Low anisotropy of the upper critical field in a strongly anisotropic layered cuprate Bi{}_{2.15}Sr{}_{1.9}CuO{}_{6+\delta}: Evidence for a paramagnetically limited superconductivity.

Low anisotropy of the upper critical field in a strongly anisotropic layered cuprate BiSrCuO: Evidence for a paramagnetically limited superconductivity.

S. O. Katterwe    Th. Jacobs Department of Physics, Stockholm University, AlbaNova University Center, SE-10691 Stockholm, Sweden    A. Maljuk Leibniz Institute for Solid State and Materials Research IFW Dresden, Helmholtzstr. 20, D-01171 Dresden, Germany    V. M. Krasnov Vladimir.Krasnov@fysik.su.se
July 14, 2019

We study angular-dependent magnetoresistance in a low layered cuprate BiSrCuO. The low  K allows complete suppression of superconductivity by modest magnetic fields and facilitate accurate analysis of the upper critical field . We observe an universal exponential decay of fluctuation conductivity in a broad range of temperatures above and propose a new method for extraction of from the scaling analysis of the fluctuation conductivity at . Our main result is observation of a surprisingly low anisotropy , which is much smaller than the effective mass anisotropy of the material . We show that the anisotropy is decreasing with increasing field and saturates at a small value when the field reaches the paramagnetic limit. We argue that the dramatic discrepancy of high field and low field anisotropies is a clear evidence for paramagnetically limited superconductivity.

74.72.Gh 74.55.+v 74.72.Kf 74.62.-c

I Introduction

The upper critical field is one of the key parameters of type-II superconductors SaintJames (). It is particularly important for understanding unconventional superconductivity Gurevich2007 (); Prozorov2012 (). However, estimation of for high-temperature superconductors is a notoriously difficult task. The high leads to an extended region of thermally activated flux-flow. The complex physics of anisotropic pinning and melting of the vortex lattice Blatter () makes it hard, if at all possible Vedeneev1999 (), to confidently obtain from flux-flow characteristics at .

The high in combination with a strong coupling leads to a large superconducting energy gap meV KrTemp (); Hoffman2002 (); Fischer2007 (); SecondOrder (); ARPESreview (); Ideta_ARPES () and T Obrian2000 (); Sekitani2004 (); MR (); Vedeneev2006 (); Li2007 (); Ramshaw2012 (); Taillifer2012 (); Grissonnanche_2014 (). Such strong fields may alter the ground state of the material. For cuprates and pnictides the parent state is antiferromagnetic. It has been demonstrated that relatively weak fields can induce a canted ferromagnetic order in strongly underdoped cuprates Ando_2003 (). Furthermore, the normal state of underdoped cuprates is characterised by the presence of the pseudogap (PG), which probably represents a charge/spin or orbital density wave order coexisting and competing with superconductivity Tallon2001 (); SecondOrder (); ARPESreview (); Vishik2012 (); Onufrieva2012 (); d_density (); Varma2002 (); Weber2009 (); Orenstein2013 (); Gabovich2013 (). Suppression of superconductivity by magnetic field may enhance the competing PG, as follows from observation of a charge density wave in a vortex core Hoffman2002 (). But even stronger magnetic fields of several hundred tesla suppress the PG KElbaum2005 (); Jacobs2012 (). Thus, both superconducting and normal state properties are affected by strong magnetic fields and separation of the two contributions is highly non-trivial and controversial. Disentanglement of superconducting and PG characteristics is difficult even above due to presence of profound superconducting fluctuations Dubroka_2011 (); Kaminski_NP2011 (); Varlamov (). Therefore, principal new questions, which do not appear for low- superconductors, are to what extent alters the abnormal normal state of high- superconductors and how to define the non-superconducting background in measured characteristics.

For many unconventional superconductors the measured exceeds the paramagnetic limit of the BCS theory SaintJames (); Clogston (). This has been reported for organic Zuo2000 (); Singleton2000 (); Lee2000 (), cuprate Vedeneev2006 (); Li2007 (), pnictide Cho2011 (); Khim2011 (); Burger2013 () and heavy fermion Radovan2003 (); Kakuyanagi2005 (); Matsuda2007 () superconductors. Such an overshooting is an important hint in a long standing search for exotic spin-triplet and Fulde-Ferrell-Larkin-Ovchinnikov states (for review see e.g. Ref.Matsuda2007 ()). Yet, the overshooting is not a proof of unconventional pairing because the paramagnetic limit is rather flexible. It is increasing in the presence of spin-orbit interaction SaintJames () and in the two-dimensional (2D) case and is lifted in the one-dimensional (1D) case Buzdin1996 (); Matsuda2007 (). Unconventional superconductors are usually anisotropic. Some of them have quasi-2D, or possibly even quasi-1D structure.Many have a significant spin-orbit interaction between localized spins and itinerant charge carriers. Consequently, one needs a more robust criterion for the paramagnetically (un)limited superconductivity in search for exotic states of matter.

