Theory of heterogeneous circuits with stochastic memristive devices
TTheory of heterogeneous circuits with stochastic memristive devices
Valeriy A. Slipko a) and Yuriy V. Pershin b) Institute of Physics, Opole University, Opole 45-052, Poland Department of Physics and Astronomy, University of South Carolina, Columbia,SC 29208 USA (Dated: 1 February 2021)
We introduce an approach based on the Chapman-Kolmogorov equation to model heterogeneous stochasticcircuits, namely, the circuits combining binary or multi-state stochastic memristive devices and continuumreactive components (capacitors and/or inductors). Such circuits are described in terms of occupation prob-abilities of memristive states that are functions of reactive variables. As an illustrative example, the seriescircuit of a binary memristor and capacitor is considered in detail. Some analytical solutions are found. Ourwork offers a novel analytical/numerical tool for modeling complex stochastic networks, which may find abroad range of applications.
I. INTRODUCTION
The well-known cycle-to-cycle variability of memris-tive devices is one of the major obstacles for their use innon-volatile memories and other potential applications.While, traditionally, the memristive devices (memristors)and their circuits have been mostly described in terms ofdeterministic models , there is a growing evidence thatat least in a certain large group of memristive devices(such as the electrochemical metallization (ECM) cells )the resistance switching occurs stochastically, namely, ata random time interval from the pulse edge . Theo-retically, the randomness in the resistance switching wasimplemented in several models .Mathematically, stochastic processes can be describedby introducing a noise term into the equations of mem-ristor dynamics. Usually these noise terms correspond toa white or, in a more complicated case, colored noise.In the case of linear equations (with respect to the in-ternal state variables), the noisy dynamics can be effec-tively analysed or even solved analytically. For non-linearstochastic differential equations, mainly numerical solu-tions can be found. This complicates tremendously theanalysis of general properties of processes that such equa-tions describe. In the cases when the stochastic processesare markovian, the evolution of probability distributionscan be described by a master equation, which was firstapplied to stochastic memristive networks by the presentauthors together with V. Dowling . However, the ap-plication range of such approach is limited to circuitscomposed of binary or multi-state stochastic memristors,current and/or voltage sources, and non-linear compo-nents such as diodes.The purpose of the present paper is to generalize themethod of master equation to more complex circuitsthat include also capacitors and/or indictors. Our maingoal is to understand the behavior of complex stochasticcircuits on average, as each particular realization of the a) [email protected] b) [email protected] circuit dynamics is not representative of the circuit be-havior overall (as it depends on the particular realizationof switching probabilities). The difficulty stems from thefact that in any particular realization of circuit dynam-ics, the switching of memristive components depends onthe values of reactive variables, and vice-versa. There-fore, the statistical description is also necessary for thedescription of reactive variables. For this purpose, weutilize a set of probability distribution functions.This paper is organized in the following sections. InSec. II, the model of binary and multi-state stochas-tic memristors is presented. Section III is the mainpart of this work, where the evolution equations fora binary memristor-capacitor circuit are derived fromthe Chapman-Kolmogorov equation (Subsec. III A), andtheir analytical solutions are found (Subsec. III B). Theapplication of Sec. III approach to more complex circuitsis discussed in Sec. IV that concludes. II. STOCHASTIC MEMRISTOR MODEL
In the present paper we consider discrete stochasticmemristors, whose resistance is characterized by G values R i , i = 0 , .., G − . The transitions between two states, i and j , are described in terms of the voltage-dependenttransition rates, γ i → j ( V M ) ≡ γ ij ( V M ) , where V M is thevoltage across the memristor. Only transitions betweenthe adjacent states are allowed (e. g., → , → , ...,at V M > , etc.). The transition from one boundary stateto another occurs sequentially through all intermediatestates.The binary stochastic memristors are the simplestcase of discrete stochastic memristors. Experiments haveshown that in some electrochemical metallization cellsthe switching can be described by a probabilistic modelwith voltage-dependent switching rates γ → ( V M ) = (cid:26) (cid:0) τ e − V M /V (cid:1) − , V M > otherwise , (1) γ → ( V M ) = (cid:26) (cid:0) τ e −| V M | /V (cid:1) − , V M < otherwise . (2) a r X i v : . [ c s . ET ] J a n Here, τ and V are constants. The interpretationof the above equations is following: The probability toswitch from R off ≡ R (state 0) to R on ≡ R (state 1)within the infinitesimal time interval ∆ t is γ → ( V M )∆ t .The probability to switch in the opposite direction is de-fined similarly. Equations similar to Eqs. (1)-(2) can beused for multi-state ( G > ) stochastic memristors . III. BINARY MEMRISTOR-CAPACITOR CIRCUIT
