Extrinsic size effect in piezoresponse force microscopy of thin films

The extrinsic size effect in piezoresponse force microscopy of ferroelectric and piezoelectric thin films on nonpolar dielectric substrate with matching elastic properties is analyzed. Analytical expressions for effective piezoresponse, object transfer function components, and Rayleigh two-point resolution are obtained. These results can be broadly applied for effective piezoelectric response calculations in thin piezoelectric and ferroelectric films as well as surface polar layers in, e.g., organic materials and biopolymers. In particular, the effective piezoresponse strongly decreases with film thickness, whereas the sharpness of domain stripe image increases due to the object transfer function spectrum broadening.

Figure 1

(Color online) (a) Coordinate systems in the PFM experiment. Electrostatic potential cross sections $\phi (x_1,0,0)$ (main plots) and $\phi (0,0,x_3)$ (insets) for the (b) sphere-plane and (c) disk models of tip apex with size $R_0$ in contact with surface. Solid curves: numerical calculations at $\gamma =1$, $\kappa =30$, and ${\epsilon }_e=1$. Dashed curves: total point charge model. (d) Dependence of effective distance $d∕R_0$ on the thickness $h∕R_0$ calculated for the spherical tip apex in contact with $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$ film ($\kappa =121$, $\gamma =0.87$) on $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$ substrate $({\kappa }_b=260)$ at ${\epsilon }_e=1$ and ${\epsilon }_e=10$.

Figure 2

(Color online) (a) Effective piezoelectric response $d_{33}^{\mathit{eff}}$ via the thickness $h$ of $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$ film ($\kappa =121$, $\gamma =0.87$) capped on substrate with the same elastic properties $(\nu =0.3)$ and different dielectric constants ${\kappa }_b=3$, 30, 300, and $3\times {10}^3$ (curves 1, 2, 3, and 4 respectively); ${\epsilon }_e\le 10$. Solid curves: exact series given in Ref. 21. Dashed curves: approximations given by Eq. (5). (b) $d_{33}^{\mathit{eff}}$ via the thickness $h$ of $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$ film on $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$ substrate $({\kappa }_b=260)$ at ${\epsilon }_e=10$ and different tip sizes $R_0=5$, 15, 50, and $100\mathrm{nm}$ (curves 1, 2, 3, and 4 from the top to bottom) for the flattened (dotted curves, $d=2R_0∕\pi$) and spherical [solid curves, $d\left(h\right)$ shown in Fig. 1d] tip apex.

Figure 3

(Color online) (a) PFM profile of periodic stripe domain structure (b) in $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$ film ($\kappa =121$, $\gamma =0.87$) on $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$ substrate (${\kappa }_b=260$, $\gamma =1$) for different film thicknesses $h∕d=0.03$, 0.1, 0.3, and 1 (curves 1, 2, 3, and 4). (c) Maximal piezoresponse vs film thickness for different stripe periods $a∕d=0.03$, 0.1, 0.3, 1, and 3 (curves 1, 2, 3, 4, and 5). (d) Dependence of response on stripe period for different film thicknesses $h∕d=0.03$, 0.1, 0.3, and 1 (curves 1, 2, 3, and 4).

Figure 4

(Color online) [(a) and (b)] Information limit for maximal piezoresponse $\max \left\{d_{33}^{\mathit{eff}}(a,h)\right\}$ of the domain stripes with period $a$ in the $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$ film of thickness $h$ on $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$ substrate for (a) different noise levels $n$ denoted by figures near the curves in pm/V and (b) for noise level $n=5\mathrm{pm}∕\mathrm{V}$, ${\epsilon }_e=40$, and different tip apex curvatures $R_0$. The inset shows 3D plot of maximal piezoresponse $\max \left\{d_{33}^{\mathit{eff}}(a,h)\right\}$.