Generalized Spin-Orbit Interaction and Its Manifestation in Two-Dimensional Electron Systems
Keywords:spin-orbit interaction, Dirac equation, Schr¨odinger equation, 2D electron gas, quantum well, spin Hall effect
In frame of Dirac quantum field theory that describes electrons and positrons as elementary excitations of the spinor field, the generalized operator of the spin-orbit interaction is obtained using non-relativistic approximation in the Hamilton operator of the spinor field taking into account the presence of an external potential. This operator is shown to contain a new term in addition to the known ones. By an example of a model potential in the form of a quantum well, it is demonstrated that the Schr¨odinger equation with the generalized spin-orbit interaction operator describes all spin states obtained directly from the Dirac equation. The dependence of the spin-orbit interaction on the spin states in quasi-two-dimensional systems of electrons localized in a quantum well is analyzed. It is demonstrated that the electric current in the quantum well layer induces the spin polarization of charge carriers near the boundary surfaces of the layer, with the polarization of the charge carriers being opposite at the different surfaces. This phenomenon appears due to the spin-orbit interaction and is known as the spin Hall effect, which was observed experimentally in heterostructures with the corresponding geometry.
H.A. Bethe. Intermediate Quantum Mechanics (W.A. Benjamin, 1964).
A.S. Davydov. Quantum Mechanics (Pergamon Press, 1976).
L.H. Thomas. The motion of the spinning electron. Nature 117, 514 (1926). https://doi.org/10.1038/117514a0
J. Frenkel. Zur Theorie der Elastizit?atsgrenze und der festigkeit kristallinischer K?orper. Z. Phys. 37, 243 (1926). https://doi.org/10.1007/BF01397292
A. Eremko, L. Brizhik, V. Loktev. Spin states of Dirac equation and Rashba spin-orbit interaction. Ann. Phys. 361, 423 (2015). https://doi.org/10.1016/j.aop.2015.07.007
A. Eremko, L. Brizhik, V. Loktev. General solution of the Dirac equation for quasi-two-dimensional electrons. Ann. Phys. 369, 85 (2016). https://doi.org/10.1016/j.aop.2016.03.008
N.N. Bogoliubov, D.V. Shirkov. Quantized Fields (Nauka, Moscow, 1980) (in Russian).
V.B. Berestetskii, E.M. Lifshitz, L.P. Pitaevskii. Relativistic Quantum Theory (Pergamon Press, 1982).
A.A. Sokolov, I.M. Ternov. Relativistic Electron (Nauka, 1974) (in Russian).
A.A. Eremko, L.S. Brizhik, V.M. Loktev. On the theory of the full set of relativistic corrections for Schr?odinger equation. Low Temp. Phys. 44, 734 (2018). https://doi.org/10.1063/1.5037561
A.A. Eremko, V.M. Loktev. On the theory of eigen spin states and spin-orbit interaction of quasi-two-dimensional electrons. Fiz. Nizk. Temp. 43, 456 (2017) (in Russian). https://doi.org/10.1063/1.4980861
J. M. Ziman. Principles of the Theory of Solids (Cambridge Univ. Press, 1979).
G. Bihlmayer, J. Rader, K. Winkler. Focus on the Rashba Effect. New J. Phys. 17, 050202 (2015). https://doi.org/10.1088/1367-2630/17/5/050202
Y.K. Kato, R.C. Myers, A.C. Gossard, D.D. Awschalom. Observation of the spin Hall effect in semiconductors. Science 306, 1910 (2004). https://doi.org/10.1126/science.1105514
J. Sinova, S.O. Valenzuela, J. Wunderlich, C.H. Back, T. Jungwirth. Spin Hall effects. Rev. Mod. Phys. 87, 1213 (2015). https://doi.org/10.1103/RevModPhys.87.1213
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