Variation and Stochasticity in Polycrystalline HZO based MFIM: Grain-Growth Coupled 3D Phase Field Model based Analysis
Revanth Koduru, Atanu Saha, Mengwei Si, X. Lyu, P. D. Ye, S. K. GUPTA
Abstract
We present the first 3D phase-field simulation framework for ferroelectric (FE) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\text{Hf}_{1-\mathrm{x}}\text{Zr}_{\mathrm{x}}\mathrm{O}_{2}$</tex> (HZO) based MFIM stack that self-consistently couples grain-growth, multi-domain (MD) polarization ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P$</tex> ) switching and electrostatics. Based on 3D grain-growth models, time-dependent Ginzburg-Landau (TDGL) equations and Poisson's equations, our framework captures the effects of polycrystalline grain size/shape distribution and inter- and intra-grain domain interactions. Using our model, we first analyze the implication of multi-domain <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P$</tex> -switching in a single grain signifying the correlation of different macroscopic properties with respect to the crystal angle. Then we analyze, how the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$F$</tex> -switching in one grain affects the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P$</tex> -switching in the neighboring grains. Next, we focus on the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P$</tex> -switching in a multi-grain scenario and analyze the sample-to-sample variations and cycle-to-cycle stochasticity in a small area FE sample. Finally, we show how FE thickness scaling (from 7nm to 3nm) can reduce variation and stochasticity in an MFIM stack by approximately 30%.