Borisyuk, V.N.

Holovchenko, A.I.

Kassi, Jassim

SocioEconomic Challenges Journal (Sumy State University)

 
 
 
J. Nano- Electron. Phys. 
3 (2011) No4, P. 20-26 
 2011 SumDU 
(Sumy State University) 
 
 
20 
PACS numbers: 05.45.Df, 68.37.Хх 
 
GENERALIZED HURST EXPONENT OF THE CARBON THIN FILM 
SURFACE  
 
V.N. Borisyuk, Jassim Kassi, A.I. Holovchenko 
 
 Sumy State University,  
 2, Rimskii-Korsakov Str., 40007 Sumy, Ukraine 
 E-mail: vad2@ukr.net 
 
Self-similar structure of the carbon thin film surface, obtained by magnetron 
sputtering, is investigated numerically. Statistical parameters are calculated within 
two dimensional multifractaldetrended fluctuation analysis. The numerical model of 
the surfaces under investigation was build from the SEM images of the carbon thin 
film. It is shown that the self-similarity in surface structure preserves through 
different resolutions of the SEM images. 
 
Keywords: SELF-SIMILARITY, CARBON THIN FILMS, HURST EXPONENT, 
FRACTAL DIMENSION. 
 
(Received 23 August 2011, published online 30 December 2011) 
 
1. INTRODUCTION 
 
Self-similar or self-affine structures are widely presented in all areas of the 
nature science [1]. The self-similarity means that the each segment of the 
initial set has the same structure as the whole object. The properties of such 
systems can be described by special parameters, like fractal dimension (or set 
of dimensions in case of complex structures [2]), Hurst exponent and other [1, 
2]. Most common examples of the geometric fractals are the Cantor and 
Serpinsky sets, the Koch curve and many others objects [2]. Self-similarity 
appears not only in topological structure of the objects, also the phase space 
of the complex stochastic systems often has a hierarchical structure with non 
participant regions [3].The stochastic fractals can be illustrated through the 
Lorenz attractor sand nonstationary time-series [4, 5]. 
 Self-similar surfaces play the important role in surface contacts. 
Morphologies of these surfaces can appear to be quite different depending on 
the scale with which they can be observed. It can be shown that concepts like 
roughness are replaced by exponents that refer not to the roughness, but to 
the fashion in which the roughness changes when the observation scale itself 
changes [6]. The self-similarity of surfaces can be produced by the fracture or 
by the mechanical processing. Surface can be formed as a result of a 
deposition process, shrink due to erosion or etching, propagate through 
inhomogeneous media.Fractal dimension of such structures is usually 
determined by obtaining the photographs taken at various zoom levels, with 
subsequent application of the photograph at a square grid and numerical 
analysis. 
 In proposed article we present the investigation of the self-similar 
structure of the carbon film surface by numerical methods of scaling 
analysis. Our calculations are based on the algorithm mofmulti fractal 
 
 
 
 GENERALIZED HURST EXPONENT OF THE CARBON… 21 
detrended fluctuation analysis (MF-DFA) [7], that differs from the standard 
box-counting based procedures of the fractal dimension estimation [8].The 
standard methodic gives precise results only for deterministic self-similar 
objects with explicit numerical model, and has a large error when applying 
to the surface models with discreet range of the coordinates [9]. The MF-
DFA algorithm initially was developed for investigation of the time series as 
a one-dimensional self-similar set [4], and later generalized for more 
complicated objects analysis [7]. Applying MF-DFA algorithm to the self-
similar surface model as to two-dimensional time series allows one to obtain 
a quantitative characteristic of the surface structure. 
 In the opening section of the paper we briefly describe the object of our 
research – carbon thin film, obtained through magnetron sputtering, and 
the possible way for numerical simulation of its surface structure. Next we 
refer the main steps of the MF-DFA algorithm and present the results of our 
calculations, comparing them for different parts of the surface sample. Our 
works ends with summation of the performed investigations. 
 
2. THIN FILMS UNDER INVESTIGATION 
 
 Carbon condensates were obtained by deposition of the ion-sputtering 
substance onto (001) KCl facets according to original technique [10-14]. As a 
result the high porous carbon thin films were obtained. The SEM images of 
the condensates at different scale resolution are shown in Fig. 1. 
 
