Advertisement

Localization of Electronic States in Amorphous Materials: Recursive Green’s Function Method and the Metal-Insulator Transition at E ≠ 0

  • Alexander Croy
  • Rudolf A. Römer
  • Michael Schreiber
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 52)

Abstract

Traditionally, condensed matter physics has focused on the investigation of perfect crystals. However, real materials usually contain impurities, dislocations or other defects, which distort the crystal. If the deviations from the perfect crystalline structure are large enough, one speaks of disordered systems. The Anderson model [1] is widely used to investigate the phenomenon of localisation of electronic states in disordered materials and electronic transport properties in mesoscopic devices in general. Especially the occurrence of a quantum phase transition driven by disorder from an insulating phase, where all states are localised, to a metallic phase with extended states, has led to extensive analytical and numerical investigations of the critical properties of this metal-insulator transition (MIT) [2–4]. The investigation of the behaviour close to the MIT is supported by the one-parameter scaling hypothesis [5, 6]. This scaling theory originally formulated for the conductance plays a crucial role in understanding the MIT [7]. It is based on an ansatz interpolating between metallic and insulating regimes [8]. So far, scaling has been demonstrated to an astonishing degree of accuracy by numerical studies of the Anderson model [9–13]. However, most studies focused on scaling of the localisation length and the conductivity at the disorder-driven MIT in the vicinity of the band centre [9, 14, 15]. Assuming a power-law form for the d.c. conductivity, as it is expected from the one-parameter scaling theory, Villagonzalo et al. [6] have used the Chester-Thellung-Kubo-Greenwood formalism to calculate the temperature dependence of the thermoelectric properties numerically and showed that all thermoelectric quantities follow single-parameter scaling laws [16, 17].

