Unrestricted Hartree-Fock Analysis of Sr${}_{3-x}$Ca${}_x$Ru${}_2$O${}_7$

We investigated the electronic and magnetic structure of Sr${}_{3-x}$Ca${}_x$Ru${}_2$O${}_7$ ($0\le x\le 3$) on the basis of the double-layered three-dimensional multiband Hubbard model with spin-orbit interaction. In our model, lattice distortion is implemented as the modulation of transfer integrals or a crystal field. The most stable states are estimated within the unrestricted Hartree-Fock approximation, in which the colinear spin configurations with five different spin-quantization axes are adopted as candidates. The obtained spin structures for some particular lattice distortions are consistent with the neutron diffraction results for Ca${}_3$Ru${}_2$O${}_7$. Also, some magnetic phase transitions can occur due to changes in lattice distortion. These results facilitate the comprehensive understanding of the phase diagram of Sr${}_{3-x}$Ca${}_x$Ru${}_2$O${}_7$.