- Email: [email protected]
Wavelet characterization of the submicron surface roughness of anisotropically etched silicon
Surface Science 470 (2000) L57±L62
www.elsevier.nl/locate/susc
Surface Science Letters
Wavelet characterization of the submicron surface roughness of anisotropically etched silicon Z. Moktadir *, K. Sato Department of Microsystems Engineering, Nagoya University, Nagoya, Japan Received 9 June 2000; accepted for publication 25 September 2000
Abstract The roughness of etched Si(1 1 0) surfaces in tetra-methyl ammonium hydroxide has been characterized using the wavelet transform formalism. Wavelet coecients corresponding to the experimental surface pro®les have been calculated and the roughness exponent has been derived using the scalegram method. Its value has been found to be 0.5. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Atomic force microscopy; Etching; Silicon; Surface structure, morphology, roughness, and topography
1. Introduction Silicon is a material widely used in microsystem technologies and exhibits anisotropic etching when it is etched by using tetra-methyl ammonium hydroxide (TMAH) or potassium hydroxide (KOH). That is, the rate of etching depends on the silicon orientation. The Si(1 0 0) plane for example is etched faster than the Si(1 1 1) plane when KOH is used. The fastest etched planes in KOH are the (h k 0) planes [1]. Anisotropic etching can be used to make a variety of complicated micromachined structures (see Ref. [2] for details) and the roughness of the structures obtained this way is something that should be investigated more extensively. The per-
* Corresponding author. Tel.: +81-52-789-5289; fax: +81-52789-5032. E-mail address: [email protected] (Z. Moktadir).
formance of ¯uidic microdevices for example depends very much on the quality of their etched surfaces. The surface quality of micromirrors obtained by etching silicon is also of crucial importance. The surface roughness of chemically etched single crystal silicon has already been analyzed qualitatively [3]. These investigations focused on the global aspect of surface texture (which is a function of silicon orientation) and the standard measurement of the surface roughness (known as Ra). Although the mechanisms involved in growth of vapor deposited ®lms, have been investigated extensively [4,5], until recently the mechanisms of anisotropic etching of silicon have received relatively little attention. Because of the application of dynamic scaling theories to anisotropic etching of silicon has not yet been established, it is important that the roughening of etched structures be investigated quantitatively. Such investigations may reveal important properties enabling both technological
0039-6028/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 6 0 2 8 ( 0 0 ) 0 0 8 9 5 - 5
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Z. Moktadir, K. Sato / Surface Science 470 (2000) L57±L62
and theoretical advances. Proposed mechanisms of the anisotropic etching of silicon [6,7] have not been con®rmed. In this paper, we will show that when silicon is etched using TMAH, it exhibits self-ane fractal surfaces. Studying the surface topography at the submicron scale is essential to understand the physical processes that takes place during anisotropic chemical etching. In the present work, silicon wafers oriented (1 1 0) were etched with a 20% solution of TMAH, and the etching temperature was 80°C. The etched surfaces were investigated by using an atomic force microscope (TOPOMETRIX EXPLORER). The scan tips were changed many times and the measurements were carried ex situ immediately after etching. 2. Wavelets characterization of etched surfaces
Fig. 1. Upper image, a 700 700 nm2 picture taken with AFM, below a representative line scan of the surface height.
