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Growth mechanism and stress relief patterns of Ni films deposited on silicone oil surfaces
Applied Surface Science 255 (2009) 8352–8358
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Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
Growth mechanism and stress relief patterns of Ni films deposited on silicone oil surfaces PingGen Cai a,b,*, SenJiang Yu c, XiaoJun Xu a, MiaoGen Chen c, ChengHua Sui a, Gao-Xiang Ye c a
Department of Applied Physics, Zhejiang University of Technology, Hangzhou 310014, PR China Graduate School of Information Science and Technology, Hokkaido University, Sapporo 060-0814, Japan c Department of Physics, Zhejiang University, Hangzhou 310027, PR China b
A R T I C L E I N F O
A B S T R A C T
Article history: Received 24 December 2008 Received in revised form 23 May 2009 Accepted 24 May 2009 Available online 23 June 2009
The growth mechanism and stress relief patterns of nickel (Ni) films, deposited on silicone oil surfaces by a thermal evaporation method, have been studied systematically. Our experiment shows that the growth mechanism of the Ni films approximately obeys a two-stage growth model. Characteristic cracks with sinusoidal appearance resulted from the internal stress can be frequently observed in the continuous Ni films after the samples are removed from the vacuum chamber. Several crack modes including the regularly sinusoidal cracks, zigzag cracks, attenuation cracks and self-similar cracks are described and analyzed by using the general theory of buckling of plates in detail. The internal stress and propagating velocity of the sinusoidal cracks are also discussed in this paper. ß 2009 Elsevier B.V. All rights reserved.
PACS: 68.55.a 83.85.St 62.20.mt Keywords: Thin film growth Stress pattern Crack Liquid substrate
1. Introduction Under suitable conditions, both single atoms and atomic clusters can randomly diffuse and aggregate on solid substrates, which may result in the formation of ramified atomic aggregates with characteristic structures [1–4]. As the nominal film thickness increases, the ramified aggregates grow gradually and connect with each other, and finally a continuous thin film forms. Generally, thin films fabricated by various deposition methods, such as vapor deposition and sputtering etc., often develop residual intrinsic stress during deposition process [3]. Additional residual thermal stress may also be induced during the cooling process when a thermal expansion mismatch exists between the film and the substrate. The films containing large residual stresses are susceptible to generating buckles or cracks [4]. To date, many interesting stress relief patterns have been observed, including the well-known sinusoidal cracks, straight wrinkles, circle bubbles and telephone cord buckles [5,6].
* Corresponding author at: Graduate School of Information Science and Technology, Hokkaido University, Sapporo 060-0814, Japan. Tel.: +81 011 706 7695; fax: +81 011 706 7592. E-mail addresses: [email protected], [email protected] (P. Cai). 0169-4332/$ – see front matter ß 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2009.05.088
For the films on solid substrates, it is very difficult to observe the real stress distributions existing in the films from the appearance of the cracks directly since the crack appearance is determined not only by the stress distributions in the films, but also by the strong adhesive interaction between the films and the solid substrates. However, for the films on liquid substrates, these characteristic stress distributions can be detected following the crack traces since the adhesive interaction between the film and the liquid substrate is very small. On the other hand, because of the characteristic solid/liquid boundary condition and interactions among the atoms and atomic clusters on the liquid surfaces, anomalous stress distributions may exist in the nearly free sustained films. Thin films on liquid substrates provide us with an ideal physical system to study the internal stress distributions in the films. In this paper, we report the growth mechanism and sinusoidal stress relief patterns of a Ni film system deposited on silicone oil surfaces. Our experiment shows that the formation mechanism of the Ni films approximately obeys a two-stage growth model, which was firstly concluded from non-magnetic film systems on liquid substrates [7]. The sinusoidal crack pattern observed in the continuous Ni films represents a sinusoidal stress distribution in the films. The evolution of the crack appearance with the time in the Ni film system gives a physical picture that the internal stress
P.G. Cai et al. / Applied Surface Science 255 (2009) 8352–8358
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Fig. 1. Typical morphology of ramified nickel (Ni) aggregates on a silicone oil surface. f = 0.12 nm/s, d = 7 nm, Dt = 0.5 h. Image size is 125 100 mm2.
distribution evolves and the strain energy releases gradually after deposition. 2. Experiment The Ni film samples were prepared by thermal evaporation of 99.5% pure nickel in a vacuum of 6 104 Pa at room temperature. Commercial silicone oil (DOW CORNING 705 Diffusion Pump Fluid) with a vapor pressure below 108 Pa was painted onto a frosted glass surface, which was fixed 120 mm below the filament (tungsten). The resulting oil substrate with an area about 10 10 mm2 had a uniform thickness of 0.5 mm. The deposition rate f and the nominal film thickness d were determined by a quartz-crystal balance, which was calibrated by a profilemeter (astep 200 profilemeter, TENCOR). After deposition, the sample was kept in the vacuum chamber (in vacuum condition) for time interval Dt. Then the chamber was filled with air slowly and finally the sample was removed from the chamber. All images for the surface morphologies of the films were taken with an optical microscope (Leica DMLM). 3. Results Fig. 1 shows the typical morphology of ramified nickel (Ni) aggregates on silicone oil surfaces in the early stage of film growth. The bright and dark parts correspond to the Ni aggregates and uncovered oil surface, respectively. It is noted that the appearance of the Ni ramified aggregates is quite similar to those of Au and Ag films deposited on the liquid substrates [7,8]. However, the average branch width of the Ni aggregates measured in the experiment is about 3–5 mm, which is several times larger than those of the Au and Ag aggregates. According to the experimental observations, we draw a conclusion that the formation process of the Ni aggregates can be generally traced to a two-stage model. Firstly, Ni atoms deposited on the liquid surface nucleate and form atomic compact clusters with the sizes of about 3–5 mm. Then, during the subsequent stage, the compact clusters diffuse and rotate randomly on the liquid surface, which leads to formation of the ramified Ni aggregates. The fact that the growth behaviors of the Ni aggregates are quite similar to those of non-magnetic films gives evidence that the magnetic interaction between the Ni atoms and clusters does not play a key role in this stage. As the nominal film thickness increases, the ramified Ni aggregates grow gradually and connect with each other, and finally a continuous Ni film forms on the liquid substrate.
