MODELLING AND THE STUDY OF THE MEMRISTOR

Thesis

Submitted to

The School of Engineering of the

UNIVERSITY OF DAYTON

In Partial Fulfillment of the Requirements for

The Degree of

Master of Science in Electrical Engineering

By

MohanUjwal Bandaru

UNIVERSITY OF DAYTON

Dayton, Ohio

December, 2012 MODELLING AND THE STUDY OF THE MEMRISTOR

Name: Bandaru, MohanUjwal

APPROVED BY:

John Loomis, Ph.D. Raul Ordonez, Ph.D. Advisor Committee Chairman Committee Member Associate Professor Associate Professor Electrical & Computer Engineering Electrical & Computer Engineering

Monish Chatterjee, Ph.D. Committee Member Professor Electrical & Computer Engineering

John G. Weber, Ph.D. Tony E. Saliba, Ph.D. Associate Dean Dean, School of Engineering School of Engineering & Wilke Distinguished Professor

ii ABSTRACT

MODELLING AND THE STUDY OF THE MEMRISTOR

Name: Bandaru, MohanUjwal University of Dayton

Advisor: Dr. John Loomis

Till date the circuitry world has known three fundamental circuit elements - capacitor, resistor and inductor. These circuit elements are defined by the relation between two of the four fundamental circuit variables- current, voltage, charge and

flux. Way back in 1971, Prof. Leon Chua proposed on the grounds of symmetry that there should be a fourth fundamental circuit element which gives the relation between

flux and charge. He named this the memristor, which is the short of memory resistor.

This theory was then practically modeled, in May 2008 when the researchers at HP

Labs published a paper announcing a model for a physical realization of a memristor.

This report mainly focuses on the model of memristor and its applications. All the simulation results are studied using LTSpice.

iii This thesis is dedicated to my parents Mrs. B Indira and CDR. B V S Rao.

iv ACKNOWLEDGMENTS

Many minds and hands have helped me in the successful completion of my research. I would like to thank my advisor and mentor Dr. John Loomis for his constant support and guiding me in the right direction throughout my research. I am also grateful to Dr. Guru Subramanyam for being a part of the committee. I am extremely grateful to Dalibor Biolek, member of he CAS/COM Czech National Group of IEEE for the timely response to my queries through e-mail and help me model the memristor.I would like to take this opportunity to thank my parents Mrs. B. Indira and CDR. B V S Rao and my friends for their valuable support and motivation.I would finally like to thank the Department of ECE for increasing my knowledge and making me feel comfortable during my time at the University of Dayton.

v TABLE OF CONTENTS

Page

ABSTRACT ...... iii

DEDICATION ...... iv

ACKNOWLEDGMENTS ...... v

LIST OF FIGURES ...... viii

CHAPTER:

1. INTRODUCTION ...... 1

2. BASIC THEORY OF MEMRISTORS ...... 3

2.1 Origin of Memristors ...... 3 2.2 Definition of Memristor ...... 4 2.3 Analogy of a Memristor ...... 6 2.4 Characteristics of Memristors ...... 8 2.4.1 ϕ-q Characteristics of Memristor ...... 8 2.4.2 i-v Characteristics of a Memristor ...... 8

3. ELECTROMAGNETIC INTERPRETATION OF MEMRISTOR . . . . 10

4. MODEL AND WORKING OF THE MEMRISTOR FROM HP LABS . 14

5. LTSpice MODEL OF MEMRISTOR ...... 17

5.1 Simulation Results-Using LTSpice Model ...... 19

vi 6. APPLICATIONS ...... 22

6.1 Integrator ...... 22 6.2 Voltage Divider ...... 23 6.3 Potential Drive Waveforms ...... 24 6.4 Arithmetic Optimization ...... 26 6.5 Pattern Recognition ...... 27

