Observation of anomalous admixture of superconducting and magnetic fractions in BaFe2−xCoxAs2 single crystals

Increasing doping concentration (x) in pnictide superconductors leads to a loss of long range magnetic order and the emergence of superconductivity. In these doped compounds the role of magnetic fluctuations and their effect on superconductivity is not well understood. Below Tc, in optimally doped BaFe2−xCoxAs2 single crystals at low applied magnetic fields (H), we observe a coexistence of regions with both positive and negative (diamagnetic) magnetization response. With increasing H the positive magnetization response weakens and the diamagnetic signal associated with superconductivity enhances. Above Tc, resistivity and magnetization measurements in the normal state of the material reveals the presence of a weak diamagnetic response admixed with the magnetic response of a magnetized state. Magneto-optical imaging studies in the optimally doped sample shows a spatially inhomogeneous local magnetic response in the normal state with local regions having diamagnetic like response existing alongside magnetized regions with short range magnetic order. Above Tc, the estimated volume fraction with diamagnetic fluctuations is found to be maximum in the optimally doped sample. We present a doping dependent H–T diagram identifying different fluctuation regimes. We propose our results suggest the admixture of magnetic and superconducting fractions in BaFe2−xCoxAs2 and a close correlation between them.


Introduction
The doping phase diagram in Pnictide superconductors has magneto-structural transformation boundaries interrupted by a superconducting dome [1][2][3][4][5][6][7][8]. Unraveling the intricate relationship between superconductivity, magnetism and structural transformation in pnictides is a topic of current interest. In general it has been found that with increasing doping, long range magnetic order gets suppressed and superconductivity emerges in these compounds. The bulk superconducting transition temperature (T c ) is maximum at optimal doping concentration (x). MuSR studies on Ba(Fe 1−x Co x ) 2 As 2 have shown that while Co doping results in loss of long range magnetic order, short range order may still be present in the system [9][10][11][12]. Iron isotope replacement studies in doped 1111 and 122 pnictides show that not only the bulk superconducting transition temperature (T c ) but the magnetic ordering temperature (T N ) also gets affected, suggesting a close interplay between magnetism and superconductivity [13] in these compounds. Studies on underdoped pnictides show the coexistence of superconductivity (SC) along with magnetic fluctuations [14][15][16][17][18][19][20] below T c . Theories have also suggested that magnetism plays an important role in mediating strong superconducting pairing correlations in pnictides [21][22][23][24][25][26][27][28][29]. It is plausible that magnetism of some form is important for sustaining superconducting correlations in these doped compounds. ARPES studies in a BaFe 2 (As 1−x P x ) 2 suggest the presence of local superconducting correlations present above T c [30]. Recent NMR studies suggest evidence for a pseudogap like state in Ca(Fe 1−x Co x ) 2 As 2 compound [31]. However STM study in Ba(Fe 1−x Co x ) 2 As 2 did not find evidence of superconducting gap like features above T c [32]. Another study in FeSe crystals suggested the presence of preformed pairs above T c [33]. From the above it also appears that evidence for the presence of superconducting fluctuations above T c in pnictides is not a well resolved issue. In this article we present our results on the behavior of local and bulk magnetization response measured in, under, optimally and over, doped, superconducting BaFe 2−x Co x As 2 single crystals. At low applied magnetic field (H), for T<T c , local magnetization measurements show the presence of weak positive magnetization response in the optimally doped sample. The local positive magnetic response transforms into a diamagnetic response with increasing H. Just above T c , in all the samples the bulk magnetization response becomes positive and exhibits a maxima, with a long tail extending upto high T. Analysis of magnetization and electrical conductivity in the normal state (above T c ) reveals the presence of a weak diamagnetic response buried within the response of a magnetic background which has short range magnetic order. Sensitive magneto-optical imaging studies performed at T>T c reveals a spatially inhomogeneous magnetic response present in the normal state of the sample. The inhomogeneous response is associated with local regions exhibiting weak diamagnetic response existing alongside regions with magnetic fluctuations which exhibit positive magnetic response. We observe that the sample with optimum doping concentration exhibits the strongest effect of magnetic fluctuations and the same sample also has the highest volume fraction with diamagnetic response above T c . Using the data gathered for each doping concentration, we construct the H-T diagrams in which different fluctuation regimes are identified. We believe our results suggest a coexistence of both magnetic and superconducting fluctuations above and below T c and a close correlation exists between them in mediating superconductivity in the BaFe 2−x Co x As 2 system.

