Understanding residual stress in polycrystalline thin films through real-time. measurements and physical models. Eric Chason and Pradeep R.

Transcription

1 Understanding residual stress in polycrystalline thin films through real-time measurements and physical models Eric Chason and Pradeep R. Guduru School of Engineering Brown University, Providence, RI Abstract Residual stress is a long-standing issue in thin film growth. Better understanding and control of film stress would lead to enhanced performance and reduced failures. In this work, we review how thin film stress is measured and interpreted. The results are used to describe a comprehensive picture that is emerging of what controls stress evolution. Examples from multiple studies are discussed to illustrate how the stress depends on key parameters (e.g., growth rate, material type, temperature, grain size, morphology, etc.). The corresponding stress-generating mechanisms that have been proposed to explain the data are also described. To develop a fuller understanding, we consider the kinetic factors that determine how much each of these processes contributes to the overall stress under different conditions. This leads to a kinetic model that can predict the dependence of the stress on multiple parameters. The model results are 1

2 compared with the experiments to show how this approach can explain many features of stress evolution. Table of contents 1. Introduction 2. Measurement of thin film stress using wafer curvature 2.1 The Stoney equation: average and incremental stress 2.2 Substrate curvature change due to stresses in polycrystalline films 2.3 Measurement techniques and multi-beam optical stress system (MOSS) 3. Phenomenology: examples from wafer curvature measurements 3.1 Stress evolution in different materials systems 3.2 Dependence on kinetic and processing parameters 3.3 Reversible and irreversible stress evolution during growth interrupts 3.4 Summary of observations from curvature measurements 4. Mechanisms of stress generation 4.1 Compressive stress in the pre-coalescence regime 4.2 Tensile stress induced by grain boundary formation 4.3 Compressive stress induced by insertion of atoms at the grain boundary 4.4 Effects of grain growth 4.5 Other stress-inducing effects outside of growth kinetics Lattice parameter mismatch 2

3 4.5.2 Thermal stress Energetic deposition effects 5. Kinetic model for intrinsic stress during film growth 5.1 Physical basis for model and evolution equations 5.2 Application to evolution of stress with thickness 5.3 Application to stress in the post-coalescence regime Effect of grain size evolution Effect of growth rate 5.4 Relationship to growth interrupts 6. Discussion and summary 7. Acknowledgements 3

4 1. Introduction Thin films are instrumental in a large number of technologies. Their uses are too numerous to describe, but critical applications include optical coatings, semiconductor devices, optoelectronics and wear-resistant coatings. By adding only a small amount of material to the surface, thin films make efficient use of high performance materials that wouldn t have the same benefits in the bulk. Thin film solar cells only need to be as thick as the absorption depth of light; any additional material is wasted. Coatings of metal-nitride layers over metal tool-bits greatly increase their wear performance 1 but bulk versions of these materials would be too brittle to use. Since deposition is a non-equilibrium process, the resulting material may not be in its most stable state. This may be useful; many metallic films would ball up if the film could achieve its equilibrium structure, as seen when they are heated too high. However, the non-equilibrium state of the film can also mean that the atoms are not in their fully relaxed positions, which creates stress in the layer. Hard coatings have tensile stress that can lead to cracking 2 or limit the usable thickness of the layer. Compressive stress can lead to buckling, blistering and delamination 3, 4 while stress in interconnects or coatings on conductors can lead to the emergence of hillocks and whiskers that can cause short circuit failures 5, 6. Consequently, there is a long history of studying stress in thin films. Over 100 years ago there was already concern with the delamination of reflectors 7 and the deformation of mirrors by stress in the layers 8. Yet despite their importance, a simple understanding of stress has been elusive. Many trends have been identified, but also 4

5 many exceptions. One of the difficulties is that many factors have been shown to play a role in stress evolution growth rate, temperature, microstructure, material type, etc. so that it is difficult to make correct conclusions if not all the factors are known or controlled. However, we believe that a comprehensive picture is beginning to emerge as is described in this article. The aim of this work is to provide an understanding of the current state of knowledge regarding the origins and evolution of stress in thin films. In particular, we would like to show how multiple features of stress evolution may be understood within a simple kinetic framework based on underlying physical mechanisms. From this point of view, phenomena that appear to be inconsistent may be the result of interaction among multiple parameters that are changing and perhaps not being monitored. We focus primarily on results in which the deposited species have no additional energy beyond their thermal energy (e.g., evaporation and electrodeposition). However, some of the additional effects that can be induced by deposition of energetic species (e.g., sputter deposition) are discussed for comparison. The manuscript is organized in the following way. The wafer curvature method for determining thin film stress is described first, with emphasis on how the curvature is related to the distribution of stress throughout the thickness of the layer. The analysis is extended to include consideration of curvature from stress in polycrystalline or nonuniform films. A description of the multi-beam optical stress (MOSS) measuring technique and its sensitivity is also discussed. The following section describes phenomenology that has been seen for a number of studies (i.e., different materials systems, different processing parameters and stress changes during growth interrupts). 5

6 The results from these studies are used to identify common trends for the dependence of the stress on growth rate, atomic mobility, temperature, surface morphology, grain size, etc. These results are used to motivate the discussion of underlying stress generating mechanisms in the following section. The emphasis is on explaining intrinsic stress, i.e., stress related to the processes of film growth and microstructural evolution, but some other mechanisms (thermal expansion, epitaxial mismatch and energetic deposition) are considered as well. In the fifth section, a kinetic model is described that combines the growth-related mechanisms into a set of rate equations to describe the dependence of the stress evolution on the parameters considered in the previous sections. The model is compared with experimental results to show how its predictions are consistent with observations in a number of different systems. This also highlights the need for systematic studies in order to determine the range of validity of the model and effects that it cannot explain. 2. Measurement of thin film stress using wafer curvature Wafer curvature measurement is a common technique used to characterize stress in thin films. It works by measuring the bending that a stressed film induces in an elastic substrate. Because such measurements can be performed relatively easily in situ and in real-time, much of our knowledge of stress evolution has been obtained from this method. X-ray diffraction (XRD) can also be used to measure the deviation in lattice spacing induced by biaxial stress in the film 9. Although XRD is a powerful method for 6

7 determining stress it is not easy to use for in situ monitoring so the following discussion will focus on wafer curvature. 2.1 The Stoney equation: average and incremental stress The geometry of the film and substrate for calculation of the curvature is shown schematically in figure 1a. The figure shows a segment of film removed from a larger sample; the substrate is not completely shown and is assumed to be much thicker than the film. The stress that is measured by the curvature is the component acting normal to the edge of the film ( xx (z)) which is assumed to be uniform in the x-direction but can vary with the depth, z. The total force per unit width acting on the edge (i.e., membrane force) is given by f h f hf 0 ( z) dz xx f S f I (1) where the integral is over the thickness of the film (h f ). The terms f S and f I represent additional contributions from the surface stress and interfacial stress 10. The surface/interface stress contributions do not change with thickness and for relatively thick films can be ignored. However, they may play an important role in very thin films or in the early stages of film growth 11, 12. 7

8 This expression allows us to define the average stress ( ) as the membrane force divided by h f. The product h f is often referred to as the stress-thickness. If the stressthickness is the same everywhere, the substrate curvature is uniform. When the substrate is relatively thick compared to the film, its curvature is given by the formula often referred to as the Stoney equation 7, 13 : 6 h M h s f 2 s (2) where h s is the substrate thickness and M s is the biaxial modulus of the substrate (given by M s =E s /(1- s where E s is the bulk modulus and s is Poisson s ratio). Eq. 2 shows that knowledge of the film thickness is needed to determine the average stress from curvature measurements. Derivation of the Stoney equation can be found in ref. 13, which involves a number of assumptions. Although the stress may vary with depth, the film is taken as being uniform in thickness and isotropic in its properties. The stress state is considered to be equi-biaxial and the thickness of the film is small relative to that of the substrate so that stress relaxation in the film due to curvature change is negligible. The Stoney equation can be extended to consider other idealized film geometries. For films that consist of multiple layers, the total membrane force normal to the edge is equal to a superposition of the forces (f i ) from each of the i layers 13 : 8

9 6 M h s 2 s i f i i Stoneyi, (3) where the sum is over the number of layers in the film. The measured curvature is equal to the curvature that would be induced by each of the layers individually (Δκ Stoney,i ) if deposited on the substrate. If the individual layers are extremely thin, then the resulting curvature can have a significant component from the interfacial stress; measurements of multilayers have been used to study the interfacial stress between layers This type of analysis has also been used to stress in systems with moving interfaces due to lithiation 17 or phase formation 18.. For a film that is not isotropic and develops a biaxial stress state with the in-plane principal stresses x and y oriented along the x and y axes respectively, the curvature may be different in the x- and y- directions. Defining f x and f y as the membrane forces in the two directions, the corresponding curvatures are given by 13 x 6( f x s f E h s 2 s y ) (4a) y 6( f y s f E h s 2 s x ) (4b) 9

