# Electromagnetic Analysis of Vertical Resistive Memory with a Sub-nm Thick Electrode
## 3 Results and Discussion
In addition, there is considerable variability in the _I-V_ response of both devices, and it is important to note that such stochasticity in the operation is usually understood as an intrinsic phenomenon of filamentary-type RRAM devices. In general, the temporal and spatial stochasticity in filamentary memory devices, including the studied VRRAMs with graphene and metal plane electrodes, is mostly related to the low probability of reproducing the exact the same number of nanoparticles (oxygen vacancies) and shape of conductive filaments in each consecutive programming cycle and device, respectively [26]. Additionally, the presence of the defects accommodated in the insulator that trap the charging electrons also contributes to the variability of the devices. Figure 2: _I-V_ response of the (**a**) Pt- and (**b**) graphene-based VRRAMs with multiple DC cycles. Blue and gray regions represent SET and RESET operations in each device, respectively. For the electromagnetic analysis, we assume that Pt and graphene electrodes are placed apart from the TiN electrode by a distance \(d\) (Figure 3). The thickness of the Pt electrode (5 nm) is much larger than the graphene thickness (3 A). When a programming voltage is applied on Pt and graphene electrodes while the TiN electrode is grounded, we compute and compare the electric field appearing in the dielectric. It is well known that in a static condition, the electric field \(E\) is expressed as the gradient of the electric potential \(V\):  
\[E=-\
abla V \tag{1}\]  
From [PERSON]'s law, the static electric potential satisfies the Laplacian equation in a source-free region:  
\[\
abla^{2}V=0 \tag{2}\]  
The electric potential is subject to the boundary condition, which enforces that the potential is zero (grounded) for the TiN electrode (x = 0 nm), while the Pt or graphene electrode has the applied voltage according to the programming settings. There are many methods to solve the Laplacian equation with the boundary condition, such as the finite element method [27] or method of moments [28]. Among those, in this work, the variational method [29] is chosen to solve the field distribution due to its simplicity in implementation. According to the variational method, solving the Laplacian equation is equivalent to the problem that minimizes an integral \(U\) defined as  
\[\mathbf{U}=\frac{\varepsilon}{2}\int\left(\
abla V\right)^{2}\mathbf{d}\mathbf{v}=\frac{ \varepsilon}{2}\int\left(\frac{\mathbf{d}V}{\mathbf{d}\mathbf{x}}\right)^{2}+\left(\frac{ \mathbf{d}V}{\mathbf{d}\mathbf{y}}\right)^{2}\mathbf{d}\mathbf{v} \tag{3}\]  
subject to the same boundary condition imposed on the potential, where \(\varepsilon\) is the permittivity of the dielectric memory cell between the electrodes. It consists of high dielectric medium HfO\({}_{2}\) (\(\varepsilon_{\mathrm{r}}\) = 22) in the region of (\(x<d\)) and SiO\({}_{2}\) (\(\varepsilon_{\mathrm{r}}\) = 3.7) in the other region (\(x>d\)) (Figure 3a-f). In other words, the variational method has converted the Laplacian equation into an optimization problem. The 2-dimensional function that minimizes the objective function (3) is the electric potential that satisfies the Laplacian Equation (2). Since the objective function (3) that we aim to minimize has a convex property, it can be solved via convex optimization [30]. Lastly, the electric field is obtained from the negative gradient of the electric potential. Figure 3a-d demonstrates the applied potentials with electric flux overlaid and electric field distributions for Pt and graphene electrodes under programming conditions used during the experimental characterization.