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Critical sample sizes for apperance and disappearance of ferroelectricity in nanosized hafnia-zirconia: Landau-type theory

Anna N. Morozovska1,*,†, Eugene A. Eliseev2,*, Sergei V. Kalinin3,‡, and Maksym V. Strikha4,5,§

  • *These authors contributed equally to this work.
  • †Contact author: anna.n.morozovska@gmail.com
  • ‡Contact author: sergei2@utk.edu
  • §Contact author: maksym.strikha@gmail.com
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Phys. Rev. B 113, 174102 – Published 4 May, 2026

DOI: https://doi.org/10.1103/hcm2-x9nj

Abstract

Nanosized hafnia-zirconia HfxZr1−xO2 in the form of thin films, multilayers, and heterostructures are indispensable silicon-compatible ferroelectric materials for advanced electronic memories and logic devices. The distinctive feature of nanoscale hafnia-zirconia are the critical sizes of ferroelectricity appearance, whereas the critical sizes of ferroelectricity disappearance exist in other ferroelectrics. Using the Landau-Ginzburg-Devonshire free energy functional with higher powers, trilinear and biquadratic couplings of polar, nonpolar, and antipolar order parameters, we calculated analytically the strain-dependent critical sizes of the ferroelectricity appearance and disappearance, analyzed how the size effect and mismatch strains influence the phase diagrams and polarization switching barrier in epitaxial HfO2 thin films and nano-islands with the out-of-plane spontaneous polarization. We have shown that the critical thickness/height of out-of-plane spontaneous polarization disappearance is determined by the size dependence of the depolarization field and correlation effects. The critical thickness/height of the ferroelectricity appearance is determined by the size dependence of the effective mismatch strain considering possible appearance of misfit dislocations and lateral relaxion of strains. Derived analytical expressions can be generalized for HfxZr1−xO2 solid solutions, providing that corresponding parameters of the free energy are known from the first-principles calculations.

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References (70)

  1. J. P. B. Silva et al., Roadmap on ferroelectric hafnia and zirconia-based materials and devices, APL Mater. 11, 089201 (2023).
  2. K.-H. Kim, I. Karpov, R. H. Olsson III, and D. Jariwala, Wurtzite and fluorite ferroelectric materials for electronic memory, Nat. Nanotechnol. 18, 422 (2023).
  3. X. Tao, L. Liu, L. Yang, and J.-P. Xu, Impacts of HfZrO2 thickness and anneal temperature on performance of MoS2 negative-capacitance field-effect transistors, Nanotechnology 32, 445201 (2021).
  4. W. W. Yang, C. X. Yu, H. L. Li, M. Q. Fan, X. J. Song, H. L. Ma, Z. Zhou, P. Y. Chang, P. Huang, F. Liu, X. Y. Liu, and J. F. Kang, Ferroelectricity of hafnium oxide-based materials: Current statuses and future prospects from physical mechanisms to device applications, J. Semicond. 44, 053101 (2023).
  5. A. Paul, G. Kumar, A. Das, G. Larrieu, and S. De, Hafnium oxide-based ferroelectric field effect transistors: From materials and reliability to applications in storage-class memory and in-memory computing, J. Appl. Phys. 138, 010701 (2025).
  6. M. H. Park, Y. H. Lee, H. J. Kim, T. Schenk, W. Lee, K. Do Kim, F. P. G. Fengler, T. Mikolajick, U. Schroeder, and C. S. Hwang, Surface and grain boundary energy as the key enabler of ferroelectricity in nanoscale hafnia-zirconia: A comparison of model and experiment, Nanoscale 9, 9973 (2017).
  7. N. Afroze, H. Fahrvandi, G. Ren, P. Kumar, C. Nelson, S. Lombardo, M. Tian, et al., Sub-nm2 ferroelectric domains via charged 180 degree walls in ZrO2, arXiv:2507.18920.
