- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Strain distribution in zinc-blende and wurtzite GaAs nanowires bent by a one-sided (In,Al)As stressor shell: Consequences for torsion, chirality, and piezoelectricity

Yiannis Hadjimichael1,*, Oliver Brandt2, Christian Merdon1, Costanza Lucia Manganelli3, and Patricio Farrell1

  • *Contact author: yiannis.hadjimichael@wias-berlin.de
PDF

Phys. Rev. B 113, 165303 – Published 14 April, 2026

DOI: https://doi.org/10.1103/pjq7-8gll

Abstract

We present a finite-strain model that is capable of describing the large deformations in bent nanowire heterostructures. The model incorporates a nonlinear strain formulation derived from the first Piola-Kirchhoff stress tensor, coupled with an energy functional that effectively captures the lattice-mismatch-induced strain field. We use the finite element method to solve the resulting partial differential equations and extract cross-sectional maps of the full strain tensor for both zinc-blende and wurtzite nanowires with lattice-mismatched core and one-sided stressor shell. In either case, we show that the bending is essentially exclusively determined by ɛzz. However, the distinct difference in shear strain has important consequences with regard to both the mechanical deformation and the existence of transverse piezoelectric fields in the nanowires.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (57)

  1. E. P. O'Reilly, Valence band engineering in strained-layer structures, Semicond. Sci. Technol. 4, 121 (1989).
  2. E. P. O'Reilly and A. R. Adams, Band-structure engineering in strained semiconductor lasers, IEEE J. Quantum Elect. 30, 366 (1994).
  3. C. K. Maiti, N. B. Chakrabarti, and S. K. e. Ray, Strained silicon heterostructures: Materials and devices (Institution of Engineering and Technology, London, 2001).
  4. A. R. Adams, Strained-layer quantum-well lasers, IEEE J. Select. Top. Quantum Electron. 17, 1364 (2011).
  5. C. Schriever, C. Bohley, J. Schilling, and R. B. Wehrspohn, Strained silicon photonics, Materials 5, 889 (2012).
  6. Z. Dai, L. Liu, and Z. Zhang, Strain engineering of 2D materials: Issues and opportunities at the interface, Adv. Mater. 31, 1805417 (2019).
  7. P. Zubko, G. Catalan, and A. K. Tagantsev, Flexoelectric effect in solids, Annu. Rev. Mater. Res. 43, 387 (2013).
  8. P. V. Yudin and A. K. Tagantsev, Fundamentals of flexoelectricity in solids, Nanotechnology 24, 432001 (2013).
  9. B. Wang, Y. Gu, S. Zhang, and L.-Q. Chen, Flexoelectricity in solids: Progress, challenges, and perspectives, Prog. Mater. Sci. 106, 100570 (2019).
  10. Y. Xia, W. Qian, and Y. Yang, Advancements and prospects of flexoelectricity, ACS Appl. Mater. Interfaces 16, 9597 (2024).
  11. D. Du, J. Hu, and J. K. Kawasaki, Strain and strain gradient engineering in membranes of quantum materials, Appl. Phys. Lett. 122, 170501 (2023).
  12. S. Maity, L. Shao, Y.-I. Sohn, S. Meesala, B. Machielse, E. Bielejec, M. Markham, and M. Lončar, Spectral alignment of single-photon emitters in diamond using strain gradient, Phys. Rev. Appl. 10, 024050 (2018).
  13. G. Stan, S. Krylyuk, A. V. Davydov, I. Levin, and R. F. Cook, Ultimate bending strength of Si nanowires, Nano Lett. 12, 2599 (2012).
  14. H. Zhang, J. Tersoff, S. Xu, H. Chen, Q. Zhang, K. Zhang, Y. Yang, C.-S. Lee, K.-N. Tu, J. Li, and Y. Lu, Approaching the ideal elastic strain limit in silicon nanowires, Sci. Adv. 2, e1501382 (2016).
  15. F. Boxberg, N. Søndergaard, and H. Q. Xu, Elastic and piezoelectric properties of zincblende and wurtzite crystalline nanowire heterostructures, Adv. Mater. 24, 4692 (2012).
