Approaching the ultimate superconducting properties of (Ba,K)Fe2As2 by naturally formed low-angle grain boundary networks

The most effective way to enhance the dissipation-free supercurrent in presence of magnetic field for type II superconductors is the introduction of defects that acts as artificial pinning centres (APCs) for the vortices. For instance, the in-field critical current density of doped BaFe2As2 (Ba122), one of the most technologically important Fe-based superconductors, has been improved over the last decade by APCs created by ion-irradiation. The technique of ion-irradiation has been commonly implemented to determine the ultimate superconducting properties. However, this method is rather complicated and expensive. Here, we report on a surprisingly high critical current density and strong pinning efficiency close to the crystallographic c-axis for a K-doped Ba122 epitaxial thin film without APCs, achieving performance comparable to ion-irradiated K-doped Ba122 single crystals. Microstructural analysis reveals that the film is composed of columnar grains having width around 30-60 nm. The grains are rotated around the b- (or a-) axis by 1.5 degree and around the c-axis by -1 degree, resulting in the formation of low-angle grain boundary networks. This study demonstrates that the upper limit of in-field properties reached in ion-irradiated K-doped Ba122 is achievable by grain boundary engineering, which is a simple and industrially scalable manner.

). The zero-resistivity temperature Tc,0 is 33 K, corresponding to the onset temperature of the diamagnetic signal measured by the temperature dependence of susceptibility. Therefore, the transition width, defined as Tc,90 -Tc,0 is 2.2 K.
To determine the upper critical field Hc2 and the irreversibility field Hirr, the temperature dependence of resistivity was measured in field up to 16 T ( fig. 2a and b). As increasing applied magnetic fields, a clear shift of Tc to lower temperatures together with a broadening of the superconducting transition is observed for both main crystallographic orientations. The broadening of the transition is more obvious for H || c than H || ab, however, such broadening is not so significant compared with LnFeAsO 21 due to the weak thermal fluctuation. It is also worth mentioning that the foot structure in the vicinity of zero resistance arising from the presence of  fig. 3c, the activation energy U0 for both H || c and || ab shows the same power law relation H -a in low fields up to 2 T: the exponent a is ~0.05-0.07, which indicates that the single vortex pinning prevails. In this regime, U0 for both directions are 12000~13000 K, whereas the respective values of the Ba0.72K0.28Fe2As2 single crystal with Tc=32 K (i.e. underdoped sample) for H || ab and || c at 1 T are 8500 K and 5000 K 26 . Above 2 T, for H || ab, a ~0.5 is consistent with a plastic pinning regime 27 . On the other hand, for H || c, a is 0.68, which is located between 0.5 and 1, where the exponent a=1 is the theoretical prediction for collective pinning 28 . It is interesting to note that for high field regime (i.e. 13~16 T) U0 of our film is comparable to the single crystals 26 .
The relationship between ln[r0] and U0 for both orientations is shown in fig. 3d Field dependence of Jc obtained from the transport and magnetisation measurements Figure 4a shows the in-field Jc properties for the K-doped Ba122 thin film measured by the I-V (or current density J -electric field E) characteristics at various temperatures. E-J curves for H || c are shown in Supplementary fig. S5. At 30 K for both H || c and || ab, Jc gradually decreases with increasing fields. However, below 25 K Jc is almost insensitive against applied magnetic fields and a high Jc above 2×10 5 A cm -2 is maintained on the entire investigated field range. The most striking feature is that Jc for H || c is exceeding that for H || ab with decreasing temperature, opposite to the expected intrinsic behaviour related to the anisotropy of Hc2. Similar features with inverse anisotropy caused by strong c-axis correlated defects were observed before, for instance in Co-doped Ba122 29 and REBCO 24, 30, 31 . These results infer that the strong c-axis pinning is active at T≤ 25 K. It is worth mentioning that Jc peak for H || c is prominent at high temperatures for REBCO but it strongly suppresses with decreasing temperature 32 , which is different from FBS.
To prevent overheating of the contact leads/pads and possible sample damage, the E-J characterisation was limited at low fields and temperatures. Hence, for completeness, the field dependence of magnetisation to extract Jc was measured on a rectangular sample cut from the same film used for transport measurements on a wider temperature range (Supplementary fig. S6).
Jc calculated from the Bean model is shown in fig. 4b. Except for 28 K, Jc has a weak field dependence, which is consistent with the transport Jc. At 4 K, self-field Jc reaches 14.4 MAcm -2 , corresponding to a ~9% of the depairing current density Jd 16 . Temperature dependence of Jc measured by electrical transport measurements follow well the magnetization Jc ( fig. 4c), although the electric field criterion Ec of the former is higher than the latter. The data at 30 K slightly deviating from the trend is likely due to the fluctuations close to the Tc. 5e), increasing to ~1.6 at low temperatures. This is a clear indication that the strong pinning around H || c is activated between 30 and 25 K. As increasing applied magnetic fields, a full evolution of the angular dependence of Jc/Jc ab can be observed from a roughly regular behaviour with maximum at 180º for H || ab (e.g. 16 T and 25 K) to an almost isotropic one (e.g. 10 T and 25 K as well as 16 T and 20 K), and finally to a behaviour strongly affected by c-axis correlated pinning at the lowest temperatures.

