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Experimental observation of ordered stress patterns in iron films deposited on silicone oil surfaces
Physics Letters A 318 (2003) 457–462 www.elsevier.com/locate/pla
Experimental observation of ordered stress patterns in iron films deposited on silicone oil surfaces Quan-Lin Ye, Xiao-Jun Xu, Ping-Gen Cai, A-Gen Xia, Gao-Xiang Ye ∗ Department of Physics, Zhejiang University, Hangzhou 310027, PR China Received 5 July 2003; accepted 3 September 2003 Communicated by J. Flouquet
Abstract We report the observations of large ordered stress patterns, namely bands, in iron films deposited on silicone oil surfaces. The bands grow from the sample edges and then extend into the central regions of the films after deposition. The total length of one band may be more than 5 mm. If two bands meet on their ways, they may cross each other without interference or coalesce harmoniously. The inverse situation that one band bifurcate into two bands is also observed. The X-ray diffraction experiment shows that the nearly free sustained iron films still exhibit polycrystalline structures and our further experiment indicates that the bands result from the ordered aggregation of the iron crystal grains. 2003 Elsevier B.V. All rights reserved. PACS: 62.40.+i; 68.55.-a; 68.90.+g Keywords: Thin film; Stress pattern; Liquid substrate
1. Introduction
Vapor phase deposition of metals on liquid substrates was studied in a number of recent investigations [1–4]. The experiments show that as the nominal film thickness increases, silver or gold atoms deposited on liquid surfaces nucleate and form compact clusters first. Then they diffuse and rotate randomly, which leads to the formation of branched atomic aggregates.
* Corresponding author.
E-mail address: [email protected] (G.-X. Ye). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.09.015
Finally the aggregates connect one another and a continuous film forms. It is expected that the films deposited on liquid substrates would exhibit anomalous properties because of their characteristic microstructures and internal stresses. Here we report a large ordered stress pattern existing in an iron film system deposited on silicone oil surfaces. The phenomena we describe for these nearly free sustained iron films show a distinctive effect of the weak interaction between the metallic films and the liquid substrates, which will in principle affect the growth mechanisms, microstructures and physical properties of various magnetic film systems (multilayer films and superlattices, for instance) not only on liquid substrates but also on soft polymer substrates [5,6].
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Q.-L. Ye et al. / Physics Letters A 318 (2003) 457–462
2. Experiment The samples were prepared by thermal evaporation of 99.58% pure iron in a vacuum of 6 × 10−4 Pa at room temperature. Commercial silicone oil (DOW CORNING 705 Diffusion Pump Fluid) with a vapor pressure below 10−8 Pa was painted onto a frosted glass surface, which was fixed 125 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 profilometer (α-step 200 profilometer, TENCOR). After deposition, the sample was kept in the vacuum chamber (in vacuum condition) for time interval t and then removed from the chamber. All images for the surface morphologies of the samples were taken with an optical microscope (Leica DMLM), equipped with a CCD camera (Leica DC 300). In our experiment, the ranges of d = 5– 60 nm and f = 0.1–1.8 nm/s. In these ranges, the iron films are almost transparent if they are glanced with the naked eye. Fortunately, they can be observed clearly by the optical microscope. Images for the top surface of an iron film were taken by the optical microscope directly before the film was separated from the oil substrate. In order to characterize the bottom surface of the iron film, i.e., the surface in contact with the oil substrate during deposition, a special method was developed to separate the film from the oil substrate: a clean surface
of a piece of glass was used to carefully touch the top surface of the film, which would stick immediately on the glass surface. After washed with acetone and ethanol, a clean bottom surface of the film would appear on the glass surface. Therefore, images for the bottom surfaces of the films could be taken by the optical microscope after they were separated from the oil substrates.
3. Result and discussion The typical ordered stress pattern in the iron films is shown in Fig. 1, where it can be seen that the band-shaped structures, namely bands, are composed of a large number of parallel key-shaped domains, namely keys. Generally, the neighboring keys possess different width w but nearly uniform length L (see Fig. 1). The maximum values of L and w observed in our experiment are Lmax ≈ 5 × 102 µm and wmax ≈ 3.6 × 102 µm, respectively. The total length of one ordered band may be more than 5 mm. The morphologies of the bands in both the top and bottom surfaces of the films resemble each other in appearance (see Fig. 1) and, in many cases, the iron films break along the outlines of the keys, indicating that Fig. 1 shows not only the ordered surface morphology but also the ordered bulk structure of the film. An interesting and unexpected phenomenon is that, if two bands meet on their ways, they may cross each other without interference [Fig. 2(a)]. According to the growth
Fig. 1. An ordered stress pattern in an iron film deposited on silicone oil surface. t = 19 h, f = 0.1 nm/s, d = 15 nm. Each image has a size of 75 × 55 µm2 . (a) Image of a band on the top surface; (b) image of a band on the bottom surface.
