Abstract
The theoretical investigation of electrons subject to both a periodic electrostatic potential and a magnetic field has a long tradition [1]. On the one hand, this situation can be realized in each crystalline solid. On the other hand, it constitutes a highly nontrivial system due to breaking the rotational symmetry of the magnetic field by the (discrete) translational invariance of the periodic potential. A simple version of the problem is obtained in the tight-binding approximation for the electrons or in perturbation theory for very weak periodic potential. In that case Schrüdinger’s equation reduces to Harper’s equation [2], the solution of which has been visualized in the form of the well-known Hofstadter butterfly [3]. The spectrum then critically depends on the number of magnetic flux quanta threading one unit cell of the lattice and becomes a fractal object, a Cantor set, if this number is an irrational one [4]. Experimental studies of such systems promise to provide a confirmation of the theoretical predictions and perhaps new interesting insights into the consequences of the spectral properties on their physical behaviour. However, they turned out to be impossible with conventional crystals because the necessary magnetic fields of several 1000 T for the observation of the peculiarities of the spectrum are still out of the reach of present-day technology.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Peierls, R. (1933) Z. Phys. 80, 763. Fischbeck, H. J. (1970) Phys. Stat. Sol. 38, 11. Zak, J. (1972) Solid State Physics (eds H. Ehrenreich, F. Seitz and W. Turnbull), Academic Press, New York, Vol. 27, p. 1.
Harper, P. G. (1955) Proc. R. Soc. Lond. A 68, 874. Azbel, M. Ya. (1964) Sov. Phys. JETP 19, 634.
Hofstadter, D. R. (1976) Phys. Rev. B 14, 2239.
Bellissard, J. and Simon, B. J. (1982) Funct. Anal. 48, 408.
Wagenhuber, J., Geisel, T., Niebauer, P. et al. (1992) Phys. Rev. B 45, 4372.
Sinai, Y. G. (1959) Dokl Akad. Nauk. 25, 768.
Gutzwiller, M. C. (1990) Chaos in Classical and Quantum Mechanics, Springer, New York.
Haake, F. (1991) Quantum Signatures of Chaos, Springer, New York.
Mehta, M. L. (1991) Random Matrices, Academic Press, Boston.
Geisel, T., Ketzmerick, R. and Petschel, G. (1991) Phys. Rev. Lett. 66, 1651.
Geisel, T., Ketzmerick, R. and Petschel, G. (1991) Phys. Rev. Lett. 67, 3635.
Petschel, G. and Geisel, T. (1993) Phys. Rev. Lett. 71, 239.
Mucciolo, E. R., Capaz, R. B., Altshuler, B. L. et al. (1994) Phys. Rev. B 50, 8245.
Weiss, D., von Klitzing, K., Ploog, K. et al. (1989) Europhys. Lett. 8, 179.
Gerhardts, R. R., Weiss, D. and von Klitzing, K. (1989) Phys. Rev. Lett. 62, 1173.
Winkler, R. W., Kotthaus, J. P. and Ploog, K. (1989) Phys. Rev. Lett. 62, 1177.
Beenakker, C. W. J. (1989) Phys. Rev. Lett. 62, 2020.
Beton, P. H., Alves, E. S., Main, P. C. et al. (1990) Phys. Rev. B 42, 9229.
Müller, G., Streda, P., Weiss, D. et ai (1994) Phys. Rev. B 50, 8938.
Gerhardts, R. R., Weiss, D. and Wulf, U. (1991) Phys. Rev. B 43, 5192.
Gerhardts, R. R. (1992) Phys. Rev. B 45, 3449.
Pfannkuche, D. and Gerhardts, R. R. (1992) Phys. Rev. B 46, 12 606.
Marcus, C. M., Rimberg, A. J., Westervelt, R. M. et al. (1992) Phys. Rev. Lett. 69, 506. Marcus, C. M., Westervelt, R. M., Hopkins, P. F. et al. (1993) Phys. Rev. B 48, 2460. Marcus, C. M., Westervelt, R. M., Hopkins, P. F. et al. (1993) Chaos 3, 643. Marcus, C. M., Westervelt, R. M., Hopkins, P. F. et al. (1994) Surf. Sci. 305, 480. Berry, M. J., Katine, J. A., Westervelt, R. M. et al. (1994) Surf. Sci. 305, 495. Keller, M. W., Millo, O., Mittel, A. et al. (1994) Surf Sci. 305, 501. Berry, M. J., Baskey, J. H., Westervelt, R. M. et al. (1994) Phys. Rev. B 50, 8857. Chang, A. M., Baranger, H. U., Pfeiffer, L. N. et al. (1994) Phys. Rev. Lett. 73, 2111. Chan, I. H., Clarke, R. M., Marcus, C. M. et al. (1995) Phys. Rev. Lett. 74, 3876. Keller, M. W., Mittal, A., Sleight, J. W. et al. (1996) Phys. Rev. B 53, R1693.
