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Possible States for a Three-Dimensional Electron Gas in a Strong Magnetic Field
Open accessAbstract
If a three-dimensional semimetal or doped semiconductor is placed in a sufficiently strong magnetic field, then a change in its transport properties will occur. If the electron-impurity interaction is dominant, then the magnetic field will produce localization of the electron wavefunctions, sometimes described as magnetic freezeout. If the electron-electron interaction is more important, then some type of collective transition may occur. Spin-density waves, charge-density waves, valley-density waves, excitonic insulators, and Wigner crystallization have been proposed to occur under various circumstances. As a generalization to three-dimensions of the integral quantized Hall effect, we show that for electrons in periodic or quasiperiodic potential, when the Fermi level lies in an energy gap, the zero temperature conductivity tensor is given by σ i j =(e 2/2π h)∑ k ε i j k G k , where \vec G is a reciprocal-lattice-vector of the potential. We discuss the effect of impurities and dislocations on this result.
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“As γ → 1, the entire Brillouin zone is filled with unit Chern number along the k z direction, and a three dimensional version of the integer quantum Hall state is realized (Halperin, 1987).”
From Weyl and Dirac semimetals in three-dimensional solids (2018) · cites this paper as Halperin, 1987
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“This triad of two-dimensional topological invariants leads to a linear three-dimensional quantum Hall effect, calculated by extending Eq. ( 10) to three spatial dimensions (Halperin, 1987;Montambaux and Kohmoto, 1990;Kohmoto, Halperin, and Wu, 1992).”
From Topological photonics (2019) · cites this paper as Halperin, 1987
In the citing paper
“A more physical way to view this is that a three-dimensional integer quantum Hall system consists of a single Chern number and a reciprocal lattice vector that describes the 'stacking' of integer quantum Hall layers (73).”
From Three-Dimensional Topological Insulators (2011) · cites this paper as [73]
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Research works printed but not linked: 11
- [1]The Quantum Hall Effect, ed. R.E. Prange and S.M. Girvin (Springer-Verlag, New York, 1987).
- [9]Z. Tesanović and B.I. Halperin, Phys. Rev. B (in press).
- [11]B.I. Halperin.and T.M. Rice, in Solid State Physics, ed. F. Seitz, D. Turnbull and H. Ehrenreich (Academic Press, New York, 1968), Vol. 21, pp. 116–192.
- [10]The possibility of a field-induced superconducting state has been suggested by M. Rasolt, Phys. Rev. Lett. 58 (1987) 1482; Z. Tesanović and M. Rasolt, preprint.
- [13]Cf. M. Ya. Azbel and N.B. Brandt, Zh. Eksp. Teor. Fiz 48 (1965) 1902 [Sov. Phys.-JETP 21 (1965) 804]; N. Miura, G. Kido and S. Chikazumi, in Application of High Magnetic Fields in Semiconductor Physics ed. G. Landwehr (Springer, New York, 1983), p. 505; S. Nakajima and D. Yoskioka, J. Phys. Soc. Jpn. 40 (1976) 328.
- [16]See for example, K. Seeger, Semiconductor Physics, 2nd ed. (Springer-Verlag, Berlin, 1982).
- [22]G. Nimtz and J. Gebhardt, in Proceedings of the 18th International Conference on the Physics of Semiconductors (Stockholm, 1986), ed. O. Engström (World Scientific, Singapore, 1987), Vol. 2, p. 1197.
- [24]R.M. Westervelt, M.J. Burns, P.F. Hopkins, A.J. Rimberg and G.A. Thomas, preprint for Proceedings of the International Symposium on Anderson Localization, Tokyo, August 1987.
- [25]N.F. Mott and E.A. Davis, Electronic Processes in Non-crystalline Materials (Clarendon Press, Oxford, 1979); B.I. Shklovskii and A.L. Efros, Electronic Properties of Doped Semiconductors (SpringerVerlag, Berlin, 1984).
- [30]B.I. Halperin, in Condensed Matter Theories, ed. P. Vashishta, et al. (Plenum Press, in press), Vol. 2.
- [36]See D. Poilblanc, G. Montambaux, M. Héritier and P. Lederer, Phys. Rev. Lett 58 (1987) 270; M. Ya. Azbel, P. Bak and P.M. Chaikin (preprint) and references therein.
Cited by
Sorted by citations
Weyl and Dirac semimetals in three-dimensional solids
Armitage, N. P., Mele, E. J., & Vishwanath, A. (2018) · Reviews of Modern Physics · doi:10.1103/revmodphys.90.015001 · Cited by 4,528
“This is physically related to the property that the Hall conductance calculated by this formula are quantized modulo Hall conductances arising from filled bands which may lead to three dimensional quantum Hall states (Halperin, 1987) which are also characterized by a reciprocal lattice vector G.”
BackgroundOzawa, T., Price, H. M., Amo, A., Goldman, N., Hafezi, M., Lu, L., Rechtsman, M. C., Schuster, D., Simon, J., Zilberberg, O., et al. (2019) · Reviews of Modern Physics · doi:10.1103/revmodphys.91.015006 · Cited by 3,722
Lu, L., Joannopoulos, J. D., & Soljačić, M. (2014) · Nature Photonics · doi:10.1038/nphoton.2014.248 · Cited by 3,655
Topological field theory of time-reversal invariant insulators
Qi, X.-L., Hughes, T. L., & Zhang, S.-C. (2008) · Physical Review B · doi:10.1103/physrevb.78.195424 · Cited by 3,329
Reflection-Symmetric Second-Order Topological Insulators and Superconductors
Langbehn, J., Peng, Y., Trifunovic, L., von Oppen, F., & Brouwer, P. W. (2017) · Physical Review Letters · doi:10.1103/physrevlett.119.246401 · Cited by 990
Metadata
- Authors
- Halperin, B. I.
- Affiliations
- Harvard University (US)
- Corresponding author
- Halperin, Bertrand I.
- Venue
- Japanese Journal of Applied Physics
- Publisher
- IOP Publishing
- Published
- 1987-01-01
- Volume
- 26
- Issue
- S3-3
- Pages
- 1913
- ISSN
- 0021-4922 (ISSN-L)
- 1347-4065
- OpenAlex
- W1994659789
- Cited by
- 321
- References
- 0
- Citations by year
- 2026: 8 · 2025: 16 · 2024: 18 · 2023: 19 · 2022: 14 · 2021: 27 · 2020: 16 · 2019: 15 · 2018: 17 · 2017: 14 · 2016: 13 · 2015: 5 · 2014: 6 · 2013: 2 · 2012: 4
- Type
- conference abstract
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Japanese Journal of Applied Physics (publishedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Quantum and electron transport phenomena
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Physics of Superconductivity and Magnetism
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Topological Materials and Phenomena
- Keywords
- three-dimensional electron gas, high magnetic fields, electron localization, electron-impurity interaction, electron-electron interaction, collective transitions, spin density wave, charge density wave, excitonic insulator, Wigner crystallization, conductivity tensor, quasiperiodic potentials, dislocations, impurities
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