Q.-L. Ye et al. / Physics Letters A 318 (2003) 457–462
459
Fig. 2. Images taken on the top surfaces of two iron films. Each image has a size of 90 × 65 µm2 . (a) A crossing of two bands. t = 10 h, f = 1.8 nm/s, d = 35 nm; (b) a coalesce of two bands. t = 10 h, f = 1.8 nm/s, d = 35 nm; (c) a beginning of a band at the edge of the substrate. t = 8 h, f = 1.5 nm/s, d = 45 nm; (d) an end of a band in the central region of the film. t = 8 h, f = 1.5 nm/s, d = 45 nm.
direction of the bands (see the description below), we can identify that two bands may coalesce or one band may bifurcate harmoniously [Fig. 2(b)]. In addition, in each band, the length of the keys near the sample edge is generally longer than that of the keys in the central region of the sample, however it seems that the change of the average key width w¯ is not quite obvious [Figs. 2(c) and (d)]. In our experiment, in the nominal film thickness range of d = 5–60 nm and the deposition rate range of f = 0.1 and 1.8 nm/s, all the phenomena described above are observed in the samples. For the samples with d < 5 nm, however, the band structure disappears. It is noted that the keys in each band do not exhibit periodic structures precisely. Instead, from the width distribution of the keys shown in Fig. 3, we find that, n, the number of the keys with the width w in each band, changes with the value of w and, besides
the main peak (peak I), a subordinate peak, which is marked as peak II in Fig. 3, is frequently observed. The subordinate peak may become more obvious if all the keys counted are from a suitable segment of the band. In many cases, w2 is twice or thrice as large as w1 , where w1 and w2 represent the corresponding key widths of peak I and peak II (see Fig. 3), respectively. In addition, the corresponding internal angles of the keys in each band, namely θ , are quite similar, as shown in Figs. 1 and 2. Although the keys in different bands present different values of θ , most of the bands exhibit θ ≈ 90◦ (Figs. 1 and 2). It was found that all the bands or keys formed after deposition but before the samples were removed from the vacuum chamber. In other words, the keys cannot grow obviously under ambient conditions. In order to identify the formation process of the bands, a series of experiments was performed in which the time t was
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Q.-L. Ye et al. / Physics Letters A 318 (2003) 457–462
Fig. 3. Width distributions of the keys in two bands, respectively. I and II are labeled for the main and subordinate peaks, respectively. (a) t = 40 h, f = 1.7 nm/s, d = 15 nm; (b) t = 7 h, f = 0.9 nm/s, d = 20 nm.
varied between t = 0.5 and 40 h, since we found that the phenomena shown in Figs. 1 and 2 can be understood qualitatively with the aid of experiments comparing different samples prepared with different time t. The experimental results show that, if t 2 h, neither bands nor keys can be observed in the iron films. For t > 2 h, however, keys start to appear at the edges of the samples. As the time t goes on, the number of the keys increases and the bands grow in both the width and length directions and extend into the central regions of the samples, as shown in Figs. 2(c) and (d). Some of the bands may grow steadily and cross the whole samples. If t 10 h, the growth process of the bands almost stops, indicating that the main stress energy in the films is released completely ten hours after deposition and the free energy approaches its minimum. Consequently, the films reach the stable states. The phenomena shown in Figs. 1, 2 and 3 imply that an ordered microstructure may exist in the iron films. The X-ray diffraction patterns show that the samples with d = 30 nm and f = 1.5 nm/s still exhibit polycrystalline structures, which is in agreement with that of the Ag films on silicone oil surfaces [7], and the average size of the crystal grains in the films is about 20 nm. This conclusion is also confirmed by the SEM (scanning electronic microscope) images for the bottom surfaces of the films. In addition, the magnetic powder pattern experiment [8] shows that the keys in the iron film samples do not exhibit detectable mag-
netic fields. These experimental results give evidence that the keys in the bands are neither single crystals nor single magnetic domains. Generally, ordered stress patterns in films depend closely on the interactions in the system [9]. We propose that the ordered bands may result from the effects of both the liquid substrate and the interactions among the iron crystal grains (or crystal grain clusters). In our experiment, since the interaction between the iron film and the silicone oil surface in the tangent direction is small, the iron film on the liquid substrate is nearly a free sustained film and therefore it can shape the microstructure so that the whole free energy approaches its minimum [10]. On the other hand, crystal grains in the iron film