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Grain growth stagnation in thin films due to shear-coupled grain boundary migration
Scripta Materialia 180 (2020) 83–87
Contents lists available at ScienceDirect
Scripta Materialia journal homepage: www.elsevier.com/locate/scriptamat
Grain growth stagnation in thin films due to shear-coupled grain boundary migration Eugen Rabkin a,∗, David J. Srolovitz b,∗∗ a b
Department of Materials Science and Engineering, Technion – Israel Institute of Technology, 3200003 Haifa, Israel Department of Materials Science and Engineering, City University of Hong Kong, Kowloon, Hong Kong SAR
a r t i c l e
i n f o
Article history: Received 27 November 2019 Revised 10 January 2020 Accepted 17 January 2020 Available online 5 February 2020 Keywords: Thin films Grain growth Grain boundary migration Disclinations
a b s t r a c t Normal grain growth in thin films attached to a substrate has been considered. It has been shown that shear-coupled grain boundary migration results in the build-up of elastic stresses in the film. A semi-quantitative model of grain growth combining the elements of disclinations theory with the BurkeTurnbull model of normal grain growth has been proposed. It has been shown that shear-coupled grain boundary migration may lead to stagnation of normal grain growth, whereas for high coupling factor values, the grains in the film do not grow at all. Finally, the mechanisms of stress relaxation enabling some grain growth are discussed. © 2020 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
The kinetics of normal grain growth in polycrystalline solids is commonly described via a parabolic law [1]. However, it is wellknown that grain growth in thin polycrystalline films is rarely parabolic and, in most cases, stagnates after a short growth period [2]. This normal grain growth stagnation often results in abnormal grain growth [1,2]. Several reasons for normal capillary-driven grain growth stagnation in thin films of pure materials have been proposed; these may be summarized as follows. (i) Thermal grain boundary (GB) groove drag [3]: Mullins showed that the dragging of surface grooves at GB/surface intersections leads to a force in the direction opposed to GB migration. Grain growth will stagnate when this drag force exceeds the grain-size-dependent capillary driving force; the critical grain size is 2–3 times the film thickness (assuming isotropy). However, grain growth stagnation is still observed when thin films are encapsulated by much more refractory overlayers which impedes diffusional groove formation (e.g., Al thin films covered by its native oxide) [4]. (ii) Stress generation associated with the GB volume defect in constrained (thin film) geometries [5]: since GBs are commonly
∗ Corresponding authors: Prof. Eugen Rabkin, Department of Materials Science and Engineering, Technion – Israel Institute of Technology, 320 0 0 03 Haifa, Israel. ∗∗ Co-Corresponding author: Prof. David J. Srolovitz, Department of Materials Science and Engineering, City University of Hong Kong, Kowloon, Hong Kong SAR. E-mail addresses: [email protected] (E. Rabkin), [email protected] (D.J. Srolovitz).
https://doi.org/10.1016/j.scriptamat.2020.01.019 1359-6462/© 2020 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
of lower density than their delimiting grains, a decrease in total GB area during grain growth leads to the development of tensile stresses in thin films (where attachment to the substrate restricts contraction of the polycrystal). When the resulting increase in elastic energy exceeds the decrease in total GB energy associated with normal grain growth, grain growth will stagnate [5]. However, analysis of this mechanism shows that it should only be effective for very small initial grain sizes. However, experiments show that grain growth stagnation occurs when the initial grain sizes are much larger than predicted on this basis [2]. (iii) Increase of vacancy concentration in the film during grain growth associated with the GB volume defect [6,7]: if the density difference between the GB and the grains is accommodated by vacancy generation, the vacancy free energy contribution to the overall thermodynamics will decrease the driving force for grain growth [6,7]. Analysis of this proposal suggests that this will initially retard grain growth but normal parabolic grain growth should occur at late time. (iv) Presence of low mobility GBs [8]: the presence of a broad distribution of GBs mobilities (with some near zero mobility GBs) ultimately limits the mean grain size. “Smooth” GBs at temperatures below the roughening transition temperature were identified as exhibiting low mobilities, and atomistic computer simulations indicated that even a small fraction of such low-mobility GBs is sufficient to cause normal grain growth stagnation [8]. Though this model represents a very
84
E. Rabkin and D.J. Srolovitz / Scripta Materialia 180 (2020) 83–87
general approach to grain growth stagnation, applicable to both two- and three-dimensional polycrystals, experimental evidence for such an abrupt change in GB mobility at the roughening transition is still lacking. Several additional explanations of normal grain growth stagnation in thin films (such as anisotropy of the GB energy, solute drag, and triple junctions drag) were considered by Barmak et al. [9]. It was shown that none of the above proposals explains the difference between grain size distributions observed in experiments and deduced from the results of numerical simulations. In this respect, it should be noted that drag on moving GBs arising from mobile solute atoms, second phase particles, or triple junctions changes the growth law, but does not cause complete grain growth stagnation [10,11]. This short overview suggests that existing models for grain growth stagnation in thin films are of limited applicability/cannot be considered universal. Here, we propose an alternative approach to explain grain growth stagnation in thin films on substrates. In particular, we demonstrate how shear-coupling [12–16] necessarily leads to stagnation. It is now well-established, based on theory [12], atomistic computer simulations [12–14], and experiments [15,16], that a shear stress applied parallel to a GB plane can drive GB migration. Conversely, GB migration produces a shear across the GB [12–16]. This shear-coupled GB migration is commonly characterized by a coupling factor β , which is the ratio of the rate of shear translation across a GB to the GB migration rate. β depends on GB crystallography and how GB motion is driven [14]. The GB crystallography effects are most robustly described in terms of line defects along GBs known as disconnections [14,17]. For low angle GBs, β is a function of the misorientation angle [12–16]. It was recently shown that shear coupling plays a fundamental role during grain growth in bulk polycrystalline materials: the incompatibility of grain displacements at triple junctions during the GB migration leads to a build-up of elastic stresses that may result in grain growth retardation or stagnation [13]. These stress concentrations can be relaxed by the formation of additional GB disconnection modes (or emission of dislocations into the grains or by twinning). These additional disconnection modes are associated with the opposite sign coupling factor. In what follows, we consider grain growth in a polycrystalline thin film that adheres to an inert substrate (i.e., continuity of tractions and displacements) and include the effect of shear-coupled GB migration. Consider a single GB migrating in a film on a rigid substrate, as illustrated in Fig. 1a. If the film were not attached to the substrate, the displacement of the GB by a distance d may result in the vertical translation of the grain on the right hand side, as depicted in Fig. 1b (the vertical translation distance is β d). In this picture, we have assumed that the shear is in the direction perpendicular to the nominal film/substrate interface. Of course, the shear coupling may also lead to shear in the plane of the film (parallel to the interface). Hence, the value of β should include the projection of shear in the direction perpendicular to the interface. Now, since the film is attached to the substrate, we must elastically deform the film in Fig. 1b, back into contact with the substrate (to satisfy compatibility and continuity of tractions and displacements). We describe the resultant deformation in terms of a wedge disclination dipole at the film/substrate interface, as illustrated in Fig. 1c. The spacing between the disclinations is the GB migration distance d, and the magnitude of the strength of the wedge disclinations is characterized by the angle ω such that tan(ω) = β . It should be noted that disclination dipole should be viewed as a convenient mean of describing the elastic field associated with a wedge of material inserted into/removed from the film/substrate interface. This does not imply that the disclinations
Fig. 1. The GB in thin film migrating to the right (a); the imaginary shape of the film assuming no bonding to the substrate after the GB migrated by a distance d. The gap between the film and the substrate on the right has formed due to the shear coupling (b). Elastic deformation of the film ensuring its attachment to the substrate, which can be represented by a dipole of two disclinations of opposite signs (c).
are necessarily physically observable linear defects. The idea of using such a description is akin to describing a Mode I crack as an array of edge dislocations or a Mode III crack as an array of screw dislocations [18], or a grain boundary (GB) diffusion wedge as an array of infinitesimal edge dislocations [19–20]. The elastic energy (per unit length of the thin film bicrystal in the direction normal to the plane of the illustrations in Fig. 1) of this disclination dipole is Ed [21],
Ed =
Dω 2 d 2 ln 1 + 4
≈
2t 2
−2Ax2 ln x2
1 A 1− 2 2x
d
= Ax2 ln 1 +
x1
1 2 x
(1)
x 1,
where the film and the substrate have the same elastic constants, D = μ/[2π (1 − ν )], x = d/2t, A = Dω2 t2 and μ and ν are the shear modulus and Poisson’s ratio, respectively. This solution is exact within linear elasticity outside the disclination core regions for an elastically isotropic solid. In the x1 limit, this result reduces to the case a straight edge dislocation of Burgers vector β d, perpendicular to a flat, free surface. The drag force Fdrag on the GB, associated with this elastic energy, is in the direction opposite to the GB migration and may be
E. Rabkin and D.J. Srolovitz / Scripta Materialia 180 (2020) 83–87
85
expressed as:
Fdrag =
≈
x
1 ∂ Ed ∂ Ed ∂ x 1 = =A ln 1 + 2 − ∂d ∂x ∂d t x 1 + x2 ⎧ x ⎨A x1 [2 ln(x ) + 1] t ⎩A 1 2t x3
(2)
x 1.
In a columnar microstructure, the GB runs from interface to surface (nominally perpendicular to both) such that the drag pressure acting on the GB, Pdrag , is:
Fdrag x = A 2 ln 1 + t t ⎧ ⎨A x [2ln(x ) + 1] t2 ≈ A 1 ⎩ 2t 2 x3
Pdrag =
1 x2
−
1 1 + x2
(3)
x1 x 1.
