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An internal stress pattern in free standing films
Physics Letters A 312 (2003) 119–122 www.elsevier.com/locate/pla
An internal stress pattern in free standing films Ping-Gen Cai, Sen-Jiang Yu, Quan-Lin Ye, Jin-Sheng Jin, Gao-Xiang Ye ∗ Department of Physics, Zhejiang University, Hangzhou 310028, PR China Received 22 January 2003; received in revised form 28 February 2003; accepted 11 April 2003 Communicated by J. Flouquet
Abstract We report a sinusoid appearance of cracks existing in a continuous nickel (Ni) film system deposited on silicone oil surfaces. The sinusoid cracks start from the sample edges or from other cracks, then their oscillatory amplitudes decrease gradually as they extend and finally disappear. One crack may bifurcate into two or three cracks, or two cracks may coalesce harmoniously. The sinusoid appearance of the cracks represents a sinusoid stress pattern in the Ni films, which mainly results from the characteristic boundary condition and interactions among the atoms in these free sustained films. 2003 Elsevier Science B.V. All rights reserved. PACS: 62.40.+i; 68.55.-a; 68.37.-d Keywords: Thin film growth; Stress pattern; Crack; Liquid substrate
1. Introduction Beautiful ripples on a calm lake made by a gentle wind represent a surface tension diagram on the water surface. Thin films with a characteristic corrugated morphology are often observed on hot milk surfaces, mirroring the pattern of the internal stress in the films. Similar corrugated surface morphology was also observed in a silver film system deposited on silicone oil surfaces by sputtering method [1], implying an oscillatory pattern of the internal stress and surface tension in the system [2]. Here we report a sine-like appearance of cracks existing in a continuous nickel (Ni) film system deposited on silicone oil surfaces. The experiment indicates that a sinusoid internal stress pat* Corresponding author.
E-mail address: [email protected] (G.-X. Ye).
tern really forms in the Ni films. This sinusoid stress pattern results from the characteristic boundary condition and the intrinsic interactions among the atoms in these nearly free sustained films, which will in principle affect the growth mechanisms, microstructures and properties of various film systems (multilayer films, for instance) not only on liquid substrates, but also on soft and solid substrates [3–6].
2. Experiment The Ni film samples were prepared by thermal evaporation of 99.5% pure nickel in a vacuum of 6 ×10−4 Pa at room temperature. Commercial silicone oil (DOW CORNING 705 Diffusion Pump Fluid) with −8 a vapor pressure below 10 Pa was painted onto a frosted glass surface, which was fixed 120 mm below
0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00612-1
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P.-G. Cai et al. / Physics Letters A 312 (2003) 119–122
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 (αstep 200 profilemeter, TENCOR). After deposition, the sample was kept in the vacuum chamber (in vacuum condition) for time interval t. 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 and discussion Fig. 1 shows the surface morphologies of two samples, in which continuous Ni films (white) with various cracks (black) are observed. The most exciting and unexpected result is that the appearance of the cracks exhibits a periodic structure, which is quite similar to a sinusoid curve. In many cases, a number of parallel sinusoid cracks may appear closely with similar period T and amplitude A (see Fig. 1(a)) and no fixed correlation among their oscillatory phases can be concluded. Generally, the sinusoid cracks start from the sample edges or from other cracks, then their amplitudes decrease gradually as they extend and finally disappear. It is observed frequently that one crack may bifurcate into two or three cracks, or two cracks may coalesce harmoniously, as shown in Fig. 1. In our experiment, in the nominal film thickness range of d = 8–80 nm, the deposition rate range of f = 0.06–0.86 nm/s and the time interval t range of t = 0.5–3 hours, all the phenomena described above can be observed and no obvious dependence between the experimental conditions and the appearance of the sinusoid cracks (i.e., T and A) can be detected. The maximum values of T and A observed in our experiment are Tmax ≈ 5.50 × 102 µm and Amax ≈ 5.80 × 102 µm, respectively. The longest sinusoid crack observed in our samples is over 1.320 × 103 µm and with more than 13 oscillatory periods. Here we define a parameter kc ≡ A/T and the maximum value of which in our experiment is about 0.98. For the samples with d < 8 nm, however, the sinusoid appearance of the cracks disappears.
