FEM Analysis of Buckled Dielectric Thin-Film Packaging Based on 3D Direct Numerical Simulation

This paper presents a 3D direct numerical simulation of buckled thin-film packaging based on transferred elastic thin-film wrinkling bonded on a compliant polymer ring. The mode change of the fabricated thin-film cap is found by measuring the thin-film cap shape at different times after Si substrate debonding. The conventional linear and nonlinear buckling simulations are not adequate to understand the behavior of the thin-film buckling mechanism creating such packaging cap mode change. Direct buckling simulation is recently reported as an easy and useful numerical wrinkling simulation method. A novel 3D FEM model of a thin-film package suitable for direct 3D buckling simulation is built to reduce the mode mixture between different buckling modes. Buckling modes of the packaging cap are investigated in terms of elastic moduli of package materials and applied strain due to thermal expansion coefficient difference. Based on the simulation results, it is found that there are two main modes in the fabricated thin-film buckling package determining the shape of the transferred thin-film packaging cover depending on the elasticity ratio between the cap and sealing ring materials. The mode shift from wrinkling cap mode to out-of-plane cap mode due to applied strain along a polymeric sealing ring is found.


Introduction
Microelectromechanical system (MEMS) devices need specific packages to house and protect the packaged device during wafer handling, wafer dicing, or plastic molding [1]. Transfer packaging techniques attract more and more interest because it delivers packaging caps from a carrier wafer to a device wafer. As packaging caps are separately prepared from the front-end process, the transfer packaging technique introduces minimal process compatibility issues compared to conventional packaging technologies [2,3]. Various transfer packaging technologies have been reported based on interface energy modification methods utilizing plasma-deposited fluorocarbon film [4], self-aligned monolayer (SAM) [5,6], solder transfer layer [7], etc. Various packaging cap materials such as metal, polymer, and Si materials can be transferred to the target device wafer by metallic or polymeric bonding techniques and debonding of the temporary carrier wafer. Because there are lots of materials associated with electronic packaging, FEM (Finite Element Method) simulations have been frequently used to find mechanical stress, strain, and reliability parameters of the electronic packages [8][9][10]. For example, FEM simulations of MEMS packaging and BCB (Benzocyclobutene) bonding have been presented with experimental verifications [11,12]. It is known that the die stress developed during the attachment can make the silicon substrate warp, and hence it seems to influence the performance of MEMS devices. Concerning the zero-level packaging, the capping near the device may generate significant stresses depending on the sealing and cap materials. Similarly, the BCB thin-film cap zero-level packaging can also have unwanted effects on the MEMS devices due to stress development resulting in package cap deformation and device chip deformation [13]. The principal source of this stress is the BCB cap's residual stress developed at the bonding step, mainly due to the barrier of the transfer technique is to achieve clean debonding of the carrier substrate after the bonding process. To this end, a thin dielectric film has been transferred by using a SAM coating in order to achieve post-it-like debonding for wafer-level packaging technology.
