Studies of two-dimensional material resistive random-access memory by kinetic Monte Carlo simulations

This study performed experiments and simulations on WS₂, MoS₂, and h-BN RRAM. The device structures are shown in Figs. 1(a), 1(b), and 1(c), respectively, using Au/Ti for both the top electrode (TE) and bottom electrode (BE). To understand the impact of metal electrodes on device characteristics, the WS₂ was fabricated with asymmetric electrodes, and the MoS₂ and h-BN were fabricated with symmetric electrodes.

The active layer of the devices had various WS₂ thicknesses. Layers of 60 nm Au and 10 nm Ti were deposited on a heavily doped p-type silicon substrate as the BE of the RRAM device by electron-beam evaporation. The WS₂ flakes were obtained from single-crystal bulk material via mechanical exfoliation. The flakes were then transferred above the BE using polydimethylsiloxane. Afterward, the Au/Ti of 60/10 nm as the TE was deposited on the WS₂ flake to form an asymmetric WS₂ RRAM device. The cross-sectional area of the RRAM device was controlled by overlapping the TE and BE, as shown in Fig. 1(f). The vertical structure of the device was observed and measured via transmission electron microscopy and atomic force microscopy, as shown in Figs. 1(g) and 1(h).

The switching state of the RRAM module was simulated to plot the I-V curves using the Ginestra^TM software. This software applies KMC modeling[11, 12] to simulate the generation, diffusion, and recombination of defects (vacancies and ions) in the RRAM device. The physics solver calculates the current value, including charge transport, temperature dependence, and 3D-space defects distribution. The KMC model simulates the dynamic defect distribution of the active layer. The physical models of the WS₂, MoS₂, and h-BN RRAM are based on experimental data to extract the physical parameters of the three 2D materials.

Figures 1(d) and 1(e) illustrate the device structures of the low resistance state (LRS) and high resistance state (HRS), respectively. The former has a complete conductive filament (CF), and the latter CF becomes sparser or even fractured because ions and vacancies recombine due to applied bias. Some defects that assist in carrier transport are recombined, causing the current to drop after reset operations. The current drop trend is highly correlated with the ion drift and diffusion in the lattice space. To shorten the RRAM simulation time, we only form one CF in the small cross-sectional area model, which causes the current drop to be discontinuous during reset operations. Therefore, the reset I-V curve in each device is the result of averaging many simulation curves. There are multiple CFs formed on the large cross-sectional device, and the conduction states of each CF differ. Statistical averaging can be closer to the conduction state of the actual device.

The ion distribution in Figs. 1(d) or 1(e) suggests that the ions gradually diffuse around CF, and the HRS resistance decreases progressively with subsequent resistance state switching. Therefore, ion diffusion in the in-plane direction is highly related to the device endurance. The 2D material has polarity, which is a typical bipolar switching mode in RRAM. Usually, the forming operation is applied with a positive bias, the set operation is in the forward bias, and the reset operation is in the reverse bias. If a negative bias is applied in the forming process, the set/reset switching voltages are reversed. Both positive and negative biases were applied in the forming operation of each device for the experiments, so the set/reset bias directions were opposite for various devices. A positive bias was uniformly applied in the simulations for the forming operations to facilitate subsequent analyses.

The vacancies and ions are generated in the device during forming or set operations by continuously increasing the applied bias. The generation rate is dependent on the 3D electric field in the device.[13, 14] The associated Arrhenius equation is defined as

$$\\ \\ R_{A,G}(X,Y,Z)=\nu\ exp[-\frac{E_{A,G}-p_{0}(2+\varepsilon_{r})/3\cdot F(X,Y,Z)}{k_{B}T}],$$

(1)

where $\nu$ is a frequency prefactor, $E_{A,G}$ is the zero-field generation activation energy, $p_{0}$ is the polarizability, $\varepsilon_{r}$ is the relative permittivity, $k_{B}$ is Boltzmann’s constant, and $T$ is the temperature. The electric field causes sulfur ions to drift in the active layer, while the diffusion rate depends on the local effective electric field along the diffusion direction. This Arrhenius equation is defined as

$$\\ \\ R_{A,D}(X)=\nu\ exp[-\frac{E_{A,D}(X)-\gamma F_{X}}{k_{B}T}],$$

(2)

$$\\ \\ R_{A,D}(Y)=\nu\ exp[-\frac{E_{A,D}(Y)-\gamma F_{Y}}{k_{B}T}],$$

(3)

$$\\ \\ R_{A,D}(Z)=\nu\ exp[-\frac{E_{A,D}(Z)-\gamma F_{Z}}{k_{B}T}],$$

(4)

where $E_{A,D}(X)$, $E_{A,D}(Y)$, and $E_{A,D}(Z)$ are respectively the diffusion activation energy in the X, Y, and Z direction, $\gamma$ is the field acceleration factor, and $F_{X}$, $F_{Y}$, and $F_{Z}$ are respectively the local electric field component of X, Y, and Z.

The retention time is how long a memory unit can retain a bit state at a specific temperature. For non-volatile memory, the most stringent requirement is retaining data for more than 10 years (about $\rm 3.1536\times 10^{8}$ s) at operating temperatures up to 85 °C. To shorten the detection time, a device is placed in a high-temperature environment, and changes in time and resistance are recorded while baking. The Arrhenius diagram can be drawn by changing the temperature to record the retention failure time of the device and extract the activation energy. This is extrapolated to the working temperature to obtain the retention time at the considered temperature. An experimental report by Gao et al.[15] indicates that the retention time of the device is also calculated through the generation of the activation energy and theoretical formulas. The generation probability of the unbiased voltage is defined as

$$\\ \\ p=exp(-E_{a}/k_{B}T),$$

(5)

where $E_{a}$ is the generation activation energy. The retention failure time of the device is defined as

$$\\ \\ t_{E}=t_{0}/(n|ln(1-p)|)\approx t_{0}/np,$$

(6)

where $t_{0}$ is the oscillation period of lattice atoms, and $n$ is the number of escape directions for ions inside the lattice (for a cube, $n$ is 6). Usually, the generation probability will be far less than 1, so we apply a Taylor expansion to the original Eq. (6) formula to obtain the first-order term. This prevents the dilemma where the denominator is zero and cannot be solved.

This work considered three 2D materials (WS₂, MoS₂, and h-BN) for comparison. More detailed studies on WS₂ materials have been made to build accurate models for KMC simulations, which can be used for device optimizations. Figure 2(a) provides the measured and simulated set/reset current characteristics with 33-nm-thick WS₂ RRAM (44 layers), and the set/reset switching voltages are 0.5 and -0.6 V, respectively. The experimental data are the result of measuring 100 set/reset operation cycles, and the simulated set operation establishes a compliance current of $10^{-3}$ A. If the current is larger than $10^{-3}$ A during set switching, the system terminates the simulation and records the last data point. So, the simulated current after set operations is slightly larger than in the experimental data.

The retention failure time is calculated by substituting the atomic oscillation period and the generation activation energy extracted from the simulations to Eq. (6). The extracted generation activation energy of WS₂ is 1.11 eV, coinciding with the calculated formation energy of S vacancy by density functional theory (DFT).[16] The oscillation period is 18 fs,[17] and the retention failure time is $\rm 1.23\times 10^{4}$ s at room temperature, or about 3 h. The experimental retention time of WS₂ can be maintained to $\rm 10^{4}$ s. The experimental results of other researchers also show that the retention time of WS₂ can reach $\rm 10^{4}$ s.[18, 19]

The simulation results indicate that the presence of an asymmetric electrode structure generates an internal electric field due to the utilization of different types of contact electrodes, thereby exerting an influence on the symmetry of the I-V curves. The work function of the contact surface with Au (BE) is larger than that of the Ti contact surface (TE) [21]. Consequently, when the reverse bias voltage is swept, the internal electric field partially offsets the external electric field, leading to a lower current under the same bias voltage than the forward bias voltage.

