  The equilibrium ON and OFF states of resistive random access memory (RRAM)
are due to formation and destruction of a conducting filament. The laws of
thermodynamics dictate that these states correspond to the minimum of free
energy. Here, we develop a numerical model that, through the minimization of
free energy at a given voltage, determines the filament parameters and thus the
electric current. Overall, it simulates the current-voltage (I-V)
characteristics of RRAM. The model describes mutual transformations of RRAM
states through SET (ON to OFF) and RESET (OFF to ON) processes. From the
modeling perspectives, these states and processes constitute four programming
modules constructed here in COMSOL Multiphysics software tackling the
electrodynamic and heat transfer equations and yielding RRAM energy and I-V.
Our modeling uniquely reproduces the observed I-V varying with voltage
ramp-rates. That is achieved by accounting for the ramp-rate dependent
activation energy of conduction. The underlying mechanism is due to the
deformation interaction caused by the double well atomic potentials universally
present in amorphous materials and having exponentially broad distribution of
relaxation times. As another unique feature, our modeling reproduces the
observed cycle-to-cycle variations of RRAM parameters attributed to the lack of
self-averaging in small ensembles of double well potentials and electronic
states in geometrically small (nano-size) RRAM structures.
