Skip to main content
SpringerLink
Log in

Size-dependent effect of the flexoelectronics in a composite beam

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

In this paper, a sandwich composite structure consisting of two piezoelectric layers and a flexoelectric semiconductor layer is proposed, where piezoelectric layers are poled in opposite direction, either tail-to-tail (T–T) or head-to-head (H–H). The macroscopic theory for piezoelectric and flexoelectric semiconductors is utilized to establish the field equations and constitutive relations of the composite beam. For the composite beam with T–T design, the electromechanical behaviors of the flexoelectric semiconductor exhibit a significant size-dependent effect. In this design, the piezoelectric field competes with the flexoelectric field, resulting in a net electric field reversal at a certain thickness. For the composite beam with H–H design, the electromechanical response of the semiconductor is enhanced due to the synergistic interaction of these two polarization fields. Furthermore, this study shows that flexoelectric-like effect from piezoelectric mimicry can induce highly appreciable electromechanical responses at both macroscopic and microscopic scales. It could be foreseen that this research provides fundamental guidance for the structural design of flexoelectric-like semiconductor devices.

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
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Yudin, P.V., Tagantsev, A.K.: Fundamentals of flexoelectricity in solids. Nanotechnology 24(43), 432001 (2013)

    CAS  PubMed  ADS  Google Scholar 

  2. Huang, S., Qi, L., Huang, W., Shu, L., Zhou, S., Jiang, X.: Flexoelectricity in dielectrics: materials, structures and characterizations. J. Adv. Dielectr. 08(02), 1830002 (2018)

    CAS  ADS  Google Scholar 

  3. Zubko, P., Catalan, G., Tagantsev, A.K.: Flexoelectric effect in solids. Annu. Rev. Mater. Res. 43(1), 387–421 (2013)

    CAS  ADS  Google Scholar 

  4. Shayeganfar, F., Torkashvand, Z., Mirabbaszadeh, K., Shahsavari, R.: Flexoelectric effects in corrugated boron nitride nanoribbons. J. Electron. Mater. 48(7), 4515–4523 (2019)

    CAS  ADS  Google Scholar 

  5. Mela, I., Poudel, C., Anaya, M., Delport, G., Frohna, K., Macpherson, S., Doherty, T.A.S., Scheeder, A., Stranks, S.D., Kaminski, C.F.: Revealing nanomechanical domains and their transient behavior in mixed-halide perovskite films. Adv. Funct. Mater. 31(23), 2100293 (2021)

    CAS  Google Scholar 

  6. Maier, F.J., Schneider, M., Schrattenholzer, J., Artner, W., Hradil, K., Artemenko, A., Kromka, A., Schmid, U.: Flexoelectricity in polycrystalline TiO2 thin films. Acta Mater. 190, 124–129 (2020)

    CAS  ADS  Google Scholar 

  7. Celano, U., Popovici, M., Florent, K., Lavizzari, S., Favia, P., Paulussen, K., Bender, H., di Piazza, L., Van Houdt, J., Vandervorst, W.: The flexoelectric effect in Al-doped hafnium oxide. Nanoscale 10(18), 8471–8476 (2018)

    CAS  PubMed  Google Scholar 

  8. Brennan, C.J., Koul, K., Lu, N., Yu, E.T.: Out-of-plane electromechanical coupling in transition metal dichalcogenides. Appl. Phys. Lett. 116(5), 053101 (2020)

    CAS  ADS  Google Scholar 

  9. Nguyen, B.H., Zhuang, X., Rabczuk, T.: NURBS-based formulation for nonlinear electro-gradient elasticity in semiconductors. Comput. Meth. Appl. Mech. Eng. 346, 1074–1095 (2019)

    MathSciNet  ADS  Google Scholar 

  10. Morozovska, A.N., Vasudevan, R.K., Maksymovych, P., Kalinin, S.V., Eliseev, E.A.: Anisotropic conductivity of uncharged domain walls in BiFeO3. Phys. Rev. B 86(8), 085315 (2012)

    ADS  Google Scholar 

  11. Eliseev, E.A., Morozovska, A.N., Svechnikov, G.S., Maksymovych, P., Kalinin, S.V.: Domain wall conduction in multiaxial ferroelectrics. Phys. Rev. B 85(4), 045312 (2012)

    ADS  Google Scholar 

  12. Catalan, G., Lubk, A., Vlooswijk, A.H., Snoeck, E., Magen, C., Janssens, A., Rispens, G., Rijnders, G., Blank, D.H., Noheda, B.: Flexoelectric rotation of polarization in ferroelectric thin films. Nat. Mater. 10(12), 963–967 (2011)

