PHYSICAL REVIEW B 84, 092104 (2011) Geometric resonances in far-infrared reflectance spectra of PbTiO 3 ceramics J. Hlinka * and E. Simon Institute of Physics, Academy of Sciences of the Czech Republic Na Slovance 2, 182 21 Prague 8, Czech Republic C. Bogicevic, F. Karolak, and P.-E. Janolin Laboratoire Structures, Propri´ et´ es et Mod´ elisation des Solides, ´ Ecole Centrale Paris, CNRS-UMR8580, Chˆ atenay-Malabry Cedex, France (Received 11 June 2011; revised manuscript received 21 August 2011; published 13 September 2011) The complex dielectric permittivity of PbTiO 3 ceramics in the THz frequency range has been investigated theoretically and by a far-infrared reflectance technique. Besides the well-known polar modes of bulk PbTiO 3 , the experiment reveals several additional modes identified as geometrical resonances (i.e., extraneous hybrid excitations created by inhomogeneous depolarization fields). A comparison of the experiment and model calculations suggests that the strong geometrical modes located near 300 and 500 cm −1 are associated with the presence of 90 ◦ ferroelectric walls. DOI: 10.1103/PhysRevB.84.092104 PACS number(s): 77.80.B−, 77.22.Gm, 63.20.D− Ferroelectric ceramics are often used as materials with high dielectric permittivity. Procedures allowing to predict their macroscopic permittivity from the dielectric tensor of constituent single crystal material are of interest for many practical purposes. Effective medium theories suggest that the macroscopic permittivity of a dense ceramic material can be obtained by substitution of the single crystal dielectric tensor in a suitable “mixing formula,” which itself reflects the necessary information about the mutual geometrical arrangements of its single-domain crystalline constituents (i.e., grains and structural domains). For example, the authors of Ref. 1 argued that the effective permittivity of a regular BaTiO 3 coarse grain ceramics should reflect the characteristic 90 ◦ twinning and they proposed a formula based on detailed investigations 2 of the actual domain microstructure in BaTiO 3 ceramics. More often, however, simple approximations are employed, such as quite general Bruggeman 3 or Lichtenecker 4 formulas. As has been realized in the past, such “mixing formulas” should be valid also for the calculation of the frequency- dependent permittivity in the phonon dispersion region be- cause the typical grain and domain sizes are sufficiently smaller than the wavelength of the probing far-infrared electromagnetic field in ceramic materials. 5–7 Moreover, the far-infrared permittivity of ceramics provides an opportunity for testing the validity of the mixing formulas because the magnitude and the anisotropy of the single crystal dielectric permittivity tensor is strongly enhanced around the polar phonon resonances. Previously, simple mixing formulas were applied to the analysis of reflectivity of other heterogenous ferroelectric perovskites, 8–10 but these results were not well suited for a critical comparison of distinct effective medium models since the polar phonon modes in these materials were too broad or their intrinsic frequencies were not well known. The aim of the present study is to discuss the applicability of several known mixing formulas to the effective permittivity of the PbTiO 3 ceramics in the THz frequency range. The advantage of the PbTiO 3 system is the combination of the low damping of polar phonon modes and the high anisotropy of its dielectric tensor in this spectral range. The Brief Report is organized as follows. First, we select three plausible models for the effective dielectric response of PbTiO 3 and the anticipated dielectric functions are calculated. Then, we present the actual experimental measurement on PbTiO 3 ceramics by the far-infrared (IR) reflectance technique. Finally, we compare the experimental results with the model predictions and discuss the physical significance of the intriguing “geometrical resonances” 7,8 identified in the course of the present study. The dielectric tensor of PbTiO 3 crystal in the phonon frequency range is fairly well known. 11 Tetragonal symmetry allows two independent nonzero components corresponding to the permittivity along and perpendicular to the tetragonal axis, respectively (ε 33 and ε 11 = ε 22 ). The equatorial complex permittivity ε 11 is determined by four E-symmetry polar phonon modes 11 and its frequency dependence can be cast in a standard form ε 11 = ε ′ 11 + iε ′′ 11 = ε 11∞ 4  i =1 ω 2 LOi − iωŴ LOi − ω 2 ω 2 TOi − iωŴ TOi − ω 2 , (1) where ω TOi and ω LOi , i = 1–4, are the transverse and longitu- dinal frequencies of the E-symmetry modes, Ŵ TOi and Ŵ LOi , i = 1–4 are the corresponding damping factors, and ε 11∞ is the high frequency limit of ε 11 . Similarly, ε 33 (ω) is determined by the three A 1 -symmetry modes 11 (i = 5–7) and ε 33∞ ε 33 = ε 33∞ 7  i =5 ω 2 LOi − iωŴ LOi − ω 2 ω 2 TOi − iωŴ TOi − ω 2 . (2) For the purpose of this study, the damping factors were set to 25 cm −1 for all the modes, and otherwise we have adopted the well established set of phonon frequencies from the single crystal Raman study of Ref. 11 (ω TOi = 87.5,218.5,289,505,148.5,359.5, and 647 cm −1 , ω LOi = 128,289,440.5,687,194,465, and 795 cm −1 , i = 1–7) along with ε 11∞ = 6.63 and ε 33∞ = 6.64. The real parts of the ε 11 and ε 33 defined in this way are shown in Fig. 1. Let us note that the so-called “silent” E mode (near 289 cm −1 ) does not have an appreciable contribution to the dielectric constant at room temperature and, in fact, its contribution to ε 11 completely vanishes when the mode parameters of Ref. 11 are adopted. The above description for the intrinsic dielectric response of PbTiO 3 single crystal has been then used to predict the effective dielectric permittivity of isotropic ceramic samples, 092104-1 1098-0121/2011/84(9)/092104(4) ©2011 American Physical Society