††thanks: Author to whom correspondence should be addressed<EMAIL_ADDRESS>
# Studies of two-dimensional material resistive random-access memory by
kinetic Monte Carlo simulations

Ying-Chuan Chen  Institute of Photonics and Optoelectronics, National Taiwan
University, Taipei City 10617, Taiwan  Corporate Research, Taiwan
Semiconductor Manufacturing Co., Ltd., Hsinchu City 30078, Taiwan  Yu-Ting
Chao  Institute of Photonics and Optoelectronics, National Taiwan University,
Taipei City 10617, Taiwan  Edward Chen  Corporate Research, Taiwan
Semiconductor Manufacturing Co., Ltd., Hsinchu City 30078, Taiwan  Chao-Hsin
Wu  Author to whom correspondence should be addressed<EMAIL_ADDRESS>Graduate Institute of Photonics and Optoelectronics, National Taiwan
University, Taipei City 10617, Taiwan  Yuh-Renn Wu  Graduate Institute of
Photonics and Optoelectronics, National Taiwan University, Taipei City 10617,
Taiwan

###### Abstract

Resistive memory based on two-dimensional (2D) tungsten disulfide (WS2),
molybdenum disulfide (MoS2), and hexagonal boron nitride (h-BN) materials is
studied via experiments and simulations. The influence of the active layer
thicknesses is discussed, and the thickness with the best On/Off ratio is
found for 2D resistive random-access memory (RRAM). This work reveals
fundamental differences between a 2D RRAM and conventional oxide RRAM.
Furthermore, the physical parameters extracted using the kinetic Monte Carlo
(KMC) model indicate that 2D materials have a lower diffusion activation
energy along the vertical direction, where a smaller bias voltage and a
shorter switching time can be achieved. The diffusion activation energy from
the chemical vapor deposition (CVD)-grown sample is much lower than for
mechanically exfoliated samples. The results suggest that MoS2 has the fastest
switching speed among the three considered 2D materials.

††preprint: APS/123-QED

##  I Introduction

The demand for storage devices is also growing along with the increasing
popularity of artificial intelligence and the Internet of Things. However,
traditional storage devices cannot meet the demand. For example, flash memory
suffers from insufficient durability, the cache memory capacity is too small,
etc. The most attention-grabbing group of new memory is called storage class
memory, characterized by good access speed and improved cache memory capacity.
RRAM is one example. Compared with traditional memory, RRAM has the advantages
of high memory density ($\sim$2.5 times that of NOR flash memory (an
electronic non-volatile memory made by the NOR logic gate)), high switching
speed ($<10$ ns), and better durability ($>10^{6}$ times).[1, 2]

Commonly used materials for RRAM are primarily transitional metal oxides,
which are fabricated as a sandwich structure of metal/insulator/metal. This
establishes the characteristic index of RRAM. The search for new RRAM
materials has progressed in recent years to the utilization of 2D materials,
such as graphene, h-BN, MoS2, WS2, and molybdenum ditelluride.[3, 4, 5, 6, 7]
Due to their low thermal budget, these 2D materials have the potential for
back-end-of-line devices and monolithic three-dimensional integrated
circuits.[8] The WS2 and MoS2 are currently the most promising 2D candidates
for logic applications due to their high mobility and large bandgap. To
effectively suppress the delay time and the power consumption between the
logic and memory layers, the embedded memory made by WS2/MoS2 RRAM matches the
requirements and can be further used for in-memory and neuromorphic
computing.[9]

Here, WS2 RRAM of different thicknesses is fabricated using gold/titanium
(Au/Ti) contacts. The MoS2 and h-BN 2D materials are also made for comparison.
We applied the KMC method developed from Ginestra[10] for experimental fitting
to extract the material properties, which helps further determine the
optimized structures in RRAM designs. Unlike analytic models, which often
oversimplify the conduction mechanism, the physics-based KMC method coupled
with Poisson and drift-diffusion solvers can extract the diffusion activation
energy and obtain the defect formation along with their distributions. This
provides an estimate of the thickness-dependent switching behaviors. The
retention failure time and breakdown voltage predicted by the physics-based
model agree well with the experiments. Furthermore, we find the thickness
having the best On/Off ratio through the extracted WS2 parameters and obtain
the temperature-dependent characteristics. The ion transport properties for
RRAM fabricated via CVD and mechanical exfoliation are compared. Finally, we
consider the differences in the material properties of WS2, MoS2, and h-BN
RRAM and discuss the feasibility of their application as 2D RRAM.

