Description

The kinetic Monte Carlo (kMC) method is a select-and-update algorithm applied to a finite set of rate equations that define individual processes between microstates within a system. At any simulation timestep, a higher rate results in a greater likelihood or probability of selection for that process. Once selected, the simulation time is updated according to the magnitude of the rate and the state of the system is updated to reflect the final state of the particular transition.

kMC can be used to simulate transient, time-dependent phenomena or steady-state behavior. The practical limitation is that the simulation time step must be shorter than the fastest process which can occur. This is intuitive since, if simulating a collection of particles with independent degrees of freedom, e.g. diffusion of electrons, in a real time step it is possible for all particles to move so the global simulation time step must be incremented at a scale that is proportional to the number of particles that can move times the maximum rate per particle. In this way, the simulation time step will be congruent with real time as has been shown (see Fichthorn).

When implemented correctly, the temporal dynamics of complex physical systems can be simulated from a condensed set of physical transitions. I have used kMC to study resistive switching in oxide memristors. Examples are shown herein.

Filament electrostatics

The kMC algorithm allows the state of the system to evolve with time. If we have an electrostatic model for the components of the system, then we can also evaluate how the electrostatics of the system evolve with time. The Figures below illustrate how the conductive filament, which is modeled as ionized acceptors, results in a change to the electrostatic potential, electric field and energy band diagram. The large negative charge in these regions can be large enough to make certain regions appear as a negative potential.

Current-voltage characteristics

Along with predicting the state of the system as it evolves with time, we can pair the electrostatic model with a transport model to calculate the current-voltage characteristics in the device. This was done using a proprietary software Synopsis TCAD Sentaurus, the gold standard in electronic device simulation. The physical model library of Sentaurus is unparalleled and well-documented.

Below, the current in a hafnium oxide memristor is evaluated for different stages of programming -- as-prepared state, low resistance state (LRS), high resistance state (HRS) and forming.

• The as-prepared state is the state of the system as it has been prepared through a physical process, e.g. atomic layer deposition, sputtering, pulsed laser deposition.

• forming refers to the initial formation of a conductive filament or a low resistance state of the system.

• reset refers to the recovery of a high resistance state of the system. 

All resistance levels between the as-prepared state and the low-resistance state are accessible in a memristor, giving it the potential for multiple bits in a single memory device.