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Electron transport mechanism of a two-dimensional infinite slab subjected to Rashba spin-orbital coupling is studied in this paper. We calculate the Hall conductance and the longitudinal resistance of the integer quantum Hall effect (IQHE). In a strong magnetic field, the Landau levels of electrons increase rapidly at large wave vectors due to the constraint of the two edges of the sample while they remain flat at small wave vectors. Although the Zeeman effect can split the energy levels of spin degeneracy under a strong magnetic field, the spacing between the Landau levels is exactly equal to the spin splitting, thus the spin degeneracies have not been fully resolved. The spin-orbital coupling fully resolves the spin degeneracies of the energy levels. This is the key to reproducing the IQHE. Electrons with rapid increasing energies are localized at the two edges of the sample and transport along the edges to form separated currents with opposite directions. In this case, back scattering of electrons is prohibited due to the localization of these two branches. Since the electrons on the upper and lower edges originate respectively from the left and right electrode, they also have the chemical potentials of the electrons in those electrodes, respectively. The computation result shows that the Hall conductance appears as plateaus at integer times of e2/h. Temperature influences the accuracy of the Hall plateaus. As an international resistance standard, exceeding a critical temperature can produce significant errors to the Hall plateaus. Below the critical temperature, the accuracy can reach 10–9. Finally the mechanism of the longitudinal resistance of the IQHE is discussed and computed numerically. It is shown that only the wave-functions with opposite and small wave vectors have a significant overlap in the bulk of the sample and thus contribute to the longitudinal resistance. Due to the separation of currents in different directions in space, the longitudinal resistance does vanish at the Hall plateaus but it appears when the Hall conductance jumps from one plateau to another one.
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[2] Tsui D C, Stormer H L, Gossard A C 1982 Phys. Rev. Lett. 48 1559
Google Scholar
[3] Laughlin R B 1983 Phys. Rev. Lett. 50 1395
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[8] 倪光炯, 陈苏卿 2004 高等量子力学 (第2版) (上海: 复旦大学出版社) 第270页
Ni G J, Chen S Q 2004 Advanced Quantum Mechanics (2nd Ed.) (Shanghai: Fudan University Press) p270 (in Chinese)
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Google Scholar
[10] Ezawa Z F 2012 Quantum Hall Effects: Field Theoretical Approach and Related Topics (2nd ed.) (Beijing: Peking University Press) p182
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[12] Lagendijk A, Wiersma D 2009 Phys. Today 62 24
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[15] Falko V I 1992 Phys. Rev. B 46 4320
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Google Scholar
[17] Yang W, Chang K 2006 Phys. Rev. B 74 193314
Google Scholar
[18] Luo J, Munekata H, Fang F F, Stiles P J 1990 Phys. Rev. B 41 7685
Google Scholar
[19] Petritz R, Scanlon W 1955 Pbs. Phys. Rev. 97 1620
Google Scholar
[20] 阎守胜 2003 固体物理基础 (第2版) (北京: 北京大学出版社) 第126页
Yan S S 2003 Basis On Solid State Physics (2nd Ed.) (Beijing: Peking University Press) p126 (in Chinese)
[21] 黄昆, 韩汝琦 1988 固体物理学 (第1版) (北京: 高等教育出版社) 第305—307页
Huang K, Han R Q 1988 Solid State Physics (1st Ed.) (Beijing: Higher Education Press) pp305−307 (in Chinese)
[22] Li H C, Sheng L, Shen R, Wang B G, Sheng D N, Xing D Y 2013 Phys. Rev. Lett. 110 266802
Google Scholar
[23] Ridley B K 2013 Quantum Processes in Semiconductors (Oxford: Oxford University Press) pp66−88
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图 1 (a)自旋简并的3个朗道能级在磁场中分裂, 但仍有能级简并; (b)第一朗道能级波函数的模, 参数
$\alpha=0.06$ ,$\lambda_{\rm h}=0$ ,$W=50$ ,$\mu_{1}=0.49$ ,$\mu_{2}=0.46$ Figure 1. (a) The lowest three spin degenerate Landau levels are split apart in the strong magnetic field, but the degeneracy is not fully resolved; (b) the modulus of the wave functions of the first Landau level for
$\alpha=0.06$ ,$\lambda_{\rm h}=0$ ,$W=50$ ,$\mu_{1}=0.49$ ,$\mu_{2}=0.46$ .
图 2 (a)在均匀磁场中通电的长条形, 箭头表示电子的输运方向; (b)自旋简并下霍尔电导的奇数霍尔平台. 参数
$k_{\rm B}T=0.002$ ,$\mu_{1}=0.49$ ,$\mu_{2}=0.46$ Figure 2. (a) An long slab with current flows in a strong magnetic field, arrows indicate the direction of electron transport; (b) odd plateaus of Hall conductance due to the spin degeneracy.
