Analysis of the electromechanical coupling characteristics of piezoelectric semiconductor PN junction shell structures

doi:10.1007/s10483-025-3259-6

Abstract

Abstract:

Based on the nonlinear drift-diffusion (NLDD) model, the coupled behavior between the mechanical and electrical fields in piezoelectric semiconductor (PS) PN junctions under two typical loading conditions is investigated. The governing equations for the general shell structure of the PS PN junction are derived within the framework of virtual work principles and charge continuity conditions. The distributions of the electromechanical coupling field are obtained by the Fourier series expansion and the differential quadrature method (DQM), and the nonlinearity is addressed with the iterative method. Several numerical examples are presented to investigate the effects of mechanical loading on the charge carrier transport characteristics. It is found that the barrier height of the heterojunction can be effectively modulated by mechanical loading. Furthermore, a nonlinearity index is introduced to quantify the influence of nonlinearity in the model. It is noted that, when the concentration difference between the two sides is considerable, the nonlinear results differ significantly from the linear results, thereby necessitating the adoption of the NLDD model.

Key words: piezoelectric semiconductor (PS), PN junction, shell structures, nonlinear drift-diffusion (NLDD) model, potential barrier (well)

2010 MSC Number:

O472⁺.91

Tiqing WANG, Feng ZHU, Peng LI, Zelin XU, Tingfeng MA, I. KUZNETSOVA, Zhenghua QIAN. Analysis of the electromechanical coupling characteristics of piezoelectric semiconductor PN junction shell structures. Applied Mathematics and Mechanics (English Edition), 2025, 46(6): 1167-1186.

TrendMD

Figures/Tables 13

Fig. 1

Table 1

Material parameters of ZnO"

Parameter	ZnO	Parameter	ZnO	Parameter	ZnO
$c 11$ /GPa	209.7	$e 15 / (C ⋅ m − 2)$	$− 0.48$	$μ 11 / (m 2 ⋅ V − 1 ⋅ s − 1)$	1
$c 33$ /GPa	210.9	$e 31 / (C ⋅ m − 2)$	$− 0.537$	$D 11 / (m 2 ⋅ s − 1)$	0.026
$c 12$ /GPa	121.1	$e 33 / (C ⋅ m − 2)$	1.32	$n 0 l (p 0 r) / m − 3$	$1 × 1021$
$c 13$ /GPa	105.1	$ε 11 / (F ⋅ m − 1)$	$7.57 × 10 − 11$	$p 0 l (n 0 r) / m − 3$	$0.7 × 1021$
$c 44$ /GPa	42.47	$ε 33 / (F ⋅ m − 1)$	$9.03 × 10 − 11$	$q / C$	$1.602 × 10 − 19$

Table 1

Table 2

Values of φ (in mV) at (−θ/2, L/2) with different N and M"

