Memristor Behavioural Modeling and Simulationsusing Verilog-AMS

Haulisson Jody Batista da CostaEducation and Technology Department

Federal Institute of Rio Grande do NorteMossoro, RN, Brazil

E-mail: haulisson.jody@ieee.org

Francisco de Assis Brito FilhoElectrical Engineering Department

Sao Paulo University - USPSao Paulo, SP, Brazil

E-mail: britofilho@ieee.org

Pedro Ivo de Araujo do NascimentoEducation and Technology Department

Federal Institute of Rio Grande do NorteMossoro, RN, Brazil

E-mail: pedroivo@ieee.org

Abstract— This paper presents a behavioural model of amemristor and a methodology to perform simulations using themodel in memristive circuits. The model was implemented inVerilog-AMS and simulated in Mentor Graphics Eldo with AMSsupport. A case study of a memristive cell is presented to validatethe model in a mixed circuit containing memristor model andSPICE components. This model can be applied in mixed circuitsimulation of memory architectures, neuromorphic circuits andmemristor based analog and RF circuits.

I. INTRODUCTION

A new step was given to state of the art with the experimen-tal verification of the existence of the memristor device in thelaboratory of Hewlett-Packard in past 2008 [1]. First theorizedby Leon Chua in 1971 [2] as the fourth fundamental circuitelement the memristor has been characterized as a device thatretains the information as conductance. Although this devicewas first implemented as part of RRAM (Resistive Random-Access Memory) its analogy to biological synapses makesthis device a promise to research in the field of neuromorphiccircuits [3]-[5]. Also,the memristor can be applied to analogand radio frequency circuits [6]-[7].

In order to contribute with efforts to overcome the chal-lenges that circuit designers have to use the memristor in theirdesigns many models have been proposed in the literatureto describe the behaviour of the memristor device. The mainmodels are based on SPICE equivalent circuits that behaveslike a memristor [9]-[11]. Other studies have focused ondescribing the memristor equations from physical constituentsas in [12].

In this paper is used the hardware description languageVerilog-AMS to modeling the memristor device in order tosupport a high level of abstraction with fidelity to their physicalcharacteristics. The Verilog-AMS memristor model is easy toimplement and has high level of detail in the physical equa-tions. Unlike modeling methods recently reported [9],[12], thismodel allows greater integration with SPICE and can be usedboth for current-controlled memristor as voltage-controlledmemristor allowing greater flexibility of application and non-linear dynamical.

The paper is organized as follows. Section II is a brief sum-mary of the information about memristor physical equationsas published in [4] and [8]. Section III gives a description

of memristor behaviour modeling approach in Verilog-AMS.Then a detailed comparison between physics equations andsimulation results are presented in Section IV. Finally theconclusions will be made in Section V.

II. MEMRISTOR PHYSICAL MODELING

In this section, the physical relationships of the memristorwill be introduced. Memristor is an element that has resistivememory. The operating principle is the logical completenessrelationship between flux ϕ to the charge q as despicted in thefigure 1 [1]-[8].

Fig. 1. Fundamental Relationships of the Passive Components.

In figure 1 is possible to see a scheme that defines thefundamental relationships of passive components includingmemristor. This scheme shows the relationship of Resistance(dv=Rdi), capacitance (dq=Cdv), inductance (dϕ=Ldi) andalso memristance M which may be expressed as in equation1.

dϕ = M.dq (1)

This relationship is similar to resistance, however the resis-tor does not have a dynamic function as shown in equation2.

V (t) = RM (x(t)).I(t) (2)

Fig. 2. Illustration of a Memristor structure (doped and undoped regions asa series resistor association).

Figure 2 illustrates the structure of the domain-wall motionbased memristor [1]. The resistance is now dependent of thedoping regions x(t). The function x(t) is a normalized ratiogiven by

x(t) =(

w(t)D

)(3)

where D is the physical length of the component. Due to eachregion had a different resistivity the memristance may describethis behaviour as two resistors in series as given by equation4 and shown in figure 2.

RM (x) = x(t)Ron + [1− x(t)].Roff (4)

The figure 2 shows that resistance variation depends on themigration of charges in the material. And it can only movein the opposite direction whether the current flux is reversedand undiminished. An important point of this movement isthe charge mobility µv that influence the velocity memristorresponse that may be described by equation 5 [8]:

vD =dw(t)

dt=

µDRon

DI(t) (5)

Thus, analyzing the function w(t) it appears that it is afunction of the charge.

Then the domain-wall effect shows the dependence of theaccumulation of charge which leads to

w(t) = w0 +µDRon

Dq(t), (6)

By inserting the equation 6 into equation 4 it shows ex-plicitly that memristance is charge dependent as predicted in[8].

