Synchronization coexistence in a Rulkov neural network based on locally active discrete memristor

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Abstract

At present, many neuron models have been proposed, which can be divided into discrete neuron models and continuous neuron models. Discrete neuron models have the advantage of faster simulation speed and the ease of understanding complex dynamic phenomena. Due to the properties of memorability, nonvolatility, and local activity, locally active discrete memristors (LADMs) are also suitable for simulating synapses. In this paper, we use an LADM to mimic synapses and establish a Rulkov neural network model. It is found that the change of coupling strength and the initial state of the LADM leads to multiple firing patterns of the neural network. In addition, considering the influence of neural network parameters and the initial state of the LADM, numerical analysis methods such as phase diagram and timing diagram are used to study the phase synchronization. As the system parameters and the initial states of the LADM change, the LADM coupled Rulkov neural network exhibits synchronization transition and synchronization coexistence.

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Nanobatteries in redox-based resistive switches require extension of memristor theory

I Valov, C Linn, S. Tappertzhofen … (2013)

Resistive switching memories are nanoionic-based electrochemical systems with a simple metal–ion conductor (insulator)–metal structure. These devices exhibit low power consumption, response times in the nanosecond range and scalability down to the atomic level1 2 3. They demonstrate excellent prospects for the application in modern information technology, in particular for novel logical devices4 and artificial neuromorphic systems5. The link6 between redox-based nanoionic-resistive memories7, memristors8 and memristive devices9 has further intensified the research in this important area and inspired the introduction of new concepts10 11 12 13 and material systems2 3. Efforts are now focused on a microscopic understanding of the physicochemical processes responsible for resistive switching14 and on meeting the challenges of circuit design15. The system properties and the material performance of redox-based nanoionic resistive memory cell (ReRAM) devices are modulated by quantum effects, excess surface free energy of atomic clusters and nonlinear mesoscopic transport phenomena, owing to the nanodimensions of the devices in both lateral and vertical direction, that are often comparable to space charge layer lengths. The borders of well-known definitions, such as those for ion conductors and insulators, become blurred at the nanoscale, and various classes of materials starting from RbAg4I5, AgI (as bulk ion conductors) and continuing to SiO2 and Ta2O5 (as bulk insulators) are all used and termed ionic or mixed ionic–electronic electrolytes at room temperature16 17 18 19 20. The cation-migration-based electrochemical metallization memory (ECM) cells are a class of ReRAMs that use Ag or Cu as an active electrode and, for example, Pt, Ir or W as an inert counter electrode. A variety of oxide, chalcogenide and halide thin films have been suggested for the solid electrolyte2. Applying a positive voltage between the active and the counter electrode leads to an oxidation (dissolution) of the active electrode’s material and to a deposition of metal (Ag or Cu) at the counter electrode. Owing to the high electric field in the order of 108 V m−1, the metallic deposit propagates in a filamentary form and short circuits the cell, thus defining a low-resistive ON state. The filament can be dissolved by applying a voltage of opposite polarity to return the cell to a high-resistive OFF state. The anion-migration-based valence change cells (VCM) typically use a high work function electrode (for example, Pt and TiN), an oxygen-affine, lower work function electrode and a metal oxide as the electrolyte. These cells rely on the formation of oxygen-deficient, mixed ionic–electronic conducting filaments and the nanoionic modification of the potential barrier between the tip of the filament and the electrode to define the ON and OFF states. The ON and OFF states are then used to read the Boolean 1 and 0, respectively. The functionality of ReRAM cells and the kinetics of the filament formation/dissolution are the subject of intensive studies from both, academia and industry, leading to a development of empirical or semi-empirical models for the operating principles21, expanding to include concepts of multibit memories22 and memristive systems6 10 15. From the circuit theory’s point of view, ReRAM cells are regarded as memristive elements or real memristors23. They are defined by two simple equations, the state-dependent Ohm’s law, and the state equation