$N$	$M = 5$	$M = 50$	$M = 100$
10	2.334	2.291	2.291
30	2.422	2.382	2.382
50	2.429	2.389	2.389
80	2.432	2.391	2.391

Table 2

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Fig. 11

References 54

[1]	WANG, Z. L. Nanopiezotronics. Advanced Materials, 19(6), 889–892 (2007)
[2]	QIN, Y., WANG, X., and WANG, Z. L. Microfibre-nanowire hybrid structure for energy scavenging. nature, 451(7180), 809–813 (2008)
[3]	WANG, X., SONG, J., LIU, J., and WANG, Z. L. Direct-current nanogenerator driven by ultrasonic waves. Science, 316(5821), 102–105 (2007)
[4]	LEE, E., PARK, J., YIM, M., KIM, Y., and YOON, G. Characteristics of piezoelectric ZnO/AlN-stacked flexible nanogenerators for energy harvesting applications. Applied Physics Letters, 106(2), 023901 (2015)
[5]	WANG, L. F., LIU, S. H., GAO, G. Y., PANG, Y. K., YIN, X., FENG, X. L., ZHU, L. P., BAI, Y., CHEN, L. B., XIAO, T. X., WANG, X. D., QIN, Y., and WANG, Z. L. Ultrathin piezotronic transistors with 2 nm channel lengths. ACS Nano, 12(5), 4903–4908 (2018)
[6]	ORSINI, A. and FALCONI, C. ZnO nanowires for feedback-assisted tuning of electromechanical resonators. ACS Applied Nano Materials, 5(10), 15817–15825 (2022)
[7]	QI, J., ZHANG, H., LU, S., LI, X., XU, M., and ZHANG, Y. High performance indium-doped ZnO gas sensor. Journal of Nanomaterials, 2015, 954747 (2015)
[8]	VISHNIAKOU, S., CHEN, R., RO, Y. G., BRENNAN, C. J., LEVY, C., YU, E. T., and DAYEH, S. A. Improved performance of zinc oxide thin film transistor pressure sensors and a demonstration of a commercial chip compatibility with the new force sensing technology. Advanced Materials Technologies, 3(3), 1700279 (2018)
[9]	MASMANIDIS, S. C., KARABALIN, R. B., DE VLAMINCK, I., BORGHS, G., FREEMAN, M. R., and ROUKES, M. L. Multifunctional nanomechanical systems via tunably coupled piezoelectric actuation. Science, 317(5839), 780–783 (2007)
[10]	XU, C., WEI, P., WEI, Z., and GUO, X. Effects of corrugated boundaries on Rayleigh waves in a piezoelectric semiconductor substrate covered with a metal layer. Applied Mathematical Modelling, 125, 110–124 (2024)
[11]	SHUKLA, S. and SAHU, S. A. Shear wave velocity in a functionally graded piezoelectric semiconductor plate clamped on a rigid base. Smart Materials and Structures, 33(10), 105006 (2024)
[12]	YANG, L., ZAPPINO, E., CARRERA, E., and DU, J. Rotation effects on propagation of shear horizontal surface waves in piezomagnetic-piezoelectric semiconductor layered structures. Applied Mathematical Modelling, 129, 494–508 (2024)
[13]	LI, D. Z., LI, S. P., MA, N. N., WANG, H. M., ZHANG, C. L., and CHEN, W. Q. Propagation characteristics of elastic longitudinal wave in a piezoelectric semiconductor metamaterial rod and its tuning. International Journal of Mechanical Sciences, 266, 108977 (2024)
[14]	LI, Y., YAN, S., ZHAO, M., and REN, J. Fracture analysis of planar cracks in 3D thermal piezoelectric semiconductors. International Journal of Mechanical Sciences, 273, 109212 (2024)
[15]	CHENG, R., ZHANG, C., CHEN, W., and YANG, J. Piezotronic effects in the extension of a composite fiber of piezoelectric dielectrics and nonpiezoelectric semiconductors. Journal of Applied Physics, 124(6), 064506 (2018)
[16]	ZHAO, L., DENG, T., JIN, F., and SHAO, Z. Nonlinear analysis on electro-elastic coupling properties in bended piezoelectric semiconductor beams with variable cross section. Applied Mathematical Modelling, 133, 20–40 (2024)
[17]	FANG, K., LI, P., LI, N., LIU, D., QIAN, Z., KOLESOV, V., and KUZNETSOVA, I. Model and performance analysis of non-uniform piezoelectric semiconductor nanofibers. Applied Mathematical Modelling, 104, 628–643 (2022)
