- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Reviews of Modern Physics

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Percolation and Conduction

Scott Kirkpatrick

Scott Kirkpatrick

  • IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598

PDF

Rev. Mod. Phys. 45, 574 – Published 1 October, 1973

DOI: https://doi.org/10.1103/RevModPhys.45.574

Abstract

Extensions of percolation theory to treat transport are described. Resistor networks, from which resistors are removed at random, provide the natural generalization of the lattice models for which percolation thresholds and percolation probabilities have previously been considered. The normalized conductance, G, of such networks proves to be a sharply defined quantity with a characteristic concentration dependence near threshold which appears sensitive only to dimensionality. Numerical results are presented for several families of 3D and 2D network models. Except close to threshold, the models based on bond percolation are accurately described by a simple effective medium theory, which can also treat continuous media or situations less drastic than the percolation models, for example, materials in which local conductivity has a continuous distribution of values. The present calculations provide the first quantitative test of this theory. A "Green's function" derivation of the effective medium theory, which makes contact with similar treatments of disordered alloys, is presented. Finally, a general expression for the conductance of a percolation model is obtained which relates G to the spin-stiffness coefficient, D, of an appropriately defined model dilute ferromagnet. We use this relationship to argue that the "percolation channels" through which the current flows above threshold must be regarded as three dimensional.

View figure in article

References (66)

