1. Memristor – The Fourth Fundamental Circuit Element

2. Introduction
Currently known fundamental passive elements – Resistors, Capacitors & Inductors.
Does a 4th passive element exist..?
Leon O. Chua formulated Memristor theory in his paper “Memristor-The Missing Circuit Element” in
Memistors are passive two terminal circuit elements.
Behaves like a nonlinear resistor with memory.

3. History Of Memristor
Four fundamental circuit variables- current i, voltage v, charge q, and flux linkage φ
Six possible combinations of these four variables
Five already defined as
Resistor(dv=Rdi), Capacitor(dq=Cdv), Inductor(dφ=Ldi), q(t)=∫i(T)dT, φ(t)=∫v(T)dT
The 6th relation defines memristance as dφ=Mdq

4. Relation between fundamental circuit elements and variables

5. So what is Memristance?
Memristance is a property of an electronic component.
When charge flows in one direction, its resistance increases, and if direction is reversed, resistance decreases.
When v=0, charge flow stops & component will ‘remember’ the last resistance it had.
When the flow of charge regains, the resistance of the circuit will be the value when it was last active.

6. Memristor Theory
Two terminal device in which magnetic flux Φm between its terminals is a function of amount of electric charge q passed through the device.
M(q) = dΦm/dq
M(q) = [dΦm/dt] / [dq/dt] = V/I
V(t) = M(q(t))I(t)
The memristor is static if no current is applied.
If I(t)=0, then V(t)=0 and M(t) is a constant. This is the essence of the memory effect.

7. Physical analogy for a memristor
Resistor is analogous to a pipe that carries water.
Water(charge q), input pressure(voltage v), rate of flow of water(current i).
In case of resistor, flow of water is faster if pipe is shorter and/or has a larger diameter.
Memristor is analogous to a special kind of pipe that expands or shrinks when water flows through it
The pipe is directive in nature.
If water pressure is turned off, pipe will retain its most recent diameter, until water is turned back on.

8. Titanium dioxide memristor
On April 30, 2008, a team at HP Labs led by the scientist R. Stanley Williams announced the discovery of a switching memristor.
It achieves a resistance dependent on the history of current using a chemical mechanism.
The HP device is composed of a thin (5nm) Titanium dioxide film between two Pt electrodes.
Initially there are two layers, one slightly depleted of Oxygen atoms, other non-depleted layer.
The depleted layer has much lower resistance than the non- depleted layer.

9. Microscopic image of memristor row
An atomic force microscope image of a simple circuit with 17 memristors lined up in a row.  Each memristor has a bottom wire that contacts one side of the device and a top wire that contacts the opposite side.  The devices act as 'memory resistors', with the resistance of each device depending on the amount of charge that has moved through each one. The wires in this image are 50 nm wide, or about 150 atoms in total width.

10. v-i characteristics

11. v-i chara..(cont.)
The most common v-i trace is a ‘figure 8’ or a ‘pinched loop’
For this current i=0, when voltage v=0.
On the application of electric field, oxygen vacancies drift, changing boundary between high & low resistance layers.
Memristance is only displayed when the doped layer & depleted layer both contribute to resistance.
The device enters hysteresis when enough charge has passed through memristor & ions can no longer move.

12. Contribution of HP Labs
HP Lab scientists were first to observe the ‘memristive behaviour’ in materials.
Introduced the titanium dioxide memristor.
Introduced memristance formula for devices.

13. Memristance formula
For linear ionic drift in a uniform field with average ion mobility µv,
The 2nd term in the parentheses which contribute more to memristance becomes larger when D is in the nanometer range.
Thus memristance is important characteristics of a device when critical dimension shrink to nanometer scale.

14. Operation as a switch
For some memristors, applied current or voltage will cause a great change in resistance.
The semiconductor film has a region of high conc. of dopants having low resistance RON & remaining portion having zero dopant conc. and much higher resistance ROFF.
By application of external bias, we can move the boundary to adjust the device resistance from RON to ROFF.

15. Applications & Advantages
can now think about fabricating a non-volatile random access memory (RAM) – or memory chips that don't forget the data when a computer is shut off. Memristors carries a memory of its past.
Replace today’s commonly used dynamic random access memory (DRAM).
Denser cells allow memristor circuits to store more data than flash memory.
The Hewlett-Packard team has successfully created working circuits based on memristors that are as small as 15 nanometers. Ultimately, it will be possible to make memristors as small as about four nanometers.

16. Applications & Advantages..(cont.)
A memristor circuit requires lower voltage, less power and less time to turn on than competitive memory like DRAM and flash.
It does not require power to maintain its memory.
The ability to store and retrieve a vast array of intermediate values also pave the way to a completely different class of computing capabilities like an analog computer in which you don't use 1s and 0s only.

17. Practical limitations of memristor
The most significant limitation is that the memristors functions at about one-tenth the speed of today’s DRAM memory cells.
The graphs in William’s report shows switching operation at only 1Hz.
Although small dimension of device seems to imply fast operation, the charge carriers move very slowly.

18. Conclusion
The rich hysteretic v-i characteristics detected in many thin film devices can now be understood as memristive behaviour.
This behaviour is more relevant as active region in devices shrink to nanometer thickness.
It takes a lot of transistors and capacitors to do the job of a single memristor.
No combination of R,L,C circuit could duplicate the memristance.
So the memristor qualifies as a fundamental circuit element.