Skip to main content
SpringerLink
Log in

Torsion of a flexoelectric semiconductor rod with a rectangular cross section

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

We study the torsion of a flexoelectric semiconductor rod with a rectangular cross section. The macroscopic theory of flexoelectric semiconductors is used. A one-dimensional model is established from the three-dimensional theory using double power series expansion of the coordinates within the cross section. The angle of twist of the rod and warping of the cross section are taken into consideration by retaining the proper lower-order terms in the expansion. Solutions of wave propagation in an unbounded rod and static torsion of a finite rod are presented, showing that electromechanical couplings exist when warping varies along the axis of the rod. The torsional wave is essentially nondispersive, but the warping wave is dispersive with a cutoff frequency. The mobile charges concentrate at the corners of the cross section. The electric potential and charge carrier distributions produced by mechanical loads are sensitive to the geometric and physical parameters of the system. The results are potentially useful for making flexotronic devices based on torsional modes.

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
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Wang, Z.L.: Piezotronics and Piezo-Phototronics. Springer, Berlin (2012)

    Book  Google Scholar 

  2. Wang, Z.L., Wu, W.Z.: Piezotronics and piezo-phototronics-fundamentals and applications. Natl. Sci. Rev. 1, 62–90 (2014). https://doi.org/10.1093/nsr/nwt002

    Article  Google Scholar 

  3. Liu, Y., Zhang, Y., Yang, Q., Niu, S.M., Wang, Z.L.: Fundamental theories of piezotronics and piezo-phototronics. Nano Energy 14, 257–275 (2015). https://doi.org/10.1016/j.nanoen.2014.11.051

    Article  Google Scholar 

  4. Wang, Z.L., Wu, W.Z., Falconi, C.: Piezotronics and piezo-phototronics with third-generation semiconductors. MRS Bull. 43, 922–927 (2018). https://doi.org/10.1557/mrs.2018.263

    Article  Google Scholar 

  5. Zhang, Y., Leng, Y., Willatzen, M., Huang, B.: Theory of piezotronics and piezo-phototronics. MRS Bull. 43, 928–935 (2018). https://doi.org/10.1557/mrs.2018.297

    Article  Google Scholar 

  6. Wauer, J., Suherman, S.: Thickness vibrations of a piezo-semiconducting plate layer. Int. J. Eng. Sci. 35, 1387–1404 (1997). https://doi.org/10.1016/s0020-7225(97)00060-8

    Article  MATH  Google Scholar 

  7. Jiao, F.Y., Wei, P.J., Zhou, Y.H., Zhou, X.L.: Wave propagation through a piezoelectric semiconductor slab sandwiched by two piezoelectric half-spaces. Eur. J. Mech. A. Solids 75, 70–81 (2019). https://doi.org/10.1016/j.euromechsol.2019.01.007

    Article  MathSciNet  MATH  Google Scholar 

  8. Jiao, F.Y., Wei, P.J., Zhou, Y.H., Zhou, X.L.: The dispersion and attenuation of the multi-physical fields coupled waves in a piezoelectric semiconductor. Ultrasonics 92, 68–78 (2019). https://doi.org/10.1016/j.ultras.2018.09.009

    Article  Google Scholar 

  9. Sladek, J., Sladek, V., Pan, E., Wuensche, M.: Fracture analysis in piezoelectric semiconductors under a thermal load. Eng. Fract. Mech. 126, 27–39 (2014). https://doi.org/10.1016/j.engfracmech.2014.05.011

    Article  Google Scholar 

  10. Tian, R., Liu, J.X., Pan, E., Wang, Y.S., Soh, A.K.: Some characteristics of elastic waves in a piezoelectric semiconductor plate. J. Appl. Phys. 126, 125701 (2019). https://doi.org/10.1063/1.5116662

    Article  Google Scholar 

  11. Zhao, M.H., Pan, Y.B., Fan, C.Y., Xu, G.T.: Extended displacement discontinuity method for analysis of cracks in 2D piezoelectric semiconductors. Int. J. Solids Struct. 94–95, 50–59 (2016). https://doi.org/10.1016/j.ijsolstr.2016.05.009

    Article  Google Scholar 

  12. Qin, G.S., Lu, C.S., Zhang, X., Zhao, M.H.: Electric current dependent fracture in GaN piezoelectric semiconductor ceramics. Materials 11, 2000 (2000). https://doi.org/10.3390/ma11102000

    Article  Google Scholar 

  13. Afraneo, R., Lovat, G., Burghignoli, P., Falconi, C.: Piezo-semiconductive quasi-1D nanodevices with or without anti-symmetry. Adv. Mater. 24, 4719–4724 (2012). https://doi.org/10.1002/adma.201104588

