Current-Controlled Negative Differential Resistance Due to Joule Heating In Tio$_2$

We show that Joule heating causes current-controlled negative differential resistance (CC-NDR) in TiO2 memristive systems by constructing an analytical model of the current-voltage characteristics based on polaronic transport for Ohm's law and Newton's law of cooling and fitting this model to experimental data. This threshold switching is he ``soft breakdown'' observed during electroforming in TiO2 and other transition-metal oxide based memristors, as well as a precursor to ``ON'' or ``SET'' switching of unipolar memristors from their high to their low resistance states. The shape of the V-I curves is a sensitive indicator of the nature of the polaronic conduction, which apparently follows an adiabatic regime [1]. \\[4pt] [1] A.S. Alexandrov, A.M.Bratkovsky, B.Bridle, S.E.Savel'ev, D. Strukov, and R.S.Williams, Appl. Phys. Lett. 99, xxx (2011).