Fig. 2. Typical morphologies of sinusoidal cracks in the continuous Ni films deposited on a silicone oil surface. f = 0.46 nm/s, d = 55 nm, Dt = 1.33 h. Each image has a size of 1440 1065 mm2.
Our experiment shows that cracks are frequently observed in the continuous Ni films after the samples were removed from the vacuum chamber. The typical morphologies of cracks in the Ni films are shown in Fig. 2. The exciting and unexpected result is that the appearance of the cracks exhibits a periodic structure, which is quite similar to a sinusoidal curve. Furthermore, the parallel cracks appear closely with similar period T and amplitude A but no fixed correlation among their oscillatory phases can be concluded (see Fig. 2). Generally, the sinusoidal cracks start from the sample edges or from other broad cracks. According to the propagation directions of the cracks, one crack may bifurcate into two or three cracks, or two cracks may also coalesce harmoniously. Moreover, it can be seen from Fig. 2 that the amplitudes decrease gradually when the cracks are propagating towards the free edges. Finally, the cracks become straight lines perpendicular to the free edges and the sinusoidal feature diminishes. In our experiment, in the nominal film thickness range of d = 8– 80 nm and the deposition rate range of f = 0.06–0.86 nm/s, the sinusoidal cracks shown in Fig. 2 can be observed and no obvious dependence between the experimental conditions and the appearance of the sinusoidal cracks (i.e., T and A) can be detected. The maximum values of T and A observed in our experiment are about 580 and 275 mm, respectively. The longest sinusoidal crack observed in our samples can be over 1.32 mm and with more than 13 oscillatory periods. The length of the crack was measured from crack one end to the other end directly. Here we define a parameter: A kc 2 ; T
(1)
the maximum value of kc in our experiment is about 0.98. In order to identify the formation process of the cracks, a series of experiments was performed in which the time Dt was varied between Dt = 0.5 and 42 h. The experimental results show that, if
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0.5 Dt 3 h, most of the cracks appeared in the Ni films exhibit the sinusoidal appearance; if Dt > 6 h, however, all the cracks extend irregularly and the sinusoidal appearance disappears. This experimental result indicates that the sinusoidal cracks formed during the period when the vacuum chamber was filled with air. Therefore, we suggest that the sinusoidal cracks result from the internal stress in the films with the help of the pressure of the atmosphere and it is independent of the oxidation of the film in the air. Furthermore, it is noted that the number density of the cracks in the samples with Dt > 6 h is smaller than that in the samples with Dt 3 h. The evolution behaviors of the number density and the appearance of the cracks with Dt give the information that the internal stress field evolves and the stress energy releases gradually with the time. In order to further confirm the formation mechanism of the sinusoidal cracks described above, a simple test was taken: two Ni films with identical f and d but different Dt (Dt = 0.5 h, 12.5 h) were broken with a sharp pin. We found that the sinusoidal cracks are only observed in the film with Dt = 0.5 h, which again indicates that the sinusoidal internal stress patterns really exist in the free sustained films and they disappear gradually with the time Dt. On the other hand, similar sinusoidal cracks may also be observed in the aluminum (Al) film system deposited on the silicone oil surfaces and the average period of the cracks is less than those observed in the Ni film system. This phenomenon indicates that the sinusoidal appearance of the cracks should not be related to the magnetic interactions among the nickel atoms (or clusters). We suggest that the internal stresses of the Ni and Al films should be responsible for the sinusoidal cracks. In principle, crack patterns caused by contraction under internal tensile stress are common in both natural and manmade systems, including dried out fields, rocks, tectonic plates, paintings and various thin films. According to the previous studies, the propagation of a crack actually occurs only if the energy relieved by the advance of the crack is enough to overcome the fracture energy needed to elongate the crack and this propagation reduces the total energy [9,10]. Under quasi-static advance the energy reduction during propagation is infinitesimally small, and typically, there is only one possible direction of propagation for each crack. All other propagation directions would produce an increase in the total energy of the system. Because the stress near an existing crack is generally parallel to the crack edge, any crack nucleating at or growing to meet the preexisting crack will be perpendicular to the crack edge. Therefore, the sinusoidal cracks observed in the Ni films generally align perpendicular to the broad cracks or film edges, as shown in Fig. 2. Near the film edges, the stress component perpendicular to the edge is relaxed, and only the stress component parallel to the edge exists. Therefore, the sinusoidal cracks always change into straight lines perpendicular to the free edges during the propagation process (see Fig. 2). 4. Discussion 4.1. General theory According to the previous studies [5–7,11,12], strong residual internal stress always exists in thin films deposited on solid and liquid substrates and it releases gradually in order to minimize the total free energy of the systems. Therefore, characteristic stress relief patterns can be frequently observed in the films. Previous works have proved that the favourable direction in which the crack extends is perpendicular to the greatest local tensile stress [13], which provides us with a method to study the residual internal stress distributions based on the appearances of the cracks in the films.
Several works have attempted to theoretically understand the sinusoidal stress patterns in thin films [5,6,11,12,14]. The validity of the models put forward using the general theory of buckling of plates and the equation is given by [5]: !
