# On the Thermal Models for Resistive Random Access Memory Circuit Simulation

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{8}

^{9}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Description of RRAM Thermal Effects

#### 2.1. Heat Equation

_{th}—stands for the thermal conductivity (W/(m K)). This parameter depends on the temperature and geometry (assuming different material layers, which is the case for the usual RRAM architecture), c—stands for the specific heat or specific heat capacity (J/(kg K)). It is assumed that we are considering the specific heat capacity at constant pressure, which is why it is also denoted as c

_{p}, $\rho $—stands for the material density (kg/m

^{3}) and ${\dot{e}}_{generated}$—stands for the power density generated (rate of heat generation by means of Joule heating, per unit volume inside the system we are considering). It can be calculated as σ(

**r**)E

^{2}(

**r**), where σ(

**r**) is the local electrical conductivity and E(

**r**) is the local electric field [18,38]; i.e., the product of the field and the current density.

- (a)
- Constant thermal conductivity, i.e., k
_{th}(x,T) = k_{th}. Neither geometric nor temperature dependencies are considered. In most cases, the CF thermal conductivity is the one considered. - (b)
- A single temperature in the whole conductive filament [38,78] is taken into account (this means a strong simplifying approach). Some models for circuit simulation can account for two different temperatures [79], this is a good strategy since the key (higher) temperature at the CF narrowing, where the CF is ruptured, is decoupled from the main CF bulk temperature; this latter temperature does not increase in the same manner. See Figure 4c in [80], where the CF temperature along the dielectric is plotted for different voltages. It is clear that the temperature is much higher in the CF narrowing while it shows a different behavior for the main CF body. The model with two different CF temperatures is more complex, hence, this issue has to be taken into account when dealing with circuits including hundreds or thousands of components.

#### 2.2. A Numerical Approach for the Heat Equation

^{+}(bottom electrode) and Ni (top electrode). Consequently, the SD thermal boundary is shifted from the dielectric/electrode interfaces to the electrodes, as commented above. A 40 nm (X axis) × 40 nm (Y axis) × 30 nm (Z axis, vertical axis running from the Si layer to the Ni layer) SD is considered, see Figure 2.

_{th}, and the negative local temperature gradient ($\mathit{q}=-{k}_{th}\nabla T$). Since the HfO

_{2}thermal conductivity is around 1 W/(K m), the temperature drops off rapidly around the CF. See also the effects of the dielectric-electrode boundary at z = 10 nm and z = 30 nm, the temperature reduction is different from what is seen in the CF perpendicular direction along the x-axis. The maximum temperature is obtained in the narrowest section for the symmetrical truncated-cone shaped CF, as seen in (c) and (d). At this point, the physical mechanisms behind RS are more active and, consequently, they trigger the CF rupture at this location. See the thermal connection between the CFs for the different shapes and distances in between; in this respect, in tree-branch shaped filaments, the destruction of the branches and the thermal and current redistribution in the remaining intertwined branches makes the reset a complicated process.

#### 2.3. Explicit Heat Equation Solutions

#### 2.3.1. RRAM with a Cylindrical Filament (Steady-State Operation, No Heat Transfer Term)

_{ox}, respectively, with temperatures T(x = 0) = T(x = t

_{ox}) = T

_{0}, where t

_{ox}stands for the dielectric thickness), see Figure 5.

_{RRAM}/t

_{ox}). Figure 5 shows the different elements taken into account to solve the HE. The solution for the maximum temperature in the middle of the CF, with the boundary conditions sketched in Figure 5, is:

_{max}, as shown below (see Appendix D.3 in [84]),

#### 2.3.2. RRAM with a Cylindrical Filament Including a Heat Transfer Term (Steady-State Operation)

_{CF}stands for the conductive filament radius.

_{max}, as calculated in Equation (7). We will consider this the thermal model 2, TM2 (see the Verilog-A code in Table 1).

#### 2.3.3. RRAM with a Truncated-Cone Shaped Filament Including Heat Transfer Coefficient (Steady-State Operation)

_{max}can be obtained as follows (the one we assume for the whole CF, this will be the thermal model 3, TM3, see the Verilog-A code in Table 1),

#### 2.3.4. RRAM with a Truncated-Cone Shaped Filament Including Heat Transfer Coefficient (Steady-State Operation) and Two Temperature Values to Represent the CF Thermal Behavior

#### 2.4. Energy Balance in the Device

_{0}stands for the temperature of the dielectric that surrounds the CF (usually assumed at room temperature) and κ is the inverse of the thermal resistance. In this case we do not account for a temperature distribution along the CF as in Equation (2). Our perspective here accounts for the whole CF, and even its surroundings, which is represented by a single temperature. In addition, we do not follow, as above, a scheme based on the HE solution and the association of the maximum temperature with the CF. The power generated can be calculated as V

_{RRAM}(t)I

_{RRAM}(t), although we will employ V(t)I(t) for short. We can study this differential equation accounting for different operation regimes.

#### 2.4.1. Steady-State

_{0}, Equation (18) can be written as follows:

_{th}stands for the effective thermal resistance (it depends on the device physical features and is associated with heat conduction [89]). Using this simple model (thermal model 5, TM5), the device temperature can be estimated from Equation (19) [88,90,91,92,93,94].

#### 2.4.2. Non-Steady-State Approach

_{th}, the thermal or heat capacitance. This approach is used in different RRAMs compact models (thermal model 6, TM6), [78,95,96]. We can reformulate Equation (17) to the following expression:

_{0}, we obtain,

_{i+}

_{1}= t

_{i}+ Δt:

_{i+1}= X

_{i+1}+ T

_{0}. From a circuit simulation point of view, Equation (20) can be implemented with the equivalent electrical sub-circuit shown in Figure 10. The values of the different electric elements and the role of the pins is the same as in Figure 9. However, a capacitor has been added to account for the thermal capacitance.

_{th}= 4 × 10

^{4}–3.5 × 10

^{5}K/W) and C

_{th}= 0.04–1.1 pJ/K were employed in [78]. In [95], the following values were given: C

_{th}= 0.318 fJ/K and τ

_{th}= 0.23 ns, from them the thermal resistance can be extracted R

_{th}= τ

_{th}/C

_{th}= 7.23 × 10

^{5}K/W. The thermal resistance in [91] was taken R

_{th}= 5 × 10

^{5}K/W. An estimation of 33 ps for the thermal time constant is reported in [18], which has to be compared to the electric pulse-width to assess the importance of thermal transient effects in conventional RRAM operation. The heat capacitance can be calculated as C

_{th}= C

_{p}t

_{ox}A [18], where A is the CF effective area, the effective CF length that approximately corresponds to the dielectric thickness (t

_{ox}) and C

_{p}is the volumetric heat capacity (calculated as ρ × c, Equation (2)) [89]. As detailed in [18], the thermal resistance can be calculated as follows:

_{th}= 23 Wm

^{−1}K

^{−1}, and C

_{p}= 1.92 JK

^{−1}cm

^{−3}. Therefore, the thermal time constant can be estimated as:

_{ox}= 20 nm).

_{th}= 4 × 10

^{4}K/W) can be performed if we assume an ideal device in the LRS under a steady-state regime with I

_{RRAM}= 1 mA and V

_{RRAM}= 1 V; using Equation (19), we would have T = T

_{0}+ R

_{th}× I × V = T

_{0}+ 40 K, or T = T

_{0}+ 500 K, if R

_{th}= 5 × 10

^{5}K/W.

_{RRAM}= 1 Ω, the temperature can be obtained from Equation (27) for R

_{th}= 2 × 10

^{5}K/W and C

_{th}= 0.1 fJ/K (τ

_{th}= 20 ps), C

_{th}= 0.25 fJ/K (τ

_{th}= 50 ps), C

_{th}= 0.5 fJ/K (τ

_{th}= 100 ps).

_{SET}. This is coherent with experimental findings that show that the Joule heating role is essential to describe the SET kinetics [81,99]. On average, these maximum temperatures achieved are in line with the estimations performed above for the thermal resistances considered. It is important to highlight the importance of the thermal resistance in the device design. See that higher R

_{th}values lead to lower set and reset voltages to produce RS, and hence, lower power consumption. From this viewpoint, higher thermal resistances (i.e., devices showing a character thermodynamically more adiabatic with respect to the surroundings) might be more interesting. Since the heat flux from the CF to the metallic electrodes (operating these as heat sinks, a reasonable assumption, allows to calculate the heat flux with the material 3D thermal conductivity) could be in the same order of magnitude for different RRAMs, the dielectric could be the key (save the role of the contact thermal resistance). In this respect, we have to call the reader’s attention to the fact that the usual RRAM dielectrics are grown in nanometric layers; consequently, the real thermal conductivity due to phonon quantization is far from the values corresponding to the corresponding 3D dielectric materials, as it was shown in [100,101,102].

_{th}associated to a CF in a conventional RRAM, as described in [18], could help to shed light on this issue. Let us assume a cylindrical CF radius of 5 nm in a dielectric layer of 20 nm thick, if the heat capacity of Hf is considered (C

_{p}= 1.92 JK

^{−1}cm

^{−3}) the CF thermal capacitance would be (C

_{th}= C

_{p}t

_{ox}A) 0.003 fJ/K. Therefore, a value τ

_{th}= 1.2 ps is expected for the same thermal resistance employed in Figure 13. Although this thermal time constant is short, different authors [78,95] have used thermal capacitances values that lead to devices with higher thermal constants. A thermal device model described by Equation (20) and the thermal capacitance of an average CF produces so low thermal time constants that no transient term in Equation (20) would be worth being taken into account. From the experimental viewpoint, no delays linked to thermal inertia would be seen for conventional memory pulsed signals. Nevertheless, current transients on longer time scales than the previously calculated τ

_{th}, linked to some extent to thermal effects have been reported previously [78]. In this respect, the thermal model, i.e., Equation (20), might not be enough to accurately describe RRAM thermal response.

#### 2.4.3. Non-Steady-State Approach with Two Different Temperatures Associated to the Device (Second-Order Memristor)

_{S}). The latter influences the internal device temperature T but it shows a different time evolution. The device intermediate surrounding region (at temperature T

_{S}) is characterized by an outer boundary assumed to be at room temperature (T

_{0}= 300 K). This outer boundary is considered to be far away from the RS active region. The intermediate surrounding region can include different material layers; therefore, effective thermal constants are employed to account for the heat flux between this region and the exterior zone. Besides, the coupling between the inner (CF volume) and the intermediate CF surrounding region could be modeled by an effective thermal resistance and thermal capacitance: R

_{th1}and C

_{th1}. Under this approach, the device can be described by the following two equations (we assume this procedure to be the thermal model 7, TM7; see the circuital implementation in Figure 14).

_{th2}and C

_{th2}) that accounts for the coupling between this region and the thermalized device exterior region. The approach described here is in line with the description of a second order memristor [100].

_{th2}, different CF and intermediate surrounding region temperatures transient responses are obtained, producing the corresponding effects on the device current. This is noticeable when a consecutive series of set and reset pulses are applied, as shown in Figure 16a, in which a sequence of set pulses (1.5 V with 1 ns on time and 0.1 ns rise and fall times) and reset pulses (−1.5 V with 1 ns on time and 0.1 ns rise and fall times). In the first configuration, with the thermal capacities C

_{th1}= 0.003 fJ/K, C

_{th2}= 1 fJ/K and thermal resistances R

_{th1}= R

_{th2}= 40 kK/W, the current evolution is shown in Figure 16b. Figure 16c shows the CF (T) and intermediate surrounding region (T

_{S}) evolution. In this first configuration, the corresponding transient shows low thermal inertia; after the pulse application, both temperatures reach room temperature (T = T

_{S}= T

_{0}). The devices show a slight increase in the maximum temperatures obtained in the set (T

_{SET}) and reset (T

_{RESET}) processes (Figure 16c).

_{th1}= 0.003 fJ/K, C

_{th2}= 10 fJ/K and thermal resistances R

_{th1}= R

_{th2}= 40 kK/W, the current evolution is plotted in Figure 16b. Figure 16d shows the CF (T) and intermediate surrounding region (T

_{S}) evolution. In this second configuration, the thermal inertia is higher in the second thermal circuit, after each set/reset pulse, the temperatures cannot go back to room temperature (T, T

_{S}≠ T

_{0}). As a result, each new cycle starts from a higher temperature than in the previous cycle; therefore, the maximum set (T

_{SET}) and reset (T

_{RESET}) temperatures show a growing trend (see Figure 16d). This temperature increase over the cycles implies that the device CF gap decreases in each new cycle, then the current increases (Figure 16b). This effect suggests the consideration of the temperature, in addition to the CF gap, as a state variable in line with the approach presented in [100] for second-order memristor. The temperature increase reported above could be employed with a series of pulses to tune the device conductivity in set cycles within a neuromorphic circuit context [100,105,106]. It is noteworthy that a third-order approach has been introduced in modeling devices for neuromorphic engineering [107].

#### 2.4.4. SPICE-Based Circuital Models with Two or More CF Temperatures

_{Thl1}and R

_{Thl2}. For the sake of generality, it has been split off into two contributions in order to make easier the connection with other thermal sub-circuits and to build more complex thermal models. Their values are given by [89]:

_{th}is the CF thermal conductivity, L

_{i}is the length of the portion of the filament modeled by the sub-circuit and r

_{i}, its radius (index i refers to the cylinder or sub-circuit number 1 or 2 in Figure 17a). On the other hand, R

_{Thn}accounts for the lateral heat dissipation and it is calculated following this expression [89]:

_{2}/Si-n

^{+}resistive switching devices [80,109,110]. The QPC block has also been added. The two cylinders-TM8 model results have been compared with those provided by a finite differences simulator that was used to fit the experimental data [80]. The lateral heat dissipation parameter, h, has been changed in order to check its influence on the i-v curve. As can be seen, more heat dissipation requires a higher voltage to reach the thermally triggered reset transition, a well-known effect in RRAMs.

_{2}/Si-n

^{+}resistive switching device [110] when a 3 V reset pulse is applied (for 100 ns) [37]. Several values of the thermal capacities have been considered. The simulation context here is different from the one shown in Section 2.4.3 since all the modeling components are linked to the device conductive filaments. Note also that only values higher than 0.2 fJ/K influence the device response. Fixed and variable thermal capacities have been used in order to analyze the role of size-dependent thermal capacities, which evolve at simulation time. As expected, variable thermal capacitors, whose value is reduced during a reset process, produce lower thermal inertia.

## 3. General Memristor Modeling Framework with Thermal Effects Emphasis

**X**stands for a set of extra state variables, including all the necessary physical magnitudes according to the implemented memristive system; indicatively, they could be the device internal temperature, the conducting filament radius, or any other non-electrical variable influencing the memristor state that ultimately affects the device charge conduction. Apparently, the dynamics of the state variables

**X**are governed by g

_{Q}(Q,i,

**X**) and Equation (35). It is noted that the importance of the class of extended memristors comes from the fact that all real-world memristor devices known until now are indeed extended memristors. Notice that from Equation (34), we can define the memristance as follows:

_{0}); this implying a possible energy input in some cases if T

_{0}is not constant. In addition, as it has been discussed in Section 2, it is difficult to determine a single value for the device internal temperature (some models, as shown above, include two different temperatures to better describe the device operation). We can write the equations by separating the temperature as follows:

_{0}. Obviously, there is no equation governing T

_{0}dynamics since it can be considered as an external signal. Temperature, T, may be a position-dependent temperature T(x,y,z), as already presented in Section 2.1. In this case, Equation (43) would correspond to the heat equation. As an additional note, it is important to highlight that a device will not present long-term memory characteristics associated solely with temperature, since the device will tend to reach thermal equilibrium with the external medium. In absence of any external electrical input, this would mean that the POP equation related to the evolution of the internal temperature is not zero in the general case. This does not preclude, however, that the system may have other internal variables that do present a long-term memory capability. As an example, we can think if the case of a phase change memory (PCM), where a phase change is activated by temperature, and it remains even after the device has cooled back to room temperature. A similar situation occurs when a RRAM conductive filament is ruptured because of an enhanced diffusion process favored by a temperature rise [24,80].

#### Example of Application

_{0}is the room temperature, and T the device internal temperature. The rest of the symbols are parameters of the thermistor model and can be considered as constants for all practical purposes. At this point, it is noteworthy to point out that these equations bear exactly the same form as Equations (41) and (43) and, thus, identify the thermistor as a memristor. That is, the thermistor is a device whose resistance depends on its electrical history, and it has an internal state variable that governs the overall behaviour (the device internal temperature). Thus, the device can be classified as an extended memristor.

^{−3}W K

^{−1}, C = 60 × 10

^{−3}J K

^{−1}, B = 3950 K, T

_{0}= 298 K, R

_{0}= 10 kΩ. Additionally, and in accordance with a typical thermistor datasheet, we have set a maximum current of 4.5 mA, and we have also used 5 different ramp slopes, as plotted in Figure 22.

_{0}, are bound to fall on the same surface, as seen in Figure 26. Using this representation may provide very interesting insights into the device dynamics. Considering our thermistor, if a trajectory goes from a state with positive derivative to another characterized by a negative derivative with increasing temperature, then it will reach a stable equilibrium point at the temperature where the derivative nullifies. An equilibrium point is a state where the device tends to remain at even if it drifts from it in its operation; this idea resembles a similar concept related to the DC quiescent point in circuit theory, or memory in memristors. In the opposite case, when the trajectory goes from negative to positive, an equilibrium point might seem to come up at the zero-crossing temperature, but it is unstable, which means that the slightest change will force the system to come out of it.

## 4. RRAM Quantum Point Contact Modeling, Thermal Effects

_{0}= 2 e

^{2}/h, where e is the electron charge and h the Planck’s constant [124]. In terms of resistance, this unit is R

_{0}= 1/G

_{0}= 12.9 KΩ. The experimental conductance values for many cycles measured at a fixed bias, or the conductance measured at consecutive steps in one cycle at different or constant biases are often displayed using histogram plots with the x-axes normalized to G

_{0}. In many cases, these histograms reveal a peak structure which is interpreted as an indicator of the number of channels available for conduction or as the occurrence of preferred atomic configurations for the CF [125]. Although the detection of peaks in the device histograms is recognized as the signature of quantum point-contact conduction, it should be taken into account that measurements can be seriously affected by a number of factors such as the existence of multiple conduction paths, series resistance, roughness and scattering caused by the granularity of matter, in general, non-adiabatic (non-smooth) potential profiles. Caution should also be exercised with the use of the term conductance quantization: only for simple s-electron metals, the transmission probability for the conductance channels is expected to open close to integer values [126]. For this reason, observations in the field of RRAM should be more appropriately considered to be in the quantum (rather than in the quantized) regime of conductance [125].

_{x}[128,129,130,131], HfO

_{2}[132,133,134,135,136,137,138,139], Ta

_{2}O

_{5}[140,141,142], NiO [143,144], ZnO [145,146], a-Si:H [147], TiO

_{2}[148], V

_{2}O

_{5}[149], YO

_{x}[150], and BiVO

_{4}[151]. Nonlinear effects in HfO

_{2}were also reported by Degraeve et al. [152] and in CeO

_{x}/SiO

_{2}-based structures by Miranda et al. [153]. From the point of view of theory, it is worth mentioning that the CF formation in monoclinic- and amorphous-HfO

_{2}was investigated from first principles by Cartoixa et al. [154] and by Zhong et al. [155]. The filamentary paths are built from oxygen vacancies and using a Green’s function formalism coupled to a density functional theory code, the conductance of filaments of different lengths was calculated. According to the obtained results, even the thinnest CFs can sustain conductive channels exhibiting signs of quantum conduction.

_{0}and with a linear I-V curve (not to be confused with Ohmic behavior). In this case, the device conductance can reach values from 10 to 100 times G

_{0}which indicates the large number of atoms participating in the filament formation. On the other hand, HRS is associated with conductance values G < G

_{0}and with a non-linear I-V curve (mainly with exponential behavior). This state is characterized by a gap or potential barrier which acts as a blocking element for the electron flow. As the starting point for the inclusion of the thermal effects in RRAMs, the Buttiker-Landauer approach for quantum point contacts is considered [156]. Importantly, the analysis does not discriminate between CBRAMs and OxRAMs, so they are treated on equal grounds.

**ν**is a coefficient related to the curvature of the potential barrier and φ the height of the potential barrier that represents the confinement effect (see Figure 27). For T = 0 K, (47) and (48) yield [159] (see Figure 27):

_{S}in (49), where R

_{S}is a series resistance. Equation (49) can be modified so as to include many parallel conducting channels [138]. For LRS, we can consider that there is no blocking element along the CF so that assuming φ→−∞ (D→1) in (49), we obtain:

_{S}(T) = R

_{S0}·[1 + α

_{T}(T − T

_{0})], where R

_{S0}= R

_{S}(T

_{0}), α

_{T}is a temperature coefficient, and T

_{0}the room temperature. In this case, the I-V characteristic still follows a linear relationship but with a lower slope given by:

_{T}is a positive coefficient, as expected for a metallic-like conductor, the current decreases as the temperature increases (see Figure 28). This behavior is in agreement with the experimental observations [134]. Notice that here the emphasis is put on the connection of the ballistic region with the rest of the device (internal or external) and in particular with the contacts. Nevertheless, R

_{S}can also be viewed as the momentum relaxation factor along the filamentary structure. If we move to the opposite limit, for HRS, and we consider specifically the case E << φ, (48) reads:

_{2}[161], the confinement potential barrier height φ can be parameterized as φ(T) = φ

_{0}-θ(T − T

_{0}), where φ

_{0}

_{=}φ(T

_{0}) and θ > 0 is a linear temperature coefficient. This correction term arises from the thermal movement of ions/vacancies in the CF around their equilibrium positions. In this case, as the temperature increases, the tunneling current increases because of the reduction of the effective barrier height (see Figure 28). The temperature effect on the barrier profile was recently investigated in detail in [139] using inverse modeling in combination with the WKB approximation for the tunneling probability.

_{0}. Notice that the current density flowing through a nanoscale CF can be extraordinarily high. The question can be summarized as, where is power dissipated in a RRAM system exhibiting quantum properties? This is a fundamental question in mesoscopic physics [123]. Let us consider here the progressive increase of the current flow as a function of time when the device is subjected to a constant voltage stress after electroforming. This process corresponds to the transition HRS→LRS which arises because of the CF widening. Following [162], we can write first the following phenomenological equation for the current evolution:

_{C}is the power dissipated at the constriction bottleneck. For the simplest case of a constant applied bias V, Equation (54) expresses that the current levels off in the long run because power dissipation first increases and then progressively transfers from the constriction to the electrodes. Second, according to Landauer’s formula, the transmission probability D (average) can be expressed as a function of the current flowing through the structure as:

_{C}occurring at the constriction using:

_{0}V. The solution to Equation (57) for a constant bias reads:

_{0}and I(t = ∞) = G

_{0}V, the initial and stationary conditions, respectively. Equation (58) expresses that, when a mesoscopic channel with conductance G

_{0}is formed, the power fundamentally dissipates at the electrodes and not at the constriction’s bottleneck (see Figure 29). Power is indeed dissipated at the constriction during the CF formation as discussed in the next section. Of course, this is a simplistic view of a much more complex process.

## 5. Thermometry of Conducting Filaments

_{2}, SiO

_{X}N

_{Y}[167] to innovative 2D dielectrics, such as h-BN [168], passing through high-k materials such as Al

_{2}O

_{3}and HfO

_{2}[169].

_{2}/TiN or with Hf/HfO

_{2}/TiN devices, in which the top electrode (Ti or Hf) acted as cathode. Clear evidence of the formation of a metallic filament made of, respectively, Ti or Hf was reported by using electron energy loss spectroscopy (EELS) imaging [172,173]. In the case of 2D h-BN (CVD) dielectric layers, there is strong evidence that the CFs are formed by metal ions that penetrate from the electrodes into the h-BN stack under the action of the electric field [174,175].

_{2}-based RRAM cells after forming and cycling [172,181] show comparable microstructural changes in the oxide, suggesting the diffusion of the anodic atomic species into the oxide layer in both cases. Thus, these two phenomena share not only similar electrical characteristics, but also generate comparable microstructural changes, suggesting a common underlying physical mechanism. In such scenario, we propose to model the results for the SET event in RRAM devices similarly to the gate-oxide BD in MOSFETs.

^{2}, the current density can reach a few MA/cm

^{2}[184,185,186]) would contribute to the generation and enlargement of the BD filament connecting the electrodes of the stack, enabling the promotion of the electro-migration of the fastest available atomic species. Since this technique unambiguously relate the transition rate (dI

_{Tr}/dt) to the heat dissipation properties during the atomic diffusion of the cathode or anode atoms into the gate dielectric in the region of the percolation path, it is possible estimate the CF temperature. Considering the model reported in [169], we can express the current transition rate (marked as TR) as:

_{ox}is the dielectric thickness, k

_{B}is the Boltzmann constant, D is the diffusion constant of the atomic species responsible for the generation of CF, I

_{SET}is the current level at the onset of the transition, and f

_{1}= n

_{e}λ

_{e}σ

_{e}, with n

_{e}being the electron density, λ

_{e}the electron mean free path and σ

_{e}the cross-section for the electron-atom collision (responsible for the momentum transfer). V is the applied voltage across the BD spot which has been assumed to be equal to the overall externally applied bias between the metal contacts of the stack. f

_{1}value is around the unity since the defect concentration in the CF is most likely very high [172]. According to Equation (59), dI

_{Tr}/dt is proportional to D × I

_{SET}. This means that the BD growth rate rises either by increasing the dominant diffusivity D of the fastest atomic species or by increasing the charge carrier flux.

_{SET}-V curve is usually modeled by assuming a simple analytical dependence as described in [190].

_{SET}× V, k

_{th}is the thermal conductivity of the dielectric, T

_{0}is the room temperature and f

_{2}is the fraction of the energy lost at the constriction:

_{1}, f

_{2}, D

_{0}, and E

_{act}; (taking into consideration that D = D

_{0}*exp(-E

_{act}/k

_{B}T)) and they describe the main features of the progressive BD effect on different stacks [190].

_{2}film sandwiched between Ti and TiN electrodes [193]. During the HRS to LRS transition the current (I

_{Tr}) increases gradually with time evidencing the progressive nature of the SET event (see Figure 30a,b). It is a noisy and progressive process well in agreement with the literature [176,177,178,194] whose duration shows a strong voltage dependence and dispersion. The time evolution of the HRS to LRS transition is quantified by the slope dI

_{Tr}/dt, as defined in [169,185,195]. TR values were experimentally evaluated through measurements such as those shown in Figure 30a,b (approximately 100 measurements for each voltage value).

_{ox}reduction (t

_{ox}considered is equal to t

_{gap}~2 nm due to the forming step) and the increase in diffusivity (D

_{0}is in the order of ~10

^{−6}cm

^{2}/sec as other species are considered to complete the CF, i.e., oxygen vacancies). The rest of the parameters involved remain as previously mentioned in the literature (E

_{act}~0.3–0.7 eV, f

_{2}~0.1 and f

_{1}~1) [193].

_{ox}reduction after the forming step was considered. First, a forming operation (a controlled dielectric BD) creates the CF through the fresh oxide layer. Then, the switching mechanism is driven by the creation of a gap (RESET) and the restoration of the CF (SET). In the case under study, it has been demonstrated using the statistics of the SET switching time (t

_{Set}) (i.e., the time to complete the HRS to LRS transition) that t

_{gap}≈2 nm is a reasonable value [193].

_{2}represents the fraction of energy lost by the carriers, which ranges from 0 to 1. This parameter also depends on the temperature, mainly because of phonon-electron scattering [183]. Therefore, f

_{2}is a function of voltage and temperature whose behavior is found by a best fitting procedure. The influence of f

_{2}on the temperature is shown in Figure 32 for different f

_{2}values.

_{0}is in the order of 3 × 10

^{−6}cm

^{2}/s and E

_{act}= 0.52 eV as indicated in [193]. In HfO

_{2}-based RRAM devices, the SET event is explained as the completion of the gap due to the migration of O

_{2}-ions through a field-assisted and thermally activated effect, which creates the oxygen vacancies that fill the gap along the CF [38,44,53,195,196]. This is quite a relevant point to notice, as the diffusivity of oxygen vacancies (OVs), in a HfO

_{2}layer of thickness like t

_{gap}spread over a range similar to the fitted diffusivity for the TR [197] (see Figure 33).

_{2}and high-k (Al

_{2}O

_{3}or HfO

_{2}) stacks with metal electrodes are of the order of 10

^{−}

^{13}cm

^{2}/s at 1000 K, with activation energies ranging from 0.3 to 0.7 eV [169], where such values are in a range compatible with the diffusivity of metals in dielectrics (see in Figure 33. the case of Cu diffusion into SiO

_{2}layers [198]).

_{act}is also much larger (E

_{act}= 1.3 eV) [184,189]. Such discrepancies may lay on the fact that the particular species involved in the electromigration and/or diffusion process may change, depending on the severity of the SET event (volatile and non-volatile), as it occurs between the BD and SET events in HfO

_{2}stacks. While in the BD event in HfO

_{2}the diffusing ion species are considered to be the metallic ions from the electrodes (D

_{0}= 1 × 10

^{−13}cm

^{2}/sec, E

_{act}= 0.3–0.7 eV [159,169,189]), the migration of oxygen vacancies from the TMO layer are responsible of the SET transition event (D

_{0}= 1 × 10

^{−6}cm

^{2}/s, E

_{act}= 0.52 eV) [188].

_{ox}reduction and the diffusivity increase (D

_{0}), alternative fitting values were used to plot curves N° 2, N° 3 and N° 4 in Figure 31. Curve N°4 coincides with the TR for gate-oxide BD, as it considers diffusivity of metals in oxide layers and no t

_{ox}reduction. The comparison of curves N° 1 and N° 4 evidence that TR is significantly higher than the TR expected for the voltage range considered. It is important to point out that TR calculated considering only a t

