Applied Physics A

, 124:813 | Cite as

Symmetry lowering and surface elasticity effects on Young’s modulus and Poisson’s ratio of nanofilms

  • Jiangang LiEmail author
  • Bai Narsu
  • Guohong Yun
  • Aoxuan Wang
  • Zhixiang GaoEmail author


Physical and mechanical properties of nanosized materials and structures are strongly affected by surface effects. In this paper, a self-consistent theoretical scheme for describing the elastic properties of nanofilms was proposed. The Young’s modulus, biaxial modulus and Poisson’s ratio of nanofilms were obtained analytically with considerations of symmetry lowering, surface elasticity, elastic parameter splitting and additional elastic coefficient. Applications of present theory to elastic systems such as Si nanofilm Young’s modulus, Cu nanofilm biaxial modulus and Poisson’s ratio yield good agreement with previous calculated results. We found that Young’s modulus and Poisson’s ratio were split due to symmetry lowering, and this splitting confirms the symmetry lowering. For a nanofilm with a given thickness, Young’s modulus and biaxial modulus increase with surface elastic coefficients increase except \(c_{{12}}^{{\alpha ,s}}\). The larger positive \(c_{{12}}^{{\alpha ,s}}\) drives Young’s modulus towards smaller abnormally. The present study in this paper is envisaged to provide useful insights for the design and application of nanofilm-based devices.



The authors acknowledge the financial support of the National Natural Science Foundation of China under Grant nos. 11072104, 11464037, 50901039, and 11447122, the Program for Innovative Research Team of Inner Mongolia University under Grant no. 10013-12110605, the Inner Mongolia Natural Science Foundation under Grant no. 2014BS0102. BN acknowledges support from NJYT-12-B07, and ZG acknowledges support from Higher Innovation Project of Shanxi Province under Grant No. 2015177.

Supplementary material

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Supplementary material 1 (DOCX 49 KB)


  1. 1.
    K. Kiani, Vibrations and instability of pretensioned current-carrying nanowires acted upon by a suddenly applied three-dimensional magnetic field. Mater. Chem. Phys. 162, 531–541 (2015)CrossRefGoogle Scholar
  2. 2.
    K. Kiani, Stability and vibrations of double parallel current-carrying nanowires immersed in a longitudinal magnetic field. Phys. Lett. A 379, 348–360 (2015)CrossRefGoogle Scholar
  3. 3.
    Y. Yao, S.-H. Chen, Surface effect in the bending of nanowires. Mech. Mater. 100, 12–21 (2016)CrossRefGoogle Scholar
  4. 4.
    Z. Yan, Modeling of a nanoscale flexoelectric energy harvester with surface effects. Physica E 88, 125–132 (2017)ADSCrossRefGoogle Scholar
  5. 5.
    J.-J. Li, K.-D. Zhu, All-optical mass sensing with coupled mechanical resonator systems. Phys. Rep. 525, 223–254 (2013)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    W.-M. Zhang, K.-M. Hu, B. Yang, Z.-K. Peng, G. Meng, Effects of surface relaxation and reconstruction on the vibration characteristics of nanobeams. J. Phys. D: Appl. Phys. 49, 165304 (2016)ADSCrossRefGoogle Scholar
  7. 7.
    J.-G. Guo, Y.-P. Zhao, The size-dependent elastic properties of nanofilms with surface effects. J. Appl. Phys. 98, 274306 (2005)Google Scholar
  8. 8.
    J.-G. Guo, Y.-P. Zhao, The size-dependent bending elastic properties of nanobeams with surface effects. Nanotechnology 18, 295701 (2007)CrossRefGoogle Scholar
  9. 9.
    K. Kiani, Axial buckling analysis of a lender current-carrying nanowires acted upon by a magnetic field using the surface energy approch. J. Phys. D: Appl. Phys. 48, 245302 (2015)ADSCrossRefGoogle Scholar
  10. 10.
    H. Sadeghian, J.F.L. Goosen, A. Bossche, B.J. Thijsse, F.V. Keulen, Effects of size and surface on the elasticity of silicon nanoplates: Molecular dynamics and semi-continuum approaches. Thin Solid Films 520, 391–399 (2011)ADSCrossRefGoogle Scholar
  11. 11.
    R. Dingreville, J.-M. Qu, M. Cherkaoui, Surface free energy and its effect on the behavior of nano-sized particles, wires and films. J. Mech. Phys. Solids 53, 1872–1854 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    F.H. Streitz, K. Sieradzki, R.C. Cammarata, Elastic properties of thin fcc films. Phys. Rev. B 41, 12285–12287 (1990) (R)ADSCrossRefGoogle Scholar
  13. 13.
    K. Kiani, Column buckling of magnetically affected stocky nanowires carrying electric current. J Phys. Chem. solids 83, 140–151 (2015)ADSCrossRefGoogle Scholar
  14. 14.
    K. Kiani, Surface and shear energy effects on vibrations of magnetically affected beam-like nanostructures carrying direct currents. Int. J Mech. Sci. 113, 221–238 (2016)CrossRefGoogle Scholar
  15. 15.
    K. Kiani, Elastic buckling of current-carrying double-nanowire systems immersed in a magnetic field. Acta Mech. 227, 3549–3570 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    K. Kiani, A refined integro-surface energy-based model for vibration of magnetically actuated double-nanowire-system carrying electric current. Phys. E 86, 225–236 (2017)CrossRefGoogle Scholar
  17. 17.
    K. Kiani, Dynamic interactions between double current-carrying nanowires immersed in a longitudinal magnetic filed: Novel integro-surface energy-based models. Int. J Mech. Sci. 107, 98–133 (2016)zbMATHGoogle Scholar
  18. 18.
    J.-G. Li, B. Narsu, G.-H. Yun, H.-Y. Yao, Elasticity theory of ultrathin nanofilms. J. Phys. D Appl. Phys. 48, 285301 (2015)CrossRefGoogle Scholar
  19. 19.
    H. Sadeghian, C.-K. Yang, J.F.L. Goosen, A. Bossche, U. Staufer, P.J. French, F.V. Keulen, Effects of size defects on the elasticity of silicon nanocantilevers. J. Micromech. Miroeng. 20, 064012 (2010)CrossRefGoogle Scholar
  20. 20.
    S.G. Nilsson, X. Borrisé, L. Montelius, Size effect on Young’s modulus of thin chromium cantilevers. Appl. Phys. Lett. 85, 3555–3557 (2004)ADSCrossRefGoogle Scholar
  21. 21.
    C.-Y. Nam, P. Jaroenapibal, D. Tham, D.E. Luzzi, S. Evoy, J.E. Fischer, Diameter-dependent electromechanical properties of GaN nanowires. nano Lett. 6, 53–158 (2006)CrossRefGoogle Scholar
  22. 22.
    G.Y. Jing, H.L. Duan, X.M. Sun, Z.S. Zhang, J. Xu, Y.D. Li, J.X. Wang, D.P. Yu, Surface effects on elastic properties of silver nanowires: Contact atomic-force microscopy. Phys. Rev. B 73, 235409 (2006)ADSCrossRefGoogle Scholar
  23. 23.
    E.P.S. Tan, Y. Zhu, T. Yu, L. Dai, C.H. Sow, V.B.C. Tan, C.T. Lim, Crystallinity and surface effects on Young’s modulus of CuO nanowires. Appl. Phys. Lett. 90, 163112 (2007)ADSCrossRefGoogle Scholar
  24. 24.
    G. Stan, S. Krylyuk, A.V. Davydov, M. Vaudin, L.A. Bendersky, Cook, R F, Surface effects on the elastic modulus of Te nanowires. Appl. Phys. Lett. 92, 241908 (2008)ADSCrossRefGoogle Scholar
  25. 25.
    C.Q. Chen, Y. Shi, Y.S. Zhang, J. Zhu, Y.J. Yan, Size dependent of Young’s modulus in ZnO nanowires. Phys. Rev. Lett. 96, 075505 (2006)ADSCrossRefGoogle Scholar
  26. 26.
    R.E. Miller, V.B. Shenoy, Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139–147 (2000)ADSCrossRefGoogle Scholar
  27. 27.
    A. Ahadi, S. Melin, Size dependence of the Poisson’s ratio in single-crystal fcc cupper nanobeams. Comput. Mater. Sci. 111, 322–327 (2016)CrossRefGoogle Scholar
  28. 28.
    F. Hao, D.-N. Fang, Modeling of magnetoelectric effects in flexural nanobilayers: The effects of surface stress. J. Appl. Phys. 113, 104103 (2013)ADSCrossRefGoogle Scholar
  29. 29.
    X. Liang, S.-L. Hu, S.-P. Shen, Effects of surface and flexoelectricity on a piezoelectric nanobeam. Smart. Mater. Struct. 23, 035020 (2014)ADSCrossRefGoogle Scholar
  30. 30.
    G. Stan, C.V. Ciobanu, P.M. Parthangal, R.F. Cook, Diameter-dependent radial and tangential elastic moduli on ZnO nanowires. nano Lett 7, 3691–3697 (2007)ADSCrossRefGoogle Scholar
  31. 31.
    H.-Y. Yao, G.-H. Yun, B. Narsu, J.-G. Li, Surface elasticity effect on the size-dependent elastic property of nanowires. J. Appl. Phys. 111, 083506 (2012)ADSCrossRefGoogle Scholar
  32. 32.
    H.-Y. Yao, G.-H. Yun, B. Narsu, Influence of exponentially increasing surface elasticity on the piezoelectric potential of a bent ZnO nanowires. J. Phys. D: Appl. Phys. 45, 285304 (2012)ADSCrossRefGoogle Scholar
  33. 33.
    du Rremolet de Lacheisserie E, Definition and measurement of the magnetoelastic coupling coefficients. Phys. Rev. B 51, 15925 (1995)ADSCrossRefGoogle Scholar
  34. 34.
    du Rremolet de Lacheisserie E, Magnetostriction Theory and Applications of Magnetoelasticity (Boca Raton, FL CRC, 1993)Google Scholar
  35. 35.
    X. Lei, B. Narsu, G.-H. Yun, J.-G. Li, H.-Y. Yao, Axial buckling transverse vibration of ultrathin nanowires: low symmetry and surface elastic effect. J. Phys. D: Appl. Phys. 49, 175305 (2016)ADSCrossRefGoogle Scholar
  36. 36.
    D. Sander, The correlation between mechanical stress and mechanic anisotropy in ultrathin films. Rep. Prog. Phys. 62, 809–858 (1999)ADSCrossRefGoogle Scholar
  37. 37.
    S. Izumi, S. Hara, T. Kumagai, S. Sakai, A method for calculating surface stress and surface elastic constants by molecular dynamics: application to the surface of crystal and amorphous silicon. Thin Solid Films 467, 253–260 (2004)ADSCrossRefGoogle Scholar
  38. 38.
    V.B. Shenoy, Atomic calculations of elastic properties of metallic fcc crystal surfaces. Phys. Rev. B 71, 094104 (2005)ADSCrossRefGoogle Scholar
  39. 39.
    C. Kittel, Introduction to Solid State Physics 7th edn (Wiley, New York, 1997)Google Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Physics and Electronic ScienceShanxi Datong University and Shanxi Province Key Laboratory of Microstructure Electromagnetic Functional MaterialsDatongPeople’s Republic of China
  2. 2.College of Physics and Electronic InformationInner Mongolia Normal University and Inner Mongolia Key Laboratory of Physics and Chemistry of Functional MaterialsHohhotPeople’s Republic of China
  3. 3.Inner Mongolia Key Lab of Nanoscience and Nanotechnology and School of Physical Science and TechnologyInner Mongolia UniversityHohhotPeople’s Republic of China
  4. 4.Committee of the Communist Youth LeagueShanxi Datong UniversityDatongPeople’s Republic of China

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