Effect of the shape of a nano-object on quantum-size states

  • Vladimir Dzyuba
  • Yurii Kulchin
  • Valentin Milichko
Research Paper


In this paper, we propose an original functional method that makes it easy to determine the effect of any deviation in the shape of a nano-object from the well-studied shape (e.g., spherical) on the quantum characteristics of charge localized inside the nano-object. The maximum dimension of the object is determined by the magnitude of influence of quantum-size effects on quantum states of charge, and is limited by 100 nm. This method is ideologically similar to the perturbation theory, but the perturbation of the surface shape, rather than the potential, is used. Unlike the well-known variational methods of theoretical physics, this method is based on the assumption that the physical quantity is a functional of surface shape. Using the method developed, we present the quantum-size state of charges for two different complex shapes of nano-objects. The results from analyzing the quantum-size states of charge in the nano-objects with a deformed spherical shape indicated that the shape perturbations have a larger effect on the probability density of locating a particle inside the nano-object than on the surface energy spectrum and quantum density of the states.


Theoretical approach Nano-object Quantum-size states Energy and wave functions 



This work was supported by Grant Nos. 12-I-0-02-010 and 12-I-0-02-055 of Presidium of Russian Academy of Science.


  1. Auvinen S, Alatalo M, Haario H, Jalava JP, Lamminmaki RJ (2011) Size and shape dependence of the electronic and spectral properties in TiO2 nanoparticles. J Phys Chem C 115:8484–8493CrossRefGoogle Scholar
  2. Dvoyan KG, Kazaryan EM, Petrosyan LS (2005) Electronic states in quantum dots with ellipsoidal symmetry. Phys E 28:333–338CrossRefGoogle Scholar
  3. Dzyuba VP (2006) Scalar-vector methods of theoretic acoustics. Dalnauka, VladivostokGoogle Scholar
  4. Forestiere C, Miano G, Boriskina S, Negro L (2009) The role of nanoparticle shape and deterministic aperiodicity for the design of nanoplasmonic arrays. Opt Express 17(12):9648–9660CrossRefGoogle Scholar
  5. Kulchin YuN, Dzyuba VP, Shcherbakov AV (2009) Optical transmittance spectra of insulator nanoparticles in bulk heterocomposites. Semiconductors 43(3):331–339CrossRefGoogle Scholar
  6. Kulchin YuN, Dzyuba VP, Voznesenskiy SS (2011) Threshold optical nonlinearity of dielectric nanocomposite. In: Reddy B (ed) Advances in diverse industrial applications of nanocomposites. InTech, Rijeka, pp 261–288Google Scholar
  7. Kwang-Hyon Kim, Hisakou A, Herrman J (2010) Linear and nonlinear optical characteristics of composites containing metal nanoparticles with different sizes and shapes. Opt Express 18(7):7488–7496CrossRefGoogle Scholar
  8. Landau LD, Lifshitz EB (2002) Quantum mechanics. Fizmatlit, MoscowGoogle Scholar
  9. Li J, Wang L-W (2003) Shape effects on electronic states of nanocrystals. Nano Lett 3(10):1357–1363CrossRefGoogle Scholar
  10. Li L, Hu J, Yang W, Paul AA (2011) Band gap variation of size- and shape-controlled colloidal CdSe quantum rods. Nano Lett 1(7):349–351CrossRefGoogle Scholar
  11. Lin Z, Franceschetti A, Lusk MT (2011) Size dependence of the multiple exciton generation rate in CdSe quantum dots. ACS Nano 5(4):2503–2511CrossRefGoogle Scholar
  12. Matthew CB, Turner MG, Schmuttenmaer CA (2002) Size-dependent photoconductivity in CdSe nanoparticles as measured by time-resolved terahertz spectroscopy. Nano Lett 2(9):983–987CrossRefGoogle Scholar
  13. Migdal AB (1975) Qualitative methods in quantum theory. ScienceGoogle Scholar
  14. Moreels I, Lambert K, Smeets D, De Muynck D, Nollet T, Martins JC, Vanhaecke F, Vantomme A, Delerue C, Allan G, Hens Z (2009) Size-dependent optical properties of colloidal PbS quantum dots. ACS Nano 3(10):3023–3030CrossRefGoogle Scholar
  15. Mors PM, Feshbach H (1953) Methods of theoretical physics. McGraw-Hill, New YorkGoogle Scholar
  16. Shmelev AB (1972) Wave scattering by statistically uneven surfaces. Physics Uspekhi 15:173–183CrossRefGoogle Scholar
  17. Stetz AW (2006) Advanced quantum mechanicsGoogle Scholar
  18. Stroyuk OL, Dzhagan VM, Shvalagin VV, Kuchmiy SYa (2010) Size-dependent optical properties of colloidal ZnO nanoparticles Charged by photoexcitation. J Phys Chem C 114:220–225CrossRefGoogle Scholar
  19. Sucu S, Mese AI, Okan SE (2008) The role of confinement and shape on the binding energy of an electron in a quantum dot. Phys E 40:2698–2702CrossRefGoogle Scholar
  20. Xu H, Chen X, Ouyang S, Kako T, Ye J (2012) Size-dependent Mie’s scattering effect on TiO2 spheres for the superior photoactivity of H2 evolution. J Phys Chem C 116:3833–3839CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  • Vladimir Dzyuba
    • 1
    • 2
  • Yurii Kulchin
    • 1
    • 2
  • Valentin Milichko
    • 1
    • 2
  1. 1.Institution of Russian Academy of Science IACP FEB RASVladivostokRussia
  2. 2.Far Eastern Federal UniversityVladivostokRussia

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