1. 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
I

2. 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)
II

3. 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.
III

4. 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
IV

5. 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
V

6. 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.
1

7. Memristors
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.
2

8. Memristors
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.
3

9. Memristors
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.
4

10. Memristors
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
5

11. Memristors
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
6

12. Memristors
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
7

13. Memristors
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
8

14. Memristors
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
9

15. Memristors
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,
10

16. Memristors
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
11

17. Memristors
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
12

18. Memristors
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.
13

19. Memristors
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.
14

20. Memristors
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.
15

21. Memristors
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.
16

22. Memristors
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.
17

23. Memristors
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‖.
18

24. Memristors
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
19

25. Memristors
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
20

26. Memristors
Multiplication
Division
Fig. 13 shows how the various arithmetic operations can be achieved using memristors.
21

27. Memristors
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.
22

28. Memristors
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.
23

29. 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/
24