$k_{\rm B}T=0.002$ ,$\mu_{1}=0.49$ ,$\mu_{2}=0.46$ .
图 3 (a)自旋轨道耦合作用下最低的6个朗道能级; (b)自旋轨道耦合作用下的霍尔电导(蓝线)和纵向电阻(红线). 参数
$\alpha=0.06$ ,$\lambda_{\rm h}=0.7$ ,$\mu_{1}=-0.02$ ,$\mu_{2}=-0.04$ ,$k_{\rm B}T=0.002$ Figure 3. (a) The lowest six Landau levels under spin-orbit coupling; (b) corresponding Hall plateaus of the conductance (blue) and the longitudinal resistance (red).
$\alpha=0.06$ ,$\lambda_{\rm h}=0.7$ ,$\mu_{1}=-0.02$ ,$\mu_{2}=-0.04$ ,$k_{\rm B}T=0.002$ .
图 4 (a)温度对霍尔平台的影响; (b)第二朗道能级上几个波矢
$k$ 对应的波函数的模(上自旋分量). 参数$\lambda_{\rm h}=0.7$ ,$\mu_{1}=-0.02$ ,$\mu_{2}=-0.04$ Figure 4. (a) The effect of temperature on Hall plateaus; (b) modulus of the wave functions of a few
$k$ vectors on the second Landau level.$\lambda_{\rm h}=0.7$ ,$\mu_{1}=-0.02$ ,$\mu_{2}=-0.04$ . -
[1] Klitzing K V, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494
Google Scholar
[2] Tsui D C, Stormer H L, Gossard A C 1982 Phys. Rev. Lett. 48 1559
Google Scholar
[3] Laughlin R B 1983 Phys. Rev. Lett. 50 1395
Google Scholar
[4] Tao R, Thouless D J 1983 Phys. Rev. B 28 1142
Google Scholar
[5] Xue Q K, Chang C Z, Zhang J S, et al. 2013 Science 340 167
Google Scholar
[6] Thouless D J, Kohmoto M, Nightingale M P 1982 Phys. Rev. Lett. 49 405
Google Scholar
[7] Niu Q, Thouless D J, Wu Y S 1985 Phys. Rev. B 31 3372
Google Scholar
[8] 倪光炯, 陈苏卿 2004 高等量子力学 (第2版) (上海: 复旦大学出版社) 第270页
Ni G J, Chen S Q 2004 Advanced Quantum Mechanics (2nd Ed.) (Shanghai: Fudan University Press) p270 (in Chinese)
[9] Aoki H 1987 Rep. Prog. Phys. 50 655
Google Scholar
[10] Ezawa Z F 2012 Quantum Hall Effects: Field Theoretical Approach and Related Topics (2nd ed.) (Beijing: Peking University Press) p182
[11] Prange R E 1981 Phys. Rev. B 23 5632
Google Scholar
[12] Lagendijk A, Wiersma D 2009 Phys. Today 62 24
[13] Datta S 1995 Electronic Transport in Mesoscopic Systems (Cambridge: Cambridge University Press) pp181−185
[14] Bychkov Y A, Rashba E I 1984 J. Phys. C 17 6039
Google Scholar
[15] Falko V I 1992 Phys. Rev. B 46 4320
Google Scholar
[16] Schliemann J, Egues J C, Loss D 2003 Phys. Rev. B 67 085302
Google Scholar
[17] Yang W, Chang K 2006 Phys. Rev. B 74 193314
Google Scholar
[18] Luo J, Munekata H, Fang F F, Stiles P J 1990 Phys. Rev. B 41 7685
Google Scholar
[19] Petritz R, Scanlon W 1955 Pbs. Phys. Rev. 97 1620
Google Scholar
[20] 阎守胜 2003 固体物理基础 (第2版) (北京: 北京大学出版社) 第126页
Yan S S 2003 Basis On Solid State Physics (2nd Ed.) (Beijing: Peking University Press) p126 (in Chinese)
[21] 黄昆, 韩汝琦 1988 固体物理学 (第1版) (北京: 高等教育出版社) 第305—307页
Huang K, Han R Q 1988 Solid State Physics (1st Ed.) (Beijing: Higher Education Press) pp305−307 (in Chinese)
[22] Li H C, Sheng L, Shen R, Wang B G, Sheng D N, Xing D Y 2013 Phys. Rev. Lett. 110 266802
Google Scholar
[23] Ridley B K 2013 Quantum Processes in Semiconductors (Oxford: Oxford University Press) pp66−88
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