## 4 Discussion  
The simulation results show that the region of the highest temperature (Figure 4), current density (Figure 5) and hydrostatic stress (Figure 6) is common in the structure considered, meaning that high values of these parameters are all present in approximately the same region, leading to the accumulation of the effects. This can also be seen in terms of the distribution of vacancy fluxes shown from Figures 8-11. The regions with the highest values of vacancy are in almost the same location, leading to the total vacancy flux, shown in Figure 12, to be concentrated at the left side of the structure in the curved attachment area. The transported vacancy concentration presented in Figure 7 shows that the accumulation of vacancies is present at the place where the failure of the experimental device occurs, as shown in Figure 3.  
The simulation results of vacancy fluxes in Table 7 summarize that vacancy flux is dominated by electromigration. The second largest contribution to vacancy flux is due to diffusion and then, at a very similar magnitude, the flux is due to the hydrostatic stress gradient. Their influence can be treated as equally important. The flux that has the smallest contribution is caused by the temperature gradient and its influence is approximately four thousand times smaller compared to the flux due to the electric field. Based on these results, real-life device failure is most likely driven by electromigration resulting from high current densities, resulting in the driving force of atomic migration, as the visible material transport in Figure 3 originates at the area proven most affected by the simulated flux migration. Additionally, Figure 11 shows the vacancy flux due to the hydrostatic stress gradient, with the vacancies moving from the positive regions of the hydrostatic stress to the negative regions. This corresponds to the notching effect observed in the heater arm in Figure 3c,d, indicating that vacancies accumulate at the region of negative hydrostatic stress, leading to void formation.  
Besides the importance of the main driving forces in the formula for total vacancy flux (Equation (10)), it can be seen that the magnitude of each term is dependent on diffusivity and vacancy concentration. In terms of those two terms, we can see in the formulas presented in Table 5 that they are dependent on temperature and hydrostatic stress. Vacancy formation is facilitated when hydrostatic stress is tensile compared to when it is compressive, and high temperature is also a favorable factor for vacancy creation as thermal energy is needed to overcome energy gap \(E_{A}\). That means that temperature and hydrostatic stress are crucial parameters related to atomic migration.  
Void growth is the most concerning failure case in the described system as void creation and growth will locally increase the current density, leading to hotspot formation and stress increase, causing the further expansion of the voids, as presented in Figure 3d, and this leads to further hotspot formation that causes the ultimate device failure (Figure 3e). The uneven stress distribution caused by the voids exacerbates mechanical failure, weakening the structural integrity of the heater arm.  
Electromigration is shown to be the leading cause of failure in the tested and simulated tungsten microhotplates. To relate the effect of atomic migration caused by high current density, the equation of MTTF (Equation (1)) can be transformed to include current density \(j\)[20] as follows:  
\[\mathrm{MTTF}=\frac{A_{0}}{j^{2}}\mathrm{exp}\bigg{(}\frac{E_{a}}{kT}\bigg{)} \tag{11}\]  
The MTTF has, in this case, not only exponential dependence on the temperature but also inverse-square dependence on the current density. This shows that the lifetime and reliability of the device will decrease significantly in the presence of high temperature and high current density.