Experimental details
The single crystals of BaFe 2−x Co x As 2 with x=0.10, 0.14, and 0.20 were prepared using self-flux method [34]. The bulk T c was identified from the onset of diamagnetism in a T dependent magnetization M (T) measurement at low H and also from the sharp drop in zero field resistance, R(T), data (T c 's are indicated by arrows in figures 1(a)-(c)). The T c 's for the three samples with different doping concentrations are tabulated in table 1. The T c 's determined from M(T) and R(T) differ slightly as they were measured in different cryogenic systems. Based on the variation of T c with Co doping concentration (x) (see table 1), we label the sample with x=0.14 as optimally doped (with the highest T c (0)=26.80 K), x=0.10 as underdoped and x=0.20 as overdoped sample. In table 1 a summary of the electron probe micro analysis (EPMA) investigation indicates a small error in x EPMA value of Co concentration, suggesting negligible local variation in the concentration of Co across the samples. The T c (x) in table 1 is consistent with earlier reports [1] (note in [1], x is defined as Ba(Fe 1−x Co x ) 2 As 2 while we define x as BaFe 2−x Co x As 2 ). The RRR for x=0.10, x=0.14 and x=0.20 samples were measured to be 1.92, 2.45 and 2.71, respectively, and resistivity at 30 K was about 100 μΩ cm. The local magnetic field distribution B z (x, y) (where the co-ordinates along the sample surface are (x, y) and z is along the vertical direction) at different H and T was measured using high sensitivity magneto-optical (MO) imaging [35][36][37] technique. In all our measurements, H was applied parallel to crystallographic c-axis of the crystals (H || c). The details of our MOI set up have been presented elsewhere [37]. Briefly, in the MOI technique (performed in the reflection mode), linearly polarized light is reflected from the sample surface on which a thin layer of Faraday active material (YIG) with high Verdets constant is placed. The reflected light is Faraday rotated by an angle proportional to the local B z on the sample. By sensitively imaging and calibrating the reflected Faraday rotated light intensity measured across the sample (I(x, y)), one infers the local B z (x, y) across the sample surface, as I(x, y)∝[B z (x, y)] 2 [35].
We also employ a differential magneto-optical (DMO) imaging technique [38][39][40] to image the differential changes in B z produced in the sample in response to modulation of the externally applied field, H. 3. Results and discussion 3.1. Zero magnetic field transport measurement Inset of figure 1(a) shows the normalized R(T) (normalized with R at 300 K) behavior for all three samples at H=0 Oe. The normalized R(T) for x=0.10 (underdoped) sample shows an anomalous upturn in R at 67 K, which is associated with the magneto-structural transition found in underdoped pnictides [1]. For the optimally doped and overdoped samples, the absence of any such anomaly in R(T) suggests the suppression of long range magnetic order with increasing Co doping concentration.    - [41] (for a rectangular plate like crystal geometry, with in-plane sample dimensions a, b (b>a)) was estimated to be about 8×10 5 Amp cm −2 , which is comparable with similar values reported in BaFe 2−x Co x As 2 samples [42]. At first glance, the shape of the positive M(T) behavior can be mistaken for the Wohlleben or the Paramagnetic Meissner Effect (PME) [43,44]. The PME is a positive M(T) response with a peak like feature found in superconductors which have been field cooled (FC) in a small H. Also in PME, the (positive) peak value of M is known to decrease in height with increasing H [43,44]. Non-uniform field cooling of a superconductor or surface superconductivity effects generate magnetic flux compression leading to the positive PME peak [43,44]. However, we argue that the features in figures 2(a)-(c) are not associated with PME because, (i) the peak at T p is found in a ZFC-M(T) measurement rather than in an FC measurement and (ii)  figure 3(e) one observes that deep inside the sample B z -H is positive (as B z (at r=100 μm) ∼35 G) for low fields (0H40 Oe). As H is increased upto 705 Oe (see figure 3(f)), B z -H turns negative (note that a negative B z -H is a characteristic signature of diamagnetic shielding response in a superconductor). Thus at low H (in the range of few tens of Gauss) the local magnetization response is positive corresponding to positively magnetized regions in the sample. At higher H the magnetic response is dominated by the diamagnetic response of the superconductor.

