Memistors, Memristors, and Memresistors



In 2008 researchers at Hewlett-Packard announced the physical realization of the “memristor” which was theoretically predicted as a fundamental non-linear circuit element by Leon Chua in 1971. Since that time there have been numerous scientific papers applying the concept of memory resistors to a wide range of thin film materials used for a new type of non-volatile memory called ReRAM (resistive random-access memory). It has also been noted that memory resistors may be applicable to explain behavior of biological neurons and some research groups have developed circuit designs exploiting memory resistors as components of neuromorphic electronics. This article reviews the historical background of various forms of memory resistors including the Widrow-Hoff memistor, Chua’s memristor model, and a mem-resistor model I developed to correct some of the deficiencies in HP’s memristor model. The potential future of memory resistors with respect to artificial intelligence and robotics is briefly discussed.

For information on why the HP’s memristor might be considered scientific fraud the reader is referred to this link.

1960: Memistors and ADALINE circuitry

The earliest example of research in memory resistors can be credited to Bernard Widrow of Stanford University and his graduate student Ted Hoff in 1960 . The “memistor” (memory resistor) was developed as a component of ADALINE (ADAptive LInear NEuron), one of the first examples of an artificial neural network and was originally developed as an electroplating cell including the feature of storing the history of the voltage (or current) in the form of the electrical resistance. While Widrow did not develop a mathematical theory to explain the behavior of the memistor he explained the operation in a technical report published in 1961 entitled ” Birth, Life, and Death in Microelectronics Systems .” The top of page 23 of this report reads:

“Like the transistor, the memistor is a 3-terminal element. The conductance between two of the terminals is controlled by the time integral of the current in the third, rather than its 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 milliamperes 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.”

So if memistors were physically realized back in 1960 why don’t we have any memistor-based electronics today? At least a partial answer to that question can be found in the 1961 report entitled ” Birth, Life, and Death in Microelectronics Systems .” Page 1 of this report noted that:

“Even more elegant, although speculative, techniques have been proposed in which large numbers of components are formed en masse in thin film patterns on appropriate substrates by evaporative or ion-beam deposition or by electron-beam micromachining.”

Basically what this is describing is the rise of the integrated circuit concept originally developed by Jack Kilby of Texas Instruments and Robert Noyce , who later founded Intel. The integrated circuit has dominated electronicsever since the 1960’s. Unfortunately, the memistor of Widrow was based on an electrochemical circuit element rather than solid-statecircuit element as demanded by the integrated circuit concept. It is an interesting side note that one of the co-developers of the memistor was Ted Hoff who went on to work for Intel and developed the microprocessor in the 1970’s leading to the personel computer and modern computing (see the book Talking Nets in which Bernard Widrow describes the history of the memistor and the ADALINE circuit in more detail).

So since electrochemical memistors as used by Widrow in the 1960’s conflicted with the emerging trend of solid state integrated circuitry they were eventually discarded. Research in artificial intelligence eventually began relying on integrated circuit designs, the microprocessor, and the software used to run programs on microprocessors. However, the production of a solid-state version of the memistor compatible with the integrated circuit may offer the possibility of continuing the development of ADALINE or similar artificial intelligence circuit architectures.

1971: The memristor as the fourth fundamental circuit element (or is it?)

At the time of Chua’s original paper and the later paper by Chua and Kang the memristor was considered a “fourth fundamental circuit element.” The other three fundamental circuit elements are the resistor, the capacitor, and the inductor. The resistance of resistors is defined by a functional relationship between current and voltage. The capacitance of capacitors is defined by the functional relationship between charge and voltage. The inductance of an inductor is defined by the functional relationship between current and magnetic flux linkage (the time integral of voltage).The mathematical justification for the existence of the memristor as a fourth fundamental circuit element was based on symmetry with the other three circuit elements in which a fourthfunctional relationship should exist between the electrical quantities of magnetic flux linkage and charge. Thus the mathematical functional relationship defining the four basic elements(according to Chua) is as follows:

