P H Y SI C A L 8 E VI E%' 8 VOL UM E 10, NUMB EB 8 15 OCTOBER 1974 Coupled modes with A, symmetry in tetragonal BaTi03 A. Chavcs, * R. S. Katiyar, & and S. P. S. Porto Departments of Physics and Electrical Engineering, University of Southern California, I. os Angeles, California 90007 (Received 2 July 1973) The infrared properties of a system of first-order-coupled phonons are analyzed. The dielectric function and the Raman line shape of the polariton modes are derived. The parameters involved in the theory can be obtained from the Raman spectra of the TO and LO modes or from the Raman spectrum of the TO modes plus infrared-reflectivity measurements. It is shown that we can objectively distinguish real coupling from imaginary, contrary to the current belief. Numerical calculations are performed for the A, -symmetry modes of tetragonal BaTiO„with good agreement for the polariton shapes and complete disagreement for the infrared reflectivity; damage at the crystal surface is pointed out as the probable cause of the discrepancy. In addition to the coupling between the lowest and the middle mode, previously known, a much larger coupling between the middle mode and the highest is shown to exist. Both couplings are shown to be real or nearly so. The discrepancy between the dielectric constant created by the resonant modes (electronic plus phonons) and the value obtained by electrical measurements is interpreted as a new indication that the crystal has a dynamical disorder; this disorder could also be cause of an anomalous broadening observed in the lowest polariton. I. INTRODUCTION The Hamiltonian of an isolated system can al- ways be brought to a diagonal form, which means that any coupling of the quantum states is reduci- ble. However, no system above the temperature 0 K is isolated, for we have at least the black- body radiation connecting it with the environment. Although there is no true stationary state in such a situation, in many cases the interlevel transi- tions of the system, in the absence of external drive, are so random that the correlation in the dynamics of any two levels is undetectable. A good description of such systems is obtained by just adding a characteristic imaginary component to each element of the diagonalized unperturbed Hamiltonian. In some cases, however, the Hamil- tonian of the system is intrinsically nondiagonal. The profile of the energy spectrum of such systems is not composed of a set of Lorentzian peaks, but contains asymmetric interfering features. Since the occurrence of coupling in the lattice modes was recognized by Barker and Hopfield' to explain the infrared ref lectivity of some perov- skites, a handful of spectral anomalies in data on Raman, Brillouin, and neutron scattering were ob- served and associated with phonon-phonon cou- pling. As the coupling phenomenon is a tem- perature-induced effect, it seems probable that its occurrence will be more frequent in crystal show- ing other thermal anomalies in the phonon behavior. In fact, most of the crystals in which the effect proved to occur, as BaTiO&, SrTiQs, AlPO4, quartz, potassium dihydrogen phosphate (KDP) and cesium dihydrogen arsenate (CsDA), undergo a structural phase transition at some temperature not far from where the interference starts to be ob- servable. BaTiQ3 presents three structural phase transitions, at — 80, 6, and 130 C. In the tetrag- onal phase between 6 and 130 'C the dynamics of the crystal is complicated. All the three A, modes of vibration are strongly coupled and two of them are heavily damped. The lowest optical E mode is overdamped in all that range of temperatures and is coupled with the acoustical modes. Further- more, the values of both components of the dielec- tric constant tensor are in disagreement with the values predicted on the basis of the phonon f requen- cies and infrared strengths. These complications prevented the clear understanding of the mecha- nism leading to the phase transitions; opinions in the literature are divided concerning important questions, such as whether they are order-dis- order or displacive phase transitions. The A, phonons have created some polemic in the past and the question is still not completely answered. The Raman spectrum for the A, (TO) phonons shows three peaks, two very broad and one (the lowest one) very sharp, but these peaks have many strange properties. One is the striking asymmetry of the lines; another is the permanence of the two broad peaks in the phase above 130 C, where the 0& symmetry ascribed to the crystal does not allow any first-order Raman scattering. The thirddifficulty is the failure of theA~ phonons, if we assign these three peaks as the A, (TO), to explain the low-frequency value of the dielectric function along the ferroelectric axis. Prior in- vestigators' '" interpreted the two broad peaks as coming from second-order scattering, and their interpretation found some support in the fact that most perovskites show strong second-order scat- tering but they~o'x~ did not pay any attention to the very weak intensity of the peaks for experiments in 10 3522