@4 w @4 w @4 w @2 w @2 w D þ 2 þ s d þ s d þ x y @x4 @x2 @y2 @y4 @x2 @y2 þ 2t xy d
@2 w þ F ¼ 0; @x@y
(2)
where D = Ed3/12(1 m2) is the moment of inertia of the film; d is the film thickness; E is Young’s modulus and m is the Poisson’s ratio; x and y are the coordinates related to the substrate, w is the film deflection as defined in the elastic theory, sx and sy are the internal stresses, txy is the shear stress and F is the external force. One type of solutions of Eq. (2) which is physically acceptable is: w ¼ 1 þ cosðkx þ qyÞ:
(3)
Solution (3) implies that the direction of the highest stress is perpendicular to straight lines [5]: kx þ qy ¼ 2np;
n ¼ 0; 1; 2; . . . :
(4)
Introducing solution (3) into Eq. (2) shows that for each k value there are two permissible values of q: q¼
sx d D
2
k
1=2 ;
(5)
Therefore, two classes of these line families with slopes of jk/qj cross each other [5]. The stress relief pattern described by Eq. (4) can also be approximated by the expression [5]: qy ¼ pcosðkxÞ:
(6)
In fact, relation (6) can be satisfied by a series of real values of k and q, which may account for various sinusoidal stress relief patterns. We will show below that the stress relief patterns observed in the Ni films can be well explained by making use of this model. 4.2. The growth modes of the cracks In most cases, the sinusoidal cracks observed in our experiment generally exhibit the regular sine curves with well-defined period and amplitude, as shown in Fig. 3(a). The regularly sinusoidal crack mode can be well explained by using of the general theory of buckling of plates described above. If k and q are constants, Eq. (4) consists of two families of parallel lines with the slopes of jk/qj, as sketched in Fig. 3(b). Once the crack process begins, it propagates in a certain direction. When it reaches a point where two lines cross each other, it may jump to the line with different slope. This process goes on and finally the regularly sinusoidal pattern forms, as shown in Fig. 3(b). It is noted that the local internal stress in the Ni films is equibiaxial, and the two components of stress, defined as sx and sy, are equal. Therefore, the propagation distances of the crack along the two crossing lines are uniform and alternate, which permits to release the stress energy effectively in these two directions. In some cases, distorted sinusoidal cracks (like zigzag motif) can be observed in the Ni films, as shown in Fig. 4(a). Here we define the two neighboring side lengths of the zigzag crack as l1 and l2, respectively (see Fig. 4(a)). In order to further understand the propagating characteristics of the zigzag crack, the lengths l1 and l2 have been measured for the samples with different experiment conditions, and the results are shown in Fig. 4(b). We find that the length l2 increases linearly with the length l1, which can be well
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Fig. 3. Appearance of regularly sinusoidal cracks in the Ni films. (a) f = 0.14 nm/s, d = 55 nm, Dt = 0.5 h. Image size is 285 210 mm2; (b) sketch drawn for (a) as two classes of the line families with fixed slopes of jk/qj. The dark line indicates the growth process of the crack.
fitted by the function l2 = 2.5 + 1.95l1. To a good approximation, the relation can be expressed as: l2 2l1 :
(7)
Be similar to the regularly sinusoidal mode, two classes of parallel lines with fixed slopes of jk/qj also exist in the Ni films. However, the two components of the biaxial stress, sx and sy, in this case are not equal. In order to balance the stress energy of the system, the propagation distance of the crack along one direction is always larger than the other one, and therefore the zigzag crack mode appears, as shown in Fig. 4(c). Our experiment shows that when the sinusoidal cracks extend towards the free edges of the sample, the amplitudes of the cracks always attenuate gradually and finally change into linear cracks (see Fig. 2). The typical attenuation modes of the cracks are shown in Fig. 5(a) and (b). It is noted that although the amplitudes of the cracks decrease obviously during propagation, the periods of the cracks are almost unchanged. Because Eq. (6) can be satisfied by a series of different real values of k and q as mentioned in the general theory, k and q are not required to be constants in a sample. We suggest that the values of k and q are location dependent, especially near the free edges of the sample. When the cracks propagate towards the free edges, the values of jk/qj decreases gradually, as shown in Fig. 5(c). The attenuation of jk/qj implies that the stress component sx, perpendicular to the free edges, decreases gradually. Correspondingly, the equi-biaxial stress changes into axial stress gradually. Therefore, the cracks always align perpendicular to the film edges and the amplitudes decrease gradually during propagation. The experiment also shows that when the sinusoidal cracks extend in the Ni films, a gradual change in the propagation
Fig. 4. A motif of zigzag crack in the Ni films. (a) f = 0.10 nm/s, d = 16 nm, Dt = 1.5 h. Image size is 720 535 mm2; (b) sketch drawn for the figure above as two classes of the line families with fixed slopes of jk/qj. The dark line indicates the growth process of the crack. (c) The length l2 of zigzag crack as a function of the length l1. The solid line represents a linear fit of the experimental data.
direction occurs occasionally, as shown in Fig. 6(a). The maximum changed angle of the cracks measured in the experiment can be larger than 908. It is noted from Fig. 6(a) that the neighboring sinusoidal cracks remain parallel to each other. Another interesting phenomenon observed in the experiment is shown in Fig. 6(b), where we find that a big sinusoidal crack is composed of a series of small sinusoidal cracks with nearly uniform period and amplitude. The sinusoidal cracks are quite similar under different scales, showing the self-similarity of the crack morphologies in this case. The scaling factor is about 10. The formation process of the cracks shown in Fig. 6(a) and (b) can be sketched in Fig. 6(c). In these cases, the two components of the stress are not equal and evolve when the cracks propagate, which finally results in formation of the comparatively complex crack modes shown in Fig. 6(a) and (b).
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Fig. 5. Typical attenuation modes of the cracks in the Ni films. (a) f = 0.16 nm/s, d = 23 nm, Dt = 1.5 h. Image size is 360 265 mm2; (b) f = 0.09 nm/s, d = 35 nm, Dt = 0.5 h. Image size is 720 535 mm2; (c) sketch drawn for the figures above as the value of jk/qj undergoing a gradual decrease. The dark line indicates the growth process of the cracks.
