Homogenization of the Three-dimensional Hall Effect and Change of Sign of the Hall Coefficient



The notion of a Hall matrix associated with a possibly anisotropic conducting material in the presence of a small magnetic field is introduced. Then, for any material having a microstructure we prove a general homogenization result satisfied by the Hall matrix in the framework of the H-convergence of Murat–Tartar. Extending a result of Bergman, we show that the Hall matrix can be computed from the corrector associated with the homogenization problem when no magnetic field is present. Finally, we give an example of a microstructure for which the Hall matrix is positive isotropic almost everywhere, while the homogenized Hall matrix is negative isotropic.


  1. 1.
    Ancona A.: Some results and examples about the behavior of harmonic functions and Green’s functions with respect to second order elliptic operators. Nagoya Math. J. 165, 123–158 (2002)MATHMathSciNetGoogle Scholar
  2. 2.
    Bauman P., Marini A., Nesi V.: Univalent solution of an elliptic system of partial differential equations arising in homogenization. Indiana Univ. Math. J. 50(2), 747–757 (2001)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bergman, D.J.: Self duality and the low field Hall effect in 2D and 3D metal–insulator composites. In: Deutscher, G., Zallen, R., Adler, J. (eds.) Percolation Structures and Processes, pp. 297–321, 1983Google Scholar
  4. 4.
    Briane M., Manceau D., Milton G.W.: Homogenization of the two-dimensional Hall effect. J. Math. Ana. App. 339, 1468–1484 (2008)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Briane M., Milton G.W., Nesi V.: Change of sign of the corrector’s determinant for homogenization in three-dimensional conductivity. Arch. Rational Mech. Anal. 173, 133–150 (2004)MATHCrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Colombini F., Spagnolo S.: Sur la convergence de solutions d’équations paraboliques. J. Math. Pures et Appl. 56, 263–306 (1977)MATHMathSciNetGoogle Scholar
  7. 7.
    Dacorogna B.: Direct Methods in the Calculus of Variations, in Applied Mathematical Sciences 78. Springer, Berlin (1989)Google Scholar
  8. 8.
    Lakes R.: Cellular solid structures with unbounded thermal expansion. J. Mater. Sci. Lett. 15, 475–477 (1996)Google Scholar
  9. 9.
    Landau L., Lifchitz E.: Électrodynamique des Milieux Continus. Éditions Mir, Moscou (1969)Google Scholar
  10. 10.
    Meyers N.G.: An L p-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Sc. Norm. Sup. Pisa 17, 189–206 (1963)MATHMathSciNetGoogle Scholar
  11. 11.
    Murat F., Tartar L.: H-convergence, Topics in the Mathematical Modelling of Composite Materials. In: Cherkaev, L., Kohn, R.V.(eds) Progress in Nonlinear Differential Equations and their Applications, pp. 21–43. Birkaüser, Boston (1998)Google Scholar
  12. 12.
    Ali Omar, M.: Elementary Solid State Physics. Addison Wesley, Reading, MA, World Student Series Edition, 1975Google Scholar
  13. 13.
    Sigmund O., Torquato S.: Composites with extreme thermal expansion coefficients. Appl. Phys. Lett. 69, 3203–3205 (1996)CrossRefADSGoogle Scholar
  14. 14.
    Sigmund O., Torquato S.: Design of materials with extreme thermal expansion using a three-phase topology optimization method. J. Mech. Phys. Solids 45, 1037–1067 (1997)CrossRefMathSciNetADSGoogle Scholar
  15. 15.
    Stroud D., Bergman D.J.: New exact results for the Hall-coefficient and magnetoresistance of inhomogeneous two-dimensional metals. Phys. Rev. B (Solid State) 30, 447–449 (1984)ADSGoogle Scholar
  16. 16.
    Levi–Civita symbol, Wikipedia, the free encyclopedia. http://en.wikipedia.org/wiki/Levi-Civita_symbol

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Centre de MathématiquesINSA de Rennes & IRMARRennes CedexFrance
  2. 2.Department of MathematicsThe University of UtahSalt Lake CityUSA

Personalised recommendations