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Fractal properties of gold, palladium and gold–palladium thin films on InP
Vacuum 84 (2010) 247–250
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
Fractal properties of gold, palladium and gold–palladium thin films on InP ¨ rmo¨s a, Szilvia Nagy b, *, Imre Mojzes a, ! Bernadett Varga a, Antal U a b
Budapest University of Technology and Economics, Department of Electronics Technology, Goldmann Gy. te´r 3, H-1111 Budapest, Hungary }r, Hungary ´n University, Department of Telecommunication, Egyetem te´r 1, H-9026 Gyo Sze´chenyi Istva
a b s t r a c t Keywords: Compound semiconductors Thin films Metallization
Thermal interaction of indium phosphide (InP) bulk compound semiconductor with thin gold metal films was investigated in the course of the present work. The interaction of the InP/Au system resulted in a pattern showing fractal dimensions. The temperature dependence of the fractal parameters was investigated in a broad temperature range from 200 to 600 C. No significant temperature dependence of the fractal dimension was observed. The same calculations will be presented for Au/InP and AuPd/InP systems. Our calculations show that the Pd-based contacts have a different behaviour than AuGe metallization where a strong temperature dependence of the fractal number was observed earlier. Another topology measure, the structural entropy is also calculated for the samples. The structural entropy is usually applied for determining the type of the localization of charge distributions, but it can also be used for generalized charges, such as the lightness of the pixels of an electron microscopy picture. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Indium phosphide (InP) is a key material for both optoelectronics and microwave devices. Ohmic and Schottky contacts are substantial parts of these compound semiconductor devices. High power density, high operational speed and reliability are the most important factors determining the materials science issues of the components used in these thin metal based layers. Palladium appears to be an important component in contact metallization [1,2]. For contacting p-InP palladium can be codeposited with gold and zinc [3]. After an appropriate heat treatment this metallization results in an Ohmic contact. For forming Schottky contacts titanium metallization can be applied [4]. For the n-InP Ohmic contact AuGe, AuPtTi and AuPtTiNiAuGe multilayers can be used [5]. For forming Schottky barriers on n-InP palladium [6] and gold [7] can be applied. If gold is combined with a single monolayer of Sb a large barrier height can be obtained [7]. In the device technology the deposition of multilayer metallization is followed by a heat treatment, usually under non-equilibrium conditions. During this process, an interaction between the compound semiconductor and the metallization results in out-
* Corresponding author. Tel.: þ3696613693; fax: þ3696613646. E-mail address: [email protected] (S. Nagy). Prof. Imre Mojzes, prominent member of the nanoscience community in Hungary, passed away on 18 April, 2009. !
0042-207X/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2009.06.004
evaporation of the volatile component; in the case of a ternary compound As, P and Sb will diffuse out [8]. During the deposition and the subsequent annealing, palladium reacts with the InP and a separate phase formation of Pd2InP [1] or Pd5InP [9] takes place. The surface of the heat-treated metallization often shows a fractal character [10,11]. The aim of this study was a detailed investigation of the fractal behaviour of the Au, Pd and AuPd metallization on the bulk InP crystals after the heat treatment. These patterns are the results of the interaction of thin (50–85 nm) layers with bulk compound semiconductor materials. 2. Experimental The gold, palladium and gold–palladium layers were deposited on n-(100) InP substrates. The wafers were cleaned and degreased in a 1:1 solution of HCl and H2O2, and, immediately before the metallization, etched in a 1% Br in methanol solution. The heat treatment was carried out in the working chamber of a scanning electron microscope (SEM) [12]. A linear heat treatment rate of 150 C/min was applied. During the heating an in-situ observation of the out-evaporation of the volatile component was observed and SEM pictures were taken at different temperatures. Some examples of the studied images are shown in Fig. 1. The calculation of the fractal dimension was carried out by a box counting algorithm of the FracLac program of the Image package. Box counting is a method for determining the fractal
248
B. Varga et al. / Vacuum 84 (2010) 247–250
Fig. 2. Structural entropy versus the filling factor of the 460 C Au (60 nm)/In P system. Five images were studied, each divided to 9 sub-pictures in order to generate more samples. The different signs mean different magnifications. Gaussian localization can be observed.
gradient of the straight line fitting to the resulting points. This definition is generalized for grayscale images, with various box shapes, but we remain with the original, black and white version, thus all the images have to be converted to black and white, by setting a threshold level. The filling factor and the structural entropy [15] form another measure for characterizing the topology of the images, however, the structural entropy is zero for black and white pictures. These two quantities can give information about the shape of the two dimensional distribution, the grayscale image. The structural entropy Sstr versus filling factor q plots of the given types of square integrable functions (like a Gaussian, an exponential peak) form a well defined line for each type of function. Comparing the Sstr(q) point of the studied picture with these known curves, the localization type of the forms in the picture can be determined. The application of the filling factor and the structural entropy has a solid state physics basis. They were used for characterizing the localization of electron density distributions on a regular grid [15–20]. They are defined for grid distributions, like the pixel intensity {Ii, i ¼ 1,.,N} of an SEM image, normalized according to
Fig. 1. Typical SEM picture of (A) the Au (60 nm)/InP(111) system at thermal treating temperature 525 C, (B) the Pd (50 nm)/In P(100) system at T ¼ 510 C, and (C) the Au(85 nm) Pd (50 nm)/In P(100) system at 540 C with 10 mm scale.
dimension according to its original definition by Hausdorff [13] and Mandelbrot [14]. Adopting the Hausdorff measure to black and white images results in the following steps. Let us count, how many boxes of diameter n are necessary for covering the white pixels. Next, cut the diameter of the box to n/2 (thus take the quarter of the area of it), and count the needed boxes again. Repeat these steps a few times (not after reaching the one pixel diameter) and plot the logarithm of the number of the boxes versus the number of steps (more precisely, the logarithm of the inverse-diameter of the boxes). The fractal dimension is the
Fig. 3. The fractal dimension d and localization factor a of the 60 nm Au film on (111)InP surface as a function of the heat-treating temperature T.
B. Varga et al. / Vacuum 84 (2010) 247–250
249
used for analyzing the shape of complex electron system with multifractal behaviour or high disorder [18–20]. A typical Sstr(q) map of x SEM images can be seen in Fig. 2. 3. Results
Fig. 4. The fractal dimension d and localization factor a of the 50 nm Pd film on (100)InP surface as a function of the heat treating temperature T.
0 Ii ; for i ¼ 1; .; N; and
N X
Ii ¼ 1;
(1)
i¼1
where N is the total number of pixels. The filling factor, defined as
q ¼
N
1 PN
(2)
2 i¼1 Ii
gives the ratio of the light pixels to all the pixels. The structural entropy is
Sstr ¼
N X i¼1
1 Ii ln Ii ln PN
2 i¼1 Ii
The fractal dimension calculations of the gold-InP systems are summarized in Fig. 3. The average localization type of the sample’s SEM image is also given the following way. The samples usually match the exp(jxja) curves, thus the approximate factor a is given by means of the heat treating temperatures. The fractal dimension of this material system slightly decreases with rather wild oscillations. The localization types of the 2D functions, i.e., the images of the samples are Gaussian or a little bit slower than it. Although, for some temperatures this localization exponent is plotted higher or lower than the average exp(jxj2), these deviations have no tendency. For the real images, they can have various background patterns due to the photography techniques, and these disturbing functions can distort the localization maps. Some photographic methods result in extremely high contrast, which does not disturb the fractal dimension calculations, but completely ruin the localization analysis; probably this causes the dominance of some extremely slowly decreasing background patterns. The results of the Pd/InP samples are shown in Fig. 4. Both analyzing techniques result in more or less constant behaviour, although in the images neither the average cluster size nor its variance remains constant. However, the images seem to be quite similar to the one in Fig. 1 B, this may cause the non-varying results. The AuPd layers’ topological data are given in Fig. 5. As a tendency, one could derive, that both the describing parameters decrease with increasing temperature, however, the localization parameter has a large oscillation at both ends. 4. Conclusions
;
(3)
where the first part of the expression can be recognized as the Shannon entropy. Both Sstr and ln q are differences of two generalized entropies, the so called Re´nyi entropies [16–18]. They are
The fractal dimension of the gold and gold–palladium systems tend to decrease as the treating temperature increases, while that of the palladium layer does not vary significantly. The localization type shows no real change or a very slight growing tendency for each of the systems. All the fractal dimensions are between 1.7 and 1.85, so the pictures show clear fractal behaviour. The localization type is typically Gaussian or slower, which can be a result of a slowly varying background pattern of the images. Acknowledgements The authors are grateful to Dr. La´szlo´ Dobos for providing the SEM pictures. This work was supported by the Bolyai Research Fellowship of the Hungarian Academy of Sciences. References
Fig. 5. The fractal dimension d and localization factor a of the 85 nm Au–50 nm Pd films on (100)-InP surface as a function of the heat treating temperature T.
[1] Ivey DG, Zhang L, Jian P. Interfacial reactions between palladium thin films and InP. J Mater Sci: Mater Electron 1991;2:21–7. [2] Mojzes I, Kova´cs B, Kun I, Ma´te´ L, Schuszter M, Dobos L. Proceedings of NATO Workshop. In: Jones WK, Harsa´nyi G, editors. Surface pattern formation and the volatile component loss of heat treated Pd/InP and Au/InP samples. Kluwer Academic Publishers; 1996. p. 307–14. [3] Bruce R, Clark D, Eicher S. Low resistance Pd/Zn/Pd Au ohmic contacts to P-type GaAs. J Electron Mat 1990;19:225–9. ¨ , Abay B, Tu¨ru¨t A, Yilmaz A. Temperature-dependent [4] Asubay S, Gu¨llu¨ O behavior of Ti/p-InP/ZnAu Schottky barrier diodes. Semicond Sci Tech 2008;23:035006. [5] Morais J, Fazan TA, Landers R, SatoEAS. Ohmic contacts formation on n-InP. J Appl Phys 1996;79:7058–61. [6] Kumar AA, Janardhanam V, Reddy VR, Reddy PN. Temperature dependence of current-voltage characteristics of the Pd/InP Schottky contacts. J Optoelectron Adv Mat 2007;9:3877–80.
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B. Varga et al. / Vacuum 84 (2010) 247–250
[7] Yamada M, Wahi AK, Kendelewicz T, Spicer WE. Large Schottky barrier heights on n-InP–A novel approach. Appl Phys Lett 1991;58:2701–3. [8] Dobos L, Kova´cs B, Mojzes I, Malina V, Pe´cz B, Kara´nyi J. The volatile component loss and the surface morphology of the gold-palladium metallizations to the compound semiconductor structure. Res Soc Symp Proc 1992;260:641–6. [9] Caron-Popowich R, Washburn J. Phase formation in the Pd-InP system. J Appl Phys 1988;64:4909–13. [10] Mojzes I, Kova´cs B, Schuszter M, Ma´te´ L, Kun I, Dobos L, et al. Fractal behaviour of the surface of in situ heat treated metal-InP contacts. Thin Solid Films 1998;317:69–72. [11] Dobos L, Kara´nyi J, Pe´cz B, Kova´cs B, Mojzes I, Malina V. Palladium based contacts to GaAs and InP. Periodica Polytechnica: Ser El Eng 1993;37:21–30. [12] Schuszter M, Bodna´r Z, Dobos L, Mojzes I. A novel evaluation method to determine fractal dimension of SEM images: a study of Au/Pd/GaAs contacts during the heat treatments. Phys Stat Sol (C) 2003;0:1051–4. [13] Hausdorff F. Dimension und a¨ußeres Maß. Mathematische Annalen 1919;79:157–79.
[14] Mandelbrot B. The Fractal geometry of nature. Lecture notes in mathematics 1358. San Francisco: W.H. Freeman, ISBN 0716711869; 1982. [15] Pipek J, Varga I. Universal classification scheme for the spatial localization properties of one-particle states in finite d-dimensional systems. Phys Rev A 1992;46:3148–64. [16] Pipek J, Varga I. Mathematical characterization and shape-analysis of localized, fractal and complex distributions in extended systems. Int J Quantum Chem 1994;51:539–53. [17] Pipek J, Varga I. Scaling behavior of energy functionals of highly complex distributions. Int J Quantum Chem 1998;70:125–31. [18] Varga I, Pipek J. Re´nyi entropies characterizing the shape and the extension of the phase space representation of quantum wave functions in disordered systems. Phys Rev E 2003;68:026202. [19] Varga I, Hofstetter E, Pipek J. One-parameter Superscaling at the Metal-Insulator Transition in Three Dimensions. Phys Rev Lett 1999;82:4683–6. [20] Varga I, Pipek J. The generalized localization lengths in one dimensional systems with correlated disorder. J Phys: Cond Mat 1998;10:305–11.
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
Fractal properties of gold, palladium and gold–palladium thin films on InP ¨ rmo¨s a, Szilvia Nagy b, *, Imre Mojzes a, ! Bernadett Varga a, Antal U a b
Budapest University of Technology and Economics, Department of Electronics Technology, Goldmann Gy. te´r 3, H-1111 Budapest, Hungary }r, Hungary ´n University, Department of Telecommunication, Egyetem te´r 1, H-9026 Gyo Sze´chenyi Istva
a b s t r a c t Keywords: Compound semiconductors Thin films Metallization
Thermal interaction of indium phosphide (InP) bulk compound semiconductor with thin gold metal films was investigated in the course of the present work. The interaction of the InP/Au system resulted in a pattern showing fractal dimensions. The temperature dependence of the fractal parameters was investigated in a broad temperature range from 200 to 600 C. No significant temperature dependence of the fractal dimension was observed. The same calculations will be presented for Au/InP and AuPd/InP systems. Our calculations show that the Pd-based contacts have a different behaviour than AuGe metallization where a strong temperature dependence of the fractal number was observed earlier. Another topology measure, the structural entropy is also calculated for the samples. The structural entropy is usually applied for determining the type of the localization of charge distributions, but it can also be used for generalized charges, such as the lightness of the pixels of an electron microscopy picture. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Indium phosphide (InP) is a key material for both optoelectronics and microwave devices. Ohmic and Schottky contacts are substantial parts of these compound semiconductor devices. High power density, high operational speed and reliability are the most important factors determining the materials science issues of the components used in these thin metal based layers. Palladium appears to be an important component in contact metallization [1,2]. For contacting p-InP palladium can be codeposited with gold and zinc [3]. After an appropriate heat treatment this metallization results in an Ohmic contact. For forming Schottky contacts titanium metallization can be applied [4]. For the n-InP Ohmic contact AuGe, AuPtTi and AuPtTiNiAuGe multilayers can be used [5]. For forming Schottky barriers on n-InP palladium [6] and gold [7] can be applied. If gold is combined with a single monolayer of Sb a large barrier height can be obtained [7]. In the device technology the deposition of multilayer metallization is followed by a heat treatment, usually under non-equilibrium conditions. During this process, an interaction between the compound semiconductor and the metallization results in out-
* Corresponding author. Tel.: þ3696613693; fax: þ3696613646. E-mail address: [email protected] (S. Nagy). Prof. Imre Mojzes, prominent member of the nanoscience community in Hungary, passed away on 18 April, 2009. !
0042-207X/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2009.06.004
evaporation of the volatile component; in the case of a ternary compound As, P and Sb will diffuse out [8]. During the deposition and the subsequent annealing, palladium reacts with the InP and a separate phase formation of Pd2InP [1] or Pd5InP [9] takes place. The surface of the heat-treated metallization often shows a fractal character [10,11]. The aim of this study was a detailed investigation of the fractal behaviour of the Au, Pd and AuPd metallization on the bulk InP crystals after the heat treatment. These patterns are the results of the interaction of thin (50–85 nm) layers with bulk compound semiconductor materials. 2. Experimental The gold, palladium and gold–palladium layers were deposited on n-(100) InP substrates. The wafers were cleaned and degreased in a 1:1 solution of HCl and H2O2, and, immediately before the metallization, etched in a 1% Br in methanol solution. The heat treatment was carried out in the working chamber of a scanning electron microscope (SEM) [12]. A linear heat treatment rate of 150 C/min was applied. During the heating an in-situ observation of the out-evaporation of the volatile component was observed and SEM pictures were taken at different temperatures. Some examples of the studied images are shown in Fig. 1. The calculation of the fractal dimension was carried out by a box counting algorithm of the FracLac program of the Image package. Box counting is a method for determining the fractal
248
B. Varga et al. / Vacuum 84 (2010) 247–250
Fig. 2. Structural entropy versus the filling factor of the 460 C Au (60 nm)/In P system. Five images were studied, each divided to 9 sub-pictures in order to generate more samples. The different signs mean different magnifications. Gaussian localization can be observed.
