Power semiconductor devices and modules are critical components in various applications, including electric vehicles, renewable energy systems, and industrial automation. These devices often employ thin film technologies to achieve high performance and compact designs [1,2,3]. However, their reliability can be compromised over time due to fatigue phenomena, particularly in environments characterized by high temperatures and cyclic loading. Understanding fatigue mechanisms in thin films within the context of power semiconductor devices is essential for ensuring their long-term stability and efficiency [4,5,6].

In power semiconductor devices and modules, such as insulated gate bipolar transistors (IGBTs) and metal-oxide-semiconductor field-effect transistors (MOSFETs), thermal stress arises from the operation and dissipation of heat during switching events and continuous operation [7,8,9,10,11,12]. The cyclic nature of these temperature fluctuations subjects thin films within the devices to mechanical strain, leading to fatigue-related phenomena [13]. Groove formation, a characteristic manifestation of thermal stress, can occur on the surfaces of critical components such as substrates, interconnects, and passivation layers [14,15,16]. These grooves typically emerge along regions subjected to maximum thermal stress, such as the interfaces between different material layers or near localized heat sources. Over time, the repetitive stress cycles deepen and widen these grooves, potentially compromising the electrical and thermal performance of the semiconductor device. Additionally, the presence of grooves can increase the risk of electrical breakdown and thermal hotspots, further exacerbating reliability issues [17,18].

IGBT power modules are intricately structured with multiple material layers, subject to diverse sources of stress that can induce gradual degradation through intrinsic (chip-related) and extrinsic (package-related) failure mechanisms. Figure 1 provides illustrations of several of these mechanisms. However, the predominant causes of degradation primarily stem from thermomechanical factors, owing to temperature fluctuations experienced during the module's operational lifespan. These fluctuations lead to progressive deterioration between successive layers, particularly pronounced in layers exhibiting significant mismatches in coefficient of thermal expansion (CTE). Additionally, the degradation of metallization and the interface between metallization and bond wires represent prominent contributors to power module failures, further underscoring the multifaceted nature of degradation processes in these modules.

In the context of power semiconductor devices, groove formation due to thermal stress is intricately linked to evaporation and diffusion phenomena. Elevated temperatures within the device can accelerate the evaporation of volatile species from thin film materials or encapsulants, altering their composition and mechanical properties [19,20]. This loss of material may exacerbate stress concentrations and promote the initiation of grooves along vulnerable regions of the device [21,22,23]. Furthermore, diffusion processes play a crucial role in redistributing material within the thin films of power semiconductor devices. Diffusion-induced phenomena, such as interdiffusion at material interfaces or dopant migration within semiconductor layers, can influence the mechanical and electrical properties of the device [25,26]. The interplay between diffusion and thermal stress contributes to the evolution of grooves and the overall degradation of device performance over time [27,28].

Fatigue in thin films within power semiconductor devices and modules poses significant challenges to their reliability and performance under demanding operating conditions [30,31]. The formation of grooves due to thermal stress, coupled with evaporation and diffusion phenomena, represents complex interdependencies that must be addressed to enhance device durability [31,32,33,34,35].

The problem of grain boundary grooving in polycrystalline thin films was studied by many researchers [14,15,16,36,37,38,39,40,41,42,43,44,45,46,47,48,49]. The works of Mullins [14,15,16] were devoted to solve the problem governing the profile of grain boundary grooving. The mathematical formulation of this problem was developed by several scientists by focusing on the evaporation-condensation and surface diffusion by adopting several non-linear methods [37,38,39,40,41,42,43,44,45,46,47,48] due to the non-linear partial differential equation previously formulated by Mullins [14,15,16,50].

By advancing our understanding of these mechanisms and developing robust materials, designs, and manufacturing processes, it is possible to mitigate the effects of fatigue and improve the reliability of power semiconductor devices. Through interdisciplinary research and innovation, we can ensure the continued advancement and widespread adoption of power electronics technology in critical applications.