    CAS  PubMed  ADS  Google Scholar 

  13. Tan, D., Willatzen, M., Wang, Z.L.: Out-of-plane polarization in bent graphene-like zinc oxide and nanogenerator applications. Adv. Funct. Mater. 30(5), 1907885 (2019)

    Google Scholar 

  14. Oshman, C., Opoku, C., Dahiya, A.S., Alquier, D., Camara, N., Poulin-Vittrant, G.: Measurement of spurious voltages in ZnO piezoelectric nanogenerators. J. Microelectromech. Syst. 25(3), 533–541 (2016)

    CAS  Google Scholar 

  15. Kumar, M., Lim, J., Park, J.Y., Seo, H.: Controllable, self-powered, and high-performance short-wavelength infrared photodetector driven by coupled flexoelectricity and strain effect. Small Methods 5(7), e2100342 (2021)

    PubMed  Google Scholar 

  16. Hirakata, H., Fukuda, Y., Shimada, T.: Flexoelectric properties of multilayer two-dimensional material MoS2. J. Phys. D-Appl. Phys. 55(12), 125302 (2021)

    ADS  Google Scholar 

  17. Kumar, M., Lim, J., Park, J.-Y., Seo, H.: Flexoelectric effect driven colossal triboelectricity with multilayer graphene. Curr. Appl. Phys. 32, 59–65 (2021)

    ADS  Google Scholar 

  18. Morozovska, A.N., Eliseev, E.A., Shevliakova, H.V., Lopatina, Y.Y., Dovbeshko, G.I., Glinchuk, M.D., Kim, Y., Kalinin, S.V.: Correlation between corrugation-induced flexoelectric polarization and conductivity of low-dimensional Transition metal dichalcogenides. Phys. Rev. Appl. 15(4), 044051 (2021)

    CAS  ADS  Google Scholar 

  19. Olson, K.P., Mizzi, C.A., Marks, L.D.: Band bending and ratcheting explain triboelectricity in a flexoelectric contact diode. Nano Lett. 22(10), 3914–3921 (2022)

    CAS  PubMed  ADS  Google Scholar 

  20. Pandey, T., Covaci, L., Milošević, M.V., Peeters, F.M.: Flexoelectricity and transport properties of phosphorene nanoribbons under mechanical bending. Phys. Rev. B 103(23), 235406 (2021)

    CAS  ADS  Google Scholar 

  21. Serry, M., Sakr, M.A.: Graphene-metal-semiconductor composite structure for multimodal energy conversion. Sens. Actuators A: Phys. 245, 169–179 (2016)

    CAS  Google Scholar 

  22. Chaudhary, P., Lu, H., Loes, M., Lipatov, A., Sinitskii, A., Gruverman, A.: Mechanical stress modulation of resistance in MoS(2) junctions. Nano Lett. 22(3), 1047–1052 (2022)

    CAS  PubMed  ADS  Google Scholar 

  23. Wang, L., Liu, S., Feng, X., Zhang, C., Zhu, L., Zhai, J., Qin, Y., Wang, Z.L.: Flexoelectronics of centrosymmetric semiconductors. Nat. Nanotechnol. 15(8), 661–667 (2020)

    CAS  PubMed  ADS  Google Scholar 

  24. Narvaez, J., Vasquez-Sancho, F., Catalan, G.: Enhanced flexoelectric-like response in oxide semiconductors. Nature 538(7624), 219–221 (2016)

    CAS  PubMed  ADS  Google Scholar 

  25. Yang, G., Du, J., Wang, J., Yang, J.: Electromechanical fields in a nonuniform piezoelectric semiconductor rod. J. Mech. Mater. Struct. 13(1), 103–120 (2018)

    MathSciNet  Google Scholar 

  26. Yang, G., Du, J., Wang, J., Yang, J.: Electromechanical fields in PN junctions with continuously graded doping in piezoelectric semiconductor rods. Arch. Appl. Mech. 92(1), 325–333 (2022)

    ADS  Google Scholar 

  27. Fan, S., Hu, Y., Yang, J.: Stress-induced potential barriers and charge distributions in a piezoelectric semiconductor nanofiber. Appl. Math. Mech.-Engl. Ed. 40(5), 591–600 (2019)

    MathSciNet  Google Scholar 

  28. Ramirez, F., Heyliger, P.R., Pan, E.: Free vibration response of two-dimensional magneto-electro-elastic laminated plates. J. Sound Vibr. 292(3–5), 626–644 (2006)

    ADS  Google Scholar 

  29. Guo, M., Li, Y., Qin, G., Zhao, M.: Nonlinear solutions of PN junctions of piezoelectric semiconductors. Acta Mech. 230(5), 1825–1841 (2019)

