Flexocoupling-induced phonons and ferrons in van der Waals ferroelectrics

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>CuIn</a:mi><a:msub><a:mi mathvariant="normal">P</a:mi><a:mn>2</a:mn></a:msub><a:msub><a:mi mathvariant="normal">S</a:mi><a:mn>6</a:mn></a:msub></a:mrow></a:math> 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 <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>CuIn</d:mi><d:msub><d:mi mathvariant="normal">P</d:mi><d:mn>2</d:mn></d:msub><d:msub><d:mi mathvariant="normal">S</d:mi><d:mn>6</d:mn></d:msub></d:mrow></d:math> 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 <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi>CuIn</g:mi><g:msub><g:mi mathvariant="normal">P</g:mi><g:mn>2</g:mn></g:msub><g:msub><g:mi mathvariant="normal">S</g:mi><g:mn>6</g:mn></g:msub></g:mrow></g:math> at low temperatures.