  Epitaxial strain has emerged as a powerful tool to tune magnetic and
ferroelectric properties in functional materials such as in multiferroic
perovskite oxides. Here, we use first-principles calculations to explore the
evolution of magnetic interactions in the antiferromagnetic multiferroic
BiFeO$_3$ (BFO), one of the most promising multiferroics for future technology.
The epitaxial strain in BFO(001) oriented film is varied between
$\varepsilon_{xx,yy}$ $\in$ $[-2\%, +2\%]$. We find that both strengths of the
exchange interaction and Dzyaloshinskii-Moriya interaction (DMI) decrease
linearly from compressive to tensile strain whereas the uniaxial
magnetocrystalline anisotropy follows a parabolic behavior which lifts the
energy degeneracy of the (111) easy plane of bulk BFO. From the trends of the
magnetic interactions we can explain the destruction of cycloidal order in
compressive strain as observed in experiments due to the increasing anisotropy
energy. For tensile strain, we predict that the ground state remains unchanged
as a function of strain. By using the domain wall (DW) energy, we envision the
region where isolated chiral magnetic texture might occur as function of strain
i.e. where the DW and the spin spiral energy are equal. This transition between
$-1.5\%$ and $-0.5\%$ of strain should allow topologically stable magnetic
states such as antiferromagnetic skyrmions and merons to occur. Hence, our work
should trigger experimental and theoretical investigations in this range of
strain.