In this Section we consider the circuit shown inFig. 1(a), where M is a stochastic binary memristor (de-scribed by Eqs. (1) and (2)), and C is a linear capaci-tor. It is assumed that the circuit is driven by a time-dependent voltage V ( t ) . A. Dynamical equations
The circuit description is based on the probability dis-tribution functions, p i ( q, t ) , i = 0 , . The meaning ofthese functions is that p i ( q, t )∆ q is the probability tofind the memristor in state R M = R i , and capacitorcharge q in the interval from q to q + ∆ q at time t . Forthe full probabilistic description of memristive switchingit is necessary to introduce the transition probabilities P ( R k , q , t | R j , q , t ) ≡ P kj ( q , t | q , t ) from the mem-ristor state R j and capacitor charge q at time t to thememristor state R k and capacitor charge q at time t .The Chapman-Kolmogorov equation describing theMarkov process of stochastic switching can be written as p k ( q, t + τ ) = M − (cid:88) j =0 (cid:90) + ∞−∞ P kj ( q, t + τ | r, t ) p j ( r, t ) d r. (3)In order to satisfy the normalization condition at anytime t M − (cid:88) k =0 (cid:90) + ∞−∞ p k ( q, t ) d q = 1 , (4)the transition probabilities must satisfy for any r , j , t ,and τ > the corresponding conditions M − (cid:88) k =0 (cid:90) + ∞−∞ P kj ( q, t + τ | r, t ) d q = 1 . (5)It is easy to find the transition probabilities for in-finitesimally small τ > . There are four transition prob-abilities as the memristive system is binary. For example,the transitional probability for switching from R to R is equal to P ( q, t + τ | r, t ) = γ → (cid:16) V ( t ) − rC (cid:17) τ × δ (cid:18) q − r − τR (cid:16) V ( t ) − rC (cid:17)(cid:19) , τ → +0 , (6) (a) MC V ( t ) (b) qq p p D q FIG. 1. (a) Schematics of the binary memristor-capacitorcircuit. (b) Evolution scheme for p ( q, t )∆ q . The interval ∆ q is defined by two vertical lines. The arrows represent the flowof probability density. where V ( t ) − r/C is the voltage drop across the memris-tor, and the Dirac delta function represents the changein the capacitor charge that is defined by Kirchhoff’s law I ≈ q − rτ = 1 R (cid:16) V ( t ) − rC (cid:17) , τ → +0 . (7)Then, for the transitional probability from R to R (no switching) we can write (similarly to Eq. (6)) P ( q, t + τ | r, t ) = (cid:104) − γ → (cid:16) V ( t ) − rC (cid:17) τ (cid:105) × δ (cid:18) q − r − τR (cid:16) V − rC (cid:17)(cid:19) , τ → +0 . (8)The transitional probabilities P and P are obtainedby replacing → and → in Eqs. (6) and (8). Notethat such expressions for the transitional probabilitiessatisfy normalization conditions (5). Also we should notethat Eqs. (6) and (8) are valid only up to the first orderof magnitude of time τ , i.e. we should omit τ -terms orhigher while using these equations.By substituting Eqs. (6) and (8) into the Chapman-Kolmogorov equation (3) with k = 0 , and expanding itwith respect to τ up to the first order in magnitude,we get the following partial differential equation for theprobability distribution function p ( q, t ) ∂p ( q, t ) ∂t + ∂∂q (cid:18) V M R p ( q, t ) (cid:19) = γ → ( V M ) p ( q, t ) − γ → ( V M ) p ( q, t ) , (9)where V M = V ( t ) − q/C is the voltage across the mem-ristor.The other equation is obtained by replacing → and → in Eq. (9): ∂p ( q, t ) ∂t + ∂∂q (cid:18) V M R p ( q, t ) (cid:19) = γ → ( V M ) p ( q, t ) − γ → ( V M ) p ( q, t ) . (10)The system of Eqs. (9) and (10) must be supplementedwith the initial distributions: p ( q, t = 0) ≡ f ( q ) and p ( q, t = 0) ≡ g ( q ) .We note that Eqs. (9) and (10) can be consideredas generalized continuity equations. Schematically, theinterpretation of various terms in Eq. (9) is presentedin Fig. 1(b). The second term in the left-hand side ofEq. (9) describes the flow of probability density throughthe boundaries of a small charge interval ∆ q (the horizon-tal arrows in Fig. 1(b)). The right-hand side of Eq. (9)represents the flow of probability density between p ( q, t ) and p ( q, t ) (the vertical arrows in Fig. 1(b)). B. Analytical solutions
While the fundamental solution of Eqs. (9) and (10)delivers the complete description of circuit behavior, itcannot be found analytically in a closed form for an ar-bitrary V ( t ) . Thus we confine ourselves to some impor-tant particular cases, when such solution can be found.They are i ) the limit of small voltages when no transi-tion occurs, and ii ) unidirectional switching case, whentransitions go only from one state to another.
