Coherent ferroelectric switching by atomic force microscopy

A classical energy approach is applied to study polarization reversal induced by voltage-modulated atomic force microscopy in ferroelectric single crystals and thin films. Formations of a reversed domain nucleus, its forward growth in a bulk, and sidewise expansions in a film are described by analyzing the domain energetics under an increasing external bias. It is shown that (i) the nucleation is possible even at low driving voltages, (ii) contact instability is typical for a domain in approaching an electrode, and (iii) different stages of the switching process are delimited on the voltage scale.

Figure 1

Nucleation in $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$ crystal. (a) Energy surface $\Delta W(r,l)$ near the saddle point $A$ at bias $U=1.6\mathrm{V}$ on the tip $TB$ (top view). Values in eVs are given on curves. (b) Activation energy versus applied bias. Barrier heights (max) at critical loads $(U_{\mathrm{cr}}+0)$ on both tips $(TS,TB)$ are marked.

Figure 2

(a) Energy versus domain length at radius $r=10\mathrm{nm}$ for $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$ bulk crystal (dashed lines, open symbols) and film of thickness $100\mathrm{nm}$ (solid lines, filled symbols). Maxima and bulk minima are depicted by squares and circles, respectively, and denoted by the letters in 2D problem. Voltages on the tip $TS$ are given and 1D bottom barrier at $10\mathrm{V}$ is shown. (b) Bulk growth in the crystal (cross-section). Reversed domains formed at definite biases are confined by solid curves and the shadow covers the domain of axes$(r^{*},l^{*})$ at $U=30\mathrm{V}$. A part of the tip $TS$ is displayed.

Figure 3

Four stages of the AFM switching in $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$ film $(H=100\mathrm{nm})$ under an increasing bias $U$ on the tip $TB$. Reversed domains are shadowed and their boundaries are drawn by solid or dot-dash curves. Legends with the domain-growth time $t_{\mathrm{gr}}$ show what process restricts the rate of a repolarization stage on the time scale ($t_w$: thermal fluctuation; $t_a$: sound propagation; $t_r$: Maxwell relaxation).