A Nonlinear Drift Memristor Model with a Modified

electronics

Article A Nonlinear Drift Memristor Model with a Modiﬁed Biolek Window Function and Activation Threshold

Valeri Mladenov and Stoyan Kirilov * ID

Department of Theoretical Electrical Engineering, Technical University Sofia, Faculty of Automatics, 1000 Sofia, 8 St. Kl. Ohridski Blvd, Bulgaria; valerim@tu-sofia.bg * Correspondence: s_kirilov@tu-sofia.bg; Tel.: 00359-2-965-33-19

Received: 28 August 2017; Accepted: 29 September 2017; Published: 3 October 2017

Abstract: The main idea of the present research is to propose a new memristor model with a highly nonlinear ionic drift suitable for computer simulations of titanium dioxide memristors for a large region of memristor voltages. For this purpose, a combination of the original Biolek window function and a weighted sinusoidal window function is applied. The new memristor model is based both on the Generalized Boundary Condition Memristor (GBCM) Model and on the Biolek model, but it has an improved property—an increased extent of nonlinearity of the ionic drift due to the additional weighted sinusoidal window function. The modified memristor model proposed here is compared with the Pickett memristor model, which is used here as a reference model. After that, the modified Biolek model is adjusted so that its basic relationships are made almost identical with these of the Pickett model. After several simulations of our new model, it is established that its behavior is similar to the realistic Pickett model but it operates without convergence problems and due to this, it is also appropriate for computer simulations. The modified memristor model proposed here is also compared with the Joglekar memristor model and several advantages of the new model are established.

Keywords: memristor; nonlinear ionic drift; modiﬁed Biolek window function; sinusoidal weighted window function; sensitivity threshold

1. Introduction Background and concept. The memristor (abbreviated from memory + resistor) is the fourth electrical passive two-terminal element alongside with the resistor, inductor, and capacitor [1,2]. The memristor is a nonlinear electrical element [1]. It directly relates the electric charge q and the magnetic flux linkage Ψ [1,2]. Leon Chua forecast the memristor element in 1971 in accordance to symmetry considerations and the relationships between the four basic electrical quantities—voltage, current, charge, and magnetic flux [1]. The memristor element has the property for memorizing the amount of charge passed through its intersection (in given limits depending on the geometrical dimensions of the element) and its respective resistance (conductance), which is proportional to the electric charge accumulated in the memristor when the electric sources are switched off [2]. The Weber-Coulomb relationship of the memristor is a nonlinear single-valued curve, while its respective current-voltage relationship is a pinched-hysteresis loop, which shape and area depend on the amplitude and on the frequency of the applied voltage [2]. Due to the effect of memorizing the resistance of the memristor after switching off the electrical sources, it could be used as a non-volatile memory element [2]. Its physical prototype made by titanium dioxide [2–4] has physical dimensions in the nano scale range and its power consumption is several times lower than the existing flash memories [2]. These facts are good preconditions for the future applications of the memristor matrices in computer memories [2]. The memristor element is a good candidate for application

Electronics 2017, 6, 77; doi:10.3390/electronics6040077 www.mdpi.com/journal/electronics Electronics 2017, 6, 77 2 of 15 in the neuromorphic engineering, in the synapses in the artificial neural networks, in the analog and digital electronics as well [2]. The first physical prototype of the memristor element predicted by L. Chua in 1971 [1] was invented in the HP research labs by Stanley Williams research team in 2008 [2]. After this flash, many scientific papers associated with memristors and memristive devices have been published and several basic memristor models have been proposed [2–10]. Each of the basic memristor models—the linear drift model proposed by Strukov and Williams [2] and the nonlinear drift models made by Joglekar [3], Pickett [4,5], and Biolek [6]—is appropriate for specific electrical modes for the operation of the memristor elements. The linear ionic drift model [2] is used for low memristor voltages and the nonlinear drift models of Biolek and Joglekar are able to represent the memristor nonlinear drift behavior for high voltages when the memristor element has highly nonlinear performance [3,6]. The Pickett memristor model [4,5] is based on laboratory experiments and measurements, and on the current flow through a tunnel barrier. It has the highest accuracy and sometimes it is used as a reference model [4,5], but it is very complicated and it is not always appropriate for computer simulations due to convergence problems. The Boundary Condition Memristor (BCM) Model is with linear dopant drift and switch-based window function for representation the boundary effects [7]. It is able to represent both the soft-switching and hard-switching memristor modes [7]. The Generalized BCM model (GBCM) created by Ascoli, Corinto, and Tetzlaff, is almost identical with the BCM model but the difference is the use of activation threshold not only for the boundaries but also for every value of the state variable [8]. The lack of a realistic versatile memristor model suitable for simulations, having a nonlinear dopant drift and accuracy near to the Pickett model was the motivation of the present research. The new model proposed in this paper is based both on Biolek and on the GBCM models but it uses a combination of a Biolek window function and an additional weighted sinusoidal window function for increasing the nonlinearity of the ionic drift. The new ideas of the memristor model proposed here are that the weight coefficient in front of the additional sinusoidal window function component and the respective extent of nonlinearity of the memristor dopant drift could be changed and adjusted till the basic relationship of the memristor are almost near to these obtained by the Pickett model for the same conditions. If the activation threshold and the coefficient in front of the weighted sinusoidal window function are equal to zero the original Biolek model is obtained. The Biolek memristor model is a special case of the modified Biolek model. The boundary effects observable for hard-switching mode are represented using the switch-based algorithm of the GBCM model. After comparison with the Pickett memristor model the modified Biolek model was adjusted so its current-voltage and the Weber-Coulomb relationships were obtained similar to the Pickett reference model. The precision of the modified Biolek model is near to the accuracy of the Pickett memristor model but the computer tests confirmed that the modified Biolek model does not have convergence problems and it is appropriate for computer simulations of memristors and memristive circuits. In Section2, the description of the titanium-dioxide memristor element according to the Pickett Tunnel Barrier model is presented and the main relationships are obtained. The basic ideas and the major relationships of the modified Biolek model are shown in Section3. The pseudo-code based algorithm of the modified Biolek model for computer simulations is presented in Section4. The comparison with the Pickett memristor model, the adjusting process of the modified Biolek model, and the basic results are presented and discussed in Section5. A comparison of the model proposed here with Joglekar memristor model is presented and discussed in Section6. The concluding remarks are presented in Section7.

2. A Brief Description of the Memristor According to the Pickett Tunnel Barrier Model The structure of a memristor cell according to the Pickett’s model [5] is shown in Figure1. The electrodes are made of platinum and the insulating layer is made of pure titanium dioxide. The conducting channel in the memristor cell is formed by a thin layer of doped with oxygen vacancies Electronics 2017, 6, 77 3 of 15

titanium dioxide material. There is also a thin tunnel barrier with a length of w. The resistance of the Electronics conducting2017, 6, 77 layer is approximately equal to Rs = 215 Ω [4,5]. 3 of 14

FigureFigure 1.Memristor 1. Memristor structure structure according according to to the the Pickett Pickett model. model.

When theWhen memristor the memristor is switched is switched in in open open state ( i(i>>0), 0), thethe differential differential equation equation has to has be expressed to be expressed with (1) with[5]. Formulas (1) [5]. Formulas (2)–(11) (2)–(11) are arealso also according according to [5[5].]. ! dw i w − ao f f |i| w dw= fo f f sinh  iexp −exp wa − i − w (1) dt f sinhio f f exp  exp wc off b  wc off   (1) dt ioff w c b w c The parameters used in (1) and their corresponding values [5] are given in Table1.

The parameters used in (1)Table and 1.theirQuantities corresponding for open switched values memristor. [5] are given in Table 1.

