Local polarization dynamics in ferroelectric materials

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Rep. Prog. Phys. 73 (2010) 056502 for the PFM response of materials with low symmetry, derive analytical expressions for the PFM resolution function and domain wall profiles, and interpret PFM spectroscopy data, as described below. In the decoupling approximation, the PFM signal, i.e. the surface displacement u i (x, y) at location x induced by the tip at position y = (y 1 ,y 2 ) is given by ∫ ∞ ∫ ∞ ∫ ∞ ∂G ij (x 1 −ξ 1 ,x 2 −ξ 2 ,ξ 3 ) u i (x,y)= dξ 1 dξ 2 dξ 3 ∂ξ k −∞ −∞ 0 ×E l (ξ)c kj mn d lnm (y 1 +ξ 1 ,y 2 +ξ 2 ,ξ 3 ). (2.6) Here, coordinate x = (x 1 ,x 2 ,z) is linked to the indentor apex, coordinates y = (y 1 ,y 2 ) denote indentor position in the sample coordinate system, y. Coefficients d lnm (y) and c kj mn are position dependent components of the piezoelectric strain and elastic stiffness tensors, respectively. E l (x) is the electric field strength distribution produced by the probe. Green’s function for a semi-infinite medium G 3j (x − ξ) links the eigenstrains d lnm c kj mn E l (x) to the displacement field, and thus allows calculating the nonlocal displacement field from local field-induced strains. For most inorganic ferroelectrics, the elastic properties are weakly dependent on orientation and hence can be approximated as elastically isotropic, and integrals in equation (2.6) can be evaluated in the closed form. The corresponding Green’s tensor, G ij (x − ξ), for an elastically isotropic half-plane is given by Lur’e [187] and Landau and Lifshitz [188]. This approach is rigorous for materials with small piezoelectric coefficients. A simple validity estimation is based on the value of the square of the dimensionless electromechanical coupling coefficients kij 2 = (d ij ) 2 /(s jj ε ii ). For instance, for barium titanate (BTO) single crystal: k15 2 ≈ 0.32, k31 2 ≈ 0.10 and k33 2 ≈ 0.31, for the ceramics lead zirconate titanate (PZT) [PZT6B grade]: k15 2 ≈ 0.14, k31 2 ≈ 0.02 and k33 2 ≈ 0.13 and for a quartz single crystal: k11 2 ≈ 0.01, suggesting that the error in the decoupling approximation does not exceed ∼30% for strongly piezoelectric materials and is on the order of ∼1% for weak piezoelectrics. 2.2.2.1. Approximate solution for homogeneous media. The use of the decoupling approximation reduces an extremely complex coupled contact mechanics problem with mixed boundary conditions to the solutions of much simpler electrostatic and mechanical Green’s function problems, and numerical or analytical integration of the result. We note that the dielectric and particularly elastic properties described by positively defined second- and fourth-rank tensors (invariant with respect to 180 ◦ rotations) are necessarily more isotropic than the piezoelectric properties described by third-rank tensors (anti-symmetric with respect to 180 ◦ rotations). Hence, elastic and dielectric properties of a material can often be approximated as isotropic. In this case, integrals in equation (2.6) can be evaluated, and surface displacement can be written in the form u i (x) = V Q W ij lk (x)d kj l , where the tensor W ij lk (x) is symmetrical on the transposition of S V Kalinin et al the indices j and l. In Voigt notation, the displacements are [184] u 1 (0)=V Q (W 111 d 11 +W 121 d 12 +W 131 d 13 +W 153 d 35 +W 162 d 26 ), (2.7a) u 2 (0)=V Q (W 121 d 21 +W 111 d 22 +W 131 d 23 +W 153 d 34 +W 162 d 16 ), (2.7b) u 3 (0)=V Q (W 313 (d 31 +d 32 )+W 333 d 33 +W 351 (d 24 +d 15 )). (2.7c) The non-zero elements of the tensor W iαk are W 111 =−(13+4ν)/32, W 121 = (1 − 12ν)/32, W 131 =−1/8, W 153 =−3/8, (2.8a) W 162 =−(7 − 4ν)/32, W 313 =−(1+4ν)/8, W 333 =−3/4, W 351 =−1/8. (2.8b) Here V Q is the electrostatic potential at the point x = (0, 0, 0) produced by a probe represented by the set of image charges located on a vertical line. Thus, the response is shown to be proportional to the potential induced by the tip on the surface. The latter fundamental result was generalized as response theorems [185]. Response theorem 1. For a transversally isotropic piezoelectric solid in an isotropic elastic approximation and an arbitrary point charge distribution in the tip (not necessarily constrained to a single line), the vertical surface displacement is proportional to the surface potential induced by tip charges in the point of contact. Response theorem 2. For an anisotropic piezoelectric solid in the limit of dielectric and elastic isotropy, the vertical and lateral PFM signals are proportional to the potential on the surface induced by the tip if the tip charges and the point of contact are located on the same line along the surface normal. Equations (2.8a) and (2.8b) thus constitute approximate PFM contrasts for anisotropic materials in the limit of elastic and dielectric isotropy. For lower material symmetries, the analytical expressions of the displacement fields induced by the point charge allow response calculations for asymmetric tips or probes with special geometry [185]. 