Bulk magnetization response in the superconducting state
In figures 3(g)-(i), for H=0 Oe and T>T c , we observe a bright magneto-optical intensity (B z >0 G) uniformly present across the sample surface albeit its intensity decreases as T→220 K (figure 3(i)). Figure 3(e) inset shows B z (r)∼0 G (as measured across the line in the MO image in figure 3(i)) at 220 K and 0 Oe, indicating that the non-zero B z present across the sample above T c weakens and disappears by 220 K. We associate the positive B z value observed above T c at 0 Oe (figures 3(g)-(i)) with the presence of short range magnetic order in the optimally doped sample. Neutron diffraction and μSR studies on Co doped BaFe 2 As 2 system had already suggested that while doping destroys long range magnetic correlations between Fe moments, short range magnetic order may still be preserved locally [9][10][11]. In our local and bulk magnetization measurements, we believe that the weak positive magnetic response found above T c is associated with short range magnetic order due to the following observations:  figure 3(j) along with weak hysteretic behavior suggests, that above T c , the local magnetic response is unlike a conventional paramagnetic response (section I of supplementary information shows the T dependence of the local magnetization hysteresis loops above T c . From the behavior of the hysteresis loops especially at high T, we argue the presence of short range magnetic order present in the system above T c ).

Identification of characteristic temperatures
Based on the above observations in figures 2 and 3, we attribute the positive magnetization response found above T c with the presence of short range magnetic order at T T mf on  (mf: magnetic fluctuations). From figures 2(d)-(f), we identify T mf on as the temperature below which one observes a paramagnetic Curie like 1/T dependence of M. However, the observed local hysteresis and nonlinear M(H) dependence in x=0.14 sample, as discussed in figure 3(j) (and in section I of supplementary information), suggests that it is not possible to attribute the Curie like behavior below T mf on (>T c ) to a conventional paramagnetic response. We believe that the regions possessing short range magnetic correlations fluctuate due to thermal fluctuations at finite T resulting a Curie like 1/T dependence in M(T) above T c as seen in figures 2(d)-(f). (Such a feature of a system with short range magnetic order exhibiting paramagnetic like features at a finite temperature is not uncommon. In the context of nanomagnetic systems one observes the superparamagnetic effect where above a temperature called the blocking temperature, the magnetic orientation in these systems is not stable due to thermal fluctuations being large compared to the magnetic anisotropy energy. In such low dimensional systems the magnetic response appears like a paramagnetic system above the blocking temperature while the shape of the magnetization versus field curve is unlike a paramagnet [45]). Despite the paramagnetic like behavior, the inherent presence of short range magnetic correlation in the system leads to the observation of local hysteresis at lower T (see supplementary information section I) and the nonlinear M(H) dependence as seen in figure 3(j). Therefore T H mf on ( ) identifies the onset of magnetization response due to stabilization of short range magnetic correlations in the samples. Figure 7(a)-(c) show the T H mf on ( ) line for underdoped, optimally doped and overdoped samples respectively. In figures 2(d)-(f), beyond T scf on we observe a large deviation of M from the linear Curie behavior. We propose this large deviation in M below T scf on is due to the onset of a weak diamagnetic response above T c which suppresses the paramagnetic like response in this temperature regime. In subsequent sections 3.7 and 3.8, we will present further evidence (via local magnetization and bulk resistivity measurements) for the presence of diamagnetic fluctuations above T c in the optimally doped sample.
3.6. Local positive magnetization response not associated with impurity effects The EPMA data (x EPMA ) in table 1 shows the average Co concentration sensitively measured across 12-13 regions (each region of size 10 μm×10 μm) distributed across the sample. The small error in x EPMA values shows that across the samples the fluctuations in, Co concentration and the overall sample stoichiometry (data not shown) are minimal. Furthermore, to rule out the possibility that the origin of positive magnetization response in the sample is predominantly associated with ferromagnetic inclusions in the sample (due to impurity effects), we measured the local iso-field magnetization response in ZFC state at various T's at 200 Oe in optimally doped sample. Figure 4 figure 2(b)) above T c which turns strongly diamagnetic for T<T c . We argue that, if the positive magnetization feature observed above T c at the dot shaped region were to be associated with local ferromagnetic impurities then the superconducting response in this region should have been significantly suppressed, as ferromagnetism is known to strongly suppress the superconducting order parameter [46]. However this dot shaped region shows the onset of a significant diamagnetic response below T c (H), which is not expected if this region had ferromagnetic impurities. We also observe that the bulk and local (at the dot location) T c 's are similar, further suggesting uniformity of T c across the sample. These suggest that local ferromagnetic impurities do not play any significant role in determining the magnetization response of our samples. Infact the following two observations suggest that the positive magnetization response coexists along with superconductivity uniformly across the sample rather than suppress it (we shall also present more evidences for this in the next section); (1) above T c the positive magnetization response is present uniformly over the entire sample rather than in any specific locations of the sample (see figures 3(g)-(i)), (2) below T c , the same regions which exhibit positive magnetization response also show a robust bulk superconducting magnetization response (see figures 1(a)-(d)). We suggest that the results in the paper cannot be attributed to effects of local magnetic impurities present in the samples. Further evidence pertaining to this issue is presented in section II of the supplementary information.
