Orbital and spin effects for the upper critical field in As deficient disordered Fe pnictide superconductors

We report B_c2 data for LaO_{0.9}F_{0.1}FeAs_{1-delta} in a wide T and field range up to 60 Tesla. The large slope of B_c2 approx ~ -6 Tesla/K near an improved T_c = 28.5 K of the in-plane B_c2(T) contrasts with a flattening starting at 23 K above 30 Tesla we regard as the onset of Pauli-limited behavior (PLB) with B_c2(0) about 65 Tesla. We interpret a similar hitherto unexplained flattening of the B_c2(T) curves reported for at least three other disordered closely related systems as also as a manifestation of PLB. Their Maki parameters have been estimated analyzing their B_c2(T) data within the WHH approach. The pronounced PLB of (Ba,K)Fe_2As_2 single crystals from a tin-flux is attributed also to a significant As deficiency. Consequences of our results are discussed in terms of disorder effects within conventional (CSC) and unconventional superconductivity (USC). USC scenarios with nodes on individual Fermi surface sheets (FSS), can be discarded for our samples. The increase of dB_c2/dT|_{T_c} by sizeable disorder provides evidence for an important intraband (intra-FSS) contribution to the orbital upper critical field. We suggest that it can be ascribed either to an impurity driven transition from s_{+-} USC to CSC of an extended s_{++}-wave state or to a stabilized s_{+-}-state provided As-vacancies cause predominantly strong intraband scattering in the unitary limit. We compare our results with B_c2 data from the literature with no PLB for fields below 60 to 70 Tesla probed so far. A novel disorder related scenario of a complex interplay of SC with two different competing magnetic instabilities is suggested.


Introduction
The recently achieved relatively high superconducting transition temperatures T c up to 57 K and last but not least remarkably high upper critical fields B c2 (0) exceeding often at least 70 T of Fe pnictides followed their discovery in the system LaO 1−x F x FeAs [1,2] has opened the door to a fascinating world of novel superconductors. Naturally, shortly after the discovery of these novel FeAs based superconductors, the underlying pairing mechanism and many basic physical properties both in the superconducting and in the normal state are still not well understood. In this context one can only agree with the statement "There is a serious need to identify and address relatively straightforward questions, in addition to broader investigations to compare and contrast " all the Fe pnictides and related materials " to identify trends that might provide a clue" [3]. Such general questions are: (i) Is this superconductivity (SC) based on Cooper pairs or on bipolarons? (ii) What is the symmetry of its order parameter ? (iii) How is the SC affected by disorder? A study of the upper critical field B c2 (T ) as a fundamental quantity of the SC is expected to provide valuable insight into the nature of the interaction responsible for the formation of Cooper pairs and to help us to answer questions (i)-(iii) in near future. Since the usual el-ph mechanism has been ruled out by a much too weak coupling strength λ ≤ 0.2 [4], a variety of nonstandard mechanisms mostly involving spin fluctuations has been proposed [5,6,7]. This way it might also provide constraints for proposed unconventional scenarios [8] based on repulsive interactions. Concerning the symmetry of the superconducting order parameter we note its robustness or sensitivity to various scattering processes. In this context the so-called sign reversal isotropic [9] s ± interband scenario [5,7,11,12,13,14,15,16] is of special interest. Here a repulsive interband interaction between disconnected nearly nested hole-type (h) and electron-type (el) Fermi surface sheets (FSS) is suggested to be nearly as effective in creating superconductivity as a standard attractive one. This, as predominantly assumed magnetic interaction is thought to be responsible for opposite signs of the superconducting order parameter (gap) on h-and el FSS centred around the Γ-point and the corners of the Brillouin zone, respectively. The resonance peak observed recently below T c ≈ 38 K near a transferred energy of 14 meV and transferred momentum of Q = 1.15Å −1 in recent inelastic neutron scattering measurements on the 122 system Ba 0.6 K 0.4 Fe 2 As 2 [17] has been regarded as evidence for an unconventional s ± -pairing state. However, the strength of that interband coupling compared with the intraband interactions and its related stability against a competing conventional s ++pairing triggered by an enhanced s ± -pairbreaking due to a possibly enlarged interband nonmagnetic impurity scattering and/or a reduced interband coupling due to a smeared nesting remains unclear [18]. More sophisticated studies along these lines are necessary to settle these questions.
In such a confusing situation a combination of several approaches seems to be necessary: (i) a detailed study of selected well-defined systems with a controlled amount of deviations of stoichiometry and other kinds of disorder, (ii) a systematic comparison of various members of the fastly growing FeAs family with the so far discovered three classes of 1111, 122, and 111 systems (see the other contributions of this volume and below) including also ferroselenides, -tellurides, -phosphides and other related layered compounds (iii) a comparison also with other exotic superconductors.
In general, the area of very high magnetic fields is experimentally not well studied due to the large necessary technical efforts and restricted mainly to resistance measurements for pulsed field magnets where the highest fields exceeding 40 T are achieved so far [19]. For that reason fields up to 60 T as reported here are available in few laboratories worldwide, only. For completeness it should be noticed that theoretically a rich variety of unexpected phenomena has been predicted [20] for ultra-high magnetic fields being a challenge for further future studies.
Controlled disorder provides insight into relevant scattering processes and in the symmetry of the pairing since very often an unconventional pairing in the sense of T c and dB c2 /dT | Tc [21,22] is expected to be more or less strongly suppressed by disorder. Adopting for instance the so-called self-consistent Born approximation valid for relatively weak scattering, one has for T c [23,24,25,26]: where ψ(x) is the digamma function and β is the strong-coupling pair-breaking parameter β = Ω 2 p ρ 0 /8π(1 + λ)T c0 which is related to the residual resistivity ρ 0 and the plasma energy Ω p in the (a, b)-plane. However, it should be noted that the T c -suppression in the opposite limit of strong scattering (unitary limit) is less prononounced. In particular, it has been suggested that in this unitary limit for a two-band superconductor in the unconventional s ± -regime, a weaker pair-breaking interband scattering (compared with non-pair-breaking intraband one) will practically drop out [15,16]. Anyhow, the relevance of this approach to the As-vacancy case considered here remains unclear and further theoretical studies of the scattering properties of As vacancies as well as for other impurities such as Co and Ni on Fe-sites are highly desirable. Some, but much weaker, suppression might occur also in the anisotropic or multiband conventional s ++ -wave case since the scattering may smear out the gap anisotropy. However, it will be shown that surprisingly just the opposite, namely, an enhancement of T c happens in our case.
For low applied fields rather different slopes dB c2 /dT ≈ -1.6 T/K up to -2 T/K at T c ≈ 26 K [27,28] and up to -4 T/K at T c ≈ 20 K [29] have been reported for the As stoichiometric La based compounds. Here, we report with dB c2 /dT ≈ -5.4 to -6.6 T/K, to our knowledge one of the highest slopes of B c2 near T c observed so far for the La-series. Another interesting issue of high-field studies considered here is the possibility to observe Pauli-limiting behavior (PLB). Triplet p-wave pairing or strong coupling (B c2 (0) ≥ 60 T) would naturally explain the reported absence of PLB [28]. In this context it is important to note that we succeeded in detecting PLB for our specific sample. It points to B c2 (0)values being much below often used WHH (Werthamer-Helfand-Hohenberg) [30] based estimates. After presenting various data which deviate from those of Ref. [28] as well as from our As-non-deficient quasi-clean samples [31,32,34,33,35], we will discuss our B c2 (T ) data in the light of these more general issues.

