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Elastic and acoustooptic properties of Sn2P2S6 crystals: Effect of ferroelectric phase transition
Optical Materials 35 (2012) 168–174
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Elastic and acoustooptic properties of Sn2P2S6 crystals: Effect of ferroelectric phase transition O. Mys, B. Zapeka, I. Martynyuk-Lototska, R. Vlokh ⇑ Institute of Physical Optics, 23 Dragomanov St., 79005 Lviv, Ukraine
a r t i c l e
i n f o
Article history: Received 23 March 2012 Received in revised form 3 July 2012 Accepted 16 July 2012 Available online 6 October 2012 Keywords: Acoustooptic figure of merit Sn2P2S6 crystals Tricritical point Critical exponent Phase diagrams
a b s t r a c t We report on the studies for temperature dependences of elastic stiffness coefficients in Sn2P2S6 crystals. Basing on the construction of acoustic velocity surfaces, we have determined the parameters of the slowest acoustic wave that propagates in Sn2P2S6 crystals. The acoustooptic figure of merit for the case of acoustooptic interaction with this wave is estimated as 0.8 1012 s3/kg. We have shown that Sn2P2S6 is very close to the conditions of tricritical point on the (x, T)- and (p, T)-phase diagrams of the solid solutions Sn2P2(SexS1x)6. The critical exponent a of the heat capacity for the Sn2P2S6 crystals is equal to 0.42 ± 0.03. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Tin thiohypodiphosphate crystals Sn2P2S6 are wide-gap semiconductors that manifest a proper, second-order paraelectric-toferroelectric phase transition, with the change 2m M m of their point symmetry at TC = 337 K [1]. These crystals are transparent in a wide spectral range (from k = 0.53 lm to k = 8.0 lm [1], have high enough electrooptic coefficients r11 = 1.74 1010 m/V at the room temperature and the light wavelength of k = 633 nm [2,3]), and manifest conspicuous photorefractive properties [4,5]. Moreover, the tin thiohypodiphosphate represents a promising magnetooptic material (the corresponding Verdet constant amounts to 115 rad/T m [6]). In our recent works [7–10] we have experimentally studied and analyzed acoustic properties of Sn2P2S6 at the room temperature and demonstrated that the material has great potentials for acoustooptic (AO) applications. The AO figure of merit (AOFM) of Sn2P2S6 calculated on the basis of our experimental AO data reaches very high values M2 = (1.7 ± 0.4) 1012 s3/kg [9]. In other words, the crystals of Sn2P2S6 reveal one of the highest AOFMs known for the AO materials operating in the visible spectral range (see, e.g., the study of AOFM for Hg2Br2 crystals [11]). This is why Sn2P2S6 can be widely used in optoelectronics, in particular in devices for AO operation of optical radiation. It is necessary to note that the AOFM value mentioned above has been obtained for the conditions of AO interaction with the ⇑ Corresponding author. Tel.: +38 032 2611488; fax: +38 032 2611483. E-mail address: [email protected] (R. Vlokh). 0925-3467/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optmat.2012.07.025
acoustic wave which still is not the slowest one. Being more precise, we have dealt with the waves having the velocities v21 = 2100 m/s and v23 = 2610 m/s (here the first and the second indices denote the directions of the wave vector q and the polarization vector e of the acoustic wave, respectively) [9]. The AOFM is inversely proportional to the cube of the acoustic wave velocity and so becomes extremely high with lowering velocity (M 2 ¼ n6 p2ef =qv 3 , with n being the refractive index, pef the effective elastooptic coefficient, and q the crystal density). We have shown in our recent work [10] that the slowest wave in the Sn2P2S6 crystals is the transverse wave with the velocity 1550 ± 120 m/s at the room temperature. It has also been found that the propagation direction of this wave coincides with neither of the principal crystallographic directions and, moreover, it does not belong to the principal crystallographic planes. One can easily calculate that, for the case of AO interaction with the latter wave, the AOFM can increase by several times, when compared with the value obtained in the work [9], if only the effective elastooptic coefficient remained approximately the same. Both the velocity of the slowest wave mentioned above and its propagation direction have been derived in the study [10] using the following procedure: the elastic stiffness tensor has been rewritten in a so-called acoustic eigen coordinate system rotated around b axis by the angle h = [arctan(2C46/(C66 C44))]/ 2 = (16 ± 5)° with respect to the initial coordinate system, in order to meet the conditions C 046 ¼ C 015 ¼ C 025 ¼ C 035 0 for the elastic stiffness tensor components written in this new system. In fact, such an ‘eigen acoustic coordinate system’ does not exist at all. Really, only the tensor component C 046 0 in this system, while
O. Mys et al. / Optical Materials 35 (2012) 168–174
the other coefficients C 015 ; C 025 and C 035 remain nonzero, though being small enough. Since the parameters of the slowest wave (the directions of its propagation and polarization) for Sn2P2S6 have been obtained in the work [10] with assuming of exact equality to zero of the coefficients C 015 ; C 025 and C 035 , these parameters could serve as a rough approximation only. On the other hand, a decrease in the acoustic wave velocities occurring with approaching the phase transition temperature TC, as revealed for some of the longitudinal waves propagating in Sn2P2S6 [12], can in principle lead to further increase in the AOFM. However, in our recent work [13] reporting the temperature dependences of both the longitudinal and transverse acoustic wave velocities in the Sn2P2S6 crystals, we have shown that the above temperature dependences should impose no notable variation of the AOFM, though the relevant results are also approximated. For determining the propagation direction of the slowest acoustic wave and possible variations of its parameters with temperature, the acoustic velocity surfaces are to be constructed for different temperatures. In order to find the minimal velocity, it would be necessary to cut these surfaces by a plane that rotates, e.g., around the b axis and then determine the minimal velocities within all of these planes. Prior to this, the temperature dependences of all the elastic stiffness coefficients for Sn2P2S6 crystals are needed. On the other hand, efficient applications of these crystals should be based on high enough stiffness coefficients in a wide temperature range, including the point TC = 337 K which is close to the room temperature. In connection with this, it is worthwhile to notice that some of the stiffness coefficients for the Sn2P2(Se0.28S0.72)6 crystals tend almost to zero near the phase transition temperature [14], thus making the crystals inconvenient for practical applications. Following from the above considerations, the aim of the present work is to study the temperature behavior of all the stiffness coefficients in the Sn2P2S6 crystals in the vicinity of the Curie temperature. Besides, we will also touch the problem of critical exponents for the Sn2P2S6 crystals, which are close to a so-called tricritical point both on ‘selenium concentration–temperature’ (x, T) [14] and ‘hydrostatic pressure–temperature’ (p, T) [15,16] phase diagrams peculiar for the solid solutions of Sn2P2(SexS1x)6. Issuing from the temperature dependences of the elastic stiffness coefficients, we will determine the critical exponent a describing temperature dependence of the heat capacity Cp (TC T)a. It is important that the latter index is close to 0.5 for the case of Sn2P2(Se0.28S0.72)6 crystals, which corresponds to thermodynamic system in the vicinity of its tricritical point [14].
