Spatially Inhomogeneous Collapse of Superconducting Gaps on the Nanoscale: Connection to Macroscopic Measurements on Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+x}$*

Using spatially resolved STM spectroscopy, we have mapped the superconducting correlations in Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+x}$ to show that these correlations collapse in a spatially inhomogeneous manner with increasing temperature. These experiments provide valuable insight for understanding the results of spatially averaged measurements such as angle-resolved photoemission, vortex Nernst and field-induced diamagnetism on the same material system. The connection between nanoscale measurements and bulk parameters such as T$_{c}$ and T* is clarified. Finally, high-resolution STM measurements provide a method to characterize the processes by which superconductivity is destroyed with increasing temperature in samples with various hole doping levels.