7. CONCLUSION ...... 34

BIBLIOGRAPHY ...... 35

APPENDIX: SPICE MODEL OF A MEMRISTOR

vii LIST OF FIGURES

Figure Page

1.1 The four fundamental two-terminal circuit elements...... 2

2.1 The relation between the circuit elements...... 4

2.2 Aristotle’s Theory of Matter...... 4

2.3 Simple Model of Memristor...... 5

2.4 Analogy of Memristor...... 7

2.5 ϕ-q curve of Memristor...... 8

2.6 The pinched hysteresis loop ...... 9

4.1 Schematic of HP MR, where D is the device channel length and w is

the length of the doped region...... 15

4.2 Behavior of HP memristor when positive and negative voltages are

applied...... 16

5.1 Symbol of the model in LTSpice...... 17

5.2 HP Memristor...... 18

5.3 Memristor Model Circuit...... 19

5.4 Input Voltage applied to the memristor...... 20

5.5 The current through memristor...... 20

5.6 Current-verses-voltage...... 21

viii 6.1 The integrator with memristor...... 22

6.2 Output current waveform...... 23

6.3 Output waveform...... 24

6.4 Resistor-memristor divider...... 24

6.5 Memristor-memristor divider...... 25

6.6 Input and the output waveforms...... 25

6.7 Input and the output waveforms for the memristor divider...... 26

6.8 The output sequence based on input a...... 27

6.9 The output sequence based on input b...... 27

6.10 The output sequence based on input c...... 28

6.11 The output sequence based on input d...... 28

6.12 Analog output representative of 2+4...... 29

6.13 Analog output representative of 1+3...... 29

6.14 Memristor Crossbar Pattern Comparison Circuit (write mode) a. . . . 29

6.15 Memristor Crossbar Pattern Comparison Circuit (write mode) b. . . . 30

6.16 Memristor Crossbar Pattern Comparison Circuit (write mode) c. . . . 30

6.17 Memristor Crossbar Pattern Comparison Circuit (write mode) d. . . . 31

6.18 Memristor Crossbar Pattern Comparison Circuit (detect mode) a. . . 31

6.19 Memristor Crossbar Pattern Comparison Circuit (detect mode) b. . . 32

6.20 Memristor Crossbar Pattern Comparison Circuit (detect mode) c. . . 32

6.21 Memristor Crossbar Pattern Comparison Circuit (detect mode) d. . . 33

ix CHAPTER 1

INTRODUCTION

In circuit theory there are three basic two-terminal devices namely-resistor, capacitor,

and inductor. These elements are defined by the relation between two of the four

fundamental circuit variables- current i, voltage v, charge q and flux ϕ, where the

time derivative of charge q is current i and according to Faradays laws voltage v is the

time derivative of flux ϕ. The resistor is defined with the relation between the voltage

v and current i as dv = Rdi. The capacitor is defined with the relation between the

charge q and voltage v as dq = Cdv. The inductor is defined with the relation between

the flux ϕ and current i as dϕ = Ldi. The discovery of the existence of the fourth

fundamental circuit element came to light in 1971 when Prof. Leon Chua proposed

the missing relation between charge q and the flux ϕ through symmetry in figure

1.1 [1].

Prof. Leon Chua named this element memristor, which is the short for memory resistor. The memristor has a memristance M and the functional relation between

flux ϕ and charge q is given by dϕ = Mdq. The already known passive two- terminal devices are the basic building blocks of modern electronics and are therefore ubiquitous in circuits.But we know that though some elements store information they eventually decay out. Even if the state of one element changes, the information about

1 Figure 1.1: The four fundamental two-terminal circuit elements.

the new state would be lost once the power is turned off and wait sometime. This basic point may seem irrelevant but fundamentally this is very crucial. The capacity to store information without the need of a power source would represent a paradigm change.

2 CHAPTER 2

BASIC THEORY OF MEMRISTORS

2.1 Origin of Memristors

Prof. Leon Chua noted that there are six different mathematical relations connecting

pairs of the four fundamental circuit variables current i, voltage v, charge q and fluxϕ.

