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The Fourth Element: Characteristics, Modelling, and Electromagnetic Theory of the Memristor
O. Kavehei, A. Iqbal, Y. S. Kim, K. Eshraghian, S. F. Al-Sarawi, D. Abbott
TL;DR
Memristor research addresses the realization and modeling of a fourth circuit element linking charge and flux, with potential relevance to dense nonvolatile devices and nanoelectronics. This paper reviews device properties, develops behavioral and SPICE models, and relates the device to Maxwell’s equations. The models reproduce expected memristor behavior and agree with the reported HP device, while nonlinear transport and process variation remain modeling challenges.
Problem
Memristor device and circuit modeling remains limited by unclear device behavior, nonlinear ionic transport, and process and defect variability.
Method
The paper surveys memristor properties, introduces behavioral and SPICE macro-models, and examines Chua’s theory through a quasi-static expansion of Maxwell’s equations.
Results
The SPICE macro-model agrees with the actual memristor reported by Strukov et al. and produces expected results when combined with resistors, capacitors, or inductors.
Takeaways & Limitations
Memristor capabilities motivate further nanoelectronics and nano-computing research, including device and circuit modeling for future architectures.
Takeaways & Limitations
Nonlinear drift models improve simulation accuracy but require more complicated equations, while behavioral models still need greater effort to accommodate defects and process variation.
Abstract
from arXiv · showhide
In 2008, researchers at HP Labs published a paper in {\it Nature} reporting the realisation of a new basic circuit element that completes the missing link between charge and flux-linkage, which was postulated by Leon Chua in 1971. The HP memristor is based on a nanometer scale TiO$_2$ thin-film, containing a doped region and an undoped region. Further to proposed applications of memristors in artificial biological systems and nonvolatile RAM (NVRAM), they also enable reconfigurable nanoelectronics. Moreover, memristors provide new paradigms in application specific integrated circuits (ASICs) and field programmable gate arrays (FPGAs). A significant reduction in area with an unprecedented memory capacity and device density are the potential advantages of memristors for Integrated Circuits (ICs). This work reviews the memristor and provides mathematical and SPICE models for memristors. Insight into the memristor device is given via recalling the quasi-static expansion of Maxwell's equations. We also review Chua's arguments based on electromagnetic theory.
1. Introduction
The paper introduces the memristor as a fourth fundamental circuit element and reviews its device properties, models, applications, and unresolved device-level challenges.
- Motivation: 2019 projections estimated DRAM capacity around 46 GB/cm2, while memristors promised more than 110 GB/cm2.The comparison assumes 100% DRAM area efficiency.
- Motivation: Memristors offer nonvolatile operation and were proposed for molecular electronics, computing, and ionic programmable resistive neural-network devices.Their nonvolatility is compared with flash memory, and programmable resistive behavior is linked to synaptic applications.
- Limitations: Device-level limitations include unclear leakage mechanisms, limited nonvolatile retention in flexible devices, slow switching, and poor endurance relative to DRAM.The HP cross-bar volatility-to-switching-speed ratio is around 10^3, compared with 10^6 for DRAM cells.
- Historical context: The paper positions the HP device as the first fabricated memristor explicitly connected to Chua’s 1971 prediction of a fourth fundamental circuit element.The authors distinguish this connection from earlier reports of devices with similar behavior.
- Scope and approach: The paper reviews memristor device properties, introduces linear and nonlinear drift models, develops a resistor modulation index, and presents SPICE macro-models and electromagnetic analysis.The review is organized around device characteristics, modeling, circuit simulation, and a quasi-static expansion of Maxwell’s equations.
2. Memristor Device Properties
The memristor is defined through the missing charge–flux relationship and realized as a state-dependent resistive device whose resistance changes with ionic motion and applied charge.
- Related devices: Earlier devices showed hysteretic or variable-resistance behavior, but devices modeled as energy-storage elements do not satisfy the memristor interpretation.The cited silicon-dioxide device switched from high to low impedance in about 100 ns but was classified differently because of its charge-storage model.
- Definition: A charge-controlled or flux-controlled memristor is defined by a functional relationship between ϕ and q, with incremental memristance measured in ohms and memductance in siemens.The corresponding forms are ϕ = fM(q) and q = gM(ϕ).
- Definition: The memristor completes the missing relationship between magnetic flux and electric charge among the four circuit variables.Resistors, capacitors, and inductors relate voltage–current, charge–voltage, and current–flux, respectively.
- Characteristics: Memristance is the slope of the ϕ-q curve, and a piecewise curve can provide two resistance values suitable for binary logic.The simple case uses two different slopes of the ϕ-q characteristic.
- Characteristics: For M(q) ≥ 0, instantaneous power p(i) = M(q)(i(t))2 is positive, so the memristor is passive and its ϕ-q curve is monotonically increasing.The passage identifies this behavior as consistent with the HP memristor.
- State variable: The device state is the total charge passing through the memristor, not charge stored as in a capacitor, making it a nonlinear resistor with charge as a state variable.This distinction also means the memristor is not an energy-storage element.
- Physical realization: The HP memristor uses a TiO2 thin film between platinum contacts, with oxygen-vacancy doping creating high-resistance undoped and low-resistance doped regions.The device is described within the broader memristive-systems form v(t) = R(w)i(t), with w as an internal state.
(a) Linear drift model
The linear drift model relates memristor state evolution to charge under a uniform electric field, then derives time-dependent memristance and current–voltage behavior. It also examines polarity, frequency, parameter sensitivity, and variability-aware modeling.
- Linear drift assumptions: Under a uniform electric field, the model assumes a linear relationship between ionic drift-diffusion velocity and net electric field.The state equation describes the memristor’s internal mechanism and boundary evolution.
- State evolution: The drift boundary moves according to applied charge, with Q_D representing the charge required to move the boundary across the device.The model relates boundary motion from w(t0) toward w(tD) to the passed charge and conductive-channel requirement.
- Memristance and circuit relations: The resulting memristance and current–voltage relations depend on device geometry and resistance parameters, including R_ON, R_OFF, and the initial memristance.When R_OFF ≫ R_ON, the initial memristance is approximately R_OFF, simplifying the derived relation.
- Scaling behavior: The inverse-square dependence on TiO2 thickness D indicates improved memristance characteristics for smaller D, while at micrometer scale the flux-dependent term becomes negligible.When the relevant term is negligible, the current–voltage relation reduces to that of a resistor.
- Frequency response: For sinusoidal excitation, the hysteresis loop crosses the origin, and increasing frequency drives the response toward a linear resistor.The flux amplitude varies inversely with frequency in the stated sinusoidal case.
- Parameter sensitivity and variability: Changing model parameters substantially alters hysteresis and memristor characteristics, motivating variability-aware approaches such as Monte Carlo simulation and corner analysis.The reported parameter sets produce ω0 values of approximately 50 kHz and 4 GHz, with memristance ratios around 2 and 120, respectively.
(b) Nonlinear drift model
The nonlinear drift model augments memristor state equations with window functions to represent strongly nonlinear ionic transport and boundary behavior. Simulations show that nonlinear modeling produces realistic boundary hysteresis that linear drift cannot reproduce, while existing window functions retain important limitations.
- Physical motivation: A few volts across a nanometer-scale film create large electric fields, producing significant nonlinearity in ionic transport.The model introduces a window function multiplied by the state equation to represent this behavior.
- Simulation behavior: Nonlinear drift simulations produce realistic boundary hysteresis that linear drift modeling cannot achieve under the same boundary conditions.Linear models require constraints on initial current, initial voltage, and the state range, which can substantially change the output.
- Window-function modeling: Window functions modify the state equation to model nonlinear current-voltage behavior at memristor boundaries.The window function is an additional term on the right-hand side of the state equation.
- Window-function modeling: The Joglekar and Biolek functions provide alternative nonlinear windows, with Biolek's adding memristor current as a parameter.Joglekar's function uses a control parameter p, whereas Biolek's function depends on x, p, and current i.
- Model limitations: Existing window functions have serious limitations: state-only dependence implies charge-history memory, and Biolek's function lacks continuity at the boundaries.The paper argues that a device may instead remember the boundary position rather than the total charge passed through it.
3. SPICE Macro-Model of memristor
The paper presents a readable SPICE macro-model that represents memristive behavior through an M-R mutator, a nonlinear resistor, and switching characteristics, then tests it across several circuit configurations.
- Model approach: SPICE macro-models are selected because they are readable and available across SPICE versions.The paper notes convergence problems but still considers SPICE the most appropriate modelling approach.
- Model approach: The M-R mutator combines an integrator, CCVS, differentiator, and VCCS, while a switched resistor sets the branch resistance.The branch resistance is 1 kΩ for V < 2 Volt and 2 kΩ for V ≥2 Volt.
- Single-device behavior: The simulated device exhibits pinched hysteresis and switches from 1 kΩ to 2 kΩ when flux reaches 2 Wb.Before the critical flux, the current-voltage slope is 1 mA/V; afterward it is 0.5 mA/V.
- Single-device behavior: A step input changes the memristance from 1 kΩ to 2 kΩ, reducing the current from 1 mA to 0.5 mA at 2 Wb.The transition occurs after 2 s, corresponding to 50 time steps in the illustrated simulation.
- Single-device behavior: At 1 kHz the sinusoidal input does not reach 2 Wb, whereas at 10 Hz it reaches that threshold within 30 ms.The frequency therefore determines whether the model remains at 1 kΩ or switches to 2 kΩ.
- Network validation: Series and parallel memristors behave like resistor counterparts, with equivalent memristances Mequ = 2M and Mequ = M/2, respectively.The model is also tested in RM, LM, and CM networks, where the memristance transition changes circuit responses and time constants.
4. Interpreting memristor in Electromagnetic Theory
The paper interprets the memristor through a quasi-static expansion of Maxwell’s equations, extending the electromagnetic framework used to explain classical circuit elements. Chua’s argument identifies a device associated with a direct relationship between first-order electric and magnetic fields, although physical realisability is not established by the argument alone.
- Quasi-static expansion: Quasi-static expansion uses successive approximations to study slowly varying electromagnetic fields, as in electric circuits.The fields are expanded using a time-rate parameter α and family time τ = αt.
- Quasi-static expansion: The expansion expresses electromagnetic fields as power series in α, with separate zero-order and first-order Maxwell equations.The coefficients of each power of α are set separately to zero.
- Quasi-static expansion: Quasi-static fields retain the zero- and first-order terms while neglecting higher-order terms.The approximation is written as E ≈ E0 + E1, D ≈ D0 + D1, H ≈ H0 + H1, B ≈ B0 + B1, and J ≈ J0 + J1.
- Circuit-element interpretation: Resistors, inductors, and capacitors correspond to combinations of zero-order and first-order quasi-static electromagnetic solutions.This provides the circuit-theory context for seeking an additional element.
- Chua’s argument: Chua’s proposed fourth device assumes negligible zero-order fields and related first-order fields in nonlinear material.The assumed device has D1 related instantaneously to B1.
- Chua’s argument: Combining the field relations yields an instantaneous D1-B1 relationship that the paper identifies as realisable in a memristor.This completes Chua’s quasi-static electromagnetic argument, but the argument itself only hints at physical existence.
5. Conclusion
The paper surveys memristors, presents behavioral and SPICE modeling, and reviews their electromagnetic basis. It concludes that further device and circuit modeling is needed to address practical variation and defect issues.
- The paper surveys key aspects of the memristor as a promising nano-device and discusses nanoelectronics and nano-computing implications.
- It introduces behavioral and SPICE macro-models, evaluates the SPICE model in PSpice, and presents a tested op-amp-based memristor.
- The paper reviews Chua’s argument for the memristor through a quasi-static expansion of Maxwell’s equations.
- Further device and circuit modeling is needed, because physical curve-fitting models require many parameters and nonlinear-drift models require more complicated equations.
- Future modeling should accommodate defect and process-variation issues and may support mapping memristors to neuromorphic systems.