2.1. Wavelet transforms of the surface pro®les When silicon is etched with TMAH, it exhibits an anisotropy not only of etch rate but also of surface roughness. That is, the surface roughness depends on the crystallographic orientation [3]. In the present work we analyzed data in dierent scales: 10±103 nm. Fig. 1 shows a representative atomic force microscopy (AFM) picture of a 700 nm square area of an etched surface (below is the corresponding vertical line scan). It is clear that the surface shows a nonEuclidean pro®le and presents a highly random structure. To detect the self-anity of the etched silicon surface, we analyzed various surface pro®les by using the wavelet transforms (WTs) method. Wafers were scanned in the x and y directions. To ensure that the surface roughness is saturated, we measured the surface roughness (rms) at dierent times for 1000 nm square area. For times exceeding 100 min, the value of the roughness was constant and was about 2 nm. This is the characteristic of a self-ane surfaces where the surface roughness w scales with linear system size L and the time t, according to the dynamic scaling hypothesis [8]: w La f t=La=b
1
Here f is a scaling function f x xb for x 1 and f x const: for x 1, and a and b are universal exponents respectively known as the roughness exponent and the growth exponent. Obtaining these scaling exponents helps to identify the universality class of the growth or the etching process. In the following, we will describe the wavelet formalism used to characterize the surface roughness, and to determine the roughness exponent a. It is well known that a function can be expanded into a base function to represent it in a form adapted to a particular question. In harmonic analysis for example, the signals obtained are sinusoids and thus the suitable method of expansion is Fourier transformation. The obtained signals in Fig. 1 have no sinusoidal but rather triangular shapes having a largely linear rise to maximum and a subsequent linear decay. A Fourier transform is thus obviously not an ideal tool for the statistical description surface pro®les we obtained. Another type of transformation using a suitable base functions should yield a more satisfying description. These base functions must have a shape intrinsically similar to that of the quick variations of the signal, permitting a
CE N IE SC RS CE E A TT RF LE SU
Z. Moktadir, K. Sato / Surface Science 470 (2000) L57±L62
resolution in frequency and space. An ideal candidate is the WT based on an appropriately chosen mother wavelet. The dierence between the WTs and the Fourier transforms is that wavelet functions are localized in space, whereas the sine and cosine are not. In the following we will give a very brief description of the wavelet transforms formalism. A detailed description is beyond the scope of this paper. See, for example, Ref. [9] for more information about the theory of wavelets. Wavelet analysis is capable of revealing aspects of data that other signal analysis techniques miss: aspects like trends, breakdown points, discontinuities in higher derivatives, and self-similarity. From an intuitive point of view, wavelet decomposition measures the degree of ``resemblance'' between the wavelet and the original signal. It is therefore suitable for detecting self-similarity or fractality of the signal. This degree of ``resemblance'' is represented by wavelet coecients. If the signal (surface pro®le) is self-similar, then wavelet coecients will be similar at dierent scales. Wavelet analysis is the breaking up of a signal into shifted and scaled versions of the original (or mother) wavelet. That is,
1 xÿb wa;b p w a a
2
where a a 6 0 and b are respectively the scale parameter and the dilatation parameter and x is the space variable, and w is the mother wavelet. These basis functions vary in scale by slicing the data space using dierent scale sizes. The continuous wavelet transform (CWT) is de®ned as the sum over all of the surface pro®le multiplied by the shifted and scaled mother wavelet: Z 3 c a; b h xwa;b x dx c a; b are the wavelet coecients which are a function of scale and position. Then, CWT describes the surface pro®le in a given position b and a given frequency a (or scale). In general, the parameter b is chosen to be equidistant in x, and a
L59
equidistant in log 1=x. For example one can choose: a
1 ; 2s
b
k 2s
4
where s and k are integers. This will enable the representation P of the obtained surface signal as a sum h cs;k ws;k . This is known as the discrete wavelet transform. Of course the choice of mother wavelet adapted to a speci®c data is important. This choice is possible because we have many wavelets to choose from Ref. [10]. This an advantage of using the WT rather than Fourier transform when the signals are non sinusoidal. We tried some known wavelets to see if the result depends on the choice of the mother wavelet. The wavelets we tried were the following: Haar wavelet, Daubechies wavelet, biorthogonal spline wavelet, and Morlet wavelet. For more information and the de®nition of these wavelets, see Ref. [10]. We found that the result did not depend on the choice of the mother wavelet. The mother wavelet we are using here as an example is the Meyer mother wavelet w shown in Fig. 2. The set of base wavelets is then derived: x ws;k 2ÿs=2 w s ÿ k 2
5
Fig. 3 shows a two-dimensional (2-D) plot of the wavelet coecients that correspond to the etched surface pro®les. The value of the coecients is averaged over 12 AFM images and each image consists on 300 line scans. The obtained map is characteristic of self-ane pro®les. At dierent scales the features of the map look similar. The data can be characterized quantitatively by using the scalegram method. This is the object of the next section. 2.2. Scalegram of surface pro®les Scargle et al. [11] have used the scalegram to characterize the stochastic data obtained from Sco X-1 binary stars X-ray source. The scalegram is de®ned as (using the same notation as above):
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Z. Moktadir, K. Sato / Surface Science 470 (2000) L57±L62
Fig. 2. Graphic reprensentation of Meyer wavelet of degree 3. The degree of Meyer wavelet is the degree of the polynomial used for windowing the wavelet. See Ref. [10] for details. The choice of this parameter does not aect the result.
S
2s X 2 c N k c;k
6
where N is the number of data points. The scale index s is related to the scale xs by xs 2s Dx, where Dx is the data sampling. The scalegram S describes
statistically the variations of the surface pro®le within the interval 2s Dx; 2s1 Dx. For a self-ane surface pro®le h, the following relation holds statistically: h kx ka h x
where k is a scaling factor and a, the roughness exponent which characterizes the self-ane rough surface quantitatively [4]. The relation (7) means that the probability distribution of an anisotropically rescaled surface pro®le is the same as the original one (and so the mean and the variance). Using the relation (3) that de®nes cs;k and relation (7), one can show that S scales as: 2a
S 2s
Fig. 3. Absolute s 1; 2; 3; 4; . . .
values
of
the
coecients
cs;k
for
7
8
Thus, the value of the roughness exponent a can be determined by plotting this relation on a logarithmic scale. Fig. 4 is a log±log plot of the scalegram versus 2s for the vertical cut of the surface substrate. The obtained curve is a straight line, as one would expect for a self-ane pro®le. The slope is twice the value of a, which means that a 0:50. The power law behavior is clearly visible and extends to at least s 128. We calculated the scale-
CE N IE SC RS CE E A TT RF LE SU
Z. Moktadir, K. Sato / Surface Science 470 (2000) L57±L62
L61
Fig. 4. Averaged scalegram for the eteched surface pro®les in the log±log sacle: the power law behavior is clearly visible and extends to s 128.
gram for both substrate directions (x-direction and y-direction), and got the same roughness exponent. Roughening of surfaces is often a random process. The origin of the surface ¯uctuations (or the reason for any particular value of a) is not always known. Investigators interested in the roughening of growing surfaces and interfaces, have developed many computer models and theoretical approaches in the last two decades. In general, the roughening of a growing surface is described by Langevin type equations which are stochastic partial dierential equations. The roughness exponent is calculated by solving these equations numerically [12] or by using the renormalization group technique [13]. We have shown that etched Si(1 1 0) surface pro®les are self-ane in both substrate directions, and we have determined the corresponding roughness exponent using the scalegram method. To identify the universality class of the chemical anisotropic etching process, one needs to determine both the roughness exponent and the growth exponent (in Eq. (1)) in two dimensions. This is possible by using the 2-D WT calculation and will be the subject of further investigations.
3. Conclusion and future work The wavelets characterization of anisotropically etched silicon presented here is a powerful tool for detecting the self-anity of the etched surface pro®les. It is in many ways advantageous over the power spectrum density analysis which uses Fourier transformations. We have determined the value of the roughness exponent of the pro®les of etched Si(1 1 0) surfaces and found that it is close to 1=2. The physical phenomena determining this value are not known and more studies are needed to understand the reason for the roughening of anisotropically etched silicon. Acknowledgements This work was supported by the Japan Society for the Promotion of Science (JSPS). References [1] K. Sato, M. Shikida, T. Yamashiro, M. Tsunekawa, S. Ito, Sensors Actuat. 73 (1999) 122±130. [2] K. Petersen, Proc. IEEE 70 (5) (1982) 420±457.
SU RF A LE CE TT S ER CIE S NC E
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Z. Moktadir, K. Sato / Surface Science 470 (2000) L57±L62
[3] K. Sato, M. Shikida, T. Yamashiro, M. Tsunekawa, S. Ito, Sensors Actuat. 73 (1999) 122±130. [4] P. Meakin, Fractal, Scaling and Growth Far From Equilibrium, Cambridge University Press, Cambridge, MA, 1998. [5] A.L. Barabarasi, H.E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, MA, 1995. [6] R.M. Finne, D.L. Klein, J. Electrochem. Soc. 114 (1967) 965. [7] H. Seidel, L. Csepregi, A. Heuberger, H. Baumgartel, J. Electrochem. Soc. 137 (1990) 3612. [8] F. Family, Physica A 168 (1990) 561.
[9] P. Wojtaszczyk, A Mathematical Introduction to Wavelets, Cambridge University Press, Cambridge, MA, 1997. [10] I. Daubechis, Ten lectures on Wavelets, CBMS, NSF Series in Applied Mathematics, vol. 61, SIAM Publications, Philadelphia, PA, 1992. [11] J.D. Scargle, T.Y. Steinman-Cameron, K. Young, D.L. Donoho, J.P. Crutch®eld, J. Imamura, Astrophys. J. 411 (1993) L91±L94. [12] S. Das Sarma, S.V. Ghaisas, Phys. Rev. Lett. 69 (26) (1992) 3762. [13] M. Kardar, G. Parisi, Y.C. Zhang, Phys. Rev. Lett. 56 (1986) 889.
www.elsevier.nl/locate/susc
Surface Science Letters
Wavelet characterization of the submicron surface roughness of anisotropically etched silicon Z. Moktadir *, K. Sato Department of Microsystems Engineering, Nagoya University, Nagoya, Japan Received 9 June 2000; accepted for publication 25 September 2000
Abstract The roughness of etched Si(1 1 0) surfaces in tetra-methyl ammonium hydroxide has been characterized using the wavelet transform formalism. Wavelet coecients corresponding to the experimental surface pro®les have been calculated and the roughness exponent has been derived using the scalegram method. Its value has been found to be 0.5. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Atomic force microscopy; Etching; Silicon; Surface structure, morphology, roughness, and topography
1. Introduction Silicon is a material widely used in microsystem technologies and exhibits anisotropic etching when it is etched by using tetra-methyl ammonium hydroxide (TMAH) or potassium hydroxide (KOH). That is, the rate of etching depends on the silicon orientation. The Si(1 0 0) plane for example is etched faster than the Si(1 1 1) plane when KOH is used. The fastest etched planes in KOH are the (h k 0) planes [1]. Anisotropic etching can be used to make a variety of complicated micromachined structures (see Ref. [2] for details) and the roughness of the structures obtained this way is something that should be investigated more extensively. The per-
* Corresponding author. Tel.: +81-52-789-5289; fax: +81-52789-5032. E-mail address: [email protected] (Z. Moktadir).
formance of ¯uidic microdevices for example depends very much on the quality of their etched surfaces. The surface quality of micromirrors obtained by etching silicon is also of crucial importance. The surface roughness of chemically etched single crystal silicon has already been analyzed qualitatively [3]. These investigations focused on the global aspect of surface texture (which is a function of silicon orientation) and the standard measurement of the surface roughness (known as Ra). Although the mechanisms involved in growth of vapor deposited ®lms, have been investigated extensively [4,5], until recently the mechanisms of anisotropic etching of silicon have received relatively little attention. Because of the application of dynamic scaling theories to anisotropic etching of silicon has not yet been established, it is important that the roughening of etched structures be investigated quantitatively. Such investigations may reveal important properties enabling both technological
0039-6028/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 6 0 2 8 ( 0 0 ) 0 0 8 9 5 - 5
SU RF A LE CE TT S ER CIE S NC E
L58
Z. Moktadir, K. Sato / Surface Science 470 (2000) L57±L62
and theoretical advances. Proposed mechanisms of the anisotropic etching of silicon [6,7] have not been con®rmed. In this paper, we will show that when silicon is etched using TMAH, it exhibits self-ane fractal surfaces. Studying the surface topography at the submicron scale is essential to understand the physical processes that takes place during anisotropic chemical etching. In the present work, silicon wafers oriented (1 1 0) were etched with a 20% solution of TMAH, and the etching temperature was 80°C. The etched surfaces were investigated by using an atomic force microscope (TOPOMETRIX EXPLORER). The scan tips were changed many times and the measurements were carried ex situ immediately after etching. 2. Wavelets characterization of etched surfaces
Fig. 1. Upper image, a 700 700 nm2 picture taken with AFM, below a representative line scan of the surface height.
