Local polarization dynamics in ferroelectric materials

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Rep. Prog. Phys. 73 (2010) 056502 S V Kalinin et al (a) d 3 Q 1 Q 2 Q 3 V Q (b) (c) 0.8 µm Piezoresponse, a.u. 4 3 2 1 0 -1 -2 (d) single data set one charge model 0 1000 2000 3000 4000 5000 Distance, nm Figure 19. (a) Representation of a realistic tip by a set of image charges. (b) Surface topography and (c) PFM image of domain wall in LNO. (d) Fit of the extrapolated data set with equal weighting for all points. Reproduced from [218]. Copyright 2007, American Institute of Physics. Table 2. Effective image charge parameters for different ferroelectrics. Wall R 0 for R 0 for Material width (nm) Q d (nm) R d (nm) ε e = 1(µm) ε e = 80 (nm) LiNbO 3 96 1000 92 58.6 4.8 60 Epitaxial PZT 107 2550 125 79.6 5 62.5 PZT in air 58 723 86.5 55.1 44 541 PZT in liquid 6 104 11.8 7.5 75 charges can be deconvoluted from the experimental domain wall profile by minimizing the functional ∫ ( 1 F [u 3 ] = PR(a) − 2πε 0 (ε e + κ) N∑ m=0 ) 2 Q m ũ 3 (s m ) da d m (2.29) with respect to the number, N, magnitude and charge–surface separation of a set of image charges {Q i ,d i } N representing the tip. Here PR(a) is the measured piezoresponse and integration is performed over all available a values. The dielectric constant of the medium can be fixed to the value of free air (ε e = 1) or water in the tip–surface junction or imaging in liquid (ε e = 80). The output of the fitting process is the set of reduced charges q i = Q i /2πε 0 and the charge surface separation, d i . Note that the charges and the dielectric constants cannot be determined independently, since only Q m /(ε e + κ)ratios enter equation (2.29). Shown in figures 19(b)–(d) is an example of a domain wall profile and the corresponding fit by equation (2.29) with N = 1 for LNO. The corresponding image charge parameters are summarized in table 2. To improve the fit quality, more complex fitting functions with N = 2 and N = 3 were attempted. However, independent of the choice of the initial values of the image charge, the fit converged to a single image charge, i.e. d i = d and ∑ Q i = Q. Similar behavior was observed for other domain walls. This analysis suggests that the electrostatic field produced by the tip is consistent with a single point charge positioned at large separation from the surface, contrary to the behavior anticipated in contact mode imaging. The analysis was extended to the case of the sphere–plane (radius of curvature R 0 ) and disk–plane (radius R d ) models. The sphere parameters are calculated both from ambient and water environment to account for possible capillary condensation effects. From the data, it is clear that the use of the sphere/air model leads to implausibly large radii. Hence, experimental data are consistent either with the presence of a capillary water film in the sphere model, or conductive disk model [218]. 2.4.2. Resolution for non-zero contact area. Extensive quantitative analysis of domain wall width, taking into account the finite contact tip–surface contact area, was reported by the Gopalan group [183]. The geometric structure of more than 100 SPM tips was ascertained using scanning electron microscopy to yield effective tip sizes [212], see figures 20(a) and (b). The domain wall width in periodically poled lithium niobate was measured as a function of tip radius. In parallel, the domain wall width was calculated using a numerical finite element analysis package, figure 20. The comparison of the experimentally measured PFM signal and domain wall width as a function of tip radius with numerical and analytical theory predictions is shown in figure 21. The data clearly suggest that the PFM signal is independent of contact area, in agreement with the theory of Karapetian [175, 176]. At the same time, the domain wall width saturates at ∼100 nm, corresponding to a tipindependent domain wall width. This can be attributed both to the effect of an ambient environment on imaging, and to wall broadening, necessitating further studies in an ultra-high vacuum environment. 21
Rep. Prog. Phys. 73 (2010) 056502 S V Kalinin et al Figure 20. (a) Field emission (FE) SEM image of a used PFM tip with a circular disk-like end with a radius r. (b) FE-SEM image of a sphere-like PFM tip. The radius r of the contact circle for a weak indentation (h ∼ 1–2 nm or 1 unit cell depth) is used to characterize the radius r of the tip as shown. (h is not to scale in the figure). (c) Finite element simulations of surface potential on LN surface under a 50 nm radius disk tip in contact with sample with 5 V applied. (d) FEM simulated piezoelectric displacements, U z of a LNO z-surface. Displacements for three different tip locations are shown: tip located on the wall and away from the wall on either side. Reproduced from [212]. Copyright 2008, American Institute of Physics. Figure 21. The maximum amplitude, |U z |, away from the wall as a function of tip radius, r, is shown. Also shown overlapped are the analytical theory and finite element method (FEM) predictions. (b) PFM wall width as a function of tip radius for sphere-like and disk-like PFM tips. Also shown are analytical theory predictions for different intrinsic wall half-widths. Reproduced from [212]. Copyright 2008, American Institute of Physics. 2.5. Implications for PFM data analysis Below, we summarize the implications of the contrast formation mechanism and resolution function analyses for the interpretation of PFM imaging and spectroscopy. The effect of finite resolution is that quantitative properties such as domain-specific effective electromechanical response can be measured only for domains much larger than the RTR value. At the same time, a domain will be visible above the information limit, but no quantitative measurements can be obtained. A good measure of the RTR is the domain wall width in the mixed PFM signal, while the IL can be determined from the domain wall width in the phase image. 22
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Local polarization dynamics in ferroelectric materials

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