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Memristors by Quantum Mechanics
Simulations of switching the HP memristor at 1 GHz support the quantum mechanics restriction that resistive heating is conserved by the creation of excitons thereby explaining the observed hysteretic behavior of resistance in I-V curves
By: Thomas Prevenslik
In 1971, Chua  claimed a two-terminal circuit element existed having a resistance that depended on the charge from time integral of the current through the device. Based on symmetry arguments alone, Chua argued that the three common elements comprising the resistor, capacitor, and inductor in electronic circuitry were incomplete and a fourth element called a memristor was required for completeness. But lacking a prototype, the memristor lay dormant for almost 40 years until a group at Hewlett-Packard (HP) in 2008 announced  the development of a switching memristor based on a thin film of titanium dioxide sandwiched between platinum electrodes. See PR at http://www.prlog.org/
Currently, HP memristor theory assumes positive charge from oxygen vacancies is the source of switching, but the theory is phenomenological lacking a physical basis for extensions to other memristors without vacancies. In fact, many nanodevices reported over the past 50 years show memristor behavior, e.g., sandwiched molecular layers between gold electrodes, and modification of electrical conduction in solid electrolytes, all of which exclude positive charge in vacancies. But sandwiched material between electrodes is not even necessary for memristor behavior. Recently, memristors are formed in a single material without electrodes, e.g., gold and silicon nanowires. Lacking vacancies, explanations of memristor behavior assume the presence of space charge, but the mechanism by which the space charge is produced is not identified.
Memristor behavior relying on the creation of charge finds commonality in the QM of QED radiation anytime EM energy is absorbed inside nanodevices, e.g., as in nanoscale heat transfer. QM stands for quantum mechanics, QED for quantum electrodynamics, and EM for electromagnetic. Indeed, the applicability of QM is supported by the fact that only at submicron sizes is the memristor behavior detectable. Supramicron memristors behave just like ordinary resistors, where resistance is equal to the voltage divided by the current.
Contrarily, Chua argued  in 2003 that QM is not applicable to memristors. Carbon nanotubes (CNTs) having submicron diameters were thought consistent with QM, but QM was thought invalidated by their supramicron lengths. However, the EM confinement of QED photons in any one direction is sufficient to justify the validity of QM, say in submicron CNT diameters. See numerous experiments of heat transfer in nanowires in http://www.nanoqed.org, 2009-2011.
The QM effect in memristors was simulated using the HP experimental data  for titanium dioxide. Resistive heating dissipated in the memristor is conserved with QED radiation inside the memristor. Since the Planck energy of QED photons exceeds the bandgap of 3.2 eV, all dissipated heating is conserved by the creation of excitons. Upon creation, the excitons comprising pairs of electron and positive holes act to promptly decrease the resistance of the memristor. But in the same switching cycle, excitons are destroyed upon being neutralized by the polarity of the voltage terminals. To understand how excitons are created and destroyed, a pair of nonlinear differential equations was formulated for the electrons and holes. Simplifications were made by assuming only the positive charged holes contribute to the resistance of the memristor. Simulations assumed 1 GHz memristor switching. Only the hole mobility was necessary to fit the HP data, i.e., the mobility of electrons and holes in titanium dioxide was assumed in the range form 800 – 1000 cm^2/ V-s. The simulations give the typical memristor hysteretic I-V curve illustrated in the above figure. Simulation details are given in http://www.nanoqed.org, at “Memristors - the Fourth Element?,” 2011.
1. The 1971 paper by Chua and numerous papers to date are classical approaches to explaining memristor behavior. Modern day electronics was developed based on macroscale response of resistors, but a QM approach is suggested at the nanoscale where memristive effects are observed.
2. QED radiation developed for nanoscale heat transfer based on QM is directly applicable to memristors by precluding any temperature increases to conserve electrical resistive heat. Conservation proceeds by the creation of QED photons inside the memristor that create excitons, the positive charged holes of which change the resistance to produce the memristive effect.
3. Explanations of memristive effects need not rely on oxygen vacancies, electromigration thinning, space charge, and the like.
4. The memristor as the missing fourth element to provide completeness for the symmetry for the resistor, capacitor, and inductor is valid provided QM is included in the argument.
5. More study is required to refine the application of QED radiation to memristors.
 Chua, L. O., Memristor - the missing circuit element. IEEE Trans. Circuit Theory 18, 507 (1971).
 Strukov, D. B., et al., The missing memristor found, Nature 453, 7191 (2008).
 Chua, L., Nonlinear Circuit Foundations for Nanodevices, Part I: The Four-Element Torus, Proc. IEEE, 91, 1830 (2003).
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About QED Indcued EM Radiation: Classically, absorbed EM energy is conserved by an increase in temperature. But at the nanoscale, temperature increases are forbidden by quantum mechanics. QED radiation explains how absorbed EM energy is conserved at the nanoscale by the emission of nonthermal EM radiation.