  
  
 
Fig. 1 – SEM images of the carbon thin films at different scale resolutions [14] 
 
 In order to investigate the self similarity in the structure of the surfaces 
we select the four SEM pictures of the surface at different microscopic 
 
 
 
22 V.N. BORISYUK, JASSIM KASSI, A.I. HOLOVCHENKO  
zoom, specified at the photos in Fig. 1. For each SEM image a numerical 
model was build. The image was presented by a surface in Cartesian 
coordinates, where each point defined by the numbers and brightness of the 
pixel in related image. Thus, the effective height of the surface model varies 
in the [0, 1] range, where zero height presents the absolutely dark (i.e. black) 
pixel. Example of the surface image and corresponded numerical model are 
presented in figure 2.Making use of the developed model allows for 
calculation of the generalized Hurst exponent within the two-dimensional MF-
DFA algorithm [7]. 
 
 
 
 
Fig. 2 – Fragment of the SEM image of the carbon thin film (upper panel) and 
related numerical model (lower panel) 
 
3. IMAGE ANALYSYS METHODOLOGY 
 
 All surfaces were investigated within two dimensional multifractal detren-
ded fluctuation analyses (MF-DFA) methodology [7]. This algorithm allows for 
calculation of the main parameter of the self similar structure, such as 
generalized Hurst exponent through the following numerical procedure [7]. 
 Self-similar surface is considering as a two-dimensional data array X (i, 
j), where in dices i, j has a discrete values i  1,2,…, M and j  1,2,…, N. 
This surface fragmented into Ms  Ns non overlapping segments of the size 
s  s, where Ms  [M/s] and Ns  [N/s] are integer numbers.  
 Denoting each segment through v, , the cumulative sum  
 
 
1 2
1 2
1 1
( , ) ( , ),1 , ,
ji
k k
u i j X k k i j s  (1) 
 
is calculated for all segments v, .  
 Next step is a detrending procedure for the obtained surface ( , )u i j . The 
trend can be removed by the fitting procedure, when it determines by some 
smooth polynomial function ( , )u i j . There are many possible expression of 
the ( , )u i j
 
function [7], we chose the simplest one, in order to reduce the 
computational time: 
 
 
 
 
 GENERALIZED HURST EXPONENT OF THE CARBON… 23 
 ( , )u i j ai bj c  (2) 
 
where a, b, c – coefficients, defined by list-squares fitting algorithm. It 
should be mentioned, that more complicated forms of ( , )u i j  does not 
provides any significant increasing of the precision of the method, but 
noticeably increase computational time [7]. After detrending we arrive at 
residual function: 
 
 ( , ) ( , ) ( , )i j u i j u i j , (3) 
 
and a dispersion of the v,  segment of lengths: 
 
 
2 2
2
1 1
1
( , , ) ( , )
s s
i j
F s i j
s
 (4) 
 
Dispersion of the all segments is calculated through averaging over all 
surfaces: 
 
 
1/
1 1
1
( ) [ ( , , )]
s s
q
M N
q
q
s s
F s F s
M N
, (5) 
 
where q is a deformation parameter, initialized to increase the role of the 
segments with small (when q < 0) or high (q > 0) fluctuations  
F2(v, , s) respectively [4]. A tq  0 expression (5) takes form [6]: 
 
 
0
1 1
1
( ) exp ln[ ( , , )]
s sM N
s s
F s F s
M N
, (6) 
 
according to l'Hôpital's rule. For statistically correct result s values must be 
varied in the range from smin  6tosmax  min(M, N) / 4.The dispersion (5) 
and the segment size s connected through a scaling relation: 
 
 ( )( ) ~
h q
qF s s
, (7) 
 
where, h(q) is a generalized Hurst exponent. If the object under investigation 
has a self similar structure, the dependence (7) must be linear in logarithmic 
scales. Examples of relation (7) for the surfaces of the carbon thin films are 
presented in Fig. 3. 
 
 
 
Fig. 3 – Relation (7) in logarithmic scales for the surface of the carbon thin films 
plotted at different negative (a) and positive (b) values of the deformation parameter q 
 
 
 
24 V.N. BORISYUK, JASSIM KASSI, A.I. HOLOVCHENKO  
As it follows from the figure, dependence (7) has a clear linear form only 
for q > 0 values, while in opposite range the calculations are expected to 
give a large error.  
 
4. RESULTS AND DISCUSSION 
 
The generalized Hurst exponent was calculated for all surfaces, shown in 
Fig. 1. According to the structure of the fractal objects, the Hurst exponent 
of the self-similar surface are expected to be independent from the 
resolution scale. The results of the calculations are shown in Fig. 3. 
 
