Memory effects in complex materials and nanoscale systems

Yuriy V. Pershin; Massimiliano Di Ventra

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Memory effects in complex materials and nanoscale systems

Yuriy V. Pershin, Massimiliano Di Ventra

arXiv:1011.3053v1Nov 12, 2010cond-mat.mes-hall

TL;DR

Memory effects in complex materials and nanoscale systems depend on the history of structural, charge, and spin dynamics. This review synthesizes these effects through memristive, memcapacitive, and meminductive frameworks, examines theoretical and experimental progress, and discusses applications. It concludes that these circuit-element categories provide a general description of observed or expected memory behaviors and point toward applications in storage, learning, programmable circuits, and biologically inspired systems.

Problem
The review addresses how history-dependent behavior arises across complex materials and nanoscale systems and how diverse resistive, capacitive, and inductive effects can be described coherently.
Method
The paper reviews microscopic mechanisms and theoretical and experimental work, organizing systems as memristive, memcapacitive, meminductive, or combined elements.
Results
The review finds that memory effects related to charge, spin, and structural dynamics can generally be categorized as memristive, memcapacitive, meminductive, or combinations of these.
Takeaways & Limitations
The circuit-element framework connects underlying physical memory mechanisms with applications in information storage, learning and programmable circuits, and biologically inspired systems.
Takeaways & Limitations
The review does not exhaust all physical systems and devices that may exhibit memory because of space limitations.

Abstract

from arXiv · show

Memory effects are ubiquitous in nature and are particularly relevant at the nanoscale where the dynamical properties of electrons and ions strongly depend on the history of the system, at least within certain time scales. We review here the memory properties of various materials and systems which appear most strikingly in their non-trivial time-dependent resistive, capacitative and inductive characteristics. We describe these characteristics within the framework of memristors, memcapacitors and meminductors, namely memory circuit elements whose properties depend on the history and state of the system. We examine basic issues related to such systems and critically report on both theoretical and experimental progress in understanding their functionalities. We also discuss possible applications of memory effects in various areas of science and technology ranging from digital to analog electronics, biologically-inspired circuits, and learning. We finally discuss future research opportunities in the field.

I. INTRODUCTION

The review defines memory as storing a system state for later access and examines how structural, charge, and spin dynamics produce memory in complex materials and nanoscale systems. It organizes these behaviors through memristive, memcapacitive, and meminductive systems, including deterministic, continuous, and stochastic formulations.

Motivation: Memory is the ability to store a system’s state at one time and access that information later.The physical storage mechanism can involve synapses, capacitor or transistor-gate charge, or magnetic spin polarization.
Motivation: The review examines microscopic mechanisms for information storage while noting that mechanisms in some materials remain unclear or disputed.Correlated oxides are cited as cases where structural transitions and electronic correlated states may both contribute.
General framework: Memory systems are classified as memristive, memcapacitive, and meminductive, or combinations of these, according to their constitutive circuit-variable relations.Current–voltage, charge–voltage, and flux–current relations define the three classes, respectively.
General framework: The general memory-element model uses an input–output relation and internal state equations, with state variables supplying history dependence.For discrete systems, y(t) = g(x,u,t)u(t) and ẋ = f(x,u,t); continuous systems replace x with a state function X(s,t).
General framework: The framework also includes continuous and stochastic memory elements, although stochastic memory elements have not been thoroughly studied.Continuous states can include coordinate-dependent spin-polarization density, while stochastic states follow differential equations with noise and autocorrelation.

B. Hysteresis loops

Hysteresis loops are a distinctive signature of memory devices, with their shape and crossing behavior determined by device properties, input amplitude, frequency, and response symmetry. The framework also links loop behavior to nonvolatile storage requirements, read/write times, and physical implementations such as thermistors.

General properties: Hysteresis-loop shape depends on both device properties and the amplitude and frequency of the applied input.Experimental loop reports should therefore fully specify the input u(t).
Crossing behavior: Even symmetry in g(x,u) and f(x,u) often accompanies non-self-crossing loops, whereas odd f(x,u) more commonly accompanies self-crossing y-u curves.The symmetry condition is not necessary: unipolar resistance switching can self-cross despite even functions.
Nonvolatile memory: Nonvolatile storage requires at least two steady-state solutions at u0 = 0, with small read-induced state changes and finite write-induced changes.The magnitude of f estimates internal-state rates, while read-time expressions estimate state alteration and the allowable number of read operations.
Memristive-system properties: Positive resistance makes memristive systems passive, and their zero-current-at-zero-voltage property prevents energy storage or discharge like a capacitor or inductor.The review also distinguishes real memristive systems from ideal memristors, whose state depends only on charge or integrated voltage.
Physical example: Thermistors exhibit type-II, non-self-intersecting pinched I-V hysteresis because their internal-state equation is symmetric under voltage-sign reversal.Their resistance decreases with temperature, while temperature depends on dissipated power through heat transfer.

E. Memcapacitors and Memcapacitive Systems

Memcapacitive systems generalize capacitors by allowing capacitance to depend on internal state, charge, voltage, and time. The review develops their axiomatic framework and examines energy behavior, frequency response, realizations, and an elastic example.

Definitions: Memcapacitive systems are defined by charge–voltage relations with internal state variables governed by their own dynamics.Voltage-controlled forms use q(t) = C(x, V, t)V(t) and ẋ = f(x, V, t); charge-controlled forms use the inverse memcapacitance.
Definitions: Ideal memcapacitors are reserved for systems whose capacitance depends on the full history of voltage or charge.The review distinguishes these ideal devices from the broader class of memcapacitive systems.
Energy and hysteresis: Memcapacitive systems can be passive, dissipative, or active because their hysteresis-loop energy may be added to or removed from the system.For the illustrated loop, shaded areas represent half-period energy exchanges; U1 + U2 > 0 indicates dissipation, whereas U1 + U2 < 0 indicates activity.
Physical realizations: Capacitance memory can arise from geometric changes or history-dependent material permittivity, with external control energy enabling active operation.An elastic capacitor changes plate separation mechanically, while quantum-mechanical carrier and bound-charge behavior can produce history-dependent permittivity.
Frequency response: Their behavior is typically linear at infinite frequency and nonlinear at zero frequency when the state equation has a steady-state solution.At low frequencies the system adjusts to changing bias, while at high frequencies it cannot follow rapid oscillations.
Elastic memcapacitive system: The elastic memcapacitive system is second-order and dissipative when γ > 0, but non-dissipative when γ = 0.Its memory comes from the dynamics of an elastically suspended capacitor plate whose separation changes as charge is applied.

F. Meminductors and Meminductive Systems

Meminductive systems extend inductive behavior by making inductance depend on internal state, current or flux, and time. The review describes their energy properties and an elastic-wire model exhibiting frequency-dependent type-II hysteresis.

Definitions: Current-controlled meminductive systems obey φ(t) = L(x, I, t)I(t) with state dynamics ẋ = f(x, I, t), while flux-controlled systems use inverse meminductance.The two formulations are distinguished by whether current or flux is the control variable.
General properties: Meminductive systems share hysteresis, low-frequency nonlinearity, high-frequency linearity, energy storage, and possible passive, dissipative, or active behavior with memcapacitive systems.Their inductance may also diverge or become negative in some cases.
Energy: Time-dependent inductance contributes an additional induced-voltage term and modifies the stored-energy expression beyond UL = LI^2/2.The extra contribution can represent processes controlling meminductance through internal state variables, including elastic energy.
Energy: A meminductive system is passive when initialized in its minimal-energy state and satisfies UL(t) ≥ 0 at all later times.The inequality accounts for energy storage and dissipative processes.
Elastic meminductive system: The elastic meminductive system uses two parallel, oppositely current-carrying wires, one of which moves under a spring and damping force.Its inductance increases as the wire separation d0 + y increases, and the interaction is repulsive.
Elastic meminductive system: The elastic meminductive system is second-order and dissipative when γ > 0, while its simulated φ–I and L–I curves show type-II hysteresis.The φ–I curves do not self-cross, whereas the L–I curves do self-cross.

G. Combination of memory features

Real nanoscale systems can exhibit memristive, memcapacitive, and meminductive behavior simultaneously. Combining these elements can yield compact effective descriptions with non-algebraic dependence on control parameters.

Concurrent memory features: At the nanoscale, multiple memory features may appear simultaneously because related physical processes can share the same state variables.The review notes that memristive behavior is often accompanied by some capacitive behavior, even if small.
Concurrent memory features: Nanoscale memristive systems may show I–V hysteresis that does not exactly cross the origin at frequencies near the inverse characteristic time.This follows from capacitive contributions accompanying memristive behavior.
Combinations: Parallel and series combinations of memory elements can be analyzed together with standard circuit elements to describe systems with multiple memory mechanisms.The combined state vector may include separate state variables when the constituent elements do not share the same memory variables.
Combinations: A parallel voltage-controlled memristive and memcapacitive system behaves as an effective voltage-controlled memristive system with non-algebraic bias dependence.The effective memductance combines the memductive contribution with the time derivative of the capacitive response.
Applications and related models: The review relates simultaneous memristive and memcapacitive behavior to thin-film memory materials through memadmittance models and conductive-bridge area changes.These models derive relations between conductive-bridge cross-sectional area and capacitance shifts.

H. Are memristors, memcapacitors and meminductors fundamental circuit elements?

Ideal memristors, memcapacitors, and meminductors cannot be reproduced by finite combinations of time-independent standard circuit elements because those combinations cannot retain the relevant full dynamics. The review nevertheless distinguishes ideal elements from realizable memory systems whose internal states may be simulated with ordinary components.

Ideal memory elements: Ideal memristors cannot be simulated by any finite combination of standard two-terminal, time-independent resistors, capacitors, and inductors.Their dynamical properties require memory beyond what such fixed elements can reproduce.
Ideal memory elements: Ideal memcapacitors and meminductors likewise cannot be represented by standard-element combinations because time-independent components cannot retain full charge or current dynamics.The review separately notes that using standard capacitors or inductors does not reproduce the required memory behavior.
Fundamentality: The authors reject calling memcapacitors and meminductors fifth and sixth fundamental circuit elements, preferring three fundamental elements with or without memory.This is presented as a terminology and classification preference rather than a denial of their distinct functionality.
Realizability: Unlike ideal memory elements, some memristive, memcapacitive, and meminductive systems can be simulated by standard nonlinear components because their internal states have physical realizations.Examples include capacitors with nonlinear resistors and nonlinear resistors with negative differential resistance.

III. MEMRISTIVE SYSTEMS

Memristive behavior arises from diverse physical mechanisms across materials and nanoscale systems, with resistance switching classified as bipolar, unipolar, or irreversible. The review connects these systems to device structures, physical mechanisms, and memory applications.

III. Memristive systems: Memristive behavior occurs in systems with widely varying physical mechanisms, including thermistors, ionic systems, nanostructures, and diverse resistive-switching materials.Examples include binary oxides, nanogap systems, metal nanogap junctions, and perovskite-type oxides.
Resistance-switching mechanisms: Electroforming applies a high-voltage pulse that creates conductive filaments through soft breakdown, with compliance current helping control the switching process.The electroforming step can define the device’s subsequent operation.
Resistance-switching types: Resistance switching is generally classified as bipolar, unipolar, or irreversible according to the observed I−V behavior.Bipolar switching requires both voltage polarities to move between ON and OFF states, whereas irreversible switching proceeds only toward a different state.
Applications and device performance: Reported devices combine nanoscale switching with practical memory characteristics, including fast writing, endurance, retention, high ON/OFF ratios, and mechanical flexibility.A flexible TiO2 device showed an ON/OFF ratio above 10^4, storage potential of about 10^6 s, and operation after 4000 flexes; another crossbar technology reported writing below 10 ns, endurance above 10^5 cycles, and retention near 7 years.
III. Memristive systems: Resistance-switching devices commonly use metal electrodes separated by an insulating or solid-electrolyte film, and can be integrated into crossbar arrays.Nanoionic memories use materials such as NiO, SiO2, SrTiO3, Ag2S, and GexSe1−x.
Molecular memory devices: Molecular junctions can show reversible bistable switching, with switching near ±1.5 V, conductance ratios up to 50, and high-state storage exceeding 22 hours.The reported interpretation attributes the behavior most likely to concerted charge delocalization and molecular conformational change above a critical voltage.

D + ROFF

Memristive models describe resistance changes through internal state variables, threshold behavior, and boundary conditions. The review contrasts idealized analytical models with experimentally observed memristive-system behavior and applies a threshold model to learning.

Model assumptions: The ideal current-controlled memristor derivation assumes current-controlled drift and constant mobility but omits boundary conditions.The review cautions that these assumptions may not hold under actual experimental conditions.
Model interpretation: Experimentally observed switching curves correspond to memristive systems rather than necessarily to ideal memristors.This distinction follows from the limitations of the idealized derivation and its underlying assumptions.
Boundary-constrained models: The Joglekar–Wolf nonlinear drift model introduces a window function satisfying F(0)=F(1)=0 to prevent state drift at the boundaries.The model uses the family Fp(y)=1−(2y−1)2p for positive integer p.
Threshold models: The threshold memristive model assumes resistance changes slowly below VT and rapidly above VT while enforcing limiting resistance values.Its state variable is the memristive resistance, and α and β set the rates below and above the threshold.
Learning applications: The threshold model was used to describe learning in simple biological organisms and in other studies, with emulator-generated I−V curves resembling experimental memristive curves.The emulator curves were compared with curves such as those shown for crossbar devices.

C. Phase-change memory cells

Phase-change memory stores information by switching a material between amorphous and crystalline phases with distinct physical properties. Electrical heating enables SET and RESET operations, and memristive models capture the cell’s coupled electrical and thermal behavior.

Phase-change memory cells: Phase-change memory uses amorphous and crystalline material phases with distinct resistive and optical properties.The technology is also known as PCM, PRAM, PCRAM, Ovonic Unified Memory, Chalcogenide RAM, and C-RAM.
Switching operation: Joule heating electrically induces switching between the two material phases.RESET melts and rapidly quenches the material into the amorphous state, while SET heats it above its crystallization temperature for sufficient time.
Switching operation: SET and RESET operations use unipolar pulses whose amplitude and duration determine the thermal transformation.The crystalline state is low-resistive, whereas the amorphous state is highly resistive.
Modeling: A memristive phase-change model represents the active-layer temperature and amorphous ratio while incorporating electrical power and thermal parameters.The amorphous ratio varies between 0 and 1, and the model includes environmental temperature, thermal resistance, thermal capacitance, and threshold voltage.

D. Metal-insulator phase transition memristive systems

Metal-insulator transitions produce history-dependent resistance changes that can be modeled as memristive behavior. VO2 provides experimentally demonstrated pulse-controlled switching and multibit storage, while related systems show temperature- and frequency-dependent effects.

VO2 memristive behavior is driven by a metal-insulator phase transition involving coupled charge, lattice, and spin degrees of freedom.
50V voltage pulses locally heat VO2, progressively promoting the insulator-to-metal transition and producing stepwise resistance changes.Each pulse changes the resistance through the evolving thermal and phase state.
The experimentally observed VO2 response is an even-function current-controlled memristive system because resistance changes depend on dissipated power rather than signal polarity.
VO2 devices have shown memory durations exceeding several hours and storage of up to 2^10 resistance values in one device.
The VO2 model uses temperature and resistance as state variables, with hysteretic resistance branches and fast relaxation toward the appropriate branch.The branch functions are displaced by 2∆, representing the hysteresis width; the selected functions are not unique.
La0.225Pr0.4Ca0.375MnO3 exhibits bidirectional pulse-controlled resistance switching and nonvolatile states below 30K, but its hysteresis is unsuitable for room-temperature applications.
Spintronic systems: Spintronic systems extend memristive behavior beyond structural and charge variables, including transverse-voltage hysteresis and domain-wall motion.Spin memory is discussed as relevant to potentially low-dissipation, reliable operation under ac bias.

F. Ionic channels

Ion-channel conductances in neuronal membranes can be represented as history-dependent memristive systems. The Hodgkin-Huxley framework maps channel gating dynamics onto circuit elements with voltage-controlled, frequency-dependent responses.

The Hodgkin-Huxley membrane model represents voltage-dependent sodium and potassium channel conductances as history-dependent memristive systems.
Gating variables n, m, and h vary between 0 and 1 and determine the time-dependent conductances of the ion channels.
Potassium channels are first-order voltage-controlled memristive systems, whereas sodium channels are second-order voltage-controlled memristive systems.
Simulations under ac voltage produce frequency-dependent I−V curves typical of memristive systems, including a self-crossing feature for sodium channels at 100Hz and negative voltages.The cited passage notes that experimental ac-bias results for these systems were not known to the authors.
More detailed neuronal models may contain many memristive systems because they include multiple ion-channel types distributed across spatially extended neurons.

IV. MEMCAPACITIVE SYSTEMS

Memcapacitive behavior arises when a capacitor’s geometry or dielectric permittivity changes over time. The review organizes these mechanisms and discusses mathematical models and SPICE modeling for such systems.

Memcapacitive memory can originate from time-dependent changes in capacitor geometry or in the permittivity of the intervening dielectric.
Geometrical mechanisms involve changing plate morphology, while permittivity-related mechanisms include delayed dielectric response and switching mechanisms.
The review presents physical systems and mathematical models for different memcapacitance mechanisms and notes a methodology for SPICE modeling.

A. Geometrical memcapacitive systems

Geometrical memcapacitive systems use mechanical motion, bistable membranes, or internal charge redistribution to create memory-dependent capacitance. Examples include MEMS devices, strained membranes, superlattices, and ionic nanopores.

Micro- and nano-electro-mechanical systems: MEMS and NEMS capacitors couple mechanical and electrical properties and serve as variable capacitors in RF applications such as tunable filters and voltage-controlled oscillators.
Micro- and nano-electro-mechanical systems: A MEMS capacitor exhibits capacitance hysteresis because voltage displaces a suspended top plate against a spring.
Elastic memcapacitive systems: A strained-membrane capacitor has two stable equilibrium configurations that can provide nonvolatile storage of one bit.
Superlattice memcapacitive systems: Superlattice memcapacitors embed N internal metal layers in an insulator, allowing internal charge redistribution while preventing transport between the external plates and layers.
Superlattice memcapacitive systems: The superlattice system is an N−1 order charge-controlled memcapacitive system with charge dynamics governed by tunneling currents between internal layers.
Superlattice memcapacitive systems: Simulations of a two-layer superlattice show non-pinched hysteresis, negative and diverging capacitance, and dissipation from tunneling energy loss.
Ionic memcapacitive systems: Ionic memcapacitive behavior can result from delayed ion redistribution in nanopore systems, although the review reports no known theoretical or experimental work for biological membranes.

C. Permittivity-switching memcapacitive systems

Memcapacitive behavior arises when a system’s capacitance depends on ionic redistribution, phase transitions, ferroelectric polarization, or charge stored in nanocrystals. The reviewed examples show programmable, hysteretic, and persistently tunable capacitance responses.

1. Polymer-based memcapacitive systems: Ionic redistribution in doped polymers produces nonvolatile changes in dielectric properties and programmable capacitance.Voltage pulses inject or extract iodine anions; resulting ionic dipoles change polymer permittivity and device capacitance.
2. Phase-transition memcapacitive systems: A vanadium-dioxide metamaterial exhibits persistent electrical tuning because voltage pulses progressively drive its metal-to-insulator transition.The transition increases permittivity and raises the metamaterial capacitance with each pulse.
2. Phase-transition memcapacitive systems: 20% redshifted the resonant frequency from ω0 = 1.65THz in the vanadium-dioxide metamaterial.Its equivalent circuit contains both memristive and memcapacitive elements, whose properties coexist during device operation.
1. Ferroelectric memcapacitive systems: Ferroelectric Pt/PZT/Pt capacitors show hysteretic C−V behavior with a characteristic butterfly shape and type-II memcapacitive response.The PZT dielectric constant ranged from εr = 83ε0 to 330ε0, with peak capacitance at ±2.5V corresponding to the coercive field.
1. MOS capacitors with nanocrystals: MOS capacitors containing Ge nanocrystals store charge that shifts their C−V curves, with transferred charge acting as the capacitance state variable.The displacement depends on charge per nanocrystal and device structural parameters, while charge transfer also introduces dissipation represented by a series resistor.

V. MEMINDUCTIVE SYSTEMS

Meminductive systems store energy while their inductance depends on past dynamics, through either geometry changes or history-dependent permeability. The reviewed examples emphasize mechanically and thermally tunable MEMS inductors and broader mixed-element memory systems.

V. MEMINDUCTIVE SYSTEMS: Meminductors can store magnetic and other energy while their inductance depends on past dynamics.Memory can originate from geometric variation or from the permeability response of the core material.
1. Bimorph meminductive systems: Bimorph MEMS inductors exploit coupled mechanical, electrical, magnetic, and thermal properties to create tunable meminductive behavior.The bimorph effect uses composite materials whose differing thermal expansion produces structural deflection.
1. Bimorph meminductive systems: 8% tuning range was reported for a gold/silicon-nitride device whose current heating deflects a movable inner inductor.The structure contains fixed outer and movable inner inductors connected in parallel.
1. Bimorph meminductive systems: 32% inductance tuning was reported for an a-Si/Al meminductive system that changes from three-dimensional at 0V to flat at 2V.The device’s geometry changes under applied bias, producing the reported tuning.
1. Bimorph meminductive systems: Under simplified single-temperature assumptions, bimorph systems are first-order current-controlled meminductive systems with type-II hysteresis under periodic stimulation.The state and response functions are even functions of current.
V. MEMINDUCTIVE SYSTEMS: History-dependent permeability provides another route to meminductive behavior, exemplified by ferromagnetic materials with magnetic hysteresis.The review also distinguishes more complex memory systems requiring combinations of basic circuit elements.

B. Memristive component in Josephson junctions

Josephson junctions contain a small memristive component associated with interference among quasiparticle pairs, while memory-element applications extend to logic, nonvolatile storage, and analog systems. The review presents both demonstrated capabilities and unresolved experimental limitations.

B. Memristive component in Josephson junctions: A realistic Josephson-junction model combines a resistor, capacitor, nonlinear inductor, and memristor in parallel.The memristor represents a small current component caused by interference among quasiparticle pairs.
B. Memristive component in Josephson junctions: The Josephson-junction memristive component is generally small but constitutes a rare example of an ideal voltage-controlled memristor.The component can therefore often be neglected in practical applications despite its theoretical significance.
A. Digital applications: Memristive systems support digital nonvolatile memory, signal processing, programmable logic, analog circuits, learning, and neuromorphic applications.The review notes that applications involving memcapacitive and meminductive systems remain at an early stage.
A. Digital applications: Material implication together with reset is functionally complete, allowing computation of any Boolean function with memristive systems.The operation stores m1 →m2 in m2, using threshold behavior and state-dependent resistance.
A. Digital applications: A memcapacitive-assisted circuit experimentally demonstrated NOT, AND, OR, and one-bit addition using memristor emulators with fewer steps than implication-based arithmetic.The modified circuit charges the memcapacitive element through input memristive systems and discharges it through output systems.
A. Digital applications: No experimental or theoretical work was identified for fuzzy-logic operations despite memory elements’ intrinsically analog behavior.The review distinguishes this gap from patented hybrid layouts for related operations.

B. Analog applications

The review presents analog applications of memory circuit elements in neuromorphic, associative-memory, quantum, programmable, and chaotic circuits. These systems use history-dependent resistance, capacitance, or inductance to implement learning, tunable interactions, sustained oscillations, and nonlinear dynamics.

Neuromorphic circuits: Memristive systems can serve as synapses in neuromorphic circuits, whose operation mimics the human or animal brain.Their small solid-state size may support synapse-like device densities in chips.
Neuromorphic circuits: Spike-timing-dependent plasticity changes synaptic strength according to the relative timing and polarity of pre- and post-synaptic signals.The model constrains excitation by voltage thresholds and bounds memristance between Rmin and Rmax.
Associative memory: A three-neuron network with two memristive synapses experimentally demonstrated associative learning: paired food and sound inputs produced an output when only sound was later applied.The association develops through a decrease in the relevant synaptic resistance.
Quantum computing with memory circuit elements: Non-dissipative memcapacitive and meminductive elements could tune coupling between superconducting qubits, enabling many interaction schemes within one circuit architecture.The proposed elements should avoid additional qubit relaxation or decoherence, at least within certain time scales.
Programmable and nonlinear circuits: Memristive systems support programmable analog circuits, while related emulators can produce effective memcapacitive and meminductive behavior.Reported programmable properties include threshold, gain, and frequency; increasing memristive resistance can reduce damping and maintain LC oscillations longer.
Programmable and nonlinear circuits: Replacing Chua’s diode with an active memristor produces novel chaotic behavior, while chaotic dynamics also occur in passive or memcapacitive systems under some conditions.Such circuits have been considered for secure communications with chaos.

VIII. CONCLUSIONS AND OUTLOOK

The review concludes that memory effects in complex materials and nanoscale systems can be organized as memristive, memcapacitive, and meminductive behavior or combinations thereof. It highlights broad application opportunities while acknowledging that the surveyed examples do not exhaust the possible systems and devices exhibiting memory.

Conclusions: Memory effects arise from charge, spin, and structural dynamics and can generally be categorized as memristive, memcapacitive, meminductive, or combined behavior.This classification connects microscopic physical mechanisms with history-dependent circuit elements.
Outlook: The reviewed examples are not exhaustive because space limitations prevented coverage of all physical systems and devices that show memory.The authors expect continued device miniaturization to reveal additional memory systems.
Outlook: The review links memory circuit elements to applications including information storage, learning and programmable circuits, and biologically inspired systems.The authors also anticipate potential commercial applications and further impacts on technology and fundamental science.