Figure 1 shows a developed device technology for bi-layer oxide memristive devices. In Figure 1A cross-sections of the fabricated TiN(50 nm)/TiO _x (30 nm)/HfO _x (2–8 nm)/Au(50 nm) bi-layer memristive devices with Al(300 nm) contact pads are sketched. They are fabricated on a 4-inch oxidized silicon wafer (1 μm of thermal SiO _x ) in the full device technology. This technology is overviewed in Figure 1B and contains around 40,000 single devices, including test structures for the device development (see microscope images in Figure 1B). This allows the investigation of various device parameters, such as the active device area (six different area sizes are realized, as shown in Figure 1B), the thickness of the active HfO _x layer, and the material compositions over the wafer for a targeted development of memristive devices. For a variation of the latter parameter, two different sputtering methods for the HfO _x layer were used. In particular, a variation in oxygen vacancies is required to achieve a desired resistive switching process in this class of memristive devices (He et al., 2017). Here, using a sputtering system equipped with three confocal source targets, two methods are employed for the deposition of the HfO _x which are referred to as method 1 and 2. During the deposition, the substrate is rotated to obtain a uniform film thickness, while a wedge film is formed without a rotation. The wedge is formed only along one direction. For method 1 the HfO _x layer was deposited on the wafer under optimal conditions, i.e., rotation of the substrate within a confocal sputtering arrangement. For method 2 the wafer was not rotated during the sputtering of the HfO _x layer. This leads to a reduced layer quality, but also a wedge over the wafer as shown in Figure 1A (further details are discussed below). As a result, we obtained two distinctive bi-layer oxide memristive device structures, which are referred to in the following as device R and device F.

In more detail, the TiO _x /HfO _x bi-layer was deposited on an inertial reactive sputtered TiN bottom-electrode via DC magnetron sputtering, where O₂/Ar of reactive gas was adjusted with the ratio of 10/40 and 10/29 for the TiO _x and HfO _x film, respectively. After the TiO _x was sputtered, the thickness of the HfO _x was controlled using the two discussed sputtering methods 1 (for device type R) and 2 (for device type F): as seen in Figure 1A a wedge layer with a variation of the HfO _x thickness from 2 to 8 nm was obtained for device F, where devices were fabricated along the axis (x-direction) perpendicular to the axis of the wedge (y-direction). Device R has a 2 nm uniform HfO _x layer. The layer deposition was finalized with an Au top-electrode layer. Thereafter, the material stack was patterned using photolithography and reactive ion etching for device R, while a lift-off in Dimethylsulfoxide (DMSO) was used for device F. The lift-off process was carried out due to the thickness variation of HfO _x in device F. Here, the investigation of the switching behaviors was preceded after we confirmed that the two patterning methods scarcely affected electrical characteristics. All devices were insulated with SiO₂ layers from the ambient air to avoid the influence of moisture in switching behaviors (Tsuruoka et al., 2012; Zhou et al., 2018; Zhou et al., 2020) (Figure 1A), and Al contact pads were deposited by e-beam evaporation.

The development of the memristive devices was supported by a material characterization accompanying the manufacturing process. The thickness and the composition of the layers were characterized by ellipsometry measurements (SE500, Sentech) and surface profile measurements (Dektak 150, Veeco).

For a detailed material investigation, unstructured HfO _x films were deposited on silicon substrates. Therefore, the two described sputtering methods 1 and 2 were employed to deposit 37 nm thick HfO _x films. On those films ellipsometry measurements were performed at 632.8 nm at 70^° of incidence. As the results, refractive indices of n = 1.9889 and n = 2.0285 were measured for, respectively, the uniform (method 1) and the wedge- (method 2) deposited HfO _x films. Thus, in agreement with previous investigations (Martínez et al., 2007) the film can be assumed to have amorphous crystallinities. However, the obtained n value from the uniform deposited film was higher than n of the wedge deposited HfO₂ film, which can be attributed to a reduced packing density (Gao et al., 2016).