[18]	LUO, Y., ZHANG, C., CHEN, W., and 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)
[19]	GAO, S., ZHANG, Z., NIE, G., LIU, J., and CHEN, W. Indentation responses of piezoelectric semiconductors. International Journal of Solids and Structures, 290, 112682 (2024)
[20]	GAO, S., ZHANG, L., LIU, J., NIE, G., and CHEN, W. Indentation behavior of a semi-infinite piezoelectric semiconductor under a rigid flat-ended cylindrical indenter. Applied Mathematics and Mechanics, 45(4), 649–662 (2024)
[21]	SLADEK, J., SLADEK, V., REPKA, M., and PAN, E. Size effect in piezoelectric semiconductor nanostructures. Journal of Intelligent Material Systems and Structures, 33(11), 1351–1363 (2021)
[22]	SUN, L., XIAO, Z., ZHANG, C., and CHEN, W. Bending analysis of circular piezoelectric semiconductor plates incorporating flexoelectricity. Acta Mechanica Solida Sinica, 37(4), 613–621 (2024)
[23]	QU, Y., JIN, F., and YANG, J. Temperature-induced potential barriers in piezoelectric semiconductor films through pyroelectric and thermoelastic couplings and their effects on currents. Journal of Applied Physics, 131(9), 094502 (2022)
[24]	QU, Y., PAN, E., ZHU, F., JIN, F., and ROY, A. K. Modeling thermoelectric effects in piezoelectric semiconductors: new fully coupled mechanisms for mechanically manipulated heat flux and refrigeration. International Journal of Engineering Science, 182, 103775 (2023)
[25]	YANG, Y. and HU, Y. Energy transfer, conversion and mechanical regulation in graded piezoelectric semiconductor bipolar junction transistor. International Journal of Heat and Mass Transfer, 232, 125946 (2024)
[26]	YANG, Z., SUN, L., ZHANG, C., ZHANG, C., and GAO, C. Analysis of a composite piezoelectric semiconductor cylindrical shell under the thermal loading. Mechanics of Materials, 164, 104153 (2022)
[27]	XIAO, Z., SUN, L., CHEN, W., and ZHANG, C. Nonlinear multi-field coupling modeling of multilayer-stacked piezoelectric semiconductor structures. International Journal of Mechanical Sciences, 288, 110025 (2025)
[28]	XIAO, Z., LIU, J., ZHANG, C., and CHEN, W. Nonlinear analysis of thermo-deformation-polarization-carrier coupling response of piezoelectric semiconductor plates. European Journal of Mechanics-A/Solids, 111, 105560 (2025)
[29]	YANG, Z., ZHANG, Z., LIU, C., GAO, C., CHEN, W., and ZHANG, C. Analysis of a hollow piezoelectric semiconductor composite cylinder under a thermal loading. Mechanics of Advanced Materials and Structures, 30(10), 2037–2046 (2023)
[30]	WANG, T., ZHU, F., LI, P., XU, Z., MA, T., KUZNETSOVA, I., DONG, B., and QIAN, Z. Two-dimensional analysis of electrical behavior in composite semiconductor shell structures with magnetic field tuning. Journal of Applied Physics, 136(11), 115701 (2024)
[31]	SZE, S. M., LI, Y., and NG, K. K. Physics of Semiconductor Devices, John Wiley and Sons, New Jersey (2021)
[32]	FAN, S., YANG, W., and HU, Y. Adjustment and control on the fundamental characteristics of a piezoelectric PN junction by mechanical-loading. Nano Energy, 52, 416–421 (2018)
[33]	LIU, Y., ZHANG, Y., YANG, Q., NIU, S., and WANG, Z. L. Fundamental theories of piezotronics and piezo-phototronics. Nano Energy, 14, 257–275 (2015)
[34]	YANG, G., YANG, L., DU, J., WANG, J., and YANG, J. PN junctions with coupling to bending deformation in composite piezoelectric semiconductor fibers. International Journal of Mechanical Sciences, 173, 105421 (2020)