Here we investigate the anisotropy of in a strongly anisotropic layered BiSrCuO (Bi-2201) cuprate with a low K. The low and the associated large disparity of superconducting and pseudogap scales Jacobs2012 () allow simple and accurate estimation of without complications typical for high- cuprates. We present a detailed analysis of angular dependence of in-plane and out-of-plane magnetoresistances (MR) and demonstrate that they exhibit very different behavior. We observe an universal approximately exponential decay of the in-plane fluctuation para-conductivity above and propose a method for extraction of from a new type of a scaling analysis of fluctuations at . It obviates the complexity of the flux-flow phenomena and allows unambiguous extraction of . Remarkably, we obtained that the anisotropy of the upper critical field is much smaller than the anisotropy of the effective mass Vinnikov (). This discrepancy clearly indicates that parallel to the CuO planes is cut-off by the paramagnetic limit.

Cuprates have homologous families with different number of CuO planes per unit cell. Cuprates within the homologous family have similar carrier concentrations, resistivities, anisotropies and layeredness, but largely different . For Bi-based cuprates the three-layer compound BiSrCaCuO (Bi-2223) has a maximum of K, the two-layer compound BiSrCaCuO (Bi-2212) has a K and a single-layer compound BiSrCuO (Bi-2201) has an optimal (with respect to Oxygen doping) that ranges from K for Bi/Pb and Sr/La substituted crystals Lavrov () to just few K in the pure Bi-2201 compound Vedeneev1999 (); Sonder1989 (); Maljuk (); Luo2014 (). According to Ref. Maljuk () the stoichiometric Bi-2201 compound is non-superconducting and a finite appears only in off-stoichiometric BiSrCuO compounds with . Thus, the Bi/Sr off-stoichiometry allows fine tuning of the maximum Maljuk (); Luo2014 ().

Development of high magnetic field techniques in recent years has lead to a significant progress in studies of in high- superconductors Sekitani2004 (); Vedeneev2006 (); Taillifer2012 (); Grissonnanche_2014 (). But the problem of disentanglement of superconducting and PG magnetic responses remains. It leads to a lack of clear criteria for extraction of from measurement at T. This problem is avoided in low- cuprates because the relative disparity between superconducting and pseudogap scales is increasing with decreasing Jacobs2012 (). Therefore, analysis of in low- cuprates should provide an unambiguous information about the superconducting state, not affected by interference with the co-existing PG. This is the main motivation of the present work.

Figure 1: (color online) (a) Scanning electron microscopy image of the sample OP(4.3). (b) Temperatures dependence of the -axis resistance at and 14 T. The compound has a low and the pseudogap onset temperature K. (c) Current-voltage characteristic of a small mesa at K and .

Ii Experimental

Studied crystals are parts of one pristine BiSrCuO single crystal with K. Growth and characterization of crystals is described in Ref. Maljuk (). Oxygen doping was consecutively decreased by soft annealing in vacuum, which does not affect the crystal quality Sonder1989 (). We present data for a slightly overdoped (with respect to oxygen content) K [OD(4.0)] and a nearly optimally doped K [OP(4.3)] crystals.

Figure 1 (a) shows an image of the studied sample OP(4.3). The sample consists of ten micron-size mesa structures (two big and eight small) with attached gold electrodes. In-plane resistance is measured with a lock-in technique in a four-probe configuration by sending an ac-current through the left and right current contacts (big mesas), and measuring the longitudinal voltage between a pair of small mesas. The -axis transport is measured in a three-probe configuration by sending a probe current through one of the small mesas to one of the current contacts. The voltage is measured with respect to unbiased contact pad. Details of sample fabrication and measurement setup can be found in Ref. Jacobs2012 ().

