- 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

Flexocoupling-induced phonons and ferrons in van der Waals ferroelectrics

Anna N. Morozovska1,*, Eugene A. Eliseev2, Oleksiy V. Bereznikov1, Mykola Ye. Yelisieiev3, Guo-Dong Zhao4, Yujie Zhu5, Venkatraman Gopalan4,†, Long-Qing Chen4,‡, Jia-Mian Hu5,§ et al.

Yulian M. Vysochanskii6,∥

  • *Contact author: anna.n.morozovska@gmail.com
  • †Contact author: vxg8@psu.edu
  • ‡Contact author: lqc3@psu.edu
  • §Contact author: jhu238@wisc.edu
  • ∥Contact author: vysochanskii@gmail.com
PDF

Phys. Rev. B 112, 014110 – Published 21 July, 2025

DOI: https://doi.org/10.1103/mr9v-khtj

Abstract

The contribution of flexoelectric coupling to the long-range order parameter fluctuations in ferroics can be critically important to the ferron dispersion and related polar, pyroelectric, and electrocaloric (EC) properties. Here, we calculate analytically the dispersion relations of soft optic and acoustic flexocoupling-induced phonons and ferrons by incorporating the flexoelectric coupling, damping, and higher elastic gradients in the Landau-Ginzburg-Devonshire free-energy functional using the van der Waals uniaxial ferrielectric CuInP2S6 as an example. We analyze the changes in the flexocoupling-induced phonon and ferron spectra arising from the appearance of spatially modulated phases induced by the flexoelectric coupling. We show that the free-energy landscape of CuInP2S6 determines the specific features of its phonon spectra and ferron dispersion. We also discuss the contributions of optic and acoustic flexocoupling-induced ferrons to the pyroelectric and EC responses of CuInP2S6 at low temperatures.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (67)

  1. A. K. Tagantsev, Piezoelectricity and flexoelectricity in crystalline dielectrics, Phys. Rev. B 34, 5883 (1986).
  2. M. S. Majdoub, P. Sharma, and T. Cagin, Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect, Phys. Rev. B 77, 125424 (2008).
  3. S. Krichen and P. Sharma, Flexoelectricity: A perspective on an unusual electromechanical coupling, J. Appl. Mech. 83, 030801 (2016).
  4. D. Lee, A. Yoon, S. Y. Jang, J.-G. Yoon, J.-S. Chung, M. Kim, J. F. Scott, and T. W. Noh, Giant flexoelectric effect in ferroelectric epitaxial thin films, Phys. Rev. Lett. 107, 057602 (2011).
  5. P. V. Yudin and A. K. Tagantsev, Fundamentals of flexoelectricity in solids, Nanotechnology 24, 432001 (2013).
  6. P. Zubko, G. Catalan, and A. K. Tagantsev, Flexoelectric effect in solids, Annu. Rev. Mater. Res. 43, 387. (2013).
  7. A. N. Morozovska, E. A. Eliseev, M. D. Glinchuk, Long-Qing Chen, and Venkatraman Gopalan, Interfacial polarization and pyroelectricity in antiferrodistortive structures induced by a flexoelectric effect and rotostriction, Phys. Rev. B. 85, 094107 (2012).
  8. E. A. Eliseev, A. N. Morozovska, Y. Gu, A. Y. Borisevich, L.-Q. Chen, V. Gopalan, and S. V. Kalinin, Conductivity of twin Walls–surface junctions in ferroelastics: Interplay of deformation potential, octahedral rotations, improper ferroelectricity and flexoelectric coupling, Phys. Rev. B 86, 085416 (2012).
  9. E. A. Eliseev, S. V. Kalinin, Y. Gu, M. D. Glinchuk, V. Khist, A. Borisevich, V. Gopalan, L.-Q. Chen, and A. N. Morozovska, Universal emergence of spatially-modulated structures induced by flexo-antiferrodistortive coupling in multiferroics, Phys. Rev. B 88, 224105 (2013).
  10. P. V. Yudin, R. Ahluwalia, and A. K. Tagantsev, Upper bounds for flexocoupling coefficients in ferroelectrics, Appl. Phys. Lett. 104, 082913 (2014).
  11. A. Kvasov and A. K. Tagantsev, Dynamic flexoelectric effect in perovskites from first-principles calculations, Phys. Rev. B 92, 054104 (2015).
  12. A. N. Morozovska and E. A. Eliseev, Surface and finite size effect on fluctuations dynamics in nanoparticles with long-range order, J. Appl. Phys. 107, 044101 (2010).
  13. G. Shirane, J. D. Axe, J. Harada, and J. P. Remeika, Soft ferroelectric modes in lead titanate, Phys. Rev. B 2, 155 (1970).
  14. G. Shirane, J. D. Axe, J. Harada, and A. Linz, Inelastic neutron scattering from single-domain BaTiO3, Phys. Rev. B 2, 3651 (1970).
  15. G. Shirane, B. C. Frazer, V. J. Minkiewicz, J. A. Leake, and A. Linz, Soft optic modes in barium titanate, Phys. Rev. Lett. 19, 234 (1967).
  16. W. Cochran, Dynamical, scattering and dielectric properties of ferroelectric crystals, Adv. Phys. 18, 157 (1969).
  17. J. Hlinka, I. Gregora, and V. Vorlıcek, Complete spectrum of long-wavelength phonon modes in Sn2P2S6 by Raman scattering, Phys. Rev. B 65, 064308 (2002).
  18. R. M. Yevych, Yu. M. Vysochanskii, M. M. Khoma, and S. I. Perechinskii, Lattice instability at phase transitions near the Lifshitz point in proper monoclinic ferroelectrics, J. Phys.: Condens. Matter 18, 4047 (2006).
  19. A. N. Morozovska, Y. M. Vysochanskii, O. V. Varenyk, M. V. Silibin, S. V. Kalinin, and E. A. Eliseev, Flexocoupling impact on the generalized susceptibility and soft phonon modes in the ordered phase of ferroics, Phys. Rev. B 92, 094308 (2015).
  20. A. N. Morozovska, E. A. Eliseev, C. M. Scherbakov, and Yu. M. Vysochanskii, The influence of elastic strain gradient on the upper limit of flexocoupling strength, spatially-modulated phases and soft phonon dispersion in ferroics, Phys. Rev. B 94, 174112 (2016).
  21. A. N. Morozovska, M. D. Glinchuk, E. A. Eliseev, and Yu. M. Vysochanskii, Flexocoupling-induced soft acoustic mode and the spatially modulated phases in ferroelectrics, Phys. Rev. B 96, 094111 (2017).
  22. G. E. W. Bauer, R. Iguchi, and K.-I. Uchida, Theory of transport in ferroelectric capacitors, Phys. Rev. Lett. 126, 187603 (2021).
  23. G. E. W. Bauer, P. Tanga, R. Iguchie, and K.-I. Uchida, Magnonics vs. ferronics, J. Magn. Magn. Mater. 541, 168468 (2022).
  24. P. Tang, R. Iguchi, K. Uchida, and G. E. W. Bauer, Excitations of the ferroelectric order, Phys. Rev. B 106, L081105 (2022).
  25. G. E. W. Bauer, P. Tang, R. Iguchi, J. Xiao, K. Shen, Z. Zhong, T. Yu, S. M. Rezende, J. P. Heremans, and K. Uchida, Polarization transport in ferroelectrics, Phys. Rev. Appl. 20, 050501 (2023).
  26. X. H. Zhou, C. Cai, P. Tang, R. L. Rodríguez-Suárez, S. M. Rezende, G. E. W. Bauer, and T. Yu, Surface ferron excitations in ferroelectrics and their directional routing, Chin. Phys. Lett. 40, 087103 (2023).
  27. R. L. Rodríguez-Suárez, X. H. Zhou, C. Y. Cai, P. Tang, T. Yu, G. E. W. Bauer, and S. M. Rezende, Surface and volume modes of polarization waves in ferroelectric films, Phys, Rev. B 109, 134307 (2024).
  28. S. H. Zhuang and J.-M. Hu, Role of polarization-photon coupling in ultrafast terahertz excitation of ferroelectrics, Phys. Rev. B 106, L140302 (2022).
  29. T. Chen, B. Wang, Y. Zhu, S. Zhuang, L.-Q. Chen, and J.-M. Hu, Analytical model and dynamical phase-field simulation of terahertz transmission across ferroelectrics, Phys. Rev. B 109, 094305 (2024).
  30. B. L. Wooten, R. Iguchi, P. Tang, J. S. Kang, K. Uchida, G. E. W. Bauer, and J. P. Heremans, Electric field–dependent phonon spectrum and heat conduction in ferroelectrics, Sci. Adv. 9, eadd7194 (2023).
  31. F. Yang and L. Q. Chen, Microscopic theory of displacive ferroelectricity: Applications to quantum criticality and classical phase transitions, arXiv:2412.04308.
  32. G. D. Zhao, F. Yang, and L. Q. Chen, Thermal properties revisited in displacive ferroelectrics, arXiv:2501.17833.
  33. S. Rowley, L. Spalek, R. Smith, M. Dean, M. Itoh, J. Scott, G. Lonzarich, and S. Saxena, Ferroelectric quantum criticality, Nat. Phys. 10, 367 (2014).
  34. E. K. H. Salje, B. Wruck, and H. Thomas, Order-parameter saturation and low-temperature extension of Landau theory, Z. Phys. B 82, 399 (1991).
  35. Ju. Banys, A. Dziaugys, K. E. Glukhov, A. N. Morozovska, N. V. Morozovsky, and Yu. M. Vysochanskii, Van der Waals Ferroelectrics: Properties and Device Applications of Phosphorous Chalcogenides (John Wiley & Sons, Weinheim, 2022).
  36. S. M. Neumayer, E. A. Eliseev, M. A. Susner, B. J. Rodriguez, S. Jesse, S. V. Kalinin, M. A. McGuire, A. N. Morozovska, P. Maksymovych, and N. Balke, Giant negative electrostriction and dielectric tunability in a van der Waals layered ferroelectric, Phys. Rev. Mater. 3, 024401 (2019).
  37. Y. M. Vysochanskii, V. A. Stephanovich, A. A. Molnar, V. B. Cajipe, and X. Bourdon, Raman spectroscopy study of the ferrielectric-paraelectric transition in layered CuInP2S6, Phys. Rev. B 58, 9119 (1998).
  38. N. Sivadas, P. Doak, and P. Ganesh, Anharmonic stabilization of ferrielectricity in CuInP2Se6, Phys. Rev. Res. 4, 013094 (2022).
  39. J. A. Brehm, S. M. Neumayer, L. Tao, A. O’Hara, M. Chyasnavichus, M. A. Susner, M. A. McGuire, S. V. Kalinin, S. Jesse, P. Ganesh, and S. Pantelides, Tunable quadruple-well ferroelectric van der Waals crystals, Nat. Mater. 19, 43 (2020).
  40. A. N. Morozovska, E. A. Eliseev, S. V. Kalinin, Y. M. Vysochanskii, and P. Maksymovych, Stress-induced phase transitions in nanoscale CuInP2S6, Phys. Rev. B 104, 054102 (2021).
  41. A. N. Morozovska, E. A. Eliseev, M. E. Yelisieiev, Y. M. Vysochanskii, and D. R. Evans, Stress-induced transformations of polarization switching in CuInP2S6 nanoparticles, Phys. Rev. Appl. 19, 054083 (2023).
  42. A. N. Morozovska, E. A. Eliseev, L. P. Yurchenko, V. V. Laguta, Y. Liu, S. V. Kalinin, A. L. Kholkin, and Y. M. Vysochanskii, The strain-induced transitions of the piezoelectric, pyroelectric and electrocaloric properties of the CuInP2S6 films, AIP Advances 13, 125306 (2023).
  43. A. N. Morozovska, E. A. Eliseev, A. Ghosh, M. E. Yelisieiev, Y. M. Vysochanskii, and S. V. Kalinin, Anomalous polarization reversal in strained thin films of CuInP2S6, Phys. Rev. B 108, 054107 (2023).
  44. A. N. Morozovska, E. A. Eliseev, Y. Liu, K. P. Kelley, A. Ghosh, Y. Liu, J. Yao, N. V. Morozovsky, A. L. Kholkin, Y. M. Vysochanskii et al., Bending-induced isostructural transitions in ultrathin layers of van der Waals ferrielectrics, Acta Mater. 263, 119519 (2024).
  45. A. N. Morozovska, S. V. Kalinin, E. A. Eliseev, S. Kopyl, Y. M. Vysochanskii, and D. R. Evans, Ferri-ionic coupling in CuInP2S6 nanoflakes: Polarization states and controllable negative capacitance, Phys. Rev. Appl. 22, 034059 (2024).
  46. A. N. Morozovska, E. A. Eliseev, Y. M. Vysochanskii, S. V. Kalinin, and M. V. Strikha, Size effect of negative capacitance state and subthreshold swing in van der Waals ferrielectric field-effect transistors, Adv. Electron. Mater. 11, 2400495 (2024).
  47. I. Zamaraite, V. Liubachko, R. Yevych, A. Oleaga, A. Salazar, A. Dziaugys, J. Banys, and Yu. Vysochanskii, Quantum paraelectric state and critical behavior in Sn(Pb)2P2S(Se)6 ferroelectrics, J. Appl. Phys. 128, 234105 (2020).
  48. K. Moriya, H. Kuniyoshi, K. Tashita, Y. Ozaki, S. Yano, and T. Matsuo, Ferroelectric phase transitions in Sn2P2S6 and Sn2P2Se6 crystals, J. Phys. Soc. Jpn. 67, 3505 (1998).
  49. V. Liubachko, A. Oleaga, A. Salazar, A. Kohutych, K. Glukhov, A. Pogodin, and Yu. Vysochanskii, Cation role in the thermal properties of layered materials M1+M3+P2(S,Se)6 (M1+=Cu, Ag; M3+=In, Bi), Phys. Rev. Mater. 3, 104415 (2019).
  50. V. Maisonneuve, V. B. Cajipe, A. Simon, R. Von Der Muhll, and J. Ravez, Ferrielectric ordering in lamellar CuInP2S6, Phys. Rev. B 56, 10860 (1997).
  51. Y. Fagot-Revurat, X. Bourdon, F. Bertran, V. B. Cajipe, and D. Malterre, Interplay between electronic and crystallographic instabilities in the low-dimensional ferroelectric CuInP2Se6, J. Phys.: Condens. Matter 15, 595 (2003).
  52. Y. Vysochanskii, V. Liubachko, K. Glukhov, R. Yevych, A. Kohutych, V. Hryts, A. Dziaugys, and J. Banys, Ordering/displacive ferrielectricity in 2D CuInP2S6, Ferroelectrics 618, 906 (2024).
  53. V. Samulionis, J. Banys, A. Dziaugys, M. I. Gurzan, I. P. Pritz, and Y. Vysochanskii, Piezoelectric and ultrasonic Studies of new lamellar crystals of CuInP2S6 type, Ferroelectrics 419, 97 (2011).
  54. A. Kohutych, V. Liubachko, V. Hryts, Yu. Shiposh, M. Kundria, M. Medulych, K. Glukhov, R. Yevych, and Yu. Vysochanskii, Phonon spectra and phase transitions in van der Waals ferroics MM′P2X6, Mol. Cryst. Liq. Cryst. 747, 14 (2022).
  55. C. Chen, H. Liu, Q. Lai, X. Mao, J. Fu, Z. Fu, and H. Zeng, Large-scale domain engineering in two-dimensional ferroelectric CuInP2S6 via giant flexoelectric effect, Nano Lett. 22, 3275 (2022).
  56. W. Ming, B. Huang, S. Zheng, Y. Bai, J. Wang, J. Wang, and J. Li, Flexoelectric engineering of van der Waals ferroelectric CuInP2S6, Sci. Adv. 8, eabq1232 (2022).
  57. L. Wang, D. Zhang, Z.-D. Luo, P. Sharma, and J. Seidel, Flexoelectric and electrostatic effects on mechanical properties of CuInP2S6, Appl. Mater. Today 35, 101981 (2023).
  58. X. Wu, L. Qi, M. A. Iqbal, S. Dai, X. Weng, K. Wu, C. Kang, Z. Li, D. Zhao, W. Tang et al., Revealing strong flexoelectricity and optoelectronic coupling in 2D ferroelectric CuInP2S6 via large strain gradient, ACS Appl. Mater. Interfaces 16, 14038 (2024).
  59. L. D. Landau and E. M. Lifshitz, Theory of Elasticity. Theoretical Physics (Butterworth-Heinemann, Oxford, 1998), Vol. 7.
  60. G. A. Smolenskii, V. A. Bokov, V. A. Isupov, N. N. Krainik, R. E. Pasynkov, and A. I. Sokolov, Ferroelectrics and Related Materials (Gordon and Breach, New York, 1984).
  61. See Supplemental Material at http://link.aps.org/supplemental/10.1103/mr9v-khtj for calculation details and material parameters.
  62. B. A. Strukov and A. P. Levanyuk, Ferroelectric Phenomena in Crystals: Physical Foundations (Springer-Verlag, Berlin, 2012).
  63. A. N. Morozovska, E. A. Eliseev, M. D. Glinchuk, H. V. Shevliakova, G. S. Svechnikov, M. V. Silibin, A. V. Sysa, A. D. Yaremkevich, N. V. Morozovsky, and V. V. Shvartsman, Analytical description of the size effect on pyroelectric and electrocaloric properties of ferroelectric nanoparticles, Phys. Rev. Mater. 3, 104414 (2019).
  64. R. Rao, B. S. Conner, J. Jiang, R. Pachter, and M. A. Susner, Raman spectroscopy study of pressure-induced phase transitions in single crystal CuInP2S6, J. Chem. Phys. 159, 224706 (2023).
  65. X. Jiang, X. Wang, X. Wang, X. Zhang, R. Niu, J. Deng, S. Xu, Y. Lun, Y. Liu, T. Xia et al., Manipulation of current rectification in van der Waals ferroionic CuInP2S6, Nat. Commun. 13, 574 (2022).
  66. https://www.wolfram.com/mathematica.
  67. https://notebookarchive.org/2025-03-8a3v8ii.

Outline

Information

Phys. Rev. B 112, 014110– Published 21 July, 2025
  • Received 15 March 2025
  • Revised 31 May 2025
  • Accepted 27 June 2025
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