Skip to main content

Computer simulation of the processes of formation of microclusters on the basis of scaling invariance of random walk


On the basis of scaling invariance of the process of random walk, a discrete three-dimensional algorithm is developed and implemented for computer simulation of the multistage processes of formation of porous clusters in a crystal matrix. A software package is worked out that provides the simulation of dynamic processes of clustering deep in the crystals with allowance for surface processes, applied external fields, and chemical reactions accompanying the processes of clustering. The morphological pattern of pores formed by the simulation is correlated with the behavior of the current-voltage characteristic of anodizing.

This is a preview of subscription content, access via your institution.


  1. 1.

    Nallet, P., Chassaing, E., Walls, M.G., and Hytch, M.J., Interface Characterization in Electrodeposited Cu-Co Multilayers, J. Appl. Phys., 1996, vol. 79, no. 9, pp. 6884–6889.

    Article  Google Scholar 

  2. 2.

    Aravamudhan, S., Luongo, K., Poddar, P., Srikanth, H., and Bhatsali, S., Porous Silicon Templates for Electrodeposition of Nanostructures, Appl. Phys., A, 2007, vol. 83, no. 4, pp. 773–780.

    Article  Google Scholar 

  3. 3.

    Feder, J., Fractals, New-York: Plenum, 1988.

    MATH  Google Scholar 

  4. 4.

    Fractals in Physics. Pietronero, L. and Tosatti, E., Eds., Amsterdam: North Holland, 1986.

    Google Scholar 

  5. 5.

    Peitgen, H.-O. and Richter, P.H., The Beauty of Fractals. Images of Complex Dynamical Systems. Berlin: Springer-Verlag, 1986.

    MATH  Google Scholar 

  6. 6.

    Ross, J.C., Fractal Surfaces, New-York: Plenum, 1994.

    Google Scholar 

  7. 7.

    Hockney, R.W., and Eastwood, J.W., Computer Simulation Using Particles, New York: McGraw-Hill, 1981.

    Google Scholar 

  8. 8.

    Fujita, H., Kobayashi, K., Kawai, T., and Shiga, K., Hall Effect of Photoelectrons in Cadmium Sulfide, J. Phys. Soc. Jpn., 1965, vol. 20, no. 1, pp. 109–122.

    Article  Google Scholar 

  9. 9.

    Gould, H. and Tobochnik, J., An Introduction to Computer Simulation Methods: Applications to Physical Systems, Reading, Mass.: Addison-Wesley, 1988.

    Google Scholar 

  10. 10.

    Smith, R.L., Chuang, S.-F., and Collins, S.D., A Theoretical Model of the Formation Morphologies of Porous Silicon, J. Electr. Mater., 1988, vol. 17, no. 6, pp. 533–541.

    Article  Google Scholar 

  11. 11.

    Smith, R.L. and Collins, S.D., Porous Silicon Formation Mechanisms, J. Appl. Phys., 1992, vol. 71, no. 8, pp. R1–R22.

    Article  Google Scholar 

  12. 12.

    Chuang, S.-F., Collins, S.D., and Smith, R.L., Preferential Propagation of Pores During the Formation of Porous Silicon: a Transmission Electron Microscopy Study, Appl. Phys. Lett., 1989, vol. 55, no. 7, pp. 675–677.

    Article  Google Scholar 

  13. 13.

    Kaplii, S.A., Prokaznikov, A.V., and Rud’, N.A., Clustering in determinate and stochastic fields, Zh. Tekh. Fiz., 2004, vol. 74, no. 5, pp. 6–11 [Tech. Phys. (Engl. Transl.), vol. 49, no. 5, pp. 526–531].

    Google Scholar 

  14. 14.

    Kaplii, S.A., Prokaznikov, A.V., and Rud’, N.A., Clusterization of Stochastically Wandering Particles in Potential Fields, Izv. Vyssh. Uch. Zaved. Fiz., 2004, no. 6, pp. 31–38 [Russ. Phys. J. (Engl. Transl.), vol. 47, no. 6, pp. 609–616].

  15. 15.

    Kvasnikov, I. A., Termodinamika i Statisticheskaya Fizika. Teoriya Neravnovesnykh sistem (Thermodynamics and Statistical Physics. Theory of Nonequilibrium Systems), Moscow: Mosk. Gos. Univ., 1987.

    Google Scholar 

  16. 16.

    Isihara, A., Statistical Physics, New York: Academic, 1971.

    Google Scholar 

  17. 17.

    Landau, L.D. and Lifshits, E.M., Gidrodinamika (Hydrodynamics), Moscow: Nauka, 1988 [Hydrodynamics (Engl. Transl.), Oxford: Pergamon, 1990)].

    Google Scholar 

  18. 18.

    Mozhaev, A.V., Buchin, E.Yu., and Prokaznikov, A.V., Dynamic Model of Three-Dimensional Cluster Formation, Pis’ma Zh. Tekh. Fiz., 2008, vol. 34, no.10, pp. 53–60 [Tech. Phys. Lett. (Engl. Transl.), vol. 34, no. 5, pp. 431–434].

    Google Scholar 

  19. 19.

    Klyatskin, V.I. and Gurarie, D., Coherent Phenomena in Stochastic Dynamical Systems, Usp. Fiz. Nauk, 1999, vol. 169, no. 2, 171–207 [Phys.-Usp. (Engl. Transl.), vol. 42, no. 2, pp. 165–198].

    Article  Google Scholar 

  20. 20.

    Kaplii, S. A., Prokaznikov, A.V., and Rud’, N.A., A Discrete Model of Adsorption with Three States, Pis’ma Zh. Tekh. Fiz., 2004, vol. 30, no.14, pp. 46–52 [Tech. Phys. Lett. (Engl. Transl.), vol. 30, no. 7, pp. 595–597].

    Google Scholar 

  21. 21.

    Kaplii, S.A., Prokaznikov, A.V., and Rud’, N.A., Discrete Model of Adsorption with a Finite Number of States, Zh. Tekh. Fiz., 2005, vol. 75, no. 12, pp. 1–9 [Tech. Phys. (Engl. Transl.), vol. 50, no. 12, pp. 1535–1543].

    Google Scholar 

  22. 22.

    Vanag, V.K., Study of Spatially Extended Dynamical Systems Using Probabilistic Cellular Automata, Usp. Fiz. Nauk, 1999, vol. 169, no. 5, 481–505 [Phys.-Usp. (Engl. Transl.), vol. 42, no. 5, p. 413–434, 1999].

    Article  Google Scholar 

  23. 23.

    Prokaznikov, A.V. and Svetovoy, V.B., Fluorine Penetration through the Whole Silicon Wafer during Anodization in HF Solution, Phys. Low-Dim. Structures, 2002, vol. 9/10, pp. 65–69.

    Google Scholar 

  24. 24.

    Lehmann, V., The Physics of Macropore Formation in Low-Doped n-Type Silicon, J. Electrochem. Soc., 1993, vol. 140, no. 10, pp. 2836–2843.

    Article  Google Scholar 

  25. 25.

    Buchin, E.Yu. and Prokaznikov, A.V., The Mechanisms of Formation of Pores of Different Morphology, Mikroelectronika, 1998, vol. 27, no. 2, pp. 107–113.

    Google Scholar 

  26. 26.

    Sokolov, A.V., Mathematical Models and Algorithms of Optimal Control of Dynamic Data Structures, Extended Abstract of Doctoral (Phys.-Math.) Dissertation, Saint Petersburg, St. Petersburg Gos. Univ., 2006.

    Google Scholar 

  27. 27.

    Shkarupa, E.V., Error Estimation and Optimization of the Functional Algorithms of a Random Walk on a Grid Which Are Applied to Solving the Dirichlet Problem for the Helmholtz Equation, Sibir. Math. Zh., 2003, vol. 44, no. 5, pp. 1163–1182 [Siber. Math. J. (Engl. Transl.), vol. 44, no. 5, pp. 908–925].

    MATH  MathSciNet  Google Scholar 

  28. 28.

    Bisi, O., Osicini, S., and Pavesi, L., Porous Silicon: a Quantum Sponge Structure for Silicon Based Electronics, Surf. Sci. Reports, 2000, vol. 38, no.1–3, pp. 1–126.

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to A. V. Prokaznikov.

Additional information

Original Russian Text © A.V. Mozhaev, A.V. Prokaznikov, 2009, published in Mikroelektronika, 2009, Vol. 38, No. 5, pp. 323–330.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mozhaev, A.V., Prokaznikov, A.V. Computer simulation of the processes of formation of microclusters on the basis of scaling invariance of random walk. Russ Microelectron 38, 291–298 (2009).

Download citation


  • 82.75.-z