2. Data processing The lattice parameters of Sn2P2S6 are equal to a = 0.9378 nm, b = 0.7448 nm, c = 0.6513 nm, and b = 91.15° (at the room temperature and the atmospheric pressure), where the crystallographic axis b is perpendicular to the symmetry mirror plane [17], while the spontaneous polarization vector is almost parallel to the a axis. The elastic stiffness coefficients have been determined with respect to the crystallographic axes a = 1, b = 2, and c = 3. Here we report on complete determination of the elastic matrix Cklmn = Cij (with i, j = 1–6; 1 = 11, 2 = 22, 3 = 33, 4 = 23, 5 = 13 and 6 = 12), via measurements of the phase ultrasonic velocities along different crystallographic directions. The temperature dependences of the velocities of longitudinal and transverse ultrasonic waves have earlier been obtained with a standard pulse-echo overlap method and presented in our recent work [13].1 Since Sn2P2S6 belongs to the 1 The Sn2P2S6 crystals studied by us have been grown with a vapor-transport technique at the Uzhgorod National University (Ukraine).
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monoclinic point symmetry group m at the room temperature, there are 13 independent nonzero elastic coefficients (C11, C22, C33, C44, C55, C66, C12, C13, C23, C15, C25, C35, and C46), and the same holds true of the point group 2/m describing the paraelectric phase. Simple expressions linking the velocities and the elastic coefficients are available only for the four of the elastic stiffness matrix components (C 22 ¼ qv 22 ; C 44 ¼ qv 232 ; C 66 ¼ qv 212 , and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 2 2 C 46 ¼ 0:5 qv 21 qv 23 ðC 44 C 66 Þ ). All the other coefficients are linked by more complicated relations or by complex systems of equations. Then cumbersome, both in experimental and computational aspects, procedures for deriving the remaining nine elastic coefficients are required for the monoclinic system [18,19]. To determine those coefficients, one should measure the velocities of quasi-longitudinal (QL) and quasi-transverse (QT) ultrasonic waves for at least six different directions ([1 0 0], [0 1 0], [0 0 1], [1 1 0], [1 0 1], and [0 1 1]). As a result, the errors for the elastic stiffness coefficients would be high enough (15%). The acoustic velocity surfaces have been constructed using the well known Christoffel equation, the software packages ‘Origin 7.0’, ‘Mathcad’ and ‘Maple 13’, and the experimental data for the elastic stiffness coefficients. The exact propagation direction and the velocity of the slowest wave have been determined after rotating the elastic stiffness tensor around the crystallographic axis b (the step of 10°) and constructing cross sections of the acoustic velocity surfaces for each rotation angle. 3. Results and discussion 3.1. Temperature dependences of elastic stiffness coefficients Under normal conditions, the elastic stiffness coefficients obtained in the present work for Sn2P2S6 agree well with the data obtained in our recent work [10]. The temperature dependences of the elastic stiffness coefficients are presented in Fig. 1. As seen from Fig. 1, only some of the coefficients reveal notable temperature dependence in the vicinity of TC, namely C11, C22, C23, C12, and C13. The other coefficients (C33, C15, C25, C35, and C66) have a step-like anomaly at the phase transition temperature, while the coefficients C44, C55, and C46 are almost temperature-independent. 3.2. Acoustic wave velocity surfaces The cross sections of the acoustic velocity surfaces calculated for different temperatures are presented in Fig. 2. The velocity of the slowest wave QT2 (with the wave vector belonging to the crystallographic coordinate plane ac) is equal to 1407 m/s. The wave vector of this wave is rotated by 55° with respect to the a axis, while the polarization vector belongs to the ac plane (see Fig. 2a). As follows from the cross sections of the acoustic wave velocity surfaces by the plane a0 b (where the angle between the a and a0 axes is equal to 55° – see Fig. 3), this wave is found to be the slowest acoustic wave for the Sn2P2S6 crystals. There are four different propagation directions of the slowest wave in Sn2P2S6 (see Fig. 2). Inside the temperature range of the ferroelectric phase, the direction of propagation of this wave remains almost the same (see Fig. 2a–c), while in the paraelectric phase the q vector is rotated by 39° with respect to the a axis (see Fig. 2d and e). This fact is caused by a jump-like rotation of the surface for the QT2 wave around the b axis occurring at the phase transition. It is seen from Fig. 4 that the velocity of the slowest wave manifests very weak temperature dependence in the both phases. The slowest transverse wave propagates along the direction rotated by 55° with respect to the a axis in the ac plane and has its polarization lying in the same plane. It should excite the
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Fig. 1. Temperature dependences of elastic stiffness coefficients for the Sn2P2S6 crystals: (a) C11 – open circles, C22 – open squares, C33 – open triangles; (b) C12 – semi-open circles, C13 – semi-open squares, C23 – semi-open triangles; (c) C15 – full circles, C25 – full squares, C35 – full triangles, C46 – full diamonds; and (d) C44 – strikethrough circles, C55 – strikethrough squares, C66 – strikethrough triangles.
mechanical strains u1 = cos 55° sin 55°u = 0.47u, u3 = cos 35° cos 55°u = 0.47u, and u5 = cos °55°u = 0.33u, where u denotes the strain caused by the incident acoustic wave. Let us remind that the crystallographic plane ac represents a plane of the intermediate (nm = n1 = 3.0256 and the largest ng = n3 = 3.0982 refractive indices, while the angle between the a axis and the direction corresponding to the principal refractive index n3 is equal to 43.3 ± 0.3° (k = 632.8 nm) [20]. If we consider the AO interaction in the ac plane with the slowest wave, three different cases are possible (see Fig. 5). These are the case of isotropic interaction described by the relation in ðaÞ Ed2 ¼ DB22 Din 2 ¼ ð0:47ðp23 p21 Þ þ 0:33p25 ÞuD2 ;
the case of the ‘mixed’ isotropic and anisotropic AO interaction given by
( ðbÞ
in Ed3 ¼ DB33 Din 3 þ DB31 D1
Ed1
¼
DB13 Din 3
þ
DB11 Din 1
( )
in Ed3 ¼ ½ð0:47ðp33 p31 Þ þ 0:33p53 ÞDin 3 þ ð0:47ðp53 p51 Þ þ 0:33p55 ÞD1 u in Ed1 ¼ ½ð0:47ðp53 p51 Þ þ 0:33p55 ÞDin 3 þ ð0:47ðp13 p11 Þ þ 0:33p15 ÞD1 u
and the case of anisotropic AO interaction described by the relations
( ðcÞ
Ed1 ¼ DB12 Din 2 Ed3 ¼ DB32 Din 2
:
d Here Din i is the electric displacement of the incident optical wave, Ej the electric field of the diffracted wave, and DBij the increment of the optical impermeability tensor due to photoelastic perturbation. Since all of the relevant photoelastic coefficients (p61, p63, p65, p41, p43, and p45) are equal to zero in the last case, this anisotropic AO diffraction cannot be implemented. Unfortunately, we cannot make the exact calculation of the AOFM using the relations mentioned above, because the photoelastic coefficients for Sn2P2S6 are unknown. Indeed, in spite of the fact that the modules of the piezooptic coefficients for the Sn2P2S6 crystals have already been determined [8], their recalculation to photoelastic ones is impossible, since the signs of these coefficients are unknown. Besides, any calculations of the photoelastic tensor components based on the piezooptic coefficients would lead to high errors even for many systems with high symmetries (see,
;
e.g., [8]), thus making the corresponding calculations of the AOFM meaningless. It is the more so as Sn2P2S6 represents a low-symmetry crystal, for which the experimental conditions for measuring the piezooptic tensor components become very complicated, and the
O. Mys et al. / Optical Materials 35 (2012) 168–174
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Fig. 2. Cross sections of acoustic wave velocity surfaces by the crystallographic planes ab (left), ac (middle) and bc (right) for different temperatures: (a) 295, (b) 325, (c) 329, (d) 340, and (e) 350 K. Squares correspond to quasi-longitudinal wave QL, open circles to quasi-transverse wave QT1, and open triangles to quasi-transverse wave QT2.
nonzero tensor components of the piezooptic and elastic stiffness coefficients are numerous, producing still larger errors of both
the measurements and the consequent recalculations. Of course, the information on the photoelastic coefficients could, in principle,
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O. Mys et al. / Optical Materials 35 (2012) 168–174 Table 1 Fitting parameters of temperature dependences of the elastics stiffness coefficients. Component of elastic stiffness tensor
Elastic stiffness coefficient
C22
2.31 ± 0.03 2.44 ± 0.03 [14] 3.04 ± 0.05 1.53 ± 0.05
C11 C13 Mean value
0
Fig. 3. Cross sections of acoustic wave velocity surfaces by the plane (a b) at 0 T = 295 K (the angle between the a and (a ) axes is equal to 55°).