Sample preparation and lattice constants
Polycrystalline samples of LaF 0.1 O 0.9 FeAs were prepared from pure components (3N or better) using a two-step solid state reaction method similar to that described by Zhu et al. [36]. In the first step, Fe powder and powdered As particles were milled, mixed and pressed into pellets under Ar atmosphere, and annealed at 500 • C for 2 h and at 700 • C for 10 h in an evacuated silica tube. In the second step, the Fe-As pellets were milled and mixed with La powder, annealed La 2 O 3 powder, and anhydrous LaF 3 powder and subsequently pressed into pellets under a well-defined pressure. Then, the samples were heated in an evacuated silica tube at 940 • C for 2 h and at 1150 • C for  •a-axis data taken from reference [31], ×, × -clean reference sample (x = 0.1), , -As-deficient sample; △, △ -nearly equivalent non-deficient sample with x ≈ 0.05 for comparison. Right: As-positions in the non-deficient ( ) and in the deficient ( ) samples.
48 h. To improve the homogeneity, the 940 • C annealing step was prolonged. Some samples have been wrapped in a Ta foil during the annealing procedure of the second step (see also reference [37]). Ta acts as an As getter at high temperatures forming a solid solution of about 9.5 at.% As in Ta with a small layer of Ta 2 As and TaAs on top of the foil. This leads to an As loss in the pellets. The annealed pellets were ground and polished, and the local composition of the resulting samples was investigated by wavelength-dispersive and energy-dispersive x-ray spectroscopy (WDX and EDX, respectively) in a scanning electron microscope (SEM). The amount of impurity phases does not exceed the x-ray diffraction resolution limit of ∼ 5 %. According to the EDX analysis, an As/Fe ratio of about 1.0 was found in the reference sample annealed without a Ta foil to be compared with 0.90 to 0.95 in the As-deficient sample. A powder-x-ray diffraction study with a Rietveld refinement of the main phase yields enhanced lattice constants of a = 0.4028 nm and c = 0.8724 nm for the As deficient sample compared to a = 0.402 nm and c = 0.8696 nm for the reference sample [37]. In figure 1 (left), the lattice parameters of these two samples are included in the dependence of the lattice parameters on the nominal F content found for non-As-deficient LaF x O 1−x FeAs samples studied at the IFW Dresden [31]. We note that for x > 0.04 the nominal fluorine concentration practically coincides with that determined from the WDX-analysis. A continuous decrease of the lattice parameters with increasing F content is observed consistently with other reports. Whereas the lattice parameters of our reference sample well agree with the data for x = 0.1 shown in figure 1 (left), the lattice constants of the As deficient sample are close to those for underdoped samples near the border of magnetism and superconductivity at x = 0.05 for stoichiometric samples. The reduced charges of the anionic As and that of cationic Fe layers causes less attraction between them and is thereby responsible for the increase of the As-position (see figure 1 (right)). According to an analysis of the reflectivity a similar amount of charge gives rise to an additional optical absorption we ascribed to bound electrons localised in the Fe-plane near the As-vacancies [38]. This explains why our As-deficient sample is not strongly overdoped by electrons as one might expect at first glance and what is the microscopic reason for the strongly enhanced scattering of the quasiparticles bearing the transport and the superconductivity. The same electrostatic argument explains a slight flattening of the LaO 0.9 F 0.1 -bilayer as compared with the more elongated As-Fe 2 -As trilayer and the resulting increase of c.

Measurements and used experimental techniques
The electrical resistance and the Hall effect were measured for plate-like samples using the standard four-point method. These measurements were done in a Physical Property Measurement System (PPMS, Quantum Design) in fields up to 14 T. In addition, resistance measurements were performed in the pulsed-field facilities of the IFW Dresden and the FZD up to 50 T and 60 T, respectively. Gold contacts (100 nm thick) were made by sputtering in order to provide a low contact resistivity and to avoid possible heating effects in the pulsed field measurements. Furthermore, some magnetic properties of our samples were studied by muon spin relaxation (µSR) measurements which were performed at the Paul-Scherrer-Institute, Villigen, at zero field and in transverse applied fields both in the superconducting state at 1.6 K and in the normal state at 40 K.

Resistivity data
In figure 2, the resistivity data for the clean reference and the As-deficient sample are compared. The resistivity of the As-deficient sample in the normal state at 31 K, with about 0.6 µΩcm, exceeds that of the clean reference sample by a factor of about three. Since each As site is surrounded by four Fe sites, the effect of even a few As-vacancies might be drastic. Thus, a substantial shortening of the mean free path due to an Asdeficieny of about 0.1 seems to be quite reasonable. In spite of the resulting disorder in the FeAs layer, the As-deficient sample is found to exhibit, with T c = 29.0 K, a higher transition temperature than the optimally doped reference sample (T c = 27.7 K) and a relatively sharp transition width (see figure 2 (right)) which excludes an anomalous inhomogeneity. Compared with underdoped LaF 0.05 O 0. 95 FeAs samples for which T cvalues of 26.3 K [39] and 20.6 K [35] were reported, the increase of T c due to Asdeficiency is even more pronounced. The unexpected increase of T c in the As-deficient sample might be caused by the suppression of the nesting related AFM fluctuations due to disorder effects and a possible additional non-phononic attractive coupling induced by the localized vacancy related electronic states [38] and/or by the suppression of pair-breaking interband scattering due to enhanced intraband scattering within an Asvacancy stabilized s ± -scenario [16]. The ρ(T )-dependence of the As-deficient sample resembles that of underdoped stoichiometric samples [35] only at high T whereas it becomes similar to that of optimally doped samples at T > T c as shown in figure 3. In particular, the pronounced low-temperature (T < 60 K) upturn of ρ(T ) characterizing underdoped stoichiometric samples is not observed for our As-deficient samples. For Temperature (K) Figure 3. T -dependence of the normalized resistivity for the As-deficient (•), the reference sample (•), and an underdoped sample (•) taken from reference [39].  more details see the data in supplementary part.

