Duality and similarity properties of the effective permittivity of two-dimensional heterogeneous medium with inclusion of fractal geometry

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 73, Issue 3, 2006, Pages

  Mejdoubi, Abdelilah ^a   Brosseau, Christian ^a  

^a UNIVERSITÉ DE BRETAGNE OCCIDENTALE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

FRACTAL GEOMETRY; INTRINSIC COMPLEXITY; MAXWELL GARNETT ANALYSIS; SURFACE FRACTION DEPENDENCE;

DIELECTRIC DEVICES; FRACTALS; GEOMETRY; HETEROJUNCTIONS; MICROSTRUCTURE; NUMERICAL METHODS;

PERMITTIVITY;

EID: 33645036271     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.73.031405     Document Type: Article

References (45)

1. C. Brosseau, J. Phys. D JPAPBE 0022-3727 (to be published).
2. C. Brosseau and A. Beroual, Prog. Mater. Sci. PRMSAQ 0079-6425 10.1016/S0079-6425(02)00013-0 48, 373 (2003).
3. M. Sahimi, Heterogeneous Materials I: Linear Transport and Optical Properties (Springer, New York, 2003).
4. S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties (Springer, New York, 2002).
5. T. Hanai, H. Z. Zhang, K. Sekine, K. Asaka, and K. Asami, Ferroelectrics FEROA8 0015-0193 86, 191 (1988).
6. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1983).
7. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications (Wiley, New York, 1990).
8. J. B. Keller, J. Appl. Phys. JAPIAU 0021-8979 10.1063/1.1729580 34, 991 (1963);
9. see also J. B. Keller, J. Math. Phys. JMAPAQ 0022-2488 10.1063/1.1704146 5, 548 (1964).
10. A. M. Dykhne, Zh. Eksp. Teor. Fiz. ZETFA7 0044-4510 59, 110 (1970)
11. A. M. Dykhne, [Sov. Phys. JETP SPHJAR 0038-5646 32, 63 (1970)].
12. K. S. Mendelson, J. Appl. Phys. JAPIAU 0021-8979 46, 918 (1975);
13. K. S. Mendelson, J. Appl. Phys. JAPIAU 0021-8979 10.1063/1.321549 46, 4740 (1975).
14. B. Ya. Balagurov, Zh. Eksp. Teor. Fiz. ZETFA7 0044-4510 81, 665 (1981)
15. B. Ya. Balagurov, [Sov. Phys. JETP SPHJAR 0038-5646 54, 355 (1981)].
16. G. W. Milton, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.38.11296 38, 11296 (1988);
17. G. W. Milton, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.46. 542 46, 542 (1981);
18. G. W. Milton, J. Appl. Phys. JAPIAU 0021-8979 10.1063/1.329386 52, 5294 (1981);
19. in Physics and Chemistry of Porous Media, edited by, D. L. Johnson, and, P. N. Sen, (American Institute of Physics, New York, 1984);
20. G. W. Milton, Commun. Math. Phys. CMPHAY 0010-3616 10.1007/BF01217763 111, 281 (1987).
21. P. P. Durand and L. H. Ungar, Int. J. Numer. Methods Eng. IJNMBH 0029-5981 10.1002/nme.1620261108 26, 2487 (1988).
22. K. A. Schulgasser, Int. Commun. Heat Mass Transfer IHMTDL 0735-1933 10.1016/0735-1933(92)90047-L 19, 639 (1992).
23. H. E. Stanley, in Fractals and Disordered Systems, edited by, A. Bunde, and, S. Havlin, (Springer-Verlag, Berlin, 1991).
24. J. Feder, Fractals (Plenum, New York, 1988).
25. Although the issue considered in the current work is valuable in its own right, it is also of practical importance since many pseudo-2D systems, such as colloid clusters, collapsed through deposition onto a substrate, deposited thin films, or surfaces roughened by evaporation or etching, are self-affine. See, e.g., M. Kardar, G. Parisi, and Y. C. Zhang, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.56.889 56, 889 (1986);
26. W. M. Tong and R. S. Williams, Annu. Rev. Phys. Chem. ARPLAP 0066-426X 10.1146/annurev.physchem.45.1.401 45, 401 (1994);
27. C. Douketis, Z. Wang, T. L. Haslett, and M. Moskovits, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.51.11022 51, 11022 (1995).
28. A. Mejdoubi and C. Brosseau, J. Appl. Phys. JAPIAU 0021-8979 (to be published).
29. A. Taflove, Computational Electrodynamics-The Finite-Difference Time-Domain Method (Artech House Inc., Norwood MA, 1995);
30. K. S. Yee, IEEE Trans. Antennas Propag. IETPAK 0018-926X AP-14, 303 (1966);
31. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time Domain Method, 2nd ed. (Artech House, Boston, 2000);
32. K. S. Kunz and R. J. Luebbers, The FDTD Method for Electromagnetics (CRC Press, Boca Raton, 1993).
33. O. Pekonen, K. K. Kärkkäinen, A. H. Sihvola, and K. I. Nikoskinen, J. Electromagn. Waves Appl. JEWAE5 0920-5071 13, 67 (1999);
34. K. K. Kärkkäinen, A. H. Sihvola, and K. I. Nikoskinen, IEEE Trans. Geosci. Remote Sens. IGRSD2 0196-2892 10.1109/36.843023 GE-38, 1303 (2000).
35. Z. Hashin and S. Shtrikman, J. Appl. Phys. JAPIAU 0021-8979 10.1063/1.1728579 33, 3125 (1962). HS lower and upper bounds for any d -dimensional two-phase isotropic mixture in which ε2 ≥ ε1 are: εL = ε1 1 + ε2 2 - 1 2 (ε2 - ε1) 2 ε1 2 + ε2 1 + (d-1) ε1 and εU = ε1 1 + ε2 2 - 1 2 (ε2 - ε1) 2 ε1 2 + ε2 1 + (d-1) ε2, respectively.
36. A. Mejdoubi and C. Brosseau (unpublished).
37. During the past decade, a large amount of experimental evidence has accumulated showing that even in biomedical sciences fractal patterns could be observed, see, e. g., G. A. Losa and T. F. Nonnenmacher, Mod. Pathol. 0893-3952 9, 174 (1996).
38. S. Mallegol, C. Brosseau, P. Queffelec, and A. M. Konn, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.68.174422 68, 174422 (2003);
39. C. Brosseau, J. B. Youssef, P. Talbot, and A. M. Konn, J. Appl. Phys. JAPIAU 0021-8979 10.1063/1.1570935 93, 9243 (2003);
40. C. Brosseau and P. Talbot, IEEE Trans. Dielectr. Electr. Insul. ITDIES 1070-9878 10.1109/TDEI.2004.1349787 11, 819 (2004);
41. C. Brosseau, S. Mallegol, P. Queffelec, and J. B. Youssef, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.70.092401 70, 092401 (2004).
42. V. Myroshnychenko and C. Brosseau, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.71.016701 71, 016701 (2005).
43. V. Myroshnychenko and C. Brosseau, J. Appl. Phys. JAPIAU 0021-8979 10.1063/1.1835544 97, 044101 (2005).
44. C. Puente, J. Romeu, R. Pous, and A. Cardama, in Fractals in Engineering, edited by, J. L. Véhel, E. Lutton, and, C. Tricot, (Springer, New York, 1997).
45. These technologies will require ultralow permittivity interlevel dielectrics with effective ε<2.1, which must incorporate porosity (ε=1). See, e.g., International Technology Roadmap for Semiconductors, http://public.itrs.net/.