# Oxygen vacancies kinetics in TaO2-h/Ta2O5-x memristive interfaces

C. Ferreyra1 R. Leal Martir2,3 D. Rubi 1 María José Sánchez2,3 1Instituto de
Nanociencia y Nanotecnología (INN), CONICET-CNEA, nodo Buenos Aires,
Argentina.
2Instituto de Nanociencia y Nanotecnología (INN),CONICET-CNEA, nodo
Bariloche,8400 San Carlos de Bariloche, Río Negro, Argentina.
3Centro Atómico Bariloche and Instituto Balseiro (Universidad Nacional de
Cuyo), 8400 San Carlos de Bariloche, Río Negro, Argentina.
<EMAIL_ADDRESS>
###### Abstract

Oxygen vacancies (OV) are pervasive in metal oxides and play a pivotal role in
the switching behaviour of oxide-based memristive devices. In this work we
address, through a combination of experiments and theoretical simulations, OV
dynamics in Pt/TaO2-h/Ta2O5-x/TaO2-y/Pt devices. In particular, we focus on
the RESET transition (from low to high resistance), induced by the application
of electrical pulse(s), by choosing different initial OV profiles and studying
their kinetics during the mentioned process. We demonstrate that by selecting
specific OV profiles it is possible to tune the characteristic time-scale of
the RESET. Finally, we show that the implementation of gradual RESETs, induced
by applying many (small) successive pulses, allows estimating the activation
energies involved in the OV electromigration process. Our results help paving
the way for OV engineering aiming at optimizing key memristive figures such as
switching speed or power consumption, which are highly relevant for
neuromorphic or in-memory computing implementations.

Recent advances in big data manipulation and machine learning demand for low-
power in-memory computing operations. Despite substantial progress in software
and processing algorithms centered on artificial neural networks [1, 2, 3],
the evolution of hardware architectures aiming to avoid the Von Neumann
bottleneck continues to encounter substantial challenges. In this scenario,
electronic hardware that could mimic the structure and information processing
mechanisms of biological systems is emerging as a promising alternative [4, 5,
2]. In particular memristors are playing a essential role [6, 7], contributing
to the realization of artificial synapses and in-memory processing and aiming
to eliminate the physical separation between memory and processing units
characteristic of the usual Von Neumann architectures [8, 9, 10, 11, 12, 13].
Memristors are metal/insulator/metal capacitor-like structures that exhibit
the resistive-switching (RS) phenomena, i.e. the reversible change of the
electrical resistance upon the application of a electrical stress [14, 15, 16,
17, 18, 19].

In Non-Volatile RS (NVRS) the resistive states persist in time after the
electrical stimulus is turned off. Hysteresis current–voltage switching loops
are a fingerprint of this behavior. On the other hand, in Volatile RS (VRS)
the attained resistive states relax once the electrical stimulus is turned off
[20, 13]. Thus, several synaptic and neuron functionalities can be
electrically emulated based on the lifespan of the resistance states. In the
past years, extensive demonstrations of RS in oxides were realized, mostly on
binary oxides such TiO2, HfO2, and Ta2O5 [21, 22, 7, 23]. In particular TayOx
based devices stand out among others, exhibiting exceptional figures as ON-OFF
ratios up to $10^{5}$, retention times above 10 years and endurances of up to
$10^{10}$ cycles [21, 24]. RS has been reported as bipolar and its origin
related to the formation and retraction/rupture of nanoscale conducting oxygen
vacancies (OV) filaments upon polarity reversal of the applied stimulus,
ruling the SET (high to low resistance) and RESET (low to high resistance)
transitions respectively [25, 26, 27, 28, 29].

The development of neuromorphic devices based on memristors requires a
comprehensive knowledge of the mechanisms ruling the RS. Understanding the
role of OV and their dynamics is crucial for extensive optimization (improve
the switching speed, minimize the energy consumption, enlarge the endurance)
and design of oxide-based memories for neuromorphic applications. Numerous
reports show that OV, present as extrinsic or intrinsic defects, play a
central role in the RS mechanism [16, 25, 30]. In particular the RESET process
in TaOx cross-bar arrays was recently analyzed employing time-resolved pulsed
experiments and attributed to the OV motion governed by drift and diffusion
processes [31]. Despite the obtention of qualitative knowledge, it remains
unclear in which way specific OV profiles and their kinetics affect the RS
properties of the devices under electrical bias. In addition, the intricate
dynamics involved in OV transport remains incompletely elucidated. Thus,
obtaining evidence pertaining to the motion of OV proves challenging and
usually involve complex in-operando experiments [32, 33, 34]. In order to pave
the way along this central issue, in this work we investigate- using a much
simpler experimental approach- the transient electrical response in a
Pt/TaO2-h/Ta2O5-x/TaO2-y/Pt device during the RESET process.