Enhanced paramagnetism: Muon spin rotation measurements
The observed PLB of the upper critical field at low T and high external fields reported below in section 4.2 should be caused by enhanced paramagnetism. To confirm this presumption, we performed zero field (ZF) and transverse field (TF) muon spin relaxation measurements (µSR) on our clean reference and the As-deficient samples.
In figure 4 we show ZF-µSR data at 1.6 K. For the clean reference sample a weak Gaussian Kubo-Toyabe-like [40] (KT) decay of the muon spin polarization is observed. This relaxation can be traced back to the tiny magnetic fields originating from nuclear moments. In contrast, for the As-deficient sample an additional exponential relaxation due to electronic magnetic moments is superimposed on the weak nuclear relaxation. Longitudinal field (LF) experiments in the normal state at 40 K clearly prove a static nature of the electronic relaxation (see inset in figure 4). Therefore we conclude that the disorder in the As-deficient sample gives rise to the formation of dilute quasistatic paramagnetic spin clusters. In a high external magnetic field these spin clusters can give rise to additional internal fields which reduce the upper critical field B c2 as we have found experimentally. An enhanced presence of paramagnetic electronic moments in the As-deficient sample is also clearly visible in the field dependence of the TF-µSR relaxation rate depicted in figure 5. As well in the paramagnetic state (at 30 and 40 K, respectively) as in the superconducting state at 1.6 K the relaxation rate for the As-deficient sample is much stronger than for the nominal composition. In particular, the increase of the relaxations rate with increasing field is typical for a paramagnetic system.

Resistance for applied static and pulsed fields
In figure 6 the electrical resistance of the studied As-deficient sample is plotted against the temperature for applied dc fields up to 14 T. With increasing applied field, the onset of superconductivity is found to be shifted to lower temperatures. Additionally, a substantial broadening of the transition curves is observed at high applied fields which mainly stems from the large anisotropy of the upper critical field B c2 which is expected for the layered Fe oxypnictide superconductors [5,41]. We performed measurements in pulsed high magnetic fields up to 60 T in some cases. Resistance data obtained in fields up to 50 T for the As-deficient sample are plotted in figure 7. Gold contacts (100 nm thick) were made by sputtering in order to provide a low contact resistivity and, therefore, to avoid possible heating effects in the high-field measurements which might seriously distort the shape of the transition curves affecting the B c2 -values to be derived. The magnetic field generated by the employed IFW's pulsed field magnet rises within about 10 ms to its maximum value B max (which can be varied up to 50 T) and decreases afterwards to zero within the same time. The resistance data shown in figure 7 were taken for descending field using B max = 47 T. Additionally, resistance data were collected for B max = 29 T at several selected temperature. The agreement between the resistance data for B max = 47 T and 29 T confirms that our data are not affected by sample heating. Again, a pronounced broadening of the transition curves is observed at high magnetic fields which is associated with the large anisotropy γ H = B ab c2 /B c c2 . For polycrystalline samples, only the higher B ab c2 is accessible. Since it is related to those grains oriented with their ab-planes along the applied field, B ab c2 can be determined from the onset of superconductivity. The onset of first dissipation in the resistive transition curves can be roughly associated with grains of the lower B c c2 . However, one has to taken into account that the estimation of the anisotropy γ H = B ab c2 /B c c2 from resistance measurements provides only a lower limit of γ H . Adopting a simple Ginzburg-Landau picture with mass anisotropy, values of γ predicted from the local-density approximation (LDA) were found to vary between 6.2 and 15 [5,41,38] whereas those from magnetic torque measurements on SmFeAsO 0.8 F 0.2 and NdFeAsO 0.8 F 0.2 single crystals have been found between about 7 (at T c ) and 19 (for T → 0) [42].

Different criteria for the determination of B c2
The R(H)-curves in figure 7 reveal a considerable magneto-resistance of the investigated sample at high magnetic fields. In a first approach, the upper critical field B ab c2 was determined as in reference [28] from the onset of superconductivity defining it at 90 % of the resistance R N in the normal state and ignoring the magneto-resistance. This criterion corresponds to the dashed horizontal line in figure 7. The B ab c2 (T )curve of our As-deficient sample obtained for the so defined upper critical field is shown in figure 8 together with the data found for our clean reference sample and for another clean sample reported by Hunte et al. [28]. For our As-deficient sample, good agreement between dc and pulsed field measurements is obtained in the field range up to 14 T. The corresponding B c2 (T )-curve in figure 8 shows a surprisingly steep dB c2 /dT | Tc = −5.4 T/K which exceeds the slopes of B c2 (T ) of the two clean samples by more than a factor of two. This points to strong impurity scattering in the As-deficient sample in accord with its enhanced resistivity at 30 K. For the clean sample [28] the available B c2 -data up to 45 T is well described by the WHH-model [30] assuming that the B c2 (T ) is limited by orbital effects, only (see figure 10 (left)). Whereas for the As-deficient sample the WWH-model fits the experimental data up to about 30 T, only. Using dB c2 /dT = −5.4 T/K and T c = 28.5 K, this model predicts B * c2 (0) = 0.69T c (dB c2 /dT ) Tc = 106 T at T =0. However, for applied fields above 30 T or at temperatures below 23 K, increasing deviations from the WHH-curve are clearly visible both for the B c2 -data from the IFW Dresden and the FZD. The small difference between both data sets is within the error bars of both measurements. The flattening of B c2 (T ) at high fields points to its limitation by the Pauli spin paramagnetism as will be discussed in more detail in the next section. In order to check, whether the observed deviations of the experimental B c2 (T ) data from the WHH prediction are affected by the definition of the upper critical field, within a second approach in defining B c2 , the magnetoresistance in the normal state and the temperature dependence of R N were taken into account. The resistance data vs. temperature plotted in figure 9 (left) for Upper critical field (T) Temperature (K) Figure 8. B ab c2 vs T . Data for the As-deficient sample from DC (• ) and pulsed field measurements ( •-IFW Dresden, •-FZD). • and -data for our and another clean reference samples, respectively. The latter are taken from reference [28]. Solid lines: WHH-model without Pauli-limiting. Dotted line: B c2 (T ) for α = 1.31 without spin-orbit scattering, where α is the Maki parameter -see equation (6).
the As-deficient sample are taken both from dc and pulsed field measurements. Within this approach, B ab c2 was defined at 90% of the resistance R N (T, H) in the normal state. The temperature dependence of R N was approximated by R N (T ) = 7.74 + 1.7 · 10 −2 T 1.  where R N is given in mΩ and T in K. This relation was found to fit the experimental R N (T ) data between T c and 80 K very well. For this modified definition of B c2 , one gets somewhat higher B c2 -values than for the first one as shown in figure 9 (right). The slope dB c2 /dT | Tc becomes, with dB c2 /dT | Tc = −6.6 T/K even steeper resulting in an enhanced field B * c2 (0) = 131 T at T = 0 predicted by the WHH model. More importantly, the resulting difference between the measured B c2 (T ) and the extrapolated B * c2 (T ) at lower T is comparable for both definitions of the upper critical field. Hence, the flattening of the experimental B c2 (T )-curve observed at high magnetic fields is rather similar, regardless of whether the magnetoresistance is taken into account in defining B c2 or not.

