# Electromagnetic Analysis of Vertical Resistive Memory with a Sub-nm Thick Electrode
## 3 Results and Discussion
Thus, the peak electric field of the graphene-based under 0.2 V applied potential is 169 dBV/m, which is slightly less than the peak strength at the edge of the Pt electrode (180 dBV/m) under the significantly higher potential of 1.4 V. Moreover, for the applied unit voltage difference (1 V; Figure 3e,f), it can be seen clearly that the peak strength of the electric field (179.0 dBV/m; \(8.9\times 10^{8}\) V/m) for a thin graphene electrode increased by approximately 71% when compared to that of the thick Pt electrode (174.3 dBV/m; \(5.2\times 10^{8}\) V/m). A stronger electric field from the thinner and sharper graphene electrode causes the required voltage for the formation of the conductive filament with a firm core to be lowered in the dielectric memory cell. It is also worth noting that the thick Pt electrode has a peak electric field near the top and bottom edge of the electrode. This indicates that the conductive filaments (formed by oxygen vacancies) tend to form at the top or bottom corner of the Pt electrode, and this may lead to variances of switching characteristics since there are two polar fields. Figure 4a,b demonstrates the programming voltage variations for the SET and RESET operations in multiple consecutive DC sweeps for both devices. The obtained data can be useful in injecting the random noise for the single device model. Notably, the programming variation in the graphene-based memory is marginally less (11.2%) than the VRRAM with Pt electrode (12.4%), which can be related to the small switching current and the focused location of the peak electric field and its strength. Figure 3: A comparison of the electric potential for **(a)** Pt and **(b)** graphene electrodes, and the electric field for **(c)** Pt and **(d)** graphene electrodes. Similarly to the experimental characterization settings, 1.4 V of electric potential is applied to the Pt electrode and 0.2 V to the graphene electrode while the TiN electrode is grounded; **(e,f)** The electric field strength comparison under equal potential conditions (1V) for graphene and Pt-based VRRAMs. The TiN electrode is placed at the plane of x = 0 (nm) and plane electrodes are sandwiched between the SiO\({}_{2}\) dielectric layers. The figures **(a)** and **(b)** show the electric potentials with the electric flux lines overlaid. In **(c–f)**, the strength of the electric field is denoted by the colormap in the unit of (dBV/m). The red arrows also represent the electric field distribution including the direction of the field. In addition to the fact that graphene provides VRRAM to operate at considerably smaller voltage/current ranges, there is a favorable tradeoff between the programming noise and memory window. However, this does not apply for conventional metal-based RRAM, since a minor reduction in programming noise will require memory to work at depressed current ranges. This can be achieved by artificially setting a lower current compliance, which, in turn, leads to the undesirable memory window shrink. Since high current compliance will lead to the formation of the conductive filament (CF) with a greater area, more traps can be accommodated indicating the dominance of the random telegraph noise (RTN) [31]. As a result, at the depressed current range, an evident tradeoff between variability and memory window size is expected. Figure 4c demonstrates the cumulative probability of the resistance distribution in the VRRAM with conventional metal plane electrodes programmed at various compliance currents. It is worth noting that both the variability and memory window were reduced for the memory with less compliance current. The development of the conductive filament (CF) can be described by two spatial means, including vertical and horizontal growth induced by the electric field [15; 32; 33].