The relation between these variable is deduced from Faradays law of Induction. A

resistor is defined by the relationship between voltage v and current i (dv = Rdi) , the

capacitor is defined by the relationship between charge q and voltage v (dq = Cdv)

and the inductor is defined by the relationship between flux ϕ and current i dϕ = Ldi

. In addition, the current i is defined as the time derivative of the charge q and according to Faradays law, the voltage v is defined as the time derivative of the flux

ϕ. This relation is shown in the figure 2.1 [2] .

Leon Chua compared the above model to that of Aristotles theory of matter.

According to this theory all matter consists of earth, water, air and fire. Each of these elements exhibits two of the four fundamental properties moistness, dryness, coldness and hotness. This is shown in figure 2.2 [2]. So depending on the above theory he saw a striking resemblance and predicted the existence of the fourth kind of element and called it memristor.

3 Figure 2.1: The relation between the circuit elements.

Figure 2.2: Aristotle’s Theory of Matter.

2.2 Definition of Memristor

Memristor, is a contraction of memory resistor because its main function is to remember its history. The memristor is a two-terminal device whose resistance

4 depends on the magnitude and polarity of the voltage applied to it and the length of

the time that voltage has been applied. When this voltage is turned off, the memristor

remembers its most recent resistance until the next time you turn it on. The simple

model of this is as shown in figure2.3 [3].

Figure 2.3: Simple Model of Memristor.

Memristor is either said to be a charge controlled or a flux controlled depending

upon the relation between the flux ϕ and the charge q as a function of the other. For a charge controlled memristor the relation between flux and charge is expressed as a function of electric charge q [2],

ϕ = f(q) (2.1)

Differentiating (2.1)

dϕ df(q) dq (2.2) dt = dq dt

5 dq dt = i(t) (2.3) dϕ dt = v(t)

v(t) = M(q).i(t) (2.4)

df(q) (2.5) M(q) = dq

For a flux controlled memristor the relation between flux and charge is expressed as a function of flux linkage ϕ [2],

q = f(ϕ) (2.6)

Differentiating (2.6)

dq df(ϕ) dϕ (2.7) dt = dϕ dt

dq dt = i(t) (2.8) dϕ dt = v(t)

i(t) = W (ϕ).v(t) (2.9)

2.3 Analogy of a Memristor

Consider a resistor to be a pipe through which water flows in a stream line [4].

The water represents the flowing electric charge. The resistors obstruction to the flow

6 of charge is comparable to the diameter of the pipe: narrower the pipe, the greater the resistance and vice versa. The resistors have a fixed pipe diameter, but where a memristor is a pipe that changes diameter with the amount and the direction of the water flow that flows through it. If the water flow through this pipe is in one direction, it expands, becoming less resistive. Send the water in the opposite direction, the pipe shrinks making it more resistive. Another important aspect of it is that the memristor remembers its diameter when water last went through it. Turn off the flow of water and the diameter of the pipe freezes until the water is turned back on. This shows the main characteristic of a memristor making it very special. The analogy is shown in the figure 2.4 below.

Figure 2.4: Analogy of Memristor.

7 2.4 Characteristics of Memristors

2.4.1 ϕ-q Characteristics of Memristor

One of the main characteristics of a ϕ-q of a memristor is it is monotonically increasing. The memristance M(q) is the slope of the ϕ-q curve. Prof. Leon Chua postulated a passivity criterion, according to which a memristor is passive if and only if the memristance M(q) is non-negative [5]. If M(q) ≥ 0, then the instantaneous power dissipated by the memristor, p(i) = M(q).[i(t)]2 , is always positive and so the

memristor is a passive device [6].The typical curves are as shown in figure2.5

Figure 2.5: ϕ-q curve of Memristor.