Expressing normal grain growth in accordance with the classical Burke-Turnbull model [1], the capillary driving force for growth Pgrowth can be expressed as
Pgrowth = α
γb R
,
(4)
where γ b and R are the GB energy and the average grain radius (inverse mean curvature), respectively. The parameter α is a geometry-dependent constant (for simplicity, we set it to α ≈1 here). For a film with initial grain radius R0 the normal grain growth stagnation condition Pdrag = Pgrowth can be obtained by combining Eqs. (3) and (4), and writing R(t → ∞)=R0 + d∞ (where R0 is the initial grain size):
γb R0 + d∞
=
γb R0 + 2t x∞
= Dω x∞ 2
1 ln 1 + 2 x∞
1 − 1 + x2∞
(5)
where x∞ = d∞ /2t. The transcendental Eq. (5) represents the main result of the present work. It allows calculation of the maximum extent of GB migration d as a function of the initial grain size, elastic (μ and ν ), GB thermodynamic (γ b ), and GB migration (β ) properties. We numerically estimate the GB migration distance for thin films of pure Al; this material is chosen because its elastic constants are nearly isotropic and because grain growth in Al thin films on substrates has been widely reported (e.g., see [4,9,22]). In these calculations, we set μ = 26.2 GPa, ν = 0.345 [23], γ b = 0.6 J/m2 [24], and β = 1 (i.e., ω = π /4). Also, though room temperature physical vapor deposition of thin Al films often result in columnar microstructure with the grains larger than several tens of nanometers in diameter [4,9,22], varying substrate temperature during deposition, impurity content, and the nature of the plasma-forming gas allow broad variations of the film microstructure and mean grain size [25]. For R0 = 5 nm and t = 200 nm, the elastic stresses associated with shear-coupled GB motion lead to grain growth stagnation at d = 0.93 nm. For larger values of R0 , grain growth stagnates at even smaller values of d. This suggests that grain growth is only possible when tan(ω) = β is small. Re-writing the left hand side of Eq. (5) in the form (γ b /2t)/[(R0 /2t)+x] allows direct comparison of the Pgrowth and Pdrag (given by Eq. (3)), and the corresponding example for ω = π /30 π /4 and R0 = 2 nm is shown in Fig. 2a. One can see that for a film of thickness t = 10 nm, the capillary driving force for grain growth is always higher than the drag force associated with shear coupling (i.e., Eq. (5) does not have a real solution). In this case, grain growth in the film will not stagnate (at least from shear coupling), although the rate of grain growth will be slowed. On the contrary, for thicker films, the capillary driving and drag forces may balance; for t = 40 nm, stagnation occurs at x∞ ≈ 0.21. Thus,
Fig. 2. (a) The dependence of the shear-coupled related drag force Pdrag (black line) and the capillarity-related driving force for grain growth Pgrowth (colored lines), on the dimensionless GB displacement, x = d/2t. If the films is thicker than the critical thickness (t∗ = 18.4 nm in this case), grain growth will stagnate. (b) The dependence of the critical film thickness t∗ on shear-coupling angle, ω. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
for each set of parameters (R0 , ω), there exists a critical thickness t∗ below which grain growth stagnation from shear coupling will not occur and above which grain growth will stagnate. In the example shown in Fig. 2, t∗ = 18.4 nm. The dependence of t∗ on ω for R0 = 2 nm is shown in Fig. 2b. Naturally, the critical thickness decreases with increasing shear coupling parameter β or ω. The dependence of the GB migration distance d∞ on the film thickness t is shown in Fig. 3 for two different coupling factors β (i.e., ω). d∞ increases with decreasing initial grain size R0 . For the small coupling factor (ω = π /30) and small initial grain size (R0 = 2 nm) case (Fig. 3b), grains may grow larger by more than a factor of ten prior to stagnation. While d∞ increases monotonically with film thickness for large β (ω), d∞ (t) exhibits a minimum for small β (ω) (see Fig. 3). This minimum is associated with the maximum in Pdrag (x), as seen in Fig. 2a. The above analysis demonstrates that substantial grain growth may occur for thin films on substrates when the shear coupling factor is small, β 1 (ω π /4). To illustrate this point, we con-
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E. Rabkin and D.J. Srolovitz / Scripta Materialia 180 (2020) 83–87
Fig. 3. The dependence of the maximum GB displacement d∞ on the film thickness for (a) ω = π /10 and (b) ω = π /30.
