# Electromagnetic Analysis of Vertical Resistive Memory with a Sub-nm Thick Electrode
## 3 Results and Discussion
Conventionally, in a 2D cross-point RRAM structure, the vertical growth refers to the tunneling gap, whereas the horizontal increase denotes the radius of the CF; however, these notions are interchanged in the 3D vertical structure due to the peculiar location of the top (pillar) and bottom (plane) electrodes, as shown in Figure 5. It is worth noting that the unique nature of programming polarity (Figure 1b) and ultra-low switching power (Figure 2b) are highly related to the graphene edge electrode and its interface performing as an  
Figure 4: Programming voltage fluctuation for (**a**) Pt- and (**b**) graphene-based VRRAMs. Inset: _I-V_ characteristic of the 1 st and last sweep for SET and RESET. (**c**) Cumulative probability of the resistance distribution in the VRRAM with a conventional metal electrode programmed at various compliance currents. oxygen reservoir [20]. Contrastingly, in the conventional VRRAM structure, the TiN pillar electrode, particularly the TiO\({}_{\rm x}\)N\({}_{\rm 1-x}\) barrier layer at the TiN/HfO\({}_{\rm 2}\) interface, operates as an oxygen reservoir to collect or discharge oxygen ions consuming considerably large energy [17; 20]. This is because conductive filament formation and rupture are associated with oxygen migration to and from the TiN/HfO\({}_{\rm x}\) interface, respectively, expedited by the electric field and Joule heating. It should be noted that in the graphene-based VRRAM, there is a slight current increase during the RESET operation (Figure 2b), which can be related to its double structure with two potential active electrodes. This means that alternating SET and RESET operations cause a small portion of the oxygen to be stored in the TiN/HfOx interface as residual oxygen ions. Thus, during the RESET operation, these residual oxygen ions migrate towards the opposite direction, causing a slight current increase under the DC voltage sweep condition. However, it is important to note that the activation energy of the oxygen ion diffusion in graphene (\(\approx\)0.7 eV) is considerably lower than in TiN (\(\approx\)2.1 eV) [34; 35]. This implies that for the programming conditions of the graphene-based RRAM, including vastly low current compliance, SET and RESET voltages are not sufficient enough to store a greater portion of the oxygen ions in the TiN electrode, indicating stable operation of the graphene-based memory. In addition, the stochastic behavior of the RRAM devices can be explained by the random location of oxygen vacancies around the CF body [32]. There is an exponential dependence on the tunneling gap of the oxide material and linear dependence on the width of the CF [33]. In 3D vertical memory cell, the horizontal evolution of the conduction path can be modeled as the generation or rupture of the CF. The Stanford RRAM model accurately demonstrates the RRAM resistance switching process based on the tunneling gap distance variation [36; 37; 38; 39]. \[I=I_{0}\times\exp(\frac{-d_{gap}}{d_{0}})\times\sinh(\frac{V}{V_{0}}) \tag{4}\]  
where \(d_{gap}\) is the tunneling gap distance and \(V\) is the applied voltage. Several tunneling mechanisms may occur in the RRAM memory cell, where the majority demonstrates the exponential dependence of the current density on the applied electric field and tunneling distance. This may include Fowler-Nordheim (FN) field emission, direct tunneling, trap-assisted tunneling (TAT), or Poole-Frenkel emission. Figure 5: The illustration of the modeling of conductive filament evolution during SET/RESET programming operations in the VRRAM with (**a**) Pt and (**b**) graphene edge electrodes. The vertical evolution of the conduction path can be described as an increase or decrease in the width of the CF. In its turn, the geometrical shape of the CF can be modeled as a cylinder or truncated cone. As a result due to the dominant Ohmic nature of the conduction path, there is a linear dependence of current density on the diameter of the CF [15; 33].