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Circuit elements with memory: memristors, memcapacitors and meminductors
Massimiliano Di Ventra, Yuriy V. Pershin, Leon O. Chua
TL;DR
The paper asks whether memory-dependent circuit behavior can extend beyond resistors to capacitors and inductors. It formulates these elements as state-dependent systems and examines their hysteretic properties, finding pinched loops in the defining variable pairs and identifying nanoscale and neuromorphic relevance.
Problem
Memory-device concepts had been developed for resistive systems, leaving capacitive and inductive elements with state- and history-dependent properties to be generalized.
Method
The paper formulates capacitive and inductive memory devices as special cases of general state-dependent systems and analyzes their constitutive relations and simulated responses.
Results
The three device classes exhibit pinched hysteretic loops in current-voltage, charge-voltage, and current-flux variables, while memcapacitive systems can store energy and show passive simulated behavior.
Takeaways & Limitations
These memory devices and their combinations may support new electronic functionalities, including non-volatile memory and neuromorphic simulations of learning and adaptive behavior.
Abstract
from arXiv · showhide
We extend the notion of memristive systems to capacitive and inductive elements, namely capacitors and inductors whose properties depend on the state and history of the system. All these elements show pinched hysteretic loops in the two constitutive variables that define them: current-voltage for the memristor, charge-voltage for the memcapacitor, and current-flux for the meminductor. We argue that these devices are common at the nanoscale where the dynamical properties of electrons and ions are likely to depend on the history of the system, at least within certain time scales. These elements and their combination in circuits open up new functionalities in electronics and they are likely to find applications in neuromorphic devices to simulate learning, adaptive and spontaneous behavior.
I. INTRODUCTION
The paper generalizes memory-dependent circuit behavior from resistive systems to capacitive and inductive systems, motivated by state and history dependence in nanoscale devices and potential low-power, analog applications.
- Motivation: Memory-dependent circuit elements could store continuously varying information without a power source, supporting low-power computation and analog storage.The paper also connects such elements to adaptive, spontaneous, and learning-related behavior in biological systems.
- Background: Memristive systems generalize resistors whose resistance depends on the system’s internal state, with examples including thermistors, molecular systems, spintronic devices, and thin-film nanostructures.The paper notes that nanoscale resistance can depend on state and dynamical history over relevant time scales.
- Generalization: The paper defines a general nth-order memory device using complementary circuit variables u(t) and y(t), a generalized response g, and state dynamics ẋ = f(x,u,t).The state equation is assumed to have a unique solution from a given initial state.
- Generalization: Memcapacitive and meminductive systems use charge-voltage and current-flux as their defining variable pairs and can combine with memristive or standard elements to provide new electronic functionalities.The paper presents these systems as special cases of the general memory-device equations.
II. MEMRISTIVE SYSTEMS
Memristive systems are state-dependent resistive devices described by internal-state equations and exhibit frequency-dependent hysteresis, including pinched loops that collapse as frequency increases.
- Definition: A current-controlled memristive system relates voltage to current through a state-dependent memristance R(x,I,t), while a voltage-controlled system uses memductance G(x,VM,t).Both formulations include an internal state vector whose evolution is governed by a differential equation.
- Properties: For positive memristance, memristive systems are passive and cannot store energy like capacitors or inductors.Their zero-voltage/zero-current relation reflects the absence of energy storage.
- Properties: For periodic current input, a memristive system exhibits a pinched hysteretic loop in its constitutive variables.This behavior is a central dynamical signature of the system.
- Properties: At infinite frequency, a memristive system behaves as a linear resistor, whereas at zero frequency it behaves as a nonlinear resistor when the state equation has a steady-state solution.The low-frequency limit allows the state to adjust, while the high-frequency limit suppresses state evolution.
- Example: Numerical simulation of a thresholded voltage-controlled model shows a pinched hysteresis loop and hysteresis collapse as the alternating-source frequency increases.The model changes resistance between limiting values R1 and R2, with rates α and β below and above threshold voltage VT.
- Extensions: Memcapacitive and meminductive extensions share memristive characteristics but differ fundamentally because they store energy.The paper introduces these extensions after establishing the memristive framework.
