**3. Memristive metal oxides**

#### **3.1. General considerations and parameters of interest**

Different switching mechanisms are observed in metal oxides—and even other mechanisms are envisioned for other materials, such as chalcogenides or polymers, which we will not refer to:

• the drift of oxide lattice defects, mostly oxygen vacancies, with consequent change in oxide valence, either localized on a restricted area called filament (**Figure 3**), or distributed over the whole area following an interface model (valence change mechanism, VCM);

• a change in stoichiometry induced by heating (thermochemical mechanism, TCM);

**2.2. Nanotubular films**

to few hours [27, 28].

**3. Memristive metal oxides**

**Figure 2.** Top and cross-section view of TiO<sup>2</sup>

electrolytes. Adapted with permission from Ref. [29].

to:

**3.1. General considerations and parameters of interest**

In the presence of aggressive species that are capable of localized dissolution of the growing oxide, nanotubular films can be grown, as shown in **Figure 2**. The peculiar morphology is associated with the simultaneous electrochemical growth of the oxide and its chemical dissolution operated by fluoride ions or, less frequently, other halogen ions. To achieve the formation of a nanotubular layer, a potentiostatic process is applied, where the chosen cell voltage—in the range 20–120 V—is maintained constant for various times, from few minutes

These nanostructures are usually developed on valve metals for applications in fields where an enhanced specific surface area is required, that is, in photocatalysis, photovoltaics, hydrogen production and sensing, where having the largest possible number of active sites of the oxide able to interact with the surrounding environment increases the material functional efficiency [27, 28]. Nevertheless, resistive switching capabilities were identified also in these

Different switching mechanisms are observed in metal oxides—and even other mechanisms are envisioned for other materials, such as chalcogenides or polymers, which we will not refer

nanotubes grown by anodic oxidation of the titanium substrate in organic

• the drift of oxide lattice defects, mostly oxygen vacancies, with consequent change in oxide valence, either localized on a restricted area called filament (**Figure 3**), or distributed over

the whole area following an interface model (valence change mechanism, VCM);

nanostructures, as will be discussed in detail in Section 3.3.

48 Advances in Memristor Neural Networks – Modeling and Applications

• the formation of conductive filaments by migration of ions from an active electrode metal and their deposition at the counterelectrode (electrochemical metallization mechanism, ECM, also called conductive bridge, CB) under the applied electrical field [4].

The most easily occurring switching mechanisms common to all metal oxides are VCM and ECM. Yet, mixed filamentary switching mechanisms, both by electrode ions migration and metal oxide reduction due to vacancies migrations, have been observed in the literature, as shown in **Figure 4**, where the two filament formation mechanisms are described [31]. Given the wide variety and complexity of switching mechanisms observed, we suggest to refer to specific reviews for a detailed explanation of the physics behind specific resistive switching mechanisms in memristive oxides [4, 32–34].

As already mentioned in the Introduction section, resistive switching implies the modification of the metal oxide of interest from a high resistance state (HRS) to a low resistance one (LRS), and vice versa (**Figure 4**). Conventionally, a set event is described as the switch from HRS to LRS, while reset, that is, restoring the initial high resistance of the oxide, causes the passage from LRS to HRS. Both events are driven by an electrical input, and more specifically by the application of a voltage. If set and reset require the application of reverse polarity, then the switching is defined bipolar, while in unipolar switching, the direction of change in resistance state depends on voltage amplitude, not on its polarity. Yet, materials usually do not show immediately a switching behavior: a first stage called electroforming is required, operated at higher voltages, which triggers the material switching ability, making subsequent cycles easier and occurring at lower voltages [35, 36]. Indeed, reset operations only allow to recover and redistribute defects (vacancies, electrode metal ions) at the oxide-electrode interface, while a conductive path remains pre-set in the inner part of the oxide [5, 37].

**Figure 3.** Schematic diagram for the mechanism of resistive switching in Pt/ZnO/Pt devices. (a) the migration of oxygen vacancies toward the cathode (oxygen ions (O2−) toward the anode) and rearrangement of Zn-dominated ZnO1−*<sup>x</sup>* leads to the formation of a conductive filament (b). (c) the rupture of the filament by joule heating. Owing to the migration of oxygen ions, the ReRAM resets back to the off state. Reprinted with permission from Ref. [30].

