CHAPTER 1

INTRODUCTION

1.1 History

For nearly 150 years, the known fundamental passive circuit elements were limited to the

capacitor (discovered in 1745), the resistor (1827), and the inductor (1831). Then, in a

brilliant but underappreciated 1971 paper, Leon Chua, a professor of electrical engineering at the

University of California, Berkeley, predicted the existence of a fourth fundamental device, which

he called a memristor. He proved that memristor behavior could not be duplicated by any

circuit  bu i l t u s i ng on ly t he o the r   t h r ee e l emen t s , wh i ch   i s  why t he

memr i s t o r i s   t r u ly fundamental.

Memristor is a contraction of “memory resistor,” because that is exactly its function to

remember its history. A memristor is a two-terminal device whose resistance depends on the

magnitude and polarity of the voltage applied to it and the length of time that voltage has been

applied. When you turn off the voltage, the memristor remembers it’s most recent resistance until

the next time you turn it on, whether that happens a day later or a year later.

 Chua discovered a missing link in the pair wise mathematical equations that relate the

four circuit quantities-charge, current, voltage, and magnetic flux-to one another. These can be

related in six ways. Two are connected through the basic physical laws of electricity and

magnetism, and three are related by the known circuit elements: resistors connect voltage and

current, inductors connect flux and current, and capacitors connect voltage and charge. But one

equation is missing from this group: the relationship between charge moving through a circuit

and the magnetic flux surrounded by that circuit.

Chua demonstrated mathematically that his hypothetical device would provide a

relationship between flux and charge similar to what a nonlinear resistor provides between

voltage and cur- rent. In practice, that would mean the device’s resistance would vary according

to the amount of charge that passed through it. And it would remember that resistance value

even after the current was turned off.

1

1.2 Theory

 

Fig 1.2.1

The memristor is formally defined as a two-terminal element in which the

magnetic flux Φm 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. 

Noting from Faraday's law of induction that magnetic flux is simply the time integral of

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

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). If M (q (t)) is

nontrivial, 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

V (t) = M (q (t)) I (t)

This equation reveals memristance defines a linear relationship between current and

voltage, as long as M does not vary with charge. Of course, nonzero current implies time varying

charge.

2

Alternating current, however, may reveal the linear dependence in circuit operation by

inducing a measurable voltage without net charge movement as long as the maximum change in

q does not cause much change in M. 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, I 2R.

As long asM (q(t )) varies little, such as under alternating current, the memristor will appear as a

resistor. If M (q(t )) increases rapidly, however, current and power consumption will quickly

stop.

3

CHAPTER2

DETAILED DISCRIPTION

2.1 Structure of Titanium Diode Memristor

Fig 2.1.1

4

The HP device is composed of a thin (50 nm) titanium dioxide film between two 5nm

thick electrodes, two platinum wires. Initially, there are two layers to the titanium dioxide film,

TiO2 and TiO2- x. The upper layer has a slight depletion of oxygen atoms.

The oxygen vacancies are donors of electrons which makes

the vacancies themselves positively charged. Stoichiometric TiO2 act as a

insulator. (It is a semiconductor) but oxygen deficient TiO2-x is a conductor and have lower

resistance than the stoichiometric compound.

2.2 Working

Fig 2.2.1

If a positive voltage is applied to the top electrode of the device, it will repel the (also

positive)oxygen vacancies in the TiO2-xlayer down into the pure TiO2 layer. That turns the

TiO2 layer into TiO2- x and makes it conductive, thus turning the device on. A negative voltage

has the opposite effect: the vacancies are attracted upward and back out of the TiO2, and thus the

thick- ness of the TiO2 layer increases and the device turns off.

The oxygen deficiencies in the TiO2-x manifest as “bubbles” of oxygen vacancies

scattered throughout the upper layer. A positive voltage on the switch repels

the (positive)oxygen deficiencies in the metallic upper TiO2- x layer, sending them into the

insulating TiO2layer below. That causes the boundary between the two materials to move down,

increasing the percentage of conducting TiO2- x and thus the conductivity of the entire

switch. The more positive voltage is applied, the more conductive the cube becomes.

5

A negative voltage on the switch attracts the positively charged oxygen bubbles, pulling

them out of the TiO2. The amount of insulating, resistive TiO2 increases, thereby making the

switch as a whole resistive. The more negative voltage is applied, the less conductive the

cube becomes. What makes this switch special 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.

Resistance also depends on the length of time that voltage has been applied.

Fig 2.2.2

A memristor’s structure, shown here in a scanning tunneling microscope

image, will enable dense, stable computer memories.

