Measurements of the superconducting fluctuations in optimallydoped BaFeNiAs under high magnetic fields: Probing the 3D anisotropic GinzburgLandau approach
Abstract
The superconducting fluctuations well inside the normal state of Febased superconductors were experimentally studied through the inplane paraconductivity in several highquality optimallydoped BaFeNiAs crystals. These measurements were performed in magnetic fields with amplitudes up to 14 T, and different orientations relative to the crystals axis (, 53, and 90). The results allowed a stringent check of the applicability of a recently proposed GinzburgLandau approach for the fluctuation electrical conductivity of 3D anisotropic materials in presence of finite applied magnetic fields.
pacs:
74.25.Ha, 74.40.n, 74.70.Xa1 Introduction
The high critical temperatures () of Febased superconductors (FeSC), and the unconventional mechanism for their superconductivity (with a pairing probably mediated by spin fluctuations and involving several bands) have generated an enormous interest for these materials in the last few years.[1] A central aspect of their phenomenology is the effect of superconducting fluctuations around .[2] Mainly due to the short coherence length and high values of these materials,[1] these effects are enhanced with respect to conventional low superconductors. In fact, the Ginzburg number characterizing the width of the critical fluctuation region around , is in FeSC halfway the ones found in conventional low superconductors and high cuprates (HTSC).[3]
In addition to their intrinsic interest, the superconducting fluctuation effects are a very useful tool to characterize the nature of a superconducting transition and to obtain material parameters,[2] so that different works have already addressed their study in FeSC through observables like the magnetization, specific heat or the electric conductivity.[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] However, some fundamental aspects of the phenomenology of the fluctuation effects in these materials are still debated. One of them is their dimensionality. In these materials the transverse coherence length amplitude is close to the Fe layers periodicity length, . Thus, depending on the particular compound studied, some works report a twodimensional (2D) behavior[3, 10, 20] similar to the one found in highly anisotropic HTSC,[24] while others find threedimensional (3D) characteristics,[4, 5, 6, 7, 11, 12, 14, 15, 16, 19, 21, 22, 23] or even a 3D2D transition[8, 13, 18] when increasing the temperature above (as in optimallydoped YBaCuO)[24].^{1}^{1}1A theory for the effect of critical fluctuations around the line on different observables in superconductors with intermediate 2D3D characteristics, was developed in Ref. [9] But it was recently reported that the fluctuation electrical conductivity above of clean LiFeAs crystals seemingly follows a well defined 2D behavior (in both the amplitude and the reducedtemperature dependence), despite that for this compound nm is much larger than the Felayers periodicity length ( nm).[20] This surprising result led the authors of Ref. [20] to propose that in these multiband superconductors the fluctuating pairs above may be driven by a single 2D band. Other interesting issues that deserve attention are the possible presence of phase fluctuations (which effect was possibly observed near at low field amplitudes in the SmFeAsOF [17], but also in members of the less anisotropic 122 family as BaKFeAs [5] and Ba(FeRh)As [25]), or the behavior of fluctuation effects in the short wavelength regime appearing at high reduced magnetic fields or temperatures.
As a contribution for the understanding of the above mentioned issues, here we present detailed measurements of the fluctuationinduced inplane electric conductivity () in several high quality BaFeNiAs crystals with doping levels near the optimal one (). These experiments were performed in magnetic fields () up to 14 T applied with different angles relative to the crystals axis (, 53, and 90 degrees), thus extending previous measurements in the same compound with up to 9 T.[21, 22] The large fields used here allow to deeply penetrate into the socalled Prange fluctuation regime, and to perform a stringent check of the applicability of a recently proposed generalization of the classic AslamazovLarkin (AL) results to finite fields through a 3Danisotropic GinzburgLandau approach.[21] In turn, the use of different magnetic field orientations provides an important consistency test of the analysis and allows to obtain precise information about the system dimensionality, basic superconducting parameters (as the coherence lengths and the anisotropy factor), and the angular dependence of the upper critical field, at present another debated issue in these materials.[26]
2 Experimental details and results
We studied three BaFeNiAs single crystals with nominal doping levels near the optimum one, two with (#1 and #2) and one with (#3). Their sizes are typically mm, being the axis of the tetragonal structure ( Å, Å) perpendicular to their largest face. Details of their growth procedure and a thorough characterization may be found in Ref. [27].