Keywords

System Size Critical Exponent Thermoelectric Property Anderson Model Localisation Length 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    1. P. W. Anderson. Absence of diffusion in certain random lattices. Phys. Rev., 109:1492–1505, 1958.CrossRefGoogle Scholar
  2. 2.
    2. B. Kramer and A. MacKinnon. Localization: theory and experiment. Rep. Prog. Phys., 56:1469–1564, 1993.CrossRefGoogle Scholar
  3. 3.
    3. R. A. Römer and M. Schreiber. Numerical investigations of scaling at the Anderson transition. In T. Brandes and S. Kettemann, editors, The Anderson Transition and its Ramifications - Localisation, Quantum Interference, and Interactions, volume 630 of Lecture Notes in Physics, pages 3–19. Springer, Berlin, 2003.Google Scholar
  4. 4.
    4. I. Plyushchay, R. A. Römer, and M. Schreiber. The three-dimensional Anderson model of localization with binary random potential. Phys. Rev. B, 68:064201, 2003.CrossRefGoogle Scholar
  5. 5.
    5. J. E. Enderby and A. C. Barnes. Electron transport at the Anderson transition. Phys. Rev. B, 49:5062, 1994.CrossRefGoogle Scholar
  6. 6.
    6. C. Villagonzalo, R. A. Römer, and M. Schreiber. Thermoelectric transport properties in disordered systems near the Anderson transition. Eur. Phys. J. B, 12:179–189, 1999. ArXiv: cond-mat/9904362.CrossRefGoogle Scholar
  7. 7.
    7. E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan. Scaling theory of localization: absence of quantum diffusion in two dimensions. Phys. Rev. Lett., 42:673–676, 1979.CrossRefGoogle Scholar
  8. 8.
    8. P. A. Lee and T. V. Ramakrishnan. Disordered electronic systems. Rev. Mod. Phys., 57:287–337, 1985.CrossRefGoogle Scholar
  9. 9.
    9. K. Slevin and T. Ohtsuki. Corrections to scaling at the Anderson transition. Phys. Rev. Lett., 82:382–385, 1999. ArXiv: cond-mat/9812065.CrossRefGoogle Scholar
  10. 10.
    10. F. Milde, R. A. Römer, and M. Schreiber. Energy-level statistics at the metalinsulator transition in anisotropic systems. Phys. Rev. B, 61:6028–6035, 2000.CrossRefGoogle Scholar
  11. 11.
    11. F. Milde, R. A. Römer, M. Schreiber, and V. Uski. Critical properties of the metal-insulator transition in anisotropic systems. Eur. Phys. J. B, 15:685–690, 2000. ArXiv: cond-mat/9911029.CrossRefGoogle Scholar
  12. 12.
    12. M. L. Ndawana, R. A. Römer, and M. Schreiber. Finite-size scaling of the level compressibility at the Anderson transition. Eur. Phys. J. B, 27:399–407, 2002.CrossRefGoogle Scholar
  13. 13.
    13. M. L. Ndawana, R. A. Römer, and M. Schreiber. Effects of scale-free disorder on the Anderson metal-insulator transition. Europhys. Lett., 68:678–684, 2004.CrossRefGoogle Scholar
  14. 14.
    14. K. Slevin, P. Markoš, and T. Ohtsuki. Reconciling conductance .uctuations and the scaling theory of localization. Phys. Rev. Lett., 86:3594–3597, 2001.CrossRefGoogle Scholar
  15. 15.
    15. D. Braun, E. Hofstetter, G. Montambaux, and A. MacKinnon. Boundary conditions, the critical conductance distribution, and one-parameter scaling. Phys. Rev. B, 64:155107, 2001.CrossRefGoogle Scholar
  16. 16.
    16. C. Villagonzalo, R. A. Römer, and M. Schreiber. Transport properties near the Anderson transition. Ann. Phys. (Leipzig), 8:SI-269–SI-272, 1999. ArXiv: cond-mat/9908218.Google Scholar
  17. 17.
    17. C. Villagonzalo, R. A. Römer, M. Schreiber, and A. MacKinnon. Behavior of the thermopower in amorphous materials at the metal-insulator transition. Phys. Rev. B, 62:16446–16452, 2000.CrossRefGoogle Scholar
  18. 18.
    18. A. MacKinnon. The conductivity of the one-dimensional disordered Anderson model: a new numerical method. J. Phys.: Condens. Matter, 13:L1031–L1034, 1980.Google Scholar
  19. 19.
    19. A. MacKinnon. The calculation of transport properties and density of states of disordered solids. Z. Phys. B, 59:385–390, 1985.CrossRefGoogle Scholar
  20. 20.
    20. B. Mehlig and M. Schreiber. Energy-level and wave-function statistics in the Anderson model of localization. In K.H. Hoffmann and A. Meyer, editors, Parallel Algorithms and Cluster Computing - Implementations, Algorithms, and Applications, Lecture Notes in Computational Science and Engineering. Springer, Berlin, 2006.Google Scholar