10 Alternatively, measuring the curvature along the different directions can be used to determine the average stress in each direction. If there are shear forces along the edge, this can also lead to a twist type of deformation. Other modifications can be found in the literature that consider the effect of the substrate s properties 13, 19, patterned film layers 20 and thick films or large deformations 21. A major advantage of curvature measurements is that they can measure the evolution of the curvature while the film is being deposited. This can be used to determine the distribution of stress at different heights in the film, not just the average stress, by considering the evolution of the stress-thickness with time. Assuming an isotropic film, with equi-biaxial stress and constant surface stress, the time derivative of f is: df dt ( h xx f dh ) dt f h f 0 xx ( z, t) dz t (5) This equation describes two contributions to the curvature occurring simultaneously. The dh f first term on the right side corresponds to the addition of new layers (at the rate dt ) with stress equal to xx (h f ). We refer to xx (h f ) as the incremental or instantaneous stress. This additional top layer is shown on the schematic in figure 1a. The second term corresponds to changes in the stress that occur in the layers of the film that have already been deposited. If the stress in the existing film does not change, i.e., any relaxation mechanisms that modify the film stress are slow compared to the deposition time scale, 10

11 then the second term can be ignored. In this case, we can determine the incremental stress from the slope of the stress-thickness vs. thickness measurements as ( h xx f d( h f ) dt d( h f ) ) (6) dh f dh f dt An example of a real-time stress thickness measurement made during the deposition of Ag on SiO 2 at room temperature 22 is shown in figure 2d. The average stress in the film at thickness h f is equal to the slope of the line from the origin to the stress-thickness at this point. It is clear from the figure that the average stress changes with the thickness, otherwise the stress-thickness vs. thickness would be a straight line. If we assume that the stress does not change in the film that has already been deposited, then the incremental stress is equal to the slope of the stress-thickness curve at h f (represented by the tangent line). In figure 2d, for the value of h f shown, the average stress in the film is positive (tensile) but the incremental stress is negative (compressive). However, it is important to note that changes in the stress in the existing film cannot always be ignored. Grain growth, plastic deformation (e.g., dislocation motion) or diffusion of solute atoms/vacancies can change the stress in the layers below the surface. This is discussed further below with respect to the contribution of grain growth to stress evolution (section 4.4). These stress-inducing effects can modify the evolution of the stress-thickness so that the measured change in the stress-thickness cannot always be attributed solely to the addition of new layers with incremental stress. To separate the 11

12 stress effects occurring by the addition of new layers at the surface from changes occurring in the bulk of the film, measurements that combine growth with specific interrupts of the growth process (discussed further in section 3.3) as well as characterization of grain structure evolution can be used. Moreover, many samples change their temperature during deposition (e.g., due to radiation from the deposition source) and thermal expansion mismatch can significantly change the stress in the film after it has cooled. Therefore, the stress state measured after growth has finished and the sample has cooled can be significantly different from the stress in the film during growth. 2.2 Substrate curvature change due to stresses in polycrystalline films The preceding discussion of the Stoney equation assumed that the stress at each height (z), xx (z), in the layer is uniform in the plane of the film. However, many thin films are polycrystalline so that the distribution of stress can be more complex. In particular, stresses can originate from processes within the interior of the grains throughout the film or they can arise from processes at the grain boundaries. Examples of the former include stresses due to thermal expansion coefficient mismatch, lattice constant mismatch (e.g. in case of epitaxial films) and distributed vacancy/interstitial populations. For these cases, the term f (given by Eq. 1) has the same value everywhere in the plane of the film and the resulting substrate curvature is given by the Stoney equation. Examples of the latter include stresses due to island coalescence and insertion 12

13 of atoms along the grain boundary (these mechanisms are discussed in more detail in the subsequent sections). In these cases where the film stress is due to processes occurring at the grain boundaries, use of the Stoney equation may require additional considerations. To estimate the effect that the grain structure has on the resulting curvature, consider an idealized polycrystalline film shown in figure 1b in which stress arises at the grain boundaries due to mechanisms mentioned above. The grain width is L and its thickness is h f. The substrate thickness is large compared to L and h f. The grain boundary stress is designated as xx, which in general can be a function of z. Variations in the direction normal to the plane of the figure are not considered here for the sake of simplicity. An approximate free body diagram of a single grain is shown in Figure 1c for additional clarity. Each vertical edge is subjected to a normal traction xx (z) and a shear traction zx (x) is induced on the bottom face (z = 0). There may also be a selfequilibrating normal traction on the bottom face, which is not shown for the sake of simplicity. Note that, if the film stress is uniform everywhere (e.g. as in figure 1a, there would be no shear traction on the bottom face of the grain (except near the outer boundary of the film). In contrast, an essential feature of grain boundary stress is that the film stress is non-uniform in the x-direction; the average normal stress (σ xx (x)) decreases and reaches a minimum value at the center of the grain, mid-way between two neighboring grain boundaries. Now, if all the grain boundaries are imagined to be cut and the new surfaces are rendered traction-free, the resulting film-substrate system would be stress-free. The film would resemble an array of stress-free islands, or equivalently a film with a periodic array of cracks. If a normal traction xx (z) is applied on each crack face, the cracks close perfectly, recovering the configuration in figure 1b. In other words, 13

14 the problem in figure 1b can be reduced to that of a film with a periodic array of cracks (of crack spacing L), with each crack face subjected to a normal traction xx (z). An approximate expression for the substrate curvature in the reduced problem can be obtained by using the result of Freund and Suresh 13, and is given by corr 6 f M gb 2 shs (7) h f where f ( x 0, z) dz is evaluated along the grain boundary and gb 0 xx corr l L tanh l L (8) In the above equation, l is a length scale parameter that is proportional to h f and the proportionality constant depends on the mismatch of elastic constants between the film and the substrate through the Dundur s parameters. For example, when there is no elastic constant mismatch, l = h f. More accurate values for the correction factor corr obtained from finite element calculations are plotted in Figure 3. The above Eq. 7 for curvature can be viewed as a modification of the usual Stoney equation in which the net grain boundary force f gb takes the place of the uniform force in the film, along with a correction factor corr which accounts for the nonuniform stress distribution in the film due to the film/substrate interaction. It is instructive 14

15 to examine the effect of grain size L on the correction factor corr in Figure 3. It can be seen from the plot that for small grain sizes, i.e., L h f, one can use the net force on the grain boundary to calculate the substrate curvature with an error of 10% or less. However, for larger grain sizes, L 2h f, such an approximation leads to a significant error (>18%) and the correction factor corr would become more important in interpreting the experimental results. In many experiments, the grain sizes are small enough relative to the thickness to allow ignoring the correction factor without significant error. Similar considerations apply for the measurement of incremental stress (Eq. 6) as well. It is worth noting that, since the model considered in Figure 1b is for a one dimensional (1-D) film, the actual values of corr would be somewhat different for a 2-D polycrystalline film. This may be the case, for instance, in the early stages of growth when the film consists of coalescing islands with non-uniform shape. A schematic of a hemispherical island that has started to coalesce with a neighboring island is shown in figure 4. The flat regions with semicircular shape at the edge of the island are where the grain boundaries between islands form. In the same manner as for the 1-D case, we can calculate the value of the forces on these faces (f x,gb, f y,gb ) by integrating the normal stress over the boundary area: f x, gb h xx f 1 L y Ly 2 L y 2 L h( x, y ) 2 0 xx ( Lx 2, y, z) dzdy (9a) f y, gb yy h f 1 L x L x 2 L x 2 Ly h( x, ) 2 0 ( x, yy L y 2, z) dzdx (9b) 15

16 The limits of the integral in the z-direction correspond to the height of the boundary at the edge of the periodic islands. The correction factors for the Stoney equation for nonuniform 2-D islands depend on the specific geometry and they remain to be calculated. However, the broad conclusions regarding the length scales and the errors described above will remain the same, i.e., the corrections will be less severe when the thickness is comparable to or larger than the island dimensions. 2.3 Measurement techniques and multi-beam optical stress system (MOSS) A number of techniques have been developed for measuring wafer curvature. The ones that are most useful for studying the fundamentals of film stress enable the curvature to be measured during deposition in real-time. Various methods have been developed for measuring the deflection of a cantilever or substrate while a film is being deposited on it. The deflection has been measured by the change in capacitance between a cantilever and a sensor or the piezoresistive response in microfabricated cantilevers 27. Optical methods enable the sample to be remotely measured inside a growth chamber without affecting the growth process. This can be done by interferometry 28 or monitoring the deflection of a reflected beam. The deflection methods may scan a beam along the sample surface 14, 29, 30 or monitor a single beam 31, 32 ; these require good stability in the system to prevent drift. Alternatively, the spacing between the reflections 16

17 of multiple beams can be monitored which reduces the sensitivity to sample position and therefore drift. A schematic of a multi-beam optical stress system (MOSS) is shown in figure 5. In this technique, an optical element is used to create an array of parallel laser beams that is aimed at the sample surface. The spacing between the reflected beams is measured using a CCD camera and digitized with a framer grabber. The lens focuses the beams on the detector to achieve small spot sizes with a spacing on the order of several mm. The optics can be easily configured to illuminate the sample through the viewport of a vacuum chamber, so that the measurements can be made during film processing. The relationship between the change in the sample curvature and the change in the beam spacing (δd) is given by: d o d o cos 2L camera (10) where κ o is the initial curvature in the sample (if any), d o is the initial spacing of the beam reflections, is the angle between the incident beam and the sample normal and L camera is the distance from the sample to the camera. The sensitivity of the system is determined by the resolution of the camera for measuring the beam spacing; in our lab, using a typical camera the relative change in the spacing (δd/d o ) can be determined with a resolution of 0.008%. If the camera is spaced 1 m from the sample (also a typical value), this corresponds to a minimum resolvable 17

