Memristors
Table of Contents
Introduction ..................................................................................................................... 3
Background ..................................................................................................................... 3
The HP Memristor ........................................................................................................... 8
Electrical Conductivity of Titanium Dioxide .................................................................... 11
Anion Migration in Titanium Dioxide .............................................................................. 14
Applications of the Memristor ........................................................................................ 18
Report Summary ........................................................................................................... 21
References .................................................................................................................... 22
2
Introduction
Every undergraduate
electrical
engineering
student
extensively studies
three
fundamental, two terminal circuit elements; the resistor, the capacitor and the inductor.
Students are taught that every passive electrical circuit can be defined as a combination
of these three fundamental elements. However, in 1971, Leon Chua, a young UC
Berkeley Professor, hypothesized that mathematically, a fourth fundamental passive
circuit element could exist. This report will examine Chua’s fourth circuit element, which
he dubbed the Memristor and how it was finally constructed in 2008.
Background
Passive electrical circuits can be thought of as a set of relationships between four
electromagnetic quantities, summarized in Table 1 [1]. Voltage can be defined as the
change magnetic flux with respect to time. Current can be defined as the change in
electric charge with respect to time. Resistance can be defined as a linear relationship
between voltage and current. Capacitance can be defined as a linear relationship
between voltage and electric charge. Inductance can be defined as a linear relationship
between magnetic flux and current.
Symbol
Name
v
Voltage
i
Current
q
Electric Charge
Φ
Magnetic Flux
Table 1: Basic Electromagnetic Quantities
3
These relationships can best be described mathematically in equations (1)-(5) below [2].
Thus, the three fundamental circuit elements and the definitions of voltage and current
are defined using almost all possible pairings between two items from Table 1. The only
two electromagnetic quantities for which there are no pairings are magnetic flux and
electric charge. Chua and his contemporaries knew of no existing physical process or
electronic device that created this remaining relationship [1].
(1)
Definition of Current :
(2)
Definition of Voltage :
(3)
Definition of Resistance:
(4)
Definition of Capacitance:
(5)
Definition of Inductance:
Identifying this omission, Chau mathematically defined a circuit element that would
provide a relationship between magnetic flux and electric charge. Chua developed
equation (6) in order to characterize this hypothetical electronic device [1].
(6)
Definition of New Circuit Element:
As (6) is a little vague when attempting to ascertain qualitative features of this new
device, some mathematical analysis is required. By integrating (6) and representing the
right hand side as a function of electric charge (q), equation (7) is realized. Equation (7)
expresses the fact that the magnetic flux of the device (φ) is a function of the cumulative
electric charge that has passed through the device.
(7)
4
If equation (7) is differentiated with respect to time, equation (8) is realized.
(8)
Recognising the definitions of voltage and current from equations (1) and (2) above,
equation (9) can be developed.
(9)
Furthermore, equation (10) is a simple way of expressing equation (9), identifying that
the voltage and current of the device are related in a manner similar to that of
resistance. However, this new device exhibits a resistance that is a function of the
electric charge that has passed through the device [1].
(10)
,
Considering that this device must have a memory of the charge that has passed
through it, (10) can be expressed as (11), where, instead of charge, a state variable w
represents the current memory state of the device.
(11)
Thus, qualitatively, Chua’s new circuit element would have an electrical resistance that
is a function of the amount of electric charge that has passed through its terminals.
Furthermore, the device would exhibit memory; its resistance value would be retained
even while not in use. Chua dubbed his new device the Memristor, a combination of the
words memory and resistor. Chua then gave Memristor the circuit symbol found in the
top left hand corner of Figure 1 [1].
5
In order to help define this new circuit element, a figure known as the bowtie diagram,
shown in Figure 1, is often employed. Figure 1 shows that the resistance of the new
device changes with applied voltage and frequency, the memory characteristic of the
device featuring prominently as a hysteretic curve. One important feature of the
Memristor is that the hysteretic curve tightens as the applied ac frequency increases [1].
Thus, for high frequencies, the resistance of the Memristor remains unchanged.
Figure 1: Memristor Bowtie Diagram (Source: [3])
Chua went on to hypothesize that these devices could have several applications in
digital memory, analog electronics and neural networks. Furthermore, Chua showed
that several passive and active circuit elements would be required in order to mimic the
device’s behaviour, as shown in Figure 2.