Bow Ties:

Leon Chua’s original graph of the hypothetical memristor’s behavior is shown at top right;

the graph of R. Stanley William’s experimental results  are shown below. The loops

map the switching behavior of the device: it begins with a high resistance, and as the voltage

increases, the current slowly increases. As charge flows through the device, the resistance drops,

and the current increases more rapidly with increasing voltage until the maximum is reached.

Then, as the voltage decreases, the current decreases but more slowly, because charge is flowing

through the device and the resistance is still dropping. The result is an on-switching loop. When

6

the voltage turns negative, the resistance of the device increases, resulting in an off-switching

loop.

 

Fig 2.2.3

2.3 Other Types of Memristors

 

Spintronic Memristor

Concept of Spintronic memristor is given as, resistance is caused by the spin of electrons

in one section of the device pointing in a different direction than those in another section,

creating a "domain wall," a boundary between the two states. Electrons flowing into

the device have a certain spin, which alters the magnetization state of the device.

Changing the magnetization, in turn, moves the domain wall and changes the device's resistance.

7

Spin Torque Magnetoresistance

Spin Torque Transfer MRAM is a well-known device that exhibits memristive behavior.

The resistance is dependent on the relative spin orientation between two sides of

a magnetic tunnel junction. This in turn can be controlled by the spin torque induced by the

current flowing through the junction.

However, the length of time the current flows through the junction determines

the amount of current needed, i.e., the charge flowing through is the key variable. Additionally,

MgO based magnetic tunnel junctions show memristive behavior based on the drift of

oxygen vacancies within the insulating MgO layer (resistive switching).  Therefore,

the combination of spin transfer torque and resistive switching leads naturally

to a second-order memristive system with w=(w1,w2) where w1 describes the magnetic state

of the magnetic tunnel junction and w2 denotes the resistive state of the MgO barrier.

Note that in this case the change of w1 is current-controlled (spin torque is due to a high

current density) whereas the change of w2 is voltage-controlled (the drift of oxygen vacancies is

due to high electric fields).

Polymeric Memristor

Juri H. Krieger and Stuart M. Spitzer claim to have developed a polymeric memristor   

before the titanium dioxide memristor more recently announced.

The i r  work  de sc r i be s   t he  p roce s so f   dynamic  dop ing  o f   po lymer   and  

i no rgan i c dielectric-like materials in order to improve the switching characteristics and

retention required to create functioning nonvolatile memory cells. Described is the use of a

special passive layer  between electrode and active thin films, which enhances the

extraction of ions from the electrode. It  is possible to use fast ion conductor as this

passive layer, which allows to significantly decreasing the ionic extraction field.

Resonant Tunnelling Diode Memristor

1994, F. A. Buot and A. K. Rajagopal demonstrated that a 'bow-tie' current-voltage (I-V)

characteristics occurs in AlAs/GaAs/AlAs quantum-well diodes containing special doping design

of the spacer layers in the source and drain regions, in agreement with the published

experimental results This 'bow-tie' current-voltage (I-V) characteristic is sine qua non of a

8

memristor although the term memristor is not explicitly mentioned in their papers. No magnetic

interaction is involved in the analysis of the 'bow-tie' I-V characteristics.

3-Terminal Memristor

Although the memristor is defined in terms of a 2-terminal circuit element, there was an

implementation of a 3-terminal device called a memistor developed by Bernard Widrow in

1960.Memistors formed basic components of a neural network architecture

called ADALINE developed by Widrow and Ted Hoff the memistor was described as

follows:

Like the transistor, the memistor is a 3-terminal element. The conductance between two

of t he t e rmina l s i s   con t ro l l ed by   t he t ime i n t eg ra l   o f t he   cu r r en t   i n

t he   t h i rd , r a t he r   t han   i t s instantaneous value as in the transistor.

Reproducible elements have been made which are continuously variable

(thousands of possible analog storage levels), and which typically vary in resistance from 100

ohms to 1 ohm, and cover this range in about 10 seconds with several mille amperes of plating

current.

Adaptation is accomplished by direct current while sensing the neuron logical structure is

accomplished nondestructively by passing alternating currents through the arrays of memistor

cells.

Since the conductance was described as being controlled by the time integral of current as

in Chua's theory of the memristor, the memistor of Widrow may be considered as a form

of memristor having three instead of two terminals.

However, one of the main limitations of Widrow's memistor was that they were

made from an electroplating cell rather than as a solid-state circuit element. Solid-state circuit

elements were required to achieve the scalability of the integrated circuit which was gaining

popularity around the same time as the invention of memristor Windrow’s memristor.

