On the mathematical modeling of memristors

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Item type Conference Paper

Authors Radwan, Ahmed G.; Salama, Khaled N.; Zidan,Mohammed A.

Citation Radwan AG, Zidan MA, Salama KN (2010) On themathematical modeling of memristors. 2010 InternationalConference on Microelectronics.doi:10.1109/ICM.2010.5696139.

Eprint version Post-print

DOI 10.1109/ICM.2010.5696139

Publisher Institute of Electrical and Electronics Engineers (IEEE)

Journal 2010 International Conference on Microelectronics

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Link to item http://hdl.handle.net/10754/247351

On the Mathematical Modeling of MemristorsA. G. Radwan1,2, M. Affan Zidan1 and K. N. Salama1

1Electrical Engineering ProgramKing Abdullah University of Science and Technology (KAUST)

Thuwal, Kingdom of Saudi Arabia 23955-69002Department of Engineering Mathematics

Faculty of Engineering, Cairo University, EgyptEmail: ahmed.radwan, mohammed.zidan, khaled.salama@kaust.edu.sa

Abstract— Since the fourth fundamental element (Memristor)became a reality by HP labs, and due to its huge potential, itsmathematical models became a necessity. In this paper, we pro-vide a simple mathematical model of Memristors characterizedby linear dopant drift for sinusoidal input voltage, showing a highmatching with the nonlinear SPICE simulations. The frequencyresponse of the Memristor’s resistance and its bounding condi-tions are derived. The fundamentals of the pinched i-v hysteresis,such as the critical resistances, the hysteresis power and themaximum operating current, are derived for the first time.

I. INTRODUCTION

The Memristor (M ) is believed to be the fourth fundamentaltwo terminals passive element, beside the Resistor (R), theCapacitor (C) and the Inductor (L). The existence of suchelement was postulated by Leon Chua in the seventies [1].Recently a practical implementation of the Memristor usingpartially doped TiO2 was presented by Hewlett-Packard [2] asshown in Fig. 1, where w ∈ (0, D). When the applied potentialis removed the dopant boundary will remain at its locationwhich will be the initial value for any later movements.

According to [2] the memristive property naturally appearsin nanoscale devices, so the understanding of Memristance willimprove the studying and modeling of nanodevices character-istic, which includes the current-voltage hysteresis behaviorobserved in many nanodevices. The invention of the Memristorwas the key for postulating new elements by Chua such as theMemcapacitors and the Meminductors [3].

The Memristor found application in memory [4], biol-ogy [5], and spintronic [6]. Recently the Memristor wasmodeled using a linearized model of the pinched i-v hysteresisas in [7] or qualitatively as in [8], [9]. However these effortsfailed to capture many of its characteristics. A Memristorsmodel for DC, square and triangular signals is presentedin [10]. This model is used in the analysis of Memristor-basedWien-family oscillators [11].

The instantaneous resistance R of the Memristor is givenby [2],

R = x ·Ron + (1− x)Roff (1)

where x = w/D, Ron is the resistance of the completelydoped Memristor, and Roff is for the undoped Memristor.The speed dopant movement is defined as,

dx

dt= k · i (t) · f (x) (2)

−1.5 −1 −0.5 0 0.5 1 1.50

0.5

1

1.5

2x 104

Voltage

Res

ista

nce

−1.5 −1 −0.5 0 0.5 1 1.5

−1

0

1

x 10−4

Voltage

Cur

rent

0 0.5 1 1.5 2−2

−1

0

1

2

Time

Voltage Current (10−4)

1 2 3

w

D 1: Doped Region2: Unoped Region

3: Metal Contact

Fig. 1. The abstract structure of the HP’s Memristor [2], current versustime, and current and resistance versus input voltage are plotted for sinusoidalwave input voltage of 1Hz frequency and −1.5V peak voltage. Memristor’sParameters: Roff = 38kΩ, Ron = 100Ω, Ri = 4kΩ and p = 10.

where f (x) = 1 − (2x − 1)2p is the window function fornonlinear dopant drift given in [12], and k = µv · Ron/D

2.Fig. 1 shows some basic relationships between the current,voltage, and resistance for the HP’s Memristor.

In this paper the frequency response of the Memristor’sresistance is studied in the case of linear dopant drift forsinusoidal waveform for the first time. A condition on the ratioof (vo/f ) to maintain unsaturated resistance is introduced.A simple implicit equation for the pinched i-v hysteresis isderived, showing its general fundamentals such as criticalresistances, maximum current magnitude and location, andthe power under and inside this hysteresis. All mathematicalequations are compared with the nonlinear Memristor’s SPICEmodel given in [12], showing perfect matching.