**3.2. Dependence of switching behavior on metal oxide characteristics**

conduction path—which would then occupy the whole component area [34].

zirconium. Oxides are often indicated as TiO2−x, NbO<sup>x</sup>

chiometry into account.

for TiO<sup>2</sup>

to produce multistate devices.

Adapted with permission from Ref. [30].

As anticipated, this section compares the switching behavior of metal oxides that hold an interest in the frame of anodic oxidation, that is, the discussion is focused on oxides of metals that are liable to anodizing. These include titanium, niobium, tantalum, hafnium, and

On these metal oxides, either filamentary switching or interfacial valence change has been observed, depending on oxide composition, production method, and metal electrode composition. Interestingly, a unified model was proposed: to be integrated in CMOS technology, feature size will be decreased more and more, until reaching the actual size of a filamentary

When the formation of oxygen vacancies (or metal precipitates) filaments is involved, the localized current percolation path preferentially locates at grain boundaries or lattice inhomogeneities, as revealed by C-AFM and TEM measurements reported in several works (see for instance [47–49]) and represented in **Figure 5**. Moreover, multiple resistance states can be obtained and explained by considering two directional movements of vacancies: from one electrode to the other, crossing the whole oxide thickness, to generate the filament; and a lateral one, to increase filament size or create new filaments [50, 51]. From a material point of view, multiple states can be seen as a gradual increase in non-stoichiometry. As an example,

, the memristive behavior is generally ascribed to the movement of vacancies that

gradually create an oxygen depleted layer with composition TiO2−x, which gains conductivity for x > 1.5 [52], hence the higher the quantity of vacancies formed, the wider the area that reaches low resistance conditions, which allows a gradual change in LRS that can be exploited

Yet, grain boundaries and other structural inhomogeneities related to crystalline oxide structures may strongly affect actual device performances: in fact, grain boundaries not only make switching easier, as abovementioned, but also cause a decrease in Roff/Ron ratio, plus they alter performance evaluation with respect to single crystal devices of envisioned nanometric size. Hence, amorphous layers are often preferred, given their enhanced reproducibility and

**Figure 5.** A series of in situ TEM images clipped from the video. (a) At the start of recording, the ZnO was in the initial state. (b) When voltage was applied, the contrast of ZnO enhanced near both electrodes. (c) A conical-shaped filament generated near the top electrode. The white dashed line highlights the filament. The specimen was still in the highresistance state. (d) The columnar filament passed through the ZnO film connecting the top and bottom electrodes.

, TaO<sup>x</sup>

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**Figure 4.** Four different operation modes for cu/ZnO/Pt in which the resistive switching originates from the formation and rupture/annihilation of (a) Cu, (b) Zn filaments. The insets schematically show filament evolution processes. Adapted with permission from Ref. [31].

The voltage applied in the electroforming step is larger than that needed in set/reset operations (**Figure 4**): the electroforming voltage in many cases is as high as a few volts and linearly dependent on oxide thickness [38–40]. Efforts are being made to produce forming-free devices; unfortunately, this is most often obtained by decreasing film thickness, which at the same time increases its defectiveness, reducing device reliability. The voltages that are required to operate the device are relevant as well: the key parameters are the write voltage, Vwr, and the read voltage, Vrd, which determine the entity of the signals required during the whole device operation. The write voltage is less dependent—in some cases even independent—on oxide thickness, as it only needs to recreate the conductive region at the oxide-electrode interface, and should be in the order or few hundred mV to allow good device efficiency and low energy consumption, while the read voltage is usually one order of magnitude lower to avoid possible undesired changes of resistance state during read operations.

When describing and comparing materials with memristive capabilities, another fundamental parameter is the Roff/Ron ratio, that is, the ratio between material resistance in the HRS vs. LRS, which gives an indication on the efficiency and robustness of switching. Indeed, although ratios of few units are theoretically sufficient to operate a device, a Roff/Ron ratio higher than 10 is generally recommended, to avoid uncertainties in read operations and improve reliability [4].

Another important touchstone parameter is endurance, that is, the number of cycles applicable to the material without loss of switch and no (or better, limited) decay of Roff/Ron ratio.

One last characteristic can play a major role, especially in neuromorphic computing, that is, the possibility to achieve multilevel storage, which makes the difference between binary and analog switching. This can be achieved either by multiple resistance states [41–44] or by encoding information not only in the conductive filament size, which rules resistivity, but also in its orientation through complementary switching [45, 46]. These aspects will be addressed in Paragraph 4.

### **3.2. Dependence of switching behavior on metal oxide characteristics**

The voltage applied in the electroforming step is larger than that needed in set/reset operations (**Figure 4**): the electroforming voltage in many cases is as high as a few volts and linearly dependent on oxide thickness [38–40]. Efforts are being made to produce forming-free devices; unfortunately, this is most often obtained by decreasing film thickness, which at the same time increases its defectiveness, reducing device reliability. The voltages that are required to operate the device are relevant as well: the key parameters are the write voltage, Vwr, and the read voltage, Vrd, which determine the entity of the signals required during the whole device operation. The write voltage is less dependent—in some cases even independent—on oxide thickness, as it only needs to recreate the conductive region at the oxide-electrode interface, and should be in the order or few hundred mV to allow good device efficiency and low energy consumption, while the read voltage is usually one order of magnitude lower to avoid

**Figure 4.** Four different operation modes for cu/ZnO/Pt in which the resistive switching originates from the formation and rupture/annihilation of (a) Cu, (b) Zn filaments. The insets schematically show filament evolution processes.

When describing and comparing materials with memristive capabilities, another fundamental parameter is the Roff/Ron ratio, that is, the ratio between material resistance in the HRS vs. LRS, which gives an indication on the efficiency and robustness of switching. Indeed, although ratios of few units are theoretically sufficient to operate a device, a Roff/Ron ratio higher than 10 is generally recommended, to avoid uncertainties in read operations and

Another important touchstone parameter is endurance, that is, the number of cycles applicable to the material without loss of switch and no (or better, limited) decay of Roff/Ron ratio. One last characteristic can play a major role, especially in neuromorphic computing, that is, the possibility to achieve multilevel storage, which makes the difference between binary and analog switching. This can be achieved either by multiple resistance states [41–44] or by encoding information not only in the conductive filament size, which rules resistivity, but also in its orientation through complementary switching [45, 46]. These aspects will be addressed

possible undesired changes of resistance state during read operations.

improve reliability [4].

Adapted with permission from Ref. [31].

50 Advances in Memristor Neural Networks – Modeling and Applications

in Paragraph 4.

As anticipated, this section compares the switching behavior of metal oxides that hold an interest in the frame of anodic oxidation, that is, the discussion is focused on oxides of metals that are liable to anodizing. These include titanium, niobium, tantalum, hafnium, and zirconium. Oxides are often indicated as TiO2−x, NbO<sup>x</sup> , TaO<sup>x</sup> , HfO2−x, ZrO2−x to take non-stoichiometry into account.

On these metal oxides, either filamentary switching or interfacial valence change has been observed, depending on oxide composition, production method, and metal electrode composition. Interestingly, a unified model was proposed: to be integrated in CMOS technology, feature size will be decreased more and more, until reaching the actual size of a filamentary conduction path—which would then occupy the whole component area [34].

When the formation of oxygen vacancies (or metal precipitates) filaments is involved, the localized current percolation path preferentially locates at grain boundaries or lattice inhomogeneities, as revealed by C-AFM and TEM measurements reported in several works (see for instance [47–49]) and represented in **Figure 5**. Moreover, multiple resistance states can be obtained and explained by considering two directional movements of vacancies: from one electrode to the other, crossing the whole oxide thickness, to generate the filament; and a lateral one, to increase filament size or create new filaments [50, 51]. From a material point of view, multiple states can be seen as a gradual increase in non-stoichiometry. As an example, for TiO<sup>2</sup> , the memristive behavior is generally ascribed to the movement of vacancies that gradually create an oxygen depleted layer with composition TiO2−x, which gains conductivity for x > 1.5 [52], hence the higher the quantity of vacancies formed, the wider the area that reaches low resistance conditions, which allows a gradual change in LRS that can be exploited to produce multistate devices.

