I

SEMINAR

ON

“ Memristors “

Submitted by

SHISHIR S BELUR Reg No.1PI09TE089

In partial fulfillment of the requirement for the award of degree in Bachelor of

Engineering in Telecommunication for the academic session Aug –Dec 2011

Carried out at

PESIT, Bangalore

Under the guidance of

Ms. Bharati V Kalghatgi

Lecturer

Dept. of TE

PESIT, Bangalore

In the academic session Aug–Dec 2011

P E S INSTITUTE OF TECHNOLOGY

100 Feet Ring Road, BSK III Stage, Bangalore - 85

(An Autonomous Institute under VTU, Belgaum)

education for the real world

II

CERTIFICATE

This is to certify that the seminar entitled “ Memristors ” is a bonafide work carried out by

Shishir S Belur Reg. No.: 1PI09TE089

at PESIT, Bangalore

in partial fulfillment of the requirement for the award of degree in

Bachelor of Engineering in Telecommunication of Visveswaraiah

Technological University for the academic session Aug –Dec 2011.

Signature of the seminar Guide Signature of the HOD

(Name & Designation of the faculty) (Name & Designation of HOD)

III

ACKNOWLEDGEMENT

I would like to thank the faculty of the Department of Telecommunication,

PESIT for having given me this opportunity to present a seminar on

'Memristors', whose guidance, advice and assistance helped formulate and

present this seminar. My special thanks to Ms. Bharati.V.Kalghatgi for having

guided me all through my efforts but for which this would not have fructified.

IV

PAGE INDEX

ABSTRACT 1

1. INTRODUCTION 2

2. MEMRISTOR THEORY

2.1. ORIGIN OF THE MEMRISTOR 4

2.2. DEFINITION OF A MEMRISTOR 5

2.3. WHAT IS MEMRISTANCE 5

2.4. PROPERTIES OF A MEMRISTOR

2.4.1. Φ-q CURVE 7

2.4.2. CURRENT-VOLTAGE CURVE 7

3. MODEL OF THE MEMRISTOR FROM HP LABS 9

3.1. LINEAR DRIFT MODEL 10

4. BENEFITS OF USING MEMRISTORS 14

5. RESULTS AND SIMULATIONS

5.1. SIMULATION RESULTS USING SPICE MODEL 15

6. POTENTIAL APPLICATIONS OF MEMRISTOR

6.1. TWO STATE CHARGE CONTROLLED MEMRISTOR 18

6.2. MEMRISTOR MEMORY 18

6.3. BASIC ARITHMETIC OPERATIONS 19

7. CONCLUSION AND FUTURE RESEARCH

7.1. CONCLUSION 22

7.2. FUTURE RESEARCH 22

8. BIBLIOGRAPHY 24

V

FIGURE INDEX

Page No.

Figure 1 about four basic circuit elements 3

Figure 2 about the three fundamental circuit elements 4

Figure 3 about the symbol of a memristor 5

Figure 4 about V-I characteristics of a memristor 8

Figure 5 about HP memristor 9

Figure 6 about voltage applied to a memristor 15

Figure 7 about current through a memristor 16

Figure 8 about charge-flux curve of a memristor 16

Figure 9 about current-voltage curve for f=1 Hz 16

Figure 10 about current-voltage curve for f=1.5 Hz 17

Figure 11 about current-voltage curve for f=2 Hz 17

Figure 12 about adjusting the memristance 19

Figure 13 about various arithmetic operations 21

Figure 14: Circuit symbols for memcapacitor and meminductor 23

1

ABSTRACT

Since the dawn of electronics, we've had only three types of circuit

components-resistors, inductors and capacitors. But in 1971, UC Berkeley

researcher Leon Chua theorized the possibility of a fourth type of component,

one that would be able to measure the flow of electric current in his paper

Memristor-The Missing Circuit Element.