II. THE FREQUENCY RESPONSE OF THE MEMRISTOR’SRESISTANCE

The instantaneous resistance of the Memristor subjected toa sinusoidal input can be derived from equations (3) and (2),for linear dopant f (x), and can be simplified as,

R2 = R2i −

2VokRd

πfsin2 (πft) , R ∈ (Ron, Roff ) (3)

where k = µv ·Ron/D2, µv is dopant drift mobility, f is the

frequency of the input sinusoidal waveform, Ri is the initialMemristor resistance at t = 0, and Rd = Roff − Ron is

Fig. 2. Memristor’s resistance versus time −Vo plane for 1Hz sinusoidalwaveform with f = 1Hz, Roff = 16kΩ, Ron = 100Ω, and Ri = 6kΩ.

the difference between the boundary resistances. However, theinstantaneous resistance must be bounded by Ron and Roff .

The range of R depends on the sign of Vo. Hence R ∈[Ri, Roff ) in case of Vo < 0 or (Ron, Ri] for Vo > 0. For anysinusoidal signal starting with a zero voltage, the Memristor’sresistance reaches its maximum or minimum value at t =(2n+ 1)T/2; n = 0, 1, 2, · · · , such that,

R2M = R2

i −2VokRd

πf, RM ∈ (Ron, Roff ) (4)

The magnitude of RM , whether maximum or minimum, de-pends on the sign of Vo. However, at t = nT ,n = 0, 1, 2, · · · ,the resistance returns to its initial value, R = Ri. Fig. 2shows the 3D surfaces of the Memristor’s resistance versust-Vo plane.

The resistance has the shape of sin2 (·) . However for highamplitude voltage there is clipping effect due to the boundingRoff and Ron. As |Vo| increases the effect of the boundaryresistancesRoff , Ron appears as clipping effect.

Fig. 3 shows the Memristor’s resistance versus v (t)-f plane.Generally, the resistance clipping to Roff exists at lowerfrequency (from the upper saturation plane). However, asfrequency increases, the clipping interval decreases in timeuntil it vanishes at a certain frequency. Any cross-section atfixed high frequency will show an ellipse shape, and the small

Fig. 3. Memristor’s resistance versus a sinusoidal input and frequency for apeak input voltage of 3V, Roff = 38kΩ, Ron = 100Ω, and Ri = 11kΩ.

10−1 101 1030

10

20

30

40

Fequancy

Res

ista

nce

Ran

ge

Roff = 16KΩ

Roff = 38KΩ

10−1 101 1030

10

20

30

40

Fequancy

Res

ista

nce

Ran

ge

Vo = 3.0 Volt

Vo = 1.5 Volt

Roff = 38KΩVo=1.5 Volt

(a) (b)

Fig. 4. Range of Memristor’s resistance versus frequency, for sinusoidal waveinput voltage for different (a) Roff , and (b) peak voltages Vo. Memristor’sParameters: Ron = 100Ω, Ri = 4kΩ.

100 102 1041

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6x 104

Frequency

RM

ax

Calculated (Roff=16KΩ)

Calculated (Roff=25KΩ)

Simulated (Roff=16KΩ)

Simulated (Roff=25KΩ)

0.5 1 1.5 20

0.5

1

1.5

2x 104

Time

Res

ista

nce

CalculatedSimulatedRoff= 38KΩ

(b)(a)

Fig. 5. Numerical and SPICE simulations in (a) time domain, withMemristor’s Parameters: Ron = 100Ω, and Ri = 4kΩ (b) frequency domain,with Memristor’s Parameters: Ron = 100Ω, Ri = 11kΩ and p = 10.

axis of the ellipse area decreases and rotates as frequencyincrease.

Fig.4 shows the range of Memristor’s resistance for both±Vo for two different cases. Fig. 4(a) studies the effect ofRoff on the operating range of resistance. As Roff decreasesthe Memristor’s resistance saturates at Roff for wider rangeof frequencies but it saturates at Ron for less range offrequencies. However, for fixed Roff and two different valuesof Vo as shown in Fig. 4(b), as Vo increases both ranges offrequencies increase. Fig. 5(a) shows the the resistance givenby equation (3) compared to the nonlinear Memristor’s SPICEmodel given in [12].

