Chapter 5
Simulation of Intrinsic Stress Build-Up in Thin Metal Films

Thin films in microelectronics exhibit residual stress which can affect the performance and reliability of devices. In Chapter 4 it was demonstrated how high values of residual stress in thin films can be problematic for the mechanical stability of 3D interconnect structures, such as open TSVs. A high value of residual stress in the W layer of the TSV structure can increase the probability of delamination at the interface between W and SiO₂.

In this chapter, the mechanism behind stress development during film growth is investigated. This study is needed for the understanding of the reduction of film stress which is necessary to increase the mechanical reliability of TSVs. Usually metal films are used in TSVs to electrically connect the integrated circuit in the 3D structure. Typically, the growth process of metal films in TSVs follows the Volmer-Weber mechanism. A FEM-based model describing the stress evolution during the three steps of Volmer-Weber growth was implemented. In general, stress in the film evolves from compressive to tensile and can then return to compressive. The stress generated in the film strongly depends on processing and material properties. Results obtained from the model were compared with different experimental data. The implemented model helps to predict the intrinsic stress in thin films during the growth process and enables an analysis of the influence of fabrication conditions on the thin film stress. The presence of scallops, which appear as a consequence of deep reactive ion etching, results in reduced intrinsic stress. Film growth on a scalloped substrate was simulated. When the scallops were more pronounced, i.e. a larger height/width ratio, the intrinsic stress was further reduced. In addition film growths at different deposition temperatures and different growth rates were simulated.

5.1 Introduction

Thin films are utilized in microelectronics to build functional devices such as interconnects. The properties of a film and its applications are determined by the choice of material and its geometry. The mechanical properties of thin films with a thickness in the range of several atomic layers differ from those of bulk materials. These differences are caused by the diverse microstructures of the film, which depend greatly on the deposition technique. Most films used for microelectronics are polycrystalline materials composed of a large number of small crystallites. In order to understand the microstructure of the film and its mechanical properties, the deposition process must be studied in detail. Significant stress may develop in the film during the deposition process and during the subsequent cooling to room temperature, due to a large mismatch in the thermal expansion coefficients between the material used in the film and the surrounding layers. Stress can lead to reliability issues such as cracking and delamination within a device structure, resulting in poor reliability and ultimately early device failure. Understanding the mechanisms which generate stress in films during the deposition processes is therefore essential in order to decrease the probability of an interconnect failure.

In this chapter the intrinsic stress build-up during the deposition process using simulation methods is analyzed. The deposition process of a given film is modeled by employing well-established theories and physical models which describe film growth [89, 90, 91, 92]. Each material has different properties and consequently a different impact on the growth process of the film. The analysis is focused on W and Cu, which are the materials typically used for TSV interconnects [28, 29] (Section 1.3.2).

Although this study deals with stress build-up during deposition, for TSVs the etching step cannot be avoided. In order to fabricate the TSV structure, etching of the silicon wafer is required. This step can be performed either by ion-enhanced plasma etching or by deep reactive ion etching (DRIE) [93]. Both methods have their own reliability concerns: Problems specific to DRIE are scallop formation along the sidewall, notch formation at the bottom of the TSV, and potential step coverage concerns [94]. Etching using ion-enhanced plasma, such as SF₆/O₂, results in noticeable sidewall tapering, resulting in a significant limitation in the aspect ratio of the etched hole [95].

5.2 Theoretical Background

The mechanical properties of a thin film are highly dependent on the conditions under which the film deposition takes place. Different deposition processes are used during integrated circuit fabrication where metals, dielectrics, and semiconductors are grown. Due to thermodynamical reasons, there are three different modes describing film growth during deposition, namely Volmer-Weber (V-W), Frank-Van der Merwe, and Stranski-Krastanov [92]. Film stress development in polycrystalline metal films deposited onto polycrystalline or amorphous substrates is investigated. The atoms of the polycrystalline metal film are more strongly bound to each other than to the substrate, and this situation is well described by the V-W growth mode [42, 90]. The film growth of these materials is therefore investigated with the basic assumption that the film growth proceeds according to the V-W mode.

The stress build-up during the growth process is also influenced by the layer on top of which the deposition takes place, such as SiO₂ or the barrier layer. The geometry along the sidewalls of TSVs on top of which metal liner is deposited can have scalloped features. These features are a result of the wafer etching process, mainly the deep reactive ion etching (Bosch process [14, 24]). In this study, the effects of the scallops on the stress build-up during film growth are also analyzed.

5.2.1 Volmer-Weber Growth

In the fourth processing step (Section 1.3.1) of TSV fabrication a metal film is deposited. Polycrystalline metal films deposited onto polycrystalline or amorphous substrates are described by the V-W growth mode [42, 90]. During the V-W mode three unique growth steps can be distinguished. Each step affects the film microstructure and consequently the physical properties of the film. The initial stage of film growth is the nucleation of islands onto the substrate. During the deposition process, atoms arrive on the surface and initiate the nucleation step. The minimum nucleus, formed by the atoms on the surface, has a characteristic size associated with a critical initial radius R_i. Beyond this size, the nucleus is stable and begins to grow [90]. The microstructure of a growing film is influenced by the deposition rate v_g, the substrate temperature T_s, and the material properties. Nucleation events continue to occur during deposition as long as there exists a substrate area exposed to the atoms  [42, 90]. The first growth step consists of the nucleation of islands in the film deposition process as shown in Figure 5.1 (a). During the second growth step the islands grow larger in size and start to impinge on each other, forming a continuous polycrystalline film. This step is commonly referred to as the island coalescence process (illustrated in Figure 5.1 (b)) After island coalescence film thickening takes place.