M(q) = R0 −µDRon(Roff −Ron)

D2q(t) (7)

Combining the equation 7 with the equation 2 implies thatthe memristance is determined by Kirchoff’s voltage law givenby

v(t) = M(q)i(t) (8)

Whereas voltage in nanometric device have a strong electricfield that will produce nonlinearities [12], the memristor

physical model has a problem with the linear boundary effect.Equation 9 shows this nonlinearity effect.

vD =dw(t)

dt=

µDRon

DI(t)F (x) (9)

The function F (x) ensures that memristance will remainsin the range [Ron, Roff ] [10].

III. MEMRISTOR VERILOG-AMS MODEL

Nowadays some works have been adopted the equivalentcircuit model to simulate the memristor behaviour [8]. Thosemodels depends on the choice of resistance and capacitancevalues that can affect their numerical stability in the simula-tions [9], besides to use two circuits to emulate the current-controlled and voltage-controlled memristor. A simple andfriendly way to represent the memristor behaviour is usingVerilog-AMS hardware description language that was pro-posed for this issue in order to represent mixed-signal circuitsfaithfully to its physical behaviour.

The Verilog-AMS model ease the representation of physicalequations that describe the memristor device. With the aidof equations described in previous section was possible toimplement a simple and understandable model. The memristorVerilog-AMS model is presented in 1.

Listing 1. Verilog-AMS model1 ‘include "disciplines.h"2

3 module memristor(p,n);4 parameter real ud =1e-14;5 parameter real D = 10e-9;6 parameter real Ron = 100;7 parameter real Roff = 160008 parameter real Wo = 5e-9;9 parameter real p_drift = 1;

...

26 analog function real stp;27 input i;28 real i;29 begin30 stp=(i >= 0) ? 1:0;31 end32 endfunction33 /////////////////////////////////34 analog begin35 vd = ud*Ron*I(p,n)*f_drift/D;36 W = idt(v,Wo);37 M = (Ron*(W/D))+(Roff*(1-W/D));38 V(p,n) <+ I(p,n)*M;39 f_drift = 1-pow((W/D)-stp(-I(p,n)),(2*p_drift));40 end41 endmodule

The parameter (from lines 4 to 9) represents the memristorphysical parameters as resistivity, carrier mobility, dimensionand so on. Although the values been previously defined inthe model it can be modified before the simulation whenthe model is instantiated in the circuit. As proposed in [11]the lines from 26 to 32 represents the function that definesthe nonlinear behaviour of boundary wall [9], [11], [12] andin thereby ensuring that the resistance value remain in the

range [Ron, Roff ]. The equations 5 to 9 defines the physi-cal behaviour of memristor and its Verilog-AMS equivalentdescription is located on lines from 35 to 39.

Next section presents simulations results from this modeland some discussion points.

IV. RESULTS AND DISCUSSION

In order to validate the model developed was made atestbench based on SPICE. This testbench was prepared usinga simple electronic schematic (see figure 3) that allow tosimulate a wide range of stimulus using either current orvoltage controlled source.

Fig. 3. SPICE based testbench used to validate the Verilog-AMS model ina mixed circuit simulation platform.

With the aim evaluating the model were analyzed the mainfigures of merit that identify the memristor behaviour. Figure4 presents the Lissajous figures of I-V relationship which wasfirst identified by Chua [2]. In this representation is possibleto evaluate the memristor velocity response from the loophysteresis width that gives the signal frequency limit by thedevice resistance variation. In this analysis it appears that themobility of the material sets the limit of the carriers velocitythereby influencing the frequency range when the memristivityis valid.

Fig. 4. I-V hysteresis curve for various frequencies. The memristor parametervalues used for simulations were Ron=100Ω, Roff =1600Ω, p=1, voltageapplied vosin2(wot), fo=1Hz, vo=1V, µv=10−10cm2s−1V −1, D=10nm.

Besides showing the memristor velocity response it ispossible to verify also the memory behaviour: the present statedependence from the past states. The main figure of merit usedto verify this behaviour is shown in figure 5 which presentsthe resistivity variation due to accumulation of pulses.

-1

-0.5

0

0.5

1

0 0.5 1 1.5 2 2.5 3

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

Vs(

t)

I s(t

)(1

0-4

A)

Time (s)

voltagecurrent

Fig. 5. Resistivity variation due to accumulation of pulses. The memristorparameter values used were Ron=100Ω, Roff =1600Ω, p=1, voltage appliedvosin2(wot), fo=1Hz, vo=1V, µv=10−10cm2s−1V −1, D=10nm.

Based on paper presented by Sangho et al [9] was madea set of tests using circuits presented in figure 3. In thissimulation was possible to verify the Verilog-AMS modelconsistency. The form how the model was implemented ispossible to analyse both referential current and voltage as canbe seen in figures 6 and 7 giving greater flexibility to thedesigners to simulate their circuits using memristor.

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

V(t

)

I s(t

) (1

0-4

A)

Time (s)

voltage

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

V(t

)

I s(t

) (1

0-4

A)

Time (s)

current

Fig. 6. Memristor response when stimulated with current.

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-2.0

-1.0

0.0

1.0

2.0

3.0

Vs(

t)

i (10

-4A

)

Time (s)

voltage

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-2.0

-1.0

0.0

1.0

2.0

3.0

Vs(

t)

i (10

-4A

)

Time (s)

current

Fig. 7. Memristor response when stimulated with voltage.

In figures 6 and 7 it is possible to extract an important figureof merit in memory systems. The observed parameter refersto the stabilization time or to the velocity that charges accu-mulates in device edges, that means when the device reachesits maximum and its minimum resistance value [Ron, Roff ].As seen in figures 6 and 7 it is possible clearly determine theregions where the device behaves like a memristor or resistor.The regions are clearly identified when occurs the variation

stability and in this point is possible to see the boundarybetween the stable value (resistance) and the dynamic value(memristance) as illustrated in figure 8.

Fig. 8. Variation stability in memristor switching. The memristor parametervalues used were Ron=100Ω, Roff =1600Ω, p=1, voltage pulse applied,fo=1Hz, vo=1V, µv=10−10cm2s−1V −1, D=10nm.

The switching between memristive and resistive states pre-sented in figure 8 occurs when the potential barrier thatseparates the doped and the undoped regions moves totallyto one of sides and only returns when the stimulus is inverted.

In this section was possible to show that the Verilog-AMSmodel is suitable to use in mixed circuit simulations assuringthat the system will behave as resistance or memristanceregardless of how it is biased, ie as a physical memristor.Also, the model allow to use the memristor as a component inmulti-domain simulations due to be described in Verilog-AMSlanguage.

V. CONCLUSIONS

The memristor Verilog-AMS description was presented as asolution to the limitations from approach of using equivalentcircuits to emulate a memristive behaviour. The global modelis fully compliant and satisfies all the properties observed byHP in its first demonstrated physical memristor. Was demon-strated by simulations the model suitability when stimulatedfrom current or voltage sources assuring their limits andvalidating the memristive and resistive behaviour together.The model can be used for mixed-signal or multi-domainsimulation in circuit designs using memristor.

ACKNOWLEDGMENT

The authors would like to thank to Petrobras and to theIFRN support. The research work presented in this paper wasfunded by PFRH from Petrobras.

REFERENCES

[1] Dimitri B. Strukov, Gregory S. Snider, Duncan R. Stewart and R. S.Williams. The missing memristor found. Nature, 453:80–83, May 2008.

[2] L. Chua. Memristor-the missing circuit element. IEEE Transactions onCircuit Theory, 18(5):507–519, September 1971.

[3] G. S. Snider. Memristor synapses in a neural computing architecture.In Memristor Memristive Syst. Symp., November 2008.

[4] Y. V. Pershin, S. L. Fontain and M. D. Ventra. Memristive model ofamoeba’s learning. Nature precedings, October 2009.

[5] Sung Hyun Jo et al. Nanoscale memristor device as synapse inneuromorphic systems. Nano Letters, 2010.

[6] T. Wey and S. Benderli. Amplitude modulator circuit featuring tio2memristor with linear dopant drift. Electronics Letters, 45(22):1103–1104, October 2009.

[7] K. Witrisal. Memristor-based stored-reference receiver - the uwbsolution? Electronics Letters, 45(14):713–714, July 2009.

[8] Y. N. Joglekar and S. J. Wolf. The elusive memristor: properties of basicelectrical circuits. Eur. J. Phys., 2009.

[9] K. K. Sangho Shin and S.-M. Kang. Compact models for memristorsbased on charge-flux constitutive relationships. IEEE Transactions onComputer-Aided Design of Integrated Circuits and Systems, 29(4):590–598, April 2010.

[10] Z. Biolek, D. Biolek and V. Biolkova. Spice modeling of memristorwith nonlinear dopant drift. Radioengineering, 18(2), June 2009.

[11] D. Biolek, Z. Biolek and V. Biolkova. Spice modeling of memristive,memcapacitive, and meminductive systems. In Proc. Eur. Conf. CircuitTheory Design, August 2009.

[12] Nathan R. McDonald et al. Analysis of dynamic linear and non-linear memristor device models for emerging neuromorphic computinghardware design. In The 2010 International Joint Conference on NeuralNetworks (IJCNN), July 2010.