Fig. 5(b) shows the maximum resistance of the transientSPICE simulation results for different frequencies chosen as6 points/decade, while the solid lines are the calculated maxi-mum frequencies according to equation (4). These simulationsare repeated for two different cases showing identical matchingwith the SPICE model. Both the frequency and the voltageamplitude affect the resistance range and the rotation ofthe pinched i-v hysteresis. The boundaries for non-saturatingresistance are given by,

(Vo/f)off =π

2kRd

(R2

i −R2off

)< 0 (5a)

(Vo/f)on =π

2kRd

(R2

i −R2on

)> 0 (5b)

The difference between these boundaries, which is indepen-dent on Ri, is given by,

|Vo/f |sat =πD2

2µv(1 +Roff/Ron) (6)

Fig. 6. Intervals of the Memristor’s resistance versus (Vo/f).

which is independent of Ri. This value is considered as theminimax ratio |Vo/f | required for the Memristor’s resistanceto reach saturation, either Roff or Ron, when Ri equals toRon or Roff respectively as shown in Fig. 6.

According to equation (4), the range of the Memristor’sresistance depends on the value of RM which is inverselyproportional to the frequency (f ). Thus the resistance rangedecreases with the increase of the frequency. Consequently theMemristor’s current range decreases leading to a shrinkage ofthe pinched hysteresis. This will be further elaborated in thenext section.

III. QUALITATIVE STUDY OF THE i-v HYSTERESIS

The 3D pinched i-v hysteresis as a function of frequencyis shown in Fig. 7. As the frequency increases, with sameamplitude voltage, the pinched hysteresis shrinks and rotates.This figure describes the rotation only in very narrow bandof frequencies from 0.5Hz up to 2Hz. The dotted points inFig. 8(a) shows the SPICE model pinched i-v hysteresis forVo = −1.5 and f = 1Hz. However the solid line representsthe numerical simulation using the mathematical model atthe same input data, showing the exact matching with thenonlinear SPICE model. In this section some of the maincharacteristics of the pinched i-v hysteresis such as maximumcurrent, angle of rotation with frequency, and power inside thehysteresis are derived.

A. Implicit Resistance Equation:

The implicit equations which relate RM as function of v is,

R± =

√R2

i −VokRd

πf

(1±

√1− (v/Vo)

2

)(7)

Fig. 7. Memristor’s 3D i-v hysteresis for a sinusoidal input voltage.

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5x 10

−4

Voltage

Curr

ent

Calculated

Simulated

(a) (b)

Fig. 8. (a) i-v hysteresis at frequency of 1Hz and peak voltage of −1.5Voltusing Roff = 16kΩ, Ron = 100Ω, Ri = 4kΩ and p = 10. (b) Generalschematic of the Memristor’s pinched i-v hysteresis.

where the positive sign is applied when the voltage increasesand the negative when decreases. Thus the rate of change ofv must be known (ie memorize the previous value). Conse-quently, the implicit equations for the i-v hysteresis is givenby i1,2 = v/R±, which matches the SPICE simulations asshown in Fig. 8(a).

B. Location of the peak current:

The peak current, occurring at the point b on the pinchedi-v hysteresis, is given by i∗ = v/R∗, where v∗andR∗ are thevoltage and resistance at point b as shown in Fig. 8(b). Let,

y =

√1− (v/Vo)

2, α1 = R2

in −VokRd

πf, α1 =

VokRd

πf(8)

The critical point at which the maximum current occurs canbe obtained by evaluating, ∂i/∂v = (∂i/∂y) (∂y/∂v) = 0,which can be simplified into the following quadratic equation,

y2 ± 2α1

α2y + 1 = 0 (9a)

y = 1− πfRin (Ri −RM )

VokRd=πf (Ri −RM )

2

2VokRd(9b)

v∗ = Vo√

1− y2, R∗ =√RinRM (9c)

Fig. 9 shows the peak current values i∗ and its location v∗

versus the frequency for unsaturated conditions. The minimumvalue of the frequency is calculated by the maximum absolutevalue of the equation (5a) and (5b). In this case to avoid Mem-ristor’s resistance saturation the frequency must be greater than33.7784Hz, when Vo = 3V, Rin = 3KΩ, Ron = 100Ω, andRoff = 16KΩ.

C. Angle of the i-v hysteresis at peak voltage

As shown in Fig. 8(b), the angle αm indicates the rotationof the hysteresis curve and is given by,

αm = cot−1 (Ra) = cot−1√RinRM (10)

where Ra is the resistance at the peak voltage v = Vo as,

Ra =√

(R2i +R2

M ) /2 (11)

Fig. 10 plots the Memristor’s resistance versus frequency.This angle depends on the initial resistance Ri and the ratio

30 40 50 60 70 80 90 1001

1.5

2

2.5

3

Frequency (Hz)

Calculated V*

Exact V*

Peak current i*

Vo=3Volt, Rin=3K,Ron=100, Roff=16K

Minimum frequency=33.7784Hz

Fig. 9. Calculated peak current and its location v∗ and its exact value.