Figure 2(b) gives the measured and simulated set/reset I-V characteristics of the 20-nm-thick MoS₂ RRAM. The switching voltages are 0.9 and -0.9 V, and the current of the reset switching dropping trend is faster than that of WS₂. Thus, the field acceleration factor of MoS₂ is greater. The determined activation energy for the generation process of MoS2 stands at 1.13 eV, aligning harmoniously with the computationally determined formation energy of a sulfur (S) vacancy, as evaluated through DFT. [16] The oscillation period is 21.51 fs,[22] and the retention failure time at room temperature is estimated as $\rm 3.18\times 10^{4}$ s, or about 8 h. The experimental results of other researchers also show that the retention time of MoS₂ can reach $\rm 10^{4}$ s.[23]

Figure 2(c) provides the measured and simulated set/reset I-V characteristics of the 39.4-nm-thick h-BN RRAM. The switching voltages are 0.65 and -0.6 V, and the current of the reset switching dropping trend is smoother than that of WS₂. Thus, the field acceleration factor of h-BN is lower. Taking into account the potential replacement of a nitrogen atom by oxygen in the surrounding environment, the extracted generation activation energy of h-BN is 1.28 eV. This closes to the computationally derived formation energy of oxygen defects, as determined through the employment of the local density approximation (LDA).[24, 25] The oscillation period is 24.4 fs,[26] and the retention failure time at room temperature is estimated as $\rm 1.18\times 10^{7}$ s, or about 136 days. Previous experimental results show that the retention time of h-BN reaches $\rm 10^{7}$ s.[27] Although h-BN has a larger bandgap, the contact between h-BN and Ti causes the Fermi level of the electrode to be fixed on the p-type bandgap.[28] The defects are generated in the depth of the bandgap due to the distribution of the Fermi level, and these vacancies can only transport a relatively small current. Therefore, the difference between the HRS and LRS current is small, and the On/Off ratio of h-BN cannot attain meaningful improvements.

Table 1 gives the basic parameters used by three 2D materials in the simulations. Table 2 indicates the simulation parameters used by the 2D materials and another hafnium oxide (HfO_x) RRAM concept.[40] The WS₂ and MoS₂ have similar generation activation energies, and their retention failure times are of the same order. The diffusion activation energy of MoS₂ in the out-of-plane direction is lower than that of WS₂, and the field acceleration factor is higher. Thus, sulfur ions more easily drift and transport inside MoS₂, which is likely due to the mass densities of the material. The mass density of MoS₂ is 5.06 $\rm g/cm^{3}$,[41] and WS₂ is 7.5 $\rm g/cm^{3}$.[42] This lessens the drift resistance of sulfur ions inside MoS₂ from atomic collisions, increasing the transport speed. Therein, the reset switching current drops more rapidly.

The generation activation energy of h-BN is the highest among the three 2D materials and has a long retention failure time. Its field acceleration factor is much lower than WS₂ and MoS₂, causing the decreased reset switching to be relatively small. This parameter characteristic explains why h-BN produces threshold switching[20] in the experiments. Therefore, h-BN must be applied with sufficient operating power to completely separate ions and vacancies, forming non-volatile memory with stable CFs. Otherwise, ions and vacancies can only be stretched outward to form electric dipoles. After removing the applied electric field, ions recombine with vacancies, exhibiting volatile memory characteristics.

As HfO_x RRAM has a greater activation energy, the retention failure time at room temperature can exceed 10 years. Thus, HfO_x has been widely studied in RRAM applications. The On/Off ratio of WS₂ RRAM is about 10 at 0.1 V, while that of the HfO_x RRAM is about 50.[40] The On/Off ratio of WS₂ RRAM is five times smaller because its bandgap is smaller than HfO_x. This increases the background current (not the current transported through the defect) of WS₂, resulting in a smaller difference between the HRS and LRS. Therein, the On/Off ratio of the overall device is relatively poor. Therefore, choosing a material with a large bandgap to make an RRAM can usually provide a higher On/Off ratio.

