Effective Landau-type model of a HfxZr1−xO2 -graphene nanostructure

To describe charge-polarization coupling in the nanostructure formed by a thin ${\mathrm{Hf}}_x{\mathrm{Zr}}_{1-x}{\mathrm{O}}_2$ film with single-layer graphene as the top electrode, we develop the “effective” Landau-Ginzburg-Devonshire model. This approach is based on the parametrization of the Landau expansion coefficients for polar (ferroelectric) and antipolar (antiferroelectric) orderings in thin ${\mathrm{Hf}}_{1-x}{\mathrm{Zr}}_x{\mathrm{O}}_2$ 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 $\mathrm{Zr}/[\mathrm{Hf}+\mathrm{Zr}]$ ratio, x, in contrast to h-independent and linearly-x-dependent expansion coefficients of classical Landau energy. We explain the dependence of the Landau expansion coefficients by the strong nonmonotonic dependence of the polar properties on the ${\mathrm{Hf}}_{1-x}{\mathrm{Zr}}_x{\mathrm{O}}_2$ 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 changes to x and h in the range 0 < x < 1 and 5 nm < h < 35 nm. Using the effective free energy, we demonstrate that polarization of ${\mathrm{Hf}}_{1-x}{\mathrm{Zr}}_x{\mathrm{O}}_2$ films influences the graphene conductivity strongly, and the full correlation between the distribution of polarization and charge carriers in graphene is revealed. In accordance with our modeling, polarization of the 5–25-nm-thick ${\mathrm{Hf}}_{1-x}{\mathrm{Zr}}_x{\mathrm{O}}_2$ films, which are in ferroelectriclike or antiferroelectriclike states for chemical compositions of 0.35 ≤ x ≤ 0.95, determine the concentration of carriers in graphene and can control its field dependence. The result is promising for the creation of next-generation $\mathrm{Si}$-compatible nonvolatile memories and graphene-ferroelectric FETs, because the working voltages applied to the ${\mathrm{Hf}}_{1-x}{\mathrm{Zr}}_x{\mathrm{O}}_2$ film (which acts as a gate) can be relatively low (less than 2 V). These low voltages are sufficient to induce the pronounced hysteresis of ferroelectric polarization in the ${\mathrm{Hf}}_{1-x}{\mathrm{Zr}}_x{\mathrm{O}}_2$ gate, which, due to strong electric coupling, induces hysteresis of the graphene charge.