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To describe the charge-polarization coupling in the nanostructure formed by athin Hf$_x$Zr$_{1-x}$O$_2$ film with a single-layer graphene as a topelectrode, we develop the "effective" Landau-Ginzburg-Devonshire model. Thisapproach is based on the parametrization of the Landau expansion coefficientsfor the polar (FE) and antipolar (AFE) orderings in thin Hf$_x$Zr$_{1-x}$O$_2$films from a limited number of polarization-field curves and hysteresis loops.The Landau expansion coefficients are nonlinearly dependent on the filmthickness h and Zr/[Hf+Zr] ratio x, in contrast to h-independent and linearlyx-dependent expansion coefficients of a classical Landau energy. We explain thedependence of the Landau expansion coefficients by the strong nonmonotonicdependence of the polar properties on the Hf$_x$Zr$_{1-x}$O$_2$ film thickness,grain size and surface energy. The proposed Landau free energy with five"effective" expansion coefficients, which are interpolation functions of x andh, describes the continuous transformation of polarization dependences onapplied electric field and hysteresis loop shapes induced by the changes of xand h in the range 0 < x < 1 and 5 nm < h < 35 nm. Using the effective freeenergy, we demonstrated that the polarization of Hf$_x$Zr$_{1-x}$O$_2$ filmsinfluences strongly on the graphene conductivity, and the full correlationbetween the distribution of polarization and charge carriers in graphene isrevealed. In accordance with our modeling, the polarization of the (5 - 25) nmthick Hf$_x$Zr$_{1-x}$O$_2$ films, which are in the ferroelectric-like orantiferroelectric-like states for the chemical compositions 0.35 < x < 0.95,determine the concentration of carriers in graphene and can control its fielddependence. The result can be promising for creation of next generationSi-compatible nonvolatile memories and graphene-ferroelectric FETs.

An effective Landau-type model of Hf$_x$Zr$_{1-x}$O$_2$ thin film - graphene nanostructure