The three fundamental circuit components- resistors, inductors and

capacitors are used to define four fundamental circuit variables which are

electric current, voltage, charge and magnetic flux. Resistors are used to

relate current to voltage, capacitors to relate voltage to charge and inductors

to relate current to magnetic flux. But there was no element which could relate

charge to magnetic flux. This lead to the idea and development of memristors.

Memristor is a concatenation of “memory resistors”. The most notable

property of a memristor is that it can save its electronic state even when the

current is turned off, making it a great candidate to replace today's flash

memory. An outstanding feature is its ability to remember a range of electrical

states rather than the simplistic "on" and "off" states that today's digital

processors recognize. Memristor-based computers could be capable of far

more complex tasks.

HP has already started produced an oxygen depleted titanium

memristor.

Memristors

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1. INTRODUCTION

In circuit theory, the three basic two-terminal devices — namely the resistor,

the capacitor and the inductor are well understood. These elements are defined

in terms of the relation between two of the four fundamental circuit variables,

namely, current, voltage, charge and flux. The current is defined as the time

derivative of the charge. According to Faraday‗s law, the voltage is defined as

the time derivative of the flux. A resistor is defined by the relationship

between voltage and current, the capacitor is defined by the relationship

between charge and voltage and the inductor is defined by the relationship

between flux and current. Out of the six possible combinations of the four

fundamental circuit variables, five are defined. In 1971, Prof. Leon Chua

proposed that there should be a fourth fundamental circuit element to set up

the relation between charge and magnetic flux and complete the symmetry as

shown on the next page in Fig. 1.

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Fig.1: Four basic circuit elements

Prof. Leon Chua named this the memristor, a short for memory resistor.

The memristor has a memristance and provides a functional relation between

charge and flux. In 2008, Stanley Williams, at Hewlett Packard, announced the

first fabricated memristor.

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2. MEMRISTOR THEORY

2.1 Origin of the Memristor

There are four fundamental circuit variables in circuit theory. They are current,

voltage, charge and flux. There are six possible combinations of the four

fundamental circuit variables. We have a good understanding of five of the

possible six combinations. The three basic two-terminal devices of circuit

theory namely, the resistor, the capacitor and the inductor are defined in terms

of the relation between two of the four fundamental circuit variables. A

resistor is defined by the relationship between voltage and current, the

capacitor is defined by the relationship between charge and voltage and the

inductor is defined by the relationship between flux and current. In addition,

the current is defined as the time derivative of the charge and according to

Faraday‗s law, the voltage is defined as the time derivative of the flux. These

relations are shown in Fig. 2.

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Fig.2: The three circuit elements defined as a relation between four circuit variables

2.2 Definition of a Memristor Memristor, the contraction of memory resistor, is a passive device that

provides a functional relation between charge and flux. It is defined as a two-

terminal circuit element in which the flux between the two terminals is a

function of the amount of electric charge that has passed through the device.

Memristor is not an energy storage element. Fig. 3 shows the symbol for a

memristor.

Fig.3: Symbol of the memristor

A memristor is said to be charge-controlled if the relation between flux

and charge is expressed as a function of electric charge and it is said to be flux-

controlled if the relation between flux and charge is expressed as a function of

the flux linkage.

2.3 What is Memristance? Memristance is a property of the memristor. When charge flows in a direction

through a circuit, the resistance of the memristor increases. When it flows in

the opposite direction, the resistance of the memristor decreases. If the applied

voltage is turned off, thus stopping the flow of charge, the memristor

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6

remembers the last resistance that it had. When the flow of charge is started

again, the resistance of the circuit will be what it was when it was last active.

The memristor is essentially a two-terminal variable resistor, with

resistance dependent upon the amount of charge q that has passed between the

terminals.

To relate the memristor to the resistor, capacitor, and inductor, it is

helpful to isolate the term M(q), which characterizes the device, and write it as

a differential equation:

where Q is defined by

and ϕ is defined by

The variable Φ ("magnetic flux linkage") is generalized from the circuit

characteristic of an inductor. The symbol Φ may simply be regarded as the

integral of voltage over time.

Thus, the memristor is formally defined as a two-terminal element in

which the flux linkage (or integral of voltage) Φ between the terminals is a

function of the amount of electric charge Q that has passed through the device.

