# Memristors : Motivation, Theory, and Feasibility

Memristors, the fourth missing element after Inductance, Resistance, and capacitance, is nowadays a scorching topic of research among the electronics community. A large number of papers on its models, applications, and feasibility are now available. In this post, I’ll try to cover most of its background research:

## Content:-

• Introduction
• What are memristors?
• Characteristics of a memristor
• Transistor vs. memristor
• The polarity of a memristor and an analogy
• Memristor links to chaos theory?[TBD]
• Motivation
• well-known problems with present-day computers
• limitations of CMOS technology
• Moore’s soul at unrest
• Theory
• History
• Before HP
• After HP
• HP’s model
• Memristive systems
• Problems with Chua’s memristor theory
• Characteristics of memristor fabrication technologies
• Application
• Flow chart
• For logic
• For NV memory
• For Neuromorphic computing
• Artificial Neural Networks
• Backpropagation, feed-forward
• Spike timing delay Plasticity(STDP) – Hebbian learning
• Feasibility
• Limitations of CMOS-Memristor hybrids
• Challenges in the fabrication of oxide-based memristors

Copyright notice:- Most of the figures presented below are taken either from google images or are publicly available under GNU/ Artistic license or are drawn by the author himself. Material presented in the following text is published under the Common Creative license.

# Introduction

## What are memristors?

Based on symmetry arguments, Chua proposed the existence of the fourth fundamental, passive component after Resistor, inductor, and capacitor; he named it Memristor(Memory resistor). Memristence of a device is Non-LTI and directly links electric charge and the flux.
where M(q(t)) is the charge controlled memristance

Another interpretation can be in terms of flux as a flux controlled memconductance:

where W(phi(t)) is the flux controlled memconductance

Out of these four fundamental elements, Memristor has unique VI characteristics, which makes it suitable for the design of all sorts of Non-linear chaotic circuits.

Memristor’s electrical resistance is not constant; rather, it depends on the history of the current that previously flowed through it. And this is what makes memristor so unique! Its memory or non-volatility has many potential applications. Chua claimed that memristor’s definition could be generalized for all sorts of 2 terminal Non-volatile memory based on resistance switching effect. An example of this is ReRAM. Biological components like blood and skin also fit this definition(apart from being 2-terminal).

https://www.researchgate.net/publication/264836366_Human_blood_liquid_memristor

### Characteristics of a memristor:

There are three main characteristics of the memristors:

• Memristors V-I curve has a pinched hysteresis loop and notes that it passes through the origin.
• The area of each lobe in the hysteresis loop shrinks as the frequency of the applied signal increases.
• As frequency tends to infinity, the Hysteresis loop degenerates to a straight line through the origin, i.e., it becomes resistant.

### The polarity of a memristor and an analogy

Consider a water pipe with a flexible area of cross-section. This pipe has it’s preferred direction of water flow, say A to B. So, if water flows from A to B, a cross-section of the pipe increases, thus allowing more water to flow(i.e., conductance increases). Now, if water flows in the reverse direction, from B to A(which is against the preferred direction), the cross-section of the pipe decreases, thus providing resistance to the flow of water. Apart from this, the Area of the cross-section at any instance of time depends on the total water flow in the past.

This is exactly how memristor works. It has it’s preferred direction of current flow, which is known as it’s polarity. When charge flow is in the direction of this polarity, the resistance of the memristor decreases. And when it’s opposite resistance increases.

### Transistor vs. memristor

Apart from the number of terminals, In transistor(transfer resistance), base control current that alters the resistance between the collector and the emitter terminals. While in memristor(Memory resistance), it’s the integral of the control current that alters the resistance.

### Memristor’s link to chaos theory?[TBD]

As presented in [8], Memristor presents few important behaviors concerning chaos theory, such as great sensitivity of circuits behavior on initial conditions, the route to chaos through the mechanism of period-doubling, as well as antimonotonicity. I’m not very sure about this interesting relationship between the two. I need to dig into this a bit deeper.

# Motivation

Why is such importance suddenly given to memristors and its applications? Its concept was around the late ’90s.

### Well-known problems with current computers:

• Memory wall – Limitation on available bandwidth for memory access and the power wasted in every memory access being made.
• Instruction level parallelism – Due to difficulty in making software that can work with highly parallel hardware. Such as in Asymmetric multiprocessor systems.
• Power wall – Since Dynamic power of CMOS circuits is related as:
P_d = C.f. V^2
as CPU frequency increases, the power also dissipated increases. This put’s a limitation on CPU maximum clock speed.

### Problems with CMOS technology:

• Leakage current – As the size of a CMOS cell decreases, leakage current increases, and thus Static power dissipation increases.
• Cost per piece – Fabrication cost of a CMOS cell increases as the size of the cell decreases
• Scaling wall – Lack of supporting fabrication technology that can work with few nanometers(<1nm) without significant error rates.

### Moore’s soul at unrest:

Due to the above mentioned technological drawbacks, Moore’s curve is now getting saturated. We are in immediate need of certain alternative technologies that can keep up to the demands of advancing human civilizations. Few such promising technologies, which will be there in the market within 10-20 years,  are 3-D CMOS circuitry and Memristor based circuits. Other evolving technologies like Quantum dots, spin-wave devices still need a lot of development.

# Theory

This section mainly deals with a more elaborated overview of memristors and few kind-of-abstract topics.

## History

A lot has changed after HP’s claim in 2008 of realizing a practical memristor. There’s a complete drift in the research community.

### Before HP’s 2008 claim

• Scholars used to discuss circuitry topologies that behave like an ideal definition of memristor
• The first implementation of memristors was to build circuits with complex dynamics. They are often considered as chaotic circuits.
• Memristor based oscillators and other analog circuits were devised.

Most of the focus was given on Ideal characteristics or models of the Memristor.

### After HP’s 2008 claim

• Different approximated but pragmatic models of memristors were proposed
• Y.Hong proposed a timing storage circuit based on the memristor model. By recording the information in a pair of matched memristors, his circuit can save or retrieve analog timing information without quantizing.(Pretty cool!!)
https://ieeexplore.ieee.org/document/7080909
• A lot of focus was put on memristor-based Non-Volatile memories like RRAM
• Memristor based logic circuits like IMPLY and MAGIC were proposed
• Usage of memristors for Neuromorphic computing was found

### HP’s claim and associated memristor model

In 2008, HP claimed to have found the practical existence of a memristor. A summary of their experiment is presented in the next paragraph (there’s a lot of elaborated material available online on HP’s model so that I won’t cover in detail)

They made a 2-terminal device, made of a layer of insulated $TiO_2$ and another layer of oxygen-deficient, conductive $TiO_2$. These two materials are divided by a barrier of width ‘w.’ When a  voltage V is applied, the produced Electric field tends to alter the barrier and thus changes the resistance. There are two possible mechanisms for this: Joules heating and Ion migration.

## Memristive Systems

There’s a difference between a memristor and memristive systems. Memristive systems are an extension of standard memristor theory to a non-linear dynamical system. Similar to a memristor, A memristive device is passive, 2 terminal, and has varying resistance. The difference is how the resistance changes.

In a memristive device, the resistance depends on an internal state $x \in R^{n}$, where n is the total number of possible states. A state depends on the charge/flux history of the device and not directly on the current charge or flux. Whereas a memristor has a mathematically scalar state, a system has a vector state. The number of state variables is independent of the number of terminals.

y(t) = g(x,u,t).u(t) where u(t) is an input signal, y(t) is an output signal, the vector x represents a set of n state variables describing the device, and g is a continuous function.

For a current-controlled memristive system, the signal u(t) represents the current signal i(t), and the signal y(t) represents the voltage signal v(t). For a voltage-controlled memristive system, the signal u(t) represents the voltage signal v(t), and the signal y(t) represents the current signal i(t).

## Problems with Chua’s memristor theory

• Violation of Launderer’s principle by ideal memristor. Launderer’s principle calls for a lower limit on the amount of energy required to flip-a-bit, which inherently opposes the idea of infinite states(which an ideal memristor has).
• ReRAM’s concept is better modeled by probabilistic switching
• In resistance, the internal energy is independent of the current flowing, which is not true for a memristor. Thus, it’s not appropriate to call a memristor as a memory resistor.
• Nano-battery effect
• No accountability of linear ionic drift assumption. Rise time and fall time are assumed to be equal.

