Temperature dependence of the Brillouin spectra in Sn 2 P 2 S ( Se ) 6 ferroelectric crystals

A temperature dependence of hypersound velocities at transition from a paraelectric phase to ferroelectric phase in Sn2P2S6 crystals was investigated by Brillouin scattering spectroscopy. Based on these data, full set of elastic constants was determined. Similar measurements were also performed for Sn2P2(Se0.28S0.72)6 mixed crystals in the paraelectric phase near a Lifshitz point. The sound velocity indicatrices’ evolution at transition from Sn2P2S6 to Sn2P2(Se0.28S0.72)6 was observed and softening of transverse acoustic phonons in the paraelectric phase near the Lifshitz point was found. An instability of the acoustic phonons is induced by an interaction with a soft optic mode which is the origin of an incommensurate phase appearance in mixed crystals Sn2P2(SexS1−x)6 with x > xLP = 0.28.


Introduction
The compounds Sn 2 P 2 S 6 and Sn 2 P 2 Se 6 are proper uniaxial ferroelectrics with phase transitions (PT) in a region of a crossover from displacive to order/disorder type.A Lifshitz point is observed on a concentration phase diagram of mixed crystals Sn 2 P 2 (Se x S 1−x ) 6 with coordinates x LP ≈ 0.28, T LP ≈ 286 K.At this triple point, an incommensurate (IC) phase of the so-called type II appears.Such IC phase is thought to arise from an interaction between a ferroelectric optic soft mode and the acoustic phonons near Brillouin zone center [1][2][3].
The relationship between the lattice dynamics and the phase transitions has been studied for the Sn 2 P 2 S(Se) 6 crystals by Raman [4,5], Brillouin [6] and neutron scattering [7,8].In the paraelectric and ferroelectric phases (space groups P2 1 /c and Pc, respectively) a clear soft optic mode was observed.Its lineshape indicates the mixed displacive-order/disorder transition.This softening is accompanied with a c R.M.Yevych, S.I.Perechinskii, A.A.Grabar, Yu.M.Vysochanskii, V.Yu.Slivka central component in the Raman and neutron scattering spectra of both Sn 2 P 2 S 6 and Sn 2 P 2 Se 6 .This central peak was also observed by Brillouin scattering in Sn 2 P 2 S 6 crystals [6].
An interaction between the soft T O(X) and T A(u xz ) acoustic modes take place in the ferroelectric phase of Sn 2 P 2 S(Se) 6 , as is evident from the dispersion and the phonon lineshapes [7,8].Consistency between the sound velocities in the ferroelectric phase obtained from ultrasonic measurements [1,9,10] and from the slopes of the acoustic branches in the neutron scattering spectra [7,8] is obtained when piezoelectric corrections are included.
Neutron scattering investigations [7,8] show an important role of the interaction between optic and acoustic phonons in the lattice instability at a non-zero wave vector q i in Sn 2 P 2 Se 6 crystals.Obviously, this interaction plays the principal role at PT near the Lifshitz point and could be observed in Brillouin scattering spectra when q i → 0. Investigations of such spectra temperature variations for Sn 2 P 2 S 6 and Sn 2 P 2 (Se 0.28 S 0.72 ) 6 crystals have been carried out in this work.