In this third step the formed film continues to thicken and gain volume until the deposition process is interrupted; this step is shown in Figure 5.1 (c)-(d). During the thickening process the structure of the grains and their evolution are strongly influenced by the adatom diffusivity of the material employed in the deposition process. There are two different types of grain-structure evolution. Materials with low adatom mobility (high melting temperature) exhibit a columnar grain structure with increasing grains during thickening referred to as Type 1; this type is depicted in Figure 5.1 (c). In these materials (e.g. Si, W, Fe), the adatom diffusivity is relatively low even at high processing temperature. In materials with high adatom mobility (low melting temperature), the microstructure forms equiaxed grains which continue to evolve during film thickening. This process is typically found in Al, Ag, Au, and Cu, depicted in Figure 5.1 (d), and is referred to as Type 2.

The three V-W growth steps generate stress within the film as shown in Figure 5.2. This stress evolves from compressive to tensile and can then develop further in one of two different ways. It can either become compressive again or remain tensile [96], depending on the adatom mobility of the material used in the deposition process.

5.2.2 DRIE Scallop Formation

An essential step in TSV fabrication is the etching of silicon, previously described as the first processing step (Section 1.3.1). Most often this proceeds using DRIE which is a combination of multiple cycles of the isotropic deposition of a polymer and an ion-enhanced plasma etching of the polymer and underlying silicon wafer.

The polymer protects the sidewalls from the chemical etch, while it is removed from the trench bottom. As the bottom becomes exposed, chemical etching proceeds there. This ensures a vertical etch, but with the presence of a rough, scalloped sidewall. In Figure 5.3 (a) a simple schematic of the DRIE process is depicted, while Figure 5.3 (b) shows the resulting trench with scalloped sidewalls after 8 cycles.

The deposition of the thin polymer layer is usually performed in a C_xF_y gas environment, while the subsequent etching step is performed in an ion-enhanced plasma environment, usually using SF₆ gas [97]. By controlling the individual times for the deposition and etch steps of the cycles, the geometry of the resulting scallops can be manipulated. In this work several scallop structures are examined in order to determine which factors have the greatest influence on the intrinsic stress generation. With these results the process sequence proceeds in a way to minimize the resulting stress in the thin film can be ensured.

5.3 Volmer-Weber Model for Thin Films

During the V-W growth of the film the produced stress plays an important role in the growth process itself. In the following sections the stress-generation mechanisms is described. First, the geometric conditions used for the simulation of the growth mechanism, followed by a theoretical model implemented with the FEM are described.

The model is capable of simulating V-W growth for materials with both high and low adatom mobility. Although the grain size in films is not uniform, in this work layers having only a single grain size which corresponds to the average grain size of the deposited film are considered. This assumption does not correspond to the real condition; however, a negligible influence of the other grain sizes on the final stress generation is expected. The main purpose of this study is to analyze the interaction between single grains.

Further, it is important to note that the crystal orientation, the elastic anisotropy, and the possible plastic relaxation were not taken into account. These material properties can influence the stress generation during film growth; however, the intent of the developed model is to describe qualitative trends of growth in confined geometries and to describe the effects of fabrication conditions on the stress evolution.

The cylindrical geometry of TSVs could impact and influence the magnitude and concentration of the film stress. A cause of the complexity to simulate the 3D TSV geometry only two-dimensional geometries are considered. Further studies are necessary to understand the impact of the cylindrical geometry of TSV.

5.3.1 Geometry

The growth process is analyzed for material islands with a hemicylindrical shape [90]. The material island is represented as a half-cylindrical cut along the z-axis, which lies on a substrate (Figure 5.4). The length l of the cylinder with a circular-cap cross section is assumed to be infinite. This assumption is implemented by using the plane strain condition (i.e. strains in z-direction are zero) in two-dimensional simulations. A thin film as a periodic array of islands is modeled, and therefore the symmetry boundary condition is imposed on outward normals of the geometry. A non-slip interface condition between the islands and the substrate is employed. A schematic representation of the islands is shown in Figure 5.4. FEM model to simulate V-W film growth is implemented in COMSOL Multiphysics [62] and described in the sections which follow. A stationary parametric sweep is used to simulate the growth steps. With the radius parameter R (Figure 5.5) the island geometry is modified as it grows. At each value of the parameter R a new geometry is created and the stress generated due to island growth is computed. The stress calculated for every value of R is used as the initial stress condition (in terms of the stress tensor component σ_xx and σ_yy) for the subsequent simulation step. It is assumed that all materials behave elastically and that all islands are of the same size. Furthermore, an initial island with a critical radius (corresponding to a small R_i value) already nucleated on the substrate is considered. For every simulation step, R is assumed to increase linearly in time t according to

(5.1)

where R_i is the initial radius of the island and h_f is the film thickness. In this way R progressively grows as a function of v_g.

The average grain size of a film depends on the deposition parameters. In the implemented model, r is the radius of the island at the coalescence point which corresponds to the average grain size of the film.

The comparison with experimental data is carried out by computing the average stress in the film ⟨σ_xx⟩ using a FEM simulation, followed by the evaluation of the stress-thickness product ⟨σ_xx⟩×h_f [90]. The stress-thickness product ⟨σ_xx⟩×h_f is often called the “force/width” and is frequently used in measurements of stresses through thin films in order to highlight that the stress can have a through-the-thickness variation which cannot be determined solely by post-deposition curvature measurement [98].

