## Introduction

As an emerging material system, two-dimensional (2D) material has attracted much attention in the last two decades due to its unique electronic, optical, mechanical and thermal properties1,2,3,4. Since the realization of graphene, a large number of materials in this category has been uncovered, including transitional metal dichalcogenides (TMDs), hexagonal boron nitride (h-BN), and graphene-like Xenes (X refers to elements ranging from group III to group VI)5,6,7,8,9,10,11. The 2D atomic sheets are defined as atomically thin, layered solids with intralayer covalent bonding and interlayer van der Waals bonding, representing the thinnest unsupported crystalline that can be realized7,12. Graphene has been utilized in electronic devices mainly as the conducting materials since it is a zero-gap semiconductor, with a remarkable electron mobility in excess of 15,000 cm2 V−1 s−1 at room temperature in clean samples5,13. Molybdenum disulfide (MoS2), a representative TMD material, has a ~1.8 eV direct band gap in monolayer form, thus is suitable to be used in opto-electronic devices or as the channel material in field-effect transistors3,12,14,15. In addition, h-BN, a 2D insulator, has a large band gap of ~5.9 eV, which has been used extensively as a dielectric or substrate material, or to form heterostructures with other 2D materials in flexible devices10,16,17. With the collection of 2D materials rapidly expanded to several hundreds, featuring diverse properties, the semiconductor industry is expected to be boosted by these emerging materials18.

Among various applications, memory devices have been regarded as the technology driver in the recent years19,20. Non-volatile resistive switching (NVRS) devices, also known as memristors, stand out as a promising candidate as the next-generation non-volatile memory21,22,23. The operation of NVRS devices is based on the resistive switching phenomenon of the active layer, typically a bulk metal-oxide film, such as SiO2 or HfO2, in a metal-insulator-metal (MIM) stacking structure24,25,26,27. By applying external electrical bias, the device can be repeatably switched between a high-resistance state and a low-resistance state, which can be subsequently sustained without power supply.

In the last few years, various 2D materials have been reported to exhibit NVRS phenomenon, expanding to a broad collection of the ultrathin layered crystalline films28. As a zero-gap semiconductor, graphene may not be suitable as the resistive switching layer. Graphene oxide, on the other hand, has been reported to show NVRS phenomenon in memory devices29,30,31. MoS2, the most studied semiconducting 2D material, has been found to show memristive effect in various forms, including 1T-phase nanosheets, solution-processed films, heterostructure with metal oxide, double-transferred layers and oxidized multi-layer MoS232,33,34,35,36,37,38,39. Other 2D materials, such as multi-layer h-BN or oxidized black phosphorus, have also been demonstrated to exhibit NVRS characteristics40,41,42.

Resistive switching was not initially thought to be accessible in monolayer atomic sheets due to excessive leakage current43. Recently, researchers have found that grain boundaries in monolayer MoS2 could produce NVRS behavior based on planar structure44,45. However, the vertical MIM devices are more favorable for practical integration with smaller footprint and 3D stacking capability. In 2018, it was firstly reported that monolayer chemical vapor deposition (CVD)-grown MoS2, together with other TMDs (MoSe2, WS2, and WSe2), shows intrinsic NVRS phenomenon in the desired vertical MIM configuration46,47. The demonstration of monolayer h-BN memristors further set the record for the thinnest material with NVRS to ~0.3 nm48. These devices can be collectively labelled as atomristors (memristive effect in atomically thin sheets), featuring low switching voltages, large on/off ratio, fast switching speed, and forming-free characteristics. However, the 2D-based NVRS devices still face the challenges of relatively low yield and endurance. Our previous first-principles simulation study and atomic-scale scanning tunneling microscopy (STM) results indicate that the interaction between sulfur vacancies on MoS2 film and metal ions from electrodes can result in the NVRS phenomenon49,50. In addition, a recent theoretical study proposed that the vacancy defect positions can be modulated by an electric field that can produce NVRS51. Thus, the defect properties in 2D materials are expected to play a significant role in the switching behavior.

Defects in MoS2, such as vacancies, traps, and grain boundaries, can be introduced in material preparation or followed fabrication process52,53. These additional defects lead to mobility degradation compared to the theoretical predictions54. On the other hand, with further understanding of these defects, defect engineering is explored to modulate the properties of 2D materials, which enables some novel applications. For example, defects could act as carrier donors. MoS2 can exhibit both n-type and p-type conduction with intrinsic defects55. While the intrinsic defects typically lead to n-type conduction in MoS2, Nb-doping is found can effectively shift the Fermi level of MoS2 and thus induce p-type doping MoS256. Near-Infrared Opto-electronic application can be achieved in MoS2 with induced sulfur vacancies as well57. Chemically reactive defects can also be induced in MoS2 with defect engineering, which can improve the catalyst efficiency of MoS2 on hydrogen production58. For those reasons, defects engineering on 2D materials has been attracting much attention59.

## Results

### Sample preparation and device fabrication

The 2D NVRS devices are fabricated using continuous CVD-grown monolayer MoS2 film sandwiched between Au top and bottom electrodes (BE) in the vertical MIM structure. We use the crossbar to define the active device area as the overlapped region between two electrodes. Figure 1a, b show the schematic and optical image of the MoS2 NVRS devices, respectively. The active layer thickness is only 0.7 nm for the monolayer MoS2, highlighting the breakthrough of the memristive effect into atomically thin materials. The inert metal, gold, is used as electrode material to eliminate possible effect of metal oxides at the metal-2D material interface46. The fabrication process is illustrated in Fig. 1c. After MoS2 transferring, the film is irradiated by electron beam in a Raith eLINE lithography system with various radiation doses up to 105 μC/cm2. Our previous studies have demonstrated that the resistive switching phenomenon in the fabricated 2D devices is intrinsically originated from the MoS2 monolayer. The potential parasitic effect from the MoS2 outside of the device area can be excluded by the thermal release tape transfer device showing similar NVRS phenomenon in Supplementary Fig. 1. The difference in film properties caused by the electron irradiation process is expected to toggle the switching performance. Note that the TE e-beam lithography process may have certain portion of electrons penetrating through the resist62,63. However, these electrons are applied equally to all the exposed regions and devices. Thus, it can be regarded as an identical offset to all doses and does not influence the trends in the following analysis.

### Characterization of monolayer MoS2 after irradiation

Material characterizations, PL and Raman spectroscopy, have been conducted on the samples to evaluate the monolayer MoS2 after irradiation. Normally the PL and Raman data were measured on three different locations for each radiation dose for confidence and accuracy on the variation of signals per evaluated area. Irradiation doses in two different range of interest are investigated with two monolayer MoS2 samples on different batches. Figure 2a and Supplementary Fig. 2a show the PL of the monolayer MoS2 sample with different radiation doses. The PL peak position for unirradiated MoS2 is located at ~1.85 eV, in good agreement with the reported optical band gap64. Two major trends, showing a red shift of peak position and a decrease of peak intensity as dose increases, are shown in Fig. 2b, c, respectively. The red shift of PL peak position is consistent with previous report on e-beam exposed monolayer MoS2, which can be attributed to the sulfur vacancies generated by the electron irradiation process65. The decrease of intensity is believed to be due to defects generated in the mid-gap that modify the recombination process66. Note that the red shift of peaks cannot be observed in PL quenching associated with oxygen defect sites, indicating negligible oxygen defects introduced during the irradiation process66.

### “Virtual” conductive points

Although the Raman and PL analysis can only indicate the creation of sulfur vacancies, it has been widely reported by other researchers that e-beam irradiation process can produce sulfur vacancies, which has been confirmed by various methods including direct observation using HR-TEM61,69,76. With the material analysis, we have confirmed that increasing irradiation dose would create more sulfur vacancies on the monolayer MoS2 film. Post-irradiation process (TE-related patterning and deposition) may have certain effect on the surface property. However, referring to other reports with similar process77,78,79 and our previous “transferred TE” devices (no TE patterning and deposition involved) that show similar NVRS phenomenon49, the sulfur vacancies likely persist and play an important role in switching behavior in the MIM devices. The illustration of the irradiation effect is shown in Fig. 5. Our material characterizations strongly support that the high-energy electrons can break the bonds between Mo and S atoms in MoS2 and remove sulfur from the lattice. With the subsequent cleaning and the immediate TE deposition after irradiation in the device fabrication process, sulfur vacancies are introduced to the monolayer MoS2 film, which would impact the device performance. Based on our previous studies, conductive points —similar to conductive filaments in metal-oxide devices, can be formed through these induced vacancies and result in the resistive switching phenomenon in 2D-based devices49,50. A key difference is that the conductive point is ‘virtual’, i.e., not a physical connection that bridges the van der Waals gap and directly connects the top electrode to the bottom electrode, unlike conductive filaments which are direct physical connections. Benefiting from the ultrathin layer thickness, the resistance can have a noticeable change with only one side of vacancy filled by metal through the virtual conductive points. The existence of multiple conductive points further amplifies the resistance ratio between HRS and LRS. The comparison between “real” conductive filaments and “virtual” conductive points is presented in the supporting information as Supplementary Fig. 7. The virtual conductive point model is supported by our STM studies showing direct observation of the substitution of gold atoms into the sulfur vacancies, which leads to the HRS to LRS switching, indicating the essential role of sulfur vacancies in NVRS phenomenon49,50. In these studies, a combination of STM and local transport studies is performed on 2D MoS2 film on gold surfaces. The STM images recorded the initial state of the sulfur vacancy before voltage is applied on the tip and the state after RESET (corresponds to the HRS), and the state of the sulfur vacancy is substituted by a gold atom after SET (corresponds to the LRS). These images and the I–V curves between the states provide direct evidence in experiments to prove the virtual conductive point mechanism that the resistive switching phenomenon can be attributed to a single metal atom migration from electrodes to vacancy.

### Representative non-volatile resistive switching behavior

Representative bipolar non-volatile resistive switching I–V curves of the MoS2 devices are shown in Fig. 6. For an unirradiated device as an example (Fig. 6a), it is initially at a high-resistance state (HRS) at pristine state. By applying a positive voltage sweeping, the current suddenly increases at ~1.25 V, indicating a transition from a HRS to a low-resistance state (LRS), which is commonly referred to as a “SET” process. To RESET the device, a negative voltage sweeping is applied in a bipolar operation, where the current decreases at ~−0.48 V, indicating a transition from LRS back to HRS. A compliance current is applied during the SET process to prevent the device from irreversible breakdown due to unconstrained high current, while for RESET process the compliance current is not needed. After SET or RESET, the device can remain at LRS or HRS without external power supply, demonstrating the non-volatility of the switching behavior. Electro-forming process, a prerequisite that commonly exists in bulk oxide-based devices to initialize a soft dielectric breakdown for filament formation, is not required for 2D-based atomristors, which is expected to reduce the power consumption in practical applications25. With MoS2 film irradiated by electron-beam, the devices exhibit similar resistive switching characteristics in the I–V curves. Figure 6b shows a representative switching curve for a MoS2 device under 250 μC/cm2 irradiation dose, with a SET voltage at ~1.32 V and a RESET voltage at ~−0.68 V. More switching curves for devices with irradiation dose ranging from 10 to 10,000 μC/cm2 are included in Supplementary Fig. 8 in supporting information. Higher doses up to 106 μC/cm2 have also been applied on the MoS2 films. However, those devices cannot show resistive switching behavior, possibly due to the excessive amount of induced sulfur vacancies which lead to an electrical short of the device. Current levels at LRS and HRS are commonly used as the key metrics in the study of memristors to obtain the ON/OFF current ratio and estimate the power consumption21,25. However, considering the interest of current density from a wider range of researchers, we use current density (current normalized to the lateral area of the device) in the I–V curves shown in Fig. 6 and Supplementary Fig. 8. The typical ON-state current density is calculated to be ~2.5 × 105 A/cm2 (or 2.5 × 10−3 A/μm2), which is within the reasonable range compared with both other emerging 2D-based devices and traditional bulk oxide-based devices40,80,81. The relatively high current level/current density has been widely recognized and studied for memristors25. For the MoS2-based devices, the high current density can partly be originated from the extremely thin active layer thickness. This can be explained by reviewing the formula of current density: J = σ*E, where J is current density, σ is conductivity, and E is electric field. The electric field E is further determined by voltage and distance between two electrodes (E = V/d). With the same external bias, monolayer MoS2 is only ~0.7 nm in thickness, which results in a larger electric field and thus a higher current density. Another factor that may attribute to the ON-state current is the property of the virtual conductive points, which determines the conductivity σ. According to our previous studies on the temperature-dependent conduction current47, the LRS shows an Ohmic-like metallic conduction behavior, which is consistent with our proposed model that electrons transport through the adsorbed metal atoms at the defect locations. Moreover, comparing to single conductive filament in bulk materials-based devices, the current density is expected to be larger with multiple conductive paths formed in MoS2-based devices. Note that although the current density of memristors can be quite high, it is still far less than that of typical metals under the same conditions. The current density of a typical metal - copper, if placed at active layer in our NVRS device, is calculated to be J = σ*E = σ*(V/d) = 4.26 × 1012 A/cm2, where V is determined by the voltage where highest current typically happens at 0.5 V and the conductivity of copper at 20 °C is 5.96 × 107 S/m82. This is much higher than our calculated ON-state current density for MoS2 devices at ~2.5 × 105 A/cm2. At LRS, the high current flowing through the active layer produces Joule heating that is believed to drive metal atoms/ions out of the vacancies and result in the RESET process. On the other hand, the high current stress may also contribute to cycling failure of the device, which requires future studies to elucidate.

### Statistical analysis on the reliability

Yield is another concern in MoS2-based NVRS devices that limits the potential applications. Figure 7b shows the yield of the fabricated devices with different irradiation doses. For each dose, we have tested ~15 devices to get the statistical results. The yield for unirradiated devices is 47%, which is consistent with our expectation for pristine monolayer MoS2 memristors. With the application of e-beam irradiation, the yield gradually increases to 71% for the devices at 250 μC/cm2 and 500 μC/cm2. The increasing yield can be explained by the increasing amount of sulfur vacancies that promote more conductive points formation in the resistive switching. With increasing S vacancies, there is less possibility that a device has no conductive point formed, this in turn increasing the overall yield. Similar to the case of DC cycles, as the irradiation dose further increases, the yield would decrease to a relatively low level at 63%. This phenomenon can be attributed to the large number of vacancies in the films, which results in a high possibility of cluster formation that makes some of the devices fail. The profile of yield and average DC cycle number under different irradiation doses has been included in the supporting information as Table S1.

### Monte Carlo simulations on electron irradiation effect

Based on these reliability test results, it has been found that the optimum performance can be achieved with a moderate irradiation dose (at ~250 μC/cm2 for monolayer MoS2-based memristors). A weak e-beam irradiation treatment cannot provide sufficient vacancies for a stable switching phenomenon, while excessive irradiation treatment would introduce overwhelming vacancies that lead to a cluster of defects or a high conduction current, which causes an irreversible breakdown. To support our hypothesis, we performed simulation works using Monte Carlo method to illustrate the effect of the induced vacancies on the electrical performance of the devices. Owing to the stochastic nature of resistive switching phenomenon, Monte Carlo method has been used as one of the most suitable tools to mimic the random process for NVRS devices92,93. First, we studied the electron irradiation effect on device yield. We have learned that more sulfur vacancies exist in monolayer MoS2 as radiation dose increases. As a result, the possibility of forming conductive points will be higher, which is beneficial to obtain a higher yield. According to our previous STM measurements on single sulfur vacancy, resistive switching phenomenon can be observed in 5–10% of all tested vacancies. In our simulation, we assume a certain percentage of vacancies can work as a conductive point to produce the resistive switching behavior. As long as the device has at least one vacancy serving as conductive point, it is deemed as a working device. The rate of conduction with the consideration of conductive point formation (Rcp) can be theoretically expressed as (Eq. (1)):

$$R_{cp} = 1 - (1 - P)^n$$
(1)

where P = 0.05 is the possibility of a sulfur vacancy working as a conductive point, and n is the number of vacancies. The simulated Rcp (Supplementary Fig. 9a) is consistent with Eq. (1). It can be seen that Rcp, which can also be referred to as the ratio of the working devices to all devices, increases with higher amount of vacancies. With more switchable vacancy cites, there is less chance for irreversible hard breakdown during the SET process due to lack of available sulfur vacancies. Nevertheless, as vacancy number increases, the possibility of forming clusters (multiple vacancies closely located) also increases, which is another factor that must be taken into consideration. This cluster effect would lead to significant metal diffusion in this region and result in a direct conduction between electrodes. A supporting evidence for this assumption is that most failed devices with high irradiation dose are tested to be short-circuit in experiments, possibly due to the direct conduction through the cluster of vacancies. Monte Carlo simulations were performed to visualize the cluster effect. Figure 8a shows a typical top-view distribution of 15 sulfur vacancies randomly placed in the device area. No obvious cluster can be seen in the image where most vacancies are far from each other. If the number of vacancies is increased to 40, a few cluster regions will be observed as noted in Fig. 8b. A clear trend of generation of vacancy clusters with excessive vacancies is suggested by this qualitative Monte Carlo simulations. Though specific vacancy number does not necessarily correspond to real defect density, qualitative Monte Carlo simulations enable deeper understanding of potential effects of vacancies number on reliability of devices. For simplicity of statistics, the device area is equally divided into a 4 × 4 grid, and the presence of cluster is defined to be the scenario that at least 5 vacancies are located in the same 1 × 1 cell. The rate of cluster (Rcl), as a parameter to characterize failed devices, is included in Supplementary Fig. 9b, showing an increasing possibility of clusters formation in MoS2 as vacancy number increases. The overall yield (Y) can thus be expressed by [Eq. 2]:

$$Y = R_{cp} \cdot (1 - R_{cl}) = (1 - \left( {1 - P} \right)^n) \cdot (1 - R_{cl})$$
(2)

which is shown in Fig. 8c based on the simulation results with the two opposite effects (conductive points and clusters) in the model together. To have a confidence interval of over 95%, 1000 times of iteration were made for each number of vacancies. Compared with the experimental results in Fig. 7b, the simulated yield exhibits similar trend that the yield first increases and then decreases, indicating the conductive points effect dominates at low radiation dose (less vacancies), while the opposite cluster effect dominates at high radiation dose (more vacancies).

## Discussions

Other switching characteristics have been studied through statistical measurements, including DC cycling variability (Supplementary Fig. 10), SET/RESET voltage (Supplementary Fig. 11), ON/OFF resistance (Supplementary Fig. 12), and RESET current (Supplementary Fig. 13) with different irradiation doses. No clear trends were detected as function of irradiation doses, with a total of 67 working devices for 1575 DC measured cycles. This vacancy number independence can be related to the localized nature of virtual conductive-point49,50, which has been observed for conductive filaments94,95,96. However, due to the stochastic nature of the switching phenomenon as discussed above, it is also possible that the difference of these switching parameters can be hidden behind the device-to-device and cycle-to-cycle variation. Currently, the stochastic nature of conductive-point/filament remains challenging to memristor research and commercialization, which requires more future efforts for optimization.

The endurance of 2D MoS2 memristors in this work is typically tested/operated with DC voltage sweep(SET/RESET), which has been extended employed in our previous studies and is relatively well understood46,47,48,49,50. Besides the material/defect engineering, endurance can also be improved with moderate program/erase operation. In general, pulse tests would induce less electric stress compared to DC cycling, and thus can produce higher endurance. In Graphene oxide-based memristor, a higher number of switching cycles (450 cycles) was achieved and reported with pulse bias, than 120 cycles with DC bias97. Endurance can be further improved with optimized pulse test conditions98. More than 106 switching cycles were obtained on HfO2-based RRAM with reported new pulse operation method, while the endurance with the traditional pulse method was less than 103 cycles. Optimizing the pulse cycling test condition for endurance enhancement is out of the scope of this work. However, for practical applications, it is an important matter that will be the focus of future studies.

The monolayer-based memristors, at an early development stage, is not yet competitive compared with the well-studied metal-oxide memristors. However, the 2D monolayers offer extreme vertical scalability (high speed and low power consumption owing to thinner active layer thickness) and better flexibility (potential applications in wearable electronics). In addition, the monolayer MoS2 memristors have already shown comparable or even better switching speed, ON/OFF ratio, switching voltage, and the forming-free characteristics (forming process is mostly required in oxide-based memristors). Taking advantage of the low ON resistance, we have proposed and realized a new application of RF switch based on monolayer memristor, which exhibits the record performance over all other emerging switches99,100. With the defect engineering methodology that has been discussed in this work, and the advances in 2D material growth, mechanism understanding, device structure & process flow optimization, testing technology, etc., it is expected that the vacancy property can be precisely controlled to achieve a better performance for the 2D NVRS devices. Table S2 shows the comparison among representative emerging 2D material-based memristors, where our monolayer-based devices have exhibited superior performance over others.

In summary, we report an effective method by electron irradiation to investigate the defect properties on monolayer MoS2, which considerably improves the reliability of the 2D-based memory devices. Material characterizations, including Raman, PL and XPS measurements, have been performed on the irradiated film and clearly show that sulfur vacancies are introduced through the irradiation process. The statistical electrical studies results suggest a moderate radiation dose is desired to achieve the highest yield and DC cycling endurance. Monte Carlo simulations have been conducted to elucidate the relationship between the induced sulfur vacancies and the enhanced reliability. The irradiation effect on device characteristics can be explained by the virtual conductive-point model that the resistive switching behavior is attributed to the metal ion migration into the sulfur vacancy. Monte Carlo simulations further illustrate the physical process behind the irradiation effect and reveal device failure mechanisms of clustering or excessive Joule heating. The results reported here provide additional insights into defect engineering for improving the performance, yield and reliability of 2D NVRS devices for various applications including flexible non-volatile memory, neuromorphic computing, and RF switches.

## Methods

### Device fabrication

The monolayer CVD-grown MoS2 films on SiO2/Si substrate were purchased from Sixcarbon Technology. All the fabricated devices in this work are in crossbar MIM structure. The BE were patterned by e-beam lithography and deposited by e-beam evaporator (2 nm Cr/80 nm Au) on a 285 nm SiO2/Si substrate. The monolayer MoS2 film was then transferred on the target substrate with BE on it using polydimethylsiloxane (PDMS) stamp transfer method. In this method, the MoS2 film is brought into conformal contact with PMDS stamp. Then, the subtrate-MoS2-PDMS system was soaked into deionized water for about 30 min. Since SiO2 on the substrate is hydrophilic, it is easy for water to diffuse into the interface between MoS2 and SiO2. The MoS2-PDMS film was separated from SiO2/Si substrate and brought into contact with the target substrate. The PDMS stamp was subsequently peeled off, leaving MoS2 on the target substrate. After transferring, TE were patterned and deposited using the same methods as BE. The device area is defined as the overlapped region between TE and BE, which is 2 × 2 μm2 in this work. Electron-beam irradiation was performed using a Raith eLINE electron beam lithography system with Raith NanoSuite software to control the exposing dose and area. The electron beam voltage is 20 kV, and the beam current is ~5 nA with 120 um aperture on different locations.

### Material and electrical characterization

Raman spectroscopy and PL were performed to evaluate the MoS2 film before and after transferring on a Renishaw in-Via system using a 532 nm wavelength source. The material characterizations before transferring are only used to confirm the quality of the purchased 2D films (not shown in this paper). The Raman and PL data presented in this paper (including the unirradiated and irradiated samples) are all collected after transferring. XPS spectra were recorded using a commercial X-ray photoelectron spectrometer (Kratos Axis Ultra DLD, Manchester, U.K.), utilizing a monochromated Al-Ka X-ray source (hv = 1486.5 eV), hybrid optics (employing a magnetic and electrostatic lens simultaneously) and a multi-channel plate coupled to a hemispherical photoelectron kinetic analyzer. The spectrometer was calibrated using the Cu 2p3/2, Ag 3d5/3, and Au 4f7/2 peaks. Samples were stored and transported to the XPS lab in a vial under ambient air. Spectra were collected with two separate pass energies of 20 and 40 eV at 0.1 eV per step, and 4 sweeps. Areas of analysis on the sample were collected using the aperture of a circular area with a 110 um diameter. Since samples were charging during data acquisition, the charge neutralizer was turned on, shifting all peaks to lower binding energies by 2.0–2.5 eV. Consequently, the binding energy of the peaks were calibrated with respect to the adventitious hydrocarbon C 1 s at 284.8 eV. Casa XPS analysis software was used for peak analysis and the stoichiometry of samples was determined using the Kratos’ library for the relative sensitivity factors of the elements of interest. The DC characteristics of the devices were taken on a Cascade probe station with an Agilent 4156 semiconductor parameter analyzer under ambient conditions.

### Monte Carlo simulation

For simulations of yield corresponding with the distribution of vacancies, we used a 2D Monte Carlo model. Vacancies are distributed in a 4*4 grid. In this model, several assumptions are made: (1) The abscissa and ordinate of each vacancy follow uniform distribution. (2) Each vacancy has the same probability to become a virtual conductive point. Here, we use 5% as the rate of being conductive to be consistent with previous STM results. (3) When there are at least 5 vacancies in a 1*1 area within the simulated area, it will be defined as a cluster. In this simulation, the number of vacancies varies from 5 to 40. To have a confidence interval of over 95%, 1000 times of iteration were made for each number of vacancies. The device with at least one conductive point and without cluster formed is defined as a working device. The yield is calculated to be the portion of working devices in all the 1000 simulated devices for each dose.

For Monte Carlo simulations of endurance (cycle number) correlated with conductive points, we assumed that, at each cycle, each conductive point has the same possibility (35%) to be not working in this cycle. In this simulation, the number of conductive points varies from 1 to 8. Similarly, to have a confidence interval of over 95%, 1000 times of iteration were made for each number of conductive points. If in one cycle, the device has no conductive point working, or has at least 5 conductive points working (a high-current state), the device is defined as failed and the cycle number is recorded as the endurance for this device. Otherwise, the device enters next cycle for another evaluation, until the device fails at certain cycle. The overall endurance for each number of conductive points is the average of cycle number for all the 1000 simulated devices.