A gradient theory is applied to the mechanical constitutive equations for piezoelectric semiconductor nanostructures. This is achieved by considering the strain gradients in the constitutive equation with high-order stresses and electric displacements in advanced continuum model. The C1 continuous interpolations of displacements or a mixed formulation is required in the finite element method (FEM) due to the presence of the second-order derivative on the elastic displacements. A mixed FEM is then developed from the principle of virtual work. Numerical examples clearly show the significant effect of flexoelectricity on the induced electric potential and electric current in the piezoelectric semiconductor nanostructures.
AgrawalREspinosaHD (2011) Giant piezoelectric size effects in zinc oxide and gallium nitride nanowires. A first principles investigation. Nano Letters11(2): 786–790.
2.
AuldBA (1973) Acoustic Fields and Waves in Solids. New York: John Wiley and Sons, pp. 357–382.
3.
BimbergDGrundmannMLedentsovNN (1999) Quantum Dot Heterostructures. New York: John Wiley & Sons.
4.
BühlmannSDwirBBaborowskiJ, et al. (2002) Size effect in mesoscopic epitaxial ferroelectric structures: Increase of piezoelectric response with decreasing feature size. Applied Physics Letters80(17): 3195–3197.
5.
DaiSGharbiMSharmaP, et al. (2011) Surface piezoelectricity: Size effects in nanostructures and the emergence of piezoelectricity in non-piezoelectric materials. Journal of Applied Physics110(10): 104305.
6.
DaiXZhuFQianZ, et al. (2018) Electric potential and carrier distribution in a piezoelectric semiconductor nanowire in time-harmonic bending vibration. Nano Energy43: 22–28.
7.
DengFDengQYuW, et al. (2017) Mixed finite elements for flexoelectric solids. Journal of Applied Mechanics84(8): 0810041–810112.
8.
DongLAtluriSN (2011) A simple procedure to develop efficient & stable hybrid/mixed elements, and voronoi cell finite elements for macro- & micromechanics. CMC: Computers, Materials & Continua24(1): 61–104.
9.
EbrahimiFBaratiMR (2017) Buckling analysis of piezoelectrically actuated smart nanoscale plates subjected to magnetic field. Journal of Intelligent Material Systems and Structures28(11): 1472–1490.
10.
FanSLiangYXieJ, et al. (2017) 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 Energy40: 82–87.
11.
GaoYWangZL (2009) Equilibrium potential of free charge carriers in a bent piezoelectric semiconductive nanowire. Nano Letters9(3): 1103–1110.
12.
GhasemiHParkHSRabczukT (2017) A level-set based IGA formulation for topology optimization of flexoelectric materials. Computer Methods in Applied Mechanics and Engineering313: 239–258.
13.
GhasemiHParkHSRabczukT (2018) A multi-material level set-based topology optimization of flexoelectric composites. Computer Methods in Applied Mechanics and Engineering332: 47–62.
14.
GitmanIMAskesHKuhlE, et al. (2010) Stress concentrations in fractured compact bone simulated with a special class of anisotropic gradient elasticity. International Journal of Solids and Structures47(9): 1099–1107.
15.
HuSLShenSP (2009) Electric field gradient theory with surface effect for nano-dielectrics. CMC: Computers, Materials & Continua13(1): 63–87.
16.
HutsonARWhiteDL (1962) Elastic wave propagation in piezoelectric semiconductors. Journal of Applied Physics33(1): 40–47.
17.
HuYZengYYangJ (2007) A mode III crack in a piezoelectric semiconductor of crystals with 6mm symmetry. International Journal of Solids and Structures44(11–12): 3928–3938.
18.
KundalwalSIShingareKBGuptaM (2020) Flexoelectric effect on electric potential in piezoelectric graphene-based composite nanowire: Analytical and numerical modelling. European Journal of Mechanics –A/Solids84: 104050.
19.
LiangXZhangRHuS, et al. (2017) Flexoelectric energy harvesters based on Timoshenko laminated beam theory. Journal of Intelligent Material Systems and Structures28(15): 2064–2073.
20.
LiangYFanSChenX, et al. (2018) Nonlinear effect of carrier drift on the performance of an n-type ZnO nanowire nanogenerator by coupling piezoelectric effect and semiconduction. Beilstein Journal of Nanotechnology9(1): 1917–1925.
21.
LiangYYangWYangJ (2019) Transient bending vibration of a piezoelectric semiconductor nanofiber under a suddenly applied shear force. Acta Mechanica Solida Sinica32(6): 688–697.
22.
LiTLiYTQinWW, et al. (2015) Piezoelectric size effects in a zinc oxide micropillar. Nanoscale Research Letters10(1): 394.
23.
LiZHeYLeiJ, et al. (2018) A standard experimental method for determining the material length scale based on modified couple stress theory. International Journal of Mechanical Sciences141: 198–205.
24.
LiZHeYZhangB, et al. (2019) Experimental investigation and theoretical modelling on nonlinear dynamics of cantilevered microbeams. European Journal of Mechanics –A/Solids78: 103834.
25.
LuoYZhangCChenW, et al. (2017) An analysis of PN junctions in piezoelectric semiconductors. Journal of Applied Physics122(20): 204502.
26.
MarangantiRSharmaNDSharmaP (2006) Electromechanical coupling in nonpiezoelectric materials due to nanoscale nonlocal size effects: Green’s function solutions and embedded inclusions. Physical Review B74(1): 014110.
27.
MorozovskaANEliseevEAGenenkoYA, et al. (2016) Flexocoupling impact on size effects of piezoresponse and conductance in mixed-type ferroelectric semiconductors under applied pressure. Physical Review B94(17): 174101.
28.
MouraAGErturkA (2018) Combined piezoelectric and flexoelectric effects in resonant dynamics of nanocantilevers. Journal of Intelligent Material Systems and Structures29(20): 3949–3959.
29.
NguyenBHNanthakumarSSZhuangX, et al. (2018) Dynamic flexoelectric effect on piezoelectric nanostructures. European Journal of Mechanics –A/Solids71: 404–409.
30.
NguyenBHZhuangXRabczukT (2019) NURBS-based formulation for nonlinear electro-gradient elasticity in semiconductors. Computer Methods in Applied Mechanics and Engineering346: 1074–1095.
31.
OatesWS (2017) Flexoelectricity, strain gradients, and singularities in ferroelectric nanostructures. Journal of Intelligent Material Systems and Structures28(20): 3091–3105.
32.
PanEAlbrechtJDZhangY (2007) Elastic and piezoelectric fields in quantum wire semiconductor structures – A boundary integral equation analysis. Physics Status Solida (b)244(6): 1925–1939.
33.
PanEZouYChungPW, et al. (2009) Strain-induced variations of electronic energy band edges of embedded semiconductor quantum dots in half-space substrates. Journal of Applied Physics106(7): 073504.
34.
PatronVZKudryavtsevBA (1988) Electromagnetoelasticity, Piezoelectrics and Electrically Conductive Solids. New York, NY: Gordon and Breach Science Publishers.
35.
SharmaJNSharmaKKKumarA (2011a) Modelling of acousto-diffusive surface waves in piezoelectric-semiconductor composite structures. Journal of Mechanics of Materials and Structures6(6): 791–812.
36.
SharmaJNSharmaKKKumarA (2011b) Acousto-diffusive waves in a piezoelectric-semiconductor-piezoelectric sandwich structure. World Journal of Mechanics01(05): 247–255.
37.
SladekJSladekVPanE, et al. (2014a) Dynamic anti-plane crack analysis in functional graded piezoelectric semiconductor crystals. CMES-Computer Modeling in Engineering & Sciences99(4): 273–296.
38.
SladekJSladekVPanE, et al. (2014b) Fracture analysis in piezoelectric semiconductors under a thermal load. Engineering Fracture Mechanics126: 27–39.
39.
SladekJBishayPLRepkaM, et al. (2018a) Analysis of quantum-dot systems under thermal loads based on gradient elasticity. Smart Materials and Structures27(9): 095009.
40.
SladekJSladekVJusM (2018b) The MLPG for crack analyses in composites with flexoelectricity effects. Composite Structures204: 105–113.
41.
SladekJSladekVBishayPL, et al. (2015) Influence of electric conductivity on intensity factors for cracks in functionally graded piezoelectric semiconductors. International Journal of Solids and Structures59: 79–89.
42.
SladekJSladekVStanakP, et al. (2017) Fracture mechanics analysis of size-dependent piezoelectric solids. International Journal of Solids and Structures113-114: 1–9.
43.
TagantsevAK (1986) Piezoelectricity and flexoelectricity in crystalline dielectrics. Physical Review B34(8): 5883–5889.
44.
TagantsevAKMeunierVSharmaP (2009) Novel electromechanical phenomena at the nanoscale: Phenomenological theory and atomistic modeling. MRS Bulletin34(9): 643–647.
45.