2.1. Wavelet transforms of the surface pro®les When silicon is etched with TMAH, it exhibits an anisotropy not only of etch rate but also of surface roughness. That is, the surface roughness depends on the crystallographic orientation [3]. In the present work we analyzed data in dierent scales: 10±103 nm. Fig. 1 shows a representative atomic force microscopy (AFM) picture of a 700 nm square area of an etched surface (below is the corresponding vertical line scan). It is clear that the surface shows a nonEuclidean pro®le and presents a highly random structure. To detect the self-anity of the etched silicon surface, we analyzed various surface pro®les by using the wavelet transforms (WTs) method. Wafers were scanned in the x and y directions. To ensure that the surface roughness is saturated, we measured the surface roughness (rms) at dierent times for 1000 nm square area. For times exceeding 100 min, the value of the roughness was constant and was about 2 nm. This is the characteristic of a self-ane surfaces where the surface roughness w scales with linear system size L and the time t, according to the dynamic scaling hypothesis [8]: w La f t=La=b
1
Here f is a scaling function f x xb for x 1 and f x const: for x 1, and a and b are universal exponents respectively known as the roughness exponent and the growth exponent. Obtaining these scaling exponents helps to identify the universality class of the growth or the etching process. In the following, we will describe the wavelet formalism used to characterize the surface roughness, and to determine the roughness exponent a. It is well known that a function can be expanded into a base function to represent it in a form adapted to a particular question. In harmonic analysis for example, the signals obtained are sinusoids and thus the suitable method of expansion is Fourier transformation. The obtained signals in Fig. 1 have no sinusoidal but rather triangular shapes having a largely linear rise to maximum and a subsequent linear decay. A Fourier transform is thus obviously not an ideal tool for the statistical description surface pro®les we obtained. Another type of transformation using a suitable base functions should yield a more satisfying description. These base functions must have a shape intrinsically similar to that of the quick variations of the signal, permitting a
CE N IE SC RS CE E A TT RF LE SU
Z. Moktadir, K. Sato / Surface Science 470 (2000) L57±L62
resolution in frequency and space. An ideal candidate is the WT based on an appropriately chosen mother wavelet. The dierence between the WTs and the Fourier transforms is that wavelet functions are localized in space, whereas the sine and cosine are not. In the following we will give a very brief description of the wavelet transforms formalism. A detailed description is beyond the scope of this paper. See, for example, Ref. [9] for more information about the theory of wavelets. Wavelet analysis is capable of revealing aspects of data that other signal analysis techniques miss: aspects like trends, breakdown points, discontinuities in higher derivatives, and self-similarity. From an intuitive point of view, wavelet decomposition measures the degree of ``resemblance'' between the wavelet and the original signal. It is therefore suitable for detecting self-similarity or fractality of the signal. This degree of ``resemblance'' is represented by wavelet coecients. If the signal (surface pro®le) is self-similar, then wavelet coecients will be similar at dierent scales. Wavelet analysis is the breaking up of a signal into shifted and scaled versions of the original (or mother) wavelet. That is,
1 xÿb wa;b p w a a
2
where a a 6 0 and b are respectively the scale parameter and the dilatation parameter and x is the space variable, and w is the mother wavelet. These basis functions vary in scale by slicing the data space using dierent scale sizes. The continuous wavelet transform (CWT) is de®ned as the sum over all of the surface pro®le multiplied by the shifted and scaled mother wavelet: Z 3 c a; b h xwa;b x dx c a; b are the wavelet coecients which are a function of scale and position. Then, CWT describes the surface pro®le in a given position b and a given frequency a (or scale). In general, the parameter b is chosen to be equidistant in x, and a
L59
equidistant in log 1=x. For example one can choose: a
1 ; 2s
b
k 2s
4
where s and k are integers. This will enable the representation P of the obtained surface signal as a sum h cs;k ws;k . This is known as the discrete wavelet transform. Of course the choice of mother wavelet adapted to a speci®c data is important. This choice is possible because we have many wavelets to choose from Ref. [10]. This an advantage of using the WT rather than Fourier transform when the signals are non sinusoidal. We tried some known wavelets to see if the result depends on the choice of the mother wavelet. The wavelets we tried were the following: Haar wavelet, Daubechies wavelet, biorthogonal spline wavelet, and Morlet wavelet. For more information and the de®nition of these wavelets, see Ref. [10]. We found that the result did not depend on the choice of the mother wavelet. The mother wavelet we are using here as an example is the Meyer mother wavelet w shown in Fig. 2. The set of base wavelets is then derived: x ws;k 2ÿs=2 w s ÿ k 2
5
Fig. 3 shows a two-dimensional (2-D) plot of the wavelet coecients that correspond to the etched surface pro®les. The value of the coecients is averaged over 12 AFM images and each image consists on 300 line scans. The obtained map is characteristic of self-ane pro®les. At dierent scales the features of the map look similar. The data can be characterized quantitatively by using the scalegram method. This is the object of the next section. 2.2. Scalegram of surface pro®les Scargle et al. [11] have used the scalegram to characterize the stochastic data obtained from Sco X-1 binary stars X-ray source. The scalegram is de®ned as (using the same notation as above):
SU RF A LE CE TT S ER CIE S NC E
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Z. Moktadir, K. Sato / Surface Science 470 (2000) L57±L62
Fig. 2. Graphic reprensentation of Meyer wavelet of degree 3. The degree of Meyer wavelet is the degree of the polynomial used for windowing the wavelet. See Ref. [10] for details. The choice of this parameter does not aect the result.
