Behavioral Model of Molecular Gap-Type Atomic Switches and Its SPICE Integration

Hiroshi Kubota^1*, Tsuyoshi Hasegawa², Megumi Akai-Kasaya¹, Tetsuya Asai¹

¹Graduate School of Information Science and Technology, Hokkaido University, Sapporo, Japan.
²School of Advanced Science and Engineering, Waseda University, Tokyo, Japan.

DOI: 10.4236/cs.2022.131001                

Abstract

Atomic switches can be used in future nanodevices and to realize conceptually novel electronics in new types of computer architecture because of their simple structure, ease of operation, stability, and reliability. The atomic switch is a single solid-state switch with inherent learning abilities that exhibits various nonlinear behaviors with network devices. However, previous studies focused on experiments and nonvolatile memory applications, and studies on the application of the physical properties of the atomic switch in computing were nonexistent. Therefore, we present a simple behavioral model of a molecular gap-type atomic switch that can be included in a simulator. The model was described by three simple equations that reproduced the bistability using a double-well potential and was able to easily be transferred to a simulator using arbitrary numerical values and be integrated into HSPICE. Simulations using the experimental parameters of the proposed atomic switch agreed with the experimental results. This model will allow circuit designers to explore new architectures, contributing to the development of new computing methods.

Keywords

Memristor, Atomic Switch, Behavioral Model, SPICE

1. Introduction

Atomic switches are a new type of nanodevices [1]. Recently, atomic switches have been developed for nonvolatile memory devices. Typically, they are generated simply by crossing Pt wires and Ag₂S-coated Ag wires and can be created with materials such as Cu₂S [2] [3]. Atomic switches can be controlled by applying a voltage between the electrodes, and their switching characteristics can be switched on and off at room temperature and in the air [4] - [9]. In addition, atomic switches have a large ratio of the highest resistance state to the lowest resistance state and can maintain their states for more than a week [10] [11]. Their working speeds are as fast as those of conventional electronic devices, and this will enable the development of a new type of computer architecture beyond von Neumann computers [12] [13] [14]. Atomic switches have the characteristic of a memristor, which means that devices memorize the input voltage, and the resistance of the device decreases as the number of inputs increases or the input interval is short [15] [16] [17]. Because atomic switches have synapse-like characteristics, their network hardware, which consists of neuromorphic devices such as reservoirs, has been studied [18] [19] [20] [21] [22]. Physical and chemical phenomena occurring in atomic switches, such as redox reactions, ion diffusion, and their nucleation, have also been studied both theoretically and experimentally [23] [24] [25]. Experiments on atomic switches in physics, nonvolatile memory applications such as programmable buses, applications as a randomly generated circuit [26] [27], and equations in physics have been proposed, but there has been no research on the circuit models for SPICE. Hence, the precise physical model of the atomic switch is required, however, before completion of the precise model, a behavioral model of atomic switches that can easily be used in SPICE is also required by circuit engineers developing memristor-based novel circuits and systems, aiming at exploring novel values of memristors, instead of optimizing circuit performances by using precise physical models. In this paper, we propose simple differential equations that relate the input voltage of an atomic switch to the resistance of the device. These equations were added to HSPICE. This enables engineers, mathematicians, and circuit designers who have not used these devices until now, for example, to explore new computing methods. When voltage is applied to a molecular gap-type atomic switch, the number of ions in the electrode is reduced, and atoms are deposited in the gap between the electrodes. This switch has two states: one in which a metal filament grows between the electrodes, and the other in which the filament growth is completed and the filament that bridges the two electrodes becomes stable. This bistability is represented by a double potential well, which can represent the states of metal atoms between electrodes as an equation. In addition, we developed an equation for metal ions in the electrode and mapped the states of metal atoms and device resistances. Device operations can be described using these equations.