Each memristor is characterized by its memristance function describing the

charge-dependent rate of change of flux with charge.

Substituting that the flux is simply the time integral of the voltage, and

charge is the time integral of current, we may write the more convenient form

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7

It can be inferred from this that memristance is simply charge-

dependent resistance. If M(q(t)) is a constant, then we obtain Ohm's

law R(t) = V(t)/ I(t).However, the equation is not equivalent

because q(t) and M(q(t)) will vary with time.

Solving for voltage as a function of time we obtain

This equation reveals that memristance defines a linear relationship

between current and voltage, as long as M does not vary with charge.

Furthermore, the memristor is static if no current is applied. If I(t) = 0, we

find V(t) = 0 and M(t) is constant. This is the essence of the memory effect.

The power consumption characteristic recalls that of a resistor, I2R

As long as M(q(t)) varies little, such as under alternating current, the

memristor will appear as a constant resistor.

2.4 Properties of a Memristor

2.4.1 Φ-q Curve of a Memristor

The Φ-q curve of a memristor is a monotonically increasing. The memristance

M(q) is the slope of the Φ-q curve. According to the memristor passivity

condition, a memristor is passive if and only if memristance M(q) is non-

negative. If M(q) ≥ 0, then the instantaneous power dissipated by the

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8

memristor, , is always positive and so the memristor is a

passive device. The memristor is purely dissipative, like a resistor.

2.4.2 Current–Voltage Curve of a Memristor

An important fingerprint of a memristor is the pinched hysteresis loop current

voltage characteristic. For a memristor excited by a periodic signal, when the

voltage v(t) is zero, the current i(t) is also zero and vice versa. Thus, both

voltage v(t) and current i(t) have identical zero-crossing. Another signature of

the memristor is that the ―pinched hysteresis loop‖ shrinks with the increase in

the excitation frequency. Figure 4 shows the ―pinched hysteresis loop‖ and an

example of the loop shrinking with the increase in frequency. In fact, when the

excitation frequency increases towards infinity, the memristor behaves as a

normal resistor.

Fig. 4: The pinched hysteresis loop and the loop shrinking with the increase in

frequency

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3. MODEL OF THE MEMRISTOR FROM HP LABS

In 2008, thirty-seven years after Chua proposed the memristor, Stanley

Williams and his group at HP Labs realized the memristor in device form. To

realize a memristor, they used a very thin film of titanium dioxide (TiO2). The

thin film is sandwiched between two platinum (Pt) contacts and one side of

TiO2 is doped with oxygen vacancies. The oxygen vacancies are positively

charged ions. Thus, there is a TiO2 junction where one side is doped and the

other side is undoped. The device established by HP is shown in Fig. 5.

Fig. 5: Schematic of HP memristor

In Fig.5, D is the device length and w is the length of the doped region.

Pure TiO2 is a semiconductor and has high resistivity. The doped oxygen

vacancies make the TiO2-x material conductive. The working of the memristor

Memristors

10

established by HP is as follows. When a positive voltage is applied, the

positively charged oxygen vacancies in the TiO2-x layer are repelled, moving

them towards the undoped TiO2 layer. As a result, the boundary between the

two materials moves, causing an increase in the percentage of the conducting

TiO2-x layer. This increases the conductivity of the whole device. When a

negative voltage is applied, the positively charged oxygen vacancies are

attracted, pulling them out of TiO2 layer. This increases the amount of

insulating TiO2, thus increasing the resistivity of the whole device. When the

voltage is turned off, the oxygen vacancies do not move. The boundary

between the two titanium dioxide layers is frozen. This is how the memristor

remembers the voltage last applied.

The simple mathematical model of the HP memristor is given by

where has the dimensions of magnetic flux. is the average drift

velocity and has the units cm2/sV; D is the thickness of titanium-dioxide film;

and are on-state and off- state resistances; and q(t) is the total

charge passing through the memristor device.