Fig. 1 (b) shows the -axis resistance versus temperature at and T along the -axis. It is seen that exhibits an upturn at K, indicating opening of the -axis PG. According to previous studies Lavrov (); Yurgens_Bi2201 (); Jacobs2012 () such a corresponds to a near optimally doped (OP) (slightly underdoped) Bi-2201. A superconducting transition occurs at a much lower K. The -axis field of 14 T completely suppresses the superconducting transition but does not change significantly the PG characteristics due to a large disparity of superconducting and PG scales in this low- compound Jacobs2012 ().

The large -axis resistance corresponds to a non-metallic resistivity cm Jacobs2012 (), which is much larger than the in-plane resistivity cm Ando1996 (). The anisotropy of resistivity and the corresponding effective mass anisotropy is very large Vinnikov (), similar to Bi-2212 Watanabe1997 () and Bi-2223 Suzuki_Bi2223 () cuprates. This reflects a layered 2D structure of Bi-based cuprates with mobile electrons localized on atomic CuO planes. The -axis transport is caused by interlayer tunneling. Below this leads to appearance of an intrinsic Josephson effect Kleiner (), observed in all Bi-based cuprates KrTemp (); Katterwe2009 (); SecondOrder (); MR (); Suzuki_Bi2223 (), including Bi-2201 Yurgens_Bi2201 (); MQT_Bi2201 (); Jacobs2012 (). Interlayer tunneling creates the basis for the intrinsic tunneling spectroscopy technique KrTemp (); Suzuki_Bi2223 (); SecondOrder (); MR (); Jacobs2012 () and facilitates simultaneous magneto-transport and spectroscopic measurements, beneficial for analysis of MR (). Fig. 1 (c) shows the current-voltage - characteristics of a small mesa at K. A detailed analysis of intrinsic tunneling characteristics of our Bi-2201 crystals can be found in Ref. Jacobs2012 (). Small area of our mesas allows investigation of intrinsic tunneling characteristics Jacobs2012 (); SecondOrder (); MR (); Suzuki_Bi2223 (); KrTemp () without significant distortion by self-heating SecondOrder ().

Iii In-plane and out-of-plain magnetoresistance

iii.1 A. In-plane magnetoresistance

Figures 2 (a) and (b) show temperature dependencies of the in-plane resistance at different magnetic fields (a) perpendicular and (b) parallel to the planes for the OP(4.3) sample. For , reaches the normal state value already at T. For the field of 17 T still does not completely suppress superconductivity. The difference is both due to the anisotropy and due to different contributions from flux-flow and orbital effects. The Lorentz force density , where is the transport current density and is the magnetic induction, acts both on vortices and mobile charge carriers. In Fig. 2 (a) the Lorentz force is at maximum and effectively drives pancake vortices Blatter () along CuO planes. Therefore, is dominated by the flux-flow contribution at . In case of Fig. 2 (b) there is no Lorentz force and the flux-flow contribution should be minimal.

Figure 2: (Color online) -dependencies of the in-plane resistance at magnetic fields (a) perpendicular and (b) parallel to the -planes. (c) Comparison of the data from (a) and (b) for zero and 17 T. For the is shifted to lower temperatures. For it is completely suppressed and is shifted upwards, indicating presence of a positive orbital magnetoresistance in the normal state. (d) MR in a perpendicular field below and just above . Note that the saturation field is decreasing with . Panels (e) and (f) show MR for both field orientations (e) below and (f) above . The positive MR at is caused both by suppression of superconducting fluctuations and an additional orbital normal state MR.

Fig. 2 (c) represents a detailed comparison of at and 17 T for the two field orientations. We notice that the resistive transition at is simply shifted towards a lower due to suppression of . On the other hand at is also shifted upwards, even at . It indicats that there is an additional positive MR in the normal state ( at T). Thus, there are two different mechanisms of positive in-plane MR. At it is mostly due to suppression of superconductivity. Such MR saturates at . Fig. 2 (d) shows field-dependence of at K and at . It is seen that saturation of occurs at significantly lower field for K, consistent with reduction of at . In the normal state the tendency is reversed. With increasing the saturation field is increasing. This can be seen from Figs. 2 (e) and (f), which show field-dependence of in perpendicular (circles) and parallel (squares) magnetic fields at K and K, respectively. Such behavior can be partly attributed to superconducting fluctuations, for which the characteristic field is increasing with Varlamov (). However, fluctuations do not explain the increment of the saturation value of , which is visible at and is significant only for , see Fig. 2 (c). Consequently, there is an additional normal state MR, caused by orbiting of mobile electrons in magnetic field Ziman (). This leads to a positive MR with saturation at , where is the cyclotron frequency and is the scattering time. Since becomes shorter with increasing , the saturation field is increasing with increasing . Due to the quasi-2D electronic structure of Bi-2201, the orbital MR should appear only at , consistent with our observation.