Fig. 1. Surface morphologies of two Ni films deposited on silicone oil surfaces. (a) f = 0.14 nm/s, d = 55 nm, t = 0.5 h. Image size is 285 × 210 µm2 ; (b) f = 0.46 nm/s, d = 55 nm, t = 1.33 h. Image size is 1440 × 1065 µm2 .
In order to identify the formation process of the cracks, a series of experiments was performed in which the time t was varied between t = 0.5 and 42 hours. The experimental results show that, if t 3 hours, most of the cracks appeared on the Ni film surfaces exhibit the sinusoid appearance; if t > 6 hours, however, all the cracks extend irregularly and the sinusoid appearance disappears (Fig. 2). This experimental result implies that the sinusoid cracks formed during the period when the vacuum chamber was filled with air. Furthermore, it is noted that the crack density in the sample shown in Fig. 2 is smaller than that of the samples in Fig. 1. The appearance of the cracks depends closely on the internal stress pattern in the homogeneous Ni films. The favourable direction that the crack extends is perpendicular to the greatest local tensile stress [7]. Therefore, the sinusoid appearance of the cracks in Fig. 1 represents the sinusoid pattern of the internal stress in the Ni films. The amplitudes, periods and the correlations among the oscillatory phases of different
P.-G. Cai et al. / Physics Letters A 312 (2003) 119–122
121
Fig. 2. Appearance of the cracks on a Ni film sample. f = 0.15 nm/s, d = 25 nm, t = 15.5 h. Image size is 360 × 265 µm2 .
cracks (see Fig. 1) show the diagram of the internal stress field in the films. The decrease of the crack density with the time t described above gives the information that the internal stress field evolves and the strain energy releases gradually with the time t. In order to confirm the phenomena above, a simple test was taken: two Ni films with identical f and d but different t were broken with a sharp pin and the crack appearances of them are shown in Fig. 3. It should be noted that the sinusoid cracks are only observed in the film with t = 0.5 hour (Fig. 3(a)), which again indicates that the sinusoid internal stress really exists in the free standing films and it disappears gradually with the time t, as shown in Fig. 3(b). On the other hand, we find that the sinusoid appearance of the cracks should not relate to the magnetic interactions among the nickel atoms since the phenomenon shown in Fig. 3 was also observed in aluminum (Al) films deposited on silicone oil surfaces. Based on the experimental observations, we propose that the sinusoid stress pattern mainly results from the characteristic boundary condition and interactions among the atoms in these free sustained films. Several works have attempted to theoretically understand these stress patterns in thin films [8–10]. The validity of the models put forward using the general theory of buckling of plates and the buckling equation is given by [8] 4 ∂ 4w ∂ 4w ∂ 2w ∂ 2w ∂ w + 2 + d + σ d + σ D x y ∂x 4 ∂x 2 ∂y 2 ∂y 4 ∂x 2 ∂y 2 + 2τxy d
∂ 2w + F = 0, ∂x∂y
(1)
Fig. 3. Surface morphologies of two broken Ni films. Each image has a size of 285 × 210 µm2 . (a) f = 0.15 nm/s, d = 38 nm, t = 0.5 h; (b) f = 0.15 nm/s, d = 38 nm, t = 12.5 h.
where D is the moment of inertia of the film, d is the film thickness, x and y are the coordinates relative to the substrate, w is the film coordinate as defined in the elastic theory, σx and σy are the internal compressive stresses, τxy is the shear stress and F is the external force. One type of solutions of Eq. (1) which is physically acceptable is w = 1 + cos(kx + qy).
