High-frequency investigation of polarization effects in nanoscale hafnium-based ferroelectrics

In this paper, we present for the first time a rigorous multi-physics approach in the time domain to investigate the polarization effects in ultra-thin (i.e., with a thickness smaller than 10 nm) layers of hafnium-based ferroelectrics at GHz frequencies. Starting from the Ginzburg-Landau theory, we have taken into account the full coupling between the material polarization and the Maxwell system of equations, in order to capture the nonlinear effects and the hysteretic residual behavior occurring at high frequencies. A thorough comparison with the measured low-frequency polarization-electric field loops constitutes a basis for the subsequent high-frequency model of the fabricated metal-ferroelectric-metal (MFM) capacitor. Then, by resuming to an electromagnetic model of the MFM capacitor embedded into a coplanar waveguide line, we demonstrate the effects of the ferroelectric hysteresis up to 10 GHz and for an input power of 30 dBm. Finally, we investigate the effect of the conductor losses, that results in a finite field decay length (skin depth) and a spatially distributed charge stored near to the contact interface with the ferroelectric. The proposed methodology can be of great use in predicting and analyzing the performance of devices and components, especially fast switches and transistors, based on nanoscale ferroelectric materials.