 
 
Fig. 4 – Hurst exponent of the carbon thin film surface at different SEM images 
resolutions (different symbols relate to the different microscopic scale zoom: 
squares – 585, circles – 1910, triangles – 383 and diamonds – 1580) 
 
 As it can be seen from the figure, the calculated parameters for different 
resolution scale of the surfaces are almost coinciding in q  0 region, and 
differs from each other for q < values, i.e. self-similarity in surface 
structure exist while q  0, and breaks at q < 0. This result can be explained 
by the statistical meaning of the fluctuation function (5), when q < 0, the 
main contribution in (5) is made by the segments v,  with small fluctuation 
(4) (namely F2(v, , s) << 1), due to the power q. 
 
 
 
Fig. 5 – Density distribution of the effective heights of the surfaces at different SEM 
images resolutions (different symbols relate to the different microscopic scale zoom: 
squares – 585, circles – 1910, triangles – 383 and diamonds – 1580) 
 
 
 
 GENERALIZED HURST EXPONENT OF THE CARBON… 25 
Such F2(v, , s) values are calculated in segments where the surface height 
coordinates are maximally close to the smooth function ( , )u i j  (in general, 
F2(v, , s) depends on the segments size s , but since the whole set 
min maxs s s is considered for every q value, this dependence is 
insignificant). Noticeably, since the fluctuation F2(v, , s) is not equal to 
zero for any segment v,  [7], totally smooth areas of the surface with any 
size s  s are never appears. The segments with F2(v, , s) << 1 can be 
related to those areas of the SEM images, that appears to be enormously 
dark or bright or corrupted because of image noises and defects of the 
scanning microscopy imaging. There is no strong self-similarity in the 
structure of these areas in the surface, and this situation causes the 
difference between Hurst exponents in the q  0 region. To verify this 
assumption we calculate the distribution of the effective height hxy over 
numerical models of each surface (figure 5). As it can be seen from the 
shown dependencies, most significant differences between distributions are 
observed for most dark and bright areas of the SEM images (minimum and 
maximum heights of the surface respectively). This feature can be the 
reason of the difference between Hurst exponents at q  0. 
 
5. CONCLUSIONS 
 
Above consideration shows, that the surface of the carbon condensates has 
the self-similar structure. This conclusion follows from the approximately 
the same Hurst exponent for different scale of the surface observation at 
positive deformation parameter q. Nevertheless, when q < 0, calculated 
parameters are strongly distinguishing. We explain this situation by the 
statistical meaning of the fluctuation function (5) and q parameter, which 
negative values related to the small fluctuations of the surface roughness, 
i.e. the self – similarity parameters at q < 0 represent the small details of 
the SEM images and can include the technical noise, which are always 
present in the scanning microscopy. Hurst exponent is often used as a 
parameter of the surface roughness [15], thus, obtained dependencies also 
can be useful in studying of the physical properties of the thin films. 
 
REFERENCES 
1. J. Feder, Fractals (Plenum press, New York and London, 1998).  
2. A.I. Olemskoi, Fractal in Condensed Matter Physics, (in: Physics Reviews, ed. 
I.M. Khalatnikov, 1995 18, Part 1, 1). 
3. A.I. Olemskoi, Synergetics of complex systems: Phenomenology and statistical 
theory (Moscow: Publishing house KRASAND: 2009) (in Russian). 
4. J.W. Kanthelhardt, S.A. Zscheinger , E. Koscienly-Bunde, S. Havlin, A. Bunde, 
H.E. Stanley, Physica A 316, 87 (2002). 
5. A.V. Khomenko, I.A. Lyashenko, V.N. Borisyuk, Fluctuat. Noise Lett. 9 19, (2010). 
6. A.L. Barabasi, H.E. Stanley, Fractal concepts in surface growth, (Cambrige 
University Press, 1995). 
7. Gao-Fen Gu, Wei-Xing Zhou, Phys. Rev. E 74, 061104 (2006). 
8. T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia, B.I. Shriman, Phys. Rev. 
A 43, 2 (1991). 
9. A.I. Olemskoi, V.M. Borisyuk, I.A. Shuda, Journal of Physical Studies 13, 2002 
(2009). 
 