Ratio a2
4ðijÞ pffiffiffiffiffiffiffiffiffi ; 109 0
Critical exponent a
Coefficient of determination R2
0.44 0.44 ± 0.03 [14] 0.42 0.40 ± 0.03 0.42 ± 0.03
0.964 0.987 [14]
a10 a3
C 0ij ; 1010 N=m2 1.91 ± 0.04 1.63 ± 0.25 [14] 3.1 ± 0.1 2.55 ± 0.39
0.967 0.982
Luckily in our situation it is sufficient to use an ‘isotropic approximation’ while determining the mean value of the piezooptic coefficients. The Sn2P2S6 crystals belong to the symmetry group 2/m in their paraelectric phase, which is a subgroup of the cubic group m3m. Using specific relationships among the elastic, piezooptic and photoelastic tensor components for the above cubic group, one can write the following rough relations for the group 2/m:
C 11 ðC 11 þ C 22 þ C 33 Þ=3 ¼ 41:16 109 N=m2 ; C 12 ðC 12 þ C 13 þ C 23 Þ=3 ¼ 13:39 109 N=m2 ;
p 12 ðp12 þ p13 þ p23 þ p21 þ p31 þ p32 Þ=6 ¼ 6:8 pN=m2 ; p 11 p11 þ p22 þ p33 Þ=3 ¼ 8:9 pN=m2 ; 12 ¼ p 12 ðC 11 C 12 Þ ¼ 0:2: 11 p p
Fig. 4. Temperature dependence of velocity of the slowest acoustic wave in Sn2P2S6 crystals.
Fig. 5. Scheme of AO interaction that occurs in the ac plane of Sn2P2S6 crystals.
be obtained when using a so-called Dixon-Cohen method [21,22], or Brillouin scattering method all though they are comparatively complex. It is necessary to notice that, in the case of low-symmetry crystals, the Dixon-Cohen and Brillouin scattering methods yield only the combination of photoelastic coefficients. A separate determination of such coefficients as p46, p15, p25 p35, p51, p52, p53 and p64 is problematic. At the same time, solving a system of equations in order to evaluate the separate coefficients basing on the data obtained from the experiment, leads to increased error in determination of these coefficients.
Then the efficiency of the AO interaction described by the case (a) should be very small, because we have p23 p21 0 and p25 0 in the isotropic approximation. In the frame of the same approximation, the corresponding relations for the case of isotropic dif11 p 12 ÞDin fraction (b) can be rewritten as Ed3 ¼ 0:47ðp and 3 12 p 11 ÞDin Ed1 ¼ 0:47ðp . In the both cases, we have the effective 1 photoelastic coefficient equal to pef 0.1. Then the AOFM can be 2ef =qv 3 0:8 1012 s3 =kg (n 6 p ¼ 3:02; q ¼ estimated as M2 ¼ n 3:54 103 kg=m3 [20], and v = 1407 m/s). As for the AO interaction with the longitudinal wave propagating along the a axis, which is characterized by more than two times ef ¼ p 11 ¼ p 11 ðC 11 þ larger velocity (v11 = 3697 m/s), we have p 2C 12 Þ þ 3p12 ðC 11 þ C 12 Þ 0:5. Then the AOFM is higher (1012 s3/kg) than that for the case of AO interaction with the slowest wave. Hence, the AOFM for the Sn2P2S6 crystals is mainly determined by the values of their photoelastic coefficients and high enough values of the refractive indices. Indeed, Sn2P2S6 is not a ferroelastic material, where low velocities of the acoustic waves are caused by softening of some acoustic phonons, due to acoustic instability of crystalline lattice near the phase transition point [23]. As a result, the velocities of the acoustic waves in Sn2P2S6 are not low enough that they should have a principled effect on the AOFM, as occurs for some well-known ferroelastic AO materials (e.g., TeO2 [24,25] or H2Br2 crystals [26,27]). 3.3. Thermodynamic analysis Now let us analyze the temperature dependences of the elastic stiffness coefficients. This can be done following from the thermodynamic potential (1) written in the form, which is general for proper ferroelectric phase transitions occurring from centrosymmetric paraelectric phases into polar ferroelectric ones:
F ¼ F0 þ
a1 2 a2 4 a3 6 a4 2 c P þ P þ P þ P e þ e2 : 2 2 4 6 2
ð1Þ
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O. Mys et al. / Optical Materials 35 (2012) 168–174
parameters such as a hydrostatic pressure or a concentration of substituting chemical elements. For example, the change in the sign of the coefficient a2 at some thermodynamic coordinates would lead to changing order of the phase transition and so appearance of the tricritical point on the phase diagram. Notice that the coefficients a1, a2 and a3 are fully symmetric tensors respectively of the ranks two, four, and six, while a4 and c represent tensors with the internal symmetries [V2]2 and [[V2]2]. After minimizing the free energy, @F=@P ¼ 0, and taking the relation a01 ¼ ða1 þ a4 eÞ ¼ a010 ðT C TÞ into account, one can write the spontaneous polarization as
a01 a010 ðT C TÞ ¼ : a2 a2
P2s ¼
ð2Þ
In other words, the critical exponent of the spontaneous polarization is equal to 0.5. The elastic stiffness coefficients are defined by the relation
X @2F @2F C mn jP–0 C mn jP¼0 ¼ DC mn ¼ vlk ; @em @Pl @en @Pk k;l
ð3Þ
where a so-called ‘clamped‘ susceptibility is given by
@2F @Pl @Pk
vlk ¼
!1 ð4Þ
:
For example, in case of Sn2P2S6 we have v1 ¼ ð4a010 ðT C TÞÞ1 at T < TC. Taking nonzero strain tensor components e1, e2, e3, and e5 into consideration, we write out the coupling energy density F c ¼ a24 P 2 e in the matrix presentation as
Fc ¼
P 21 ða4ð11Þ e1 þ a4ð12Þ e2 þ a4ð13Þ e3 þ a4ð15Þ e5 Þ; 2
ð5Þ
where a4(ij) are the components of a forth-rank tensor. Considering formulae (2)–(5), one can obtain the following relations for the elastic stiffness increments in the Sn2P2S6 crystals:
DC 11 ¼ DC 13
a24ð11Þ
;
DC 22 ¼
4a2 a4ð11Þ a4ð13Þ ¼ ; 4a2
DC 25 ¼
a4ð12Þ a4ð15Þ ; 4a2
a24ð12Þ 4a2
;
DC 33 ¼
a24ð13Þ 4a2
;
DC 12 ¼
DC 23 ¼
a4ð12Þ a4ð13Þ ; 4a2
DC 15 ¼
DC 35 ¼
a4ð13Þ a4ð15Þ ; 4a2
DC 55
a4ð11Þ a4ð12Þ ; 4a2
a4ð11Þ a4ð15Þ ; 4a2 2 a4ð15Þ ¼ : 4a2 ð6Þ
The elastic stiffness tensor components appearing in formulae (6) should reveal a step-like behavior at the Curie point, whereas the components DC44 = DC66 = DC46 = 0 should remain unchanged while passing the temperature of the phase transition. Let us consider the thermodynamic potential given by formula (1) under the conditions of tricritical point a2 ¼ 0 at x = xTC, where xTC = 0.28 is the concentration of Se that corresponds to the tricritical point [14]). If a2 = 0, the relation for the spontaneous polarization may be written as Fig. 6. Dependences of elastic stiffness coefficients in the ferroelectric phase on the relative temperature (TC T): (a) C11, (b) C13, and (c) C22. Solid curves are fits according to formula (10).