  21. 21.
    21. P. Karmann, R. A. Römer, M. Schreiber, and P. Stollmann. Fine structure of the integrated density of states for Bernoulli-Anderson models. In K.H. Hoffmann, and A. Meyer, editors, Parallel Algorithms and Cluster Computing - Implementations, Algorithms, and Applications, Lecture Notes in Computational Science and Engineering. Springer, Berlin, 2006.Google Scholar
  22. 22.
    22. B. Bulka, B. Kramer, and A. MacKinnon. Mobility edge in the three dimensional Anderson model. Z. Phys. B, 60:13–17, 1985.CrossRefGoogle Scholar
  23. 23.
    23. B. Bulka, M. Schreiber, and B. Kramer. Localization, quantum interference, and the metal-insulator transition. Z. Phys. B, 66:21, 1987.CrossRefGoogle Scholar
  24. 24.
    24. T. Ohtsuki, K. Slevin, and T. Kawarabayashi. Review on recent progress on numerical studies of the Anderson transition. Ann. Phys. (Leipzig), 8:655–664, 1999. ArXiv: cond-mat/9911213.zbMATHCrossRefGoogle Scholar
  25. 25.
    25. T. Ando. Numerical study of symmetry effects on localization in two dimensions. Phys. Rev. B, 40:5325, 1989.CrossRefMathSciNetGoogle Scholar
  26. 26.
    26. P.Cain, R. A. Römer, and M. Schreiber. Phase diagram of the three-dimensional Anderson model of localization with random hopping. Ann. Phys. (Leipzig), 8:SI-33–SI-38, 1999. ArXiv: cond-mat/9908255.Google Scholar
  27. 27.
    27. F. Milde, R. A. Römer, and M. Schreiber. Multifractal analysis of the metalinsulator transition in anisotropic systems. Phys. Rev. B, 55:9463–9469, 1997.CrossRefGoogle Scholar
  28. 28.
    28. H. Stupp, M. Hornung, M. Lakner, O. Madel, and H. v. Löhneysen. Possible solution of the conductivity exponent puzzle for the metal-insulator transition in heavily doped uncompensated semiconductors. Phys. Rev. Lett., 71:2634–2637, 1993.CrossRefGoogle Scholar
  29. 29.
    29. S. Waffenschmidt, C. Pffeiderer, and H. v. Löhneysen. Critical behavior of the conductivity of Si:P at the metal-insulator transition under uniaxial stress. Phys. Rev. Lett., 83:3005–3008, 1999. ArXiv: cond-mat/9905297.CrossRefGoogle Scholar
  30. 30.
    30. F. Wegner. Electrons in disordered systems. Scaling near the mobility edge. Z. Phys. B, 25:327–337, 1976.CrossRefGoogle Scholar
  31. 31.
    31. D. Belitz and T. R. Kirkpatrick. The Anderson-Mott transition. Rev. Mod. Phys., 66:261–380, 1994.CrossRefGoogle Scholar
  32. 32.
    32. R. A. Römer, C. Villagonzalo, and A. MacKinnon. Thermoelectric properties of disordered systems. J. Phys. Soc. Japan, 72:167–168, 2002. Suppl. A.Google Scholar
  33. 33.
    33. C. Villagonzalo. Thermoelectric Transport at the Metal-Insulator Transition in Disordered Systems. PhD thesis, Chemnitz University of Technology, 2001.Google Scholar
  34. 34.
    34. P. Cain, F. Milde, R.A. Römer, and M. Schreiber. Applications of cluster computing for the Anderson model of localization. In S.G. Pandalai, editor, Recent Research Developments in Physics, volume 2, pages 171–184. Transworld Research Network, Trivandrum, India, 2001.Google Scholar
  35. 35.
    35. P. Cain, F. Milde, R. A. Römer, and M. Schreiber. Use of cluster computing for the Anderson model of localization. Comp. Phys. Comm., 147:246–250, 2002.zbMATHCrossRefGoogle Scholar
  36. 36.
    36. B. Kramer and M. Schreiber. Transfer-matrix methods and .nite-size scaling for disordered systems. In K. H. Hoffmann and M. Schreiber, editors, Computational Physics, pages 166–188, Springer, Berlin, 1996.Google Scholar
  37. 37.
    37. A. Eilmes, R. A. Römer, and M. Schreiber. The two-dimensional Anderson model of localization with random hopping. Eur. Phys. J. B, 1:29–38, 1998.CrossRefGoogle Scholar
  38. 38.
    38. U. Elsner, V. Mehrmann, F. Milde, R. A. Römer, and M. Schreiber. The Anderson model of localization: a challenge for modern eigenvalue methods. SIAM J. Sci. Comp., 20:2089–2102, 1999. ArXiv: physics/9802009.zbMATHCrossRefGoogle Scholar
  39. 39.
