Charge disproportionation (CD) and spin differentiation in Na$_{ 1/2}$CoO$_2$ are studied using the correlated band theory approach. The simultaneous CD and gap opening seen previously is followed through a first order charge disproportionation transition 2Co$^{ 3.5+} \to$ Co$^{ 3+}$+Co$^{ 4+}$, whose ionic identities are connected more closely to spin (S=0, S=1/2 respectively) than to real charge. Disproportionation in the Co $a_g$ orbital is compensated by opposing charge rearrangement in other 3d orbitals. At the transition large and opposing discontinuities in the (all-electron) kinetic and potential energies are slightly more than balanced by a gain in correlation energy. The CD state is compared to characteristics of the observed charge-ordered insulating phase in Na$_{ 1/2}$CoO$_2$, suggesting the Coulomb repulsion value $U$ is concentration-dependent, with $U(x=1/2)\simeq$3.5 eV.
Categories: cond-mat.str-el cond-mat.supr-con