Etude des anomalies paraelectriques quantiques de perovskites derivees de KTaO_3
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.
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
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)
called ferroelectric mode which softens (tends to zero frequency) as the temperature decreases. This phonon is associated with the polarization fluctuations induced
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
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,
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
ferroelectric displacements. This corresponds to a quantum paraelectric state which remains stable down to the lowest temperatures. Moreover, the softening of the
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
the lowest temperature (and SrTiO_3 becomes tetragonal at about 105 K). It is then considered to be 'simpler' than other QPE's.
When decreasing temperature in KTaO_3 crystals, some unexpected features appear on Brillouin scattering spectra : (i) a broad quasi-elastic central peak, first
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
central peak of QPE's, which was associated to second sound phenomena (propagation of heat) with acoustic phonons (Hehlen 1995).
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
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
'strange' behaviour showed up.
Then an extensive Brillouin characterisation of low frequency excitations (acoustic phonons, central peak, doublets) in the vicinity of Brillouin zone center was
performed in pure and Na-doped KTaO_3 crystals. Many anomalous features were reported, and the effect of doping, eventually leading to ferroelectric transition,
was also studied.
Using both neutron and Brillouin data, a phenomenological parametrisation of low frequency phonon sheets was successfully applied to the center part of Brillouin
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
velocities, some kind of 'valleys'), whereas C_3 axis are rather up-hill shaped (higher energies and group velocities).
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
center part of Brillouin zone. The phonon energy anisotropy is also visible in dampings. Comparison with experimental data shows reasonable agreement.
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
phonons : phonons whos normal dampings are greater than the secound sound frequency constitute thermal waves in which temperature is well defined and
quasi-momentum is conserved (no resistive processes : defects, Umklapps, ...). Those waves can interact with light, and show up as doublets on Brillouin scattering
spectra (Wehner and Klein, 1972). We computed second sound Brillouin zone center velocity with temperature, which is found to be in excellent agreement with
experimental measurements of doublet frequency obtained through Brillouin spectroscopy. But secound sound intensity estimations could not fit with observations
(either in value or in anisotropy).
Consecutively, we looked for an alternative interpretation of doublets in terms of phonon density fluctuations. When phonon dampings are smaller than the secound
sound frequency, the collective mode cannot propagate. But light can still couple to over-damped second sound (usual central entropy fluctuation Rayleigh peak,
diffusion of heat) and pairs of phonons (two-phonon difference scattering). The latter process was computed for low energy phonons, using energy parametrisation
and damping evaluations. The resulting Brillouin spectra are in excellent agreement with measured ones, in all directions and for all temperatures below 100 K.
The doublet and the broad central peak can then be associated to two-phonon difference scattering processes from transverse acoustic phonons according to their
dampings and group velocities. Optical phonons contribution are very broad, and rather appear as a quasi-constant background, while longitudinal acoustic
phonons produce very small contributions when compared to others.
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
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)
called ferroelectric mode which softens (tends to zero frequency) as the temperature decreases. This phonon is associated with the polarization fluctuations induced
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
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,
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
ferroelectric displacements. This corresponds to a quantum paraelectric state which remains stable down to the lowest temperatures. Moreover, the softening of the
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
the lowest temperature (and SrTiO_3 becomes tetragonal at about 105 K). It is then considered to be 'simpler' than other QPE's.
When decreasing temperature in KTaO_3 crystals, some unexpected features appear on Brillouin scattering spectra : (i) a broad quasi-elastic central peak, first
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
central peak of QPE's, which was associated to second sound phenomena (propagation of heat) with acoustic phonons (Hehlen 1995).
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
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
'strange' behaviour showed up.
Then an extensive Brillouin characterisation of low frequency excitations (acoustic phonons, central peak, doublets) in the vicinity of Brillouin zone center was
performed in pure and Na-doped KTaO_3 crystals. Many anomalous features were reported, and the effect of doping, eventually leading to ferroelectric transition,
was also studied.
Using both neutron and Brillouin data, a phenomenological parametrisation of low frequency phonon sheets was successfully applied to the center part of Brillouin
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
velocities, some kind of 'valleys'), whereas C_3 axis are rather up-hill shaped (higher energies and group velocities).
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
center part of Brillouin zone. The phonon energy anisotropy is also visible in dampings. Comparison with experimental data shows reasonable agreement.
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
phonons : phonons whos normal dampings are greater than the secound sound frequency constitute thermal waves in which temperature is well defined and
quasi-momentum is conserved (no resistive processes : defects, Umklapps, ...). Those waves can interact with light, and show up as doublets on Brillouin scattering
spectra (Wehner and Klein, 1972). We computed second sound Brillouin zone center velocity with temperature, which is found to be in excellent agreement with
experimental measurements of doublet frequency obtained through Brillouin spectroscopy. But secound sound intensity estimations could not fit with observations
(either in value or in anisotropy).
Consecutively, we looked for an alternative interpretation of doublets in terms of phonon density fluctuations. When phonon dampings are smaller than the secound
sound frequency, the collective mode cannot propagate. But light can still couple to over-damped second sound (usual central entropy fluctuation Rayleigh peak,
diffusion of heat) and pairs of phonons (two-phonon difference scattering). The latter process was computed for low energy phonons, using energy parametrisation
and damping evaluations. The resulting Brillouin spectra are in excellent agreement with measured ones, in all directions and for all temperatures below 100 K.
The doublet and the broad central peak can then be associated to two-phonon difference scattering processes from transverse acoustic phonons according to their
dampings and group velocities. Optical phonons contribution are very broad, and rather appear as a quasi-constant background, while longitudinal acoustic
phonons produce very small contributions when compared to others.
Les cristaux para'electriques quantiques sont des
di'electriques qui devraient ?etre ferro'electriques
en dessous de la temp'erature de CurieWeiss Tc .
Que se passetil ? A basse temp'erature, les
fluctuations ferro'electriques sont contr?ol'ees par
un mode de vibration transverse optique (TO),
dit ferro'electrique qui devient mou (sa fr'equence
diminue) au centre de la zone de Brillouin lorsque
la temp'erature d'ecro?it. Mais pour T ! T q ¸ Tc ,
le fort mouvement quantique de point z'ero de l'ion
central (Ta dans KTaO3) emp?eche la condensation
du mode ferro'electrique, dont la fr'equence se sta
bilise `a basse 'energie. La constante di'electrique
ffl sature `a une tr`es grande valeur, et un fort cou
plage avec le mode transverse acoustique devient
visible.
En 1991, K.A. M¨uller, d'apr`es une 'etude RPE,
envisage une transition de phase d'un nouveau
type dans ces mat'eriaux. B. Hehlen et E.
Courtens s'int'eressent alors aux propri'et'es basses
fr'equences du SrTiO3 et du KTaO3 . Les 'etudes
de diffusion neutronique, mais surtout Brillouin,
montrent alors, parmi les nombreuses anomalies,
une nouvelle excitation tr`es basse fr'equence, qu'ils
attribuent au second son, la propagation ondula
toire de la chaleur.
Ce travail pr'esente des r'esultats originaux
obtenus en diffusion neutronique et Brillouin `a
basse temp'erature (5300 K). Une param'etrisation
des nappes de phonons dans KTaO3 a servi de
base au calcul de la largeur normale des phonons,
de la vitesse th'eorique du second son, dans le cas
o`u il serait effectivement observable, et du spectre
complet de diffusion Brillouin par des processus de
diff'erence de deux phonons. La comparaison des
simulations avec l'exp'erience montre un bon ac
cord pour le calcul des largeurs et des nappes, et le
processus physique observ'e en spectroscopie Bril
louin est probablement celui de diff'erence de deux
phonons. Les diff'erents aspects de la dynamique
des phonons basse fr'equence peuvent ?etre unifi'es
au sein d'une th'eorie de diffusion Brillouin au sec
ond ordre.
Mots clefs : phonons, para'electriques quan
tiques, second son, anharmonicit'e, diffusion neu
tronique, spectroscopie Brillouin, mod'elisation et
simulation num'erique, KTaO3 , SrTiO3 .
di'electriques qui devraient ?etre ferro'electriques
en dessous de la temp'erature de CurieWeiss Tc .
Que se passetil ? A basse temp'erature, les
fluctuations ferro'electriques sont contr?ol'ees par
un mode de vibration transverse optique (TO),
dit ferro'electrique qui devient mou (sa fr'equence
diminue) au centre de la zone de Brillouin lorsque
la temp'erature d'ecro?it. Mais pour T ! T q ¸ Tc ,
le fort mouvement quantique de point z'ero de l'ion
central (Ta dans KTaO3) emp?eche la condensation
du mode ferro'electrique, dont la fr'equence se sta
bilise `a basse 'energie. La constante di'electrique
ffl sature `a une tr`es grande valeur, et un fort cou
plage avec le mode transverse acoustique devient
visible.
En 1991, K.A. M¨uller, d'apr`es une 'etude RPE,
envisage une transition de phase d'un nouveau
type dans ces mat'eriaux. B. Hehlen et E.
Courtens s'int'eressent alors aux propri'et'es basses
fr'equences du SrTiO3 et du KTaO3 . Les 'etudes
de diffusion neutronique, mais surtout Brillouin,
montrent alors, parmi les nombreuses anomalies,
une nouvelle excitation tr`es basse fr'equence, qu'ils
attribuent au second son, la propagation ondula
toire de la chaleur.
Ce travail pr'esente des r'esultats originaux
obtenus en diffusion neutronique et Brillouin `a
basse temp'erature (5300 K). Une param'etrisation
des nappes de phonons dans KTaO3 a servi de
base au calcul de la largeur normale des phonons,
de la vitesse th'eorique du second son, dans le cas
o`u il serait effectivement observable, et du spectre
complet de diffusion Brillouin par des processus de
diff'erence de deux phonons. La comparaison des
simulations avec l'exp'erience montre un bon ac
cord pour le calcul des largeurs et des nappes, et le
processus physique observ'e en spectroscopie Bril
louin est probablement celui de diff'erence de deux
phonons. Les diff'erents aspects de la dynamique
des phonons basse fr'equence peuvent ?etre unifi'es
au sein d'une th'eorie de diffusion Brillouin au sec
ond ordre.
Mots clefs : phonons, para'electriques quan
tiques, second son, anharmonicit'e, diffusion neu
tronique, spectroscopie Brillouin, mod'elisation et
simulation num'erique, KTaO3 , SrTiO3 .