Imaging the Vortex Liquid State in Bi$_{2}$Sr$_{2}$CuO$_{6+\delta}$

We use a low temperature scanning tunneling microscope (STM) to study the vortex state of the high-T$_c$ superconductor Bi$_{2}$Sr$_{2}$CuO$_{6+\delta}$ in magnetic fields up to 9 T. At a temperature of 6 Kelvin, we find no localized vortices down to H = 0.25 T. However, the gap depth from the spatially averaged dI/dV spectrum decreases with increasing magnetic field, which indicates a vortex liquid state. By tracking atomically resolved locations at different magnetic fields, we apply a normalization technique to remove inhomogeneities in the underlying density of states, revealing a more homogeneous superconducting state.