3.7. Imaging the spatially inhomogeneous state with diamagnetic and paramagnetic response above T c in x=0.14 sample using DMO  Figure 4(d) thus displays the inhomogeneous nature of the superconducting state at T<T c where the blue shaded (superconducting) regions are interspersed with greenish (magnetic) regions. As T is increased, the color contrast in the image changes from one which mainly has a bluish shade (superconducting) at 11 K (figure 4(d)) to one which predominantly has greenish shade (magnetic) (figure 4(e)) above T c at 35 K. From figures 4(e) and (f) it is interesting to observe that above T c although most of the regions are positively magnetized (green-yellowish contrast), they co-exist along with regions with weak superconducting diamagnetic response (dark blue contrast regions with negative B B H .  figure 2(e)) the density of the bluish (superconducting) regions have significantly reduced although a small fraction still survives. It may be worthwhile mentioning that similar to the stripe like features we observe in figure 4, an earlier study [47] had found that below T c stripe like regions with enhanced diamagnetic shielding response are located over twin boundaries in underdoped BaFe 2−x Co x As 2 samples. It is possible that the stripe like regions with local diamagnetism we observe above T c , are located around extended defects in these samples. In our optimally doped sample, the regions with weak diamagnetic  3.8. Behavior of normal state conductivity, evidence of weak diamagnetism in the normal state Figure 5(a) shows resistance (R) versus T/T c (0) plot of x=0.14 sample at 0 and 12 T measured upto T/ T c (0)∼4. Using this R(T) data, we plot the conductivity (σ) difference between the 0 T and 12 T data, σ 0T−12T , versus T/T c (0). The shaded region (above T c (0)) in figure 5(b) uncovers the excess conductivity (σ 0T >σ 12T ) at 0 T. The origin of the excess conductivity above T c at 0 T is due to the presence of superconducting fluctuation which get suppressed by a large field of 12 T. In figure 5(b) one observes that the excess conductivity regime exists upto T/T c (0)∼2.5, which is close to T 60 K scf on~f or x=0.14 sample. We analyze the 0 T conductivity data within the Azlamazov-Larkin (AL) [48] theory, which proposes the presence of fluctuating Cooper pairs (namely, superconducting fluctuations) above T c at zero magnetic field. The existence of superconducting fluctuations causes the conductivity to be higher than that in the normal state of a superconductor without any superconducting fluctuation. In three dimensions, the temperature dependence above T c of the zero magnetic field excess conductivity also called Paraconductivity is given by the AL expression [49] e 32 0 , 2 and ξ(0) is the coherence length of the superconductor. (To calculate Δσ : the σ 0T (T) data above 95 K is fitted (not shown here). This fit represents the normal state temperature dependence of σ 0T in the absence of superconducting fluctuations. Using the fitting parameters we calculate the expected normal state conductivity (in the absence of any fluctuations) over the entire temperature range of our measurement, i.e., σ n (T). The excess conductivity due to the presence of diamagnetic fluctuations then is Δσ=σ 0T (T)−σ n (T)). indicating a weaker temperature dependence of Δσ in x=0.14 sample compared to equation (2). This suggests the presence of significant diamagnetic fluctuations sustained in the sample upto T well above T c , and may also indicate the pairing interactions could be strong leading to weak T dependence above T c . In figure 5(c) we see that above the arrow locating T , scf on the Δσ ( ) e drops rapidly as diamagnetic response disappears. Thus the magnetic response above T c in the x=0.14 sample has an admixture of magnetic response associated with magnetic fluctuations and diamagnetic response (related to superconducting fluctuation). The above observation concurs well with the observation of the presence of stripes with diamagnetic shielding response present above T c at 35 and 70 K in figures 4(e) and (f).
3.9. Comparison of local magnetization in x=0.14 and x=0.20 samples In figure 6, we compare the local B z distribution in x=0.14 and x=0.20 samples measured at T=11 K (<T c ) and H=0 Oe. In figure 6(a) the x=0.14 sample exhibits a larger positive B z response than x=0.20 sample. Although small and much weaker than x=0.14 sample, the B z distribution deep inside the overdoped (x=0.20) sample is not zero but finite with B z ∼2 G (see inset figure 6(a)). This is consistent with the observation in figures 2(a)-(c) that the positive M features in bulk M(T) are also present in x=0.20 sample although they are much weaker as compared to x=0.14 sample. Figure 6(b) shows the B z (r) measured across a line shown in inset of figure 6(b) for x=0.20 sample at 11 K and different H. The B z (r) profiles for x=0.20

Discussion
The close relationship between magnetic fluctuations and superconductivity can be inferred from the following observation: as magnetic fluctuation effects get stronger as in x=0.14 sample (see section II of supplementary information), simultaneously the superconducting volume fraction above T c (v sf =0.346%) and the superconducting transition temperature also increases. With weakening of magnetic fluctuations as in x=0.20 sample (see section II of supplementary information), v sf (=0.008%) and T c decreases. To explain these features, we speculate that magnetic fluctuations perhaps play an important role in mediating superconducting pairing above T c in this compound. As seen from Neutron diffraction and muSR measurements [9][10][11], Co doping destroys long range SDW magnetic order in the parent compound however the doping generates local short range magnetic order between Fe moments present uniformly throughout the sample. Above T c , the presence of this short range magnetic order (magnetic fluctuations) perhaps mediates superconducting pairing [21][22][23][24][25][26][27][28] resulting in our observation of diamagnetic response sustained upto temperatures well above T c (see figures 2(d)-(f), 4(d)-(f), 5(b), (c)). Recent experimental studies have found evidence for the presence of magnetic fluctuations along with superconductivity below T c [51,52]. Studies in FeSe samples have shown evidence for BCS-BEC crossover in a Fermi system with strong spin imbalance, where the presence of preformed pairs in the crossover regime [33] was reported, namely, the possibility of superconducting fluctuations present above T c . The work suggested the possibility that in such systems with spin imbalance, the superconducting state may be exotic with spin triplet pairing. While we do observe the presence of diamagnetic fluctuations above T c coexisting along with magnetic fluctuations, however in our present measurements we do not find evidence of exotic spin triplet pairing in BaFe 2−x Co x As 2 . We believe that in our BaFe 2−x Co x As 2 samples below T c , the extent of free energy reduction due to onset of superconductivity is higher than an increase in energy due to Zeeman contributions from any magnetized state surviving below T c (even in the presence of H). Therefore, energetically the lower energy superconducting state gets favoured in the presence of a field H as compared to the magnetic state (see figure 3(f)) and we find diamagnetic signal in the sample enhances with increasing H below T c . In BaFe 2 −x Co x As 2 compounds, as the same set of electrons need to choose between participating in pairing or contributing to magnetism near T c , perhaps the above energy considerations favor a magnetic fluctuation mediated pairing [13,53,54], leading to the superconducting state being preferred over a magnetic state. While these considerations could partially explain the observation in figure 3(f) of the diamagnetic response strengthening as H is increased within a moderate field range, however we do not fully understand the origin of this phenomenon. Below T c the magnetic phase may remain as a microscopic phase coexisting along with the superconducting phase.

Summary and conclusions
We have presented evidence for the presence of a two component magnetic response in single crystals of BaFe 2−x Co x As 2 with different Co-doping concentration. While one component of the response is diamagnetic, the other component is magnetic with the relative strengths of these two signals varying depending on whether the temperature is below T c or above it. Presence of excess conductivity above T c in the optimally doped sample is related to diamagnetic fluctuations above T c . The normal magnetized state in this system is spatially inhomogeneous. The local susceptibility response suggests the inhomogeneous regions is related to regions with superconducting fluctuations coexisting with regions having short range magnetic order. We find the optimally doped sample with highest T c exhibits the strongest effect of magnetic fluctuations and it also possesses the highest volume fraction with diamagnetic response above T c . Below T c along with bulk superconductivity, weak magnetic fluctuations are sustained although the response from regions with magnetic fluctuations get suppressed with the application of a magnetic field. The study suggests that in BaFe 2−x Co x As 2 systems magnetic fluctuations perhaps play an important role in generating local superconducting pairing correlations above T c .