Magnetic phase diagram ofCa2−xSrxRuO4governed by structural distortions

We constructed, by the first-principles calculations, a magnetic phase diagram of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ in the space spanned by structural distortions. Our phase diagram can qualitatively explain the experimental one for ${\mathrm{Ca}}_{2-x}{\mathrm{Sr}}_x{\mathrm{RuO}}_4.$ We found that the rotation and the tilting of the ${\mathrm{RuO}}_6$ octahedron are responsible for the ferro- and antiferromagnetism, respectively, while the flattening of ${\mathrm{RuO}}_6$ is the key factor to stabilize those magnetic ground states. Our results imply that the magnetic and the structural instabilities in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ are closely correlated cooperatively rather than competitively.