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What is memristance? 



Memristance is a property of an electronic component. If charge flows in one direction through a circuit, the resistance of that component of the circuit will increase, and if charge flows in the opposite direction in the circuit, the resistance will decrease. If the flow of charge is stopped by turning off the applied voltage, the component will 'remember' the last resistance that it had, and when the flow of charge starts again the resistance of the circuit will be what it was when it was last active.


Why is memristance important? 



It turns out that memristance is becoming stronger as the feature sizes in circuits are getting smaller. At some point as we scale into the realm of nanoelectronics, it will be necessary to explicitly take account of memristance in our circuit models in order to simulate and design electronic circuits properly.


Have people seen memristance before? 



Yes, we are aware of over 100 published papers going back to at least the early 1960's in which researchers observed and reported unusual 'hysteresis' in their currentvoltage plots of various devices and circuits based on many different types of materials and structures. In retrospect, we can understand that those researchers were actually seeing memristance, but they were apparently not aware of it.


What is a memristor? 



An ideal memristor is a passive twoterminal electronic device that is built to express only the property of memristance (just as a resistor expresses resistance and an inductor expresses inductance). However, in practice it may be difficult to build a 'pure memristor,' since a real device may also have a small amount of some other property, such as capacitance (just as any real inductor also has resistance).


What is an analogy for a memristor? 



A common analogy for a resistor is a pipe that carries water. The water itself is analogous to electrical charge, the pressure at the input of the pipe is similar to voltage, and the rate of flow of the water through the pipe is like electrical current. Just as with an electrical resistor, the flow of water through the pipe is faster if the pipe is shorter and/or it has a larger diameter. An analogy for a memristor is an interesting kind of pipe that expands or shrinks when water flows through it. If water flows through the pipe in one direction, the diameter of the pipe increases, thus enabling the water to flow faster. If water flows through the pipe in the opposite direction, the diameter of the pipe decreases, thus slowing down the flow of water. If the water pressure is turned off, the pipe will retain it most recent diameter until the water is turned back on. Thus, the pipe does not store water like a bucket (or a capacitor) – it remembers how much water flowed through it.


Who first predicted the existence of memristance and memristors? 



Prof. Leon Chua had just moved to the Electrical Engineering Department of UC Berkeley when he published his seminal paper, "Memristor  The missing circuit element." IEEE Trans. Circuit Theory CT18, 507519 (1971). In this paper, Prof. Chua proved a number of theorems to show that there was a 'missing' twoterminal circuit element from the family of "fundamental" passive devices: resistor, capacitor and inductor (e.g. elements that do not add energy to a circuit). He proved that no combination of nonlinear resistors, capacitors and inductors could duplicate the properties of a memristor. The most recognizable signature of a memristor is that when an AC voltage is applied to the device, the currentvoltage (IV) plot is a Lissajous figure (the curve formed by combining two oscillations that are perpendicular to each other). The most commonly observed IV trace is a 'figure 8', or a 'pinched loop' for which the current is zero when the voltage is zero. This inability to duplicate the properties of a memristor with the other passive circuit elements is what makes the memristor fundamental. However, this original paper requires a considerable effort for a nonexpert to follow. In a later paper, Prof. Chua introduced his 'periodic table' of circuit elements. This was a visually pleasing illustration that we borrowed and modified for our Nature paper on finding memristors.


Aren't there other fundamental passive devices that don't add energy to a circuit? What about diodes? 



No, there are only four fundamental types of passive circuit elements. Diodes are just nonlinear resistors  the resistance of a diode changes with the applied voltage, but if you turn off the voltage and start back at 0 volts, the resistance of the diode is the same as it was before at 0 volts, not what it was when the voltage was turned off. This is also true of a resistor that heats up and increases its resistance because of a temperature increase. Thus, neither a diode nor a heated resistor 'remember' their history. However, each type of fundamental circuit element is actually a family of devices with essentially an infinite number of higher order members. To see all the members of the four families of fundamental devices, see the following paper: Leon O. Chua, "Nonlinear Circuit Foundations for Nanodevices, Part I: The FourElement Torus," Proc. IEEE 91, 18301859 (2003). This is a very educational paper, but requires a significant investment in effort to appreciate. Note: Part II has not appeared in the literature yet.


What was the contribution of HP Labs? 



We were the first to understand that the hysteresis that was being observed in the IV curves of a wide variety of materials and structures was actually the result of memristance and something more general that can be called 'memristive behavior' [see L.O. Chua & S. M. Kang, "Memristive devices and systems," Proc. IEEE 64, 209223 (1976)]. We then went on to create an elementary circuit model that was defined by exactly the same mathematical equations as those predicted by Chua for the memristor, with the exception that this model had an upper bound to the resistance (which means that at large bias or long times, it is a memristive device). We then showed that this simple model could reproduce a wide variety of eccentric and complex IV curves that have been observed and reported over the years by many researchers, including ourselves. Most of these did not look much like the 'figure 8' curves of Chua, but rather 'S' and 'N' curves that have erroneously been attributed to negative differential resistance, which is one reason why the connection to memristive behavior had not been made earlier. We also showed that in a highly simplified form appropriate for a general audience journal like Nature or for a basic undergraduate course, the equations for the drift of oxygen vacancies in TiO2 and their influence on the electronic conduction in the material were also identical with our equivalent circuit model, and thus Chua's memristor equations. From this, we could for the first time write down a formula for the memristance of a device in terms of material and geometrical properties of the device (just as the resistance is the resistivity of the material times the length divided by the cross sectional area of the resistor). Our memristance formula immediately showed that the size of the most important term in the memristance gets larger the smaller the device – thus showing that it was not very important for micronscale electronics but is becoming very important for nanoscale devices. We have developed more sophisticated and accurate models that will be published at a future date, and we have used our models to design and build better memristors.


What types of applications could memristors have? 



We see two types of applications for memristors and memristive devices.
The first, as the name "memory resistor" implies, is for a type of nonvolatile random access memory, or NVRAM. Such a memory would have very useful properties, in that it would not 'forget' the data that it stores when the power is turned off. We think that NVRAM made with the types of memristor materials that are currently being studied by many groups around the world could be a strong competitor to the flash memory market in about five years. The great thing is that the various metal oxides that have been identified as having a memory function are highly compatible with present chip fabrication facilities, so they can be made in existing foundries without a lot of changes being required. The major contribution of our work to this effort at this point is to make the connection to the nonlinear circuit theory of Leon Chua – without the fundamental understanding that comes from his circuit equations, the devices themselves are fairly useless.
Another interesting application is as an 'artificial synapse' in a circuit designed for analog computation. Prof. Chua himself pointed out the connection between the properties of his proposed memristor and those of a synapse in his earliest papers, and he has performed a lot of research in the area of neural computing. We also think that this is a very interesting and potentially valuable research direction.
However, as experience shows, the most valuable applications of memristors will most likely come from some young student who learns about these devices and has an inspiration for something totally new.