sider the experiments of Barmak et al. on grain growth in thin Al films [4]. Here, the grain radius in the 100 nm-thick Al film was initially R0 ≈ 34 nm, which upon annealing doubles before stagnation. Employing Eq. (5) for this set of data yields ω ≈ β ≈ π /56. While small coupling factors are expected for low angle GBs, there is also a tendency for coupling factors to decrease (towards zero) with increasing temperature in curvature-driven (normal) grain growth [14,26]. This is consistent with the results of Ref. [4] where grain growth temperature was 400 °C, which is ∼0.72Tm (Tm is the melting point). The variation in the coupling factor between different grains may result in the evolution of the crystallographic texture during grain growth; GBs characterized by small values of β will quickly disappear from the microstructure, leaving behind GBs with large coupling factors. We now consider what will happen once grain growth stagnates. The stress concentration associated with disclination dipoles (shear coupling) can be relaxed by grain boundary [19] and/or film-substrate interface [27] diffusion. In both cases, the apparent activation energy for grain growth will be closer to the activation energy of GB diffusion than to the activation energy of intrinsic GB migration (typically significantly smaller than that for diffusion)
[28]. The relaxation of the internal stresses associated with GB migration by diffusion along the film-substrate interface results in material accretion at the interface and may lead to the formation of hillocks [29], and/or to GB sliding normal to the substrate and formation of characteristic steps at GBs [30–31]. GB sliding associated with shear-coupled GB migration is well-documented in freestanding thin films [31], whereas in films attached to a rigid substrate GB sliding (normal to the interface) can only occur in conjunction with interface diffusion [30], plastic deformation within the grains or fracture. Another mechanism for stress relaxation was reported recently [13] in which secondary GB disconnection modes, with the sign of the coupling factor opposite that of the primary disconnection modes, are formed. The formation of such secondary disconnection modes typically involves a larger barrier than that of the primary mode. Since the secondary disconnection modes will have opposite sign β than that of the primary mode, the effective value of β should decrease with increasing temperature. Secondary disconnection modes may have higher or lower formation energies than that associated with diffusion. The disconnection nucleation scenario is consistent with the jerky, “stop-and-go” motion of GBs during grain growth in thin films [22]: each “stop” stage may be associated with the nucleation time of appropriate GB disconnection. Thus, we conclude that the kinetics of grain growth in thin films on substrates is determined by the kinetics of relaxation of stresses associated with shear-coupled GB motion; either by GB or interface diffusion or from the formation of secondary disconnection modes. As a result of the differences in barriers for disconnection modes and the activation energies for these different types of diffusion, the corresponding activation energy for grain boundary migration should vary widely between GBs. The grain growth stagnation mechanism proposed in the present work is quite general and is applicable to thin films of pure materials, alloys and compounds; i.e., whenever GB migration is coupled to a shear across the GB. It can be easily distinguished from the GB grooving mechanism [3] by depositing a thin refractory passivation layer on the surface of the film. Such a passivating layer should impede surface self-diffusion of metal atoms and greatly reduce deep GB thermal groove formation, thus leading to less GB drag and larger grains at the end of annealing. In the case of the shear-coupled mechanism proposed here, supressing surface diffusion should slow the relaxation of stresses accumulated due to the coupled motion [19] and thus impede GB migration, resulting in a smaller final grain size. We considered the effect of shear-coupled GB migration on grain growth in thin films strongly adhered to a substrate. We demonstrated that for high values of coupling factor β ≥ 1, shear coupling can lead to grain growth stagnation in thin films. For β 1 the competition between the drag force associated with shear-coupling and the capillary driving force for grain growth depends on film thickness. In films with thicknesses greater than a threshold t∗ , grain growth stagnates at a limiting grain size, while in thinner films grain growth will not stagnate The limiting grain size increases with decreasing initial grain size. We proposed that for long annealing times the kinetics of grain growth in large thickness films (t > t∗ ) is controlled by the kinetics of the relaxation of stresses accumulated as a result of shear-coupled GB migration. A lower bound on the corresponding activation energy is determined by the activation energy of the GB self-diffusion.
