Praseodymium doping effect on the superconducting properties of FeSe$_{0.5}$Te$_{0.5}$ bulks under ambient and high-pressure growth conditions

A series of Pr-doped FeSe$_{0.5}$Te$_{0.5}$ (Fe$_{1-x}$Pr$_x$Se$_{0.5}$Te$_{0.5}$; $x = 0$ to 0.3) bulks are prepared by conventional synthesis process at ambient pressure (CSP), and high gas pressure and high temperature synthesis (HP-HTS) methods. These bulks are well characterized by structural and microstructural analysis, Raman spectroscopy, transport, and magnetic measurements. The HP-HTS process of the parent bulks has enhanced the onset transition temperature ($T_c^{\rm onset}$) by 1.5 K and the critical current density ($J_c$) by two orders of magnitude compared to the CSP method. Pr-doped FeSe$_{0.5}$Te$_{0.5}$ up to 10% doping content prepared, either CSP or HP-HTS, slightly increases the unit cell volume, and high-pressure growth produces an almost pure superconducting phase, which confirms the successful Pr-doping at Fe sites. Raman spectroscopy measurements and DFT calculations suggest the substitution of Pr-atoms in the interlayer spacing of Fe(Se,Te) lattice. High-pressure growth of Fe$_{1-x}$Pr$_x$Se$_{0.5}$Te$_{0.5}$ also makes the sample less dense compared to the parent sample grown by HP-HTS. Transport and magnetic measurements depict that Tconset is almost unaffected by Pr-doping, whereas $J_c$ of Pr-doped FeSe$_{0.5}$Te$_{0.5}$ is enhanced by one order of magnitude relative to the parent sample developed by CSP but lower than that of the parent sample grown by HP-HTS. Hence, Pr-doping at Fe sites preserves $T_c^{\rm onset}$ and improves $J_c$ of FeSe$_{0.5}$Te$_{0.5}$ regardless of the doping contents and growth conditions. These results are promising for the practical application of iron-based superconductors to improve $J_c$ properties without affecting $T_c^{\rm onset}$ through CSP process and congruent with discoveries from other superconductors, like cuprates and MgB$_2$.