sábado, 24 de julio de 2010


KNb03 continues t o be of a p a r t i c u l a r i n t e r e s t among f e r r o e l e c t r i c (FE) materials because it undergoes three successive phase t r a n s ~ t i o n s , whose mechanism is t h e object of many controversies. I n e l a s t i c neutron scattering measurements1 performed i n the tetragonal phase (T = 245OC) reveal the presence of a low-frequency E(T0) zone center mode together with an high anisotropy in the lowest phonon branches which can be related t o the existence of strong correlations along the <100> axes between t h e l a r g e motions associated t o the soft modes. Recent infrared r e f l e c t i v i t y measurements clearly show the softening of the lowest-frequency Flu phonon with decreasing temperature in the e n t i r e range of the cubic phase (425"C<910°C).<>
Lattice-dynamical calculations are c a r r i e d out for the cubic (0; space group) and the tetragonal ( c : ~sp ace group) phases in order t o explain the main experimental features. The aim of t h i s paper is to show, f o r t h e f i r s t time, that the dispersion curves for both phases can be described s a t i s f a c t o r i l y by the same harmonic model.

Description of the model.
In t h i s work for the cubic phase we use the model developped by cowley3 for SrTi03 with axially symmetric short-range force constants A and B between nearest neighbours (K-0, Nb-0, 0-0). In addition we allow an anisotropy in the oxygen core-shell coupling constant k(O) as previously suggested by Migoni e t a l . for both SrTi03 and KTa03. Therefore the components of the tensor 5 (0) in the directions of neighbouring K and Nb ions are denoted k(0-K) and k(0-Nb) respectively. This leads to a 15 parameters model f o r t h e cubic phase (Table 1) .
Since the symmetry is lower in the tetragonal phase (Figure I ) , the number of parameters is, i n principle, larger than in the cubic phase. Using simple geometry arguments we are able to derive the parameters of the tetragonal phase from those of the cubic structure. Consequently, the short-range constants of the tetragonal phase are expressed as functions of the l a t t i c e parameters c and a, of the spontaneous ionic displacements 6, measured by ~ e w a t ~an,d of the s e t of parameters A and B, already defined i n the cubic phase. Furthermore, in the tetragonal phase, we distinguish two kinds of oxygen ions, 0(1) (located on the polar axis passing through the Nb ion) and 0(2) or 0(3) ( i n t h e plane perpendicular) . Consequently we allow the
parameter k (0(1) -Nb) to be different from the parameters k(0 -Nb) and k(0(3)-Nb) .

Results and discussion.
For the tetragonal phase, the dispersion curves are calculated with parameters adjusted to the results of i n e l a s t i c neutron1 and ama an^ scattering measurements. The experimental data are q u i t e s a t i s f a c t o r i l y reproduced by the model calculations (Fig.2) except for the A2 (TOhpranchnearthe zoneboundary This f a i l u r e may be due to the large anharmonic coupling between acoustic and s of toptic phonons in t h i s direction. The strong anisotropy in the lowest dispersion branches is interesting to notice together with an anticrossing and eigenvector exchange between optic and acoustic A2 branches, which take place approximately a t 1/3 of the Brillouin zone edge. The s o f t E(T0) phonon a t qZO is essentiallyckiaracterized by a large vibrationnal amplitude of the niobium ion relative to the oxygen ion located along the mode polarization direction. Moreover the Einstein o s c i l l a t o r response of t h i s mode in the whole Brillouin zone confirms the presence of dynamical correlation chains Nb-0-Nb directed along the [loo] and [010] axes1. This shows t h e v a l i d i -
ty of t h e l i n e a r chain model for describing the s o f t mode behaviour in perovskites, as emphasized by Bilz e t a1.7 for KTa03.

For the cubic phase, the phonon branches are obtained with the parameter values previously used in the tetragonal phase, except for the coupling k(0-Nb) (Table 1). In figure 2 the calculations are compared with the experimental data of Nunes e t a1.8 In addition to t h e l a r g e anisotropy in the phonon dispersion surface, we can also note the softening of the lowest Flu phonon with a r e l a t i v e l y s l i g h t change in the value of k(0-Nb). The other zone-centre modes are nearly insensitive to this variation.

At the cubic-tetragonal phase transition, therefore, the splitting and the frequency shift6 of these phonons depend only on the geometrical effect in the force constants. On the contrary, the large separation of the FE soft Flu phonon into a soft E component (1.55 THz) and a hardened A1 (8.35 Mz) essentially originates from a k (0 -Nb) value larger than k (0 -Nb) (Table 1) . (1) (2) All these results emphasize the role of the intraionic oxygen polarizability in the phase transition mechanism of KNb03. This polarizability is dynamically enhanced by the hybridization of oxygen 2p states with niobium 4d states9 in the directdn of the chain like coupling Nb-0-Nb. This leads to a softening of the lowest cubic Flu(TO) and tetragonal E(TO1 modes. At the phase transition, the disappearance of the [001] correlation due to the asymmetry of the Nb-0-Nb bound is related to the abrupt change in the value of k(0-Nb) along the polar direction, and therefore to the stabilization of the ferroelectric Al(T0) component.
In order to specify the behaviour of the oxygen polarizability, calculations with a model including the temperature dependence of the soft mode are actually in progress.
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