Here P means the electric polarization, e the mechanical strain, a1 = a10(TC T) the linear electric susceptibility a2 and a3 being the nonlinear susceptibility coefficients), a4 the coupling coefficient associated with electrostriction, and c the elastic stiffness coefficient determined under the condition of P = 0. All of the coefficients are invariant with respect to symmetry operations of the point group of paraelectric phase and almost temperature-independent in the range of paraelecric phase, except of a1. Of course, these coefficients can be dependent on some other thermodynamic
P2s ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a010 ðT C TÞ : a3
ð7Þ
Now we analyze more closely the relation (6) for the temperature dependences of the elastic stiffness coefficients in case of the tricritical point (xTC = 0.28). Then the temperature behavior of the electric susceptibility is given by
v11 ¼ 4a010 ðT T C Þ
1
:
ð8Þ
Accounting for formulae (3), (7), and (8), one gets the following relations for the increments of elastic stiffness coefficients observed at the tricritical point:
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O. Mys et al. / Optical Materials 35 (2012) 168–174
a24ð11Þ a24ð12Þ a24ð13Þ ; DC 22 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; DC 33 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; DC 11 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 0 a10 a3 ðT T C Þ a10 a3 ðT T C Þ a10 a3 ðT T C Þ a4ð11Þ a4ð12Þ a4ð11Þ a4ð13Þ a4ð12Þ a4ð13Þ ; DC 13 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; DC 23 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; DC 12 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a010 a3 ðT T C Þ a010 a3 ðT T C Þ a010 a3 ðT T C Þ a4ð11Þ a4ð15Þ a4ð12Þ a4ð15Þ a4ð13Þ a4ð15Þ ; DC 25 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; DC 35 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; DC 15 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a010 a3 ðT T C Þ a010 a3 ðT T C Þ a010 a3 ðT T C Þ
transitions, we have shown that the Sn2P2S6 crystals are very close to the tricritical point. The latter is realized at xTC = 0.28 on the (x, T)-phase diagram of Sn2P2(SexS1x)6 solid solutions and at pTC = 4.3 kbar on their (p, T)-phase diagram. We have also found that the critical exponent a for the Sn2P2S6 crystals is equal to 0.42 ± 0.03.
a24ð15Þ : DC 55 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a010 a3 ðT T C Þ
Acknowledgements ð9Þ
In other terms, all of these elastic stiffness components should behave in the ferroelectric phase according to the power law DCij (T TC)0.5 and go to infinity at T = TC. Only the increments DC44 = DC66 = DC46 = 0 should remain unchanged when the phase transition point is being passed. We have fitted the temperature dependences of the elastic stiffness coefficients in the ferroelectric phase by the function
a24ðijÞ ffi ðT T C Þa ; C ij ¼ C 0ij þ pffiffiffiffiffiffiffiffiffiffiffi a010 a3
ð10Þ
where C 0ij is a so-called ‘background’ elastic stiffness, which is independent of temperature (see Table 1, Fig. 6). We have found that the critical exponent a is equal to 0.42 ± 0.03. Thus, the phase transition in Sn2P2S6 is close enough to the tricritical point, which should occur at xTC = 0.28 on the (x, T)-phase diagram and at pTC = 4.3 kbar on the (p, T)-phase diagram of the Sn2P2(SexS1x)6 solid solutions. 4. Conclusions In the present work we have constructed the acoustic velocity surfaces and determined the parameters describing the slowest acoustic wave that can propagate in the Sn2P2S6 crystals. We have revealed that, under the normal conditions, this wave is transverse, it has the velocity of 1407 m/s, propagates in the crystallographic plane ac along the direction defined by the angle of 55° with respect to the a axis, and its polarization vector belongs to the ac plane. The velocity of this wave is almost temperature-independent in the temperature region of the ferroelectric phase transition. The AOFM calculated for the case of AO interaction with this wave is equal to 0.8 1012 s3/kg. It is smaller than that experimentally obtained in our recent work [9] and even smaller than the value peculiar for the AO interaction with the longitudinal wave revealing higher velocity (1012 s3/kg). This is explained as being due to the fact that the Sn2P2S6 crystals are not a ferroelastic material, for which low velocities of the acoustic waves are caused by softening of some acoustic phonons, as a result of acoustic instability of crystalline lattice near the phase transition point. As a result, the acoustic wave velocity (even that characteristic of the slowest wave) is not low enough for increasing sufficiently the AOFM value. Basing on our analysis of temperature dependences of the elastic stiffness coefficients in the frame of mean-field theory of phase
The authors acknowledge financial support of the present study Ministry of Education, Science, Youth and Sports of Ukraine. We thank Prof. Yu. Vysochanskii and Prof. A. Grabar for supplying the crystals. References [1] Y.M. Vysochanskii, T. Janssen, R. Currat, R. Folk, J. Banys, J. Grigas, V. Damulionis, Phase Transitions in Ferroelectric Phosphorous Chalcogenide Crystals, Vilnius University Publishing House, Vilnius, 2006. [2] R.O. Vlokh, Yu.M. Vysochanskii, A.A. Grabar, A.V. Kityk, V.Yu. Slivka, Sov. Phys.:Izv. Akad. Nauk SSSR, Ser. Neorg. Mater. 27 (1991) 689–692. [3] D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Günter, A.A. Grabar, I.M. Stoika, Yu.M. Vysochanskii, Opt. Commun. 215 (2003) 333–343. [4] S.G. Odoulov, A.N. Shumelyuk, U. Hellwig, R. Rupp, A.A. Grabar, I.M. Stoyka, J. Opt. Soc. Am. B. 13 (1996) 2352–2360. [5] M. Jazbinsek, G. Montemezzani, P. Gunter, A.A. Grabar, I.M. Stoika, Y.M. Vysochanskii, J. Opt. Soc. Am. B. 20 (2003) 1241–1256. [6] O. Krupych, D. Adamenko, O. Mys, A. Grabar, R. Vlokh, Appl. Opt. 47 (2008) 6040–6045. [7] O. Mys, I. Martynyuk-Lototska, A. Grabar, Yu. Vysochanskii, R. Vlokh, Ukr. J. Phys. Opt. 7 (2006) 124–128. [8] O.G. Mys, I.Yu. Martynyuk-Lototska, A.A. Grabar, Yu.M. Vysochanskii, R.O. Vlokh, Ferroelectrics 352 (2007) 171–175. [9] I.Yu. Martynyuk-Lototska, O.G. Mys, A.A. Grabar, I.M. Stoika, Yu.M. Vysochanskii, R.O. Vlokh, Appl. Opt. 47 (2008) 52–55. [10] O. Mys, I. Martynyuk-Lototska, A. Grabar, R. Vlokh, J. Phys.: Condens. Matter. 21 (2009) 265401–265407. [11] M. Gottlieb, T.J. Isaacs, J.D. Feichtner, G.W. Roland, J. Appl. Phys. 45 (1974) 5145–5151. [12] V. Samulionis, Ultragarsas 4 (2002) 7–12. [13] O. Mys, I. Martynyuk-Lototska, R. Vlokh, Ukr. J. Phys. Opt. 12 (2011) 166–170. [14] I. Martynyuk-Lototska, O. Mys, B. Zapeka, R. Vlokh, Phil. Mag. 91 (2011) 3519– 3546. [15] I. Martynyuk-Lototska, A. Say, O. Mys, R. Vlokh, Phil. Mag. 91 (2011) 4293– 4301. [16] I. Martynyuk-Lototska, A. Say, B. Zapeka, O. Mys, R. Vlokh, Ukr. J. Phys. Opt. Suppl. 3 (3) (2012) S1–S10. [17] C.D. Carpentier, R. Nitsche, Mater. Res. Bull. 9 (1974) 401–410. [18] H.B. Huntington, S.N. Gangoli, J. Chem. Phys. 50 (1969) 3844–3849. [19] S. Prawer, T.F. Smith, T.R. Finlayson, Aust. J. Phys. 38 (1985) 63–83. [20] D. Haertle, A. Guarino, J. Hajfler, G. Montemezzani, P. Günter, Opt. Express. 13 (2005) 2047–2057. [21] R.W. Dixon, M.G. Cohen, Appl. Phys. Lett. 8 (1966) 205–207. [22] R.W. Dixon, J. Appl. Phys. 38 (1967) 5149–5153. [23] R. Vlokh, I. Martynyuk-Lototska, Ukr. J. Phys. Opt. 10 (2009) 89–99. [24] P.S. Peercy, I.J. Fritz, Phys. Rev. Lett. 32 (1974) 466–469. [25] D.B. McWhan, R.J. Birgeneau, W.A. Bonner, H. Taub, J.D. Ace, J. Phys. C: Solid State Phys. 8 (1975) L81–L85. [26] N.B. Singh, W.M.B. Duval, Growth kinetics of physical vapour transport processes: crystal growth of opto-electronic material mercurous chloride, NASA Technical Memorandum, vol. 103788, 1991. [27] B. Singh, R.H. Hopkins, D.R. Suhre, L.H. Taylor, W. Rosch, M. Gottlieb, W.M.B. Duval, M.E. Glicksman, Northrop Grumman, The 128th TMS Annual Meeting & Exhibition, San Diego, California USA Technical Program, vol. 36, 1999.