2.4.2 i-v Characteristics of a Memristor

An important characteristic of a memristor is the pinched hysteresis loop current-

voltage characteristics [2]. For a memristor excited by a periodic signal, when the

8 voltage v(t) is 0, the current i(t) is zero and vice versa. If any device has the above characteristics, then the device is either a memristor or a memristive device.The characteristics are as shwon in figure 2.6 [7]

Figure 2.6: The pinched hysteresis loop

9 CHAPTER 3

ELECTROMAGNETIC INTERPRETATION OF

MEMRISTOR

It is well know that the circuit theory is a special case of electromagnetic field

theory. The characterization of the three classical circuit elements can be given

an elegant electromagnetic interpretation in terms of the quasi-state expansion of

Maxwells equations [8]. The main objective is to give an analogous interpretation for

the characterization of Memristors. The Maxwell equations in the differential form is

given by

∂B (3.1) curlE = − ∂t

∂D (3.2) curlH = J + ∂t

Where E and H are the electric and magnetic field intensity, D and B are the electric and magnetic flux density, and J is the current density. It has been shown that circuit theory belongs to the realm of quasi-static fields and can be seen with the help of the zero-order and the first-order Maxwells equations in quasi-state form

[8]

Zero-Order Maxwells Equations,

10 curlE0 = 0 (3.3)

curlH0 = J0 (3.4)

First-Order Maxwells Equations,

∂B0 (3.5) curlE1 = − ∂t

∂D0 (3.6) curlH1 = J1 + ∂t

A resistor has been identified to be an electromagnetic system whose first-order

fields are negligible compared to its zero-order fields, so that its characterization can be interpreted as an instantaneous relationship between the two zero-order fields E0 and H0 . An inductor has been identified to be an electromagnetic system where only the first-order magnetic field is negligible. A capacitor has been identified to an electromagnetic system where only first-order electric field is negligible. The case where both the first-order fields are not negligible has been dismissed as having no corresponding situation in circuit theory [8]. This missing combination was interpreted to have the characterization of a memristor.

To realize the above case it is needed to show that under certain conditions the instantaneous value of the first-order electric flux density D1 is related to the instantaneous value of the first-order magnetic flux density B1 . As the surface integral of the electric flux density D1 is proportional to the charge q(t), and the

11 surface integral of the magnetic flux density B1 is proportional to the flux ϕ(t). This would be possible if the two-terminal device is postulated with the following two properties, 1) Both zero-order fields are negligible compared to the first-order fields:

E ≈ E1 ,H ≈ H1 ,D ≈ D1 ,B ≈ B1 , and J ≈ J1 . 2) The material from which the device is made is nonlinear. To make it more generalized the nonlinear relationship is given as follows

J1 = J(E1) (3.7)

B1 = B(H1) (3.8)

D1 = D(E1) (3.9)

Where J(.), B(.), and D(.) one-to-one functions from R onto R. Under these assumptions, (3.6) and (3.7) can be combined to give

curlH1 = J(E1) (3.10)

From the above equation under any specified boundary condition appropriate for the device, the first-order electrical field E1 is related to the first-order magnetic field

H1 by a relation

E1 = f(H1) (3.11)

12 If we substitute (3.11) for E1 in (3.9) and then substitute the inverse function of

B(.) from (3.8) into the resulting expressing we get [5]

−1 D1 = D◦f ◦ [B (B1)] ≡ g(B1) (3.12)

Thus the physical mechanism characterizing a memristor device must come from the instantaneous interaction between the first-order electric field and the first-order magnetic field of a fabricated device which have the above mentioned two properties.

This implied that a physical memristor device is essentially an a.c device, or else the d.c electromagnetic fields would give rise to non-negligible zero-order fields.

13 CHAPTER 4

MODEL AND WORKING OF THE MEMRISTOR FROM

HP LABS

Stanley Williams and his group at the HP Labs realized the memristor in a device form [4]. Their basic model had two platinum electrodes on the either end of the device. The surface of the bottom platinum wire was oxidized to make an extremely thin layer of platinum dioxide, which is highly conducting. Then they assembled a dense film, only one molecule thick, of specially designed switching molecules. Over this layer they deposited 2-3 nm layer of Ti metal and the final layer was the top platinum electrode. After years of experiments, they finally realized that what they have was a memristor, as their i-v characteristic are similar to the one proposed by