Assisted by the Voltage-Enhanced Oxygen Vacancy model (VEOV) [35, 36, 37, 38],
we identify the specific OV profile at the origin of the RESET transition and
we determine its time evolution, providing an unique insight into OV dynamics
along this process. In addition, through the study of successive partial
RESETs, we were able to estimate the activation energy for OV migration, which
is a key issue for controlling the RS effect and designing efficient operation
electrical protocols.

###  .1 EXPERIMENTAL SET UP

A bilayer Ta-oxide thin film was deposited on commercially platinized silicon
using Pulsed Laser Deposition (PLD) -the laser fluence was fixed at 1.5 J
cm-2\- from a Ta2O5 ceramic target at room temperature. The deposition was
made in 2-steps,using oxygen pressures of 0.01 and 0.1 mbar, respectively. We
obtained two layers, a more reduced with a thickness of 35 nm and a more
oxidized with a 15 nm thickness, as shown in the sketch of Fig. 1(a). It is
usually considered that more reduced layers behave as OV source/sink that
favors the reduction/oxidation of more oxidized ones, driving the memristive
effect [39].

Top Pt electrode was made by sputtering and micro-structured using optical
lithography [37]. On the other hand, the Pt of the substrate acts as the
bottom electrode, which was grounded. The electrical stimulus (voltage) was
applied to the top Pt electrode.

The pristine device starts in a low resistance state of $\approx$ 100
$\Omega$. By applying a negative stimulus of - 3V during 1 ms, an intermediate
resistance state of the order of k$\Omega$ was attained, awakening the
memristive response. The electroformed device results in the stack
Pt/TaO2-h/Ta2O5-x/TaO2-y/Pt [37], as sketched in Fig.1 (a). The active zone
for RS comprises the more oxidized layer, Ta2O5-x, sandwiched between the two
reduced layers, defining a quasi-symmetric configuration with two active
memristive interfaces: the top interface (TI) TaO2-h/Ta2O5-x and the bottom
interface (BI) Ta2O5-x/TaO2-y, respectively. On the other hand Pt/TaO2-h(y) is
an ohmic interface playing a passive role in the RS. The memristive behavior
of the device was studied in detailed in Ref.37 and can be understood in terms
of OV electromigration between the central zone, Ta2O5-x, and the interfacial
regions under the applied stimulus.

Figure 1:  (a) Sketch of the post formed device. The region enclosed by the
dashed rectangle comprises the Ta2O5-x layer and the two active interfaces,
TaO2-h/Ta2O5-x (TI) and Ta2O5-x/TaO2-y (BI) respectively. (b) Experimental set
up employed for the electrical characterization. (c) Sketch of the device and
the active zone employed in the numerical modelling. See text for more details

The electrical characterization was performed in two steps. Firstly, using a
Keithley 2612 source-meter, we applied the RS protocol (described below in
Sec..2) to check the correct RS behavior and tune the desired resistive state
of the sample. Then, with an Agilent 81110A pulse generator, a single RESET
pulse is applied. A homemade matrix allowed to select the needed instrument in
each case. The RESET current response was measured using a shunt connected to
a Tektronix TDS5032B Oscilloscope. The experimental setup is shown in Fig.1
(b).

###  .2 RESULTS

The RS protocol consists of a pulsed ramp that alternates between write
pulses, which changes the resistance state of the device, and read pulses that
capture their remanent resistance states (see inset of Fig.2 (a)). From these
pulses, we obtain respectively, the dynamic I-V curves (not shown) and the so-
called Hysteresis Switching Loop (HSL) [35] which displays the remanent
resistance as a function of the applied (write) voltage. Depending on the
polarity and the symmetry of the voltage protocol, defined by the maximum and
minimun intensities $V_{max}$ ($\geq$0) and $V_{min}$ ($\leq$0), different
HSLs with shapes and chiralities -reflecting the activation of one memristive
interface, the other or both- can be obtained [37]. For the present device,
the HSLs presents a squared-like shape with a clockwise (CW) chirality for
$|V_{max}|<|V_{min}|$, and a counter-clockwise (CCW) chirality for the
inverted voltage excursion i.e $|V_{max}|>|V_{min}|$. In addition, when the
voltage excursions are enlarged and symmetrized to $V_{max}=-V_{min}$, the HSL
displays a table with legs (TWL) -like shape [37]. The squared HSLs correspond
to a single active interface while the TWL corresponds to two (complementary)
active interfaces; that is, when one switches from low to high resistance the
other one does it inversely [35]. Therefore, each memristive interface can be
selectively activated or deactivated according to the polarity and intensity
of the applied stimuli [37, 38].

Figure 2: (a) Clockwise Hysteresis switching loop (HSL): Blue dots
(experimental), red dots (simulations). The arrows indicate the circulation of
the HSL. The coloured squares numbered $1-6$ refer to selected values of
resistances used for the analysis of the RESET transition. Inset: experimental
electrical protocol displaying the write and read pulses respectively. (b) OV
distributions obtained from the VEOV model along the active region comprising
the TI, C and BI, for each of the resistance values labelled $1-6$ in the HSL
of panel (a). The vertical dashed lines indicate the boundaries of the TI and
the BI respectively. See text for more details.

Starting with a post-forming device, we prepared the device in a low
resistance state by following the CW HSL, using a non-symmetrical protocol
with $V_{min}=-1.8$V and $V_{max}=1.2$ V. The sample was initially cycled
several times to ensure stability and repeatability of the CW HSL, displayed
in Fig.2 (a) (blue dots), with well-defined LRS ($R_{L}=$1.05 k$\Omega$), HRS
($R_{H}=$1.65 k$\Omega$) and SET and RESET voltages, respectively. Notice that
the LRS remains almost flat for voltages in the range [1.2 V, -1.2 V],
reflecting the time-stability (non-volatility) of the resistance states. We
start by analysing the RESET process under the application of single
electrical pulses of different intensities. To this end, the device was
prepared in the low-resistance state (LRS) just before reaching the RESET
threshold. In particular, we chose as the initial state (IS) the resistance
$R_{IS}=$ 1.05 k $\Omega$, labelled $\\#6$ (magenta square) in the CW HSL of
Fig.2 (a).

Figure 3(a) shows, in continuous lines, the evolution of the resistance with
time during the RESET process for three different voltages applied during 100
$\mu$s, starting from the chosen IS. A higher voltage induces a complete RESET
attaining the HRS in a shorter time, as expected. For -2.8 V, it takes
$\approx$ 30$\mu$s to complete the RESET process, while for -2.6 V it takes
$\approx$ 100$\mu$s. This response is consistent with the fact that for
negative voltages, OV -as positive defects- move from the central region
towards the TI (TaO2-h/Ta$2$O5-x), which is active throughout the entire
(RESET) process. Thus, for higher applied voltages, OV (electro) migration is
enhanced and therefore the RESET takes place in shorter time scales. In
addition, the non-linearity of the resistance as a function of time during the
RESET increases with the intensity of the voltage - a characteristic of
activated processes previously reported [40, 41]-.

Figure 3: (a) Resistance vs. time for different single negative bias pulses of
100 $\mu$s, starting from the IS (magenta square). Black line: -2.6 V, red
line: -2.7 V, green line: -2.8 V. Experimental data are shown with solid
lines, and simulations with dashed lines. The complete RESET is finished when
the HRS, $R_{H}\approx$1.65 k$\Omega$, is attained. (b) Power consumption as a
function of time. Different curves correspond to voltages amplitudes -2.6V
(black line ), -2.7V (red line ) and -2.8V (green lines), respectively. Inset:
Associated energy consumption (shaded area below each of the previous curves).
See text for details.

It is interesting to estimate and compare the energy consumption involved in
the RESET processes. To this end, we followed Refs. 42, 37 and compute in
Fig.3(b) the power consumption as $P=V^{2}/R$, where $R$ values correspond to
the measured resistances displayed in Fig.3(a). In addition, the inset shows
the energy (computed as the shaded area under each of the curves when the
power variation is below 5$\%$) for the three voltage values. It is noteworthy
that the energy involved in the RESET process is lower for higher pulses
intensities, even though the power consumption is larger. This is a clear
evidence of the non linear dependence of the RESET process with the applied
voltage, in agreement with the voltage-time dilemma typically present in RS
devices [41] .

To model the experimental response we used the 1d Voltage-enhanced Oxygen
Vacancies Oxygen (VEOV) migration model [35, 36, 37, 38] that describes the
resistive change in oxide based memristors as a result of the migration of the
ubiquitous OV under the application of external electrical stimuli.

The VEOV model considers a 1d chain of N nanodomains which can contain a
specific total amount of OV. Each domain defines a site in the chain and has
an associated resistivity, which depends locally on the OV content. In binary
oxides of transition metals, OV act as n-type dopants supplying free electrons
to the conduction band and thereby reducing the electrical resistance. In
agreement with this, we define the resistivity of each nanodomain $i$ as
$\rho_{i}=\frac{\rho_{0}}{(1+A_{\alpha}\delta_{i})}$, being $\delta_{i}$ the
OV concentration at site $i$ (i.e. the ratio between the number of OV at site
i over the total amount of OV), $A_{\alpha}$ is a parameter that depends on
the zone where the domain is located (i.e. interface or not-interface zones)
and $\rho_{0}$ is the resistivity of the oxide in the absence of OV [36]. The
total resistance of the device is computed as $R=c\sum_{i=1}^{N}\rho_{i}$ with
c being a geometric pre-factor.

For the present case, and due to the aforementioned interfaces, the chain of
nanodomains is divided into three regions: TI and BI (associated with the
interfaces) which allocate $N_{TI}$ and $N_{BI}$ sites respectively and region
C that represents the bulk zone of the device, as depicted in Fig. 1 (c). For
a given external stimulus $V$, at each simulation step the OV concentrations
are updated with the transfer rate $p_{i,i\pm 1}=\delta_{i}(1-\delta_{i\pm
1})\exp({(-V_{0}+\Delta V_{i})}/{(KT)})$, from site $i$ to site $i\pm 1$,
which is proportional to the OV concentration $\delta_{i}$ at site $i$ and to
the availability of the neighbouring sites $i\pm 1$. The exponential Arrhenius
factor contains- given in units of thermal energy $KT$\- the activation energy
for OV diffusion $V_{0}$ (that could vary along different regions of the
chain) and the local potential drop at each site, $\Delta V_{i}\propto
V\rho_{i}/R$, due to the external electrical stimulus. With the new OV’s
concentrations, the local resistivities and the total resistance are therefore
updated. In order to emulate the electrical response of the device during the
RESET we established an equivalence between experimental and numerical
parameters (see the Suppl. Mat.).

Following the described procedure we computed the remanent resistance as a
function of $V(t)$ and obtained the HSL depicted (in red) in Fig.2 (a), which
is in good agreement with the experimental (in blue) one. In particular, the
simulated HRS and LRS are almost indistinguishable from the experimental
values.

A key outcome of the VEOV model are the OV profiles associated to the
different resistance states [35, 36, 43, 37, 38]. In particular, each of the
states labelled from 1 to 6 in Fig.2(a) has associated the OV profiles
displayed in Fig.2(b)-labelled with the same numbers as their resistance
counterparts. These profiles could be taken as different initial condition to
simulate the RESET process by applying a single electrical pulse. However, as
we already mentioned, the experiment was performed starting from the IS which,
according to the VEOV model, has associated the OV profile 6 in Fig. 2 (b).
The simulated RESET process for this IS is displayed in Fig.3 (a) in dashed
lines for the three voltages considered. The agreement with the experimental
results is quite good and, in particular, the simulations follow nicely the
experimental curves for $V$= -2.7 and -2.8 V. We notice that IS profile 6 is
Gaussian-like, providing a clue about the actual OV distribution prior to the
RESET transition.

Figure 4:  Snapshots of the OV´s profiles along the simulated RESET process
for $V$=-2.7 V and for times 0, 4, 24, and 46 $\mu s$.(a) Starting from
profile 6 (IS). (b) Starting from profile 1.

Figure 4 (a) shows the OV profiles for the IS (at time 0 $\mu s$) and for
times 4, 24, and 46 $\mu s$, selected along the simulated RESET displayed in
Fig.5 (red-dashed line), for $V=$ -2.7 V. Notice that, as time evolves, OV
profiles move to the TI and accumulate there, acquiring a shape that strongly
departs from the initial Gaussian-like shape, as expected for a non diffusive
dynamics driven by an external field. In particular, the OV profile for 80
$\mu s$ (not shown) -once the complete RESET is attained- is extremely sharp
and localized along the first few sites.

For the sake of comparison, we plot in Fig. 4 (b) the OV profiles obtained
from the simulated RESET for $V$ =-2.7 V - displayed in Fig.5 (red-dotted
lines)- starting from the low resistance state labelled 1 in Fig.2 (a).
Despite the fact that the initial and final resistance values for profiles 1
and 6 are barely distinguishable , their transient dynamics are clearly
different. Profile 1 completes the RESET faster than profile 6, as depicted in
Fig. 5. For profile 1, the simulated RESET is completed in 46 $\mu s$, while
neither the experimental data (red solid red) nor the simulated evolution
starting from the IS (red dashed line) -which nicely follows the experimental
curve- have attained the RESET for that time. This is a direct consequence of
the fact that the initial OV distribution for profile 1 -which is slightly
non-Gaussian- has its maximum closer to the TI than the IS profile 6.
Therefore, for profile 1 it should be possible, by lowering the external
stimulus, to slow OV dynamics in order to have a resistance vs. time evolution
that matches the one obtained for the initial IS stimulated with a $V=$-2.7V
pulse. Indeed, we depict in Fig. 5 (grey-dotted line) the simulated RESET
obtained from profile 1 for $V=$-2.65V, which follows the experimental RESET
curve almost with the same trend as the IS for $V=$-2.7V.

Thus, a clever strategy to reduce the RESET time and, additionally, optimize
the power consumption, could be to engineer the initial OV profile [44], as
barely different OV distributions with quite similar resistance values give
rise to experimentally accessible lower RESET times for the same applied
stimulus or, alternative, similar RESET curves for lower stimulus.

Figure 5: Experimental (red solid curve) and simulated RESET evolution
starting from the IS profile 6 (red dashed line) and from profile 1 (red
dotted line), in all the cases for $V=$ -2.7V. The grey-dotted line shows the
simulated RESET evolution starting from profile 1 but for $V$=-2.65 V. Square
symbols indicate different resistance states, whose associated OV profiles-
excepted for time 80 $\mu s$\- are depicted respectively in Fig.4 (a) and (b).

The VEOV model was recently extended from 1d to 2d (2d-VEOV), as a tool to
describe how OV electromigration is affected by the presence of structural or
morphological defects, such as dislocations or grain boundaries. For the
present analysis, we have not taken into account those defects, as we do not
have experimental evidence pointing to any strong influence of such type of
morphological defects, neither from the electrical measurements nor the
structural characterization perfomed for this sample [37]. However, we
analyzed the transient dynamics with the 2d VEOV model in the $x-y$ plane for
different initial OV distributions, when an external stimulus is applied along
the $x$ direction (see Supp. Mat.). Although the OV evolution during the
transient process depends on the shape and location of the initial OV
distribution, after a characteristic time scale an uniform OV profile is
attained along the $y$ direction while a Gaussian-like profile is obtained in
the $x$ direction, for the studied initial conditions. The uniformization in
the $y$ direction takes place for time scales much shorter than the time
needed in the experiment for cycling until the stable HSL depicted in Fig. 2
(a) is obtained. Therefore, under this result, we safely simulate the actual
RESET process in terms of many parallel OV 1d channels along the $x$
direction, employing the 1d version of the VEOV model, which consumes
significantly less computing time.

As previously discussed, it is clear that reducing the amplitude of the
applied voltage could lead to a situation where a single electrical pulse
could not be enough to produce a complete RESET, i.e. the device does not
reach its HRS. An interesting avenue to explore is to apply many consecutive
pulses of lower intensities [45, 46], each producing a partial RESET. In Fig.6
we show the experimental evolution of the remanent resistance for six
successive single pulses of V= -2.4 V with a duration of 100 $\mu$s applied
starting from the IS. Each pulse produces a partial RESET that progressively
increases the resistance. In this protocol, the attained resistance for a
given pulse is taken as the initial state for the subsequent pulse. Notice
that once the complete RESET is achieved (in the present case after six
pulses), the HRS resistance remains constant in time -besides the existence of
some experimental fluctuations-. The simulated partial RESETs reproduce quite
accurately the experimental ones, as shown in Fig.6 with solid lines following
the experimental trend. Moreover, when the amplitude of the pulses is reduced
to $V=$ -2.3 V, the initial resistance value remains almost unchanged after
the sequence of six pulses ended, as it is shown by the horizontal dashed
lines. We stress that this response persists even after increasing the number
of applied pulses.

