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Foldy–Wouthuysen Transformation and Structured States of a Graphene Electron in External Fields and Free (2 + 1)-Space

  • PHYSICS OF SOLID STATE AND CONDENSED MATTER
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Abstract

The relativistic Foldy–Wouthuysen transformation is used for an advanced description of a free and interacting planar graphene electron. The exact Foldy–Wouthuysen Hamiltonian of a graphene electron in a uniform and a nonuniform magnetic field is derived. The exact energy spectrum agreeing with experimental data and exact Foldy–Wouthuysen wave eigenfunctions are obtained. These eigenfunctions describe structured states in (2 + 1)-space. It is proven that the Hermite–Gauss beams exist even in the free space. In the structured Hermite–Gauss states, graphene electrons acquire nonzero effective masses dependent on a quantum number and move with group velocities which are less than the Fermi velocity.

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REFERENCES

  1. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183 (2007).

    Article  ADS  Google Scholar 

  2. A. J. Silenko, “Foldy–Wouthuysen transformation for relativistic particles in external fields,” J. Math. Phys. 44, 2952 (2003).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. A. J. Silenko, “General method of the relativistic Foldy–Wouthuysen transformation and proof of validity of the Foldy–Wouthuysen Hamiltonian,” Phys. Rev. A 91, 022103 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  4. A. J. Silenko, Phys. Rev. A 91, 012111 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  5. L. D. Landau, “Diamagnetismus der Metalle,” Z. Phys. 64, 629 (1930).

    ADS  MATH  Google Scholar 

  6. L. D. Landau, E. M. Lifshitz, Quantum Mechanics. Non-Relativistic Theory, 3rd ed. (Nauka, Moscow, 1974; Pergamon Press, Oxford, 1977), pp. 458–461.

  7. D. L. Miller, K. D. Kubista, G. M. Rutter, M. Ruan, W. A. de Heer, P. N. First, and J. A. Stroscio, “Observing the quantization of zero mass carriers in graphene,” Science 324, 924 (2009).

    Article  ADS  Google Scholar 

  8. P. Cheng et al., “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).

    Article  ADS  Google Scholar 

  9. A. J. Silenko, to be published.

  10. A. K. Geim, “Graphene: status and prospects,” Science 324, 1530 (2009).

    Article  ADS  Google Scholar 

  11. L. L. Foldy and S. A. Wouthuysen, “On the Dirac theory of spin 1/2 particles and its non-relativistic limit,” Phys. Rev. 78, 29 (1950).

    Article  ADS  MATH  Google Scholar 

  12. A. J. Silenko, “Classical limit of relativistic quantum mechanical equations in the Foldy–Wouthuysen representation,” Phys. Part. Nucl. Lett. 10, 91 (2013)].

    Article  Google Scholar 

  13. A. J. Silenko, “Foldy–Wouthyusen transformation and semiclassical limit for relativistic particles in strong external fields,” Phys. Rev. A 77, 012116 (2008).

    Article  ADS  Google Scholar 

  14. Y. Sucu and N. Ünal, “Exact solution of Dirac equation in 2 + 1 dimensional gravity,” Found. Phys. 48, 052503 (2007).

    MathSciNet  MATH  Google Scholar 

  15. C. Koke, C. Noh, and D. G. Angelakis, “Dirac equation on a square waveguide lattice with site-dependent coupling strengths and the gravitational Aharonov–Bohm effect,” Phys. Rev. D 102, 013514 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  16. R. Jackiw, “Fractional charge and zero modes for planar systems in a magnetic field,” Phys. Rev. D 29, 2375 (1984).

    Article  ADS  MathSciNet  Google Scholar 

  17. Yu. E. Lozovik, S. P. Merkulova, and A. A. Sokolik, “Collective electron phenomena in graphene,” Phys. Usp. 51, 727 (2008).

    ADS  Google Scholar 

  18. K. M. Case, “Some generalizations of the Foldy-Wouthuysen transformation,” Phys. Rev. 95, 1323 (1954).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. W. Tsai, “Energy eigenvalues for charged particles in a homogeneous magnetic field–an application of the Foldy-Wouthuysen transformation,” Phys. Rev. D 7, 1945 (1973).

    Article  ADS  Google Scholar 

  20. A. J. Silenko, “Connection between wave functions in the Dirac and Foldy–Wouthuysen representations,” Phys. Part. Nucl. Lett. 5, 501 (2008).

    Article  Google Scholar 

  21. A. J. Silenko, “High precision description and new properties of a spin-1 particle in a magnetic field,” Phys. Rev. D 89, 121701(R) (2014).

  22. L. Zou, P. Zhang, and A. J. Silenko, “Paraxial wave function and Gouy phase for a relativistic electron in a uniform magnetic field, J. Phys. G: Nucl. Part. Phys. 47, 055003 (2020).

    Article  ADS  Google Scholar 

  23. F. D. M. Haldane, “Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the “parity anomaly”, Phys. Rev. Lett. 61, 2015 (1988).

    Article  ADS  MathSciNet  Google Scholar 

  24. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550 (1966).

    Article  ADS  Google Scholar 

  25. A. E. Siegman, Lasers (University Science Books, Sausalito, 1986).

    Google Scholar 

  26. F. Pampaloni and J. Enderlein, “Gaussian, Hermite-Gaussian, and Laguerre-Gaussian beams: A primer,” arXiv:physics/0410021 (2004).

  27. A. J. Silenko, P. Zhang, and L. Zou, “Relativistic quantum-mechanical description of twisted paraxial electron and photon beams,” Phys. Rev. A 100, 030101(R) (2019).

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Funding

The author acknowledges the support by the Chinese Academy of Sciences President’s International Fellowship Initiative (Grant no. 2019VMA0019).

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Authors and Affiliations

  1. Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980, Dubna, Russia

    A. J. Silenko

  2. Institute of Modern Physics, Chinese Academy of Sciences, 730000, Lanzhou, China

    A. J. Silenko

  3. Research Institute for Nuclear Problems, Belarusian State University, 220030, Minsk, Belarus

    A. J. Silenko

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  1. A. J. Silenko
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Correspondence to A. J. Silenko.

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Silenko, A.J. Foldy–Wouthuysen Transformation and Structured States of a Graphene Electron in External Fields and Free (2 + 1)-Space. Phys. Part. Nuclei Lett. 20, 1131–1134 (2023). https://doi.org/10.1134/S1547477123050680

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  • DOI: https://doi.org/10.1134/S1547477123050680

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