[Paper Review] Thickness-Dependence of the Coercive Field in Ferroelectrics

P. Chandra, Matthew Dawber+2 more|arXiv (Cornell University)|Jun 3, 2002

Ferroelectric and Piezoelectric Materials1 references3 citations

TL;DR

This paper proposes a modified Kolmogorov-Avrami model that accounts for inhomogeneous nucleation and electrode field penetration to explain the thickness-dependence of the coercive field in ferroelectrics. It shows that including Thomas-Fermi screening in metal electrodes quantitatively recovers Kay-Dunn scaling over five decades of thickness (100 μm to 1 nm), resolving long-standing discrepancies in ultrathin films like PVDF without requiring new physics.

ABSTRACT

For forty years researchers on ferroelectric switching have used the Kay-Dunn theory to model the thickness-dependence of the coercive field; it works surprisingly well, despite the fact that it is based upon homogeneous nucleation and a small-field expansion, neither of which is realized in thin films. Here we demonstrate that this result can be obtained from a more general Kolmogorov-Avrami model of (inhomogeneous) nucleation and growth. By including a correction to the switching field across the dielectric that includes Thomas-Fermi screening in the metal electrode, we show that our theory quantitatively describes the coercive fields versus thickness in several different families of ferroelectric (lead zirconate-titanate [PZT], potassium nitrate, and polyvinylidenefluoride [PVDF]) over a wide range of thickness (5 decades). This agreement is particularly satisfying in the case of PVDF, as it indicates that the switching kinetics are domain-wall limited down to 1 nanometer and thus require no new effects.

Motivation & Objective

• To resolve the long-standing discrepancy between the empirically successful Kay-Dunn scaling law and the physical inapplicability of its assumptions in thin ferroelectric films.
• To explain the observed deviation from Kay-Dunn scaling in ultrathin ferroelectric films (especially PVDF below 100 nm) by incorporating electrode field penetration and depolarization fields.
• To demonstrate that domain-wall-limited switching persists down to 1 nm, eliminating the need for novel nanoscale switching mechanisms.
• To provide a tunable framework for optimizing film thickness and electrode choice in FeRAM and DRAM applications.

Proposed method

• Adapts the Kolmogorov-Avrami model to describe inhomogeneous nucleation and growth of ferroelectric domains in confined thin films.
• Models the untransformed fraction as P(E,f) ~ exp[-N(E)(κd)], where N(E) ∝ E^{3/2} and κ is the transverse domain area fraction.
• Incorporates field penetration in electrodes via Thomas-Fermi screening, introducing a depolarization field E_dp ∝ P_s / (ε_f λ_TF) that modifies the effective switching field.
• Derives a corrected coercive field E_c(d) that includes electrode screening effects, with E_dp = P_s / (ε_f λ_TF) and λ_TF as the screening length.
• Uses experimental parameters (e.g., ε_f = 14 for PVDF, λ_TF = 0.45 Å for Al) to fit data without adjustable parameters.
• Validates the model by collapsing data from PZT, KNO₃, and PVDF onto a universal log E_c vs. log C curve when corrected for E_dp.

Experimental results

Research questions

• RQ1 Why does the Kay-Dunn scaling law E_c ∝ d^{-2/3} hold across five decades of thickness despite its reliance on unphysical assumptions like homogeneous nucleation?
• RQ2 What causes the observed deviation from Kay-Dunn scaling in ultrathin ferroelectric films (e.g., PVDF below 100 nm)?
• RQ3 To what extent do electrode field penetration and depolarization fields explain the breakdown of scaling at nanoscale thicknesses?
• RQ4 Can the coercive field in ultrathin films be quantitatively described using a unified model that recovers scaling when electrode effects are corrected?
• RQ5 How can the minimum stable film thickness be tuned via material parameters like spontaneous polarization and electrode screening length?

Key findings

• The modified model, incorporating Thomas-Fermi screening in electrodes, quantitatively describes E_c(d) across five decades of thickness (100 μm to 1 nm) for PZT, KNO₃, and PVDF with no adjustable parameters.
• For PVDF films below 100 nm, the observed deviation from Kay-Dunn scaling is quantitatively explained by electrode field penetration and depolarization fields, not new physics.
• Correcting measured E_c values for depolarization fields restores Kay-Dunn scaling universally across materials, confirming E_c ∝ d^{-2/3} when electrode effects are accounted for.
• The minimum film thickness d_min is tunable via spontaneous polarization P_s and electrode screening length λ_TF, with semiconducting electrodes yielding larger d_min than perfect conductors.
• For FeRAMs, the optimal processing window lies between 10 nm and 100 nm, where E_c is below the idealized scaling value but before depolarization instability sets in.
• For DRAMs, total capacitance is limited by electrode screening properties, not film thickness or dielectric constant, favoring high-resistivity ferroelectrics and electrodes with short λ_TF (e.g., Pt, Au over SrTiO₃).