3.1 Linear Drift Model

Let us assume a uniform electric field across the device. Therefore, there is a

linear relationship between drift-diffusion velocity and the net electric field.

The state equation can be written as

Integrating this gives,

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where is the initial length of w . The speed of drift under a uniform

electric field across the device is then given by

In a uniform field D= . In this case, defines the amount of

charge required to move the boundary from , where w 0, to distance

, where w D. Therefore, . Thus,

If

then,

The amount of charge that is passed through the channel over the

required charge for a conductive channel is given as

, then

Substituting

, we get

If we assume that the initial charge , then

and

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Where

and is the memristive value at . Thus the

Memristance at a time t is given by

,

Where . When >> , .

Substituting this in , when

we get,

)

Since

, the solution is

For

If

, then the internal state of the memristor is

The current-voltage relationship in this case is

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13

This shows the inverse-square relation between memristance and TiO2

thickness, D. Thus, for smaller values of D, the memristance shows improved

characteristics. Nowadays, memristance becomes more important for

understanding as the dimensions of electronic devices are shrinking to

nanometre scale.

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4. BENEFITS OF USING MEMRISTORS

The advantages of using memristors are as given below:

It provides greater resiliency and reliability when power is interrupted

in data centers.

Memory devices built using memristors have greater data density

Combines the jobs of working memory and hard drives into one tiny

device.

Faster and less expensive than present day devices

Uses less energy and produces less heat.

Would allow for a quicker boot up since information is not lost when the

device is turned off.

Operating outside of 0s and 1s allows it to imitate brain functions.

Eliminates the need to write computer programs that replicate small

parts of the brain.

The information is not lost when the device is turned off.

Has the capacity to remember the charge that flows through it at a given

point in time.

A very important advantage of memristors is that when used in a device, it can

hold any value between 0 and 1. However present day digital devices can hold

only 1 or 0. This makes devices implemented using memristors capable of

handling more data.

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5. RESULTS AND SIMULATIONS

5.1 Simulation Results— Using SPICE model

For this simulation, the width D of the TiO2 film is considered to be 10 nm

and the dopant mobility = . The values assumed are

=1KΩ, =100KΩ and the initial resistance required to model the

initial conditions of the capacitor is assumed to be 80KΩ. The simulation

results are shown below in Figs. 5,6,7,8,9 and 10 .

Fig. 6 An input voltage applied to the memristor.

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Fig. 7: Waveform of the current through the memristor.

Fig. 8: Charge-versus-flux curve for memristor.

Fig. 9: Current-versus-voltage curve for input frequency of 1 Hz.

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Fig. 10: Current-versus-voltage curve for input frequency of 1.5 Hz.

Fig. 11: Current-versus-voltage curve for input frequency of 2 Hz.

These results are very much consistent with the theoretical graphs which we

expect and this shows that the memristor which we have developed till now is

accurate.

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6. POTENTIAL APPLICATIONS OF MEMERISTOR

6.1 Two-state Charge-controlled Memristor

The slope of the Φ–q curve gives the memristance. The two values of the

memristance can be considered as two different states which can be used as

binary states. The memristor holds logical values as impedance state and not as

voltages. The resistance can be changed from one state to another by applying

appropriate voltage.

6.2 Memristor Memory

Memristors can be used as non-volatile memory, allowing greater data density

than hard drives. The memristor based crossbar latch memory prototyped by

HP can fit 100 gigabits within a square centimetre. HP also claims that

memristor memory can handle up to 1,000,000 read/write cycles before

degradation, compared to flash at 100,000 cycles. In addition, memristors also

consume less power.

In memristor memories, the reading operation is performed by applying a

voltage lesser than the threshold value. The memristor will conduct even at

this voltage if it is ―on‖. If it is ―off‖ then it will not conduct. To write one of

the logic levels (0 or 1) a voltage greater than the threshold value is applied.

To write the other logic level, a voltage of opposite polarity whose magnitude

is greater than the threshold voltage is applied. This turns the memristor ―off‖.

