BaFe₂As₂ single crystal undergoes strongly coupled tetragonal to orthorhombic / paramagnetic to antiferromagnetic phase transitions at 134 K and can become superconducting under doping. To study the correlations between the structural / magnetic phase transitions and the superconductivity, seven transition metal electron

doped series : Ba(Fe_1-xTM_x)₂As₂ (TM=Co, Ni, Cu, Co / Cu, Rh and Pd) were grown and characterized by microscopic, transport and thermodynamic measurements.

It was found that the structural and magnetic phase transitions (upper transitions) at 134 K in pure BaFe₂As₂ are monotonically suppressed and increasingly separated in these doped series; in addition, superconductivity can be stabilized with a superconducting temperature rising from 0 K to a maximum and then being suppressed back to 0 K again with doping. The analysis of the temperature-doping concentration (T-x) and temperature-extra electrons (T-e) phase diagrams for all these series shows whereas the upper phase transitions are suppressed in a grossly similar manner with increasing doping concentration for all these dopants, the superconducting domes act quite differently, scaling with e: the number of extra electrons added by the doping, at least on the overdoped side of the superconducting dome. Further examination of the structural, magnetic phase transition temperatures (T_s and T_m) and the superconducting temperature,T_c, in the underdoped regime for these series reveals an essentially linear relation between T_s / T_mand T_c, which indicates a clear correlation between the suppression of structural / magnetic phase transitions and the stabilization of superconductivity. These observations lead to the understanding that the suppression of the upper (structural / magnetic ) phase transitions is a necessary but not a sufficient condition for the occurrence of low temperature superconductivity. There exists a superconducting window (a limited range of the number of extra electrons added), inside which superconductivity can be stabilized if the structural and magnetic phase transitions are suppressed enough.