5.3.2 First Step: Compressive Stress

In the first step of V-W growth (cf. Figure 5.1 (a)) a compressive stress develops in the islands before the film becomes continuous. The stress subsequently emerging is mainly due to the action of the surface stress [89, 90]. In the solid nucleus the surface atoms have an equilibrium interatomic spacing which differs from those of the interior (bulk) atoms, due to the different bondings between surface and interior atoms. This difference in equilibrium spacing results in the development of surface stress f (cf. Figure 5.5). f is associated with the reversible work per unit area required to elastically strain a solid surface. Because of the surface stress f a Laplace pressure ΔP is generated inside the island. During the deposition process, the island increases in volume dV and some amount of work ΔPdV is produced. The work available for a volume increase, due to the pressure, must balance the work required for an increase of the surface, due to the surface stress f. For positive surface stress f a compressive stress due to Laplace pressure is present and measurable during the first step of the deposition process. As for a free surface, surface stress can also be associated with the solid-solid interface between island and substrate. The interface stress necessary to stretch an interface by elastically deforming the island and substrate is denoted by g (Figure 5.5) [99]. In this analysis the effect of the stress on the island-substrate interface [89, 90] are neglected.

There is a second mechanism concurrently generating a compressive stress. This stress is based on the fact that the island becomes strongly attached (non-slip) to the substrate during the first growth step. The radius size at which the island is frozen-in at the substrate is denoted as R_fr which is a function of the strength of the island-substrate bonding. If the radius of the island is less than R_fr, this mechanism does not affect the internal stress, since the island is unconstrained  [89, 90]. As the volume of the island’s nucleus grows further, the internal elastic stress tends to relax due to the Laplace pressure. Since the island is now rigidly bonded to the substrate, the difference between the lattice parameter, corresponding to R_fr, and the lattice parameter which it would have due to the Laplace pressure tends to generate a compressive stress in the islands. Considering both effects and the geometry included in the model, the compressive stress is given by

(5.2)

This equation is used in the model to describe the initial compressive stress observed at an early stage of film growth prior to island coalescence.

5.3.3 Second Step: Island Coalescence Process

The tensile stress evolution during film growth is connected to island impingement as depicted in Figure 5.1 (b). The reason for this process is that the small gaps between adjacent grains are closed by forming grain boundaries with height h_gb (Figure 5.5). The energy released through the reduction of the surface area of the islands is converted into elastic deformation energy of the participating grains. In recent years, several approaches for the computation of the tensile stress generated by island coalescence have been formulated [90, 100, 101]. In the chosen model’s geometry, the approach described by Seel [90] is considered, where the generated average tensile stress in the continuous film is given by

(5.3)

where E is the Young’s modulus, ν is the Poisson’s ratio, γ_s is the surface energy, γ_gb is the grain boundary energy, and r is the radius of the island R at the coalescence point. In the derivation of (5.3) the islands are considered to have a circular shaped cross section [90]. In this study (5.3) is used to describe the tensile stress generated during the island coalescence process. The equation can be applied to predict the average film stress by using parameters γ_s and γ_gb available in the literature [102].

5.3.4 Third Step: Thickening

In this section the third growth step for low and high adatom mobility materials is described. Usually, materials with low adatom mobility exhibit a large tensile stress which remains constant throughout the thickening process [89, 92]. For these materials, the third step under the assumption that the stress generated during the second step remains constant throughout is considered.

For high adatom mobility materials, on the other hand, compressive stress can re-emerge. Different theories have been developed to describe compressive stress generation during the third step of V-W growth. Compressive stress development can be due to the movement of atoms toward the edges of the film [103], excess adatom concentration on the surface [104], capillarity effects [105], or movement of atoms into the grain boundaries [91]. The compressive stress generation by the diffusion of atoms into the grain boundaries during film growth is considered. This kinetic model for the evolution of residual stress was validated with measurements in several studies [1, 2, 91]. The theory assumes that the film during its deposition is composed of multiple thin layers. The film is thereby represented by parallel atomic layers in the grains with a grain boundary between them. During the deposition process, the grain boundary height h_gb changes as the atoms fill the edge in between the grains. For this process islands are considered with circular arcs between the grain boundaries with the same center for all thicknesses as shown in Figure 5.6. h_gb,i denotes the grain boundary height, where the ith layer is grown. The atoms can move into the grain boundary, changing the grain boundary height from h_gb,i to h_gb,i+1 [91]. Film stress therefore changes as a function of the grain boundary velocity v_gb. The value of v_gb is not constant, as it does not only depend on the constant growth velocity v_g, but also on the geometrical transformation of the islands. Figure 5.7 shows that a higher v_g results in higher grain boundary velocities and consequently changes the magnitude of the generated compressive stress due to the shorter time required for the atoms to move into the grain boundary.

The atoms can move into the grain boundary, because the film surface is not in equilibrium during the deposition process, which leads to an atomic flux. The chemical potentials on the surface and in the grain boundaries are different from their equilibrium [2, 91].

The increase in the chemical potential on the surface due to non-equilibrium conditions is defined as δμ_s. The non-equilibrium condition is caused by the flux of impinging atoms, leading to a supersaturation of adatoms or to a change in surface morphology. The non-equilibrium value of the chemical potential in the grain boundary is a cause of the stress in the layer. It is defined as -σ_iΩ where σ_i is the stress in the ith layer and Ω is the atomic volume. The chemical potential is negative, because the chemical potential is lower than the equilibrium value in the presence of tensile stress. The stress in the layer, and therefore the chemical potential in the grain boundary, changes with time due to the diffusion of atoms into the grain boundary. The difference in the chemical potential between the surface μ_s and the grain boundary μ_gb is given by

(5.4)

The chemical potential depends on growth conditions and on the stress in the film. The difference in the chemical potential Δμ is the driving force behind the necessary movement of atoms N_gb from the surface to the grain boundary. When there is no growth, the chemical potential on the surface and at the grain boundary are identical, therefore no atom moves into the grain boundary. The rate of insertion of atoms into the grain boundary is defined as [1, 2, 91]