Fig. 6. Typical morphologies of self-similar cracks in the Ni films. (a) f = 0.15 nm/s, d = 25 nm, Dt = 1.5 h. Image size is 360 265 mm2; (b) f = 0.08 nm/s, d = 25 nm, Dt = 1.5 h. Image size is 360 265 mm2; (c) sketch drawn for the figures above as the unusual growth behavior of the cracks. The dark line indicates the growth process of the cracks.
P.G. Cai et al. / Applied Surface Science 255 (2009) 8352–8358
4.3. The internal stress of the cracks In Eq. (6), we can define the amplitude A = p/q and the wave vector k = 2p/T. Hence, Eq. (5) can be written as: D p2 4 1 sx ¼ þ 2 ; (8) 2 d T A then introduce the expression of D into Eq. (8) [5]
sx ¼
2 Ed p2 4 1 þ 2 ; 2 2 12ð1 m Þ T A
(9)
where the Young’s modulus E and the Poisson’s ratio m is 199.5 GPa and 0.312, respectively. In our samples, both T and 2A are about 102 mm size, thus Eq. (9) gives the order of the stress s 105 Pa. We find that the sinusoidal cracks occur spontaneously for intrinsic stresses above a threshold value. When T and 2A equal their maximum values, corresponding to d = 20 nm, the intrinsic stress reaches its threshold value which is approximately 2 103 Pa. Combining the crack morphologies and Eq. (9), we may approach a concrete analysis for the stress problems, the diagram of the stress s of the crack at the bottom of Fig. 5(b) is shown in Fig. 7. It can be seen from Fig. 7(a) that the value of the stress s increases clearly along the propagation direction of the crack. Combining Eqs. (1) and (8), the dependence of sx upon period T is given by: D p2 4 sx ¼ 2 4 þ 2 ; (10) kc dT which suggests that sx is proportional to 1/T2, indicating that there is indeed a strong dependence of crack wavelength on
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stress, which is in agreement with those proposed by Gioia and Ortiz [15]. In addition, at the tip of a sinusoidal crack, A tends to zero, therefore kc changes to zero (see Eq. (1)). Accordingly, we can conclude that sx approaches to infinite when kc trends towards zero (see Eq. (10)). This phenomenon is a consequence of the fact that as soon as a single crack extends a small distance, the stress at its tip increases, and the stresses at the tips of the other cracks decrease. This generates an unstable and rapid propagation of the single crack [16]. 4.4. The velocity of the cracks Since the total crack formation process is finished in few minutes during the period when the vacuum chamber is filled with air, the extending velocity v of the cracks is not easy to be measured experimentally. Therefore, we settle this problem in theory. The crack patterns in Fig. 5 can be fitted well by the function A exp(x/ l) cos(kx + x0), where l is the attenuation length, and the amplitude of the oscillation obeys a Ginzburg–Landan equation [17]:
@t A ¼ eA A3 ;
(11)
and making use of the relation @t A ¼ v@x A, we obtain an attenuation oscillation solution: A ¼ A0 expðxjej=vÞ;
(12)
where A denotes the amplitude of oscillation, e is the reduced control parameter, e ðv vc Þ=vc , and here x is the x position of the tip. The transition from linear to oscillating cracks happens when the velocity is a little below vc , which is the critical value of v. Combining Fig. 5 and Eq. (12), Fig. 6(b) displays the diagram of the dependence of v=ðeÞ upon x. We find that the change law of v upon x is similar to v=ðeÞ upon x since v is the increasing function of v=ðeÞ. Therefore, Fig. 6(b) shows that v drops quickly; then it increases gradually; until it reaches its secondary maximum value, finally drops again along the propagation direction of the crack. 5. Conclusion In conclusion, we have studied the formation mechanism and sinusoidal stress patterns of the nickel films deposited on silicone oil surfaces. The growth mechanism of the films can be approximately traced to the two-stage growth model and the stress relief patterns can be well explained by making use of the general theory of plates buckling. We show that the two-stage growth process and the sinusoidal stress patterns do not relate to the magnetic interactions among the atoms and they are resulted mainly from the characteristic free boundary condition of the nearly free sustained films. Unfortunately, several tough problems still remain poorly understood, such as the details of the interactions among the Ni atoms and between the films and the liquid substrates, which should be responsible for the film microstructures and the sinusoidal crack patterns. Therefore, many new avenues of investigation are still open to us. The phenomena shown in Figs. 2–6 strongly suggest that the free sustained films may possess characteristic microstructures and therefore they may exhibit anomalous physical properties. We hope that the results above would contribute to further research on theoretical and practical projects in thin film physics. Acknowledgements
Fig. 7. (a) Dependence between the stress s and the crack length l; (b) dependence between v=ðeÞ and l. This measurement was taken for the sample shown in Fig. 5(b).
This work was supported by the National Natural Science Foundations of China (grant nos. 60777034 and 10874159) and by the Natural Science Foundation of Zhejiang Province in China (grant no. Y7080042).
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T.A. Witten, L.M. Sander, Phys. Rev. Lett. 47 (1981) 1400. P. Meakin, Phys. Rev. Lett. 51 (1983) 1119. G.A.J. Amaratunga, M.E. Welland, J. Appl. Phys. 68 (1990) 5140. A.S. Argon, V. Gupta, H.S. Landis, J.A. Cornie, J. Mater. Sci. 24 (1989) 1207. D. Nir, Thin Solid Films 112 (1984) 41. Q. Wang, C.B. Wang, Z. Wang, J.Y. Zhang, D.Y. He, Appl. Surf. Sci. 255 (2008) 1836. G.X. Ye, Th. Michely, V. Weidenhof, I. Friedrich, M. Nucleation Wuttig, Phys. Rev. Lett. 81 (1998) 622. [8] G.X. Ye, A.G. Xia, G.L. Gao, Y.F. Lao, X.M. Tao, Phys. Rev. B 63 (2001) 125405.