The distinctive feature of memristive elements is a pinched characteristics at the origin of the I–V plane (that is, the I–V zero-crossing property)8 9, which is a direct result of these equations, and therefore represents the essential fingerprint23. A short excursus on the use of the term ‘memristor’ is given in Supplementary Note 1. Here, we report on non-equilibrium ON and OFF states in ReRAM cells determined by chemical potential gradients generating an electromotive force of up to a few hundred millivolts, violating the zero-crossing property. The emf may affect the retention time, and both influences and is influenced by the processes during formation and rupture of the metallic filament. We introduce additional equations to account for the emf, that is, for the non-zero-crossing hysteresis loop, thus qualitatively and quantitatively extending the memristor theory. The conclusions we draw with respect to a series of ECM systems apply to VCM-type ReRAM cells as well as other electrochemical and (neuro-)biological systems. Results Origins of electromotive force in ReRAM cells To clearly identify the individual influences of different chemical potential gradients, we studied ECM cells build from a series of materials, selected so as to ensure a transition of particular chemical and transport properties, that is, SiO2–GeS x , GeSe x –AgI (with x=2.2 and 2.3, respectively). In the as-deposited state, all materials are electronic insulators where the first and the last compounds of this series represent the two extremes. That is to say, SiO2 has a very small (but at the nanoscale not completely negligible) electronic conductivity and AgI is a stoichiometric ionic compound with considerable Ag+ ion conductivity. Neither is able to dissolve Ag chemically. In contrast, GeS x and GeSe x are able to dissolve Ag (or Cu) to different extents moving from insulators to mixed ionic–electronic electrolytes, thus displaying a transport property transition between SiO2 and AgI. We found three factors that contribute to the formation of the cell voltage V emf, as illustrated in Fig. 1: (1) the classical Nernst potential V N (Fig. 1a), (2) the diffusion potential V d (Fig. 1b) and (3) the Gibbs–Thomson potential due to the different surface free energies of macro- and nanoparticles V GT (Fig. 1c), for the case that a metallic nanofilament is formed without short-circuiting the electrodes. Note that in the case that a metallic nanofilament is formed, the measureable emf is zero owing to the short circuit (Fig. 1d). The Nernst voltage V N is given by the difference between the potential-determining half-cell reactions at each electrode/electrolyte interface: with V s′ and V s″ being the half-cell potentials at the active electrode/electrolyte (s′) and inert electrode/electrolyte (s″) interfaces, z the number of exchanged electrons and V 0 the difference in the standard potentials of these reactions. At the s′ interface, the potential-determining reaction is the same for all ECM cells: At the s″ interface, Red and Ox are general expressions in equation (3) for a species undergoing a redox process. The nature of this species is determined by the specific cell, as will be shown below. The non-equilibrium diffusion potential V d in ECM cells arises owing to excess concentrations of charged species, that is, Ag+ ions, electrons and/or OH− ions within the electrolyte film. These ions are introduced into the solid films either electrochemically during forming or SET/RESET cycles or chemically owing to a chemical dissolution of Ag. In both cases, the ratio of concentration changes across the electrolyte layer, with particularly pronounced changes in the vicinity of the electrodes. The electromotive force generated by this inhomogeneous charge distribution and mobilities is given by24: where k, T and e are the Boltzmann constant, the temperature and the elementary charge, respectively. and adenote the averaged transference number and the activity of the metal cations, respectively. The averaged transference number and the activity of negatively charged species (anions and electrons) are expressed by and a −. z i is the charge number of each species. The interfaces denoted by s′ and s″ correspond to the active electrode/electrolyte and the inert electrode/electrolyte interfaces, respectively. The potential difference generated in the neuron cells to transport electric signals has the same nature and originates in the diffusion (Donnan) potential24. The contribution of the third component V GT to the total cell voltage V emf is expected in the case of a non-contacting filament owing to the different surface free energies of the macrocrystalline active electrode and the nanosize filament in accordance with the Gibbs–Thomson equation: where γ is the surface free energy, r is the radius of the particle, V m is the