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Numerical studies on the anderson localization problem

I. Localization Edges and Critical Exponents

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Zeitschrift für Physik B Condensed Matter

Abstract

The electron localization is studied for Anderson's tight-binding model with diagonal and off-diagonal disorder for a very large square lattice (10,000 sites) and diamond lattice (27,000 sites). The numerical investigations are based on the Lanczos recursion method. The convergence of the recursion coefficientsa n ,b n is discussed with regard to the electron localization.

From Anderson's criterion and an exact real space renormalization method the energy of the localization edge is found as a function of the degree of disorder. Also the dependence of the spatial decay rate of localized wave functions on the energy and the degree of disorder is evaluated. Near the Anderson transition, where all states become localized, we get two critical exponentsv E andv W , which lead us to the tentative suggestion of multicritical scaling laws for this transition.

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References

  1. Anderson, P.W.: Phys. Rev.109, 1492 (1958)

    Google Scholar 

  2. Zittartz, J., Langer, J.S.: Phys. Rev.148, 741 (1966)

    Google Scholar 

  3. Halperin, B., Lax, M.: Phys. Rev.148, 722 (1966)

    Google Scholar 

  4. Economou, E.N., Cohen, M.H.: Phys. Rev. B5, 2931 (1972)

    Google Scholar 

  5. Nitzan, A., Freed, K.F., Cohen, M.H.: Phys. Rev. B15, 4476 (1977)

    Google Scholar 

  6. Aharony, A., Imry, Y.: J. Phys. C: Solid St. Phys.10, L487 (1977)

    Google Scholar 

  7. Schuster, H.G.: Z. Physik B31, 99 (1978)

    Google Scholar 

  8. Wegner, F.J.: Z. Physik B25, 327 (1976)

    Google Scholar 

  9. Licciardello, D.C., Thouless, D.J.: J. Phys. C: Solid St. Phys.8, 4157 (1975)

    Google Scholar 

  10. Weaire, D., Williams, A.R.: J. Phys. C: Solid St. Phys.10, 1239 (1977)

    Google Scholar 

  11. Schönhammer, K., Brenig, W.: Phys. Lett.42 A, 447 (1973)

    Google Scholar 

  12. Yoshino, S., Okazaki, M.: J. Phys. Soc. Jap.43, 415 (1977)

    Google Scholar 

  13. Kramer, B., Weaire, D.: J. Phys. C: Solid St. Phys.11, L5 (1978)

    Google Scholar 

  14. Prelovsek, P.: Phys. Rev. B18, 3657 (1978)

    Google Scholar 

  15. Thouless, D.J.: Phys. Rep.13, 75 (1974)

    Google Scholar 

  16. Eliott, R.L., Krumhansl, J.A., Leath, P.L.: Rev. Mod. Phys.46, 465 (1974)

    Google Scholar 

  17. Ishii, K.: Progr. Theor. Phys. (suppl)53, 77 (1973)

    Google Scholar 

  18. Kimball, J.C.: J. Phys. C: Solid St. Phys.11, 4347 (1978)

    Google Scholar 

  19. Mott, N.F.: Adv. Phys.16, 49 (1967) Mott, N.F., Davis, E.: Electronic Processes in Non-Crystalline Materials. Oxford: Clarendon Press 1971

    Google Scholar 

  20. Götze, W.: Solid State Commun.27, 1393 (1978) and preprint

    Google Scholar 

  21. Götze, W., Prelovsek, P., Wölfle, P.: preprint, to appear in Solid State Commun.

  22. Abrahams, E., Anderson, P.W., Licciardello, D.C., Ramakrishna, T V.: Phys. Rev. Lett.42, 673 (1979)

    Google Scholar 

  23. Licciardello, D.C., Thouless, D.J.: J. Phys. C: Solid St. Phys.11, 925 (1978)

    Google Scholar 

  24. Haydock, R., Heine, V., Kelly, M.J.: J. Phys. C: Solid St. Phys.8, 2591 (1975)

    Google Scholar 

  25. Krey, U.: J. Magn. and Magn. Mater.7, 150 (1978)

    Google Scholar 

  26. Krey, U.: Z. Physik B31, 247 (1978)

    Google Scholar 

  27. Wilkinson, H.J.: The Algebraic Eigenvalue Problem, Oxford: Clarendon Press

  28. Haydock, R.: Phil. Mag.37, 97 (1978)

    Google Scholar 

  29. Weaire, D., Hodges, C.: J. Phys. C: Solid St. Phys.11, L685 (1978) and preprint (1979)

    Google Scholar 

  30. Haydock, R.: private communication

  31. Stein, J., Krey, U.: Solid State Commun.27, 797 (1978)

    Google Scholar 

  32. Stein, J., Krey, U.: Solid State Commun.27, 1405 (1978)

    Google Scholar 

  33. Economou, E.N., Antoniou, P.D.: Solid State Commun.21, 285 (1977)

    Google Scholar 

  34. Hoshino, K., Watabe, M.: J. Phys. Soc. Jap.43, 1337 (1977)

    Google Scholar 

  35. Schreiber, J., John, W.: Phys. stat. sol. (b)78, 199 (1976)

    Google Scholar 

  36. Weaire, D., Srivastava, V.: Solid State Commun.23, 863 (1977)

    Google Scholar 

  37. Odagaki, T., Yonezawa, F.: Solid State Commun.27, 1207 (1978)

    Google Scholar 

  38. Lubensky, T.C.: Phys. Rev. B11, 3573 (1975)

    Google Scholar 

  39. Anderson, P.W.: Proc. Nat. Acad. Sci. USA69, 1097 (1972)

    Google Scholar 

  40. Abram, R.A., Edwards, S.F.: J. Phys. C: Solid St. Phys.5, 1183 (1972)

    Google Scholar 

  41. Freed, K.F.: Phys. Rev. B5, 4802 (1972)

    Google Scholar 

  42. Abram, R.A.: J. Phys. C: Solid St. Phys.6, L379 (1973)

    Google Scholar 

  43. Licciardello, D.C., Thouless, D.J.: J. Phys. C: Solid St. Phys.9, L417 (1976)

    Google Scholar 

  44. Stauffer, D.: Phys. Rep. C, in press

  45. Mott, N.F.: Commun. Phys.1, 203 (1976)

    Google Scholar 

  46. Cohen, M.H., Jortner, J.: Phys. Rev. Lett.30, 699 (1973; Phys. Rev. A10, 978 (1974)

    Google Scholar 

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Authors and Affiliations

  1. Fachbereich Physik, Universität Regensburg, Universitätsstrasse 31, D-8400, Regensburg, Federal Republic of Germany

    J. Stein & U. Krey

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  1. J. Stein
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  2. U. Krey
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Stein, J., Krey, U. Numerical studies on the anderson localization problem. Z Physik B 34, 287–296 (1979). https://doi.org/10.1007/BF01325624

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  • DOI: https://doi.org/10.1007/BF01325624

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