Furthermore, X-ray photoelectron spectroscopy (XPS) measurement was utilized to study the quantitative atomic ratio O/Hf in sputtered HfO _x layers. The XPS analysis were carried out using monochromatic Al_K-alpha radiation (excitation energy hν = 1,486.68 eV) under charge neutralization using a SPECS SAGE HR 150 XPS system equipped with a 1D delayline detector and a Phoibos 150 analyzer. The calibration of the energy scale was ensured by reference measurements on a polycrystalline silver sample. Before the measurements, HfO _x was sputtered on Si/SiO₂ wafers for 900 s using the two different sputtering method 1 and 2. As a result, a ratio of O/Hf of 1.80 / 1 was observed for deposition method 2, while a ratio of 1.98 / 1 was recorded for samples sputtered via method 1 (see Supplementary Data S1). The sputtering method 1 provides a stoichiometry close to HfO₂, while the obtained stoichiometry via method 2 leads to optimal condition for the forming of oxygen vacancy filaments (McKenna, 2014). Thus, we can conclude that sputtering method 1 leads to a reduced number of oxygen vacancies than the sputtering method 2. Hence, a HfO _x layer with a higher density of oxygen vacancies can be assumed for device type F in respect to device type R.

Current-voltage measurements (I-V curves) and voltage pulse measurements were carried out to characterize the electrical properties of TiO _x /HfO _x bi-layer memrsitive devices using a source measurement unit (Keysight b2901a). Therefore, a voltage is applied to the top-electrode of the device (bottom-electrode were grounded), while the current has been measured simultaneously. Furthermore, current compliance was imposed during the measurement to prevent the device from damage. The used current compliance was I C C = 10   μ A , I C C = − 5   m A for R and F devices, respectively. For pulse measurements the device resistance was measured at, respectively, 1 and 0.1 V for R and F devices. The switching voltage to set and reset the device resistances was 3 V/−2 V for device type R and −1 V/1.5 V for F type devices. For both devices a pulse duration of ∼10 ms was used. For a statistical evaluation of the electrical properties, median values were extracted taking into account the variability in cycle to cycle (C2C), and device to device (D2D). In the C2C investigation 10 times of DC voltage sweep cycles in one device were carried out. For reliable statistics, automated measurements of more than 10 memristive devices in each device parameter were performed, which means a total of 180 devices measured for three different thicknesses and six different area sizes. Both C2C and D2D statistics were investigated in DC conditions.

For a profound understanding of the resistive switching mechanisms and a targeted development of the devices a physics based device model was developed. In Figure 2A a sketch of this device model is shown: the model consists of two RC elements representing, respectively, the HfO _x and the TiO _x layer. The metal-semiconductor contact between the HfO _x layer and the Au electrode is considered by a Schottky diode ( D S c h o t t k y ). Thus, an external applied voltage (V) is divided into the local voltage drops at the Schottky diode ( V S c h o t t k y ), over the HfO _x layer ( V H f O x ) and the TiO _x layer ( V T i O x ) according to V = V S c h o t t k y + V H f O x + V T i O x (1) An important difference between here investigated two types of memristive devices is sketched in Figure 2B. While for the type F device a filament of oxygen vacancies is formed under the external voltage application, the type R device does not form any filaments. Essential for this is the concentration of oxygen ions and vacancies in the active HfO _x layer (Dirkmann et al., 2018). For the filamentary device F, we assumed that a number of oxygen vacancies are the mobile ions that vary between a minimum and a maximum concentration, denoted as N m i n and N m a x , respectively. In detail, for the filamentary device F we estimated N m i n = 4 ⋅ 10 24   m − 3 and N m a x = 2 ⋅ 10 27   m − 3 in accordance with the previous work (Menzel et al., 2011; Dirkmann et al., 2018). On the other hand, for the device type R we assumed a significantly lower concentration of oxygen vacancies due to a better layer quality. Here, the mobile species are oxygen ions where a concentration of N = 10 23   m − 3 was used which is in qualitative agreement with (Dirkmann et al., 2016).