(2)
From solution (2) it implies that the direction of the highest stress is perpendicular to straight lines [8]: kx + qy = 2nπ,
n = 0, ±1, ±2, . . ..
(3)
Introducing solution (2) into Eq. (1) shows that for each k value there are two permissible values of q, so two classes of these families with slopes of ±|k/q| cross each other [8]. If there exist only two classes of the families with slopes of ±|k/q|, then the crack will exhibit periodic sinusoid pattern. Once a crack develops, the crack progresses in a certain direction; when it reaches a point where the lines cross each other, it may turn onto
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either of them. The sinusoid cracks shown in Fig. 1 imply that Eq. (1) may also be used to describe the stress patterns in the films on liquid substrates. This stress pattern described by Eq. (3) can also be approximated by the expression [8] qy = π cos(kx).
(4)
The relation (4) can be satisfied by many real values of k and q. From this comparative study it is clear that one kind of the stress relates to one value of k and q. In Fig. 1, the amplitudes and periods of the cracks change obviously as they extend, which implies that the k and q are location dependence in our samples. With the development of a crack, it begins in a certain line, when it reaches a point where two lines interlace each other, it may jump into another line with different slope. Finally, the location dependent behavior of k and q results in the characteristic of crack pattern in Fig. 1 and no fixed correlation among their oscillatory amplitudes and periods can be concluded. According to the theoretical analysis, the relation between the kc and θc , is given by tan θc = 2kc ,
(5)
where we define tan θc ≡ |k/q|. In Eq. (5), one finds that θc and kc reach their maximum values together. Since the maximum value of kc observed in our experiment is about 0.98, correspondingly, the upper bound of θc equals to π/3 approximately. While θc approaches its lower bound, i.e., θc = 0, the sinusoid patterns turn into straight lines (see Figs. 2 and 3(b)). However, to make a full understanding of the phenomenon in Fig. 1, further studies, including the physical explanation for the limiting values of θc , are still needed.
4. Conclusion The phenomena shown in Figs. 1–3 imply that the interactions may be rather regular and the physical effects are quite plentiful in free standing films, which
may in principle result in characteristic microstructures and subsequently anomalous properties of the films. The experiment above also provides us with a simple but effective method to study the internal stress field existing in free standing films. Up to now, the details of the interactions among the atoms in the free standing films, which should be responsible for the exact patterns of the internal stress fields, still remain poorly understood. Therefore, many new avenues of investigation are still open to us. In particular, to explain the sinusoid pattern of the internal stress fields existing in the free standing films theoretically is a big challenge. This will allow a new branch of thin film studies that fabricate various free standing film systems on different substrates and study their properties.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 10174063) and by the Natural Science Foundation of Zhejiang Province in China (Grant No. 1997-RC9603).
References [1] G.X. Ye, Q.R. Zhang, C.M. Feng, H.L. Ge, Z.K. Jiao, Phys. Rev. B 54 (1996) 14754. [2] D.R.M. Williams, Phys. Rev. Lett. 75 (1995) 453. [3] G.X. Ye, Th. Michely, V. Weidenhof, I. Friedrich, M.N. Wuttig, Phys. Rev. Lett. 81 (1998) 622. [4] G.J. Kovacs, P.S. Vincett, Thin Solid Films 111 (1984) 65. [5] K.M. Rao, et al., J. Phys. D: Appl. Phys. 32 (1999) 2327. [6] G.X. Ye, A.G. Xia, G.L. Gao, Y.F. Lao, X.M. Tao, Phys. Rev. B 63 (2001) 125405. [7] N.E. Frost, K.J. Marsh, L.P. Pook, Metal Fatigue, Oxford Univ. Press, Oxford, 1974. [8] D. Nir, Thin Solid Films 112 (1984) 41. [9] N. Matuda, S. Baba, A. Kinbara, Thin Solid Films 81 (1981) 301. [10] S.B. Iyer, K.S. Harshavardhan, V. Kumar, Thin Solid Films 256 (1995) 94.