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Memristors can ―remember‖ even when the power is turned off. Thus, the

computers developed using memristors will have no boot up time. The

computer can be turned on, like turning on a light switch and it will instantly

display all information that was there on it when it was turned off.

6.3 Basic arithmetic operations

For performing any arithmetic operation such as addition, subtraction,

multiplication or division, at first, two operands should be represented by some

ways. In almost all of currently working circuits, signal values are represented

by voltage or current. However, as explained in previous section, analog

values can be represented by the memristance of the memristor as well. Figure

11 shows the typical circuit that can be used for adjusting the memristance of

one memristor to the predetermined input value, i.e Vin.

Fig. 12: Typical circuit for adjusting the memristance of the memristor with the

predetermined value.

In this figure, the coefficient is considered to make the dropping voltage across

the memristor to be meaningful and reasonable. The absolute value of the

voltage dropped across the memristor at any time will be aM. If aM be lower

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20

than aVin, the output of the opamp will be at its lowest value, i.e. 0 volt, which

will cause the left current

source to derive the memristor. Passing current from the memristor in this

direction will increase its memristance. On the other hand, if aM be higher

than aVin, the output of the opamp will be at its highest value, i.e. 5 volt,

which will cause the left current source to derive the memristor. Passing

current from the memristor in this direction will

decrease its memristance. As a result, final value of the voltage which drops

across the memristor, i.e aM, will be equal to aVin and therefore by this way,

the memristance of the memristor will be set to Vin . Now, this adjusted

memristor can be used as an operand for performing arithmetic operations.

Addition Subtraction

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Multiplication

Division

Fig. 13 shows how the various arithmetic operations can be achieved using memristors.

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7. CONCLUSION AND FUTURE RESEARCH

7.1 Conclusion

This report presents a detailed study of the memristor. The properties of the

memristor and the model proposed by HP are discussed. This model is

simulated by subjecting it to various input voltages and noting the results

obtained. This report also presents a brief insight into the potential applications

of the memristor.

Nanotechnology is fast emerging, and nanoscale devices automatically bring in

memristive functions. Thus, memristors might revolutionize the 21st century

as radically as the transistor in the 20th century. Memristor memories have

already been developed and the researchers at HP believe that they can offer a

product with a storage density of about 20 gigabytes per square centimetre by

2013.

Leon Chua rightly said ―It‗s time to rewrite all the Electronics Engineering

books‖.

7.2 Future Research

Recently, researchers have defined two new memdevices- memcapacitor and

meminductor, thus generalizing the concept of memory devices to capacitors

and inductors. These devices also show ―pinched‖ hysteresis loops in two

constitutive variables— charge—voltage for the memcapacitor and current—

flux for meminductor. Figure 13 shows the symbols for the memcapacitor and

the meminductor.

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Fig. 14: Circuit symbols for memcapacitor and meminductor

Memristors are not lossless devices. As non-volatile memories, memristors do

not consume power when idle but they do dissipate energy when they are

being read or written. Hence, there is a need to invent lossless non-volatile

device. Memcapacitors and meminductors are good contenders as they are

lossless devices.

24

BIBLIOGRAPHY

[1] http://www.memristor.org/

[2] Dmitri B. Strukov, Gregory S. Snider, Duncan R. Stewart & R. Stanley

Williams, ―The missing memristor found‖, Vol 453| 1 May 2008|

doi:10.1038/nature06932

[3] http://en.wikipedia.org/wiki/Memristor

[4] O. Kavehei, A. Iqbal, Y. S. Kim, K. Eshraghian, S. F. Al-Sarawi, D.

Abbott, ―The Fourth Element: Characteristics, Modelling, and Electromagnetic

Theory of the Memristor‖

[5] http://www.hpl.hp.com/news/2011/apr-jun/memristors.html

[6] http://spectrum.ieee.org/semiconductors/design/the-mysterious-memristor

[7] http://highscalability.com/blog/2010/5/5/how-will-memristors-change-

everything.html

[8] http://www.wired.com/gadgetlab/2008/04/scientists-prov/