Fermi surfaces and Phase Stability of Ba(Fe$_{1-x}$M$_x$)$_2$As$_2$ (M=Co, Ni, Cu, Zn)

BaFe$_2$As$_2$ with transition-metal doping exhibits a variety of rich phenomenon from coupling of structure, magnetism, and superconductivity. Using density functional theory, we systematically compare the Fermi surfaces (FS), formation energies ($\Delta E_f$), and density of states (DOS) of electron-doped Ba(Fe$_{1-x}$M$_x$)$_2$As$_2$ with M={Co, Ni, Cu, Zn} in tetragonal (I$4/mmm$) and orthorhombic (F$mmm$) structures in nonmagnetic (NM), antiferromagnetic (AFM), and paramagnetic (PM, disordered local moment) states. We explain changes to phase stability ($\Delta E_f$) and Fermi surfaces (and nesting) due to chemical and magnetic disorder, and compare to observed/assessed properties and contrast alloy theory with that expected from rigid-band model. With alloying, the DOS changes from common-band (Co,Ni) to split-band (Cu,Zn), which dictates $\Delta E_f$ and can overwhelm FS-nesting instabilities, as for Cu,Zn cases.