Figure 3: (Color online) Temperature dependencies of the -axis resistance for (a) and (b) . (c) Comparison of the data from (a) and (b) for and 17 T. Panels (d) and (e) show -axis MR for the two field orientations (d) below and (e) above . It is seen the normal state negative MR is present for both field orientations. (f) measured up to 65 (data from Ref. Jacobs2012 ()). It is seen that there is both a positive MR at low fields due to suppression of the supercurrent and a negative MR at high fields due to suppression of the PG.

iii.2 B. Out-of-plane magnetoresistance

Figure 3 (a) and (b) show temperature dependencies of the -axis resistance at different magnetic fields (a) perpendicular and (b) parallel to the planes. Irrespective of field orientation, there are both positive and negative contributions to -axis MR. Fig. 3 (c) represents a detailed comparison of at and at T for the two field orientations. It is seen that in the normal state there is a significant negative -axis MR for both field orientations. It is largest for and reaches almost in 17 T field.

A positive MR appears only in the superconducting state . It is due to suppression of the interlayer Josephson current with respect to the bias current Morozov2000 (); Katterwe2008 (). At there is a profound Josephson flux-flow phenomenon due to easy sliding of Josephson vortices along the -planes Katterwe2009 (); Katterwe2010 (). This also leads to a positive MR with a peak at strictly parallel to the -planes Motzkau2013 (). The negative -axis MR persists both in the superconducting MR () and the normal states and is attributed to field suppression of either the superconducting gap MR () or the pseudogap KElbaum2005 (). For high- Bi-2212 KrTemp (); SecondOrder () and Bi-2223 Suzuki_Bi2223 () cuprates the corresponding energies ( meV, meV) and fields ( T, T) are similar MR (); KElbaum2005 () and separation of the two contributions is difficult. However, in the studied low- superconductor the separation becomes trivial because, as shown in Ref. Jacobs2012 (), all PG characteristics remain similar to high- materials, but all superconducting characteristics scale down with Ideta_ARPES (), leading to a large disparity of superconducting and PG characteristics.

Figs. 3 (d,e) show -axis MR for different field orientations and temperatures (d) below and (e) above . It is seen that the negative MR persists at T and at and is due to field suppression of the PG KElbaum2005 (); Jacobs2012 (). Fig. 3 (f) shows pulsed field measurements of at K up to 65 T for a slightly underdoped crystal from the same batch (data from Ref. Jacobs2012 ()). It is seen that at high fields is approximately linear in the semi-logarithmic scale. An extrapolation to the normal resistance yields the PG closing field T. It corresponds to the Zeeman energy meV Jacobs2012 ().

iii.3 C. Angular magnetoresistance at

Figure 4: (Color online) (a) and (b) Angular dependencies of in-plane resistances for rotation around two orthogonal axes in the -plane. Dashed lines in (b) represent scaled data from (a) Note1 (). (c) Angular dependence of the -axis resistance. The peak at is due to onset of Josephson flux-flow. (d) Theoretical angular dependencies of flux-flow resistances for a 2D model with an anisotropy . Note that the cusp at becomes sharper at because superconductivity survives only in a narrow range of angles around . A similar narrowing is seen in panels (a-c). Panels (e) and (f) represent comparison of (e) in-plane and (f) out-of-plane angular MR (symbols) with resistances at the corresponding perpendicular (solid lines) and parallel field components at K and T.

Angular dependence of the upper critical field is given by the following equations:


for a three-dimensional (3D) superconductor and


for the 2D case. In the simplest case of an isotropic superconductor the flux-flow resistivity can be approximately estimated from the Bardeen-Stephen model Stephen (),


It connects the angular MR with . The main qualitative difference between 3D and 2D cases is that has a smooth minimum in the 3D case and a sharp cusp-like dip in the 2D case Naughton1988 ().

Figures 4 (a) and (b) show angular dependencies of the in-plane resistance at K measured upon rotation around two orthogonal axes in the -plane (a) perpendicular and (b) parallel to the current. In both cases corresponds to , . But , corresponds to either (a) the Lorentz force-free configuration , or (b) to the case when the Lorentz force is acting on Josephson vortices in the direction perpendicular to layers. Dashed lines in (b) represent properly scaled data from panel (a) Note1 (). It is seen that the behavior in both cases is very similar. Therefore, at the flux-flow contribution to is small either due to zero Lorentz force or a strong intrinsic pinning in the layered superconductor Tachiki (); Kwok1991 (); NbCu1996 (), which prevents motion of Josephson vortices across the planes.