molar volume and μ Ag is the chemical potential of Ag. V GT can be observed, for instance, for high-resistive (R>12.9 kΩ) ON states (no metallic short circuit) typically caused by filaments in ECM cells without galvanic contact to the active electrode (see Supplementary Fig. S1). It should be noted that the contribution of V GT could not be clearly distinguished from the contribution of V d in the particular systems we have studied. ReRAM cells based on initially insulating thin films In Fig. 2, we show the time evolution of the open cell voltage V cell and of the short circuit current I for Ag/SiO2/Pt cells. Figure 2a displays the simplified equivalent circuit of the nanobattery where the cell voltage is Here, t ion represents the total ionic transfer number, R i is the total resistance of the ionic current path and R e the total electronic resistance of the cell. Figure 2b shows V cell after establishing a defined concentration c ion of Ag+ and OH− ions for unformed cells. We utilized the fact that the amount of ions generated at the s′ and s″ interfaces during cell operation is adjusted by the pulse length and height or the sweep rate25. The decrease of V cell over a few hundred seconds is caused by the equilibration of the Ag+ and OH− concentration gradient, which has an initial maximum immediately at the metal/electrolyte interface. Furthermore, Fig. 2b (red line) shows V cell immediately after a RESET operation and switching to the open cell measuring conditions. In this case, the Ag+ concentration in the immediate vicinity to the Ag electrode is depleted (in contrast to unformed cells) on the one hand side owing to the reduction process (that is, the RESET process) and on the other hand side owing to slow supply/diffusion of ions from layers apart from the interface (diffusion limited). Thus, in this situation V cell increases with time but relaxes to the same value as the unformed cells after equilibration. The potential-determining reaction at the inert electrode, for example, Pt, depends on the material properties of the solid electrolyte and we distinguish two cases. The system Ag/SiO2/Pt represents the first case for which the solid SiO2 film contains no initial Ag+ and, hence, reaction (4) cannot be potential-determining. Instead, as shown in Tsuruoka et al.26, moisture is typically incorporated into this electrolyte during fabrication and hence protons provide the required counter reaction at the inert electrode, for example, and the Nernst voltage takes the form: The total emf of the ECM cell is a combination of equations (5) and (9) and is expressed by: with V 0=V 0+const. (see Supplementary Note 2). In Fig. 2c we evaluate V cell after initial relaxation. The slope of the line provides the pre-exponential term and we were thus able to determine the ionic transference number by using equation (10). The x-axes intercept corresponds to the condition ln(a ion)=0 and provides the value of V 0=0.17 V. Both values are used as parameters in the device modelling below. Figure 2d illustrates the short-circuit currents of a Ag/SiO2/Pt cell after RESET operation. In the inset, an extended discharge time of 20,000 s (~5.5 h) is shown. The measurements were carried out until they approached the resolution limit of our measuring system (~10 fA). The current is proportional to the electrode area (see Supplementary Fig. S4). An integration of current over time reveals a total charge of ~5 nC released during the discharge of the battery, which corresponds to a conversion of ~2 monolayers Ag. In comparison, the dielectric discharge of the cell capacitor is ~0.4 pC, clearly demonstrating that the discharge phenomenon is of an electrochemical nature. Please note that retention time of the cell (here: in the OFF state) is not immediately related to the relaxation time of the emf voltage. While the first has to be >10 years for universal non-volatile memories, the latter may be in the order of minutes to days as shown. In other words, the discharge of the ReRAM nanobattery does not lead to a loss of the OFF state (in a similar manner as the discharge of any other rechargeable battery does to end up in a filamentary short between the electrodes). Instead, typically only a shift of the particular R OFF value is observed upon the relaxation of the emf. Such a shift has been reported, for example, by Miao et al.27 and Choi et al.28 for both ECM and VCM systems, respectively. More detailed discussion is provided in Supplementary Note 3. ReRAM cells based on mixed conducting thin films In the second case condition for establishing a Nernst voltage according to equation (3), the electrolyte contains mobile Me z+ ions dissolved by electrochemical and/or chemical processes with almost homogeneous distribution, that is, (a Me z+ )s′~(a Me z+ )s″. Then the potential-determining half-cell reaction at the s″ interface will be the same as at s′, that is, reaction (4), and, because no chemical potential gradient of the charged species is assumed, equation (3) can be simplified to: where (a Me)s′ denotes the activity of the active metal Me at interface s′, that is, (a Me)s′=1, if a pure metal is used, and (a Me)s″ is the activity of Me at the inert counter electrode s″, that is, (a Me)s″ is typically very low. is the ion transference number averaged throughout the electrolyte thickness. As (a Me)s′ is fixed, the Nernst voltage is a function of (a Me)s″ alone. We succeeded in demonstrating a Nernst emf in accordance with equation (11) (Fig. 1a) for the systems Ag/Ag–GeS2.2/Pt and Ag/Ag–GeSe2.3/Pt, which contain a significant amount of chemically dissolved Ag+ ions (a solubility of up to 35 at% Ag has been reported29). We applied a positive external voltage of 1 V to the inert electrode, thus removing the residual Ag atoms from the s″ interface and lowering their activity. In the coupled cathodic reaction at the Ag interface s′, Ag dendrites are deposited as shown in the optical image in Fig. 3. Thus, apart from the different concentration of Ag+ ions at both interfaces, we generated a difference in the activities of Ag. The emf of the system was then monitored over time and correlated to the evolution of the dendrite morphology further recorded by optical images. The time evolution shown in Fig. 3 arises from the interplay of a Nernst potential and a diffusion potential. In the voltage-modified state (t 1 in Fig. 3), the contribution of the (positive) diffusion potential fades, leading to a dominating (negative) Nernst potential, which results to an emf value of ~−450 mV owing to the contribution of equation (11). The emf value corresponds to a difference in the Ag activities at both electrodes, (a Ag)s″≈2.5 × 10−8(a Ag)s′. As monolayer(s) of Ag begin to form at the Pt electrode (in parallel to the dissolution of the dendrites at the Ag electrode), the emf increases again simultaneously and relatively fast because of (a Ag)s′ ≈ (a Ag)s″. But owing to slower diffusion of the Ag+ ions a(Ag+)s′>a(Ag+)s″ the positive diffusion potential governs V cell. Thus, the control over V cell changes from a situation mainly determined by equation (11) to a situation dominated by equation (5). Emfs for different types of VCM and ECM memory cells In the course of our emf studies, we investigated a series of different ECM cells as well as selected VCM cells. We were able to prove that in all types of ReRAM cells tested significant (electro)chemical potential gradients are generated by the operation of the cells. Inevitably, these gradients give rise to an emf, and, hence, the cells show the characteristics of nanosize batteries with cell voltages in the range displayed in Fig. 4a (for details see Supplementary Table S1). After forming, the cell voltages of VCM cells are much lower than for ECM cells owing to the significantly higher electronic conductance in the OFF state originating from the stub of the highly conductive filament. Discussion The non-equilibrium states reflected by the emf voltage may affect both the retention and the device operation. Although OFF and intermediate states always experience emf voltages, for ON states with metallic contact the cell voltage is generally close to zero owing to the electronic conductance of the filament. Apart from that, chemical potential gradients, size effects and electrolyte non-stoichiometry can result in a chemical dissolution of the filament30. Thus, the retention of the ON state is strongly dependent on the filament characteristics and only indirectly dependent on the emf voltage. Being an intensive state property, V emf is independent on the cell size. However, the ionic and electronic resistances, R i and R e, and hence V cell may scale in a non-trivial manner depending on the ReRAM type, for example, the properties of the filament stub and the gap between the filament tip and the electrode in the case of VCM cells. This is further illustrated in Supplementary Figs S5,S6. Our results strongly suggest that in all bipolar ReRAM devices a nanobattery with dedicated emf voltages is present. This fact has far-reaching consequences on the application of the theory of memristive elements and on the device modelling in general. The emf is a state property corresponding to a non-zero-crossing I–V characteristic (compare Fig. 4b), and we correspondingly expanded the memristive equations to obtain an active device that can be considered an extended memristive element. In fact, any electrochemical system is active by nature, thus two-terminal nanoionic-resistive switches cannot be pure passive memristors. The same is true for neurobiological systems, which are also active, offering passive memristive elements only as internal elements31. However, in contrast to the emf of ReRAM cells, the biological resting membrane potential is modulated only by V d (eq. (5)) and not by V N (eq. (3)). By abandoning the zero-crossing property, a large number of dynamical devices can be added to the framework of memristive, memcapacitive and meminductive devices. For example, ferroelectric capacitors as well as ferromagnetic inductors can thus be regarded as extended memcapacitive and meminductive elements, respectively (Fig. 5a). By means of this expanded framework, a ReRAM cell is still a memristive device, as shown in Fig. 5a. The starting point for a practical memristive model of a ReRAM cell is the memristive model from Strukov et al.6, to which we add the nanobattery in parallel. The resulting model, which we term an extended memristive model, consists of a voltage source V emf and a nonlinear internal resistance R i—which together form the nanobattery—and a parallel resistor R el whose state variable x is controlled by the nanobattery (Fig. 5b and Supplementary Fig. S11). The state-dependent resistance of the electronic current path (R el) is a nonlinear function of the applied voltage, for example a tunnelling equation32. The electronic leakage current is accounted by a further parallel resistance R leak, which is state-independent and already present in the pristine device. Both contributions, R el and R leak, in parallel represent R e in Fig. 2a. The resistance of the ionic current path (R i) is defined by another nonlinear equation, determined by the Butler–Volmer equation and/or the high-field drift equation. The state-dependent Ohm’s law for the extended memristive device reads: and the state equation (compare Fig. 5b) reads: where d is the thickness of the active layer and K 1 is a constant while I ion offers a highly nonlinear voltage dependence, responsible for the pronounced nonlinearity of the switching kinetics. To fit the observed changes in the emf measurements (Fig. 2), a second-state variable, the ion concentration c ion, is required. In this case, the emf equation (10) with the experimentally determined V 0=0.17 V for Ag/SiO2/Pt (Fig. 2c) reads: Modelling details are described in the Supplementary Note 4 and corresponding simulation results are depicted in Fig. 5c. The inset clearly shows the non-zero-crossing I–V behaviour. Thus, ReRAM cells are non-zero-crossing devices, and therefore, the original memristor theory must be significantly extended in order to accommodate redox-based resistive switching systems. Besides modelling accuracy, the emf also has direct impact on future memory device development and corresponding circuitry. First, the stability of intermediate resistive states (for example, required in multilevel memory and neuromorphic applications) must be carefully considered in terms of the emf. Moreover, the effect of the emf within ultra-dense passive crossbar arrays may become relevant owing to possible device-to-device interactions. Second, the READ voltage of future generation memory elements will be in the range of 100 to 200 mV (ref. 33). This voltage is in the same order of magnitude as the emf voltage of some of the ReRAM types studied here. Third, the READ current for a ReRAM cell in a memory matrix will be in order of 100 nA and the length of the READ pulse 1014 Ω), a Keithley 617 electrometer (>200 TΩ) and a Keithley 2636A SourceMeter (>1014 Ω) for comparison. Throughout the paper, the right electrode in cells denoted M′/I/M′′ was used as reference electrode for all measurements. We used triaxial cables and electrostatic shielding to avoid RFI effects. The offset voltage measured across a 10 MΩ resistor has been proven to be within the device specification (below 1 μV accuracy). Details on the measurement resolution and accuracy are shown in Supplementary Fig. S12 and Supplementary Table S2, respectively. Author contributions I.V. conceived the idea, designed the experiments, interpreted the data and wrote the manuscript; E.L. performed the memristive simulations, co-wrote the manuscript and contributed to data interpretation; S.T. prepared the ECM cells, performed the measurements and contributed to data interpretation; S.S. performed measurements on VCM cells; J.v.d.H. prepared GeS x cells; F.L. prepared the WO x cells and performed measurements; R.W. initiated and supervised the research, and contributed to the concept of the study. All authors discussed the results and implications at all stages, and contributed to the improvement of the manuscript text. Additional information How to cite this article: Valov, I. et al. Nanobatteries in redox-based resistive switches require extension of memristor theory. Nat. Commun. 4:1771 doi: 10.1038/ncomms2784 (2013). Supplementary Material Supplementary Information Supplementary Figures S1-S12, Supplementary Tables S1-S2, Supplementary Notes 1-4 and Supplementary References.