We proposed in previous works [51,52,53] analytical solutions to the mathematical problem in the case of the evaporation-condensation in polycrystalline thin films by resolving the corresponding second non-linear partial differential equation [51,52] without any approximation, when materials are submitted to thermal and mechanical stress, and fatigue effects. One proved the non-validity of Mullins approximation that neglected the first derivative in the mathematical equation associated to the evaporation case. Whereas, in a recent study, we studied the problem of diffusion in thin polycrystalline films, resolved the fourth-order partial differential equation of diffusion and giving an analytical solution to the equation $\frac{\partial y}{\partial t}+B\frac{{\partial }^{4}y}{\partial x^4}=0$, obtained by admitting the approximation ${y\text{'}}^2\ll 1$.

In this paper, we studied the effect of the simultaneous combination of the evaporation-condensation and diffusion on the grain groove profile by resolving the fourth-order partial differential equation associated to these two combined effects. An analytical solution was proposed and detailed in this present work and compared to the solution obtained separately in the diffusion and evaporation-condensation cases.

The differential equation describing the grain boundary grooving when evaporation and diffusion simultaneously act in polycrystalline thin films, is given by Eq 1 or 2: or where $y^{\text{'}}=\frac{\partial y}{\partial x}$ and $y^{\text{'}\text{'}}=\frac{{\partial }^{2}y}{\partial x^2}$ and the boundary conditions are given by Eq 3:

Where $x$ and $y(x,t)$ represent the coordinates of a point at the surface along the axis normal to the initial flat surface at a time $t$. Whereas, $C$ and $B$ are two constants relative respectively to the evaporation and diffusion phenomena. $C$ and $B$ are given by Eqs 4 and 5: where $\gamma$ is the isotropic surface energy or tension of the metal/vapor interface, $P_0\left(T\right)$the vapor pressure at temperature $T$ in equilibrium with the plane surface of the metal, $\omega$ the atomic volume, $m$ the molecular mass of the metal, k the Boltzmann constant, $D_s$ is the surface diffusivity, and $N_S$ the number of diffusing atoms per unit area.

Equation 2 can be written as follows:

By taking the following variable changes (Eq 7):

The different derivatives of $y\left(x, t\right)$and $u\left(x, t\right)$ are given by Eq 8:

One deduced the following general equation of the grain groove profile:

The above partial differential equation cannot analytically be resolved without approximation in the case of both evaporation and diffusion processes. In the following sections, we resumed the essential results obtained in previous studies [51,52,53].

The evaporation-condensation problem is governed by the following mathematical equation:

Equation 9 can be transformed with the same notations to the following equation: or

To resolve the non-linear differential equation 11, Mullins did the approximation of small slope by supposing $\left|y\text{'}\right|\ll 1$. He wrote:

The integration of Eq 12 gave:

With A a constant of the problem determined by the condition boundary $y\text{'}\left(0, t\right)=m$:

The solution of the differential equation (12) obtained by the approximated Mullins problem is given by Eq 13:

In previous study {51,52], we corrected the solution given by Mullins by considering the general equation 10 without any approximation and obtained the following equation:

And the final solution is given by Eq 15:

Where $\theta$ is the groove angle.

The diffusion case previously studied [53] was relative to the mathematical solution of the formation of grain boundary grooving in polycrystalline thin films by taking the approximation ${y\text{'}}^2\ll 1$. The following fourth differential equation (Eq. 16) was obtained:

by the resolution of the fourth differential equation formulated by Mullins that supposed ${y\text{'}}^2\ll 1$

Satisfying the previous boundary conditions.

The analytical solution of the fourth order differential equation (Eq 16) was given as follows.

With the explicit expressions of the solution:

It was proved that the solution given by Hamieh et al. [53] revealed a damped sinusoidal groove profile in the case of electronic power devices. The expressions of zeros, minima, and maxima of the profile as a function of the order number, as well as detailed information about the groove profile y(x) and its derivatives were given. The comparison of this new solution with Mullins’ results showed an overestimation by Mullins of the geometric characteristics of the groove, exceeding the actual values by more than 2.5 times. Additionally, valuable insights into the diffusion behavior of various metals gained through this study. The new expressions relative to the diffusion were used to study several metals such as Cu, Al, Sr, Li, Cs, Ti, Co, Ga and Tl and giving the geometric parameters such as the depth $h_{Max}$ and the width $w_{Max}$ of the grain groove.