Chua. They observed that under the molecular layer, instead of platinum dioxide, there was only pure platinum. Above the molecular layer instead of titanium, they found an unexpected and unusual layer of titanium dioxide. The titanium dioxide was not just the regular titanium dioxide, it has split itself up into two chemically different layers. Adjacent to the molecules, the oxide was stoichiometric titanium dioxide, meaning the ratio of oxygen to titanium was perfect, exactly 2 to 1. But closer to the platinum electrode, the titanium oxide was missing a tiny amount of

14 its oxygen, and it was called oxygen-deficient T iO2−x where x is about 0.05. The schematic of the above description is as shown in figure 4.1 [6]

Figure 4.1: Schematic of HP MR, where D is the device channel length and w is the length of the doped region.

The operation that was observed which could explain the above mechanism is

that if a positive voltage is applied to the top electrode of the device, it will repel the

oxygen vacancies in the T iO2−x layer down into pure T iO2, that turns the T iO2 layer

to T iO2−x and makes it conductive, thus turning the device on. When a negative

voltage is applied the vacancies are attracted upward and back out of the T iO2 and

thus the thickness of T iO2 layer would increase and turn off the device.This schematic

is shown in figure 4.2 [4] What makes this special-memristive- is that when the voltage

is turned off, positive or negative, the oxygen bubbles do not migrate. They stay where

they are, which means that the boundary between the two titanium dioxide layers is

frozen. That is how the memristor remembers how much voltage was last applied.

The mathematical model [9] of the HP Memristor is given by

15 Figure 4.2: Behavior of HP memristor when positive and negative voltages are applied.

Ron (4.1) M(q) = Roff [1 − β q(t)]

D2 where β = has the dimensions of magnetic flux. µD is the average drift velocity µD

2 and has the units cm /sV : D is the thickness of titanium-oxide film: Ron and Roff

are on-state and off-state resistances: and q(t) is the total charge passing through the meristor device.

16 CHAPTER 5

LTSpice MODEL OF MEMRISTOR

Biolek provided the Spice model of the memristor. Though PSpice and LTSpice are similar in the nature of their analysis, but even though there wasn’t any analysis done in LTSpice. So I tried using LTSpice to be the basis of my simulation. The listing for the LTSpice model is included in the appendix. The model generated by me in LTSpice is as shown in figure 5.1

Figure 5.1: Symbol of the model in LTSpice.

The physical model of the memristor is shown in figure 5.2 [6]. The total resistance of the doped and undoped regions is given by

Rmem(x) = Ronx + Roff (1 − x) (5.1)

= Roff x − (Roff − Ron)x

w Where x = D ∈ (0, 1) is the width of the doped region, referenced to the total length D of the T iO2 layer, and Roff and Ron are the limit values of the memristor

17 Figure 5.2: HP Memristor.

resistance for w=0 and w=D. The ratio of the two resistances usually given as 102

103 [10]. The speed of the movement of the boundary between the doped and undoped regions depend on the resistance of the doped area, on the passing current, and on other factors according to the equation given

dx dt = k.i(t).f(x) (5.2) µvRon k = D2

−14 2 −1 −1 where µv ≈ 10 m s V is the dopant mobility. As mentioned in [9], in nanoscale devices, small voltages can yeild enormous electric fields, which can secondarily produce significant nonlinearities in ionic transport. These nonlinearities manifest themselves particularly at the thin film edges, where the speed of boundary between the doped and undoped regions gradually decrease to zero. This phenomenon, called nonlinear dopant drift, can be modeled by the so-called window function f(x) on the right of (5.2). A concrete window function that would correspond to the memristor from HP labs is not available at this moment [11].

18 The paper [10] proposes the window function in the following from:

f(x) = 1 − (2x − 1)2p (5.3)

where p is a positive integer. The differences between the models with linear and nonlinear drift disappears when p increases.

5.1 Simulation Results-Using LTSpice Model

For the simulation of the memristor for its desired characteristics, the width D of

−14 2 −1 −1 the T iO2 film is considered to be 10nm and the dopant mobility µv ≈ 10 m s V .