Figure 1: Schematic diagrams for the (a) WS2, (b) MoS2, and (c) h-BN resistive
random-access memory (RRAM) structures with plots for the 33 nm (d) WS2 RRAM
low resistance state (LRS) device structure and (e) WS2 RRAM high resistance
state (HRS) device structure. (f) The RRAM was fabricated by exfoliated WS2,
and the top electrode (TE)/bottom electrode (BE) overlap determined the
conducting area, as shown in scanning electron microscopy analysis. (g) The
WS2 RRAM cross-section was analyzed via transmission electron microscopy. (h)
The WS2 thickness from the atomic force microscopy line scan.

##  II Methods

###  II.1 Device fabrication and measurements

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).

###  II.2 Simulation methodology

The switching state of the RRAM module was simulated to plot the I-V curves
using the GinestraTM 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.

##  III Results and Discussion

###  III.1 Analysis of 2D RRAM characteristics

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]

Figure 2: (a) 33 nm WS2, (b) 20 nm MoS2,[20] and (c) 39.4 nm h-BN RRAM
set/reset I-V characteristics.[20]

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: Basic parameters of the WS2, MoS2 and h-BN. Parameter | Description | WS2 | MoS2 | h-BN
---|---|---|---|---
$\epsilon_{r}$ | Relative permittivity |  $6$[29] |  $7.1$[30] |  $5.65$[31]
$E_{g}$ (eV) | Bandgap |  $1.54$[29] |  $1.23$[32] |  $5.97$[28]
$E_{a}$ (eV) | Electron affinity |  $3.92$[29] |  $4.2$[33] |  $0.8$[28]
$k_{th}$ ($\rm W\cdot{cm}^{-1}\cdot K^{-1}$) | Thermal conductivity |  $1.21$[34] |  $0.035$[35] |  $7.51$[36]
$m_{e}$ ($\rm m_{0}$) | Electron density of states effective mass |  $0.631$[37] |  $0.73$[38] |  $0.93$[39]
$m_{h}$ ($\rm m_{0}$) | Hole density of states effective mass |  $0.832$[37] |  $0.78$[38] |  $0.77$[39]
Table 2: Simulation parameters and switching times of the WS2, MoS2, h-BN, and HfOx[40] RRAM. Parameter | Description | WS2 | MoS2 | h-BN | HfOx[40]
---|---|---|---|---|---
$E_{A,G}$ (eV) | Generation activation energy | $1.11$ | $1.13$ | $1.28$ | $2.9$
$E_{A,D}$(X/Y) (eV) | Ion diffusion activation energy in the X/Y directions | $0.7$ | $0.7$ | $0.7$ | $0.7$
$E_{A,D}$(Z) (eV) | Ion diffusion activation energy in the Z direction | $0.39$ | $0.2$ | $0.38$ | $0.7$
$E_{T}$ (eV) | Thermal ionization energy of vacancy | $0.4\pm 0.04$ | $0.3\pm 0.04$ | $3.3\pm 0.05$ | –
$\nu$ (Hz) | Frequency prefactor | $4.5\times 10^{13}$ | $4.5\times 10^{13}$ | $4.5\times 10^{13}$ | $7\times 10^{13}$
$p_{0}$ (eÅ) | Polarizability | $9$ | $29$ | $75$ | $5.2$
$\gamma$ (eÅ) | Field acceleration factor | $0.2$ | $0.4$ | $0.01$ | $0.2$
$t_{R}$ (sec) | Retention failure time | $1.23\times 10^{4}$ | $3.18\times 10^{4}$ | $1.18\times 10^{7}$ | –
$t_{S}$ (ns) | Switching time | $14.83$ | $7.33$ | $12.28$ | $21.33$

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 (HfOx) 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.