Declaration of Competing Interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
E. Rabkin and D.J. Srolovitz / Scripta Materialia 180 (2020) 83–87
Acknowledgements The authors gratefully acknowledge the support of the (DJS) US National Science Foundation grant 1609267, and of the (ER) USIsrael Binational Science Foundation grant 2015680. Helpful discussions with Dr. L. Klinger are heartily appreciated. References [1] F.J. Humphreys, M. Hatherly, Recrystallization and Related Annealing Phenomena, Pergamon, 1995. [2] C.V. Thompson, Annual Rev. of Mater. Sci. 20 (1990) 245–268. [3] W.W. Mullins, Acta Metall 6 (1958) 414–427. [4] K. Barmak, J. Kim, C.-.S. Kim, W.E. Archibald, G.S. Rohrer, A.D. Rollett, D. Kinderlehrer, S. Ta’asan, H. Zhang, D.J. Srolovitz, Scripta mater 54 (2006) 1059–1063. [5] P. Chaudhari, J. Vac. Sci. Technol. 9 (1972) 520–522. [6] M. Upmanyu, D.J. Srolovitz, L.S. Shvindlerman, G. Gottstein, Interface Sci. 6 (1998) 287–298. [7] Y. Estrin, G. Gottstein, E. Rabkin, L.S. Shvindlerman, Acta mater 49 (2001) 673–681. [8] E.A. Holm, S.M. Foiles, Science 328 (2010) 1138–1141. [9] K. Barmak, E. Eggeling, D. Kinderlehrer, R. Sharp, S. Ta’asan, J. Kim, A.D. Rollett, K.R. Goffey, Progr. in Mater. Sci. 58 (2013) 987–1055. [10] E. Rabkin, Scripta mater 42 (20 0 0) 1199–1206. [11] G. Gottstein, A.H. King, L.S. Shvindlerman, Acta mater 48 (20 0 0) 397–403. [12] J.W. Cahn, Y. Mishin, A. Suzuki, Acta mater 54 (2006) 4953–4975.
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Contents lists available at ScienceDirect
Scripta Materialia journal homepage: www.elsevier.com/locate/scriptamat
Grain growth stagnation in thin films due to shear-coupled grain boundary migration Eugen Rabkin a,∗, David J. Srolovitz b,∗∗ a b
Department of Materials Science and Engineering, Technion – Israel Institute of Technology, 3200003 Haifa, Israel Department of Materials Science and Engineering, City University of Hong Kong, Kowloon, Hong Kong SAR
a r t i c l e
i n f o
Article history: Received 27 November 2019 Revised 10 January 2020 Accepted 17 January 2020 Available online 5 February 2020 Keywords: Thin films Grain growth Grain boundary migration Disclinations
a b s t r a c t Normal grain growth in thin films attached to a substrate has been considered. It has been shown that shear-coupled grain boundary migration results in the build-up of elastic stresses in the film. A semi-quantitative model of grain growth combining the elements of disclinations theory with the BurkeTurnbull model of normal grain growth has been proposed. It has been shown that shear-coupled grain boundary migration may lead to stagnation of normal grain growth, whereas for high coupling factor values, the grains in the film do not grow at all. Finally, the mechanisms of stress relaxation enabling some grain growth are discussed. © 2020 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
The kinetics of normal grain growth in polycrystalline solids is commonly described via a parabolic law [1]. However, it is wellknown that grain growth in thin polycrystalline films is rarely parabolic and, in most cases, stagnates after a short growth period [2]. This normal grain growth stagnation often results in abnormal grain growth [1,2]. Several reasons for normal capillary-driven grain growth stagnation in thin films of pure materials have been proposed; these may be summarized as follows. (i) Thermal grain boundary (GB) groove drag [3]: Mullins showed that the dragging of surface grooves at GB/surface intersections leads to a force in the direction opposed to GB migration. Grain growth will stagnate when this drag force exceeds the grain-size-dependent capillary driving force; the critical grain size is 2–3 times the film thickness (assuming isotropy). However, grain growth stagnation is still observed when thin films are encapsulated by much more refractory overlayers which impedes diffusional groove formation (e.g., Al thin films covered by its native oxide) [4]. (ii) Stress generation associated with the GB volume defect in constrained (thin film) geometries [5]: since GBs are commonly
∗ Corresponding authors: Prof. Eugen Rabkin, Department of Materials Science and Engineering, Technion – Israel Institute of Technology, 320 0 0 03 Haifa, Israel. ∗∗ Co-Corresponding author: Prof. David J. Srolovitz, Department of Materials Science and Engineering, City University of Hong Kong, Kowloon, Hong Kong SAR. E-mail addresses: [email protected] (E. Rabkin), [email protected] (D.J. Srolovitz).