Experimental results
Brillouin scattering spectra stimulated by He-Ne laser irradiation were studied using three pass Fabry-Perot scanned by pressure interferometer with a fineness of 35 and 2.52 cm −1 free spectral range.Right angle scattering mode was employed.The crystalline samples were placed in a UTREX cryostat where the temperature was stabilized with an accuracy of 0.25 K.The spectra lines were fitted by Lorentzian.
Sound velocities V and attenuation α were calculated using formulas: where Ω 0 is a half-width of the Brillouin component, ω 0 is frequency of He-Ne-laser, θ is scattering angle, n 0 , n S are indices of refraction for stimulated and scattered light.The refractive indices were determined by prism method and were taken: n 0 = n S = 3.0 and 3.25 for pure crystals and solid solutions, respectively.The accuracy was about 3% for sound velocities and about 10% for attenuation.Elastic modules and velocity indicatrices were calculated on the basis of the experimental hypersound velocities by using the known Kristoffel relation where c ijkl are the elastic constants, n i are the unit wave vector components, ρ is the density, v is the velocity and p i are the unit polarization vector components.
We used a crystallographic setup when a Cartesian axis Y coincides with [010] direction, perpendicular to a symmetry plane (010) of a 2/m and m point groups.An axis X was placed along [100] direction, and thus the Z axis deviated for 1.15 • from [001] direction.These axes, X and Z, are closed to the directions of a spontaneous 0,0 0,5 polarization in the ferroelectric phase and to the directions of a wave vector of modulation in the IC phase, respectively.
The Brillouin scattering spectra obtained in Sn 2 P 2 S 6 crystals at two different directions are shown in figure 1.Here, the conventional scattering geometry was used when the directions of the input and the scattered light are indicated, and parenthesis contain their polarizations.Spectral lines of longitudinal LA and transverse T A acoustic phonons are clearly seen.
3.9, 2.4, 1.8 --  This way, the hypersound velocities were obtained for six different directions of the phonon propagation in Sn 2 P 2 S 6 (table 1).Based on these data a full set of elastic constants was calculated for room temperature (table 2).
The Brillouin scattering spectra were also investigated at different temperatures in both paraelectric and ferroelectric phases of the Sn 2 P 2 S 6 crystals.It was found that near the second order PT at T 0 ≈ 337 K the temperature dependencies of longitudinal hypersound velocity and attenuation have strong anomalies along [010] (see for example figure 2) and [011] directions.These anomalies are similar to the ones observed earlier [6] in Brillouin scattering for Sn 2 P 2 S 6 in [010] direction and satisfy the known Landau-Khalatnikov relations [11].The transverse hypersound velocities propagated in the symmetry plane (010) of Sn 2 P 2 S 6 crystals, especially in the [001] direction, do not have a pronounced anomaly at PT from the paraelectric phase to the ferroelectric one (figure 3).
The Brillouin scattering spectra for Sn 2 P 2 (Se 0.28 S 0.72 ) 6 crystals at room temperature are shown in figure 4. The polarization of scattered light was not analyzed due to a weak intensity of Brillouin components.As the Lifshitz point in Sn 2 P 2 (Se 0.28 S 0.72 ) 6 occurs at the T LP ≈ 286 K, the hypersound velocities and elastic constants collected in table 3     The examples of velocity indicatrices for the paraelectric and the ferroelectric phases of the Sn 2 P 2 S 6 crystals and for the paraelectric phase of the Sn 2 P 2 (Se 0.28 S 0.72 ) 6 crystals are shown in figure 5. We can see a transformation of the orientation dependence of sound velocities in the paraelectric phase near T 0 or T LP at transition from Sn 2 P 2 S 6 to Sn 2 P 2 (Se 0.28 S 0.72 ) 6 .Herein, a softening of the quasi-transverse acoustic phonons, that are polarized and propagated in (010) plane and related to the u xz elastic deformation, was found in the paraelectric phase near the Lifshitz point.

Discussion
The results of Brillouin scattering investigations for Sn 2 P 2 Se 6 crystals are generalized based on the transverse acoustic phonon instability found in the paraelectric phase near the Lifshitz point.Let us consider the origin of this instability for proper ferroelectrics Sn 2 P 2 (Se x S 1−x ) 6 .
The interaction of the order parameter fluctuation with elastic degrees of freedom plays an essential role in the mechanism of the PT to the IC phase in proper ferroelectrics.A shift of the rigidity dispersion minimum of the order parameter η from the Brillouin zone center results in the rise of the coupling of a soft optical phonon with the acoustic ones.In the case of Sn 2 P 2 S 6 -like crystals along q y , the linear inter-action between the soft T O mode and the transverse acoustic T A mode polarized along the X axis is involved.This interaction is proportional to the wave vector modulus, and both modes attain the similar B symmetry at q y = 0. Moving away from the Brillouin zone center along q z , the soft T O mode and mixed quasi-longitudinal and quasi-transverse acoustic vibrations polarized in the XZ plane attain the same A symmetry which permits a linear interaction.These situations are characterized by the occurrence of (∂η/∂Y ) • u xy -like or (∂η/∂Z) • u xz -like gradient invariants.Thus, critical wave vector of the IC phase and its temperature range, which are defined by a form of generalized rigidity dispersion, depend considerably on the elastic modules.If dispersion surface of the soft optical mode is close to isotropic one, the direction characterized by the smallest velocity of a transverse elastic wave is most favorable for modulation wave production.
Note that in Sn 2 P 2 S 6 -like crystals along q y , the T O mode interacts linearly only with the lower T A branch, whereas the interaction with LA branch occurs along q z too, and this branch may serve as a mediator which gains the linear coupling between the soft T O and T A phonons.Moreover, due to a full-symmetric character of u xz shift, the η 2 u xz invariant is available.A term η 2 u xy is forbidden by symmetry, and thus, for transverse acoustic phonons, propagating along Z axis, non-linear interaction with the order parameter fluctuations is allowed in the lowest order (for longitudinal phonons this holds true for η 2 u ii ).Due to the increase of the fluctuations on the approach to the PT from the paraelectric phase, this circumstance promotes the reduction of the velocity of T A phonons with q z .Fluctuational variation of the velocity of T A phonons with q y must be much smaller since only biquadratic nonlinear coupling of η 2 u 2 xy is allowed.These facts agree qualitatively with softening of quasitransverse acoustic phonons that are polarized in (010) plane and propagate near the direction of the modulation wave vector in the IC phase of Sn 2 P 2 (Se x S 1−x ) 6 crystals (figure 5).
Earlier, the relationship between the softening observed by neutron scattering optic and acoustic branches and the incommensurate transition for Sn 2 P 2 Se 6 crystal was considered in a simple model for the dispersion curves and interaction strengths, valid in the low q continuum limit [8].Such a model is suggested by the analysis of the spectra with the assumption of real coupling between optic and acoustic phonons.It was assumed that all of the soft fluctuation behavior is contained in a "bare" temperature dependent optic mode (polarization P x ), which interacts with a temperature independent acoustic mode (strain u xz ) via a temperature independent and real interaction strength.In this interaction model, a small change in the material's parameters (an increase of the soft optic mode dispersion and acoustic phonons velocity or a decrease of their interaction parameter) will change the position of the instability from q i = 0 to q i = 0 [8].The sensitivity of the reciprocal-space position of the instability to the material parameters permits to investigate the origin of the Lifshitz point on the temperature -composition phase diagram of Sn 2 P 2 (Se x S 1−x ) 6 crystals.However, to carry out the quantitative analysis, the investigations of temperature dependence of the mixed soft optic-acoustic phonons spectra near the PT are required.
It is important to notice that the PT from the paraelectric to IC phase in Sn 2 P 2 Se 6 simulated in continuous approximation occurs at too low temperature [8].This may result from the presence of a central peak, which stabilizes the soft branch above T i and related to the order/disorder component of the phase transition [3].Obviously, the mixed optic-acoustic phonons do not completely condense, but only trigger the phase transition.At the same time a mechanism of strong interaction between the soft optic and acoustic branches near the Brillouin zone center in Sn 2 P 2 Se 6 crystals could be related to the disorder of the tin atoms in the paraelectric phase [12].Taking into account that the condensed mode has got a mixed optic-elastic character and contains both oscillation and relaxation components, further spectroscopic investigations of the complex spectra of polarization dynamics are required.It is also interesting to refine the displacement and disordering components together with polarization and elastic contributions into the modulation wave in the IC phase of Sn 2 P 2 (Se x S 1−x ) 6 crystals.

Conclusions
For Sn 2 P 2 S 6 and Sn 2 P 2 (Se 0.28 S 0.72 ) 6 crystals, the sound velocities were obtained for several propagation directions and a full set of elastic constants for the ferroelectric and the paraelectric phases, respectively, was determined.In the region of the second order ferroelectric phase transition, the strong temperature dependence of sound velocities was observed along [010] and [011] directions, while the weak dependence along [100] direction takes place.Transverse acoustic modes along [001] direction did not show a clear temperature dependence.There was found a transformation of the orientation dependence of sound velocities at transition from pure Sn 2 P 2 S 6 to mixed crystal Sn 2 P 2 (Se 0.28 S 0.72 ) 6 .The softening of transverse acoustic phonons propagated and polarized in the (010) symmetry plane was found in paraelectric phase near the Lifshitz point in Sn 2 P 2 (Se 0.28 S 0.72 ) 6 crystal.This instability is related to the origin of the IC phase appearance in the Sn 2 P 2 (Se x S 1−x ) 6 crystals and obviously determines the peculiarities of the modulation wave.

Figure 2 .
Figure 2. Temperature dependencies of velocity and attenuation of the longitudinal hypersound propagating along [010] direction in Sn 2 P 2 S 6 crystals.
Brillouin spectra of Sn 2 P 2 S 6 crystals for: q

Table 2 .
Elastic constants of Sn 2 P 2 S 6 crystals at room temperature.
and table 4 reflect the elastic anisotropy of the investigated crystals in the paraelectric phase at approximately T LP + 8 K.The bulk compressibility calculated from the above data for the Sn 2 P 2 S 6 and Sn 2 P 2 (Se 0.28 S 0.72 ) 6 crystals at room temperature is equal approximately to 4.2 • 10 −11 m 2 /N for both compounds.