Three-Dimensional Simulation of Bipolar Resistive Switching Memory with Embedded Conductive Nanocrystals in an Oxide Matrix

Abstract

In this work, the simulation of deoxidation–oxidation of oxygen vacancies (V_Os) in an oxide matrix with embedded conductive nanocrystals (c-NCs) is carried out for the development of bipolar resistive switching memories (BRSMs). We have employed the three-dimensional kinetic Monte Carlo (3D-kMC) method to simulate the RS behavior of BRSMs. The c-NC is modeled as fixed oxygen vacancy (f-V_O) clusters, defined as sites with zero recombination probability. The three-dimensional oxygen vacancy configuration (3D-V_OC) obtained for each voltage step of the simulation is used to calculate the resistive state and the electrical current. It was found that the c-NC reduces the voltage required to switch the memory state from a high to a low resistive state due to the increase in a nonhomogeneous electrical field between electrodes.

1. Introduction

Resistive switching memory (RSM) is a device with an active layer situated between two conductive electrodes. This layer enables the device to alter and record its resistivity [1,2], and it typically comprises an oxide, such as ZnO [3], TiO₂ [4], HfO₂ [5], SnO₂ [6], SiO₂ [7], and others. The RSM devices are classified according to the polarization voltage; if the writing or SET process (decrement of resistivity) and erasing or RESET process (increment of resistivity) occur without changing the polarization voltage, the resistive switching (RS) is classified as unipolar mode. On the contrary, if the SET and RESET processes occur at different polarization voltages, the RS is classified as a bipolar mode [2]. The values of SET and RESET voltage depend on the active material used in the RSM and on their thickness. Lower SET/RESET voltages are required to obtain a low power consumption. One alternative to reduce these voltage values is the use of conductive nanocrystals (c-NCs) embedded in the active layer [8]. These structures can be experimentally fabricated using alternated layers in the form of dielectric/c-NCs/dielectric, deposited by sputtering, where c-NCs are formed by a post-annealing process [9]. Therefore, this work is focused on the simulation of the effect of c-NCs in bipolar resistive switching memories (BRSMs).

The actual computing systems require the development of devices that can store and process information in a short time with low energy consumption [10]. In recent years, there has been a proposal to use RSM as a fast and low-power alternative to two main types of technology: (i) non-volatile random-access memory (NVRAM) and (ii) artificial neuronal networks. Regarding technology (i), the RSMs retain their resistivity state without requiring energy, which is a non-volatile feature. Additionally, the continuous writing and erasing process of this element enables the function of RAM [11]. Regarding technology (ii), the actual neuronal networks are artificially implemented in classical computing hardware where the central processing unit (CPU) is separated from the memory unit. This architecture, known as the Von Neumann model, has created a bottleneck between the units. Therefore, the RSM has been proposed to overcome this issue, since it can emulate the synapse process between neurons or nodes. This feature helps to avoid the information transmission between the CPU and the memory unit, thanks to the implicit capacity of the RSM to memorize its resistive state [12,13,14].

There are different methods in which the simulation of RS phenomena can be approached. (i) Quantum chemistry: This method simulates the quantum properties that determine the conduction at different resistive states in the active layer. This approach considers the electronic properties and configuration of defects that change during the SET or RESET processes [8,15]. (ii) The finite element method: This approach solves the physical and chemical equations at the level of the selected mesh elements to simulate the conduction and RS processes [16,17,18]. (iii) Kinetic Monte Carlo models: This approach avoids the complex molecular dynamics by calculating the probability of defect generation and recombination, which are pivotal in determining the resistivity change in the active layer. Due to the stochastic nature of these processes, the Monte Carlo method refers to the use of random values to compare with these probabilities [19,20,21,22]. (iv) Compact models: The theory of the memristor is used as a four-passive electrical element to establish a relation between the electrical charge and magnetic flux. The primary objective of these models is to simulate the RSM in electrical circuit models, as proposed by Leon Chua in his memristor theory [23]. (v) Other potential alternatives that combine (i), (ii), (iii), and (iv) or can be complemented by other approximations, such as random circuit breaker networks, cylindrical conductive filaments, hourglass quantum-point contact, or others [24,25,26,27].

Approaches (i) and (ii) require significant computational resources and specialized software, while model (iii) requires medium computational resources and can be implemented at the lattice constant level in two dimensions (2D) or three dimensions (3D) [28,29]. In contrast, approach (iv) typically has the lowest computational cost, but it does not consider properties at the lattice constant level of the active layer.

In this work, based on the previous simulation in 2D, we have used approach (iii) in 3D, using constant values derived from other investigations based on approaches (i) and (ii) [9,28]. We integrated a straightforward algorithm into the 3D-kMC to calculate the resistive state and the electrical current ($I$) at different charge conduction mechanisms.

2. Deoxidation–Oxidation Model of Bipolar Resistive Switching

The resistivity of RSM devices with oxides as active layers can be modified by the diffusion of metallic ions, which form metallic filaments. These kinds of devices are known as electrochemical metallization memories. On the other hand, devices where the valence state of metallic atoms alters the resistivity of metallic oxides through the deoxidation–oxidation process are known as valence change memories, where oxygen ions (O_ions) move under the action of an electric field and create oxygen vacancies (V_Os) that form a conductive filament (CF) [11].

Figure 1 illustrates the structure of a BRSM device. It comprises a top conductive film (purple), an active layer or oxide, a substrate (black) with a much lower resistivity than the oxide, and a bottom conductive film (gray). The top conductive film is referred to as the top electrode, and the set of substrate–bottom conductive film is referred to as the bottom electrode of the BRSM.

Figure 1a shows the movement of O_ions under the influence of an electric field $F$. These ions are accumulated at the top electrode and then leave or generate V_Os. These defects form complete chains of V_Os or CFs that connect both electrodes. Therefore, the subsequent state of the BRSM following deoxidation is the low resistive state (LRS) or ON state. This indicates that deoxidation is essential to obtain the SET or the writing step of the BRSM. Figure 1b illustrates how a change in voltage polarization enables the return of O_ions towards the oxide, enhancing the recombination or oxidation of V_O, and then the interruption of CFs near the top electrode. This process, known as oxidation, effectively allows the RESET process or the erasing of the BRSM, returning the device to a high resistive state (HRS) or OFF state [30].

3. Three-Dimensional Kinetic Monte Carlo Simulation

One of the optimal approaches to simulate the behavior of c-NCs embedded in oxides is to model them as 3D clusters. The 3D coordinate system used for the simulation is shown in Figure 2.

The lattice constant of the oxide ($a$) is the level selected to simulate the distributions of V_Os. The oxide is divided into $MNP$ sites with a thickness of $L=Pa$. The area $\left(Ma\right)\left(Na\right)$ is selected to represent the structure of the oxide with a total area $A$. By selecting $\left(Ma\right)\left(Na\right)\ll A$, the computational cost of the simulation can be reduced. When a positive voltage $V$ is applied to the top electrode, the electric field $F$ is positive.

The kinetic Monte Carlo simulation employs the harmonic transition state theory or Vyneyard’s theory to calculate the generation and recombination probabilities of V_Os. This theory streamlines the molecular dynamic simulation of O_ions, assuming harmonic oscillation modes at the shallow and deep levels of potential wells. Therefore, the statistical physics of these vibrations is represented by an effective frequency of harmonic transition state theory or escape frequency ($1/t_0$) with a value ${10}^{13}$ Hz [31,32,33].

The generation probability ($P_{ijk}^G$) is calculated using Arrhenius’s relation, Equation (1). The O_ion accumulates potential energy during a time $t$; subsequently, they could escape from the site $\left(i,j,k\right)$, generating a new V_O [20].

$$P_{ijk}^G={\displaystyle \frac{t}{t_0}}\mathit{exp}\left[-{\displaystyle \frac{E_{Oe}-q\gamma a{F}_{ij}^{nH}}{k{T}_r}}\right]$$

(1)

where $E_{Oe}$ represents the required energy to move an O_ion from its equilibrium position. This value can be derived from chemical studies, typically through quantum chemistry simulations. $q$ is the elementary charge and $\gamma$ is a correction factor. $F_{ij}^{nH}$ is the nonhomogeneous electric field in the vertical column $\left(i,j\right)$, $k$ is the Boltzmann’s constant, and $T_r$ is the room temperature ($300$ K). While $F_{ij}^{nH}$ is defined by

$$F_{ij}^{nH}={\displaystyle \frac{\left|V\right|}{L-a{{\displaystyle \sum }}_{k=1}^P{\delta }_{Vo,ijk}}}$$

(2)

where ${\delta }_{Vo,ijk}$ is the Kronecker’s delta, which takes the value of 1 when the site $\left(i,j,k\right)$ is a V_O and 0 in the opposite case.

Regarding the recombination probability of V_O, i.e., the RESET process, the oxidation depends on the speed of O_ions. Based on the rigid ion-point model of [34], the velocity of O_ions is calculated by

$$v={\displaystyle \frac{a}{t_0}}\mathit{exp}\left(-{\displaystyle \frac{E_{Om}}{k{T}_J}}\right)\mathit{sinh}\left[{\displaystyle \frac{q{\phi }_{drift}a\left(-F^H\right)}{k{T}_J}}\right]$$

(3)

where $E_{Om}$ is the potential barrier for migration of O_ions. This value can be obtained through quantum chemistry studies. ${\phi }_{drift}$ is an enhancement factor of the drift of O_ions. In this model, the migration of O_ions is produced by the homogeneous electric field $F^H=V/L$. $T_J$ is the temperature due to Joule heating of the CFs, and it is modeled using results of finite element simulations for [35], as

$$T_J=T_r+\left(IV\right)R_{th}$$

(4)

The thermal resistance of the CFs, $R_{th}$, is assumed to be a constant value. The recombination probability between the O_ions and V_Os in the site $\left(i,j,k\right)$ is modeled in accordance with the methodology set forth by [20]

$$P_{ijk}^R={\displaystyle \frac{t}{t_0}}{\beta }_R{f}_k\exp \left(-{\displaystyle \frac{\left|v\right|t}{L_O}}\right)\exp \left(-{\displaystyle \frac{E_{Oe}}{k{T}_J}}\right)$$

(5)

where ${\beta }_R$ denotes a correction factor for recombination, while $L_O$ represents the decaying length of the oxygen concentration. The pre-factor $f_k$ is included to take into account the comparison between the distance of the site $\left(i,j,k\right)$ to the top electrode ($z_k=ak$) and the distance reached for the O_ion ($vt$):

$$f_k=\left\{\begin{array}{cc}1,& z_k\le vt\\ 0.3,& vt<z_k\le vt+a\\ 0.1,& vt+a<z_k\le vt+3a\\ 0,& z_k>vt+3a\end{array}\right.$$

(6)

It is important to note that the Joule heating is concentrated in the CFs, which must be considered during their shortening [36]. Therefore, $T_J$ affects the temperature for the calculation of $P_{ijk}^R$ and $v$.

Figure 3a shows the algorithm for each mesh site $\left(i,j,k\right)$ in consideration of the stochastic process of RS phenomena. The generation and recombination probabilities are calculated using Equations (1) and (5), respectively. These values are then compared with a random value ($R_{ijk}$), where the site can change to V_O or non-V_O depending on its initial state. Figure 3b illustrates the flow chart for each value of voltage sweep. By using $V$, $I$, and $T_J$, the 3D-V_OC, or distribution of V_O, can be calculated. The estimation of the resistive state to simulate $I$ is explained in the next section.

4. The Resistive State and the Simulation of the Electrical Current

It has been reported that experimental results exhibit different charge conduction mechanisms during the ON and OFF states, which are crucial insights for RS studies [37]. Based on the Mott hopping rate model, which is used to calculate the charge transport through traps, we propose to estimate the contribution of each column $\left(i,j\right)$ to the conductivity of oxide by the expression [20,38]

$$G_{ij}=\exp \left({\displaystyle \frac{a{{\displaystyle \sum }}_{k=1}^P{\delta }_{Vo,ijk}-L}{a_0}}\right)$$

(7)

with $a_0$ as the attenuation length of the electron wave function in a trap [20]. Therefore, the calculation proposed for the state $N_S$ of the BRSM is

$$N_S=\ln \left({{\displaystyle \sum }}_{i=1}^N{{\displaystyle \sum }}_{j=1}^M{G}_{ij}\right)$$

(8)

The charge conduction mechanism (current density in function of $F^H$) in BRSMs depends on the material used for the electrodes as well as on the active layer [37]. Our proposal is to consider different charge conduction mechanisms at the OFF and ON states. This is based on the pre-factors $f_{OFF}$ and $f_{ON}$, which are shown in the piecewise functions in Figure 4 [9]. The $N_{FS}$ state represents the FRESH state of the BRSM, while the $N_{OFF}$ state indicates the OFF or HRS, while the $N_{ON}$ state is the ON or LRS.

Some experimental studies about RS have reported the Poole–Frenkel (P-F) conduction mechanism during the OFF state and the space-charge-limited current (SCLC) conduction mechanism in the ON state [3,37]. The P-F density current is modeled as follows [39]:

$$J_{PF}=K_{PF}\left|F^H\right|\exp \left({\displaystyle \frac{\beta \sqrt{\left|F^H\right|}-q{\varphi }_t}{k{T}_r}}\right)$$

(9)

The constant $\beta$ is given by the formula $\sqrt{q^3/\pi \epsilon }$, where $\epsilon$ is the dielectric permittivity of the oxide, $q{\varphi }_t$ is the depth of conduction traps, and $K_{PF}$ depends on the effective density of states and the mobility of the conduction charge. The SCLC, including the Frenkel effect due to a high electrical field, is [40]

$$J_{SCLC}={\displaystyle \frac{K_{SCLC}}{N_S-N_{FS}}}{\displaystyle \frac{{\left(F^H\right)}^2}{L}}\exp \left({\displaystyle \frac{0.891\beta \sqrt{\left|F^H\right|}-q{\varphi }_t}{k{T}_J}}\right)$$

(10)

The temperature used for this calculation corresponds to the Joule heating, $T_J$, through the CFs. $K_{SCLC}$ depends on the effective density of states, the mobility of the conduction charge, and the density of the traps. For simplicity, the values $K_{PF}$ and $K_{SCLC}$ are assumed to be constant to be adjusted based on the experimental results.

The proposed method for calculating the electrical current is as follows:

$$I=A\left(J_{PF}{f}_{OFF}+J_{SCLC}{f}_{ON}\right)$$

(11)

This expression can be adapted to other conduction mechanisms distinct from P-F or SCLC.

It is a standard practice to use a limit of the electrical current to maximum values when measuring the electrical properties of RSM. This current compliance (CC) prevents the irreversible dielectric breakdown of the oxide. In our previous semi-empirical work about the 2D kinetic Monte Carlo simulation of a BRSM, we used an algorithm to emulate the CC by reducing the time $t$ (Equations (1) and (5)) at electrical currents greater than the experimental CC [9]. In the present work we have used the same algorithm.

5. Simulation and Results

To simulate the bipolar switching through c-NCs, we proposed emulating these nanostructures using clusters of fixed oxygen vacancies (f-V_Os), which are V_Os that cannot recombine with O_ions. With the objective of evaluating the effects of c-NCs using the mathematical expressions explained above, we simulated the BRSM assuming the properties of a ZnO film as the active layer. The values used for the simulations are specified in

Table 1. Values used for simulations.

Parameter	Value
$\mathit{t}$	5 μs
$\mathbf{1}\mathbf{/}{\mathit{t}}_{\mathbf{0}}$	1013 Hz
${\mathit{E}}_{\mathit{O}\mathit{e}}$	1 eV
$\mathit{\gamma }$ during the SET	3.0
$\mathit{\gamma }$ during the RESET	0.4
$\mathit{a}$	0.325 nm
M × N × P	10 × 10 × 14
${\mathit{\beta }}_{\mathit{R}}$	6 × 106
${\mathit{L}}_{\mathit{O}}$	5.4 $a$
${\mathit{E}}_{\mathit{O}\mathit{m}}$	1 eV
${\mathit{\phi }}_{\mathit{d}\mathit{r}\mathit{i}\mathit{f}\mathit{t}}$	8
${\mathit{a}}_{\mathbf{0}}$	0.33 nm
${\mathit{K}}_{\mathit{P}\mathit{F}}$	7.5 × 10−15 A/V-cm
$\mathit{\epsilon }$	8.5 (8.85 × 10−14) F/cm
$\mathit{q}{\mathit{\varphi }}_{\mathit{t}}$	0.1 eV
${\mathit{K}}_{\mathit{S}\mathit{C}\mathit{L}\mathit{C}}$	3.5 × 10−19 A-cm/V2
$\mathit{A}$	1 mm2
${\mathit{R}}_{\mathit{t}\mathit{h}}$	8 × 103 K/W
${\mathit{N}}_{\mathit{O}\mathit{N}}$	4.0
${\mathit{N}}_{\mathit{O}\mathit{F}\mathit{F}}$	0.0
${\mathit{N}}_{\mathit{F}\mathit{S}}$	−9.0

