Charge/Orbital Ordered Phases of La$_{2-2x}$Sr$_{1+2x}$Mn$_{2}$O$_{7-\delta }$

Our studies have significantly modified the conventionally-held view of the phase diagram of La$_{2-2x}$Sr$_{1+2x}$Mn$_{2}$O$_{7-\delta }$ for two compositions exhibiting charge (and orbital) order (CO), i.e., at hole doping levels, h=x-$\delta $, of $\sim $0.5 and $\sim $0.6. These CO states are stable over very narrow doping ranges ($\Delta $h$\sim \pm $0.005) at the lowest temperatures, but those ranges increase at higher temperatures (to $\Delta $h$\sim \pm $0.02) in a manner consistent with simple entropy considerations. Such narrow ranges dictate the crucial need for crystal homogeneity. Attesting to such homogeneity is a conductivity ratio of $>$10$^{10}$ upon crossing the first-order phase boundary from CO at h=0.60 to AAFM at h$\sim $0.59 or h$\sim $0.61 plus two findings that were missed in the existing literature: that these CO phases are the ground state at the lowest temperatures and, for h$\sim $0.5, that coexistence of the CO and AAFM phase is absent at any temperature.