  8. B. Mukherjee, N. S. Fedorova, and J. Íñiguez-González, Extrinsic nature of the polarization in hafnia ferroelectrics, arXiv:2508.00372.
  9. F. Delodovici, P. Barone, and S. Picozzi, Finite-size effects on ferroelectricity in rhombohedral HfO2, Phys. Rev. B 106, 115438 (2022).
  10. F. Delodovici, P. Barone, and S. Picozzi, Trilinear-coupling-driven ferroelectricity in HfO2, Phys. Rev. Mater. 5, 064405 (2021).
  11. S. Jung and T. Birol, Triggered ferroelectricity in HfO2 from hybrid phonons and higher-order dynamical charges, Adv. Mater. 38, e21602 (2026).
  12. S. Jung and T. Birol, Electric polarization from nonpolar phonons, arXiv:2512.00628.
  13. J. Holakovsky, A new type of the ferroelectric phase transition, Physica status solidi (b) 56, 615 (1973).
  14. S. Zhou, J. Zhang, and A. M. Rappe, Strain-induced antipolar phase in hafnia stabilizes robust thin-film ferroelectricity, Sci. Adv. 8, eadd5953 (2022).
  15. S. Estandìa, N. Dix, J. Gazquez, I. Fina, J. Lyu, M. F. Chisholm, J. Fontcuberta, and F. Sánchez, Engineering ferroelectric Hf0.5Zr0.5O2 thin films by epitaxial stress, ACS Appl. Electron. Mater. 1, 1449 (2019).
  16. S. Estandìa, N. Dix, M. F. Chisholm, I. Fina, and F. Sánchez, Domain-matching epitaxy of ferroelectric Hf0.5Zr0.5O2(111) on La2/3Sr1/3MnO3(001), Cryst. Growth Des. 20, 3801 (2020).
  17. S. Dutta, P. Buragohain, S. Glinsek, C. Richter, H. Aramberri, H. Lu, U. Schroeder, E. Defay, A. Gruverman, and J. Íñiguez, Piezoelectricity in hafnia, Nat. Commun. 12, 7301 (2021).
  18. H.-J. Lee, K.-J. Go, P. Kumar, C. H. Kim, Y. Kim, K. Lee, T. Shimizu, S. C. Chae, H. Jin, M. Lee, U. Waghmare, S.-Y. Choi, J. H. Lee, Phonon-pair-driven ferroelectricity causes costless domain-walls and bulk-boundary duality, arXiv:2403.01415.
  19. Y. Tamura, K. Masuda, and Y. Kumagai, On-demand phase-field modeling: Three-dimensional Landau energy for HfO2 through machine learning, arXiv:2512.16207.
  20. Y. Qi and K. M. Rabe, Competing phases of HfO2 from unstable flat phonon bands of an unconventional high-symmetry structure, Phys. Rev. Lett. 135, 046101(2025).
  21. Y. Qi, S. Singh, and K. M. Rabe, Polarization switching in ferroelectric HfO2 from first-principles lattice mode analysis, Phys. Rev. B 111, 134106 (2025).
  22. K. Ooe, Y. Shen, K. Shitara, R. Maki, S. Kobayashi, Y. Shimakawa, D. Kan, and J. Etheridge, Direct observation of cation-dependent polarisation switching dynamics in fluorite ferroelectrics, Nat. Commun. 17, 2660 (2026).
  23. K. Fujimoto, Y. Sato, Y. Fuchikami, R. Teranishi, and Kenji Kaneko, Orthorhombic-like atomic arrangement in hafnium-oxide-based nanoparticles, J. Am. Ceram. Soc. 105, 2823 (2022).
  24. S. Kang, W.-S. Jang, A. N. Morozovska, O. Kwon, Y. Jin, Y. H. Kim, H. Bae, C. Wang, S. H. Yang, A. Belianinov, and S. Randolph, Highly enhanced ferroelectricity in HfO2-based ferroelectric thin film by light ion bombardment, Science 376, 731 (2022).
  25. K. P. Kelley, A. N. Morozovska, E. A. Eliseev, Y. Liu, S. S. Fields, S. T. Jaszewski, T. Mimura, J. F. Ihlefeld, and S. V. Kalinin, Ferroelectricity in hafnia controlled via surface electrochemical state, Nat. Mater. 22, 1144 (2023).
  26. M. D. Glinchuk, A. N. Morozovska, A. Lukowiak, W. Stręk, M. V. Silibin, D. V. Karpinsky, Y. Kim, and S. V. Kalinin, Possible electrochemical origin of ferroelectricity in HfO2 thin films, J. Alloys Compd. 830, 153628 (2020).
  27. L.-Y. Ma and S. Liu, Structural polymorphism kinetics promoted by charged oxygen vacancies in HfO2, Phys. Rev. Lett. 130, 096801 (2023).
  28. T. K. Paul, A. K. Saha, and S. K. Gupta, Oxygen vacancy-induced monoclinic dead layers in ferroelectric HfO2 with metal electrodes, J. Appl. Phys. 137, 144102 (2025).
  29. G. Zhang, H. Wu, X. Xu, S. Lin, Z. Zhang, Z. Yan, X. Lu, G. Yuan, and J.-M. Liu, Enhanced reliability of Hf0.5Zr0.5O2 ferroelectric memory through WOx buffer layer to minimize oxygen vacancies, Appl. Phys. Lett. 127, 112102 (2025).
  30. O. S. Pylypchuk, V. V. Vainberg, V. N. Poroshin, O. V. Leshchenko, V. N. Pavlikov, I. V. Kondakova, S. E. Ivanchenko, L. P. Yurchenko, L. Demchenko, A. O. Diachenko, M. V. Karpets, M. P. Trubitsyn, E. A. Eliseev, and A. N. Morozovska, A colossal dielectric response of HfxZr1−xO2 nanoparticles, Phys. Rev. Mater. 9, 114412 (2025).
  31. O. S. Pylypchuk, I. V. Fesych, V. V. Vainberg, Y. O. Zagorodniy, V. I. Styopkin, J. M. Gudenko, I. V. Kondakova, L. P. Yurchenko, V. N. Pavlikov, A. O. Diachenko, M. M. Koptiev, M. D. Volnyanskii, V. V. Laguta, E. A. Eliseev, M. P. Trubitsyn, and A. N. Morozovska, Resistive switching and charge accumulation in Hf0.5Zr0.5O2 nanoparticles, J. Phys. Chem. C 129, 14299 (2025).
  32. A. N. Morozovska, M. V. Strikha, K. P. Kelley, S. V. Kalinin, and E. A. Eliseev, Effective Landau-type model of a HfxZr1−xO2-graphene nanostructure, Phys. Rev. Appl. 20, 054007 (2023).
  33. E. A. Eliseev, Y. O. Zagorodniy, V. N. Pavlikov, O. V. Leshchenko, H. V. Shevilakova, M. V. Karpec, A. D. Yaremkevych, O. M. Fesenko, S. V. Kalinin, and A. N. Morozovska, Phase diagrams and polarization reversal in nanosized HfxZr1−xO2−y, AIP Adv. 14, 055224 (2024).
  34. R. Blinc and B. Zeks, Soft Mode in Ferroelectrics and Antiferroelectrics (North-Holland, Amsterdam, 1974).
  35. K. M. Rabe, Antiferroelectricity in oxides: A reexamination, in Functional Metal Oxides (John Wiley & Sons, New York, 2013), pp 221–244.
  36. J. Hlinka, T. Ostapchuk, E. Buixaderas, C. Kadlec, P. Kuzel, I. Gregora, J. Kroupa, M. Savinov, A. Klic, J. Drahokoupil, et al., Multiple soft-mode vibrations of lead zirconate, Phys. Rev. Lett. 112, 197601 (2014).
