  To describe the charge-polarization coupling in the nanostructure formed by a
thin Hf$_x$Zr$_{1-x}$O$_2$ film with a single-layer graphene as a top
electrode, we develop the "effective" Landau-Ginzburg-Devonshire model. This
approach is based on the parametrization of the Landau expansion coefficients
for 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 film
thickness h and Zr/[Hf+Zr] ratio x, in contrast to h-independent and linearly
x-dependent expansion coefficients of a classical Landau energy. We explain the
dependence of the Landau expansion coefficients by the strong nonmonotonic
dependence 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 and
h, describes the continuous transformation of polarization dependences on
applied electric field and hysteresis loop shapes induced by the changes of x
and h in the range 0 < x < 1 and 5 nm < h < 35 nm. Using the effective free
energy, we demonstrated that the polarization of Hf$_x$Zr$_{1-x}$O$_2$ films
influences strongly on the graphene conductivity, and the full correlation
between the distribution of polarization and charge carriers in graphene is
revealed. In accordance with our modeling, the polarization of the (5 - 25) nm
thick Hf$_x$Zr$_{1-x}$O$_2$ films, which are in the ferroelectric-like or
antiferroelectric-like states for the chemical compositions 0.35 < x < 0.95,
determine the concentration of carriers in graphene and can control its field
dependence. The result can be promising for creation of next generation
Si-compatible nonvolatile memories and graphene-ferroelectric FETs.
