Coexistence of the Phonon and Relaxation Soft Modes in the Terahertz Dielectric Response of Tetragonal BaTiO 3 J. Hlinka, 1 T. Ostapchuk, 1 D. Nuzhnyy, 1 J. Petzelt, 1 P. Kuzel, 1 C. Kadlec, 1 P. Vanek, 1 I. Ponomareva, 2 and L. Bellaiche 2 1 Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 18221 Praha 8, Czech Republic 2 Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA (Received 29 April 2008; published 15 October 2008) The dielectric response to infrared waves polarized along the tetragonal axis of a ferroelectric single- domain crystal of BaTiO 3 was determined by time-domain THz spectroscopy and Fourier-transform infrared reflectivity techniques. In addition to the three well-known polar lattice modes, the experiment shows an additional mode of the relaxation type in the THz spectral region, which accounts for the Curie- Weiss behavior of the c-axis dielectric constant. A comparison of experimental results with ab initio based effective-Hamiltonian simulations allows us to elucidate its relation to the order-disorder model of Comes, Lambert, and Guinier [Solid State Commun. 6, 715 (1968)]. DOI: 10.1103/PhysRevLett.101.167402 PACS numbers: 78.30.j, 63.20.e, 77.80.e The nature of certain structural phase transitions is frequently disputed: order-disorder (OD) or displacive one? In the displacive case, atoms remain associated with their average positions, and phase transition occurs as the positional pattern changes its symmetry. In the OD case, the structural model involves partially occupied sites, and the transition occurs as the symmetry of the occupational distribution is broken. Although it has been shown [1] that the purely displacive and OD cases are merely limiting situations, they are typically thought to describe a majority of the known ferroelectric phase transitions [2]. In the spectroscopic community, it is admitted that the transition type is determined by the behavior of the order- parameter spectral response around the phase transition temperature T c . If the corresponding phonon mode fre- quency decreases to zero near T c , it is a signature of a displacive case (‘‘soft mode behavior’’). In the OD case, the relevant phonon frequency stays temperature indepen- dent, while another strongly temperature dependent exci- tation of the relaxation type (central mode, CM) occurs (‘‘critical slowing down’’). Interestingly, the CM and the soft mode (SM) are found to coexist in a number of real ferroelectrics [3]. The coexistence of displacive and OD features was also advocated by theory [1,4,5], and BaTiO 3 is often quoted as an example in this context [6–8]. To which extent do we understand order-parameter dynamics in this textbook sub- stance? In its cubic phase, SM was observed by several methods [9]. The possibility of an additional CM was vividly discussed but the considerable damping of the SM did not allow to draw a convincing conclusion. The fit of recent IR reflectivity data [10] was improved if two modes were assumed, but ab initio based calculations were needed to prove the relevance of these two modes [10]. As we will see, it turns out that the coexistence is completely obvious in the  c dielectric response of BaTiO 3 in its tetragonal phase (probing field parallel to the spontaneous polarization, E k P S ). There are three A 1 type polar phonon modes which should contribute to  c according to the standard factor- group analysis, well known from previous Raman and IR reflectivity studies: it is the so-called Last (L), Slater (S) and Axe (A) mode [11–14]. Frozen displacements in the tetragonal BaTiO 3 correspond to the S-mode, and fre- quency of this mode shows indeed some softening [12– 14]. Nevertheless, contribution of these modes to the clamped static permittivity  c given by Lyddane-Sachs- Teller(LST) relation is barely temperature dependent and much smaller than the clamped static permittivity obtained by direct capacitance measurements [13,14]. This discrep- ancy suggests [13,14] an additional strongly temperature dependent contribution to the  c spectrum below 2 THz, which was not revealed so far [15]. The present experiments and calculations provide for the first time direct access to the  c complex dielectric function of BaTiO 3 in the THz frequency range, required for ob- servation of this missing CM. The analysis elucidates its intrinsic nature, its relation to the order-disorder phase transition model of Ref. [16] and its coupling to the SM. This Letter thus settles a discussion that lasted over several decades, and suggests a path towards deeper understanding of structural phase transitions in general. As a sample we used colorless single-domain crystal plate of BaTiO 3 , top-seeded solution grown (T c ¼ 403:5K) at Shanghai Institute of Ceramics. Preliminary THz measurements were performed on an as-received, 120 m thick, 7  7 mm large, bothside polished plate, with in-plane oriented tetragonal axis. Then the sample was etched in 85% H 3 PO 4 acid at 100  C in order to achieve an optimal thickness for the experiment. Most measurements shown here were done with a 32  1 m thick sample. THz transmission measurements were per- formed using a custom-made THz time-domain spec- trometer based on an amplified fs Ti:sapphire laser system (Quantronix, Odin) for generation of THz pulses via optical rectification in ZnTe crystal [17]. It allows us to PRL 101, 167402 (2008) PHYSICAL REVIEW LETTERS week ending 17 OCTOBER 2008 0031-9007= 08=101(16)=167402(4) 167402-1 Ó 2008 The American Physical Society