  We study the heat transport in filamentary RRAM nano-sized devices by
comparing the accurate results of COMSOL modeling with simplified analytical
models for two complementary mechanisms: one neglecting the radial heat
transfer from the filament to the insulating host, while the other describing
the radial transport through the dielectric in the absence of the filament heat
transfer. For the former, we find that the earlier assumed simplification of
the electrodes being ideal heat conductors is insufficient; a more adequate
approximation is derived where the heat transport is determined by the adjacent
proximities of the filament tips in the electrodes.We find that both
complementary mechanisms overestimate the maximum temperature yet offering
acceptable results. However, the two in parallel provide a better analytical
approximation. In addition, we show that the Wiedemann-Franz-Lorenz law helps
the analysis when the Lorenz parameter is chosen from the actual data. We
present an approximate expression for the SET voltage possessing a high degree
of universality and predicting that filament materials with low Lorenz numbers
can be good candidates for the future low set voltage devices.