Fig. 4 (c) shows angular dependencies of the -axis resistance. Apart from the dip at due to the anisotropy of , the has an additional sharp maximum at due to onset of the Josephson flux-flow phenomenon Motzkau2013 (). In this case the Lorentz force is directed along the -planes and easily drags Josephson vortices with low pinning and viscosity Katterwe2010 (). The shape of at large angles is visibly affected by the negative normal state MR, which causes a shallow minimum of at at large fields.

From Figs. 4 (a-c) it is seen that exhibits a cusp at , indicating the 2D-nature of superconductivity in CuO planes. The cusp becomes narrower and sharper with increasing field. This is in a qualitative agreement with calculations for the 2D model using Eqs.(2,3), shown in Fig. 4 (d). The sharpening of the cusp at occurs when the field becomes larger than . In this case the sample is in the normal state with a flat for angles at which . As the field approaches , superconductivity survives only in a narrow range of angles . Therefore, a significant narrowing of the cusp at T in Fig. 4 (a-c) indicates that is close to 17 T.

The anisotropy of can be analyzed from comparison of angular-dependent with MR at the corresponding parallel and perpendicular field orientations. If one of the field components is smaller than the corresponding , adding of an orthogonal component will contribute to suppression of superconductivity. But if the field component is larger than , than an extra field component will not give a significant contribution to MR. In Figs. 4 (e) and (f) we perform such the comparison at K. Black symbols in Figs. 4 (e) represent at T from Fig. 4 (a) as a function of . The solid red line represents the MR in solely the perpendicular field component . The dashed blue line represents a sum of resistances in the corresponding perpendicular and parallel field components , shown in Fig. 2 (e). It is seen that at the angular MR is determined almost entirely by and an additional does not contribute significantly to MR. This angle corresponds to T, as indicated by a vertical arrow in Fig. 4 (e). At larger angles superconductivity is already suppressed because and MR becomes insensitive to an additional parallel field component. Such the analysis confirms that T. At smaller angles and does contribute to MR, although not additively.

Fig. 4 (f) represents a similar comparison for the out-of-plane resistance. Solid and dashed lines represent the MR solely in perpendicular and parallel fields from Fig. 3 (d). Apparently, is not determined by a single field component. The most pronounced feature of is a rapid drop at , which reflects the corresponding behavior of . Therefore, the crystal still maintains some superconductivity at T, but it is rapidly suppressed by a small additional component upon a slight rotation of the crystal. Consequently, is slightly larger than 17 T. On the other hand, since T, there is no similar drop at .

Figure 5: (Color online) Angular dependence of (a) in-plane and (b) -axis resistances at T for different temperatures. The cusp at is vanishing at for and at for . (c) Angular dependent MR above , normalized by . At K the MR is varying in a smooth 3D-manner. (d) Temperature dependencies of absolute values of angular MR amplitudes normalized by the field (in the semi-logarithmic scale). Note that the in-plane MR decays at a scale and the out-of plane MR at the PG temperature K. Panels (e) and (f) represent comparison of (e) in-plane and (f) out-of-plane angular MR at K with the MR at the corresponding parallel and perpendicular field components. The in-plane angular MR at not too small angles is dominated by the perpendicular field component (solid line). The -axis angular MR is given by the additive contribution of the two field components (dashed lines).

Iv Fluctuation magnetoresistance

From comparison of Figs. 4 (a), (b) and (d) it is clear that Eqs. (2) and (3) only explain the narrowing of the cusp, but do not fit the data. This demonstrates inappropriateness of Eq. (3) for layered superconductors because it does not take into consideration transformation of the vortex structure, the pinning strength and the Lorentz force upon rotation of the crystal. Furthermore, Eq. (3) assumes that the resistance always reaches the normal state value at and thus neglects the remaining fluctuation para-conductivity at Varlamov (). As discussed above, should have minimal flux-flow contribution either due to zero Lorentz force, or presence of a strong intrinsic pinning. Consequently, the dip in resistance at in Fig. 4 (a) and the major part of the resistive transition at in Fig. 2 (b) are due to fluctuation conductivity, rather than flux-flow. Without flux-flow, would correspond to the onset of resistivity , rather than . This has been demonstrated by simultaneous tunneling and transport measurements for conventional superconductors MR (). Without exact knowledge of the flux-flow contribution it is impossible to confidently extract from data at . The lack of criteria for obscures estimation of Vedeneev1999 (). Therefore, in the remaining part of the manuscript we will focus on the analysis of fluctuation part of MR at . As we will demonstrate, such data do not suffer from ambiguity associated with flux-flow phenomenon and facilitate confident extraction of .