Orbital and paramagnetic upper critical field
Standard superconductivity as described by the BCS-or the more sophisticated Eliashberg-theory rests on Cooper-pairs. They consist of two electrons with opposite spins (non-p-wave) and momenta. Hence, there are two magnetic channels to affect a superconducting state: (i) by the Lorentz-force acting via the charge and the opposite momenta (the phases) on the paired electrons (usually called diamagnetic or orbital effect) and (ii) the spin channel (called also paramagnetic effect) where a singlet pair is transfered in a practically unbound triplet, i.e. it is broken by the Zeeman effect. Orbital and spin pair breaking (of a singlet Cooper pair) in the presence of a magnetic field are illustrated in figure 10 (left). Usually, at high temperature below T c , the suppression in the orbital channel is more effective. Then the paramagnetic effects may be visible at low-temperature and high fields, only. Thus, at sufficiently large magnetic field, the superconductivity is destroyed by orbital and spin pair breaking. (Contrarily, in the hypothetical case of bipolarons B c2 (0) is essentially unlimited or extremely large [43].) According to the WHH approach the orbital limited upper critical field B * c2 (0) is related both to T c and to the slope of B c2 (T ) at T c . The relation between the orbital and spin paramagnetically limited upper critical fields is illustrated in figure 10 (right), where the Gibbs free energy is plotted against the applied magnetic field. By applying a magnetic field, the free energy of the superconductor in the superconducting state, F s , increases, whereas its free energy in the normal state, F N , is lowered by an amount of 0.5χ sp B 2 /µ 0 with χ sp being the Pauli spin susceptibility. The field at which F N (B) becomes equal to the condensation energy of the superconductor defines the Pauli limiting field B p which is for weakly coupled superconductors given by  at T = 0, as was pointed out by Clogston [44] and Chandrasekhar [45], where 2∆ 0 denotes (the isotropic) gap in the adopted there s-wave picture. Including strong coupling corrections to BCS due to el-boson and el-el interactions, one gets where η ∆ describes the strong coupling intraband correction for the gap, η ef f (λ) = (1 + λ) ε η ib (1 − I) with I as the Stoner factor I = N(0)J [46,47], N(0) is the electronic density of states (DOS) per spin at the Fermi level E F , J is an effective exchange integral, and η ib has been introduced to describe phenomenologically the effect of gap anisotropy, multi-band character, energy dependence of states etc., which are possibly present also in FeAs based superconductors (see for instance the references [48,49,50]). It is assumed to be strong enough to compensate the effect of the unknown Stoner factor. The so called strong coupling (i.e. finite el-boson coupling) correction factor η ef f (λ) scales with (1 + λ) ε , where the exponent ε amounts either 0.5 or 1 according to references [46,47], respectively. The paramagnetically limited upper critical field, B p c2 , corresponds to that field at which F N (B) and F s (B) are equal. B p c2 is always lower than both B * c2 and B p as illustrated in figure 10 (right). According to Maki [51], the paramagnetically limited field B p c2 reads where the Maki parameter, α, is given by The Maki parameter α provides a convenient measure for the relative strength of orbital and spin pair-breaking. Within the WHH approach the shape of B c2 (T ) depends sensitively on the magnitude of α, namely, with increase of α an increasing flattening of the B c2 (T )-curve is predicted. Introducing a second auxilary parameter λ s0 ascribed to spin-orbit scattering the strong effect of α could be partially reduced. Anyhow, since in our As-deficient sample the effect of spin-orbit scattering on B c2 (T ) is expected to be rather weak, only, it has been ignored in our analysis [52].

5.2.
As-deficient LaO 0.9 F 0.1 FeAs 1−δ In figure 8, the B c2 (T ) data of our As-deficient samples are analyzed within the WHH model. A satisfying fit of the experimental data to this model was obtained for α =1.31. Using B * c2 (0) =106 T (and 131 T in the second approach to determine B c2 in which the magneto-resistance was taken into account) one obtains B p c2 (0) = 63 T (and 68 T within the second approach) for the upper critical field at T = 0 and B p (0) =114 T (and 141 T within the second approach) for the Pauli limiting field from equations (5) and (6), respectively. We used λ >0.6 [34,38] (estimated for the clean sample with a bulk T c of about 26 K [31]) for a representative value of the el-boson coupling constant for LaO 0.9 F 0.1 FeAs 1−δ and T c =28.5 K and estimated a rather large value of η ∆ η ef f =2.09 from equation (4). The dotted B p c2 (T ) line plotted in figure 8 is based on equation (5) and was obtained by replacing B * c2 (0) entering both its numerator and α by B * c2 (T ) of the WHH model for α = 0. This rough approximation of B p c2 (T ) has been used to illustrate the T -dependence of the upper critical field due to PLB in the studied As-deficient samples.