Quantity f i a b w Table 1. Quantitiesoff for openoff switchedoff memristorc . Dimension µm/s µA nm µA pm Value 3.5 115 1.20 500 107 Quantity foff ioff aoff b wc Dimension µm/s µA nm µA pm WhenValue the memristor is3.5 switched in closed115 state (i < 0)1.20 [5], the differential500 equation is:107 dw i w − aon |i| w = fonsinh exp − exp − − (2) When the memristor isdt switched in iclosedon state (i< 0)w [5]c , theb differentialwc equation is:

The parameters and their values [5] are given in Table2.  dw iwa on i w Tablefon 2.sinhQuantities exp for closed  exp switched memristor.   (2) dt ion w c b w c

Quantity fon ion aon b wc The parameters and their valuesDimension [5] areµm/s givenµA in Table nm 2µ.A pm Value 40 8.9 1.80 500 107

Table 2. Quantities for closed switched memristor.

Quantity fon ion aon b wc Dimension µm/s µA nm µA pm Value 40 8.9 1.80 500 107

The current flowing through the tunnel barrier of the memristor element [5] is:

jA i0  exp  B     u exp  B   u (3) w2 1 1  1gg  1  where ug is the voltage drop over the tunnel barrier [5]. This voltage drop could be expressed using the Kirchhoff’s Voltage Law (KVL) and the voltage u over the memristor element [5,11]:

ugs u iR (4)

Electronics 2017, 6, 77 4 of 15

The current ﬂowing through the tunnel barrier of the memristor element [5] is:

j0 A h p p i i = Φ exp −B Φ − Φ + ug exp −B Φ + ug (3) ∆w2 1 1 1 1 where ug is the voltage drop over the tunnel barrier [5]. This voltage drop could be expressed using the Kirchhoff’s Voltage Law (KVL) and the voltage u over the memristor element [5,11]:

ug = u − iRs (4)

The constant j0 [5] could be expressed as follows:

e 1.6 · 10−19 j = = = 3.84 · 1013, [C/(J · s)] (5) 0 2πh 2 · 3.14 · 6.63 · 10−34 where e is the elementary charge of the electron, h is the Planck’s constant. The area of the tunnel junction [5] is equal to A = 104 nm2. The deviation of the length of the tunnel junction [5] is:

∆w = w2 − w1 (6)

The second term of (6) [5] is: λw 2 w1 = 1.2 , nm (7) Φ0 where Φ0 = 0.95 V is the height of the potential barrier. The quantity λ [5] from Equation (7) is expressed as follows: e ln 2 = λ −9 , V (8) 8πkε0w · 10 −12 where k = 5 is the permittivity of the titanium dioxide material, ε0 = 8.85·10 F/m [5] is the absolute permittivity of the vacuum space. After transformation of (7) it is obtained that w1 = 0.126 nm. Using (6) and according to [5] the following Equation (9) is derived: ! 9.2 · λ w2 = w1 + w 1 − , nm (9) 3Φ0 + 4λ − 2 ug

The quantity B [5] is expressed as follows: √ −9 4π∆w · 10 2me − 1 B = , V 2 (10) h where m is the electron weight. The quantity Φ1 [5] is: w1 + w2 1.15 · λw w2(w − w1) Φ1 = Φ0 − ug − ln , V (11) w ∆w w1(w − w2)

The Pickett memristor model [5] is fully described with Equations (1)–(11). A Simulation Program with Integrated Circuit Emphasis (SPICE) code of Pickett model [5] is created using (1)–(11) and it is used for simulation of the memristor element. The memristor circuit under test is presented in Figure2. It is created of a sinusoidal voltage source, a series resistor with very small resistance (R1 = 1 Ω), a difference devices, and integrators. The magnetic ﬂux linkage is obtained as a time integral of the voltage drop across the memristor, and the electric charge is the time integral of the memristor current and, respectively, the voltage drop across the resistor R1 [11]. ElectronicsElectronics 2017, 6, 201777 , 6, 77 5 of 15 5 of 14

U1 1 2 PLUSMINUS

MODELMEMRISTOR

V1 VOFF = 0 IN1 Electronics 2017, 6, 77 VAMPL = 0.6 IN2 OUT 1.0 psi 5 of 14 R1 FREQ = 1 1 0V AC = 0

U1 1 2 PLUSMINUS

MODELMEMRISTOR 0

IN1 V1 VOFF = 0 IN1 IN2 OUT 1.0 q VAMPL = 0.6 IN2 OUT 1.0 psi R1 0V FREQ = 1 1 0V AC = 0 Figure 2.FigureMemristor 2. Memristor circuit circuit for forsimulation simulation in in Simulation Simulation Program Program with with Integrated Integrated Circuit Circuit Emphasis Emphasis (SPICE)(SPICE) environment environment.. 0

IN1 Figure3 was the Weber-Coulomb relationshipIN2 OUT according1.0 to Pickettq model and the I-V relationship of the memristor element according to Pickett model. The sinusoidal0V voltage signal generated by the voltageFigure source 2.Memristor and used circuit for thefor simulation computer in simulation Simulation is: Programu(t) = 0.6with·sin(2 Integrated·π·1·t). Circuit For the Emphasis simulation OrCAD(SPICE PSpice) environment ver. 16.3. Demo was used.

(a) (b) Figure 3.(a) Weber-Coulomb relationship according to Pickett model; (b) I-V relationship of the memristor element according(a) to Pickett model, u(t) = 0.6∙sin(2∙π∙1∙t). (b) Figure 3.(a) Weber-Coulomb relationship according to Pickett model; (b) I-V relationship of the Figurememristor 3. ( elementa) Weber-Coulomb according to relationship Pickett model, according u(t) = 0.6∙sin(2∙π∙1∙ to Pickett model;t). (b) I-V relationship of the 3. Basic Ideasmemristor and the element Main according Relationships to Pickett model, of theu(t) Modified= 0.6·sin(2·π· 1Biolek·t). Model The3. modifiedBasic Ideas memristorand the Main model Relationships proposed of the here Modified will Biolek be discussed Model using the titanium-dioxide 3. Basic Ideas and the Main Relationships of the Modiﬁed Biolek Model memristor nanostructureThe modified memristor described model in proposed [2] by Strukov here will beand discussed Williams. using A the graph titanium of -dioxide the memristor structurememristor accordingThe modiﬁed nanostructure to the memristor model described proposed model in proposed [2] by by Strukov Strukov here will and and be Williams discussedWilliams. usingAis presented graph the of titanium-dioxide the in memristor Figure 4. The left memristorstructure according nanostructure to the described model proposed in [2] by by Strukov Strukov and and Williams. Williams A graphis presented of the memristorin Figure 4. structure The left region of the TiO2 memristor structure with a length of l is partially doped with oxygen vacancies accordingregion of the to the TiO model2 memristor proposed structure by Strukov with anda length Williams of l is presentedpartially doped in Figure with4. Theoxygen left regionvacancies of using an initial electroforming process with a high constant voltage and has low resistance [2]. The theusing TiO an2 memristor initial electroforming structure with process a length with of lais high partially constant doped voltage with oxygen and has vacancies low resistance using an [2]. initial The 2 second subelectroformingsecond-layer sub -oflayer the process of memristor the with memristor a high element constantelement isis voltage preparedprepared and of has of pure lowpure TiO resistance TiO2 and andit [2has]. Theit very has second high very resistance sub-layer high resistance [2]. The of[2].length the The memristor lengthof the of wholeelement the whole memristor is preparedmemristor ofnanostructure nanostructure pure TiO2 and is it is denoted has denoted very with high with resistanceD and D hasand [a2 ].valuehas The a lengthof value 10 nm of of [2]. the 10 nm [2]. whole memristor nanostructure is denoted with D and has a value of 10 nm [2].

Figure 4. Titanium dioxide memristor nanostructure.