    MathSciNet  Google Scholar 

  30. Luo, Y., Zhang, C., Chen, W., Yang, J.: Piezopotential in a bended composite fiber made of a semiconductive core and of two piezoelectric layers with opposite polarities. Nano Energy 54, 341–348 (2018)

    CAS  Google Scholar 

  31. Fang, K., Qian, Z., Yang, J.: Piezopotential in a composite cantilever of piezoelectric dielectrics and nonpiezoelectric semiconductors produced by shear force through e15. Mater. Res. Express 6(11), 115917 (2019)

    ADS  Google Scholar 

  32. Cheng, R., Zhang, C., Chen, W., Yang, J.: Piezotronic effects in the extension of a composite fiber of piezoelectric dielectrics and nonpiezoelectric semiconductors. J. Appl. Phys. 124(6), 064506 (2018)

    ADS  Google Scholar 

  33. Liang, Y., Yang, W., Yang, J.: Transient bending vibration of a piezoelectric semiconductor nanofiber under a suddenly applied shear force. Acta Mech. Solida Sin. 32(6), 688–697 (2019)

    Google Scholar 

  34. Dai, X., Zhu, F., Qian, Z., Yang, J.: Electric potential and carrier distribution in a piezoelectric semiconductor nanowire in time-harmonic bending vibration. Nano Energy 43, 22–28 (2018)

    CAS  Google Scholar 

  35. Wang, G., Liu, J., Liu, X., Feng, W., Yang, J.: Extensional vibration characteristics and screening of polarization charges in a ZnO piezoelectric semiconductor nanofiber. J. Appl. Phys. 124(9), 094502 (2018)

    ADS  Google Scholar 

  36. Zhang, C.L., Wang, X.Y., Chen, W.Q., Yang, J.S.: Propagation of extensional waves in a piezoelectric semiconductor rod. AIP Adv. 6(4), 045301 (2016)

    ADS  Google Scholar 

  37. Yang, W., Hu, Y., Yang, J.: Transient extensional vibration in a ZnO piezoelectric semiconductor nanofiber under a suddenly applied end force. Mater. Res. Express 6(2), 025902 (2018)

    ADS  Google Scholar 

  38. Cheng, R., Zhang, C., Chen, W., Yang, J.: Electrical behaviors of a piezoelectric semiconductor fiber under a local temperature change. Nano Energy 66, 104081 (2019)

    CAS  Google Scholar 

  39. Cheng, R., Zhang, C., Yang, J.: Thermally induced carrier distribution in a piezoelectric semiconductor fiber. J. Electron. Mater. 48(8), 4939–4946 (2019)

    CAS  ADS  Google Scholar 

  40. Luo, Y., Zhang, C., Chen, W., Yang, J.: Thermally induced electromechanical fields in unimorphs of piezoelectric dielectrics and nonpiezoelectric semiconductors. Integr. Ferroelectr. 211(1), 117–131 (2020)

    CAS  ADS  Google Scholar 

  41. Chu, L., Dui, G., Mei, H., Liu, L., Li, Y.: An analysis of flexoelectric coupling associated electroelastic fields in functionally graded semiconductor nanobeams. J. Appl. Phys. 130(11), 115701 (2021)

    CAS  ADS  Google Scholar 

  42. Li, H., Chu, L., Li, Y., Dui, G., Deng, Q.: Study on PN heterojunctions associated bending coupling in flexoelectric semiconductor composites considering the effects of size-dependent and symmetry-breaking. J. Appl. Phys. 132(12), 125701 (2022)

    CAS  ADS  Google Scholar 

  43. Chen, Y., Yan, Z.: Investigation of pull-in behaviors of a nanoswitch tuned by piezoelectric and flexoelectric effects. Int. J. Mech. Sci. 161, 105032 (2019)

    Google Scholar 

  44. Ren, C., Wang, K.F., Wang, B.L.: Adjusting the electromechanical coupling behaviors of piezoelectric semiconductor nanowires via strain gradient and flexoelectric effects. J. Appl. Phys. 128(21), 215701 (2020)

    CAS  ADS  Google Scholar 

  45. Zhao, L., Jin, F.: The adjustment of electro-elastic properties in non-uniform flexoelectric semiconductor nanofibers. Acta Mech. 234(3), 975–990 (2022)

    MathSciNet  Google Scholar 

  46. Qu, Y., Jin, F., Yang, J.: Torsion of a flexoelectric semiconductor rod with a rectangular cross section. Arch. Appl. Mech. 91(5), 2027–2038 (2021)

    ADS  Google Scholar 

  47. Zhang, G.Y., Guo, Z.W., Qu, Y.L., Gao, X.L., Jin, F.: A new model for thermal buckling of an anisotropic elastic composite beam incorporating piezoelectric, flexoelectric and semiconducting effects. Acta Mech. 233(5), 1719–1738 (2022)