  37. S. Lim, Y. Ahn, B. Won, S. Lee, H. Park, M. Kumar, and H. Seo, Effects of substrate and annealing conditions on the ferroelectric properties of non-doped HfO2 deposited by rf plasma sputter, Nanomaterials 14, 1386 (2024).
  38. A. Tagantsev, L. E. Cross, and J. Fousek, Domains in Ferroic Crystals and Thin Films (Springer, New York, 2010).
  39. E. A. Eliseev, S. V. Kalinin, and A. N. Morozovska, Ferro-ionic states and domains morphology in HfxZr1−xO2 nanoparticles, J. Appl. Phys. 137, 034103 (2025).
  40. G. B. Stephenson and M. J. Highland, Equilibrium and stability of polarization in ultrathin ferroelectric films with ionic surface compensation, Phys. Rev. B 84, 064107 (2011).
  41. M. J. Highland, T. T. Fister, D. D. Fong, P. H. Fuoss, C. Thompson, J. A. Eastman, S. K. Streiffer, and G. B. Stephenson, Equilibrium polarization of ultrathin PbTiO3 with surface compensation controlled by oxygen partial pressure, Phys. Rev. Lett. 107, 187602 (2011).
  42. E. A. Eliseev, I. V. Kondakova, Y. O. Zagorodniy, H. V. Shevliakova, O. V. Leshchenko, V. N. Pavlikov, M. V. Karpets, L. P. Yurchenko, and A. N. Morozovska, The origin of the ferroelectric-like orthorhombic phase in oxygen-deficient HfO2−y nanoparticles, Semicond. Phys. Optoelectron. Quantum Electron. 28, 134 (2025).
  43. See Supplemental Material at http://link.aps.org/supplemental/10.1103/hcm2-x9nj for material parameters and calculation details.
  44. X. Zhao and D. Vanderbilt, First-principles study of electronic and dielectric properties of ZrO2 and HfO2, MRS Online Proc. Lib. 747, 72 (2002).
  45. R. Bindu, Structural studies of LaSrMnO3 (x = 0.1-1.0), Eur. Phys. J. B 37, 321 (2004).
  46. D. R. Tilley, Finite-size effects on phase transitions in ferroelectrics, in Ferroelectic Thin Films, edited by C. Paz de Araujo, J. F. Scott, and G. W. Teylor (Gordon and Breach, Amsterdam, 1996), pp. 11–45.
  47. M. D. Glinchuk, A. N. Morozovska, and E. A. Eliseev, Correlation radius in thin ferroelectric films, Ferroelectrics 400, 243 (2010).
  48. E. A. Eliseev and A. N. Morozovska, General approach to the description of the size effect in ferroelectric nanosystems, J. Mater. Sci. 44, 5149 (2009).
  49. A. K. Tagantsev and G. Gerra, Interface-induced phenomena in polarization response of ferroelectric thin films, J. Appl. Phys. 100, 051607 (2006).
  50. I. Stolichnov, A. K. Tagantsev, E. Colla, N. Setter, and J. S. Cross, Physical model of retention and temperature-dependent polarization reversal in ferroelectric films, J. Appl. Phys. 98, 084106 (2005).
  51. A. K. Tagantsev, G. Gerra, and N. Setter, Short-range and long-range contributions to the size effect in metal-ferroelectric-metal heterostructures, Phys. Rev. B 77, 174111 (2008).
  52. I. S. Vorotiahin, E. A. Eliseev, Q. Li, S. V. Kalinin, Y. A. Genenko, and A. N. Morozovska, Tuning the polar states of ferroelectric films via surface charges and flexoelectricity, Acta Mater. 137, 85 (2017).
  53. N. A. Pertsev, A. G. Zembilgotov, and A. K. Tagantsev, Effect of mechanical boundary conditions on phase diagrams of epitaxial ferroelectric thin films, Phys. Rev. Lett. 80, 1988 (1998).
  54. A. N. Morozovska, E. A. Eliseev, A. Ghosh, M. E. Yelisieiev, Yu. M. Vysochanskii, and S. V. Kalinin, Anomalous polarization reversal in strained thin films of CuInP2S6, Phys. Rev. B 108, 054107 (2023).
