# Electromagnetic Analysis of Vertical Resistive Memory with a Sub-nm Thick Electrode
## 3 Results and Discussion
Figure 2:0  
where \(d_{CF}\) is the diameter and \(\rho\) is the resistivity of the CF, and \(t_{OX}\) is the oxide (memory cell) thickness. Thus, the model shown in (5) depicts RRAM device resistance switching relying on diameter evolution. It should be noted that both the evolution rate of tunneling gap (\(d_{gap}\)) and the diameter (\(d_{CF}\)) of the CF are related to the probability of the oxygen ions to obtain enough energy to exceed the energy barrier (\(E_{a}\)) following the Arrhenius law [40]. Figure 2:1  
Figure 2:2  
Figure 2:3  
where \(a_{0}\) is the ion hopping distance and \(\gamma\) is a local field enhancement factor that regards distinct characteristics for tunneling gap-based bipolar (SET/RESET) switching in oxide material [41; 42]. In contrast, \(\xi\) in (8) is a field dependence factor which accounts for the decrease in the electric field effect after the formation of the CF due to radial diffusion of oxygen ions induced by the local temperature and Coulomb repulsion (attraction) [15; 33; 43]. \(K\), \(v_{0}\), \(\alpha\), \(\gamma_{0}\), \(\beta\) are parameters that have a considerable effect on the \(I\)-\(V\) curve and can be extracted upon experiment results [41]. By applying the time integral, the evolution of both tunneling gap and diameter can be determined in (9) and (10), respectively. Furthermore, the effect of the thermal noise can be evaluated given that the variation is estimated to a tolerable level. Figure 2:4  
Figure 2:5  
For the effective implementation of a compact model for the VRRAM with metal and graphene electrodes, some notions should be considered. In virtue of the dominant switching process based on the increase/decrease in the horizontal length of the CF in the VRRAM with the Pt electrode (Figure 5a), the Verilog-A model can be simplified to focus on the evolution of the tunneling gap. In the graphene-based VRRAM, as there is lack of an interfacial layer (oxygen reservoir) in the TiN/HfO\({}_{x}\) interface and the thickness of the graphene plane electrode is substantially smaller in comparison with the active pillar electrode, for reasons of simplicity, the switching process in the Verilog-A model may prevalently depend on the radial evolution of the CF (Figure 5b). Consequently, the \(I\)-\(V\) characteristics of the VRRAM device model with conventional metal and graphene edge electrodes are as shown in Figure 6a,b. Furthermore, the derived model inherently has a statistical aspect, and during the experimental measurement, it was noted that both device responses have a specific variability common to filamentary-type RRAMs. The temporal variations shown in Figure 4a,b can be further injected into the model as the random noise in the field enhancement and dependence factors leading to the variability of the programming voltages. In addition, the read noise, predominantly caused by the RTN, is extracted from the experimental results and can be modeled as the fluctuation of the conductive filament diameter/tunneling gap and the current read. Figure 6c,d demonstratesthe resistance distribution from the experimental result and the model with injected stochasticity. The simulated models are in good tolerance with experimental data; thus, it is believed that the obtained results will be useful in the estimation of sub-nm graphene material in neuromorphic computing applications.