Thin-film transfer packaging is depicted in Figure 1: (a) Thin elastic film is deposited on a Si substrate with hydrophobic surface modification. The BCB ring has been patterned on a Si substrate as a sealing material, which has only been soft-baked at 90 • C. (b) The two Si substrates have started the bonding at 250 • C with a Karlsuss bonding machine. Temperature and pressure profiles applied for the bonding are found in Ref. [27]. (c) During BCB curing at 250 • C for 1 h, the BCB ring is fully cured and, thus, its volume shrinkage becomes substantial. A mechanical mismatch between the thin direct film and the BCB ring creates a wrinkling pattern on the sealing ring part. (d) After the bonding process is completed, the Si substrate with a thin dielectric film is mechanically debonded. Because there is SAM (Self-Aligned Monolayer) for Si surface modification, the debonding of the Si substrate is quickly accomplished due to the cracking of the interface between the Si substrate and the dielectric film.
14, x FOR PEER REVIEW 3 of 12 and polymer have been transferred by various transfer methods based on hydrophobic coating, solder, and SAM (Self-Aligned Monolayer) coating. The most critical technological barrier of the transfer technique is to achieve clean debonding of the carrier substrate after the bonding process. To this end, a thin dielectric film has been transferred by using a SAM coating in order to achieve post-it-like debonding for wafer-level packaging technology. Thin-film transfer packaging is depicted in Figure 1: (a) Thin elastic film is deposited on a Si substrate with hydrophobic surface modification. The BCB ring has been patterned on a Si substrate as a sealing material, which has only been soft-baked at 90 °C. (b) The two Si substrates have started the bonding at 250 °C with a Karlsuss bonding machine. Temperature and pressure profiles applied for the bonding are found in Ref. [27]. (c) During BCB curing at 250 °C for 1 h, the BCB ring is fully cured and, thus, its volume shrinkage becomes substantial. A mechanical mismatch between the thin direct film and the BCB ring creates a wrinkling pattern on the sealing ring part. (d) After the bonding process is completed, the Si substrate with a thin dielectric film is mechanically debonded. Because there is SAM (Self-Aligned Monolayer) for Si surface modification, the debonding of the Si substrate is quickly accomplished due to the cracking of the interface between the Si substrate and the dielectric film.  Figure 2a shows the transferred buckling thin film on the BCB ring, and the surface wrinkling of the sealing ring was confirmed by the overhang of the thin film in the figure on the left. The thin-film cap shape change was found by measuring the thin-film cap profiles at different time intervals after Si substrate debonding, as shown in Figure 2b. It was also found that wrinkling of the thin dielectric film from the dielectric overhang occurred due to a mechanical mismatch between the dielectric film and the BCB ring. Interestingly, thin-film buckling occurred even if a flat Si substrate had been pressed against the sealing ring. As this wrinkling pattern is the initiating force for the buckling thin-film packaging, it was used for the modeling of the fabricated buckling thin-film package.   Figure 2a shows the transferred buckling thin film on the BCB ring, and the surface wrinkling of the sealing ring was confirmed by the overhang of the thin film in the figure on the left. The thin-film cap shape change was found by measuring the thin-film cap profiles at different time intervals after Si substrate debonding, as shown in Figure 2b. It was also found that wrinkling of the thin dielectric film from the dielectric overhang occurred due to a mechanical mismatch between the dielectric film and the BCB ring. Interestingly, thin-film buckling occurred even if a flat Si substrate had been pressed against the sealing ring. As this wrinkling pattern is the initiating force for the buckling thin-film packaging, it was used for the modeling of the fabricated buckling thin-film package.

Theory of Thin-Film Buckling on Polymer Substrate
The buckled thin-film packaging is driven by wrinkling of the thin dielectric film bonded to the BCB ring, and thus, it was modeled and simulated with the direct 3D FEM method instead of 2D modeling. First, the theoretical background of elastic thin-film wrinkling will be briefly explained in the following.
The force balance approach for surface buckling instability of a thin film on a compliant substrate is briefly introduced here [28]. Considered a semi-infinite substrate under plane strain deformation, the classical equation for bending an elastic film on a compliant elastic substrate is given by Equation (1): where = E/(1 − ν 2 ) is the plane-strain modulus, E is the Young's modulus, v is the Poisson's ratio, I = wh 3 /12 is the moment of inertia (where w is the width of the film and h is its thickness), F is the uniaxially applied force or load, and k is the Winkler's modulus of an elastic half-space (k = / ). The subscripts, f and s, denote the film and substrate, respectively. As the buckling instability of interest here is the first sinusoidal mode, the film deflection can be described by Equation (2).

Theory of Thin-Film Buckling on Polymer Substrate
The buckled thin-film packaging is driven by wrinkling of the thin dielectric film bonded to the BCB ring, and thus, it was modeled and simulated with the direct 3D FEM method instead of 2D modeling. First, the theoretical background of elastic thin-film wrinkling will be briefly explained in the following.
The force balance approach for surface buckling instability of a thin film on a compliant substrate is briefly introduced here [28]. Considered a semi-infinite substrate under plane strain deformation, the classical equation for bending an elastic film on a compliant elastic substrate is given by Equation (1): where E = E/(1 − ν 2 ) is the plane-strain modulus, E is the Young's modulus, v is the Poisson's ratio, I = wh 3 /12 is the moment of inertia (where w is the width of the film and h is its thickness), F is the uniaxially applied force or load, and k is the Winkler's modulus of an elastic half-space (k = E s wπ/λ). The subscripts, f and s, denote the film and substrate, respectively. As the buckling instability of interest here is the first sinusoidal mode, the film deflection can be described by Equation (2).
Substituting Equation (2) into Equation (1) and solving for the applied force in the thin film on compliant substrate gives Equation (3).
The film buckling wavelength can be found by minimizing F with respect to λ (or ∂F/∂λ = 0): It should be noted here that the wavelength is only a function of the thickness of the film and the elastic properties of the film and substrate. Thus, the wrinkling wavelength can be used to determine the material properties of the thin film if the substrate material properties are known.
Under finite polymer substrate thickness in a thin film on an elastic substrate, the buckling wavelength derived in Equation (4) should be modified considering the limited substrate thickness [29,30]. For a buckled bilayer of a thin film shown in Figure 3, the total free energy is given by the sum of the bending energy of the metal layer and the deformation energy of the underlying polymer layer. For a sinusoidal deformation in one direction, the low amplitude (t m k << 1) wave profile w(x) can be written as w(x) = A sin(kx), where tm is the metal thickness, A is the amplitude of the buckling wave, the buckling wavelength λ is 2π/k, and k is the wave number. Referring to the geometry of the thin film on a polymer layer given below, the free energy per unit area to bend a metal thin film having thickness t m , elastic modulus E m , and Poisson's ratio ν m is given by Micromachines 2023, 14, x FOR PEER REVIEW ( ) = sin 2 Substituting Equation (2) into Equation (1) and solving for the applied thin film on compliant substrate gives Equation (3).
The film buckling wavelength can be found by minimizing F with resp / = 0): It should be noted here that the wavelength is only a function of the thick film and the elastic properties of the film and substrate. Thus, the wrinkling w can be used to determine the material properties of the thin film if the substra properties are known.
Under finite polymer substrate thickness in a thin film on an elastic su buckling wavelength derived in Equation (4) should be modified considering substrate thickness [29,30]. For a buckled bilayer of a thin film shown in Figure free energy is given by the sum of the bending energy of the metal layer and mation energy of the underlying polymer layer. For a sinusoidal deformation rection, the low amplitude (tmk << 1) wave profile w(x) can be written as w(x) where tm is the metal thickness, A is the amplitude of the buckling wave, th wavelength λ is 2π/k, and k is the wave number. Referring to the geometry of t on a polymer layer given below, the free energy per unit area to bend a met having thickness tm, elastic modulus Em, and Poisson's ratio νm is given by For the free energy of deformation of an isotropic elastic film, the free ene sion approximated by the sum of the energy in the long wavelength limit (tp that in the short-wavelength limit (tpk >> 1) is where the subscript p indicates the polymer layer. For the small amplitude (A consideration, the external strain U can be approximated by (Ak/2) 2 . Then tot can be written as  For the free energy of deformation of an isotropic elastic film, the free energy expression approximated by the sum of the energy in the long wavelength limit (t p k << 1) and that in the short-wavelength limit (t p k >> 1) is where the subscript p indicates the polymer layer. For the small amplitude (A << λ) under consideration, the external strain U can be approximated by (Ak/2) 2 . Then total energy F t can be written as The intrinsic buckling wavelength can be obtained by minimizing the free energy with respect to the wave number and in a dimensionless form:

Uniaxial Loading of Thin Film/Polymeric Ring Stack
As previously shown in experimental results, the BCB sealing ring had created one-dimensional linear buckling due to its aspect ratio. Therefore, it was first checked with linear buckling modeling and simulation. The model and its boundary conditions of the dielectric film and BCB sealing ring stack are shown in Figure 4. As direct 3D modeling uses imperfections directly defined in the elements of the model, such as the previous work [31], the sealing ring was modeled and simulated to find the effect of imperfections on the wrinkling pattern. The model for the 3D direct simulation was composed of two layers, a thin dielectric film, and a BCB sealing ring, as shown in Figure 4a.

Uniaxial Loading of Thin Film/Polymeric Ring Stack
As previously shown in experimental results, the BCB sealing ring had created dimensional linear buckling due to its aspect ratio. Therefore, it was first checked w linear buckling modeling and simulation. The model and its boundary conditions of dielectric film and BCB sealing ring stack are shown in Figure 4. As direct 3D mode uses imperfections directly defined in the elements of the model, such as the prev work [31], the sealing ring was modeled and simulated to find the effect of imperfect on the wrinkling pattern. The model for the 3D direct simulation was composed of layers, a thin dielectric film, and a BCB sealing ring, as shown in Figure 4a.  The model had a length of Lx = Ly = 125 µm, a thin-film thickness of tf = 0.5 µm, a polymer thickness of tp = 50 um, and the boundary conditions were Ux = 0 at x = 0, Uy = Uz = 0 at z = 0. The imperfection was defined at the center of the elements on top of the BCB sealing ring. Note that the mesh size was kept as a constant equal to λcr/3, where λcr is the critical wavelength when Es = 1GPa. Material properties of the model were Ef = 5 MPa, νf = 0.3, Es = 2 GPa, and νs = 0.4. As expected, the sinusoidal wrinkling pattern occurred by uniaxially applied displacement load, as shown in Figure 4b. It should be noted that the thin substrate effect was included in the comparison. The wavelength of the wrinkling pattern was studied as a function of the elasticity of the polymeric material substrate. The wavelength obtained from the 3D direct simulation had good agreement with the analytical calculation, as shown in Figure 5. For example, the elasticity of the thin film, 1 GPa, created a wavelength of 13 µm at given conditions. Afterward, the imperfection defined in a polymer substrate was essential for the 3D direction modeling and simulation, and thus, the dependence of the number of imperfections on the wrinkling pattern was investigated. As the sealing ring had a rectangular shape, the imperfections were defined along the width at its center line, as shown in Figure 6a. The number of imperfections 1, 3, and 5 were equally spaced imperfections that were manually selected and given with metallic material properties. The wrinkling patterns had sinusoidal waveforms for all the imperfection cases, and the wavelengths of the wrinkling patterns had no dependency on the number of imperfections, as presented in Figure 6b.
Micromachines 2023, 14, x FOR PEER REVIEW and thus, the dependence of the number of imperfections on the wrinkling patt investigated. As the sealing ring had a rectangular shape, the imperfections were along the width at its center line, as shown in Figure 6a. The number of imperfec 3, and 5 were equally spaced imperfections that were manually selected and giv metallic material properties. The wrinkling patterns had sinusoidal waveforms fo imperfection cases, and the wavelengths of the wrinkling patterns had no depend the number of imperfections, as presented in Figure 6b.

Buckled Thin-Film Package Transferred on Polymer Sealing Ring
As explained in the packaging process, the thin film transferred on the polymer sealing ring has shown two different profiles at certain time intervals. Evidently, there are two sources of the forces to determine the buckled thin film: (i) polymer sealing ring shrinkage resulting in wrinkling thin film and (ii) residual stress of the transferred thin film. Hence, it is necessary to study the effect of the two elements on the thin-film profile using FEM modeling. The buckled thin-film package has been modeled and simulated with a quarter model to reduce simulation time. Figure 7 shows the quarter models with the applied boundary conditions as follows. The symmetry planes create boundary conditions of Ux = 0 at x = 0 and Uy = 0 at y = 0, and the bottom surface has been constrained Uz = 0 to avoid rigid body motion. As explained in the previous section, the key parameter for the thin-film bucking is displacement (or strain) load which should be applied in the sealing ring of the package model. Therefore, the model has been slightly modified to avoid complex buckling modes by decoupling the buckling mode at the corner of the sealing ring, as shown in Figure 7b. The dimension and material properties of the thin film and sealing ring for the modeling are summarized in Table 1.