.

Figure 5a,d illustrate the structures used to simulate 3D-CV_O without and with c-NCs, respectively. During the ON state, as illustrated in Figure 5b,e, the primary effect of c-NCs is the reduction in CFs, i.e., the preferential formation of V_O chains through the c-NC as compared to other vertical columns where the c-NC is not present. The deoxidation–oxidation model, which forms the basis of this work, suggests that the cluster of f-V_O enables the reduction in the equivalent distance between the electrodes, as indicated in (2). Figure 5c,f demonstrates how the RESET process produces these OFF states through the recombination of V_Os near the top electrode.

Figure 6a shows the simulation of $N_S$ through five voltage sweeps: (1) FORMING of the first CFs, (2) ON state during the return to $0$ V at positive voltage, (3) erase or RESET process, (4) OFF state during the return to $0$ V at negative voltage, and (5) write or SET process. As we can see, both BRSMs with and without a c-NC exhibit three distinct stable states near to $N_{FS}$, $N_{OFF}$, and $N_{ON}$ (Table 1). Furthermore, the simulation results demonstrate the closed-loop functionality of the BRSM, encompassing the SET and RESET processes. Initially, during sweep 1 before the FORMING, the BRSMs with c-NCs have a stable state $N_S$ greater than BRSMs without c-NCs. This effect can be explained because the conductivity of the cluster of f-V_O reduces the resistivity of the active layer of ZnO, as compared to the BRSM without the c-NC. On the other hand, the c-NC enables the possibility of the FORMING of first CFs at a lower voltage and $N_S$. The simulation of the RESET process produces the recombination of V_O, which is located in the top middle of the oxide as illustrated in Figure 5c,d. Therefore, the state $N_S$ during the OFF state for both BRSMs is similar, as shown in Figure 6a.

As observed in Figure 6b, the simulation of the electrical behavior of both BRSMs presents the closed loop of current–voltage (I–V) curves similar to experimental I-V results of devices based in binary metal oxides or semiconductor oxides like TiO₂, ZnO, NiO, WO₃, HfO_2, or SiO_x [7,41]. It is evident that the presence of c-NCs embedded in the active layer reduces the voltage required to obtain the FORMING and SET processes thanks to the presence of a lower quantity of complete CFs that change the BRSM from the OFF to ON states with respect to the BRSM without c-NCs. Moreover, the nonhomogeneous electric field, modeled in Equation (2), in the vertical columns where the c-NCs are located is greater at the same voltage levels with respect to the BRSM without c-NCs. These results of simulation are according to the explanation of previous experimental results when nanostructures are included in the oxide matrix of RSMs [7,42,43].

6. Conclusions

The harmonic transition state theory and the rigid ion-point model are complemented with the Monte Carlo method to simulate the stochastic process of generation–recombination of V_Os in BRSMs based on the deoxidation–oxidation mechanism. Our approach to the resistive state ($N_S$) through 3D-V_OC allows us to take into account different charge conductive mechanisms at ON and OFF states. The emulation of c-NCs as bulk f-V_O clusters embedded in oxides helps to take account of these nanostructures in the 3D-kMC to simulate BRS phenomena. Thus, the oxides with c-NC-based BRSMs are an important alternative for the development of BRSMs since they allow the formation of focalized CFs through c-NCs. This results in fewer V_Os reaching the ON state of the BRSM, which can decrease the defects produced for the applied electric fields and then reduce the voltage required for the FORMING and the SET processes.