Sodium ion ordering of Na0.77CoO2 under competing multi-vacancy cluster, superlattice and domain formation

Hexagonal superlattice formed by sodium multi-vacancy cluster ordering in Na$_{0.77}$CoO$_2$ has been proposed based on synchrotron X-ray Laue diffraction study on electrochemically fine-tuned single crystals. The title compound sits closely to the proposed lower end of the miscibility gap of x ~ 0.77-0.82 phase separated range. The average sodium vacancy cluster size is estimated to be 4.5 Na vacancies per layer within a large superlattice size of sqrt{19}a*sqrt{19}a*3c. The exceptionally large Na vacancy cluster size favors large twinned simple hexagonal superlattice of sqrt{19}a, in competition with the smaller di-, tri- and quadri-vacancy clusters formed superlattices of sqrt{12}a and sqrt{13}a. Competing electronic correlations are revealed by the observed spin glass-like magnetic hysteresis below ~ 3K and the twin, triple and mono domain transformations during thermal cycling between 273-373K.