1. No switching case
For those moments of time t , capacitor charge q , andapplied voltage V ( t ) when no switching practically oc-curs, Eqs. (9)-(10) can be simplified to the followingindependent equations: ∂p ( q, t ) ∂t + ∂∂q (cid:20) V M R p ( q, t ) (cid:21) = 0 , (11) ∂p ( q, t ) ∂t + ∂∂q (cid:20) V M R p ( q, t ) (cid:21) = 0 . (12)The general solution of Eqs. (11)-(12) can be found bythe method of characteristics and presented as p ( q, t ) = e tCR f qe tCR − t (cid:90) e τCR V ( τ ) R d τ , (13) p ( q, t ) = e tCR g qe tCR − t (cid:90) e τCR V ( τ ) R d τ , (14)where f ( q ) and g ( q ) are two arbitrary functions (initialconditions). We reiterate that Eqs. (13) and (14) with p ( q,
0) = f ( q ) and p ( q,
0) = g ( q ) provide the full solu-tion in the case when the transitions between the statescan be neglected.To illustrate the above solution, we consider the step-like initial probability distribution p ( q,
0) = (cid:40) q β − q α , for q ∈ ( q α , q β ) , , otherwise , (15) p ( q,
0) = 0 . (16)Then in accordance with Eq. (13) we get the followingexpression for the charge probability distribution at any moment of time: p ( q, t ) = (cid:40) e tCR q β − q α , for q ∈ ( q α ( t ) , q β ( t )) , , otherwise , (17)(18)where q i ( t ) = q i e − tCR + (cid:90) t e τ − tCR V ( τ ) R d τ, (19)with i = α, β . Eq. (19) is similar to the time-dependenceof capacitor charge in the classical RC-circuit subjectedto the voltage V ( t ) .From Eq. (18) we see that the dissipative nature ofmemristor-capacitor circuit leads to the exponential nar-rowing of the probability distributions with time. At longtimes, t (cid:29) CR , the distribution approaches Dirac deltafunction p ( q, t ) = δ q − t (cid:90) e ( τ − t ) CR V ( τ ) R d τ . (20)It is clear Eq. (20) is valid for any initial distribution f ( q ) in the absence of resistance switching events.Moreover, the initially deterministic state will remaindeterministic in the absence of switchings. For q ( t = 0) = q and p ( q, t = 0) = 0 , Eqs. (13) and (14) lead to p ( q, t ) = δ q − q e − tCR − t (cid:90) e ( τ − t ) CR V ( τ ) R d τ , (21) p ( q, t ) = 0 . (22)According to Eq. (21), the evolution of the capacitorcharge is q ( t ) = q e − tCR + t (cid:90) e ( τ − t ) CR V ( τ ) R d τ. (23)
2. Unidirectional switching
Next we consider the region in q − t phase plane, wherethe transitions from R to R are forbidden, but the op-posite transitions may occur. In this situation Eqs. (9)and (10) can be rewritten as ∂p ( q, t ) ∂t + ∂∂q (cid:20) V M R p ( q, t ) (cid:21) = − γ → ( V M ) p ( q, t ) , (24) ∂p ( q, t ) ∂t + ∂∂q (cid:20) V M R p ( q, t ) (cid:21) = γ → ( V M ) p ( q, t ) . (25)Eqs. (24) and (25) are valid when γ → ( V M ) = 0 . Thegeneral solution of Eqs. (24) and (25) can be derived byusing the method of characteristics . As a result we get p ( q, t ) = e tCR f qe tCR − t (cid:90) e τCR V ( τ ) R d τ exp − t (cid:90) γ → V (˜ t ) − qC e ( t − ˜ t ) CR − e − ˜ tCR CR t (cid:90) t e τCR V ( τ ) d τ d ˜ t , (26) p ( q, t ) = e tCR g (cid:18) qe tCR − (cid:90) t e τCR V ( τ ) R d τ (cid:19) + t (cid:90) d ˜ tγ → V (˜ t ) − qC e ( t − ˜ t ) CR − CR e − ˜ tCR ˜ t (cid:90) t d τ e τCR V ( τ ) e ( t − ˜ t ) CR p qe ( t − ˜ t ) CR + ˜ t (cid:90) t d τ e ( τ − ˜ t ) CR V ( τ ) R , ˜ t , (27) p (cid:6) t (cid:7) p (cid:6) t (cid:7) t (cid:6) s (cid:7) p i FIG. 2. Probabilities to find the memristive devices in thestate 0 and state 1 in the binary memristor-capacitor circuitsubjected to a constant voltage V a . This plot was obtainedusing Eq. (30) with parameter values C = 1 µ F, R = 100 k Ω , V a = 0 . V, V = 0 . V, τ = 3 · s, q = 0 . which are valid in the region q < CV ( t ) , where γ → ( V M ) = 0 , and f ( q ) and g ( q ) are two arbitrary func-tions.By the interchanging indexes → and → in Eqs.