(5.5)

where C_s is the fractional coverage, D is the effective diffusivity, a is the height of an atomic step (Ω^1∕3), k is the Boltzmann constant, and T is the deposition temperature. The atomic step is the amount of space necessary to close two adjacent layers, forming a bond and an additional grain boundary. The atoms can only migrate after a grain boundary is formed, after which a tensile stress, generated by the island coalescence process, is already present. The insertion of atoms into the grain boundary decreases the tensile stress in the film. Thus, the generated stress σ_i at the ith layer can be described as [1, 2]

(5.6)

where σ_T is the tensile stress generated due to island coalescence, ν is the Poisson ratio, and L is the grain size (equal to 2 ⋅r (cf. Figure 5.5)). The second term in the right-hand-side of (5.6) represents the compressive stress generated by the movement of atoms into the grain boundary.

The stress evolution for the ith layer is obtained by differentiating (5.6) with respect to time. Using (5.5), the rate of stress change is given by

(5.7)

By solving (5.7), Chason et al. [1, 2] evaluated the stress in the ith layer to

(5.8)

where Δt_i is the time period in which the atoms can move into the grain boundary. The other terms are defined as

σ c = - ,	(5.9a)
= .	(5.9b)

The time increment Δt_i is defined [1, 2] by

(5.10)

When the layer forms (t = t_i), only the term σ_T , which is due to coalescence, is relevant.

The velocity v_gb has an important influence on compressive stress release. At the beginning, v_gb is high and the atoms have a short time in which to migrate into the grain boundary. The stress in the layer therefore becomes more tensile. When the thickness increases, v_gb decreases and converges to the average deposition rate; consequently the tensile stress in the film decreases.

To include (5.8) in the model, a reformulation is performed. The velocity v_gb can be obtained from the geometry; therefore, the stress for discrete layers can be replaced by a continuum description. By using (5.1), the term h_gb,i can be written as:

(5.11)

The grain boundary velocity is given by

(5.12)

In [1, 2] the stress-thickness product of the film was calculated by summing the stress contributions from each layer and multiplying by the layer thickness

(5.13)

In this model the average stress is calculated as a function of film thickness h_f with (5.8) and (5.12) integrating over the interval [h_gb,1,h_gb,i]

(5.14)

where

(5.15)

The parameter β has the dimension of a velocity, and its value influences the exponential term in (5.14). The average stress generated in the film depends on the number of mobile atoms adjusted by the parameters β and v_g. The value σ_T can be calculated theoretically using the stress generated at the coalescence point [90] or by fitting to experimental data [1, 2].

Equation (5.14) is implemented in the model and enables to simulate growth processes for different boundary conditions. At every film thickness the integral (5.14) is calculated and its value is used as the initial stress in the film for the next simulation step. Using experimental data, the model can be calibrated and evaluated for different boundary and geometric conditions.

5.4 Results

W and Cu are materials frequently used for TSV interconnects [28, 29]. W is a material with a low adatom mobility, and it exhibits a high value of intrinsic tensile stress. Cu is a material with high adatom mobility and can exhibits compressive or tensile stress depending on the film thickness and the conditions of deposition. For open TSVs, W [29] or Cu [28] are usually used for metalization.

The development of compressive or tensile stress in the thin film in an open TSV can induce delamination or cracking (Chapter 4). Usually, tensile stress can generate surface cracking, channeling, spalling, and debonding of the substrate. If the stress in the film is compressive, the dominant failure modes are buckling-driven interface or edge delamination [45]. In the following sections an investigation of the factors which influence the evolution of stress in the film in order to ascertain how to reduce the probability of mechanical failure is studied. The capabilities of the model are demonstrated for W and Cu films.

As described in Section 5.2.1 during the first step and the second step of the V-W growth, materials with low and high adatom mobility exhibit the same physical behavior. Therefore, it is important to clarify that in this work the results regarding the first and second step are valid for both types of materials. A complete analysis of the three steps of the V-W for low adatom mobility is reported. In contrast only the third step of the V-W for high adatom mobility is presented, since the first two steps are identical to the low adatom mobility. During these two steps proper material parameters have to be used in (5.2) and (5.3) [89, 90].

5.4.1 Low Adatom Mobility Analysis

5.4.1.1 Sample Description

In the following sections the mechanical effects of the V-W growth on a full plate sample and at the sidewall area of an open TSV structure are examined.

Residual stress measurements of W layers, either deposited as blanket films or grown along the sidewall of TSVs etched in silicon, were investigated in [86]. In the full plate sample described in [86], a specific stack was deposited on the silicon substrate. The stack was composed of SiO₂ (500 nm), Ti (40 nm), TiN (55 nm) and W (200 nm). The layer sequence of Ti/TiN/W was repeated twice. In the W layer on the top of the sample structure a grain size of 300 nm (2⋅r in Figure 5.5) was measured. Since the V-W growth impacts mainly only the first layers, the implement model was simplified by considering one Ti/TiN/W stack.

For the full plate sample the residual stress measured [86] at room temperature was 1.6 GPa. In the TSV sidewall it was four times smaller 0.4 GPa. This large difference could be caused by delamination, difference in the residual stress, non equi-biaxial state of stress or by the presence of rippled sidewalls in the TSV [86]. The measured residual stress is basically thermal stress generated from the different thermal expansion coefficients between film and substrate (cf. Section 2.3.2). The V-W growth occurs during the deposition process where the film stress is different than the value measured at room temperature. Therefore, a FEM elastic thermo-mechanical simulation was performed to obtain the value of intrinsic stress during the deposition process. By considering a deposition process at 400 ^∘C, the obtained film stress was 0.64 GPa. In the following simulations the calibration of the full plate sample with a grain size of 300 nm was performed by using an intrinsic film stress of 0.64 GPa.