The values assumed for Ron = 100Ω, Roff = 10kΩ and the initial resistance Rinit =

1kΩ. The circuit symmetry is shown in figure 5.3

Figure 5.3: Memristor Model Circuit.

The simulation results are shown in figure 5.4,figure 5.5 and figure 5.6

19 Figure 5.4: Input Voltage applied to the memristor.

Figure 5.5: The current through memristor.

20 Figure 5.6: Current-verses-voltage.

21 CHAPTER 6

APPLICATIONS

6.1 Integrator

Figure 6.1 shows an integrator using a memristor and an operational amplifier [7].

When a square wave is applied to the input, the output would be a ramp signal, which is in fact the integral of the input voltage.The output waveform is shown figure 6.2

Figure 6.1: The integrator with memristor.

If the period or the amplitude of the input square wave increases, the output is

not a ramp anymore. Waveform shown in figure 6.3

22 Figure 6.2: Output current waveform.

In the figure 6.3 both the amplitude and the period were changed to see the combined effect.

6.2 Voltage Divider

I made an attempt to see how the memristors would behave in the case of a voltage divider. And surprisingly memristors at infinitesimal time periods behave like a resistor. In-order to show this behavior I have made a combination of the resistor-memristor and a memristor-memristor. The circuit is shown in figure 6.4 and

figure 6.5 respectively and also their output are shown in figure 6.6 and figure 6.7 respectively.

23 Figure 6.3: Output waveform.

Figure 6.4: Resistor-memristor divider.

6.3 Potential Drive Waveforms

Timing modulation and amplitude modulation circuits implemented in hardware can require complex circuitry and have limitations in adaptability and range of 24 Figure 6.5: Memristor-memristor divider.

Figure 6.6: Input and the output waveforms.

possible waveforms. Software based solutions require a microprocessor which can be difficult/expensive to miniaturize for several portable electronics applications. With

25 Figure 6.7: Input and the output waveforms for the memristor divider.

the memristor crossbar, drive waveform circuits could have a better performance.

The circuitry are shown in figure 6.8, figure 6.9, figure 6.10, figure 6.11.

Both the timing and amplitude of the output waveform can be adjusted by binary switching of the memristance states in the crossbar. Combining with genetic algorithms the circuitry has the potential for self-optimization drive waveforms.

6.4 Arithmetic Optimization

In the present systems segmentation between memory and the processor circuitry may produce a bottleneck in speed. The logic based arithmetic is inefficient for some network optimization problems, which involve repeated recalculation. Using the memristor crossbar, the arithmetic circuit is as shown figure 6.12, figure 6.13.

26 Figure 6.8: The output sequence based on input a.

Figure 6.9: The output sequence based on input b.

Although op-amps are slower than logic circuits, the combination of memory and processing in a single circuit reduces the reliance on a clock.

6.5 Pattern Recognition

Usually the software based pattern recognition requires time for data transfer between memory and processor circuits which cause a lag in responsiveness. Hardware solutions can be faster but have limits in adaptability and limits in the range of

27 Figure 6.10: The output sequence based on input c.

Figure 6.11: The output sequence based on input d.

patterns that can be classified. Memristors offer a route to pattern recognition solution combining both memory storage and data processing in a common circuit.

The circuitry for write mode are shown in figure 6.14, figure 6.15, figure 6.16, figure

6.17.

The circuitry for the detect mode are shown in figure 6.18, figure 6.19, figure 6.20,

figure 6.21.

28 Figure 6.12: Analog output representative of 2+4.

Figure 6.13: Analog output representative of 1+3.

Figure 6.14: Memristor Crossbar Pattern Comparison Circuit (write mode) a.

29 Figure 6.15: Memristor Crossbar Pattern Comparison Circuit (write mode) b.

Figure 6.16: Memristor Crossbar Pattern Comparison Circuit (write mode) c.

30 Figure 6.17: Memristor Crossbar Pattern Comparison Circuit (write mode) d.

Figure 6.18: Memristor Crossbar Pattern Comparison Circuit (detect mode) a.

