Abstract
In this paper, we develop a nonlinear mixed finite element method for flexoelectric semiconductors and analyze the mechanically tuned redistributions of free carriers and electric currents through flexoelectric polarization in typical structures. We first present a macroscopic theory for flexoelectric semiconductors by combining flexoelectricity and nonlinear drift-diffusion theory. To use C0 continuous elements, we derive an incremental constrained weak form by introducing Langrage multipliers, in which the kinematic constraints between the displacement and its gradient are guaranteed. Based on the weak form, we established a mixed C0 continuous nine-node quadrilateral finite element as well as an iterative process for solving nonlinear boundary-value problems. The accuracy and convergence of the proposed element are validated by comparing linear finite element method results against analytical solutions for the bending of a beam. Finally, the nonlinear element method is applied to more complex problems, such as a circular ring, a plate with a hole, and an isosceles trapezoid. Results indicate that mechanical loads and doping levels have distinct influences on electric properties.
Graphical Abstract Figure
References
1.
Li
, Q.
, Li
, S.
, Zhou
, L.
, Cao
, X.
, Lan
, Y.
, Xu
, X.
, Huang
, Y.
, et al, 2023
, “Optimal Design Towards High Performance of Sandwich Flexible Piezoelectric Energy Harvesters
,” ASME J. Appl. Mech.
, 90
(6
), p. 061007
. 2.
Qi
, L.
, Shi
, Y.
, Hunag
, C.
, Li
, Z.
, and Gao
, C.-F.
, 2021
, “Electrical-Driven Delamination of Flexible Piezoelectric System and Its Applications in Surface Energy Testing
,” ASME J. Appl. Mech.
, 88
(6
), p. 061007
. 3.
Jamneala
, T.
, Bradley
, P.
, Shirakawa
, A.
, Thalhammer
, R. K.
, and Ruby
, R.
, 2016
, “An Investigation of Lateral Modes in FBAR Resonators
,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control
, 63
(5
), pp. 778
–789
. 4.
Li
, N.
, Qian
, Z.
, and Yang
, J.
, 2017
, “Two-Dimensional Equations for Piezoelectric Thin-Film Acoustic Wave Resonators
,” Int. J. Solids Struct.
, 110–111
, pp. 170
–177
. 5.
Yang
, J.
, 2005
, “A Review of Analyses Related to Vibrations of Rotating Piezoelectric Bodies and Gyroscopes
,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control
, 52
(5
), pp. 698
–706
. 6.
Yang
, J.
, 2006
, Analysis of Piezoelectric Devices
, World Scientific
, Singapore
.7.
Maranganti
, R.
, Sharma
, N. D.
, and Sharma
, P.
, 2006
, “Electromechanical Coupling in Nonpiezoelectric Materials Due to Nanoscale Nonlocal Size Effects: Green’s Function Solutions and Embedded Inclusions
,” Phys. Rev. B
, 74
(1
), p. 014110
. 8.
Shen
, S.
, and Hu
, S.
, 2010
, “A Theory of Flexoelectricity With Surface Effect for Elastic Dielectrics
,” J. Mech. Phys. Solids
, 58
(5
), pp. 665
–677
. 9.
Liu
, Y.
, Chen
, L.
, Wang
, B.
, Yang
, S.
, and Sharma
, P.
, 2021
, “Tuning Crumpled Sheets for an Enhanced Flexoelectric Response
,” ASME J. Appl. Mech.
, 89
(1
), p. 011011
. 10.
Qu
, Y. L.
, Guo
, Z. W.
, Zhang
, G. Y.
, Gao
, X.-L.
, and Jin
, F.
, 2022
, “A New Model for Circular Cylindrical Kirchhoff–Love Shells Incorporating Microstructure and Flexoelectric Effects
,” ASME J. Appl. Mech.
, 89
(12
), p. 121010
. 11.
Wang
, Z. L.
, and Song
, J.
, 2006
, “Piezoelectric Nanogenerators Based on Zinc Oxide Nanowire Arrays
,” Science
, 312
(5771
), pp. 242
–246
. 12.
Jiang
, X.
, Huang
, W.
, and Zhang
, S.
, 2013
, “Flexoelectric Nano-generator: Materials, Structures and Devices
,” Nano Energy
, 2
(6
), pp. 1079
–1092
. 13.
Hickernell
, F. S.
, 2005
, “The Piezoelectric Semiconductor and Acoustoelectronic Device Development in the Sixties
,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control
, 52
(5
), pp. 737
–745
. 14.
15.
Luo
, Y.
, Zhang
, C.
, Chen
, W.
, and Yang
, J.
, 2019
, “Piezotronic Effect of a Thin Film With Elastic and Piezoelectric Semiconductor Layers Under a Static Flexural Loading
,” ASME J. Appl. Mech.
