Doping induced spin state transition in Li$_x$CoO$_2$ as studied by the GGA$+$DMFT calculations

The magnetic properties of Li$_x$CoO$_2$ for $x=0.94, 0.75, 0.66$, and $0.51$ are investigated within the method combining the generalized gradient approximation with dynamical mean field theory (GGA$+$DMFT). A delicate interplay between Hund's exchange energy and $t_{2g}-e_g$ crystal field splitting is found to be responsible for the high-spin to low-spin state transition for Co$^{4+}$ ions. The GGA$+$DMFT calculations show that the Co$^{4+}$ ions at a small doping level adopt the high-spin state, while delithiation leads to an increase in the crystal field splitting and low-spin state becomes preferable. The Co$^{3+}$ ions are found to stay in the low-spin configuration for any $x$ values.