Resistor: v = R(i)

Capacitor: q = C(v)

Inductor: ϕ= L(i)

Memristor: ϕ = M(q)

where v = voltage, i = current, q = charge, ϕ = magnetic flux linkage

However, there are a few problems with considering the memristor as a fourth fundamental circuit element. The first problem is that for linear circuit elements in which the memristive function is a linear function there is no effective mathematical difference between the resistor equation and the memristor equation (i.e. differentiating the memristive equation produces the resistor equations). Another problem is that even non-linear memristors would have the same unit as resistors (i.e. ohms) in contrast to capacitors and inductors which each have their own respective unit (i.e. farads, henries).

A more recent reason for memristors to be considered other than a fourth fundamental circuit element results from a 2009 publication entitled “Circuit elements with memory: memristors, memcapacitors, and meminductors,” authored by Massimiliano Di Ventra, Yuriy Pershin, and Leon Chua. This paper generalizes the concept of memristive systems to include memcapacitors and meminductors. Mathematical equations for memristive, memcapacitive, and meminductive systems may be summarized as follows:

Memristive system: v = R(x,i) i and dx/dt = m(x,i)

Memcapacitive system: q = C(x,v) v and dx/dt = g(x,v)

Meminductive system: ϕ = L(x,i) i and dx/dt = h(x,i)

where x is a state variable (or, more generally, state vector), dx/dt is the time derivative of the state variable, and m(x,i), g(x,v), and h(x,i) are continuous functions of the respective variables. It is noted that these equations reduce to that of non-linear resistors, capacitors, and inductors provided that dx/dt = 0.

Based on this interpretation it is more reasonable to consider resistors having memory effects as a dynamic generalization of non-linear resistors rather than as a “fourth fundamental circuit element.” However, Chua continues to argue that the memristor should be considered a fundamental circuit element since no combination of non-linear resistor, non-linear capacitor, and non-linear inductor can reproduce the dynamic properties of a memristor. This is a fallacious argument since by the same reasoning a diode should be considered a fundamental circuit element because no combination of linear resistor, linear capacitor, and linear inductor can replicate the non-linear properties of a diode. A diode is not considered as a “fundamental” circuit element but rather as one example of a non-linear resistor. In the same fashion Chua’s memristor should not be considered as a “fundamental” circuit element but rather as one (and not the only) possible example of a dynamic resistor.

2008: HPLabs creates the first physical model of the memristor (or did they?)

The recent interest in the memristor theory of Chua was sparked by an article in the journal Nature published by researchers at HPLabs in 2008 reporting memristive effects in TiO2. The title of the article was “The missing memristor found” in reflection of the title of Chua’s original 1971 paper on the memristor theory which proposed the memristor as a “missing” circuit element. Apparently (at the time) neither Chua or the HPLabs group were aware of the earlier work of Widrow using the similarly named memistor in 1960 (I have confirmed this via private e-mail with Leon Chua). The HPLabs group did recognize that various materials exhibiting memristive characteristics were noted in the literature (see references 2-4 of the HPLabs Nature article). It is also notable that uncited work on TiO2 was performed in 1967 including substantially similar results (see F.Argall, “Switching Phenomena in Titanium Oxide Thin Films,” Solid-State Electronics, vol. 11, issue 5).

What the HP researchers attempted to accomplish in their article was to show that the memristor theory could be used to model the resistive switching effects found in a variety of thin film materials. A memristive systems model may be mathematically formulated as:

v = R(x) i and dx/dt = m(i)

where x= state variable, v = voltage, i = current, R(x)=memristance function, m(i) = rate of change of the state variable.