The evaporation/condensation and diffusion phenomena in polycrystalline thin films were studied separately in literature. We did not find a complete and rigorous development of the simultaneous combination of the evaporation/condensation and diffusion. In this section we developed a new mathematical solution of this case by supposing that ${y\text{'}}^2\ll 1$.

In this case, equation 6 can be written as

By using the reduced function $g\left(u\right)$, equation 19 became as:

The ratio $\frac{C}{B}$ is given by:

Where M is the molar mass of the metal and R the perfect gas constant.

Equation 20 can be also written as follows:

By considering the development of $g\left(u\right)$ in whole series as a function $u$, one writes:

The different derivatives of the above whole series are given by Eq. 30:

Equation 28 can be then transformed into the following equation for all values of $u$ by replacing all derivatives by their developments:

And one obtains a recurrent relation between the different terms $a_i$ of the series representing the function $g\left(u\right)$:

The complete determination of coefficients $a_i$ can be obtained by knowing the first four terms of the series $a_0$, $a_1$, $a_2$, and $a_3$. The other terms $a_i, \forall i\ge 4$can be deduced by recurrent relations. The boundary conditions of the grain grooving profile are used to determine the different terms of the series. Two types of boundary conditions can be applied: the first one was proposed by Mullins [14,15] and the second one by Amram et al. [54].

The use of the boundary conditions obtained by Mullins led to the following relations:

Using relation $a_2=\frac{a_0}{8}$ and Eq. 31, one obtained the expression of $a_4$ as a function of the evaporation/diffusion coefficient $a$:

The calculations proved that the odd terms of the series are zero and one writes:

The even terms are given by the following recurrent relation:

Therefore, the analytical solution $g\left(u\right)=\sum _{i=0}^{\infty }{a}_i{u}^i$ was fully determined and obviously convergent towards zero.

By using the boundary conditions proposed by Amram et al. [54], one writes Eqs. 35

This allowed to give the value of $a_4=-\frac{a_0}{96}$ and the recurrent formula giving all values $a_n$:

It was found that the boundary conditions proposed by Mullins [12,13] and Amram et al. [54] gave very large difference between the two solutions. Figure 7 showed that the maximum of the groove with the boundary conditions of Amram et al. was four times smaller than that obtained by using the boundary conditions of Mullins, whereas the grain groove deep was twice deeper than that of Mullins’s conditions. All geometric characteristics of the grain groove profile are affected by the change of the boundary conditions. It seems that the classical Mullins-type groove growing by surface diffusion alone cannot be satisfied by the presence of evaporation/condensation process. The correction made by this present work represents a key microstructural element of thin metal polycrystalline films and in many cases participates to the determination of their physical, mechanical and functional properties

A new analytical solution of the partial differential equation $\frac{\partial y}{\partial t}=C{y}^{\text{'}\text{'}}-B{y}^{\text{'}\text{'}\text{'}\text{'}}$ was proposed in the combined cases of evaporation/condensation and diffusion in polycrystalline thin films. A detailed mathematical method was given. The found solution showed a damped sinusoidal groove profile in the case of electronic power devices, less attenuated than the case of the diffusion alone. The determination of the groove profile $y\left(x, t\right)$ and its derivatives allowed to conclude that the validity of the new mathematical solution can be extended to the domain located out of that relative to the approximation ${y\text{'}}^2\ll 1$. Indeed, the obtained solution is valid even if the contact angle of the groove is greater than 30° for distance x greater than 6 mm. The values of the maxima and minima obtained in the combination cases of evaporation and diffusion were found less than those obtained with diffusion alone proving the capital role of the evaporation/condensation phenomenon. The analytical expressions of the groove profile and its derivatives led to the determination of all geometric characteristics of the grain groove profile and can be therefore used to study the effect of the combined effects of evaporation and diffusion on metals. Another analytical solution based on series development was found. All terms of the series representing the function $g\left(u\right)$ and then $y(x,t)$ were determined by using the Mullins and Amram et al. boundary conditions. The obtained results showed the important role of these conditions on the geometric characteristics of the groove profile.