 
 
26 V.N. BORISYUK, JASSIM KASSI, A.I. HOLOVCHENKO  
10. V.I. Perekrestov, Yu.A. Kosminskaya, JETP Lett. 78, 223 (2003). 
11. V.I. Perekrestov,  Yu.A. Kosminskaya, Tech. Phys. Lett. 31, 830 (2005).  
12. V.I. Perekrestov, A.S. Kornyushchenko,Yu.A. Kosminskaya, Tech. Phys. Lett. 32, 
868 (2006). 
13. V.I. Perekrestov, A.I. Olemskoi, A.S. Kornyushchenko, Yu.A. Kosminskaya, 
Phys. Solid State. 51, 1060 (2009). 
14. A.I. Olemskoi, V.I. Perekrestov, I.A. Shuda, .M. Borisyuk, A.A. Mokrenko, 
Metallofiz. Noveyshie Technol. 31, 11 (2009). 
15. U. Tartaglino, V.N. Samoilov, B.N.J. Persson, J. Phys.: Condens. Matter 18, 
4143 (2006). 


English

Generalized Hurst Exponent of The Carbon Thin Film Surface

Electronic Sumy State University Institutional Repository

   J. Nano- Electron. Phys. 3 (2011) No4, P. 20-26  2011 SumDU (Sumy State University)   20 PACS numbers: 05.45.Df, 68.37.Хх  GENERALIZED HURST EXPONENT OF THE CARBON THIN FILM SURFACE   V.N. Borisyuk, Jassim Kassi, A.I. Holovchenko   Sumy State University,   2, Rimskii-Korsakov Str., 40007 Sumy, Ukraine  E-mail: vad2@ukr.net  Self-similar structure of the carbon thin film surface, obtained by magnetron sputtering, is investigated numerically. Statistical parameters are calculated within two dimensional multifractaldetrended fluctuation analysis. The numerical model of the surfaces under investigation was build from the SEM images of the carbon thin film. It is shown that the self-similarity in surface structure preserves through different resolutions of the SEM images.  Keywords: SELF-SIMILARITY, CARBON THIN FILMS, HURST EXPONENT, FRACTAL DIMENSION.  (Received 23 August 2011, published online 30 December 2011)  1. INTRODUCTION  Self-similar or self-affine structures are widely presented in all areas of the nature science [1]. The self-similarity means that the each segment of the initial set has the same structure as the whole object. The properties of such systems can be described by special parameters, like fractal dimension (or set of dimensions in case of complex structures [2]), Hurst exponent and other [1, 2]. Most common examples of the geometric fractals are the Cantor and Serpinsky sets, the Koch curve and many others objects [2]. Self-similarity appears not only in topological structure of the objects, also the phase space of the complex stochastic systems often has a hierarchical structure with non participant regions [3].The stochastic fractals can be illustrated through the Lorenz attractor sand nonstationary time-series [4, 5].  Self-similar surfaces play the important role in surface contacts. Morphologies of these surfaces can appear to be quite different depending on the scale with which they can be observed. It can be shown that concepts like roughness are replaced by exponents that refer not to the roughness, but to the fashion in which the roughness changes when the observation scale itself changes [6]. The self-similarity of surfaces can be produced by the fracture or by the mechanical processing. Surface can be formed as a result of a deposition process, shrink due to erosion or etching, propagate through inhomogeneous media.Fractal dimension of such structures is usually determined by obtaining the photographs taken at various zoom levels, with subsequent application of the photograph at a square grid and numerical analysis.  In proposed article we present the investigation of the self-similar structure of the carbon film surface by numerical methods of scaling analysis. Our calculations are based on the algorithm mofmulti fractal     GENERALIZED HURST EXPONENT OF THE CARBON… 21 detrended fluctuation analysis (MF-DFA) [7], that differs from the standard box-counting based procedures of the fractal dimension estimation [8].The standard methodic gives precise results only for deterministic self-similar objects with explicit numerical model, and has a large error when applying to the surface models with discreet range of the coordinates [9]. The MF-DFA algorithm initially was developed for investigation of the time series as a one-dimensional self-similar set [4], and later generalized for more complicated objects analysis [7]. Applying MF-DFA algorithm to the self-similar surface model as to two-dimensional time series allows one to obtain a quantitative characteristic of the surface structure.  In the opening section of the paper we briefly describe the object of our research – carbon thin film, obtained through magnetron sputtering, and the possible way for numerical simulation of its surface structure. Next we refer the main steps of the MF-DFA algorithm and present the results of our calculations, comparing them for different parts of the surface sample. Our works ends with summation of the performed investigations.  2. THIN FILMS UNDER INVESTIGATION   Carbon condensates were obtained by deposition of the ion-sputtering substance onto (001) KCl facets according to original technique [10-14]. As a result the high porous carbon thin films were obtained. The SEM images of the condensates at different scale resolution are shown in Fig. 1.       Fig. 1 – SEM images of the carbon thin films at different scale resolutions [14]   In order to investigate the self similarity in the structure of the surfaces we select the four SEM pictures of the surface at different microscopic    22 V.N. BORISYUK, JASSIM KASSI, A.I. HOLOVCHENKO  zoom, specified at the photos in Fig. 1. For each SEM image a numerical model was build. The image was presented by a surface in Cartesian coordinates, where each point defined by the numbers and brightness of the pixel in related image. Thus, the effective height of the surface model varies in the [0, 1] range, where zero height presents the absolutely dark (i.e. black) pixel. Example of the surface image and corresponded numerical model are presented in figure 2.Making use of the developed model allows for calculation of the generalized Hurst exponent within the two-dimensional MF-DFA algorithm [7].     Fig. 2 – Fragment of the SEM image of the carbon thin film (upper panel) and related numerical model (lower panel)  3. IMAGE ANALYSYS METHODOLOGY   All surfaces were investigated within two dimensional multifractal detren-ded fluctuation analyses (MF-DFA) methodology [7]. This algorithm allows for calculation of the main parameter of the self similar structure, such as generalized Hurst exponent through the following numerical procedure [7].  Self-similar surface is considering as a two-dimensional data array X (i, j), where in dices i, j has a discrete values i  1,2,…, M and j  1,2,…, N. This surface fragmented into Ms  Ns non overlapping segments of the size s  s, where Ms  [M/s] and Ns  [N/s] are integer numbers.   Denoting each segment through v, , the cumulative sum    1 21 21 1( , ) ( , ),1 , ,jik ku i j X k k i j s  (1)  is calculated for all segments v, .   Next step is a detrending procedure for the obtained surface ( , )u i j . The trend can be removed by the fitting procedure, when it determines by some smooth polynomial function ( , )u i j . There are many possible expression of the ( , )u i j function [7], we chose the simplest one, in order to reduce the computational time:      GENERALIZED HURST EXPONENT OF THE CARBON… 23  ( , )u i j ai bj c  (2)  where a, b, c – coefficients, defined by list-squares fitting algorithm. It should be mentioned, that more complicated forms of ( , )u i j  does not provides any significant increasing of the precision of the method, but noticeably increase computational time [7]. After detrending we arrive at residual function:   ( , ) ( , ) ( , )i j u i j u i j , (3)  and a dispersion of the v,  segment of lengths:   2 221 11( , , ) ( , )s si jF s i js (4)  Dispersion of the all segments is calculated through averaging over all surfaces:   1/1 11( ) [ ( , , )]s sqM Nqqs sF s F sM N, (5)  where q is a deformation parameter, initialized to increase the role of the segments with small (when q < 0) or high (q > 0) fluctuations  F2(v, , s) respectively [4]. A tq  0 expression (5) takes form [6]:   01 11( ) exp ln[ ( , , )]s sM Ns sF s F sM N, (6)  according to l'Hôpital's rule. For statistically correct result s values must be varied in the range from smin  6tosmax  min(M, N) / 4.The dispersion (5) and the segment size s connected through a scaling relation:   ( )( ) ~h qqF s s, (7)  where, h(q) is a generalized Hurst exponent. If the object under investigation has a self similar structure, the dependence (7) must be linear in logarithmic scales. Examples of relation (7) for the surfaces of the carbon thin films are presented in Fig. 3.    Fig. 3 – Relation (7) in logarithmic scales for the surface of the carbon thin films plotted at different negative (a) and positive (b) values of the deformation parameter q    24 V.N. BORISYUK, JASSIM KASSI, A.I. HOLOVCHENKO  As it follows from the figure, dependence (7) has a clear linear form only for q > 0 values, while in opposite range the calculations are expected to give a large error.   4. RESULTS AND DISCUSSION  The generalized Hurst exponent was calculated for all surfaces, shown in Fig. 1. According to the structure of the fractal objects, the Hurst exponent of the self-similar surface are expected to be independent from the resolution scale. The results of the calculations are shown in Fig. 3.    