Figure 3: (a) 12 nm WS2 RRAM forming I-V characteristics,[20] (b) WS2 RRAM
retention characteristics, and (c) WS2 RRAM HRS temperature-dependent I-V
characteristics.

As HfOx RRAM has a greater activation energy, the retention failure time at
room temperature can exceed 10 years. Thus, HfOx 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 HfOx RRAM is about 50.[40] The On/Off ratio of WS2 RRAM is five
times smaller because its bandgap is smaller than HfOx. 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 (HfOx 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 HfOx RRAM. This indicates that ions
easily diffuse along the out-of-plane direction in 2D material compared to
HfOx. 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 HfOx, 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.

###  III.2 Comparison of WS2 RRAM experiments and simulations

####  III.2.1 Forming operations

Figure 3(a) provides the measured and simulated forming current
characteristics of the RRAM device made from 12-nm-thick WS2 (16 layers)
flakes. The device begins to break down at 2.3 V, and the defects inside the
material generate larger quantities, causing the current to rise rapidly. In
the experiments, a compliance current prevents the over-reaction of the
material and causes the device to burn. The current is limited to $10^{-4}$ A
when the applied voltage exceeds 2.3 V. The simulations calculate the current
characteristics of the device region that has not been burned, so the current
continues to increase. After the forming operation, the device model is output
on the compliance current to continue the reset and set simulations.

####  III.2.2 Retention time

Figure 3(b) gives the measured and simulated data at a reading voltage of 0.1
V. The solid dots are the HRS and LRS resistances measured in the experiments,
which only recorded to $\rm 10^{4}$ s. The retention time of WS2 was
maintained at least to $\rm 10^{4}$ s. The solid line is the resistance
calculated in the simulations. The actual device reduces the HRS resistance
due to switching multiple times. Therefore, it is reasonable that the
simulated On/Off ratio is slightly larger than the experimental result. In
Fig. 3(b), the simulation results predict a significant drop in the HRS
resistance after $\rm 10^{4}$ s. The resistance state cannot maintain
stability. This agrees well with previous retention failure times ($\rm
1.23\times 10^{4}$ s) calculated by Eq. (6). It is speculated that the
resistance of the HRS experimental data falling to 6 k$\Omega$ has a high
probability due to experimental measurement error. The LRS experimental and
simulation data show a stable resistance state at 1 k$\Omega$.

####  III.2.3 Temperature-dependent analysis

The electrical characteristics of the 24-nm-thick WS2 (32 layers) were
measured in the temperature-dependent experiments, where the state of this
device was HRS. The specific heat density and thermal conductivity used to
simulate the heat conduction are 1.19 $\rm J\cdot{cm}^{-3}\cdot K^{-1}$ and
1.21 $\rm W\cdot{cm}^{-1}\cdot K^{-1}$.[34] The simulation and experimental
data agree well, and the 360 K data missing from the experiment were added to
Fig. 3(a) via simulation. The experimental trend of the adjacent temperature
(340 and 380 K) indicates consistency in the corrected data. In the ambient
temperature range from 300 to 400 K, the current increases proportionally with
the temperature. When the applied bias increases, the current difference in
the temperature in adjacent regions also increases. Thus, the temperature-
dependent characteristics significantly affect the current. Therefore, we must
consider changes in the ambient temperature on the current when WS2 RRAM works
under high bias.

####  III.2.4 CVD-grown WS2 RRAM analysis

This section considers differences in the electrical characteristics and
material properties of WS2 RRAM fabricated via mechanical exfoliation and CVD.
The previous WS2 RRAM was mechanically exfoliated to obtain the 2D flakes.
Here, the device used CVD-grown WS2 flakes, which are ultra-thin devices with
a thickness of 3 nm (four layers), as shown in Fig. 4(a). Therefore, it is
necessary to fine-tune some parameters in the simulations. We changed the
basic parameters of the WS2 bulk material to three or four layers. For
example, the bandgap increased from the original 1.54 eV to 2.76 eV[43] to
simulate the carrier properties for the few-layer semiconductor device. The
experimental and simulation results are shown in Fig. 4(b). The ultra-thin
device strengthens the tunneling effect, so the overall current increases
significantly with the applied bias.

Table 3: Simulation parameters and switching time of WS2 RRAM with mechanical exfoliation and CVD. Parameter | Mechanical Exfoliation | CVD
---|---|---
$E_{A,G}$ (eV) | $1.11$ | $1.11$
$E_{A,D}$(X/Y) (eV) | $0.7$ | $0.7$
$E_{A,D}$(Z) (eV) | $0.39$ | $0.36$
$E_{T}$ (eV) | $0.4\pm 0.04$ | $0.75\pm 0.04$
$\nu$ (Hz) | $4.5\times 10^{13}$ | $4.5\times 10^{13}$
$p_{0}$ (eÅ) | $9$ | $9$
$\gamma$ (eÅ) | $0.2$ | $0.4$
$t_{R}$ (sec) | $1.23\times 10^{4}$ | $1.23\times 10^{4}$
$t_{S}$ (ns) | $14.83$ | $11.06$
Figure 4: (a) TEM was employed to analyze the CVD-grown large area transferred
WS2 RRAM. (b) CVD WS2 RRAM set/reset I-V characteristics, and (c) correlation
between the thickness and breakdown voltage of the WS2 RRAM.

Table 3 details the simulation parameters used in mechanical exfoliation and
CVD WS2 RRAM samples. The generation activation energy, polarizability, and
frequency prefactor of the two samples are the same. This indicates that the
generation of defects is independent of the arrangement state between layered
molecules. There is a high correlation between the molecules’ bond breaking
and the induced electric field between polar molecules. The most apparent
difference between the two fabrication methods is the drift and diffusion of
sulfur ions inside the device. The WS2 samples obtained via mechanical
exfoliation are primarily single crystals, and the WS2 grown via CVD is
primarily polycrystalline. Therefore, CVD WS2 has smaller domains and more
defects. Each layer has more gaps and vacancies, and the probability of sulfur
ion drifting and diffusion between the upper and lower layers is significantly
increased.

Comparing the simulation results indicates that the CVD WS2 RRAM has a lower
out-of-plane diffusion activation energy and a higher field acceleration
factor. Thus, it has a faster reset switching speed and a lower switching
voltage. The CVD WS2 device also has several defects. Therein, the required
voltage for the forming operation is lower, and the switching time is shorter.
In other words, the switching power consumption is lower. The experiments also
confirm these results. Therefore, WS2 obtained via CVD is more suitable for
making RRAM devices than mechanical exfoliation.

####  III.2.5 Breakdown electric field

The experiment measured the breakdown voltages of five devices with
thicknesses of 10, 20, 30, 40, and 50 nm for the RRAM devices from the
simulations. We assume that the device is an ideal defect-free material with
WS2 parameters from Table 2. The simulation results are consistent with the
experimental data, as shown in Fig. 4(c). The breakdown electric field is
extracted through the WS2 thickness and the breakdown voltage. The
experimental and simulation results give a breakdown electric field for WS2 of
155 MV/m. Defects are generated in the device when the internal electric field
is greater than 155 MV/m, which causes the current to increase rapidly, and
the initial forming operation is performed. We also used an initial model with
few defects to simulate the forming operations of natural materials. The
results indicate that the internal electric field generated by adding a few
defects is insufficient to affect the breakdown voltage.

###  III.3 Simulation predictions

Figure 5: Set/reset I-V characteristics of the (a) WS2 and (b) MoS2 RRAM
devices with various thicknesses.