    Article  Google Scholar 

  14. Fan, S.Q., Liang, Y.X., Xie, J.M., Hu, Y.T.: Exact solutions to the electromechanical quantities inside a statically-bent circular ZnO nanowire by taking into account both the piezoelectric property and the semiconducting performance: part I-linearized analysis. Nano Energy 40, 82–87 (2017). https://doi.org/10.1016/j.nanoen.2017.07.049

    Article  Google Scholar 

  15. Liang, Y.X., Fan, S.Q., Chen, X.D., Hu, Y.T.: Nonlinear effect of carrier drift on the performance of an n-type ZnO nanowire nanogenerator by coupling piezoelectric effect and semiconduction. Nanotechnology 9, 1917–1925 (2018). https://doi.org/10.3762/bjnano.9.183

    Article  Google Scholar 

  16. Sharma, J.N., Sharma, K.K., Kumar, A.: Acousto-diffusive waves in a piezoelectric-semiconductor-piezoelectric sandwich structure. World J. Mech. 1, 247–255 (2011). https://doi.org/10.4236/wjm.2011.15031

    Article  Google Scholar 

  17. Zhang, C.L., Luo, Y.X., Cheng, R.R., Wang, X.Y.: Electromechanical fields in piezoelectric semiconductor nanofibers under an axial force. MRS Adv. 2, 3421–3426 (2017). https://doi.org/10.1557/adv.2017.301

    Article  Google Scholar 

  18. Yang, J.S.: Analysis of Piezoelectric Semiconductor Structures. Springer Nature, Cham (2020)

    Book  Google Scholar 

  19. Yang, M.M., Kim, D.J., Alexe, M.: Flexo-photovoltaic effect. Science 360, 904–907 (2020). https://doi.org/10.1126/science.aan3256

    Article  Google Scholar 

  20. Zou, H., Zhang, C., Xue, H., Wu, Z., Wang, Z.L.: Boosting the solar cell efficiency by flexophotovoltaic effect? ACS Nano 13, 12259–12267 (2019). https://doi.org/10.1021/acsnano.9b07222

    Article  Google Scholar 

  21. Zhao, M.H., Liu, X., Fan, C.Y., Lu, C.S., Wang, B.B.: Theoretical analysis on the extension of a piezoelectric semi-conductor nanowire: effects of flexoelectricity and strain gradient. J. Appl. Phys. 127, 085707 (2020). https://doi.org/10.1063/1.5131388

    Article  Google Scholar 

  22. Qu, Y.L., Jin, F., Yang, J.S.: Effects of mechanical fields on mobile charges in a composite beam of flexoelectric dielectrics and semiconductors. J. Appl. Phys. 127, 194502 (2020). https://doi.org/10.1063/5.0005124

    Article  Google Scholar 

  23. Wang, L.F., Liu, S.H., Feng, X.L., Zhang, C.L., Zhu, L.P., Zhai, J.Y., Qin, Y., Wang, Z.L.: Flexoelectronics of centrosymmetric semiconductors. Nat. Nanotechnol. 15, 661–667 (2020). https://doi.org/10.1038/s41565-020-0700-y

    Article  Google Scholar 

  24. Zhang, R., Liang, X., Shen, S.P.: A Timoshenko dielectric beam model with flexoelectric effect. Meccanica 51, 1181–1188 (2015). https://doi.org/10.1007/s11012-015-0290-1

    Article  MathSciNet  MATH  Google Scholar 

  25. Deng, Q., Kammoun, M., Erturk, A., Sharma, P.: Nanoscale flexoelectric energy harvesting. Int. J. Solids Struct. 51, 3218–3225 (2014). https://doi.org/10.1016/j.ijsolstr.2014.05.018

    Article  Google Scholar 

  26. Hu, Y.T., Wang, J.N., Yang, F., Xue, H., Hu, H.P., Wang, J.: The effect of first-order strain gradient in micro piezoelectric-bimorph power harvester. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 58, 849–852 (2011). https://doi.org/10.1109/tuffc.2011.1878

    Article  Google Scholar 

  27. Zhou, Z.D., Yang, C.P., Su, Y.X., Huang, R., Lin, X.L.: Electromechanical coupling in piezoelectric nanobeams due to the flexoelectric effect. Smart Mater. Struct. 26, 095025 (2017). https://doi.org/10.1088/1361-665x/aa7936

    Article  Google Scholar 

  28. Dhaba, A.E., Gabr, M.E.: Flexoelectric effect induced in an anisotropic bar with cubic symmetry under torsion. Math. Mech. Solids 25, 820–837 (2020). https://doi.org/10.1177/1081286519895569

    Article  MathSciNet  MATH  Google Scholar 

  29. Jordi, M.M.: Flexoelectricity in nanobeams under torsion. Bachelor thesis, UPC (2016). http://www.hdl.handle.net/2117/91478

  30. Tagantsev, A.K.: Theory of flexoelectric effect in crystals. Sov. Phys. JETP 61, 1246–1254 (1985)

    Google Scholar 

  31. Tagantsev, A.K., Meunier, V., Sharma, P.: Novel electromechanical phenomena at the nanoscale: phenomenological theory and atomistic modeling. MRS Bull. 34, 643–647 (2009). https://doi.org/10.1557/mrs2009.175

    Article  Google Scholar 

  32. Maranganti, R., Sharma, N.D., Sharma, P.: Electromechanical coupling in nonpiezoelectric materials due to nanoscale nonlocal size effects: Green’s function solutions and embedded inclusions. Phys. Rev. B (2006). https://doi.org/10.1103/physrevb.74.014110

    Article  Google Scholar 

  33. Shen, S., Hu, S.: A theory of flexoelectricity with surface effect for elastic dielectrics. J. Mech. Phys. Solids 58, 665–677 (2010). https://doi.org/10.1016/j.jmps.2010.03.001

    Article  MathSciNet  MATH  Google Scholar 

  34. Xu, L., Shen, S.S.: Size-dependent piezoelectricity and elasticity due to the electric field-strain gradient coupling and strain gradient elasticity. Int. J. Appl. Mech. 05, 1350015 (2013). https://doi.org/10.1142/s1758825113500154

    Article  Google Scholar 

  35. Pierret, R.F.: Semiconductor Device Fundamentals. Pearson, Uttar Pradesh (1996)

    Google Scholar 

  36. Sze, S.M.: Physics of Semiconductor Devices. Wiley, New York (1981)

    Google Scholar 

  37. Bleustein, J.L., Stanley, R.: A dynamical theory of torsion. Int. J. Solids Struct. 6, 569–586 (1970). https://doi.org/10.1016/0020-7683(70)90031-4

    Article  MATH  Google Scholar 

  38. Dokmeci, M.C.: A theory of high frequency vibrations of piezoelectric crystal bars. Int. J. Solids Struct. 10, 401–409 (1974). https://doi.org/10.1016/0020-7683(74)90109-7

    Article  MATH  Google Scholar 

  39. Mindlin, R.D.: Low frequency vibrations of elastic bars. Int. J. Solids Struct. 12, 27–49 (1976). https://doi.org/10.1016/0020-7683(76)90071-8

    Article  MATH  Google Scholar 

  40. Chou, C.S., Yang, J.W., Huang, Y.C., Yang, H.J.: Analysis on vibrating piezoelectric beam gyroscope. Int. J. Appl. Electromagn. 2, 227–241 (1991)

    Google Scholar 

  41. Yang, J.S.: Equations for the extension and flexure of a piezoelectric beam with rectangular cross section and applications. Int. J. Appl. Electromagn. 9, 409–420 (1998). https://doi.org/10.3233/jaem-1998-121

    Article  Google Scholar 

  42. Li, P., Jin, F., Ma, J.: One-dimensional dynamic equations of a piezoelectric semiconductor beam with a rectangular cross section and their application in static and dynamic characteristic analysis. Appl. Math. Mech. 39, 685–702 (2018). https://doi.org/10.1007/s10483-018-2325-6

    Article  MathSciNet  MATH  Google Scholar 

  43. Shu, L.L., Wei, X.Y., Pang, T., Yao, X., Wang, C.L.: Symmetry of flexoelectric coefficients in crystalline medium. J. Appl. Phys. 110(10), 53 (2011). https://doi.org/10.1063/1.3662196

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 12072253), 111 Project version 2.0, and the Fundamental Research Funds for the Central Universities (xzy022020016).

Author information

Authors and Affiliations

  1. State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, China

    Yilin Qu & Feng Jin

  2. Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE, 68588-0526, USA

    Jiashi Yang

Authors
  1. Yilin Qu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Feng Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jiashi Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Feng Jin or Jiashi Yang.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qu, Y., Jin, F. & Yang, J. Torsion of a flexoelectric semiconductor rod with a rectangular cross section. Arch Appl Mech 91, 2027–2038 (2021). https://doi.org/10.1007/s00419-020-01867-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-020-01867-0

Keywords

Search

Navigation