreciable Tc-inhomogeneities in our microbridges. In fact, the possible presence of a small amount of non-superconducting domains uniformly distributed in the samples could just affect the paraconductivity amplitude, a relatively small effect that may be expected to remain well below the uncertainties associated with the microbridges dimensions. [105]∥ Some examples of the in-plane resistivity (for magnetic fields, H, applied perpendicularly to the superconducting ab-layers) are presented in Figs. 2(a) and (b). ∥Note that an incomplete superconducting volume fraction (f < 1) would affect both the electrical resistivity and the magnetization, and then the extraction of the corresponding ∆ab and ∆χ. In the case of the magnetization, independently of the spatial distribution of the non-superconducting domains, their presence will manifest only in the corresponding ∆χ(T, H) amplitude (and then only the Hc2(0) value resulting from the comparison with the GGL approach will be affected). The effects of the nonsuperconducting domains are more difficult to evaluate when extracting ∆σab(ε). Assuming that there are non-superconducting domains uniformly distributed, probably the most usual case, the effective electrical conductivity may be approximated by using an effective medium approach [105] Zσ−σeff σ+ 2σeff Q(σ)dσ= 0,(1) where Q(σ) is the conductivity volume distribution. This expression may be recast into σeff =ZσQ(σ)dσ−Z(σ−σeff )2 σ+ 2σeff Q(σ)dσ. (2) Then, to the first order in (σ−σeff )/σeff the effective conductivity will be the volume-weighted average of the local conductivity. By assuming the presence of two types of domains, one with σ1= 1/ρB+ ∆σ and a superconducting volume fraction f, and other non superconducting with σ2= 1/ρBand a volume fraction 1 −fit follows σeff =fσ1+ (1 −f)σ2. Then the effective paraconductivity would be given by ∆σeff ≡σeff −σB=f∆σi.e., it is also modulated in amplitude by f(a fact that could affect the τrel value resulting from the comparison with the experimental data, see subsection 4.2). Therefore, the quantity −∆χ(ε)H/T∆σab(ε) would be almost unaffected by a possible incomplete superconducting volume fraction. The conductivity and the magnetization around Tcin OPT Y-123... 7 Tc(K) ∆Tc(K) Tonset (K) ∆Tonset (K) σab(T) 91.1 0.6 150 15 χ(T)H→090.5 1 150 15 Table 1. The critical temperature, Tc, and the onset temperature of the rounding effects, Tonset, together with the corresponding dispersions, ∆Tcand ∆Tonset, of the two optimally doped (OPT) Y-123 samples studied in this work. The data obtained from the resistivity measurements correspond to a thin film, whereas those obtained from the magnetization correspond to a single-crystal. For details see the main text. 75 80 85 90 95 0 20 40 60 T(K) (b) ab (10 -8 m) 0 100 200 300 0 100 200 0 T 5 T 0.5 T 6 T 1 T 7 T 2 T 8 T 3 T 9 T 4 T ab (10 -8 m) (a) Figure 2. (a) Temperature dependence of the in-plane resistivity measured under magnetic fields up to 9 T applied perpendicularly to the ab-layers. The solid line is the background contribution. (b) Detail around Tc. The data of Fig. 2(a) show that above 150 K the magnetoresistivity is inappreciable. This behaviour contrasts with the important magnetoresistivity observed near Tc, as illustrated by the results summarized in Fig. 2(b). 2.2. Single-crystal and measurements of the magnetization To measure the magnetization with Happlied perpendicular to the superconducting ab-layers, M(T, H), we have used a high-quality OPT Y-123 single-crystal prepared following a previously reported self-flux technique. [12] It has a quite regular plate-like shape, with dimensions 2.3×2.9 mm2in the ab plane and a thickness of 40 µm. The mass of this sample is 1.74 mg. The magnetization measurements were performed with The conductivity and the magnetization around Tcin OPT Y-123... 8 a magnetometer based in the superconducting quantum interference (Quantum Design, model MPMS-XL). A first magnetic characterization of our crystal was obtained through the temperature dependence of the zero-field-cooling (ZFC) magnetic susceptibility (χ) under a low magnetic field (0.25 mT) perpendicular to the ab layers (Fig. 3(a)). From this curve one may obtain the diamagnetic transition temperature, Tc, as the temperature at which dχ/dTis maximum (Tc= 90.5 K). Also, the diamagnetic transition width, ∆Tc, is estimated as the width at half maximum of the dχ/dT peak, and for this example ∆Tc= 1 K. These results are also summarized in Table I. The differences between these Tcand ∆Tcvalues and the ones corresponding to the microbridge studied in the previous subsection are well within the dispersion expected for different high-quality OPT Y-123 samples. [8–20,23–34] The magnetic susceptibility rounding above Tc, that following the conventional denomination will already be called precursor diamagnetism, is orders of magnitude smaller than in the Meissner region. [3, 8] Therefore, to measure these roundings one must apply magnetic fields orders of magnitude larger than those used in the ZFC measurements. Figure 3(b) provides an example