Tahaei YaghoubiSMousaviSMPaavolaJ (2017) Buckling of centrosymmetric anisotropic beam structures within strain gradient elasticity. International Journal of Solids and Structures109: 84–92.
46.
TianRLiuJPanE, et al. (2019) Some characteristics of elastic waves in a piezoelectric semiconductor plate. Journal of Applied Physics126(12): 125701.
47.
TianRLiuJPanE, et al. (2020) SH waves in multilayered piezoelectric semiconductor plates with imperfect interfaces. European Journal of Mechanics – A/Solids81: 103961.
48.
TianXSladekJSladekV, et al. (2021) A collocation mixed finite element method for the analysis of flexoelectric solids. International Journal of Solids and Structures217–218: 27–39.
49.
VinyasM (2021) Computational analysis of smart magneto-electro-elastic materials and structures: Review and classification. Archives of Computational Methods in Engineering28: 1205–1248.
50.
WangKFWangBL (2018) Electrostatic potential in a bent piezoelectric nanowire with consideration of size-dependent piezoelectricity and semiconducting characterization. Nanotechnology29(25): 255405.
51.
WangZPanXHeY, et al. (2015) Piezoelectric nanowires in energy harvesting applications. Advances in Materials Science and Engineering2015: 1–21.
52.
WhiteDL (1962) Amplification of ultrasonic waves in piezoelectric semiconductors. Journal of Applied Physics33(8): 2547–2554.
53.
WuWWangZL (2016) Piezotronics and piezo-phototronics for adaptive electronics and optoelectronics. Nature Reviews Materials1(7): 16031.
54.
YangBPanE (2003) Elastic fields of quantum dots in multilayered semiconductors: A novel Green’s function approach. ASME Journal of Applied Mechanics70(2): 161–168.
55.
YangJS (2006) The Mechanics of Piezoelectric Structures. World Scientific, New York. p.328.
56.
YangJSZhouHG (2005a) Amplification of acoustic waves in piezoelectric semiconductor plates. International Journal of Solids and Structures42(11–12): 3171–3183.
57.
YangJSZhouHG (2005b) Propagation and amplification of gap waves between a piezoelectric half-space and a semiconductor film. Acta Mechanica176(1–2): 83–93.
58.
YangJSZhouHG (2005c) Wave propagation in a piezoelectric ceramic plate sandwiched between two semiconductor layers. International Journal of Applied Electromagnetics and Mechanics22(1-2): 97–109.
59.
YangWHuYYangJ (2018b) Transient extensional vibration in a ZnO piezoelectric semiconductor nanofiber under a suddenly applied end force. Materials Research Express6(2): 025902.
60.
YangWHuYPanE (2019) Tuning electronic energy band in a piezoelectric semiconductor rod via mechanical loading. Nano Energy66: 104147.
61.
YudinPVTagantsevAK (2013) Fundamentals of flexoelectricity in solids. Nanotechnology24(43): 432001.
62.
ZhangCWangXChenW, et al. (2017) An analysis of the extension of a ZnO piezoelectric semiconductor nanofiber under an axial force. Smart Materials and Structures26(2): 025030.
63.
ZhangCLWangXYChenWQ, et al. (2018) Bending of a cantilever piezoelectric semiconductor fiber under an end force. In: Generalized Models and Non-classical Approaches in Complex Materials 2. Springer, Cham. pp.261–278.
64.
ZhaoMLiXLuC, et al. (2020a) Nonlinear analysis of a crack in 2D piezoelectric semiconductors with exact electric boundary conditions. Journal of Intelligent Material Systems and Structures32: 632–639.
65.
ZhaoMLiuXFanC, et al. (2020b) Theoretical analysis on the extension of a piezoelectric semi-conductor nanowire: Effects of flexoelectricity and strain gradient. Journal of Applied Physics127(8): 085707.
66.
ZhaoMPanYFanC, et al. (2016) Extended displacement discontinuity method for analysis of cracks in 2D piezoelectric semiconductors. International Journal of Solids and Structures94–95: 50–59.
67.
ZhaoMYangCFanC, et al. (2018) Extended displacement discontinuity method for analysis of penny-shaped cracks in three-dimensional thermal piezoelectric semiconductors. European Journal of Mechanics –A/Solids70: 23–36.
68.
ZhaoYZhouCZhaoM, et al. (2017) Penny-shaped cracks in three-dimensional piezoelectric semiconductors via Green’s functions of extended displacement discontinuity. Journal of Intelligent Material Systems and Structures28(13): 1775–1788.
