To improve the J_c performance of K-doped BaFe₂As₂ (K-Ba122) to be used in wire fabrication, we recently performed systematic studies on the synthesis routes of bulk samples so as to understand what limits its grain-to-grain connectivity. We found that high purity materials and working in a high performance glovebox are essential to obtain clean grain boundaries and high J_c exceeding 10⁴ A/cm² at 10 T and 4.2 K [1]. In this work we investigate the distribution of properties of high purity K-Ba122 by performing in-field specific heat characterizations [2] on samples that underwent two heat treatments (HTs): here we varied either the 1^st or the 2^nd HT temperature. Increasing the 1^st HT temperature between 600 and 825°C, we found a non-monotonic trend of the T_c-distribution, which shows the highest-temperature peak at ~37.3 K for the 1^st HT at 750°C (2^nd HT at 600°C) despite a broader distribution (peak width s~0.7 K). Increasing the 2^nd HT temperature (1^st HT at 750°C) has, instead, detrimental effect on the T_c-distribution that, despite a narrower transition, shows a clear peak shift at low temperature. We also observed signs of a Schottky-like anomaly at low temperature, particularly marked in the best sample (750°C+600°C), which also contains traces of a low-T_c secondary phase. Although all samples are weakly affected by applying a magnetic field of 16 T, the highest H_c2 was found for the 750°C+600°C sample, where the maximum of the T_c-distribution was shifted by less than 1.4 K. Here we will compare and correlate these results to magnetic characterizations and micro-structural/micro-chemical analyses to reveal grain size and connectivity effects. We will show that, despite the low-T_c phase and marked Schottky-like anomaly could be signs of current-blocking phases, the 750°C+600°C sample has the best J_c performance. We will also discuss routes for further J_c improvements obtainable via heat treatment and composition optimization.

Acknowledgements
This work is supported by the US Department of Energy Office of High Energy Physics under the grant no. DE-SC0018750 and by the National High Magnetic Field Laboratory, which is supported by the National Science Foundation under NSF/DMR-1644779 and by the State of Florida.
[1] C. Pak et al., Supercond. Sci. Technol. 33, 084010 (2020)
[2] C. Tarantini et al., in preparation

Keywords: BaFe2As2, Specific heat, Critical current density, Iron-based superconductor