Model for the commensurate charge-density waves in under-hole-doped cuprate superconductors

A simple model of the commensurate charge-density wave (CCDW) portion of the underdoped pseudogap regions of monolayer Bi$_2$Sr$_{2-x}$La$_x$CuO$_{6-x}$ (Bi2201), bilayer Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi2212), and trilayer Bi$_2$Sr$_2$Ca$_2$Cu$_3$O$_{10+\delta}$ (Bi2223) cuprate superconductors is presented and studied. Above the superconducting transition temperature $T_c$ but below the pseudogap transition temperature $T_p > T_c$, the CCDW forms on the oxygen sites in the CuO$_2$ layers with excess charges of $\pm\delta e$, where $e$ is the electronic charge, forming on alternating oxygen sites. This model is equivalent to $N$-layer versions of the two-dimensional Ising model for spins on a square lattice with repulsive interactions $J' , J>0$ between near-neighbor inter- and intralayer sites, respectively. For strong coupling, we show analytically for sections of $L\times M\times N$ sites that the partition function in the $J'\rightarrow\pm \infty$ limits reduces to that for an effective single layer with $L\times M$ sites and $J$ replaced by $NJ$. The CCDW is therefore strongly enhanced and stabilized by multilayer structures, likely accounting for the enhanced THz emission observed from the intrinsic Josephson junctions in underdoped Bi2212 mesas and for the many experiments on Bi2212 and related compounds purporting to provide evidence for a superconducting order parameter with $d_{x^2-y^2}$-wave symmetry.