Theory of conductive filaments in threshold switches
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Theory of conductive filaments in threshold switches V. G. Karpov, M. Nardone, and M. Simon Department of Physics and Astronomy, University of Toledo, Toledo, OH 43606, USA ABSTRACT We show that the average parameters of conductive filaments and the related characteristics of threshold switches can be described thermodynamically based on the system free energy. In particular, we derive analytical expressions for the filament radius as a function of applied bias, and its current-voltage characteristics, the observations of which have remained without mathematical description for about 30 years. Our theory is extendible to filament transients and allows for efficient numerical simulations of arbitrary switching structures. This new understanding may be important in the advancement of novel technologies that combine threshold switches with phase change memory, such as 3D architectures. INTRODUCTION Chalcogenide glasses exhibit reversible switching between highly resistive (amorphous) and conductive (crystalline) phases when subjected to appropriate voltage pulses. This phenomenon recently regained interest in connection with phase change memory (PCM) applications [1]. Another application is found with threshold switches (TS), which require a minimum holding voltage or current to sustain the conductive state [2]. A new type of chalcogenide devices combines TS with phase change memory [3]. It is known that upon switching, a high-current filament forms [4], the radius of which increases with current as [5] r ∝ I 1/2 . This relation is often cited but remains poorly understood; at present, there is no theory relating the filament properties to material parameters. An approach based on the principle of least entropy production [6] did not lead to specific predictions. The validity of that principle remains questionable [7]. Here, we introduce a thermodynamic theory of a steady state conductive filaments. It predicts the filament radius vs. the electric current and material parameters as well as the corresponding current-voltage (IV) characteristics. THEORY Whether electronic [9, 8] or crystalline [10], the conductive filament represents a domain of different phase, thus calling upon the analysis of phase equilibrium. Our theory starts with the kinetic Fokker-Planck equation (see e.g. Ref. 11, p. 428) in the space of cylinder radii r , ∂f ∂s ∂f ∂ ⎛ f ⎞ = − , s ≡ − B + Af = − Bf 0 ⎜⎜ ⎟⎟. (1) ∂t ∂r ∂r ∂r ⎝ f 0 ⎠ Here, f is the distribution function so that f (r )dr gives the concentration of filaments in the interval (r , r + dr ) ; s is the flux in radii space (s −1 cm −3 ). B is the `filament radius diffusion coefficient'; A is connected with B by a relationship which follows from the fact that s = 0 for the equilibrium distribution f 0 (r ) ∝ exp(− F (r )/kT ) , where F is the free energy, k is Boltzmann's constant, and T is the temperature.
We note that the concept of free energy F that appears with the equilibrium distribution f 0 is not compromised by the fact that electric current flows through the filament, since tha
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