2. Behavioral Model of Atomic Switches

The equations represent the qualitative behavior of these atomic switches. When voltage is applied to this element, metal ions drift to the negative side of the electrode and deposit as atoms between the electrodes, causing an element resistance change.

2.1. Ion Density Gradient

The deposition of atoms is caused by the reduction in the number of ions, and the electrochemical potential of ions in the electrode changes the deposition speed [28]. We used molecular gap atomic switches with one electrode containing ions and the other without ions. We introduced a variable ${\sigma }_r$ that shows the entire status with respect to the ion shift in the electrode when voltage is applied. ${\sigma }_r$ is positive when the electrochemical potential of ions advances the atom deposition; otherwise, it is negative and is described as

${\tau }_r{\stackrel{\dot{}}{\sigma }}_r=-\left({\sigma }_r-\alpha v\right),$ (1)

where v is the applied voltage, alpha is the gain, and ${\tau }_r$ is the time constant. This equation gives the value of ${\sigma }_r$ as per $\alpha v$ with a time delay.

2.2. Growth of Metal Filament

Deposited atoms form a metal filament that grows from a positively biased electrode to a negatively biased electrode, forming a bridge between the electrodes. Under reduced or reversed voltages, the atoms are ionized and dissolved into the electrode, where they initially existed. An atomic switch has two stable states: an OFF state in which ions exist in an electrode under lower voltage, and an ON state in which the bridge formation is completed. These two states are stable even after removing applied bias. This demonstrates bistability. However, if the ON state element is applied with reversed voltage over the threshold, it changes to the OFF state [27].

The status of an atomic filament is represented by variable ${\sigma }_0$. We assume that the bistable dynamics are obtained by the following double-well function:

$H\left({\sigma }_0\right)\equiv \frac{{\sigma }_0^4}{4}-\frac{{\left({\sigma }_0-V_d\right)}^2}{2}-{\sigma }_0{f}_{\text{ext}}\left({\sigma }_r\right),$ (2)

where ${\sigma }_0$ represents the two states ON/OFF ( ${\sigma }_0\ge 1/\sqrt{3}$ : ON and ${\sigma }_0\le -1/\sqrt{3}$ : OFF).

$f_{\text{ext}}\left({\sigma }_r\right)$ is the external force, which is a function of ${\sigma }_r$ [because the deposition or ionization of the bridging atoms is dominated by the electrochemical potential of ions ( ${\sigma }_r$ )].

$V_d$ is a constant representing the asymmetry of the amounts of the external forces [ $f_{\text{ext}}\left({\sigma }_r\right)$ ] for ON → OFF and OFF → ON transitions. From the stable condition given by $\text{d}{\sigma }_0/\text{d}t=-\nabla H$, we obtain

${\tau }_0{\stackrel{\dot{}}{\sigma }}_0=-\left({\sigma }_0^3-\left({\sigma }_0-V_d\right)-f_{\text{ext}}\left({\sigma }_r\right)\right),$ (3)

where ${\tau }_0$ is a time constant, which is time dependent. We define $f_{\text{ext}}\left({\sigma }_r\right)\equiv p{\sigma }_r$, where p (>0) is a constant.

In Equation (3), when the condition is $-2\times 3^{-1.5}\le {\sigma }_r-V_d$, the state changes from the OFF state to the ON state; and when the condition is $-2\times 3^{-1.5}\ge {\sigma }_r-V_d$, the state changes from the ON state to the OFF state. In addition, in Equation (1), ${\sigma }_r=\alpha v$ when time passes. Hence, we obtain

$\alpha =\frac{4}{3\sqrt{3}}\frac{1}{V_{\text{on}}-V_{\text{off}}},$ (4)

$V_d=\frac{2}{3\sqrt{3}}\frac{V_{\text{on}}+V_{\text{off}}}{V_{\text{on}}-V_{\text{off}}},$ (5)

where $V_{\text{on}}$ is the OFF → ON threshold voltage, and $V_{\text{off}}$ is the ON → OFF threshold voltage.

2.3. Equation of Resistance

The atom deposition changes the element resistance, and the resistance exponentially decreases with an increase in the number of atoms deposited between the electrodes. Therefore, the resistance function $R\left({\sigma }_0\right)$ is defined by the exponential function as

$R\left({\sigma }_0\right)\equiv k_R\exp \left\{-\beta \times f\left({\sigma }_0\right)\right\},$ (6)

where $k_R$ and $\beta$ are the fitting parameters, and $f\left({\sigma }_0\right)$ is a monotonically increasing function that represents high resistance when ${\sigma }_0$ is negative and low resistance when ${\sigma }_0$ is positive.

3. Experimental and Simulation Results

3.1. Target Device and Features

Pt/PTCDA¹/Ta₂O₅/Ag molecular atomic switch is our target device [Figure 1(a)] [27]. It is composed of Ag (active electrode), Ta₂O₅ (ionic transfer layer), PTCDA (gap layer), and Pt (inert electrode). When a positive voltage is applied to the Ag electrode, Ag⁺ ions move in the transfer layer toward the gap layer and deposit as Ag atoms that subsequently bridge the gap.

Figure 1(b) shows the V–I characteristics of this device. Voltage is applied to the bottom electrode as 0 V → 0.3 V → −0.3 V → 0 V with a compliance current of 50 μA. Initially, the atomic switch was in the OFF state, then it changed to the ON state at 0.29 V and reset to the ON state at –0.16 V. We extracted the parameter values of $V_{\text{off}}$ and $V_{\text{on}}$, which are the threshold voltages that cause transitions of the device states in Equations (4) and (5), from Figure 1(b) as $V_{\text{off}}=-0.16\text{V}$, $V_{\text{on}}=0.29\text{V}$.

Figure 1(c) and Figure 1(d) show the behavior of another device under the application of voltage pulses (0.36 V in height and 1.0 s. in width) at different intervals of (c) 4.0 s and (d) 1.0 s. Short-interval pulses can change an OFF state device to the ON state, but long interval pulses cannot.

The measurement method and the conditions were the same as in this paper [27].

3.2. Determination of Resistance Function

We defined the OFF state as ${\sigma }_0<s_{th}$, the ON state as ${\sigma }_0>s_{th}$, where $s_{th}$ is the local maximum of $H\left({\sigma }_0\right)$ representing the peak of the intervening barrier when $f_{\text{ext}}\left({\sigma }_r\right)=0$, as shown in the upper part of Figure 2, $R_{\text{high}}$ as the high resistance when the device is OFF state, and $R_{\text{low}}$ as the low resistance when

Figure 1. (a) Target atomic switch device. (b) V-I characteristics of device (John Wiley and Sons, license # 5255060486672) (c) and (d) Conductance of device when 10 pulses are applied in the atomic switch. Pulse heights, widths, and intervals of (c) are 0.36 V, 1.0 s, and 4.0 s; and (d) 0.36 V, 1.0 s, and 1.0 s, respectively [27].

Figure 2. Potential $H\left({\sigma }_0\right)$ [Equation (2)] and device resistance function $R\left({\sigma }_0\right)$ [Equation (6) with Equation (7)] vs. ${\sigma }_0$.

the device is ON state. In addition, we assume that the monotonically increasing function $f_{\text{ext}}\left({\sigma }_r\right)$ as a function of the device resistance [Equation (6)] is a resistance function that changes smoothly with ${\sigma }_0$ ; hence, we used a hyperbolic tangent function. In addition, $R\left({\sigma }_0\right)$ decreases as ${\sigma }_0$ increases, which is the filament growth; thus, we used a linear increasing function. To sum up, we defined $f\left({\sigma }_0\right)$ in Equation (6) as follows:

$f\left({\sigma }_0\right)\equiv \left({\sigma }_0-\theta \right)\times g_R+\tanh \left\{\left({\sigma }_0-\theta \right)\times s_R\right\}$, (7)

where $\theta$ is the inflection point value in $f\left({\sigma }_0\right)$, $g_R\left(>0\right)$ is the gradient of the linear increasing function, and $s_R$ is the gradient at the inflection point in $f\left({\sigma }_0\right)$. The parameters $\theta$ and $s_R$ are defined below.

First, parameter $\theta$ is explained as follows. As shown in Figure 1(c) (changes in the conductance of the atomic switch device in its OFF state with periodic voltage pulse inputs), the device is not completely turned on (the conductance is temporarily increased at the pulse onset [positive pulse amplitude], but suddenly decreased to zero when the pulse amplitude becomes zero). However, the conductance is significantly changed, that is, the device resistance changes between $R_{\text{high}}$ and $R_{\text{low}}$ even though the device remains in the OFF state. Therefore, $s_{th}$ cannot directly represent the resistance change inflection points.

Here, we introduce two new parameters— $\theta$, representing the resistance change inflection point, as shown in the lower part of Figure 2, and $d_R$, representing the difference between the threshold value of the ON/OFF state transition ( $s_{th}$ ) and $\theta$, that is, $d_R\equiv s_{th}-\theta$.

Second, the parameter $g_R$ is explained. Figure 1(d) shows that the device resistance decreases when pulses are applied, even after the transition to the ON state. Because $f\left({\sigma }_0\right)\approx \left({\sigma }_0-\theta \right)\times g_R+1$ in the ON state and the increase in ${\sigma }_0$ with pulse input increases $f\left({\sigma }_0\right)$ [decreasing $R\left({\sigma }_0\right)$ ], we can fit the parameter values of $\theta$ and/or $g_R$ with the experimental results. For simplicity, we chose $g_R$ as the tuning parameter.

In our simulations, we determined the parameters $s_R=15$ and $\theta =-0.38$. When $f\left({\sigma }_0\right)$ is defined and the parameters of are determined as outlined above,

$R_{\text{high}}=R\left(-1\right)\approx k_R\exp \left[-\beta \left\{\left(-1-\theta \right)\times g_R-1\right\}\right]$, (8)

$R_{\text{low}}=R\left(0\right)\approx k_R\exp \left\{-\beta \left(-\theta \times g_R+1\right)\right\}$ (9)

are derived.

We determined $R_{\text{high}}=10\text{M}\Omega$ because when the gap layer has no atoms, the tunneling current flow and the device resistance are approximately 10 MΩ. We determined $R_{\text{low}}$ from experimental results. From this, we obtained the following parameters:

$\beta =\frac{\log \left(\frac{R_{\text{high}}}{R_{\text{low}}}\right)}{g_R+2}$, $k_R=R_{\text{high}}\exp \left[-\beta \left\{\left(1+\theta \right)\times g_R+1\right\}\right]$. (10)

The lower part of Figure 2 shows this resistance curve as outlined above with $\theta =-0.38$, $g_R=0.1$, $s_R=15$, $R_{\text{high}}=10\text{M}\Omega$, and $R_{\text{low}}=2.8\text{k}\Omega$.

3.3. HSPICE Netlists

Figure 3 shows HSPICE netlists of our atomic switch model based on Equations (1), (3), and (6) with Equation (7). Figure 3(a) is the atomic switch subcircuit.

Figure 3. HSPICE netlists of behavioral atomic switch model. (a) Atomic switch subcircuit. (b) Simulation circuit for V-I characteristics. (c) Subcircuit of compliance current. (d) Simulation circuit with pulses.

Both differential Equations (1) and (3) are represented by a circuit of capacitors (“c” elements) and voltage-controlled current source (“g” elements). Details for implementing ordinary differential equations in SPICE can be found in the general guidelines [29] [30]. The device parameters were defined as variables in “.params”. The ends of the device are nodes “n1” and “n2”. The resistance was represented by a current source that supplies $V_{\text{n}1,\text{n}2}/R\left({\sigma }_0\right)$, where $V_{\text{n}1,\text{n}2}$ is the voltage applied to the device, and $R\left({\sigma }_0\right)$ is the resistance of the device calculated in this subcircuit.

