Résumé
The object of the work was to study Quantum Paraelectric anomalies in KTaO_3 pure and Na-doped crystals, with E. Courtens and A.K. Tagantsev. <br /><br />What are quantum paraelectrics (QPE, also called incipient ferroelectrics)? They are crystals (such as KTaO_3 and SrTiO_3) that should be ferroelectric under a<br />certain Curie-Weiss temperature T_c, but they aren't ! At low temperature, the ferroelectric fluctuations are controlled by a zone-center transverse optic mode (TO)<br />called ferroelectric mode which softens (tends to zero frequency) as the temperature decreases. This phonon is associated with the polarization fluctuations induced<br />by the vibration of the cell-center ion (Ta or Ti) inside the octaedral oxygen cage. Around T_q (37 K for SrTiO3 and about 10 K for KTaO3), the related variation of<br />the dielectric constant epsilon deviates from the normal Curie-Weiss divergence for ferroelectrics, and stabilizes at a large but finite value below T_q. Thus,<br />ferroelectricity is not achieved. It is usually considered to be prevented by zero-point quantum fluctuations which are about the same amplitude as the would-be<br />ferroelectric displacements. This corresponds to a quantum paraelectric state which remains stable down to the lowest temperatures. Moreover, the softening of the<br />TO branch depresses the transverse acoustic phonon (TA), due to a strong TO-TA coupling. KTaO_3 was chosen for this study because it remains cubic down to<br />the lowest temperature (and SrTiO_3 becomes tetragonal at about 105 K). It is then considered to be 'simpler' than other QPE's. <br /><br />When decreasing temperature in KTaO_3 crystals, some unexpected features appear on Brillouin scattering spectra : (i) a broad quasi-elastic central peak, first<br />reported by Lyons and Fleury (1976), which is usually interpreted as second order scattering, and (ii) a new doublet recently observed over the quasi-elastic<br />central peak of QPE's, which was associated to second sound phenomena (propagation of heat) with acoustic phonons (Hehlen 1995). <br /><br />I first measured precisely inelastic low energy phonons along high symetry axis (C_2, C_3 and C_4) in pure KTaO_3 (IN14, with B. Hehlen and R. Currat), in order<br />to look for those up-mentionned anomalies near Brillouin zone center by neutron scattering experiments. But, except for the strong TA-TO coupling, no particular<br />'strange' behaviour showed up. <br /><br />Then an extensive Brillouin characterisation of low frequency excitations (acoustic phonons, central peak, doublets) in the vicinity of Brillouin zone center was<br />performed in pure and Na-doped KTaO_3 crystals. Many anomalous features were reported, and the effect of doping, eventually leading to ferroelectric transition,<br />was also studied. <br /><br />Using both neutron and Brillouin data, a phenomenological parametrisation of low frequency phonon sheets was successfully applied to the center part of Brillouin<br />zone (|q| < 0.3 rlu). This emphasized the unusual anisotropy of phonons, specially for acoustic phonons along C_2 and C_4 axis (low energies and group<br />velocities, some kind of 'valleys'), whereas C_3 axis are rather up-hill shaped (higher energies and group velocities). <br /><br />This model was first used in order to compute a lower value of three phonon electrostrictive normal processes dampings for the five lower energy phonons over the<br />center part of Brillouin zone. The phonon energy anisotropy is also visible in dampings. Comparison with experimental data shows reasonable agreement. <br /><br />Then, the second sound hypothesis for the origin of doublet was tested. We here notice than second sound is the collective propagation mode of thermalised<br />phonons : phonons whos normal dampings are greater than the secound sound frequency constitute thermal waves in which temperature is well defined and<br />quasi-momentum is conserved (no resistive processes : defects, Umklapps, ...). Those waves can interact with light, and show up as doublets on Brillouin scattering<br />spectra (Wehner and Klein, 1972). We computed second sound Brillouin zone center velocity with temperature, which is found to be in excellent agreement with<br />experimental measurements of doublet frequency obtained through Brillouin spectroscopy. But secound sound intensity estimations could not fit with observations<br />(either in value or in anisotropy). <br /><br />Consecutively, we looked for an alternative interpretation of doublets in terms of phonon density fluctuations. When phonon dampings are smaller than the secound<br />sound frequency, the collective mode cannot propagate. But light can still couple to over-damped second sound (usual central entropy fluctuation Rayleigh peak,<br />diffusion of heat) and pairs of phonons (two-phonon difference scattering). The latter process was computed for low energy phonons, using energy parametrisation<br />and damping evaluations. The resulting Brillouin spectra are in excellent agreement with measured ones, in all directions and for all temperatures below 100 K. <br /><br />The doublet and the broad central peak can then be associated to two-phonon difference scattering processes from transverse acoustic phonons according to their<br />dampings and group velocities. Optical phonons contribution are very broad, and rather appear as a quasi-constant background, while longitudinal acoustic<br />phonons produce very small contributions when compared to others.