https://doi.org/10.1016/j.scriptamat.2020.01.019 1359-6462/© 2020 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
of lower density than their delimiting grains, a decrease in total GB area during grain growth leads to the development of tensile stresses in thin films (where attachment to the substrate restricts contraction of the polycrystal). When the resulting increase in elastic energy exceeds the decrease in total GB energy associated with normal grain growth, grain growth will stagnate [5]. However, analysis of this mechanism shows that it should only be effective for very small initial grain sizes. However, experiments show that grain growth stagnation occurs when the initial grain sizes are much larger than predicted on this basis [2]. (iii) Increase of vacancy concentration in the film during grain growth associated with the GB volume defect [6,7]: if the density difference between the GB and the grains is accommodated by vacancy generation, the vacancy free energy contribution to the overall thermodynamics will decrease the driving force for grain growth [6,7]. Analysis of this proposal suggests that this will initially retard grain growth but normal parabolic grain growth should occur at late time. (iv) Presence of low mobility GBs [8]: the presence of a broad distribution of GBs mobilities (with some near zero mobility GBs) ultimately limits the mean grain size. “Smooth” GBs at temperatures below the roughening transition temperature were identified as exhibiting low mobilities, and atomistic computer simulations indicated that even a small fraction of such low-mobility GBs is sufficient to cause normal grain growth stagnation [8]. Though this model represents a very
84
E. Rabkin and D.J. Srolovitz / Scripta Materialia 180 (2020) 83–87
general approach to grain growth stagnation, applicable to both two- and three-dimensional polycrystals, experimental evidence for such an abrupt change in GB mobility at the roughening transition is still lacking. Several additional explanations of normal grain growth stagnation in thin films (such as anisotropy of the GB energy, solute drag, and triple junctions drag) were considered by Barmak et al. [9]. It was shown that none of the above proposals explains the difference between grain size distributions observed in experiments and deduced from the results of numerical simulations. In this respect, it should be noted that drag on moving GBs arising from mobile solute atoms, second phase particles, or triple junctions changes the growth law, but does not cause complete grain growth stagnation [10,11]. This short overview suggests that existing models for grain growth stagnation in thin films are of limited applicability/cannot be considered universal. Here, we propose an alternative approach to explain grain growth stagnation in thin films on substrates. In particular, we demonstrate how shear-coupling [12–16] necessarily leads to stagnation. It is now well-established, based on theory [12], atomistic computer simulations [12–14], and experiments [15,16], that a shear stress applied parallel to a GB plane can drive GB migration. Conversely, GB migration produces a shear across the GB [12–16]. This shear-coupled GB migration is commonly characterized by a coupling factor β , which is the ratio of the rate of shear translation across a GB to the GB migration rate. β depends on GB crystallography and how GB motion is driven [14]. The GB crystallography effects are most robustly described in terms of line defects along GBs known as disconnections [14,17]. For low angle GBs, β is a function of the misorientation angle [12–16]. It was recently shown that shear coupling plays a fundamental role during grain growth in bulk polycrystalline materials: the incompatibility of grain displacements at triple junctions during the GB migration leads to a build-up of elastic stresses that may result in grain growth retardation or stagnation [13]. These stress concentrations can be relaxed by the formation of additional GB disconnection modes (or emission of dislocations into the grains or by twinning). These additional disconnection modes are associated with the opposite sign coupling factor. In what follows, we consider grain growth in a polycrystalline thin film that adheres to an inert substrate (i.e., continuity of tractions and displacements) and include the effect of shear-coupled GB migration. Consider a single GB migrating in a film on a rigid substrate, as illustrated in Fig. 1a. If the film were not attached to the substrate, the displacement of the GB by a distance d may result in the vertical translation of the grain on the right hand side, as depicted in Fig. 1b (the vertical translation distance is β d). In this picture, we have assumed that the shear is in the direction perpendicular to the nominal film/substrate interface. Of course, the shear coupling may also lead to shear in the plane of the film (parallel to the interface). Hence, the value of β should include the projection of shear in the direction perpendicular to the interface. Now, since the film is attached to the substrate, we must elastically deform the film in Fig. 1b, back into contact with the substrate (to satisfy compatibility and continuity of tractions and displacements). We describe the resultant deformation in terms of a wedge disclination dipole at the film/substrate interface, as illustrated in Fig. 1c. The spacing between the disclinations is the GB migration distance d, and the magnitude of the strength of the wedge disclinations is characterized by the angle ω such that tan(ω) = β . It should be noted that disclination dipole should be viewed as a convenient mean of describing the elastic field associated with a wedge of material inserted into/removed from the film/substrate interface. This does not imply that the disclinations
Fig. 1. The GB in thin film migrating to the right (a); the imaginary shape of the film assuming no bonding to the substrate after the GB migrated by a distance d. The gap between the film and the substrate on the right has formed due to the shear coupling (b). Elastic deformation of the film ensuring its attachment to the substrate, which can be represented by a dipole of two disclinations of opposite signs (c).
are necessarily physically observable linear defects. The idea of using such a description is akin to describing a Mode I crack as an array of edge dislocations or a Mode III crack as an array of screw dislocations [18], or a grain boundary (GB) diffusion wedge as an array of infinitesimal edge dislocations [19–20]. The elastic energy (per unit length of the thin film bicrystal in the direction normal to the plane of the illustrations in Fig. 1) of this disclination dipole is Ed [21],
Ed =
Dω 2 d 2 ln 1 + 4
≈
2t 2
−2Ax2 ln x2
1 A 1− 2 2x
d
= Ax2 ln 1 +
x1
1 2 x
(1)
x 1,
where the film and the substrate have the same elastic constants, D = μ/[2π (1 − ν )], x = d/2t, A = Dω2 t2 and μ and ν are the shear modulus and Poisson’s ratio, respectively. This solution is exact within linear elasticity outside the disclination core regions for an elastically isotropic solid. In the x1 limit, this result reduces to the case a straight edge dislocation of Burgers vector β d, perpendicular to a flat, free surface. The drag force Fdrag on the GB, associated with this elastic energy, is in the direction opposite to the GB migration and may be
E. Rabkin and D.J. Srolovitz / Scripta Materialia 180 (2020) 83–87
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expressed as:
Fdrag =
≈
x
1 ∂ Ed ∂ Ed ∂ x 1 = =A ln 1 + 2 − ∂d ∂x ∂d t x 1 + x2 ⎧ x ⎨A x1 [2 ln(x ) + 1] t ⎩A 1 2t x3
(2)
x 1.
