Perfect memristor: non-volatility, switching and negative differential resistance

We propose a simple model of a nanoswitch as a memory resistor. The resistance of the nanoswitch is determined by electron tunneling through a nanoparticle diffusing around one or more potential minima located between the electrodes in the presence of Joule's heat dissipation. In the case of a single potential minimum, we observe hysteresis of the resistance at finite applied currents and a negative differential resistance. For two (or more) minima the switching mechanism is non-volatile, meaning that the memristor can switch to a resistive state of choice and stay there. Moreover, the noise spectra of the switch exhibit $1/f^2\rightarrow 1/f$ crossover, in agreement with recent experimental results.