31 Figure 6.19: Memristor Crossbar Pattern Comparison Circuit (detect mode) b.

Figure 6.20: Memristor Crossbar Pattern Comparison Circuit (detect mode) c.

32 Figure 6.21: Memristor Crossbar Pattern Comparison Circuit (detect mode) d.

33 CHAPTER 7

CONCLUSION

The impact that the memristor can have on the existing technology is colossal.

This report presents the detail study of memristors and also shows some worked out applications. The memristor is modeled in LTSpice using the existing model proposed by HP Labs. Memristors has certainly showed a lot of promise and also has the potential to be another milestone in the path of evolution of technology for better. It was rightly said by Prof. Leon Chua that All the engineering text books have to be re-written.

34 BIBLIOGRAPHY

[1] J. Mullians, “Memristive minds,” in http://nature.berkeley.edu/Memristor.pdf.

[2] “Memristor and memristive systems symposium,” in

http://www.youtube.com/watch?v=QFdDPzcZwbs.

[3] A. K. S. Kvatinsky, E.G. Friedman and U. Weiser, “Memristor and related

application.”

[4] R. Williams, “How we found the missing memristor,” in IEEE Spectrum, vol. 45,

2008, pp. 28–35.

[5] L. Chua, “Memristor- the missing circiut element,” in IEEE Trans. Circuit

Theory, vol. CT-18, 1971, pp. 507–519.

[6] A. I. K. E. S. A.-S. O. Kavehei, Y.S. Kim and D. Abbott, “The fourth element:

Characteristics, modelling, and electromagnetic theory of the memristor,” in

http://arxiv.org/abs/1002.3210.

[7] A. C. P. M. Mahuash, “A memristor spice model for designing memristor

circuits.”

35 [8] L. C. R.M. Fano and R. Adler, “Electromagnetic fields, energy and forces,” in

Proceedings of IEEE Congress on Evolutionary Computation (CEC-2002)., New

York, USA., 1960.

[9] D. S. D.B. Strukov, G.S. Snider and R. Williams, “The missing memristor

found,” in Nature., vol. 453, 2008, pp. 80–83.

[10] Y. Joglekar and S. Wolf, “The elusive memristor: properties of basic electrical

circuits,” in arXiv:0807.3994, vol. 2, 2009, pp. 1–24.

[11] D. B. Z. Biolek and V. Biolkova, “Spice model of memristor with nonlinear

dopant drift,” in Radioengineering J., vol. 18, 2009, pp. 210–214.

36 APPENDIX

SPICE MODEL OF A MEMRISTOR

*Modified HP Memristor SPICE Model

**************************

* Ron, Roff - Resistance in ON / OFF States

* Rinit - Resistance at T=0

* D - Width of the thin film

* uv - Migration coefficient

* p - Parameter of the WINDOW-function

* for modeling nonlinear boundary conditions

* x - W/D Ratio, W is the actual width

* of the doped area (from 0 to D)

*

.SUBCKT memristor plus minus PARAMS:

+ Ron=100 Roff=10K Rinit=1k D=10N uv=10F p=10

***********************************************

* DIFFERENTIAL EQUATION MODELING *

***********************************************

Gx 0 x value={ I(Emem)*uv*Ron/D**2*f(V(x),p)}

37 Cx x 0 1 IC={(Roff-Rinit)/(Roff-Ron)}

Raux x 0 1T

* RESISTIVE PORT OF THE MEMRISTOR *

*******************************

Emem plus aux value={-I(Emem)*V(x)*(Roff-Ron)}

Roff aux minus {Roff}

***********************************************

*Flux computation*

***********************************************

Eflux flux 0 value={SDT(V(plus,minus))}

***********************************************

*Charge computation*

***********************************************

Echarge charge 0 value={SDT(I(Emem))}

***********************************************

* WINDOW FUNCTIONS

***********************************************

*window function, according to Joglekar

.func f(x,p)={1-(2*x-1)**(2*p)}

*proposed window function

;.func f(x,i,p)={1-(x-sttp(-i))**(2*p)}

.ENDS memristor

38