(26) and (27) we can also find the solutions of Eqs. (9)and (10) for the region q > CV ( t ) , where γ → ( V M ) = 0 .Note that if there are no transitions at all ( γ → ≡ ),then the results (26) and (27) coincide with Eqs. (13) and(14).As an example of the theory above let us consider thecase of deterministic initial conditions and constant ap-plied voltage: q ( t = 0) = q , p ( q, t = 0) = 0 , and V ( t ) = V a . Then, Eq. (26) simplifies to p ( q, t ) = δ ( q − q ( t )) e − CR τ (cid:20) Ei (cid:18) Va − qCV e tCR (cid:19) − Ei (cid:18) Va − qCV (cid:19)(cid:21) , (28)where Ei ( x ) is the exponential integral function and q ( t ) = q e − tCR + V a C (cid:16) − e − tCR (cid:17) . (29)The integral of Eq. (28) over q from minus to plus infinitygives the probability to find the memristor in the state 0at time t : p ( t ) = e − CR τ (cid:20) Ei (cid:18) Va − q CV (cid:19) − Ei (cid:18) Va − q CV e − tCR (cid:19)(cid:21) , (30) where q ( t ) is defined by Eq. (29). We note that p ( t ) canbe found from p ( t ) + p ( t ) = 1 .Fig. 2 shows the evolution of p ( t ) and p ( t ) = 1 − p ( t ) found with the help of Eq. (30). While it may look thatthe memristor switching is incomplete on average, in fact,the circuit dynamics is a two-time-scale process where thefast initial relaxation (such as the visible one in Fig. 2)transforms into a very slow one occurring on the timescale of τ (as a consequence of Eq. (1)). Clearly, thefast initial relaxation decelerates with time as the voltagebuilds up across the capacitor.The mean switching time for the dynamics in Fig. 2can be calculated using (cid:104) T (cid:105) = 1 p ( t ∗ ) t ∗ (cid:90) t d p ( t ) d t d t, (31)where t ∗ is the characteristic saturation time for the fastinitial dynamics of p i ( t ) . Taking t ∗ = 1 s, one can findthat (cid:104) T (cid:105) = 5 . ms, which is in perfect agreement withthe behavior in Fig. 2. The exponential relaxation withthe mean switching time (cid:104) T (cid:105) provides an excellent ap-proximation for the initial interval of fast evolution of p ( t ) , see Fig. 3. On long times, the relaxation can beapproximated by p ( t ) (cid:39) p ( t ∗ ) exp ( − t/τ ) .Moreover, by using Eq. (30) one can find p ( t ∗ ) in themost interesting parameter region ( V a − q /C ) (cid:29) V corresponding to the initial interval of fast evolution.This is accomplished employing the asymptotic expan-sion of the exponential integral function Ei ( x ) = e x / ( x − (cid:0) O (1 /x ) (cid:1) , when x → + ∞ . For the time interval (cid:104) T (cid:105) (cid:46) t (cid:28) τ , we can omit the second term in the squarebrackets in Eq. (30) and use the asymptotic expansion forthe first one. This leads us to the following expressionfor the probability of no switching event p ( t ∗ ) = exp − CR τ e (cid:16) Va − q /CV (cid:17) V a − q /CV − . (32)For the same parameter values as in Fig. 2, Eq. (32)gives p ( t ∗ ) ≈ . , which is in an excellent agreementwith the result p ( t ∗ ) = 0 . obtained numerically fromEq. (30) . p (cid:6) t (cid:7) Exp (cid:17)(cid:10) t (cid:12)(cid:13) T (cid:15)(cid:18)(cid:6) (cid:10) p (cid:6) t (cid:8) (cid:7)(cid:7)(cid:9) p (cid:6) t (cid:8) (cid:7) t (cid:6) s (cid:7) p i FIG. 3. The probability to find the memristive device in state0, p ( t ) is well approximated by an exponential decay curvebased on (cid:104) T (cid:105) (Eq. (31)). The exponential decay curve wasobtained using (cid:104) T (cid:105) = 5 . ms and t ∗ = 1 s. The probability p ( t ) curve is the same as in Fig. 2. IV. DISCISSION