5.4.1.2 Island Nucleation and Expansion

During the first step of the V-W growth (5.2) was applied to simulate the growth of W islands already nucleated on a Ti/TiN substrate.

The stress σ_xx distribution due to the Laplace pressure during the first step of the V-W growth is shown in Figure 5.8. A low tensile stress acts on the surface of the islands and a compressive stress at the center of the islands.

The islands exhibit a compressive average stress ⟨σ_xx⟩ (negative value) which is reported as a function of the film thickness in Figure 5.9 for different grain sizes. The magnitude of the generated stress is influenced by the grain sizes of the islands. Thin films having big grain size produce less compressive stress compared to small grain size. Because R_fr=10 nm was set the compressive stress starts from the thickness of 10 nm (cf. Figure 5.9).

The generated stress during this first step of the V-W growth can influence the substrate (Figure 5.8). If the substrate is thin and the generated stress is high, the probability of crack or delamination at the interface increases (Chapter 4).

R_fr is a function of the strength of the island-substrate bonding and in this configuration it was assumed to be in a range between 1 nm and 10 nm [90]. In Figure 5.10 results for three different R_fr values and an island grain size of 300 nm are shown. The compressive stress decreases with increasing R_fr, and the smallest R_fr produces a compressive stress around 300 MPa. Stress in this range can influence the mechanical stability of the system and in particular in the thin layers.

The magnitude of the generated compressive stress is determined by the surface stress f, the frozen-in radius R_fr, and the island radius R. Due to the simulation conditions, islands having a large grain size do not impinge rapidly. At the initial simulation step, the distance between islands corresponds at the simulated grain size (2 ⋅r in Figure 5.5). When a large grain size is considered, the initial distance between islands is wide. Therefore, islands of a depositing film whose expected grain size is large do not impinge rapidly. The substrate is therefore exposed to compressive stress for an extended period of time. If R_fr is small, the strength of the island-substrate bonding is strong and compressive stress can reach a high value, critical to the mechanical stability of the system. In this study the model was applied for a low adatom mobility material. It should be clarified that the model can also be adapted to suit high adatom mobility materials by adjusting the parameters f and R_fr in (5.2).

5.4.1.3 Island Coalescence and Grain Formation

During the coalescence process the film stress is gradually changing from compressive to tensile. (5.3) was used at the first contact point between islands where the tensile stress is generated due to the impinge of them. During the coalescence process grain boundaries are formed due to the reduction of the surface area of the islands.

The simulation was calibrated by using the intrinsic stress value of 0.64 GPa (cf. Section 5.4.1.1). The term (2γ_s-γ_gb) in (5.3) was varied until the desired average ⟨σ_xx⟩ at the film thickness of 200 nm was obtained. The values used are shown in Table 5.1.

The film stress ⟨σ_xx⟩ as a function of the film thickness is shown in Figure 5.11. The simulations were carried out until a thickness of 500 nm was reached. Initially, the film stress is compressive when coalescence starts the compressive stress decreases until it changes its sign and becomes tensile. Subsequently, the film stress increases until it saturates and reaches a constant value. The end of the coalescence process corresponds to the first peak value of the tensile stress. In Figure 5.11 the influence of the grain size on the film stress is reported. Small grain size produces high stress in the film, on the other hand big grain size produces a small film stress [90] as described in Section 5.3.3.

The stress-thickness ⟨σ_xx⟩× h_f is mainly tensile and increases nearly linearly with film thickness. The normalized stress-thickness for different grain size is displayed in Figure 5.12. The constant slopes for thicknesses above 50 nm (grain size=50 nm), 100 nm (grain size=100 nm), and 150 nm (grain size=150 nm) indicate the independence of the film stress on the film thickness.

5.4.1.4 Effects of the Island Shapes

The island coalescence process assuming three different nuclei shapes was investigated. In this analysis, stress forms in the film due only to the island coalescence process.

The three samples studied are shown in Figure 5.13. Sample 2 shows an island with a circular shape, whereas Sample 1 and Sample 3 depict islands of elliptical shape. The island height b is a function of the grain radius (cf. Figure 5.5). For all three samples (5.3) was applied at the first contact point between the islands. With the exception of the grain height, all parameters were kept constant for the simulations of all three samples.

The difference in island geometries leads to a different stress evolution during V-W growth. Figure 5.13 shows the stress components σ_xx generated, when coalescence takes place. Due to the different height to width ratio, coalescence is initiated at different film thicknesses. The stress increase associated with film coalescence therefore starts at different times for Sample 1, 2, and 3 (Figure 5.14).

When the film converges to a homogeneous height, the surface irregularity reduces, and the islands’ initial shapes have no influence on the intrinsic stress in the film. This phenomenon can be observed in Figure 5.14, where the normalized average stress ⟨σ_xx⟩ as a function of film thickness is shown to be constant after film coalescence and equal for all three island shapes.

The maximum stress develops in the film, when the surface roughness is low, specifically when the grain boundary height is equal to the film thickness of a homogeneous film. During the film growth after island collision, the grain boundary height changes as the film thickness increases (h_gb in Figure 5.6). At the point of coalescence between islands the grain boundary height is small, but as the film becomes thicker the grain boundary height grows reaching approximately the film thickness. Tensile stress develops due to a reduction in the surface areas of the islands [90, 100, 101] and the maximum tensile stress is generated when the grain boundary height is equal to the film thickness. This process is consistent with the results reported in [106], which suggests that tensile stress due to coalescence evolves during grain boundary formation and reaches its maximum value, when the film develops into a homogeneous film. From this analysis, it is possible to see how the microstructure influences ⟨σ_xx⟩ at different film thicknesses. In particular, the grains’ size influences stress as well as the thickness of the film and the islands’ shapes.

5.4.1.5 Influence of Scallops

For low adatom mobility materials the stress generated at the end of the coalescence process remains constant during the third step of V-W growth. The model was verified for W V-W growth on a full plate sample and at the sidewall of an open TSV. The V-W growth simulation of W on a full plate sample was calibrated using corresponding experimental results Section 5.4.1.1. Subsequently, the calibrated model was used to simulate the V-W growth of W at the sidewall of an open TSV.

The third simulation step is dominated by the film-thickening. The deposited metal (W) is a low-mobility material, therefore the generated intrinsic stress remains tensile during the growth [89]. In Figure 5.11 the results of the V-W growth on a full plate sample are shown. The mechanical stress in the film is constant and tensile after the end of the coalescence process. As the grain size of the film decreases the intrinsic film stress increases.

The formation of surface roughness along the TSV sidewall, called scallops, due to the etch process is characteristic of open TSVs. This particular structure modifies the distribution of stress in the W layer. The CVD growth process was reproduced on a scalloped surface. As reported in [86] the film stress measured at room temperature along the sidewall of an open TSV was four times smaller than the stress in a full plate sample. To understand the possible effects of the TSV geometry a more detailed investigation on the influence of a scalloped substrate on the intrinsic film stress was carried out.

In Figure 5.15 the 1 μm/0.1 μm structure is depicted, where w_s and h_s indicate the width and the height of the scallops, respectively. The W layer is grown on top of a 40 nm TiN layer which sits on top of a SiO₂ substrate; this structure corresponds to the sidewall of an open TSV before W deposition [29]. During V-W growth the generated stress also influences the substrate where the thin film is grown. In order to investigate how the mechanical stress affects the substrate, a compressive residual stress of 1GPa was set in the TiN layer [67]. Usually the SiO₂ substrate contains inconsequential amounts of residual stress [67] and it was therefore considered stress free in this study. In all simulations the same material properties were used and the grain size of the W islands 2 ⋅ r was set to 200 nm.

The W layer exhibits a low adatom mobility; therefore (5.3) was used to describe the film growth. The effects of the scalloped structure on the normalized average stress were investigated (cf. Figure 5.15). This particular structure can produce a high concentration of stress during film thickening, which leads to mechanical failure. In Figure 5.15 the normalized stress distributions at film thicknesses of ~0 nm, 200 nm, and 400  nm are displayed. The highest tensile stress is observed at the intersection points of the grains. In turn, the highest compressive stress is observed at the intersection points of the scallops. These locations are the critical areas in which a crack in the film or delamination can occur.

The compressive stress in the TiN layer is influenced by the film growth. Initially high values of compressive stress are concentrated only at the intersection points of the scallops. However, when the film is formed, the compressive stress in the TiN layer spreads toward the center of the scallops.

5.4.1.6 Effects of the Scallop Geometry (Width & Height)

By simulating different scallop sizes important information about the stress generation during film growth on a confined geometry was obtained. The etch process can be designed in order to produce scallops with different geometries. Scallops with a width/height of 0.75 μm / 0.1 μm, 1 μm / 0.2 μm, and 1.25 μm/0.3 μm were considered. As previously reported the average film stress is strongly dependent on the film thickness. In the scalloped structure, compared to the full plate sample, the tensile stress due to coalescence increases at a reduced rate as the thickness increases.

Figure 5.16, Figure 5.17, and Figure 5.18, are the simulation results of a W film grown on scalloped geometries with height/width of 0.75 μm/0.1 μm, 1 μm/0.2 μm, and 1.25 μm/0.3 μm, respectively. In all the figures the green line indicates the normalized average stress during the V-W growth on a full plate sample. The other lines represent the normalized average stress during V-W growth for different scalloped structures where h_s and w_s were varied. For the full plate sample, during the third V-W growth step, the average stress remains constant, as is characteristic of low adatom mobility materials.

In Figure 5.16 (a), Figure 5.17 (a) and Figure 5.18 (a) the width w_s of the scallops was kept constant at 0.75 μm, 1 μm, and 1.25 μm, respectively, and the height h_s of the scallops was varied between 0.1 μm, 0.2 μm, and 0.3 μm at every width. In all samples the same stress evolution is observable. The width w_s of the scallops influences the stress at small thicknesses; subsequently, as the thickness increases, the film stress evolves similarly for all samples. In Figure 5.16 (a) it is evident that the width of the scallops affects the film stress evolution until a thickness around 200 nm is reached. Similar is the case in Figure 5.17 (a) and in Figure 5.18 (a), where the film stress evolution is unaffected by scallops until thicknesses of 225 nm and 250 nm are reached, respectively. In all figures it is evident that a sharp stress increase takes place during thickening. This increase is connected to the overtaking of grains by neighboring grains. As the scallop width w_s increases this process is delayed, leading to the sharp stress increase at different film thicknesses. This phenomenon of overtaking grains is further described in the next subsection.

In all the samples with h_s=0.1 μm and a varied w_s the normalized average stress does not significantly vary with the scallop width. This is depicted in Figure 5.16 (b), where in all the scalloped structures the tensile stress evolves similarly and converges at the same maximum value for all samples. In this configuration the scallop heights do not significantly change the film geometry which is still essentially flat and therefore no extreme stress variations are observable.

In Figure 5.17 (b) h_s was set to 0.2 μm and w_s was varied. Here, between film thicknesses of 125 nm to 250 nm different stress behaviors are observed. In this thickness range the film assumes the scalloped shape and its stress is influenced by the geometry. In all the results a sharp stress increase is evident due to grains which are overtaken by neighboring grains. The smallest generated stress is in the structure having w_s=0.75 μm due to the scallop curvatures. In general, as the curvature of the scallops increases (big h_s and small w_s), the film stress decreases.

In Figure 5.18 (b) the scallop structures have h_s=0.3 μm and the width w_s was varied. These samples correspond to the structures having the most curved scallops. In particular the sample with w_s=0.75 μm has the most pronounced scallop curvature and a small average film stress develops before finally reaching the maximum stress. Also here the sharp stress increase at a thickness between 200 nm and 250 nm is due to the overtaking of grains by neighboring grains. This stress behavior is related to the curvature of the scallops that influence the grain growth during film thickening. The grain boundary height h_gb increases slowly, unlike in a full plate sample. During the deposition process the film stress build-up evolves. The maximum tensile stress is reached only when the grain boundary height is equal to the film thickness [106]. Due to the scallop curvatures a homogeneous film (for low roughness the grain boundary height corresponds to the film thickness) is formed at high thicknesses; therefore, at small thicknesses the grain boundary height is small and a low amount of intrinsic tensile stress is generated.

In all samples a stress decrease was observed for thin films (below ~200-250 nm), when a notable curvature is present in the scallops, such as at high h_s or at small w_s. The generated stress in these samples is less than that produced in a full plate sample. When the grains are grown on curved surfaces, they have less contact area, thereby producing less stress.

This simulation approach can be used to investigate the stress evolution for arbitrary geometries.

5.4.1.7 Overtaking of Grains

The sharp stress increase observed in Figure 5.16, Figure 5.17, and Figure 5.18 of the previously subsection can be described by considering the overtaking of grains. As the film thickness increases a grain can be overtaken by the neighboring grains as depicted in Figure 5.19. As described previously the intrinsic stress build-up is an evolving process dependent on the grain boundary height. During overtaking the contact area (grain boundary height) between grains rapidly increases and the film starts to be homogeneous, leading to a rapid stress increase.

The stress in low adatom mobility films grown on a full plate sample stays constant during the third step of the V-W growth. However, when films are grown on a scalloped structure this is no longer the case and two phases of film thickening can be clearly identified:

• The first phase identifies the stage before the overtaking of grains. The film stress has a small value compared to a film grown on a full plate sample. In this phase the intrinsic stress is significantly lower than that in flat samples. This explains the reduced stress in thin films from published experiments [86].
• The second phase indicates the stage after the overtaking of grains. A homogeneous film is forming and a rapid stress increase in the film increase characterizes this stage is observed. As thickening proceeds, the stress approaches the value found in flat samples.

The two-phase stress development during thickening suggests that the scallops only help reduce stress in thin films. For thick films the stress eventually reaches the levels observed in flat samples.

5.4.2 High Adatom Mobility Analysis

5.4.2.1 Temperature Effects

The implemented model, which simulates the generation of compressive stress during the third V-W growth step for high adatom mobility materials, was verified by using stress measurements of Ag films grown on SiO₂. In [1, 2] the stress-thickness product as a function of thickness was measured by changing the deposition temperature. Since the adatom mobility depends on temperature, the shape of the curve at higher temperatures becomes less tensile (i.e. more compressive) as the thickness increases. This experimentally derived behavior [1, 2] is shown in Figure 5.20. All samples were grown using a fixed growth rate of v_g = 0.2 nm∕s.

With the FEM model the experimental data were reproduced by using (5.14), where σ_T = 1550 MPa and σ_C = -200 MPa. All simulations were started with an assumed film thickness of 20 nm. The model reproduces the experimental results at different temperatures by varying β (the employed β values are in the inset of Figure 5.20).

In the stress-thickness versus thickness plot (cf. Figure 5.20) a tensile behavior is only observable at low temperatures (green line), and at high temperatures the transition from tensile to compressive stress can be seen clearly (red line). In these simulations is assumed that the generated compressive stress is driven only by adatom mobility, allowing the atom to move into the grain boundary. This suggests to vary the atom diffusivity with β. Higher values of β correspond to higher adatom mobilities, whereas smaller values stand for lower adatom mobilities. The simulations produced results with qualitatively good agreement to experimental data and it provided an understanding of the transition from tensile to compressive stress. The best fittings are obtained at low deposition temperatures. This observation is ascribed to a constant distribution of grain sizes under these growth conditions. At higher temperatures the simulation results of the model fit quantitatively better to experimental results at high film thickness. Especially for temperatures between -20 ^∘C and 30 ^∘C (cf. black and red lines in Figure 5.20) the simulations are in quantitatively good agreement for thicker films, where the model predicts the generation of compressive stress. For small thicknesses the model underestimates the generation of tensile stress. It is supposed that the film roughness and the different sizes of the islands for small film thicknesses have a significant influence on the stress. These effects are not included in the model, due to the impossibility of tracking them experimentally.

⟨σ_xx⟩ was calculated in the film as a function of film thickness and the results are depicted in Figure 5.21. The magnitude of the stress is strongly dependent on the deposition temperature. The transition from tensile to compressive stress at high temperatures is clearly observable. Therefore, the model can be used for the simulation at different deposition temperatures.

As far as interconnect design is concerned, the knowledge of ⟨σ_xx⟩ as function of film thickness can be used to reduce the probability of mechanical failure. In this study the capability of the model to reproduce the deposition process for different temperatures were verified.

