**3. Thermal Simulation and Results Analysis**

### *3.1. Material Characterization and Initial Simulation Results*

A thermal simulation of the test structure was prepared by using MathWorks®Matlab environment and proposed author's approximation scheme for the DPL model. The used computational node includes 4-core, 8-threads Intel® Core™ i7 2.6 GHz (3.6 GHz in Turbo mode) CPU, 32 GB DDR4 memory supported by 265 GB of swap file. In order to obtain simulation results, parameters' values presented in Table 2 were taken into consideration.

The most problematic issue is related to establishing parameters of the air layer, being an ambient of investigated test structure. In particular, the air between two platinum resistors is crucial in presented investigation. Moreover, the contact layer between platinum resistors and the oxide layer also needs special consideration. In order to emphasize the problem, a sample analysis of temperature rises over the time were prepared for different values of the DPL model parameters, as well as for different values of a thermal conductivity for the mentioned part of the air layer.

The first part of the simulation process does not include the investigation demonstrated in Section 2.4. Two sets of reference DPL model parameters were considered during the simulation process:


The results, as demonstrated in Figures 3 and 4, were additionally normalized in order to make their analysis easier. The normalization was prepared in the following way:

$$T\_k^{norm}(t) = \frac{T\_k(t)}{\max\_{t,k} \{ T\_k(t) \}} \qquad \qquad k \in \{ 1, 2, \dots, n\_x \cdot n\_y \}, \ t \in R\_+ \cup \{ 0 \} \tag{29}$$

**Figure 3.** Average temperature rises over the time inside heat source (excluding investigation in Section 2.4).

**Figure 4.** Average temperature rises over the time inside temperature sensor (excluding investigation

in Section 2.4). As it can be seen, the average temperature rise inside the heater is less than 95%

 of the maximal recorded temperature rise, while the temperature rise in the platinum sensor is characterized by a slightly more than 61% of the highest observed temperature rise. Differences in temperature rise values yielded for analyzed sets of DPL model parameters are observed between 1 ps and 1 ns.

Time shifts between observed lines in Figures 3 and 4 were plotted in Figure 5. In order to show differences between results, excluding and including air conductivity investigation presented in Section 2.4, a time shift analysis over the time, demonstrated in Figure 6, was carried out. In this figure, "new kair" means the thermal conductivity of the air layer between platinum resistors calculated based on the analysis shown in Section 2.4. As a comparison, the simulation results obtained for both DPL time lags equal to zero for the air layer are also included.

Taking into consideration the sensitivity of yielded results to even small changes of the air layer material parameters, in the second part of the simulation, values, calculated considering the investigation described in Section 2.4, were employed only.

### *3.2. Final Simulation Results and Comparison to Real Measurements of the Test Structure*

The analysis in the previous subsections shows that there is a need for calculation of the proper material parameters for the air layer, especially between platinum resistors of the test structure. Moreover, results of further research sugges<sup>t</sup> using the fractional order of the space derivative of the temperature rise function for the investigated region, as well as this one between platinum resistors and the silicon dioxide, according to the theory described in Section 2.4. Taking into consideration these facts, the simulation of the temperature distribution in the cross-sectional area of the real test structure was carried out.

**Figure 5.** Time shifts over the time for temperature rises in heat source and temperature sensor (excluding investigation in Section 2.4).

**Figure 6.** Time shifts over the time for temperature rises observed in the heat source and temperature sensor (including analysis in Section 2.4).

It was assumed that differences between mesh nodes are equal to 10 nm in both axes. Furthermore, values of DPL model time-lag parameters were calculated according to the theory presented in Zubert et al. [8]. Thus, for Platinum resistors, heat-flux time lag was set at approximately 550 ps, while the considered value of the temperature time lag was equal to 15.6 ns. In the case of other materials, investigated parameters were equal to 18 and 480 ns, respectively. Simulation results and their comparison to the real measured data (collected and described in [32,33]) are presented in Figures 7 and 8, for the transient and steady state analyses, respectively. Moreover, in Figure 7, simulation results received by using the classical F–K model are also included, for comparison purposes.

**Figure 7.** Comparison of normalized temperature rises in heater and thermometer.

**Figure 8.** Steady-state temperature distribution in cross-sectional area of investigated structure.

In Figure 7, the results for the heated platinum resistor were marked by dashed and dotted lines, while those ones obtained for the temperature sensor were plotted with dashed lines. Moreover, outputs for the heater and thermometer were marked by the red and blue curves, respectively. On the other hand, measurement data (collected and described in Janicki et al. [32,33]) were plotted, using green lines, while results obtained by using the F–K model were marked by the black color.

The maximal temperature rise above the ambient temperature was observed at the top surface of the heat source, i.e., in the first platinum resistor (layer 3). Its value is nearly 60 K. On the other hand, the temperature rise recorded at the surface of the temperature sensor, i.e., the second platinum resistor (layer 4), is about 6 K, which states approximately 10% of the temperature rise observed in the heat source.

Both of the investigated temperature rises coincide almost exactly with measurements of the real structure (green curves in Figure 7). Parameter α*x* allows for changing the result curves' slopes, while DPL model parameters τ*q* and τ*T* cause the curves to shift over the time. The mentioned change is proportional to the parameters' values.

In order to check the quality of the obtained results for used the discretization mesh and mesh nodes distances (10 nm), which were described in Section 2.2, the simulation curves' fitting to the measured one was considered. Each curve was plotted based on 703 points logarithmically distributed over the time. Then, such metrics as coefficient of determination (R2), root mean squared error (RMSE), sum of squared estimate error (SSE), and mean squared error (MSE) were calculated. Moreover, to assess the goodness of recognition as a volatility over the time, a Pearson correlation coefficient (corr) was also considered. Determined metrics values are presented in Table 3.


**Table 3.** Evaluation of goodness of fitting of simulation results to real data.

As it can be seen, both the heater and thermometer curves are characterized by relatively small values of MSE, RMSE, and SSE values. Moreover, the coefficient of determination and correlation coefficient sugges<sup>t</sup> a proper recognition of a shape of the measured curve by a simulated one. Generally, it can be stated that calculated metrics (MSE, RMSE, SSE, coefficient of determination, and correlation coefficient) for the simulation fitting to the real data confirm highly accurate simulation results. This situation clearly shows that the proposed approach based on the theory described in Section 2 allows for the production of outputs reflecting the real thermal phenomena observed at the nanoscale. Moreover, as it was shown in Figure 7, the classical F–K model should not be used in the case of electronic nanosized structures.