## Characteristics of Memristor’s fabrication technologies

• Most of them are fabricated as oxides sandwiched between metals(Metal-insulator-Metal, MIM) with minimal size
• Produced memristor has high write endurance, low switching energy, high fabrication density
• The electric field causes ionic movements and local structural changes, which in turn can cause a measurable change in device resistance.

# Applications

Memristors found a huge application in both analog and digital areas. They are mainly used in two forms, discrete and crossbar formation.

I’ll elaborate some of its important applications:

### For Non-Volatile Memories:

All 2 terminals NV memory devices based on resistance switching are said to be memristor-based.

They usually have faster write time, large resistance ratio (Ron/Roff), and small power consumption. It is said that with 3D memristor crossbar fabrication, we can even have 1TB of storage on a single chip. Following categorization of resistive memory, tend to replace DRAM:

• Phase change memory(PCM) – based on Joules heating effect
• Electrostatic effect memory(EEM) – based on purely electronic/electro-static effects
• Redox based – based on REDuction, OXidation(REDOX) chemical effect

### For Digital Logic:

There are three types of digital logic that can be implemented with memristors:

1. Boolean logic – Like AND/OR/NOT. It has further classifications:
1. CMOS-like Memristor based logic(CLM)
2. Memristor ratioed logic with CMOS hybrid(MRL)
3. MAGIC – Memristor only logic family
4. Crossbar memristor logic – for carrying out complex computation
2. Implication logic – Logics which determine propositions like A implies B. Stateful logical state is determined by the resistance of the memristor. CRS – Complementary Resistive Switch is one of the types, which can also be used for implementing boolean logic.
3. Threshold logic – If this weighted sum of all the inputs is greater than or equal to the threshold value, T then only the output, Y will be equal to one. Programmable CMOS memristor logic and hybrid current mirror logic can be used to implement threshold logic.

### For Neuromorphic computing:

Neural networks used for neuromorphic applications are of two types:

• Spiking Neural networks like Spike Timing Delay Plasticity(STDP)
• Here, Synapse plasticity depends on latency between spikes from the previous and current neuron.
• Weight of synapse increase when the lag between two neurons decreases and vice versa
• Like, Hebbian learning network, which strengthens the connection between neutron whose activities are casually related.
• STDP exploits the threshold effects observed in switching characteristics of several types of memristors
• It’s closer to the actual working of the brain
• Artificial neural networks like feedforward, backpropagation, etc.
• Here, Synapse plasticity depends on the weights assigned to the synapse network.
• Since learning in ANN is iterative, it can adjust for mismatch in memristive synaptic elements.

# Feasibility

## Challenges in CMOS-Memristor system integration

• Power: CMOS has scrubbed scaling induced power dissipation by lowering operating voltages. However, memristor-based resistive memories need the energy to read/write them.
• Fabricating and Electroforming these devices need a high Electric field to be applied across the device terminal. This might break down dielectric and need a high voltage tip of the fabricating machine.
• Large Fan-in and Fan-out for memristors is a big problem
• Challenges in scaling up the size of a memristor include non-uniform resistance profiling, resistance drift, an inherent device to device, and cycle to cycle variation.

## Challenges in the fabrication of Oxide-based memristor

• Difficult to control oxygen concentration in Titanium oxides – It’s practically very problematic to control the concentration of oxygen in TiO2, what’s even worse is that no two devices will have the same oxygen concentration. Thus characteristics vary from device to device.
• Poor state retention

TBD:-

• Series and parallel combinations
• Different window function

## References:-

1. Wikipedia- memristors
2. https://www.researchgate.net/publication/325710976
3. Memristor based circuits and architectures – Thesis
4. Memristors for computing: Myth or reality? – Research paper
5. Study of  memristors models and applications – Thesis
6. Book- randhavan  chapter 2-3
7. Composite characteristics of memristor circuits: series or parallel – research paper
8. Memristor: A new approach to non-linear circuit design – research paper
9. https://pdfs.semanticscholar.org/presentation/6bb4/6b8db3eb8aab1664831fc70f72808e9b1a00.pdf
10. https://knowm.org/the-trouble-with-oxide-based-memristors/
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