Compared with the benchmark device (HfO_x RRAM[40]), this work shows that 2D RRAM has a lower generation activation energy to generate defects at a smaller bias. The diffusion activation energy of 2D RRAM along the out-of-plane (Z) direction is also lower than that of HfO_x RRAM. This indicates that ions easily diffuse along the out-of-plane direction in 2D material compared to HfO_x. No chemical bond between layers causes this effect. Under the 2D layered molecular arrangement, the torque perpendicular to the plane is more likely to cause molecular bond breaking and form defects, which act as channels for ion transport. Therefore, the set/reset switching voltage of 2D RRAM is lower than that of HfO_x, which means 2D RRAM has a faster resistance switching speed. The data in Table 2 indicate that MoS₂ has the shortest switching time. Thus, MoS₂ is the most suitable for making RRAM devices among the three considered 2D materials. To verify the reliability of the KMC model, we conduct a series of experiments and simulation comparisons for WS₂ in the next section.

The simulations are performed with the extracted WS₂ parameters to optimize the On/Off ratio as a function of thickness. The simulation results attain a stable I-V characteristic curve, as shown in Fig. 5(a). The stopping voltage and compliance current for each thickness are set to -1.5 V and $10^{-3}$ A. The TE and BE biases and the vertical electric field ($F=V/d$) indicate that thinner devices have smaller switching voltages, which is consistent with the simulation results. The current of thinner devices drops more rapidly during reset switching. The vertical electric field for thicker devices increases less from the sweeping bias, and the drift distance of ions in the vertical direction inside the device is longer. Thus, the overall ion drift speed is slower, and the switching time is longer, which gives a smoother current reduction curve. After all the ions drift from the TE to the BE, no more ions or vacancies exist for bonding recombination reactions. The current of the element then gradually increases with the bias voltage.

The current trend for each thickness in Fig. 5(a) suggests that the current of the thicker device drops during reset operations because there are more defects. Both the differences in the currents and in the number of defects between the HRS and LRS increase. As the thickness exceeds 40 nm, ion diffusion in the in-plane direction and the vertical distance both increase, which reduces the difference in the number of defects. Thus, the HRS current tends to be at the same level as the LRS, and the On/Off ratio difference is small. Table 4 provides the current and On/Off ratio of each thickness at -0.1 and 0.1 V biases. The 40-nm-thick device has the best On/Off ratio as it minimizes bit reading errors. Thinner devices have faster switching speeds and lower energy consumption, but their On/Off ratios are smaller, which explains why few-layer 2D materials are not commonly made into RRAM devices.

As MoS₂ has the fastest switching speed among the three considered 2D materials, we conduct electrical simulations for MoS₂ RRAM with various thicknesses. The simulation method is the same as in the previous section. The difference is the stopping voltage, which is set based on the state of the reset operations for each thickness. The stopping voltages for thicknesses of 10 to 50 nm are -0.5, -1.0, -1.5, -2.0, and -2.5 V so that the devices can be fully reset to attain the maximum on/off ratio.

Figure 5(b) shows thicker devices have more extensive On/Off ratios. The On/Off ratios for each thickness are given in Table 5. The 50 nm MoS₂ RRAM has the largest On/Off ratio, which is much larger than that of WS₂ RRAM. Therefore, MoS₂ is more suitable for RRAM devices in terms of switching speed and On/Off ratios.

The thickness of the RRAM should be determined based on the given operating voltage (V_DD) and power consumption constraints on the overall circuit system. Table 5 shows that thicker devices have greater switching voltages, which leads to increased power consumption. The stopping voltage of the 50 nm devices is set as a reverse bias voltage greater than 2.0 V. The current noise should be reduced to less than the device On/Off ratio to avoid memory misinterpretation.

The authors have no conflicts to disclose.

Yu-Ting Chao fabricated and measured the 2D devices. Ying-Chuan Chen simulated various devices and analyzed the data and wrote the manuscript. Chao-hsin Wu conduct the experiment work and Yuh-Renn Wu supervises the simulation. All the authors discussed the results and explanations.