Fig. 4 – Hurst exponent of the carbon thin film surface at different SEM images resolutions (different symbols relate to the different microscopic scale zoom: squares – 585, circles – 1910, triangles – 383 and diamonds – 1580)   As it can be seen from the figure, the calculated parameters for different resolution scale of the surfaces are almost coinciding in q  0 region, and differs from each other for q < values, i.e. self-similarity in surface structure exist while q  0, and breaks at q < 0. This result can be explained by the statistical meaning of the fluctuation function (5), when q < 0, the main contribution in (5) is made by the segments v,  with small fluctuation (4) (namely F2(v, , s) << 1), due to the power q.    Fig. 5 – Density distribution of the effective heights of the surfaces at different SEM images resolutions (different symbols relate to the different microscopic scale zoom: squares – 585, circles – 1910, triangles – 383 and diamonds – 1580)     GENERALIZED HURST EXPONENT OF THE CARBON… 25 Such F2(v, , s) values are calculated in segments where the surface height coordinates are maximally close to the smooth function ( , )u i j  (in general, F2(v, , s) depends on the segments size s , but since the whole set min maxs s s is considered for every q value, this dependence is insignificant). Noticeably, since the fluctuation F2(v, , s) is not equal to zero for any segment v,  [7], totally smooth areas of the surface with any size s  s are never appears. The segments with F2(v, , s) << 1 can be related to those areas of the SEM images, that appears to be enormously dark or bright or corrupted because of image noises and defects of the scanning microscopy imaging. There is no strong self-similarity in the structure of these areas in the surface, and this situation causes the difference between Hurst exponents in the q  0 region. To verify this assumption we calculate the distribution of the effective height hxy over numerical models of each surface (figure 5). As it can be seen from the shown dependencies, most significant differences between distributions are observed for most dark and bright areas of the SEM images (minimum and maximum heights of the surface respectively). This feature can be the reason of the difference between Hurst exponents at q  0.  5. CONCLUSIONS  Above consideration shows, that the surface of the carbon condensates has the self-similar structure. This conclusion follows from the approximately the same Hurst exponent for different scale of the surface observation at positive deformation parameter q. Nevertheless, when q < 0, calculated parameters are strongly distinguishing. We explain this situation by the statistical meaning of the fluctuation function (5) and q parameter, which negative values related to the small fluctuations of the surface roughness, i.e. the self – similarity parameters at q < 0 represent the small details of the SEM images and can include the technical noise, which are always present in the scanning microscopy. Hurst exponent is often used as a parameter of the surface roughness [15], thus, obtained dependencies also can be useful in studying of the physical properties of the thin films.  REFERENCES 1. J. Feder, Fractals (Plenum press, New York and London, 1998).  2. A.I. Olemskoi, Fractal in Condensed Matter Physics, (in: Physics Reviews, ed. I.M. Khalatnikov, 1995 18, Part 1, 1). 3. A.I. Olemskoi, Synergetics of complex systems: Phenomenology and statistical theory (Moscow: Publishing house KRASAND: 2009) (in Russian). 4. J.W. Kanthelhardt, S.A. Zscheinger , E. Koscienly-Bunde, S. Havlin, A. Bunde, H.E. Stanley, Physica A 316, 87 (2002). 5. A.V. Khomenko, I.A. Lyashenko, V.N. Borisyuk, Fluctuat. Noise Lett. 9 19, (2010). 6. A.L. Barabasi, H.E. Stanley, Fractal concepts in surface growth, (Cambrige University Press, 1995). 7. Gao-Fen Gu, Wei-Xing Zhou, Phys. Rev. E 74, 061104 (2006). 8. T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia, B.I. Shriman, Phys. Rev. A 43, 2 (1991). 9. A.I. Olemskoi, V.M. Borisyuk, I.A. Shuda, Journal of Physical Studies 13, 2002 (2009).    26 V.N. BORISYUK, JASSIM KASSI, A.I. HOLOVCHENKO  10. V.I. Perekrestov, Yu.A. Kosminskaya, JETP Lett. 78, 223 (2003). 11. V.I. Perekrestov,  Yu.A. Kosminskaya, Tech. Phys. Lett. 31, 830 (2005).  12. V.I. Perekrestov, A.S. Kornyushchenko,Yu.A. Kosminskaya, Tech. Phys. Lett. 32, 868 (2006). 13. V.I. Perekrestov, A.I. Olemskoi, A.S. Kornyushchenko, Yu.A. Kosminskaya, Phys. Solid State. 51, 1060 (2009). 14. A.I. Olemskoi, V.I. Perekrestov, I.A. Shuda, .M. Borisyuk, A.A. Mokrenko, Metallofiz. Noveyshie Technol. 31, 11 (2009). 15. U. Tartaglino, V.N. Samoilov, B.N.J. Persson, J. Phys.: Condens. Matter 18, 4143 (2006). 

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Generalized Hurst Exponent of The Carbon Thin Film Surface

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