The concentration of the oxygen ions and vacancies has a particular effect on the active area used for resistance switching (cf. Figure 2B). Thus, for the filamentary device F only the filament area is relevant for the switching effect, i.e., A = A f i l (see upper drawing in Figure 2B). For the type R device the whole device area is involved in the switching mechanism, i.e., A = A d e v i c e (see lower drawing in Figure 2B). Both, the active area and the oxygen ion/vacancy concentration, are relevant for the resistance of the HfO _x layer: R H f O x = d H f O x e ⋅ z v 0 ⋅ A ⋅ μ n ⋅ N (2) where μ n is the electron mobility, z v 0 is the ion charge number, and e is the elementary charge (Hardtdegen et al., 2018).

The layer thickness of TiO _x is significantly larger than that of HfO _x . Therefore, a much lower local electrical field strength is assumed ( E = V l a y e r / d l a y e r ) . Thus, under an external bias voltage oxygen ion drift is suppressed within the TiO _x layer and the resistance R T i O x of the TiO _x layer can be assumed to be constant. Nevertheless, the TiO _x layer plays a crucial role in the functionality of the bi-layer oxide structure: (i) it serves as a reservoir for oxygen vacancies in filament devices, and (ii) it stabilizes the switching process for both types of devices (Stathopoulos et al., 2017; Hardtdegen et al., 2018; Mikhaylov et al., 2020). For the latter point, the electronic contribution of the TiO _x layer is particularly important and has to be captured in the model. In general, the electronic charge transport through metal oxide layers can be determined by various transport mechanisms. It has been shown that a good approximation for the electron current is given by the following voltage realization (Jiang et al., 2016): I T i O x = j 0 ⋅ A ⋅ s i n h ( V T i O x ) (3) where j 0 is a fit parameter that has to be adapted to the real devices. The layer capacitances are given by C l a y e r = ε A d l a y e r (4) where ε = ε r ε 0 is the permittivity of the respective layer.

The starting point of the switching model is the memristive behavior caused by a temporal and spatial change of the oxygen ions in the HfO _x layer. This effect is taken into account in the device model via an average ion velocity. d x d t = c d r i f t ⋅ I I o n (5) where x is the memristive state variable, i.e., the average position of oxygen ions or length of the filament in the HfO _x layer (cf. Figure 2B) and I I o n is the ionic current of the oxygen ions. Furthermore, c d r i f t describes the resulting drift constant of the system, which is defined as c d r i f t = μ n ⋅ R m e a n d H f O x ⋅ A (6) Here, R m e a n is the mean resistance of the HfO _x layer, which is given by R m e a n = 1 2 ⋅ [ R m i n + R m a x ] for devices of type F and R m e a n = R H f O x for the devices of type R. In particular, for devices of type F the resistance of the HfO _x layer can be specified as a function of the memristive state variable x: R H f O x = d H f O x e ⋅ z v 0 ⋅ A ⋅ μ n ⋅ [ 1 N m a x ⋅ x + 1 N m i n ⋅ ( 1 − x ) ] (7) An essential important property of ionic based memristive devices is the back diffusion of the ions. The back diffusion determines the reliability and the storage time of the memristive device and is crucial parameter for a precise adjustment of multiple resistance states. In order to consider this behavior in the model, a further term was added to Eq. 6: c d r i f t = μ n ⋅ R m e a n d H f O x ⋅ A − c b a c k ⋅ [ 1 − ( 2 x − 1 ) 2 ] (8) Here c b a c k is a parameter that describes the strength of the back diffusion and must be adapted to the measured data.

The ion current can be written in the following form using the law of Mott and Gurney (Hardtdegen et al., 2018): I I o n = 4 A e N m e a n a ν 0 ⋅ exp ( − Δ W V T ) ⋅ sinh ( a ⋅ E H f O x V T ) (9) where Δ W is the diffusion barrier, which is reduced by the electric field E H f O x . Furthermore, V T is the thermal voltage, a the hopping distance, and ν 0 is the attempt frequency. N m e a n determines the mean concentration of mobile ion species in the HfO _x layer, i.e., N m e a n = 1 2 ⋅ ( N m a x + N m i n ) , while A is the active area of the device, which depends on the device type (cf. Figure 2B). Thus A = A f i l for the filamentary device and A = A d e v i c e for the interface based switching device (cf. Figure 2B).