6
Figure 2: A Memristor Built From Other Circuit Elements (Source: [1])
Nevertheless, while mathematically sound and very intriguing, Chua’s work had very
little impact within the electrical engineering community. The three fundamental existing
circuit elements can be easily and reliably constructed. A capacitor can be constructed
from two parallel plate conductors. An inductor can be constructed from a coil of wire. A
resistor can be constructed from a block of material exhibiting low electrical conductivity.
At the time, nobody had observed a material or physical structure that exhibited the
characteristics of a Memristor. Thus, Chau’s work sat in obscurity, a mathematical
novelty with tremendous potential for real world applications. This situation remained
until 2008, when a small research team at HP Labs used TiO2 to build Chua’s Memristor
[4].
7
The HP Memristor
In 2008, a research team led by R.S. Williams working at HP Labs in Palo Alto, revealed
that they had fabricated a device which behaved in a manner satisfying the
requirements of Chua’s Memristor [4].
The HP Memristor, as shown in Figure 3, is a device composed of thin films of TiO2 and
TiO2-x placed between two platinum electrodes. This structure can be thought of as a
continuous medium of TiO2 but with the TiO2-x region doped with oxygen vacancies.
Thus, in the TiO2-x region, the ratio between titanium atoms and oxygen atoms has been
altered such that there is less oxygen than in a regular TiO2 sample [4]. The width of the
doped region is represented by w and the width of the entire structure is represented by
D. An essential characteristic of this arrangement is that resistances of the doped and
undoped regions differ significantly. As will be discussed below, TiO2-x exhibits a much
lower electrical resistance than TiO2 [5]. In the extreme cases, the electrical resistance
of the structure will produce either a high resistance or a low resistance corresponding
to w being D or 0 respectively. The resistance of the device when w = D will be
designated RON and when w = 0 the resistance will be designated as ROFF. This
convention is shown in Figure 4.
Figure 3: Vertical Cross Section of the HP Memristor (Source: [6])
8
Figure 4: Resistance Naming Convention (Source: [6])
Given this naming convention, the entire resistance of the structure can be thought of as
series combinations of two resistances, the values of which are dependant the value of
w. This simplified structure is as shown in Figure 5.
Figure 5: Effective Electrical Structure of the HP Memristor (Source: [6])
(12)
(
(
))
From Figure 5, the effective IV behaviour of the structure can be represented as
equation (12) [6]. At first glance, this arrangement does not appear to be functionally
different than a regular resistor; it appears to simply be a compound device made of two
resistors. However, with an applied electric field, the oxygen vacancies behave rather
strangely. The vacancies will drift from the doped region into the undoped region [6].
This drift will increase the effective width of the doped region (w) at a rate defined by
9
Equation (13) [7]. Equation (13) assumes that the vacancies exhibit an average mobility
of μv and the voltage across RON is designated VON.
(13)
Rearranging equation (13) to be expressed in terms of current, equation (14) can be
obtained [7]. If equation (14) is then integrated with respect to time equation (15) can be
realized [7].
(14)
(15)
Substituting equation (15) into equation (12) and assuming that Ron << Roff, equation
(16) can be realized [7]. Equation (16) shows that the incremental resistance of the
device is a function of the charge that has passed through it [6]. The device should
therefore behave as a Memristor.
(16)
(
)
The memristance effect is a strong function of the q-dependant term on the right-hand
side of Equation (16). This gives us a clue as to why such a simple mechanism has only
recently been discovered. Looking at this term, it can be seen that the magnitude of the
memristance effect is inversely proportional to the square of the manufacturing feature
size (D). Thus, the effect is 1,000,000 times more pronounced at the nanometre scale
than at the micrometer scale [6]. This result makes sense, as the mechanism relies on
an electric field to cause motion in oxygen vacancies; the higher the electric field, the
greater the motion. For a given voltage, the electric field is inversely proportional to the
10
distance over which the voltage is applied. Thus, in older, larger processes, there were
insufficient electric field strengths being generated in order to produce a noticeable
memristance effect. Furthermore, the mechanism by which memristance is achieved
does not involve a magnetic field, which is possibly why researchers specifically hunting
for a memristance process had never stumbled upon this structure [4].
Some have hypothesised that it might be possible to explain some of the spurious
characteristics exhibited by some devices at the nanometre scale using memristance
[4]. The implication of these predictions is that memristance may be an essential
characteristic to take into account when designing new integrated circuit processes at
the nanometre scale.
Electrical Conductivity of Titanium Dioxide
The HP Memristor relies on the relative electrical conductivity of TiO2 and TiO2-x. Thus,
some attention must be given to why there is a difference in conductivity between these
two very similar materials. The equilibrium phase of TiO2 at all temperature is known is
Rutile and exhibits a tetragonal crystal structure as shown in Figure 6 with a = 4.4923 A
and
c = 2.8930 A [5]. Furthermore, the crystal is known to exhibit different electrical
characteristics along these axes. For example, the dielectric constant along the a axis
has been measured to be 89 whereas the along the c axis it has been measured to be
173 [5]. TiO2-x is simply TiO2 that has had oxygen removed from this crystal, such that
the new material has slightly less than two oxygen atoms per titanium atom [5].
11
Figure 6: Crystal Structure of Rutile TiO2 (Source: [5])
By measuring the Hall Effect coefficient, [5] was able to determine the electron carrier
concentration in various samples of TiO2-x. These experiments subsequently showed
that excess electrons are generated as oxygen vacancies are introduced [5].
Figure 7 shows the carrier concentration (
) as an inverse function of temperature (1/T)
for various samples of TiO2-x. Oxygen was removed by reduction; the samples were
heated to very high temperatures (600K to 900K) in an H2 atmosphere and
subsequently left to cool before measuring carrier concentrations. Higher reduction
temperatures create TiO2-x samples with less oxygen and thus larger values of x [5].
Figure 7 shows that as oxygen is removed (and x increased), the electron carrier
concentration increases. This effect is most visible for the ceramic samples.
Furthermore, the experiments indicated that majority carriers are in fact electrons [5].
Additionally, the resistivity was measured by [5] and is shown below in Figure 8. These
results show that as the reduction temperature is increased (and x increased) the
resistivity of ceramic TiO2 becomes smaller. From the [5] results, it is quite clear that as
TiO2 is doped with oxygen vacancies it begins to behave as an n-type semiconductor
[5].
12
Figure 7: Majority Carrier Concentration in Reduced TiO2 samples (Source: [5])
Figure 8: Resistivity in Ceramic ReducedTiO2 Samples (Source: [5])
13
This n-type behaviour is believed to occur because the oxygen vacancies behave as
donor centers. The vacancies thus raise the Fermi Energy and add electrons to the
conduction band [5]. Attempting to quantify their results, [5] found that Equation (17)
could be used to represent all of the carrier concentration curves. This equation uses
four parameters:
.
(17)
When the values for these parameters are selected for a given curve, a pattern
emerges:
and
[5].This result suggests that there are two types of
donor centers present, one with very low activation energy and one with a much larger
activation energy [5]. As oxygen vacancies provide two donor electrons, there must two
distinct energies associated with ionizing an electron into the conduction band [5]. Thus,
this result is viewed as a confirmation that oxygen vacancies are behaving as electron
donors.
Anion Migration in Titanium Dioxide
Another characteristic of TiO2 leveraged by the Memristor is an effect known as resistive
switching. Resistive switching is the tendency for certain materials to exhibit a
resistance that can be changed electrically and thus switched from one value to
another. Generally, resistive switching takes on two possible forms: unipolar switching
and bipolar switching [8]. In both cases, the resistance of the material changes when
certain transition voltages are reached. For unipolar switching, the polarities of the “on”
and “off” transition voltages are identical [8]. If a positive voltage turns on the device, a
positive voltage is required to turn it off. For bipolar switching, once a certain state has
14
been achieved, the polarity of the device must be reversed in order to switch the device
[8]. Examples of both unipolar and bipolar switching are shown in Figure 9 with time
indicated by the arrows.
Figure 9: IV Curves for Unipolar (left) and Bipolar (right) Switching (Source: [8])
It should be quite obvious that the IV curve for bipolar switching closely resembles the
bowtie diagrams predicted by Chua for the Memristor. Indeed, the material used for the
HP Memristor is TiO2, a material that exhibits bipolar resistive switching.
The exact cause of resistive switching is still not completely understood and is in fact
believed to be caused by different mechanisms for different materials [8]. For the case
of TiO2, the mechanism of anion migration is suspected to be the cause of resistive
switching [8]. Anion migration occurs when charged anions within the material begin to
drift in the presence of an electric field [8]. For TiO2, the anions that are drifting are the
positively charged oxygen vacancies.
A description of TiO2 during anion migration is shown in Figure 10. The process begins
when an electric field is applied and oxygen vacancies are formed at the anode. Oxygen
15
vacancy generation is governed by the reaction described by equation (18) [8]. In this
reaction, bonded oxygen (Oo) turns into oxygen gas molecules that escape into the
anode while leaving behind positively charged oxygen vacancies (
) in the crystal
structure [8]. These vacancies then drift across the material towards the cathode as a
result of the applied electric field. Assuming that the cathode blocks ion exchange
reactions, the oxygen vacancies will begin to accumulate near the cathode [8]. The
oxygen vacancies will then trap electrons emitted by the cathode, obeying equation (19)
[8]. This reaction forms TiO2-x at the cathode (where n = x - 2), which exhibits a much
lower resistance than TiO2. As the TiO2-x has a low resistance, its accumulation
effectively moves the cathode to the right, forming a virtual cathode as indicated in
Figure 10 [8].
( )
(18)
(19)
V
- +
Cathode
TiO2-x
TiO2
Anode
nV-o
-
ne
n2en(½)O2
Virtual Cathode
Figure 10: Anion Migration in Titanium Dioxide with an Applied Electric Field
16
As shown in Figure 10 and equation (18), oxygen vacancies are created at the anode
when oxygen gas molecules are absorbed by the anode itself. Thus, the anode must be
constructed of a material that exhibits fairly high oxygen mobility but will not react easily
with oxygen [8]. Furthermore, a cathode material is required that does not absorb
oxygen vacancies, allowing them to “pile up” at the cathode [8]. It is for these reasons
that Platinum is often selected for both the anode and cathode in a Memristor. Selecting
the same material for the anode and the cathode is important as it is a requirement for
bipolar switching that the structure be symmetric [8].
An interesting characteristic of anion migration in TiO2 is the formation of conducting
filaments. Essentially, the process described above does not happen evenly across the
material but instead tends to occur in narrow channels of TiO2-x [8]. Using a conductivetip atomic force microscope (C-AFM), these filaments can be observed by scanning the
surface of titanium dioxide. Such a measurement is shown in Figure 11; on the left, the
material was scanned during a low resistance state, on the right the same material was
scanned during a high resistance state. The white spots correspond to the low resistivity
filaments, there being significantly more of them during the low impedance state.
Figure 11: Surface Resistivity of TiO2 Measured Using a C-AFM (Source: [9])
17
Applications of the Memristor
The Memristor has the potential to augment or enhance several areas of integrated
circuit design and computing. Extensive literature regarding applications of the
Memristor has been produced since the HP announcement but a few developments are
particularly noteworthy.
One highly pervasive area where Memristors may be applied is that of non-volatile
digital memory [10]. The Memristor is seen as having significant potential in this area as
the device exhibits memory, does not require continuous power draw and consumes
little physical area.
column[2]
column[1]
column[0]
row[0]
row[1]
row[2]
Figure 12: Example of a Memristor Crossbar Memory Topology
For digital memory applications, one bit of information can be stored using a single
Memristor. This can be achieved by forcing the Memristor to its extreme resistance
values (RON and ROFF), each state corresponding to either a 1 or a 0. DC voltages are
18
used to set the resistance of a Memristor element [10]. In order to read stored data, AC
signals are utilized so that the stored data is not disturbed [10]. A crossbar arrangement
is often used for this memory architecture, as shown in Figure 12 [10]. The crossbar
topology is composed of a grid of horizontal and vertical traces. At each intersection, a
vertical trace is connected to a horizontal trace by a Memristor. Thus, an individual
Memristor can be programmed or read by applying a voltage to the necessary
horizontal and vertical traces while letting the remaining traces float.
Figure 13: Programmable Gain Differential Amplifier (Source: [3])
While the Memristor can be used at its extreme resistance values in order to provide
digital memory, it can also be made to behave in an analog manner. One potential
application of this behaviour is that of a dynamically adjustable electric load [3]. Thus,
existing electronic circuit topologies with characteristics that depend on a resistance can
be made with Memristors that behave as variable, programmable resistances. This
19
design approach has the potential to create electronically adjustable filters and
amplifiers as shown in Figure 13.
A common operation involved in image recognition and other image processing
techniques is that of edge detection. Edge detection identifies where large changes in a
digital image occur, such as the outline of an object. However, edge detection is
notoriously computationally intensive. One application of Memristors identified by [10] is
that of performing edge detection using a memristance grid. An example of this
operation can be seen below in Figure 14. The light intensity of the original photo is
applied as voltages to points on a grid of Memristors (a). The resulting changes in
resistance across the grid over time are then shown in the frames in (b). Thus, after the
system has been allowed to settle, the edge detected image can be recovered by
measuring the resistance of each element in the grid [10].
Figure 14: Memristor Based Edge Detection (Source: [10])
20
Report Summary
The mathematical properties of a fourth fundamental passive circuit element were
predicted 40 years ago by Leon Chua. This new circuit element was predicted to
behave as a resistor but with a resistance that changed with the electric charge that had
passed through the device. Dubbed the Memristor, it was not until 2008 that such a
device was discovered by a team working at HP labs. The HP Memristor is a simple
device constructed of a TiO2 thin film sandwiched between platinum electrodes. The HP
Memristor is easy to manufacture but requires a modern nanometre scale process.
The HP Memristor relies on two physical characteristics of titanium dioxide. The first
characteristic is the sensitivity of TiO2 conductivity to oxygen depletion. TiO2 is normally
an insulator but behaves as an n-type semiconductor when oxygen vacancies are
introduced, forming TiO2-x. The second characteristic that the HP Memristor relies on is
that of anionic migration, the tendency for oxygen vacancies within TiO2 to drift with an
applied electric field.
The Memristor has the potential for several important applications in the electronics and
computing industries. These applications include but are not limited to: non-volatile
memory, analog electronics and image processing. These applications take advantage
of the Memristors ability to store digital and analog information in a simple and power
efficient manner.
21
References
[1]
Chua, L.; , "Memristor-The missing circuit element," Circuit Theory, IEEE
Transactions on , vol.18, no.5, pp. 507- 519, Sep 1971
[2]
Albo-Canals, Jordi; Pazienza, Giovanni E.; , "How to teach memristors in EE
undergraduate courses," Circuits and Systems (ISCAS), 2011 IEEE International
Symposium on , vol., no., pp.345-348, 15-18 May 2011
[3]
Sangho Shin; Kyungmin Kim; Sung-Mo Kang; , "Memristor Applications for
Programmable Analog ICs," Nanotechnology, IEEE Transactions on , vol.10, no.2,
pp.266-274, March 2011
[4]
Williams, R.; , "How We Found The Missing Memristor," Spectrum, IEEE , vol.45,
no.12, pp.28-35, Dec. 2008
[5]
Breckenridge, Robert G.; and Hosler, William R.; , " Electrical Properties of
Titanium Dioxide Semiconductors," American Physical Society , vol. 91, pp.793-802,
Aug. 1953
[6]
D. Strukov, G. Snider, D. Stewart, and R. Williams, "The missing memristor
found," Nature, vol. 453, no. 7191, pp. 80-83, 2008.
[7]
Kavehei, O.; Yeong-Seuk Kim; Iqbal, A.; Eshraghian, K.; Al-Sarawi, S.F.; Abbott,
D.; , "The fourth element: Insights into the memristor," Communications, Circuits and
Systems, 2009. ICCCAS 2009. International Conference on , vol., no., pp.921-927, 2325 July 2009
[8]
R. Waser and M. Aono, “Nanoionics-based resistive switching memories,” Nat.
Mater., vol. 6, no. 11, pp. 833–840, 2007.
[9]
Choi, B. J.; Jeong, D. S.; Kim, S. K.; Rohde, C.; Choi, S.; Oh, J. H.; Kim, H. J.;
Hwang, C. S.; Szot, K.; Waser, R.; Reichenberg, B.; Tiedke, S.; , "Resistive switching
mechanism of TiO2 thin films grown by atomic-layer deposition," Journal of Applied
Physics , vol.98, no.3, pp.033715-033715-10, Aug 2005
[10] Prodromakis, T.; Toumazou, C.; , "A review on memristive devices and
applications," Electronics, Circuits, and Systems (ICECS), 2010 17th IEEE International
Conference on , vol., no., pp.934-937, 12-15 Dec. 2010
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