Surface pinning of fluctuating charge order: An extraordinary surface phase transition

We study the mean-field theory of charge-density wave (CDW) order in a layered system, including the effect of the long-range Coulomb interaction and of screening by uncondensed electrons. We particularly focus on the conditions necessary for an ``extraordinary'' transition, in which the surface orders at a higher temperature, and is more likely to be commensurate, than the bulk. We interpret recent experiments on ${\mathrm{Ca}}_{2\ensuremath{-}x}{\mathrm{Na}}_{x}\mathrm{Cu}{\mathrm{O}}_{2}{\mathrm{Cl}}_{2}$ as indicating the presence of commensurate CDW at the surface that is not present in the bulk. More generally, we show that poor screening of the Coulomb interaction tends to stabilize incommensurate order, possibly explaining why the CDW order in ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$ and $\mathrm{Nb}{\mathrm{Se}}_{2}$ remains incommensurate to $T\ensuremath{\rightarrow}0$, despite the small magnitude of the incommensurability.