S
2s X 2 c N k c;k
6
where N is the number of data points. The scale index s is related to the scale xs by xs 2s Dx, where Dx is the data sampling. The scalegram S describes
statistically the variations of the surface pro®le within the interval 2s Dx; 2s1 Dx. For a self-ane surface pro®le h, the following relation holds statistically: h kx ka h x
where k is a scaling factor and a, the roughness exponent which characterizes the self-ane rough surface quantitatively [4]. The relation (7) means that the probability distribution of an anisotropically rescaled surface pro®le is the same as the original one (and so the mean and the variance). Using the relation (3) that de®nes cs;k and relation (7), one can show that S scales as: 2a
S 2s
Fig. 3. Absolute s 1; 2; 3; 4; . . .
values
of
the
coecients
cs;k
for
7
8
Thus, the value of the roughness exponent a can be determined by plotting this relation on a logarithmic scale. Fig. 4 is a log±log plot of the scalegram versus 2s for the vertical cut of the surface substrate. The obtained curve is a straight line, as one would expect for a self-ane pro®le. The slope is twice the value of a, which means that a 0:50. The power law behavior is clearly visible and extends to at least s 128. We calculated the scale-
CE N IE SC RS CE E A TT RF LE SU
Z. Moktadir, K. Sato / Surface Science 470 (2000) L57±L62
L61
Fig. 4. Averaged scalegram for the eteched surface pro®les in the log±log sacle: the power law behavior is clearly visible and extends to s 128.
gram for both substrate directions (x-direction and y-direction), and got the same roughness exponent. Roughening of surfaces is often a random process. The origin of the surface ¯uctuations (or the reason for any particular value of a) is not always known. Investigators interested in the roughening of growing surfaces and interfaces, have developed many computer models and theoretical approaches in the last two decades. In general, the roughening of a growing surface is described by Langevin type equations which are stochastic partial dierential equations. The roughness exponent is calculated by solving these equations numerically [12] or by using the renormalization group technique [13]. We have shown that etched Si(1 1 0) surface pro®les are self-ane in both substrate directions, and we have determined the corresponding roughness exponent using the scalegram method. To identify the universality class of the chemical anisotropic etching process, one needs to determine both the roughness exponent and the growth exponent (in Eq. (1)) in two dimensions. This is possible by using the 2-D WT calculation and will be the subject of further investigations.
3. Conclusion and future work The wavelets characterization of anisotropically etched silicon presented here is a powerful tool for detecting the self-anity of the etched surface pro®les. It is in many ways advantageous over the power spectrum density analysis which uses Fourier transformations. We have determined the value of the roughness exponent of the pro®les of etched Si(1 1 0) surfaces and found that it is close to 1=2. The physical phenomena determining this value are not known and more studies are needed to understand the reason for the roughening of anisotropically etched silicon. Acknowledgements This work was supported by the Japan Society for the Promotion of Science (JSPS). References [1] K. Sato, M. Shikida, T. Yamashiro, M. Tsunekawa, S. Ito, Sensors Actuat. 73 (1999) 122±130. [2] K. Petersen, Proc. IEEE 70 (5) (1982) 420±457.
SU RF A LE CE TT S ER CIE S NC E
L62
Z. Moktadir, K. Sato / Surface Science 470 (2000) L57±L62
[3] K. Sato, M. Shikida, T. Yamashiro, M. Tsunekawa, S. Ito, Sensors Actuat. 73 (1999) 122±130. [4] P. Meakin, Fractal, Scaling and Growth Far From Equilibrium, Cambridge University Press, Cambridge, MA, 1998. [5] A.L. Barabarasi, H.E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, MA, 1995. [6] R.M. Finne, D.L. Klein, J. Electrochem. Soc. 114 (1967) 965. [7] H. Seidel, L. Csepregi, A. Heuberger, H. Baumgartel, J. Electrochem. Soc. 137 (1990) 3612. [8] F. Family, Physica A 168 (1990) 561.
[9] P. Wojtaszczyk, A Mathematical Introduction to Wavelets, Cambridge University Press, Cambridge, MA, 1997. [10] I. Daubechis, Ten lectures on Wavelets, CBMS, NSF Series in Applied Mathematics, vol. 61, SIAM Publications, Philadelphia, PA, 1992. [11] J.D. Scargle, T.Y. Steinman-Cameron, K. Young, D.L. Donoho, J.P. Crutch®eld, J. Imamura, Astrophys. J. 411 (1993) L91±L94. [12] S. Das Sarma, S.V. Ghaisas, Phys. Rev. Lett. 69 (26) (1992) 3762. [13] M. Kardar, G. Parisi, Y.C. Zhang, Phys. Rev. Lett. 56 (1986) 889.