  55. J. S. Speck and W. Pompe, Domain configurations due to multiple misfit relaxation mechanisms in epitaxial ferroelectric thin films. I. Theory, J. Appl. Phys. 76, 466 (1994).
  56. A. N. Morozovska, M. D. Glinchuk, and E. A. Eliseev, Phase transitions induced by confinement of ferroic nanoparticles, Phys. Rev. B 76, 014102 (2007).
  57. C. L. Jia, V. Nagarajan, J. Q. He, L. Houben, T. Zhao, R. Ramesh, K. Urban, and R. Waser, Unit-cell scale mapping of ferroelectricity and tetragonality in epitaxial ultrathin ferroelectric films, Nat. Mater. 6, 64 (2007).
  58. H. S. Park, J. C. Shin, K. Do Kim, S. Jae Shin, J. Hee Song, S. Kyu Ryoo, I. Soo Lee, S. Hyun Lee, H. Nam, and C. S. Hwang, Enhancing ferroelectric properties of Hf0.5Zr0.5O2 thin films using the HfN/TiN and W/TiN bi-layer bottom electrodes, J. Materiom. 11, 101109 (2025).
  59. W. Ye and B. Chen, Elastic relaxation in 3D epitaxial nanoisland with strain-dependent surface stress effect, J. Cryst. Growth 410, 59 (2015).
  60. C-H. Chiu and C. T. Poh, Strain energy of nanoislands on strained film–substrate systems, Phys. Rev. B 71, 045406 (2005).
  61. V. Mitin, V. Kochelap, and M. Stroscio, Introduction to Nanoelectronics: Science, Nanotechnology, Engineering, and Applications (Cambridge University Press, Cambridge, 2012), p. 129, p. 346.
  62. H. Luth, Solid Surfaces, Interfaces and Thin Films (Springer, Berlin, 2010).
  63. K. Oura, V. G. Lifshits, A. Saranin, and A. V. Zotov, Mitsuhiro Katayama: Surface Science–An Introduction (Springer, Berlin, 2004).
  64. J. A. Venables, Introduction to Surface and Thin Film Processes (Cambridge University Press, Cambridge, 2003).
  65. K. W. Kolasinski, Surface Science (Wiley, New York, 2008).
  66. J. Shin, H. Seo, K. Hee Ye, Y. Ho Jang, D. Seon Kwon, J. Lim, T. Kyun Kim, et al., Understanding phase evolution of ferroelectric Hf0.5Zr0.5O2 thin films with Al2O3 and Y2O3 inserted layers, J. Mater. Chem. C 12, 5035 (2024).
  67. Y. Qi, X. Xu, I. Krylov, and M. Eizenberg, Ferroelectricity of as-deposited HZO fabricated by plasma-enhanced atomic layer deposition at 300 °C by inserting TiO2 interlayers, Appl. Phys. Lett. 118, 032906 (2021).
  68. T. Wang, H. Zhang, Y. Xie, D. Sheng, S.Du, Z. Liu, S. Wang, H. Li, Q. Zhang, K. Wang, and B. Xu, Ferroelectricity in undoped HfO2 down to one-unit-cell on Si substrate, arXiv:2501.12616.
  69. R. Materlik, C. Künneth, and A. Kersch, The origin of ferroelectricity in Hf1−xZrxO2: A computational investigation and a surface energy model, J. Appl. Phys. 117, 134109 (2015).
  70. E. Eliseev and A. Morozovska, Reentrant ferroelectricity in nanosized hafnia-zirconia: Landau-type theory from the Notebook Archive (2026), https://www.wolfram.com/mathematica, https://notebookarchive.org/2026-01-60qfsh4

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Phys. Rev. B 113, 174102– Published 4 May, 2026
  • Received 9 January 2026
  • Revised 4 March 2026
  • Accepted 13 April 2026
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