Yet, grain boundaries and other structural inhomogeneities related to crystalline oxide structures may strongly affect actual device performances: in fact, grain boundaries not only make switching easier, as abovementioned, but also cause a decrease in Roff/Ron ratio, plus they alter performance evaluation with respect to single crystal devices of envisioned nanometric size. Hence, amorphous layers are often preferred, given their enhanced reproducibility and

**Figure 5.** A series of in situ TEM images clipped from the video. (a) At the start of recording, the ZnO was in the initial state. (b) When voltage was applied, the contrast of ZnO enhanced near both electrodes. (c) A conical-shaped filament generated near the top electrode. The white dashed line highlights the filament. The specimen was still in the highresistance state. (d) The columnar filament passed through the ZnO film connecting the top and bottom electrodes. Adapted with permission from Ref. [30].

better long term stability with respect to polycrystalline ones: these properties, ascribed to the material structural homogeneity, nicely match with an easier production with respect to single crystal oxides [53–55].

low power equipment (voltage scale 0–30 V, current scale 0–100 mA) in a neutral solution of non-aggressive salts [17]. Another advantage is the possibility to use the same metal substrate as a back-end material, that is, one of the two electrodes is intrinsically integrated in the component. Currently, the biggest drawback that limits applications of anodic oxidation is the minimum device size: the technique is generally employed on full surfaces, thus not allowing the growth of space-confined nanometric or sub-micrometric pads, and the only method to reduce the size of the anodized spot is to apply insulating masks that avoid electronic contact

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Keeping in mind these important general features, we here summarize current research on

The first indication of anodic oxides presenting memristive behavior was recorded on titanium oxides grown in a water-glycerol-based ammonium fluoride solution at 30 V. No clear morphological characterization of the oxide is made; yet, although the presence of fluorides may indicate typical conditions of nanotubes production, the short anodizing times applied allow to presume the growth of a compact oxide, some tens of nanometers thick. Interestingly, annealing has a detrimental effect on the memristive behavior. This is ascribed to the exceeding formation of oxygen vacancies that creates ohmic contacts; in addition, annealing is known to induce crystallization in anodic oxides which—in the anodizing conditions considered—would show amorphous structure in the as-prepared state [82]. This may have as well a role in the degradation of resistive switching. Other compact oxide films showing memristive behavior were then grown on titanium as well as on niobium and tantalum: anodizing in diluted phosphoric acid at 25 V, corresponding to an oxide thickness of approximately 60 nm,

The cited works all based their considerations on macroscale samples. A nanoscale characterization of anodic titanium oxides was performed by means of conductive atomic force microscopy (C-AFM), which allowed to assess the electrical properties of nanometer-size spots on the oxide surface: results indicated that oxide properties are far from being homogeneous, with resistive switching spots embedded in a nonconductive matrix and located mostly at grain boundaries [49]. More recently, efforts were made in the direction of producing real devices and testing the material at the microscale. Anodizing was performed on tantalum [81] and on titanium [84] metallic films deposited on glass, in borate buffer solution or in diluted phosphoric acid, respectively, at cell voltages of 5–20 V. Micrometer-size conductive metal pads (either Pt or Cu) were then deposited by lithography, allowing better characterization of

In all abovementioned cases, the anodic oxides showed parameters compatible with requirements identified for resistive switching materials: high Roff/Ron ratio (> 10, with best values in the order of 80), set/reset values lower than 1 V and possibility to obtain multilevel switching [81]. Moreover, in several works, the oxides produced were electroforming-free: this can be ascribed to the anodic oxidation process itself, which is known to generate non-stoichiometric oxides, therefore the content of oxygen vacancies natively present in the oxide is already suf-

was found to allow the achievement of the best switching behavior [83].

the devices, which also included endurance evaluation.

ficient to produce the switching [49, 83, 84].

of the metal with the electrolyte.

anodic oxides showing memristive characteristics.