, 86
(5
), p. 051003
. 16.
Qu
, Y.
, Jin
, F.
, and Yang
, J.
, 2022
, “Electromechanical Interactions in a Composite Plate With Piezoelectric Dielectric and Nonpiezoelectric Semiconductor Layers
,” Acta Mech.
, 233
(9
), pp. 3795
–3812
. 17.
Qu
, Y.
, Pan
, E.
, Zhu
, F.
, Jin
, F.
, and Roy
, A. K.
, 2023
, “Modeling Thermoelectric Effects in Piezoelectric Semiconductors: New Fully Coupled Mechanisms for Mechanically Manipulated Heat Flux and Refrigeration
,” Int. J. Eng. Sci.
, 182
, p. 103775
. 18.
Gokhale
, V. J.
, and Rais-Zadeh
, M.
, 2014
, “Phonon–Electron Interactions in Piezoelectric Semiconductor Bulk Acoustic Wave Resonators
,” Sci. Rep.
, 4
(1
), p. 5617
. 19.
Xu
, C.
, Wei
, P.
, Wei
, Z.
, and Guo
, X.
, 2023
, “Rayleigh Wave in Layered Piezoelectric Semiconductor With Consideration of PN Junction Effects
,” Math. Mech. Solids
, 28
(8
), pp. 1817
–1833
. 20.
Wang
, L.
, Liu
, S.
, Feng
, X.
, Zhang
, C.
, Zhu
, L.
, Zhai
, J.
, Qin
, Y.
, and Wang
, Z. L.
, 2020
, “Flexoelectronics of Centrosymmetric Semiconductors
,” Nat. Nanotechnol.
, 15
(8
), pp. 661
–667
. 21.
Qu
, Y.
, Jin
, F.
, and Yang
, J.
, 2021
, “Buckling of Flexoelectric Semiconductor Beams
,” Acta Mech.
, 232
(7
), pp. 2623
–2633
. 22.
Qu
, Y.
, Jin
, F.
, and Yang
, J.
, 2022
, “Bending of a Flexoelectric Semiconductor Plate
,” Acta Mech. Sol. Sin.
, 35
(3
), pp. 434
–445
. 23.
Sun
, L.
, Zhu
, L.
, Zhang
, C.
, Chen
, W.
, and Wang
, Z.
, 2021
, “Mechanical Manipulation of Silicon-Based Schottky Diodes Via Flexoelectricity
,” Nano Energy
, 83
, p. 105855
. 24.
Ren
, C.
, Wang
, K. F.
, and Wang
, B. L.
, 2021
, “Analysis of Piezoelectric PN Homojunction and Heterojunction Considering Flexoelectric Effect and Strain Gradient
,” J. Phys. D: Appl. Phys.
, 54
(49
), p. 495102
. 25.
Abdollahi
, A.
, Peco
, C.
, Millán
, D.
, Arroyo
, M.
, and Arias
, I.
, 2014
, “Computational Evaluation of the Flexoelectric Effect in Dielectric Solids
,” J. Appl. Phys.
, 116
(9
), p. 093502
. 26.
Thai
, T. Q.
, Rabczuk
, T.
, and Zhuang
, X.
, 2018
, “A Large Deformation Isogeometric Approach for Flexoelectricity and Soft Materials
,” Comput. Methods Appl. Mech. Eng.
, 341
, pp. 718
–739
. 27.
Zhuang
, X.
, Nguyen
, B. H.
, Nanthakumar
, S. S.
, Tran
, T. Q.
, Alajlan
, N.
, and Rabczuk
, T.
, 2020
, “Computational Modeling of Flexoelectricity—A Review
,” Energies
, 13
(6
), p. 1326
. 28.
Amanatidou
, E.
, and Aravas
, N.
, 2002
, “Mixed Finite Element Formulations of Strain-Gradient Elasticity Problems
,” Comput. Methods Appl. Mech. Eng.
, 191
(15
), pp. 1723
–1751
. 29.
Mao
, S.
, Purohit
, P. K.
, and Aravas
, N.
, 2016
, “Mixed Finite-Element Formulations in Piezoelectricity and Flexoelectricity
,” Proc. R. Soc. A
, 472
(2190
), p. 20150879
. 30.
Deng
, F.
, Deng
, Q.
, Yu
, W.
, and Shen
, S.
, 2017
, “Mixed Finite Elements for Flexoelectric Solids
,” ASME J. Appl. Mech.
, 84
(8
), p. 081004
. 31.
Tian
, X.
, Sladek
, J.
, Sladek
, V.
, Deng
, Q.
, and Li
, Q.