Comparison with other samples: slope of B c2 (T ) near T c , disorder, and paramagnetism
Superconductivity in ROFeAs (R= La, Pr, Sm, Nd, Gd) can be induced by carrier doping due to the suppression of the magnetic order and the structural phase transition observed for the parent compounds. This can be done by substituting F for O (in the case of R= La, Pr, Sm, Nd, Gd), by partial removing of O (for R= Sm, ) or, in the case of GdOFeAs and LaOFeAs, by substituting Th for Gd and Sr for La, respectively. High values of T c = 55 K have been reported both for SmO 0.9 F 0.1 FeAs and SmO 0.85 FeAs. A slightly higher T c of 56 K was reported for Gd 1−x Th x OFeAs. Superconductivity can be obtained also by direct carrier doping into the conducting FeAs planes which are essentially for superconductivity substituting Co [53,54,55,56,57,58,59] (or Ni [60,61,62] or Zn [63]) for Fe, as was demonstrated for LaOFeAs, CaFFeAs, BaFe 2 As 2 , and EuFe 2 As 2 . Replacing Fe with Co (or Ni) is expected not only to carrier doping, but also to introducing disorder in the FeAs layer. It is remarkable that these superconducting compounds can tolerate considerable disorder in the FeAs layers.
In compounds with enhanced disorder in the FeAs layers, a strong increase of B c2 (T ) and its slope near T c was found. We suggest that this results from a reduced mean free path and an enhanced intraband scattering like in usual dirty s-wave superconductors such as NbTi [64]. For instance, a relatively large slope of dB c2 /dT = −4.9 T/K near T c has been reported for the Co-doped Ba(Fe 0.9 Co 0.1 ) 2 As 2 system [53], which is only slightly below the value of -5.4 T/K for our As-deficient La-1111 sample. Disorder due to As vacancies seems to be responsible also for the large slope of dB c2 /dT = −6.3 T/K near T c reported for (Ba 0.55 K 0.45 )Fe 2 As 2 [65] as will be discussed below in more detail. The above mentioned slopes of B c2 (T ) near T c for these somehow disorded systems are included in table 1. For completeness we note that the rare earth-1111 systems exhibit large slopes dB c2 /dT < ∼ −9 T/K [66,67]. In some cases also an onset of a flattening like for our sample has been observed [66] (see figure 11 right and the discussion below). In our opinion this should be caused due to paramagnetic effects of unkown microscopic origin. A remarkable relatively large isotropic slope of -5.96 T/K for both directions, to the basal plane and ⊥ it at a low T c of 12.4 K has been reported also for a Fe 1.03 Te single crystal [68] caused by excess Fe ions (i.e. excess Fe impurities at minor Fe(2) positions). The weak anisotropy of about 1.6 is also noteworthy. Another puzzling observation to be understood is the fact that many of the known FeAs-based SCs show almost no Pauli-limiting behaviour up to 70 T as examined at present including even systems with relatively low T c -values. For example, the B c2 (T ) data reported for LaO 0.93 F 0.07 FeAs [69] only slightly deviate from the WHH curve as shown in figure 11 (left). In contrast, few more or less strongly disordered systems exhibit clear deviations from the WHH curves, qualitatively similar to our findings reported above, see figure 11 (right).
For the As-stoichiometric reference family LaO 1−x F x FeAs it has been reported that the Pauli paramagnetic susceptibility within this series is affected by the F doping Temperature (K) Figure 11. B c2 vs T -data of an As-deficient LaO 0.9 F 0.  showing a maximum around x = 0.05 [70]. We found for our As-deficient samples also indications for a strongly enhanced Pauli paramagnetism from µSR experiments as was discussed above. This explains the flattening of B c2 (T ) observed for this sample at applied fields above 30 T. We analysed the B c2 (T ) data shown in figure 11 in order to determine the different strength of the paramagnetic pair-breaking in these samples. For LaO 0.93 F 0.07 FeAs (see figure 11 (left)), a small value of the Maki-parameter of α =0.25 is derived. A sizable paramagnetic pair-breaking effect is expected for larger values of the Maki-parameter, i.e. according to equation (6) for large orbital B * c2 (0) and/or low Pauli-limiting fields B p (0). Indeed, the value of the As-deficient sample is, with α =1.31, by more than a factor of 5 larger than estimated for the LaO 0.93 F 0.07 FeAs sample. This is mainly due to the low Pauli limiting field estimated for the As-deficient sample which is, with B p (0) =114 T almost three times smaller than B p (0) estimated for LaO 0.93 F 0.07 FeAs (see table 1). The low value of B p (0) of the As-deficient sample can be explained by its enhanced Pauli spin susceptibility. Additionally, its large orbital B * c2 (0) contributes to the observed Pauli-limiting behaviour of this sample which is enhanced by the large values of both T c and dB c2 /dT at T c . The paramagnetic pair-breaking of selected compounds is compared in figure 12 where the normalized upper critical field h * = B c2 / [T c (dB c2 /dT ) Tc ] is plotted against the reduced temperature t = T /T c . Besides our data for an As deficient La-1111 sample data from other systems obtained in references [53,65,66] have been included. The Maki parameter was found to increase from α =0.25 for LaO 0.93 F 0.07 FeAs [69] over α = 0.9 for Ba(Fe,Co) 2 As 2 [53] and α = 1.31 for our As-deficient La-1111 sample up to about α = 3.5 for NdO 0.7 F 0.3 FeAs [66]. Table 1. Upper critical field data of selected Fe pnictides. B * c2 (0) and B p c2 (0) denote the orbital and the paramagnetically limited B c2 at T = 0, respectively; α is the fitted Maki parameter; B p (0) denotes the Pauli limiting field.