Electronics 2017, 6, 77 5 of 14

U1 1 2 PLUSMINUS

MODELMEMRISTOR

V1 VOFF = 0 IN1 VAMPL = 0.6 IN2 OUT 1.0 psi R1 FREQ = 1 1 0V AC = 0

IN1 IN2 OUT 1.0 q 0V Figure 2.Memristor circuit for simulation in Simulation Program with Integrated Circuit Emphasis (SPICE) environment.

(a) (b) Figure 3.(a) Weber-Coulomb relationship according to Pickett model; (b) I-V relationship of the memristor element according to Pickett model, u(t) = 0.6∙sin(2∙π∙1∙t).

3. Basic Ideas and the Main Relationships of the Modified Biolek Model The modified memristor model proposed here will be discussed using the titanium-dioxide memristor nanostructure described in [2] by Strukov and Williams. A graph of the memristor structure according to the model proposed by Strukov and Williams is presented in Figure 4. The left region of the TiO2 memristor structure with a length of l is partially doped with oxygen vacancies using an initial electroforming process with a high constant voltage and has low resistance [2]. The second sub-layer of the memristor element is prepared of pure TiO2 and it has very high resistance Electronics 2017, 6, 77 6 of 15 [2]. The length of the whole memristor nanostructure is denoted with D and has a value of 10 nm [2].

FigureFigure 4. Titanium 4. Titanium dioxide dioxide memristor nanostructure. nanostructure.

The normalized length of the doped layer, also known as the state variable x of the memristor [2,3],

could be deﬁned with the following formula (12):

l x = (12) D The equivalent resistance of the memristor element could be expressed using the assumption Electronics for2017 series, 6, 77 connection of the doped and the un-doped regions [2,3] and the substituting circuit given 8 of 14 in Figure5: If m = 0 and if the activationR = Rthresholddoped + Run− ofdoped the= RmemristorON x + ROFF u(1thr−= x0) the modified Biolek(13) model is transformed into the original Biolek model. So it could be concluded that the original Biolek where RON =100 Ω and ROFF = 16 kΩ are the memristances of the memristor for fully-closed and memristorfully-open model statesis a special [2,7], respectively, case of the for modifiedx = 1 or x = Biolek 0. model proposed in this paper.

Figure Figure 5. A s5.ubstitutingA substituting circuit circuit of of the the titanium titanium dioxide dioxide memristor memristor element element presenting presenting the the state-dependentstate-dependent resistances resistances of the of the memristor memristor layers layers..

4. A Pseudo-TheCode current-voltage Algorithm relationship for Simulation could be of expressed the Modified using (13) Biolek and Ohm’s Model law [11]:

The pseudo-code algorithm presentedu = Ri = [ R belowON x + R isOFF based(1 − x )] oni Equation (27) and its(14) numerical solution with the use of the finite differences method in MATLAB [13]. The readers could use this The voltage drop across the doped region of the memristor element ul [11] is: code if they want to simulate memristors. l u = R i = R xi = R i (15) 1. The basic code used for simulations ofl thedoped modifiedON Biolek modelON D 2. Constants and parameters: qmax=0.0001; eta=1; um=3.3; f=2.01; psiu=-pi/3; Ron=100; 3. Roff=16000; deltaR=Ron-Roff; mu=1e-14; D=10e-9; k=(mu*Ron)/(D^2); x0=0.3; 4. xmin=0; xmax=1; [u,t,deltat,tmin,tmax,N]=sine_gen(um,f,psiu); 5. n=1:1:N+1; flux=integr(u,deltat,tmin,tmax); 6. [x,biolekwinmod]=biolek_mod(deltat,u,k,eta,deltaR,Roff,x0,xmax,xmin,N); 7. Req=deltaR*x+Roff; iM=u./Req; q=qmax*x; 8. Function for generation of sinusoidal voltage signal 9. function [u,t,deltat,tmin,tmax,N] = sine_gen(um,f,psiu) 10. Time domain: T=1/f; tmin=0; tmax=8*T; deltat=(tmax-tmin)/1e5; t1=tmin:deltat:tmax; 11. omega=2*pi*f ;f1=um*sin(omega*t1+psiu); u=f1;t=t1;N=tmax/deltat; end function. 12. Function for integrating the memristor voltage 13. function psi=integr(u,deltat,tmin,tmax) 14. N=(tmax-tmin)/deltat; n=1:1:N+1; psi=[ ]; 15. for n=1; psi1(n)=0; end; 16. for n=2:1:N+1; psi1(n)=psi1(n-1)+u(n-1)*deltat; end; 17. psi=[psi psi1]; end function. 18. Function for deriving the state variable and the modified Biolek window 19. function [x,biolekwinmod]=biolek_mod(deltat,u,k,eta,deltaR,Roff,x0,xmax,xmin,N) 20. Parameters: p=2; m=0.2; x=[ ]; biolekwinmod=[ ]; for n=1; x1=x0; biolekmod1=1; end; 21. for n=2:1:N+1; A=deltaR*x1(n-1)+Roff; if abs(u(n))<0.1; then x1(n)=x1(n-1); 22. if u(n)<=0; then biolekmod1(n)=(((1-(x1(n)-1)^(2*p))+m*(sin(pi*x1(n)))^2))/(1+m); 23. else biolekmod1(n)=(((1-(x1(n))^(2*p))+m*(sin(pi*x1(n)))^2))/(1+m); end; 24. else; if u(n)<=0; then x1(n)=x1(n-1)+(eta*k*u(n-1)*deltat*(1/A)*(((1-(x1(n-1)-1)^(2*p))+... 25. m*(sin(pi*x1(n-1)))^2))/(1+m)); biolekmod1(n)=(((1-(x1(n)-1)^(2*p))+m*(sin(pi*x1(n)))^2))/(1+m); 26. if x1(n)<=xmin & u<=0; then x1(n)=xmin; else;x1(n)=x1(n-1)+(eta*k*u(n-1)*deltat*(1/A)*(((1-(x1(n-1)-1)... 27. ^(2*p))+m*(sin(pi*x1(n-1)))^2))/(1+m)); biolekmod1(n)=(((1-(x1(n)-1)^(2*p))+m*(sin(pi*x1(n)))^2))/(1+m); 28. end; else x1(n)=x1(n-1)+(eta*k*u(n-1)*deltat*(1/A)*(((1-(x1(n-1))^(2*p))+m*(sin(pi*x1(n-1)))^2))/(1+m)); 29. biolekmod1(n)=(((1-(x1(n))^(2*p))+m*(sin(pi*x1(n)))^2))/(1+m); if x1(n)>=xmax & u(n)>0; x1(n)=xmax; 30. else x1(n)=x1(n-1)+(eta*k*u(n-1)*deltat*(1/A)*(((1-(x1(n-1))^(2*p))+m*(sin(pi*x1(n-1)))^2))/(1+m)); 31. biolekmod1(n)=(((1-(x1(n))^(2*p))+m*(sin(pi*x1(n)))^2))/(1+m); end; 32. end; end; end; x=[x x1]; biolekwinmod=[biolekwinmod biolekmod1]; end function.