of these measurements, performed under a field amplitude of 1 T. To assure a complete temperature stabilization, data are taken after waiting ∼5 minutes since the temperature was within 0.5% the target temperature. In the case of the isothermal curves, it is waited around 5 minutes after setting the magnetic field in permanent mode. In both cases, the measurements were performed with the reciprocating sample option (RSO), that performs sinusoidal oscillations of the sample about the centre of the detection system, and improves the resolution with respect to the conventional dc option. Each data point is the average of 4-6 individual measurements, each one consisting in 10 cycles at 1 Hz. The final resolution in magnetic moment was in the 10−8emu range. Two aspects of these measurements deserve to be commented. First, as earlier stressed elsewhere, [62] the presence in our crystal of a relatively small amount on non-superconducting domains with large characteristic lengths (much larger than the superconducting coherence length amplitude) will, independently of their spatial distribution, just affect the amplitude of the magnetization rounding observed above Tc,i.e., the precursor diamagnetism amplitude. Therefore, to simplify the analyses presented in the next sections (in particular when comparing with the paraconductivity) these possible effects will be neglected. When comparing with the GGL approaches, this approximation will in turn mainly affect the value of the amplitude of the upper critical field, but the corresponding uncertainties are expected to be comparable with those associated with the sample volume. Note also that the linear temperature dependence of the normal-state magnetic susceptibility observed well above Tcin the inset of Fig. 3(b) further confirms the high quality of our crystal. These last data correspond to measurements under an applied magnetic field of 1 T but, as shown by the data in Fig. 4, this linearity remains for field amplitudes up to 5 T. The conductivity and the magnetization around Tcin OPT Y-123... 9 85 90 95 100 -1.0 -0.5 0.0 100 140 180 0 2 4 100 200 0 2 4 T C ZFC T C 0 H=0.25 mT d/dT region fitting (b) (10 -5 ) 0 H=1 T (a) T onset T low onset 100 150 200 250 0 5 10 15 (c) T (K) T onset T low onset d /dT (10 -7 K -1 ) 120 150 180 0 1 2 T onset T low onset Figure 3. (a) Detail around the superconducting transition of the magnetic susceptibility measured in the single crystal described in the main text. These measurements were performed by using a low amplitude magnetic field (0.25 mT) applied perpendicularly to the ab-layers after zero-field-cooling (ZFC). The solid curve is dχ/dT in arbitrary units. The corresponding Tc, and the transition width, ∆Tc, are also indicated. (b) An example of the magnetic susceptibility versus temperature measured with a 1 T magnetic field applied perpendicularly to the ab-layers. An overview up to room temperature is shown in the inset. The background contribution is represented as a solid line. As in the example for ρab(T) shown in Fig. 1, the background fitting region extends up to well above the onset of the fluctuation effects (Tonset). It is also indicated here, as Tlow onset, the lower limit for this onset. (c) Temperature dependence of dχ/dT for µ0H= 1 T, and the corresponding Tonset and Tlow onset. Inset: Detail around Tonset. 3. Characterization of the rounding effects: Model- and background-independent results 3.1. Onset temperature of the rounding effects As shown in the precedent Section, well above Tcand up to at least 250 K both the normal-state resistivity and the magnetic susceptibility depend linearly on temperature. The conductivity and the magnetization around Tcin OPT Y-123... 16 0.0 0.2 0.4 0.6 0.8 1.0 1.1 1.2 1.3 0 1 2 3 4 5 -45 -30 -15 0 h/ - /T (10 -7 K -1 ) 120 K110 K 105 K 97.5 K 95 K 92.5 K 91.5 K 90.5 K M (A/m) H (T) 90 K 0.2 0.4 0.6 0.8 1.0 -150 -100 -50 0 93 K 92.5 K 92.5 K (c) (b) 110 K H (T) (10 -7 ) (a) Figure 7. (a) Magnetic field dependence of the fluctuation magnetization for several temperatures around Tc. The lines were calculated from Eq. 9 by using the same parameters as in the previous analyses (see the main text). (b) Detail to better appreciate the linear behavior of ∆χ(H) below 1 T. (c) Examples of the −∆χ/T dependence on h/ε. The lines are the GGL prediction under a total-energy cutoff for the Prange regime [Eq. 9] with the same ξc(0), ξab(0) and εcvalues as in the Schmidt limit. magnetic fields in the critical region around Tc(H).¶Detailed descriptions of these ¶The relevance of other contributions to the in-plane paraconductivity and magnetoconductivity in Y-123 (Maki-Thomson, regular Maki-Thomson, and DOS, see Ref. [111]) were studied in detail in Ref. [112]. It was concluded that there is no consistent set of values for the parameters involved in these contributions that produce a simultaneous agreement with the experiments. Therefore, these contributions do not seem to give a good description of the effects of fluctuations in planar transport, regardless of whether they are useful or not to describe transport in the direction perpendicular to the planes, which is the purpose of the DOS effects as initially proposed in Ref. [113]. For detailed The conductivity and the magnetization around Tcin OPT Y-123... 17 theoretical results may be seen in the original works. [4–10, 30, 34, 37, 115, 116] An extension of the GGL calculations to the finite magnetic field regime will be summarized in Appendix A. Here we will just comment briefly on some aspects of the GL scenario that are still controversial, as the influence of the multilayering or the adequacy of the total-energy cutoff. 4.1. Onset temperature of the superconducting fluctuations As earlier discussed in the pioneering work of Gollub and coworkers [117] (see also Refs. [3] and [8]), the mean-field GGL approaches are formally valid only in the ε-region εLG ≤ε≪1, where εLG is the so-called Levanyuk-Ginzburg reduced temperature. Since then, different attempts have been proposed to extend to high-εthese meanfield descriptions, including the introduction of different versions of the conventional momentum cutoff, [8–19,36,43–59,61,87–89,91–95,118] or an ad hoc penalization (not a cutoff) of the short-wavelength fluctuation modes that already takes into account the quantum localization. [119, 120] However, none of these earlier proposals lead to the vanishing of all fluctuation modes above a well-defined temperature, Tonset. Such a Tonset was proposed in Ref. [34] by just taking into account the limits imposed by the Heisenberg uncertainty principle to the shrinkage, when the temperature increases above Tc, of the superconducting coherence length, ξ(T), that cannot be smaller than ξT=0K, the actual coherence length at zero temperature first introduced by Pippard, [3, 121] i.e., ξ(T)> ∼ξT=0K.(7) This condition directly leads to a well-defined Tonset, given by ξ(Tonset) = ξT=0K, above which all fluctuation modes vanish, independently of their wavelength. The corresponding cutoff reduced temperature is then just εc≡ln(Tonset/Tc). Note also that the above condition is indeed compatible with the existence of superconductivity in samples with sizes smaller than the value of ξT=0K in the bulk. This is because, as earlier commented elsewhere (second paper in Ref. [34]), in such small superconductors the Pippard coherence length looses its conventional meaning (from a crude point of view, in these small superconductors the coherence length amplitude is also reduced with respect to the bulk value). As also stressed in Ref. [34], the above condition is general, and must apply to any theoretical description of the superconducting transition. Only the value of εcwill depend, through the temperature dependence of ξ(T) and the relationship between ξ(T) and ξT=0K, of each particular approach. A relevant example will correspond to the combination of the mean-field temperature dependence of the coherence length, ξ(T) = ξ(0)ε−1/2[and then εc= (ξ(0)/ξT=0K)2], with the relationship between ξ(T) and ξT=0K proposed by the mean-field BCS theory, which in the clean limit is ξ(0)= 0.74ξT=0K. [121] This leads then to εc≈0.55, i.e.,Tonset ≈1.7Tc, in excellent measurements of the fluctuation conductivity normal to the ab layers see Ref. [114]. Other aspects are commented in subsection 5.1. The conductivity and the magnetization around Tcin OPT Y-123... 18 0.5 1.0 1.5 100 150 200 0 2 4 0 2 4 6 8 10 1.0 1.5 2.0 2.5 T onset (c) (b) (a) d ab /dT(10 -8 m/K) 0 T 9 T T onset 1 T 5 T d /dT (10 -7 K -1 ) T (K) data from ab ( ,H) data from ( ,H) T onset /T c 0 H (T) Figure 8. (a)-(b) Temperature dependence of dρab/dT and dχ/dT from measurements under two different magnetic fields. These data illustrate that Tonset is field independent, up to 9 T and, respectively, 5 T. The dotted lines delimit the noise level (see the main text for details). (c) Tonset/Tcfor the two observables as a function of the applied magnetic field amplitude. The solid line corresponds to Tonset/Tc= 1.7. agreement with our present experimental results for the onset of the resistivity and the magnetization roundings in the zero-field limit presented in Figs. 1(c) and, respectively, 3(c), and summarized in Table I. The total-energy cutoff reduced-temperature, εc, may also be affected by the application of a magnetic field large enough as to reduce the superconducting coherence length amplitude, i.e., of the order of Hc2(0). [34] This effect was observed experimentally in low-Tcsuperconductors, [106] and in a low-Tccuprate. [122] However, in our experiments the maximum field amplitude was 9 T, much lower than the µ0Hc2(0) value for OPT Y-123, and thus one might expect that εcwill remain field independent. The results summarized in Fig. 8 fully confirm this prediction. Other aspects of εcwill be commented in the next Section. The conductivity and the magnetization around Tcin OPT Y-123... 