9

2.4Array Based Multilevel Memory of Memristor

The proposed method has the operating point of the memristor be maintained its desired

location (or resistance value) utilizing a set of pre-determined multiple resistance levels.

Fig.2.4.1 shows the basic idea of the proposed method, where the resistance array to be

referenced and the memristor to be programmed (tuned) are shown.

The goal is to have the memristor keep any of the resistance level selected from the

resistance array. If a predetermined magnitude of the current pulse Is (t) is applied to the

resistance array, different levels of voltages V k will appear at each node of the resistance array.

The same current pulse is (t) is also applied to the memristor.

Fig 2.4.1

The programming (tuning) of the memristor is performed by applying additional

current pulses to the memristor with the appropriate directions until the voltage of the memristor

equals to that of the selected node voltage in the resistance array. If the voltage of the memristor

reaches that of the selected node, the resistance value of the memristor becomes the same as the

partial sum of the resistance from the ground to the selected node of the resistance array.

10

This idea is employed in both the “write-in” and the “read-out/restoration” circuits.

Detailed description of these circuits will be presented in the following sections.

2.5 Memristor Write-In Circuit

Fig 2.5.1

The memristor write-in circuit is used to bias the memristor at a desired resistance level.

The critical write-in circuit is shown in Fig 2.5.1.

The first step is to choose the write-in memristor and the resistance value to be

memorized by turning on one of the switches in switch array S1 of Fig2.5.1. and the

corresponding switch pair in switch array S4 respectively.

Then, an initial current pulse I s(t) is applied at the drain of the transistor Q1

so that its mirrored current pulses appear at transistors Q2 and Q3. With this current pulse,

11

negative voltages appear at both the selected reference nodes and at the output terminal V out of

the memristors.

Suppose the selected memristance M j is less than the referenced sum of the resistances

Rk  sum , in this particular case, Diff k+ is smaller than Diff k- since Vout (Tp) is less negative

than that of Vk (Tp). These Diff  k outputs caused the comparator C1 to generate

a positive pulse. Note that the negative and the positive output terminals of Diff  k are linked to

the positive and the negative input terminals of C1 respectively.

As a consequence, switch S3 is turned on. Ø such increased flux Ø, the increment of the

memristance can be obtained with a monotonically increasing function via the R vs. Ø

graph in Fig 3.2.3 As a consequence, the memristor voltage decreases toward the selected

reference level.

The processing from the above voltage difference computation repeats until

the difference between Vk (Tp) and Vout (Tp) becomes zero, thereby completing the

“write-in” processing of the reference resistance Rk sum.

On the other hand, when the selected memristance M j is larger than the referenced sum

 Rk sum of the resistances, the memristance of the selected memristor is decreased and the

memristor voltage is increased toward the selected reference level through the opposite

procedure mentioned above.

The above comparison between the voltages and the adjustment of the memristance are

repeated until the memristor voltage is equal to its selected reference voltage level.

12

2.6 Memristor Read-Out/Restoration Circuit

Fig 2.6.1

The memristor read-out/ restoration circuit is used to read the content of the memristor by

applying an appropriate integrating current or voltage. The critical function of this

circuit is to guarantee the memristor will stay at a set of fixed values without being perturbed

when a read- out voltage or a noise voltage is applied across the memristor. To achieve this goal,

a single compensating pulse is applied to have the memristance changed toward the closest

reference resistance after the initial read-out pulse is applied.

The read-out circuit is the same as the write- in circuit except the negative signal

excluding circuit (N_Excld), MIN A and MIN B circuits as shown in Fig.2.6.1. The N_Excld is

the circuit to choose only the positive signals from Diffk+ or Diff k- using the negative

signal excluding circuit N_Excld by comparing between the DC voltage and the

13

output of the Diff circuit. The circuits MIN A and MIN B together with the comparator

C1 are used to choose the smallest absolute value among all Diff k+ and Diff k- signals.

If the output of MIN A is smaller than that of MIN B, the memristor voltage is

higher than that of its closest reference voltage (with M j< Rk sum).

In this case, the memristance M j should be increased. On the other hand, if the

output of the MIN A is larger than that of MIN B, then the memristor voltage is smaller than

that of its closest reference voltage(with M j>Rk sum). In this case, M j should be decreased.

The above adjustment of the memristor is executed only once during each read-out

processing.

14

CHAPTER 3

ADVANTAGES AND DISADVANTAGES

3.1Advantages

When you turn off the voltage, the memristor remembers its most recent resistance until

the next time you turn it on, whether that happens a day later or a year later. This freezing

property suits memristors brilliantly for computer memory. The ability to indefinitely store

resistance values means that a memristor can be used as a nonvolatile memory. That might not

sound like very much, but go ahead and pop the battery out of your laptop, right now—no

saving, no quitting, nothing. You’d lose your work, of course. But if your laptop were built

using a memory based on memristors, when you popped the battery back in, your

screen would return to life with everything exactly as you left it: no lengthy reboot, no half-

dozen auto-recovered files.