Finally, we show now how the already displayed partial RESET transitions can
be used to estimate the OV activation energies. If we aim to reproduce the
multiple partial RESETs using number of pulses $n$ -each one of 100$\mu s$\-
given by the experiments, we notice that the VEOV model present two possible
control parameters, $-V_{0}$ and $\Delta V_{i}$, which are added in the
argument of the exponential function that defines the Arrhenius factor that
controls the OV transfer rate $p_{i,i\pm 1}$. Lower $V_{0}$ values (higher $V$
values) imply that fewer pulses are needed to reach the HRS, while higher
$V_{0}$ values (lower $V$ values) require more pulses to achieve the same HRS.
It is noteworthy that the same response could be achieved by adjusting both
parameters, provided that the quantity $(-V_{0}+\Delta V_{i})$ remains
unchanged. Additionally, as we have shown, there is an external threshold
voltage $V_{th}$ below which the resistance exhibits no evolution or resistive
changes regardless of the number of applied pulses. Above this threshold, the
system starts to exhibit resistive changes. With this threshold voltage we can
define an effective potential barrier $V_{{ef}}\equiv(-V_{0}+\Delta V_{th})$,
where for simplicity we assume a linear voltage drop with $\Delta V_{th}\equiv
V_{th}/N$.

Thus, given a value of $V_{0}$, in the simulations we can tune the external
voltage in order to determine $V_{{th}}$, and consequently compute $V_{{ef}}$.
In the present case, we varied the external voltage until for six consecutive
pulses the resistive change was below 5$\%$ of full RESET. With this criteria
we obtained $V_{{th}}=-2.3V$ for the example depicted by the dashed lines in
Fig. 6. We assume that the experimental threshold voltage, $V_{the}$, is equal
to numerical one, $V_{th}$. Increments of $\pm 0.1$ V are fixed by the
accuracy of the pulse generator, imposing an uncertainty in $V_{the}$.

By increasing the external voltage to $V=-2.4V\equiv V_{6r}$, we reproduce the
six experimental partial RESETs, depicted by solid lines in Fig.6, and non-
trivial characteristics of the observed behavior. In this context, we have
consider the difference $V_{6r}-V_{{th}}$ as the experimental uncertainty.

With these ingredients at hand we can estimate an effective potential barrier
for OV electromigration representative of the memristive system under study,
by comparing simulations with experimental data. Following this reasoning, for
the present case we use $V_{the}=$-2.3 V ($\Delta V_{th}\sim 0.02V$) and from
the value of $V_{0}=0.12V$ employed in the simulations (see Sup. Material) we
estimate e$V_{efe}=(0.10\pm 0.01)$eV, in consistency with reported values
[47].

Figure 6: Resistance vs time evolution for six consecutive single pulses of V=
-2.4 V and each one of 100 $\mu$s. The experimental results and the simulated
ones- both starting from the IS profile 6 are plotted in solid curves (the
simulations correspond to the curves with no fluctuations). In dashed lines
are the simulations for a sequence of 6 pulses of V= -2.3 V, which altogether
do not induce any noticeable change in the resistance.

##  I Concluding remarks

We investigated the RESET process and the associated OV dynamics for a
Pt/TaO2-h/Ta2O5-x/TaO2-y/Pt memristive device, using a combination of
experimental methods and theoretical simulations based on the VEOV model. We
focus on the complete RESET process, which involves the transition from LR to
the HR state under electrical stress, providing insights into the distribution
and kinetics of OVs throughout the device during this process. An interesting
outcome of our analysis is that quite similar initial (low) resistance values
can have associated different OV profiles the corresponding RESET transitions
are completed in different times. Thus, it appears that engineering the OV
profiles could be a possible route to change the time scale of this process.

Additionally, by applying several successive pulses inducing gradual RESETs,
we were able to estimate the activation energies associated with OV
electromigration, resulting in very good agreement with reported values [47].
This further evidences that the VEOV model is powerful enough to extract
relevant energy scales and system´s parameter values. In addition, we notice
that by using the 2d VEOV model it should be possible to extend the present
study to systems including different types of extended defects. This will
allow determining the influence of these defects on the memristive behaviour
of oxide-based devices. We notice that this could be the key to obtain
physical information about the origin of cycle-to-cycle and/or device-to-
device variations, which are usually present in RS devices and are often
attributed to the present of (uncontrolled) defects, stabilized during the
device fabrication process. [44, 48, 49, 50]. This topic deserves further
consideration and we leave it for a future work.

We acknowledge support from ANPCyT (PICT2019-02781, PICT2019-0654,
PICT2020A-00415), UNCuyo(06/C026-T1) and EU-H2020-RISE project MELON (Grant
No. 872631)

Supplementary Material

See Supplementary Material for additional information on the electrical
characterization, system parameters and for the analysis of the OV dynamics
employing the 2d VEOV model.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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