Discussions
Through microstructural analyses and electrical transport measurements, "c-axis correlated defect" in our K-doped Ba122 thin film is identified as low angle grain boundary (LAGB). On the assumption that the mean distance d of correlated pinning is identical to that of the width of Kdoped Ba122 grains (i.e. 30-60 nm), the matching field Bf~f0/d 2 is around 2 T at which a kink of The tilted growth of K-doped Ba122 is due presumably to the geometrical configuration of the deposition sources together with the deposition without rotating substrates. In our setup, vapour flux arrives at the substrate with an oblique angle. Additionally, adatoms are expected to diffuse relatively slow on the substrate, since the substrate temperature was low compared with the melting temperature of K-doped Ba122 (832°C) 34 . Hence, the shadowing effect 35 , which limits the formation of new nuclei during the deposition behind initially formed-nuclei, is pronounced, resulting in the inclined columnar growth.
A pinning force density Fp of 114 GNm -3 is recorded even at 15 K and 14-16 T (obtained from the transport measurement) and exceeds 200 GNm -3 at 4 K and field above 6 T (the data at 4 K is obtained from the magnetisation measurements in fig. 4b). In fig. 6, the field dependence of Fp for our K-doped Ba122 thin film is plotted. For comparison, we also plotted the following data of pinning enhanced Ba122 single crystals and thin films with different dopants: K-doped Ba122 single crystals with Pb-ions irradiation measured at 5 K 17 , Co-doped Ba122 thin film with large amounts of stacking faults measured at 4.2 K 35 , Co-doped Ba122 thin film with 3mol% BaZrO3 (BZO) measured at 5 K 36 , and P-doped Ba122 with 3mol% BZO measured at 4.2 K and 15 K 37 .
As can be seen, up to 4 T, Fp of our K-doped Ba122 thin film is the highest among the pinning enhanced Ba122. Albeit the Fp data above 7 T are missing, the extrapolated value at 9 T for the K-doped Ba122 thin film is comparable to the highest value reported for Co-doped Ba122 thin film.
Huge improvement of the superconducting properties of our K-doped Ba122 thin film without APCs is due to a high density of correlated pinning centres created by LAGB networks. Unlike Co-and P-doped Ba122 thin films, the growth temperature of K-doped Ba122 thin films is quite low (~400°C). This low temperature synthesis may lead to small grain size, and hence, the increase of the density of LAGB. It is worth mentioning that the dislocation density increases with increasing the grain boundary angle. Hence, further improvement of in-field Jc is possible by enlarging the texture spread within the critical angle qc. Grain boundary engineering present in this study highlights the possible novel approach to improve the superconducting properties, which is a simple and industrially scalable manner.               The distribution of g shows that a large fraction lies 43º-45º. Residual resistivity ratio (RRR), defined as (300 K)/ ! ( ! : normal state resistivity, see below), is 7.5. Large RRR, low resistivity at 300 K and a sharp superconducting transition of 2.2 K ( ",$% − ",% , ",$% : Temperature reaches 90% of ! , ",% : Temperature reaches zero resistivity) compared to the polycrystalline films [S1] indicate that our K-doped Ba122 film is free of current-blocking impurities and with a good inter-grain connectivity. b, Temperature dependence of the normalised c in the vicinity of the superconducting transition for field cooling (FC) and zero field cooling (ZFC) showed a Tc of 33 K. An external field of 1 mT was applied for H || ab. This temperature coincides with ",% . c, Enlarged view of ( ) in the vicinity of the transition. The blue arrow defines the normal state resistivity ! below which the resistivity deviates from the linear fit to the normal state (red line).