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The electric ﬁeld intensity in the doped layer of the memristor element El is [11,12]:

u R E = l = i ON (16) l l D The rate of moving the boundary between the doped and the un-doped layers of the memristor element [12] is: dl d dx v = = (xD) = D = µE (17) dt dt dt l where µ = 1·10−14 m2/(V·s) is the mobility of the oxygen vacancies [2,7]. After transformation of (17) and using (16) the following Equation (18) is derived:

dx µR = ON i = ki (18) dt D2 where k is a constant dependent only on memristor parameters . Using (18), the maximal amount of electric charges qmax that the memristor accumulates could be calculated [11] when x = 1:

τ 1 2 Z 1 Z 1 D2 10 · 10−9 = ( ) = = = = = · −4 qmax i t dt dx −14 1 10 C (19) k k µRON 1 · 10 · 100 0 0 where the time for charging the memristor τ depends on the current intensity i. Using (18) and (19) it could be derived that the electric charge accumulated in the memristor element is proportional to the maximal amount of electric charge qmax and the state variable x [11]:

τ1<τ x Z 1 Z q(x) = i(t)dt = dy = q x = 10−4 · x (20) k max 0 0 where in the integral in the right side of (20) the state variable x is substituted with the variable y, and x is the upper limit of the deﬁnite integral. When two or more memristor elements are connected in an electric circuit [7,8] formula (18) is modiﬁed and Equation (21) is derived:

dx µR = η ON i = ηki (21) dt D2 where η is a polarity coefﬁcient [7,8]. When the memristor element is forward-biased then η = 1. For a reverse-biased memristor η = −1 [7,8]. Formula (21) is valid only for very small electrical currents and memristor voltages [2]. For representing the phenomenon of a nonlinear ionic dopant drift in the general case an additional nonlinear window function f(x) has to be used in the right side of Equation (21) [3,6,7] and the following formula is derived (22):

dx µR = η ON i f (x) = ηki f (x) (22) dt D2 Equations (14) and (22) fully describe the general memristor model. Several different basic window functions are used in the memristor modeling [3,6,7]. A very important and frequently used window function is [6]: 2p fB(x, i) = 1 − [x − stp(−i)] (23)

The function expressed with (23) is presented for ﬁrst use by Biolek in [6], and it is also known as a Biolek window function [6]. The function stp(i) [6] used in (23) is: Electronics 2017, 6, 77 8 of 15

( 1, i f i ≥ 0 (u ≥ 0) stp(i) = (24) 0, i f i < 0 (u < 0)

After substitution of (24) [6] in (23) the following equation is derived:

2p fB(x) = 1 − (x − 1) , u(t) ≤ 0, [i(t) ≤ 0] 2p (25) fB(x) = 1 − x , u(t) > 0, [i(t) > 0]

The original Biolek model [6] is fully described with Equations (14), (22), and (25). Here a simple modification of (25) is made. To increase the nonlinearity extent of the modified Biolek model, the authors proposed an additional weighted sinusoidal window function of the state variable x: 2p 1−(x−1) +m(sin2(πx)) fBM(x) = 1+m , u(t) ≤ 0 (26) 1−x2p+m(sin2(πx)) fBM(x) = 1+m , u(t) > 0 where m is in the interval [0, 1] and fBM is the modified window function. The modified Biolek model proposed in this paper is fully described with the following Equation (27):

2p 2 dx 1−(x−1) +m(sin (πx)) dt = ηki 1+m , u(t) ≤ 0

2p 2 dx 1−x +m(sin (πx)) (27) dt = ηki 1+m , u(t) > 0

u = Ri = [RON x + ROFF(1 − x)]i

If m = 0 and if the activation threshold of the memristor uthr = 0 the modiﬁed Biolek model is transformed into the original Biolek model. So it could be concluded that the original Biolek memristor model is a special case of the modiﬁed Biolek model proposed in this paper.

4. A Pseudo-Code Algorithm for Simulation of the Modiﬁed Biolek Model The pseudo-code algorithm presented below is based on Equation (27) and its numerical solution with the use of the ﬁnite differences method in MATLAB [13]. The readers could use this code if they want to simulate memristors.

1. The basic code used for simulations of the modiﬁed Biolek model 2. Constants and parameters: qmax=0.0001; eta=1; um=3.3; f=2.01; psiu=-pi/3; Ron=100; 3. Roff=16000; deltaR=Ron-Roff; mu=1e-14; D=10e-9; k=(mu*Ron)/(Dˆ2); x0=0.3; 4. xmin=0; xmax=1; [u,t,deltat,tmin,tmax,N]=sine_gen(um,f,psiu); 5. n=1:1:N+1; ﬂux=integr(u,deltat,tmin,tmax); 6. [x,biolekwinmod]=biolek_mod(deltat,u,k,eta,deltaR,Roff,x0,xmax,xmin,N); 7. Req=deltaR*x+Roff; iM=u./Req; q=qmax*x; 8. Function for generation of sinusoidal voltage signal 9. function [u,t,deltat,tmin,tmax,N] = sine_gen(um,f,psiu) 10. Time domain: T=1/f; tmin=0; tmax=8*T; deltat=(tmax-tmin)/1e5; t1=tmin:deltat:tmax; 11. omega=2*pi*f ;f1=um*sin(omega*t1+psiu); u=f1;t=t1;N=tmax/deltat; end function. 12. Function for integrating the memristor voltage 13. function psi=integr(u,deltat,tmin,tmax) 14. N=(tmax-tmin)/deltat; n=1:1:N+1; psi=[ ]; 15. for n=1; psi1(n)=0; end; 16. for n=2:1:N+1; psi1(n)=psi1(n-1)+u(n-1)*deltat; end; Electronics 2017, 6, 77 9 of 15

17. psi=[psi psi1]; end function. 18. Function for deriving the state variable and the modified Biolek window 19. function [x,biolekwinmod]=biolek_mod(deltat,u,k,eta,deltaR,Roff,x0,xmax,xmin,N) 20. Parameters: p=2; m=0.2; x=[ ]; biolekwinmod=[ ]; for n=1; x1=x0; biolekmod1=1; end; 21. for n=2:1:N+1; A=deltaR*x1(n-1)+Roff; if abs(u(n))<0.1; then x1(n)=x1(n-1); 22. if u(n)<=0; then biolekmod1(n)=(((1-(x1(n)-1)ˆ(2*p))+m*(sin(pi*x1(n)))ˆ2))/(1+m); 23. else biolekmod1(n)=(((1-(x1(n))ˆ(2*p))+m*(sin(pi*x1(n)))ˆ2))/(1+m); end; 24. else; if u(n)<=0; then x1(n)=x1(n-1)+(eta*k*u(n-1)*deltat*(1/A)*(((1-(x1(n-1)-1)ˆ(2*p))+... 25. m*(sin(pi*x1(n-1)))ˆ2))/(1+m)); biolekmod1(n)=(((1-(x1(n)-1)ˆ(2*p))+m*(sin(pi*x1(n)))ˆ2))/(1+m); 26. if x1(n)<=xmin & u<=0; then x1(n)=xmin; else;x1(n)=x1(n-1)+(eta*k*u(n-1)*deltat*(1/A)*(((1-(x1 (n-1)-1)... 27. ˆ(2*p))+m*(sin(pi*x1(n-1)))ˆ2))/(1+m)); biolekmod1(n)=(((1-(x1(n)-1)ˆ(2*p))+m*(sin(pi*x1(n)))ˆ2))/ (1+m); 28. end; else x1(n)=x1(n-1)+(eta*k*u(n-1)*deltat*(1/A)*(((1-(x1(n-1))ˆ(2*p))+m*(sin(pi*x1(n-1)))ˆ2))/ (1+m)); 29. biolekmod1(n)=(((1-(x1(n))ˆ(2*p))+m*(sin(pi*x1(n)))ˆ2))/(1+m); if x1(n)>=xmax & u(n)>0; x1(n)=xmax; 30. else x1(n)=x1(n-1)+(eta*k*u(n-1)*deltat*(1/A)*(((1-(x1(n-1))ˆ(2*p))+m*(sin(pi*x1(n-1)))ˆ2))/(1+m)); Electronics31. biolekmod1(n)=(((1-(x1(n))ˆ(2*p))+m*(sin(pi*x1(n)))ˆ2))/(1+m); 2017, 6, 77 end; 9 of 14 32. end; end; end; x=[x x1]; biolekwinmod=[biolekwinmod biolekmod1]; end function. 5. A Comparison with the Pickett Memristor Model, Adjusting of the Modified Biolek Model and the5. A Basic Comparison Results Obtained with the Pickett by the MemristorComputer Simulations Model, Adjusting of the Modified Biolek Model and the Basic Results Obtained by the Computer Simulations 5.1. Adjusting the Modified Memristor Model Using Comparison with the Pickett Model 5.1. Adjusting the Modified Memristor Model Using Comparison with the Pickett Model Using the pseudo-code presented above for different values of the exponent p and of the coefficientUsing m the, discussed pseudo-code in the presentedprevious section above, several for different simulations values were of the made exponent in MATLABp and R2010b of the environmentcoefficient m, discussed[13]. After in the adjusting previous the section, coefficients, several simulations the basic were results made were in MATLAB derived. R2010b The Weberenvironment-coulomb [13 ].relation Afteradjusting shown in the Figure coefficients, 6a is almost the basic similar results to those were of derived. the Pickett The Weber-coulombmodel—Figure 3relationa. The respective shown in Figurecurrent6a-voltage is almost relationship similar to thoseis presented of the Pickett in Figure model—Figure 6b and it is 3almosta. The identical respective to thosecurrent-voltage of the Pickett relationship model given is presented in Figure in3b. Figure The adjusted6b and it values is almost of the identical coefficients to those are ofp = the 7 and Pickett m = 0.2.model given in Figure3b. The adjusted values of the coefficients are p = 7 and m = 0.2.

