Mapping the Superconducting Anti-ferromagnetic C$_{\mathrm{4}}$ Phase in Iron-Pnictides*

Following the discovery of the microscopic coexistence of antifermagnetic spin density waves and superconductivity in Ba$_{\mathrm{1-x}}$K$_{\mathrm{x}}$Fe$_{\mathrm{2}}$As$_{\mathrm{2\thinspace }}$and the low temperature re-entrance to the novel magnetic C$_{\mathrm{4}}$~tetragonal phase in Ba$_{\mathrm{1-x}}$Na$_{\mathrm{x}}$Fe$_{\mathrm{2}}$As$_{\mathrm{2}}$, there has been significant interest in developing an understanding of the properties and formation of these phases and analyzing their dependence on temperature and composition in hole-doped 122 alkaline earth metal/iron-pnictides. We describe the mapping of various Ba, Sr, and Ca 122 phase diagrams with systematically controlled levels of hole-doping of alkaline metal onto the alkaline earth metal site, which was investigated via x-ray and neutron diffraction. Our elaborate synthesis, diffraction work, and analysis maps and firmly establishes the C$_{\mathrm{4}}$ phase space in these ternary diagrams as well as the boundary lines that separate the individual phases, and provides natural clues as well as a framework to investigate the stability and formation of the C$_{\mathrm{4\thinspace }}$domes that shift location with doping contents in the phase diagrams.