18 curvature of 4 x 10-5 m -1 or a maximum radius of curvature of 25 km. The sensitivity of the stress-thickness measurement additionally depends on the thickness and modulus of the substrate. For Si wafers with a thickness of 200 μm, a minimum change of 0.05 N/m in stress-thickness can be observed. Because of the dependence of the curvature on the substrate properties in eq. 2, using a substrate that is thinner or less stiff can increase the sensitivity further. Because curvature measures the product of stress and thickness, the value of the smallest stress that can be measured depends on the thickness of the film. With a stressthickness system sensitivity of 0.05 N/m, a stress change of 0.05 MPa can be measured in a film that is 1 μm thick. For a film that is only 1 nm thick, the minimum stress that can be measured is 0.05 GPa. This sensitivity is sufficient to measure the stress from less than a monolayer of Ge deposited on a Si surface, so even effects of single atomic layers can be observed with the curvature technique 33, 35. In addition to the good stress sensitivity, a significant advantage of the multibeam technique is its relative insensitivity to rigid motion of the sample. Sample motion can be a problem for in situ measurements in deposition systems where there may be substantial amounts of mechanical noise (e.g., vibrations due to vacuum pumps or sample drift during experiments that last hundreds of minutes). This can be challenging to control because the sample must be free to bend during the measurements and cannot be clamped down. For single beam or scanning beam measurement techniques, motion-induced change in the reflected beam can be difficult to separate from the effects of stress-induced curvature changes. Therefore, for these systems the overall resolution can be limited by the mechanical stability of the sample during the measurement. 18

19 In the multi-beam technique, however, the spacing between different beams is used to determine the curvature, not the absolute position of the beam. Sample vibration or motion causes all of the reflected beams to move together, which does not affect the difference between the measured positions. Therefore motion that does not change the shape of the substrate does not affect the measured curvature. This can be seen in measurements of reflected beams from a Si sample in a vacuum chamber (shown in figure 6) that were obtained over a period of 100 s. The upper part of the figure shows the position of a single beam reflected from the surface; the standard deviation of the measured value is 0.43 pixels. In comparison, the spacing between two reflected beams is shown in the lower part of the figure. The standard deviation of the spacing is only 0.09 pixels, showing that the noise on the spacing (and hence the curvature) is less than the noise on the individual beam position by a factor of almost 50. The absolute curvature of a sample (instead of just the change in the curvature) can be determined by using a standard such as a high quality flat mirror, which provides the spacing of the incident beams (d o ). Multiple measurements of a flat mirror show that the absolute curvature can be determined with a standard deviation of less than 0.1% of the measured value. Because of its versatility, the multi-beam technique has been used to study many different film systems including PVD (physical vapor deposition) 22, 36, electrodeposition 37, sputter deposition 38, MOCVD (metallo-organic chemical vapor deposition) 39, MBE (molecular beam epitaxy) 40 ion bombardment 41, 42, pulsed laser deposition 43 and electrochemical modification 17. All that is necessary for its implementation is a viewport from which specular reflections from the surface can be monitored. In cases where this is 19

20 not possible, mirrors have been mounted inside the chamber to illuminate the surface and collect the reflected beams. If a fluid layer fills the space between the sample and the viewport, the second term on the right hand side of Eq. 10 acquires the refractive index of the fluid in the denominator. If the growth process results in a rough surface that reduces the quality of the specular refection (as in electrodeposition), reflections from the back side of the sample can be monitored instead of from the film side. 3. Phenomenology: examples from wafer curvature measurements With many years of studying thin film stress, there is a large literature of measurements characterizing many materials systems under many different conditions. It is not within the scope of this manuscript to review all previous work (some useful comparative studies can be found in , 49. Instead, we will describe the results of several studies that are illustrative of the range of phenomena that have been observed and have led to our current level of understanding of the origins of stress. Of course, to the film user it may not matter what the origin of the stress is. But understanding its origin may be useful in determining how to control it or mitigate it. The examples discussed in the following sections are chosen to focus on several different mechanisms of intrinsic stress generation. By intrinsic stress we mean the stress created by the processes of film growth. The examples are taken from studies in which the deposited atoms have near-thermal energy (evaporation and electrodeposition). The first set of examples (setion 3.1) are chosen to highlight the difference in stress evolution 20

21 due to the atomic mobility of the material. Section 3.2 focuses on changes in the stress evolution that can be induced by changing the processing conditions (growth rate and temperature). Section 3.3 describes effects that occur when the growth is interrupted. The underlying intrinsic stress-generating mechanisms suggested by these studies are discussed in section Stress evolution in different materials systems As described by Abermann 50, the evolution of the curvature for different materials typically falls into two categories based on the atomic mobility (or equivalently the melting point). High melting point/low mobility materials have curvature evolution that indicates only tensile stress in the film. Referred to as type I behavior, examples are shown in figure 7a for Cr, Fe and Ti films 45 deposited at room temperature onto MgF 2 - coated glass substrates at a rate of nm/s. The stress-thickness has a nearly constant slope showing that the incremental stress reaches a steady-state value that does not change with increasing thickness. Because the low mobility suppresses grain growth, these types of films typically have a columnar morphology with a grain size that does not increase substantially with the thickness. Consequently, each additional layer of the film is grown on underlying layers with a similar microstructure. The absence of change in the underlying layers is further supported by the observation that the stress-thickness does not change after the growth is terminated (at the time indicated by the vertical line in the 21

22 figure). A discussion of the origin of the tensile stress in type I films due to the formation of grain boundaries is in section 4.2. The evolution of the stress-thickness for materials with lower melting point/higher mobility (called type II behavior) is shown in figure 7b for layers of Ag, Cu and Au deposited at room temperature onto mica substrates at a rate of 0.1 nm/s 51. The incremental stress is observed to change significantly as a function of thickness (based on the slope of the stress-thickness). In the very early stages (< 10 nm) the incremental stress is low or slightly compressive. As the film gets thicker (from nm), the slope becomes increasingly positive, corresponding to tensile stress in the film. With continued growth, the stress-thickness ultimately reaches a maximum and the incremental stress changes from tensile to compressive. This is not just a relaxation of the tensile stress because with additional thickness the average stress in the film becomes compressive. To understand the transitions of the incremental stress with thickness for type II materials, measurements of the surface morphology that corresponds to the different stages in the stress evolution have been performed 23, 45, 52, 53 An example is shown in the micrographs in figure 2a-c taken after 10, 22 and 40 nm of deposited thickness 54. The black circles in the stress-thickness measurement (figure 2d) indicate the thickness where the micrographs were taken. In the early stages of growth when the incremental stress is small, the film consists of individual islands (pre-coalescence). As the islands grow larger (figure 2b), the impinging islands start to coalesce and form grain boundaries. The incremental stress in this region is tensile, and the stress maximum is often found to be associated with the point where the film becomes completely coalesced. For greater 22

23 thickness, the incremental stress becomes compressive and the morphology is that of a uniform polycrystalline film. In the steady-state regime, the slope of the stress-thickness may reach a relatively constant value (as in figure 7a) or it may continue to change with thickness (as in figure 7b). The decreasing slope of the stress-thickness for type II materials indicates a tendency for the incremental stress to become less compressive as the film gets thicker. For high mobility materials, the microstructure of the film may also change with thickness 55. The grain size may change at the surface of the film, or grain growth may occur throughout the film s thickness. This can lead to a corresponding change in the stress (discussed in section 4.4) that can be used to explain the observed evolution of the stress-thickness. The observation of compressive stress in type II materials is attributed to the higher atomic mobility. A proposed mechanism for the compressive stress involving the insertion of atoms into the grain boundary is discussed in section 4.3. The effect of higher mobility is also seen by the large transient in the stress-thickness to more tensile values when the growth is terminated (at the time indicated by the vertical line). However, the type of stress evolution does not just depend on the material but also on the processing conditions as discussed in the next section. Therefore, it is useful to consider the stress under different conditions as the result of a kinetic balance between tensile and compressive mechanisms which leads to a spectrum of possible behaviors. 3.2 Dependence on kinetic and processing parameters 23

24 The measurements for different materials described above were interpreted to indicate that the mobility of atoms on the surface affects the evolution of the stress and microstructure. However, this begs the question of what is meant by high and low mobility. To probe this, performing measurements of stress at different temperatures enables the mobility of the atoms to be modified. An example is shown in figure 8a of stress-thickness measurements made during the growth of Ag on oxidized Si at 0.2 nm/s for the different temperatures indicated in the figure 56. As the growth temperature is raised and the mobility is increased, the evolution changes from low-mobility type I behavior to high-mobility type II. At the lowest temperature (-80 o C), the incremental stress at the largest thickness remains highly tensile, while at 30 o C the final incremental stress is compressive. Similar behavior has been seen in other materials, albeit with a different temperature scale. For example, compare the measurements of Ag with Fe films (figure 8b) deposited at different temperatures (27, 75, 90 and 200 o C) and a similar growth rate (0.087 nm/s) 49, 57. For the Fe films, the final stress in the sample deposited at 27 o C is highly tensile whereas the stress in the Ag film was compressive for the same temperature. Compressive incremental stress in Fe is only observed at the highest growth temperature (200 o C). The observed temperature dependence and its relationship to the melting point of the material suggests that the stress evolution is determined by diffusion-mediated processes in the material. However, the steady-state stress does not exhibit a simple Arrhenius temperature dependence, indicating that its relationship to the diffusivity is more complex than simply linear. 24