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Optical Materials journal homepage: www.elsevier.com/locate/optmat
Elastic and acoustooptic properties of Sn2P2S6 crystals: Effect of ferroelectric phase transition O. Mys, B. Zapeka, I. Martynyuk-Lototska, R. Vlokh ⇑ Institute of Physical Optics, 23 Dragomanov St., 79005 Lviv, Ukraine
a r t i c l e
i n f o
Article history: Received 23 March 2012 Received in revised form 3 July 2012 Accepted 16 July 2012 Available online 6 October 2012 Keywords: Acoustooptic figure of merit Sn2P2S6 crystals Tricritical point Critical exponent Phase diagrams
a b s t r a c t We report on the studies for temperature dependences of elastic stiffness coefficients in Sn2P2S6 crystals. Basing on the construction of acoustic velocity surfaces, we have determined the parameters of the slowest acoustic wave that propagates in Sn2P2S6 crystals. The acoustooptic figure of merit for the case of acoustooptic interaction with this wave is estimated as 0.8 1012 s3/kg. We have shown that Sn2P2S6 is very close to the conditions of tricritical point on the (x, T)- and (p, T)-phase diagrams of the solid solutions Sn2P2(SexS1x)6. The critical exponent a of the heat capacity for the Sn2P2S6 crystals is equal to 0.42 ± 0.03. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Tin thiohypodiphosphate crystals Sn2P2S6 are wide-gap semiconductors that manifest a proper, second-order paraelectric-toferroelectric phase transition, with the change 2m M m of their point symmetry at TC = 337 K [1]. These crystals are transparent in a wide spectral range (from k = 0.53 lm to k = 8.0 lm [1], have high enough electrooptic coefficients r11 = 1.74 1010 m/V at the room temperature and the light wavelength of k = 633 nm [2,3]), and manifest conspicuous photorefractive properties [4,5]. Moreover, the tin thiohypodiphosphate represents a promising magnetooptic material (the corresponding Verdet constant amounts to 115 rad/T m [6]). In our recent works [7–10] we have experimentally studied and analyzed acoustic properties of Sn2P2S6 at the room temperature and demonstrated that the material has great potentials for acoustooptic (AO) applications. The AO figure of merit (AOFM) of Sn2P2S6 calculated on the basis of our experimental AO data reaches very high values M2 = (1.7 ± 0.4) 1012 s3/kg [9]. In other words, the crystals of Sn2P2S6 reveal one of the highest AOFMs known for the AO materials operating in the visible spectral range (see, e.g., the study of AOFM for Hg2Br2 crystals [11]). This is why Sn2P2S6 can be widely used in optoelectronics, in particular in devices for AO operation of optical radiation. It is necessary to note that the AOFM value mentioned above has been obtained for the conditions of AO interaction with the ⇑ Corresponding author. Tel.: +38 032 2611488; fax: +38 032 2611483. E-mail address: [email protected] (R. Vlokh). 0925-3467/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optmat.2012.07.025
acoustic wave which still is not the slowest one. Being more precise, we have dealt with the waves having the velocities v21 = 2100 m/s and v23 = 2610 m/s (here the first and the second indices denote the directions of the wave vector q and the polarization vector e of the acoustic wave, respectively) [9]. The AOFM is inversely proportional to the cube of the acoustic wave velocity and so becomes extremely high with lowering velocity (M 2 ¼ n6 p2ef =qv 3 , with n being the refractive index, pef the effective elastooptic coefficient, and q the crystal density). We have shown in our recent work [10] that the slowest wave in the Sn2P2S6 crystals is the transverse wave with the velocity 1550 ± 120 m/s at the room temperature. It has also been found that the propagation direction of this wave coincides with neither of the principal crystallographic directions and, moreover, it does not belong to the principal crystallographic planes. One can easily calculate that, for the case of AO interaction with the latter wave, the AOFM can increase by several times, when compared with the value obtained in the work [9], if only the effective elastooptic coefficient remained approximately the same. Both the velocity of the slowest wave mentioned above and its propagation direction have been derived in the study [10] using the following procedure: the elastic stiffness tensor has been rewritten in a so-called acoustic eigen coordinate system rotated around b axis by the angle h = [arctan(2C46/(C66 C44))]/ 2 = (16 ± 5)° with respect to the initial coordinate system, in order to meet the conditions C 046 ¼ C 015 ¼ C 025 ¼ C 035 0 for the elastic stiffness tensor components written in this new system. In fact, such an ‘eigen acoustic coordinate system’ does not exist at all. Really, only the tensor component C 046 0 in this system, while
O. Mys et al. / Optical Materials 35 (2012) 168–174
the other coefficients C 015 ; C 025 and C 035 remain nonzero, though being small enough. Since the parameters of the slowest wave (the directions of its propagation and polarization) for Sn2P2S6 have been obtained in the work [10] with assuming of exact equality to zero of the coefficients C 015 ; C 025 and C 035 , these parameters could serve as a rough approximation only. On the other hand, a decrease in the acoustic wave velocities occurring with approaching the phase transition temperature TC, as revealed for some of the longitudinal waves propagating in Sn2P2S6 [12], can in principle lead to further increase in the AOFM. However, in our recent work [13] reporting the temperature dependences of both the longitudinal and transverse acoustic wave velocities in the Sn2P2S6 crystals, we have shown that the above temperature dependences should impose no notable variation of the AOFM, though the relevant results are also approximated. For determining the propagation direction of the slowest acoustic wave and possible variations of its parameters with temperature, the acoustic velocity surfaces are to be constructed for different temperatures. In order to find the minimal velocity, it would be necessary to cut these surfaces by a plane that rotates, e.g., around the b axis and then determine the minimal velocities within all of these planes. Prior to this, the temperature dependences of all the elastic stiffness coefficients for Sn2P2S6 crystals are needed. On the other hand, efficient applications of these crystals should be based on high enough stiffness coefficients in a wide temperature range, including the point TC = 337 K which is close to the room temperature. In connection with this, it is worthwhile to notice that some of the stiffness coefficients for the Sn2P2(Se0.28S0.72)6 crystals tend almost to zero near the phase transition temperature [14], thus making the crystals inconvenient for practical applications. Following from the above considerations, the aim of the present work is to study the temperature behavior of all the stiffness coefficients in the Sn2P2S6 crystals in the vicinity of the Curie temperature. Besides, we will also touch the problem of critical exponents for the Sn2P2S6 crystals, which are close to a so-called tricritical point both on ‘selenium concentration–temperature’ (x, T) [14] and ‘hydrostatic pressure–temperature’ (p, T) [15,16] phase diagrams peculiar for the solid solutions of Sn2P2(SexS1x)6. Issuing from the temperature dependences of the elastic stiffness coefficients, we will determine the critical exponent a describing temperature dependence of the heat capacity Cp (TC T)a. It is important that the latter index is close to 0.5 for the case of Sn2P2(Se0.28S0.72)6 crystals, which corresponds to thermodynamic system in the vicinity of its tricritical point [14].
