Enhanced Optical Gap in Bi-layered Manganites La$_{2-2x}$Sr$_{1+2x}$Mn$_{2}$O$_{7}$ near $x=0.4$

We have systematically investigated the optical conductivity spectra of La$_{2-2x}$Sr$_{1+2x}$Mn$_{2}$O$_{7}$ ($0.3 \leqslant x \leqslant 0.5$). We find that just above the magnetic ordering temperatures, the optical gap shows an enhancement up to $\sim 0.3$ eV near $x=0.4$. Based on a $x$-dependent comparison of the nesting vector of the hypothetical Fermi surface and the superlattice wave-vector, we suggest that the peculiar $x$-dependence of the optical gap can be understood in terms of charge and lattice correlation enhanced by the charge density wave instability in nested Fermi surface.