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

Nanosized hafnia-zirconia <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:msub> <a:mi mathvariant="normal">Hf</a:mi> <a:mi>x</a:mi> </a:msub> <a:msub> <a:mi mathvariant="normal">Zr</a:mi> <a:mrow> <a:mn>1</a:mn> <a:mo>−</a:mo> <a:mi>x</a:mi> </a:mrow> </a:msub> <a:msub> <a:mi mathvariant="normal">O</a:mi> <a:mn>2</a:mn> </a:msub> </a:mrow> </a:math> 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:msub> <e:mi>HfO</e:mi> <e:mn>2</e:mn> </e:msub> </e:mrow> </e:math> 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 <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:msub> <f:mi mathvariant="normal">Hf</f:mi> <f:mi>x</f:mi> </f:msub> <f:msub> <f:mi mathvariant="normal">Zr</f:mi> <f:mrow> <f:mn>1</f:mn> <f:mo>−</f:mo> <f:mi>x</f:mi> </f:mrow> </f:msub> <f:msub> <f:mi mathvariant="normal">O</f:mi> <f:mn>2</f:mn> </f:msub> </f:mrow> </f:math> solid solutions, providing that corresponding parameters of the free energy are known from the first-principles calculations.