Buckled Thin-Film Package Transferred on Polymer Sealing Ring
As explained in the packaging process, the thin film transferred on the polymer sealing ring has shown two different profiles at certain time intervals. Evidently, there are two sources of the forces to determine the buckled thin film: (i) polymer sealing ring shrinkage resulting in wrinkling thin film and (ii) residual stress of the transferred thin film. Hence, it is necessary to study the effect of the two elements on the thin-film profile using FEM modeling. The buckled thin-film package has been modeled and simulated with a quarter model to reduce simulation time. Figure 7 shows the quarter models with the applied boundary conditions as follows. The symmetry planes create boundary conditions of Ux = 0 at x = 0 and Uy = 0 at y = 0, and the bottom surface has been constrained Uz = 0 to avoid rigid body motion. As explained in the previous section, the key parameter for the thin-film bucking is displacement (or strain) load which should be applied in the sealing ring of the package model. Therefore, the model has been slightly modified to avoid complex buckling modes by decoupling the buckling mode at the corner of the sealing ring, as shown in Figure 7b. The dimension and material properties of the thin film and sealing ring for the modeling are summarized in Table 1.    In a similar way to the previous section, imperfections in the sealing ring have been defined to perform the 3D direct buckling simulation. In this case, all the meshed elements of the top center of the sealing ring have been defined with imperfections, as shown in Figure 8. Given the imperfection, the applied displacements indicated in Figure 7b create the wrinkling of the thin nitride film due to a mechanical mismatch between the sealing ring and the thin film. Such wrinkling pattern on the sealing ring is one of the driving forces of the buckled thin-film packaging. The residual stress of the thin film should be   In a similar way to the previous section, imperfections in the sealing ring have been defined to perform the 3D direct buckling simulation. In this case, all the meshed elements of the top center of the sealing ring have been defined with imperfections, as shown in Figure 8. Given the imperfection, the applied displacements indicated in Figure 7b create the wrinkling of the thin nitride film due to a mechanical mismatch between the sealing ring and the thin film. Such wrinkling pattern on the sealing ring is one of the driving forces of the buckled thin-film packaging. The residual stress of the thin film should be another important factor for the final shape of the buckled thin-film package. The applied displacement was −1 µm in the x and y directions. In a similar way to the previous section, imperfections in the sealing ring have defined to perform the 3D direct buckling simulation. In this case, all the meshed elem of the top center of the sealing ring have been defined with imperfections, as show Figure 8. Given the imperfection, the applied displacements indicated in Figure 7b c the wrinkling of the thin nitride film due to a mechanical mismatch between the se ring and the thin film. Such wrinkling pattern on the sealing ring is one of the dr forces of the buckled thin-film packaging. The residual stress of the thin film shou another important factor for the final shape of the buckled thin-film package. The ap displacement was −1 μm in the x and y directions.

Elasticity Ratio Ef/Es
Buckling patterns of the thin film depending on the ratio of Ef/Es have been investigated by varying the elasticity of the sealing ring. Figure 8 shows the buckled thin film depending on the elasticity of the sealing ring. It varies from 5 MPa to 40 MPa when the Young's modulus of the thin film is 2 GPa. The wrinkling frequency of the thin film is inversely proportional to the Ef/Es ratio, as is the theoretical calculation. The increase in the wrinkling frequency has also been validated with the wrinkling patterns obtained at AA in Figure 8, as shown in Figure 9. This wrinkling pattern has been found just after Si carrier wafer debonding, as previously presented in Figure 2b. The measured wrinkling pattern is quite close to that of the simulation with Es is 5 MPa which is also validated from the cap profile at the center of the thin-film package shown in Figure 10b.

