First-principles investigation of the cooperative Jahn-Teller effect for octahedrally coordinated transition-metal ions

Fundamental aspects of the cooperative Jahn-Teller effect are investigated using density functional theory in the generalized gradient approximation. ${\mathrm{LiNiO}}_2,$ ${\mathrm{LiMnO}}_2,$ and ${\mathrm{LiCuO}}_2$ are chosen as candidate materials as they possess small, intermediate, and large cooperative Jahn-Teller distortions, respectively. The cooperative distortion is decomposed into the symmetrized-strain modes and $k=0\mathrm{}$ optical phonons, revealing that only the $E_g$ and $A_{1g}$ strain modes and $E_g$ and $A_{1g}$ $k=0\mathrm{}$ optical-phonon modes participate in the cooperative distortion. The first-principles results are then used to find values for the cooperative Jahn-Teller stabilization energy and the electron-strain and electron-phonon coupling. It is found that the dominant source of anisotropy arises from the third-order elastic contributions, rather than second-order vibronic contributions. Additionally, the importance of higher-order elastic coupling between the $E_g$ and $A_{1g}$ modes is identified, which effectively causes expansion of $A_{1g}$-type modes and allows for a larger $E_g$ distortion. Finally, the strain anisotropy induced by the antiferromagnetically ordered states is shown to cause a significant difference in the cooperative Jahn-Teller stabilization energy for the different orientations of the cooperative distortion.