Fractional Quantum Hall States for Filling Factors 2/3 < ν < 2
Author(s)
Shosuke Sasaki*
Affiliation(s)
Center for Advanced High Magnetic Field Science, Graduate School of Science, Osaka University, Osaka, Japan.
Center for Advanced High Magnetic Field Science, Graduate School of Science, Osaka University, Osaka, Japan.
ABSTRACT
Fractional quantum Hall effect (FQHE) is investigated by employing normal electrons and the fundamental Hamiltonian without any quasi particle. There are various kinds of electron configurations in the Landau orbitals. Therein only one configuration has the minimum energy for the sum of the Landau energy, classical Coulomb energy and Zeeman energy at any fractional filling factor. When the strong magnetic field is applied to be upward, the Zeeman energy of down-spin is lower than that of up-spin for electrons. So, all the Landau orbitals in the lowest level are occupied by the electrons with down-spin in a strong magnetic field at 1 < ν < 2. On the other hand, the Landau orbitals are partially occupied by up-spins. Two electrons with up-spin placed in the nearest orbitals can transfer to all the empty orbitals of up-spin at the specific filling factors
and so on. When the
filling factor ν deviates from ν0, the number of allowed transitions
decreases abruptly in comparison with that at ν0. This mechanism creates the energy
gaps at ν0. These energy gaps yield the
fractional quantum Hall effect. We compare the present theory with the
composite fermion theory in the region of 2/3 < ν < 2.
Fractional quantum Hall effect (FQHE) is investigated by employing normal electrons and the fundamental Hamiltonian without any quasi particle. There are various kinds of electron configurations in the Landau orbitals. Therein only one configuration has the minimum energy for the sum of the Landau energy, classical Coulomb energy and Zeeman energy at any fractional filling factor. When the strong magnetic field is applied to be upward, the Zeeman energy of down-spin is lower than that of up-spin for electrons. So, all the Landau orbitals in the lowest level are occupied by the electrons with down-spin in a strong magnetic field at 1 < ν < 2. On the other hand, the Landau orbitals are partially occupied by up-spins. Two electrons with up-spin placed in the nearest orbitals can transfer to all the empty orbitals of up-spin at the specific filling factors
and so on. When the
filling factor ν deviates from ν0, the number of allowed transitions
decreases abruptly in comparison with that at ν0. This mechanism creates the energy
gaps at ν0. These energy gaps yield the
fractional quantum Hall effect. We compare the present theory with the
composite fermion theory in the region of 2/3 < ν < 2.
Cite this paper
Sasaki, S. (2015) Fractional Quantum Hall States for Filling Factors 2/3 < ν < 2. Journal of Modern Physics, 6, 584-600. doi: 10.4236/jmp.2015.65064.
Sasaki, S. (2015) Fractional Quantum Hall States for Filling Factors 2/3 < ν < 2. Journal of Modern Physics, 6, 584-600. doi: 10.4236/jmp.2015.65064.
References
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http://dx.doi.org/10.1017/CBO9780511607561 [2] Das Sarma, S. (1996) Localization, Metal-Insulator Transitions, and Quantum Hall Effect. In: Das Sarma, S. and Pinczuk, A., Perspectives in Quantum Hall Effects: Novel Quantum Liquids in Low-Dimensional Semiconductor Structures, Wiley, New York, 1-36.
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http://dx.doi.org/10.1142/9789812815989_0001 [4] Jain, J.K. (1989) Physical Review Letters, 63, 199-202.
http://dx.doi.org/10.1103/PhysRevLett.63.199 [5] Kamilla, R.K., Wu, X.G. and Jain, J.K. (1996) Physical Review Letters, 76, 1332.
http://dx.doi.org/10.1103/PhysRevLett.76.1332 [6] Jain, J.K. and Kamilla, R.K. (1997) Physical Review B, 55, R4895.
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http://dx.doi.org/10.1103/PhysRevLett.80.4237 [8] Stormer, H.L. and Tsui, D.C. (2007) Composite Fermions in the Fractional Quantum Hall Effect. In: Das Sarma, S. and Pinczuk, A., Eds., Perspectives in Quantum Hall Effects: Novel Quantum Liquids in Low-Dimensional Semiconductor Structures, Wiley, New York, 385-421.
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http://dx.doi.org/10.1103/RevModPhys.75.1101 [12] Sitko, P., Yi, K.-S. and Quinn, J.J. (1997) Physical Review B, 56, 12417-12421.