gradient of the straight line fitting to the resulting points. This definition is generalized for grayscale images, with various box shapes, but we remain with the original, black and white version, thus all the images have to be converted to black and white, by setting a threshold level. The filling factor and the structural entropy [15] form another measure for characterizing the topology of the images, however, the structural entropy is zero for black and white pictures. These two quantities can give information about the shape of the two dimensional distribution, the grayscale image. The structural entropy Sstr versus filling factor q plots of the given types of square integrable functions (like a Gaussian, an exponential peak) form a well defined line for each type of function. Comparing the Sstr(q) point of the studied picture with these known curves, the localization type of the forms in the picture can be determined. The application of the filling factor and the structural entropy has a solid state physics basis. They were used for characterizing the localization of electron density distributions on a regular grid [15–20]. They are defined for grid distributions, like the pixel intensity {Ii, i ¼ 1,.,N} of an SEM image, normalized according to
Fig. 1. Typical SEM picture of (A) the Au (60 nm)/InP(111) system at thermal treating temperature 525 C, (B) the Pd (50 nm)/In P(100) system at T ¼ 510 C, and (C) the Au(85 nm) Pd (50 nm)/In P(100) system at 540 C with 10 mm scale.
dimension according to its original definition by Hausdorff [13] and Mandelbrot [14]. Adopting the Hausdorff measure to black and white images results in the following steps. Let us count, how many boxes of diameter n are necessary for covering the white pixels. Next, cut the diameter of the box to n/2 (thus take the quarter of the area of it), and count the needed boxes again. Repeat these steps a few times (not after reaching the one pixel diameter) and plot the logarithm of the number of the boxes versus the number of steps (more precisely, the logarithm of the inverse-diameter of the boxes). The fractal dimension is the
Fig. 3. The fractal dimension d and localization factor a of the 60 nm Au film on (111)InP surface as a function of the heat-treating temperature T.
B. Varga et al. / Vacuum 84 (2010) 247–250
249
used for analyzing the shape of complex electron system with multifractal behaviour or high disorder [18–20]. A typical Sstr(q) map of x SEM images can be seen in Fig. 2. 3. Results
Fig. 4. The fractal dimension d and localization factor a of the 50 nm Pd film on (100)InP surface as a function of the heat treating temperature T.
0 Ii ; for i ¼ 1; .; N; and
N X
Ii ¼ 1;
(1)
i¼1
where N is the total number of pixels. The filling factor, defined as
q ¼
N
1 PN
(2)
2 i¼1 Ii
gives the ratio of the light pixels to all the pixels. The structural entropy is
Sstr ¼
N X i¼1
1 Ii ln Ii ln PN
2 i¼1 Ii
The fractal dimension calculations of the gold-InP systems are summarized in Fig. 3. The average localization type of the sample’s SEM image is also given the following way. The samples usually match the exp(jxja) curves, thus the approximate factor a is given by means of the heat treating temperatures. The fractal dimension of this material system slightly decreases with rather wild oscillations. The localization types of the 2D functions, i.e., the images of the samples are Gaussian or a little bit slower than it. Although, for some temperatures this localization exponent is plotted higher or lower than the average exp(jxj2), these deviations have no tendency. For the real images, they can have various background patterns due to the photography techniques, and these disturbing functions can distort the localization maps. Some photographic methods result in extremely high contrast, which does not disturb the fractal dimension calculations, but completely ruin the localization analysis; probably this causes the dominance of some extremely slowly decreasing background patterns. The results of the Pd/InP samples are shown in Fig. 4. Both analyzing techniques result in more or less constant behaviour, although in the images neither the average cluster size nor its variance remains constant. However, the images seem to be quite similar to the one in Fig. 1 B, this may cause the non-varying results. The AuPd layers’ topological data are given in Fig. 5. As a tendency, one could derive, that both the describing parameters decrease with increasing temperature, however, the localization parameter has a large oscillation at both ends. 4. Conclusions
;
(3)
where the first part of the expression can be recognized as the Shannon entropy. Both Sstr and ln q are differences of two generalized entropies, the so called Re´nyi entropies [16–18]. They are
The fractal dimension of the gold and gold–palladium systems tend to decrease as the treating temperature increases, while that of the palladium layer does not vary significantly. The localization type shows no real change or a very slight growing tendency for each of the systems. All the fractal dimensions are between 1.7 and 1.85, so the pictures show clear fractal behaviour. The localization type is typically Gaussian or slower, which can be a result of a slowly varying background pattern of the images. Acknowledgements The authors are grateful to Dr. La´szlo´ Dobos for providing the SEM pictures. This work was supported by the Bolyai Research Fellowship of the Hungarian Academy of Sciences. References
Fig. 5. The fractal dimension d and localization factor a of the 85 nm Au–50 nm Pd films on (100)-InP surface as a function of the heat treating temperature T.
