Brazilian Journal of Physics

, Volume 49, Issue 6, pp 801–807 | Cite as

Energy Spectrum of a Dirac Particle with Position-Dependent Mass Under the Influence of the Aharonov-Casher Effect

  • R. R. S. OliveiraEmail author
  • V. F. S. Borges
  • M. F. Sousa
Condensed Matter


In this paper, we investigate the influence of the Aharonov-Casher (AC) effect on the relativistic and nonrelativistic energy spectra of a neutral Dirac particle with position-dependent mass (PDM). To exactly solve our system, we use the projection operators left-handed and right-handed. Next, we explicitly determine the energy spectra for the bound states of the particle. As a result, we verify that the relativistic spectrum depends on the quantum numbers n and ml, AC quantum phase ΦAC generated by AC effect and of the parameter κ that characterize the PDM. In addition, this spectrum is a periodic function and increase in absolute values with the increase of ΦAC. We also verify that the energies of the particle are minors that of the antiparticle, and in the limit of the constant mass (κ → 0) the rest energy is recovered. However, in the absence of the AC effect (ΦAC → 0), the spectrum still remains quantized in terms of n and ml. Finally, we analyze the nonrelativistic limit of our work, where we obtain an energy spectrum with some characteristics similar to the relativistic case. Making an analogy with some works of the literature, in particular with the hydrogen atom, we note that our nonrelativistic spectrum provides the so-called binding energies, while that its absolute values provides the so-called ionization energies.


Dirac particle Position-dependent mass Aharonov-Casher effect Relativistic energy spectrum Nonrelativistic energy spectrum 


Funding Information

This work was financially supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).