-5 -4 x 10 Weber-Coulomb relationship x 10 I-V relationship 10 2.5

9.5 2

9 1.5 8.5 1 8 0.5 7.5 0 7

Memristor charge, C Memristor current, A -0.5 6.5

-1 6

5.5 -1.5

5 -2 -0.05 0 0.05 0.1 0.15 0.2 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Magnetic flux linkage, Wb Memristor voltage, V (a) (b)

FigureFigure 6. 6.(a) Weber-Coulomb Weber-Coulomb relationship relationship according according to the to modiﬁed the modified Biolek model; Biolek ( b) model I-V relationship; (b) I-V relationshipof the memristor of the element memristor according element to according the modiﬁed to the Biolek modified model, Bioleku(t) = model 0.6·sin(2, u(·t)π ·=1 ·0.6∙sin(2∙π∙1∙t). t).

The respective time diagrams of the memristor voltage and state variable are presented in Figure 7a. The state variable x does not reach its boundary values so it could be concluded that the memristor operates in a soft-switching mode. The relationship between the modified window function and the state variable is presented in Figure 7b. The operating point of the memristor in the field of the coordinate system moves between two branches of the window function in accordance with the sign of the memristor voltage.

Relationship between x and f(x) 1

0.9

0.8

0.7

0.6

0.5

Modified window function Modified window 0.4

0.3

0.2 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 State variable x (a) (b)

Figure 7.(a) Time diagrams of memristor voltage and state variable according to the modified Biolek model; (b) The relationship between the modified window function and the state variable of the memristor according to the modified Biolek model, u(t)= 0.6∙sin(2∙π∙1∙t).

Electronics 2017, 6, 77 9 of 14

5. A Comparison with the Pickett Memristor Model, Adjusting of the Modified Biolek Model and the Basic Results Obtained by the Computer Simulations

5.1. Adjusting the Modified Memristor Model Using Comparison with the Pickett Model Using the pseudo-code presented above for different values of the exponent p and of the coefficient m, discussed in the previous section, several simulations were made in MATLAB R2010b environment [13]. After adjusting the coefficients, the basic results were derived. The Weber-coulomb relation shown in Figure 6a is almost similar to those of the Pickett model—Figure 3a. The respective current-voltage relationship is presented in Figure 6b and it is almost identical to those of the Pickett model given in Figure 3b. The adjusted values of the coefficients are p = 7 and m = 0.2.

-5 -4 x 10 Weber-Coulomb relationship x 10 I-V relationship 10 2.5

9.5 2

9 1.5 8.5 1 8 0.5 7.5 0 7

Memristor charge, C Memristor current, A -0.5 6.5

-1 6

5.5 -1.5

5 -2 -0.05 0 0.05 0.1 0.15 0.2 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Magnetic flux linkage, Wb Memristor voltage, V Electronics 2017, 6, 77 (a) (b) 10 of 15

Figure 6.(a) Weber-Coulomb relationship according to the modified Biolek model; (b) I-V relationship of the memristor element according to the modified Biolek model, u(t) = 0.6∙sin(2∙π∙1∙t). The respective time diagrams of the memristor voltage and state variable are presented in Figure7a.The respective state variable time diagramsx does of thenot memristor reach its voltage boundary and state values variable so are it could presented be in concluded that the memristorFigure 7a. The operates state variable in a soft-switching x does not reach mode.its boundary The relationshipvalues so it could between be concluded the modiﬁed that the window memristor operates in a soft-switching mode. The relationship between the modified window function and the state variable is presented in Figure7b. The operating point of the memristor in the function and the state variable is presented in Figure 7b. The operating point of the memristor in the ﬁeld of thefield coordinate of the coordinate system system moves moves between between twotwo branches of the of thewindow window function function in accordance in accordance with the signwith of the the sign memristor of the memristor voltage. voltage.

Relationship between x and f(x) 1

0.9

0.8

0.7

0.6

0.5

Modified window function Modified window 0.4

0.3

0.2 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 State variable x (a) (b)

Figure 7.(a) Time diagrams of memristor voltage and state variable according to the modified Biolek Figure 7. (modela) Time; (b) diagrams The relationship of memristor between the voltage modified and window state variable function and according the state tovariable the modified of the Biolek model; (b)memristor The relationship according to betweenthe modified the Biolek modified model, u(t) window= 0.6∙sin(2∙π∙1∙ functiont). and the state variable of the memristor according to the modified Biolek model, u(t)= 0.6·sin(2·π·1·t).

5.2. Testing the New Model for Hard-Switching Mode Electronics 2017, 6, 77 10 of 14 The adjusted memristor model was tested for a voltage signal with several times higher amplitude 5.2. Testing the New Model for Hard-Switching Mode and with negative initial voltage phase—u(t) = 3.6·sin(2·π·1·t − 120◦).The respective Weber-Coulomb relationship isThe presented adjusted in memristor Figure 8 modela. In this was case tested it for is a a multi-valued voltage signal with hysteresis several curve times higher which reaches amplitude and with negative initial voltage phase—u(t) = 3.6∙sin(2∙π∙1∙t − 120°).The respective the boundaryWeber values-Coulomb of relati theonship accumulated is presented in thein Figure memristor 8a. In this charge. case it is Then a multi it- couldvalued behysteresis concluded that the memristorcurve which operates reaches in the a hard-switchingboundary values of the mode. accumulated The respective in the memristor current-voltage charge. Then it could relationship is presentedbe in concluded Figure8 b.that In the this memristor case, theoperates resistance in a hard of-switching the memristor mode. Thefor respective forward current biasing-voltage is low and the respectiverelationship resistance is presented for negative in Figure voltages 8b. In this is very case, high. the resistan So itce could of the be memristor said that for in forward hard-switching biasing is low and the respective resistance for negative voltages is very high. So it could be said that mode the memristorin hard-switching behaves mode like the amemristor rectifying behaves diode. like The a rectifying time diagrams diode. The of time the diagrams memristor of the voltage and state variablememristor for hard-switching voltage and state variable are presented for hard-swit inching Figure are9 presenteda. In this incase, Figure the 9a. In state this variablecase, the reaches its limitingstate values—0 variable reaches and 1. its The limiting respective values— relationship0 and 1. The respective between relationship the modiﬁed between window the modified function and the state variablewindow function is shown and in the Figure state variable9b. In thisis shown case, in theFigure window 9b. In this function case, the values window are function in the interval values are in the interval between 0 and 1. The operating point of the memristor in the field of the between 0coordinate and 1. The system operating moves onpoint all the of the length memristor of the two in branches the ﬁeld in of accordance the coordinate to the memristor system moves on all the lengthvoltage of thesign. two branches in accordance to the memristor voltage sign.

-4 x 10 Weber-Coulomb relationship I-V relationship 1 0.03

0.9 0.025 0.8

0.7 0.02

0.6 0.015 0.5 0.01 0.4

Memristor charge, C

Memristor current, A 0.3 0.005

0.2 0 0.1

0 -0.005 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 -4 -3 -2 -1 0 1 2 3 4 Magnetic flux linkage, Wb Memristor voltage, V (a) (b)

Figure 8.(a) Weber-Coulomb relationship according to the modified Biolek model; (b) I-V relationship Figure 8. (a) Weber-Coulomb relationship according to the modiﬁed Biolek model; (b) I-V relationship of the memristor element according to the modified Biolek model, u(t) = 3.6∙sin(2∙π∙1∙t− 120°). of the memristor element according to the modiﬁed Biolek model, u(t) = 3.6·sin(2·π·1·t− 120◦).

Time diagrams of u and x Relationship between x and f(x) 4 1 State variable 0.9 3 Memristor voltage 0.8 2 0.7 1 0.6

0 0.5

0.4 -1

Voltage and stateVoltage and x

Modified window function Modified window 0.3 -2 0.2 -3 0.1

-4 0 0 1 2 3 4 5 6 7 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time, s State variable x

(a) (b)

Figure 9.(a) Time diagrams of memristor voltage and state variable according to the modified Biolek model; (b) The relationship between the modified window function and the state variable of the memristor according to the modified Biolek model, u(t) = 3.6∙sin(2∙π∙1∙t − 120°).

The test of the modified Biolek model for a pseudo-sinusoidal voltage signal with exponentially increasing amplitude confirms the expected behavior of the model for voltages lower and higher

Electronics 2017, 6, 77 10 of 14

5.2. Testing the New Model for Hard-Switching Mode The adjusted memristor model was tested for a voltage signal with several times higher amplitude and with negative initial voltage phase—u(t) = 3.6∙sin(2∙π∙1∙t − 120°).The respective Weber-Coulomb relationship is presented in Figure 8a. In this case it is a multi-valued hysteresis curve which reaches the boundary values of the accumulated in the memristor charge. Then it could be concluded that the memristor operates in a hard-switching mode. The respective current-voltage relationship is presented in Figure 8b. In this case, the resistance of the memristor for forward biasing is low and the respective resistance for negative voltages is very high. So it could be said that in hard-switching mode the memristor behaves like a rectifying diode. The time diagrams of the memristor voltage and state variable for hard-switching are presented in Figure 9a. In this case, the state variable reaches its limiting values—0 and 1. The respective relationship between the modified window function and the state variable is shown in Figure 9b. In this case, the window function values are in the interval between 0 and 1. The operating point of the memristor in the field of the coordinate system moves on all the length of the two branches in accordance to the memristor voltage sign.

-4 x 10 Weber-Coulomb relationship I-V relationship 1 0.03

0.9 0.025 0.8

0.7 0.02

0.6 0.015 0.5 0.01 0.4

Memristor charge, C

Memristor current, A 0.3 0.005

0.2 0 0.1

0 -0.005 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 -4 -3 -2 -1 0 1 2 3 4 Magnetic flux linkage, Wb Memristor voltage, V (a) (b)

Electronics 2017Figure, 6, 8. 77(a) Weber-Coulomb relationship according to the modified Biolek model; (b) I-V relationship 11 of 15 of the memristor element according to the modified Biolek model, u(t) = 3.6∙sin(2∙π∙1∙t− 120°).

Time diagrams of u and x Relationship between x and f(x) 4 1 State variable 0.9 3 Memristor voltage 0.8 2 0.7 1 0.6

0 0.5

0.4 -1

Voltage and stateVoltage and x

Modified window function Modified window 0.3 -2 0.2 -3 0.1

-4 0 0 1 2 3 4 5 6 7 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time, s State variable x

(a) (b)

Figure 9.(a) Time diagrams of memristor voltage and state variable according to the modified Biolek Figure 9. (a) Time diagrams of memristor voltage and state variable according to the modified Biolek model; (b) The relationship between the modified window function and the state variable of the model; (b) The relationship between the modified window function and the state variable of the memristor according to the modified Biolek model, u(t) = 3.6∙sin(2∙π∙1∙t − 120°). memristor according to the modified Biolek model, u(t) = 3.6·sin(2·π·1·t − 120◦). The test of the modified Biolek model for a pseudo-sinusoidal voltage signal with exponentially ElectronicsincreasingThe 2017 test, 6, of 77amplitude the modified confirms Biolek themodel expected for behavior a pseudo-sinusoidal of the model voltagefor voltages signal lower with and exponentially higher11 of 14 increasing amplitude confirms the expected behavior of the model for voltages lower and higher than thanthe activation the activation threshold thresholduthr. Theuthr. respective The respective Weber-Coulomb Weber-Coulomb relationship relationship in this case in this is presented case is presentedin Figure in10 a.Figure In the 10 beginning,a. In the beginning the operation, the operation point of the point memristor of the memristor in the field in of the the field coordinate of the coordinatesystem moves system on amoves straight on line a straight because line the because state variable the state does variable not change does for not voltages change lowerfor voltages that the lowersensitivity that the threshold. sensitivity When threshold. the voltage When is higherthe voltage than theis higher activation than threshold,the activation the operatingthreshold, point the operatingstarts to move point onstarts a different to move branch on a different of the Weber-coulomb branch of the relationship,Weber-coulomb which relationship, is a curve line, which and is the a curvestate variable line, and changes the state its value variable in accordance changes its with value the in flux accordance linkage. The with respective the flux current-voltage linkage. The respectiverelationship current is shown-voltage in Figure relationship 10b. In is the shown beginning, in Figure it is a10 straightb. In the line beginning and for, voltages it is a straight higher line than andthe for sensitivity voltages threshold, higher than it is the a multi-valuedsensitivity threshold, pinched hysteresisit is a multi loop.-valued pinched hysteresis loop.

-5 -4 x 10 Weber-Coulomb relationship x 10 I-V relationship 9 4

8 3

7 2

6 1 5 0 4 -1

Memristor charge, C 3 Memristor current, A

-2 2

1 -3

0 -4 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Magnetic flux linkage, Wb Memristor voltage, V (a) (b)

FigureFigure 10.((aa)) Weber-CoulombWeber-Coulomb relationship relationship according according to the to modified the modified Biolek model; Biolek (b ) I-V model relationship; (b) I-V of relationshipthe memristor of element the memristor according to element the modified according Biolek model, to theu(t) = modified 0.05·exp(0.51 Biolek·t)·sin(2 model·π·1·t,− 120u(t)◦=). 0.05∙exp(0.51∙t)∙sin(2∙π∙1∙t− 120°). The respective time diagrams of the memristor voltage and the state variable are presented in The respective time diagrams of the memristor voltage and the state variable are presented in Figure 11a. The voltage signal is pseudo-sinusoidal with exponentially increasing amplitude. In the Figure 11a. The voltage signal is pseudo-sinusoidal with exponentially increasing amplitude. In the beginning for voltages lower than the sensitivity threshold the state variable does not change its beginning for voltages lower than the sensitivity threshold the state variable does not change its value but for higher voltages it starts to change. For voltages lower than the activation threshold, value but for higher voltages it starts to change. For voltages lower than the activation threshold, the the memristor behaves like a linear resistor. If the signal becomes higher than the activation threshold memristor behaves like a linear resistor. If the signal becomes higher than the activation threshold then the state variable changes and the element behaves like a memristor. The relationship between the modified window function and the state variable is presented in Figure 11b.