[1] Ivey DG, Zhang L, Jian P. Interfacial reactions between palladium thin films and InP. J Mater Sci: Mater Electron 1991;2:21–7. [2] Mojzes I, Kova´cs B, Kun I, Ma´te´ L, Schuszter M, Dobos L. Proceedings of NATO Workshop. In: Jones WK, Harsa´nyi G, editors. Surface pattern formation and the volatile component loss of heat treated Pd/InP and Au/InP samples. Kluwer Academic Publishers; 1996. p. 307–14. [3] Bruce R, Clark D, Eicher S. Low resistance Pd/Zn/Pd Au ohmic contacts to P-type GaAs. J Electron Mat 1990;19:225–9. ¨ , Abay B, Tu¨ru¨t A, Yilmaz A. Temperature-dependent [4] Asubay S, Gu¨llu¨ O behavior of Ti/p-InP/ZnAu Schottky barrier diodes. Semicond Sci Tech 2008;23:035006. [5] Morais J, Fazan TA, Landers R, SatoEAS. Ohmic contacts formation on n-InP. J Appl Phys 1996;79:7058–61. [6] Kumar AA, Janardhanam V, Reddy VR, Reddy PN. Temperature dependence of current-voltage characteristics of the Pd/InP Schottky contacts. J Optoelectron Adv Mat 2007;9:3877–80.
250
B. Varga et al. / Vacuum 84 (2010) 247–250
[7] Yamada M, Wahi AK, Kendelewicz T, Spicer WE. Large Schottky barrier heights on n-InP–A novel approach. Appl Phys Lett 1991;58:2701–3. [8] Dobos L, Kova´cs B, Mojzes I, Malina V, Pe´cz B, Kara´nyi J. The volatile component loss and the surface morphology of the gold-palladium metallizations to the compound semiconductor structure. Res Soc Symp Proc 1992;260:641–6. [9] Caron-Popowich R, Washburn J. Phase formation in the Pd-InP system. J Appl Phys 1988;64:4909–13. [10] Mojzes I, Kova´cs B, Schuszter M, Ma´te´ L, Kun I, Dobos L, et al. Fractal behaviour of the surface of in situ heat treated metal-InP contacts. Thin Solid Films 1998;317:69–72. [11] Dobos L, Kara´nyi J, Pe´cz B, Kova´cs B, Mojzes I, Malina V. Palladium based contacts to GaAs and InP. Periodica Polytechnica: Ser El Eng 1993;37:21–30. [12] Schuszter M, Bodna´r Z, Dobos L, Mojzes I. A novel evaluation method to determine fractal dimension of SEM images: a study of Au/Pd/GaAs contacts during the heat treatments. Phys Stat Sol (C) 2003;0:1051–4. [13] Hausdorff F. Dimension und a¨ußeres Maß. Mathematische Annalen 1919;79:157–79.
[14] Mandelbrot B. The Fractal geometry of nature. Lecture notes in mathematics 1358. San Francisco: W.H. Freeman, ISBN 0716711869; 1982. [15] Pipek J, Varga I. Universal classification scheme for the spatial localization properties of one-particle states in finite d-dimensional systems. Phys Rev A 1992;46:3148–64. [16] Pipek J, Varga I. Mathematical characterization and shape-analysis of localized, fractal and complex distributions in extended systems. Int J Quantum Chem 1994;51:539–53. [17] Pipek J, Varga I. Scaling behavior of energy functionals of highly complex distributions. Int J Quantum Chem 1998;70:125–31. [18] Varga I, Pipek J. Re´nyi entropies characterizing the shape and the extension of the phase space representation of quantum wave functions in disordered systems. Phys Rev E 2003;68:026202. [19] Varga I, Hofstetter E, Pipek J. One-parameter Superscaling at the Metal-Insulator Transition in Three Dimensions. Phys Rev Lett 1999;82:4683–6. [20] Varga I, Pipek J. The generalized localization lengths in one dimensional systems with correlated disorder. J Phys: Cond Mat 1998;10:305–11.