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http://dx.doi.org/10.1103/PhysRevB.29.636 [15] Sasaki, S. (2011) Binding Energy, Polarization of Fractional Quantum Hall State. Proceedings of the 25th International Conference on the Physics of Semiconductors, Osaka, 17-22 September 2000, 925-926. [16] Sasaki, S. (2003) Surface Science, 532-535, 567-575.
http://dx.doi.org/10.1016/S0039-6028(03)00091-8 [17] Sasaki, S. (2004) Surface Science, 566-568, 1040-1046.
http://dx.doi.org/10.1016/j.susc.2004.06.101 [18] Sasaki, S. (2005) Binding Energies and Spin Polarizations of Fractional Quantum Hall States. In: Norris, C.P., Ed., Surface Science: New Research, Nova Science Publishers, Hauppauge, 103-161. [19] Sasaki, S. (2013) ISRN Condensed Matter Physics, 2013, Article ID: 489519.
http://dx.doi.org/10.1155/2013/489519 [20] Sasaki, S. (2012) Advances in Condensed Matter Physics, 2012, Article ID: 281371.
http://dx.doi.org/10.1155/2012/281371 [21] Sasaki, S. (2014) ISRN Condensed Matter Physics, 2014, Article ID: 468130. [22] Sasaki, S. (2000) Physica B: Condensed Matter, 281-282, 838-839.
http://dx.doi.org/10.1016/S0921-4526(99)00840-6 [23] Sasaki, S. (2008) Journal of Physics: Conference Series, 100, Article ID: 042021.
http://dx.doi.org/10.1088/1742-6596/100/4/042021 [24] Sasaki, S. (2008) Journal of Physics: Conference Series, 100, Article ID: 042022. [25] Sasaki, S. (2010) E-Journal of Surface Science and Nanotechnology, 8, 121-124.
http://dx.doi.org/10.1380/ejssnt.2010.121 [26] Sasaki, S. (2013) Journal of Modern Physics, 4, 1-7.
http://dx.doi.org/10.4236/jmp.2013.49A001 [27] Sasaki, S. (2007) Calculation of Binding Energies for Fractional Quantum Hall States with Even Denominators.
http://arxiv.org/abs/cond-mat/0703360 [28] Sasaki, S. (2007) Energy Spectra for Fractional Quantum Hall States.
http://arxiv.org/abs/0708.1541 [29] Sasaki, S. (2008) Consideration of ac Josephson Effect in Fractional Quantum Hall States.
http://arxiv.org/abs/0807.0288 [30] Sasaki, S. (2008) Frequency Dependence of Diagonal Resistance in Fractional Quantum Hall Effect via Periodic Modulation of Magnetic Field.
http://arxiv.org/abs/0803.0615 [31] Jain, J.K. (2014) Indian Journal of Physics, 88, 915-929.
http://dx.doi.org/10.1007/s12648-014-0491-9
[1] Jain, J.K. (2007) Composite Fermions. Cambridge University Press, New York.
http://dx.doi.org/10.1017/CBO9780511607561 [2] Das Sarma, S. (1996) Localization, Metal-Insulator Transitions, and Quantum Hall Effect. In: Das Sarma, S. and Pinczuk, A., Perspectives in Quantum Hall Effects: Novel Quantum Liquids in Low-Dimensional Semiconductor Structures, Wiley, New York, 1-36.
http://dx.doi.org/10.1002/9783527617258.ch1 [3] Jain, J.K. and Kamilla, R.K. (1998) Composite Fermions: Particles of the Lowest Landau Level. In: Heinonen, O., Ed., Composite Fermions: A Unified View of the Quantum Hall Regime, World Scientific, New York, 1-90.
http://dx.doi.org/10.1142/9789812815989_0001 [4] Jain, J.K. (1989) Physical Review Letters, 63, 199-202.
http://dx.doi.org/10.1103/PhysRevLett.63.199 [5] Kamilla, R.K., Wu, X.G. and Jain, J.K. (1996) Physical Review Letters, 76, 1332.
http://dx.doi.org/10.1103/PhysRevLett.76.1332 [6] Jain, J.K. and Kamilla, R.K. (1997) Physical Review B, 55, R4895.
http://dx.doi.org/10.1103/PhysRevB.55.R4895 [7] Park, K. and Jain, J.K. (1998) Physical Review Letters, 80, 4237.
http://dx.doi.org/10.1103/PhysRevLett.80.4237 [8] Stormer, H.L. and Tsui, D.C. (2007) Composite Fermions in the Fractional Quantum Hall Effect. In: Das Sarma, S. and Pinczuk, A., Eds., Perspectives in Quantum Hall Effects: Novel Quantum Liquids in Low-Dimensional Semiconductor Structures, Wiley, New York, 385-421.
http://dx.doi.org/10.1002/9783527617258 [9] Jain, J.K. (2000) Physics Today, 53, 39.
http://dx.doi.org/10.1063/1.883035 [10] Halperin, B.I. (2003) Physica E: Low-Dimensional Systems and Nanostructures, 20, 71-78.
http://dx.doi.org/10.1016/j.physe.2003.09.022 [11] Murthy, G. and Shankar, R. (2003) Reviews of Modern Physics, 75, 1101-1158.
http://dx.doi.org/10.1103/RevModPhys.75.1101 [12] Sitko, P., Yi, K.-S. and Quinn, J.J. (1997) Physical Review B, 56, 12417-12421.
http://dx.doi.org/10.1103/PhysRevB.56.12417 [13] Tao, R. and Thouless, D.J. (1983) Physical Review B, 28, 1142-1144.
http://dx.doi.org/10.1103/PhysRevB.28.1142 [14] Tao, R. (1984) Physical Review B, 29, 636-644.
http://dx.doi.org/10.1103/PhysRevB.29.636 [15] Sasaki, S. (2011) Binding Energy, Polarization of Fractional Quantum Hall State. Proceedings of the 25th International Conference on the Physics of Semiconductors, Osaka, 17-22 September 2000, 925-926. [16] Sasaki, S. (2003) Surface Science, 532-535, 567-575.
http://dx.doi.org/10.1016/S0039-6028(03)00091-8 [17] Sasaki, S. (2004) Surface Science, 566-568, 1040-1046.
http://dx.doi.org/10.1016/j.susc.2004.06.101 [18] Sasaki, S. (2005) Binding Energies and Spin Polarizations of Fractional Quantum Hall States. In: Norris, C.P., Ed., Surface Science: New Research, Nova Science Publishers, Hauppauge, 103-161. [19] Sasaki, S. (2013) ISRN Condensed Matter Physics, 2013, Article ID: 489519.
http://dx.doi.org/10.1155/2013/489519 [20] Sasaki, S. (2012) Advances in Condensed Matter Physics, 2012, Article ID: 281371.
http://dx.doi.org/10.1155/2012/281371 [21] Sasaki, S. (2014) ISRN Condensed Matter Physics, 2014, Article ID: 468130. [22] Sasaki, S. (2000) Physica B: Condensed Matter, 281-282, 838-839.
http://dx.doi.org/10.1016/S0921-4526(99)00840-6 [23] Sasaki, S. (2008) Journal of Physics: Conference Series, 100, Article ID: 042021.
http://dx.doi.org/10.1088/1742-6596/100/4/042021 [24] Sasaki, S. (2008) Journal of Physics: Conference Series, 100, Article ID: 042022. [25] Sasaki, S. (2010) E-Journal of Surface Science and Nanotechnology, 8, 121-124.
http://dx.doi.org/10.1380/ejssnt.2010.121 [26] Sasaki, S. (2013) Journal of Modern Physics, 4, 1-7.
http://dx.doi.org/10.4236/jmp.2013.49A001 [27] Sasaki, S. (2007) Calculation of Binding Energies for Fractional Quantum Hall States with Even Denominators.
http://arxiv.org/abs/cond-mat/0703360 [28] Sasaki, S. (2007) Energy Spectra for Fractional Quantum Hall States.
http://arxiv.org/abs/0708.1541 [29] Sasaki, S. (2008) Consideration of ac Josephson Effect in Fractional Quantum Hall States.
http://arxiv.org/abs/0807.0288 [30] Sasaki, S. (2008) Frequency Dependence of Diagonal Resistance in Fractional Quantum Hall Effect via Periodic Modulation of Magnetic Field.
http://arxiv.org/abs/0803.0615 [31] Jain, J.K. (2014) Indian Journal of Physics, 88, 915-929.
http://dx.doi.org/10.1007/s12648-014-0491-9