Compound
T Except for the last sample, the experimental data can be well described by the WHH model using the approximation mentioned above. As expected, the deviation of h * (t) at low temperatures from h * (t) for α = 0 increases with α due to rising paramagnetic pair-breaking. The suppression of h * (t) at low temperature is mostly pronounced for NdO 0.7 F 0.3 FeAs. We estimated for this compound an upper critical field of B p c2 (0) ∼80 T at T = 0 (see table 1). The importance of paramagnetic effects on the B c2 (T ) data of NdO 0.7 F 0.3 FeAs was also pointed out by Jaroszynski et al. [66]. They analysed their B c2 (T )-data within a two-band model for dirty-limit superconductors [71] and estimated B p c2 (0) ∼ 130 T. Within this approach, the kink in B c2 (T ) at about 30 T which is clearly visible for H a [66] could not perfectly described [72]. However, we consider this kink as indication for the influence of paramagnetic effects on B c2 (T ).
According to figure 12, the paramagnetic pair-breaking in Ba 0.55 K 0.45 Fe 2 As 2 is stronger than in the, at first glance, more disordered Ba(Fe,Co) 2 As 2 sample and comparable with that in our As-deficient La-1111 sample. This apparent discrepancy can be resolved by taking into account the real stoichiometry of the investigated Ba 0.55 K 0.45 Fe 2 As 2 single crystals which has been analysed by wavelength dispersive x-ray spectroscopy (WDX) [73]. In the single crystals which were grown by the flux method using a tin flux, about 5% As vacancies and ∼ 0.66% Sn were found, the latter being most likely incorporated on As sites [73]. Most probably, the As vacancies are created during the preparation process of the single crystal in the tin flux due to the stronger solubility of As in Sn than of the other parts of the compound therein. Thus, the strong paramagnetic pair-breaking in the Ba 0.55 K 0.45 Fe 2 As 2 single crystal seems to be attributed to the pronounced As deficiency of these samples which is comparable with that in our As-deficient La-1111 sample.

Aspects of anisotropy and multiband superconductivity
In general it should be noticed that on one hand a sizable anisotropy both for the upper critical fields B c2 as well as of the penetration depths is not surprising in view of the layered structure of the Fe pnictide superconductors. On the other hand the different temperature dependence of the anisotropy for the penetration depth and the upper critical field has been claimed to be a great puzzle [74,75]. Albeit we are still not able to explain all strange anisotropies, at least several most striking observations regarding the upper critical field with an almost vanishing,"confluence"-like, anisotropy at lowtemperature such as in SrFe 2−x Co x As 2 [76] H (a,b). Noteworthy, a similar effect with γ B ≈ 1.9,only, has been reported for the one-layer cuprate superconductor Bi 2 Sr 2 CuO 6+δ by Vedeneev et al. [78]. We note that the reported deviation between their observed upper critical field B ab c2 =52 T and the theoretical one can be removed introducing a moderate strong coupling correction 1 + λ, with λ = 0.625, for B p as considered above.
The striking "confluence"-like behaviour deserves special attention. In a simple paramagnetic picture like that used here a more or less sharp turn to a common flattening of B c2 (T ) for lower T would be expected. If in contrary a significant further common increase of all B c2 -components at lower T and higher fields will be detected in future measurements, a redistribution of electrons between the charge carrier subsystem, the electronic "glue" of the pairing interaction, and the limiting paramagnetic subsystem should be envisaged. Alternatively, when the superconducting gap becomes comparable with the energy of a coupled bosonic mode, or for a strongly anharmonic lattice system the electron-lattice interaction and/or polaronic effects might strengthen resulting in a stronger coupled superconducting state.
The different anisotropy ratio of the penetration depths is probably a multiband effect related to "heavy" holes being mainly responsible for B c2 ∝ v −2 F and "fast" electrons which dominate the penetration depths. In fact, adopting for a crude estimate the averaged Fermi velocities v F for the hole and electron bands as calculated by Singh and Du [41] for LaOFeAS one has v h F,ab = 0.81 · 10 5 m/s and v h F,c = 0.34 · 10 5 m/s for the hole bands as well as v el F,ab = 2.39 · 10 5 m/s and v el F,c = 0.35 · 10 5 m/s for the electron bands. Ignoring for the sake of simplicity the interband interaction (possibly important for the high-T c value), we estimate the anisotropy ratio of the upper critical fields at very low temperature in the clean limit γ B (0) ∼ v h F,ab /v h F,c = 2.38 in accord with about 2 estimated from torque measurements (extrapolated to T = 0 from γ B (34 K)= 5.21 for NdO 0.7 F 0.3 FeAs) in figure 4 of reference [42]. Since this estimate is based on the orbital B c2 ignoring the possible PLB, it should be regarded as an upper bound. For the analogous penetration depth quantity one estimates γ λ L = v el F,ab 1 + δ el c / v el F,c (1 + δ el ab ) = 6.82 1 + δ el c / 1 + δ el ab , where δ i ∼ ξ 0,i /l i is a parameter which measures the effect of disorder on the anisotropic penetration depth λ L,i (0), ξ 0,i is the anisotropic coherence length and l i denotes the corresponding free mean path [79]. In order to reproduce the large experimental value of 19 [42], significantly anisotropic scattering rates δ z ≫ δ ab must be naturally assumed. Noteworthy, a similar anisotropy and moderate dirtyness has been observed for NbSe 2 according to its analysis given by Bulayevski [80]. There the in-plane mean free path exceeds the inter-plane one by a factor of two to four. A more detailed analysis is hampered by the lacking information of the experimental partial transverse plasma frequencies. If the introduced disorder affects the interband scattering, a slightly reduced T c would be expected. However, the suppressing of remnants of (fluctuating) SDW antiferromagnetism and a possible additional pairing attributed to the polarization of charges localized near the As vacancies [38] might even overcompensate the former effect. Even if the disorder will not seriously affect the interband scattering due to different symmetry of the states which form the electron and the hole FSS as suggested in [15,16], there must be a relevant intraband scattering mechanism which is responsible for the strong increase of B c2 (T ) at high T near T c and relatively low external fields (below 30 T).
At least for highly disordered samples with strong enough interband scattering an unconventional scenario is very unlikely. The limiting Pauli-field B p can be estimated in a two-band situation by which generalizes equation (4). Here and before the occurence of a first order transition related to a FFLO-type state (Fulde-Ferrel-Larkin-Ovchinnikov) at low T has been ignored for the sake of simplicity and the fact that such a situation seems to be still beyond our available high-field range. Its influence has been considered in the weak coupling limit for the case of a two-band superconductor with very weak interband coupling by Dias [81]. Finally, we note that similarly as in reference [82] devoted to the study of highly anisotropic intercalated transition metal dichalcogenide layered compounds, the formation of some microshorts between the superconducting layers cannot be ruled out. Such microshorts might contribute to the interlayer coupling and to a somewhat reduction of field penetration in the interlayer spacing for parallel external fields. This way the anisotropy ratio γ B would be also reduced. In the context of weakly coupled superconducting layers the observation of a crossover from 3D to 2D-fluctuations with increasing x from underdoped to optimally doped cases and a huge slope of B ab c2 (T ) near T c up to -11 to -12 T/K for SmO 1−x F x FeAs [83] is noteworthy. Whereas 3D-fluctuations above T c have been reported in reference [84] for Ba 1−x K x Fe 2 As 2 single crystals.