The interface between the HfO _x /Au is assumed to be the relevant interface for the resistive switching process in both types of devices. In the simulation, this interface is modeled as a Schottky diode with variable Schottky barrier ( ϕ B ). Using the thermionic emission theory, the charge transport over a Schottky barrier can be described in the following equation (Sze and Ng, 2006): I S = I R ( exp e V n V T − 1 ) (10) Where n is the ideality factor, which describes the deviation from an ideal diode characteristic, and I R the reverse current, which is given by: I R = A * A T 2 ⋅ exp − e ϕ B V T (11) where A * is the effective Richardson constant, which is 1.20173 ⋅ 10 6 A m − 2 K − 2 , T the local temperature, and A the active area. Under negative voltage polarities, however, the reverse current decreases gradually with the applied bias voltage. Therefore, on this polarity the reverse current is (Sze and Ng, 2006): I R , v < 0 = − A * A T 2 ⋅ exp − e ϕ B V T exp − e α r | V | V T (12) Here α r is a device dependent parameter. In our model we assumed that both quantities n and ϕ B depend on the concentration of moved ions at the Au/HfO _x interface. A higher concentration of the negatively charged oxygen ions at that interface in R type devices increases the electron concentration locally. For devices of type F an increased concentration of oxygen vacancies increases the amount of acceptor states for electrons at the interface and thus there is also an accumulation of electrons at the interface. Thus, for both type of devices a reduction of the Schottky barrier is expected, which in turn has a significant effect on the charge transport through the complete device. In the model this was considered by a state variable dependency of those quantities: ϕ B ( x ) = ϕ B L R S ⋅ x x m a x + ϕ B H R S ⋅ ( 1 − x x m i n ) (13) n ( x ) = n L R S ⋅ x x m a x + n H R S ⋅ ( 1 − x x m i n ) (14) the values for n L R S and n H R S , as well as ϕ B H R S and ϕ B L R S were obtained from the experimental I-V curves using Eq. 10. Another important parameter influencing the ion movement within the memristive device is the local temperature change. This includes mainly Joule heating and plays a crucial role particularly in filamentary-based device structures. This was taken into account in the simulation model as follows (Ielmini and Milo, 2017). T = I ⋅ V ⋅ R t h e r m + T 0 (15) Here, R t h e r m is the effective thermal resistance and T 0 is the room temperature. The temperature along the filament is assumed to be relatively homogeneous and thus a uniform filament temperature can be assumed (Ielmini and Milo, 2017).

The device parameters have been carefully collected from measurements and literature and are summarized in Table 1. The I-V curves simulated with the model are shown in Figure 2C and compared with the measurement curves determined experimentally. As can be seen from this figure, the model presented here shows very good agreement with the experiment. A more detailed description of the results follows in the next chapter.

In Figure 3A typical obtained I-V curves of the two kinds of memristive devices (named as F and R) are shown. Common for both device types is that they show bipolar resistive switching with a gradual resistance change. A major difference between both types of devices is their voltage polarity. While type R devices require a positive voltage (applied to the top electrode) to set the device, type F devices require a negative voltage to be applied for the set process. The different polarity behaviors are originated from differences in concentration and species of mobile ions, which will be discussed in Concentration of Mobile Icon. Furthermore, while a highly rectifying memristive behavior is observed for device type R, a more symmetric memristive behavior is found for devices of type F together with a 3 times higher current level as compared to type R devices (cf. Figure 3A). In some more detail: the rectifying behavior of devices of type R can be quantified by the ratio between the maximum and minimum current r a s y m = | I m a x / I m i n | at a voltage of ±0.5 V. Here we were able to determine r a s y m = 70 for an active device area of 100   μ m 2 which, however, has a strong area dependence. In particular, for an area of 625   μ m 2 the asymmetry ratio r a s y m is reduced from 70 to 4 (further information is provided in Supplementary Figure 3).