, 2021
, “A Collocation Mixed Finite Element Method for the Analysis of Flexoelectric Solids
,” Int. J. Solids Struct.
, 217–218
, pp. 27
–39
. 32.
Tian
, X.
, Zhou
, H.
, Deng
, Q.
, Yan
, Z.
, Sladek
, J.
, and Sladek
, V.
, 2024
, “Modeling the Flexoelectric Effect in Semiconductors Via a Second-Order Collocation MFEM
,” Int. J. Mech. Sci.
, 264
, p. 108837
. 33.
Li
, Y.
, Zhou
, Q.
, Wu
, J.
, Xu
, J.
, Shi
, W.
, Su
, C.
, Chen
, D.
, and Shao
, Z.
, 2022
, “Controllably Grown Single-Crystal Films as Flexoelectric Nanogenerators for Continuous Direct Current Output
,” Npj Flex. Electron.
, 6
(1
), p. 88
. 34.
Guo
, D.
, Guo
, P.
, Ren
, L.
, Yao
, Y.
, Wang
, W.
, Jia
, M.
, Wang
, Y.
, Wang
, L.
, Wang
, Z. L.
, and Zhai
, J.
, 2023
, “Silicon Flexoelectronic Transistors
,” Sci. Adv.
, 9
(10
), p. eadd3310
. 35.
Zhao
, M.
, Yang
, C.
, Fan
, C.
, and Zhang
, Q.
, 2020
, “A Shooting Method for Nonlinear Boundary Value Problems in a Thermal Piezoelectric Semiconductor Plate
,” Z. Angew. Math. Mech.
, 100
(12
), p. e201900302
. 36.
Zhao
, M.
, Zhang
, Q.
, and Fan
, C.
, 2019
, “An Efficient Iteration Approach for Nonlinear Boundary Value Problems in 2D Piezoelectric Semiconductors
,” Appl. Math. Model.
, 74
, pp. 170
–183
. 37.
Zhao
, L.
, Jin
, F.
, Shao
, Z.
, and Wang
, W.
, 2023
, “Nonlinear Analysis on Electrical Properties in a Bended Composite Piezoelectric Semiconductor Beam
,” Appl. Math. Mech.-Engl. Ed.
, 44
(12
), pp. 2039
–2056
. 38.
Han
, C.
, Lu
, C.
, Zhao
, M.
, and Zhang
, Q.
, 2023
, “Nonlinear Finite Element Analysis of Electromechanical Behaviors in a Piezoelectric Semiconductor Beam
,” Int. J. Non-Linear Mech.
, 149
, p. 104311
. 39.
Auld
, B. A.
, 1973
, Acoustic Fields and Waves in Solids
, Рипол Классик
, New York
.40.
Pierret
, R. F.
, 1996
, Semiconductor Device Fundamentals
, Pearson Education India
, Uttar Pradesh, India
.41.
Gao
, X.-L.
, and Park
, S. K.
, 2007
, “Variational Formulation of a Simplified Strain Gradient Elasticity Theory and Its Application to a Pressurized Thick-Walled Cylinder Problem
,” Int. J. Solids Struct.
, 44
(22
), pp. 7486
–7499
. 42.
Yang
, J.
, 2020
, Analysis of Piezoelectric Semiconductor Structures
, Springer
, New York
.43.
Reddy
, J. N.
, 2019
, Introduction to the Finite Element Method
, McGraw-Hill Education
, New York
.44.
Wu
, H.-P.
, Shang
, Y.
, Cen
, S.
, and Li
, C.-F.
, 2023
, “Penalty C0 8-Node Quadrilateral and 20-Node Hexahedral Elements for Consistent Couple Stress Elasticity Based on the Unsymmetric Finite Element Method
,” Eng. Anal. Bound. Elem.
, 147
, pp. 302
–319
. 45.
Wang
, X.
, 2003
, Finite Element Method
, Tsinghua University Press
, Beijing
.46.
Bathe
, K.-J.
, 2006
, Finite Element Procedures
, Klaus-Jurgen Bathe
, New Jersey
.47.
Lazar
, M.
, Maugin
, G. A.
, and Aifantis
, E. C.
, 2006
, “Dislocations in Second Strain Gradient Elasticity
,” Int. J. Solids Struct.
, 43
(6
), pp. 1787
–1817
. 48.
Sze
, S. M.
, Li
, Y.
, and Ng
, K. K.
, 2021
, Physics of Semiconductor Devices
, John Wiley & Sons
, New York
.49.
Soh
, A.-K.
, and Wanji
, C.
, 2004
, “Finite Element Formulations of Strain Gradient Theory for Microstructures and the C0–1 Patch Test
,” Int. J. Numer. Methods Eng.
, 61
(3
), pp. 433
–454
. 