More recently, low-cost processes have been successfully employed to produce resistive switching oxides, including solution processing—sol-gel, hydrothermal synthesis—and electrochemical techniques, both electrodeposition and anodic oxidation. The former set of techniques has the advantage of producing oxides free of substrate, hence they can be deposited on any substrate, including flexible ones [56–58]. Production of the oxides generally involves mild temperatures and ambient pressures in case of sol-gel [57, 59–62], or the use of a pressurized vessel, specific for hydrothermal treatments [63–65], which in all cases represent low-cost alternatives to low pressure, high-temperature chemical or physical deposition processes.

On the other hand, the absence of a substrate implies an immobilization step—which can be performed by drop-coating, inkjet printing, and other methods—that may introduce a further level of inhomogeneity in the final device properties. Indeed, oxide particles need to be dispersed in a proper solvent, which must then be completely removed: defects such as porosities due to solvent removal, or even residual solvent may then arise. To improve homogeneity, often multiple deposition steps are performed, which increases overall film thickness and consequently electroforming voltage [59], while oxygen or argon plasma etching can be employed to introduce oxygen vacancies in the as-deposited materials, hence reducing electroforming voltages or even eliminating the need for this step [60, 66–68]. The possibility of applying multiple coating steps also opens the way to sol-gel processed double-layer structures [69], which brings potential benefits that span from increased endurance to reduced power consumption [44, 70–72]. For instance, in TiO<sup>2</sup> -based memristors oxygen vacancies migration can lead to oxygen gas evolution at the anode, which irreversibly compromises the oxide stoichiometry: the presence of a blocking layer can act as sink of oxygen ions and limit currents involved, avoiding oxide breakdown [3, 73, 74].

Concerning the switching type, both unipolar and bipolar switching can be observed within the same material [75, 76]: which of the two is operating can be associated at a first approximation with different reset processes, being thermal dissolution the prevailing one for unipolar behavior, and ionic migration responsible for bipolar switching ([5] and references therein: [77–79]).

#### **3.3. Anodic oxides showing memristive behavior**

The choice of anodic oxidation to produce memristive elements is driven by a number of benefits over current technologies, first of all its low cost, non-vacuum and low-temperature characteristics. Moreover, it allows to produce amorphous oxides with nonstoichiometric composition [14, 49, 80], that is, already containing a non-negligible amount of oxygen vacancies, and characterized by higher density compared with sputtered films, where residual porosity is intrinsic to the production technique [81]. Eventually, anodizing allows fast oxide growth: to obtain a metal oxide few tens of nanometers thick, the general duration of an anodizing process is in the order of few seconds, and it can be performed with a relatively low power equipment (voltage scale 0–30 V, current scale 0–100 mA) in a neutral solution of non-aggressive salts [17]. Another advantage is the possibility to use the same metal substrate as a back-end material, that is, one of the two electrodes is intrinsically integrated in the component. Currently, the biggest drawback that limits applications of anodic oxidation is the minimum device size: the technique is generally employed on full surfaces, thus not allowing the growth of space-confined nanometric or sub-micrometric pads, and the only method to reduce the size of the anodized spot is to apply insulating masks that avoid electronic contact of the metal with the electrolyte.

better long term stability with respect to polycrystalline ones: these properties, ascribed to the material structural homogeneity, nicely match with an easier production with respect to

More recently, low-cost processes have been successfully employed to produce resistive switching oxides, including solution processing—sol-gel, hydrothermal synthesis—and electrochemical techniques, both electrodeposition and anodic oxidation. The former set of techniques has the advantage of producing oxides free of substrate, hence they can be deposited on any substrate, including flexible ones [56–58]. Production of the oxides generally involves mild temperatures and ambient pressures in case of sol-gel [57, 59–62], or the use of a pressurized vessel, specific for hydrothermal treatments [63–65], which in all cases represent low-cost alternatives to low pressure, high-temperature chemical or physical deposition processes.

On the other hand, the absence of a substrate implies an immobilization step—which can be performed by drop-coating, inkjet printing, and other methods—that may introduce a further level of inhomogeneity in the final device properties. Indeed, oxide particles need to be dispersed in a proper solvent, which must then be completely removed: defects such as porosities due to solvent removal, or even residual solvent may then arise. To improve homogeneity, often multiple deposition steps are performed, which increases overall film thickness and consequently electroforming voltage [59], while oxygen or argon plasma etching can be employed to introduce oxygen vacancies in the as-deposited materials, hence reducing electroforming voltages or even eliminating the need for this step [60, 66–68]. The possibility of applying multiple coating steps also opens the way to sol-gel processed double-layer structures [69], which brings potential benefits that span from increased endurance to reduced

migration can lead to oxygen gas evolution at the anode, which irreversibly compromises the oxide stoichiometry: the presence of a blocking layer can act as sink of oxygen ions and limit

Concerning the switching type, both unipolar and bipolar switching can be observed within the same material [75, 76]: which of the two is operating can be associated at a first approximation with different reset processes, being thermal dissolution the prevailing one for unipolar behavior, and ionic migration responsible for bipolar switching ([5] and references therein:

The choice of anodic oxidation to produce memristive elements is driven by a number of benefits over current technologies, first of all its low cost, non-vacuum and low-temperature characteristics. Moreover, it allows to produce amorphous oxides with nonstoichiometric composition [14, 49, 80], that is, already containing a non-negligible amount of oxygen vacancies, and characterized by higher density compared with sputtered films, where residual porosity is intrinsic to the production technique [81]. Eventually, anodizing allows fast oxide growth: to obtain a metal oxide few tens of nanometers thick, the general duration of an anodizing process is in the order of few seconds, and it can be performed with a relatively


power consumption [44, 70–72]. For instance, in TiO<sup>2</sup>

currents involved, avoiding oxide breakdown [3, 73, 74].

**3.3. Anodic oxides showing memristive behavior**

[77–79]).

single crystal oxides [53–55].

52 Advances in Memristor Neural Networks – Modeling and Applications

Keeping in mind these important general features, we here summarize current research on anodic oxides showing memristive characteristics.

The first indication of anodic oxides presenting memristive behavior was recorded on titanium oxides grown in a water-glycerol-based ammonium fluoride solution at 30 V. No clear morphological characterization of the oxide is made; yet, although the presence of fluorides may indicate typical conditions of nanotubes production, the short anodizing times applied allow to presume the growth of a compact oxide, some tens of nanometers thick. Interestingly, annealing has a detrimental effect on the memristive behavior. This is ascribed to the exceeding formation of oxygen vacancies that creates ohmic contacts; in addition, annealing is known to induce crystallization in anodic oxides which—in the anodizing conditions considered—would show amorphous structure in the as-prepared state [82]. This may have as well a role in the degradation of resistive switching. Other compact oxide films showing memristive behavior were then grown on titanium as well as on niobium and tantalum: anodizing in diluted phosphoric acid at 25 V, corresponding to an oxide thickness of approximately 60 nm, was found to allow the achievement of the best switching behavior [83].

The cited works all based their considerations on macroscale samples. A nanoscale characterization of anodic titanium oxides was performed by means of conductive atomic force microscopy (C-AFM), which allowed to assess the electrical properties of nanometer-size spots on the oxide surface: results indicated that oxide properties are far from being homogeneous, with resistive switching spots embedded in a nonconductive matrix and located mostly at grain boundaries [49]. More recently, efforts were made in the direction of producing real devices and testing the material at the microscale. Anodizing was performed on tantalum [81] and on titanium [84] metallic films deposited on glass, in borate buffer solution or in diluted phosphoric acid, respectively, at cell voltages of 5–20 V. Micrometer-size conductive metal pads (either Pt or Cu) were then deposited by lithography, allowing better characterization of the devices, which also included endurance evaluation.

In all abovementioned cases, the anodic oxides showed parameters compatible with requirements identified for resistive switching materials: high Roff/Ron ratio (> 10, with best values in the order of 80), set/reset values lower than 1 V and possibility to obtain multilevel switching [81]. Moreover, in several works, the oxides produced were electroforming-free: this can be ascribed to the anodic oxidation process itself, which is known to generate non-stoichiometric oxides, therefore the content of oxygen vacancies natively present in the oxide is already sufficient to produce the switching [49, 83, 84].

In this respect, metal electrode ions injection has also been proposed as a possible mechanism for the onset of switching, which would indicate the establishing of a CB mechanism [81]. Nevertheless, proofs of the actual onset of a VCM are provided through the observation of switching with C-AFM measurements, where no top electrode is present: analyses have been conducted both on the top surface of the anodic oxide [49], and on a lateral device, where no electrode metals are available [85].

and analyzes the adaptation of algorithms with respect to device variation and scalability. Since the discovery of memristive behavior at the nanoscale at Hewlett Packard laboratories in 2008, the scientific community has devoted a large deal of efforts to derive suitable models that capture the nonlinear dynamics of memristors. Pickett's model is a reference model that is well suited for describing the physical mechanisms at the origin of memristor dynamics. Simplified versions aiming at fitting the behavior of Pickett's model are the TEAM and V-TEAM models, Biolek's model and the boundary condition model for a comprehensive review [95]. It is worth noting that such models are not oriented toward nonlinear circuit synthesis. In order to effectively analyze the dynamic behavior of memristors, and also in view of their simulation and emulation, it is fundamental to develop circuit memristor models, that is, models obtained by interconnecting basic nonlinear blocks. This will be pursue along the lines of the general method for device modeling in [96] and exploiting recent techniques for

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There is an increasing interest in the implementation of oscillators using nanoscale devices as memristors. As remarked in [97], a source of controllable chaotic behavior that can be implemented by a single scalable electronic device and incorporated into a neural-inspired circuit may be an essential component of future computational systems. In this framework, the memristor is required to display a quasi-static voltage-current characteristic with a negative differential resistance (NDR). Various classes of relaxation oscillators displaying a tunable range of periodic and chaotic self-oscillations have been implemented during recent years and their importance in neuromorphic applications, such as pattern recognition and signal processing tasks in real time, have been demonstrated. They can also be used as core devices with a rich variety of nonlinear dynamics within the framework of reservoir computing architectures. Work so far has been mainly based on experimental and phenomenological observations of oscillations and complex phenomena, while a circuit model and a clear analytic understanding of the underlying nonlinear dynamics and bifurcations is basically missing. Recently, a new method, named Flux-Charge Analysis Method (FCAM), has been developed to effectively analyze a wide class of nonlinear circuits containing ideal memristors in the flux-charge domain [98]. FCAM permits to bring back the dynamic analysis to that of a lower-order circuit, with respect to that in the standard voltage-current domain, using flux and charge as state variables. This enables to obtain a clear picture of the dynamical behavior displayed by memristor circuits. In particular, some peculiar aspects, such as the presence of invariant manifolds and the coexistence of different dynamics for the same set of (fixed) circuit parameters, are singled out. Also, it is possible to assess the presence of a new interesting phenomenon of bifurcations which emerge without changing the system parameters, namely, bifurcations due to changing the initial conditions for the state variables for a fixed set of circuit parameters (BWP) [99]. Using FCAM, the dynamics of classes of oscillators and chaotic circuits with ideal memristors have been deeply analyzed assessing the occurrence of Hopf and period-doubling BWPs and quite rich complex dynamics. In addition, it has been shown that FCAM can be combined with techniques, such as the harmonic balance method citare, to effectively analyze and control such BWPs. Moreover, by suitably exploiting BWPs, it turned out that different chaotic dynamics in a class of Chua's oscillators can be programmed by means of suitable current or voltage pulses [100]. Synchronization aspects in

the identification of switching and PWA (piece-wise-affine) systems.

arrays of coupled oscillators have been analyzed as well [101].

Memristive nanotubes were also produced on titanium [86–88]. In these cases, either longer anodizing times (hours) and/or higher voltages (up to 120 V) are required, and thicker oxides, some hundreds of nanometers thick, are achieved. Yet, the limited adherence and mechanical stability of these oxides compared with compact ones, and the higher thickness introduced, make them less appealing for real applications.
