Competition between Jahn-Teller instability and uniaxial magnetism in Ca$_{3}$CoMO$_{6}$ (M = Mn, Co, Rh)

Ca$_{3}$CoMO$_{6}$ (M = Mn, Co, Rh) exhibits a uniaxial magnetism, because the Co$^{n+}$ ions of their CoO$_{6}$ trigonal prisms (n = 2 or 3) possess an electron configuration with unevenly filled degenerate d-states, so the Co$^{n+}$ ions have a nonzero magnetic moment only along the axis of the rotational symmetry causing the degeneracy (i.e., the 3-fold rotational axis along the CoMO$_{6}$ chain). Such ions lead to Jahn-Teller (JT) instability, and the associated distortion removes the rotational symmetry responsible for the uniaxial magnetism. We investigated how these opposing factors compete in Ca$_{3}$CoMO$_{6}$ (M = Mn, Co, Rh) on the basis of first principles DFT calculations.