are quite free to diffuse and change their locations on the liquid substrate driven by the interactions among them. They may aggregate locally in some regions of the iron film, which finally results in the keys and then the bands in the iron film. It is believed that the symmetric and parallel structure of the keys in each band (Figs. 1, 2 and 3) may balance the space deformation and then reduce the total free energy of the film [10]. If the description above is correct, the difference between the material densities in and out of the keys should be detectable. Fig. 4 shows that many holes exist in the regions around the keys and the density in the keys is obviously larger than that of the other regions of the films. Furthermore, the cracks with observable widths, as shown in Fig. 4(b), also provide
Q.-L. Ye et al. / Physics Letters A 318 (2003) 457–462
461
Fig. 4. Morphologies of the keys in two iron films (top surfaces). Each image has a size of 145 × 105 µm2 . (a) t = 15 h, f = 0.6 nm/s, d = 20 nm; (b) t = 11.5 h, f = 0.6 nm/s, d = 36 nm.
evidence that the bands and keys really result from the local aggregation of the iron crystal grains (or crystal grains clusters). The material aggregation phenomenon described above depends closely on the internal stress in the iron films. It is well known that internal stresses generally exist in films after deposition and then release gradually in order to minimize the total free energy of the films. This stress relief will give rise to what appears to be various surface patterns or morphologies in the films [11,12]. Many works have previously applied a general theory of buckling of plates to analyze the stress relief patterns [9,12,13] and the buckling equation is given by [12]
∂ 4W ∂ 4W ∂ 4W + 2 + D ∂x 4 ∂x 2 ∂y 2 ∂y 4
+ σx d
∂ 2W ∂x 2
∂ 2W ∂ 2W + F = 0, + σy d 2 + 2τxy d ∂y ∂x∂y
kx + qy = 2nπ,
(1)
(2)
n = 0, ±1, ±2, . . ..
(3)
Introducing solution (2) into Eq. (1) shows that for each k there are two permissible values of q, therefore two classes of line families with slopes of ±|k/q| cross each other. If the buckling or crack processes in a certain direction, when it reaches the point where the two lines with the slopes of ±|k/q| cross each other, it turns onto the other line with different slope, and this process may finally result in a sinusoidal stress pattern [9,12,13]. The sinusoidal stress pattern can be described approximately by the expression [12,13] y = A cos(kx),
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 usually defined in 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).
Solution (2) implies that the direction of the highest stress is perpendicular to straight lines [12]:
(4)
where A = π/q is the amplitude, k = 2π/λ is the wave vector and λ is the wavelength. In fact, Eq. (4) may be satisfied by many different real values of k and q and therefore a pair of values of k and q corresponds to one kind of sinusoidal stress pattern, which implies that there may exist a series of different kinds of sinusoidal stress patterns simultaneously in the films, just as described in Ref. [13]. The ordered band structures shown in Figs. 1, 2 and 4 support this theoretical prediction, i.e., a series of sinusoidal stress patterns with different amplitudes and frequency spectrums may exist in the iron films and therefore the combination of these sinusoidal stress patterns may domain the material in the iron films and
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Q.-L. Ye et al. / Physics Letters A 318 (2003) 457–462
finally results in the rectangular and quasi-periodic stress patterns. In order to describe the characteristic stress patterns shown in Figs. 1, 2 and 4, here we define two parameters, i.e., the fundamental amplitude A0 ≈ L/2 and the fundamental wavelength λ0 = 2π/k0 ≈ 2w, ¯ where k0 is the fundamental wave vector. In our experiment, no obvious relations between the deposition conditions and the parameters A0 and λ0 could be detected. However, the bands with large A0 or λ0 are frequently found in the samples prepared with high deposition rates or large film thicknesses. Furthermore, we find in our experiment that the maximum value of λ0 is less than 100 µm and the maximum value of A0 /λ0 is about 10, which should be related to the scales of the stress patterns in the films.
new avenues of investigation are still open to us. The formation mechanism of the characteristic stress patterns in Figs. 1, 2 and 4 presents us with an example that the films deposited on liquid substrates may possess characteristic microstructures and therefore they may exhibit anomalous physical properties. This will allow a new class of thin film studies that fabricate various magnetic film systems on different liquid substrates and study their properties.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 10174063 and 20274040) and by the Natural Science Foundation of Zhejiang Province in China (Grant No. 1997RC9603).
4. Conclusion In conclusion, we have described the ordered stress patterns in the iron films deposited on silicone oil surfaces. After deposition, the ordered bands grow from the sample edges and then extend into the central regions of the continuous iron films. Various peculiarities of these bands are also presented, like the crossing and bifurcation behaviors of the bands. Our experiments show that the ordered stress patterns result from the aggregation of the iron crystal grains in this nearly free sustained film system. Furthermore, the phenomena presented in this Letter indicate that a series of sinusoidal stress patterns with different amplitudes and frequency spectrums may exist in the iron films and therefore the combination of these sinusoidal stress patterns may domain the material in the iron films and finally results in the rectangular and quasi-periodic stress patterns. Up to now, the details of the interactions among the iron crystal grains and between the iron films and the liquid substrates, which should be responsible for the quasi-periodic stress patterns and the film microstructures, still remain poorly understood. Therefore, many
References [1] G.X. Ye, Th. Michely, V. Weidenhof, I. Friedrich, M. Wuttig, Phys. Rev. Lett. 81 (1998) 622. [2] Th. Michely, G.X. Ye, V. Weidenhof, M. Wuttig, Surf. Sci. 432 (1999) 228. [3] B. Yang, J. Scheidtmann, J. Mayer, M. Wuttig, Th. Michely, Surf. Sci. 497 (2002) 100. [4] G.X. Ye, A.G. Xia, G.L. Gao, Y.F. Lao, X.M. Tao, Phys. Rev. B 63 (2001) 125405. [5] G.J. Kovacs, P.S. Vincett, Thin Solid Films 111 (1984) 65. [6] K.M. Rao, M. Pattabi, S.R. Sainkar, A. Lobo, S.K. Kulkanrni, J. Uchil, M.S. Sastry, J. Phys. D: Appl. Phys. 32 (1999) 2327. [7] G.X. Ye, Q.R. Zhang, C.M. Feng, H.L. Ge, Z.K. Jiao, Phys. Rev. B 54 (1996) 14754. [8] K.H. Stewart, in: Ferromagnetic Domains, Cambridge Univ. Press, Cambridge, 1954, pp. 32–34, 81–92. [9] S.B. Iyer, K.S. Harshavardhan, V. Kumar, Thin Solid Films 256 (1995) 94. [10] D.R.M. Williams, Phys. Rev. Lett. 75 (1995) 453. [11] J. Muller, M. Grant, Phys. Rev. Lett. 82 (1999) 1736. [12] D. Nir, Thin Solid Films 112 (1984) 41. [13] P.G. Cai, S.J. Yu, Q.L. Ye, J.S. Jin, G.X. Ye, Phys. Lett. A 312 (2003) 119.
Experimental observation of ordered stress patterns in iron films deposited on silicone oil surfaces Quan-Lin Ye, Xiao-Jun Xu, Ping-Gen Cai, A-Gen Xia, Gao-Xiang Ye ∗ Department of Physics, Zhejiang University, Hangzhou 310027, PR China Received 5 July 2003; accepted 3 September 2003 Communicated by J. Flouquet
Abstract We report the observations of large ordered stress patterns, namely bands, in iron films deposited on silicone oil surfaces. The bands grow from the sample edges and then extend into the central regions of the films after deposition. The total length of one band may be more than 5 mm. If two bands meet on their ways, they may cross each other without interference or coalesce harmoniously. The inverse situation that one band bifurcate into two bands is also observed. The X-ray diffraction experiment shows that the nearly free sustained iron films still exhibit polycrystalline structures and our further experiment indicates that the bands result from the ordered aggregation of the iron crystal grains. 2003 Elsevier B.V. All rights reserved. PACS: 62.40.+i; 68.55.-a; 68.90.+g Keywords: Thin film; Stress pattern; Liquid substrate
1. Introduction
Vapor phase deposition of metals on liquid substrates was studied in a number of recent investigations [1–4]. The experiments show that as the nominal film thickness increases, silver or gold atoms deposited on liquid surfaces nucleate and form compact clusters first. Then they diffuse and rotate randomly, which leads to the formation of branched atomic aggregates.
* Corresponding author.
E-mail address: [email protected] (G.-X. Ye). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.09.015
Finally the aggregates connect one another and a continuous film forms. It is expected that the films deposited on liquid substrates would exhibit anomalous properties because of their characteristic microstructures and internal stresses. Here we report a large ordered stress pattern existing in an iron film system deposited on silicone oil surfaces. The phenomena we describe for these nearly free sustained iron films show a distinctive effect of the weak interaction between the metallic films and the liquid substrates, which will in principle affect the growth mechanisms, microstructures and physical properties of various magnetic film systems (multilayer films and superlattices, for instance) not only on liquid substrates but also on soft polymer substrates [5,6].
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Q.-L. Ye et al. / Physics Letters A 318 (2003) 457–462
2. Experiment The samples were prepared by thermal evaporation of 99.58% pure iron in a vacuum of 6 × 10−4 Pa at room temperature. Commercial silicone oil (DOW CORNING 705 Diffusion Pump Fluid) with a vapor pressure below 10−8 Pa was painted onto a frosted glass surface, which was fixed 125 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 profilometer (α-step 200 profilometer, TENCOR). After deposition, the sample was kept in the vacuum chamber (in vacuum condition) for time interval t and then removed from the chamber. All images for the surface morphologies of the samples were taken with an optical microscope (Leica DMLM), equipped with a CCD camera (Leica DC 300). In our experiment, the ranges of d = 5– 60 nm and f = 0.1–1.8 nm/s. In these ranges, the iron films are almost transparent if they are glanced with the naked eye. Fortunately, they can be observed clearly by the optical microscope. Images for the top surface of an iron film were taken by the optical microscope directly before the film was separated from the oil substrate. In order to characterize the bottom surface of the iron film, i.e., the surface in contact with the oil substrate during deposition, a special method was developed to separate the film from the oil substrate: a clean surface
of a piece of glass was used to carefully touch the top surface of the film, which would stick immediately on the glass surface. After washed with acetone and ethanol, a clean bottom surface of the film would appear on the glass surface. Therefore, images for the bottom surfaces of the films could be taken by the optical microscope after they were separated from the oil substrates.
3. Result and discussion The typical ordered stress pattern in the iron films is shown in Fig. 1, where it can be seen that the band-shaped structures, namely bands, are composed of a large number of parallel key-shaped domains, namely keys. Generally, the neighboring keys possess different width w but nearly uniform length L (see Fig. 1). The maximum values of L and w observed in our experiment are Lmax ≈ 5 × 102 µm and wmax ≈ 3.6 × 102 µm, respectively. The total length of one ordered band may be more than 5 mm. The morphologies of the bands in both the top and bottom surfaces of the films resemble each other in appearance (see Fig. 1) and, in many cases, the iron films break along the outlines of the keys, indicating that Fig. 1 shows not only the ordered surface morphology but also the ordered bulk structure of the film. An interesting and unexpected phenomenon is that, if two bands meet on their ways, they may cross each other without interference [Fig. 2(a)]. According to the growth
Fig. 1. An ordered stress pattern in an iron film deposited on silicone oil surface. t = 19 h, f = 0.1 nm/s, d = 15 nm. Each image has a size of 75 × 55 µm2 . (a) Image of a band on the top surface; (b) image of a band on the bottom surface.
Q.-L. Ye et al. / Physics Letters A 318 (2003) 457–462
459
Fig. 2. Images taken on the top surfaces of two iron films. Each image has a size of 90 × 65 µm2 . (a) A crossing of two bands. t = 10 h, f = 1.8 nm/s, d = 35 nm; (b) a coalesce of two bands. t = 10 h, f = 1.8 nm/s, d = 35 nm; (c) a beginning of a band at the edge of the substrate. t = 8 h, f = 1.5 nm/s, d = 45 nm; (d) an end of a band in the central region of the film. t = 8 h, f = 1.5 nm/s, d = 45 nm.
direction of the bands (see the description below), we can identify that two bands may coalesce or one band may bifurcate harmoniously [Fig. 2(b)]. In addition, in each band, the length of the keys near the sample edge is generally longer than that of the keys in the central region of the sample, however it seems that the change of the average key width w¯ is not quite obvious [Figs. 2(c) and (d)]. In our experiment, in the nominal film thickness range of d = 5–60 nm and the deposition rate range of f = 0.1 and 1.8 nm/s, all the phenomena described above are observed in the samples. For the samples with d < 5 nm, however, the band structure disappears. It is noted that the keys in each band do not exhibit periodic structures precisely. Instead, from the width distribution of the keys shown in Fig. 3, we find that, n, the number of the keys with the width w in each band, changes with the value of w and, besides
the main peak (peak I), a subordinate peak, which is marked as peak II in Fig. 3, is frequently observed. The subordinate peak may become more obvious if all the keys counted are from a suitable segment of the band. In many cases, w2 is twice or thrice as large as w1 , where w1 and w2 represent the corresponding key widths of peak I and peak II (see Fig. 3), respectively. In addition, the corresponding internal angles of the keys in each band, namely θ , are quite similar, as shown in Figs. 1 and 2. Although the keys in different bands present different values of θ , most of the bands exhibit θ ≈ 90◦ (Figs. 1 and 2). It was found that all the bands or keys formed after deposition but before the samples were removed from the vacuum chamber. In other words, the keys cannot grow obviously under ambient conditions. In order to identify the formation process of the bands, a series of experiments was performed in which the time t was
460
Q.-L. Ye et al. / Physics Letters A 318 (2003) 457–462
Fig. 3. Width distributions of the keys in two bands, respectively. I and II are labeled for the main and subordinate peaks, respectively. (a) t = 40 h, f = 1.7 nm/s, d = 15 nm; (b) t = 7 h, f = 0.9 nm/s, d = 20 nm.
varied between t = 0.5 and 40 h, since we found that the phenomena shown in Figs. 1 and 2 can be understood qualitatively with the aid of experiments comparing different samples prepared with different time t. The experimental results show that, if t 2 h, neither bands nor keys can be observed in the iron films. For t > 2 h, however, keys start to appear at the edges of the samples. As the time t goes on, the number of the keys increases and the bands grow in both the width and length directions and extend into the central regions of the samples, as shown in Figs. 2(c) and (d). Some of the bands may grow steadily and cross the whole samples. If t 10 h, the growth process of the bands almost stops, indicating that the main stress energy in the films is released completely ten hours after deposition and the free energy approaches its minimum. Consequently, the films reach the stable states. The phenomena shown in Figs. 1, 2 and 3 imply that an ordered microstructure may exist in the iron films. The X-ray diffraction patterns show that the samples with d = 30 nm and f = 1.5 nm/s still exhibit polycrystalline structures, which is in agreement with that of the Ag films on silicone oil surfaces [7], and the average size of the crystal grains in the films is about 20 nm. This conclusion is also confirmed by the SEM (scanning electronic microscope) images for the bottom surfaces of the films. In addition, the magnetic powder pattern experiment [8] shows that the keys in the iron film samples do not exhibit detectable mag-
netic fields. These experimental results give evidence that the keys in the bands are neither single crystals nor single magnetic domains. Generally, ordered stress patterns in films depend closely on the interactions in the system [9]. We propose that the ordered bands may result from the effects of both the liquid substrate and the interactions among the iron crystal grains (or crystal grain clusters). In our experiment, since the interaction between the iron film and the silicone oil surface in the tangent direction is small, the iron film on the liquid substrate is nearly a free sustained film and therefore it can shape the microstructure so that the whole free energy approaches its minimum [10]. On the other hand, crystal grains in the iron film are quite free to diffuse and change their locations on the liquid substrate driven by the interactions among them. They may aggregate locally in some regions of the iron film, which finally results in the keys and then the bands in the iron film. It is believed that the symmetric and parallel structure of the keys in each band (Figs. 1, 2 and 3) may balance the space deformation and then reduce the total free energy of the film [10]. If the description above is correct, the difference between the material densities in and out of the keys should be detectable. Fig. 4 shows that many holes exist in the regions around the keys and the density in the keys is obviously larger than that of the other regions of the films. Furthermore, the cracks with observable widths, as shown in Fig. 4(b), also provide
Q.-L. Ye et al. / Physics Letters A 318 (2003) 457–462
461
Fig. 4. Morphologies of the keys in two iron films (top surfaces). Each image has a size of 145 × 105 µm2 . (a) t = 15 h, f = 0.6 nm/s, d = 20 nm; (b) t = 11.5 h, f = 0.6 nm/s, d = 36 nm.
