Landau level broadening without disorder, non–integer plateaus without interactions – an alternative model of the quantum Hall effect

T. Kramer

Department of Physics, Harvard University, 17 Oxford Street, Cambridge, MA 02138, U.S.A. e–mail: tobias.kramer@mytum.de

Recibido el 26 de enero de 2006
Aceptado el 15 de abril de 2006

Abstract

I review some aspects of an alternative model of the quantum Hall effect, which is not based on the presence of disorder potentials. Instead, a quantization of the electronic drift current in the presence of crossed electric and magnetic fields is employed to construct a non–linear transport theory. Another important ingredient of the alternative theory is the coupling of the two–dimensional electron gas to the leads and the applied voltages. By working in a picture where the external voltages fix the chemical potential in the 2D subsystem, the experimentally observed linear relation between the voltage and the location of the quantum Hall plateaus finds an natural explanation. Also, the classical Hall effect emerges as a natural limit of the quantum Hall effect. For low temperatures (or high currents), a non–integer substructure splits higher Landau levels into sublevels. The appearence of substructure and non–integer plateaus in the resistivity is not linked to electron–electron interactions, but caused by the presence of a (linear) electric field. Some of the resulting fractions correspond exactly to half–integer plateaus.

Keywords: Quantum Hall effects; theory and modeling.

Resumen

Se revisan algunas propiedades de un modelo alternativo del efecto Hall cuántico, que no está basado en la presencia de potenciales de desorden. En cambio, se emplea una cuantización de la corriente de arrastre electrónico en la presencia de campos eléctricos y magnéticos cruzados para construir una teoría de transporte no–lineal. El acoplamiento del gas bidimensional de electrones a las guías y los voltajes aplicados es otro ingrediente importante de esta teoría alternativa. Se encuentra una explicación natural de la relación lineal que se observa experimentalmente entre el voltaje y la ubicación de los niveles Hall cuánticos. Además, el efecto Hall clásico emerge como un límite natural del efecto Hall cuántico. A temperaturas bajas (o corrientes altas), una subestructura no–entera divide los niveles Landau más altos en subniveles. La aparición de una subestructura y niveles no–enteros en la resistividad no está ligada a las interacciones electrón–electrón, sino que es causada por la presencia de un campo eléctrico (lineal). Algunas de las fracciones resultantes corresponden exactamente a niveles semi–enteros.

Descriptores: Efectos Hall cuanticos; teoría y modelos.

PACS: 73.43.Cd

DESCARGAR ARTÍCULO EN FORMATO PDF

Acknowledgments

I would like to thank the organizers T. Belyaeva, R. Bijker, and E. Martinez Quiroz for the opportunity to present this work at XXIX Symposium on Nuclear Physics in Cocoyoc, Mexico. The invitation and hospitality of the Instituto de Física, U.N.A.M., (M. Moshinsky) and the Instituto de Ciencias Nucleares, U.N.A.M., (A. Frank) are gratefully acknowledged. I appreciate helpful discussions with P. Kramer, M. Kleber, C. Bracher, and A. Frank. This work is supported by the Deutsche Forschungsgemeinschaft (grant KR 2889 [Emmy Noether Programme]) and NSEC [E. Heller, Harvard].

References

1. T. Kramer, International Journal of modern physics B 20 (2005) 1243. (arxiv: http://arxiv.org/abs/cond-mat/0509451).        [ Links ]

2. The effective g–factor is empirically deduced. I assume a constant g*, which explains the observed structures very well. Thus a dependency of g* on the magnetic field is not needed here.

3. C.W.J. Beenakker and H. van Houten, Solid state physics 44 (1991) 228.        [ Links ]

4. E.H. Hall, American Journal of Mathematics 2 (1879) 287.        [ Links ]

5. K.V. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45 (1980) 494.        [ Links ]

6. J. Hajdu, editor, Introduction to the Theory of the Integer Quantum Hall Effect (VCH, Weinheim, 1994).        [ Links ]

7. B. Kramer, S. Kettermann, and T. Ohtsuki, Physica E 20 (2003) 172.        [ Links ]

8. D.H. Cobden, C.H.W. Barnes, and C.J.B. Ford, Phys. Rev. Lett. 82 (1999) 4695.        [ Links ]

9. V. Mitin, V. Kochelap, and M. Stroscio Quantum Heterostructures: Microelectronics and Optoelectronics (Cambridge University Press, Cambridge, 1999).        [ Links ]

10. Y. Hirayama, K. Muraki, K. Hashimoto, K. Takashina, and T. Saku, Physica E 20 (2003) 133.        [ Links ]

11. T. Kramer, C. Bracher, and M. Kleber, J. Opt. B: Quantum Semiclass. Opt. 6 (2004) 21.        [ Links ]

12. A.Y. Lozovoi, A. Alavi, J. Kohanoff, and R.M. Lynden–Bell, J. Chem. Phys. 115 (2001) 1661.        [ Links ]

13. T. Kramer and C. Bracher, "Propagation in crossed magnetic and electric fields: The quantum source approach" In B. Gruber, G. Marmo, and N. Yoshinaga, editors, Symmetries in Science XI (Dordrecht, 2004, Kluwer) p 317. http://arxiv.org/abs/cond–mat/0309424.        [ Links ]

14. S. Ilani et al., Nature 427 (2004) 328.        [ Links ]

15. T. Kramer, Matter waves from localized sources in homogeneous force fields, PhD thesis, (Technische Universitat München, 2003) http://tumb1.biblio.tu-muenchen.de/publ/diss/ph/2003/kramer.html.        [ Links ]

16. S. Kawaji, K. Hirakawa, and M. Nagata Physica B 184 (1993) 17.        [ Links ]

17. S. Kawaji, Semicond. Sci. Technol. 11 (1996) 1546.        [ Links ]

18. T. Shimada, T. Okamoto, and S. Kawaji, Physica B 249–251 (1998) 107.        [ Links ]

19. M.P. Lilly, K.B. Cooper, J.P. Eisenstein, L.N. Pfeiffer, and K.W. West, Phys. Rev. Lett. 82 (1999) 394.        [ Links ]