An internal stress pattern in free standing films Ping-Gen Cai, Sen-Jiang Yu, Quan-Lin Ye, Jin-Sheng Jin, Gao-Xiang Ye ∗ Department of Physics, Zhejiang University, Hangzhou 310028, PR China Received 22 January 2003; received in revised form 28 February 2003; accepted 11 April 2003 Communicated by J. Flouquet
Abstract We report a sinusoid appearance of cracks existing in a continuous nickel (Ni) film system deposited on silicone oil surfaces. The sinusoid cracks start from the sample edges or from other cracks, then their oscillatory amplitudes decrease gradually as they extend and finally disappear. One crack may bifurcate into two or three cracks, or two cracks may coalesce harmoniously. The sinusoid appearance of the cracks represents a sinusoid stress pattern in the Ni films, which mainly results from the characteristic boundary condition and interactions among the atoms in these free sustained films. 2003 Elsevier Science B.V. All rights reserved. PACS: 62.40.+i; 68.55.-a; 68.37.-d Keywords: Thin film growth; Stress pattern; Crack; Liquid substrate
1. Introduction Beautiful ripples on a calm lake made by a gentle wind represent a surface tension diagram on the water surface. Thin films with a characteristic corrugated morphology are often observed on hot milk surfaces, mirroring the pattern of the internal stress in the films. Similar corrugated surface morphology was also observed in a silver film system deposited on silicone oil surfaces by sputtering method [1], implying an oscillatory pattern of the internal stress and surface tension in the system [2]. Here we report a sine-like appearance of cracks existing in a continuous nickel (Ni) film system deposited on silicone oil surfaces. The experiment indicates that a sinusoid internal stress pat* Corresponding author.
E-mail address: [email protected] (G.-X. Ye).
tern really forms in the Ni films. This sinusoid stress pattern results from the characteristic boundary condition and the intrinsic interactions among the atoms in these nearly free sustained films, which will in principle affect the growth mechanisms, microstructures and properties of various film systems (multilayer films, for instance) not only on liquid substrates, but also on soft and solid substrates [3–6].
2. Experiment The Ni film samples were prepared by thermal evaporation of 99.5% pure nickel in a vacuum of 6 ×10−4 Pa at room temperature. Commercial silicone oil (DOW CORNING 705 Diffusion Pump Fluid) with −8 a vapor pressure below 10 Pa was painted onto a frosted glass surface, which was fixed 120 mm below
0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00612-1
120
P.-G. Cai et al. / Physics Letters A 312 (2003) 119–122
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 (αstep 200 profilemeter, TENCOR). After deposition, the sample was kept in the vacuum chamber (in vacuum condition) for time interval t. 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 and discussion Fig. 1 shows the surface morphologies of two samples, in which continuous Ni films (white) with various cracks (black) are observed. The most exciting and unexpected result is that the appearance of the cracks exhibits a periodic structure, which is quite similar to a sinusoid curve. In many cases, a number of parallel sinusoid cracks may appear closely with similar period T and amplitude A (see Fig. 1(a)) and no fixed correlation among their oscillatory phases can be concluded. Generally, the sinusoid cracks start from the sample edges or from other cracks, then their amplitudes decrease gradually as they extend and finally disappear. It is observed frequently that one crack may bifurcate into two or three cracks, or two cracks may coalesce harmoniously, as shown in Fig. 1. In our experiment, in the nominal film thickness range of d = 8–80 nm, the deposition rate range of f = 0.06–0.86 nm/s and the time interval t range of t = 0.5–3 hours, all the phenomena described above can be observed and no obvious dependence between the experimental conditions and the appearance of the sinusoid cracks (i.e., T and A) can be detected. The maximum values of T and A observed in our experiment are Tmax ≈ 5.50 × 102 µm and Amax ≈ 5.80 × 102 µm, respectively. The longest sinusoid crack observed in our samples is over 1.320 × 103 µm and with more than 13 oscillatory periods. Here we define a parameter kc ≡ A/T and the maximum value of which in our experiment is about 0.98. For the samples with d < 8 nm, however, the sinusoid appearance of the cracks disappears.