  1. Abeles, B., and J. J. Hanak, 1970, Phys. Lett. A 34, 165
  2. Adler, D., L. P. Flora, and S. D. Senturia, 1973, Solid State Commun. 12, 9
  3. Brenig, W., P. Wölfle, and G. Döhler, 1971, Z. Phys. 246, 1
  4. Broadbent. S. R., and J. M. Hammersley, 1957, Proc. Camb. Philos. Soc. 53, 629
  5. Brown, W. F., 1956, Handb. Phys. 17, 104
  6. Bruggeman, D. A. G., 1935, Ann. Phys. (Leipz.) 24, 636
  7. Cohen, M. H., 1971, Phys. Today 24 (5), 26
  8. Cohen, M. H., and J. Jortner, 1973, Phys. Rev. Lett. 30, 696 30, 699
  9. Cutler, M., and N. F. Mott, 1969, Phys. Rev. 181, 1336
  10. Dean, P., and N. F. Bird, 1966, Mathematics Division Report Ma61 of the National Physical Laboratory, Teddington, Middlesex, England
  11. Dean, P., and N. F. Bird, 1967, Proc. Camb. Philos. Soc. 63, 477
  12. Edwards, S. F., 1961, Philos. Mag. 6, 617
  13. Eggarter, T. P., 1972, Phys. Rev. A 5, 2496
  14. Eggarter, T. P., and M. H. Cohen, 1970, Phys. Rev. Lett. 25, 807
  15. Eggarter, T. P., and M. H. Cohen, 1971, Phys. Rev. Lett. 27, 129
  16. Erdös, P., and A. Rényi, 1960, Publ. Math. Inst. Hung. Acad. Sci. 5, 17
  17. Essam, J. W., 1973, in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic, New York, 1973)
  18. Fisher, M. E., 1964, Proceedings of the IBM Scientific Computing Sumposium on Combinatorial Problems (New York, 1964)
  19. Fisher, M. E., and J. W. Essam, 1961, J. Math. Phys. 2, 609
  20. Frisch, H. L., and J. M. Hammersley, 1963, J. Soc. Ind. Appl. Math. 11, 894
  21. Frisch, H. L., J. M. Hammersley, and D. J. A. Walsh, 1962, Phys. Rev. 126, 949
  22. Fritzsche, H., 1973, in Amorphous Semiconductors, edited by J. Tauc (Plenum, New York)
  23. Godunov, S. K., and V. S. Ryabenki, 1964, Theory of Finite Difference Schemes (North-Holland, Amsterdam, 1964), Chap. 1, Sec. 4
  24. Goldin, E., and H. J. Juretschke, 1958, Trans. Met. Soc. AIME 212, 357
  25. Gurland, J., 1966, Trans. Met. Soc. AIME 236, 642
  26. Hammersley, J. M., 1957, Proc. Camb. Philos. Soc. 53, 642
  27. Harris, A. B., and P. Leath, 1973 (private communications)
  28. Herring, C., 1960, J. Appl. Phys. 31, 1939
  29. Izyumov, Y., 1966, Proc. Phys. Soc. Lond. 87, 505
  30. Juretschke, H. J., R. Landauer, and J. A. Swanson, 1957, J. Appl. Phys. 27, 836
  31. Kadanoff, L. P., and P. C. Martin, 1963, Ann. Phys. (N.Y.) 24, 419
  32. Kerner, E. H., 1956, Proc. Phys. Soc. Lond. B 69, 802 69, 808
  33. Kirkpatrick, S., 1971, Phys. Rev. Lett. 27, 1722
  34. Kirkpatrick, S., 1972a, Proceedings of the 2nd International Conference on Liquid Metals, Tokyo, Japan (Taylor and Francis, London, 1972)
  35. Kirkpatrick, S., 1972b, J. Non-Cryst. Solids 8, 160
  36. Kirkpatrick, S., 1973a, Solid State Commun. 12, 1279
  37. Kirkpatrick, S., 1973b, Lecture Notes from the Kyoto Symposium on Electrons in Disordered Materials, August 1972 Technical Report B15 (1973) of the Institute of Solid State Physics, University of Tokyo, Japan
  38. Kirkpatrick, S., and A. F. Mayadas, 1973, J. Appl. Phys. Oct. issue
  39. Krumhansl, J. H., 1973, in Amorphous Magnetism, edited by H. O. Hooper and A. M. deGraaf (Plenum, New York), p. 15
  40. Kumar, D., and A. B. Harris, 1973, Phys. Rev. B. (to be published)
  41. Landauer, R., 1952, J. Appl. Phys. 23, 779
  42. Landauer, R. 1971, Poor Man's Percolation Theory, private memorandum
  43. Last, B. J., 1972, J. Phys. C 5, 2805
  44. Last, B. J., and D. J. Thouless, 1971, Phys. Rev. Lett. 27, 1719
  45. LeLieur, J. P., 1973, J. Chem. Phys. (to be published)
  46. Lightsey, P. A., 1972, thesis, Cornell University (1972)
  47. Malliaris, A., and D. T. Turner, 1971, J. Appl. Phys. 42, 614
  48. Miller, M. N., 1969, J. Math. Phys. 10, 1988
  49. Morgan, D. J., and G. S. Rushbrooke, 1961, Mol. Phys. 4, 291
  50. Morgan, D. J., and G. S. Rushbrooke, 1963, Mol. Phys. 6, 477
  51. Morita, T., and T. Horiguchi, 1971, J. Phys. Soc. Jap. 30, 957
  52. Mott, N. F., and E. A. Davis, 1971, Electronic Processes in Non-Crystalline Materials (Oxford U. P., New York, 1971)
  53. Oguchi, T., 1969, J. Phys. Soc. Jap. 26, 580
  54. Pollak, M., 1972, J. Non-Cryst. Solids 11, 1
  55. Rushbrooke, G. S., 1970, in Critical Phenomena in Alloys, Magnets, and Superconductors (McGraw-Hill, New York, 1970), p. 155
  56. Rushbrooke, G. S., R. A. Muse, R. L. Stepenson, and K. Pirnie, 1972, J. Phys. C 5, 3371
  57. Scher, H., and R. Zallen, 1970, J. Chem. Phys. 53, 3759
  58. Shante, V. K. S., and S. Kirkpatrick, 1971, Adv. Phys. 20, 325
  59. Soven, P., 1967, Phys. Rev. 156, 809
  60. Stinchcombe, R. B., 1973, J. Phys. C 6, L 1
  61. Stone, H. L., 1968, SIAM J. Numer. Anal. 5, 530
  62. Thompson, W. A., T. Penney, S. Kirkpatrick, and F. Holtzberg, 1972, Phys. Rev. Lett. 29, 779
  63. Velický, B., S. Kirkpatrick, and H. Ehrenreich, 1968, Phys. Rev. 175, 747
  64. Vyssotsky, V. A., S. B. Gordon, H. L. Frisch, and J. M. Hammersley, 1961, Phys. Rev. 123, 1566
  65. Zallen, R., and H. Scher, 1971, Phys. Rev. B 4, 4771
  66. Ziman, J. M., 1968, J. Phys. C 1, 1532

Outline

Information

Rev. Mod. Phys. 45, 574– Published 1 October, 1973
  • Received 21 March 1973

© 1973 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation