Comparison 2D/3D: validation of the 2D model

In this study, the goal is to precisely characterize the thermo-mechanical response of the stack of different materials that constitute the module. This includes also the study of the Al metallization and the IMCs, which are very thin layers. Conducting such a study on a large 3D model would be very difficult and time consuming. Indeed, modeling very thin layers in a large 3D model means difficulties for the meshing and results in a large and complex model that requires long calculation times for the simulation of several PTC or APC cycles. But when a sensitivity study has to be performed, simulations have to be relatively quick and easily reproducible. And this would not be the case with a large and complex 3D model. Thus a simplified model able to take into account thin layers and providing reliable results has to be developed. This corresponds in our case to a 2D axisymmetrical model of one MOSFET structure. As this model does not corresponds exactly to the real geometry of one MOSFET of the B6 Bridge, a validation was first done by comparing results obtained with the 2D model to the ones obtained for different cross-sections of a 3D model.

5.1.2.1 3D models

A 3D Finite Element (FE) model of the entire B6 Bridge is shown Figure 5.3. Only one MOSFET of the entire module is analyzed in details. A study has been previously done to determine which MOSFET of the module is the most damaged and thus representing the worst case. The study revealed that all MOSFETs have approximately the same behavior, but one had slightly more stress and strain than the others. This MOSFET is the one circled in red on the Figure 5.3 and was the one loaded in the experiments. It mesh was then refined in comparison to the rest of the module. 3 different cross-sections of the 3D model were defined and compared to 2D, thus requiring 3 different 3D models with different meshes. All models had 3D tetrahedral structural solid elements with 10 nodes, called SOLID187 in Ansys. This type of element is well suited to mesh our model which was imported from a CAD program. These elements have 3 degrees of freedom at each node, which are the 3 translations in directions x, y, z, and are also compatible with plasticity, creep and large deflections which are required for our analysis.

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Figure 5.3: View of the mesh for a 3D model of the entire B6 Bridge with the hidden mold

The 3 cross-sections studied were:

- A cross-section along a straight line in the middle of the chip - A cross-section along the chip diagonal

- A cross-section along the top solder diagonal

The mesh of these 3 cross-sections is shown on Figure 5.4 and the number of elements for each corresponding model is detailed [Yua13].

Figure 5.4: Views of the mesh of the 3 different cross-sections of one MOSFET

5.1.2.2 2D model

A 2D FE model has been created for a MOSFET structure with an axisymmetry condition. A MOSFET structure refers here to one chip and its close surrounding, which means its Cu lead frame, its solder layers, its Al metallization, its clip and the surrounding mold. Such a 2D axisymmetric model with chip-midpoint as symmetry axis is possible as the center of the chip is the neutral point (NP) in terms of thermal expansion. In this FE model, a fine mesh is defined using quadratic plane elements with 8 nodes, called PLANE183 in Ansys. These elements have 2 degrees of freedom at each node, which are the 2 translations in directions x and y, and are also compatible with plasticity, creep and large deflections which are required for our analysis.

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The model counts 44 775 elements. The thinnest layers have a minimum of 4 elements in their thicknesses, which means that the smallest elements have a size of about 1 µm. Attention was also paid to refine the mesh in the most critical areas of the module. Both solder layers have fine elements with an aspect ratio of 2,8 and critical areas of the metallization and the IMC layer have elements with an aspect ratio of 6,5, which guarantee reliable results. Some elements have a bigger aspect ratio, up to 35, but these elements are not located in critical areas and thus did neither affect the convergence nor results.

Figure 5.6: Global view of the mesh in the 2D model with zooms at both solders meniscus

5.1.2.3 Comparison 2D/3D

Results obtained with the 2D axisymmetric model are compared to the ones obtained for the 3 different cross- sections of the 3D model (chip middle, chip diagonal and top solder diagonal). Here, in-plane stresses in the chip and creep strains in both solders are analyzed in order to verify that the 2D model represents well the behavior of the module. First we look at the in-plane stress along a path defined in the middle of the chip thickness and running from its center to one extremity of the layer. Thus for the 3D models, 2 paths are defined per cross-section, one for the left side and one for the right side. The in-plane stresses along all paths of 2D and 3D models at -40°C and 150°C are plotted Figure 5.7. One can see that all curves are following the same trend and are reaching very similar stress values for both temperatures. Paths do not have always the same length, as cross-sections are different. Sometimes quite high values are reached at the end of the path, which corresponds to the layer extremity. These high values may be induced by the material interface. One curve named “solder diagonal left” has a slightly different profile: an abrupt decrease in stress appears before

Figure 5.5: Schematic 2D section of the 2D model of one MOSFET structure

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the final stress increase until reaching the extremity of the layer. This comes again from the different geometries of cross-sections. Here the cross-section passes through the top solder meniscus, and ended in the corner of the chip. The decrease occurs in the area where the cross-section cut the top solder meniscus. Except this slightly different curve, one can conclude that the 2D model represents well the in-plane stress state in the chip. The cross-section the best represented by the 2D model is the cross-section through the chip diagonal, as their results are very close.

Figure 5.7: Comparison of in-plane stresses along paths in chip for 2D and 3D models. On the left at -40°C and on the right at 150°C.

Then creep strains of both solder layers are analyzed (Figure 5.8). A path is defined in both solders: for the bottom solder the path is at 1/8 of its thickness beneath the chip, and for the top solder the path is at 1/8 of its thickness above the chip. Both paths are running from the center of the cross-section to one extremity.

Figure 5.8: Creep strain in the bottom and the top solders of the cross-section through the top solder diagonal at -40°C

The creep strain in the bottom solder along paths of the different models is plotted Figure 5.9 at low and high temperatures. Here also paths have different lengths due to the different cross-sections. All curves are evolving in the same domain, but the creep strain of the 2D model has a slightly different trend line. This can be explained by the fact that the bottom solder layer is not modeled the same way in 2D and in 3D. In 3D the bottom solder is modeled as a quadratic element without meniscus, whereas in 2D the meniscus is taken into account. Nevertheless values reached for all paths are similar, and for all of them it is the extremity of the path which suffers the most from creep strain. Here also it seems like the cross-section through the chip diagonal is the cross-section the best represented by the 2D model.

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Figure 5.9: Comparison of creep strain along paths in the bottom solder for 2D and 3D models. On the left at -40°C and on the right at 150°C.

The same curves are plotted for the top solder Figure 5.10. Here more differences are visible between the different models. This comes from the way with which the top solder layer was cut by the cross-section. For the cross-section through the chip diagonal, on one side (left) the cross-section passes through the top solder meniscus, while on the other side (right) it does not. This is creating asymmetrical behavior for the 2 paths of one cross-section. Then the solder meniscus is not always located at the same distance to the center depending on the cross-section, thus peaks of creep strain are occurring at different path lengths on the graphic. Here there are also some differences in trend lines between 2D and 3D as the geometry of the top solder is simplified in 2D. Indeed, in 2D there is no Cu clip with a complex form, it is only a quadratic element, and thus the geometry of the top solder is also simple. It does not follow the form of the Cu clip as in 3D models.

Despite all these differences, the creep strain reached in 2D stays in the same order of magnitude as the values reached with 3D, and the path extremity is the most critical zone for all models.

Figure 5.10: Comparison of creep strain along paths in the top solder for 2D and 3D models. On the left at -40°C and on the right at 150°C.

Finally the accumulated (acc) creep strain for both solders is calculated by averaging the 10% of paths extremities, and is plotted Figure 5.11. Concerning the acc creep strain in the bottom solder results are quite different from one model to another. The percentage of acc creep strain reached after the first temperature change is quite different depending on the model, but at the end, almost all results are converging to a similar amount of acc creep strain: 1,8%. The 2D model reaches also the 1,8% of acc creep strain and have the second highest plateau at constant temperature. Thus the amount of acc creep strain in the bottom solder will not be underestimated by the 2D model. Results obtained for the 2D model are close to the ones from the cross- section through the chip diagonal.

Regarding the acc creep strain in the top solder, results are not as scattered as for the bottom solder. Curves from the different models are parallel to each other, and at the end almost all curves are tending to reach a value around 1% of acc creep strain. The 2D model also reaches around 1% of acc creep strain and has the second highest plateau at constant temperature. Thus, for the top solder also there is no risk that the 2D model underestimates the creep strain. The curve of the 2D model is included between both curves of the cross- section through the chip diagonal.

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Figure 5.11: Comparison of acc creep strain for 2D and 3D models. On the left for the bottom solder and on the right for the top solder.

To sum up this comparative study between 2D and 3D models, it can be stated that the 2D model provides results that correspond to the ones obtained with 3D model. The cross-section through the chip diagonal is the one that is the best represented by the 2D model. Thus, the 2D model can be further used in our study.