5.4.2.2 Growth Rate Effects

In [3] stress measurements during Cu electrodeposition on an Au substrate were carried out. Cu was deposited at room temperature with different growth rates v_g. In order to create a consistent starting layer with respect to grain size for all the considered deposition processes, 100 nm of Cu was deposited with a growth rate of 9.1 nm∕s in the first phase. The resulting grains had an average size of L = 200 nm [3]. After reaching 100 nm thickness, the second phase was initiated by changing the electrical potential, thereby producing different growth rates. Unlike the situation described in the previous section, the model applied here was verified by changing the growth rate and keeping the other parameters constant.

The electrical potential at the electrodes influences the nucleation density and the growth rate in the electrodeposition process.

The experimental stress-thickness product as a function of thickness was reproduced by simulation. The parameters σ_T = 230 MPa, σ_c = -195 MPa, and β = 4.125 nm∕s were utilized in the simulation. In Figure 5.22, the obtained experimental values for the stress-thickness product between -0.45 V to -0.65 V were compared with the simulation results. An average grain size of 200 nm was used as it fits well to experimental data, even though the grain size for thicknesses over 100 nm was not measured for this experiment [3]. A change in the average grain size during growth may alter the film stress, however, the model qualitatively reproduces the experimental results well with the assumed grain size. The model accurately fits the experimental results for growth rates between 9.1 and 23.5 nm∕s. For this range of deposition rates the grain sizes generated by the initial deposition process for all samples were kept constant in the subsequent process phases as the growth rates are increased. At a growth rate of 5.6 nm∕s the model does not properly predict the stress for high material thicknesses possibly due the fact that the growth rate in the initial phase surpasses the growth rate of the thickening phase. This can lead to the generation of additional grains leading to a higher roughness of the film. These effects, which might influence the stress in the film, are not included in the model. The growth rate has a significant influence on the residual stress in the film (cf. Figure 5.23). For all the simulated deposition conditions, as the growth rate decreases, the film stress reduces. Slow growth rates correpond to a low grain boundary velocity and therefore the film stress becomes less tensile and more compressive. At slow growth rates the adatoms have ample opportunity to move into the grain boundary, generating compressive stress, while at high growth rates the stress remains almost constant throughout thickening.

5.5 Summary

The intrinsic stress generated during the deposition processes of thin metal films was investigated. Since the metalization of wafers in microelectronics is usually performed at the "back end" of the production line, it is important to understand the origin and behavior of stress during film deposition in order to increase reliability and to prevent mechanical failure of the interconnects. The three growth steps responsible for the evolution of stress were identified, and each of them was simulated by FEM.

By applying the concepts of Laplace pressure and of the frozen-in radius, compressive stress generation in the islands during the first growth step was simulated. The estimation of the magnitude of the stress is a very challenging task due to the lack of experimentally confirmed parameters. Nevertheless, the model helps to understand, how stress caused by volume increase develops at the substrate and inside the islands. Information regarding stress distribution in the substrate is useful, when it comes to adjusting the deposition process in order to limit mechanical failures (e.g. delamination, cracks, etc.).

By using Seel’s approach the second step regarding the process of island coalescence was examined. At small thicknesses of the film, the shape of the island, grain sizes, and film roughness are factors which influence the stress build-up in a film. For thick films the film is homogeneous (low roughness) and the stress is equally distributed. During the growth process, the maximum stress is reached, when a homogeneous film is formed. For a thin film the island shapes change the stress distribution and thereby the average stress in the film. Therefore, different stress values can be obtained in thin films. Smaller grains exhibit higher stresses, whereas bigger grains produce lower stress in the film.

The stress behavior due to low adatom mobility was analyzed during W growth on a scalloped structure and on a full plate sample. The results demonstrated that the substrate shape influences the film stress for thin films. Thin films grown on a scalloped surface with a high curvature develop a small intrinsic stress. Due to the curved substrate the thickness necessary to reach a homogeneous film increases leading to a small intrinsic stress generation. The grain boundary height on a scalloped structure grows differently than in a flat sample. During the film grows, height and width of the scallops influence the contact areas between islands and consequently the tensile stress generated. As film thickness increases the film becomes homogeneous (when the grain boundary height correspond to film thickness) and the tensile stress converges at the stress value found in a full plate. A new phase in the stress evolution during film thickening on scalloped surfaces was identified. The first phase indicates thin film growth at the stage before the overtaking of grains, while the second phase occurs after grain overtaking and is similar to the traditional V-W thickening during deposition on flat surfaces. This information can be used during deep reactive ion etching for TSV fabrication. By controlling the etching parameters, the appropriate scallop size necessary to minimize mechanical failure can be achieved. Scallops having a larger height/width ratio induce small intrinsic stress generation. The scallop geometry permits to reduce considerably the film stress. Deposition on scallops having the largest height/width ratio (h_s=0.3 μm / w_s=0.75 μm) produce ~70% less film stress compared to a full plate sample. On the other hand, films grown on scallops having the smallest height/width ratio (h_s=0.1 μm / w_s=1.25 μm) produce ~20% less of stress compared to a full plate sample. Scallops influence the stress evolution between the coalescence process and the overtaking of islands. As the film becomes thicker the scallop geometry does not impact the stress evolution.

A high adatom mobility model was implemented on the basis of the theory described by Chason. Compressive stress in a film is generated due to the movement of atoms into the grain boundary. The implemented model is able to reproduce measured data from two different experiments. In the first, the effects of the deposition temperature on film stress and in the second the effects of different growth rates were replicated. High deposition temperatures and slow growth rate produce small intrinsic tensile stress. The calibrated model can be applied to investigate the stress evolution for future applications. In open TSVs, Cu is still deposited by electrodeposition. This process often shows an abnormal grain growth. For such cases the described model is not able to properly handle the deposition process. Future technologies generations demanding higher aspect ratio TSVs might require a CVD Cu deposition. However, Cu films deposited by CVD are still not worthy manufacturing [107].