An important property of memristive devices and another difference between the here considered devices is the initial electroforming process. While no initial electroforming step was necessary for type R devices, type F devices had to be electroformed at the beginning. For a more precise discussion of the electroforming process of type F devices, the median of the required voltages as a function of the thickness of the HfO _x layer is depicted in Figure 3B. In detail, electroforming voltages of 2.35, 2.42, and 2.52 V have been observed for, respectively, a 2, 5, and 8 nm thick HfO _x layer. Thus, the electroforming voltage shows moderate thickness scalability. After the electroforming process type F device are operated typically at a maximum (minimum) voltage ±0.75 V. In terms of operating voltage, type F devices also differed from type R devices: type R devices require on average a 1.3 V higher operating voltage with a moderate area dependence (cf. Figure 3C). The operating voltages for type R devices were 2.2 V/−0.42 V (SET/RESET) for the smallest area size and 1.7V/0 V for the largest area size. However, the type R devices show a more gradual transition from the inertial high resistive state (HRS) to the low resistive state (LRS) (cf. Figure 3A).

A crucial property of memristive devices is their retention time. Furthermore, a detailed investigation of the retention characteristic already provides important conclusions about the underlying resistive switching mechanism (Hansen et al., 2015). The retention behavior for the here discussed two types of memristive bi-layer structures are shown in Figure 3D. For the measurement of the retention characteristics, the two types of devices were initially set to the low resistance state and then the resistance value of the devices was determined at regular intervals by means of voltage pulses. As can be seen in the figure, the two types of devices show quite different retention behaviors. For device type R, diffusive characteristics were observed (see red data points in Figure 3D), while much higher retention is observed for device type F. In order to analyze the retention characteristics in some more detail the retention curves were fitted using a power law according to the Curie-von Schweidler equation (Mikheev et al., 2014; Goossens et al., 2018): R = R o n / R o f f ∝ t α (16) where α is a fit parameter, which is between 0 and 1 (Yang et al., 2010). As a result, α = 0.3 is observed for devices of type R, whereas α = 0.02 best reflects the experimental data for devices of type F. While α = 0.02 describes a very good retention time for devices of type F, α = 0.3 shows clearly lower retention for devices of type R. This difference can be explained by the different ion dynamics between the two devices. While in the type R device the filamentary structures are suppressed and mobile oxygen ions are shifted toward the electrode, in the type F devices oxygen vacancies are organized in filamentary structures. This leads to different activation energies of the ion dynamics. It has been shown that the activation energy of oxygen vacancies is in the range of 6–8 eV inside filaments (McKenna, 2014), while it is less than 1 eV outside filaments (Dirkmann et al., 2016; Dirkmann et al., 2018). Furthermore, it is worth mentioning that localized electronic states at the Au/HfOx interface can also contribute to the observed switching mechanism. The localized electronic states are filled or emptied depending on the applied bias voltage polarity (Hansen et al., 2015; Zhou et al., 2016). Even if the exact mechanism underlying the switching effect cannot be clearly explained by the presented measurements alone, the strong difference in the retention times and the different voltage polarity indicate that oxygen vacancies dominate the respective switching behavior in type F devices, while mobile oxygen ions lead to resistive switching in type R devices.

In order to be able to make suitable statements about possible applications of the memristive devices in neuromorphic systems and to tailor the device characteristics accordingly, a statistical investigation of relevant device parameters is required. As relevant device parameters we considered the thickness of the active HfO _x layer ( d H f O x ), the active area size A, and the concentration of mobile ions N. The results of that investigation are shown in Figure 4. In the cumulative distribution function (CDF) of the resistance of type R devices (Figure 4A) and of type F devices (Figure 4B) the resistances were obtained from voltage sweep measurements by calculating the corresponding median values and the standard deviations. The resistance obeyed a lognormal distribution for all examined devices. For devices of type R (cf. Figure 4A) the resistance distributions for area sizes from 100 to 1,225 µm² are shown. For the devices of type F the different curves in Figure 4B originate from the different d H f O x . As a result, we found that for devices of type R the resistance window decreased with increasing active area size A which can be attributed to the decreasing rectifying ratio (further details are in the supplement). Furthermore, the relatively small width of the CDF curve was observed for type R devices indicating a high device uniformity. For the devices of type F, the low resistant states show a steeper change in the CDF curve than the high resistant states. Even though the found variations in the resistances are small, the devices with a HfO _x thickness of 5 nm show here the best variability.

To be able to make detailed statements about the requirements to be met by the physical device parameters, the influence of the variability in relation to the physical parameters must be examined. Therefore, normalized standard deviations of the devices were determined and plotted as a function of the active volume, i.e., the layer thickness of the active HfO _x layer times the device area. The obtained results for both types of devices are shown in Figure 4C. The figure shows the different measured variabilities for the devices of type R (triangular data points) and the devices of type F (circular data points) as a function of the active volume of the device. For devices of type R it appears that the variability is only weakly affected by increased area size. Here the normalized standard deviation of 0.2 is quite constant over the investigated area sizes (see the dashed black line in Figure 4C). However, for devices of type F a parabolic curve best describes the found trend which indicates a clear optimum at approximately 2.6 ⋅ 10 3   μ m 2 ⋅ n m for the HRS and 2.73 ⋅ 10 3   μ m 2 ⋅ n m for the LRS. This means that a reduction from d H f O x = 8   n m to d H f O x = 2   n m increases the optimal device area from A d e v i c e = ( 18 × 18 )   μ m 2 to A d e v i c e = ( 36 × 36 )   μ m 2 . Thus, the trend can be observed that with extremely small layer thicknesses, a larger area leads to a more stable device behavior.

A sound understanding of the resistive switching mechanism is important to enable a targeted design of the memristive devices for application in neuromorphic systems. For this reason, the device model described in Physics Based Device Model was used to interpret the experimental results described above. The obtained results are shown in Figure 2C. Therein the experimental I-V curves are compared with simulated curves. The used simulation parameters are summarized in Table 1. In both cases, i.e., in the case of the filamentary (type F device) and the interface-based device (type R device), one can see quite good agreement between simulation model and experiment. The main difference between the two I-V curves in the simulation model comes from (i) differences in concentration and species of mobile ions due to stoichiometric differences between Hf and O, (ii) different active areas that are responsible for the switching behavior (cf. Figure 2C), and (iii) the lowering of the Schottky barrier and the change of the ideality factor. In order to understand more exactly the underlying switching mechanisms that lead to the different device characteristics, the mentioned points (i–iii) will be discussed in the following in more detail.

The emulation of synaptic plasticity processes with memristive devices is one of the most important application fields of memristive devices in neuromorphic systems (Ziegler et al., 2018). In particular, this requires the design of suitable learning and training processes (Ielmini and Ambrogio, 2020), which needs a targeted adjustment of the resistance states of individual memristive devices in networks. In the following section, it is presented that type F devices fulfill requirements for machine learning based algorithms, whereas type R devices for neurobiologically inspired learning schemes.