evidence that the bands and keys really result from the local aggregation of the iron crystal grains (or crystal grains clusters). The material aggregation phenomenon described above depends closely on the internal stress in the iron films. It is well known that internal stresses generally exist in films after deposition and then release gradually in order to minimize the total free energy of the films. This stress relief will give rise to what appears to be various surface patterns or morphologies in the films [11,12]. Many works have previously applied a general theory of buckling of plates to analyze the stress relief patterns [9,12,13] and the buckling equation is given by [12]
∂ 4W ∂ 4W ∂ 4W + 2 + D ∂x 4 ∂x 2 ∂y 2 ∂y 4
+ σx d
∂ 2W ∂x 2
∂ 2W ∂ 2W + F = 0, + σy d 2 + 2τxy d ∂y ∂x∂y
kx + qy = 2nπ,
(1)
(2)
n = 0, ±1, ±2, . . ..
(3)
Introducing solution (2) into Eq. (1) shows that for each k there are two permissible values of q, therefore two classes of line families with slopes of ±|k/q| cross each other. If the buckling or crack processes in a certain direction, when it reaches the point where the two lines with the slopes of ±|k/q| cross each other, it turns onto the other line with different slope, and this process may finally result in a sinusoidal stress pattern [9,12,13]. The sinusoidal stress pattern can be described approximately by the expression [12,13] y = A cos(kx),
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 usually defined in 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).
Solution (2) implies that the direction of the highest stress is perpendicular to straight lines [12]:
(4)
where A = π/q is the amplitude, k = 2π/λ is the wave vector and λ is the wavelength. In fact, Eq. (4) may be satisfied by many different real values of k and q and therefore a pair of values of k and q corresponds to one kind of sinusoidal stress pattern, which implies that there may exist a series of different kinds of sinusoidal stress patterns simultaneously in the films, just as described in Ref. [13]. The ordered band structures shown in Figs. 1, 2 and 4 support this theoretical prediction, i.e., a series of sinusoidal stress patterns with different amplitudes and frequency spectrums may exist in the iron films and therefore the combination of these sinusoidal stress patterns may domain the material in the iron films and
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Q.-L. Ye et al. / Physics Letters A 318 (2003) 457–462
finally results in the rectangular and quasi-periodic stress patterns. In order to describe the characteristic stress patterns shown in Figs. 1, 2 and 4, here we define two parameters, i.e., the fundamental amplitude A0 ≈ L/2 and the fundamental wavelength λ0 = 2π/k0 ≈ 2w, ¯ where k0 is the fundamental wave vector. In our experiment, no obvious relations between the deposition conditions and the parameters A0 and λ0 could be detected. However, the bands with large A0 or λ0 are frequently found in the samples prepared with high deposition rates or large film thicknesses. Furthermore, we find in our experiment that the maximum value of λ0 is less than 100 µm and the maximum value of A0 /λ0 is about 10, which should be related to the scales of the stress patterns in the films.
new avenues of investigation are still open to us. The formation mechanism of the characteristic stress patterns in Figs. 1, 2 and 4 presents us with an example that the films deposited on liquid substrates may possess characteristic microstructures and therefore they may exhibit anomalous physical properties. This will allow a new class of thin film studies that fabricate various magnetic film systems on different liquid substrates and study their properties.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 10174063 and 20274040) and by the Natural Science Foundation of Zhejiang Province in China (Grant No. 1997RC9603).
4. Conclusion In conclusion, we have described the ordered stress patterns in the iron films deposited on silicone oil surfaces. After deposition, the ordered bands grow from the sample edges and then extend into the central regions of the continuous iron films. Various peculiarities of these bands are also presented, like the crossing and bifurcation behaviors of the bands. Our experiments show that the ordered stress patterns result from the aggregation of the iron crystal grains in this nearly free sustained film system. Furthermore, the phenomena presented in this Letter indicate that a series of sinusoidal stress patterns with different amplitudes and frequency spectrums may exist in the iron films and therefore the combination of these sinusoidal stress patterns may domain the material in the iron films and finally results in the rectangular and quasi-periodic stress patterns. Up to now, the details of the interactions among the iron crystal grains and between the iron films and the liquid substrates, which should be responsible for the quasi-periodic stress patterns and the film microstructures, still remain poorly understood. Therefore, many
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