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.

Table 4: On/Off ratio for WS2 RRAM at $\pm$0.1 V for various thicknesses. Thickness | On/Off Ratio | On/Off Ratio
---|---|---
(nm) | at -0.1 V | at 0.1 V
$10$ | $5.56$ | $3.20$
$20$ | $6.14$ | $7.89$
$30$ | $7.05$ | $14.43$
$40$ | $10.13$ | $17.43$
$50$ | $6.38$ | $17.10$
Table 5: On/Off ratio for MoS2 RRAM at $\pm$0.1 V for various thicknesses. Thickness | On/Off Ratio | On/Off Ratio | Stopping Voltage
---|---|---|---
(nm) | at -0.1 V | at 0.1 V | (V)
$10$ | $12.98$ | $13.47$ | $-0.4$
$20$ | $18.73$ | $32.66$ | $-0.86$
$30$ | $51.39$ | $137.54$ | $-1.2$
$40$ | $111.94$ | $149.16$ | $-1.6$
$50$ | $208.68$ | $253.84$ | $-2.0$

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 (VDD) 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.

##  IV Summary

We built physical models of WS2, MoS2, and h-BN RRAM through KMC simulations
and experimental data to extract the physical parameters of these 2D
materials. Theoretical formulas provided the retention failure times of WS2,
MoS2, and h-BN RRAM as $\rm 1.23\times 10^{4}$, $\rm 3.18\times 10^{4}$, and
$\rm 1.18\times 10^{7}$ s, respectively. Compared with the HfOx RRAM
(benchmark device), the models show that ions easier drift in the out-of-plane
direction due to the molecular arrangement of the 2D material. The 2D RRAM has
a lower threshold voltage, increasing the switching speed. In particular, the
switching speed of MoS2 RRAM is the fastest of the three 2D RRAM devices. To
verify the reliability of the KMC model, we comparatively analyzed the
temperature-dependent experiments and simulations for WS2 RRAMs. The current
characteristics at high bias voltage are more significantly affected by
temperature change. We discuss the physical attributes of RRAM made from 2D
materials obtained via mechanical exfoliation and CVD. The results show that
2D materials grown via CVD have better device characteristics and are more
suitable for making RRAM. Finally, electrical analysis and simulations for
different active layer thicknesses provide the breakdown electric field of WS2
as 155 MV/m. The thickness models simulate the On/Off ratio and switching
voltage of WS2 and MoS2 RRAM at various thicknesses for reference in future 2D
RRAM device designs.

If high-retention RRAM devices are desired, we suggest that the active layer
be a material with high-generation activation energy (i.e., bond-dissociation
energy) to improve retention. Although h-BN has a high retention failure time,
its current difference is too low due to its low electron affinity. When h-BN
is in contact with a metal, the Fermi level will be pinned to the p-type
level. Thus, the h-BN RRAM can only generate deep-level defects to make a low
On/Off ratio, indicating it is not an ideal RRAM material. The RRAM materials
with better retention should be outside the scope of 2D materials due to the
high bandgap, high electron affinity, and high bond-dissociation energy. To
develop 2D RRAM devices, rapid ion drift in 2D materials helps manufacture
RRAM devices with faster switching speeds. This advantage can solve the slow
switching speed for traditional solid-state disks and the volatile memory
property of DRAM. 2D RRAM can make up for the speed gap of the memory
hierarchy.[44] To find faster switching speed RRAMs, 2D RRAM should be
considered with more in-depth experimental and simulation studies on MoS2 in
the future.

## ACKNOWLEDGMENTS

This work was supported by the National Science and Technology Council under
Grant Nos. NSTC 111-2221-E-002-075, 111-2622-8-002-001, 112-2119-M-002-013,
and 111-2218-E-002-025. The authors would like to thanks Tzu-Heng Wang for his
early work on preparing MoS2 and h-BN samples.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

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.

## DATA AVAILABILITY

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## references

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