Experimental observation of magic doping fractions and two-dimensional charge ordering in La$_{2-x}$Sr$_x$CuO$_4$

Competing order is currently an issue of controversy in the study of high temperature superconductivity. In LSCO, neutron scattering experiments have found one-dimensional spin stripes; however, it is unclear whether LSCO has some sort of charge-ordered state. If there is a charge-ordered state in LSCO, the charge ordering tendency is expected to be pronounced near certain doping levels where the charge modulation is commensurate with the underlying lattice; therefore, to examine the nature of possible charge ordering in LSCO, we have carefully measured the hole-doping dependence of the in-plane resistivity using a series of high-quality single crystals. Our detailed measurements find a tendency toward charge ordering at particular rational hole doping fractions of 1/16, 3/32, 1/8, and 3/16. This observation is most consistent with a recent theoretical prediction of the checkerboard-type ordering of the Cooper pairs at rational doping fractions $x=(2m+1)/2^n$, with integers $m$ and $n$.