Crystal  (K)  (K)  (nm)  (nm)  

6%  2%  8%  
#1  19.8  0.3  1.28  2.57  2.00 
#2  19.7  0.2  1.28  2.54  1.98 
#3  19.8  0.2  1.24  2.52  2.02 
The resistivity along the layers, , was measured with a Quantum Design’s Physical Property Measurement System (PPMS) in presence of magnetic fields up to 14 T with different orientations relative to the axis (, 53, and 90 degrees). For that we used a standard fourprobe method with a low contact resistance (less than 1 Ohm) and a current of 1 mA. The data were obtained by sweeping the temperature at a rate of 0.3 K/min. An example of the behavior (corresponding to crystal #1) around is presented in Fig. 1. The value was determined from the transition midpoint for the data, and the transition width was estimated as . The corresponding values for the three samples studied are compiled in Table 1. The small values (about ) confirm the excellent stoichiometric quality of the crystals. An overview of in absence of field and up to is presented in the inset of Fig. 1(c) for all samples studied. As it may be seen, is almost temperature independent from few degrees above up to . This is an important experimental advantage to determine the conductivity induced by superconducting fluctuations (or paraconductivity), which is given by
(1) 
where is the normalstate or background contribution. The procedure to estimate is illustrated in the example of Fig. 2, also corresponding to crystal #1: In the region K (corresponding to , where fluctuation effects are expected to be negligible [21]) the resistivity is linear with the temperature up to the largest field used in the experiments. Besides, as it is shown in the inset of that figure, the magnetoresistivity in the normal state is roughly quadratic in the applied magnetic field. This allows to parametrize the background resistivity as
(2) 
where
(3) 
The coefficients , , , and were obtained by linear fittings to the curves measured with and 14 T. An example (corresponding to crystal #1) of the resulting dependence on the reduced temperature, , is presented in Fig. 3.
3 Data analysis
3.1 Inplane paraconductivity in the lowfield limit
In the absence of a magnetic field and for temperatures close to , it is expected that will follow the classical AslamazovLarkin result, which for 3D superconductors may be written as[28]
(4) 
where is the electron charge, is the reduced Planck constant, and is the axis coherence length amplitude. As it may be seen in the inset of Fig. 3, for reducedtemperatures below a critical exponent close to is observed, in agreement with Eq. (4). Above this value a rapid falloff of the fluctuation effects is observed and a welldefined critical exponent is no longer observed, a behavior that may be attributed to shortwavelength fluctuation effects.[2, 29, 30] For completeness, in the same inset we present the prediction of the 2DAL result
(5) 
where Å is the FeAs layers periodicity length. As it may be clearly seen, it overestimates the experimental data by almost two orders of magnitude, which is well beyond the experimental uncertainties, including those associated to the determination of the normalstate background.
In the presence of a finite magnetic field, roughly above the socalled ghost critical field (which is the symmetric above of the corresponding line [31]), is expected to be significantly reduced with respect to Eq. (4).[2, 32, 33] As it may be seen in Fig. 3, in particular in the representation of panels (df), such a reduction is clearly observed with the field amplitudes used in our experiments and, as expected, it is more prominent for temperatures close to (i.e., for ). It is also noticeable the dependence on the field orientation relative to the crystals axis, which is a direct consequence of the anisotropy of the upper critical field in the studied compound (see below).
3.2 Comparison with the GL approach for the finitefield or Prange regime
In what follows we will present an analysis of the experimental data in terms of the 3Danisotropic GinzburgLandau (GL) approach developed in Ref. [21]. It adapts to this dimensional case the model proposed by A. Schmid, which is based on a combination of the standard Gaussian GLexpression of the thermallyaveraged current density with the generalized Langevin equation of the order parameter.[34] Since the energy of the fluctuation modes increases with , this finitefield approach includes an energy cutoff in the fluctuations spectrum as proposed in Refs. [29, 30]. For perpendicular to the layers, it leads to
(6) 
where , is the linear extrapolation to K of the upper critical field for (i.e., ), and is a cutoff constant which value is expected to be about .[29, 30] In the zerofield limit (for ), Eq. (6) is transformed into
(7) 
which at low reduced temperatures () reduces to the conventional AL expression, Eq. (4). Following the scaling transformation for anisotropic materials developed in Refs. [35, 36, 37], Eq. (6) may be generalized to an arbitrary field orientation by just replacing by
(8) 
being the upper critical field (linearly extrapolated to K) for an arbitrary angle between and the crystal axis.
Equation (6) is fitted to the complete set of data for each field orientation with only three free parameters: the upper critical field , the amplitude term (directly related to ), and the cutoff constant . As it may be seen in Figs. 3(ac) the agreement is excellent, extending down to a fielddependent temperature below that may be close to the upper bound of the critical region.^{2}^{2}2In this region fluctuation effects are so important that the Gaussian approximation [used to derive Eq. (6)] is no longer applicable, see Ref. [2]. The agreement is also excellent in the representation of Figs. 3(df), which is focused on temperatures above . The resulting values are presented in Fig. 4 for all samples studied. These data follow the behavior expected for 3Danisotropic materials (solid lines) [38],
(9) 
which represents an important consistency check of the present results. The and values are within the ones obtained in the literature in the same material from the shift of the resistive transition induced by the field,[39, 40, 41, 42] although the rounding associated to fluctuation effects makes this procedure strongly dependent on the criterion used (generally a given % of the normalstate resistivity). In the present case, as it may be seen in Fig. 5, the 50% criterion gives values in good agreement with the ones resulting from the analysis of fluctuation effects.
As the amplitude may be affected by the uncertainties associated with the finite size of the electrical contacts and with the crystals geometry, the amplitude term in Eq. (6) is not used to determine . Instead, the GL coherence length amplitudes are obtained from the values in Fig. 4 according to
(10) 
and
(11) 
where the anisotropy factor is obtained from the ratio
(12) 
The values corresponding to each sample are compiled in Table 1.
Regarding the cutoff constant, it resulted to be in the range for all samples. This value is close to the ones found in previous works about fluctuation effects above in FeSC, in particular on the paraconductivity at high values in the same compound,[21] and on the precursor diamagnetism in optimallydoped BaKFeAs.[14] It is also close to the cutoff constant found in other superconducting families, including high cuprates,[43, 44, 45, 46] low metallic elements and alloys,[47, 48] and compounds like MgB or NbSe.[49, 50] Our results confirm the proposal in Refs. [29, 30] about an universal value close to , associated with the limits encountered at high or to the shrinkage of the superconducting wavefunction to lengths of the order of the pairs size.
Let us finally comment on the applicability of a GL approach to a twoband superconductor as BaFeNiAs. In principle, the analysis of fluctuation effects in multiband superconductors, in particular when they involve two or more weaklycoupled bands with different anisotropy, would require a specific multiband functional that takes into account the nonlocal effects arising from having an effective coherence length in one of the crystallographic directions of the system much smaller than the associated to one of the bands. It has been proposed that this is the case of MgB [51], although a good description of fluctuation effects in terms of GL approaches was also found for this compound [49]. The applicability of a GL approach to BaFeNiAs would suggest that the interband coupling in this compound is larger than that in MgB. This is consistent with the fact that the relative band interaction constant defined in Ref. [51], , is in MgB of the order of , while in optimally doped BaFeNiAs we find (i.e., four times larger) by using the coupling parameters reported in Ref. [52].