Possible origin for the Pauli-limiting behaviour
According to the classical wisdom the Pauli susceptibility χ sp is the central physical quantity being responsible for the pair-breaking of singlet Cooper-pairs [44,45,46,47,80] (see also figure 10 (right) for a schematical view). Its enhancement causes a lowering of the Pauli limiting field B p . In fact, in our As-deficient sample a significantly enhanced induced magnetic moment has been observed in the normal state above T c applying weak external magnetic fields [85]. Reducing the electron doping a similar increase of χ(T ) by a factor of two relatively to optimally doped LaO 0.9 F 0.1 FeAs with a maximum just at the boundary between the AFM commensurate SDW and the paramagnetic state has been observed at x = 0.05 by Nomura et al. [70]. Its suppression for smaller x is a natural consequence of the opening of the SDW-gap and the corresponding lost of density of states (DOS) at the Fermi energy N(E F ) ∝ χ sp (T = 0). Suppressing the nesting induced competing antiferromagnetism and its fluctuations by the As vacancies, the ferromagnetic fluctuations measured directly by χ sp (being the response function for ferromagnetic ordering) may further increase beyond the value achieved in As stoichiometric samples at x = 0.05. In this context it is noteworthy that the observation of ferromagnetic spin fluctuations has been reported by Kohama al. [86]. This might explain the missing Pauli-limiting behaviour in stoichiometric but underdoped samples. Microscopically, the disorder caused by As-vacancies will affect the total Fe 3d exchange integrals due to a modulation of the superexchange admixture involving the As-4p orbitals. The theoretical difficulty of such a scenario consists in its essential physics beyond the mean-field type behaviour favoured in the present case by the quasi-2D nature of the antiferromagnetism. The latter manifests itself in the increasing χ(T ) with an expected maximum somewhere above 300 K for the clean samples [87] in contrast with an decreasing χ(T ) at the mentioned above high-level for our sample at least for 375 K ≥ T > T c [85]. Further theoretical and experimental work is necessary to make this qualitative scenario more quantitative to be checked in detail. In this context the observation of very strong FM spin fluctuations and possibly also of an unconventional spin triplet p or f wave pairing as in superfluid 3 He or in Sr 2 RuO 4 for the closely related LaOFeP system by Kohama et al. [88] is noteworthy. An unconventional pairing symmetry different from that in other pnictide superconductors has been also proposed by Fletcher et al. [89] based on a linear T -dependence of the penetration depth down to 100 mK. The close vicinity of this remarkable system to a multiple critical point is also illustrated by the absence of any ordering, neither superconducting nor FM or AFM one but again by the observation of some FM spin fluctuations (being, however, relatively weak, if a free electron Landau diamagnetic contribution to the total spin susceptibility is adopted) according to reference [90]. Here the missing p-wave superconductivity might attributed to relatively strong disorder reflected by the residual resistivity exceeding by an order of magnitude the values reported for superconducting samples with T c ≈ 6 K [91,92]. Without doubt, inspite of its low T c , LaOFeP is one of the most challenging pnictide superconductors due to the vicinity of at least four competing phases. The general necessity to improve the LDA-calculations for such systems with respect to fluctuations has been pointed out for instance in recent references [93,94,95].
Thus, LaOFeP seems to show no AFM ordering due to a less pronounced nesting and/or a stronger influence of competing FM fluctuations compared with LaOFeAs. In our case of As-deficient LaFe oxyarsenide a corresponding p or f pairing can be excluded due to our observation of a PLB. Microscopically, it is prevented by the strong disorder resulting from the As vacancies.
Alternatively, the suppressed upper critical field at low T and high external fields might be attributed to some enhanced local exchange field acting on the superconducting charge carriers. Such a local field might be caused by an increasing alignment of localized extrinsic magnetic moments. In the present context localized magnetic moments might occur on a microscopic level just in the vicinity of As vacancies as shown by Lee et al. [3] for the analogous case of Se-vacancies for the closely related system FeSe 0.85 , albeit a Se-vacancy represents probably a weaker perturbation than the As counterpart due its smaller ionic charge of -2 compared with -3 for the latter. Remarkably, the DOS at the Fermi-level is enhanced introducing vacancies according to reference [3]. Hence, even on a one-particle level the spin susceptibility χ(0) will be enhanced. A similar effect resulting in an enhanced spin susceptibility χ sp (0) and an onset of paramagnetic effects already above about 2 T, we would suggest also for LiFeAs [96] in order to explain its unusual negative curvature of B c2 near T c and the extraordinary large slope of B c2 ∼ −8 T/K at a relatively low T c -value of 18 K, only. Returning to our As-deficient sample we note, that localized magnetic moments might occur also by a possible electronic phase separation. The formation of a nonsuperconducting paramagnetic phase up to a volume fraction as large as 49 at.% for Ba 1−x K x Fe 2 As 2 and only about 25 at.% for the superconducting phase has been reported by Park et al. [97] to be compared with a significantly larger > ∼ 40 % fraction of the superconducting phase in our case. The simplest possibility for an enhanced local field would be given by a contribution from a present strongly paramagnetic, AFM or FM secondary phases coexisting with the superconducting main phase. For instance the AFM compound Fe 2 As or others might be in high fields converted into a highly polarized magnetic state. Irrespective of its relation to the suppressed upper critical fields, the last two 'extrinsic' cases might be explain the observation of static magnetic moments reported above in 3.2 devoted to muon spin relaxation spectroscopy.
Again, further work is necessary to distinguish between these three possibilities. Independently on the true microscopic origin we are confronted with a very unusual situation for competing superconductivity and various magnetic instabilities or effects unknown to best of our knowledge for other competing families such as the magnetic borocarbides [98] and ternary compounds [99]. Namely, an enhancend superconductivity and upper critical fields (enlarged slopes of B c2 near T c see also references [100,101]) at high temperature und relative low fields below 30 T are opposed by a weakened superconductivity at high fields above 30 T and low temperature below 23 K.