[35]	LUO, Y., ZHANG, C., CHEN, W., and YANG, J. An analysis of PN junctions in piezoelectric semiconductors. Journal of Applied Physics, 122(20), 204502 (2017)
[36]	LIANG, C., ZHANG, C., CHEN, W., and YANG, J. Effects of magnetic fields on PN junctions in piezomagnetic-piezoelectric semiconductor composite fibers. International Journal of Applied Mechanics, 12(8), 2050085 (2020)
[37]	CHENG, R., ZHANG, C., CHEN, W., and YANG, J. Temperature effects on PN junctions in piezoelectric semiconductor fibers with thermoelastic and pyroelectric couplings. Journal of Electronic Materials, 49(5), 3140–3148 (2020)
[38]	LUO, Y., CHENG, R., ZHANG, C., CHEN, W., and YANG, J. Electromechanical fields near a circular PN junction between two piezoelectric semiconductors. Acta Mechanica Solida Sinica, 31(2), 127–140 (2018)
[39]	MAGUIRE, J. R., MCCLUSKEY, C. J., HOLSGROVE, K. M., SUNA, A., KUMAR, A., MCQUAID, R. G. P., and GREGG, J. M. Ferroelectric domain wall p-n junctions. Nano letters, 23(22), 10360–10366 (2023)
[40]	CHEN, R., WANG, D., FENG, B., ZHU, H., HAN, X., MA, J., XIAO, H., and LUAN, C. High responsivity self-powered DUV photodetectors based on β-Ga₂O₃/GaN heterogeneous PN junctions. Vacuum, 215, 112332 (2023)
[41]	WEN, X., WU, W., DING, Y., and WANG, Z. L. Piezotronic effect in flexible thin-film based devices. Advanced Materials, 25(24), 3371–3379 (2013)
[42]	VISHNIAKOU, S., LEWIS, B. W., NIU, X., KARGAR, A., SUN, K., KALAJIAN, M., PARK, N., YANG, M., JING, Y., BROCHU, P., SUN, Z., LI, C., NGUYEN, T., PEI, Q., and WANG, D. Tactile feedback display with spatial and temporal resolutions. Scientific Reports, 3, 2521 (2013)
[43]	OH, H., YI, G. C., YIP, M., and DAYEH, S. A. Scalable tactile sensor arrays on flexible substrates with high spatiotemporal resolution enabling slip and grip for closed-loop robotics. Science Advances, 6(46), eabd7795 (2020)
[44]	ZHU, C., FANG, X., and LIU, J. Nonlinear free vibration of piezoelectric semiconductor doubly-curved shells based on nonlinear drift-diffusion model. Applied Mathematics and Mechanics (English Edition), 44(10), 1761–1776 (2023) https://doi.org/10.1007/s10483-023-3039-7
[45]	ZHANG, G., KONG, X., and MI, C. Mechanically manipulated in-plane electric currents and thermal control in piezoelectric semiconductor films. Acta Mechanica, 235(6), 3463–3481 (2024)
[46]	WANG, T., ZHU, F., LI, P., XU, Z., MA, T., KUZNETSOVA, I., and QIAN, Z. Analysis and modeling of two-dimensional piezoelectric semiconductor shell theory. European Journal of Mechanics-A/Solids, 106, 105331 (2024)
[47]	YANG, J. Analysis of Piezoelectric Semiconductor Structures, Springer Nature, Switzerland (2020)
[48]	SOEDEL, W. Vibrations of Shells and Plates, Marcel Dekker, Inc., New York (2004)
[49]	REDDY, J. N. Theory and Analysis of Elastic Plates and Shells, CRC press, Florida (2006)
[50]	GERMAIN, P. The method of virtual power in the mechanics of continuous media, I: second-gradient theory. Mathematics and Mechanics of Complex Systems, 8(2), 153–190 (2020)
[51]	MINDLIN, R. High frequency vibrations of piezoelectric crystal plates. International Journal of Solids and Structures, 8(7), 895–906 (1972)
[52]	BERT, C. W. and MALIK, M. Differential quadrature method in computational mechanics: a review. Applied Mechanics Reviews, 49(1), 1–28 (1996)
[53]	AULD, B. A. Acoustic Fields and Waves in Solids, John Wiley and Sons, New York (1973)
[54]	FANG, K., LI, P., LI, N., LIU, D., QIAN, Z., KOLESOV, V., and KUZNETSOVA, I. Impact of PN junction inhomogeneity on the piezoelectric fields of acoustic waves in piezo-semiconductive fibers. Ultrasonics, 120, 106660 (2022)