_{ox}reduction or an increase in diffusivity (D

_{0}) cannot meet the experimental data. This can be interpreted as that the two factors determine the dependence with the TR voltage, since none of them can separately adjust the results independently.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Lanza, M.; Wong, H.-S.P.; Pop, E.; Ielmini, D.; Strukov, D.; Regan, B.C.; Larcher, L.; Villena, M.A.; Yang, J.J.; Goux, L.; et al. Recommended Methods to Study Resistive Switching Devices. Adv. Electron. Mater.
**2019**, 5, 1800143. [Google Scholar] [CrossRef] [Green Version] - Pan, F.; Gao, S.; Chen, C.; Song, C.; Zeng, F. Recent progress in resistive random access memories: Materials, switching mechanisms, and performance. Mater. Sci. Eng. R Rep.
**2014**, 83, 1–59. [Google Scholar] [CrossRef] - Villena, M.A.; Roldán, J.B.; Jiménez-Molinos, F.; Miranda, E.; Suñé, J.; Lanza, M. SIM
^{2}RRAM: A physical model for RRAM devices simulation. J. Comput. Electron.**2017**, 16, 1095–1120. [Google Scholar] [CrossRef] - IRDS. The International Roadmap for Devices and Systems: More Moore IEEE; IRDS: New York, NY, USA, 2020. [Google Scholar]
- Carboni, R.; Ielmini, D. Stochastic memory devices for security and computing. Adv. Electron. Mater.
**2019**, 5, 1900198. [Google Scholar] [CrossRef] [Green Version] - Puglisi, F.M.; Larcher, L.; Padovani, A.; Pavan, P. A Complete Statistical Investigation of RTN in HfO
_{2}-Based RRAM in High Resistive State. IEEE Trans. Electron Devices**2015**, 62, 2606–2613. [Google Scholar] [CrossRef] - Wei, Z.; Katoh, Y.; Ogasahara, S.; Yoshimoto, Y.; Kawai, K.; Ikeda, Y.; Eriguchi, K.; Ohmori, K.; Yoneda, S. True random number generator using current difference based on a fractional stochastic model in 40-nm embedded ReRAM. In Proceedings of the 2016 IEEE International Electron Devices Meeting (IEDM), IEEE, San Francisco, CA, USA, 3–7 December 2016; pp. 4.8.1–4.8.4. [Google Scholar]
- Puglisi, F.M.; Zagni, N.; Larcher, L.; Pavan, P. random telegraph noise in resistive random access memories: Compact modeling and advanced circuit design. IEEE Trans. Electron Devices
**2018**, 65, 2964–2972. [Google Scholar] [CrossRef] - Lanza, M.; Wen, C.; Li, X.; Zanotti, T.; Puglisi, F.M.; Shi, Y.; Saiz, F.; Antidormi, A.; Roche, S.; Zheng, W.X.; et al. Advanced data encryption using two-dimensional materials. Adv. Mater.
**2021**, in press. [Google Scholar] - Yao, P.; Wu, H.; Gao, B.; Tang, J.; Zhang, Q.; Zhang, W.; Yang, J.J.; Qian, H. Fully hardware-implemented memristor convolutional neural network. Nat. Cell Biol.
**2020**, 577, 641–646. [Google Scholar] [CrossRef] [PubMed] - Merolla, P.A.; Arthur, J.V.; Alvarez-Icaza, R.; Cassidy, A.S.; Sawada, J.; Akopyan, F.; Jackson, B.L.; Imam, N.; Guo, C.; Nakamura, Y.; et al. A million spiking-neuron integrated circuit with a scalable communication network and interface. Science
**2014**, 345, 668–673. [Google Scholar] [CrossRef] - Yu, S.; Gao, B.; Fang, Z.; Yu, H.; Kang, J.; Wong, H.-S.P. A neuromorphic visual system using RRAM synaptic devices with Sub-pJ energy and tolerance to variability: Experimental characterization and large-scale modeling. In Proceedings of the 2012 International Electron Devices Meeting, San Francisco, CA, USA, 10–13 December 2012; pp. 10.4.1–10.4.4. [Google Scholar]
- Zidan, M.A.; Strachan, J.P.; Lu, W.D. The future of electronics based on memristive systems. Nat. Electron.
**2018**, 1, 22–29. [Google Scholar] [CrossRef] - Prezioso, M.; Merrikh-Bayat, F.; Hoskins, B.D.; Adam, G.C.; Likharev, K.K.; Strukov, D.B. Training and operation of an integrated neuromorphic network based on metal-oxide memristors. Nature
**2015**, 521, 61–64. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Romero-Zaliz, R.; Pérez, E.; Jiménez-Molinos, F.; Wenger, C.; Roldán, J. Study of Quantized Hardware Deep Neural Networks Based on Resistive Switching Devices, Conventional versus Convolutional Approaches. Electronics
**2021**, 10, 346. [Google Scholar] [CrossRef] - Quesada, E.P.-B.; Romero-Zaliz, R.; Pérez, E.; Mahadevaiah, M.K.; Reuben, J.; Schubert, M.; Jiménez-Molinos, F.; Roldán, J.; Wenger, C. Toward Reliable Compact Modeling of Multilevel 1T-1R RRAM Devices for Neuromorphic Systems. Electronics
**2021**, 10, 645. [Google Scholar] [CrossRef] - Mead, C.; Ismail, M. Analog VLSI Implementation of Neural Systems; Springer: Berlin/Heidelberg, Germany, 1989. [Google Scholar]
- Ielmini, D. Resistive switching memories based on metal oxides: Mechanisms, reliability and scaling. Semicond. Sci. Technol.
**2016**, 31, 063002. [Google Scholar] [CrossRef] - Ielmini, D.; Milo, V. Physics-based modeling approaches of resistive switching devices for memory and in-memory computing applications. J. Comput. Electron.
**2017**, 16, 1121–1143. [Google Scholar] [CrossRef] [Green Version] - Huang, P.; Gao, B.; Chen, B.; Zhang, F.; Liu, L.; Du, G. Stochastic simulation of forming, SET and RESET process for transition metal oxide-based resistive switching memory. Proc. SISPAD
**2011**, 2012, 312–315. [Google Scholar] - Aldana, S.; Roldán, J.B.; García-Fernández, P.; Suñe, J.; Romero-Zaliz, R.; Jiménez-Molinos, F.; Long, S.; Gómez-Campos, F.; Liu, M. An in-depth description of bipolar resistive switching in Cu/HfO
_{x}/Pt devices, a 3D Kinetic Monte Carlo simulation approach. J. Appl. Phys.**2018**, 123, 154501. [Google Scholar] [CrossRef] - Garcia-Redondo, F.; Gowers, R.P.; Crespo-Yepes, A.; Lopez-Vallejo, M.; Jiang, L. SPICE Compact modeling of bipolar/unipolar memristor switching governed by electrical thresholds. IEEE Trans. Circuits Syst. I Regul. Pap.
**2016**, 63, 1255–1264. [Google Scholar] [CrossRef] [Green Version] - Dirkmann, S.; Kaiser, J.; Wenger, C.; Mussenbrock, T. Filament growth and resistive switching in hafnium oxide memristive devices. ACS Appl. Mater. Interfaces
**2018**, 10, 14857–14868. [Google Scholar] [CrossRef] - Aldana, S.; García-Fernández, P.; Rodríguez-Fernández, A.; Romero-Zaliz, R.; González, M.B.; Jiménez-Molinos, F.; Campabadal, F.; Gómez-Campos, F.; Roldán, J.B. A 3D kinetic monte carlo simulationstudy of resistive switching processes in Ni/HfO
_{2}/Si-n+-based RRAMs. J. Phys. D**2017**, 50, 335103. [Google Scholar] [CrossRef] - Jagath, A.L.; Nandha Kumar, T.; Almurib, H.A.F. Modeling of Current Conduction during RESET Phase of Pt/Ta
_{2}O_{5}/TaO_{x}/Pt Bipolar Resistive RAM Devices. In Proceedings of the 2018 IEEE 7th Non-Volatile Memory Systems and Applications Symposium (NVMSA), Hakodate, Japan, 28–31 August 2018; pp. 55–60. [Google Scholar] - Fang, X.; Yang, X.; Wu, J.; Yi, X. A Compact SPICE model of unipolar memristive devices. IEEE Trans. Nanotechnol.
**2013**, 12, 843–850. [Google Scholar] [CrossRef] - González-Cordero, G.; González, M.B.; García, H.; Campabadal, F.; Dueñas, S.; Castán, H.; Jiménez-Molinos, F.; Roldán, J.B. A physically based model for resistive memories including a detailed temperature and variability description. Microelectron. Eng.
**2017**, 178, 26–29. [Google Scholar] [CrossRef] - Karpov, V.; Niraula, D.; Karpov, I. Thermodynamic analysis of conductive filaments. Appl. Phys. Lett.
**2016**, 109, 093501. [Google Scholar] [CrossRef] - Maestro-Izquierdo, M.; Gonzalez, M.B.; JimenezMolinos, F.; Moreno, E.; Roldan, J.B.; Campabadal, F. Unipolar resistive switching behavior in Al
_{2}O_{3}/HfO_{2}multilayer dielectric stacks: Fabrication, characterization and simulation. Nanotechnology**2020**, 31, 135202. [Google Scholar] [CrossRef] [PubMed] - Larentis, S.; Nardi, F.; Balatti, S.; Gilmer, D.C.; Ielmini, D. Resistive Switching by Voltage-Driven Ion Migration in Bipolar RRAM—Part II: Modeling. IEEE Trans. Electron Devices
**2012**, 59, 2468–2475. [Google Scholar] [CrossRef] - Jimenez-Molinos, F.; Villena, M.A.; Roldan, J.B.; Roldan, A.M. A SPICE compact model for unipolar RRAM reset process analysis. IEEE Trans. Electron Devices
**2015**, 62, 955–962. [Google Scholar] [CrossRef] - Menzel, S.; Kaupmann, P.; Waser, R. Understanding filamentary growth in electrochemical metallization memory cells using kinetic Monte Carlo simulations. Nanoscale
**2015**, 7, 12673–12681. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Picos, R.; Roldán, J.B.; al Chawa, M.M.; García-Fernández, P.; García-Moreno, F.J.Y.E. Semiempirical modeling of reset transitions in unipolar resistive-switching based memristors. Radioeng. J.
**2015**, 24, 420–424. [Google Scholar] [CrossRef] - Vandelli, L.; Padovani, A.; Larcher, L.; Bersuker, G. Microscopic modeling of electrical stress-induced breakdown in poly-crystalline hafnium oxide dielectrics. IEEE Trans. Electron Devices
**2013**, 60, 1754–1762. [Google Scholar] [CrossRef] - Blasco, J.; Ghenzi, N.; Suñé, J.; Levy, P.; Miranda, E. Equivalent circuit modeling of the bistable conduction characteristics in electroformed thin dielectric films. Microelectron. Reliab.
**2015**, 55, 1–14. [Google Scholar] [CrossRef] - Maldonado, D.; Gonzalez, M.B.; Campabadal, F.; JimenezMolinos, F.; Al Chawa, M.M.; Stavrinides, S.G.; Roldan, J.B.; Tetzlaff, R.; Picos, R.; Chua, L.O. Experimental evaluation of the dynamic route map in the reset transition of memristive ReRAMs. Chaos Solitons Fractals
**2020**, 139, 110288. [Google Scholar] [CrossRef] - Jiménez-Molinos, F.; González-Cordero, G.; Cartujo-Cassinello, P.; Roldán, J.B. SPICE modeling of RRAM thermal reset transition for circuit simulation purposes. In Proceedings of the Spanish Conference on Electron Devices, Barcelona, Spain, 8–10 February 2017. [Google Scholar]
- Bocquet, M.; Deleruyelle, D.; Aziza, H.; Muller, C.; Portal, J.-M.; Cabout, T.; Jalaguier, E. Robust compact model for bipolar oxide-based resistive switching memories. IEEE Trans. Electron. Devices
**2014**, 61, 674–681. [Google Scholar] [CrossRef] - al Chawa, M.M.; Picos, R.; Tetzlaff, R. A Simple Memristor Model for Neuromorphic ReRAM Devices. In Proceedings of the 2020 IEEE International Symposium on Circuits and Systems (ISCAS), Seville, Spain, 10–21 October 2020. [Google Scholar]
- al Chawa, M.M.; Picos, R. A simple quasi-static compact model of bipolar ReRAM memristive devices. IEEE Trans. Circuits Syst. II
**2020**, 67, 390–394. [Google Scholar] [CrossRef] - Panda, D.; Sahu, P.P.; Tseng, T.Y. A collective study on modeling and simulation of resistive random access memory. Nanoscale Res. Lett.
**2018**, 13, 1–48. [Google Scholar] [CrossRef] [PubMed] - Reuben, J.; Biglari, M.; Fey, D. Incorporating Variability of Resistive RAM in Circuit Simulations Using the Stanford–PKU Model. IEEE Trans. Nanotechnol.
**2020**, 19, 508–518. [Google Scholar] [CrossRef] - Mikhaylov, A.; Guseinov, D.; Belov, A.; Korolev, D.; Shishmakova, V.; Koryazhkina, M.; Filatov, D.; Gorshkov, O.; Maldonado, D.; Alonso, F.; et al. Stochastic resonance in a metal-oxide memristive device. Chaos Solitons Fractals
**2021**, 144, 110723. [Google Scholar] [CrossRef] - Aldana, S.; Pérez, E.; JimenezMolinos, F.; Wenger, C.; Roldán, J.B. Kinetic Monte Carlo analysis of data retention in Al:HfO
_{2}-based resistive random access memories. Semicond. Sci. Technol.**2020**, 35, 115012. [Google Scholar] [CrossRef] - Roldán, J.B.; Alonso, F.J.; Aguilera, A.M.; Maldonado, D.; Lanza, M. Time series statistical analysis: A powerful tool to evaluate the variability of resistive switching memories. J. Appl. Phys.
**2019**, 125, 174504. [Google Scholar] [CrossRef] - Miranda, E.; Mehonic, A.; Ng, W.H.; Kenyon, A.J. Simulation of cycle-to-cycle instabilities in SiO
_{x}-based ReRAM devices using a self-correlated process with long-term variation. IEEE EDL**2019**, 40, 28–31. [Google Scholar] [CrossRef] - Kvatinsky, S.; Ramadan, M.; Friedman, E.G.; Kolodny, A. VTEAM: A general model for voltage-controlled memristors. IEEE Trans. Circuits Syst. II
**2015**, 62, 786–790. [Google Scholar] [CrossRef] - Picos, R.; Roldan, J.B.; Al Chawa, M.M.; JimenezMolinos, F.; Garcia-Moreno, E. A physically based circuit model to account for variability in memristors with resistive switching operation. In Proceedings of the 2016 Conference on Design of Circuits and Integrated Systems (DCIS), Granada, Spain, 23–25 November 2016; pp. 1–6. [Google Scholar]
- al Chawa, M.M.; de Benito, C.; Picos, R. A simple piecewise model of reset/set transitions in bipolar ReRAM memristive devices. IEEE Trans. Circuits Syst. I
**2018**, 65, 3469–3480. [Google Scholar] [CrossRef] - al Chawa, M.M.; Tetzlaff, R.; Picos, R. A Simple Monte Carlo Model for the Cycle-to-Cycle Reset Transition Variation of ReRAM Memristive Devices. In Proceedings of the 9th International Conference on Modern Circuits and Systems Technologies (MOCAST), Bremen, Germany, 7–9 September 2020. [Google Scholar]
- Alonso, F.J.; Maldonado, D.; Aguilera, A.M.; Roldan, J.B. Memristor variability and stochastic physical properties modeling from a multivariate time series approach. Chaos Solitons Fractals
**2021**, 143, 110461. [Google Scholar] [CrossRef] - Pérez, E.; Maldonado, D.; Acal, C.; Ruiz-Castro, J.E.; Alonso, F.J.; Aguilera, A.M.; Jiménez-Molinos, F.; Wenger, C.; Roldán, J.B. Analysis of the statistics of device-to-device and cycle-to-cycle variability in TiN/Ti/Al:HfO
_{2}/TiN RRAMs. Microelectron. Eng.**2019**, 214, 104–109. [Google Scholar] [CrossRef] - Aldana, S.; García-Fernández, P.; Romero-Zaliz, R.; González, M.B.; Jiménez-Molinos, F.; Gómez-Campos, F.; Campabadal, F.; Roldán, J.B. Resistive switching in HfO
_{2}based valence change memories, a comprehensive 3D kinetic Monte Carlo approach. J. Phys. D**2020**, 53, 225106. [Google Scholar] [CrossRef] - Guy, J.; Molas, G.; Blaise, P.; Bernard, M.; Roule, A.; Le Carval, G.; Delaye, V.; Toffoli, A.; Ghibaudo, G.; Clermidy, F.; et al. Investigation of forming, SET, and data retention of conductive-bridge random-access memory for stack optimization. IEEE Trans. Electron Devices
**2015**, 62, 3482–3489. [Google Scholar] [CrossRef] - Villena, M.A.; Roldán, J.B.; González, M.B.; González-Rodelas, P.; Jiménez-Molinos, F.; Campabadal, F.; Barrera, D. A new parameter to characterize the charge transport regime in Ni/HfO
_{2}/Si-n^{+}-based RRAMs. Solid State Electron.**2016**, 118, 56–60. [Google Scholar] [CrossRef] - González-Cordero, G.; Roldán, J.B.; Jiménez-Molinos, F. SPICE simulation of RRAM circuits. A compact modeling perspective. In Proceedings of the 2017 Spanish Conference on Electron Devices, Barcelona, Spain, 8–10 February 2017; pp. 26–29. [Google Scholar]
- Huang, P.; Zhu, D.; Chen, S.; Zhou, Z.; Chen, Z.; Gao, B.; Liu, L.; Liu, X.; Kang, J. Compact model of HfO
_{X}-based electronic synaptic devices for neuromorphic computing. IEEE Trans. Electron. Devices**2017**, 64, 614–621. [Google Scholar] [CrossRef] - Kwon, S.; Jang, S.; Choi, J.-W.; Choi, S.; Jang, S.-J.; Kim, T.-W.; Wang, G. Controllable switching filaments prepared via tunable and well-defined single truncated conical nanopore structures for fast and scalable SiO
_{x}memory. Nanoletters**2017**, 17, 7462–7470. [Google Scholar] [CrossRef] - Villena, M.; Gonzalez, M.B.; Roldán, J.; Campabadal, F.; Jiménez-Molinos, F.; Gómez-Campos, F.; Suñé, J. An in-depth study of thermal effects in reset transitions in HfO
_{2}based RRAMs. Solid-State Electron.**2015**, 111, 47–51. [Google Scholar] [CrossRef] - Lohn, A.J.; Mickel, P.R.; Marinella, M.J. Analytical estimations for thermal crosstalk, retention, and scaling limits in filamentary resistive memory. J. Appl. Phys.
**2014**, 115, 234507. [Google Scholar] [CrossRef] - Sun, P.; Lu, N.; Li, L.; Li, Y.; Wang, H.; Lv, H.; Liu, Q.; Long, S.; Liu, S.; Liu, M. Thermal crosstalk in 3-dimensional RRAM crossbar array. Sci. Rep.
**2015**, 5, 13504. [Google Scholar] [CrossRef] [PubMed] - Deshmukh, S.; Islam, R.; Chen, C.; Yalon, E.; Saraswat, K.C.; Pop, E. Thermal modeling of metal oxides for highly scaled nanoscale RRAM. In Proceedings of the 2015 International Conference on Simulation of Semiconductor Processes and Devices (SISPAD), Washington, DC, USA, 9–11 September 2015; Volume 2015, pp. 281–284. [Google Scholar]
- Wang, D.-W.; Chen, W.; Zhao, W.-S.; Zhu, G.-D.; Kang, K.; Gao, P.; Schutt-Aine, J.E.; Yin, W.-Y. Fully Coupled Electrothermal Simulation of Large RRAM Arrays in the “Thermal-House”. IEEE Access
**2018**, 7, 3897–3908. [Google Scholar] [CrossRef] - Rodríguez, N.; Roldán, F.G.y.J.B. Modeling of inversion layer centroid and polysilicon depletion effects on ultrathin-gate-oxide MOSFET behaviour: The influence of crystallographic orientation. IEEE Trans. Electron. Devices
**2007**, 54, 723–732. [Google Scholar] [CrossRef] - González, B.; Roldán, J.; Iniguez, B.; Lazaro, A.; Cerdeira, A. DC self-heating effects modelling in SOI and bulk FinFETs. Microelectron. J.
**2015**, 46, 320–326. [Google Scholar] [CrossRef] - Roldán, J.B.; Gámiz, F.; JimenezMolinos, F.; Sampedro, C.; Godoy, A.; Rodríguez, N. An analytic I-V model for surrounding-gate MOSFET including quantum and velocity overshoot effects. IEEE Trans. Electron. Devices
**2010**, 57, 2925–2933. [Google Scholar] [CrossRef] - Blanco-Filgueira, B.; Roldán, P.L.Y.J.B. Analytical modeling of size effects on the lateral photoresponse of CMOS photodiodes. Solid State Electron.
**2012**, 73, 15–20. [Google Scholar] [CrossRef] - Blanco-Filgueira, B.; Roldán, P.L.y.J.B. A closed-form and explicit analytical model for crosstalk in CMOS photodiodes. IEEE Trans. Electron Devices
**2013**, 60, 3459–3464. [Google Scholar] [CrossRef] - Gámiz, F.; Godoy, A.; Donetti, L.; Sampedro, C.; Roldán, J.B.; Ruiz, F.; Tienda, I.; Jiménez-Molinos, N.R.Y.F. Monte Carlo simulation of nanoelectronic devices. J. Comput. Electron.
**2009**, 8, 174–191. [Google Scholar] [CrossRef] - Ielmini, D.; Waser, R. Resistive Switching: From Fundamentals of Nanoionic Redox Processes to Memristive Device Applications; Wiley-VCH: Hoboken, NJ, USA, 2015. [Google Scholar]
- Corinto, F.; Civalleri, P.P.; Chua, L.O. A theoretical approach to memristor devices. IEEE J. Emerg. Sel. Top. Circuits Syst.
**2015**, 5, 123–132. [Google Scholar] [CrossRef] [Green Version] - Chua, L.O. Everything you wish to know about memristors but are afraid to ask. Radioengineering
**2015**, 24, 319–368. [Google Scholar] [CrossRef] - James, A.P. A hybrid memristor–CMOS chip for AI. Nat. Electron.
**2019**, 2, 268–269. [Google Scholar] [CrossRef] - Volos, C.K.; Kyprianidis, I.M.; Stavrinides, S.G.; Stouboulos, I.N.; Anagnostopoulos, A.N. Memristors: A new approach in nonlinear circuits design. In Proceedings of the 14th WSEAS International Conference on Communication, Cape Town, South Africa, 23–27 May 2010; pp. 25–30. [Google Scholar]
- Li, Y.; Wang, Z.; Midya, R.; Xia, Q.; Yang, J.J. Review of memristor devices in neuromorphic computing: Materials sciences and device challenges. J. Phys. D Appl. Phys.
**2018**, 51, 503002. [Google Scholar] [CrossRef] - Padovani, A.; Larcher, L.; Pirrotta, O.; Vandelli, L.; Bersuker, G. Microscopic Modeling of HfO
_{x}RRAM operations: From forming to switching. IEEE Trans. Electron. Devices**2015**, 62, 1998–2006. [Google Scholar] [CrossRef] - Cazorla, M.; Aldana, S.; Maestro, M.; González, M.B.; Campabadal, F.; Moreno, E.; Jiménez-Molinos, F.; Roldán, J.B. A thermal study of multilayer RRAMs based on HfO
_{2}and Al_{2}O_{3}oxides. J. Vac. Sci. Technol. B**2019**, 37, 012204. [Google Scholar] [CrossRef] - Guan, X.; Yu, S.; Wong, H.-S.P. A SPICE compact model of metal oxide resistive switching memory with variations. IEEE Electron Device Lett.
**2012**, 33, 1405–1407. [Google Scholar] [CrossRef] - González-Cordero, G.; Roldan, J.B.; Jiménez-Molinos, F.; Suñé, J.; Liu, S.L.y.M. A new model for bipolar RRAMs based on truncated cone conductive filaments, a Verilog-A approach. Semicond. Sci. Technol.
**2016**, 31, 115013. [Google Scholar] [CrossRef] - Villena, M.A.; González, M.B.; Jiménez-Molinos, F.; Campabadal, F.; Roldán, J.B.; Suñé, J.; Romera, E.; Miranda, E. Simulation of thermal reset transitions in resistive switching memories including quantum effects. J. Appl. Phys.
**2014**, 115, 214504. [Google Scholar] [CrossRef] - Von Witzleben, M.; Fleck, K.; Funck, C.; Baumkötter, B.; Zuric, M.; Idt, A.; Breuer, T.; Waser, R.; Böttger, U.; Menzel, S. Investigation of the impact of high temperatures on the switching kinetics of redox-based resistive switching cells using a high-speed nanoheater. Adv. Electron. Mater.
**2017**, 3, 1700294. [Google Scholar] [CrossRef] - Lantos, N.; Nataf, F. Perfectly matched layers for the heat and advection–diffusion equations. J. Comput. Phys.
**2010**, 229, 9042–9052. [Google Scholar] [CrossRef] [Green Version] - Moreno, E.; Hemmat, Z.; Roldan, J.B.; Pantoja, M.F.; Bretones, A.R.; Garcia, S.G.; Faez, R. Implementation of open boundary problems in photo-conductive antennas by using convolutional perfectly matched layers. IEEE Trans. Antennas Propag.
**2016**, 64, 4919–4922. [Google Scholar] [CrossRef] - González-Cordero, G. Compact Modeling of Memristors Based on Resistive Switching Devices. Ph.D. Thesis, Universidad de Granada, Granada, Spain, 2020. [Google Scholar]
- Villena, M.A.; Jiménez-Molinos, F.; Roldan, J.B.; Suñe, J.; Long, S.; Lian, X.; Gamiz, F.; Liu, M. An in-depth simulation study of thermal reset transitions in resistive switching memories. J. Appl. Phys.
**2013**, 114, 144505. [Google Scholar] [CrossRef] - Guan, X.; Yu, S.; Wong, H.S.P. On the Variability of HfO
_{x}RRAM: From Numerical Simulation to Compact Modeling Technical. In Proceedings of the 2012 NSTI Nanotechnology Conference and Expo, NSTI-Nanotech, Santa Clara, CA, USA, 18–21 June 2012; Volume 2, pp. 815–820. [Google Scholar] - Jiang, Z.; Yu, S.; Wu, Y.; Engel, J.H.; Guan, X.; Wong, H.-S.P. Verilog-A compact model for oxide-based resistive random access memory (RRAM). In Proceedings of the 2014 International Conference on Simulation of Semiconductor Processes and Devices (SISPAD), Yokohama, Japan, 9–11 September 2014; pp. 41–44. [Google Scholar]
- Jiang, Z.; Wu, Y.; Yu, S.; Yang, L.; Song, K.; Karim, Z.; Wong, H.-S.P. A compact model for metal–oxide resistive random access memory with experiment verification. IEEE Trans. Electron Devices
**2016**, 63, 1884–1892. [Google Scholar] [CrossRef] - Moran, M.J.; Shapiro, H.N.; Munson, B.R.; Dewitt, D.P.; Wiley, J.; Hepburn, K.; Fleming, L. Introduction to Thermal Systems Engineering: And Heat Transfer; John Wiley & Sons Inc.: New York, NY, USA, 2003; Volume 169, ISBN 0471204900. [Google Scholar]
- Sheridan, P.; Kim, K.-H.; Gaba, S.; Chang, T.; Chen, L.; Lu, W. Device and SPICE modeling of RRAM devices. Nanoscale
**2011**, 3, 3833–3840. [Google Scholar] [CrossRef] [PubMed] - Huang, P.; Liu, X.Y.; Chen, B.; Li, H.T.; Wang, Y.J.; Deng, Y.X.; Wei, K.L.; Zeng, L.; Gao, B.; Du, G.; et al. A physics-based compact model of metal-oxide-based RRAM DC and AC operations. IEEE Trans. Electron Devices
**2013**, 60, 4090–4097. [Google Scholar] [CrossRef] - Li, H.; Huang, P.; Gao, B.; Chen, B.; Liu, X.; Kang, J. A SPICE Model of Resistive Random Access Memory for Large-Scale Memory Array Simulation. IEEE Electron Device Lett.
**2013**, 35, 211–213. [Google Scholar] [CrossRef] - Li, H.; Jiang, Z.; Huang, P.; Wu, Y.; Chen, H.Y.; Gao, B.; Liu, X.Y.; Kang, J.F.; Wong, H.S. Variation-Aware, Reliability-Emphasized Design and Optimization of RRAM using SPICE Model. In Proceedings of the Design, Automation & Test in Europe Conference & Exhibition, Grenoble, France, 9–13 March 2015; pp. 1425–1430. [Google Scholar]
- Chen, A. A review of emerging non-volatile memory (NVM) technologies and applications. Solid State Electron.
**2016**, 125, 25–38. [Google Scholar] [CrossRef] - Chen, P.-Y.; Yu, S. Compact modeling of RRAM devices and its applications in 1T1R and 1S1R array design. IEEE Trans. Electron. Devices
**2015**, 62, 4022–4028. [Google Scholar] [CrossRef] - Chiang, M.-H.; Hsu, K.-H.; Ding, W.-W.; Yang, B.-R. A predictive compact model of bipolar RRAM cells for circuit simulations. IEEE Trans. Electron. Devices
**2015**, 62, 2176–2183. [Google Scholar] [CrossRef] - Kwon, J.; Sharma, A.A.; Chen, C.M.; Fantini, A.; Jurczak, M.; Herzing, A.A.; Bain, J.A.; Picard, Y.N.; Skowronski, M. Transient thermometry and high-resolution transmission electron microscopy analysis of filamentary resistive switches. ACS Appl. Mater. Interfaces
**2016**, 8, 20176. [Google Scholar] [CrossRef] - Sharma, A.A.; Noman, M.; Skowronski, M.; Bain, J.A. Technology, Systems and Applications (VLSI-TSA). In Proceedings of the 2014 International Symposium on VLSI Technology, Systems and Applications, Hsinchu, Taiwan, 28–30 April 2014; p. 1. [Google Scholar]
- Nishi, Y.; Menzel, S.; Fleck, K.; Boettger, U.; Waser, R. Origin of the SET kinetics of the resistive switching in tantalum oxide thin films. IEEE Electron. Device Lett.