    39. M. Schreiber, F. Milde, R. A. Römer, U. Elsner, and V. Mehrmann. Electronic states in the Anderson model of localization: benchmarking eigenvalue algorithms. Comp. Phys. Comm., 121–122:517–523, 1999.CrossRefGoogle Scholar
  40. 40.
    40. E. N. Economou. Green's Functions in Quantum Physics. Springer-Verlag, Berlin, 1990.Google Scholar
  41. 41.
    41. G. Czycholl, B. Kramer, and A. MacKinnon. Conductivity and localization of electron states in one dimensional disordered systems: further numerical results. Z. Phys. B, 43:5–11, 1981.CrossRefGoogle Scholar
  42. 42.
    42. J. L. Cardy. Scaling and Renormalization in Statistical Physics. Cambridge University Press, Cambridge, 1996.Google Scholar
  43. 43.
    43. M. Büttiker. Absence of backscattering in the quantum Hall effect in multiprobe conductors. Phys. Rev. B, 38:9375, 1988.CrossRefGoogle Scholar
  44. 44.
    44. A. Croy. Thermoelectric properties of disordered systems. M.Sc. thesis, University of Warwick, Coventry, United Kindgom, 2005.Google Scholar
  45. 45.
    45. B. K. Nikolić. Statistical properties of eigenstates in three-dimensional mesoscopic systems with o.-diagonal or diagonal disorder. Phys. Rev. B, 64:14203, 2001.CrossRefGoogle Scholar
  46. 46.
    46. D. Boese, M. Lischka, and L.E. Reichl. Scaling behaviour in a quantum wire with scatterers. Phys. Rev. B, 62:16933, 2000.CrossRefGoogle Scholar
  47. 47.
    47. P. Cain, M. L. Ndawana, R. A. Römer, and M. Schreiber. The critical exponent of the localization length at the Anderson transition in 3D disordered systems is larger than 1. 2001. ArXiv: cond-mat/0106005.Google Scholar
  48. 48.
    48. J. X. Zhong, U. Grimm, R. A. Römer, and M. Schreiber. Level spacing distributions of planar quasiperiodic tight-binding models. Phys. Rev. Lett., 80:3996–3999, 1998.CrossRefGoogle Scholar
  49. 49.
    49. U. Grimm, R. A. Römer, and G. Schliecker. Electronic states in topologically disordered systems. Ann. Phys. (Leipzig), 7:389–393, 1998.CrossRefGoogle Scholar
  50. 50.
    50. U. Grimm, R. A. Römer, M. Schreiber, and J. X. Zhong. Universal level-spacing statistics in quasiperiodic tight-binding models. Mat. Sci. Eng. A, 294–296:564, 2000. ArXiv: cond-mat/9908063.CrossRefGoogle Scholar
  51. 51.
    51. A. Eilmes, R. A. Römer, and M. Schreiber. Critical behavior in the twodimensional Anderson model of localization with random hopping. phys. stat. sol. (b), 205:229–232, 1998.CrossRefGoogle Scholar
  52. 52.
    52. P. Biswas, P. Cain, R. A. Römer, and M. Schreiber. O.-diagonal disorder in the Anderson model of localization. phys. stat. sol. (b), 218:205–209, 2000. ArXiv: cond-mat/0001315.CrossRefGoogle Scholar
  53. 53.
    53. A. Eilmes, R. A. Römer, and M. Schreiber. Localization properties of two interacting particles in a quasi-periodic potential with a metal-insulator transition. Eur. Phys. J. B, 23:229–234, 2001. ArXiv: cond-mat/0106603.CrossRefGoogle Scholar
  54. 54.
    54. R. A. Römer and A. Punnoose. Enhanced charge and spin currents in the onedimensional disordered mesoscopic Hubbard ring. Phys. Rev. B, 52:14809–14817, 1995.CrossRefGoogle Scholar
  55. 55.
    55. M. Leadbeater, R. A. Römer, and M. Schreiber. Interaction-dependent enhancement of the localisation length for two interacting particles in a one-dimensional random potential. Eur. Phys. J. B, 8:643–652, 1999.CrossRefGoogle Scholar
  56. 56.
    56. R. A. Römer, M. Schreiber, and T. Vojta. Disorder and two-particle interaction in low-dimensional quantum systems. Physica E, 9:397–404, 2001.CrossRefGoogle Scholar
  57. 57.
    57. C. Schuster, R. A. Römer, and M. Schreiber. Interacting particles at a metalinsulator transition. Phys. Rev. B, 65:115114–7, 2002.CrossRefGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • Alexander Croy
    • 1
  • Rudolf A. Römer
    • 2
  • Michael Schreiber
    • 1
  1. 1.Institut für PhysikTechnische Universität ChemnitzChemnitzGermany
  2. 2.Centre for Scientific Computing and Department of PhysicsUniversity of WarwickCoventryUnited Kingdom

Personalised recommendations