25 The determination of high or low mobility is also relative to the rate at which atoms are deposited on the surface. There are surprisingly few studies that have looked systematically at the effect of growth rate on the stress, perhaps because the results have not been easy to interpret; some examples of materials studied include Cu 14, 58-60, Ag 61, Ni 37, 62, 63 Sn 64, B 65 and AlN 66. The results from several studies 62, 63, 65, 66 for the 22, 58, post-coalescence steady-state stress vs deposition rate are shown in figure 9. Parameters for each case (material, method, temperature and grain size) are given in table 1; further details can be found in the individual references. These studies were chosen for the figure because the film s grain size was measured or controlled to not change significantly during the stress measurement. Grain size changes can also influence the stress, as discussed in section 4.4. Although the data in the figure is obtained using different growth methods and temperatures, several common features can be seen. The stress is generally higher for the high melting point materials (AlN, B) than that for the low melting point metal (Ni). In addition, in all cases, the stress becomes more tensile/less compressive as the growth rate increases. The tensile stress has a tendency to saturate at high growth rates, though this cannot be seen for all the materials. Quantifying the effect of temperature and growth rate on the stress is difficult because changing one of these parameters can also affect the grain size and microstructural evolution. Hence, it is important to fully characterize the samples to determine the grain size, growth rate and temperature during the growth in order to relate the measurements to the underlying fundamental processes. Even so, interpreting the dependence of the stress on the growth rate and temperature is difficult to do without a 25

26 model. In the following section, we use the measurements described above (and others) to elucidate the individual mechanisms that play a role in stress evolution during film growth. In the final section, we describe a model for the stress evolution that incorporates these various mechanisms into rate equations that predict the dependence of the stress on the kinetic and microstructural parameters. 3.3 Reversible and irreversible stress evolution during growth interrupts To understand the origins of residual stress, it is important to distinguish between stress changes occurring due to the addition of new layers (incremental stress) and those happening in the layers that have already been deposited (bulk relaxation). Insight into these different processes has been obtained by studying the changes when the growth rate is interrupted and when it is resumed after interruption. As shown in figure 7a, for low mobility materials there is very little relaxation when the growth is stopped due to the lack of surface diffusion and grain growth. However, for higher mobility materials (figure 7b), the stress relaxes significantly when the growth is terminated. When the stress was allowed to relax fully and growth was then resumed 67, the stress evolution continued in a manner that was similar to the uninterrupted growth. Additional studies have been performed by others to look at the effect of growth interrupts on the stress evolution 14, 25, 48, 64, If the period of the interrupt is not too long, the stress evolution after the growth is resumed can quickly return to follow the same behavior as for growth that wasn t interrupted. This has been shown to happen for 26

27 interrupts in different stages of growth, i.e., before or after the islands have coalesced into a continuous film. The reversibility has been interpreted to indicate that the stress change results from a change in the near-surface region and not in the bulk of the film 14, 71. If the growth is resumed at a different growth rate 22, 25, then the stress returns to a different value that depends on the growth rate, indicating that the surface conditions play a major role in determining the incremental stress. Recently, Yu et al. 70 looked at interrupts performed for different lengths of time in the growth of Ni and Au films. Their results are shown schematically in figure 10. If the period of interruption is short (5 min.), the stress evolution returns to the same behavior when the growth is resumed. However, if the period is long (24 hrs), then there are irreversible changes in the stress evolution. The slope (incremental stress) may return to a similar value, but there is an offset in the stress that corresponds to changes in the bulk of the film. They interpreted their results in terms of two relaxation mechanisms, a fast, reversible one that corresponds to relaxation in the near-surface region and a slow, irreversible one that corresponds to grain growth in the film. As discussed in section 4.4, by coupling their measurements of the curvature with characterization of the microstructural evolution they have been able to separate the effects of grain growthinduced stress from the incremental growth stress. Leib et al. 69 also showed that the grain size in Au films affects the magnitude of the stress relaxation when the growth is terminated. For epitaxial films with few grain boundaries, there was essentially no relaxation when the growth was stopped. Similar results have been reported by others 72. Shin et al. 64 measured stress relaxation during electrodeposition of very high mobility Sn films. They found that the stress relaxed 27

28 completely when the growth was interrupted and then returned to the same evolution when growth was resumed. In addition, the relaxation time when the growth was interrupted was found to depend on the grain size and film thickness. These results provide strong evidence that the grain boundaries play a role in generating and relaxing stress in high mobility films. 3.4 Summary of observations from curvature measurements The wafer curvature studies described above indicate several features of intrinsic stress in thin films that can be summarized as follows. The stress can change from tensile to compressive with thickness, indicating a dependence on microstructure and surface morphology. The stress can be compressive or tensile depending on the growth kinetics and conditions (i.e., material type, growth rate, temperature). The general trends are that higher mobility/higher temperature/lower growth rates lead to more compressive/less tensile stress while the opposite leads to more tensile/less compressive stress. The stress can change when the growth is interrupted. The reversible component of the stress change suggests that it is due to processes that are occurring in the near-surface region while the layer is growing, while the irreversible component suggests that stress changes occur in the layers that have already been deposited (e.g., grain growth). The dependence of the relaxation on grain size and thickness further suggests that the grain boundaries play a role in the stress evolution. 28

29 4. Mechanisms of stress generation In this section, we describe some of the mechanisms that have been proposed to explain thin film stress. Each of the mechanism may make a contribution to the stress and they should not be thought of as acting alone. Several of them may operate simultaneously so that the measured stress is the result of a dynamic balance among them mediated by kinetic processes in the film. The first parts of this section are devoted to several processes of intrinsic stress generation that are directly related to the kinetics of growth and microstructural evolution. They are intended to shed light on the type of results described in the wafer curvature studies of the previous section. The final part of this section has a brief discussion of other important stress-inducing mechanisms beyond the ones related to growth kinetics (lattice mismatch, thermal stress, energetic deposition) that can have a large effect on the stress. They are included because they can have a strong impact on the stress generation and on its modification, both during film growth and after. 4.1 Compressive stress in the pre-coalescence regime Curvature changes are observed to occur in some films when the film still consists of isolated islands, before they have coalesced into a continuous film. The stress is observed to be compressive, indicating that the islands are more dense (i.e., have a 29

30 smaller lattice parameter) than the equilibrium value. One proposed mechanism for this is that the surface stress in small islands induces compression of the island, analogous to the Laplace pressure induced by the surface on small spherical droplets 12, 45, 73. The difference between thin film islands and droplets is that as unconstrained droplets get bigger, the density approaches the bulk density. In thin film islands, however, it is argued that at some size the island gets firmly attached to the substrate and its lattice parameter retains its compressed value as the island gets bigger. Subsequent layers of the film are presumed to grow epitaxially on top of this compressed layer with the same compressed value of the lattice parameter. As the effect of the surface stress decreases at larger island size, the substrate applies the force to compress the island which leads to the measured curvature change. This argument suggests that the stress is not large enough to induce relaxation of the stress by dislocation motion at the base of the island. Although physically appealing, this mechanism has not been firmly established and concerns about it have been raised 13. An alternative has been proposed by Friesen and Thompson 68 that the stress is induced by the modification of surface stress by adatoms or other defects on the surface. This model was motivated by their observations that the pre-coalescence stress was relaxed during growth interrupts and then returned to the same values when the growth was resumed; this is difficult to reconcile with the surface stress model described above. However, molecular dynamics simulations 74 suggest that the stresses associated with surface defects are not high enough to produce the observed stresses. Therefore, it is fair to say that the origin of stress in the pre-coalescence regime is still not well understood. 30

31 4.2 Tensile stress induced by grain boundary formation As the film grows and islands start to coalesce into a uniform film, the incremental stress is observed to become tensile. The correspondence between the stress increase and microstructural evolution suggests that it has its origin in grain boundary formation. Such a mechanism was proposed by Hoffman 75 who argued that neighboring islands could reduce their interfacial energy by forming a grain boundary between them, thus effectively reducing the number of broken bonds on the free surface of the isolated islands. Grain boundary formation can occur even if it induces some elastic strain (i.e., by closing a small gap between the islands) provided the total energy of the system decreases. For the case of square islands with an edge length equal to L, energy minimization predicts that the maximum tensile stress ( T ) generated by this coalescence mechanism is 76 T M s 2 L 1 2 (11) where Δγ is the change in interfacial energy associated with the grain boundary formation: 1 s gb 2 (12) 31

32 γ s is the surface energy, γ gb is the grain boundary energy and the factor of ½ reflects the fact that two free surfaces are converted into one grain boundary. The resulting stress is larger for smaller grain sizes because of the larger number of interfaces created. The physical basis of the Hoffman model is similar to the Griffith s theory for crack propagation in which a crack opens when the energy required for creating two new crack surfaces is less than the amount of mechanical potential energy relieved for an incremental crack advance. In the Hoffman picture, the inverse process occurs and the gap between two adjacent islands closes (i.e., forms a grain boundary) as long as the reduction in the surface energy is more than the increase in the strain energy. Although this formulation is based on the grain boundary forming all at once, similar processes may occur on a layer-by-layer basis during deposition as adjacent islands grow towards each other to form new sections of grain boundary. Among the different stress inducing mechanisms, there appears to be the greatest consensus that grain boundary formation is the origin of the tensile stress during coalescence (although this statement is difficult to justify quantitatively). Other calculations have been performed that consider the effects of different geometries 77 and contact mechanics 76 ; these find a similar power-law dependence on the inverse grain size. Finite element analysis calculations 52, 78, 79 including cohesive zone methods 80, 81 have also been used to calculate the stress due to grain boundary formation. In all these cases, the driving force for the tensile stress to develop is the energy reduction due to grain boundary formation. It is useful to note that the tensile stress predicted by many calculations is much larger than the measured values, especially for metal films 48,

33 This may be due to the effect of other stress-generating mechanisms that counterbalance or relax this stress, such as the compressive mechanisms discussed below. Alternatively, factors such as surface roughness and contact angle 79 may play a role in reducing the stress of coalescence. 4.3 Compressive stress induced by insertion of atoms at the grain boundary While grain boundary formation is somewhat accepted as the driving force for tensile stress, the origin of the compressive stress is much more controversial 71, 82, 83. The development of compressive stress as the film gets thicker suggests that the density of the growing film is higher than it would be in equilibrium. One suggestion has been that the film inherits the compressed lattice parameter established in the pre-coalescence regime 45, 73. This compressive stress becomes evident again after the coalescence stage is completed and the tensile stress has saturated. A major difficulty with this model is that it is hard to explain the presence of the reversible stress changes during growth interrupts. As pointed out by several authors 14, 71, the development of tensile stress from grain growth and recrystallization is not reversible and cannot give rise to the observed reversible in the stress-thickness changes during growth interrupts. In order for the film to become more dense than its equilibrium value (i.e., compressively strained), there must be more atoms per unit area on the substrate than there would be for the unstressed film. Accommodating additional atoms into the film s crystal structure in the form of interstitial defects is energetically unfavorable and they 33

34 would be ejected rapidly in materials with high atomic mobility. However, the grain boundaries offer sites where additional atoms can be added to the film without breaking the symmetry and creating highly stressed defects. Nix and Clemens 77 proposed addition of atoms into the grain boundaries as a way to relax the tensile stress in the film. Spaepen 84 further suggested that trapping of additional atoms at growing ledges on the crystal surface could trap extra atoms into the film, creating compressive stress. Chason et al. 78, 85 have also suggested a mechanism for compressive stress generation due to insertion of atoms into the grain boundary during thin film growth. They propose that this is a consequence of the non-equilibrium nature of the surface during deposition (i.e., supersaturation). This raises the chemical potential of atoms on the surface, which acts as a driving force for them to flow into the grain boundary and induce compressive stress. The grain boundary insertion model has the advantage of naturally explaining the reversible growth interrupts as being due to atoms flowing into and out of the grain boundaries as the driving force (growth flux) is turned on and off. Experimental evidence for this mechanism can be found in the dependence of the stress relaxation during growth interrupts in Sn on the thickness discussed above. The insertion mechanism can explain why the stress relaxes completely when the growth is terminated and then returns to the same value for different film thicknesses. It can also explain why the time constant for relaxation depends on the thickness and the grain size. Leib et al. 69 used this mechanism to explain why the amount of relaxation during interrupts depends on the inverse grain size, with no relaxation for epitaxial films without grain boundaries. Flototto et al. 53 also attributed the compressive stress to diffusion of atoms into the grain boundaries from studying systems with different surface/grain 34

35 boundary diffusivities. Other support comes from molecular dynamics (MD) calculations by Pao et al. 86, 87 that show that Ni atoms are inserted into the grain boundary during simulations of the growth of bicrystals. Buehler et al. 88 also used MD simulations to explain plastic deformation in polycrystalline films via diffusion of atoms into the grain boundary Effects of grain growth As shown by eq. 5, the stress-thickness can change due to processes happening at the surface or in the bulk of the film. An important process that may affect the stress in both these ways is grain growth. Changing the size of the grains at the surface may change the stress in the new layer being deposited (i.e., incremental stress), e.g., according to the Hoffman picture the tensile stress of grain boundary formation decreases as the grain size gets larger. The compressive stress induced by grain boundary insertion may also depend on the grain size 85. Alternatively, if the grain size changes in the bulk of the film, this can lead to tensile stress 45, 46, 73, 89 by shrinking the film s dimensions while it is constrained by the substrate. The grain structure can change in very different ways as the film grows depending on the atomic mobility and the growth rate and this has important consequences for the stress evolution. The structure zone model of Thornton 90 describes how the microstructure evolves as the thickness changes for different deposition temperatures (T s ) relative to the melting point (T m ). Several possible modes of microstructure evolution are 35

36 shown schematically in figure 11 (from ref. 55 ). If the reduced temperature (T s /T m ) is low, the grain size that is established in the early stages of coalescence may not change much during the subsequent growth. In this case (referred to as zone I), the grains are columnar (i.e., with primarily vertical boundaries) and the grain size doesn t change with thickness. This kind of microstructural evolution is typical for materials referred to as lowmobility/type I in section 3.1; in these films the corresponding stress achieves a tensile steady-state value that doesn t change with thickness. A second type of evolution occurs in films with slightly higher mobility (zone T). As these films are grown, the grain size changes at the surface but the grain size in the previously deposited layers (i.e., in the bulk of the film) does not change appreciably. This leads to a microstructure that has a variation in grain size with thickness, with small grains at the bottom of the film and larger grains near the top. The increasing grain size may change the stress in the new layer being added to the surface, but the stress in the bulk is not expected to change significantly. In films with even higher reduced temperature, the grain size can change in the bulk of the film (zone II). This also results in a columnar microstructure but one in which the grain size changes with the thickness, unlike the zone I growth described above where the grain size retains its original value. The growth of a zone II microstructure indicates that the grain size is evolving throughout the thickness of the layer. Since grain boundaries are regions of low atomic density relative to the bulk, the elimination of grain boundaries leads to a decrease in the dimensions of an unconstrained film. Because the film is attached to the substrate, it cannot change its size in the plane and the film 36

37 develops an elastic strain to satisfy this constraint 46, 89, 91. The average stress in the film ( gg ) due to increasing the grain size to L from its initial value (L o ) is given by 62 : gg M s 1 a L o 1 L (13) where Δa is the excess volume associated with a unit area of grain boundary. This relation predicts that an additional tensile stress is developed as the grain size increases for zone II growth. The effect of grain structure evolution on the stress-thickness can be seen by comparison of Ni films grown by electrodeposition 37 and UHV evaporation 62. The electrodeposited Ni films were grown from a sulfamate bath at 40 o C over a 150 nm Au seed layer. The evaporated films were deposited using electron-beam evaporation under ultra-high vacuum conditions at 373 K. Additional details of the growth conditions can be found in the original papers. For the electrodeposited Ni, grain growth is suppressed so that they grow with a zone I or zone T microstructure. The corresponding stressthickness evolution is shown in figure 12a for growth rates indicated on the figure. As the thickness increases, the slope is observed to reach a constant value. Since the grain size isn t changing, we can interpret this slope as the incremental stress at the film surface. In contrast, the evaporated Ni films grow with a zone II microstructure with a grain size that is proportional to the film thickness, e.g., Ni films grown to a thickness of 35 nm had a uniform grain size of approximately 15 nm while films with a thickness of 37

38 70 nm had a uniform grain size of approximately 25 nm. The stress-thickness evolution is shown in figure 12b for several different growth rates. Unlike the electrodeposited Ni, the slope of the stress-thickness does not reach a constant value at larger thickness. However, the slope of the stress-thickness is not equal to the incremental stress for zone II growth. Because the grain size is changing with thickness, the bulk stress in the film is also changing and this must be taken into account when interpreting the stress-thickness. By measuring the microstructural evolution as well as the curvature, Yu and Thompson were able to separate the contribution of grain growth from the incremental stress in the stress-thickness evolution. Their results show how the changing grain size provides a tensile contribution to the stress-thickness evolution. They use the measured grain size evolution to remove the grain growth contribution from the curvature and thus determine the incremental stress at the film surface. This work clearly shows how the effect of grain growth must be considered on the stress-thickness evolution for films that have grain growth in the bulk of the film. 4.5 Other stress-inducing effects outside of growth kinetics The emphasis of this manuscript is on describing intrinsic stress effects related to film growth, but there are other stress generation mechanisms due to processes besides those discussed above. Several stress-related issues (lattice mismatch, thermal stress and energetic deposition processes) are discussed below because they may have a large impact on residual stress or can be used for stress modification. 38

39 4.5.1 Lattice parameter mismatch Stress may arise due to a difference in the lattice parameter between the film material and the layer it is grown on (either substrate or underlayer). For crystalline materials, the film will initially grow pseudomorphically with the same orientation and lattice parameter as the substrate (called epitaxial growth). This leads to elastic mismatch strain, i.e. epitaxial strain and consequently stress in the deposited film (at least in the early stages) given by σ = M f ε = M f (a s a f)/ a f (14) where a s and a f are the equilibrium lattice parameters of the substrate and film, respectively. There is a potential for lattice mismatch stress whenever a film is grown over a material with a different crystal structure or lattice parameter. For fundamental studies of intrinsic stress, amorphous substrates are often used to remove this effect. Epitaxial mismatch can be reduced by the use of different substrates. On the other hand, the stress may be desired as a way to modify the optical and electronic properties of the deposited material (e.g., in strained-layer heteroepitaxy 92 ). Significant study has been devoted to understanding how the stress is relieved by dislocation formation as the film gets thicker. 39

40 4.5.2 Thermal stress Thermal stress is similar to lattice mismatch stress in that it arises from the difference in the coefficients of thermal expansion (CTE) of the film and the substrate. Changes in sample temperature during processing result in different thermal strains in the film and the substrate. Since a thin film is constrained to remain bonded to the much thicker substrate, the resulting thermal stress is proportional to the change in temperature and the difference between the CTE of the substrate (α s ) and film (α f ): Δσ thermal = M f ε = M f (α s α f )ΔT (15) Thermal stress is often not considered to be intrinsic, but in many cases it can be the dominant stress-inducing mechanism in the material. For instance, many materials are grown at high temperatures to get better properties. In other cases (e.g., sputter deposition) the samples often heat up during deposition due to exposure to the plasma. After growth is finished and the sample is cooled, the temperature change can lead to large thermal stress. As an example, the thermal expansion coefficient difference between a Ni film and a silicon substrate is on the order of 1 x 10-5 / o C. Therefore, changing the temperature of the sample by 200 o C can induce strains on the order of 0.2 %. With a biaxial modulus of 280 GPa, this would induce a tensile stress of 560 MPa. Such stress may be much larger than any intrinsic stress generated during growth. It can be large enough to induce cracking or delamination if the film is thick enough. 40

41 Alternatively, thermally-induced stress can be used to measure the mechanical properties of thin films 46. The different coefficient of expansion between the film and the substrate induces a strain that is proportional to the temperature as the sample is heated. The difference between the applied thermal strain and the elastic strain determined from the wafer curvature is due to plastic relaxation processes in the film. This has been used, for example, to measure the plastic deformation in Al-Si films used for interconnects 29 and power-law creep in Sn layers over Cu conductors Energetic deposition effects Sputter deposition is a commonly used technique for thin film growth. Sputtering is different from the other deposition methods discussed above in that both the deposited species and inert gas ions may have kinetic energies significantly above the thermal energy. Although a full discussion of stress in sputter-deposited films is outside the scope of this review, because the stress can vary significantly from non-energetic deposition a brief description of some relevant aspects is given here. Sputter deposition creates films with high uniformity and density that can be deposited rapidly and with good adhesion. It is an efficient method for growing films of refractory materials (e.g., metals and oxides/nitrides 1 ) that are difficult to deposit by other means. The energetic component of the sputtered species adds an additional factor that can be used to modify the film stress significantly. For instance, sputtered W films 94 can have stress that changes from highly tensile (appr. 2.1 GPa) to highly compressive (- 2.3 GPa) by changing the energy of the incoming species. Indeed one of the benefits of 41

42 sputter deposition over evaporation is to counteract the very large tensile stresses that arise during the evaporation of refractory materials. The stress modification has been attributed to a process referred to as atomic peening 97 in which momentum transfer from the energetic particle drives the atoms in the film into more dense configurations 98 or creates stress-inducing defects. This may occur through processes at the grain boundaries or interaction with the bulk of the film 44, Models have been proposed 102, that use the momentum transfer and trapping of defects to explain the contribution of energetic particles to the stress. Surface processes may also enhance adatom mobility 109 and hence contribute to diffusional incorporation of atoms into the grain boundaries. In addition to modifying the stress during deposition, energetic beams can also be used to modify the stress in films after deposition 110, 111. An example of real-time stress measurements during sputtering of Mo films made by Fillon et al. 96 is shown in figure 13. These studies looked at the effect of both pressure and growth rate on the stress (as indicated in the figure); the grain size was reported to be the same for the different conditions. In all cases, the slope of the stressthickness changes with thickness, going through a maximum in incremental tensile stress at less than 10 nm and then becoming less tensile or compressive in the steady-state regime. This is similar to the behavior seen in non-energetic growth for high-mobility type II materials. Without the energetic deposition, however, Mo would be expected to behave like a type I material and have a large tensile stress. Lowering the pressure has the effect of increasing the energy of the impinging species by reducing the amount of scattering in the gas phase. For lower pressure at the same growth rate, the slope in the steady-state regime is more negative (i.e., more 42

43 compressive stress in the film). This shows how the energetic component of the deposition can greatly modify the stress. For the growth rate of 006 nm/s, the steadystate stress could be changed from 1 GPa at high pressure to less than -1 GPa at low pressure. The growth rate dependence of the stress is also different than what is seen for non-energetic deposition. For higher growth rates at the same pressures, the slopes in figure 13 become more negative (i.e., less tensile/more compressive). This is the opposite of what is usually seen during non-energetic growth where higher growth rates typically make films more tensile. It has been proposed that the stress-inducing effects from energetic bombardment act independently from those from growth 112 so that it may be possible to consider the resulting stress as the superposition of intrinsic growth stress and contributions from the energetic species. 5. Kinetic model for intrinsic stress during film growth The preceding sections describe observations based on curvature measurements and the mechanisms that have been suggested to play a role in the stress evolution. Although these mechanisms can help explain some trends, they do not allow us to understand their relative importance under different conditions. Therefore, they aren t sufficient to predict how the stress will evolve in different systems. To address this, these mechanisms have been integrated into a comprehensive model for stress evolution that can address different observations. By comprehensive, we 43

44 do not mean that the model has been shown to explain all the observed phenomena. Instead, we mean that the model includes many of the relevant parameters (growth rate, diffusivity, grain size, surface morphology and material properties) that enable it to make predictions about what will happen under different conditions. Comparison of the model with experiments is still needed to determine if the mechanisms in the model explain how the stress develops in different cases. The model has been described previously 49, 85 so only a brief description of the physical basis for the model and the equations that describe the stress evolution is given here. The model is used to analyze several sets of measurements to show how its predictions compare with the measured dependence on thickness, grain size and growth rate. 5.1 Physical basis for model and evolution equations Based on insight gained from experiments, the model focuses on the stress that develops at the top of the grain boundary as it forms between two adjacent islands during film growth. A schematic of this geometry is shown in figure 14. The large image shows the morphology of a polycrystalline film while the encircled region shows a magnified view of the top of a single grain boundary (referred to as the triple junction). The grain boundary separates the layers in adjacent grains that have grown together to form the bulk film. The triple junction marks the transition between the bulk of the film and the surface; the index i indicates the atomic layer in the film that is being considered. As additional atoms are deposited, the edges of terraces grow across the surface until they 44

45 reach the end of the grain. Assuming that the film is dense (i.e., not highly porous), the edges ultimately get close enough to the adjacent grain to form new segments of grain boundary, at which point the triple junction moves up. The model calculates the stress that develops in the triple junction as the film grows. Several assumptions are made to simplify the model and allow analytical solutions to be obtained. We assume that the stress in each layer is independent of the stress in the layers above and below it (which Guduru et al. 113 refer to as the linear spring model). We also assume that the stress is only generated in the triple junction and not in lower layers of the film, equivalent to assuming that there is minimal diffusivity in the grain boundary. Higher grain boundary diffusion could be considered 113 but the equations would have to be solved numerically. A variation of the model that considers systems with very high atomic mobility (i.e., uniform stress in the grain boundary) is described in ref. 22. First we consider the stress that is generated due to the processes of grain boundary formation. Based on the Hoffman picture, we assume that the atoms at the edge of the layers in adjacent islands are attracted to each other as they grow closer together. When the gap between them becomes small enough, they form a bond (new segment of grain boundary) which induces a tensile stress (σ T ) in the layer. At this point, the triple junction moves up by one layer (from layer i to i+1). After the new segment of grain boundary forms, atoms can be inserted into the triple junction to modify the stress, as described above. This can only happen for an interval of time (Δt i ) until the triple junction moves up to the next layer, i.e., the current top layer is covered over by the growth of the subsequent layer. This time interval is 45

46 dh gb equal to a/ dt dh gb where a is the nominal height of the layer and dt is the rate at which the grain boundary height increases. The addition of N i atoms into the triple junction modifies the initial tensile stress in that layer: i T M f N a i L (16). Note that we assume that atoms can only be inserted into the triple junction in this layer while it is at the surface. The driving force for insertion of atoms into the triple junction is the difference in chemical potential between atoms on the surface ( s ) and atoms in the triple junction ( tj ). The chemical potential on the surface during film growth is assumed to be higher than it is at equilibrium by an amount defined as s (due to the non-equilibrium conditions or supersaturation). The chemical potential in the triple junction is also modified by the product of the stress in the layer (σ i ) and the nominal atomic volume ( ); the sign is chosen so that tensile stress lowers the chemical potential of the triple junction. The difference between these effects is the driving force (Δ): S tj S i (17) 46

47 The chemical potential difference drives a flux of atoms from the surface into the triple junction of this layer while it is at the surface (during the time interval Δt i ). The rate at which atoms flow in (dn i /dt) is equal to dni dt D 4Cs a kt 2 (18) where C s is the concentration of mobile species on the surface and D is an effective diffusivity for transitions from the surface into the triple junction. Putting these mechanisms together results in a differential equation for the stress 85. The stress in the i th layer is determined by evaluating the solution to this equation at Δt i, when the triple junction moves up to the next layer: i C ( T D C ) exp L dh gb dt (19a) where the parameter is defined as CS M 4 kt f (19b) 47

48 and c is the compressive stress due to the supersaturation of the surface layer (equal to - s /). The corresponding stress-thickness can be calculated from the stress in each layer by integrating it over the height of the grain boundary interface. The solution in eq. 19 depends on the rate at which new grain boundary is being dh gb added to the film ( dt ). This is because the balance between the tensile and compressive mechanisms is mediated by the rate at which new grain boundary is being formed relative dh gb to the rate at which atoms are inserted into it. For large values of dt, the stress becomes dh gb more tensile. Conversely, for smaller values of dt the stress becomes less tensile/more compressive. The growth rate is scaled by the values of D and L so that changing the diffusivity or the grain size can also modify the incremental stress. 5.2 Application to evolution of stress with thickness The dependence of the stress on dh gb dt in the model can be used to understand the transition from tensile to compressive incremental stress with thickness. This is because dh gb dt is not constant with thickness, even if the film is grown at a constant rate. This is illustrated in figure 15 for islands that are semi-circular in cross-section. The corresponding grain boundary velocity (normalized by the average growth rate) is shown vs. the average film thickness (normalized by the island spacing) on the right side of the figure. The velocity is highly non-uniform and changes for different stages in the film growth. For small island radii (ρ less than half of the island separation, L), the islands 48

49 dh gb are separate and there is no grain boundary. Therefore dt is zero and the stress predicted by the model is zero. As the islands begin to coalesce (ρ>l/2), the high contact angle between the islands leads to rapid growth of the grain boundary and a high grain dh gb boundary velocity. As the film continues to grow, dt asymptotically approaches the average growth rate of the film (R). The changing rate of grain boundary growth with thickness explains many features of the transition of stress with thickness. Before coalescence, the model predicts that there is no stress (this model does not include the effect of the surface stress in the pre-coalescence regime). As the coalescence begins, the grain boundary grows rapidly which creates a tensile stress, as observed in experiments. At larger thicknesses, the rate slows down which leads to less tensile/more compressive stress, also consistent with observations (assuming that the grain size isn t changing). The value of the incremental stress after coalescence can be tensile or compressive, depending on the steady-state growth rate. This picture is sufficient to explain qualitatively how the film morphology can induce a transition from tensile to compressive incremental stress with thickness. By using the grain boundary velocity for islands with circular cross-sections (the form shown in figure 15), it has been shown to reproduce many of the features observed in the evolution of stress-thickness in Ag shown in figure However, a more quantitative analysis is difficult because the actual shape and spacing of the islands is not known in those experiments. To address this, recent experiments were performed in which Ni films were grown by electrodeposition on patterned substrates 114, 115. Because growth by 49

50 electrodeposition occurs normal to the surface, the islands in these films grow in the form of hemispherical caps. Scanning electron microscope (SEM) images of the surface morphology (figure 16a) show this geometry for several different layer thicknesses. Because the morphology of the islands is known, it is possible to calculate the velocity of the grain boundary at different thickness for use in evaluating the stress in the growth model. Measurements of the stress-thickness vs. time for several growth rates is shown in figure 16b. The corresponding velocity of the grain boundary, calculated from the island geometry, is shown directly below the stress-thickness. For all the rates, the stressthickness is small before the islands grow large enough to intersect (this occurs at different times for the different growth rates). The stress-thickness rises rapidly when the islands start to coalesce, indicating that the formation of a boundary between the islands is inducing a large stress. The incremental stress is tensile, corresponding to the rapid velocity of the grain boundary. For longer times, the grain boundary velocity decreases to a steady-state value and the stress becomes less tensile or compressive, depending on the final value. For quantitative comparison with the experiments, the incremental stress at the top of the grain boundary can be calculated from eq. 19 as the island s size increases, using the velocity shown in the lower part of figure 16b. The corresponding grain boundary forces can be obtained from eqs. 9a-b for comparison with the measured stressthickness (note that these data were not corrected for the effects of the island spacing discussed in section 2.2). The predictions of the model for the stress evolution are shown as the solid lines in the figure. The model parameters were obtained from a non- 50

51 linear least squares fitting of the model to the data. Essentially the same fitting parameters were used for all the calculations (σ T = 64.9 MPa, σ C = MPa while βd was allowed to vary slightly between data sets with an average value of = / nm 2 /s). The pattern spacing of 22 μm is used for the value of L (i.e., the stress is assumed to be generated at the boundaries between the patterned islands). The only difference between each of the calculated curves is the growth rate that is input to the model. The different stress-thickness evolution for each growth rate appears to be explained satisfactorily by the different evolution of the grain boundary velocity for each growth rate. 5.3 Application to stress in the post-coalescence regime After the film has fully coalesced, the growth rate of the grain boundary approaches the average growth rate of the film (R). In this regime, the model predicts that the incremental stress is equal to inc C ( T D C ) exp LR (20) The calculated stress depends exponentially on the parameter D/RL (there is also an additional dependence of the parameter σ T on 1/L ½ ). For small values of D/RL (occurring 51

52 for low diffusivity, large grain size or high growth rates), this predicts that the incremental stress becomes more tensile. Conversely, for large D/RL (high diffusivity, small grains or low growth rates) the stress becomes less tensile or compressive. This is consistent with many of the trends observed with temperature, growth rate and material type described above. However, the coupling between the parameters also explains why it is so difficult to interpret many studies. For instance, changing the temperature also changes the nucleation rate and the grain size evolution, so that L may change as a function of temperature and as the thickness evolves. This shows why it is important to fully characterize the sample structure and consider all the parameters in order to interpret the stress evolution Effect of grain size evolution In this section, we consider the effect of the grain size on the stress-thickness predicted by the model for different modes of microstructural evolution. If the grain size does not change with thickness, (zone I growth), the incremental stress predicted by eq. 20 reaches a constant value in the post-coalescence regime. However, this is not the case if the grain size changes with thickness. For zone T growth (where the grain size changes only at the surface), the dependence of eq. 20 on L predicts how the incremental stress changes with the grain size. For zone II growth (where the grain size also changes throughout the film), then an additional contribution from grain growth should be added to the stress-thickness based on eq

53 Illustrative calculations of the stress-thickness evolution for these different modes dh gb are shown in figure 17 assuming that the film is fully coalesced (i.e., dt is equal to the average growth rate R). The calculations start from a thickness of 15 nm using parameters of σ T = 5 GPa, σ C = -1.4 GPa, βd = 3.2 nm 2 /s and R = 0.1 nm/s, chosen to give results similar to the measurements of Yu and Thompson 62. For the calculation of zone I growth, the grain size was kept at a constant value (L=10 nm). The resulting incremental stress (from eq. 20) doesn t change with thickness and the slope of the stressthickness has a constant value. The zone T calculations are done with the same parameters as zone I, but with an increasing grain size that is proportional to the thickness (i.e., L = L o + αh f where α is a constant), consistent with the measurements. The calculated stress-thickness has a slope that becomes more positive with the increasing thickness/grain size, corresponding to less compressive incremental stress. For the zone II growth calculation, a tensile contribution from the grain growth was added to the stress-thickness. For each layer of the film at height z, the stress due to grain growth is equal to ( z) M gg s 1 1 a Lo ( z) L( h f ) (21) where L o (z) is the grain size when the layer was deposited and L(h f ) is the grain size when the film has thickness h f. Integrating this over the thickness for different values of h f gives the contribution of grain growth to the stress-thickness. This contribution is in 53

54 addition to the thickness-dependent incremental stress calculated in zone T. For the calculations shown, the value for M s Δa was taken to be 15 N/m. The model for incremental stress, with the addition of grain growth stress, appears to be able to produce behavior similar to what is shown in figure 12 for the electrodeposited Ni (zone I) and evaporated Ni (zone II). The very different behavior for each type of microstructural evolution (zone I,T and II) shows that grain growth can have a large effect on the measured stress evolution. It should be noted that the results shown here for grain growth are only qualitative; further analysis is ongoing to account quantitatively for the measured grain size dependence. The grain size dependence predicted by the model is also consistent with measurements of incremental stress in Fe at different grain sizes by Koch et al. 82, Effect of growth rate For another comparison of the model with experimental data, we consider the dependence of the incremental stress in the steady-state regime on the growth rate. The AlN, B and electrodeposited Ni data shown in figure 9 have been fit to the form in eq. 20 and the results are shown as the solid lines in the figure. The predicted exponential dependence on the inverse growth rate is seen to provide good agreement with the data. The calculated stress is more tensile/less compressive for higher growth rates and becomes less tensile/more compressive at lower growth rates. 54

55 The AlN and electrodeposited Ni samples had nearly constant grain sizes so that grain growth did not affect the incremental stress measurement or calculation for these samples. The data for evaporated Ni was not analyzed with the model because the incremental stress is affected by the extensive grain growth as described in the previous section. Interestingly, the amorphous B sample can also be fit with this model. This is consistent with previous work on other amorphous films 36, 116 which showed that they follow similar stress evolution as polycrystalline samples. Presumably accommodation of surface atoms at the boundaries between different islands plays a role similar role to the grain boundary in polycrystalline films. These fitting results were only taken from a small collection of data so the parameters are not reliable and therefore we do not show them here. Further study is needed over a wider range of growth rates to obtain parameters that can be compared across different materials systems and operating conditions. Therefore, these fits should only be considered as examples of how the model predicts a dependence on growth rate that is similar to the experimental results. 5.4 Relationship to growth interrupts The measurements of stress relaxation when the growth is interrupted (described in section 3.3) also provide insight into the mechanisms used in the model. Since the model is based on the incorporation of atoms into the triple junction during growth, the equations described above do not directly predict relaxation behavior. However, the 55

56 underlying physical mechanism of diffusion in and out of the grain boundary is consistent with stress relaxation when the growth flux is interrupted. During interrupts in growth, the non-equilibrium conditions established by the deposition flux are no longer present and the chemical potential on the surface is expected to decrease to its equilibrium value. This creates a driving force for atoms to diffuse out of the stressed grain boundary region and relieve the compressive stress in the film. The rate of out-diffusion of atoms from the grain boundary can be modeled analytically for the case of very high grain boundary mobility 49 (i.e., when the gradients are small and the stress changes uniformly throughout the thickness of the film). This has been used to analyze the relaxation kinetics during interrupts in Sn electrodeposition. The equations predict that the rate of relaxation depends on the film thickness, in agreement with what is seen in the experiments. Because the mobility is high, the stress relaxes completely when the growth is turned off. This approach also explains the development of tensile stress during etching of the Sn film due to a decrease of the surface chemical potential relative to the equilibrium value (the opposite of the excess chemical potential that was used to account for compressive stress during deposition). For the opposite case of very low mobility systems, the stress does not relax when the growth is stopped (as in figure 7a), presumably indicating that the mobility is too low for atoms to leave the grain boundary. Many metal systems (e.g., figure 7b) lie between these extremes of very high and very low mobility. The magnitude of the relaxation indicates that not all the stress is relaxed, but it is too large to be explained by diffusion only out of the surface layer. One possible explanation is that atoms are diffusing along the grain boundary and then leaving 56

57 at the surface. This process could be modeled by numerically solving for the stress evolution in the grain boundary, using the appropriate boundary conditions when the flux is terminated 113. However this solution would assume that the shape of the grain boundary doesn t change when growth is terminated. As pointed out by Yu and Thompson 70 the surface structure around the grain boundary can change during the relaxation time. Therefore, although the grain boundary insertion mechanism in principle allow reversible stress changes during growth and interrupts, the simultaneous effect of morphology changes around the grain boundary must also be considered. 6. Discussion and summary The evolution of stress seen in individual studies can be complex, but many features are becoming clear. As described above, low mobility/high growth rate tends to produce films with tensile stress and vice versa. Stress changes during growth interrupts show that effects at the surface (reversible) and in the bulk (irreversible) must be considered. Correlation of the stress with the corresponding film microstructure shows that grain size evolution, which also depends on the mobility, can have a significant effect on the resulting stress in the film. From these studies, different mechanisms of stress generation have been proposed. The formation of grain boundaries between adjacent islands is proposed to induce tensile stress in the layer. Although the mechanism for compressive stress is more controversial, insertion of atoms into the grain boundary can explain much of the phenomenology 57

58 including the stress relaxation when the growth is interrupted. Grain growth in the bulk of the film leads to additional tensile stress in the layer, which must also be considered. These different mechanisms have been integrated into a model that focuses on the stress that gets generated in new segments of grain boundary that form between adjacent islands. The resulting equations predict the dependence of the incremental stress on the rate at which the grain boundary moves up (which is proportional to the growth rate) as well as the diffusivity, the grain size and other materials parameters. The model compares favorably with the measured dependence on the thickness in patterned films, the dependence on the grain size for different types of growth and the dependence on the growth rate in several systems. An important benefit of the model is that it provides a quantitative framework with which to analyze stress evolution under different conditions. It shows how different parameters may interact so that complex behavior may possibly be understood in terms of the underlying physical mechanisms. For instance, the model suggests that measuring the steady-state incremental stress vs. growth rate is a useful way to understand different aspects of the stress (with corrections for grain growth, if necessary). The stress at high growth rate highlights the tensile grain-boundary induced stress mechanism while the stress at low growth rate shows the asymptotic compressive stress. The kinetic parameters that determine the balance between these mechanisms can be obtained from the transition between tensile and compressive stress with the growth rate. Although the results discussed above are encouraging, a systematic comparison with experimental studies is needed to determine the model s validity and its range of applicability. Deviations from the behavior predicted by the model will be useful for 58

59 determining when the model assumptions break down (e.g., the independence of the stress in the layers or the dependence of the tensile stress on the grain size). If the model is found to work over multiple systems, obtaining reliable parameters will enable them to be compared with more fundamental studies. For instance, measurements of the temperature-dependence of the stress will be able to determine the activation energy for the effective diffusivity. This will be useful for comparison with the surface and grain boundary diffusivity to determine what controls the insertion of atoms into the grain boundary. A deeper understanding would enable these parameters to be calculated and allow stress to be predicted over a wide range of conditions. It is also possible to imagine extending the approach described here to other types of film growth. It has been suggested that the effects of ion bombardment are additive to those of non-energetic film growth 112 and we are working on extending the model to include energetic effects. Alloy films are another area that is technologically very important but where the fundamental understanding of stress is limited The model will be extended to alloys in the future by considering multiple diffusing species with different sizes. 7. Acknowledgements The work of EC was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award # DE-SC PRG acknowledges the DOE EPSCoR Implementation grant (grant # DE-SC ), which supports his work 59

60 on thin film mechanics. The authors acknowledge helpful contributions from Alison Engwall, Hang Yu, Carl Thompson, Ben Freund, Allan Bower, Joe Greene, Bill Nix, Jae Wook Shin, Sean Hearne, Jerry Floro, Brian Sheldon, Greg Abadias, Kostas Sarakinos, and Gery Stafford. 60

61 Table 1. Material method Range of rates (nm/s) temperature Grain size (nm) Ref. AlN ECR-MBE C B e-beam evaporation C amorphous 65 Ni e-beam evaporation C Ni electrodeposition RT Table and figure captions Table 1. Deposition parameters associated with the data for the steady-state stress vs. growth rate shown in figure 9. The deposition method labeled ECR-MBE stands for electron-cyclotron resonance assisted molecular beam epitaxy. Figure 1. (a) Schematic of a uniform thin film on a substrate showing stress distribution through the thickness of the film. (b) Schematic illustration of a polycrystalline film 61

62 showing the grain boundary stress xx (z). (c) An approximate free-body diagram of an individual grain is shown for clarity. Figure 2. Evolution of microstructure and stress-thickness during growth of Ag on SiO 2. a) c) TEM micrographs of film structure at thicknesses of 10, 22 and 40 nm, respectively. d) Stress-thickness measured during deposition. The slope of the dashed lines on the figure represent the average ( ) and incremental ((h f )) stress. Figure 3. The effect of grain size on the correction factor corr to the Stoney equation for stress at the grain boundaries in a polycrystalline film, for the case of no elastic mismatch between the film and the substrate. Data is from Chapter 3 of Freund and Suresh 13. Figure 4. Schematic showing the structure of a single island in a periodic, non-uniform film. The edge forces (shown as f x,gb and f y,gb ) can be determined by integrating the normal stress over the boundary between islands. Figure 5. Schematic of multi-beam optical stress system (MOSS) for measuring curvature evolution. The CCD camera measures the change in spacing (δd) between initially parallel laser beams. Figure 6. Comparison of the centroid of a single beam and the spacing between a pair of beams measured by a MOSS system over a 100 s period. The noise in the spacing 62

63 between beams is much less than the noise in the positon of the centroid, illustrating the stability of the curvature measurement with respect to sample vibration. Figure 7. Examples of stress-thickness evolution during vapor-phase growth in a) low mobility (type I) and b) high mobility (type II) materials. The materials are indicated in the figure. The data in (a) was reproduced from ref. 45 with permission. The data in (b) was reproduced from ref 51 with permission. Figure 8. Evolution of stress-thickness during evaporation of a) Ag and b) Fe at different temperatures indicated in the figure. The data is reproduced from refs. 85 and 57 with permission. Figure 9. Comparison of incremental steady-state stress as a function of growth rate for AlN, B and Ni (evaporated and electrodeposited). The growth parameters for the different materials are given in table 1. Figure 10. Schematic of stress evolution during growth interruptions of a) 5 min and b) 24 hrs. Figure reproduced from ref 70 with permission. Figure 11. Schematic showing evolution of grain size with thickness at different reduced temperatures (T s /T m ) where T s is the deposition temperature and T m is the melting point. The different behavior is associated with the structure zone model of Thornton 90. Figure reproduced from ref. 55 with permission. 63

64 Figure 12. Comparison of stress-thickness evolution in a) electrodeposited Ni and b) evaporated Ni. The data in a) was reproduced from ref. 37 with permission. The data in b) was reproduced from ref. 62 with permission. Figure 13. Measurements of stress-thickness vs thickness during sputter deposition of Mo films at different at pressures of a) 0.24 Pa and b) 0.11 Pa. Growth rates indicated in figure (figure adapted from ref. 96 ). Figure 14. Schematic of geometry used for modeling stress at triple junction, as described in text. The triple junction is the point where layers in adjacent grains grow together to form new segments of grain boundary. Figure 15. a) Schematic showing morphology of film for islands with circular crosssection. Images represent different values of the island radius () relative to the grain size (L). b) Corresponding rate (dh gb /dt) at which grain boundary height increases vs. film thickness. The rate and thickness have been normalized by the radial growth rate (R) and island spacing (L), respectively. Reproduced from 49 with permission. Figure 16. a) SEM micrographs of electrodeposited Ni film grown on patterned substrate in the form of hemispherical islands. Island radii are 9.9, 11.9 and 14.5 m from top of figure to bottom. b) Stress-thickness vs. time measured during growth of patterned films at different growth rates indicated in the figure. The lower part of the figure shows the 64

65 corresponding velocity of the grain boundary. The solid lines superimposed over the data are fits to the model described in text. The figures are adapted from the data in ref Figure 17. Calculations from model showing stress-thickness evolution for films that grow in zones I, T and II, as indicated on the figure. 65

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