Research Of The Stochastic Model Of Memristors Based On Conductive Filaments

Memristor has attracted much attention for its small size,low power consumption and non-volatile characteristics.A predictive memristor model is required for circuit design.One of the challenges to forster the practical application is the stochasticity of the memristors.In this paper,we focus on introducing a stochastic model for memristors.The main contributions are listed as follows.Firstly,the fundamentals of memristors are analyzed,including the classification,basic operations,switching mechanism and mathematical models.Among these,the boundary migration based model and conductive filament based model are studied.The mathematical formula of each model is derived.Secondly,stochastic memristor models are studied.Starting from the deterministic model,the framework of stochastic memristor model is established by substituting the state variable for the stochastic variable with certain probability distribution.Besides,the Markov assumption is made to govern that the distribution of the state variable in next time step relies only on the current time step.Based on which,the transfer matrix is introduced to describe the transition process for stochastic memristor model with discrete time and discrete status.The disrtribution in high resistance status and low resistance status along with the switching time are derived.Moreover,conductive filament based stochastic memristor model is proposed.Three microscopic forces,including electric field force,Fick force and Soret force are introduced.The transfer matrix of a conductive filament based memristor is demonstrated to be tridiagonal.The properties of the conductive filament based memristor model are analyzed and three conclusions are obtained.Firstly,the steady states are uniform distributed when the transfer matrix is symmetric.Secondly,the steady states will converge to the last state if the backward probability degenerates to zero.Thirdly,the rate of the convergence dependent on the second largest eigenvalue of the transfer matrix.Thirdly,fading memory property are analyzed for both deterministic and stochastic model.In deterministic model,the long-term evolution of the state variable is independent with its initial states.In stochastic model,the distribution after enough time steps does not rely on its initial distribution.After analyzing the property of the fading memory in DC and AC situation,the fading memory in any voltage without zero are proved to be exist.Finally,the algorithm for solving the transfer matrix of stochastic memristor model are proposed.The algorithm is divided into three parts.The first part is developed to find the eigenvalues of the transfer matrix according to the distribution of switch time.The second part creates a symmetric tridiagonal matrix with all of these eigenvalues.The last part of the algorithm aims at applying a similarity transformation to the symmetric tridiagonal matrix such that the feature vector of the matrix with the largest eigenvalue equals to the given steady state vector.The algorithm is simulated and proved to be accurate to solve the transfer matrix of the stochastic memristor model.