Defect formation, magnetic interaction and electron phonon coupling in iron selenides M$_{1-x}$Fe$_{2-y}$Se$_{2}$

We perform a systematic study to explore the electronic and magnetic structures in iron selenide superconductors M$_{1-x}$Fe$_{2-y}$Se$_{2}$ using first principles calculations. We show that there is an intimate relationship between Se-height and the underneath Fe-spin square in M$_{1-x}$Fe$_{2-y}$Se$_{2}$. A displacement of the Se atom by as small as 0.2{\AA} is enough to change the amount of charge in the Fe-plane as much as 0.7 e per Fe. The Se-height increases as the number of ferromagnetic Fe-Fe bonds increases, yielding an expansion of 2{\AA} expansion in the c-axis for fully ferromagnetic spin configuration, which indicates a giant magneto-elastic coupling in these systems. Our calculations also explain why the formation of Fe vacancies is favorable in iron selenides, but not in iron pnictides. Finally, we calculate the spin-resolved electron-phonon coupling in MFe$_{2}$Se$_{2}$ and M$_{1-x}$Fe$_{2-y}$Se$_{2}$ to shed light on the mechanism of superconductivity in these materials.