Balancing orbital effects and on-site Coulomb repulsion through Na modulations in Na x V O 2

A common Na density wave pattern is found to evolve in Na${}_{x}$TiO${}_{2}$, Na${}_{x}$VO${}_{2}$, and Na${}_{x}$CrO${}_{2}$, where the separation of antiphase boundary within the pattern continuously changes with Na composition, corresponding to the incommensurate peak shift found in the $i\phantom{\rule{0}{0ex}}n$ $s\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}u$ x-ray diffraction. More details disclosed by x-ray diffraction analysis, $a\phantom{\rule{0}{0ex}}b$ $i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o$ simulation and theoretical modeling suggest that the unique trimer structure in the P2 stacking of Na${}_{x}$VO${}_{2}$ is a delicate balance between strong electronic correlations and orbital effects, well explaining the metal insulator transition of the material. On the contrary, the remaining materials, due to the lack of such a delicate balance, share a sodium-modulated Peierls-like transition for the dimer formation instead.