Jalabert, R. A., Baranger, H. U. and Stone, A. D. (1990) Phys. Rev. Lett. 65, 2442. Baranger, H. U., Jalabert, R. A. and Stone, A. D. (1993) Phys. Rev. Lett. 70, 3876. Baranger, H. U., Jalabert, R. A. and Stone, A. D. (1993) Chaos 3, 665. Baranger, H. U. and Mello, P. A. (1994) Phys. Rev. Lett. 73, 142. Baranger, H. U. and Mello, P. A. (1995) Phys. Rev. B 51, 4703. Baranger, H. U. and Mello, P. A. (1996) Europhys. Lett. 33, 465. Pluhar, Z., Weidenmüller, H. A., Zuk, J. A. et al. (1994) Phys. Rev. Lett. 73, 2115. Baranger, R. A., Pichard, J.-L. and Beenakker, C. W. J. (1994) Europhys. Lett. 27, 255. Efetov, K. B. (1995) Phys. Rev. Lett. 74, 2299.
Roukes, M. L. and Scherer, A (1989) A. Bull. Am. Phys. Soc. 34, 633.
Fang, H., Zeller, R. and Stiles, P. J. (1989) Appl. tPhys. Lett. 55, 1433.
Ensslin, K. and Petroff, P. M. (1990) Phys. Rev. B 41, 12307.
Lorke, A., Kotthaus, J. P. and Ploog, K. (1991) Superlatt. Microstruct. 9, 103.
Weiss, D., Roukes, M. L., Menschig, A. et al. (1991) Phys. Rev. Lett. 66, 2790.
Gusev, G. M., Kvon, Z. D., Kudryashov, V. M. et al. (1991) JETP Lett. 54, 368.
Berthold, G., Smoliner, J., Rosskopf, V. et al. (1992) Phys. Rev. B 45, 11 350.
Schuster, R., Ensslin, K., Kotthaus, J. P. et al. (1992) Superlatt. Microstruct. 12, 93.
Weiss, D. (1992) Habilitationsschrift, Universität Stuttgart.
Weiss, D., Richter, K., Menschig, A.et al. (1993) Phys. Rev. Lett. 70, 4118.
Nihey, F. and Nakamura, K. (1993) Physica B 184, 398.
Gusev, G. M., Kvon, Z. D., Litvin, L. V. et al. (1992) JETP Lett. 55, 123. Gusev, G. M., Kvon, Z. D., Litvin, L. V. et al. (1992) J. Phys.: Condens. Mat. 4, L269.
Nihey, F., Hwang, S. W. and Nakamura, K. (1995) Phys. Rev. B 51, 4649.
Schuster, R., Ensslin, K. and Wharam, D. et al. (1994) Phys. Rev. B 49, 8510.
Lenssen, K.-M. H., Boonman, M. E. J., Harmans, C. J. P. M. et al. (1995) Phys. Rev. Lett. 74, 454.
Schlüsser, T., Ensslin, K. and Kotthaus, J. P. (1996) Europhys. Lett. 33, 683.
Ensslin, K., Sasa, S., Deruelle, T. et al. (1992) Surf. Sci. 263, 319.
Schuster, R., Ensslin, K., Kotthaus, J. P. et al. (1993) Phys. Rev. B 47, 6843.
Weiss, D., Richter, K., Vasiliadou, E. et al. (1994) Surf. Sci. 305, 408.
Schuster, R. and Ensslin, K. (1994) Advances in Solid State Physics 34, 195.
Tsukagoshi, K., Wakayama, S., Oto, K. et al. (1995) Phys. Rev. B 52, 8344.
Fang, H. and Stiles, P. J. (1990) Phys. Rev. B 41, 10171.
Yamashiro, T., Takahara, J., Tagagaki, Y. et al. (1991) Solid State Commun. 79, 885.
Gusev, G. M., Kvon, Z. D., Litvin, L. V. et al. (1992) JETP Lett. 56, 170.
Wenchang Lu, (1996) Phys. Rev. B 54, 8049. Wenchang Lu and Andersen, O. K. (1996) The Physics of Semiconductors (eds M. Scheffler and R. Zimmermann), World Scientific, Singapore, P. 1497.
Kubo, R. (1957) J. Phys. Soc. Jpn. 12, 570.
Fleischmann, R., Geisel, T. and Ketzmerick, R. (1992) Phys. Rev. Lett. 68, 1367.
Lorke, A., Kotthaus, J. P. and Ploog, K. (1991) Phys. Rev. B 44, 3447.
Baskin, E. M., Gusev, G. M., Kvon, Z. D. et al. (1992) JETP Lett. 55, 678.
Hofmann, A. (1995) Diplomathesis, Ludwig-Maximilians-Universität München.
Schuster, R., Ernst, G., Ensslin, K. et al. (1994) Phys. Rev. B 50, 8090.
Tsukagoshi, K., Haraguchi, M., Takaoka, S. et al. (1996) J. Phys. Soc. Jpn. 65, 811.
Aoki, K. (1996) The Physics of Semiconductors (eds M. Scheffler and R. Zimmer-mann), World Scientific, Singapore, p. 1521.
Fleischmann, R., Geisel, T. and Ketzmerick, R. (1994) Europhys. Lett. 25, 219.
Zak, J. (1964) Phys. Rev. 134, A1602.
Ferrari, R. (1990) Phys. Rev. B 42, 4598.
Silberbauer, H. (1992) J. Phys.: Condens. Mat. 4, 7355.
Rotter, P., Rüssler, U., Silberbauer, H. et al. (1995) Physica B 212, 231.
Schuster, R., Ensslin, K., Kotthaus, J. P. et al. (1997) Phys. Rev. B. 55, 2237.
Pandey, A. (1981) Ann. Phys. 134, 110. French, J. B., Kota, V. K. B., Pandey, A. et al. (1988) Ann. Phys. N. Y. 181, 198. Bohigas, O., Tomsovic, S. and Ullmo, D. (1993) Phys. Rep. 223, 43.
Brody, T. A. (1973) Lett. Nuovo Cimento 7, 482.
Berry, M. V. and Robnik, M. (1984) J. Phys. A: Math. Gen. 17, 2413.
Izrailev, F. M. (1988) Phys. Lett. A 134, 13.
Lenz, G. and Haake, F. (1990) Phys. Rev. Lett. 65, 2325. Lenz, G. and Haake, F. (1991) Phys. Rev. Lett. 67, 1.
Leyvraz, F. and Seligman, T. H. (1990) J. Phys. A: Math. Gen. 23, 1555.
Caurier, E., Grammaticos, B. and Ramani, A. (1990) J. Phys. A: Math. Gen. 23, 4903.
Leitner, D. M. (1993) Phys. Rev. B 48, 2536.
Prosen, T. and Robnik, M. (1993) J. Phys. A: Math. Gen. 26, 2371.
Simons, B. D. and Altshuler, B. L. (1993) Phys. Rev. Lett. 70, 4063. Simons, B. D. and Altshuler, B. L. (1993) Phys. Rev. B 48, 5422. Taniguchi, N. and Altshuler, B. L. (1993) Phys. Rev. Lett. 71, 4031.
Silberbauer, H., Rotter, P., Rüssler, U. et al. (1995) Europhys. Lett. 31, 393.
Silberbauer, H., Rotter, P., Suhrke, M. et al. (1994) Semicond. Sci. & Technol. 9, 1906.
Steffens, O., Rotter, P. and Suhrke, F.M. (1997) Phys. Rev. B 55, 4486.
Huang, D. and Gumbs, G. (1993) Phys. Rev. B 48, 2835. Huang, D., Gumbs, G. and MacDonald, A. H. (1993) Phys. Rev. B 48, 2843.
Oakeshott, R. B. S. and. MacKinnon, A (1993) J. Phys.: Condens. Matt. 5, 6991. Oakeshott, R. B. S. and. MacKinnon, A (1994) J. Phys.: Condens. Matt. 6, 1519.
Hongqi Xu, Zhen-Li Ji and Berggren, K.-F. (1992) Superlatt. & Microstruct. 12, 237. Hongqi Xu (1994) Phys. Rev. B 50, 8469. Hongqi Xu (1994) Phys. Rev. B 50, 12 254. Hongqi Xu (1996) The Physics of Semicondutors (eds M. Scheffler and R. Zimmer-mann), World Scientific, Singapore, p. 1489.
Zozulenko, I. V., MaaøF. A., Hauge, E. H. (1995) Phys. Rev. B 51, 7058. Zozulenko, I. V., Maaø, F. A. and Hauge, E. H. (1996) Phys. Rev. B 53, 7975. Zozulenko, I. V., Maaø, F. A. and Hauge, E. H. (1996) Phys. Rev. B 53, 7987. Zozulenko, I. V., Maaø, F. A. and Hauge, E. H. (1996) The Physics of Semicondutors (eds M. Scheffler and R. Zimmermann), World Scientific, Singapore, p. 1493.
