The spin state issue in the $R$BaCo$_2$O$_{5.5}$ cobaltates

The double perovskites $R$BaCo$_2$O$_{5+\delta}$ ($R$=rare earth, 0$\le\delta\le$1) display intriguing phenomena such as charge and orbital ordering, as well as antiferromagnetic to ferromagnetic transition, depending on the oxygen concentration. In particular, the $\delta$=0.5 system shows a giant magnetoresistance effect, and its metal-insulator transition has been often interpreted in terms of a spin-state transition [1,2], which, however, is fiercely debated [3,4]. To address the spin-state issue, we performed density-functional theory calculations which include a mean-field correction for the correlation effects caused by the Co $3d$ electrons. We have investigated various scenarios with different combinations of the low-, intermediate- and high-spin (LS, IS, and HS) states. Our results show that the pyramidally coordinated Co$^{3+}$ ions are exclusively in the HS state since [3], in disagreement with [1,2]. The octahedrally coordinated Co$^{3+}$ can be stabilized into a LS-HS ordered state if we take into account the superstructure recently reported [4]. Our results put limits as to how much spin-state transition could accompany the metal-insulator transition. [1] C. Frontera $et$ $al$., Phys. Rev. B {\bf 65}, 180405(R) (2002). [2] A. A. Taskin $et$ $al$., Phys. Rev. Lett. {\bf 90}, 227201 (2003). [3] Z. Hu $et$ $al$., Phys. Rev. Lett. {\bf 92}, 207402 (2004). [4] D. D. Khalyavin $et$ $al$., Phys. Rev. B {\bf 75}, 134407 (2007)