There are several advantages of the memristor memory over conventional

transistor- based memories. One is its strikingly small size. Though memristor is  still

at its early development stage, its size is  at most one tenths of its RAM

15

counterparts. If the fabrication technology for memristor is improved, the size and

advantage could be even more significant .

Another feature of the memristor is its incomparable potential to store analog information

which enables the memristor to keep multiple bits of information in a memory cell.

Besides these features, the memristor is also an ideal device

for implementing synaptic weights in artificial neural networks. Williams' solid-state

memristors can be combined into devices called crossbar latches, which could replace transistors

in future computers, taking up a much smaller area.

They can also be fashioned into non-volatile solid-state memory,

which would allow greater data density than hard drives with access times potentially similar

to DRAM, replacing both components HP prototyped a crossbar latch memory using

the devices that can fit 100gigabits in a square centimeter, and has designed a highly

scalable 3D design (consisting of up to1000 layers or 1 petabit in a cubic CM) has

reported that its version of the memristor is currently about one-tenth the speed

of DRAM . The devices' resistance would be read with alternating current so that they do

not affect the stored value.

3.2disadvantages

Despite many favorable features, memristors have several weaknesses in practice. One

weakness comes from the nonlinearity in the Ø vs. q curve which makes it difficult to determine

the proper pulse width for achieving a desired resistance value.

If the nonlinearity is spatially variant in the die of a chip which is common in the

fabrication process, the difficulty could be very serious. Another difficulty comes from the

property of the memristor which integrates any kind of signals including noise that appeared at

the memristor and results in the memristors being perturbed from its original pre-set values.

  Thus, the resistance can be interpreted as the slope at an operating point on the Ø-q curve.

If the Ø-q curve is nonlinear, the resistance will vary with the operating point. For instance, if the

Ø-q curve is the nonlinear function Shown in Fig. 3.2.1, its small-signal resistance can be

obtained by re-plotting it as a function of Øq in the R vs .Ø plane as in Fig. 3.2.2.Since the flux

16

Ø is obtained by integrating the voltage, the resistance of the memristor can be controlled by

applying a voltage signal across the memristor.

The principle of the memristor is based on the nonlinear property of basic circuit

elements. In the relationships defining basic circuit elements, charge is defined as the time

integral of current, namely,

Fig 3.2.1

17

Fig 3.2.2

Fig 3.2.3

The above memristance tuning method assumes an ideal operating condition.

In practice, there are some problems that must be overcome. The first problem is caused by then

nonlinearitybetween the applied voltage andthe corresponding resistance. Suppose the

resistance characteristics of the memristors is different from each other as shown in Fig. 3.2.3,where

the resistance between the applied voltage and the corresponding resistance. Suppose the

resistance characteristics of the memristors is different from each other as shown in Fig. 3.2.3,where

the resistance R d is obtained at different values of Ø such as Ø1Ø2 and Ø3. If the same

18

magnitude of voltage pulses is chosen, then the durations of the pulse widths for obtaining

the same resistance will be different depending on the characteristics of the memristors.

Another problem comes from the fact that the operating point and its

associated memristance would be changed whenever some voltage is applied across the

memristor. The voltage applied for read-out or even noise voltages would be integrated which

causes the flux Ø to be altered. Again, this causes the programmed resistance to be varied. Chua

had suggestedapplying a voltage doublet with equal positive and negative read-out pulses to

resolve such problem. However, the problem remains if the positive and the negative

pulses are not perfectly identical due to the non-ideal pulse-generation circuits.

19

CHAPTER 4

FUTURE SCOPE

Combined with transistors in a hybrid chip, memristors could radically improve

the performance of digital circuits without shrinking transistors. Using transistors more

efficiently could in turn give us another decade, at least, of Moore’s Law performance

improvement, without requiring the costly and increasingly difficult doublings of transistor

density on chips. In the end, memristors might even become the cornerstone of new analog

circuits that compute using architecture much like that of the brain. Memristors potential goes far

beyond instant-on computers to embrace one of the grandest technology challenges: mimicking

the functions of a brain. Within a decade, memristors could let us emulate, instead of merely

simulate, networks of neurons and synapses. Many research groups have been working

toward a brain in silico: IBM’s Blue Brain project, Howard Hughes Medical Institute’s

Janelia Farm, and Harvard’s Center for Brain Science are just three. However, even a mouse

brain simulation in real time involves solving an astronomical number of coupled partial

differential equations. A digital computer capable of coping with this staggering workload would

need to be the size of a small city, and powering it would require several dedicated nuclear

power plants.

Memristors can be made extremely small, and they function like synapses. Using them,

we will be able to build analog electronic circuits that could fit in a shoebox and

function according to the same physical principles as a brain. Memristors can potentially learn

like synapses and be used to build human brain-like computers.

Two CMOS circuits connected by a memristor is analogous to two neurons in the brain

connected by a synapse. It is thought that synaptic connections strengthen as the neurons

either side fire and so brain 'circuits' are established which constitutes the basis of human

learning.

20

Wei Lu, a University of Michigan scientist connected two CMOS circuits by a silver and

s i l i con Memris tor and powered the two CMOS c i rcu i t s on and of f wi th

vary ing t ime gaps  between them.

 

The memristor alters its state differently depending on the timing of the powering of the

CMOS circuits.

This is said to be the same behavior as that shown by synapses, called "spike

timing plastic dependency", which is thought to be the possible basis for memory and

learning inhuman and other mammalian brains.

The synaptic connection between neurons becomes stronger or weaker, as the time

gap between when they are stimulated becomes shorter or longer. In the same way,

the shorter the time interval the lower the resistance of the memristor to electricity flowing

across it between the two CMOS circuits.

A 20 millisecond time interval between the two CMOS circuits caused a resistance level

roughly half that of a 40 millisecond gap. Lu said: "Cells that fire together wire together... The

memristor mimics synaptic action.

"We show that we can use voltage timing to gradually increase or decrease the electrical

conductance in this memristor-based system. In our brains, similar changes in synapse

conductance essentially give rise to long term memory.

A hybrid circuit—containing many connected memristors and transistors—could help us

research actual brain function and disorders. Such a circuit might even lead to machines that can

recognize patterns the way humans can, in those critical ways computers can’t—for

example, picking a particular face out of a crowd even if it has changed significantly since

our last memory of it.

There are several advantages of the memristor memory over conventional

transistor- based memories. One is its strikingly small size. Though memristor is still

at its early development stage, its size is at most one tenths of its RAM counterparts. If the

fabrication technology for memristor is improved, the size and advantage could be even more

significant. Another feature of the memristor is its incomparable potential to store analog

information which enables the memristor to keep multiple bits of information in a memory cell.

Besides these features, the memristor is also an ideal device for implementing synaptic

21

weights in artificial neural networks.

HP already has plans to implement memristor in a new type of non-volatile memory

which could eventually replace flash and other memory systems.

Recently, a simple electronic circuit consisting of an LC network and a memristor was

used to model experiments on adaptive behavior of unicellular organisms. It was shown that the

electronic circuit subjected to a train of periodic pulses learns and anticipates the next pulse to

come, similarly to the behavior of slime moulds Physarumpolycephalum subjected to periodic

changes of environment. Such a learning circuit may find applications, e.g., in

pattern recognition.

 

22

CHAPTER 5

CONCLUSION

 

The reference resistance array-based multilevel memristor memory is

proposed in this paper. The idea has been implemented with two circuits namely the

write-in and the read-out circuits. Simulation of the write-in circuit shows that the

memristors memorize the desired discrete resistance levels regardless of their characteristic

differences. In read-out simulation, contents of the memristors move toward their original values

from the deviated ones whenever the read-out processing is performed.

The proposed multilevel idea of the memristor together with its intrinsic feature of small

size should make the memristor to be a powerful memory device. Also, if the number

of multilevel of memory is increased, the memristor could be an ideal element for synaptic

weight implementation since the synaptic multiplication can be performed simply by Ohm’s law

V=IR in the memristor.

Memristor is the fourth fundamental component. Thus the discovery of a brand new fundamental

circuit element is something not to be taken lightly and has the potential to open the

door to a brand new type of electronics. HP already has plans to implement memristors in a

new type of non-volatile memory which could eventually replace flash and other memory

systems.

23

CHAPTER 6

REFERENCES

•Hyongsuk Kim Sah , M.P .Changju Yang Chua ,L .O.”Memris tor based

mul t i l eve l memory” Cellular Nanoscale Networks and Their Applications (CNNA), 2010

12thInternational Workshop, 3-5 Feb. 2010, pp1-6.

•Memristor from Wikipedia, en.wikipedia.org/wiki/Memristor

•http://www.memristor.org/reference/research/13/what-are-memristors

•www.hpl.hp.com/news/2010/apr-jun/

memristor.html

24