I. INTRODUCTION

As the development of information and communication technologies such as internet of things (IoT), artificial intelligence (AI), and machine learning (ML) with big data, high-speed and energy-efficient computing architecture have been actively exploited. When considering the nonvolatile memory (NVM) components which play crucial roles as synaptic device as well as stand-alone memory unit, resistive-switching random-access memory (ReRAM) is regarded as one of the most promising next-generation NVMs owing to its high device scalability, fast switching speed, and stack array processing viability. Along with the stand-alone memory application, ReRAM is frequently considered as the synaptic device in the neuromorphic computing, more hardware-driven AI, which usually calls for multi-level weight tunability (electrical conductance states) helped by optimally designed conductance update algorithms ^[1]. However, circuit and system-level design lack of accurate compact ReRAM models describing the device behaviors. In order to reflect the empirically observed resistive-switching characteristics, several compact models have been constructed ^[2-4]. The majority of the past works have been dedicated to exaggeratedly simplified models solely based on filament growth with full predictability at room temperature. For higher accuracy, stochastic filament growths and temperature dependence can be further implanted in the model. By preparing an advanced ReRAM model, reliability in circuit and system designs would be substantially enhanced ^[5,6]. Efforts were dedicated to more dynamic physics-based compact models, in consideration of thermal effects, time-dependent conduction change, and stochastic behaviors of ReRAM, in a few works ^[7-9]. However, high computational expenses in terms of time and cost need to be considered when the circuit and system-level simulations based on ReRAM cells are performed. In this work, a compact ReRAM model with higher compactness and accuracy has been developed utilizing Verilog-A and HSPICE in cooperation. As the result, more exact portrayals of spatial and transient growth progress of the conductive filament have been achieved. Further, the dependency of conductance change on the level of compliance current has been reflected.

II. PHYSICS AND PARAMETERS

The process of conducting filament (CF) construction can be schematically illustrated as shown in Fig. 1(a) through (e). The switching material was assumed to be HfO$_{2}$ in compact modeling. HfO$_{2}$ has been already confirmed with its Si compatibility and brought into the massive production in logic semiconductor technologies ^[10]. It is also found in the researches on ferroelectric devices in recent times ^[11]. For these reasons, introducing HfO$_{2}$ into the RRAM application would make the over process integration more Si-friendly and provide higher viability of integration with CMOS circuits and other memory technologies on a single chip ^[12]. In the model, the CF growth is depicted by the change in length of the CF determined by the biasing condition between set and reset operations. It is known that the construction of a CF results from the migration of oxygen ions, vacancy movements, and carrier transports in the oxide-based ReRAMs. One or more conduction mechanisms for the carrier transport across the switching layer can attribute to the conduction current in low-resistance state (LRS) and high-resistance state (HRS). At the same time, the dependence of conduction current on geometric effect, the filament length (or equivalently the oxide barrier length), should be considered in modeling. The barrier length can be expressed as a function of electric field between the top electrode (TE) and the top of the tip end. It is universal to understand which conduction mechanism might have the predominance in determining the conduction current by having a close look into the formulation between the electric field and conduction current. A simple model in Fig. 2 can be the preliminary frame, in which TE and bottom electrode (BE) resistances, switching resistance, and switching layer capacitance are considered. Specifically, The TE and BE metals are TiN and Pt, respectively. The resistances of these electrode metals used in the simulation works are 5 ${\times}$ 10$^{-4}$ ${\Omega}$ and 2.1 ${\times}$ 10$^{-3}$ ${\Omega}$, respectively, and the values were brought from the previous literature ^[5-7].

Fig. 1. Schematics of a ReRAM cell under different bias conditions: (a) Before; (b) after forming operations. The blue cylinder indicates the conductive filament; (c) Reset operation partially oxidizes the CF creating a barrier (orange cylinder). The barrier length increases as the magnitude of bias gets larger over a RESET operation in (d) and (e).

Fig. 2. A widely used simple equivalent circuit of ReRAM cell composed of resistors and a single capacitor.

III. RESULTS AND DISCUSSION

A. Forming Operation and Current Compliance

Forming operation is attributed by controlled breakdown of the switching layer, which is usually accomplished by applying a high enough positive voltage while forcing a particular level of compliance current I$_{\mathrm{C}}$. The forming operation can be explained by a process in the loop of feedback between temperature and field effects causing the creation of CFs in the switching layer. The path for vacancy movements is prepared in this manner, through which the set and reset operations are realized. The read current after a set operation, LRS current, can be converted to LRS resistance depending on cases, by simply dividing the read voltage (inference voltage in the application of ReRAM cell as a synaptic device) by the LRS current. Further, the inverse of LRS resistance, electrical conductance, can be translated as the synaptic weight. In practical operation of ReRAM cells, limit in conduction current needs to be put. Otherwise, it becomes very probable to observe failure in reset operation due to the CFs excessively thickened by a high LRS current ^[13-19].

B. Reset Operation and Filament Dynamics

The reset operation, sensible by the change in resistance from LRS to HRS, is described by a dynamic fractional oxidation of the CF due to field and temperature-driven movement of oxygen atoms and their recombination with the metallic species (Hf in this work). Trap-assisted tunneling (TAT) of charges can be found in the current conduction across the barrier formed in the CF ^[17]. For a reset operation, a large enough negative voltage is applied on the TE with regard to BE in the ReRAM cell. Gap distance (g) between the filament tip and the electrode on the other side can be set as a dependent variable determined by external electric field, which eventually determines the cell conductivity. The speed of change in gap distance, or equivalently, growth speed of the CF can be described by a closed-form equation, Eq. (1), as follows:

Here, E$_{a}$ is the activation energy for vacancy generation, t$_{ox}$ is the switching layer (metal oxide) thickness, a$_{0}$ is the distance between hopping sites, and V is the applied voltage between TE and BE. ${\gamma}$ is the local field enhancement factor accounted for polarizability of the switching layer dielectric and can be modeled as a function of gap distance g as follows in Eq. (2).

where ${\gamma}$$_{0}$ and ${\beta}$ are fitting parameters. In particular, ${\beta}$ reflects the gradual nature of reset process. Accuracy and reality can be further expected by plugging a temperature function T into Eq. (1). By introducing a function T in Eq. (3), dependences on fluctuation in temperature, ${\delta}g$(T), and conduction current at a specific time are considered for more precise calculation of dg/dt as a function of T, I, V, and time, concurrently.

Here, I is the current through the ReRAM cell at a moment and R$_{\mathrm{th}}$ is the equivalent resistance. Eqs. (2) and (3) are rearranged in Eq. (1) to keep away from utilization of various conditions, which lifts up the convergence problems and greatly reduces total time in executing the Verilog-A codes. The gap distance g is calculated by considering electric field, temperature-dependent oxygen migration, and local temperature. The current flowing through an ReRAM cell can be formulated in relations with V and g as Eq. (4). g$_{0}$, ${\alpha}$, and V$_{0}$ can be extracted in fitting the modeled current to the experimentally measured value.

Table 1 summarizes the values of the primary parameters making up Eqs. (2) through (4), which have been adopted in compact modeling. The reference current I$_{0}$ of 1 A in Table 1 is surely an excessively large quantity but it makes the control over ${\alpha}$ more easily recognizable. ${\alpha}$ plays a role of scaling I$_{0}$ down to a modeled current to a realistic value in consideration of the cell area, by which a rather imaginary number of I$_{0}$ is altered a practical one. Fig. 3 shows the I-V curve from the developed compact model with I$_{\mathrm{C}}$ = 1 μA. The modeled HfO$_{2}$ ReRAM cell clearly exhibits bipolar switching behaviors with the set operation voltage at 1 V and the reset one at -1.2 V, which demonstrates low-voltage operation capability. These results show the coherence with those obtained in an earlier literature on the ReRAM device with the same material stack ^[14].

Fig. 4 shows that the weight tunability has been also modeled altering the compliance current. Controlling the compliance current can be presumed to be realized by either width or gate voltage control over the transistors at the bitline ends in the practical sense. Further reduction in power consumption can be schemed with the help of optimal structure design with adequately working scalability based on the lumped equations in the mode. The constructed model can be transplanted into the higher-level integrated circuit and system designs toward the advanced computing architecture embedding the ReRAM cell array.

I$_{0}$, g$_{0}$, V$_{0}$, ${\upsilon}$$_{0}$, ${\gamma}$$_{0}$, and ${\beta}$ are the practical parameters that effectively fit the modeled curve to the median of switching behaviors of the measured ReRAM cell. Fig. 5(a) through (e) systemically demonstrates the effects of individual fitting parameters on DC switching I-V characteristics of the ReRAM cell to be modeled. I$_{0}$, g$_{0}$, and V$_{0}$ dominate the nonlinearity in the I-V curves as can be inferred by Fig. 5(a), (c), and (e). V$_{0}$ depicts the nonlinearity in the resistance curve more specifically. I$_{0}$ mainly shifts the curve to different current levels. g$_{0}$ has the major controllability over the set operation voltage shift as shown in Fig. 5(b). ${\upsilon}$$_{0}$, ${\gamma}$$_{0}$, and ${\beta}$ are correlated with the process of filament growth, or change in gap distance g. In particular, ${\upsilon}$$_{0}$ and ${\gamma}$$_{0}$ determine the voltage where the gap starts to grow. ${\upsilon}$$_{0}$ mainly changes the reset voltage at the knee point, while ${\gamma}$$_{0}$ has an effect of adjusting set voltage. ${\beta}$ modulates the slope in the reset operation as shown in Fig. 5(d). Fig. 6 depicts the displacement current in the modeled HfO$_{2}$ ReRAM cell as a function of pulse rising time. Rising time of the input pulse in operating an ReRAM device is one of the important control variables regarding operation speed and device reliability. There can be a huge amount of surge displacement current in the short rising time regime as can be implied by Fig. 5. The displacement current stimulated by a pulse with a short rising time originates from the capacitive component depicted in Fig. 2, which exists in the actual ReRAM cell due to the switching dielectric layer, not just as a modeling component ^[15]. Fig. 6 provides a guideline for optimally shaping the operation voltage pulse.

Fig. 3. DC I-V characteristics of a HfO2ReRAM cell showing typical bipolar switching behaviors.

Fig. 4. Weight tunability by compliance with current control.

Fig. 5. Dependency of I-V characteristics of a HfO2ReRAM cell on model parameters: (a) LRS current levels at different I0’s in the positive bias region; (b) Change in set voltage by controlling γ0; (c) LRS current levels by changing g0 in the negative bias region; (d) Reset operation slope in the effect of β; (e) I-V curve slope in the positive bias region depending on V0.

Fig. 6. Displacement current in the modeled HfO2ReRAM cell as a function of rising time of a pulse.

V. CONCLUSION

In this work, we have presented a more accurate compact model of an ReRAM cell with a HfO$_{2}$ switching layer. For higher accuracy and credibility of the model, stochastic device characteristic deviations subordinate to temperature change has been considered in describing the filament growth. Bipolar resistive-switching characteristics have been plausibly reproduced by the developed model in comparison with the measurement results. Also, it has been affirmed that multiple conductance levels can be obtained from the ReRAM model by identifying the deciding parameters. The weight tunability can be put in conjunction with pulse operation schemes for potentiation and depression of a synaptic device as a future task. The more accurate and compact model will play a crucial role as a pivotal part in higher-level simulations and framework tasks for faster design of integrated circuits and systems embedding a large ReRAM array toward the neuromorphic and processing-in-memory (PIM) applications as well as high-density memory product.