In a columnar microstructure, the GB runs from interface to surface (nominally perpendicular to both) such that the drag pressure acting on the GB, Pdrag , is:
Fdrag x = A 2 ln 1 + t t ⎧ ⎨A x [2ln(x ) + 1] t2 ≈ A 1 ⎩ 2t 2 x3
Pdrag =
1 x2
−
1 1 + x2
(3)
x1 x 1.
Expressing normal grain growth in accordance with the classical Burke-Turnbull model [1], the capillary driving force for growth Pgrowth can be expressed as
Pgrowth = α
γb R
,
(4)
where γ b and R are the GB energy and the average grain radius (inverse mean curvature), respectively. The parameter α is a geometry-dependent constant (for simplicity, we set it to α ≈1 here). For a film with initial grain radius R0 the normal grain growth stagnation condition Pdrag = Pgrowth can be obtained by combining Eqs. (3) and (4), and writing R(t → ∞)=R0 + d∞ (where R0 is the initial grain size):
γb R0 + d∞
=
γb R0 + 2t x∞
= Dω x∞ 2
1 ln 1 + 2 x∞
1 − 1 + x2∞
(5)
where x∞ = d∞ /2t. The transcendental Eq. (5) represents the main result of the present work. It allows calculation of the maximum extent of GB migration d as a function of the initial grain size, elastic (μ and ν ), GB thermodynamic (γ b ), and GB migration (β ) properties. We numerically estimate the GB migration distance for thin films of pure Al; this material is chosen because its elastic constants are nearly isotropic and because grain growth in Al thin films on substrates has been widely reported (e.g., see [4,9,22]). In these calculations, we set μ = 26.2 GPa, ν = 0.345 [23], γ b = 0.6 J/m2 [24], and β = 1 (i.e., ω = π /4). Also, though room temperature physical vapor deposition of thin Al films often result in columnar microstructure with the grains larger than several tens of nanometers in diameter [4,9,22], varying substrate temperature during deposition, impurity content, and the nature of the plasma-forming gas allow broad variations of the film microstructure and mean grain size [25]. For R0 = 5 nm and t = 200 nm, the elastic stresses associated with shear-coupled GB motion lead to grain growth stagnation at d = 0.93 nm. For larger values of R0 , grain growth stagnates at even smaller values of d. This suggests that grain growth is only possible when tan(ω) = β is small. Re-writing the left hand side of Eq. (5) in the form (γ b /2t)/[(R0 /2t)+x] allows direct comparison of the Pgrowth and Pdrag (given by Eq. (3)), and the corresponding example for ω = π /30 π /4 and R0 = 2 nm is shown in Fig. 2a. One can see that for a film of thickness t = 10 nm, the capillary driving force for grain growth is always higher than the drag force associated with shear coupling (i.e., Eq. (5) does not have a real solution). In this case, grain growth in the film will not stagnate (at least from shear coupling), although the rate of grain growth will be slowed. On the contrary, for thicker films, the capillary driving and drag forces may balance; for t = 40 nm, stagnation occurs at x∞ ≈ 0.21. Thus,
Fig. 2. (a) The dependence of the shear-coupled related drag force Pdrag (black line) and the capillarity-related driving force for grain growth Pgrowth (colored lines), on the dimensionless GB displacement, x = d/2t. If the films is thicker than the critical thickness (t∗ = 18.4 nm in this case), grain growth will stagnate. (b) The dependence of the critical film thickness t∗ on shear-coupling angle, ω. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
for each set of parameters (R0 , ω), there exists a critical thickness t∗ below which grain growth stagnation from shear coupling will not occur and above which grain growth will stagnate. In the example shown in Fig. 2, t∗ = 18.4 nm. The dependence of t∗ on ω for R0 = 2 nm is shown in Fig. 2b. Naturally, the critical thickness decreases with increasing shear coupling parameter β or ω. The dependence of the GB migration distance d∞ on the film thickness t is shown in Fig. 3 for two different coupling factors β (i.e., ω). d∞ increases with decreasing initial grain size R0 . For the small coupling factor (ω = π /30) and small initial grain size (R0 = 2 nm) case (Fig. 3b), grains may grow larger by more than a factor of ten prior to stagnation. While d∞ increases monotonically with film thickness for large β (ω), d∞ (t) exhibits a minimum for small β (ω) (see Fig. 3). This minimum is associated with the maximum in Pdrag (x), as seen in Fig. 2a. The above analysis demonstrates that substantial grain growth may occur for thin films on substrates when the shear coupling factor is small, β 1 (ω π /4). To illustrate this point, we con-
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E. Rabkin and D.J. Srolovitz / Scripta Materialia 180 (2020) 83–87
Fig. 3. The dependence of the maximum GB displacement d∞ on the film thickness for (a) ω = π /10 and (b) ω = π /30.