The method used to model the memristor-capacitorcircuit above can be straightforwardly extended to othercircuits. Consider, for instance, a circuit composed of
N G -state stochastic memristors, K capacitors and J inductors. In the general case, the simulation of sucha circuit requires G N probability distribution functionsof K + J variables and time. Circuits with symmetriesmay require less functions to represent their states (seeRef. 11 for examples). The number of independent reac-tive variables can be less than K + J . For example, ifthe external voltage E ( t ) is applied directly across somecapacitor, then its charge is not an independent variable.It is anticipated that the general evolution equationcan be written similarly to Eqs. (9) and (10). Intro-ducing the sets of capacitive and inductive variables, Q and I , the evolution equation for a particular state i isformulated as ∂p i ( Q, I, t ) ∂t + (cid:88) j ∂∂Q j (cid:104) ˙ Q j p i ( Q, I, t ) (cid:105) + (cid:88) j ∂∂I j (cid:104) ˙ I j p i ( Q, I, t ) (cid:105) = (cid:88) j (cid:54) = i [ γ ji p j ( Q, I, t ) − γ ij p i ( Q, I, t )] , (33)where the memristive switching rates corresponds to theflip of a single memristor and depend on the voltageacross the same memristor in a particular circuit con-figuration. In order to close Eq. (33) we need to expressthe full derivatives ˙ Q j and ˙ I j as functions of R, Q, I byusing the Kirchhoff’s circuit laws. If there are no tran-sitions, then the RHS of Eq. (33) turns to zero and thisequation coincides with the continuity equation for theprobability density p i ( Q, I, t ) as it should be.In conclusion, we have introduced a powerful analyticalapproach to model heterogeneous stochastic circuits. Asimple example was considered in detail and the recipe toapply the approach to other circuits has been formulated.Compared to the traditional Monte Carlo simulations,the proposed approach can be used to derive analyticalexpressions describing the circuit dynamics on average.The data that support the findings of this study areavailable from the corresponding author upon reasonablerequest. L. O. Chua and S. M. Kang, Proc. IEEE , 209 (1976). S. Kvatinsky, M. Ramadan, E. G. Friedman, and A. Kolodny,IEEE Transactions on Circuits and Systems II: Express Briefs , 786 (2015). Y. V. Pershin, S. La Fontaine, and M. Di Ventra, Phys. Rev. E , 021926 (2009). J. P. Strachan, A. C. Torrezan, F. Miao, M. D. Pickett, J. J.Yang, W. Yi, G. Medeiros-Ribeiro, and R. S. Williams, IEEETransactions on Electron Devices , 2194 (2013). I. Valov, R. Waser, J. R. Jameson, and M. N. Kozicki, Nanotech-nology , 254003 (2011). S. H. Jo, K.-H. Kim, and W. Lu, Nano letters , 496 (2009). S. Gaba, P. Sheridan, J. Zhou, S. Choi, and W. Lu, Nanoscale , 5872 (2013). S. Gaba, P. Knag, Z. Zhang, and W. Lu, in (IEEE,2014) pp. 2592–2595. S. Menzel, I. Valov, R. Waser, B. Wolf, S. Tappertzhofen, andU. Böttger, in (2014) pp. 1–4. R. Naous and K. N. Salama, in
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