Uryu, S. and Ando, T. (1996) Phys. Rev. B 53, 13613. Uryu, S. and Ando, T. (1996) The Physics of Semicondutors (eds M. Scheffler and R. Zimmermann), World Scientific, Singapore, p. 1505.
Wenchang Lu and Andersen, O. K. (1996) The Physics of Semicondutors (eds M. Scheffler and R. Zimmermann), World Scientific, Singapore, p. 1497.
Silberbauer, H. and Rüssler, U. (1994) Phys. Rev. B 50, 11911.
Ishizaka, S., Nihey, F., Nakamura, K. et al. (1995) Phys. Rev. B 51, 9881.
Weiss, D., Menschig, A., von Klitzing, K. et al. (1992) Surf. Sci. 263, 314.
Schlüsser, T., Ensslin, K., Kotthaus, J. P. et al. (1996) Surf Sci. 847, 361–362.
Neudert, R., Rotter, P., Rüssler, U. et al. (1997) Phys. Rev. B. 55, 2242. Rotter, P., Rüssler, U., Suhrke, M. et al. (1996) The Physics of Semicondutors (eds M. Scheffler and R. Zimmermann), World Scientific, Singapore, p. 1517.
Thornton, T. J., Roukes, M. L., Scherer, A. et al. (1989) Phys. Rev. Lett. 63, 2128.
Richter, K. (1995) Europhys. Lett. 29, 7.
Hackenbroich, G. and von Oppen, F. (1995) Europhys. Lett. 29, 151.
Yacoby, A., Sivan, U., Umbach, C. P. et al. (1991) Phys. Rev. Lett. 66, 1938. Yacoby, A., Umansky, V., Shtrikman, H. et al. (1994) Phys. Rev. Lett. 73, 3149. Kurdak, C., Chang, A. M., Chin, A. et al. (1992) Phys. Rev. B 46, 6846.
Rotter, P., Suhrke, M. and Rüssler, U. (1996) Phys. Rev. B 54, 4452.
Lütjering, G., Weiss, D., Tank, R. et al. (1996) Surf. Sci. 925, 361–362.
Nihey, F., Kastner, M. A. and Nakamura, K. (1996) The Physics of Semiconductors (eds M. Scheffler and R. Zimmermann), World Scientific, Singapore, p. 1525.
Izawa, S., Katsumoto, S., Endo, A. et al. (1995) J. Phys. Soc. Jpn. 64, 706. Carmona, H. A., Geim, A. K., Nogaret, A. et al. (1994) Phys. Rev. Lett. 74, 3009. Ye, P. D., Weiss, D., Gerhardts, R. R. et al. (1994) Phys. Rev. Lett. 74, 3013.
Ye, P. D., Weiss, D., von Klitzing, K. et al. (1995) Appl. Phys. Lett. 67, 1441.
Vasilopoulos, P. and Peeters, F. M. (1990) Superlatt. & Microstruct. 7, 393. Xue, D. P. and Xiao, G. (1992) Phys. Rev. B 45, 5986. Peeters, F. M. and Vasilopoulos, P. (1993) Phys. Rev. B 47, 1466. Oakeshott, R. B. S. and MacKinnon, A. (1993) J. Phys.: Condens. Matt. 5, 9355.
Schmidt, G. J. O. (1993) Phys. Rev. B 47, 13 007. Schmelcher, P. and Shepelyansky, D. L. (1994) Phys. Rev. B 49, 7418. Gerhardts, R. R. (1996) Phys. Rev. B 53, 11 064.
Menne, R. and Gerhardts, R. R. (1997) Proc. 12th Int. Conf. Application of High Magnetic Fields in Semiconductor Physics (eds G. Landwehr and W. Ossau), World Scientific, Singapore, p. 323.
Chang, M. C. and Niu, Q. (1994) Phys. Rev. B 50, 10843. Gerhardts, R. R., Pfannkuche, D. and Gudmundsson, V. (1996) Phys. Rev. B 53, 9591.
Kardynal, B., Barnes, C. H. W. and Linfield, E. H. (1996) Phys. Rev. Lett. 76, 3802.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Suhrke, M., Rotter, P. (1998). Transport in systems with chaotic dynamics: Lateral superlattices. In: Schöll, E. (eds) Theory of Transport Properties of Semiconductor Nanostructures. Electronic Materials Series, vol 4. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-5807-1_8
Download citation
DOI: https://doi.org/10.1007/978-1-4615-5807-1_8
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-412-73100-6
Online ISBN: 978-1-4615-5807-1
eBook Packages: Springer Book Archive