Theoretical Background

  • Evdokiya Georgieva Kostadinova
Part of the Springer Theses book series (Springer Theses)


This chapter provides a review of some of the more fundamental developments in the theory of Anderson localization, including Andersson’s original method, Edwards and Thouless approach, and scaling theory. Section 2.1 presents a list of criteria, which have been used over the years as a qualitative distinction between localized and extended states. We emphasize the dependence of the discussed criteria on mean free path and system size. Sections 2.2 through 2.4 focus in more detail on three specific theoretical/numerical methods, which we consider the ‘backbone’ of the existing transport theory. The goal of this chapter is to outline the main features of the well-established models for localization, which will allow for a meaningful comparison with the spectral approach introduced in Chap.  3.


Anderson model Edwards and Thouless model Scaling theory 


  1. 1.
    P.W. Anderson, Absence of diffusion in certain random lattices. Phys. Rev. 109(5), 1492–1505 (1958)ADSCrossRefGoogle Scholar
  2. 2.
    S.N.F. Mott, H. Jones, The Theory of the Properties of Metals and Alloys (Courier Corporation, 1958)Google Scholar
  3. 3.
    N.F. Mott, Conduction in non-crystalline systems IX. The minimum metallic conductivity. Philos. Mag. 26(4), 1015–1026 (1972)ADSCrossRefGoogle Scholar
  4. 4.
    N. Mott, Metal-insulator transitions. Phys. Today 31(11) (1978)Google Scholar
  5. 5.
    A.F. Ioffe, Non-crystalline, amorphous, and liquid electronic semiconductors. Prog. Semicond. 4, 237–291 (1960)Google Scholar
  6. 6.
    R.J. Bell, P. Dean, Atomic vibrations in vitreous silica. Discuss. Faraday Soc. 50(0), 55–61 (1970)CrossRefGoogle Scholar
  7. 7.
    R.J. Bell, P. Dean, D.C. Hibbins-Butler, Localization of normal modes in vitreous silica, germania and beryllium fluoride. J. Phys. C Solid State Phys. 3(10), 2111 (1970)ADSCrossRefGoogle Scholar
  8. 8.
    J.T. Edwards, D.J. Thouless, Numerical studies of localization in disordered systems. J. Phys. C Solid State Phys. 5(8), 807 (1972)ADSCrossRefGoogle Scholar
  9. 9.
    D.J. Thouless, Electrons in disordered systems and the theory of localization. Phys. Rep. 13(3), 93–142 (1974)ADSCrossRefGoogle Scholar
  10. 10.
    B.J. Last, D.J. Thouless, Evidence for power law localization in disordered systems. J. Phys. C Solid State Phys. 7(4), 699 (1974)ADSCrossRefGoogle Scholar
  11. 11.
    E. Abrahams, Scaling theory of localization: absence of quantum diffusion in two dimensions. Phys. Rev. Lett. 42(10), 673–676 (1979)ADSCrossRefGoogle Scholar
  12. 12.
    D. Vollhardt, P. Wölfle, Scaling equations from a self-consistent theory of Anderson localization. Phys. Rev. Lett. 48(10), 699–702 (1982)ADSCrossRefGoogle Scholar
  13. 13.
    P.A. Lee, D.S. Fisher, Anderson localization in two dimensions. Phys. Rev. Lett. 47(12), 882–885 (1981)ADSCrossRefGoogle Scholar
  14. 14.
    G.J. Dolan, D.D. Osheroff, Nonmetallic conduction in thin metal films at low temperatures. Phys. Rev. Lett. 43(10), 721–724 (1979)ADSCrossRefGoogle Scholar
  15. 15.
    D.J. Bishop, D.C. Tsui, R.C. Dynes, Nonmetallic conduction in electron inversion layers at low temperatures. Phys. Rev. Lett. 44(17), 1153–1156 (1980)ADSCrossRefGoogle Scholar
  16. 16.
    M.P.A. Fisher, G. Grinstein, S.M. Girvin, Presence of quantum diffusion in two dimensions: universal resistance at the superconductor-insulator transition. Phys. Rev. Lett. 64(5), 587–590 (1990)ADSCrossRefGoogle Scholar
  17. 17.
    M. Y. Azbel’, “Quantum particle in a random potential: exact solution and its implications,” Phys. Rev. B, vol. 45, no. 8, pp. 4208–4216, 1992ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    D. Popović, A.B. Fowler, S. Washburn, Metal-insulator transition in two dimensions: effects of disorder and magnetic field. Phys. Rev. Lett. 79(8), 1543–1546 (1997)ADSCrossRefGoogle Scholar
  19. 19.
    S.J. Papadakis, M. Shayegan, Apparent metallic behavior at B = 0 of a two-dimensional electron system in AlAs. Phys. Rev. B 57(24), R15068–R15071 (1998)ADSCrossRefGoogle Scholar
  20. 20.
    P.T. Coleridge, R.L. Williams, Y. Feng, P. Zawadzki, Metal-insulator transition at B = 0 in p-type SiGe. Phys. Rev. B 56(20), R12764–R12767 (1997)ADSCrossRefGoogle Scholar
  21. 21.
    A. Punnoose, Metal-insulator transition in disordered two-dimensional electron systems. Science 310(5746), 289–291 (2005)ADSCrossRefGoogle Scholar
  22. 22.
    B.l. Altshuler, D.l. Maslov, V.m. Pudalov, Metal–insulator transition in 2D: Anderson localization in temperature-dependent disorder? Phys. Status Solidi B 218(1), 193–200 (2000)ADSCrossRefGoogle Scholar
  23. 23.
    S.V. Kravchenko, G.V. Kravchenko, J.E. Furneaux, V.M. Pudalov, M. D’Iorio, Possible metal-insulator transition at B = 0 in two dimensions. Phys. Rev. B 50(11), 8039–8042 (1994)ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Evdokiya Georgieva Kostadinova
    • 1
  1. 1.Center for Astrophysics, Space Physics and Engineering ResearchBaylor UniversityWacoUSA

Personalised recommendations