The challenge in the machine learning based algorithm is to create suitable local learning rules that guarantee a local change of the device state so that a requested global network functionality is enabled. Therefore, a general framework is provided by the Hebbian learning rule (Ziegler et al., 2015), which can be systematized in the following equation: d ω i j d t = f ( ω i j , A j , A i ) (17) where ω i j describes the coupling strength between the pre- and the post-synaptic neuron and A j ( i ) their activities, as sketched in Figure 8A. This formula translates Hebb’s postulate, that synaptic connections change only when the respective pre- and post-synaptic neurons are active at the same time. The choice of the function f is thus decisive for the learning or training procedure of any artificial neural network. A common way to realize the weight update according to Eq. 17 is provided by the delta rule (Kendall et al., 2020), which is at the heart of deep learning neural networks: Δ ω i j = α ⋅ ( d i − y i ) ⋅ p j (18) where the coefficient α is named learning rate and is usually positive. Furthermore, p j is the activity of the pre-neuron (input value), y i is the activity of the post-neuron (output value), and d i the desired output value for a given input p j used during learning. To convert that equation into hardware, the coupling strength ω i j can be represented by the conductance G i j of the memristive device, and y j , p j , and d j by voltage- or current-dependent functions that either increase or decrease the conductance of the memristive device (Linares-Barranco et al., 2011). Thus, for the implementation of memristive devices in neuromorphic network structures via the delta rule a precise change of the conductance in dependence on applied voltage (or current) pulses is required (Payvand et al., 2018).

In order to investigate the resistance update behavior of the devices used here under voltage pulsing, AC pulses trains were used (see the sketch in Figure 8B). Therefore, a voltage train of 20 SET pulses followed by 20 RESET pulses was applied to the devices. Furthermore, the resistance states have been determined by a readout pulse that followed each switching pulse. The results obtained are shown in Figure 8C for type F devices and in (D) for type R devices. Read pulses of 1.0 and 0.1 V with a pulse width of 10 ms have been used for R and F devices, respectively. For the reset pulse, the width was 1 ms and the amplitudes were −2 and 1.5 V for R and F devices. As a result, a gradual transition change with multiple resistance states was observed in devices of type R, while a more binary behavior was recorded for devices of type F (cf. Figures 8C,D). In order to investigate the pulse behavior of the type F devices in more detail with respect to Eq. 18, the voltage amplitudes for SET and RESET pulses were successively changed in each pulse, as sketched in the inset of Figure 8E. The therewith obtained resistance change as a function of the voltage pulse amplitudes is shown in Figure 8E. Thus, a linear change in resistance with a successive incremental increase of the voltage pulse height was recorded for both set and reset. Furthermore, the resistance change was nearly symmetric in both resistance states, presenting 0.44 and 0.56 linearity for set and reset, respectively. Hence, this behavior fulfills nicely the requirement proposed by equation 18 and makes type F devices, together with their relatively good retention, perfect candidates for the hardware realization of deep learning neural networks. In this context, bi-layer oxide memristive devices of similar types have already proven their performance (Yao et al., 2020).

While the delta rule underlies a variety of machine learning systems and allows an effective implementation of Hebb’s learning rule within artificial neural networks, there is no explicit time dependence. However, the time dependence of learning processes is an important parameter in biology and determines how the synaptic connection is strengthened or weakened (Panwar et al., 2017). Here, an important property is the memory effect of synapses which leads to a sustained strengthening of the synaptic connection after repeated (high frequency) excitation named long-term potentiation (LTP). Therefore, the respective time interval between the excitation is required. At this respect, the diffusive ionic processes of memristive devices and their memory behavior are unique properties for the emulation of bio-realistic time-dependent learning (Ziegler et al., 2018), such as spike-timing dependent plasticity (STDP) and paired-pulse facilitation (PPF), to only mention two important plasticity processes. Many ways to emulate such learning schemes have been presented in recent years with memristive devices (Wang et al., 2020). However, the challenge here is to select the correct voltage functions for the pre- and post-neurons, so that an appropriate voltage pulse is applied across the memristive device (Linares-Barranco et al., 2011; Ambrogio et al., 2013). To investigate this for the type R device, we took a closer look at the PPF scheme. The results obtained,therefore, are shown in Figure 8F. Two identical sequential SET pulses were applied using different time intervals. In this study, the PPF ratio was defined as the incremental percentage change in the resistance after the first and second pulses. As a result, we found, the longer the time interval is, the smaller the resistance changes with a linear trend. This, particularly, corresponds to the enhanced back diffusion of oxygen ions in R type devices, as discussed above, and resembles well with biology.