2.2.2.2. Orientational imaging. In piezoelectric materials, the strong orientation dependence of electromechanical response [189–194] opens the pathway to orientation imaging [148]. Briefly, the orientation of a 3D object in the laboratory coordinate system is given by three Euler angles (θ,ϕ,ψ). The relationship between the d ij k tensor in the laboratory coordinate system and the dij 0 k tensor in the crystal coordinate system is d ij k = A il A jm A kn dlmn 0 , where A ij (θ,ϕ,ψ) is the rotation matrix [195]. Experimentally, PFM measures three components of the response vector. Hence, local crystallographic orientation can be determined from the solutions of equations (2.7a)–(2.7c), which can graphically be represented as response surfaces. As an example, we compare vertical displacement, u 3 , surfaces with piezoelectric tensors, d 33 , and surfaces for tetragonal PbTiO 3 (PTO) and 7
Rep. Prog. Phys. 73 (2010) 056502 S V Kalinin et al Figure 4. The dependence of (a) displacement u 3 ,(b) piezoelectric tensor component e 33 , for PbTiO 3 (PTO, ν = 0.3 (Poisson’s ratio)) on Euler’s angle θ in the laboratory coordinate system. The dependence of (c) piezoelectric tensor component e 33 and (d) displacement u 3 for LiTaO 3 (LTO, ν = 0.25) on Euler’s angles ϕ, θ in the laboratory coordinate system. Reproduced from [185]. Copyright 2007, American Institute of Physics. trigonal LiTaO 3 (LTO) model systems in figure 4. Note that while in this analysis, the dielectric properties of the material are assumed to be close to isotropic and hence the electric field distribution is insensitive to sample orientation, similar analysis can be performed for full dielectric and elastic anisotropy. The dependence of the piezoelectric tensor component e 35 versus the orientation of the crystallographic axes with respect to the laboratory coordinate system for a LTO crystal is shown in the upper row of figure 5. The horizontal displacement below the tip versus the orientation of the crystallographic axes with respect to the laboratory coordinate system for a LTO crystal is shown in the bottom row of figure 5. A common feature of the displacement surfaces shown in figures 4 and 5 is that the u 1 angular distribution is smoother, much more symmetric and convex than the one for e 35 . Similarly to the longitudinal components of the piezoelectric tensors e 33 and d 33 , the d 35 surfaces are very similar to e 35 surfaces. 2.3. Resolution theory in PFM One of the basic parameters characterizing performance of a microscope is the spatial resolution. Despite the ubiquity of usage and ‘intuitive’ meaning, the resolution in SPM is typically defined ad hoc. A quantitative imaging theory in PFM (and other SPMs) is required in order to: • define the resolution and information limits in PFM and establish their dependence on tip geometry and materials properties, hence suggesting strategies for high-resolution imaging; • develop the pathways for calibration of tip geometry in the PFM experiment for quantitative data interpretation; • interpret the imaging and spectroscopy data in terms of intrinsic domain wall widths and the size of the nascent domain below the tip; • reconstruct the ideal image from experimental data (deconvolute tip contribution), and establish applicability limits and errors associated with such deconvolution processes. In this section, we describe the basic principles of linear imaging theory, provide definitions of resolution and information limit and describe instrumental and theoretical aspects of resolution function theory in PFM. 2.3.1. Linear imaging theory: transfer function, resolution and information limit. The definition of spatial resolution and resolution theory have originally evolved in the context of optical and electron microscopy (EM). In optics, the Rayleigh criterion [196] defines the resolution as the minimum distance by which two point scatterers must be separated in order to be discernible for a given imaging system. A commonly used alternative reading of the criterion postulates that for two Gaussian-shaped image features of similar intensity to be resolved, the dip between the two maxima should be at least 21% of the maximum. This criterion is illustrated in figure 6(a) and shows the transition of the two features from completely resolved to unresolved as a function of the separating distance. Note that the criterion is not absolute. It is possible that for a system with a sufficiently high signal-to-noise ratio, peaks separated by less than Rayleigh’s resolution can be discernible 8
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