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E.A. Jagia, Phys. Rev. E 65 (2002) 46147. M. Marder, Nature 362 (1993) 295. N. Matuda, S. Baba, A. Kinbara, Thin Solid Films 81 (1981) 301. S.J. Yu, Y.J. Zhang, P.G. Cai, Q.L. Ye, X.W. Tang, G.X. Ye, J. Phys.: Condens. Matter 16 (2004) L147. N.E. Frost, K.J. Marsh, L.P. Pook, Metal Fatigue, Oxford University Press, Oxford, 1974, pp. 320–341. B. Iyer Suman, K.S. Harshavardhan, V. Kumar, Thin Solid Films 256 (1995) 94. G. Gioia, M. Ortiz, Adv. Appl. Mech. 33 (1997) 119. H.K. Pulker, R. Buhl, J. Maser, Proc. 7th Int. Vac. Cong., Vienna, (1977), p. 1761. A. Yuse, M. Sano, Nature (London) 362 (1993) 329.
Contents lists available at ScienceDirect
Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
Growth mechanism and stress relief patterns of Ni films deposited on silicone oil surfaces PingGen Cai a,b,*, SenJiang Yu c, XiaoJun Xu a, MiaoGen Chen c, ChengHua Sui a, Gao-Xiang Ye c a
Department of Applied Physics, Zhejiang University of Technology, Hangzhou 310014, PR China Graduate School of Information Science and Technology, Hokkaido University, Sapporo 060-0814, Japan c Department of Physics, Zhejiang University, Hangzhou 310027, PR China b
A R T I C L E I N F O
A B S T R A C T
Article history: Received 24 December 2008 Received in revised form 23 May 2009 Accepted 24 May 2009 Available online 23 June 2009
The growth mechanism and stress relief patterns of nickel (Ni) films, deposited on silicone oil surfaces by a thermal evaporation method, have been studied systematically. Our experiment shows that the growth mechanism of the Ni films approximately obeys a two-stage growth model. Characteristic cracks with sinusoidal appearance resulted from the internal stress can be frequently observed in the continuous Ni films after the samples are removed from the vacuum chamber. Several crack modes including the regularly sinusoidal cracks, zigzag cracks, attenuation cracks and self-similar cracks are described and analyzed by using the general theory of buckling of plates in detail. The internal stress and propagating velocity of the sinusoidal cracks are also discussed in this paper. ß 2009 Elsevier B.V. All rights reserved.
PACS: 68.55.a 83.85.St 62.20.mt Keywords: Thin film growth Stress pattern Crack Liquid substrate
1. Introduction Under suitable conditions, both single atoms and atomic clusters can randomly diffuse and aggregate on solid substrates, which may result in the formation of ramified atomic aggregates with characteristic structures [1–4]. As the nominal film thickness increases, the ramified aggregates grow gradually and connect with each other, and finally a continuous thin film forms. Generally, thin films fabricated by various deposition methods, such as vapor deposition and sputtering etc., often develop residual intrinsic stress during deposition process [3]. Additional residual thermal stress may also be induced during the cooling process when a thermal expansion mismatch exists between the film and the substrate. The films containing large residual stresses are susceptible to generating buckles or cracks [4]. To date, many interesting stress relief patterns have been observed, including the well-known sinusoidal cracks, straight wrinkles, circle bubbles and telephone cord buckles [5,6].
* Corresponding author at: Graduate School of Information Science and Technology, Hokkaido University, Sapporo 060-0814, Japan. Tel.: +81 011 706 7695; fax: +81 011 706 7592. E-mail addresses: [email protected], [email protected] (P. Cai). 0169-4332/$ – see front matter ß 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2009.05.088
For the films on solid substrates, it is very difficult to observe the real stress distributions existing in the films from the appearance of the cracks directly since the crack appearance is determined not only by the stress distributions in the films, but also by the strong adhesive interaction between the films and the solid substrates. However, for the films on liquid substrates, these characteristic stress distributions can be detected following the crack traces since the adhesive interaction between the film and the liquid substrate is very small. On the other hand, because of the characteristic solid/liquid boundary condition and interactions among the atoms and atomic clusters on the liquid surfaces, anomalous stress distributions may exist in the nearly free sustained films. Thin films on liquid substrates provide us with an ideal physical system to study the internal stress distributions in the films. In this paper, we report the growth mechanism and sinusoidal stress relief patterns of a Ni film system deposited on silicone oil surfaces. Our experiment shows that the formation mechanism of the Ni films approximately obeys a two-stage growth model, which was firstly concluded from non-magnetic film systems on liquid substrates [7]. The sinusoidal crack pattern observed in the continuous Ni films represents a sinusoidal stress distribution in the films. The evolution of the crack appearance with the time in the Ni film system gives a physical picture that the internal stress
P.G. Cai et al. / Applied Surface Science 255 (2009) 8352–8358
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Fig. 1. Typical morphology of ramified nickel (Ni) aggregates on a silicone oil surface. f = 0.12 nm/s, d = 7 nm, Dt = 0.5 h. Image size is 125 100 mm2.
distribution evolves and the strain energy releases gradually after deposition. 2. Experiment The Ni film samples were prepared by thermal evaporation of 99.5% pure nickel in a vacuum of 6 104 Pa at room temperature. Commercial silicone oil (DOW CORNING 705 Diffusion Pump Fluid) with a vapor pressure below 108 Pa was painted onto a frosted glass surface, which was fixed 120 mm below the filament (tungsten). The resulting oil substrate with an area about 10 10 mm2 had a uniform thickness of 0.5 mm. The deposition rate f and the nominal film thickness d were determined by a quartz-crystal balance, which was calibrated by a profilemeter (astep 200 profilemeter, TENCOR). After deposition, the sample was kept in the vacuum chamber (in vacuum condition) for time interval Dt. Then the chamber was filled with air slowly and finally the sample was removed from the chamber. All images for the surface morphologies of the films were taken with an optical microscope (Leica DMLM). 3. Results Fig. 1 shows the typical morphology of ramified nickel (Ni) aggregates on silicone oil surfaces in the early stage of film growth. The bright and dark parts correspond to the Ni aggregates and uncovered oil surface, respectively. It is noted that the appearance of the Ni ramified aggregates is quite similar to those of Au and Ag films deposited on the liquid substrates [7,8]. However, the average branch width of the Ni aggregates measured in the experiment is about 3–5 mm, which is several times larger than those of the Au and Ag aggregates. According to the experimental observations, we draw a conclusion that the formation process of the Ni aggregates can be generally traced to a two-stage model. Firstly, Ni atoms deposited on the liquid surface nucleate and form atomic compact clusters with the sizes of about 3–5 mm. Then, during the subsequent stage, the compact clusters diffuse and rotate randomly on the liquid surface, which leads to formation of the ramified Ni aggregates. The fact that the growth behaviors of the Ni aggregates are quite similar to those of non-magnetic films gives evidence that the magnetic interaction between the Ni atoms and clusters does not play a key role in this stage. As the nominal film thickness increases, the ramified Ni aggregates grow gradually and connect with each other, and finally a continuous Ni film forms on the liquid substrate.
Fig. 2. Typical morphologies of sinusoidal cracks in the continuous Ni films deposited on a silicone oil surface. f = 0.46 nm/s, d = 55 nm, Dt = 1.33 h. Each image has a size of 1440 1065 mm2.
Our experiment shows that cracks are frequently observed in the continuous Ni films after the samples were removed from the vacuum chamber. The typical morphologies of cracks in the Ni films are shown in Fig. 2. The exciting and unexpected result is that the appearance of the cracks exhibits a periodic structure, which is quite similar to a sinusoidal curve. Furthermore, the parallel cracks appear closely with similar period T and amplitude A but no fixed correlation among their oscillatory phases can be concluded (see Fig. 2). Generally, the sinusoidal cracks start from the sample edges or from other broad cracks. According to the propagation directions of the cracks, one crack may bifurcate into two or three cracks, or two cracks may also coalesce harmoniously. Moreover, it can be seen from Fig. 2 that the amplitudes decrease gradually when the cracks are propagating towards the free edges. Finally, the cracks become straight lines perpendicular to the free edges and the sinusoidal feature diminishes. In our experiment, in the nominal film thickness range of d = 8– 80 nm and the deposition rate range of f = 0.06–0.86 nm/s, the sinusoidal cracks shown in Fig. 2 can be observed and no obvious dependence between the experimental conditions and the appearance of the sinusoidal cracks (i.e., T and A) can be detected. The maximum values of T and A observed in our experiment are about 580 and 275 mm, respectively. The longest sinusoidal crack observed in our samples can be over 1.32 mm and with more than 13 oscillatory periods. The length of the crack was measured from crack one end to the other end directly. Here we define a parameter: A kc 2 ; T
(1)
the maximum value of kc in our experiment is about 0.98. In order to identify the formation process of the cracks, a series of experiments was performed in which the time Dt was varied between Dt = 0.5 and 42 h. The experimental results show that, if
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0.5 Dt 3 h, most of the cracks appeared in the Ni films exhibit the sinusoidal appearance; if Dt > 6 h, however, all the cracks extend irregularly and the sinusoidal appearance disappears. This experimental result indicates that the sinusoidal cracks formed during the period when the vacuum chamber was filled with air. Therefore, we suggest that the sinusoidal cracks result from the internal stress in the films with the help of the pressure of the atmosphere and it is independent of the oxidation of the film in the air. Furthermore, it is noted that the number density of the cracks in the samples with Dt > 6 h is smaller than that in the samples with Dt 3 h. The evolution behaviors of the number density and the appearance of the cracks with Dt give the information that the internal stress field evolves and the stress energy releases gradually with the time. In order to further confirm the formation mechanism of the sinusoidal cracks described above, a simple test was taken: two Ni films with identical f and d but different Dt (Dt = 0.5 h, 12.5 h) were broken with a sharp pin. We found that the sinusoidal cracks are only observed in the film with Dt = 0.5 h, which again indicates that the sinusoidal internal stress patterns really exist in the free sustained films and they disappear gradually with the time Dt. On the other hand, similar sinusoidal cracks may also be observed in the aluminum (Al) film system deposited on the silicone oil surfaces and the average period of the cracks is less than those observed in the Ni film system. This phenomenon indicates that the sinusoidal appearance of the cracks should not be related to the magnetic interactions among the nickel atoms (or clusters). We suggest that the internal stresses of the Ni and Al films should be responsible for the sinusoidal cracks. In principle, crack patterns caused by contraction under internal tensile stress are common in both natural and manmade systems, including dried out fields, rocks, tectonic plates, paintings and various thin films. According to the previous studies, the propagation of a crack actually occurs only if the energy relieved by the advance of the crack is enough to overcome the fracture energy needed to elongate the crack and this propagation reduces the total energy [9,10]. Under quasi-static advance the energy reduction during propagation is infinitesimally small, and typically, there is only one possible direction of propagation for each crack. All other propagation directions would produce an increase in the total energy of the system. Because the stress near an existing crack is generally parallel to the crack edge, any crack nucleating at or growing to meet the preexisting crack will be perpendicular to the crack edge. Therefore, the sinusoidal cracks observed in the Ni films generally align perpendicular to the broad cracks or film edges, as shown in Fig. 2. Near the film edges, the stress component perpendicular to the edge is relaxed, and only the stress component parallel to the edge exists. Therefore, the sinusoidal cracks always change into straight lines perpendicular to the free edges during the propagation process (see Fig. 2). 4. Discussion 4.1. General theory According to the previous studies [5–7,11,12], strong residual internal stress always exists in thin films deposited on solid and liquid substrates and it releases gradually in order to minimize the total free energy of the systems. Therefore, characteristic stress relief patterns can be frequently observed in the films. Previous works have proved that the favourable direction in which the crack extends is perpendicular to the greatest local tensile stress [13], which provides us with a method to study the residual internal stress distributions based on the appearances of the cracks in the films.