Fig. 1. Surface morphologies of two Ni films deposited on silicone oil surfaces. (a) f = 0.14 nm/s, d = 55 nm, t = 0.5 h. Image size is 285 × 210 µm2 ; (b) f = 0.46 nm/s, d = 55 nm, t = 1.33 h. Image size is 1440 × 1065 µm2 .
In order to identify the formation process of the cracks, a series of experiments was performed in which the time t was varied between t = 0.5 and 42 hours. The experimental results show that, if t 3 hours, most of the cracks appeared on the Ni film surfaces exhibit the sinusoid appearance; if t > 6 hours, however, all the cracks extend irregularly and the sinusoid appearance disappears (Fig. 2). This experimental result implies that the sinusoid cracks formed during the period when the vacuum chamber was filled with air. Furthermore, it is noted that the crack density in the sample shown in Fig. 2 is smaller than that of the samples in Fig. 1. The appearance of the cracks depends closely on the internal stress pattern in the homogeneous Ni films. The favourable direction that the crack extends is perpendicular to the greatest local tensile stress [7]. Therefore, the sinusoid appearance of the cracks in Fig. 1 represents the sinusoid pattern of the internal stress in the Ni films. The amplitudes, periods and the correlations among the oscillatory phases of different
P.-G. Cai et al. / Physics Letters A 312 (2003) 119–122
121
Fig. 2. Appearance of the cracks on a Ni film sample. f = 0.15 nm/s, d = 25 nm, t = 15.5 h. Image size is 360 × 265 µm2 .
cracks (see Fig. 1) show the diagram of the internal stress field in the films. The decrease of the crack density with the time t described above gives the information that the internal stress field evolves and the strain energy releases gradually with the time t. In order to confirm the phenomena above, a simple test was taken: two Ni films with identical f and d but different t were broken with a sharp pin and the crack appearances of them are shown in Fig. 3. It should be noted that the sinusoid cracks are only observed in the film with t = 0.5 hour (Fig. 3(a)), which again indicates that the sinusoid internal stress really exists in the free standing films and it disappears gradually with the time t, as shown in Fig. 3(b). On the other hand, we find that the sinusoid appearance of the cracks should not relate to the magnetic interactions among the nickel atoms since the phenomenon shown in Fig. 3 was also observed in aluminum (Al) films deposited on silicone oil surfaces. Based on the experimental observations, we propose that the sinusoid stress pattern mainly results from the characteristic boundary condition and interactions among the atoms in these free sustained films. Several works have attempted to theoretically understand these stress patterns in thin films [8–10]. The validity of the models put forward using the general theory of buckling of plates and the buckling equation is given by [8] 4 ∂ 4w ∂ 4w ∂ 2w ∂ 2w ∂ w + 2 + d + σ d + σ D x y ∂x 4 ∂x 2 ∂y 2 ∂y 4 ∂x 2 ∂y 2 + 2τxy d
∂ 2w + F = 0, ∂x∂y
(1)
Fig. 3. Surface morphologies of two broken Ni films. Each image has a size of 285 × 210 µm2 . (a) f = 0.15 nm/s, d = 38 nm, t = 0.5 h; (b) f = 0.15 nm/s, d = 38 nm, t = 12.5 h.
where D is the moment of inertia of the film, d is the film thickness, x and y are the coordinates relative to the substrate, w is the film coordinate as defined in the elastic theory, σx and σy are the internal compressive stresses, τxy is the shear stress and F is the external force. One type of solutions of Eq. (1) which is physically acceptable is w = 1 + cos(kx + qy).
(2)
From solution (2) it implies that the direction of the highest stress is perpendicular to straight lines [8]: kx + qy = 2nπ,
n = 0, ±1, ±2, . . ..
(3)
Introducing solution (2) into Eq. (1) shows that for each k value there are two permissible values of q, so two classes of these families with slopes of ±|k/q| cross each other [8]. If there exist only two classes of the families with slopes of ±|k/q|, then the crack will exhibit periodic sinusoid pattern. Once a crack develops, the crack progresses in a certain direction; when it reaches a point where the lines cross each other, it may turn onto
122
P.-G. Cai et al. / Physics Letters A 312 (2003) 119–122
either of them. The sinusoid cracks shown in Fig. 1 imply that Eq. (1) may also be used to describe the stress patterns in the films on liquid substrates. This stress pattern described by Eq. (3) can also be approximated by the expression [8] qy = π cos(kx).
(4)
The relation (4) can be satisfied by many real values of k and q. From this comparative study it is clear that one kind of the stress relates to one value of k and q. In Fig. 1, the amplitudes and periods of the cracks change obviously as they extend, which implies that the k and q are location dependence in our samples. With the development of a crack, it begins in a certain line, when it reaches a point where two lines interlace each other, it may jump into another line with different slope. Finally, the location dependent behavior of k and q results in the characteristic of crack pattern in Fig. 1 and no fixed correlation among their oscillatory amplitudes and periods can be concluded. According to the theoretical analysis, the relation between the kc and θc , is given by tan θc = 2kc ,
(5)
where we define tan θc ≡ |k/q|. In Eq. (5), one finds that θc and kc reach their maximum values together. Since the maximum value of kc observed in our experiment is about 0.98, correspondingly, the upper bound of θc equals to π/3 approximately. While θc approaches its lower bound, i.e., θc = 0, the sinusoid patterns turn into straight lines (see Figs. 2 and 3(b)). However, to make a full understanding of the phenomenon in Fig. 1, further studies, including the physical explanation for the limiting values of θc , are still needed.
4. Conclusion The phenomena shown in Figs. 1–3 imply that the interactions may be rather regular and the physical effects are quite plentiful in free standing films, which
may in principle result in characteristic microstructures and subsequently anomalous properties of the films. The experiment above also provides us with a simple but effective method to study the internal stress field existing in free standing films. Up to now, the details of the interactions among the atoms in the free standing films, which should be responsible for the exact patterns of the internal stress fields, still remain poorly understood. Therefore, many new avenues of investigation are still open to us. In particular, to explain the sinusoid pattern of the internal stress fields existing in the free standing films theoretically is a big challenge. This will allow a new branch of thin film studies that fabricate various free standing film systems on different substrates and study their properties.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 10174063) and by the Natural Science Foundation of Zhejiang Province in China (Grant No. 1997-RC9603).
References [1] G.X. Ye, Q.R. Zhang, C.M. Feng, H.L. Ge, Z.K. Jiao, Phys. Rev. B 54 (1996) 14754. [2] D.R.M. Williams, Phys. Rev. Lett. 75 (1995) 453. [3] G.X. Ye, Th. Michely, V. Weidenhof, I. Friedrich, M.N. Wuttig, Phys. Rev. Lett. 81 (1998) 622. [4] G.J. Kovacs, P.S. Vincett, Thin Solid Films 111 (1984) 65. [5] K.M. Rao, et al., J. Phys. D: Appl. Phys. 32 (1999) 2327. [6] G.X. Ye, A.G. Xia, G.L. Gao, Y.F. Lao, X.M. Tao, Phys. Rev. B 63 (2001) 125405. [7] N.E. Frost, K.J. Marsh, L.P. Pook, Metal Fatigue, Oxford Univ. Press, Oxford, 1974. [8] D. Nir, Thin Solid Films 112 (1984) 41. [9] N. Matuda, S. Baba, A. Kinbara, Thin Solid Films 81 (1981) 301. [10] S.B. Iyer, K.S. Harshavardhan, V. Kumar, Thin Solid Films 256 (1995) 94.