The HP researchers noted that the state parameter x could be equated with the ionic distribution within a thin insulative film. In this case the first equation of the memristive system v=R(x)i may be interpreted as defining the relationship between voltage and electron flow through the thin film at a particular ionic distribution while the second equation dx/dt = m(i) may be interpreted as defining the relationship between the ionic flow and the electron flow. In this respect I believe the HP researchers are correct. In the Nature paper they propose R(x) and m(i) as:

R(x) = R ON (x/D) + R OFF (1-x/D)

m(i) = u v (R ON /D) i

where D = the thickness of the thin film, u v = the ionic mobility, R ON = the on resistance, R OFF = the off resistance.

However, there are several problems with these choice of functions and a direct application of these equations do not match observed measurement. One of the lead HP researchers named Stan Williams recognized this deficiency at the 2008 Memristor and Memristive Systems Symposium and acknowledged that this model is a so-called “freshman model” (see 1:26 at this link ). HPLabs group did also publish a more in-depth paper entitled “Memristive switching mechanism for metal/oxide/metal nanodevices” in Nature Nanotechnology which goes into more detail of the physics of the system but which does not do a very good job at connecting the physics to a memristive system model based on the physical parameters of the system such as the ionic mobility, film thickness, or ionic density. Some other problems with the HP models are the following:

1) The models do not take into account memcapacitive effects.

The metal/insulator/metal structures exhibiting memristive effects are by their nature also capacitive. This has not been formally treated and a variety of thin film materials that exhibit the memristive effect (such as perovskites) also exhibit a memcapacitive effect.

2) The models are poorly formulated for electrical engineering applications.

A working engineer is traditionally used to mathematical models based on current as the dependent variable and voltage as the independent variable. For example, this is the way the equations are formulated for the analysis of diodes or the analysis of field effect transistors. However, the formulas presented in the HP paper are based on voltage as the dependent variable making them unwieldy for convenient use.

3) The models do not properly distinguish the internal and external electric field.

If the underlying mechanism is based on ionic drift the model does not properly distinguish between the external field due to an applied voltage and an internal field due to the distribution of ions in the film.

4) The newer models do not actually represent a memristor.

This is something that I am surprised no one has really picked up on yet but the models used by HP in describing Pt-TiO 2-x -Pt have diverged from the original Nature paper and no longer describe a memristor but rather the more general class of memristive systems in which dx/dt exhibits an exponential rather than linear relationship with current. As far as I can tell there is STILL no solid state material which can legitimately be called a 2-terminal memristor as originally defined in Leon Chua’s 1971 paper.

In 2011, Chua has made an attempt to change the definition of the memristor in the article “Resistance switching memories are memristors” published in Applied Physics A. In that article the memristor is defined as follows:

“All 2-terminal non-volatile memory devices based on resistance switching are memristors, regardless of the device material and physical operating characteristics.”

However, in an earlier paper which describes the same type of 2-terminal non-volatile memory devices (“Memristive Systems and Devices” published in the Proceedings of the IEEE, vol. 64, No. 2, 1976) it was noted that:

“..there remains an even broader class of physical devices and systems whose characteristics resemble those of the memristor and yet cannot be realistically modeled by this element..”

Clearly there is an inconsistency between the memristor definitions used by Chua from 1976 compared to 2011. It appears that Chua has broadened the definition to save face for HP which has found that their device is not a memristor according to the original definition. This is arguably an example of scientific fraud and at minimum is propaganda to satisfy the business agenda of HP regarding RRAM while elevating the reputation of Chua who is attempting to use his revised memristor definition to take credit for phase change memory, CBRAM, MRAM, and all other forms of 2-terminal non-volatile memories.

2011: My mem-resistor model

The ionic memristor model of HP are deficient for failing to consider the effective mass of ions and repulsive forces between ions and for not properly considering the effects of ion transport on the built-in voltage and depletion width of Schottky junctions or the gap of tunneling junctions. The effective mass may be particularly important in cases where the ions are attached to nanoparticles or are located at the tip of a conducting filament in which case the ionic effective mass can be several orders of magnitude greater than a single ion. Below is an abstract from the article I posted on Cornell ArXiv (March 11, 2011) available at this link . A mirror copy is available at this link .

Dynamic Systems Model for Ionic Mem-Resistors based on Harmonic Oscillation


Memristive system models have previously been proposed to describe ionic memory resistors. However, these models neglect the mass of ions and repulsive forces between ions and are not well formulated in terms of semiconductor and ionic physics. This article proposes an alternative dynamic systems model in which the system state is derived from a second order differential equation in the form of a driven damped harmonic oscillator. Application is made to Schottky and tunneling barriers.

I posted another article on Mar 24, 2011 extending the harmonic oscillator model to filamentary mem-resistors available at this link .

Dynamic Systems Model for Filamentary Mem-Resistors


A dynamic systems model is proposed describing memory resistors which include a filament conductive bridge. In this model the system state is defined by both a dynamic tunneling barrier (associated with the filament-electrode gap) and a dynamic Schottky barrier (associated with the electron depletion width surrounding the filament-electrode gap). A general model is formulated which may be applicable to many different forms of memory resistor materials. The frequency response of the model is briefly discussed.

Another article from May 01, 2011 available at this link .

Set, Reset, and Retention Times for Ionic and Filamentary Mem-Resistors


A dynamic systems model has previously been proposed for mem-resistors based on a driven damped harmonic oscillator differential equation describing electron and ionic depletion widths in a thin semiconductor film. This paper derives equations for set, reset, and retention times based on the previously proposed model.

This article (link) from Jan. 30, 2012 debunks the mathematical basis used by Leon Chua and Stan Williams to claim that all forms of memory resistors are memristors.


It has been erroneously asserted by the circuit theorist Leon Chua that all zero-crossing pinched hysteresis curves define memristors. This claim has been used by Stan Williams of HPLabs to assert that all forms of RRAM and phase change memory are memristors. This paper demonstrates several examples of dynamic systems which fall outside of the constraints of memristive systems and yet also produce the same type of zero-crossing hysteresis curves claimed as a fingerprint for a memristor. This establishes that zero-crossing hysteresis serves as insufficient evidence for a memristor.

The Future: Artificial Intelligence and Robotics

It is possible that neural computer designs using materials with similar properties to that of Bernard Widrow’s 3-terminal memistor or 2-terminal memresistors will be realized within the next few years. Ironically Moore’s Law, which has driven semiconductor-based electronics over the past several decades, may be viewed as a brief interruption in the development of memory resistive electronics in the future. DARPA is currently supporting a program called SYNAPSE in which IBM, Hewlett-Packard, and HRL are being funded to create neuromorphic circuitry in which memresistor materials may play a role.

One may ask why memory resistive electronics would have any more effect on artificial intelligence than semiconductor-based microprocessors in which neural networks have been simulated in software. The use of software in artificial neural networks typically is based on a Von Neumann computer architecture which segregates memory storage and data processing. This segregation results in a limit to parallel processing of data which is one reason that pattern recognition and real-time responsiveness is limited in computer and robotic systems in comparison to biological systems. However, memory resistive systems integrate data storage and data processing capabilities in a single device which offers the potential to more closely emulate the capabilities of biological intelligence.

Of course it is unlikely that the development of memory resistive artificial intelligence or robotics achieving the capabilities of human beings will occur overnight. Since it has taken hundred of millions of yearsof evolution to develop human-level intelligence it may take a while for a similar level to occur in autonomous mem-resistive systems which model biological neurons (provided that such modeling was even possible). One way to short-circuit the amount of time necessary may be to use brain-computer interfaces to copy human brain patterns into mem-resistive circuitry. Given the popularity of A.I. driven video games such as The Sims and Second Life and companies such as Emotiv and OCZ Technology, which develop brain-computer interfaces for video game applications, such developments may not be too far in the future.

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