# Multiscale Modeling for Application-Oriented Optimization of Resistive Random-Access Memory
## 3 Results
### Effects of Forming Conditions
This is evidenced in Figure 5, which shows the low state resistance distribution exhibited by TiN/5 nm HfO\({}_{N}\)/TiO\({}_{N}\)/TiN RRAM stacks after the forming process at three different compliance currents (i.e., 1, 5, and 10 \(\upmu\)A), using a ramped voltage and an external temperature of 25 \({}^{\circ}\)C.  
Figure 4: (**a**) Experimental (symbols) and simulated (lines) cumulative distributions of TiN/5 nm HfO\({}_{N}\)/TiO\({}_{N}\)/TiN RRAM stacks’ low state resistances after forming at 25 and 125 \({}^{\circ}\)C. (**b**) Resulting oxygen ions/vacancies distribution in the device after forming at 25 \({}^{\circ}\)C. (**c**) Resulting oxygen ions/vacancies distribution in the device after forming at 125 \({}^{\circ}\)C. The forming process was performed using a ramped voltage and a 1 \(\upmu\)A compliance current. The effects of the compliance current on the defect generation processes and the resulting device resistance magnitude and variability can be effectively investigated using the presented multiscale modeling platform. The simulations of the forming processes, shown in Figure 5, reproduce the trend found in the experiments, highlighting that the lower variability of the low resistance state is in fact related to the higher density of oxygen vacancies generated at higher compliance currents. The platform confirms its effectiveness for investigating the best strategies to control the device-to-device variability for specific applications, as previously mentioned. In filamentary RRAM, the morphology of the CF is also strongly affected by the forming voltage stress mode (i.e., the time-varying profile of the forming voltage), as it determines different distributions of oxygen ions and vacancies in the device volume. Evaluating the effect of a specific voltage stress mode is highly desirable, since it allows the best strategy for a specific application to be identified. However, this task is extremely complex, as it requires the interplay between the (possibly) time-varying forming voltage, the defect generation processes driving the growth of the CF, and the oxygen ions diffusion to be considered. The presented multiscale modeling platform provides a powerful tool for such investigation. Figure 6 shows the simulation of the oxygen ions/vacancies distribution in TiN/5 nm HfO\({}_{\mathrm{x}}\)/TiO\({}_{\mathrm{y}}\)/TiN RRAM stacks after the forming process using three different voltage stress modes (i.e., constant voltage, ramped voltage, and pulsed voltage), a compliance current of 1 \(\upmu\)A, and an external temperature of 25 \({}^{\circ}\)C.  
Noticeably, the oxygen ions distribution is significantly affected by the voltage stress mode. At constant voltage forming, the oxygen ions are mostly accumulated near the top electrode, while ramped  
Figure 5: Experimental (symbols) and simulated (lines) cumulative distributions of TiN/5 nm HfO\({}_{\mathrm{x}}\)/TiO\({}_{\mathrm{y}}\)/TiN RRAM stacks’ low state resistances after forming at 1, 5, and 10 \(\upmu\)A. The forming process was performed using a ramped voltage and 25 \({}^{\circ}\)C external temperature. Figure 6: Simulated distributions of the oxygen ions (blue) and vacancies (red) in TiN/5 nm HfO\({}_{\mathrm{x}}\)/TiO\({}_{\mathrm{y}}\)/TiN RRAM stacks after forming with a (**a**) constant voltage, (**b**) ramped voltage, and (**c**) pulsed voltage. The forming process was performed using a 1 \(\upmu\)A compliance current and 25 \({}^{\circ}\)C external temperature. and pulsed voltage forming results in a more uniform distribution. The pulsed voltage forming, however, leads to a higher radial diffusion of the oxygen ions compared to the constant voltage forming. Comparing the oxygen ions and vacancies distribution shown in Figures 4 and 6, it can be noticed that the distribution corresponding to a low-temperature forming process (Figure 4b) is similar to the one of a ramped voltage forming process (Figure 6b).