19 4.2. Comparison of the fluctuation-induced conductivity, magnetoconductivity and diamagnetism in the mean-field region above Tcwith the GGL model equations As stressed above, the experimental finding that the ratio between the precursor diamagnetism and the paraconductivity does not depend on temperature, a modelindependent result, strongly suggests the absence of indirect fluctuation effects on the electrical conductivity and of non-local electrodynamic effects on the precursor diamagnetism. These effects may be then neglected, providing a first simplification when analyzing the experimental data. Another relevant simplification concerns the effective distance, s, between adjacent superconducting ab-layers. As explained in detail in Refs. [7,8,66], in the GGL approach above Tcin bilayered superconductors with two alternate interlayer distances s1and s2(such as Y-123) the effective sbecomes equal to the bilayer periodicity s1+s2only if the corresponding Josephson interlayer couplings, γ1 and γ2, differ among them by orders of magnitude, i.e.,γ1/γ2→ ∞. Instead, γ1/γ2< ∼30 in OPT Y-123, and in this case s= (s1+s2)/2 is an excellent approximation. [37,123]. Then, in what follows we will use s= (s1+s2)/2 as effective interlayer distance. The detailed analyses presented below will fully confirm the adequacy of these two starting simplifications. To probe the GGL approach we will then use the expressions for the direct fluctuation effects on the in-plane paraconductivity and the precursor diamagnetism in layered superconductors in a finite field H⊥ab, and under a total-energy cutoff. The calculations of these expressions are summarized in Appendix A. The in-plane paraconductivity is given by: ∆σab(ε, h) = τrele2 64¯hπh Zπ/s −π/s "ψ(1) h+ε+ωkz 2h!−ψ(1) h+εc+ωkz 2h!#dkz,(8) whereas the precursor diamagnetism is: ∆χ(ε, h) T=−kB 2πϕ0hHc2(0) Zπ/s −π/s "−εc+wkz 2hψ h+εc+wkz 2h! −ln Γ h+ε+wkz 2h!+ ln Γ h+εc+wkz 2h! +ε+wkz 2hψ h+ε+wkz 2h!+εc−ε 2h#dkz.(9) Here τrel ≡τ0(0)/τBCS 0is the GL amplitude of the relaxation time of the fluctuating Cooper pairs for a wave vector k= 0, expressed in adimensional units relative to the BCS amplitude τBCS 0=π¯h/(8kBTc), [124] eis the electron charge, ¯his the reduced Planck constant, kBis the Boltzmann constant, ϕ0is the magnetic flux quantum, h≡H/Hc2(0) is the reduced magnetic field, wkz=BLD(1−cos(kzs))/2 is the out-of-plane spectrum of the fluctuations, BLD = (2ξc(0)/s)2is the so-called Lawrence-Doniach (LD) parameter, Γ, ψand ψ(1) are the gamma, digamma and trigamma functions, and ξc(0) and ξab(0) are the amplitudes of the transverse and in-plane GL coherence lengths. As indicated above, for the superconducting layers’ periodicity length we will use s= (s1+s2)/2 = 0.59 nm. The conductivity and the magnetization around Tcin OPT Y-123... 20 In principle the total-energy cutoff constant, εc, could be directly determined from the Tonset obtained in the previous subsection. However, as a further check of consistency, here we will leave εcfree and we will compare the result with ln(Tonset/Tc). In the low magnetic field limit (i.e., for h≪ε, εc) Eqs. 8 and (9) reduce to ∆σab(ε) = τrele2 16¯hs "1 ε1 + BLD ε−1/2 −1 εc1 + BLD εc−1/2#,(10) and, respectively, ∆χ(ε) T=−πkBµ0ξ2 ab(0) 3ϕ2 0s"1 ε1 + BLD ε−1/2 −1 εc1 + BLD εc−1/2#,(11) where µ0is the vacuum permeability. In turn, for ε≪εc(that corresponds then to a region outside the short-wavelength fluctuation regime), Eqs. 10 and (11) reduce to ∆σab(ε) = τrele2 16¯hs 1 ε1 + BLD ε−1/2 ,(12) that is the well-known Aslamazov-Larkin (AL) expression for the paraconductivity in the LD scenario, [125] and to ∆χ(ε) T=−πkBµ0ξ2 ab(0) 3ϕ2 0s 1 ε1 + BLD ε−1/2 ,(13) that is the GGL prediction for the fluctuation diamagnetism in the Schmidt limit for layered superconductors, first proposed by Lawrence and Doniach [125] (and, independently, by Tsuzuki [126] and by Yamaji [127]). Another relevant result of the GGL scenario, which may be directly obtained by just combining Eqs. 10 and (11), is that the relationship between the precursor diamagnetism and the paraconductivity in the low field (Schmidt) limit is temperature-independent, −∆χ(ε) T∆σab(ε)=16µ0kB 3π¯h ξ2 ab(0) τrel = 2.79 ×105ξ2 ab(0) τrel (SI units),(14) a result that extends to the high-εregion our earlier proposal. [20] +Equation (14) agrees with the experimental results summarized in Section 3.5 and in Fig. 5(c). This agreement further suggests the absence of indirect fluctuation effects on the paraconductivity and of non-local electrodynamic effects on the precursor diamagnetism. [3] The solid line in Fig. 5(c) corresponds to the best fit of Eq. 14 to the data, with ξ2 ab(0)/τrel as the only free parameter. This leads to ξ2 ab(0)/τrel ≃1.1 nm2a value that provides a relevant constraint to the remaining free parameters of the GGL scenario. In order to disentangle the values of τrel and ξc(0), and also to determine the remaining free parameters entering the GGL expressions, i.e.,ξc(0) and εc, we need to simultaneous and consistently fit Eqs. (8) to (11) to our full set of ∆σab(ε), ∆σab(ε, H), ∆χ(ε) and ∆M(ε, h) data. The combined fit to all our data above ε= 0.02 is presented in Figs. 5 to 7. The values for the fitting parameters are summarized in Table II. +The relationship between ∆χ(ε)Hand ∆σab(ε), first proposed in Ref. [20], was since then applied by different authors to analyze the superconducting fluctuations around Tcin different high-Tc superconductors (see, e.g., Ref. [8]). Examples of recent works using such a relationship are the second work in Ref. [53], and Refs. [58,61]. The conductivity and the magnetization around Tcin OPT Y-123... 