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symmetry.”) Polarization switching has light sources (see (Credit: Chen, et al.; American Institute of Physics. Reprinted with-liant beams of X-rayswith tunable photonenergy that are available at synchrotron-The polarization of ferroelectric materials can be changed by chang ing the applied field. (See sidebar “Piezoelectricity, crystal structure, and profound effects on the piezoelectric sidebar “Synchrotron distortion because the piezoelectric Radiation,” p. 23). The coefficients are effectively reoriented high brilliance of the when the polarization is changed. beam allows for focus- permission.) Piezoelectricity is thus an excellent ing it to small spot marker for the interplay of mechanical sizes. The important Figure 1. Piezoelectric shift in the wavevector of the 002 and electronic phenomena responsible aspect of X-ray scatter- Bragg reflection of an 001-oriented BiFeO3 thin film during an electric-field pulse lasting 12 ns. The wavevector shift cor- for polarization switching. ing studies is that the responds to a piezoelectric strain of ~0.5%.11 The thinness, faster operating tim- intensity and location escales, and novel structural degrees of in reciprocal space of the reflections reflections appear) provide the lattice freedom available in epitaxial ferroelec- provide key information about the constant, and the variations of these tric thin films pose difficult challenges functional properties of piezoelectrics. positions as a function of the applied for characterization using conventional The positions in reciprocal space electric field determine the piezoelec- experimental methods. Researchers (derived from the angles at which X-ray tric coefficients. The strain and diffrac- have developed a series of powerful— now standard—characterization tech- Piezoelectricity, crystal structure, and symmetry—The piezoelectric niques based on measuring the displace- coefficients ment of the surface of the thin film using piezoelectric force microscopy or Piezoelectricity results from the polarization of crystals lacking inversion symmetry. In these materi- interferometry.3 Alternatively, the stress fields lead to a change in the lattice constants, referred to as the piezoelectric strain. In the limit ofals, an applied stress leads to a change in the electrical polarization, and, conversely, applied electric imparted by the piezoelectric material small strains, fields, and stresses, the piezoelectric strain is proportional to the applied electric field, can be quantified using the curvature of and the strain tensor and electric field are related by εjk = dijk∙Ei, where is the strain tensor, d is the the substrate or a cantilever.4 Another piezoelectric coefficient, and E is the applied electric field vector.20 Note that the piezoelectric tensor approach is to use focused ion-beam can lead to strains and shears along directions that are orthogonal to the applied field. The units of d, milling or selective etching to create more properly referred to as the converse piezoelectric coefficient, are distance divided by potential a bridge structure or cantilever into difference, often given in picometers per volt. The three-index notation for the piezoelectric coefficient the film by removing a section of the can be reduced to a two-index notation, dij, where the index i refers to the electric field direction in the conventional manner where 1, 2, and 3 refer to the x, y, and z directions, respectively. The underlying substrate and to observe the second index j refers to elements of the strain tensor using Voigt notation.20 The tensor of converse important limits, particularly regarding ε ε ε = d d d d d d E1 Ei:E16d15d to the electric fieldjε14d13d12d11d5ε6 relates the piezoelectric strainεijd1εpiezoelectric coefficients- These approaches have proved5 distortion of the shape of this struc ture. to be phenomenally successful, but face time resolution and the precision with ε ε ε d d d d d d E4226252423222126 which the relationship between atomic- 5 4 3 31 32 33 34 35 36 3 scale effects and the overall electrome- The symmetry of thin films and ceramics is such a strong effect that a second, engineering notation, chanical distortion of the sample can be is widely used in describing the piezoelectric coefficients. The notation and units are identical to the determined. Understanding the atomic engineering notation, piezoelectric coefficients are defined so that the z direction, corresponding toones described above, which can lead to some confusion about which definition is in use. In the origins of piezoelectricity, particularly subscript 3, is always in the direction of the applied electric field. Thus, the expansion along the field at nanosecond time scales, has proved direction is determined by the piezoelectric coefficient d33 in the engineering notation. The symmetry challenging, but new techniques based of the piezoelectric tensor also is different between the two definitions. In the crystallographic defini- on X-ray scattering address this void. tion, the piezoelectric tensor has the symmetry of the crystallographic unit cell. In the engineering definition, the tensor has the same symmetry as the overall shape of the piezoelectric thin film or X-ray diffraction ceramic solid, which is quite different from the crystallographic symmetry. X-ray diffractometry techniques pro- The multiferroic complex oxide bismuth ferrite BiFeO3 is an excellent example of the difference vide direct insight into the piezoelec- between the crystallographic and engineering definitions of piezoelectricity. Although BiFeO3 has tricity of ceramics and epitaxial oxides. rhombohedral symmetry in bulk crystals, a pseudocubic notation for the BiFeO3 piezoelectric tensor and X-ray reflections are often used to emphasize the epitaxial relationship between the BiFeO thin Several experimental approaches do film and its cubic substrate. The rhombohedral symmetry of BiFeO is not apparent from this notation,3 this by taking advantage of time- which has the side effect of complicating the expression for the piezoelectric tensor. Projecting the3 resolved scattering techniques.6–9 For piezoelectric tensor onto the 100, 010, and 001 directions of a tetragonal material forces most of example, X-ray scattering experiments the terms to be zero and makes many of the remaining coefficients identical.20 n take advantage of the highly bril- American Ceramic Society Bulletin, Vol. 92, No. 1 | www.ceramics.org 19


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