Skip to main content
SpringerLink
Log in

Energy eigenstates of position-dependent mass particles in a spherical quantum dot

  • Regular Article - Mesoscopic and Nanoscale Systems
  • Published:
The European Physical Journal B Aims and scope Submit manuscript

Abstract

We obtain the exact energy spectrum of nonuniform mass particles for a collection of Hamiltonians in a three-dimensional approach to a quantum dot. By considering a set of generalized Schrödinger equations with different orderings between the particle’s momentum and mass, the energy-bound states are calculated analytically for hard boundary conditions. The present results are of interest in atomic physics and quantum dot theory.

Graphic Abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Data availability statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: Data used in the manuscript are available in the cited bibliography.]

References

  1. P. Harrison and A. Valavanis, Quantum Wells, Wires and Dots – Theoretical and Computational Physics of Semiconductor Nanostructures. New York: John Wiley & Sons, 4th ed., 2016

  2. L. Serra, E. Lipparini, Spin response of unpolarized quantum dots. Europhys. Lett. 40(6), 667 (1997)

    Article  ADS  Google Scholar 

  3. R.A. El-Nabulsi, A generalized self-consistent approach to study position-dependent mass in semiconductors organic heterostructures and crystalline impure materials. Phys. E 124, 114295 (2020)

    Article  Google Scholar 

  4. R. Valencia-Torres, J. Avendaño, A. Bernal, J. García-Ravelo, Energy spectra of position-dependent masses in double heterostructures. Phys. Scr. 95, 075207 (2020)

    Article  ADS  Google Scholar 

  5. R.A. El-Nabulsi, A new approach to schrodinger equation with position-dependent mass and its implications in quantum dots and semiconductors. J. Phys. Chem. Sol. 140, 109384 (2020)

    Article  Google Scholar 

  6. G.H. Wannier, The structure of electronic excitation levels in insulating crystals. Phys. Rev. 52(3), 191 (1937)

    Article  ADS  Google Scholar 

  7. J.C. Slater, Electrons in perturbed periodic lattices. Phys. Rev. 76(11), 1592 (1949)

    Article  ADS  Google Scholar 

  8. J.M. Luttinger, W. Kohn, Motion of electrons and holes in perturbed periodic fields. Phys. Rev. 97(4), 869 (1955)

    Article  ADS  Google Scholar 

  9. D.J. BenDaniel, C.B. Duke, Space-charge effects on electron tunneling. Phys. Rev. 152(2), 683 (1966)

    Article  ADS  Google Scholar 

  10. T. Gora, F. Williams, Theory of electronic states and transport in graded mixed semiconductors. Phys. Rev. 177(3), 1179 (1969)

    Article  ADS  Google Scholar 

  11. Q.-G. Zhu, H. Kroemer, Interface connection rules for effective-mass wave functions at an abrupt heterojunction between two different semiconductors. Phys. Rev. B 27(6), 3519 (1983)

    Article  ADS  Google Scholar 

  12. G. Bastard, Wave mechanics applied to semiconductor heterostructures. Les Editions de Physique, 1992

  13. M.S. Cunha, H.R. Christiansen, Analytic results in the position-dependent mass Schrödinger problem. Commun. Theor. Phys. 60(6), 642 (2013)

    Article  MathSciNet  Google Scholar 

  14. H.R. Christiansen, M.S. Cunha, Solutions to position-dependent mass quantum mechanics for a new class of hyperbolic potentials. J. Math. Phys. 54(12), 122108 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  15. H.R. Christiansen, M.S. Cunha, Energy eigenfunctions for position-dependent mass particles in a new class of molecular Hamiltonians. J. Math. Phys. 55(9), 092102 (2014)

    Article  MathSciNet  ADS  Google Scholar 

  16. R. M. Lima and H. R. Christiansen, “The Kinetic Hamiltonian with Position-Dependent Mass,” Physica E: Low dimensional Systems and Nanostructures, vol. 150, p. 115688,( 2023); arXiv:2303.02507

  17. B.G. da Costa, I.S. Gomez, M. Portesi, \(\kappa \)-Deformed quantum and classical mechanics for a system with position-dependent effective mass. J. Math. Phys. 61(8), 082105 (2020)

    Article  MathSciNet  ADS  Google Scholar 

  18. C.-L. Ho, P. Roy, Generalized Dirac oscillators with position-dependent mass. Europhys. Lett. 124(6), 60003 (2019)

    Article  ADS  Google Scholar 

  19. A.G.M. Schmidt, A.L. de Jesus, Mapping between charge-monopole and position-dependent mass systems. J. Math. Phys. 59(10), 102101 (2018)

    Article  MathSciNet  ADS  Google Scholar 

  20. C. Chang-Ying, R. Zhong-Zhou, J. Guo-Xing, Exact solutions to three-dimensional Schrödinger equation with an exponentially position-dependent mass. Commun. Theoretical Phys. 43(6), 1019 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  21. H. Eleuch, P.K. Jha, Y.V. Rostovtsev, Analytical solution to position dependent mass for 3D-Schrödinger equation. Math. Sci. Lett 1, 1–6 (2012)

    Article  Google Scholar 

  22. J. Guo-Xing, X. Yang, R. Zhong-Zhou, Localization of \(s\)-wave and quantum effective potential of a quasi-free particle with position-dependent mass. Commun. Theoretical Phys. 46(5), 819 (2006)

    Article  ADS  Google Scholar 

  23. O. von Roos, Position-dependent effective masses in semiconductor theory. Phys. Rev. B 27(12), 7547 (1983)

    Article  ADS  Google Scholar 

  24. O. Mustafa, S.H. Mazharimousavi, Ordering ambiguity revisited via position dependent mass pseudo-momentum operators. Int. J. Theor. Phys. 46(7), 1786–1796 (2007)

    Article  MathSciNet  Google Scholar 

  25. T.L. Li, K.J. Kuhn, Band-offset ratio dependence on the effective-mass Hamiltonian based on a modified profile of the \(\text{ GaAs-Al}_x\text{ Ga}_{1-x}\text{ As }\) quantum well. Phys. Rev. B 47(19), 12760 (1993)

    Article  ADS  Google Scholar 

  26. R. Shankar, Principles of Quantum Mechanics. Springer, 2nd ed., (2013)

  27. M. Reed, J. Randall, R. Aggarwal, R. Matyi, T. Moore, A. Wetsel, Observation of discrete electronic states in a zero-dimensional semiconductor nanostructure. Phys Rev Lett 60, 535–537 (1988)

    Article  ADS  Google Scholar 

  28. C.-L. Ho, P. Roy, Artificial atoms. Phys. Today 46(1), 24 (1993)

    Article  Google Scholar 

  29. R. Ashoori, Electrons in artificial atoms. Nature 379, 413 (1996)

    Article  ADS  Google Scholar 

  30. M. Rakhlin, K. Belyaev, and G. e. a. Klimko, Inas/algaas quantum dots for single-photon emission in a red spectral range, Sci.Rep., vol. 8, p. 5299, (2018)

  31. I. A. S. Milton Abramowitz, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Table. Dover Publications, (1970)

  32. U. Banin, Y. Cao, D. Katz, O. Millo, Identification of atomic-like electronic states in indium arsenide nanocrystal quantum dots. Nature 400(6744), 542–544 (1999)

    Article  ADS  Google Scholar 

  33. D. C. Agrawal, Introduction to nanoscience and nanomaterials. World Scientific Publishing,( 2013)

  34. L.S. Dang, G. Neu, R. Romestain, Optical detection of cyclotron resonance of electron and holes in CdTe. Solid State Commun. 44(8), 1187–1190 (1982)

    Article  ADS  Google Scholar 

  35. M. Fanciulli, T. Lei, T. Moustakas, Conduction-electron spin resonance in zinc-blende GaN thin films. Phys. Rev. B 48(20), 15144 (1993)

    Article  ADS  Google Scholar 

  36. K. Miwa, A. Fukumoto, First-principles calculation of the structural, electronic, and vibrational properties of gallium nitride and aluminum nitride. Phys. Rev. B 48(11), 7897 (1993)

  37. W. Fan, M. Li, T. Chong, J. Xia, Electronic properties of zinc-blende GaN, AlN, and their alloys Ga1-x Al x N. J. Appl. Phys. 79(1), 188–194 (1996)

  38. S.-S. Li, J.-B. Xia, Z. Yuan, Z. Xu, W. Ge, X.R. Wang, Y. Wang, J. Wang, L.L. Chang, Effective-mass theory for InAs/GaAs strained coupled quantum dots. Phys. Rev. B 54(16), 11575 (1996)

    Article  ADS  Google Scholar 

  39. F. Long, P. Harrison, W. Hagston, Empirical pseudopotential calculations of Cd 1–x Mn x Te. J. Appl. Phys. 79(9), 6939–6942 (1996)

    Article  ADS  Google Scholar 

  40. R. de Paiva, R.A. Nogueira, C. de Oliveira, H.W. Alves, J.L.A. Alves, L.M.R. Scolfaro, J.R. Leite, First-principles calculations of the effective mass parameters of \(\text{ Al}_x\text{ Ga}_{1-x}\text{ N }\) and \(\text{ Zn}_x\text{ Cd}_{1-x}\text{ Te }\) alloys. Brazilian J. Phys. 32, 405–408 (2002)

    Article  ADS  Google Scholar 

  41. G. Margaritondo, Semiconductor, General Properties, in Encyclopedia of Condensed Matter Physics (F. Bassani, G. L. Liedl, and P. Wyder, eds.), pp. 311–321, Oxford: Elsevier, (2005)

  42. A. Wasserman, Effective Masses, in Encyclopedia of Condensed Matter Physics (F. Bassani, G. L. Liedl, and P. Wyder, eds.), pp. 1–5, Oxford: Elsevier, (2005)

Download references

Acknowledgements

The authors would like to thank Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for partial support. Data Availability Statement: The authors declare that the data supporting the findings of this study are available within the paper. Author contribution statement: The authors declare that they contributed equally to the research.

Author information

Authors and Affiliations

  1. Centro Brasileiro de Pesquisas Físicas, CEP 22290-180, Rio de Janeiro, Brazil

    R. M. Lima

  2. IFCE—Instituto Federal de Educação, Ciência e Tecnologia do Ceará, Maranguape, CEP 61940-750, Ceará, Brazil

    H. R. Christiansen

Authors
  1. R. M. Lima
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. R. Christiansen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. R. Christiansen.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lima, R.M., Christiansen, H.R. Energy eigenstates of position-dependent mass particles in a spherical quantum dot. Eur. Phys. J. B 96, 150 (2023). https://doi.org/10.1140/epjb/s10051-023-00620-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjb/s10051-023-00620-0

Search

Navigation