iv.1 A. Angular magnetoresistance at

Figure 5 (a) shows angular dependencies of the in-plane resistance at T and at different close and above K. Here corresponds to zero Lorentz force configuration . It is seen that the cusp at , characteristic for the 2D superconducting state, is rapidly diminishing with increasing . It disappears at . At K it turns into a shallow minimum, which persists to and represents the anisotropy of the positive orbital MR in the normal state.

Fig. 5 (b) shows angular dependencies of the -axis resistance below and above . Here, measurements were performed at bias above the Josephson flux-flow branch in the - so that the Josephson flux-flow peak in does not occur Motzkau2013 (). Above the cusp in completely disappears and only a shallow maximum at remains, which indicates a small angular anisotropy of the normal state MR, as seen from Fig. 3 (e). In Fig. 5 (c) we show angular dependencies of in-plane and -axis resistances, normalized by the corresponding values at . One can see a shallow 3D behavior in the normal state.

Figure 6: (Color online) Fluctuation part of the in-plane conductivity , normalized by the normal state resistance, for fields (a) perpendicular and (b,c) parallel to the -plane. Curves in panels (b) and (c) were obtained from the same data using different : (b) T), (c) linear extrapolation from high , shown by the dashed line in Fig. 2 (b). It is seen that fluctuation para-conductivity decays approximately exponentially with increasing with an almost field independent slope. Panels (d) and (e) show data from (a) and (c), respectively, shifted by . (f) Field dependence of para-conductivity at different . An approximately exponential decay is seen.

In Fig. 5 (d) we show absolute values of the angular MR amplitude MR, normalized by the magnetic field, for the in-plane and the -axis resistances. The in-plane MR (circles) is large in the superconducting state and remains significant in the fluctuation region at when the cusp in is observed, see Fig. 5 (a). With increasing , MR rapidly decreases. At K it flattens off. The remaining weakly -dependent value represents the anisotropy of the positive in-plane MR in the normal state, presumably of the orbital origin. The out-of-plane MR (squares) decreases almost exponentially with increasing temperature in a wide -range above . It becomes hardly detectable above the pseudogap opening temperature K, while the in-plane MR still remains recognizable.

A different behavior of in-plane and out-of-plane MR can be also seen from comparison of individual and combined contributions of the two field components. Symbols in Figs. 5 (e) and (f) show angular dependent (e) in-plane and (f) -axis MR at K as a function of . Dashed blue lines represent additive contributions from the two field components, , where , and and are the corresponding MR solely in perpendicular and parallel fields, shown in Figs. 2 (f) and 3 (e). It is seen that the -axis MR is well described by the simple additive contribution of the two field components, while the in-plane does not. This reflects different mechanisms of in-plane and out-of-plane magnetoresistances. The negative -axis MR is due to field suppression of the pseudogap. The applied field is much smaller than the PG closing field T Jacobs2012 (). Therefore, the -axis MR is far from saturation and is approximately linear in field, leading to additive, independent from each other, contribution from the two field components.

The positive in-plane MR at is mostly due to suppression of superconducting fluctuations with the characteristic field T, which is in the range of applied fields. This leads to saturation of MR and to non-additive contribution of the two field components. Unlike the normal state angular MR, which has a 3D character, as shown in Fig. 5 (c), superconducting fluctuations at remain quasi-2D, as seen from the cusp in in Fig. 5 (b). The solid line in Fig. 5 (e) indicates that at not too small angles the in-plane MR is determined by the -axis field component.

iv.2 B. Fluctuation conductivity

Fluctuation para-conductivity is seen as a tail of the in-plane resistive transitions from Figs. 2 (a) and (b) at K, in the same range where the cusp is seen in the angular MR, Fig. 5 (a). Figures 6 (a-c) represent normalized excess conductivities , in perpendicular and parallel magnetic fields. Here we used different approximations for : (a) , (b) , and (c) a linear extrapolation from high , shown by the dashed line in Fig. 2 (b).

It is seen that for both field orientations the fluctuation conductivity at decreases approximately linearly in the semi-logarithmic scale with almost field-independent slopes. This implies


where is some constant. A similar exponential decay has been reported for other cuprates MR (); Alloul2011 (); Luo2014 (). Even though such an exponential decay does not follow explicitly from theoretical analysis of fluctuation conductivity Varlamov (); Finkelstein2012 (), it allows an unambiguous determination of the characteristic temperature scale from the relative shift of the curves along the -axis with respect to the known . Since the curves in Figs. 6 (a-c) remain almost parallel at different , such determination of does not suffer from widening of the resistive transition, as in the flux-flow case at in Fig. 2 (a). Therefore, thus obtained has the same degree of certainty as .

Figure 7: (Color online) (a) The upper critical field perpendicular (filled squares) and parallel to layers (filled circles and rhombuses) obtained from the scaling analysis of fluctuation conductivity according to Eq. (4) at . For comparison we also show middle points of in-plane (small open circles) and out-of-plane (small open squares) resistive transitions in parallel field. The dashed line represents the dependence. Open triangles (right axis) show -dependence of the superconducting gap, obtained by intrinsic tunneling spectroscopy Jacobs2012 (). (b) -dependence of the anisotropy of the upper critical field . It is seen that at low temperature it saturates at a small value . (c) Angular anisotropy of the in-plane resistance at K as a function of magnetic field. The anisotropy rapidly decreases with increasing as soon as the field approaches the paramagnetic limit.

Equation (4) suggests that curves could be collapsed in one by shifting them along the -axis by . In Figs. 6 (d) and (e) we show such an attempt for the data from Figs. 6 (a) and (c), respectively. Even though the scaling is not always perfect, the shift parameter is determined unambiguously because: (i) The shift for the curve at is fixed by . (ii) The curves from low to intermediate fields do collapse at high enough . (iii) When the curves do not collapse, we required that fluctuation conductivity for a given should be decreasing with increasing because superconductivity is suppressed by magnetic field. This means that the curves at higher should always lie lower and should not cross the curves at smaller . In Fig. 6 (d) the curve was not shifted at all, implying that , which is consistent with our previous estimation of T.

Fig. 6 (f) represents a semi-logarithmic plot of vs for the OD(4.0) sample at K and slightly above at K. It is seen that decays almost exponentially also as a function of field at constant . In this case the relative shift along the horizontal axis provides the characteristic magnetic field scale for suppression of superconductivity . Assuming that at , we estimate from the relative shift of the two curves that T. This is consistent with , estimated from scaling in Fig. 6 (d). Thus, from the analysis of fluctuation conductivity we obtain a confident estimation of or equivalently .

iv.3 C. The upper critical field

Figure 7 (a) contains the main result of this work: -dependencies of obtained from the analysis of fluctuation conductivity, Eq.(4), at (filled symbols). Filled blue and red squares represent for OD(4.0) and OP(4.3) crystals, respectively. Horizontal and vertical error bars correspond to the accuracy of scaling of curves according to Eq. (4), as seen in Figs. 6 (d) and (f).

Estimation of at low is complicated by the lack of confident knowledge of . In Fig. 6 (b) and (c) we used two different approximations of . Filled circles and rhombuses represent for the OP(4.3) crystal, obtained from the scaling of data in Fig. 6 (b) and Figs. 6 (c, f), respectively. Up to T both approximations of give the same . Therefore, those values are confident. However, at T results start to depend on the choice of . Unfortunately, none of the two approximations is good enough at . Qualitatively, tends to underestimate because it assumes T. The linear extrapolation of tends to overestimate because it assumes that . However, without the flux-flow phenomenon MR (). This is what we expect for our Lorentz force free data at . In absence of a better way to define at low , in Fig. 7 (a) we also show fields at which middle points of resistive transitions occurs for in-plane (open circles) and -axis (open squares) resistances. Those points fall inbetween the underestimating (solid circles) and overestimating (rhombuses) analysis of fluctuation conductivity. Therefore, they provide a reasonable estimate of at lower .

From Fig. 7 (a) it is seen that and are qualitatively different. The is almost linear in the whole -range . Such a behavior is consistent with a conventional orbital upper critical field,


where is the flux quantum and is the in-plane coherence length, nm.

