A memristor (; a portmanteau of memory resistor) is a non-linear two-terminal electrical component relating electric charge and magnetic flux linkage. It was described and named in 1971 by Leon Chua, completing a theoretical quartet of fundamental electrical components which also comprises the resistor, capacitor and inductor.
Chua and Kang later generalized the concept to memristive systems. Such a system comprises a circuit, of multiple conventional components, which mimics key properties of the ideal memristor component and is also commonly referred to as a memristor. Several such memristor system technologies have been developed, notably ReRAM.
The identification of memristive properties in electronic devices has attracted controversy. Experimentally, the ideal memristor has yet to be demonstrated.
Electric component
Chua, in his 1971 paper, identified a theoretical symmetry between the non-linear resistor (voltage vs. current), non-linear capacitor (voltage vs. charge), and non-linear inductor (magnetic flux linkage vs. current). From this symmetry, he inferred the characteristics of a fourth fundamental non-linear circuit element, linking magnetic flux and charge, which he called the memristor. In contrast to a linear (or non-linear) resistor, the memristor has a dynamic relationship between current and voltage, including a memory of past voltages or currents. Other scientists had proposed dynamic memory resistors such as the memistor of Bernard Widrow, but Chua introduced a mathematical generality.
Derivation and characteristics The memristor was originally defined in terms of a non-linear functional relationship between magnetic flux linkage and the amount of electric charge that has flowed, :
The magnetic flux linkage, , is generalized from the circuit characteristic of an inductor. It represent a magnetic field here. Its physical meaning is discussed below. The symbol may be regarded as the integral of voltage over time.
In the relationship between and , the derivative of one with respect to the other depends on the value of one or the other, and so each memristor is characterized by its memristance function describing the charge-dependent rate of change of flux with charge:
Substituting the flux as the time integral of the voltage, and charge as the time integral of current, the more convenient forms are:
To relate the memristor to the resistor, capacitor, and inductor, it is helpful to isolate the term , which characterizes the device, and write it as a differential equation.
+ Differential equation relationships between resistance, capacitance, inductance, and memristance Device Symbol Characteristic property Units Unit ratio (V, A, C, Wb) Differential equation Resistor Resistance ohm () volts per ampere () Capacitor Capacitance farad () coulombs per volt () Inductor Inductance henry () webers per ampere () Memristor Memristance ohm () webers per coulomb ()
The above table covers all meaningful ratios of differentials of , , , and . No device can relate to , or to , because is the time derivative of and is the time derivative of .
It can be inferred from this that memristance is charge-dependent resistance. If is a constant function (i.e. has the same value for all ), then we obtain Ohm's law: . If is nontrivial, however, the equation is not equivalent because and thus vary with time. Solving for voltage as a function of time produces
This equation reveals that memristance defines a linear relationship between current and voltage, as long as does not vary with charge. Non-zero current implies time-varying charge. Alternating current, however, may reveal the linear dependence in circuit operation by inducing a measurable voltage without net charge movement—as long as the maximum value of does not cause much change in compared to the initial value .
Furthermore, the memristor has a constant memristance if no current is applied. If , is constant due to being constant. This is the essence of the memory effect.
Analogously, we can define a as memductance (portmanteau of memory and conductance):
Memductance with respect to flux is the inverse of memristance with respect to charge, i.e.
and therefore the unit of memductance is the same as the unit of conductance ‒ Siemens.
The power consumption characteristic recalls that of a resistor, :
As long as varies little, such as under alternating current, the memristor will appear as a constant resistor. If increases rapidly, however, current and power consumption will quickly stop.
is physically restricted to be positive for all values of (assuming the device is passive and does not become superconductive at some ). A negative value for some charge implies it acts as a power source at this charge level. A negative value for all charges would mean that it would perpetually supply energy when operated with alternating current.
Modelling and validation In order to understand the nature of memristor function, some knowledge of fundamental circuit theoretic concepts is useful, starting with the concept of device modeling.
Engineers and scientists seldom analyze a physical system in its original form. Instead, they construct a model that approximates the behaviour of the system. By analyzing the behaviour of the model, they hope to predict the behaviour of the actual system. The primary reason for constructing models is that physical systems are usually too complex to be amenable to a practical analysis.
In the 20th century, work was done on devices where researchers did not recognize the memristive characteristics. This has raised the suggestion that such devices should be recognised as memristors. Pershin and Di Ventra have proposed a test that can help to resolve some of the long-standing controversies about whether an ideal memristor does actually exist or is a purely mathematical concept.
The rest of this article primarily addresses memristors as related to ReRAM devices, since the majority of work since 2008 has been concentrated in this area.
Superconducting memristor component Dr. Paul Penfield, in a 1974 MIT technical report mentions the memristor in connection with Josephson junctions. This was an early use of the word memristor in the context of a circuit device.
One of the terms in the current through a Josephson junction is of the form:
where is a constant based on the physical superconducting materials, is the voltage across the junction and is the current through the junction.
Through the late 20th century, research regarding this phase-dependent conductance in Josephson junctions was carried out. A more comprehensive approach to extracting this phase-dependent conductance appeared with Peotta and Di Ventra's seminal paper in 2014.
Memristor circuits Study of the ideal memristor poses practical difficulties, so other electrical devices are commonly used as models for memristors. For a mathematical description of a memristive device (systems), see .
A discharge tube can be modelled as a memristive device, with resistance being a function of the number of conduction electrons .
is the voltage across the discharge tube, is the current flowing through it, and is the number of conduction electrons. A simple memristance function is . The parameters , , and depend on the dimensions of the tube and the gas fillings. An experimental identification of memristive behaviour is the pinched hysteresis loop in the plane.
Thermistors can be modeled as memristive devices:
is a material constant, is the absolute body temperature of the thermistor, is the ambient temperature (both temperatures in Kelvin), denotes the cold temperature resistance at , is the heat capacitance and is the dissipation constant for the thermistor.
A fundamental phenomenon that has hardly been studied is memristive behaviour in p-n junctions. The memristor plays a crucial role in mimicking the charge storage effect in the diode base, and is also responsible for the conductivity modulation phenomenon (which is so important during forward transients).
Criticisms In 2008, a team at HP Labs found experimental evidence for Chua's memristor based on an analysis of a thin film of titanium dioxide, thus connecting the operation of ReRAM devices to the memristor concept. According to HP Labs, the memristor would operate in the following way: the memristor's electrical resistance is not constant but depends on the current that had previously flowed through the device, i.e., its present resistance depends on how much electric charge has previously flowed through it and in what direction; the device remembers its history—the so-called . When the electric power supply is turned off, the memristor remembers its most recent resistance until it is turned on again.
The HP Labs result was published in the scientific journal Nature. Chua argued that the memristor definition could be generalized to cover all forms of two-terminal non-volatile memory devices based on resistance switching effects. Chua also argued that the memristor is the oldest known circuit element, with its effects predating the resistor, capacitor, and inductor. However, there are doubts as to whether a memristor can actually exist. Additionally, some experimental evidence contradicts Chua's generalization since a non-passive nanobattery effect is observable in resistance switching memory. A simple test has been proposed by Pershin and Di Ventra to analyze whether such an ideal or generic memristor does actually exist or is a purely mathematical concept. Up to now, there seems to be no experimental resistance switching device (ReRAM) which can pass the test.
These devices are intended for applications in nanoelectronic memory devices, computer logic, and neuromorphic/neuromemristive computer architectures. In 2013, Hewlett-Packard CTO Martin Fink suggested that memristor memory may become commercially available as early as 2018. In March 2012, a team of researchers from HRL Laboratories and the University of Michigan announced the first functioning memristor array built on a CMOS chip.
According to the original 1971 definition, the memristor is the fourth fundamental circuit element, forming a non-linear relationship between electric charge and magnetic flux linkage. In 2011, Chua argued for a broader definition that includes all two-terminal non-volatile memory devices based on resistance switching. Williams argued that MRAM, phase-change memory and ReRAM are memristor technologies. Some researchers argued that biological structures such as blood and skin fit the definition. Others argued that the memory device under development by HP Labs and other forms of ReRAM are not memristors, but rather part of a broader class of variable-resistance systems, and that a broader definition of memristor is a scientifically unjustifiable land grab that favored HP's memristor patents.
In 2011, Meuffels and Schroeder noted that one of the early memristor papers included a mistaken assumption regarding ionic conduction. In 2012, Meuffels and Soni discussed some fundamental issues and problems in the realization of memristors. They indicated inadequacies in the electrochemical modeling presented in the Nature article "The missing memristor found" because the impact of concentration polarization effects on the behavior of metal−TiO2−x−metal structures under voltage or current stress was not considered.
In a kind of thought experiment, Meuffels and Soni furthermore revealed a severe inconsistency: If a current-controlled memristor with the so-called exists in physical reality, its behavior would violate Landauer's principle, which places a limit on the minimum amount of energy required to change information states of a system. This critique was finally adopted by Di Ventra and Pershin in 2013.