Vo/f as discussed before. The maximum range of this angleis given by,

cot−1 (Roff ) < αm < cot−1 (Ron) (12)

D. Area enclosed by the i-v hysteresis:

The power under the curves C1 and C2, and the enclosedarea inside the pinched hysteresis is a key characteristic ofthe Memristive device. The area inside one loop of the i-vcharacteristic of the HP’s Memristor is,

AHysteresis = −∫C1

i1d−∫C2

i2dv (13)

where the negative signs are due to the counterclockwisedirection of the closed loop. The area under any curvesbetween v = v1 and v2 can be calculated as,

A1 = −2π2f2

k2R2d

r2=Ra∫r1=Ri

(R2

i −VokRd

πf− r2

)dr

=2π2f2

3k2R2d

(RM −Ra)(RaRM −R2

i

)(14)

30 40 50 60 70 80 90 1000

500

1000

1500

2000

2500

3000

3500

Frequency (Hz)

RM (min)Ra

R*

Rin

Fig. 10. The relationship between the initial resistance, resistance atpeak voltage, resistance at peak current, and the minimum resistance versusfrequency when Vo = 3V, Ri = 3kΩ, Ron = 100Ω, and Roff = 16kΩ.

In addition, A2 can be given by,

A2 =2π2f2

3k2R2d

(Ra −Ri)(R2

M −RiRa

)(15)

So, the hysteresis power (enclosed by the two curves) is,

AHysteresis = A2 −A1 =π2f2

3k2R2d

|Rinitial −RM |3 (16)

As the frequency increases by small amount, the differencebetween the Ri and RM decreases, which compensates thei-v hysteresis area. This decreasing will be dominant, and thisarea at high frequency can be approximated as,

AHysteresis ≈V 3o kRd

3πfR3initial

(1 +

VokRd

2πfR2initial

)(17)

These expressions are valid if and only if R ∈ (Ron, Roff ).

IV. CONCLUSION

In this paper, we present a mathematical model for Memris-tors characterized by linear dopant drift under sinusoidal inputvoltage stimulation. The model describes the Memristor’sresistance using its voltage and current without the need forthe flux (φ) or charge equations (Q). The frequency responseof the Memristor’s resistance and its bounding conditionsare derived for the first time. A simple implicit equation forthe pinched i-v hysteresis is derived, describing its generalcharacteristics such as critical resistances, maximum currentmagnitude and location, and the power inside this hysteresis.All the derived formulas are compared with the nonlinearMemristor’s SPICE model showing perfect matching.

REFERENCES

[1] L. Chua, “Memristor-the missing circuit element,” IEEE Transactionson Circuit Theory, vol. 18, no. 5, pp. 507–519, 1971.

[2] D. B. Strukov, G. S. Snider, and D. R. Stewart, “The missing memristorfound,” Nature, vol. 435, pp. 80–83, 2008.

[3] M. Di Ventra, Y. Pershin, and L. Chua, “Circuit elements with memory:Memristors, memcapacitors, and meminductors,” Proceedings of theIEEE, vol. 97, no. 10, pp. 1717–1724, 2009.

[4] P. O. Vontobel, W. Robinett, P. J. Kuekes, D. R. Stewart, J. Straznicky,and R. S. Williams, “Writing to and reading from a nano-scale crossbarmemory based on memristors,” Nanotechnology, vol. 20, no. 42, p.425204, 2009.

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[6] Y. V. Pershin and M. Di Ventra, “Spin memristive systems: Spin memoryeffects in semiconductor spintronics,” Phys. Rev. B, vol. 78, no. 11, p.113309, 2008.

[7] D. Wang, Z. Hu, X. Yu, and J. Yu, “A PWL model of memristor andits application example,” International Conference on Communications,Circuits and Systems, 2009. ICCCAS 2009, pp. 932–934, July 2009.

[8] F. Y. Wang, “Memristor for intrductory physics,” Physics. class-ph, pp.1–4, 2008.

[9] Y. N. Joglekar and S. J. Wolf, “The elusive memristor: properties of basicelectrical circuits,” European Journal of Physics, vol. 30, pp. 661–675,2009.

[10] A. G. Radwan, M. A. Zidan, and K. N. Salama, “HP Memristor math-ematical model for periodic signals and DC,” 53rd IEEE InternationalMidwest Symposium on Circuits and Systems (MWSCAS), pp. 861–864,Aug 2010.

[11] A. Talukdar, A. G. Radwan, and K. N. Salama, “Time domain oscillatingpoles: Stability redefined in memristor based wien-oscillators,” IEEEInternational Conference on Microelectronics (ICM), Dec 2010.

[12] Z. Biolek, D. Biolek, and V. Biolkova, “Spice model of memristor withnonlinear dopant drift,” Radioengineering, vol. 18, no. 2, pp. 210–214,2009.