2. Data processing The lattice parameters of Sn2P2S6 are equal to a = 0.9378 nm, b = 0.7448 nm, c = 0.6513 nm, and b = 91.15° (at the room temperature and the atmospheric pressure), where the crystallographic axis b is perpendicular to the symmetry mirror plane [17], while the spontaneous polarization vector is almost parallel to the a axis. The elastic stiffness coefficients have been determined with respect to the crystallographic axes a = 1, b = 2, and c = 3. Here we report on complete determination of the elastic matrix Cklmn = Cij (with i, j = 1–6; 1 = 11, 2 = 22, 3 = 33, 4 = 23, 5 = 13 and 6 = 12), via measurements of the phase ultrasonic velocities along different crystallographic directions. The temperature dependences of the velocities of longitudinal and transverse ultrasonic waves have earlier been obtained with a standard pulse-echo overlap method and presented in our recent work [13].1 Since Sn2P2S6 belongs to the 1 The Sn2P2S6 crystals studied by us have been grown with a vapor-transport technique at the Uzhgorod National University (Ukraine).
169
monoclinic point symmetry group m at the room temperature, there are 13 independent nonzero elastic coefficients (C11, C22, C33, C44, C55, C66, C12, C13, C23, C15, C25, C35, and C46), and the same holds true of the point group 2/m describing the paraelectric phase. Simple expressions linking the velocities and the elastic coefficients are available only for the four of the elastic stiffness matrix components (C 22 ¼ qv 22 ; C 44 ¼ qv 232 ; C 66 ¼ qv 212 , and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 2 2 C 46 ¼ 0:5 qv 21 qv 23 ðC 44 C 66 Þ ). All the other coefficients are linked by more complicated relations or by complex systems of equations. Then cumbersome, both in experimental and computational aspects, procedures for deriving the remaining nine elastic coefficients are required for the monoclinic system [18,19]. To determine those coefficients, one should measure the velocities of quasi-longitudinal (QL) and quasi-transverse (QT) ultrasonic waves for at least six different directions ([1 0 0], [0 1 0], [0 0 1], [1 1 0], [1 0 1], and [0 1 1]). As a result, the errors for the elastic stiffness coefficients would be high enough (15%). The acoustic velocity surfaces have been constructed using the well known Christoffel equation, the software packages ‘Origin 7.0’, ‘Mathcad’ and ‘Maple 13’, and the experimental data for the elastic stiffness coefficients. The exact propagation direction and the velocity of the slowest wave have been determined after rotating the elastic stiffness tensor around the crystallographic axis b (the step of 10°) and constructing cross sections of the acoustic velocity surfaces for each rotation angle. 3. Results and discussion 3.1. Temperature dependences of elastic stiffness coefficients Under normal conditions, the elastic stiffness coefficients obtained in the present work for Sn2P2S6 agree well with the data obtained in our recent work [10]. The temperature dependences of the elastic stiffness coefficients are presented in Fig. 1. As seen from Fig. 1, only some of the coefficients reveal notable temperature dependence in the vicinity of TC, namely C11, C22, C23, C12, and C13. The other coefficients (C33, C15, C25, C35, and C66) have a step-like anomaly at the phase transition temperature, while the coefficients C44, C55, and C46 are almost temperature-independent. 3.2. Acoustic wave velocity surfaces The cross sections of the acoustic velocity surfaces calculated for different temperatures are presented in Fig. 2. The velocity of the slowest wave QT2 (with the wave vector belonging to the crystallographic coordinate plane ac) is equal to 1407 m/s. The wave vector of this wave is rotated by 55° with respect to the a axis, while the polarization vector belongs to the ac plane (see Fig. 2a). As follows from the cross sections of the acoustic wave velocity surfaces by the plane a0 b (where the angle between the a and a0 axes is equal to 55° – see Fig. 3), this wave is found to be the slowest acoustic wave for the Sn2P2S6 crystals. There are four different propagation directions of the slowest wave in Sn2P2S6 (see Fig. 2). Inside the temperature range of the ferroelectric phase, the direction of propagation of this wave remains almost the same (see Fig. 2a–c), while in the paraelectric phase the q vector is rotated by 39° with respect to the a axis (see Fig. 2d and e). This fact is caused by a jump-like rotation of the surface for the QT2 wave around the b axis occurring at the phase transition. It is seen from Fig. 4 that the velocity of the slowest wave manifests very weak temperature dependence in the both phases. The slowest transverse wave propagates along the direction rotated by 55° with respect to the a axis in the ac plane and has its polarization lying in the same plane. It should excite the
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O. Mys et al. / Optical Materials 35 (2012) 168–174
Fig. 1. Temperature dependences of elastic stiffness coefficients for the Sn2P2S6 crystals: (a) C11 – open circles, C22 – open squares, C33 – open triangles; (b) C12 – semi-open circles, C13 – semi-open squares, C23 – semi-open triangles; (c) C15 – full circles, C25 – full squares, C35 – full triangles, C46 – full diamonds; and (d) C44 – strikethrough circles, C55 – strikethrough squares, C66 – strikethrough triangles.
mechanical strains u1 = cos 55° sin 55°u = 0.47u, u3 = cos 35° cos 55°u = 0.47u, and u5 = cos °55°u = 0.33u, where u denotes the strain caused by the incident acoustic wave. Let us remind that the crystallographic plane ac represents a plane of the intermediate (nm = n1 = 3.0256 and the largest ng = n3 = 3.0982 refractive indices, while the angle between the a axis and the direction corresponding to the principal refractive index n3 is equal to 43.3 ± 0.3° (k = 632.8 nm) [20]. If we consider the AO interaction in the ac plane with the slowest wave, three different cases are possible (see Fig. 5). These are the case of isotropic interaction described by the relation in ðaÞ Ed2 ¼ DB22 Din 2 ¼ ð0:47ðp23 p21 Þ þ 0:33p25 ÞuD2 ;
the case of the ‘mixed’ isotropic and anisotropic AO interaction given by
( ðbÞ
in Ed3 ¼ DB33 Din 3 þ DB31 D1
Ed1
¼
DB13 Din 3
þ
DB11 Din 1
( )
in Ed3 ¼ ½ð0:47ðp33 p31 Þ þ 0:33p53 ÞDin 3 þ ð0:47ðp53 p51 Þ þ 0:33p55 ÞD1 u in Ed1 ¼ ½ð0:47ðp53 p51 Þ þ 0:33p55 ÞDin 3 þ ð0:47ðp13 p11 Þ þ 0:33p15 ÞD1 u
and the case of anisotropic AO interaction described by the relations
( ðcÞ
Ed1 ¼ DB12 Din 2 Ed3 ¼ DB32 Din 2
:
d Here Din i is the electric displacement of the incident optical wave, Ej the electric field of the diffracted wave, and DBij the increment of the optical impermeability tensor due to photoelastic perturbation. Since all of the relevant photoelastic coefficients (p61, p63, p65, p41, p43, and p45) are equal to zero in the last case, this anisotropic AO diffraction cannot be implemented. Unfortunately, we cannot make the exact calculation of the AOFM using the relations mentioned above, because the photoelastic coefficients for Sn2P2S6 are unknown. Indeed, in spite of the fact that the modules of the piezooptic coefficients for the Sn2P2S6 crystals have already been determined [8], their recalculation to photoelastic ones is impossible, since the signs of these coefficients are unknown. Besides, any calculations of the photoelastic tensor components based on the piezooptic coefficients would lead to high errors even for many systems with high symmetries (see,
;
e.g., [8]), thus making the corresponding calculations of the AOFM meaningless. It is the more so as Sn2P2S6 represents a low-symmetry crystal, for which the experimental conditions for measuring the piezooptic tensor components become very complicated, and the
O. Mys et al. / Optical Materials 35 (2012) 168–174
171
Fig. 2. Cross sections of acoustic wave velocity surfaces by the crystallographic planes ab (left), ac (middle) and bc (right) for different temperatures: (a) 295, (b) 325, (c) 329, (d) 340, and (e) 350 K. Squares correspond to quasi-longitudinal wave QL, open circles to quasi-transverse wave QT1, and open triangles to quasi-transverse wave QT2.
nonzero tensor components of the piezooptic and elastic stiffness coefficients are numerous, producing still larger errors of both
the measurements and the consequent recalculations. Of course, the information on the photoelastic coefficients could, in principle,
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O. Mys et al. / Optical Materials 35 (2012) 168–174 Table 1 Fitting parameters of temperature dependences of the elastics stiffness coefficients. Component of elastic stiffness tensor
Elastic stiffness coefficient
C22
2.31 ± 0.03 2.44 ± 0.03 [14] 3.04 ± 0.05 1.53 ± 0.05
C11 C13 Mean value
0
Fig. 3. Cross sections of acoustic wave velocity surfaces by the plane (a b) at 0 T = 295 K (the angle between the a and (a ) axes is equal to 55°).
Ratio a2
4ðijÞ pffiffiffiffiffiffiffiffiffi ; 109 0
Critical exponent a
Coefficient of determination R2
0.44 0.44 ± 0.03 [14] 0.42 0.40 ± 0.03 0.42 ± 0.03
0.964 0.987 [14]
a10 a3
C 0ij ; 1010 N=m2 1.91 ± 0.04 1.63 ± 0.25 [14] 3.1 ± 0.1 2.55 ± 0.39
0.967 0.982
Luckily in our situation it is sufficient to use an ‘isotropic approximation’ while determining the mean value of the piezooptic coefficients. The Sn2P2S6 crystals belong to the symmetry group 2/m in their paraelectric phase, which is a subgroup of the cubic group m3m. Using specific relationships among the elastic, piezooptic and photoelastic tensor components for the above cubic group, one can write the following rough relations for the group 2/m:
C 11 ðC 11 þ C 22 þ C 33 Þ=3 ¼ 41:16 109 N=m2 ; C 12 ðC 12 þ C 13 þ C 23 Þ=3 ¼ 13:39 109 N=m2 ;
p 12 ðp12 þ p13 þ p23 þ p21 þ p31 þ p32 Þ=6 ¼ 6:8 pN=m2 ; p 11 p11 þ p22 þ p33 Þ=3 ¼ 8:9 pN=m2 ; 12 ¼ p 12 ðC 11 C 12 Þ ¼ 0:2: 11 p p
Fig. 4. Temperature dependence of velocity of the slowest acoustic wave in Sn2P2S6 crystals.
Fig. 5. Scheme of AO interaction that occurs in the ac plane of Sn2P2S6 crystals.
be obtained when using a so-called Dixon-Cohen method [21,22], or Brillouin scattering method all though they are comparatively complex. It is necessary to notice that, in the case of low-symmetry crystals, the Dixon-Cohen and Brillouin scattering methods yield only the combination of photoelastic coefficients. A separate determination of such coefficients as p46, p15, p25 p35, p51, p52, p53 and p64 is problematic. At the same time, solving a system of equations in order to evaluate the separate coefficients basing on the data obtained from the experiment, leads to increased error in determination of these coefficients.
Then the efficiency of the AO interaction described by the case (a) should be very small, because we have p23 p21 0 and p25 0 in the isotropic approximation. In the frame of the same approximation, the corresponding relations for the case of isotropic dif11 p 12 ÞDin fraction (b) can be rewritten as Ed3 ¼ 0:47ðp and 3 12 p 11 ÞDin Ed1 ¼ 0:47ðp . In the both cases, we have the effective 1 photoelastic coefficient equal to pef 0.1. Then the AOFM can be 2ef =qv 3 0:8 1012 s3 =kg (n 6 p ¼ 3:02; q ¼ estimated as M2 ¼ n 3:54 103 kg=m3 [20], and v = 1407 m/s). As for the AO interaction with the longitudinal wave propagating along the a axis, which is characterized by more than two times ef ¼ p 11 ¼ p 11 ðC 11 þ larger velocity (v11 = 3697 m/s), we have p 2C 12 Þ þ 3p12 ðC 11 þ C 12 Þ 0:5. Then the AOFM is higher (1012 s3/kg) than that for the case of AO interaction with the slowest wave. Hence, the AOFM for the Sn2P2S6 crystals is mainly determined by the values of their photoelastic coefficients and high enough values of the refractive indices. Indeed, Sn2P2S6 is not a ferroelastic material, where low velocities of the acoustic waves are caused by softening of some acoustic phonons, due to acoustic instability of crystalline lattice near the phase transition point [23]. As a result, the velocities of the acoustic waves in Sn2P2S6 are not low enough that they should have a principled effect on the AOFM, as occurs for some well-known ferroelastic AO materials (e.g., TeO2 [24,25] or H2Br2 crystals [26,27]). 3.3. Thermodynamic analysis Now let us analyze the temperature dependences of the elastic stiffness coefficients. This can be done following from the thermodynamic potential (1) written in the form, which is general for proper ferroelectric phase transitions occurring from centrosymmetric paraelectric phases into polar ferroelectric ones:
F ¼ F0 þ
a1 2 a2 4 a3 6 a4 2 c P þ P þ P þ P e þ e2 : 2 2 4 6 2
ð1Þ
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O. Mys et al. / Optical Materials 35 (2012) 168–174
parameters such as a hydrostatic pressure or a concentration of substituting chemical elements. For example, the change in the sign of the coefficient a2 at some thermodynamic coordinates would lead to changing order of the phase transition and so appearance of the tricritical point on the phase diagram. Notice that the coefficients a1, a2 and a3 are fully symmetric tensors respectively of the ranks two, four, and six, while a4 and c represent tensors with the internal symmetries [V2]2 and [[V2]2]. After minimizing the free energy, @F=@P ¼ 0, and taking the relation a01 ¼ ða1 þ a4 eÞ ¼ a010 ðT C TÞ into account, one can write the spontaneous polarization as
a01 a010 ðT C TÞ ¼ : a2 a2
P2s ¼
ð2Þ
In other words, the critical exponent of the spontaneous polarization is equal to 0.5. The elastic stiffness coefficients are defined by the relation
X @2F @2F C mn jP–0 C mn jP¼0 ¼ DC mn ¼ vlk ; @em @Pl @en @Pk k;l
ð3Þ
where a so-called ‘clamped‘ susceptibility is given by
@2F @Pl @Pk
vlk ¼
!1 ð4Þ
:
For example, in case of Sn2P2S6 we have v1 ¼ ð4a010 ðT C TÞÞ1 at T < TC. Taking nonzero strain tensor components e1, e2, e3, and e5 into consideration, we write out the coupling energy density F c ¼ a24 P 2 e in the matrix presentation as
Fc ¼
P 21 ða4ð11Þ e1 þ a4ð12Þ e2 þ a4ð13Þ e3 þ a4ð15Þ e5 Þ; 2
ð5Þ
where a4(ij) are the components of a forth-rank tensor. Considering formulae (2)–(5), one can obtain the following relations for the elastic stiffness increments in the Sn2P2S6 crystals:
DC 11 ¼ DC 13
a24ð11Þ
;
DC 22 ¼
4a2 a4ð11Þ a4ð13Þ ¼ ; 4a2
DC 25 ¼
a4ð12Þ a4ð15Þ ; 4a2
a24ð12Þ 4a2
;
DC 33 ¼
a24ð13Þ 4a2
;
DC 12 ¼
DC 23 ¼
a4ð12Þ a4ð13Þ ; 4a2
DC 15 ¼
DC 35 ¼
a4ð13Þ a4ð15Þ ; 4a2
DC 55
a4ð11Þ a4ð12Þ ; 4a2
a4ð11Þ a4ð15Þ ; 4a2 2 a4ð15Þ ¼ : 4a2 ð6Þ
The elastic stiffness tensor components appearing in formulae (6) should reveal a step-like behavior at the Curie point, whereas the components DC44 = DC66 = DC46 = 0 should remain unchanged while passing the temperature of the phase transition. Let us consider the thermodynamic potential given by formula (1) under the conditions of tricritical point a2 ¼ 0 at x = xTC, where xTC = 0.28 is the concentration of Se that corresponds to the tricritical point [14]). If a2 = 0, the relation for the spontaneous polarization may be written as Fig. 6. Dependences of elastic stiffness coefficients in the ferroelectric phase on the relative temperature (TC T): (a) C11, (b) C13, and (c) C22. Solid curves are fits according to formula (10).
Here P means the electric polarization, e the mechanical strain, a1 = a10(TC T) the linear electric susceptibility a2 and a3 being the nonlinear susceptibility coefficients), a4 the coupling coefficient associated with electrostriction, and c the elastic stiffness coefficient determined under the condition of P = 0. All of the coefficients are invariant with respect to symmetry operations of the point group of paraelectric phase and almost temperature-independent in the range of paraelecric phase, except of a1. Of course, these coefficients can be dependent on some other thermodynamic
P2s ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a010 ðT C TÞ : a3
ð7Þ
Now we analyze more closely the relation (6) for the temperature dependences of the elastic stiffness coefficients in case of the tricritical point (xTC = 0.28). Then the temperature behavior of the electric susceptibility is given by
v11 ¼ 4a010 ðT T C Þ
1
:
ð8Þ
Accounting for formulae (3), (7), and (8), one gets the following relations for the increments of elastic stiffness coefficients observed at the tricritical point:
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O. Mys et al. / Optical Materials 35 (2012) 168–174
a24ð11Þ a24ð12Þ a24ð13Þ ; DC 22 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; DC 33 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; DC 11 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 0 a10 a3 ðT T C Þ a10 a3 ðT T C Þ a10 a3 ðT T C Þ a4ð11Þ a4ð12Þ a4ð11Þ a4ð13Þ a4ð12Þ a4ð13Þ ; DC 13 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; DC 23 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; DC 12 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a010 a3 ðT T C Þ a010 a3 ðT T C Þ a010 a3 ðT T C Þ a4ð11Þ a4ð15Þ a4ð12Þ a4ð15Þ a4ð13Þ a4ð15Þ ; DC 25 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; DC 35 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; DC 15 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a010 a3 ðT T C Þ a010 a3 ðT T C Þ a010 a3 ðT T C Þ
transitions, we have shown that the Sn2P2S6 crystals are very close to the tricritical point. The latter is realized at xTC = 0.28 on the (x, T)-phase diagram of Sn2P2(SexS1x)6 solid solutions and at pTC = 4.3 kbar on their (p, T)-phase diagram. We have also found that the critical exponent a for the Sn2P2S6 crystals is equal to 0.42 ± 0.03.
a24ð15Þ : DC 55 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a010 a3 ðT T C Þ
Acknowledgements ð9Þ
In other terms, all of these elastic stiffness components should behave in the ferroelectric phase according to the power law DCij (T TC)0.5 and go to infinity at T = TC. Only the increments DC44 = DC66 = DC46 = 0 should remain unchanged when the phase transition point is being passed. We have fitted the temperature dependences of the elastic stiffness coefficients in the ferroelectric phase by the function
a24ðijÞ ffi ðT T C Þa ; C ij ¼ C 0ij þ pffiffiffiffiffiffiffiffiffiffiffi a010 a3
ð10Þ
where C 0ij is a so-called ‘background’ elastic stiffness, which is independent of temperature (see Table 1, Fig. 6). We have found that the critical exponent a is equal to 0.42 ± 0.03. Thus, the phase transition in Sn2P2S6 is close enough to the tricritical point, which should occur at xTC = 0.28 on the (x, T)-phase diagram and at pTC = 4.3 kbar on the (p, T)-phase diagram of the Sn2P2(SexS1x)6 solid solutions. 4. Conclusions In the present work we have constructed the acoustic velocity surfaces and determined the parameters describing the slowest acoustic wave that can propagate in the Sn2P2S6 crystals. We have revealed that, under the normal conditions, this wave is transverse, it has the velocity of 1407 m/s, propagates in the crystallographic plane ac along the direction defined by the angle of 55° with respect to the a axis, and its polarization vector belongs to the ac plane. The velocity of this wave is almost temperature-independent in the temperature region of the ferroelectric phase transition. The AOFM calculated for the case of AO interaction with this wave is equal to 0.8 1012 s3/kg. It is smaller than that experimentally obtained in our recent work [9] and even smaller than the value peculiar for the AO interaction with the longitudinal wave revealing higher velocity (1012 s3/kg). This is explained as being due to the fact that the Sn2P2S6 crystals are not a ferroelastic material, for which low velocities of the acoustic waves are caused by softening of some acoustic phonons, as a result of acoustic instability of crystalline lattice near the phase transition point. As a result, the acoustic wave velocity (even that characteristic of the slowest wave) is not low enough for increasing sufficiently the AOFM value. Basing on our analysis of temperature dependences of the elastic stiffness coefficients in the frame of mean-field theory of phase
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