  Implementation of antiferromagnetic compounds as active elements in
spintronics has been hindered by their insensitive nature against external
perturbations which causes difficulties in switching among different
antiferromagnetic spin configurations. Electrically-controllable strain
gradient can become a key parameter to tune the antiferromagnetic states of
multiferroic materials. We have discovered a correlation between an
electrically-written straight-stripe mixed-phase boundary and an in-plane
antiferromagnetic spin axis in highly-elongated La-5%-doped BiFeO$_{3}$ thin
films by performing polarization-dependent photoemission electron microscopy in
conjunction with cluster model calculations. Model Hamiltonian calculation for
the single-ion anisotropy including the spin-orbit interaction has been
performed to figure out the physical origin of the link between the strain
gradient present in the mixed phase area and its antiferromagnetic spin axis.
Our findings enable estimation of the strain-gradient-induced magnetic
anisotropy energy per Fe ion at around 5$\times$10$^{-12}$ eV m, and provide a
new pathway towards an electric-field-induced 90$^{\circ}$ rotation of
antiferromagnetic spin axis at room temperature by flexomagnetism.