Figure 3(b) shows the test circuit for simulating the V-I characteristics of the atomic switch model. In this simulator, the input voltage transition changes linearly with 0 V → 0.3 V → –0.3 V → 0 V with a compliance current of 50 µA. The compliance current is provided by the subcircuit shown in Figure 3(c). The subcircuit has a 50 µA current source and a voltage-controlled switch. This switch is off when 50 µA minus the value of the flowing current in this subcircuit element is negative, otherwise, it is on. This means that the current value flowing in this subcircuit is limited to 50 µA; otherwise, all the current flows through this element.

Figure 4. (a) Simulation result for V-I characteristic. Initially, OFF state is set and it is changed to ON state at 0.29 V and reset and changed to OFF state at −0.16 V. (b)-(c) Simulation results of device conductance with input pulses. Pulse heights, widths, and intervals of (b) are 0.36 V, 1.0 s, and 4.0 s; and (c) 0.36 V, 1.0 s, and 1.0 s, respectively.

Figure 3(d) is a pulse response simulator circuit with height, width, and interval of 0.36 V, 1.0 s and 4.0 s, respectively. This subcircuit was simulated with a variety of heights, widths, and intervals by changing the top “.param” card.

3.4. Simulation Results

In the simulations shown in Figures 4(a)-(c), we set $\theta =-0.38$, $g_R=0.1$, $s_R=15$, $p=1.2$, ${\tau }_r=20\text{ms}$, and ${\tau }_0=345\text{ms}$. Figure 4(a) shows the results of our atomic switch model simulated by Figure 4(b), which provides the netlist, similar to Figure 4(b). We extracted $R_{\text{low}}$ from Figure 1(b) as $R_{\text{low}}=2.8\text{k}\Omega$. This indicates that our model has two states: it changes from the OFF state to the ON state at 0.29 V and returns from the ON state to the OFF state at approximately −0.16 V. Each state has a different resistance.

We simulated this model with a pulse whose height, width, and interval in (b) are 0.36 V, 1.0 s, and 4.0 s; and (c) are 0.36 V, 1.0 s, and 1.0 s. In Figure 4(b) and Figure 4(c), we simulated our model with each pulse condition from Figure 1(c) and Figure 1(d), which are the different interval pulse inputs. We extracted $R_{\text{low}}$ from Figure 1(d) as $R_{\text{low}}=830\Omega$. When different interval pulses are applied, we obtain different transitions, which remain in the OFF state with longer interval pulses and change to the ON state with shorter interval pulses.

4. Conclusions

We proposed a behavioral model of a molecular gap-type atomic switch. It was described by two differential equations and a nonlinear function and can be integrated into a circuit simulator. We integrated on HSPICE, and the simulation represented the device characteristics. This showed two states and a pulse-dependent change. The results agreed with the experimental results for a molecular gap-type atomic switch.

By using the proposed model, various circuit engineers can use a novel nonlinear, nonvolatile memory device in their research, which may accelerate exploring new architectures for nonvolatile memory-array circuits, expansion of nonlinear science that exploits nonlinear dynamics of atomic switches as well as novel computing architectures. For example, a novel reservoir computing on an elementary model of atomic switches with high precision and excellent memory characteristics has already been proposed [31], which implies that the proposed behavioral model will be useful for exploring the extension of various analog and digital circuits and systems.

Acknowledgements

This work is partly based on the results obtained from a project (JPNP16007) subsidized by the New Energy and Industrial Technology Development Organization (NEDO).

NOTES

¹N,N'-Dioctyl-3,4,9,10-Perylenedicarboximide.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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