  Chemically exfoliated nanoscale few-layer thin Li$_x$CoO$_2$ samples are
studied as function of annealing at various temperatures, using transmission
electron microscopy (TEM) and Electron Energy Loss Spectroscopies (EELS),
probing the O-K, Co-L$_{2,3}$ spectra along with low energy interband
transitions. These spectra are compared with first-principles DFT calculations
of -Im$[\varepsilon^{-1}(q,\omega)]$ and O-2p Partial Densities of States
weighted by dipole matrix elements with the core wavefunction and including the
O-1s core-hole and with known trends of the L$_2$/L$_3$ peak ratio to average
Co valence. Trends in these spectra under the annealing procedures are
established and correlated with the structural phase changes observed from
diffraction TEM and High Resolution TEM images. The results are also correlated
with conductivity measurements on samples subjected to the same annealing
procedures. A gradual disordering of the Li and Co cations in the lattice is
observed starting from a slight distortion of the pure LiCoO$_2$ $R\bar{3}m$ to
$C2/m$ due to the lower Li content, followed by a $P2/m$ phase forming at
200$^o$C indicative of Li-vacancy ordering, formation of a spinel type
$Fd\bar{3}m$ phase around 250$^o$C and ultimately a rocksalt type $Fm\bar{3}m$
phase above 350$^o$C. This disordering leads to a lowering of the band gap as
established by low energy EELS. The O-K spectra of the rocksalt phase are only
reproduced by a calculation for pure CoO and not for a model with random
distribution of Li and Co. This indicates that there may be a loss of Li from
the rocksalt regions of the sample at these higher temperatures. The
conductivity measurements indicate a gradual drop in conductivity above
200$^o$C, which is clearly related to the more Li-Co interdiffused phases, in
which a low-spin electronic structure is no longer valid and stronger
correlation effects are expected.
