Resistance Switching Statistics and Mechanisms of Pt Dispersed Silicon Oxide-Based Memristors

Silicon oxide-based memristors have been extensively studied due to their compatibility with the dominant silicon complementary metal–oxide–semiconductor (CMOS) fabrication technology. However, the variability of resistance switching (RS) parameters is one of the major challenges for commercialization applications. Owing to the filamentary nature of most RS devices, the variability of RS parameters can be reduced by doping in the RS region, where conductive filaments (CFs) can grow along the locations of impurities. In this work, we have successfully obtained RS characteristics in Pt dispersed silicon oxide-based memristors. The RS variabilities and mechanisms have been analyzed by screening the statistical data into different resistance ranges, and the distributions are shown to be compatible with a Weibull distribution. Additionally, a quantum points contact (QPC) model has been validated to account for the conductive mechanism and further sheds light on the evolution of the CFs during RS processes.

In this work, we fabricated Pt dispersed silicon oxide-based memristors and successfully obtained their RS characteristics. In order to investigate the variability of RS parameters, the statistics of RS parameters have been analyzed by screening the statistical data into different resistance ranges in both the Reset and Set processes. Additionally, a quantum point contact model has been validated to account for the conductive mechanism and further shed light on the evolution of the conductive filaments (CFs) during RS processes.

Statistical Distributions
To investigate the variability of RS parameters in both the Set and Reset processes, the statistics of RS parameters versus the initial resistances has been done, and are shown in Figure 2. Figure 2a,b shows the Reset voltage and Reset current ( and ) versus the ON-state resistance ( ), which is calculated at a low voltage (0.1 V). According to the statistics results, we can see that is nearly independent of , whereas is inversely proportional to . This observation is compatible with the thermal-activated dissolution model [38]. In this model, the Reset event happens only when the temperature of the CFs reaches a critical value. Figure 2c,d shows the Set voltage and Set current ( and ) versus the OFF-state resistance ( ), also calculated at 0.1 V. From these two figures, it can be seen that is proportional to , whereas is nearly independent of . Through the statistics of RS parameters, we can know that the variations of and have a strong impact on the uniform distributions of RS parameters. We could improve the performance of memristors by controlling the sizes of the CFs before the Reset and Set transitions. The Current-Voltage (I-V) switching curves and resistance measurements were performed by using an Agilent B1500 semiconductor parameter analyzer. After the electroforming operation, long lasting repetitive cycling experiments were performed using voltage ramp stress for both the Set and Reset processes, and a current compliance limit of 0.5 mA was given in the Set process to avoid the breakdown. The Pt/Pt:SiO x /Ta memristors show a bipolar switching behavior, i.e., Set to the low-resistance state (LRS) under negative voltages and Reset to the high-resistance state (HRS) under positive voltages, as shown in Figure 1a. Figure 1b presents the ON and OFF resistance states of 400 cycles, and the average RS range is approximately from 1 to 10 kΩ.