  16. C. P. Dietrich, M. Lange, F. J. Klüpfel, H. Von Wenckstern, R. Schmidt-Grund, and M. Grundmann, Strain distribution in bent ZnO microwires, Appl. Phys. Lett. 98, 031105 (2011).
  17. S. Xu, W. Guo, S. Du, M. M. T. Loy, and N. Wang, Piezotronic effects on the optical properties of ZnO nanowires, Nano Lett. 12, 5802 (2012).
  18. X. Fu, G. Jacopin, M. Shahmohammadi, R. Liu, M. Benameur, J.-D. Ganière, J. Feng, W. Guo, Z.-M. Liao, B. Deveaud, and D. Yu, Exciton drift in semiconductors under uniform strain gradients: Application to bent ZnO microwires, ACS Nano 8, 3412 (2014).
  19. X. Fu, C. Su, Q. Fu, X. Zhu, R. Zhu, C. Liu, Z. Liao, J. Xu, W. Guo, J. Feng, J. Li, and D. Yu, Tailoring exciton dynamics by elastic strain-gradient in semiconductors, Adv. Mater. 26, 2572 (2014).
  20. Y. Gao and Z. L. Wang, Electrostatic potential in a bent piezoelectric nanowire. The fundamental theory of nanogenerator and nanopiezotronics, Nano Lett. 7, 2499 (2007).
  21. Z. L. Wang, Nanopiezotronics, Adv. Mater. 19, 889 (2007).
  22. Y. Gao and Z. L. Wang, Equilibrium potential of free charge carriers in a bent piezoelectric semiconductive nanowire, Nano Lett. 9, 1103 (2009).
  23. C. Liu, S. Hu, and S. Shen, Effect of flexoelectricity on electrostatic potential in a bent piezoelectric nanowire, Smart Mater. Struct. 21, 115024 (2012).
  24. Z. Zhang, D. Geng, and X. Wang, Calculation of the piezoelectric and flexoelectric effects in nanowires using a decoupled finite element analysis method, J. Appl. Phys. 119, 154104 (2016).
  25. K. F. Wang and B. L. Wang, Electrostatic potential in a bent piezoelectric nanowire with consideration of size-dependent piezoelectricity and semiconducting characterization, Nanotechnology 29, 255405 (2018).
  26. R. B. Lewis, P. Corfdir, H. Küpers, T. Flissikowski, O. Brandt, and L. Geelhaar, Nanowires bending over backward from strain partitioning in asymmetric core-shell heterostructures, Nano Lett. 18, 2343 (2018).
  27. A. Davtyan, D. Kriegner, V. Holý, A. AlHassan, R. B. Lewis, S. McDermott, L. Geelhaar, D. Bahrami, T. Anjum, Z. Ren, C. Richter, D. Novikov, J. Müller, B. Butz, and U. Pietsch, X-ray diffraction reveals the amount of strain and homogeneity of extremely bent single nanowires, J. Appl. Crystallogr. 53, 1310 (2020).
  28. S. McDermott and R. B. Lewis, Bending of GaAs–InP core–shell nanowires by ssymmetric shell deposition: Implications for sensors, ACS Appl. Nano Mater. 4, 10164 (2021).
  29. S. McDermott, T. R. Smith, R. R. LaPierre, and R. B. Lewis, Phosphorus-Controlled nanoepitaxy of asymmetric GaAs–InP core–shell bent nanowires: Implications for bottom-up nanowire transistors and sensors, ACS Appl. Nano Mater. 7, 1257 (2024).
  30. P. G. Ciarlet, Mathematical elasticity: Three-dimensional elasticity (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2021).
  31. Y. Hadjimichael, C. Merdon, M. Liero, and P. Farrell, An energy-based finite-strain model for 3D heterostructured materials and its validation by curvature analysis, Int. J. Numer. Methods Eng. 125 e7508 (2024).
  32. Y. Hadjimichael and C. Merdon, ElectroMechanicsFEM.jl (2024).
  33. See the supporting information of Ref. [26].
  34. See Supplemental Material at http://link.aps.org/supplemental/10.1103/pjq7-8gll for additional details on the bending of nanowires with one-sided stressor shells, piezoelectric polarization for the nanowires examined in the paper, and the material parameters used throughout our study, which also contains Refs. [54, 55, 56, 57].
