Three-Dimensional Non-Linear Complex Model of Dynamic Memristor Switching
Authors: Chernov Andrey A. , Islamov Damir R. , Pil'nik Andrey A. , Perevalov Timofey V. , Gritsenko Vladimir A. 
Published: February 2020

In recent years significant efforts are being made for implementation of fourth electronic element – resistor with memory called memristor. This element is a passive device whose resistance can be changed by passing through a certain electric current pulse. The memristor has a simple metal-high-k-metal structure.
Functionally the memristors can be divided into two-level (one-bit) and multilevel (multibit) ones. If the one-bit device may be used as a cell of non-volatile memory with a large storage time and high resistance to radiation, the multibit memristors on the one hand can increase the density of integration of the non-volatile memory, and on the other hand can be used to form the basis of adaptive neuromorphic (cognitive) computational systems. One of the most interesting and hitherto unsolved problems is the nature of the intermediate (discrete) multi-level states in a memristor.
The hypothesis that the effect of memristive switch is directly connected with the formation of a filament, that occurs when certain current pulse pass over a dielectric, is distributed among specialists. However, the kinetics of filament formation during the transition from the high resistance state of the memristor to a low resistive state is still not clear. Another unsolved problem in the way of developing memristor arrays is a description of the forming process of memristor elements – the first switch from the initial high resistance state to a low resistance at high voltages, accompanied by intense Joule heat release. Nature of the forming of memristor matrix is still unclear, and the absence of any plausible physical model constrains the practical implementation of these elements.
It is considered that charge transport in high-k is associated with the presence of defects. The most probable and the most widespread defects are oxygen vacancies. Properties of these vacancies are actively studied both experimentally and theoretically. However, many contradictory data can be found in literature, and the role and characteristics of oxygen vacancies in the charge transport through dielectric process are not established yet. It is obvious that rate of change in structure of dielectric (namely the generation rate of oxygen vacancies associated with conductivity) in strong electric field is mainly determined by non-stationary processes of heat and mass transfer. Thus, the kinetics of such processes determines final morphology and properties of memristor.
In the presented work we introduce thermodynamical model of filament growing when a current pulse of variable length and value flows. The model is the boundary value problem, which includes nonstationary heat conduction equation with non-linear Joule heat source, Poisson equation, and Shockley-Read-Hall equations taking into account strong electron-phonon interactions in trap ionization and charge transport processes. The charge current, which defines the heating in the model, depends on the rate of the oxygen vacancy generation. The latter depends on the local temperature. Kinetic parameters of our model are obtained from the transport and optical experiments and ab initiosimulations [V.A.Gritsenko, T.V.Perevalov, D.R.Islamov, Physics Reports
613, 1 (2016)]. The solution of the introduced problem allows us to describe the kinetics of the switch process and the final filament morphology. The resulting filamentary structure in turn may explain the memristor switch.
End-to-end solution of the task allows us to develop recommendations for optimizing the memristor technology and predict the behavior of memristor arrays, including neuromorphic electronics and non-volatile large memory arrays.
The work is supported by Russian Science Foundation (grant #16-19-00002).