  1. 1.
    Y. Aharonov, A. Casher, Topological quantum effects for neutral particles. Phys. Rev. Lett. 53, 319 (1984)ADSMathSciNetGoogle Scholar
  2. 2.
    X.G. He, B.H. McKellar, Topological phase due to electric dipole moment and magnetic monopole interaction. Phys. Rev. A. 47, 3424 (1993)ADSGoogle Scholar
  3. 3.
    J.P. Dowling, C.P. Williams, J.D. Franson, Maxwell duality, Lorentz invariance, and topological phase. Phys. Rev. Lett. 83, 2486 (1999)ADSMathSciNetzbMATHGoogle Scholar
  4. 4.
    A. Cimmino, G.I. Opat, A.G. Klein, H. Kaiser, S.A. Werner, M. Arif, R. Clothier, Observation of the topological Aharonov-Casher phase shift by neutron interferometry. Phys. Rev. Lett. 63, 380 (1989)ADSGoogle Scholar
  5. 5.
    K. Sangster, E.A. Hinds, S.M. Barnett, E. Riis, Measurement of the Aharonov-Casher phase in an atomic system. Phys. Rev. Lett. 71, 3641 (1993)ADSGoogle Scholar
  6. 6.
    W.J. Elion, J.J. Wachters, L.L. Sohn, J.E. Mooij, Observation of the Aharonov-Casher effect for vortices in Josephson-junction arrays. Phys. Rev. Lett. 71, 2311 (1993)ADSGoogle Scholar
  7. 7.
    C.R. Hagen, Exact equivalence of spin-1/2 Aharonov-Bohm and Aharonov-Casher effects. Phys. Rev. Lett. 64, 2347 (1990)ADSMathSciNetzbMATHGoogle Scholar
  8. 8.
    X.G. He, B.H. McKellar, The topological phase of the Aharonov-Casher effect and the anyon behaviour of charged particles in 2 + 1 dimensions. Phys. Lett. B. 256, 250–254 (1991)ADSGoogle Scholar
  9. 9.
    S. Bruce, L. Roa, C. Saavedra, A.B. Klimov, Unbroken supersymmetry in the Aharonov-Casher effect. Phys. Rev. A. 60, R1 (1999)ADSGoogle Scholar
  10. 10.
    S. Bruce, Neutron confinement and the Aharonov-Casher effect. J. Phys A:, Math. Gen. 38, 6999 (2005)ADSMathSciNetzbMATHGoogle Scholar
  11. 11.
    K. Li, J. Wang, The topological AC effect on non-commutative phase space. Eur. Phys. J. C. 50, 1007–1011 (2007)ADSzbMATHGoogle Scholar
  12. 12.
    E.O. Silva, F.M. Andrade, C. Filgueiras, H. Belich, On Aharonov–Casher bound states. Eur. Phys. J. C. 73, 2402 (2013)ADSGoogle Scholar
  13. 13.
    Q.G. Lin, Aharonov-bohm effect on Aharonov-Casher scattering. Phys. Rev. A. 81, 012710 (2010)ADSGoogle Scholar
  14. 14.
    F.S. Azevedo, E.O. Silva, L.B. Castro, C. Filgueiras, D. Cogollo, Relativistic quantum dynamics of a neutral particle in external electric fields: an approach on effects of spin. Ann. Phys. 362, 196 (2015)ADSMathSciNetzbMATHGoogle Scholar
  15. 15.
    K. Bakke, C. Furtado, Geometric phase for a neutral particle in rotating frames in a cosmic string spacetime. Phys. Rev. D. 80, 024033 (2009)ADSGoogle Scholar
  16. 16.
    K. Bakke, J.R. Nascimento, C. Furtado, Geometric phase for a neutral particle in the presence of a topological defect. Phys. Rev. D. 78, 064012 (2008)ADSGoogle Scholar
  17. 17.
    K. Bakke, C. Furtado, On the interaction of the Dirac oscillator with the Aharonov–Casher system in topological defect backgrounds. Ann. Phys. 336, 489 (2013)ADSMathSciNetzbMATHGoogle Scholar
  18. 18.
    R.R.S. Oliveira, R.V. Maluf, C.A.S. Almeida, Exact solutions of the Dirac oscillator under the influence of the Aharonov-Casher effect in the cosmic string background. arXiv:1810.11149 (2018)
  19. 19.
    R.R.S. Oliveira, M.F. Sousa, Relativistic quantum dynamics of a neutral Dirac fermion in the presence of an electromagnetic field. Braz. J. Phys. 49, 315–320 (2019)ADSGoogle Scholar
  20. 20.
    C. Kittel, P. McEuen. Introduction to Solid State Physics (Wiley, New York, 1996)Google Scholar
  21. 21.
