Electric-field control of magnetic orderings in the tetragonal BiFeO$_{3}$*

We present a systematic first-principles study on the magnetic properties of the tetragonal BiFeO$_{3}$ (P4mm) with various in-plane lattice constants. The Heisenberg model is applied to study the behaviors of exchange constants (J$_{1a}$, J$_{1c}$, J$_{2a}$, J$_{2c})$ under the influence of the in-plane strain. We find that in certain region of the in-plane lattice constant, switching the direction of polarization from out of plane to in-plane by electric field could result in transition of magnetic orderings, e.g., from G-type to C-type antiferromagnetic states in tetragonal BiFeO$_{3}$. This may open a new avenue to controlling magnetoresistance using electric field.