Time diagrams of u and x Relationship between x and f(x) 2.5 1

2 0.95 State variable 1.5 Memristor voltage 0.9 1 0.85 0.5 0.8 0 0.75 -0.5

Voltage and stateVoltage and x 0.7

-1 function Modified window

0.65 -1.5

-2 0.6

-2.5 0.55 0 1 2 3 4 5 6 7 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time, s State variable x (a) (b)

Figure 11.(a) Time diagrams of memristor voltage and state variable according to the modified Biolek model; (b) The relationship between the modified window function and the state variable of the memristor according to the modified Biolek model, u(t) = 0.05∙exp(0.51∙t)∙sin(2∙π∙1∙t− 120°).

Electronics 2017, 6, 77 11 of 14 than the activation threshold uthr. The respective Weber-Coulomb relationship in this case is presented in Figure 10a. In the beginning, the operation point of the memristor in the field of the coordinate system moves on a straight line because the state variable does not change for voltages lower that the sensitivity threshold. When the voltage is higher than the activation threshold, the operating point starts to move on a different branch of the Weber-coulomb relationship, which is a curve line, and the state variable changes its value in accordance with the flux linkage. The respective current-voltage relationship is shown in Figure 10b. In the beginning, it is a straight line and for voltages higher than the sensitivity threshold, it is a multi-valued pinched hysteresis loop.

-5 -4 x 10 Weber-Coulomb relationship x 10 I-V relationship 9 4

8 3

7 2

6 1 5 0 4 -1

Memristor charge, C 3 Memristor current, A

-2 2

1 -3

0 -4 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Magnetic flux linkage, Wb Memristor voltage, V (a) (b)

Figure 10.(a) Weber-Coulomb relationship according to the modified Biolek model; (b) I-V relationship of the memristor element according to the modified Biolek model, u(t)= 0.05∙exp(0.51∙t)∙sin(2∙π∙1∙t− 120°).

The respective time diagrams of the memristor voltage and the state variable are presented in Figure 11a. The voltage signal is pseudo-sinusoidal with exponentially increasing amplitude. In the beginning for voltages lower than the sensitivity threshold the state variable does not change its valueElectronics but2017 for, 6higher, 77 voltages it starts to change. For voltages lower than the activation threshold12, ofthe 15 memristor behaves like a linear resistor. If the signal becomes higher than the activation threshold thenthen the the state state variable variable changes changes and and the the element element behaves behaves like like a a memristo memristor.r. The relationship between thethe modified modiﬁed window function and the state variabl variablee is presented in Figure 11b.

Time diagrams of u and x Relationship between x and f(x) 2.5 1

2 0.95 State variable 1.5 Memristor voltage 0.9 1 0.85 0.5 0.8 0 0.75 -0.5

Voltage and stateVoltage and x 0.7

-1 function Modified window

0.65 -1.5

-2 0.6

-2.5 0.55 0 1 2 3 4 5 6 7 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time, s State variable x (a) (b)

Figure 11.(a) Time diagrams of memristor voltage and state variable according to the modified Figure 11. (a) Time diagrams of memristor voltage and state variable according to the modified Biolek Biolek model; (b) The relationship between the modified window function and the state variable of model; (b) The relationship between the modified window function and the state variable of the the memristor according to the modified Biolek model, u(t) = 0.05∙exp(0.51∙t)∙sin(2∙π∙1∙t− 120°). memristor according to the modified Biolek model, u(t) = 0.05·exp(0.51·t)·sin(2·π·1·t− 120◦).

According to the voltage sign, the operating point of the memristor in the ﬁeld of the coordinate system moves between the two branches of the characteristics.

6. A Comparison of the Modiﬁed Memristor Model with Existing Models In this section a comparison of the modiﬁed Biolek model with the Joglekar memristor model [3] for soft-switching mode will be done. According to [3] the window function used in Equation (22) is:

2p fJ(x) = 1 − (2x − 1) (28)

Equation (28) is presented for ﬁrst use by Joglekar in [3] and it is also known as Joglekar window function [3]. Using (14), (22), and (28) the Joglekar memristor model could be fully described with Equation (29) where the state-dependent Ohm’s Law is also included [3]. h i dx = ηki 1 − (2x − 1)2p dt (29) u = Ri = [RON x + ROFF(1 − x)]i

According to the voltage sign, the operating point of the memristor in the field of the coordinate system moves between the two branches of the characteristics.

6. A Comparison of the Modified Memristor Model with Existing Models In this section a comparison of the modified Biolek model with the Joglekar memristor model [3] for soft-switching mode will be done. According to [3] the window function used in Equation (22) is:

2 p fJ  x 1  2 x  1 (28)

Equation (28) is presented for first use by Joglekar in [3] and it is also known as Joglekar window function [3]. Using (14), (22), and (28) the Joglekar memristor model could be fully described with Equation (29) where the state-dependent Ohm’s Law is also included [3].

dx 2 p ki1  2 x  1 dt  (29) u Ri   RON x  R OFF 1  x  i

The Joglekar memristor model could be simulated using the code given in Section 4 but Equation (28) has to be used instead of the Biolek window function in row 6 in the code. The code of the function for calculating the state variable and the Joglekar window according to the Joglekar memristor model is presented below and it could be used by the readers for simulations [13]: 1. A function for simulating the Joglekar memristor model 2. function [x,joglwin]=jogl(deltat,u,k,eta,deltaR,Roff,x0,N) 3. Parameter p=2; x=[ ];joglwin=[ ]; 4. for n=1; x1=x0; joglwin1=1-((2*x1-1)^(2*p)); end; 5. for n=2:1:N+1; x1(n)=x1(n-1)+eta*k*deltat*u(n-1)*((1-(2*x1(n-1)-1)^(2*p))/(deltaR*x1(n-1)+Roff)); joglwin1(n)=1-((2*x1(n)-1)^(2*p)); end; Electronics6. 2017x=[x, 6x1];, 77 joglwin=[joglwin joglwin1]; end function. 13 of 15 The voltage signal used for simulation of the Joglekar model and the modified model is: u(t) = 3.3∙sin(2∙π∙2∙t− 60°). The I-V relationships of the memristor are presented in Figure 12a. In this case, the I-Vthe curveI-V curve according according to to Joglekar’s Joglekar’s model model isis double-valueddouble-valued while while the the modified modified model model represents represents multi-valuedmulti-valued curve. curve. The The state-flux state-fluxrelationships relationships are are given given in in Figure Figure 12 12b. b. According According to Joglekar to Joglekar’s’s model,model this, relationthis relation is single-valued, is single-valued which, which is an is advantage an advantage with with respect respect to the to multi-valued the multi-valued state-flux characteristicsstate-flux characteristics obtained by obtained the modified by the model. modified model.

-4 -4 x 10 I-V relationship according to Joglekar model x 10 Weber-Coulomb relation according to Joglekar model 5 1

0 0.5

Memristor charge, C

Memristor current, A -5 0 -4 -3 -2 -1 0 1 2 3 4 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Memristor voltage, V Magnetic flux linkage, Wb -4 -5 x 10 I-V relationship according to the modified model x 10 Weber-Coulomb relation according to the modified model 10 10

8 5 6 0 4

Memristor charge, C

Memristor current, A -5 2 -4 -3 -2 -1 0 1 2 3 4 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Memristor voltage, V Magnetic flux linkage, Wb

(a) (b)

Figure 12.(a) Current-voltage relations of the memristor according to Joglekar model and the Figure 12. (a) Current-voltage relations of the memristor according to Joglekar model and modified model; (b) State-flux relation according to Joglekar’s and the modified model, u(t) = the modified model; (b) State-flux relation according to Joglekar’s and the modified model, 3.3∙sin(2∙π∙2∙t− 60°). ◦ Electronicsu(t) 2017= 3.3, 6·sin(2, 77 ·π·2·t− 60 ). 13 of 14

TheThe time time diagrams diagrams of of the the Joglekar Joglekar memristor memristor model model and and of of the the modified modified model model presented presented in in FigureFigure 1313aa are are almost almost identical. identical. The The relationships relationships between between the the window window function function values values and and the the state state variablevariable xx forfor the the J Joglekaroglekar model model,, and and for for the the modified modified memristor memristor model model are are presented presented in in Figure Figure 13 13b.b. TThehe state variable of the memristor models changes in equivalent intervals. The The modified modified Biolek Biolek windowwindow function function in in this this case case provides provides higher higher nonlinearity nonlinearity of of the the memristor memristor,, which which is is an advantage ofof the the modifiedmodified model with respect to the Joglekar model. Another Another advantages advantages of of the the modified modified BiolekBiolek model model proposed proposed here here is is the the ability ability for for realistic realistic representation representation of of the the boundary boundary effects effects when when thethe memristor memristor operates operates in in a a hard hard-switching-switching mode mode and and the the use of sensitivity threshold. The The original original JoglekarJoglekar memristor memristor model model does does not not have have mechanism mechanism for for representing representing the the boundary boundary effects effects and and it it is is appropriateappropriate only only for for simulating of of memristors for soft soft-switching-switching mode.

Time diagrams of u and x according to Joglekar model Relation between x and f(x) according to Joglekar model 4 1 State variable x 2 Memristor voltage 0.9

0 0.8

-2 0.7

Voltage and stateVoltage and x

-4 function Modified window 0 0.5 1 1.5 2 2.5 3 3.5 4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time, s State variable x Time diagrams of u and x according to the modified model Relationship between x and f(x) according to the modified model 4 1 State variable x 2 Memristor voltage 0.8 0 0.6 -2

Voltage and stateVoltage and x

-4 function Modified window 0.4 0 0.5 1 1.5 2 2.5 3 3.5 4 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time, s State variable x

(a) (b)

FigureFigure 13. 13.(a)( aT)ime Time diagrams diagrams of the of memristor the memristor voltage voltage and state and variable state variable according according to Joglekar to Joglekar model andmodel the andmodified the modiﬁed memristor memristor model; ( model;b) Relationship (b) Relationship between betweenthe window the windowfunction functionvalue and value the stateand the variable state variableaccording according to Joglekar to Joglekar memristor memristor model and model the and modified the modiﬁed Biolek m Biolekodel, model,u(t) = 3.3∙sin(2∙π∙2∙u(t) = 3.3·sin(2t− ·60°).π·2·t − 60◦).

7. Conclusions After finishing the adjusting process of the modified Biolek model and obtaining the basic results, several conclusions could be made. The modified memristor model based both on Biolek model and GBCM model is a general one and it contains a modified window function, which is a sum of the original Biolek window function, and a weighted sinusoidal window function. In the special case when the coefficient in front of the sinusoidal window function and the activation threshold are chosen to be equal to zero, the original Biolek memristor model is obtained. If the modified model presented here is adjusted, we could obtain results identical to those obtained by the Pickett model. Of course, the results obtained in the present research are not exactly the same like those produced by Pickett model, which has the highest accuracy but also has many convergence problems and is not appropriate for simulations. The main advantage of the modified Biolek model with respect to the Pickett model is the lack of convergence problems and the possibility for use of the modified model for computer simulations of memristors and memristive circuits. The modified memristor model proposed here was compared with Joglekar memristor model for soft-switching mode. The state-flux relationship obtained by Joglekar model is a single-valued curve, which is an advantage of Joglekar model with respect to the modified model which state-flux characteristics are multi-valued. On the other hand, the modified memristor model proposed in this paper has several advantages with respect to Joglekar memristor model—higher nonlinearity of the dopant drift, ability for realistic representation of the boundary effects, and the use of sensitivity voltage threshold. After comparison of the modified memristor model proposed in this paper with the Pickett model it could be concluded that the Pickett model could be simulated for a narrow range of voltages and for higher voltages convergence problems occur. During the tests of Pickett model only operation in a soft-switching mode was observed. An advantage of the model proposed in this

Electronics 2017, 6, 77 14 of 15

7. Conclusions After finishing the adjusting process of the modified Biolek model and obtaining the basic results, several conclusions could be made. The modified memristor model based both on Biolek model and GBCM model is a general one and it contains a modified window function, which is a sum of the original Biolek window function, and a weighted sinusoidal window function. In the special case when the coefficient in front of the sinusoidal window function and the activation threshold are chosen to be equal to zero, the original Biolek memristor model is obtained. If the modified model presented here is adjusted, we could obtain results identical to those obtained by the Pickett model. Of course, the results obtained in the present research are not exactly the same like those produced by Pickett model, which has the highest accuracy but also has many convergence problems and is not appropriate for simulations. The main advantage of the modified Biolek model with respect to the Pickett model is the lack of convergence problems and the possibility for use of the modified model for computer simulations of memristors and memristive circuits. The modified memristor model proposed here was compared with Joglekar memristor model for soft-switching mode. The state-flux relationship obtained by Joglekar model is a single-valued curve, which is an advantage of Joglekar model with respect to the modified model which state-flux characteristics are multi-valued. On the other hand, the modified memristor model proposed in this paper has several advantages with respect to Joglekar memristor model—higher nonlinearity of the dopant drift, ability for realistic representation of the boundary effects, and the use of sensitivity voltage threshold. After comparison of the modified memristor model proposed in this paper with the Pickett model it could be concluded that the Pickett model could be simulated for a narrow range of voltages and for higher voltages convergence problems occur. During the tests of Pickett model

onlyElectronics operation 2017, 6, 77 in a soft-switching mode was observed. An advantage of the model proposed in14 of this 14 Electronics 2017, 6, 77 14 of 14 research with respect to the Pickett model is the possibility for simulating the modiﬁed model in a broaden voltage range and the observation of hard-switching mode behavior. researchresearch withwith respectrespect toto thethe PickettPickett modelmodel isis thethe possibilitypossibility forfor simulatingsimulating thethe modifiedmodified modelmodel inin aa broadenbroaden voltagevoltage rangerange andand thethe obsobservationervation ofof hardhard--switchingswitching modemode behavior.behavior.

Acknowledgments:Acknowledgments: TheThe researchresearch is is supported supported byby nationalnational CoCo Co-financing--financingfinancing (contract(contract №№ ДКОСТ01/14)ДКОСТ01/14)) of of COST COST Action № IC1401 MemoCIS. Action № IC1401 MemoCIS. AuthorAuthor Contributions: Contributions: V.M.V.M. and and S.K. S.K. made made the the modified modifiedmodified computer computer memristormemristor modelmodel andand and simulasimula simulations;tions;tions; V.M.V.M. V.M. analyzedanalyzed the the resultsresults andandand adjusted adjustedadjusted the thethe new newnew memristor memristormemristor model modelmodel in inin accordance accordanceaccordance with withwith the thethe Pickett PickettPickett model; model;model; V.M. V.M.V.M. and andand S.K. compared the modified memristor model with Joglekar model and prepared the manuscript of the present paper. S.K.S.K. comparedcompared thethe modifiedmodified memristormemristor modelmodel withwith JoglekarJoglekar modelmodel andand preparedprepared thethe manuscriptmanuscript ofof thethe presentpresent paperConflictspaper.. of Interest: The authors declare no conflict of interest. Conflicts of Interest: The authors declare no conflict of interest. ConflictsThe founding of Interest: sponsors The had authors no role d ineclare the designno conflict of the of study; interest. in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results. TheThe foundingfounding sponsorssponsors hadhad nono rolerole inin thethe designdesign ofof thethe study;study; inin thethe collection,collection, analyses,analyses, oror interpretationinterpretation ofof data;data; inin thethe writingwriting ofof thethe manuscript,manuscript, andand inin thethe decisiondecision toto publishpublish thethe results.results. References

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