Current filamentation in large Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8}$ mesas observed by luminescent and scanning laser thermal microscopy*

Self-heating is a critical issue in stacked intrinsic Josephson junction devices designed for terahertz emission. Some theoretical models, as well as experimental evidence, suggest that self-heating may indeed be helpful for maximizing THz power output. Here we study the self-heating of a Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8}$ mesa terahertz source via two techniques. We show that low-temperature scanning-laser microscopy measurements - a sensitive, but indirect probe of device temperature - agree well with direct temperature measurements obtained via a thermoluminescent imaging technique. Due to the semiconductor-like c-axis resistivity of Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8}$, we find that at low temperatures device self-heating is highly non-uniform, displaying hysteretic nucleation of narrow hotspots with elevated current density. Also, the hotspot radius grows with increasing device temperature. These behaviors are consistent with theoretical predictions for a current filament forming in a material whose resistance falls with increasing temperature.