Conclusions and Outlook
To summarize, we reported a high-field study of LaO 0.9 F 0.1 FeAs 1−δ samples with improved superconductivity near T c . At lower T and very high fields, however, a flattening of the B c2 (T )-curve points to Pauli limiting with B c2 (0) ≈ 63 T to 68 T extrapolated. A similar behaviour can be deduced also from other disordered systems. In particular, we interpret the flattening of the B c2 (T )-curve reported for Ba 0.55 K 0.45 Fe 2 As 2 single crystals in reference [65] for H a, b as a strong indication for Pauli limiting.
Disorder in these single crystals is due to about 5 at.% As vacancies appearing during their preparation in a Sn flux probably due to the stronger solubility of As in Sn than for the other parts of the compound. In view of the achieved improved dB c2 /dT near T c and the enhanced T c -value, the introduction of As vacancies or of other appropriate defects opens new routes for optimising their properties. We are confronted with a rather unusual situation not observed so far to the best of our knowledge: Improved superconductivity at high temperature and low fields and somewhat suppressed superconductivity at high-fields and low temperature. The first observation and the enhancement of T c can be understood within conventional s-wave superconductivity by enhanced disorder and by a disorder-induced suppression of nested related remnant antiferromagnetism, respectively. The relative weakening of the highfield properties strongly suggests an enhanced paramagnetism as the preceding state to a ferromagnetism phase. In other words, in disordered pnictides superconductivity seems to compete with at least two kinds of magnetism. Further investigations devoted to a more detailed study of this interplay, for instance between the actual As vacancy concentration and a variation of the electron doping by changing the F and/or the La content or introducing also oxygen vacancies are of considerable basic and technological interest. The elucidation of the microscopic reason of the observed anomalous high-field properties will be obviously helpful for the understanding of the still unknown pairing mechanism, too. In particular, the striking "confluence"-like behaviour of the anisotropy near 45 to 60 T in various 122-systems, makes a more detailed investigation especially at still higher fields strongly desirable. On the basis of our results for B c2 (T ) for relatively low magnetic fields and high temperatures, and the mentioned above increasing hints for s ± symmetry realized in the clean and quasi-clean limits, two alternative scenarios of opposite disorder influence might be suggested: scenario (i): an impurity driven change of the pairing state from unconventional s ± to conventional s ++ superconductivity and scenario (ii): a special impurity driven stabilization of the unconventional s ± state due to the suppression of remnant pair-breaking interband scattering. In scenario (i) the reduction of the clean multiband T c due to the smearing out of the multigap/anisotropic gap structure and the unfavorable negative chemical pressure (see figure  1) must be compensated by a strengthened pairing interaction and/or the suppression of competing remnant magnetic effects. Whereas in scenario (ii) the special As-vacancies are assumed to scatter predominatly within the individual FSS only and the efficiency of an unfavorable interband scattering present already in the non-As-deficient samples is reduced. For this reason T c is increased provided the scattering occurs close to the unitary limit in contrast to the weaker scattering described approximately by the selfconsistent Born approximation in case (i). Due to the present poor understanding of the scattering properties of As-vacancies, we are unable to decide which scenario is realized for our system and further experimental studies are necessary. Key experiments would be the observation/nonobservation of the spin resonance mode in inelastic neutron scattering or other signatures of the s ± state. In general, a detailed theoretical and experimental study of the scattering properties for various relevant impurities is very important for properly understanding the superconductivity in FeAs based compounds.
The Pauli limiting found her suggests that measurements should be continued at least up to 70 T-100 T in order to eludicate, whether there is still much room for increasing B c2 beyond that range. Apparently, the solution of this problem will affect the evaluation of future high-field applications of Fe based arsenides and related systems in dependence of the requested external field regime.
Note added. In preparing a revised version of the present manuscript to be compared with [102], we have learned about an upper critical field study by Terashima et al. [103] of the low-T c overdoped superconductor KFe 2 As 2 which also nicely fits our PLB scenario. For this reason we have re-analyzed their data in the same manner as for the other five systems given in table 1 adopting λ s0 = 0 and included them in figure 12, too [104]. This less complex system with T c ≈ 2.7 K is of special interest because there the multiband corrections are almost negligible [105] and the large initial slope is related mainly to the intraband scattering on the dominant hole Fermi surface sheets. The observed relatively large slope is in accord with our picture of strong intraband scattering on hole and electron FSS in the general case. Since this single KFe 2 As 2 crystal has been grown also from a tin-flux we ascribe the scattering centers also to As-vacencies.
The case of strong interband scattering and the corresponding s ++ -state has been considered by Kulić et al. [106].

SUPPLEMENTARY MATERIAL
In this supplementary part we collect some transport data which might be helpful for comparison and characterization with other As-deficient samples to be possibly prepared and investigated by other authors in the future.

S1 Hall data
The field dependency of the Hall resistivity ρ xy of the As-deficient sample shown in figure 13 (left) at T c ≤ T ≤ 110 K reveals a linear dependence of ρ xy on the applied magnetic field H and a slight temperature dependence of the slope ρ xy /H. The resulting Hall coefficient R H = ρ xy /µ 0 H deviates from that known for optimal doped samples, as shown in figure 13 (right).
From the Hall coefficient, the carrier density n H has been estimated within the single band approach both for optimally and underdoped LaF x O 1−x FeAs [S1,26,35,38] using the relation n H = (e | R H |) −1 . Within this approach a charge carrier density of n H = 0.52 · 10 21 cm −3 is estimated at T = 30 K for the As-deficient sample, which is comparable to n H = 0.55 · 10 21 cm −3 reported for underdoped LaF x O 1−x FeAs with x = 0.05 [39], but only half as large as n H ≈ 1.0 · 10 21 cm −3 [S1, 26,35] reported for optimally doped LaF x O 1−x FeAs with x = 0.1. The reduced carrier density of the As-deficient sample points to a reduced doping level of x = 0.05 of this sample instead of x = 0.1 expected from its F content. This is consistent with the observed enhancement of the lattice parameters of this sample which are close to those reported for x = 0.5.

S2 Scaling analyzis of the resistivity
In order to get more insight in relevant scattering mechanisms we performed a scaling analysis of the resistivity ρ(T ) data from T c up to 300 K. Such an analysis is also helpful to classify our sample with respect to other underdoped and overdoped samples of the  Figure 13. Field dependence of the Hall resistivity for the As-deficient sample (left). Temperature dependence of the Hall coefficient (right) for the As-deficient sample (•) and a clean sample ( )as taken from reference [39]. Table 2. Scaling analysis of the resistivity ρ(T ) according to equation (S2,1), where ρ 0 denotes the residual resistivity and ∆ is a pseudo-gap-like quantity.