    MathSciNet  Google Scholar 

  48. Qu, Y., Jin, F., Yang, J.: Magnetically induced charge redistribution in the bending of a composite beam with flexoelectric semiconductor and piezomagnetic dielectric layers. J. Appl. Phys. 129(6), 064503 (2021)

    CAS  ADS  Google Scholar 

  49. Zhang, G.Y., Guo, Z.W., Qu, Y.L., Mi, C.W.: Global and local flexotronic effects induced by external magnetic fields in warping of a semiconducting composite fiber. Compos. Struct. 295, 115711 (2022)

    CAS  Google Scholar 

  50. Ghobadi, A., Beni, Y.T., Golestanian, H.: Size dependent thermo-electro-mechanical nonlinear bending analysis of flexoelectric nano-plate in the presence of magnetic field. Int. J. Mech. Sci. 152, 118–137 (2019)

    Google Scholar 

  51. Qu, Y., Pan, E., Zhu, F., Jin, F., Roy, A.K.: Modeling thermoelectric effects in piezoelectric semiconductors: new fully coupled mechanisms for mechanically manipulated heat flux and refrigeration. Int. J. Eng. Sci. 182, 103775 (2023)

    MathSciNet  Google Scholar 

  52. Guo, Z., Qu, Y., Zhang, G., Mi, C.: Magnetically induced electromechanical fields in a flexoelectric composite microplate. Math. Mech. Solids 28(4), 1091–1110 (2022)

    MathSciNet  Google Scholar 

  53. Guinovart-Sanjuán, D., Vajravelu, K., Rodríguez-Ramos, R., Guinovart-Díaz, R., Bravo-Castillero, J., Lebon, F., Sabina, F.J., Merodio, J.: Effective predictions of heterogeneous flexoelectric multilayered composite with generalized periodicity. Int. J. Mech. Sci. 181, 105755 (2020)

    Google Scholar 

  54. Guo, D., Guo, P., Yao, Y., Ren, L., Jia, M., Wang, W., Wang, Y., Zhang, Y., Yu, A., Zhai, J.: Macroscopic flexotronics enhanced controllable piezotronic-like response by flexual semiconductor devices. Nano Energy 100, 107508 (2022)

    CAS  Google Scholar 

  55. Abdollahi, A., Vasquez-Sancho, F., Catalan, G.: Piezoelectric mimicry of flexoelectricity. Phys. Rev. Lett. 121(20), 205502 (2018)

    CAS  PubMed  ADS  Google Scholar 

  56. Zhang, X., Pan, Q., Tian, D., Zhou, W., Chen, P., Zhang, H., Chu, B.: Large flexoelectriclike response from the spontaneously polarized surfaces in ferroelectric ceramics. Phys. Rev. Lett. 121(5), 057602 (2018)

    PubMed  ADS  Google Scholar 

  57. Qu, Y., Jin, F., Yang, J.: Effects of mechanical fields on mobile charges in a composite beam of flexoelectric dielectrics and semiconductors. J. Appl. Phys. 127(19), 194502 (2020)

    CAS  ADS  Google Scholar 

  58. Mindlin, R.D., Deresiewicz, H.: Thickness-shear vibrations of piezoelectric crystal plates with incomplete electrodes. J. Appl. Phys. 25(1), 21–24 (1954)

    MathSciNet  ADS  Google Scholar 

  59. Mindlin, R.D.: Equations of high frequency vibrations of thermopiezoelectric crystal plates. Int. J. Solids Struct. 10(6), 625–637 (1973)

    Google Scholar 

  60. Mindlin, R.D.: An Introduction to the Mathematical Theory of Vibrations of Elastic Plates. World Scientific, Singapore (2006)

    Google Scholar 

  61. Auld, B.A.: Acoustic Fields and Waves in Solids. Wiley, New York (1974)

    Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge the support of the National Key Research and Development Program of China (2021YFB2011400) and National Natural Science Foundation of China (No. 51975065).

Funding

This study was funded by National Key Research and Development Program of China (2021YFB2011400) and National Natural Science Foundation of China (51975065).

Author information

Authors and Affiliations

  1. State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing, 400044, China

    Chao Wei, Jian Tang & Wenbin Huang

Authors
  1. Chao Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jian Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wenbin Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wenbin Huang.

Ethics declarations

Conflict of interest

The authors declare that they have 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

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

Wei, C., Tang, J. & Huang, W. Size-dependent effect of the flexoelectronics in a composite beam. Acta Mech 235, 925–939 (2024). https://doi.org/10.1007/s00707-023-03777-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-023-03777-y

Search

Navigation