**2013**, 35, 259–2061. [Google Scholar] [CrossRef] - Kim, S.; Du, C.; Sheridan, P.; Ma, W.; Choi, S.; Lu, W.D. Experimental demonstration of a second-order memristor and its ability to biorealistically implement synaptic plasticity. Nano Lett.
**2015**, 15, 2203–2211. [Google Scholar] [CrossRef] - Panzer, M.A.; Shandalov, M.; Rowlette, J.A.; Oshima, Y.; Chen, Y.W.; McIntyre, P.C.; Goodson, K.E. Thermal Properties of Ultrathin Hafnium Oxide Gate Dielectric Films. IEEE Electron Device Lett.
**2009**, 30, 1269–1271. [Google Scholar] [CrossRef] - Scott, E.A.; Gaskins, J.T.; King, S.W.; Hopkins, P.E. Thermal conductivity and thermal boundary resistance of atomic layer deposited high-k dielectric aluminum oxide, hafnium oxide, and titanium oxide thin films on silicon. APL Mater.
**2018**, 6, 058302. [Google Scholar] [CrossRef] [Green Version] - Roldán, A.M.; Roldán, J.B.; Reig, C.; Cubells-Beltrán, M.-D.; Ramírez, D.; Cardoso, S.; Freitas, P.P. A DC behavioral electrical model for quasi-linear spin-valve devices including thermal effects for circuit simulation. Microelectron. J.
**2011**, 42, 365–370. [Google Scholar] [CrossRef] - Busani, M.; Menozzi, R.; Borgarino, M.; Fantini, F. Dynamic thermal characterization and modeling of packaged AlGaAs/GaAs HBTs. IEEE Trans. Compon. Packag. Technol.
**2000**, 23, 352–359. [Google Scholar] [CrossRef] [Green Version] - Pedro, M.; Martin-Martinez, J.; Gonzalez, M.; Rodriguez, R.; Campabadal, F.; Nafria, M.; Aymerich, X. Tuning the conductivity of resistive switching devices for electronic synapses. Microelectron. Eng.
**2017**, 178, 89–92. [Google Scholar] [CrossRef] - González-Cordero, G.; Pedro, M.; Martin-Martinez, J.; González, M.; Jiménez-Molinos, F.; Campabadal, F.; Nafría, N.; Roldán, J. Analysis of resistive switching processes in TiN/Ti/HfO
_{2}/W devices to mimic electronic synapses in neuromorphic circuits. Solid-State Electron.**2019**, 157, 25–33. [Google Scholar] [CrossRef] - Kumar, S.; Williams, R.S.; Wang, Z. Third-order nanocircuit elements for neuromorphic engineering. Nat. Cell Biol.
**2020**, 585, 518–523. [Google Scholar] [CrossRef] - González-Cordero, G.; Jiménez-Molinos, F.; Roldán, J.B.; González, M.B.; Campabadal, F. Transient SPICE Simulation of Ni/HfO
_{2}/Si-n^{+}Resistive Memories. In Proceedings of the Design of Circuits and Integrated Systems Conference, DCIS, Granada, Spain, 23–25 November 2016. [Google Scholar] - González-Cordero, G.; Jiménez-Molinos, F.; Villena, M.A.; Roldán, J.B. SPICE Simulation of Thermal Reset Transitions in Ni/HfO
_{2}/Si-n^{+}RRAMs Including Quantum Effects. In Proceedings of the 19th Workshop on Dielectrics in Microelectronics, WoDiM, Catania, Italy, 27–30 June 2016. [Google Scholar] - González, M.B.; Rafí, J.M.; Beldarrain, O.; Zabala, M.; Campabadal, F. Analysis of the switching variability in Ni/HfO
_{2}-based RRAM devices. IEEE Trans. Device Mater. Reliab.**2014**, 14, 769–771. [Google Scholar] - Wang, W.; Laudato, M.; Ambrosi, E.; Bricalli, A.; Covi, E.; Lin, Y.-H.; Ielmini, D. Volatile Resistive Switching Memory Based on Ag Ion Drift/Diffusion—Part II: Compact Modeling. IEEE Trans. Electron Devices
**2019**, 66, 3802–3808. [Google Scholar] [CrossRef] - Chua Leon, O.; Kang, S.M. Memristive devices and systems. Proc. IEEE
**1976**, 64, 209–223. [Google Scholar] [CrossRef] - Ginoux, J.M.; Muthuswamy, B.; Meucci, R.; Euzzor, S.; Di Garbo, A.; Ganesan, K. A physical memristor based Muthuswamy-Chua-Ginoux system. Sci. Rep.
**2020**, 10, 1–10. [Google Scholar] [CrossRef] - Steinhart, J.S.; Hart, S.R. Calibration curves for thermistors. Deep Sea Res. Oceanogr. Abstr.
**1968**, 15, 497–503. [Google Scholar] [CrossRef] - Theodorakakos, A.; Stavrinides, S.G.; Hatzikraniotis, E.; Picos, R. A non-ideal memristor device. In Proceedings of the 2015 International Conference on Memristive Systems (MEMRISYS), Paphos, Cyprus, 8–10 November 2015; pp. 1–2. [Google Scholar]
- Biolek, D.; Biolek, Z.; Biolkova, V.; Kolka, Z. Some fingerprints of ideal memristors. In Proceedings of the 2013 IEEE International Symposium on Circuits and Systems (ISCAS), Beijing, China, 19–23 May 2013; pp. 201–204. [Google Scholar]
- Wagner, G.; Jones, R.; Templeton, J.; Parks, M. An atomistic-to-continuum coupling method for heat transfer in solids. Comput. Methods Appl. Mech. Eng.
**2008**, 197, 3351–3365. [Google Scholar] [CrossRef] [Green Version] - Xu, Z. Heat transport in low-dimensional materials: A review and perspective. Theor. Appl. Mech. Lett.
**2016**, 6, 113–121. [Google Scholar] [CrossRef] [Green Version] - Mosso, N.; Drechsler, U.; Menges, F.; Nirmalraj, P.; Karg, S.; Riel, H.; Gotsmann, B. Heat transport through atomic contacts. Nat. Nanotech.
**2017**, 12, 430–433. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hanggi, P.; Talkner, P.; Borkovec, H. Reaction-rate theory: Fifty years after Kramers. Rev. Mod. Phys.
**1990**, 62, 251. [Google Scholar] [CrossRef] - Chiu, F. A review on conduction mechsnisms in dielectric films. Adv. Mater. Sci. Eng.
**2014**, 2014, 578168. [Google Scholar] [CrossRef] [Green Version] - Waser, R.; Dittmann, R.; Saikov, G.; Szot, K. Redox-based resistive switching memories nanoionic mechanisms, prospects, and challenges. Adv. Mater.
**2009**, 21, 2632–2663. [Google Scholar] [CrossRef] - Datta, S. Electronic Transport in Mesoscopic Systems; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Kouwenhoven, L.P.; Van Wees, B.J.; Harmans, C.J.P.M.; Williamson, J.G.; Van Houten, H.; Beenakker, C.W.J.; Foxon, C.T.; Harris, J.J. Nonlinear conductance of quantum point contacts. Phys. Rev. B
**1989**, 39, 8040–8043. [Google Scholar] [CrossRef] [PubMed] [Green Version] - van Ruitenbeek, J.; Masis, M.M.; Miranda, E. Quantum point contact conduction. In Resistice Switching: From Fundamentals of Nanoinic Redox Processes to Memristive Device Applications; Ielmini, D., Waser, R., Eds.; John Wiley & Sons Inc.: New York, NY, USA, 2016; pp. 197–224. [Google Scholar]
- Agrait, N.; Yeyati, A.L.; van Ruitenbeek, J.M. Quantum properties of atomic-sized conductors. Phys. Rep.
**2003**, 377, 81. [Google Scholar] [CrossRef] [Green Version] - Li, Y.; Long, S.; Liu, Y.; Hu, C.; Teng, J.; Liu, Q.; Lv, H.; Suñé, J.; Liu, M. Conductance quantization in resistive random access memory. Nanoscale Res. Lett.
**2015**, 10, 420. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Suñé, J.; Miranda, E.; Nafría, M.; Aymerich, X. Point contact conduction at the oxide breakdown of MOS devices. In Proceedings of the IEEE International Electron Device Meeting (IEDM), San Francisco, CA, USA, 6–9 December 1998; p. 191. [Google Scholar]
- Suñé, J.; Miranda, E.; Nafría, M.; Aymerich, X. Modeling the breakdown spots in silicon dioxide films as point contacts. Appl. Phys. Lett.
**1999**, 75, 959–961. [Google Scholar] [CrossRef] [Green Version] - Mehonic, A.; Vrajitoarea, A.; Cueff, S.; Hudziak, S.; Howe, H.; Labbe, C.; Rizk, R.; Pepper, M.; Kenyon, A.J. Quantum conductance in silicon oxide resistive memory devices. Sci. Rep.
**2013**, 3, 2708. [Google Scholar] [CrossRef] [PubMed] - Nandakumar, S.R.; Minvielle, M.; Nagar, S.; Dubourdieu, C.; Rajendran, B. A 250 mV Cu/SiO
_{2}/W Memristor with Half-Integer Quantum Conductance States. Nano Lett.**2016**, 16, 1602–1608. [Google Scholar] [CrossRef] [PubMed] - Miranda, E.; Walczyk, C.; Wenger, C.; Schroeder, T. Model for the resistive switching effect in HfO
_{2}MIM structures based on the transmission properties of narrow constrictions. IEEE Electron. Device Lett.**2010**, 31, 609–611. [Google Scholar] [CrossRef] - Degraeve, R.; Roussel, P.; Goux, L.; Wouters, D.; Kittl, J.; Altimime, L.; Jurczak, M.; Groeseneken, G. Generic learning of TDDB applied to RRAM for improved understanding of conduction and switching mechanism through multiple filaments. In Proceedings of the 2010 International Electron Devices Meeting, San Francisco, CA, USA, 6–8 December 2010. [Google Scholar]
- Walczyk, C.; Walczyk, D.; Schroeder, T.; Bertaud, T.; Sowinska, M.; Lukosius, M.; Fraschke, M.; Wolansky, D.; Tillack, B.; Miranda, E.; et al. Impact of temperature on the resistive switching behavior of embedded HfO
_{2}-Based RRAM devices. IEEE Trans. Electron. Dev.**2011**, 58, 3124–3131. [Google Scholar] [CrossRef] - Long, S.; Lian, X.; Cagli, C.; Cartoixà, X.; Rurali, R.; Miranda, E.; Jiménez, D.; Perniola, L.; Liu, M.; Suñé, J. Quantum-size effects in hafnium-oxide resistive switching. Appl. Phys. Lett.
**2013**, 102, 183505. [Google Scholar] [CrossRef] - Prócel, L.M.; Trojman, L.; Moreno, J.; Crupi, F.; Maccaronio, V.; Degraeve, R.; Goux, L.; Simoen, E. Experimental evidence of the quantum point contact theory in the conduction mechanism of bipolar HfO
_{2}-based resistive random access memories. J. Appl. Phys.**2013**, 114, 074509. [Google Scholar] [CrossRef] - Rahavan, N. Performance and reliability trade-offs for high-K RRAM. Microelectron. Reliab.
**2014**, 54, 2253–2257. [Google Scholar] [CrossRef] - Roldán, J.B.; Miranda, E.; González-Cordero, G.; García-Fernández, P.; Romero-Zaliz, R.; González-Rodelas, P.; Aguilera, A.M.; González, M.B.; Jiménez-Molinos, F. Multivariate analysis and extraction of parameters in resistive RAMs using the Quantum Point Contact model. J. Appl. Phys.
**2018**, 123, 014501. [Google Scholar] [CrossRef] - Calixto, D.; Maldonado, E.; Miranda, J.B.; Roldán, M. Modeling of the temperature effects in filamentary-type resistive switching memories using quantum point-contact theory. J. Phys. D Appl. Phys.
**2020**, 53, 295106. [Google Scholar] [CrossRef] - Tsuruoka, T.; Hasegawa, T.; Terabe, K.; Aono, M. Conductance quantization and synaptic behavior in a Ta
_{2}O_{5}-based atomic switch. Nanotechnology**2012**, 23, 435705. [Google Scholar] [CrossRef] [PubMed] - Chen, C.; Gao, S.; Zeng, F.; Wang, G.; Li, S.; Song, C.; Pan, F. Conductance quantization in oxygen-anion-migration-based resistive switching memory devices. Appl. Phys. Lett.
**2013**, 103, 043510. [Google Scholar] [CrossRef] - Yi, W.; Savelev, S.; Medeiros-Ribeiro, G.; Miao, F.; Zhang, M.; Yang, J.; Bratkovsky, A.; Williams, R.S. Quantized conductance coincides with state instability and excess noise in tantalum oxide memristors. Nat. Commun.
**2016**, 7, 11142. [Google Scholar] [CrossRef] - Ye, J.Y.; Li, Y.Q.; Gao, J.; Peng, H.Y.; Wu, S.X.; Wu, T. Nanoscale resistive switching and filamentary conduction in NiO thin films. Appl. Phys. Lett.
**2010**, 97, 132108. [Google Scholar] [CrossRef] - Nishi, Y.; Sasakura, H.; Kimoto, T. Appearance of quantum point contact in Pt/NiO/Pt resistive switching cells. J. Mater. Res.
**2017**, 32, 2631–2637. [Google Scholar] [CrossRef] [Green Version] - Zhu, X.-J.; Shang, J.; Li, R.-W. Resistive switching effects in oxide sandwiched structures. Front. Mater. Sci.
**2012**, 6, 183–206. [Google Scholar] [CrossRef] - Zhu, X.; Su, W.; Liu, Y.; Hu, B.; Pan, L.; Lu, W.; Zhang, J.; Li, R. Observation of conductance quantization in oxide-based resistive switching memory. Adv. Mater.
**2012**, 24, 3941–3946. [Google Scholar] [CrossRef] - Hajto, J.; Rose, M.J.; Snell, A.J.; Osborne, I.S.; Owen, A.E.; Lecomber, P.G. Quantised electron effects in metal/a-Si:H/metal thin film structures. J. Non-Cryst. Solids
**1991**, 137, 499–502. [Google Scholar] [CrossRef] - Samardzic, N.; Mionic, M.; Dakic, B.M.; Hofmann, H.; Dautovic, S.; Stojanovic, G. Analysis of Quantized Electrical Characteristics of Microscale TiO
_{2}Ink-Jet Printed Memristor. IEEE Trans. Electron Devices**2015**, 62, 1898–1904. [Google Scholar] [CrossRef] - Yun, E.-J.; Becker, M.F.; Walser, R.M. Room temperature conductance quantization in V∥amorphous-V
_{2}O_{5}∥V thin film structures. Appl. Phys. Lett.**1993**, 63, 2493–2495. [Google Scholar] [CrossRef] - Petzold, S.; Piros, E.; Eilhardt, R.; Zintler, A.; Vogel, T.; Kaiser, N.; Radetinac, A.; Komissinskiy, P.; Jalaguier, E.; Nolot, E.; et al. Tailoring the Switching Dynamics in Yttrium Oxide-Based RRAM Devices by Oxygen Engineering: From Digital to Multi-Level Quantization toward Analog Switching. Adv. Electron. Mater.
**2020**, 6, 2000439. [Google Scholar] [CrossRef] - Zhao, M.; Yan, X.; Ren, L.; Zhao, M.; Guo, F.; Zhuang, J.; Du, Y.; Hao, W. The role of oxygen vacancies in the high cycling endurance and quantum conductance in BiVO
_{4}-based resistive switching memory. InfoMat**2020**, 2, 960–967. [Google Scholar] [CrossRef] [Green Version] - Degraeve, R.; Fantini, A.; Clima, S.; Govoreanu, B.; Goux, L.; Chen, Y.Y.; Wouters, D.; Roussel, P.; Kar, G.; Pourtois, G.; et al. Dynamic ‘hour glass’ model for SET and RESET in HfO
_{2}RRAM. In Proceedings of the Symposium on VLSI Technology, Honolulu, HI, USA, 12–14 June 2012. [Google Scholar] - Miranda, E.; Kano, S.; Dou, C.; Kakushima, K.; Suñé, J.; Iwai, H. Nonlinear conductance quantization effects in CeO
_{x}/SiO_{2}-based resistive switching devices. App. Phys. Lett.**2012**, 101, 012910. [Google Scholar] [CrossRef] - Cartoixa, X.; Rurali, R.; Suñé, J. Transport properties of oxygen vacancy filaments in metal/crystalline or amorphous HfO
_{2}/metal structures. Phys. Rev. B**2012**, 86, 165445. [Google Scholar] [CrossRef] - Zhong, X.; Rungger, I.; Zapol, P.; Heinonen, O. Oxygen modulated quantum conductance for ultra-thin HfO
_{2}-based memristive switching devices. Phys. Rev. B**2016**, 94, 165160. [Google Scholar] [CrossRef] [Green Version] - Büttiker, M. Quantized transmission of a saddle-point constriction. Phys. Rev. B
**1990**, 41, 7906. [Google Scholar] [CrossRef] - Hu, P. One-dimensional quantum electron system under a finite voltage. Phys. Rev. B
**1987**, 35, 4078. [Google Scholar] [CrossRef] - Senz, V.; Heinzel, T.; Ihn, T.; Lindermann, S.; Held, R.; Ensslin, K.; Wegscheider, W.; Bichler, M. Analysis of the temperature-dependent quantum point contact conductance in view of the metal-insulator transition in two dimensions. J. Phys. Cond. Mat.
**2001**, 13, 3831. [Google Scholar] [CrossRef] [Green Version] - Miranda, E.; Suñé, J. Electron transport through broken down ultra-thin SiO
_{2}layers in MOS devices. Microelectron. Reliab.**2004**, 44, 1–23. [Google Scholar] [CrossRef] - Landauer, R. Spatial variation of currents and fields due to localized scatterers in metallic conduction. IBM J. Res. Dev.
**1957**, l, 223–231. [Google Scholar] [CrossRef] - Avellán, A.; Miranda, E.; Schroeder, D.; Krautschneider, W. Model for the voltage and temperature dependence of the soft-breakdown current in ultrathin gate oxides. J. Appl. Phys.
**2005**, 97, 14104. [Google Scholar] [CrossRef] - Miranda, E. The role of power dissipation on the progressive breakdwon dynamics of ultra-thin gate oxides. In Proceedings of the IEEE Proc International Reliability Physics Simposium, Phoenix, AZ, USA, 15–19 April 2007; p. 572. [Google Scholar]
- Lombardo, S.; Stathis, J.H.; Linder, B.P.; Pey, K.L.; Palumbo, F.; Tung, C.H. Dielectric breakdown mechanisms in gate oxides. J. Appl. Phys.
**2005**, 98, 121301. [Google Scholar] [CrossRef] - Stathis, J.H. Percolation models for gate oxide breakdown. J. Appl. Phys.
**1999**, 86, 5757–5766. [Google Scholar] [CrossRef] - Stathis, J.H. Reliability limits for the gate insulator in CMOS technology. IBM J. Res. Dev.
**2002**, 46, 265–286. [Google Scholar] [CrossRef] - Dumin, D.J. Oxide Reliability: A Summary of Silicon Oxide Wearout, Breakdown, and Reliability; World Scientific: Singapore, 2002. [Google Scholar]
- Linder, B.; Lombardo, S.; Stathis, J.; Vayshenker, A.; Frank, D. Voltage dependence of hard breakdown growth and the reliability implication in thin dielectrics. IEEE Electron Device Lett.
**2002**, 23, 661–663. [Google Scholar] [CrossRef] - Palumbo, F.; Liang, X.; Yuan, B.; Shi, Y.; Hui, F.; Villena, M.A.; Lanza, M. Bimodal Dielectric Breakdown in Electronic Devices Using Chemical Vapor Deposited Hexagonal Boron Nitride as Dielectric. Adv. Electron. Mater.
**2018**, 4, 1700506. [Google Scholar] [CrossRef] - Palumbo, F.; Lombardo, S.; Eizenberg, M. Physical mechanism of progressive breakdown in gate oxides. J. Appl. Phys.
**2014**, 115, 224101. [Google Scholar] [CrossRef] [Green Version] - Tung, C.H.; Pey, K.L.; Tang, L.J.; Radhakrishnan, M.K.; Lin, W.H.; Palumbo, F.; Lombardo, S. Percolation path and dielectric-breakdown-induced-epitaxy evolution during ultrathin gate dielectric breakdown transient. Appl. Phys. Lett.
**2003**, 83, 2223–2225. [Google Scholar] [CrossRef] - Pey, K.L.; Ranjan, R.; Tung, C.H.; Tang, L.J.; Lin, W.H.; Radhakrishnan, M.K. Gate dielectric degradation mechanism associated with DBIE evolution. In Proceedings of the IEEE International Reliability Physics Symposium, Phoenix, AZ, USA, 25–29 April 2004; Volume 2004, pp. 117–121. [Google Scholar]
- Privitera, S.; Bersuker, G.; Butcher, B.; Kalantarian, A.; Lombardo, S.; Bongiorno, C.; Geer, R.; Gilmer, D.; Kirsch, P. Microscopy study of the conductive filament in HfO
_{2}resistive switching memory devices. Microelectron. Eng.**2013**, 109, 75–78. [Google Scholar] [CrossRef] - Privitera, S.; Bersuker, G.; Lombardo, S.; Bongiorno, C.; Gilmer, D. Conductive filament structure in HfO
_{2}resistive switching memory devices. Solid-State Electron.**2015**, 111, 161–165. [Google Scholar] [CrossRef] - Yang, Y.; Gao, P.; Li, L.; Pan, X.; Tappertzhofen, S.; Choi, S.; Waser, R.; Valov, I.; Lu, W.D. Electrochemical dynamics of nanoscale metallic inclusions in dielectrics. Nat. Commun.
**2014**, 5, 4232. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Pan, C.; Ji, Y.; Xiao, N.; Hui, F.; Tang, K.; Guo, Y.; Xie, X.; Puglisi, F.M.; Larcher, L.; Miranda, E.; et al. Coexistence of Grain-Boundaries-Assisted Bipolar and Threshold Resistive Switching in Multilayer Hexagonal Boron Nitride. Adv. Funct. Mater.
**2017**, 27. [Google Scholar] [CrossRef] - Rodriguez-Fernandez, A.; Cagli, C.; Perniola, L.; Suñé, J.; Miranda, E. Identification of the generation/rupture mechanism of filamentary conductive paths in ReRAM devices using oxide failure analysis. Microelectron. Reliab.
**2017**, 76–77, 178–183. [Google Scholar] [CrossRef] - Nishi, Y.; Fleck, K.; Böttger, U.; Waser, R.; Menzel, S. Effect of RESET Voltage on Distribution of SET Switching Time of Bipolar Resistive Switching in a Tantalum Oxide Thin Film. IEEE Trans. Electron Devices
**2015**, 62, 1561–1567. [Google Scholar] [CrossRef] - Palumbo, F.; Shekhter, P.; Weinfeld, K.C.; Eizenberg, M. Characteristics of the dynamics of breakdown filaments in Al
_{2}O_{3}/InGaAs stacks. Appl. Phys. Lett.**2015**, 107, 122901. [Google Scholar] [CrossRef] [Green Version] - Palumbo, F.; Miranda, E.; Ghibaudo, G.; Jousseaume, V. Formation and Characterization of Filamentary Current Paths in HfO
_{2}-Based Resistive Switching Structures. IEEE Electron Device Lett.**2012**, 33, 1057–1059. [Google Scholar] [CrossRef] - Palumbo, F.; Condorelli, G.; Lombardo, S.; Pey, K.; Tung, C.; Tang, L. Structure of the oxide damage under progressive breakdown. Microelectron. Reliab.
**2005**, 45, 845–848. [Google Scholar] [CrossRef] - Du, H.; Jia, C.-L.; Koehl, A.; Barthel, J.; Dittmann, R.; Waser, R.; Mayer, J. Nanosized conducting filaments formed by atomic-scale defects in redox-based resistive switching memories. Chem. Mater.
**2017**, 29, 3164–3173. [Google Scholar] [CrossRef] - Palumbo, F.; Eizenberg, M.; Lombardo, S. General features of progressive breakdown in gate oxides: A compact model. In Proceedings of the IEEE International Reliability Physics Symposium, Monterey, CA, USA, 19–23 April 2015; Volume 2015, pp. 5A11–5A16. [Google Scholar]
- Lombardo, S.; Wu, E.Y.; Stathis, J.H. Electron energy dissipation model of gate dielectric progressive breakdown in n- and p-channel field effect transistors. J. Appl. Phys.
**2017**, 122, 085701. [Google Scholar] [CrossRef] - Lombardo, S.; Stathis, J.H.; Linder, B.P. breakdown transients in ultrathin gate oxides: Transition in the degradation rate. Phys. Rev. Lett.
**2003**, 90, 167601. [Google Scholar] [CrossRef] [PubMed] - Pagano, R.; Lombardo, S.; Palumbo, F.; Kirsch, P.; Krishnan, S.; Young, C.; Choi, R.; Bersuker, G.; Stathis, J. A novel approach to characterization of progressive breakdown in high-k/metal gate stacks. Microelectron. Reliab.
**2008**, 48, 1759–1764. [Google Scholar] [CrossRef] - Palumbo, F.; Lombardo, S.; Stathis, J.; Narayanan, V.; Mcfeely, F.; Yurkas, J. Degradation of ultra-thin oxides with tungsten gates under high voltage: Wear-out and breakdown transient. In Proceedings of the 2004 IEEE International Reliability Physics Symposium, Phoenix, AZ, USA, 25–29 April 2004; pp. 122–125. [Google Scholar] [CrossRef]
- Larcher, L. Statistical simulation of leakage currents in mos and flash memory devices with a new multiphonon trap-assisted tunneling model. IEEE Trans. Electron Devices
**2003**, 50, 1246–1253. [Google Scholar] [CrossRef] - Nigam, T.; Martin, S.; Abusch-Magder, D. Temperature Dependence and Conduction Mechanism after Analog Soft Breakdown. In Proceedings of the 41st Annual Symposium 2003 IEEE International Reliability Physics, Dallas, TX, USA, 30 March–4 April 2003; pp. 417–423. [Google Scholar]
- Condorelli, G.; Lombardo, S.A.; Palumbo, F.; Pey, K.-L.; Tung, C.H.; Tang, L.-J. Structure and conductance of the breakdown spot during the early stages of progressive breakdown. IEEE Trans. Device Mater. Reliab.
**2006**, 6, 534–541. [Google Scholar] [CrossRef] - Palumbo, F.; Wen, C.; Lombardo, S.; Pazos, S.; Aguirre, F.; Eizenberg, M.; Hui, F.; Lanza, M. A Review on Dielectric Breakdown in Thin Dielectrics: Silicon Dioxide, High-k, and Layered Dielectrics. Adv. Funct. Mater.
**2020**, 30, 1900657. [Google Scholar] [CrossRef] - Takagi, S.; Yasuda, N.; Toriumi, A. Experimental evidence of inelastic tunneling in stress-induced leakage current. IEEE Trans. Electron Devices
**1999**, 46, 335–341. [Google Scholar] [CrossRef] - Blöchl, P.E.; Stathis, J.H. Hydrogen electrochemistry and stress-induced leakage current in Silica. Phys. Rev. Lett.
**1999**, 83, 372–375. [Google Scholar] [CrossRef] - Aguirre, F.L.; RodriguezFernandez, A.; Pazos, S.M.; Sune, J.; Miranda, E.; Palumbo, F. Study on the connection between the set transient in RRAMs and the progressive breakdown of thin oxides. IEEE Trans. Electron. Devices
**2019**, 66, 1–7. [Google Scholar] [CrossRef] - Ielmini, D. Modeling the universal set/reset characteristics of bipolar RRAM by field-and temperature-driven filament growth. IEEE Trans. Electron Devices
**2011**, 58, 4309–4317. [Google Scholar] [CrossRef] - Pazos, S.; Aguirre, F.L.; Miranda, E.; Lombardo, S.; Palumbo, F. Comparative study of the breakdown transients of thin Al
_{2}O_{3}and HfO_{2}films in MIM structures and their connection with the thermal properties of materials. J. Appl. Phys.**2017**, 121, 094102. [Google Scholar] [CrossRef] - Rodríguez-Fernández, A.; Cagli, C.; Sune, J.; Miranda, E. Switching Voltage and Time Statistics of Filamentary Conductive Paths in HfO
_{2}-Based ReRAM Devices. IEEE Electron Device Lett.**2018**, 39, 656–659. [Google Scholar] [CrossRef] - Zafar, S.; Jagannathan, H.; Edge, L.F.; Gupta, D. Measurement of oxygen diffusion in nanometer scale HfO
_{2}gate dielectric films. Appl. Phys. Lett.**2011**, 98, 152903. [Google Scholar] [CrossRef] - Shacham-Diamand, Y.; Dedhia, A.; Hoffstetter, D.; Oldham, W.G. Copper Transport in Thermal SiO
_{2}. J. Electrochem. Soc.**1993**, 140, 2427–2432. [Google Scholar] [CrossRef] - Nason, T.C.; Yang, G.; Park, K.; Lu, T. Study of silver diffusion into Si(111) and SiO
_{2}at moderate temperatures. J. Appl. Phys.**1991**, 70, 1392–1396. [Google Scholar] [CrossRef] - Kim, B.G.; Yeo, S.; Lee, Y.W.; Cho, M.S. Comparison of diffusion coefficients and activation energies for Ag diffusion in silicon carbide. Nucl. Eng. Technol.
**2015**, 47, 608–616. [Google Scholar] [CrossRef] [Green Version] - Zobelli, A.; Ewels, C.P.; Gloter, A.; Seifert, G. Vacancy migration in hexagonal boron nitride. Phys. Rev. B
**2007**, 75, 094104. [Google Scholar] [CrossRef]

**Figure 1.**Different conductive filament shapes employed in RRAM compact models for circuit simulation. We will use this CF shapes in the thermal models described below. (

**a**) CF that occupies all the modeling domain, (

**b**) cylindrical CF, (

**c**) truncated-cone shaped CF.

**Figure 2.**Simulated RRAMs. Four different LRS situations are considered, in all cases a double conductive filament was employed assuming a distance, d, between them (the bottom and top figures are the same in each case from different perspectives). The bottom electrode is assumed to be Si-n

^{+}and the top electrode is made of Ni, the dielectric consists of HfO

_{2}. The conductive filament shapes employed are shown for the different cases under study, they are assumed to be metallic-like, formed by Ni atom clusters [24,55,80]. The physical parameters are the same employed in the simulation in [29].

**Figure 3.**Temperature plots for some of the devices simulated as explained above (see the insets), the cross-section cuts corresponds to y = 20 nm in our SD. (

**a**) RRAM with two cylindrical CFs (diameter = 3 nm) separated 1 nm apart for a bias of 0.6 V. (

**b**) RRAM with two cylindrical CFs (diameter = 3 nm) separated 5.5 nm apart for a bias of 0.7 V. (

**c**) RRAM with one cylindrical CF (diameter = 6 nm) and a symmetrical truncated-cone shaped CF (low diameter = 3 nm, high diameter = 6 nm) separated 6 nm apart for a bias of 0.5 V. (

**d**) RRAM with one cylindrical CF (diameter = 6 nm) and a symmetrical truncated-cone shaped CF (low diameter = 3 nm, high diameter = 6 nm) separated 6.5 nm apart for a bias of 0.8 V. For the sake of visibility, some of the 3D plots are rotated with respect to the 2D CF scheme.

**Figure 4.**Temperature plots for some of the devices simulated as explained above (see the insets), the cross-section cuts corresponds to y = 20 nm in our SD. (

**a**) RRAM with one cylindrical CF (diameter = 3 nm) and a symmetrical truncated-cone shaped CF (low diameter = 3 nm, high diameter = 6 nm) separated 2 nm apart for a bias of 0.8 V. (

**b**) RRAM with one cylindrical CF (diameter = 3 nm) and a symmetrical truncated-cone shaped CF (low diameter = 3 nm, high diameter = 6 nm) separated 6 nm apart for a bias of 0.5 V. (

**c**) RRAM with one cylindrical CF (diameter = 3 nm) and a symmetrical truncated-cone shaped CF (low diameter = 5 nm, high diameter = 7 nm) separated 6 nm apart for a bias of 0.75 V. (

**d**) RRAM with one cylindrical CF (diameter = 3 nm) and a symmetrical truncated-cone shaped CF (low diameter = 5 nm, high diameter = 7 nm) separated 1 nm apart for a bias of 0.6 V. For the sake of visibility, some of the 3D plots are rotated with respect to the 2D CF scheme.

**Figure 5.**Schematic showing the different elements considered to solve the simplified HE in a cylindrical and homogenous conductive filament.

**Figure 6.**Sketch of a cylindrical filament with the different terms in the HE shown in Equation (6), including the heat transfer term.

**Figure 7.**(

**a**) Energy dissipation terms included in the heat equation and geometrical domain for the CF thermal description, (

**b**) cylindrical CF equivalent employed to simplify the HE solution and obtain a compact analytical expression for the CF temperature. Note that the conductive filament length (L

_{CF}) can be lower than the oxide layer (t

_{ox}), due to the gap (g) between the top electrode and the conductive filament tip (see [27,78,86,87,88]).

**Figure 8.**RRAM cell CF scheme for the thermal model based on two different CF temperatures. (

**a**) Original filament with the corresponding boundary conditions. Cylindrical CFs (shown in dashed lines) employed to compute the (

**b**) top temperature T

_{T}and (

**c**) the bottom T

_{B}.

**Figure 9.**Equivalent electric circuit for the RS device thermal model based on a thermal resistance R

_{th}.

**Figure 10.**Equivalent electric circuit of a RRAM thermal model based on a thermal resistance and capacitance to implement Equation (20).

**Figure 11.**(

**a**) Voltage applied to the device versus time, (

**b**) Temperature versus time obtained for different values of C

_{th}(assuming R

_{th}= 2 × 10

^{5}K/W).

**Figure 12.**Simulations performed making use of the Stanford model including the TM5 with different thermal resistances. (

**a**) Current versus voltage applied to the device, (

**b**) voltage signal applied to the device, (

**c**) temperature versus time.

**Figure 13.**Simulations performed with the Stanford model (including TM6) making use of different thermal capacitances, C

_{th}, assuming a common value of the thermal resistance, R

_{th}= 4 × 10

^{5}K/W. (

**a**) Voltage applied to the device versus time, (

**b**) Temperature versus time.

**Figure 14.**Equivalent electric circuit of a RRAM thermal model based on a double thermal circuit described by Equations (30) and (31).

**Figure 15.**(

**a**) Three-dimensional view of the Stanford scheme to model the different device areas (TE: top electrode, Oxide Layer, CF: conductive filament and BE: bottom electrode), (

**b**) model parameters, g: gap between the TE and the filament tip and t

_{ox}: dielectric thickness, (

**c**) subcircuit representation for the implemented model. The connection between blocks represents the states variables used: g, which depends on kinetic block and it is linked to the two temperatures (T, T

_{S}).

**Figure 16.**RRAM simulation making use of the Stanford model including a double RC thermal model R

_{th1}= 40 kK/W, R

_{th2}= 40 kK/W, C

_{th1}= 0.003 fJ/K and C

_{th2}with values 1 fJ/K and 10 fJ/K. (

**a**) Applied voltage pulses for consecutive set and reset, (

**b**) temporal current evolution, (

**c**) temporal evolution of device filament temperature (T) and the intermediate surrounding region (T

_{S}) with C

_{th2}= 1 fJ/K and (

**d**) C

_{th2}= 10 fJ/K.

**Figure 17.**Schema of the circuital compact model. A truncated-cone shaped conductive filament is represented by connected cylinders for modeling purposes (

**a**). The behaviour of each portion of the filament (cylinder) is modeled by the subcircuit inside the blue rectangle (

**b**), which has electrical connections (EC) and thermal connections (TC). Each cylinder (subcircuit) is characterized by different state variables (radius, r_cf, and temperature, Temp). The cylinder subcircuit consists of several more subcircuits: a kinetic block for calculating the transient CF evolution; an electrical block for current calculation; and, finally, the thermal subcircuit, which includes the equivalent circuit for the thermal model. As can be seen, the subcircuits (thermal, kinetic and electrical blocks) are connected all together because they are interdependent. If necessary, a last subcircuit is added in series (

**a**) in order to account for the conduction through a constriction by means of the quantum point contact model (see Section 4) [108].

**Figure 18.**Circuital equivalent for thermal model TM8. The circuit inputs are the dissipated power (pw) and the CF radius (r_cf), while the output is the temperature (T_CF). The actual values of the thermal resistances depend on the filament radius (it is assumed to be a cylinder) and, therefore, they are updated as the CF evolves. The subcircuit has been prepared for being connected to other thermal subcircuits (through TC1, TC2 and TCox) in order to obtain a more complex thermal model of the whole device (with several temperatures along the filament or different temperatures for the surrounding insulator or bulk insulator, Figure 17). If only one block is used, all the resistances are in parallel and the model is equivalent to TM5, although the thermal resistance value keeps the dependency on the filament size in TM8 and it changes during the simulation.

**Figure 19.**Simulation of a reset transition in unipolar Ni/20 nm-HfO

_{2}/Si-n

^{+}resistive switching devices [109]. The i-v curve provided by a finite differences simulator used to fit the experimental data [80] is compared with the i-v curve calculated by means of the two cylinders model with TM8. Note that with only two subcircuits (Figure 17a) and taking variable electrical and thermal resistances into account, both types of simulators provide very close results, as far as the circuital model includes variable electric and thermal resistances. Two values of the lateral heat dissipation parameter, h, have been used for the sake of comparison.

**Figure 20.**Circuital equivalent for thermal model TM9. It is similar to TM8 (Figure 18), but thermal inertia has been added by means of capacitances. As in TM8, the actual values of the thermal resistances and capacitances depend on the filament radius (it is assumed to be a cylinder) and, therefore, they are updated as the CF evolves [37].

**Figure 21.**Simulated current in a Ni/20 nm-HfO

_{2}/Si-n

^{+}resistive switching device [37,110] when a 3 V reset pulse is applied (for 100 ns). Different values of the thermal capacities have been assumed. Fixed thermal capacitances (solid lines) and size-dependent thermal capacitances (symbols) have been used [37,111].

**Figure 22.**Input current waveform versus time for five different ramps. Colours are coherently employed in the following plots.

**Figure 23.**Memristance versus time for five different input voltage signals under ramped voltage stress.

**Figure 24.**Memristance versus input current, for five different slopes. Colours are coherent with the results shown in the previous figures.

**Figure 26.**Memristor Dynamic Route Map (surface), showing as lines the five trajectories corresponding to the previous figures, using the same colour code. It can be seen that all these trajectories fall on the surface, which defines univocally the device behaviour. Notice that this surface is, in fact, a family of surfaces that depend on the room temperature T

_{0}.

**Figure 27.**Schematic of the energy structure of the conducting filament. In LRS (high current), the CF is completely formed and the confinement potential barrier is low. In HRS (low current), the filament is broken and the confinement potential barrier is high. The green arrows width indicates the electron current magnitude.

**Figure 28.**Effects of the temperature on the HRS and LRS I-V characteristics. (

**a**) log-linear axis and (

**b**) log-log axis.

**Figure 29.**Evolution of the power dissipated in the structure P (solid lines), and at the constriction P

_{C}(dashed lines). (

**a**) Corresponds to different applied voltages V, and (

**b**) corresponds to different transition rates η.

**Figure 30.**Current transient of the DUTs under constant voltage stress (CVS). (

**a**) represents the lowest –450 mV– voltage whereas (

**b**) the highest –650 mV–. Ball marker 1 points out the initial current (I

_{Init}), whereas 2 the onset of the progressive increase of current (I

_{On}) and 3 the final jump to the compliance level (I

_{End}).

**Figure 31.**TR data (square markers –ball markers represent the mean value–) fitted assuming oxygen vacancies and t

_{gap}= t

_{ox}(cyan dashed line – curve N° 1). Additionally, TR assuming literature values for t

_{ox}and D is plotted –curves N° 2, 3 and 4–. TR presents a strong dependence with applied voltage, increasing almost one order of magnitude for every 50 mV step.

**Figure 32.**Temperature estimation according to Equation (60) as function of voltage and the energy loss in the CF. The red shaded zone indicates the voltages employed for the CVS (0.45 to 0.65 V).

**Figure 33.**Reported values of diffusivity for different atomic species vs. reciprocal of temperature. The diffusivity D required for TR fitting is shown to be in the same range as the OVs diffusivity. OVs diffusivity [197] is ~10

^{4}times higher than for the metallic species as Cu:SiO

_{2}[198], Ag:SiC, Ag:Si and Ag:Al

_{2}O

_{3}[199,200]. VB diffusivity in h-BN data corresponds to [201].

Thermal Model | Verilog-A Code for the Temperatures Calculation |
---|---|

TM1 | T = T_{0} + sigma*V**2/(8.0*kth); |

TM2 | E = V/tox; alpha = tox/2.0*sqrt(2*h/(kth*rcf)); T = T _{0} + (sigma*E**2*rcf*(exp(alpha)−1)**2)/(2*h*(exp(2*alpha)+1)); |

TM3 | LCF = tox−g; rg = sqrt(rt*rb); eta = rt/rb; E = V/LCF; alpha = LCF*sqrt(2*h/(kth*rg)); T = T _{0} + rg*sigma*E**2/(eta*h)*(0.5 − (exp(alpha/2.0)/(exp(alpha) + 1)); |

TM4 | analog function real fdt_{0};real sigmat,rcf,E,alpha; input sigmat,rcf,E,alpha; begin fdt _{0} = sigmat*rcf*E**2*tanh(alpha*tox/2.0)/(sqrt(2*kth*h*rcf));end endfunction analog function real fT; real sigmat,rcf,eta,alpha,dt _{0};input sigmat,rcf,eta,alpha,dt _{0};begin fT = T _{0} + sigmat*rcf*E**2/(2*h)*(1-cosh(alpha*tox/2.0)) + dt_{0}/alpha*sinh(alpha*tox/2.0);end endfunction analog function real falpha; real rcf; input rcf; begin falpha = sqrt(2*h/(kth*rcf)); end endfunction analog function real fsigmat; real T; input T; begin fsigmat = sigma/(1 + alphat * (T − T _{0}));end endfunction |

Stanford-PKU Model Parameters | |||
---|---|---|---|

Device Parameters | Unit | Resistive Switching | |

SET | RESET | ||

V_{o} | V | 0.4 | |

I_{0} | mA | 0.2 | |

g_{0} | nm | 0.35 | |

ν_{0} | m/s | 10^{6} | |

α | - | 1 | |

β | - | 3 | |

γ_{0} | - | 10 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Roldán, J.B.; González-Cordero, G.; Picos, R.; Miranda, E.; Palumbo, F.; Jiménez-Molinos, F.; Moreno, E.; Maldonado, D.; Baldomá, S.B.; Moner Al Chawa, M.;
et al. On the Thermal Models for Resistive Random Access Memory Circuit Simulation. *Nanomaterials* **2021**, *11*, 1261.
https://doi.org/10.3390/nano11051261

**AMA Style**

Roldán JB, González-Cordero G, Picos R, Miranda E, Palumbo F, Jiménez-Molinos F, Moreno E, Maldonado D, Baldomá SB, Moner Al Chawa M,
et al. On the Thermal Models for Resistive Random Access Memory Circuit Simulation. *Nanomaterials*. 2021; 11(5):1261.
https://doi.org/10.3390/nano11051261

**Chicago/Turabian Style**

Roldán, Juan B., Gerardo González-Cordero, Rodrigo Picos, Enrique Miranda, Félix Palumbo, Francisco Jiménez-Molinos, Enrique Moreno, David Maldonado, Santiago B. Baldomá, Mohamad Moner Al Chawa,
and et al. 2021. "On the Thermal Models for Resistive Random Access Memory Circuit Simulation" *Nanomaterials* 11, no. 5: 1261.
https://doi.org/10.3390/nano11051261