Effective Landau-type model of a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mml:msub><mml:mi>Hf</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:msub><mml:mi>Zr</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:math>-graphene nanostructure

Abstract

To describe charge-polarization coupling in the nanostructure formed by a thin ${\mathrm{Hf}}_{x}{\mathrm{Zr}}_{1\ensuremath{-}x}{\mathrm{O}}_{2}$ 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 ${\mathrm{Hf}}_{1\ensuremath{-}x}{\mathrm{Zr}}_{x}{\mathrm{O}}_{2}$ 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 ${\mathrm{Hf}}_{1\ensuremath{-}x}{\mathrm{Zr}}_{x}{\mathrm{O}}_{2}$ 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 ${\mathrm{Hf}}_{1\ensuremath{-}x}{\mathrm{Zr}}_{x}{\mathrm{O}}_{2}$ 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 ${\mathrm{Hf}}_{1\ensuremath{-}x}{\mathrm{Zr}}_{x}{\mathrm{O}}_{2}$ films, which are in ferroelectriclike or antiferroelectriclike states for chemical compositions of 0.35 \ensuremath{\le} x \ensuremath{\le} 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 ${\mathrm{Hf}}_{1\ensuremath{-}x}{\mathrm{Zr}}_{x}{\mathrm{O}}_{2}$ 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 ${\mathrm{Hf}}_{1\ensuremath{-}x}{\mathrm{Zr}}_{x}{\mathrm{O}}_{2}$ gate, which, due to strong electric coupling, induces hysteresis of the graphene charge.

Topics & Concepts