Elasticity Ratio Ef/Es
Buckling patterns of the thin film depending on the ratio of Ef/Es have been investigated by varying the elasticity of the sealing ring. Figure 8 shows the buckled thin film depending on the elasticity of the sealing ring. It varies from 5 MPa to 40 MPa when the Young's modulus of the thin film is 2 GPa. The wrinkling frequency of the thin film is inversely proportional to the Ef/Es ratio, as is the theoretical calculation. The increase in the wrinkling frequency has also been validated with the wrinkling patterns obtained at AA' in Figure 8, as shown in Figure 9. This wrinkling pattern has been found just after Si carrier wafer debonding, as previously presented in Figure 2b. The measured wrinkling pattern is quite close to that of the simulation with Es is 5 MPa which is also validated from the cap profile at the center of the thin-film package shown in Figure 10b.

Applied Strain
The strain developed in the polymeric sealing ring varies as a function of time after the debonding of the Si substrate. The Si substrate for the transfer of the thin film could prevent the ring from shrinking during the bonding process. The polymeric sealing ring should be fully cured during the bonding process. It is known that the volume shrinkage of the BCB polymer sealing ring is substantial after full curing. Therefore, the effect of the applied strain of the polymeric sealing ring on the thin-film buckling modes was studied. For this purpose, the case of Es with 5 MPa in the previous section was chosen because it has a similar cap shape to the fabrication result. Based on the simulation results, it is found that the thin-film packaging cap shape has been changed due to the applied strain, which is caused by the BCB sealing ring shrinkage. Experimental results have good agreement in the packaging cap shape shift from Figure 11b-d. The profiles of the wrinkling packaging cap of the two cases are shown in Figure 12.

Applied Strain
The strain developed in the polymeric sealing ring varies as a function of time after the debonding of the Si substrate. The Si substrate for the transfer of the thin film could prevent the ring from shrinking during the bonding process. The polymeric sealing ring should be fully cured during the bonding process. It is known that the volume shrinkage of the BCB polymer sealing ring is substantial after full curing. Therefore, the effect of the applied strain of the polymeric sealing ring on the thin-film buckling modes was studied. For this purpose, the case of Es with 5 MPa in the previous section was chosen because it has a similar cap shape to the fabrication result. Based on the simulation results, it is found that the thin-film packaging cap shape has been changed due to the applied strain, which is caused by the BCB sealing ring shrinkage. Experimental results have good agreement in the packaging cap shape shift from Figure 11b-d. The profiles of the wrinkling packaging cap of the two cases are shown in Figure 12.

Conclusions and Perspectives
A 3D direct modeling and simulation have been implemented to comprehend the behavior of the Si3N4 buckled thin-film package fabricated with a transfer technique. The mechanism of the buckled thin-film package is the mechanical mismatch between the elastic thin film and the polymeric sealing ring. Such a buckling phenomenon is simulated with commercial FEM software based on eigenvalue buckling and non-linear buckling sequences. The buckling simulation method provided by the commercial FEM software may not be suitable for understanding the behavior of the thin-film cap buckling due to the complex mode mixture among basic buckling modes. In this work, a fabricated thin-film package was modeled and simulated with the existing 3D direct buckling method. To achieve an efficient model, the 3D direct FEM model for the thin-film package was built to reduce such mode mixture of the buckling thin film and, thus, the buckling thin-film cap shapes were successfully extracted depending on mechanical parameters such as mechanical elasticities of the package materials and applied strain. Through the simulation, it was found that the experimental buckling cap shape of the thin-film package was mainly changed due to applied strain caused by the shrinkage of the polymer sealing ring. In conclusion, the direct geometrical imperfection method is an easy and versatile technique to make a buckling simulation of nonconventional mechanical structures, which can also be applied to other devices based on the buckling phenomenon.
Funding: This research received no external funding.

Conflicts of Interest:
The author declares no conflict of interest.