Kramers' escape problem for white noise driven switching in ferroelectrics

TL;DR

The paper addresses whether stochastic resonance can be realized in ferroelectric devices by mapping polarization dynamics onto Kramers' escape over an energy barrier and solving a Landau-Ginzburg-Devonshire description under thermal and external noise. It combines a homogeneous TDGL framework with an Euler-Maruyama solver and Linear Response Theory to predict and verify SR, and extends the analysis to a multidomain MFM model to assess real-world feasibility. A key result is that the SR output power peaks when ${\Omega}/{2\pi} \approx {r_K}/{2}$ and the optimal external-noise level scales as $D_{ext}/\Delta F \approx 0.235$ for the explored frequency, with theory and numerics in good agreement. The multidomain simulations corroborate the single-domain findings, suggesting SR-based weak-signal detection could be feasible in CMOS-compatible ferroelectrics like Hf0.5Zr0.5O2 (HZO).

Abstract

A simulation-based study of stochastic resonance (SR) in a ferroelectric capacitor is presented. The SR phenomenon involves the detection of weak signals by adding an optimal amount noise to a non-linear system. This is linked with Kramers' escape problem, which deals with the escape of a particle undergoing Brownian motion over an energy barrier. The position of the particle is analogous to the polarisation dynamics of a ferroelectric. Within this framework, we numerically investigate SR in single domain ferroelectrics using the Landau-Ginzburg-Devonshire (LGD) theory. In addition, we use a model for multidomain ferroelectrics to demonstrate feasibility in real world applications. Our results show that SR in ferroelectrics is promising for the purpose of weak signal detection, given that these materials are widely used for various applications in the semiconductor industry.

Figures (4)

• Figure 1: Kramers' problem for a double well potential landscape in a ferroelectric. The blue(solid) line depicts the double well potential when a voltage $V=0$ is applied across the ferroelectric, while the red(dashed) line depicts the situation under an applied voltage $0<V<V_c$. $V_c$ is the coercive voltage where the potential barrier disappears. The state of polarisation denoted by the red disc is analogous to a particle trapped in a potential well. Assuming the polarisation is at point A when $V < V_c$, it can still escape the well, over point C, when subject to thermal fluctuations. Kramers time quantifies the average lifetime of system in state A.
• Figure 2: Verification of Kramers time through numerical simulations. Our results match the analytical predictions, based on eq. (\ref{['eq:krrate']}), very well.
• Figure 3: (a) Schematic showing quasi-periodic switching of a ferroelectric due to a weak sinusoidal input (amplitude below $V_c$) and noise. (b,c) SNR and Output Power as a function of normalised noise intensity ($D_{ext}/\Delta F$). The dotted line shows the approximate noise intensity value that yields maximum output power, determined from Kramers' theory (${\Omega}/{2\pi} \approx {r_K}/{2}$).
• Figure 4: Ferroelectric modelled as multiple domains which interact through the coupling constant $k$.