Several works have attempted to theoretically understand the sinusoidal stress patterns in thin films [5,6,11,12,14]. The validity of the models put forward using the general theory of buckling of plates and the equation is given by [5]: !
@4 w @4 w @4 w @2 w @2 w D þ 2 þ s d þ s d þ x y @x4 @x2 @y2 @y4 @x2 @y2 þ 2t xy d
@2 w þ F ¼ 0; @x@y
(2)
where D = Ed3/12(1 m2) is the moment of inertia of the film; d is the film thickness; E is Young’s modulus and m is the Poisson’s ratio; x and y are the coordinates related to the substrate, w is the film deflection as defined in the elastic theory, sx and sy are the internal stresses, txy is the shear stress and F is the external force. One type of solutions of Eq. (2) which is physically acceptable is: w ¼ 1 þ cosðkx þ qyÞ:
(3)
Solution (3) implies that the direction of the highest stress is perpendicular to straight lines [5]: kx þ qy ¼ 2np;
n ¼ 0; 1; 2; . . . :
(4)
Introducing solution (3) into Eq. (2) shows that for each k value there are two permissible values of q: q¼
sx d D
2
k
1=2 ;
(5)
Therefore, two classes of these line families with slopes of jk/qj cross each other [5]. The stress relief pattern described by Eq. (4) can also be approximated by the expression [5]: qy ¼ pcosðkxÞ:
(6)
In fact, relation (6) can be satisfied by a series of real values of k and q, which may account for various sinusoidal stress relief patterns. We will show below that the stress relief patterns observed in the Ni films can be well explained by making use of this model. 4.2. The growth modes of the cracks In most cases, the sinusoidal cracks observed in our experiment generally exhibit the regular sine curves with well-defined period and amplitude, as shown in Fig. 3(a). The regularly sinusoidal crack mode can be well explained by using of the general theory of buckling of plates described above. If k and q are constants, Eq. (4) consists of two families of parallel lines with the slopes of jk/qj, as sketched in Fig. 3(b). Once the crack process begins, it propagates in a certain direction. When it reaches a point where two lines cross each other, it may jump to the line with different slope. This process goes on and finally the regularly sinusoidal pattern forms, as shown in Fig. 3(b). It is noted that the local internal stress in the Ni films is equibiaxial, and the two components of stress, defined as sx and sy, are equal. Therefore, the propagation distances of the crack along the two crossing lines are uniform and alternate, which permits to release the stress energy effectively in these two directions. In some cases, distorted sinusoidal cracks (like zigzag motif) can be observed in the Ni films, as shown in Fig. 4(a). Here we define the two neighboring side lengths of the zigzag crack as l1 and l2, respectively (see Fig. 4(a)). In order to further understand the propagating characteristics of the zigzag crack, the lengths l1 and l2 have been measured for the samples with different experiment conditions, and the results are shown in Fig. 4(b). We find that the length l2 increases linearly with the length l1, which can be well
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Fig. 3. Appearance of regularly sinusoidal cracks in the Ni films. (a) f = 0.14 nm/s, d = 55 nm, Dt = 0.5 h. Image size is 285 210 mm2; (b) sketch drawn for (a) as two classes of the line families with fixed slopes of jk/qj. The dark line indicates the growth process of the crack.
fitted by the function l2 = 2.5 + 1.95l1. To a good approximation, the relation can be expressed as: l2 2l1 :
(7)
Be similar to the regularly sinusoidal mode, two classes of parallel lines with fixed slopes of jk/qj also exist in the Ni films. However, the two components of the biaxial stress, sx and sy, in this case are not equal. In order to balance the stress energy of the system, the propagation distance of the crack along one direction is always larger than the other one, and therefore the zigzag crack mode appears, as shown in Fig. 4(c). Our experiment shows that when the sinusoidal cracks extend towards the free edges of the sample, the amplitudes of the cracks always attenuate gradually and finally change into linear cracks (see Fig. 2). The typical attenuation modes of the cracks are shown in Fig. 5(a) and (b). It is noted that although the amplitudes of the cracks decrease obviously during propagation, the periods of the cracks are almost unchanged. Because Eq. (6) can be satisfied by a series of different real values of k and q as mentioned in the general theory, k and q are not required to be constants in a sample. We suggest that the values of k and q are location dependent, especially near the free edges of the sample. When the cracks propagate towards the free edges, the values of jk/qj decreases gradually, as shown in Fig. 5(c). The attenuation of jk/qj implies that the stress component sx, perpendicular to the free edges, decreases gradually. Correspondingly, the equi-biaxial stress changes into axial stress gradually. Therefore, the cracks always align perpendicular to the film edges and the amplitudes decrease gradually during propagation. The experiment also shows that when the sinusoidal cracks extend in the Ni films, a gradual change in the propagation
Fig. 4. A motif of zigzag crack in the Ni films. (a) f = 0.10 nm/s, d = 16 nm, Dt = 1.5 h. Image size is 720 535 mm2; (b) sketch drawn for the figure above as two classes of the line families with fixed slopes of jk/qj. The dark line indicates the growth process of the crack. (c) The length l2 of zigzag crack as a function of the length l1. The solid line represents a linear fit of the experimental data.
direction occurs occasionally, as shown in Fig. 6(a). The maximum changed angle of the cracks measured in the experiment can be larger than 908. It is noted from Fig. 6(a) that the neighboring sinusoidal cracks remain parallel to each other. Another interesting phenomenon observed in the experiment is shown in Fig. 6(b), where we find that a big sinusoidal crack is composed of a series of small sinusoidal cracks with nearly uniform period and amplitude. The sinusoidal cracks are quite similar under different scales, showing the self-similarity of the crack morphologies in this case. The scaling factor is about 10. The formation process of the cracks shown in Fig. 6(a) and (b) can be sketched in Fig. 6(c). In these cases, the two components of the stress are not equal and evolve when the cracks propagate, which finally results in formation of the comparatively complex crack modes shown in Fig. 6(a) and (b).
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Fig. 5. Typical attenuation modes of the cracks in the Ni films. (a) f = 0.16 nm/s, d = 23 nm, Dt = 1.5 h. Image size is 360 265 mm2; (b) f = 0.09 nm/s, d = 35 nm, Dt = 0.5 h. Image size is 720 535 mm2; (c) sketch drawn for the figures above as the value of jk/qj undergoing a gradual decrease. The dark line indicates the growth process of the cracks.
Fig. 6. Typical morphologies of self-similar cracks in the Ni films. (a) f = 0.15 nm/s, d = 25 nm, Dt = 1.5 h. Image size is 360 265 mm2; (b) f = 0.08 nm/s, d = 25 nm, Dt = 1.5 h. Image size is 360 265 mm2; (c) sketch drawn for the figures above as the unusual growth behavior of the cracks. The dark line indicates the growth process of the cracks.
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4.3. The internal stress of the cracks In Eq. (6), we can define the amplitude A = p/q and the wave vector k = 2p/T. Hence, Eq. (5) can be written as: D p2 4 1 sx ¼ þ 2 ; (8) 2 d T A then introduce the expression of D into Eq. (8) [5]
sx ¼
2 Ed p2 4 1 þ 2 ; 2 2 12ð1 m Þ T A
(9)
where the Young’s modulus E and the Poisson’s ratio m is 199.5 GPa and 0.312, respectively. In our samples, both T and 2A are about 102 mm size, thus Eq. (9) gives the order of the stress s 105 Pa. We find that the sinusoidal cracks occur spontaneously for intrinsic stresses above a threshold value. When T and 2A equal their maximum values, corresponding to d = 20 nm, the intrinsic stress reaches its threshold value which is approximately 2 103 Pa. Combining the crack morphologies and Eq. (9), we may approach a concrete analysis for the stress problems, the diagram of the stress s of the crack at the bottom of Fig. 5(b) is shown in Fig. 7. It can be seen from Fig. 7(a) that the value of the stress s increases clearly along the propagation direction of the crack. Combining Eqs. (1) and (8), the dependence of sx upon period T is given by: D p2 4 sx ¼ 2 4 þ 2 ; (10) kc dT which suggests that sx is proportional to 1/T2, indicating that there is indeed a strong dependence of crack wavelength on
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stress, which is in agreement with those proposed by Gioia and Ortiz [15]. In addition, at the tip of a sinusoidal crack, A tends to zero, therefore kc changes to zero (see Eq. (1)). Accordingly, we can conclude that sx approaches to infinite when kc trends towards zero (see Eq. (10)). This phenomenon is a consequence of the fact that as soon as a single crack extends a small distance, the stress at its tip increases, and the stresses at the tips of the other cracks decrease. This generates an unstable and rapid propagation of the single crack [16]. 4.4. The velocity of the cracks Since the total crack formation process is finished in few minutes during the period when the vacuum chamber is filled with air, the extending velocity v of the cracks is not easy to be measured experimentally. Therefore, we settle this problem in theory. The crack patterns in Fig. 5 can be fitted well by the function A exp(x/ l) cos(kx + x0), where l is the attenuation length, and the amplitude of the oscillation obeys a Ginzburg–Landan equation [17]:
@t A ¼ eA A3 ;
(11)
and making use of the relation @t A ¼ v@x A, we obtain an attenuation oscillation solution: A ¼ A0 expðxjej=vÞ;
(12)
where A denotes the amplitude of oscillation, e is the reduced control parameter, e ðv vc Þ=vc , and here x is the x position of the tip. The transition from linear to oscillating cracks happens when the velocity is a little below vc , which is the critical value of v. Combining Fig. 5 and Eq. (12), Fig. 6(b) displays the diagram of the dependence of v=ðeÞ upon x. We find that the change law of v upon x is similar to v=ðeÞ upon x since v is the increasing function of v=ðeÞ. Therefore, Fig. 6(b) shows that v drops quickly; then it increases gradually; until it reaches its secondary maximum value, finally drops again along the propagation direction of the crack. 5. Conclusion In conclusion, we have studied the formation mechanism and sinusoidal stress patterns of the nickel films deposited on silicone oil surfaces. The growth mechanism of the films can be approximately traced to the two-stage growth model and the stress relief patterns can be well explained by making use of the general theory of plates buckling. We show that the two-stage growth process and the sinusoidal stress patterns do not relate to the magnetic interactions among the atoms and they are resulted mainly from the characteristic free boundary condition of the nearly free sustained films. Unfortunately, several tough problems still remain poorly understood, such as the details of the interactions among the Ni atoms and between the films and the liquid substrates, which should be responsible for the film microstructures and the sinusoidal crack patterns. Therefore, many new avenues of investigation are still open to us. The phenomena shown in Figs. 2–6 strongly suggest that the free sustained films may possess characteristic microstructures and therefore they may exhibit anomalous physical properties. We hope that the results above would contribute to further research on theoretical and practical projects in thin film physics. Acknowledgements
Fig. 7. (a) Dependence between the stress s and the crack length l; (b) dependence between v=ðeÞ and l. This measurement was taken for the sample shown in Fig. 5(b).
This work was supported by the National Natural Science Foundations of China (grant nos. 60777034 and 10874159) and by the Natural Science Foundation of Zhejiang Province in China (grant no. Y7080042).
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