Characterization of the Metal-Insulator Transition in the Anderson Model of Localization

  • Michael Schreiber
  • Frank Milde


In this chapter we discuss three different methods in statistical physics which have been successfully implemented to determine the metal-insulator transition and to characterize the electronic states in disordered systems described by the Ander­son model of localization. First, we study the spatial decay of electronic states of the Anderson Hamiltonian along quasi-one-dimensional bars and use finite-size scaling to analyze the data and the transition in infinite three-dimensional samples. Second, we calculate the eigenfunctions and describe their spatial distribution by means of multi-fractal analysis. Third, we compute the eigenvalue spectrum and study the energy level statistics to determine the transition. Emphasis is laid on programming tricks to save computer time or to increase accuracy. As an example, some results of large-scale nu­merical investigations for anisotropic materials are presented, demonstrating that the three methods yield coinciding results. Several related topics of current research on the electronic properties of disordered materials are mentioned in which these statistical methods have been successfully applied.


Lyapunov Exponent System Size Extended State Anderson Model Localization Length 
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  1. 1.
    P.W. Anderson: Phys. Rev. 109, 1492 (1958)ADSCrossRefGoogle Scholar
  2. 2.
    P.E. Lindelof, J. Norregard, J. Hanberg: Phys. Scr. T 14, 317 (1986)Google Scholar
  3. 3.
    P.-E. Wolf, G. Maret: Phys. Rev. Lett. 55, 2696 (1985)ADSCrossRefGoogle Scholar
  4. 4.
    E.N. Economou, C.M. Soukoulis: Phys. Rev. B 28, 1093 (1983)ADSCrossRefGoogle Scholar
  5. 5.
    W. Apel, T.M. Rice: J. Phys. C 16, L1151 (1983); Q. Li, C.M. Soukoulis, E.N. Economou, G.S. Grest: Phys. Rev. B 40, 2825 (1989); I. Zambetaki, Q. Li, E.N. Economou, C.M. Soukoulis: Phys. Rev. Lett. 76, 3614 (1996); Q. Li, S. Katsoprinakis, E.N. Economou, C.M. Soukoulis: Phys. Rev. B 56, R4297 (1997), condmat/9704104Google Scholar
  6. 6.
    H. Stupp, M. Hornung, M. Lakner, O. Madel, H.V. Löhneysen: Phys. Rev. Lett. 71, 2634 (1993)ADSCrossRefGoogle Scholar
  7. 7.
    V.I. Oseledec: Trans. Moscow Math. Soc. 19, 197 (1968)MathSciNetGoogle Scholar
  8. 8.
    B. Kramer, M. Schreiber: `Transfer-Matrix Methods and Finite-Size Scaling for Disordered Systems. In: Computational Physics - Selected Methods, Simple Exercises, Serious Applications, ed. by K.H. Hoffmann, M. Schreiber ( Springer, Berlin, Heidelberg 1996 ) pp. 166 - 188Google Scholar
  9. 9.
    http://www.tu- Google Scholar
  10. 10.
    A. MacKinnon, B. Kramer: Z. Phys. B 53, 1 (1983)ADSCrossRefGoogle Scholar
  11. 11.
    A. MacKinnon: J. Phys. Condens. Matter 6, 2511 (1994)Google Scholar
  12. 12.
    F. Milde: Disorder-Induced Metal-Insulator Transition in Anisotropic Systems. Dissertation, Technische Universität Chemnitz (Chemnitz 2000)Google Scholar
  13. 13.
    K. Slevin, T. Ohtsuki: Phys. Rev. Lett. 82, 382 (1999), cond-mat/9812065Google Scholar
  14. 14.
    F. Milde, R. A. Römer, M. Schreiber, V. Uski: Eur. Phys. J. B 15, 685 (2000)Google Scholar
  15. 15.
    M. Schreiber: ‘Multifractal Characteristics of Electronic Wave Functions in Disordered Systems’. In: Computational Physics - Selected Methods, Simple Exercises, Serious Applications, ed. by K.H. Hoffmann, M. Schreiber ( Springer, Berlin, Heidelberg 1996 ) pp. 147 - 165Google Scholar
  16. 16.
    J. Cullum and R.A. Willoughby: Lanczos Algorithms for Large Symmetric Eigen-value Computations, Vol. 1: Theory. ( Birkhäuser, Boston 1985 )Google Scholar
  17. 17.
    U. Elsner, V. Mehrmann, F. Milde, R.A. Römer, M. Schreiber: SIAM J. Sci. Comp. 20, 2089 (1999)MATHCrossRefGoogle Scholar
  18. 18.
    M. Schreiber, F. Milde, R.A. Römer, U. Elsner, V. Mehrmann: Comp. Phys. Comm. 121-122, 517 (1999)ADSCrossRefGoogle Scholar
  19. 19.
    B.B. Mandelbrot: The Fractal Geometry of Nature ( Freemann, New York 1982 )MATHGoogle Scholar
  20. 20.
    H. Aoki: J. Phys. C: Solid State Phys. 16, L205 (1983)ADSCrossRefGoogle Scholar
  21. 21.
    F. Wegner: Nucl. Phys. B 316, 663 (1989)Google Scholar
  22. 22.
    M. Schreiber, H. Grussbach: J. Fractals 1, 1037 (1993)CrossRefGoogle Scholar
  23. 23.
    H. Grussbach, M. Schreiber: Phys. Rev. B 57, 663 (1995)ADSCrossRefGoogle Scholar
  24. 24.
    F. Milde, R. A. Römer, M. Schreiber: Phys. Rev. B 55, 9463 (1997)Google Scholar
  25. 25.
    M. Schreiber, U. Grimm, R.A. Römer, J.X. Zhong: Physica A 266, 477 (1999)CrossRefGoogle Scholar
  26. 26.
    U. Grimm, R.A. Römer, M. Schreiber, J.X. Zhong: Mat. Sci. and Eng. 294-296, 564 (2001), cond-mat/9908063.Google Scholar
  27. 27.
    J.X. Zhong, U. Grimm, R.A. Römer, M. Schreiber: Phys. Rev. Lett. 80, 3996 (1998)ADSCrossRefGoogle Scholar
  28. 28.
    I.K. Zharekeshev, B. Kramer: Phys. Rev. Lett. 79, 717 (1997)ADSCrossRefGoogle Scholar
  29. 29.
    G. Casati, F. Izrailev, L. Molinari: J. Phys. A 24, 4755 (1991)Google Scholar
  30. 30.
    F. Epperlein, M. Schreiber, T. Vojta: Phys. Rev. B 56, 5890 (1997)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2002

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  • Michael Schreiber
  • Frank Milde

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