Imaging Locally Oriented Charge Modulations in a Cuprate Superconductor

We use scanning tunneling microscopy to image the local orientation of the static charge modulations in Bi$_{2-y}$Pb$_y$Sr$_{2-z}$LazCuO$_{6+x}$, for samples spanning a wide range of doping. For each sample, we compute the size distribution of locally $x$-oriented and locally $y$-oriented clusters. We analyze the size distributions within a random field Ising model to obtain the fractal dimension and other critical exponents. We discuss the utility of scaling collapse to extract the critical doping $x_c$ of the smectic charge order in Bi$_{2-y}$Pb$_y$Sr$_{2-z}$LazCuO$_{6+x}$.