Material properties and material models used to describe the thermo-mechanical behavior of materials in simulations are presented in the following sections. In our simulations models, the Al metallization as well as intermetallics are taken into account in order to precisely simulate the mechanisms occurring inside the module. The solderable metallization and the metal barrier are not included in simulations as they are too thin (<1 µm) to be modeled with the other layers.
2.3.1 The copper lead frame and clip
Cu is a ductile metal with excellent electrical and thermal conductivity. His mechanical properties are quite good and it has a quite high CTE. The lead frame and clip were modeled with a bilinear kinematic hardening to simulate the elasto-plastic behavior of the Cu.
Table 2.1: Table of thermo-mechanical properties of the Cu
2.3.2 The solder
As solder degradation is an important failure mechanism in power module, it is therefore important to accurately model its thermo-mechanical behavior.
Table 2.2: Table of thermo-mechanical properties of the SAC solder alloy
Solder alloys are mainly subjected to creep deformation. The creep is typically a plasticity which occurs at high temperatures: when the temperature is greater than 0,4 Tm where Tm is the melting point in Kelvin. The creep curve at constant stress level describes this deformation as a time function until the fracture of the material (Figure 2.6). Three regions are to be distinguished: the first stage is the primary creep, where the creep rate is decreasing by increasing creep strain or time. The second stage is the secondary creep or steady state creep as the creep rate decreases to a constant value. Finally, in the tertiary creep, the creep rate increases continuously until the fracture, due to the increasing of cavities and cracks [Dép07]. Most of creep models are only describing the secondary creep stage, only a few of them are also including the primary creep.
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Figure 2.6: Creep curve at a constant stress level
To describe the solder behavior a Garofalo model which defines the secondary creep rate as a function of stress and temperature with a hyperbolic sine was chosen:
Eq. 2.1
Where is the steady-state creep rate, Qc is the activation energy for the creep in J/mol, R the universal gas constant (8,14 J/mol.K), T the temperature in K, σ the stress in MPa, and A and B are constants from experiment.
The SAC solder alloy was characterized by [Schu03], as follow:
Eq. 2.2
Where C1 through C4 are constants defined as follow:
Table 2.3: Table of values for the constants of the Garofalo creep law for the SAC solder alloy
2.3.3 The silicon chip
Si is a brittle material and has therefore a high elasticity modulus and is obviously modeled with a linear elastic model.
Table 2.4: Table of thermo-mechanical properties of the Si
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2.3.4 The aluminum metallization
The chip metallization is made of an AlCu alloys with an Cu weight percentage ranging from 0,1% to 1%. In a first approach, simulations were performed with a material model for Al with a plasticity model independent from temperature. But this model was not precise enough to be able to correctly simulate the plastic deformations of the metallization. So a literature study on material properties of Al alloys was conducted. 28 publications were reviewed [Alprop]. In those publications, the mechanical properties of different Al alloys with an Al weight percentage of about 99% (Al, AlSi, AlSiCu, AlCu) in the form of a thin film deposited on a Si substrate were investigated. The material properties were dependent on the film thickness. A thin film effect induces that thinner layers, are stronger. Our layer is 5 µm thick, but the data collected referred mainly to layers of about 1µm. The yield stress in function of the layer’s thickness was plotted Figure 2.7 for all the literature’s data: yield stress values for an Al film of 1µm thick are widespread, ranging from 20 MPa to 470 MPa. This diversity of results can be explained by the fact that the alloys studied did not always had the same composition, and the methods used to measure stress may also differ from one paper to another. For a layer thicker than 1 µm, the yield stress ranges between 50 MPa to 150 MPa. Stress-strain curves from literature were also analyzed and converted into a bilinear kinematic hardening model. Those models obtained for room temperature were compared with each other
Figure 2.8. Again, one can see that yield stresses range from 20 MPa to 470 MPa but also with different Young’s modulus and tangent modulus. Finally the temperature dependence of yield stress was investigated (Figure 2.9). The dependence was mainly studied for temperature ranging from 20°C to 150°C. Only a few studies looked at the mechanical behavior at negative temperatures. Here, the yield stress ranges from 50 MPa to 230 MPa, but the slopes of the different curves are often comparable.
Figure 2.7: Yield stress versus layer’s thickness at room temperature of literature’s data
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Figure 2.8: Bilinear kinematic models of literature’s data
Figure 2.9: Yield stress versus temperature of literature’s data
Finally, as a too high value of yield stress does not allow plastic deformations to occur, and because the yield stress of an Al layer thicker than 1 µm ranges between 50 MPa to 150 MPa, a yield stress in the vicinity of 100 MPa was chosen for room temperature. This corresponds to the model used by [Kan13]. This model also included the temperature dependence of the yield stress.
Table 2.5: Table of thermo-mechanical properties of the AlCu metallization
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Figure 2.10: Bilinear kinematic hardening temperature dependent model for the AlCu metallization
2.3.5 The intermetallics (IMC)
To determine the material properties of intermetallics (IMC) a literature study was required [Fie91, Fre94, Gro12, Häs06, Jia97, Lee07, Tsa06]. Here the IMC were chosen to correspond to Cu6Sn5. IMC are brittle materials; they have then a linear elastic model.
Table 2.6: Table of thermo-mechanical properties of the intermetallics(IMC)
2.3.6 The molding compound
The commercially available epoxy-based molding compound is a polymer, and has thus a viscoelastic behavior, meaning that the mold exhibits both viscous and elastic characteristics when undergoing deformation. Upon application of a load, the elastic deformation is instantaneous while the viscous part occurs over time. Viscoelastic deformations are then depending on load, time and temperature.
Table 2.7: Table of thermo-mechanical properties of the molding compound
The material is restricted to be ThermoRheologically Simple (TRS), which implies that the material response to a load at a high temperature over a short duration is identical to that at a lower temperature but over a longer duration. Thus, the time-temperature superposition principle can be used to determine the temperature- dependent mechanical properties of mold from known properties at a reference temperature. Indeed, the elastic moduli of typical amorphous polymers increase with loading rate but decrease when the temperature is
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increased. Fortunately, curves of the instantaneous modulus as a function of time do not change shape as temperature is changed but appear only to shift left or right. This implies that a master curve at a given temperature can be used as the reference to predict curves at various temperatures by applying a shift operation [Ima08]. Here, the master curve was implemented through the use of Prony series (
Figure 2.11) and the shift operation was defined with the Williams-Landel-Ferry (WLF) shift function (Figure 2.12). In the Prony series, the shear modulus is defined as follow:
Eq. 2.3
Where G(t) is the shear modulus at time t, is the long term modulus once the material is totally relaxed, and are the relaxation times.
The WLF shift function is an empirical equation:
Eq. 2.4
Where T is the temperature, Tr is a reference temperature chosen to construct the master curve, and C1 and C2
are empirical constants adjusted to fit the values of the superposition parameter aT.
Figure 2.11: Master curve for stress relaxation of the molding compound
Figure 2.12: Shift function of the molding compound
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3 Thermal pre-study
Before to be able to test the B6 Bridge under APC, the specific thermal behavior of the module has to be known and understood. That is why a thermal study was performed and the thermal impedance of the module was characterized. Based on this thermal study, the Design of Experiment for APC tests was defined.