Abstract
To describe charge-polarization coupling in the nanostructure formed by a thin Hf饾懃鈦r1鈭掟潙モ仮O2 film with single-layer graphene as the top electrode, we develop the 鈥渆ffective鈥� 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 Hf1鈭掟潙モ仮Zr饾懃鈦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 classical Landau energy. We explain the dependence of the Landau expansion coefficients by the strong nonmonotonic dependence of the polar properties on the Hf1鈭掟潙モ仮Zr饾懃鈦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鈥�<鈥墄鈥�<鈥�1 and 5 nm鈥�<鈥塰鈥�<鈥�35 nm. Using the effective free energy, we demonstrate that polarization of Hf1鈭掟潙モ仮Zr饾懃鈦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鈥�25-nm-thick Hf1鈭掟潙モ仮Zr饾懃鈦2 films, which are in ferroelectriclike or antiferroelectriclike states for chemical compositions of 0.35鈥夆墹鈥墄鈥夆墹鈥�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 Si-compatible nonvolatile memories and graphene-ferroelectric FETs, because the working voltages applied to the Hf1鈭掟潙モ仮Zr饾懃鈦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 Hf1鈭掟潙モ仮Zr饾懃鈦2 gate, which, due to strong electric coupling, induces hysteresis of the graphene charge.
