# Multiscale Modeling for Application-Oriented Optimization of Resistive Random-Access Memory
## Materials and Methods
### Multiscale Modeling of RRAMs
The simulation requires the material parameters, calculated using ab-initio methods, and a definition of the device geometry. The modeling platform considers the 3D potential and electric field and how they are affected by localized trapped charge and power dissipation at defects. The results are the various characteristic curves describing the device electrical response. The charge transport dynamics are strongly coupled to the different phenomena related to atomic species in the oxide (i.e., generation, diffusion, recombination). Therefore, they must be consistently calculated to effectively model the structural material modifications occurring during RRAM operations (i.e., forming, setting, and resetting). The generation of atomic species is described by a thermochemical bond breakage model [67], consisting of compact effective-energy formulas accounting for the microscopic material properties and the two main generation mechanisms: (i) the breakage of atomic bonds, enhanced by the local electric and temperature field profiles and locally favored by the possible presence of precursors [68] and other defects, and (ii) the redox reactions that occur at the interfaces, as well-favored by the local electric and temperature fields. The resulting generation rate is  
\[G(x,y,z)=G_{0,G}\exp\left[-\frac{E_{A,G}\ -\ b\cdot F(x,y,z)}{k_{B}T(x,y,z)} \right], \tag{1}\]  
where \(b=p_{0}\frac{2+k}{3}\) is the bond polarization factor related to the molecular dipole moment, \(p_{0}\) is the bond breakage activation energy, \(k\) is the material dielectric constant, \(G_{0,G}\) is the effective bond vibration frequency, \(E_{A,G}\) is the bond breakage activation energy, \(F(x,y,z)\) is the 3-D electric field, \(k_{B}\) is the Boltzmann's constant, and \(T(x,y,z_{\prime})\) is the 3-D temperature field. The material-related parameters (i.e., the effective bond vibration frequency \(G_{0,G}\), the molecular bond polarizability \(p_{0}\), and the bond breakage activation energy \(E_{A,G}\)) are calculated using ab-initio methods [54; 69]. The transport of atomic species is dominated by a DD mechanism, driven by the electric field and strongly accelerated by the local temperature, described by the equation  
\[G_{D}(x,y,z)=G_{0,D}\exp\left[-\frac{E_{A,D}\ -\gamma\cdot E(x,y,z)}{k_{B}T(x,y,z)} \right], \tag{2}\]  
where \(G_{0,D}\) is the effective bond vibration frequency, \(E_{A,D}\) is the diffusion activation energy, \(\gamma\) is the field acceleration factor, \(F(x,y,z)\) is the 3-D electric field, \(k_{B}\) is the Boltzmann's constant, and \(T(x,y,z_{\prime})\) is the 3-D temperature field. However, both the electric and temperature fields are in turn affected by the presence of atomic species, implying a strong and complex coupling between the transport mechanism and the fields in the device. For correctly modeling the DD transport, the internal device conditions (e.g., current, trapped charge distribution, electric and temperature fields) are updated every time an individual defect is generated, recombined, or moved. The stochastic nature of the mechanisms involved in RRAM is successfully accounted for by using a kinetic Monte Carlo approach, which allows a consideration of phenomena like the intrinsic variability of the forming, set, and reset processes, and the occurrence of Random Telegraph Noise [70; 71; 72; 73; 74], together with their dependence on the material properties and geometry, providing insights for their optimization. The presented modeling platform successfully reproduces the RRAM electrical responses to arbitrary voltage and current inputs, allowing for the extraction of various characteristic curves of the device (I-V, C-V, and G-V). The parameters used in all the simulations are reported in Table 2.