3.3 phase diagram for
The large number of measured isofields allowed to plot detailed phase diagrams of the amplitude for the three field orientations studied. An example for crystal #1 is shown in Fig. 6. The solid line is the upper critical field, as obtained from the and values in Fig. 4 by assuming a linear temperature dependence close to the transition. The dotted line represents the experimental limit of applicability of Eq. (6) which, as commented above, may be close to the onset of the critical region. In these phase diagrams finite field effects may be seen as deviations from the verticality of the iso curves. These effects are more prominent for and, as expected, appear for fields roughly above the corresponding ghost field, . Finally, the circles indicate the points at which falls below the noise level, which are expected to be close to the irreversibility line, . It is worth noting that the dependence of on the orientation of the applied magnetic field provides a further check of the applicability of the 3D anisotropic GL approach to the compound under study. In fact, by approximating the irreversibility line by the melting line, according to Ref. [37] it is expected that
(13) 
Then, taking into account the above dependence, the lines should scale when normalized by . As it may be seen in Fig. 7, such a scaling is observed when using the values in Table 1. Just for completeness, note that the irreversibility line for follows the temperature dependence predicted in Ref. [53], which was obtained within a 3D disordered GinzburgLandau model:
(14) 
where , , and and are fitting parameters defined in Ref. [53] representing the disorder and the strength of thermal fluctuations, respectively. Values obtained from the fitting are , and . The value of suggests that disorder is important in the reversible region below the line.
3.4 Comparison with recent works
In a recent work by RullierAlbenque et al. (Ref. [20]) it is reported that the paraconductivity of clean LiFeAs samples is 2D in nature, in spite that the superconducting parameters of this compound (similar to the ones of BaFeNiAs) would suggest a 3D behavior: nm is much larger than the interlayer distance, nm. Here we show that this conflicting result may be an artifact associated to the procedure used to determine the normalstate contribution. First of all, let us note that in clean crystals the same fluctuation effects (i.e., the same ) lead to a much weaker resistivity rounding than in dirty samples (with a much larger background resistivity): the reason is that not too close to the change in the electrical resistivity due to superconducting fluctuations may be approximated by
(15) 
In the clean crystals used in Ref. [20], m just above . Subsequently, in the case of 3D fluctuations, at intermediate reduced temperatures (e.g., ) the relative change in is expected to be about 0.06%. Even in the case of 2D fluctuations, for which is given by Eq. (5), the relative change in is expected to be smaller than 1%. Indeed, the observation of fluctuation effects in the resistivity of so clean crystals would require an extraordinary precision in the determination of the background, whatever the procedure used. In Ref. [20] is estimated by allegedly quenching fluctuation effects with magnetic fields typically above 10 T. However, it has been shown that fluctuations above survive up to fields of the order of .[33] As for LiFeAs this quantity is about 28 T [54], it is likely that fluctuation effects are still present above 10 T.
Nevertheless, in Ref. [20] it is found a good agreement (without free parameters) between Eq. (5) and the data obtained in one of the samples (named FP2). However, there is a large difference with the results obtained in the other sample (FP1): the data at 18.8 K (), not included in Fig. 4 of Ref. [20] but available from the data in Fig. 2(b), is a factor 2.5 larger than the one for sample FP2. The difference cannot be attributed to a wider superconducting transition [as it is shown in Fig. 1(a) both samples present similar transition widths], and suggests that the agreement of Eq. (5) with sample FP1 could be accidental. If the fluctuation effects in LiFeAs were actually 3D in nature, the arguments given in Ref. [20] supporting a pure behavior for transverse magnetoresistivity in this compound, should be revised.
4 Conclusions
We have presented detailed measurements of the conductivity induced by superconducting fluctuations just above the superconducting transition of three highquality optimallydoped BaFeNiAs single crystals. These measurements were performed with magnetic fields up to 14 T, which allowed to deeply penetrate into the finitefield (or Prange) fluctuation regime. The magnetic field was applied with different orientations with respect to the crystals axis (, 53, and 90), allowing to investigate the anisotropy of fluctuation effects. The analysis of the experimental data lead to solid evidence of the applicability to these compounds of a recently published Gaussian GL approach for 3D anisotropic superconductors in presence of finite applied magnetic fields. Our results contrast with the recent observation of a seemingly twodimensional paraconductivity in clean LiFeAs single crystals, in spite that the coherence length amplitudes of this compound are similar to the ones in optimally doped BaFeNiAs. The discrepancy may be attributed to the uncertainty in the LiFeAs normalstate contribution.
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