sider the experiments of Barmak et al. on grain growth in thin Al films [4]. Here, the grain radius in the 100 nm-thick Al film was initially R0 ≈ 34 nm, which upon annealing doubles before stagnation. Employing Eq. (5) for this set of data yields ω ≈ β ≈ π /56. While small coupling factors are expected for low angle GBs, there is also a tendency for coupling factors to decrease (towards zero) with increasing temperature in curvature-driven (normal) grain growth [14,26]. This is consistent with the results of Ref. [4] where grain growth temperature was 400 °C, which is ∼0.72Tm (Tm is the melting point). The variation in the coupling factor between different grains may result in the evolution of the crystallographic texture during grain growth; GBs characterized by small values of β will quickly disappear from the microstructure, leaving behind GBs with large coupling factors. We now consider what will happen once grain growth stagnates. The stress concentration associated with disclination dipoles (shear coupling) can be relaxed by grain boundary [19] and/or film-substrate interface [27] diffusion. In both cases, the apparent activation energy for grain growth will be closer to the activation energy of GB diffusion than to the activation energy of intrinsic GB migration (typically significantly smaller than that for diffusion)
[28]. The relaxation of the internal stresses associated with GB migration by diffusion along the film-substrate interface results in material accretion at the interface and may lead to the formation of hillocks [29], and/or to GB sliding normal to the substrate and formation of characteristic steps at GBs [30–31]. GB sliding associated with shear-coupled GB migration is well-documented in freestanding thin films [31], whereas in films attached to a rigid substrate GB sliding (normal to the interface) can only occur in conjunction with interface diffusion [30], plastic deformation within the grains or fracture. Another mechanism for stress relaxation was reported recently [13] in which secondary GB disconnection modes, with the sign of the coupling factor opposite that of the primary disconnection modes, are formed. The formation of such secondary disconnection modes typically involves a larger barrier than that of the primary mode. Since the secondary disconnection modes will have opposite sign β than that of the primary mode, the effective value of β should decrease with increasing temperature. Secondary disconnection modes may have higher or lower formation energies than that associated with diffusion. The disconnection nucleation scenario is consistent with the jerky, “stop-and-go” motion of GBs during grain growth in thin films [22]: each “stop” stage may be associated with the nucleation time of appropriate GB disconnection. Thus, we conclude that the kinetics of grain growth in thin films on substrates is determined by the kinetics of relaxation of stresses associated with shear-coupled GB motion; either by GB or interface diffusion or from the formation of secondary disconnection modes. As a result of the differences in barriers for disconnection modes and the activation energies for these different types of diffusion, the corresponding activation energy for grain boundary migration should vary widely between GBs. The grain growth stagnation mechanism proposed in the present work is quite general and is applicable to thin films of pure materials, alloys and compounds; i.e., whenever GB migration is coupled to a shear across the GB. It can be easily distinguished from the GB grooving mechanism [3] by depositing a thin refractory passivation layer on the surface of the film. Such a passivating layer should impede surface self-diffusion of metal atoms and greatly reduce deep GB thermal groove formation, thus leading to less GB drag and larger grains at the end of annealing. In the case of the shear-coupled mechanism proposed here, supressing surface diffusion should slow the relaxation of stresses accumulated due to the coupled motion [19] and thus impede GB migration, resulting in a smaller final grain size. We considered the effect of shear-coupled GB migration on grain growth in thin films strongly adhered to a substrate. We demonstrated that for high values of coupling factor β ≥ 1, shear coupling can lead to grain growth stagnation in thin films. For β 1 the competition between the drag force associated with shear-coupling and the capillary driving force for grain growth depends on film thickness. In films with thicknesses greater than a threshold t∗ , grain growth stagnates at a limiting grain size, while in thinner films grain growth will not stagnate The limiting grain size increases with decreasing initial grain size. We proposed that for long annealing times the kinetics of grain growth in large thickness films (t > t∗ ) is controlled by the kinetics of the relaxation of stresses accumulated as a result of shear-coupled GB migration. A lower bound on the corresponding activation energy is determined by the activation energy of the GB self-diffusion.
Declaration of Competing Interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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