The is clearly non-linear. The dashed line in Fig. 7 (a) demonstrates that . At the first glance, it resembles the behavior of in thin film multilayers Tachiki (); NbCu1996 (),


where is the thickness of superconducting layers. However, the corresponding nm is much larger than the thickness of CuO layers nm, as noted previously in Ref. Vedeneev2006 (), and is not connected to any geometrical length scale of the sample. Consequently, there is no agreement with Eq. (6).

iv.4 D. The paramagnetic limit

The upper limit of is determined by Pauli paramagnetism. The spin-singlet pairing is destroyed when the Zeeman spin-split energy becomes comparable to the superconducting energy gap . This gives SaintJames (); Clogston ()


where is the gyromagnetic ratio and is the Bohr magneton. In case of negligible spin-orbit coupling this yields  T/K for d-wave superconductors ParamagneticLimit (). Our values T/K and T/K and especially T/K and T/K clearly exceed this limit. Most importantly, does not depend on orientation of the field. Therefore, paramagnetically limited should be approximately isotropic, irrespective of the underlying effective mass anisotropy.

According to Eq.(7), is determined solely by . Open triangles in Fig. 7 (a) show -dependence measured by intrinsic tunneling spectroscopy on a slightly underdoped crystal from the same batch Jacobs2012 (). It matches nicely . Therefore, we conclude that the observed dependence is not originating from the geometrical confinement, Eq.(6), but follows the corresponding dependence of in Eq.(7).

Fig. 7 (b) shows the anisotropy of the upper critical field . Close to it diverges due to different -dependencies of the two fields. However, at it shows a tendency for saturation at . Such a low anisotropy of is remarkable for the layered Bi-2201 compound with Vinnikov ().

In Fig. 7 (c) we show magnetic field dependence of the angular anisotropy obtained from the data in Fig. 4 (a). The anisotropy is large at low fields, but rapidly decreases at T when the paramagnetic limitation starts to play a role. At high fields it tends to saturate at , consistent with in Fig. 7 (b). As mentioned above, paramagnetically limited should be isotropic. Therefore, a finite residual anisotropy indicates that only is paramagnetically limited, while is still governed by orbital effects. Finally we note that was reported for several unconventional superconductors Sekitani2004 (); Vedeneev2006 (); Zuo2000 (); Singleton2000 (). In particular, a nearly isotropic was reported for the (Ba,K)FeAs pnictide Yuan2009 () despite a quasi-2D electronic structure. It is likely that all those observations have the same origin.

V Conclusions

To conclude, we presented a comprehensive analysis of both in-plane and out-of-plane magnetoresistance in a layered cuprate BiSrCuO with a low K. We have shown that the in-plane and the out-of-plane resistances behave differently almost in all respects. The in-plane magnetoresistance has two positive contributions. The positive in-plane MR due to suppression of superconductivity (or superconducting fluctuations) is dominant at and magnetic fields T. It is clearly distinguishable by its 2D cusp-like angular dependence. At the superconducting contribution vanishes and only a weakly -dependent positive MR, presumably of orbital origin, remains. Such normal state in-plane MR has a smooth 3D-type angular dependence. The -axis MR at is dominated by a negative MR caused by suppression of the pseudogap. It decays rapidly upon approaching the PG opening temperature K and at the PG closing field T , and exhibits a smooth 3D-type angular dependence. Different behavior of the in-plane and the out-of-plane MR underlines different origins of superconductivity and the -axis pseudogap, which becomes particularly obvious from analysis of low- cuprates Jacobs2012 ().

The main focus of our work was on analysis of fluctuation conductivity at . We observed a universal, nearly exponential, decay of in-plane para-conductivity as a function of temperature and magnetic field and proposed a method for extraction of based on a new type of a scaling analysis of the fluctuation para-conductivity. This way we obtained confident values of , avoiding the complexity of flux-flow phenomena at . We observed that is following a linear -dependence , typical for limited by orbital effects. On the other hand, follows the -dependence of the superconducting gap with a characteristic dependence close to . Our main result is observation of a remarkably low anisotropy of the upper critical field , which is much smaller than the effective mass anisotropy . This demonstrates that the anisotropy of in unconventional superconductors may have nothing to do with the anisotropy of the electronic structure and the actual anisotropy of superconductivity at zero field. The large discrepancy in anisotropies serves instead as a robust evidence for paramagnetically limited superconductivity.

v.1 Acknowledgements

Technical support from the Core Facility in Nanotechnology at Stockholm University is gratefully acknowledged. We are grateful to A. Rydh and M.V. Kartsovnik for assistance in experiment and useful remarks.


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