Theory for the Magnetic Ordering in Fe$_{1+y}$Te$_{1-x}$Se$_{x}$

We develop a local spin model to explain the rich magnetic structures in the iron-based superconductors $Fe_{1+y}Te_{1-x}Se_x$. We show that our model exhibits both commensurate antiferromagnetic and incommensurate magnetic order along the crystal a-axis, the transition between which can be obtained by increasing the concentration of the excess $Fe$ to a critical value. The incommensurate modulation vector is also shown to be $Fe^2$ concentration dependent. Experimentally measurable spin-wave features and Fermi surface properties are calculated and compared to those of other Fe-based superconductors. Our model also suggests the existence of a large quantum critical region due to strong spin frustration upon increasing $Se$ concentration.