    I.O. Vakarchuk, The Kepler problem in Dirac theory for a particle with position-dependent mass. J. Phys. A:, Math. Gen. 38, 4727 (2005)ADSMathSciNetzbMATHGoogle Scholar
  22. 22.
    A. Alhaidari, Solution of the Dirac equation with position-dependent mass in the Coulomb field. Phys. Lett. A. 322, 72 (2004)ADSMathSciNetzbMATHGoogle Scholar
  23. 23.
    S.C.y Cruz, O. Rosas-Ortiz, Position-dependent mass oscillators and coherent states. J. Phys. A:, Math. Theor. 42, 185205 (2009)ADSMathSciNetzbMATHGoogle Scholar
  24. 24.
    O. Krebs, P. Voisin, Giant optical anisotropy of semiconductor heterostructures with no common atom and the quantum-confined Pockels effect. Phys. Rev. Lett. 77, 1829 (1996)ADSGoogle Scholar
  25. 25.
    O. Von Roos, Position-dependent effective masses in semiconductor theory. Phys. Rev. B. 27, 7547 (1983)ADSGoogle Scholar
  26. 26.
    P. Harrison. Quantum Wells Wires and Dots (Wiley, Hoboken, 2000)Google Scholar
  27. 27.
    L. Serra, E. Lipparini, Spin response of unpolarized quantum dots. Europhys. Lett. 40, 667 (1997)ADSGoogle Scholar
  28. 28.
    M. Barranco, M. Pi, S.M. Gatica, E.S. Hernandez, J. Navarro, Structure and energetics of mixed He-He drops. Phys. Rev. B. 56, 8997 (1997)ADSGoogle Scholar
  29. 29.
    F.A. Saavedra, J. Boronat, A. Polls, A. Fabrocini, Effective mass of one He atom in liquid He. Phys. Rev. B. 50, 4248 (1994)ADSGoogle Scholar
  30. 30.
    S.M. Ikhdair, Rotation and vibration of diatomic molecule in the spatially-dependent mass schrödinger equation with generalized q-deformed Morse potential. Chem. Phys. 361, 9 (2009)Google Scholar
  31. 31.
    A.R. Plastino, A. Rigo, M. Casas, F. Garcias, A. Plastino, Supersymmetric approach to quantum systems with position-dependent effective mass. Phys. Rev. A. 60, 4318 (1999)ADSGoogle Scholar
  32. 32.
    A. de Souza Dutra, C.A.S. Almeida, Exact solvability of potentials with spatially dependent effective masses. Phys. Lett. A. 275, 25 (2000)ADSMathSciNetzbMATHGoogle Scholar
  33. 33.
    F.S.A. Cavalcante, R.N. Costa Filho, J. Ribeiro Filho, C.A.S. Almeida, V.N. Freire, Form of the quantum kinetic-energy operator with spatially varying effective mass. Phys. Rev. B. 55, 1326 (1997)ADSGoogle Scholar
  34. 34.
    A. Alhaidari, H. Bahlouli, A. Al-Hasan, M. Abdelmonem, Relativistic scattering with a spatially dependent effective mass in the Dirac equation. Phys. Rev. A. 75, 062711 (2007)ADSGoogle Scholar
  35. 35.
    S.M. Ikhdair, R. Sever, Solutions of the spatially-dependent mass Dirac equation with the spin and pseudospin symmetry for the Coulomb-like potential. Appl. Math. Comput. 216, 545 (2010)MathSciNetzbMATHGoogle Scholar
  36. 36.
    R. Renan, M. Pacheco, C.A.S. Almeida, Treating some solid state problems with the Dirac equation. J. Phys. A. 33, L509 (2000)ADSMathSciNetzbMATHGoogle Scholar
  37. 37.
    C.S. Jia, A.S. Dutra, Extension of PT-symmetric quantum mechanics to the Dirac theory with position-dependent mass. Ann. Phys. 323, 566 (2008)ADSMathSciNetzbMATHGoogle Scholar
  38. 38.
    L.B. Castro, On the Dirac equation with PT-symmetric potentials in the presence of position-dependent mass. Phys. Lett. A. 375, 2510 (2011)ADSMathSciNetzbMATHGoogle Scholar
  39. 39.
    O. Mustafa, S.H. Mazharimousavi, (1 + 1)-dirac particle with position-dependent mass in complexified lorentz scalar interactions: effectively-symmetric. Int. J. Theor. Phys. 47, 1112 (2008)MathSciNetzbMATHGoogle Scholar
  40. 40.
    P. Alberto, C. Fiolhais, V.M.S. Gil, Relativistic particle in a box. Eur. J. Phys. 17, 19 (1996)Google Scholar
  41. 41.
    P. Alberto, S. Das, E.C. Vagenas, Relativistic particle in a three-dimensional box. Phys. Lett. A. 375, 1436 (2011)ADSzbMATHGoogle Scholar
  42. 42.
    P. Pedram, Dirac particle in gravitational quantum mechanics. Phys. Lett. B. 702, 295 (2011)ADSGoogle Scholar
  43. 43.
    N.M.R. Peres, A.C. Neto, F. Guinea, Dirac fermion confinement in graphene. Phys. Rev. B. 73, 241403 (2006)ADSGoogle Scholar
  44. 44.
    P. Carmier, D. llmo, Berry phase in graphene: semiclassical perspective. Phys. Rev. B. 77, 245413 (2008)ADSGoogle Scholar
  45. 45.
    G.W. Semenoff, V. Semenoff, F. Zhou, Domain walls in gapped graphene. Phys. Rev. Lett. 101, 087204 (2008)ADSGoogle Scholar
  46. 46.
    M. Zarenia, O. Leenaerts, B. Partoens, F.M. Peeters, Substrate-induced chiral states in graphene. Phys. Rev. B. 86, 085451 (2012)ADSGoogle Scholar
  47. 47.
    L.J.P. Xavier, D.R. da Costa, A. Chaves, J.M.Jr Pereira, G.A. Farias, Electronic confinement in graphene quantum rings due to substrate-induced mass radial kink. J. Phys. Condens. Matter. 28, 505501 (2016)Google Scholar
  48. 48.
    V. Jakubský, D. Krejčiřík, Qualitative analysis of trapped Dirac fermions in graphene. Ann. Phys. 349, 268–287 (2014)ADSMathSciNetzbMATHGoogle Scholar
  49. 49.
    R.R.S. Oliveira, A.A. Araújo Filho, R.V. Maluf, C.A.S. Almeida, The relativistic Aharonov-Bohm-Coulomb system with position-dependent mass. arXiv:1812.07756 (2018)
  50. 50.
    C.L. Ho, P. Roy, Generalized Dirac oscillators with position-dependent mass. Europhys. Lett. 124, 60003 (2019)ADSGoogle Scholar
  51. 51.
    P.R. Auvil, L.M. Brown, The relativistic hydrogen atom: a simple solution. Am. J. Phys. 46, 679 (1978)ADSGoogle Scholar
  52. 52.
    V.M. Villalba, A.R. Maggiolo, Energy spectrum of a 2D Dirac electron in the presence of a constant magnetic field. Eur. Phys. J. B. 22, 31 (2001)ADSGoogle Scholar
  53. 53.
    W. Greiner. Relativistic Quantum Mechanics, Wave Equations (Springer, Berlin, 1997)zbMATHGoogle Scholar
  54. 54.
    M. Ezawa, Intrinsic Zeeman effect in graphene. J. Phys. Soc. Jpn. 76, 094701 (2007)ADSGoogle Scholar
  55. 55.
    Y. Kluger, J.M. Eisenberg, B. Svetitsky, F. Cooper, E. Mottola, Fermion pair production in a strong electric field. Phys. Rev. D. 45, 4659 (1992)ADSGoogle Scholar
  56. 56.
    W.T. Grandy. Relativistic Quantum Mechanics of Leptons and Fields, Vol. 41 (Springer, Berlin, 2012)Google Scholar
  57. 57.
    M. Abramowitz, I.A. Stegun. Handbook of Mathematical Functions (Dover Publications Inc., New York, 1965)zbMATHGoogle Scholar
  58. 58.
    B. Zaslow, M.E. Zandler, Two-dimensional analog to the hydrogen atom. Am. J. Phys. 35, 1118–1119 (1967)ADSGoogle Scholar
  59. 59.
    X.L. Yang, S.H. Guo, F.T. Chan, K.W. Wong, W.Y. Ching, Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory. Phys. Rev. A. 43, 1186 (1991)MathSciNetGoogle Scholar
  60. 60.
    A.I. Safonov, S.A. Vasilyev, I.S. Yasnikov, I.I. Lukashevich, S. Jaakkola, Observation of quasicondensate in two-dimensional atomic hydrogen. Phys. Rev. Lett. 81, 4545 (1998)ADSGoogle Scholar

Copyright information

© Sociedade Brasileira de Física 2019

Authors and Affiliations

  1. 1.Departamento de FísicaUniversidade Federal do Ceará (UFC)FortalezaBrazil
  2. 2.Departamento de QuímicaUniversidade Federal do Piauí (UFPI)JuncoBrazil
  3. 3.Departamento de FísicaUniversidade Federal de Campina Grande (UFCG)Campina GrandeBrazil

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