21 ξc(0) ξab(0) τrel εc (nm) (nm) 0.11 ±0.02 1.1±0.2 1.0±0.2 0.55 ±0.15 Table 2. Values of the parameters arising in the mean-field Ginzburg-Landau scenario. With these values, the GL approaches explain at a quantitative level, simultaneous and consistently, the precursor diamagnetism in the low magnetic field limit (Schmidt regime) and under moderate fields (Prange regime), the paraconductivity and the fluctuation-induced magnetoconductivity. It may be illustrative to discuss some of those observables separately in specific (ε, h) ranges of particular interest. For instance, Fig. 5(a) shows that for the zero-field paraconductivity ∆σab(ε) it is not necessary to take cutoff effects into account for ϵ< ∼0.1, where Eq. 12 (dashed line in that figure) accounts for the data. This same equation fails, however, to account for the rapid fall of fluctuations at larger ε-values, for which the use of Eq. 10 is needed (continuous line in the same figure). A similar situation happens for the low-field fluctuation magnetic susceptibility, displayed in Fig. 5(b): While for ϵ< ∼0.1 Eq. 13 without a cutoff accounts for the data, at larger εvalues Eq. 11 is needed. Concerning the effects of a finite magnetic field on the fluctuation-induced observables, Figs. 6 and 7 illustrate the H-dependence of both ∆σab(ε, H) and ∆M(ε, h). It is evident in these figures that Eqs. 8 and (9) account very well for these data, using the parameter values summarized in Table II. 4.3. GL scaling of the fluctuation diamagnetism and conductivity in the critical region around Tc(H) Although the analysis of the rounding effects in OPT Y-123 in terms of the GL scenario presented in this paper focuses on the behavior in the Gaussian mean-field-like region above Tc, and they were done on the grounds of the extended GGL approximation, for completeness in this subsection we will analyze the critical region around the Tc(H) line, where the Gaussian approximation is no longer valid. For that we will use the GL approach for 3D superconductors proposed by Ullah and Dorsey, [6] based on the lowest-Landau-level (LLL) approximation. The adequacy of this GL-LLL approach to explain the fluctuation diamagnetism and conductivity observed around Tcin OPT Y- 123 was also earlier demonstrated [23–27] (see also Ref. [128]). In the framework of this approach, ∆M(T, H) and ∆σab(T, H) follow a scaling behavior, [6] ∆M(T, H)3D= (TH)2/3FM T−Tc(H) (TH)2/3!(15) and ∆σab(T, H)3D= T2 H!1/3 Fσ T−Tc(H) (TH)2/3!,(16) The conductivity and the magnetization around Tcin OPT Y-123... 22 -1.0 -0.5 0.0 M/(HT) 2/3 (10 -3 A 1/3 m -1/ 3 K -2/ 3 ) (a) 80 85 90 -0.8 -0.4 0.0 M (10 3 A/m) T(K) 0.5 T 5 T -1 0 1 2 0 5 10 [T-T c (H)]/(TH) 2/3 (10 -5 A -2/3 m 2/3 K 1/3 ) H/T 2 1/3 (10 7 -1 A 1/3 m -4/ 3 K -2/ 3 ) (b) 80 85 90 0 50 100 2 T 9 T T(K) 10 5 m -1 Figure 9. (a)-(b) 3D-LLL scaling of the magnetization (for magnetic fields between 0.5 T and 5 T in steps of 0.5 T) and of the fluctuation conductivity (for magnetic fields between 2 T and 9 T in steps of 1 T) for temperatures around Tc(H). The scaling variables were evaluated by using the superconducting parameters resulting from the analysis in the Gaussian region. The arrows indicate the limit of applicability of the scalings. Insets: Temperature dependence of the raw fluctuation magnetization and conductivity. where FMand Fσare scaling functions. [129] In applying Eqs. 15 and (16) to the experimental data, Tc(H) was determined from the GL relation Tc(H) = Tc"1−H ϕ0/2πµ0ξ2 ab(0)#,(17) by using the Tcand ξab(0) values resulting from the analysis in the Gaussian region. The resulting scalings are shown in Figs. 9(a) and 9(b), where the corresponding unscaled ∆M(T, H) and ∆σab(T, H) data are also presented in the insets. As it may be seen, the scaling is excellent for both observables down to (T−Tc(H))/(TH)2/3∼ −10−5 A−2/3m2/3K1/3(indicated by an arrow), in agreement with previous works. [23–27,128] The conductivity and the magnetization around Tcin OPT Y-123... 23 The present results nicely extend the applicability of the GL-LLL approaches to fields as large as 9 T, and represent a stringent consistency check of the analysis in the Gaussian region. It is also worth noticing the existence of self-consistent calculations of ∆Mand ∆σin layered superconductors based on the GL-LD model, [95,130,131] that agree with the experimental data near Tc(H). [95,131] 4.4. Phase diagram for the superconducting fluctuations scenario A schematic H−Tphase diagram of the GL scenario showing the different regions for the superconducting fluctuations is presented in Fig. 10. The regions covered by our measurements of the rounding effects on the electrical conductivity and magnetization (shaded area) is presented in Fig. 10(a). In Figs. 10(b) and 10(c) the color scale represents the fluctuation-induced conductivity. In Fig. 10(b), the diamonds show the experimental onset temperature for each field and the solid line near them is the estimate ξab(Tonset, H) = ξT=0K (that for the low fields in this figure corresponds to Tonset ≈ 1.7Tc). To better appreciate the critical region, Fig. 10(c) shows a detail centered on Tc(H). In this figure, squares and circles represent the lower limit of applicability of the GL-LLL scalings of ∆M(T, H) and, respectively, ∆σab(T, H). The critical region boundary (dashed line) was evaluated by fitting to these data the behavior predicted by the so-called H-dependent Ginzburg criterion for 3D superconductors, [132] |T−Tc(H)| Tc ∝ H Hc2(0)!2/3 .(18) Within the Gaussian region the dotted line represents the condition h≈0.2ε, roughly separating the Prange regime from the low-HSchmidt regime. For completeness, we also indicate the experimental irreversibility line (triangles) obtained from the temperatures at which ρab(T)H= 0 [see Fig. 2(b)]. It is worth noting that direct application of Eq. (18) leads to a null width of the critical region in the limit H→0. This arises from the failure at low fields of the LLL approximation used by Eq. (18). At H= 0, the width of the full-critical region (where the mean-field-like approaches are no longer applicable and the superconducting fluctuations are described instead in terms of the XY model) may be approximated by the well-known Ginzburg criterion originally calculated for 3D superconductors by Levanyuk [133] and Ginzburg [134] (for an extension to layered superconductors, see Ref. [135]). The precise field-dependence of the critical region at low fields, and how it merges with the low-field full-critical region, is still unknown. Other still open issues concerning the full-critical region at zero field include the behavior of the fluctuation conductivity and magnetization: As already mentioned, the uncertainties in the determination of Tc(and also the large effect of even a small Tcdistribution due to inhomogeneities, unavoidable in real samples) difficult determining the bare critical exponents in the very close vicinity of Tc(H= 0). In moderately anisotropic superconductors such as OPT Y-123, the 3D XY model is expected to provide a first approximation to the fluctuations in the full-critical region. But it could also be The conductivity and the magnetization around Tcin OPT Y-123... 24 80 85 90 95 100 0 2 4 6 8 Gaussian region vortex liquid Critical region vortex solid Schmidt Prange H C2 H irr T(K) H (T) H C2 full-critical (c) normal state with SC fluctuations 90 120 150 180 0 2 4 6 8 ab ( -1 m -1 ) ab (T,H)= T=0K vortex solid H (T) normal state (a) 10 4 10 5 10 6 10 7 Temperature 0 (b) vortex solid H C1 Meissner phase vortex liquid H C2 H irr normal state with SC fluctuations Magnetic field ab (T,H)= T=0K normal state 1.7T C T C Figure 10. (a) Schematic H−Tphase diagram of the GL scenario with superconducting fluctuations showing the regions covered by the measurements summarized in this paper (shaded area). The color scale in (b) represents the fluctuation-induced conductivity, and the diamonds show the onset temperature for each field (the solid line over them is Tonset ≈1.7Tc). A detail around Tcis presented in (c). Circles and squares are the limit of applicability of the GL-LLL scalings of ∆σab and, respectively, ∆M. The dashed line is a fit to the H-dependent Ginzburg criterion, Eq. 18. The solid line is the upper critical field given by Eq. 17. The dotted line in the Gaussian region corresponds to h≈0.2ε, and roughly separates the Prange regime and the low-Hregion (Schmidt regime). The irreversibility line (triangles) was obtained from the temperatures at which ρab(T)H= 0. Closer to Tcand in the zero field limit is the so-called full-critical region, indicated by the dashed-dotted line. expected that the layered structure may induce extra fluctuation contributions arising from vortex-antivortex effects akin to those first proposed by Berezinskii [136] and by The conductivity and the magnetization around Tcin OPT Y-123... 25 Kosterlitz and Thouless [137] for 2D superfluids (for a theoretical discussion of the effect in layered superconductors see, e.g., Refs. [75, 138,139], and for a measurement in the much more anisotropic La2−xSrxCuO4see Ref. [140]). For OPT Y-123, partial evidences of 3D XY exponents in the zero-field fluctuation conductivity for ε < 0.02 have been in fact early reported in, e.g., Ref. [12], while some indications of a behaviour of the Berezinskii-Kosterlitz-Thouless type were suggested by measurements in, e.g., Ref. [141]. Further work will be clearly needed to fully understand these aspects. 5. On the application to the rounding effects measured in the OPT Y-123 of the scenarios associated with the presence of charge density waves and with the pseudogap. As already noted in the Introduction, different unconventional (non-Ginzburg-Landau like) scenarios have been proposed for the rounding effects measured above Tcin cuprate superconductors, the most popular being those based on the possible existence of a region of phase disordering up to the temperature where a pseudogap opens. [38,40–59,61–85] Unfortunately, to the best of our knowledge, up to now the theoretical approaches to account for these rounding effects in terms of non-GL scenarios are still not well settled. Nevertheless, for completeness in this Section we will briefly present some comments on the role that could play on these effects the possible presence of charge density waves (CDW) and also the pseudogap opening. In this last case, we have already commented, in particular in Refs. [63,65–67], various aspects of their confrontation with the Gaussian-Ginzburg-Landau (GGL) scenarios for the conventional, thermodynamiclike, superconducting fluctuations. Let us stress already that our comments here will mainly address the application of these scenarios to the rounding effects around Tc in OPT Y-123 and, therefore, our conclusions will not exclude the relevance of these unconventional effects in overdoped and underdoped HTSC. In fact, the entanglement between superconducting fluctuations and pseudogap effects could arise in many of the rounding behaviors measured above Tcin underdoped HTSC, enhancing the interest of our present results when using the resistivity roundings to locate T∗in these compounds. 5.1. On the presence of charge density waves In what concerns the CDW, as shown, e.g., in Ref. [142], their coexistence and mutual interaction with the superconducting order may be described in terms of two simultaneous order parameters, namely, the homogeneous superconducting order parameter associated with the superconducting gap and the periodic order parameter characteristic of the CDW-state. However, for the CDW to have a significant influence on the SC-state, several conditions are required. First of all, if the wave vector of the CDW differs significantly from 2kF, their influence on the stability of the Cooper pairs pairing becomes negligible. [142] Once this constrain is fulfilled, the possible effects of the CDW on the superconducting phase can be described by means of a modified The conductivity and the magnetization around Tcin OPT Y-123... 32 standard GL expression for the thermal-averaged current density of the superconducting condensate with the generalized Langevin equation of the order parameter, taking into account a total-energy cutoff to include the short-wavelength effects expected to be important when ε> ∼0.1. [155] This procedure is similar to the one earlier proposed by A. Schmid in his pioneering calculations in the zero field limit and without cutoff. [116] Here, we briefly summarize a similar extension of these calculations to the 2D and the layered scenarios. We first consider the 2D case, i.e., a superconductor with thickness din the perpendicular (or z-) direction much smaller than the superconducting coherence length amplitude in that direction [i.e.,d≪ξc(0)]. Under this condition the spectrum of the fluctuations appearing in the formalism of Ref. [155] becomes frustrated along the zdirection (with the corresponding component of the fluctuations wavevector, kz, bound between −π/d and π/d and verifying ξc(0)kz≪1). When applied to Eqs. (B.17) and (B.18) of Ref. [155], this leads to ∆σ2D(ε, h, εc) = e2 32¯hd 1 h"ψ1 ε+h 2h!−ψ1 εc+h 2h!# (A.1) for the fluctuation-induced conductivity in the finite magnetic field regime with a totalenergy cutoff, and to ∆σ2D(ε, h) = e2 32¯hd 1 hψ(1) ε+h 2h!(A.2) in absence of cutoff. In the zero field limit (i.e. for h≪ε, εc), from Eq. A.1 it results that the paraconductivity with a total-energy cutoff in a 2D superconductor is ∆σ2D(ε, εc) = e2 16¯hd 1 ε−1 εc,(A.3) that in absence of cutoff leads to the well-known Aslamazov-Larkin (AL) expression for ∆σin a 2D superconductor [156] ∆σ2D(ε) = e2 16¯hdε .(A.4) Let us note here that, as done in the classical paper by Hikami and Larkin [108] for the low-ε, no-cutoff case, we could expand Eqs. A.1 and (A.2) in powers of hup to quadratic order. Our Eq. A.2 would then produce for ∆˜ ˜σan extra prefactor 2/3 with respect to the result in Ref. [108], which origin may be traced back to a different use of the Langevin equation for the order parameter in both calculations. (This prefactor difference exist also in the layered case, presented later in this Appendix). It does not seem feasible in practice to determine what use of the Langevin equation produces better accordance with experiments at low fields, given that the prefactor may be compensated with a change of only ∼10% in the value of ξab(0) [because h∝ξ2 ab(0)] and this is below the uncertainty in any determination of the experimental value of ξab(0). However, direct introduction of the cutoff in the formalism of Ref. [108] leads to some unphysical The conductivity and the magnetization around Tcin OPT Y-123... 33 inconsistencies in the resulting equations; we favor, therefore, our present formalism when analyzing the data including the ε> ∼0.1 region. We now summarize here the extension of these results to layered superconductors. [157] We shall consider here only layered superconductors with a single interlayer distance, s. For that, we first adapt Eq. (B.18) of Ref. [155] (giving the fluctuationinduced conductivity in 3D as a sum over the contributions of different Landau levels) to the layered case by introducing the appropriate out-of-plane spectrum of the fluctuations [108] (i.e., substituting ω3D kz=ξ2 c(0)k2 zby ωLD kz=BLD[1 −cos(kzs)]/2) and taking into account the structural cutoff in the z-direction through |kz| ≤ π/s. This leads to ∆σLD =e2h 16π¯hZπ/s −π/s dkzX n [ε+h(2n+ 1) + ωLD kz]−2,(A.5) where the sum over Landau-levels is to be performed up to nmax = (εc−ε)/(2h)−1.† Then, we obtain ∆σLD(ε, h, εc) = e2 64π¯hh Zπ/s −π/s dkz"ψ1 ε+h+ωLD kz 2h!−ψ1 εc+h+ωLD kz 2h!# (A.6) for the fluctuation-induced conductivity in layered superconductors under an applied magnetic field with a total-energy cutoff. The paraconductivity in the same scenario corresponds to the limit h≪ε, εcof the above expression, given by ∆σLD(ε, εc) = e2 16¯hs "1 ε1 + BLD ε−1/2 −1 εc1 + BLD εc−1/2#.(A.7) In 2D and 3D superconductors, in the long-wavelength regime (i.e., for h, ε ≪εc) Eqs. A.6 and (A.7) become cutoff independent and they reduce to ∆σLD(ε, h) = e2 64π¯h 1 hZπ/s −π/s dkzψ1 ε+h+ωLD kz 2h!,(A.8) and, respectively, ∆σLD(ε) = e2 16¯hs 1 ε1 + BLD ε−1/2 .(A.9) This last being the well-known AL-like expression for the paraconductivity in the LD- scenario. [125] As already commented in detail for the 3D case in Ref. [155], the high-εbehaviour of the zero-field paraconductivity produced by our present formalism is slightly different †Previous calculations of different fluctuation-induced observables under a total energy cutoff in layered superconductors were obtained from the corresponding 2Dresults by applying the substitution ε→ε+ωLD kzeven to the maximum Landau level index, that leads to nmax =εc −ε−ωkz 2h−1. [32,33] In the 3D limit ωLD kz→ω3D kzand the condition on nmax is somehow equivalent to cut the k-space considering only the fluctuating modes inside a sphere. However, to be consistent with our own 3D calculations of ∆σ(ε, h) and of other fluctuation-induced observables (see Ref. [155] and references therein), we have performed the total energy cutoff in the form of a finite cylinder, that leads to nmax =εc −ε 2h−1. The difference between the two ways to apply the total energy cutoff in the LD-scenario lead to small quantitative differences that can be absorbed by a small change in the cutoff amplitude in materials where BLD ≪εc, as it is the case of OPT Y-123. The conductivity and the magnetization around Tcin OPT Y-123... 34 from our previous results for ∆σab(ε, εc) in Ref. [32]. These small differences are only quantitative, not qualitative: In both cases it is predicted a rapid falloff of the fluctuation-induced effects above ε≃0.1, and a vanishing of these fluctuations at ε=εc. In fact, in our present analysis these differences could be absorbed by a small change in the εcvalue. Appendix A.2. Fluctuation-induced diamagnetism Let us here summarize the ∆Mexpressions calculated as described in Ref. [33] but under the cutoff prescriptions described in the previous subsection. We obtain for layered and 2D superconductors respectively: ∆MLD(ε, H) = −kBT 2πϕ0Zπ/s −π/s "−εc+wLD kz 2hψ h+εc+wLD kz 2h! −ln Γ h+ε+wLD kz 2h!+ ln Γ h+εc+wLD kz 2h! +ε+wLD kz 2hψ h+ε+wLD kz 2h!+εc−ε 2h#dkz,(A.10) and ∆M2D(ε, H) = −kBT ϕ0s"−εc 2hψ h+εc 2h!−ln Γ h+ε 2h! + ln Γ h+εc 2h!+ε 2hψ h+ε 2h!+εc−ε 2h#.(A.11) By applying h≪ε, εcin the above equations, we obtain the Schmidt limit under a total-energy cutoff ∆MLD(ε) = −kBT 6ϕ0sh"1 ε1 + BLD ε−1/2 −1 εc1 + BLD εc−1/2#,(A.12) and ∆M2D(ε) = −kBT 6ϕ0sh1 ε−1 εc.(A.13) In the limit ε≪εcthese expressions recover the classical expressions for the Schmidt limit for layered and 2D superconductors respectively: [7,8,33,124–127] ∆MLD(ε) = −kBT 6ϕ0s h ε1 + BLD ε−1/2 ,(A.14) and ∆M2D(ε) = −kBT 6ϕ0s h ε.(A.15) References [1] See, e.g., Norman M R 2011 Science 332 196 The conductivity and the magnetization around Tcin OPT Y-123... 35 Bari˘si´c N et al. 2013 Nat. Phys. 9761 Tafti F F, Lalibert´e F, Dion M, Gaudet J, Fournier P and Taillefer L 2014 Phys. Rev. B 90 024519 Yu G, Xia D -D, Bari˘si´c N, He R -H, Kaneko N, Sasagawa T, Li Y, Zhao X, Shekhter A and Greven M, arXiv:1210.6942v1 See also Ref. [2] and references therein. [2] Basov D N and Chubukov A V 2011 Nat. Phys. 7272 Hashimoto M, Vishik I M, He R-H, Devereaux T P and Shen Z-X 2014 Nat. Phys. 10 483 [3] For an introduction of the phenomenological Gaussian-Ginzburg-Landau (GGL) descriptions of the superconducting fluctuations see, e.g., M. Tinkham, Introduction to Superconductivity, (McGraw-Hill, New York, 1996), Chaps. 8 and 9, and references therein. A.I. Larkin and A.A. Varlamov, Theory of Fluctuations in Superconductors, Clarendon Press, Oxford, (2005) [4] Maki K and Thompson R S 1989 Phys. Rev. B39 2767 [5] Klemm R A 1990 Phys. Rev. B41 2073 For earlier theoretical results on fluctuation effects in layered superconductors see R. A. Klemm, M. R. Beasley, and A. Luther, Phys. Rev. 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