Advertisement

Journal of Statistical Physics

, Volume 143, Issue 5, pp 970–989 | Cite as

Anderson Localization Triggered by Spin Disorder—With an Application to Eu x Ca1−x B6

  • Daniel Egli
  • Jürg Fröhlich
  • Hans-Rudolf Ott
Article

Abstract

The phenomenon of Anderson localization is studied for a class of one-particle Schrödinger operators with random Zeeman interactions. These operators arise as follows: Static spins are placed randomly on the sites of a simple cubic lattice according to a site percolation process with density x and coupled to one another ferromagnetically. Scattering of an electron in a conduction band at these spins is described by a random Zeeman interaction term that originates from indirect exchange. It is shown rigorously that, for positive values of x below the percolation threshold, the spectrum of the one-electron Schrödinger operator near the band edges is dense pure-point, and the corresponding eigenfunctions are exponentially localized.

Localization near the band edges persists in a weak external magnetic field, H, but disappears gradually, as H is increased. Our results lead us to predict the phenomenon of colossal (negative) magnetoresistance and the existence of a Mott transition, as H and/or x are increased.

Our analysis is motivated directly by experimental results concerning the magnetic alloy Eu x Ca1−x B6.

Keywords

Anderson localization Spin disorder Magnetoresistance 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abrahams, E., Anderson, P.W., Licciardello, D.C., Ramakrishnan, T.V.: Scaling theory of localization: absence of quantum diffusion in two dimensions. Phys. Rev. Lett. 42(10), 673 (1979) ADSCrossRefGoogle Scholar
  2. 2.
    Aizenman, M., Lieb, E.H.: Magnetic properties of some itinerant-electron systems at t>0. Phys. Rev. Lett. 65(12), 1470–1473 (1990) ADSCrossRefGoogle Scholar
  3. 3.
    Anderson, P.W.: Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492 (1958) ADSCrossRefGoogle Scholar
  4. 4.
    Bourgain, J.: An approach to Wegner’s estimate using subharmonicity. J. Stat. Phys. 134, 969–978 (2009) MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    Bourgain, J., Kenig, C.E.: On localization in the continuous Anderson-Bernoulli model in higher dimension. Invent. Math. 161, 389–426 (2005) MathSciNetADSMATHCrossRefGoogle Scholar
  6. 6.
    Egli, D.: Two problems in transport theory: localization and friction. Ph.D. thesis, ETH Zürich (2011) Google Scholar
  7. 7.
    Elgart, A.: Lifshitz tails and localization in the three-dimensional Anderson model. Duke Math. J. 146(2), 331–360 (2009) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Fisk, Z., Johnston, D.C., Cornut, B., von Molnar, S., Oseroff, S., Calvo, R.: Magnetic, transport, and thermal properties of ferromagnetic EuB6. J. Appl. Phys. 50, 1911 (1979) ADSCrossRefGoogle Scholar
  9. 9.
    Fröhlich, J., Spencer, T.C.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88, 151–184 (1983) ADSMATHCrossRefGoogle Scholar
  10. 10.
    Fröhlich, J., Ueltschi, D.: Hund’s rule and metallic ferromagnetism. J. Stat. Phys. 118(516), 973–996 (2005) MathSciNetADSMATHCrossRefGoogle Scholar
  11. 11.
    Fröhlich, J., Martinelli, F., Scoppola, E., Spencer, T.C.: Constructive proof of localization in the Anderson tight binding model. Commun. Math. Phys. 101, 21–46 (1985) ADSMATHCrossRefGoogle Scholar
  12. 12.
    Goldsheid, I., Molchanov, S., Pastur, L.: Pure point spectrum of stochastic one-dimensional Schrödinger operators. Funct. Anal. Appl. 11, 1 (1977) CrossRefGoogle Scholar
  13. 13.
    Hislop, P.D., Klopp, F.: The integrated density of states for some random operators with non-sign definite potentials. J. Funct. Anal. 195, 12–47 (2002) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Kunes, J., Pickett, W.E.: Kondo and anti-Kondo coupling to local moments in EuB6. Phys. Rev. B 69, 165111 (2004) ADSCrossRefGoogle Scholar
  15. 15.
    Kunz, H., Souillard, B.: Sur le spectre des opérateurs aux différences finies aléatoires. Commun. Math. Phys. 78, 201–246 (1980) MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    Lieb, E.: Classical limit of quantum spin systems. Commun. Math. Phys. 31(4), 327–340 (1973) MathSciNetADSMATHCrossRefGoogle Scholar
  17. 17.
    Nagaoka, Y.: Ferromagnetism in a narrow, almost half-filled s band. Phys. Rev. 147(1), 392–405 (1966) ADSCrossRefGoogle Scholar
  18. 18.
    Pereira, V.M., Lopes dos Santos, J.M.B., Castro, E.V., Castro Neto, A.H.: Double exchange model for magnetic hexaborides. Phys. Rev. Lett. 93(14), 147202 (2004) ADSCrossRefGoogle Scholar
  19. 19.
    Spencer, T. C.: Lifshitz tails and localization. Notes (1993) Google Scholar
  20. 20.
    Thouless, D.J.: Exchange in solid 3He and the Heisenberg hamiltonian. Proc. Phys. Soc. 86, 893–904 (1965) MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    Wegner, F.: The mobility edge problem: continuous symmetry and a conjecture. Z. Phys. B 35, 207–210 (1979) ADSCrossRefGoogle Scholar
  22. 22.
    Wegner, F.: Bounds on the density of states in disordered systems. Z. Phys. B 44, 9–15 (1981) MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    Wigger, G.A., Beeli, C., Felder, E., Ott, H.R., Bianchi, A.D., Fisk, Z.: Percolation and the colossal magnetoresistance of Eu-based hexaboride. Phys. Rev. Lett. 93(14), 147203 (2004) ADSCrossRefGoogle Scholar
  24. 24.
    Wigger, G.A., Monnier, R., Ott, H.R.: Electronic transport in EuB6. Phys. Rev. B 69, 125118 (2004) ADSCrossRefGoogle Scholar
  25. 25.
    Wigger, G.A., Felder, E., Weller, M., Streule, S., Ott, H.R., Bianchi, A.D., Fisk, Z.: Percolation limited magnetic order in Eu1−xCaxB6. Eur. Phys. J. B 46, 231–235 (2005) ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsETH ZürichZürichSwitzerland
  2. 2.Laboratory for Solid State PhysicsETH ZürichZürichSwitzerland

Personalised recommendations