III. MEMCAPACITATIVE SYSTEMS
Memcapacitive systems generalize memory-dependent circuit behavior to capacitance, with charge–voltage relations governed by internal state and history. Their hysteresis, frequency-dependent response, energy behavior, and nanoscale physical mechanisms distinguish them from standard capacitors.
- Definition: A voltage-controlled memcapacitive system relates charge to voltage through a state-dependent memcapacitance and evolves its internal state dynamically.The charge-controlled formulation instead uses voltage as a function of charge and state.
- Properties: Memcapacitors can store energy: zero voltage implies zero charge, but zero current does not imply zero charge.Energy may therefore be added to or removed from the system.
- Physical mechanisms: Capacitance memory can arise from geometrical changes, history-dependent material properties, or both, while inelastic changes may dissipate heat not representable as a series resistance.Other state-control sources can also supply energy that is later released in the circuit.
- Hysteresis: A periodic voltage produces a simple q–VC loop through the origin, with at most two charges for a given voltage under the stated uniqueness assumption.The loop is also antisymmetric when the capacitance and state dynamics satisfy the specified voltage-symmetry condition.
- Dynamic response: Memcapacitive systems approach linear-capacitor behavior at infinite frequency and nonlinear-capacitor behavior at zero frequency when the state equations have a steady-state solution.The contrast reflects the system’s ability to adjust at slow frequencies and inability to respond to extremely fast oscillations.
- Example: Numerical simulations of a voltage-controlled memcapacitive circuit show capacitance hysteresis, frequency-dependent behavior, and a pinched hysteresis loop, with the modeled device behaving passively.The example varies capacitance between C1 and C2 using threshold-dependent rates α and β.
IV. MEMINDUCTIVE SYSTEMS
Meminductive systems generalize inductors by making inductance depend on internal state, current or flux, and time. They can store energy and exhibit frequency-dependent behavior and pinched current–flux loops.
- Definitions: An nth-order current-controlled meminductive system uses flux φ(t) = L(x, I, t)I(t) and state dynamics ẋ = f(x, I, t).The corresponding flux-controlled class uses the inverse meminductance.
- Definitions: Meminductive systems include current-controlled and flux-controlled subclasses, depending on whether the constitutive relation is expressed through current or flux.The paper separately defines current-controlled meminductors and flux-controlled meminductors as reduced forms of the general systems.
- Properties: A time-dependent meminductance contributes an additional term to the induced voltage, and its stored energy provides a passivity criterion when the energy remains nonnegative.For constant inductance, the energy reduces to the familiar expression UL = LI^2/2.
- Properties: Under periodic current excitation, the flux–current curve forms a loop through the origin, with antisymmetry when L and f are even in current.The loop has at most two flux values for a given current under the stated conditions.
- Properties: A meminductive system behaves as a nonlinear inductor at low frequencies and a linear inductor at high frequencies when the governing equations admit a steady-state solution.Magnetic permeability can adjust to slow current variations but not to high-frequency variations.
- Examples: A possible circuit model makes inductance increase for current in one direction and decrease when current reverses, but the paper does not report the calculation.The model is described as analogous to models for memristive and memcapacitive systems.
V. CONCLUSIONS
The paper extends memory devices to capacitive and inductive systems, identifying shared hysteretic behavior and relevance at nanoscale dimensions. It also points to non-volatile memory and neuromorphic applications while emphasizing that combined circuit operations remain largely unexplored.
- Conclusions: Memory devices show pinched hysteretic loops in current–voltage, charge–voltage, or current–flux variables for memristive, memcapacitive, and meminductive systems.These constitutive-variable pairs define the respective device classes.
- Conclusions: Nanoscale systems are especially relevant because electron and ion dynamics may depend strongly on system history within certain time scales.The conclusion links this history dependence to the increasing relevance of memory-device concepts with device miniaturization.
- Applications: Memory devices may support non-volatile memories and neuromorphic devices that simulate learning, adaptive, and spontaneous behavior.The paper also relates memristive behavior in primitive organisms and neurons to possible laboratory models.
- Conclusions: Combined operations of memory devices in electronic circuits remain largely unexplored, including their analog functionalities.The authors hope this motivates experimental and theoretical investigations.