Statistical Distributions
To investigate the variability of RS parameters in both the Set and Reset processes, the statistics of RS parameters versus the initial resistances has been done, and are shown in Figure 2. Figure 2a,b shows the Reset voltage and Reset current (V RESET and I RESET ) versus the ON-state resistance (R ON ), which is calculated at a low voltage (0.1 V). According to the statistics results, we can see that V RESET is nearly independent of R ON , whereas I RESET is inversely proportional to R ON . This observation is compatible with the thermal-activated dissolution model [38]. In this model, the Reset event happens only when the temperature of the CFs reaches a critical value. Figure 2c,d shows the Set voltage and Set current (V SET and I SET ) versus the OFF-state resistance (R OFF ), also calculated at 0.1 V. From these two figures, it can be seen that V SET is proportional to R OFF , whereas I SET is nearly independent of R OFF . Through the statistics of RS parameters, we can know that the variations of R ON and R OFF have a strong impact on the uniform distributions of RS parameters. We could improve the performance of memristors by controlling the sizes of the CFs before the Reset and Set transitions. Next, the nature of the variation of RS parameters was explored using a data screening method. The cumulative distributions of and in different ON-state resistance ranges are shown in Figure 3a,b, respectively, and the cumulative distributions of and in different OFF-state resistance ranges are shown in Figure 4a,b, respectively. In these four cases, the cumulative distributions are almost straight lines, which are compatible with the Weibull distribution. Therefore, we can use the Weibull distribution function to fit the experimental data of RS parameters in different resistance ranges to obtain the Weibull parameters. The Weibull distribution is defined as: where β is the Weibull slope or shape factor, which represents the statistical dispersion. % is the scale factor parameter, which is the value of F 63%. After fitting of the experimental data by the Weibull distribution, we can obtain the Weibull parameters of and , as shown in Figure  3c,d. The scale factor of ( % ) is independent of , and the scale factor of ( % ) is inversely proportional to , which is consistent with the scatter plots of Figure  2a,b. The Weibull slope of and is nearly independent of the ON-state resistances, which means that there are no microstructure variations of the CFs before the Reset point [38,39]. Similarly, the Weibull parameters of and can be obtained by fitting the experimental data using the Weibull distribution function, as shown in Figure 4c,d, respectively. The scale factor of ( % ) is proportional to , and the scale factor of ( % ) is independent of , which is consistent with the scatter plots of Figure 2c,d. The Weibull slopes of and are nearly independent of the OFF-state resistances, which means that there are no obvious microstructure variations of the CFs before the Set point [40]. Next, the nature of the variation of RS parameters was explored using a data screening method. The cumulative distributions of V RESET and I RESET in different ON-state resistance ranges are shown in Figure 3a,b, respectively, and the cumulative distributions of V SET and I SET in different OFF-state resistance ranges are shown in Figure 4a,b, respectively. In these four cases, the cumulative distributions are almost straight lines, which are compatible with the Weibull distribution. Therefore, we can use the Weibull distribution function to fit the experimental data of RS parameters in different resistance ranges to obtain the Weibull parameters. The Weibull distribution is defined as: where β is the Weibull slope or shape factor, which represents the statistical dispersion. x 63% is the scale factor parameter, which is the value of F ≈ 63%. After fitting of the experimental data by the Weibull distribution, we can obtain the Weibull parameters of V RESET and I RESET , as shown in Figure 3c,d. The scale factor of V RESET (V RESET63% ) is independent of R ON , and the scale factor of I RESET (I RESET63% ) is inversely proportional to R ON , which is consistent with the scatter plots of Figure 2a,b. The Weibull slope of V RESET and I RESET is nearly independent of the ON-state resistances, which means that there are no microstructure variations of the CFs before the Reset point [38,39]. Similarly, the Weibull parameters of V SET and I SET can be obtained by fitting the experimental data using the Weibull distribution function, as shown in Figure 4c,d, respectively. The scale factor of V SET (V SET63% ) is proportional to R OFF , and the scale factor of I SET (I SET63% ) is independent of R OFF , which is consistent with the scatter plots of Figure 2c,d. The Weibull slopes of V SET and I SET are nearly independent of the OFF-state resistances, which means that there are no obvious microstructure variations of the CFs before the Set point [40].
The QPC model is based on the Landauer transmission approach to calculate conduction along narrow microscopic constrictions [57,58]. According to the Landauer's approach, the current flowing through a CF with N paths can be calculated as [62]: where f is the Fermi-Dirac distribution function, E is the energy, T(E) is the transmission probability, β is the averaged asymmetry parameter (with the constraint 0 < β ≤ 1), and V is the applied voltage assumed to drop at the cathode and anode interfaces with a fraction of β and (1 − β), respectively. Assuming an inverted parabolic potential barrier, we can obtain an expression for the tunneling probability [63][64][65], where Φ is the barrier height, α = t B π 2 h −1 √ 2m * /Φ is related to the inverse of the potential barrier curvature, m * is the effective electron mass, and t B is the barrier width at the equilibrium Fermi energy, assumed to be equal to t gap . Inserting the tunneling probability into Equation (3), we can obtain: There are four parameters in Equation (4). In order to simplify the fitting process, here we fixed Φ = 0.5 eV and β = 1 by considering the asymmetry structure of the devices. Then, we extracted the number of CF paths N and the average t gap from the fitting experimental data of 400 cycles by using Equation (4) and the least squares estimation (LSE) method. The I-V fitting results are excellent in both log and linear scales, as shown in Figure 5a

Discussion
According to the screening of the statistical data into different resistance ranges, the distributions of RS parameters were shown to be compatible with a Weibull distribution. After using the Weibull distribution function to fit the experimental data of RS parameters into different resistance ranges, we can obtain that % is independent of and % is inversely proportional to , whereas % is proportional to and % is independent of , which are consistent with the experimental results. Besides, the Weibull slopes of , , , and are nearly independent of the initial resistances, which means that there are no microstructure variations of the CFs before the Reset and Set points. Furthermore, the QPC model has been validated to account for the conductive mechanism and further show the evolution of the CFs during RS processes. From the LRS to HRS, the number of CF paths would decrease, while the barrier gap would increase.
Combining the fitting results of the QPC model with the statistics of RS parameters, we now try to propose the conductive mechanisms of RS processes. During the ON switching, the RS process is mainly driven by an applied electric field, and the CFs are more likely to grow along the locations of Pt nanostructures. Cation migration and metallic CF formation in RS layers can be identified as a candidate RS mechanism due to the abrupt increase of the current in I-V curves (Figure 1a) [66,67]. During the OFF switching, cations are driven out of the CFs and thus introduce a gap between the CFs and the top Pt electrode. Therefore, the number of CF paths would decrease, while the barrier gap would increase from the LRS to the HRS. The Reset event happens only when the temperature of the CFs reaches a critical value, according to the thermal-activated dissolution model. In addition, according to the statistics, we can know that the variations of the RS parameters can be significantly reduced and the performance of memristors could be improved by controlling the sizes of the CFs before the Reset and Set transitions. That is to say, the variability of RS parameters can be reduced by doping in RS regions, where CFs can be induced to grow along the locations of impurities, or by inserting a two-dimensional material with engineered nanopores, which can modify the RS characteristics of memristors.

Discussion
According to the screening of the statistical data into different resistance ranges, the distributions of RS parameters were shown to be compatible with a Weibull distribution. After using the Weibull distribution function to fit the experimental data of RS parameters into different resistance ranges, we can obtain that V RESET63% is independent of R ON and I RESET63% is inversely proportional to R ON , whereas V SET63% is proportional to R OFF and I SET63% is independent of R OFF , which are consistent with the experimental results. Besides, the Weibull slopes of V RESET , I RESET , V SET , and I SET are nearly independent of the initial resistances, which means that there are no microstructure variations of the CFs before the Reset and Set points. Furthermore, the QPC model has been validated to account for the conductive mechanism and further show the evolution of the CFs during RS processes. From the LRS to HRS, the number of CF paths would decrease, while the barrier gap would increase.
Combining the fitting results of the QPC model with the statistics of RS parameters, we now try to propose the conductive mechanisms of RS processes. During the ON switching, the RS process is mainly driven by an applied electric field, and the CFs are more likely to grow along the locations of Pt nanostructures. Cation migration and metallic CF formation in RS layers can be identified as a candidate RS mechanism due to the abrupt increase of the current in I-V curves (Figure 1a) [66,67]. During the OFF switching, cations are driven out of the CFs and thus introduce a gap between the CFs and the top Pt electrode. Therefore, the number of CF paths would decrease, while the barrier gap would increase from the LRS to the HRS. The Reset event happens only when the temperature of the CFs reaches a critical value, according to the thermal-activated dissolution model. In addition, according to the statistics, we can know that the variations of the RS parameters can be significantly reduced and the performance of memristors could be improved by controlling the sizes of the CFs before the Reset and Set transitions. That is to say, the variability of RS parameters can be reduced by doping in RS regions, where CFs can be induced to grow along the locations of impurities, or by inserting a two-dimensional material with engineered nanopores, which can modify the RS characteristics of memristors.