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 ${\mathrm{Ca}}_{2-x}{\mathrm{Na}}_x\mathrm{Cu}{\mathrm{O}}_2{\mathrm{Cl}}_2$ 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 ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ and $\mathrm{Nb}{\mathrm{Se}}_2$ remains incommensurate to $T\to 0$, despite the small magnitude of the incommensurability.

Figure 1

Schematic one-parameter cut of the phase diagram for a CDW system with a surface: $r$ and $r_0$ measure the strength of the quadratic term in the McMillan free energy in the bulk and on the surface. The full and broken lines represent the bulk and surface phase transitions, respectively: $T_I$ and $T_C$ are the bulk critical temperatures for the disordered- incommensurate and for the bulk incommensurate-commensurate phase transitions; $T_{sI}$ and $T_{sC}$ are the corresponding surface phase transitions. The extraordinary surface transitions for $r_0∕r$ are shown. SI: surface incommensurate CDW; SC: surface commensurate CDW. Other surface orderings (e.g., “ordinary”surface transitions) are possible but are not shown.