Engineering magnetic domain wall energies in ${\mathrm{BiFeO}}_3$ via epitaxial strain: A route to assess skyrmionic stabilities in multiferroics from first principles

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 (AFM) multiferroic ${\mathrm{BiFeO}}_3$ (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 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 energy, we envision the region where isolated chiral magnetic textures might occur as a function of strain, i.e., where the collinear AFM and the spin spiral energies are equal. This transition between $-1.5$ and $-0.5\%$ of strain should allow topologically stable magnetic states such as antiferromagnetic skyrmions and/or merons to occur. Hence, our paper should trigger experimental and theoretical investigations in this range of strain.