Técnicas e instrumentos de recolección y análisis de datos

8.1.3.1 Control of convergence of the numerical solution with the Reference Phase

In addition to the procedure of mesh refinement, which is recommended to enhance the convergence of a numerical solution based e. g. on the finite element method (FEM), we have proposed and applied an alternative approach, specific for thermal wave problems, which is based on the known value (Φref = −45° or tanΦref = –1) of the phase shift of the

thermal wave at the top surface (where modulated excitation takes place) of a homogenous sample. We have demonstrated that this value of the phase shift, labelled as reference phase, is a powerful tool to judge the convergence of the numerical solution. The proposed method works according to the principle of mesh refinement but allows to obtain a more accurate solution. Thus, the convergence of the numerical solution can be established when the calculated phase shift of the thermal wave at the top surface of a homogeneous model is equal or very close to the reference phase shift in the considered frequency range. For a model consisting of a superposition of layers with different thermal properties, the poorest thermal properties (all belonging to one of the layers!) are first considered for all layers in order to fulfill the requirement of a homogenous sample. Then, first simulations are performed after the meshing of the model. If the phase of the thermal wave at the top surface (where the modulated thermal load has been applied) of the model reaches the reference phase lag within the entire frequency range, then the meshing is optimal, otherwise another meshing is required until the calculated phase is equal or very close to the reference phase. Once the agreement between the calculated phase and the reference phase is established, the thermal parameters of each participating layer are reintroduced in the program in order to run new simulations at the

8.1 Conclusions

end of which the actual phases and amplitudes of the photothermal signal are recorded and collected for discussion and interpretation.

Of course, the reference phase approach is also valid even if only one modulation frequency is concerned. If the heating modulation frequency is very high, e.g. f = 100 kHz, which means a small value of the thermal diffusion length, the mesh size at small penetration depth has to be very small while it can be considered large at large penetration depths. If the modulation frequency is rather very low, e.g. f = 1.00 Hz, then the mesh size at both the small and large penetration depths have to be considered large. However, the value Φref = -45° for

the reference phase shift is typical for a homogeneous solid regarded as semi-infinite. This is why for a small thickness of the sample (model), e.g. d = 500 µm, the phase shift deviates from Φref = -45° towards Φref = –90° in the limit of low modulation frequencies or at large

penetration depths. The reasons of these deviations have been explained with the help of the thermal thickness, a=d πf /α , which is a measure of the ability of the thermal wave to propagate throughout the entire sample. This way of judging the convergence of the numerical solution has been adopted to perform and control the simulations and the obtained results were in conformity with the expectations.

8.1.3.2 Simulation of thermal and thermo-elastic signals in micro-structured

semiconductor devices

The finite element (FE) simulations performed in this work were aimed at determining the thermal oscillations in micro-structured devices and the thermo-elastic displacements of their surface as consequence of modulated heating. To simplify the calculations, the lateral thermal strains in the sample were assumed so small in comparison with the longitudinal thermal strain (Poisson number ν negligible) that the thermo-elastic expansions were only confined in the vertical direction, along the depth of the modelled structures. By definition, the vertical thermo-elastic expansion is proportional to the integral of the temperature oscillations over the sample depth and the proportionality factor is the coefficient of linear thermal expansion (CLTE), meaning that a thermal analysis is coupled with a structural analysis. In this case, two resolution methods are possible: In the direct method, degrees of freedom (thermal oscillation and thermal expansion) of the coupled field analysis (thermal and structural) are simulated simultaneously. In the indirect method, the results of one analysis are used as entries of the following analysis. This latest method was preferred in this work.

Using the software ANSYS (MULTIPHYSICS/Thermal), versions 5.7 and 6.1, the real and the imaginary components of the modulated temperatures in the models of devices were first simulated for a given modulation frequency as a function of lateral distance from the hot spot. Then the thermo-elastic displacements of the sample surface were deduced by integrating the calculated modulated temperatures along the depth of the sample. During our

8.1 Conclusions

investigations, we found out that ANSYS offers a special command (PCALC), valid in the general post-processor (POST1), which enables such a numerical integration of the nodal temperatures over a defined path, providing that the parameters involved in this integration are clearly identified and defined by the solver, otherwise the obtained results would be hazardous. Going from the consideration that a particular location of the microstructure was subjected to an external excitation source, e.g. ac-electrical heating, modulated laser beam, the main tasks were to detect the location of the hot spot and to assess the thermal expansion of the hot area. Experimental thermal expansions performed on a HEMT-structure were compared with the results of FE simulations, and allowed to find out that the deposition of a gold film on top of the structure to coat the transistor terminals, namely the source, the gate and the drain, was not homogenous and had considerable roughness. We particularly observed that the thermal expansion of the gold film dominates the contribution of the subsurface layers of the microstructure. From this observation, we point out the fact that the thermal expansion of the gold film could be exploited to get access to the temperature of the hot spot but the way is not yet clear and the approach is under study.

Another important point was the investigation of the tip contribution to the thermo- elastic signal. For this, the tip structure was approximated by a truncated cone with a diameter of 40 nm at its smaller end. The following aspects were checked:

(i)−Nature of the contact tip-sample. Here, the tip was first considered to be in mechanical contact with the sample surface. Due to the contact, the thermal flow from sample to tip was optimal but the calculated thermal expansion of the tip was found very small (value δd = 1.3 pm) in comparison with that of the sample surface which was in the order of some Å. (ii) −Impact of the material of the tip on the heat transfer sample-to-tip. In this frame, the Si-tip was successively substituted by a Pt-tip and a W-tip. The material of poorer thermal properties (with respect to silicon), namely Pt (Platinium), contributed to increase relatively the temperature amplitude of the sample surface in the hot spot while the material of better thermal properties, namely W (Tungsten), induced a decrease of that temperature amplitude. Most important is that, with respect to the surface temperature of the Si-tip (δTmax = 2.00 K) at

the contact point, the temperature of the Pt-tip increased (δTmax = 2.47 K) while that of the W-

tip rather decreased (δTmax = 1.69 K). These observations were explained by the fact that

lower thermal effusivities lead to higher surface temperature. In fact, the change in the material of the tip brought little changes in the surface temperature of the sample only in a limited region of the hot spot, thus giving the proof that the cooling effect of the tip on the substrate material is not so severe.

(iii) −Influence of the substrate material in mechanical contact with the tip. For this case, the following observations were made: a)− Using a Si-tip and a GaAs-sample, the maximal amplitude of the thermal wave at the sample surface was given by δTmax = 2.85 K

while a temperature amplitude of δTmax = 2.0 K was recorded at the center of the contact area