  35. S. Schulz, M. A. Caro, E. P. O'Reilly, and O. Marquardt, Symmetry-adapted calculations of strain and polarization fields in (111)-oriented zinc-blende quantum dots, Phys. Rev. B 84, 125312 (2011).
  36. R. M. Martin, Relation between elastic tensors of wurtzite and zinc-blende structure materials, Phys. Rev. B 6, 4546 (1972).
  37. S. C. Cowin, Elastic symmetry restrictions from structural gradients, in Rational Continua, Classical and New (Springer, Milan, 2003), p. 51.
  38. M. Fraldi and S. Cowin, Chirality in the torsion of cylinders with trigonal symmetry, J. Elast. 69, 121 (2002).
  39. S. C. Cowin, Continuum mechanics of anisotropic materials (Springer, New York, 2013).
  40. W. S. Slaughter, Torsion of noncircular cylinders, in The Linearized Theory of Elasticity (Birkhäuser, Boston, MA, 2002), p. 305.
  41. M. Grundmann, Theory of semiconductor solid and hollow nano- and microwires with hexagonal cross-section under torsion, Phys. Status Solidi B 252, 773 (2015).
  42. R. Lakes, Elastic and viscoelastic behavior of chiral materials, Int. J. Mech. Sci. 43, 1579 (2001).
  43. S.-A. Papanicolopulos, Chirality in isotropic linear gradient elasticity, Int. J. Sol. Struct. 48, 745 (2011).
  44. C. Polizzotto, A unifying variational framework for stress gradient and strain gradient elasticity theories, Eur. J. Mech. A. Solids 49, 430 (2015).
  45. H. Askes and E. C. Aifantis, Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results, Int. J. Sol. Struct. 48, 1962 (2011).
  46. S. Lurie and Y. Solyaev, Revisiting bending theories of elastic gradient beams, Int. J. Eng. Sci. 126, 1 (2018).
  47. B. E. Abali, W. H. Müller, and F. dell'Isola, Theory and computation of higher gradient elasticity theories based on action principles, Arch. Appl. Mech. 87, 1495 (2017).
  48. X.-F. Li, B.-L. Wang, and K. Y. Lee, Size effects of the bending stiffness of nanowires, J. Appl. Phys. 105, 074306 (2009).
  49. L. Xu and S. Shen, Size-dependent piezoelectricity and elasticity due to the electric field-strain gradient coupling and strain gradient elasticity, Int. J. Appl. Mech. 05, 1350015 (2013).
  50. T. Kaiser, S. Forest, and A. Menzel, A finite element implementation of the stress gradient theory, Meccanica 56, 1109 (2021).
  51. M. Lazar, E. Agiasofitou, and T. Böhlke, Mathematical modeling of the elastic properties of cubic crystals at small scales based on the Toupin–Mindlin anisotropic first strain gradient elasticity, Continuum Mech. Thermodyn. 34, 107 (2022).
  52. S. Timoshenko, Analysis of bi-metal thermostats, J. Opt. Soc. Am. 11, 233 (1925).
  53. Y. Hadjimichael and C. Merdon, StrainDistributionNanowires.jl: Reproducibility repository for numerical simulations on core/shell zincblende and wurtzite nanowires, Zenodo (2026), https://doi.org/10.5281/zenodo.18611390.
  54. I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, Band parameters for III–V compound semiconductors and their alloys, J. Appl. Phys. 89, 5815 (2001).
  55. A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, and K. A. Persson, Commentary: The Materials Project: A materials genome approach to accelerating materials innovation, APL Mater. 1, 011002 (2013).
  56. F. Bernardini, V. Fiorentini, and D. Vanderbilt, Spontaneous polarization and piezoelectric constants of III-V nitrides, Phys. Rev. B 56, R10024(R) (1997).
  57. A. Beya-Wakata, P.-Y. Prodhomme, and G. Bester, First- and second-order piezoelectricity in III-V semiconductors, Phys. Rev. B 84, 195207 (2011).

Outline

Information

Phys. Rev. B 113, 165303– Published 14 April, 2026
  • Received 11 November 2025
  • Accepted 29 January 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation