- Email: [email protected]
Reflection and vibrational spectra of SbSCl0.1I0.9 crystals in the ferroelectric phase-transition region
Journal of Physics and Chemistry of Solids 75 (2014) 194–197
Contents lists available at ScienceDirect
Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
Reflection and vibrational spectra of SbSCl0.1I0.9 crystals in the ferroelectric phase-transition region Algirdas Audzijonis, Leonardas Žigas n, Raimundas Žaltauskas, Raimundas Sereika Department of Physics, Lithuanian University of Educational Sciences, Studentu 39, 08106 Vilnius, Lithuania
art ic l e i nf o
a b s t r a c t
Article history: Received 17 June 2013 Received in revised form 2 August 2013 Accepted 14 September 2013 Available online 20 September 2013
SbSCl0.1I0.9 crystals were grown from the vapor phase and reflection spectra were recorded using a Fouriertransform IR spectrometer. The optical parameters and optical functions along the c axis were calculated using an improved Kramers–Kronig technique with two confining spectral limits. The reflection spectra were analyzed using the oscillator parameter fitting technique for comparison of results. The vibrational frequencies ωL and ωT were evaluated. The ferroelectric phase-transition temperature was estimated as 330 K for SbSCl0.1I0.9 from experimental reflection measurements and theoretical investigation of the potential energy of Sb atoms for B1u soft normal modes of SbSI and SbSCl0.1I0.9 crystals. & 2013 Elsevier Ltd. All rights reserved.
Keywords: B. Crystal growth C. Infrared spectroscopy D. Crystal structure SbSCl D. Dielectric properties D. Optical properties
1. Introduction
2. Crystal growth
Ferroelectric semiconductors have a variety of interesting properties [1]. These include pyroelectric, piezoelectric, electromechanical, electro-optical, and other nonlinear optical effects [2]. Applications vary from infrared integrated optical devices and actuators to pyro-optic detectors and imagers [3–7]. Solid SbSI solutions have been intensively studied because they have a Curie temperature (TC) that depends on their composition. In particular, SbSClxI1 x crystals are especially valuable because TC exceeds that of pure SbSI (TC ¼293 K). A ferroelectric phase-transition temperature of 330 K was determined from measurements of the temperature-dependent dielectric permittivity of SbSClxI1–x (x ¼0.01, 0.1) crystals along the c axis [8–10]. However, there have been no experimental investigations of the optical properties and vibrational spectra of mixed SbSClxI1 x crystals to date. The aims of the present study were: (1) to grow SbSClxI1–x crystals with a mirror-like surface suitable for measurement of Fourier-transform IR ((FTIR) reflection spectra; (2) to determine reflection and vibrational spectra of SbSCl0.1I0.9 crystals in the phase transition region; and (3) to investigate the phase transition temperature from experimental reflection spectra and theoretical calculations of the potential energy of Sb atoms on B1u soft mode amplitudes of normal coordinates.
We grew SbSCl0.1I0.9 crystals from the vapor phase, since this is the best way to obtain crystals suitable for optical studies. A twozone furnace was used for growth. The temperature was 673 K in the growth zone and 713 K in the second zone. Sb2S3, I2, and Sb2Cl3 were used as starting materials. The material was synthesized in an evacuated quartz ampoule placed in a rotating furnace before the crystal growth stage. The temperature was slowly increased to 973 K and held at this level for 12 h. Then 20 g of the synthesized SbSCl0.1I0.9 was placed in a quartz ampoule of 20 mm in diameter and 200 mm in length. The ampoule was evacuated and placed in the growth furnace. This technique can yield crystals with dimensions of up to 5 5 25 mm3 within 50–60 days. The crystal surfaces are mirrorlike. The largest surface is a [1 1 0] plane and the crystal is always oriented along the c axis (Fig. 1).
n
Corresponding author. E-mail address: [email protected] (L. Žigas).
0022-3697/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jpcs.2013.09.012
3. Reflection and vibrational spectra of SbSCl0.1I0.9 in the phase transition region We measured the far-IR reflectivity of SbSI bulk crystals under polarized light at 93–353 K over the range 15–400 cm 1 [11,12]. The optical constants and dielectric functions were calculated using the Kramers–Kronig (KK) analysis method. It has been reported that in the range 15–300 cm 1, SbSI reflectivity spectra have five peaks at temperatures less that TC (293 K) for E||c of and two peaks at temperatures greater than TC [11,12].
A. Audzijonis et al. / Journal of Physics and Chemistry of Solids 75 (2014) 194–197
195
Table 1 Oscillator parameters for SbSCl0.1I0.9 crystals obtained using Kramers–Kronig (KK) and oscillator parameter (OP) fitting techniques. Phase
KK ωT (cm–1)
ωL (cm–1)
OP ωT (cm–1)
ωL (cm–1)
Paraelectric (335 K)
– 169
124 271
– 172
121 273
Ferroelectric (300 K)
– 106 146 165 243
100 124 153 216 270
– 108 147 164 245
101 116 153 216 274
The experimental error in the low-frequency range (o 75 cm 1) is 75 cm–1. The ferroelectric phase transition temperature TC ¼ 330 K for SbSCl0.1I0.9 was estimated from dielectric permittivity data [8–10]. As observed in Fig. 2, the reflection spectrum between 550 and 50 cm–1 consists of five peaks at T oTC ¼330 K and two peaks at T 4TC. We used an FTIR spectrophotometer (LAFS-1000) equipped with a small-crystal unit and a cryostat with good temperature stability (75 K) to record IR reflection spectra of small crystals with perfectly reflecting surfaces in the region ω 4 90 cm–1. Since reflection spectra could not be measured in the microwave and IR regions for ω o 90 cm–1, we used an improved KK technique with spectral limits of a ¼90 cm–1 and b¼275 cm–1 to calculate the optical constants and functions. The phase angle was calculated as b þ a þmΔω 2a þ mΔω rad ; ð1Þ Θm ¼ A ln þ Φm þ B ln b a þmΔω mΔω where
Φm ¼ Δω Fig. 1. SbSCl0.1I0.9 crystal.
yn ¼
1.0
Fit. T = 335 K PEP
R(ω)
0.8 0.6 0.4 0.2 0.0 1.0
Fit. T = 300 K FEP
R(ω)
0.8 0.6 0.4 0.2 0.0 50
100 150 200 250 300 350 400 450 500 550
ω (cm-1)
n ¼ ðb a=ΔωÞ 1
∑
n ¼ 1;n a m
a þmΔω
π
U
yn ;
a þ nΔω
ln Rn 2 2 ; a þ mΔω
Far-IR reflectivity of the SbSCl0.1I0.9 crystal along the c axis was measured at 300–335 K from 550 to 50 cm 1. We used an FTIR spectrometer equipped with a unit suitable for small crystals with perfectly reflecting surfaces. A cryostat was used to vary the crystal temperature at intervals of 10 K. Vibrational reflection spectra of SbSCl0.1I0.9 were fitted using the oscillator parameter (OP) technique at (Fig. 2). OP results for the paraelectric and ferroelectric phases are presented in Table 1.
ð3Þ
Φm is the phase angle at point m, Rn is the reflection coefficient at point n, Δω is the spectral range, and a and b are the limiting energies of the reflection spectrum. The constants A and B can be found by estimating Φm1 and Φm2, where m1 and m2 are the rad spectral points at which absorption is absent (i.e., Θm ¼ 0. From Eq. (1), we obtain two equations at m1 and m2: ( 0 ¼ A C 1 þ Φm1 þ B C 2 ; ð4Þ 0 ¼ A C 3 þ Φm2 þ B C 4 where Φm1 and Φm2 are the values of Φ at m1 and m2 and the absorption index is k ¼0. C1, C2, C3, and C4 are described by 2a þ m1 Δω ; C 2 ¼ lnb þ a þ m1 Δω ; C 1 ¼ ln b a þ m1 Δω m1 Δω b þ a þ m Δω 2a þ m2 Δω 2 ; ð5Þ C 3 ¼ ln ; C 4 ¼ ln b a þ m Δω m Δω 2
Fig. 2. Reflection spectra for SbSCl0.1I0.9 crystal according to oscillator parameter fitting at E||c at (A) 335 K (paraelectric phase) and (B) 300 K (ferroelectric phase).
ð2Þ
2
where B¼ A¼
Φm1 C 3 Φm2 C 2 C4C1 C2C3
;
ð6Þ
Φm2 þ BC 4
:ð7Þ
C3
The optical function is estimated according to Imε 1 ðωÞ ¼
ε″ðwÞ
ðε′ðwÞÞ2 þ ðε″ðwÞÞ2
:ð8Þ
A. Audzijonis et al. / Journal of Physics and Chemistry of Solids 75 (2014) 194–197
Since reflection R(ω) spectra of these crystals could not be measured at o90 cm–1, we used an improved KK technique with spectral limits of a ¼90 cm–1 and b¼275 cm–1 to calculate the optical constants and functions. Spectra of the real ε′(ω) and imaginary parts of the dielectric permittivity ε″(ω) for SbSCl0.1I0.9 in the phase transition region at E||c are shown in Figs. 3 and 4, respectively. Spectra of the optical function Im ε–1(ω) for SbSCl0.1I0.9 in the phase transition region at E||c are shown in Fig. 5. The optical frequency of transverse ωT and longitudinal ωL modes can be determined from positions of the maxima in ε″(ω) (Fig. 4) and Im ε–1(ω) spectra (Fig. 5), respectively. The KK results are presented in Table 1. As seen from Figs. 3–5, the ferroelectric phase transition temperature for SbSCl0.1I0.9 is approximately 330 K. It was previously reported that the increase in TC for SbSCl0.1I0.9 (330 K) compared to SbSI (293 K) can be attributed to substitution of individual I– with Cl– ions, which leads to large deformation of the unit cell [8,9]. The changes observed, especially the decrease in c and b for the unit cell in comparison with pure SbSI (Table 2), are in agreement with the expectation that I– ions with a larger radius are substituted by Cl– ions with a smaller radius. Therefore, we investigated the relationship between ferroelectricity and its phase transition for SbSCl0.1I0.9 and deformation of the unit cell and high anharmonicity of the soft mode due to phonon interaction.
0.5
T = 300 K T = 315 K T = 325 K T = 335 K
0.4
-Im ε-1
196
0.3 0.2 0.1 0.0
100
125
150
175
ω
200
225
250
275
(cm-1)
Fig. 5. Spectra of the optical function of SbSCl0.1I0.9 in the phase transition region at E||c.
Table 2 Lattice parameters for SbSI and SbSClxI1–x crystals. Sample
a (Å)
b (Å)
c (Å)
Temperature (K)
SbSI [14] SbSCl0.1I0.9 [8]
8.522 8.509
10.145 10.083
4.097 4.099
293 330
4. Potential energy of Sb atoms
40 35 30 25
εI
We previously proposed a method for calculating the potential energy at point r of the unit cell [13]. All 12 atoms of the unit cell induce the following potential energy at point r:
T = 300 K T = 315 K T = 325 K T = 335 K
V ðr Þ ¼
20 15 10 5 0 -5
100
125
150
175
ω
200
225
250
275
(cm-1)
Fig. 3. Real part of the dielectric permittivity for SbSCl0.1I0.9 in the phase transition region at E||c.
25
T = 300 K T = 315 K T = 325 K T = 335 K
20
εII
15 10 5 0
100
125
150
175
200
225
250
275
ω (cm-1) Fig. 4. Spectra of the imaginary part of the dielectric permittivity for SbSCl0.1I0.9 in the phase transition region at E||c.
4π
∑s 2 f α ðsÞexp i r R0;α Q α s exp½ M α ðsÞ;
Ω α;s
ð9Þ
where Ω ¼(a b c) is the unit cell volume, s is the reciprocal lattice vector, α is the number of atoms in the unit cell, R0,α is the radius vector for atoms in the unit cell, Qα is the normal coordinate, exp[ Mα(s)] is the Debye–Waller factor, which depends on the mean square amplitude of the thermal vibration of atoms, and fα(s) is a form factor for electrons, calculated as f α ðsÞ ¼ ∑ nlmexp½ iðr UsÞnlm ; ð10Þ nlm
where nlm is the set of electron quantum numbers. The potential energy of atoms in the normal mode is evaluated by substituting r with radius vectors R0,α corresponding to the positions of atoms in the unit cell. The mean potential energy of Sb atoms in the unit cell in the normal mode for SbSI and SbSCl0.1I0.9 crystals can be expressed as VP ¼
V P ðRSb1 Þ þ V P ðRSb2 Þ þ V P ðRSb3 Þ þV P ðRSb4 Þ ; 4
ð11Þ
where RSb1 ¼R0,Sb1 þQSb1; RSb2 ¼R0,Sb2 þQSb2; RSb3 ¼R0,Sb3 þQSb3; and RSb4 ¼ R0,Sb4 þ QSb4. The coordinates for S and I atoms can be obtained according to Rα ¼ R0,α þ Qα, where α ¼ S1, S2, S3, S4, I1, I2, I3, and I4. We studied the potential energy of Sb atoms of the B1u (Au) symmetry with soft mode vibration as a function of relative atom displacement from their equilibrium position z0,α along the c axis. Since atoms oscillate along the c axis in the B1u normal mode, we set Qα ¼zα in Eq. (9), where zα denotes atomic displacement from z0,α. We varied Qα in small steps from – Qα(max) toþQα(max) for numerical evaluation of V P ðzÞ according to Eq. (11). For SbSI and SbSCl0.1I0.9, the average potential energy of S and I atoms in the paraelectric phase near the phase transition is a single-well function and depends only slightly on temperature and
A. Audzijonis et al. / Journal of Physics and Chemistry of Solids 75 (2014) 194–197
SbSI T = 293 K SbSCl0.1 I0.9 T = 330 K
197
ferroelectric phase transition for SbSCl0.1I0.9 occurs at 330 K. Our V P ðzÞ results demonstrate good agreement between theoretically estimated and experimentally determined TC [8–10] for SbSCl0.1I0.9 crystals.
51,0039
5. Conclusion
V P(a.u.)
51,0026 51,5
I
51,0013
1
2
-
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
z (a.u.) Fig. 6. Mean potential energy as a function of the amplitude of the B1u soft mode for normal coordinates (relative displacement of all atoms from their equilibrium positions) at Sb atom sites along the c axis in the paraelectric phase for SbSI at 293 K and SbSCl0.1I0.9 at 330 K.
the lattice parameters. However, V P ðzÞ for Sb atoms in the paraelectric phase near the phase transition is a double-well function and strongly depends on temperature and the lattice parameters, so we only investigated this for Sb atoms. The lattice parameters for SbSI [14] and SbSCl0.1I0.9 [8] are listed in Table 2. We used atomic form factors to calculate V P ðzÞ for SbSI and SbSCl0.1I0.9 f VII ¼ f Cl Ux þ f I U ð1 xÞ;
ð12Þ
where fCl and fI are atomic form factors for Cl and I atoms, respectively. Calculations involved 5000 s vectors. The results for Sb atoms in SbSI and SbSCl0.1I0.9 are presented in Fig. 6. To investigate anharmonicity, it is useful to expand V P ðzÞ according to n
n
V P ðzÞ ¼ V 0 þ an z þ b z2 þ d z3 þ cn z4 ; n
n
n
n
ð13Þ
where a , b , d , and c are polynomial expansion coefficients. In the paraelectric phase, the polynomial expansion coefficients are an ¼dn ¼0. Calculation of the potential for SbSI at 293 K and SbSCl0.1I0.9 at 330 K reveals that coefficient bn o0. Fig. 6 shows that the barrier height between minima of the double-well potential is the same for SbSI at 293 K and SbSCl0.1I0.9 at 330 K using the lattice parameters from Table 2. Thus, the
SbSCl0.1I0.9 crystals were grown from the vapor phase and their reflection spectra were studied. Vibrational frequencies were evaluated for the paraelectric and ferroelectric phases using an improved KK technique with two confining spectral limits and the OP fitting technique. Analysis of the temperature dependence of R(ω), ε″(ω), and Im ε–1(ω) for SbSCl0.1I0.9 indicated that the phase transition temperature is at 330 K. We found that the mean potential energy as a function of the relative displacement of all atoms from their equilibrium positions at Sb atoms site along the c axis has a double-well shape for the paraelectric phase of SbSI and SbSCl0.1I0.9 crystals close to the phase transition point. V P ðzÞ depends on phonon interactions, temperature, and the lattice parameters. For SbSCl0.1I0.9 and SbSI crystals, phase transition occurs at the temperature at which the barrier height between the two potential energy minima reaches 0.0066 a.u. TC depends on soft mode anharmonicity arising from interaction between phonons and the temperature dependence of the lattice parameters. TC is higher for SbSCl0.1I0.9 (330 K) than for SbSI (295 K) because the temperature dependence of the lattice parameters differs for these crystalline materials. References [1] J. Grigas, Microwave Dielectric Spectroscopy of Ferroelectrics and Related Materials, Gordon & Breach, Amsterdam (1996) 1996; 336. [2] A. Audzijonis, L. Žigas, A. Kvedaravičius, Ferroelectrics 413 (2011) 342. [3] P. Szperlich, M. Nowak, I. Bober, J. Szala, D. Stroz, Ultrasonics Sonochem. 16 (2009) 398. [4] A. Audzijonis, R. Sereika, R. Žaltauskas, Solid State Commun. 147 (2008) 88. [5] M. Nowak, P. Szperlich, I. Bober, J. Szala, G. Moskal, D. Stroz, Ultrasonics Sonochem 15 (2008) 709. [6] P. Gupta, A. Stone, N. Woodward, V. Dierolf, J. Himanshu, Opt. Mater. Exp. 1 (2011) 652. [7] L. Cross, A. Bhalla, F. Ainger, US Patent 4994672, 1991. [8] B. Garbarz-Glos, Ferroelectrics 292 (2003) 137. [9] B. Garbarz-Glos, J. Grigas, Ferroelectrics 393 (2009) 38. [10] A. Audzijonis, L. Žigas, R. Sereika, A. Kvedaravičius, Ferroelectrics 425 (2011) 45. [11] J. Petzelt, Phys. Status Solidi b 36 (1969) 321. [12] J. Petzelt, Ferroelectrics 5 (1973) 219. [13] A. Audzijonis, L. Žigas, A. Kvedaravičius, R. Žaltauskas, Phase Transitions 83 (2010) 64. [14] K. Lukaszewicz, A. Pietraszko, J. Stepien-Damm, A. Kajokas, Pol. J. Chem. 71 (1997) 1852.
Contents lists available at ScienceDirect
Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
Reflection and vibrational spectra of SbSCl0.1I0.9 crystals in the ferroelectric phase-transition region Algirdas Audzijonis, Leonardas Žigas n, Raimundas Žaltauskas, Raimundas Sereika Department of Physics, Lithuanian University of Educational Sciences, Studentu 39, 08106 Vilnius, Lithuania
art ic l e i nf o
a b s t r a c t
Article history: Received 17 June 2013 Received in revised form 2 August 2013 Accepted 14 September 2013 Available online 20 September 2013
SbSCl0.1I0.9 crystals were grown from the vapor phase and reflection spectra were recorded using a Fouriertransform IR spectrometer. The optical parameters and optical functions along the c axis were calculated using an improved Kramers–Kronig technique with two confining spectral limits. The reflection spectra were analyzed using the oscillator parameter fitting technique for comparison of results. The vibrational frequencies ωL and ωT were evaluated. The ferroelectric phase-transition temperature was estimated as 330 K for SbSCl0.1I0.9 from experimental reflection measurements and theoretical investigation of the potential energy of Sb atoms for B1u soft normal modes of SbSI and SbSCl0.1I0.9 crystals. & 2013 Elsevier Ltd. All rights reserved.
Keywords: B. Crystal growth C. Infrared spectroscopy D. Crystal structure SbSCl D. Dielectric properties D. Optical properties
1. Introduction
2. Crystal growth
Ferroelectric semiconductors have a variety of interesting properties [1]. These include pyroelectric, piezoelectric, electromechanical, electro-optical, and other nonlinear optical effects [2]. Applications vary from infrared integrated optical devices and actuators to pyro-optic detectors and imagers [3–7]. Solid SbSI solutions have been intensively studied because they have a Curie temperature (TC) that depends on their composition. In particular, SbSClxI1 x crystals are especially valuable because TC exceeds that of pure SbSI (TC ¼293 K). A ferroelectric phase-transition temperature of 330 K was determined from measurements of the temperature-dependent dielectric permittivity of SbSClxI1–x (x ¼0.01, 0.1) crystals along the c axis [8–10]. However, there have been no experimental investigations of the optical properties and vibrational spectra of mixed SbSClxI1 x crystals to date. The aims of the present study were: (1) to grow SbSClxI1–x crystals with a mirror-like surface suitable for measurement of Fourier-transform IR ((FTIR) reflection spectra; (2) to determine reflection and vibrational spectra of SbSCl0.1I0.9 crystals in the phase transition region; and (3) to investigate the phase transition temperature from experimental reflection spectra and theoretical calculations of the potential energy of Sb atoms on B1u soft mode amplitudes of normal coordinates.
We grew SbSCl0.1I0.9 crystals from the vapor phase, since this is the best way to obtain crystals suitable for optical studies. A twozone furnace was used for growth. The temperature was 673 K in the growth zone and 713 K in the second zone. Sb2S3, I2, and Sb2Cl3 were used as starting materials. The material was synthesized in an evacuated quartz ampoule placed in a rotating furnace before the crystal growth stage. The temperature was slowly increased to 973 K and held at this level for 12 h. Then 20 g of the synthesized SbSCl0.1I0.9 was placed in a quartz ampoule of 20 mm in diameter and 200 mm in length. The ampoule was evacuated and placed in the growth furnace. This technique can yield crystals with dimensions of up to 5 5 25 mm3 within 50–60 days. The crystal surfaces are mirrorlike. The largest surface is a [1 1 0] plane and the crystal is always oriented along the c axis (Fig. 1).
n
Corresponding author. E-mail address: [email protected] (L. Žigas).
0022-3697/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jpcs.2013.09.012
3. Reflection and vibrational spectra of SbSCl0.1I0.9 in the phase transition region We measured the far-IR reflectivity of SbSI bulk crystals under polarized light at 93–353 K over the range 15–400 cm 1 [11,12]. The optical constants and dielectric functions were calculated using the Kramers–Kronig (KK) analysis method. It has been reported that in the range 15–300 cm 1, SbSI reflectivity spectra have five peaks at temperatures less that TC (293 K) for E||c of and two peaks at temperatures greater than TC [11,12].
A. Audzijonis et al. / Journal of Physics and Chemistry of Solids 75 (2014) 194–197
195
Table 1 Oscillator parameters for SbSCl0.1I0.9 crystals obtained using Kramers–Kronig (KK) and oscillator parameter (OP) fitting techniques. Phase
KK ωT (cm–1)
ωL (cm–1)
OP ωT (cm–1)
ωL (cm–1)
Paraelectric (335 K)
– 169
124 271
– 172
121 273
Ferroelectric (300 K)
– 106 146 165 243
100 124 153 216 270
– 108 147 164 245
101 116 153 216 274
The experimental error in the low-frequency range (o 75 cm 1) is 75 cm–1. The ferroelectric phase transition temperature TC ¼ 330 K for SbSCl0.1I0.9 was estimated from dielectric permittivity data [8–10]. As observed in Fig. 2, the reflection spectrum between 550 and 50 cm–1 consists of five peaks at T oTC ¼330 K and two peaks at T 4TC. We used an FTIR spectrophotometer (LAFS-1000) equipped with a small-crystal unit and a cryostat with good temperature stability (75 K) to record IR reflection spectra of small crystals with perfectly reflecting surfaces in the region ω 4 90 cm–1. Since reflection spectra could not be measured in the microwave and IR regions for ω o 90 cm–1, we used an improved KK technique with spectral limits of a ¼90 cm–1 and b¼275 cm–1 to calculate the optical constants and functions. The phase angle was calculated as b þ a þmΔω 2a þ mΔω rad ; ð1Þ Θm ¼ A ln þ Φm þ B ln b a þmΔω mΔω where
Φm ¼ Δω Fig. 1. SbSCl0.1I0.9 crystal.
yn ¼
1.0
Fit. T = 335 K PEP
R(ω)
0.8 0.6 0.4 0.2 0.0 1.0
Fit. T = 300 K FEP
R(ω)
0.8 0.6 0.4 0.2 0.0 50
100 150 200 250 300 350 400 450 500 550
ω (cm-1)
n ¼ ðb a=ΔωÞ 1
∑
n ¼ 1;n a m
a þmΔω
π
U
yn ;
a þ nΔω
ln Rn 2 2 ; a þ mΔω
Far-IR reflectivity of the SbSCl0.1I0.9 crystal along the c axis was measured at 300–335 K from 550 to 50 cm 1. We used an FTIR spectrometer equipped with a unit suitable for small crystals with perfectly reflecting surfaces. A cryostat was used to vary the crystal temperature at intervals of 10 K. Vibrational reflection spectra of SbSCl0.1I0.9 were fitted using the oscillator parameter (OP) technique at (Fig. 2). OP results for the paraelectric and ferroelectric phases are presented in Table 1.
ð3Þ
Φm is the phase angle at point m, Rn is the reflection coefficient at point n, Δω is the spectral range, and a and b are the limiting energies of the reflection spectrum. The constants A and B can be found by estimating Φm1 and Φm2, where m1 and m2 are the rad spectral points at which absorption is absent (i.e., Θm ¼ 0. From Eq. (1), we obtain two equations at m1 and m2: ( 0 ¼ A C 1 þ Φm1 þ B C 2 ; ð4Þ 0 ¼ A C 3 þ Φm2 þ B C 4 where Φm1 and Φm2 are the values of Φ at m1 and m2 and the absorption index is k ¼0. C1, C2, C3, and C4 are described by 2a þ m1 Δω ; C 2 ¼ lnb þ a þ m1 Δω ; C 1 ¼ ln b a þ m1 Δω m1 Δω b þ a þ m Δω 2a þ m2 Δω 2 ; ð5Þ C 3 ¼ ln ; C 4 ¼ ln b a þ m Δω m Δω 2
Fig. 2. Reflection spectra for SbSCl0.1I0.9 crystal according to oscillator parameter fitting at E||c at (A) 335 K (paraelectric phase) and (B) 300 K (ferroelectric phase).
ð2Þ
2
where B¼ A¼
Φm1 C 3 Φm2 C 2 C4C1 C2C3
;
ð6Þ
Φm2 þ BC 4
:ð7Þ
C3
The optical function is estimated according to Imε 1 ðωÞ ¼
ε″ðwÞ
ðε′ðwÞÞ2 þ ðε″ðwÞÞ2
:ð8Þ
A. Audzijonis et al. / Journal of Physics and Chemistry of Solids 75 (2014) 194–197
Since reflection R(ω) spectra of these crystals could not be measured at o90 cm–1, we used an improved KK technique with spectral limits of a ¼90 cm–1 and b¼275 cm–1 to calculate the optical constants and functions. Spectra of the real ε′(ω) and imaginary parts of the dielectric permittivity ε″(ω) for SbSCl0.1I0.9 in the phase transition region at E||c are shown in Figs. 3 and 4, respectively. Spectra of the optical function Im ε–1(ω) for SbSCl0.1I0.9 in the phase transition region at E||c are shown in Fig. 5. The optical frequency of transverse ωT and longitudinal ωL modes can be determined from positions of the maxima in ε″(ω) (Fig. 4) and Im ε–1(ω) spectra (Fig. 5), respectively. The KK results are presented in Table 1. As seen from Figs. 3–5, the ferroelectric phase transition temperature for SbSCl0.1I0.9 is approximately 330 K. It was previously reported that the increase in TC for SbSCl0.1I0.9 (330 K) compared to SbSI (293 K) can be attributed to substitution of individual I– with Cl– ions, which leads to large deformation of the unit cell [8,9]. The changes observed, especially the decrease in c and b for the unit cell in comparison with pure SbSI (Table 2), are in agreement with the expectation that I– ions with a larger radius are substituted by Cl– ions with a smaller radius. Therefore, we investigated the relationship between ferroelectricity and its phase transition for SbSCl0.1I0.9 and deformation of the unit cell and high anharmonicity of the soft mode due to phonon interaction.
0.5
T = 300 K T = 315 K T = 325 K T = 335 K
0.4
-Im ε-1
196
0.3 0.2 0.1 0.0
100
125
150
175
ω
200
225
250
275
(cm-1)
Fig. 5. Spectra of the optical function of SbSCl0.1I0.9 in the phase transition region at E||c.
Table 2 Lattice parameters for SbSI and SbSClxI1–x crystals. Sample
a (Å)
b (Å)
c (Å)
Temperature (K)
SbSI [14] SbSCl0.1I0.9 [8]
8.522 8.509
10.145 10.083
4.097 4.099
293 330
4. Potential energy of Sb atoms
40 35 30 25
εI
We previously proposed a method for calculating the potential energy at point r of the unit cell [13]. All 12 atoms of the unit cell induce the following potential energy at point r:
T = 300 K T = 315 K T = 325 K T = 335 K
V ðr Þ ¼
20 15 10 5 0 -5
100
125
150
175
ω
200
225
250
275
(cm-1)
Fig. 3. Real part of the dielectric permittivity for SbSCl0.1I0.9 in the phase transition region at E||c.
25
T = 300 K T = 315 K T = 325 K T = 335 K
20
εII
15 10 5 0
100
125
150
175
200
225
250
275
ω (cm-1) Fig. 4. Spectra of the imaginary part of the dielectric permittivity for SbSCl0.1I0.9 in the phase transition region at E||c.
4π
∑s 2 f α ðsÞexp i r R0;α Q α s exp½ M α ðsÞ;
Ω α;s
ð9Þ
where Ω ¼(a b c) is the unit cell volume, s is the reciprocal lattice vector, α is the number of atoms in the unit cell, R0,α is the radius vector for atoms in the unit cell, Qα is the normal coordinate, exp[ Mα(s)] is the Debye–Waller factor, which depends on the mean square amplitude of the thermal vibration of atoms, and fα(s) is a form factor for electrons, calculated as f α ðsÞ ¼ ∑ nlmexp½ iðr UsÞnlm ; ð10Þ nlm
where nlm is the set of electron quantum numbers. The potential energy of atoms in the normal mode is evaluated by substituting r with radius vectors R0,α corresponding to the positions of atoms in the unit cell. The mean potential energy of Sb atoms in the unit cell in the normal mode for SbSI and SbSCl0.1I0.9 crystals can be expressed as VP ¼
V P ðRSb1 Þ þ V P ðRSb2 Þ þ V P ðRSb3 Þ þV P ðRSb4 Þ ; 4
ð11Þ
where RSb1 ¼R0,Sb1 þQSb1; RSb2 ¼R0,Sb2 þQSb2; RSb3 ¼R0,Sb3 þQSb3; and RSb4 ¼ R0,Sb4 þ QSb4. The coordinates for S and I atoms can be obtained according to Rα ¼ R0,α þ Qα, where α ¼ S1, S2, S3, S4, I1, I2, I3, and I4. We studied the potential energy of Sb atoms of the B1u (Au) symmetry with soft mode vibration as a function of relative atom displacement from their equilibrium position z0,α along the c axis. Since atoms oscillate along the c axis in the B1u normal mode, we set Qα ¼zα in Eq. (9), where zα denotes atomic displacement from z0,α. We varied Qα in small steps from – Qα(max) toþQα(max) for numerical evaluation of V P ðzÞ according to Eq. (11). For SbSI and SbSCl0.1I0.9, the average potential energy of S and I atoms in the paraelectric phase near the phase transition is a single-well function and depends only slightly on temperature and
A. Audzijonis et al. / Journal of Physics and Chemistry of Solids 75 (2014) 194–197
SbSI T = 293 K SbSCl0.1 I0.9 T = 330 K
197
ferroelectric phase transition for SbSCl0.1I0.9 occurs at 330 K. Our V P ðzÞ results demonstrate good agreement between theoretically estimated and experimentally determined TC [8–10] for SbSCl0.1I0.9 crystals.
51,0039
5. Conclusion
V P(a.u.)
51,0026 51,5
I
51,0013
1
2
-
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
z (a.u.) Fig. 6. Mean potential energy as a function of the amplitude of the B1u soft mode for normal coordinates (relative displacement of all atoms from their equilibrium positions) at Sb atom sites along the c axis in the paraelectric phase for SbSI at 293 K and SbSCl0.1I0.9 at 330 K.
the lattice parameters. However, V P ðzÞ for Sb atoms in the paraelectric phase near the phase transition is a double-well function and strongly depends on temperature and the lattice parameters, so we only investigated this for Sb atoms. The lattice parameters for SbSI [14] and SbSCl0.1I0.9 [8] are listed in Table 2. We used atomic form factors to calculate V P ðzÞ for SbSI and SbSCl0.1I0.9 f VII ¼ f Cl Ux þ f I U ð1 xÞ;
ð12Þ
where fCl and fI are atomic form factors for Cl and I atoms, respectively. Calculations involved 5000 s vectors. The results for Sb atoms in SbSI and SbSCl0.1I0.9 are presented in Fig. 6. To investigate anharmonicity, it is useful to expand V P ðzÞ according to n
n
V P ðzÞ ¼ V 0 þ an z þ b z2 þ d z3 þ cn z4 ; n
n
n
n
ð13Þ
where a , b , d , and c are polynomial expansion coefficients. In the paraelectric phase, the polynomial expansion coefficients are an ¼dn ¼0. Calculation of the potential for SbSI at 293 K and SbSCl0.1I0.9 at 330 K reveals that coefficient bn o0. Fig. 6 shows that the barrier height between minima of the double-well potential is the same for SbSI at 293 K and SbSCl0.1I0.9 at 330 K using the lattice parameters from Table 2. Thus, the
SbSCl0.1I0.9 crystals were grown from the vapor phase and their reflection spectra were studied. Vibrational frequencies were evaluated for the paraelectric and ferroelectric phases using an improved KK technique with two confining spectral limits and the OP fitting technique. Analysis of the temperature dependence of R(ω), ε″(ω), and Im ε–1(ω) for SbSCl0.1I0.9 indicated that the phase transition temperature is at 330 K. We found that the mean potential energy as a function of the relative displacement of all atoms from their equilibrium positions at Sb atoms site along the c axis has a double-well shape for the paraelectric phase of SbSI and SbSCl0.1I0.9 crystals close to the phase transition point. V P ðzÞ depends on phonon interactions, temperature, and the lattice parameters. For SbSCl0.1I0.9 and SbSI crystals, phase transition occurs at the temperature at which the barrier height between the two potential energy minima reaches 0.0066 a.u. TC depends on soft mode anharmonicity arising from interaction between phonons and the temperature dependence of the lattice parameters. TC is higher for SbSCl0.1I0.9 (330 K) than for SbSI (295 K) because the temperature dependence of the lattice parameters differs for these crystalline materials. References [1] J. Grigas, Microwave Dielectric Spectroscopy of Ferroelectrics and Related Materials, Gordon & Breach, Amsterdam (1996) 1996; 336. [2] A. Audzijonis, L. Žigas, A. Kvedaravičius, Ferroelectrics 413 (2011) 342. [3] P. Szperlich, M. Nowak, I. Bober, J. Szala, D. Stroz, Ultrasonics Sonochem. 16 (2009) 398. [4] A. Audzijonis, R. Sereika, R. Žaltauskas, Solid State Commun. 147 (2008) 88. [5] M. Nowak, P. Szperlich, I. Bober, J. Szala, G. Moskal, D. Stroz, Ultrasonics Sonochem 15 (2008) 709. [6] P. Gupta, A. Stone, N. Woodward, V. Dierolf, J. Himanshu, Opt. Mater. Exp. 1 (2011) 652. [7] L. Cross, A. Bhalla, F. Ainger, US Patent 4994672, 1991. [8] B. Garbarz-Glos, Ferroelectrics 292 (2003) 137. [9] B. Garbarz-Glos, J. Grigas, Ferroelectrics 393 (2009) 38. [10] A. Audzijonis, L. Žigas, R. Sereika, A. Kvedaravičius, Ferroelectrics 425 (2011) 45. [11] J. Petzelt, Phys. Status Solidi b 36 (1969) 321. [12] J. Petzelt, Ferroelectrics 5 (1973) 219. [13] A. Audzijonis, L. Žigas, A. Kvedaravičius, R. Žaltauskas, Phase Transitions 83 (2010) 64. [14] K. Lukaszewicz, A. Pietraszko, J. Stepien-Damm, A. Kajokas, Pol. J. Chem. 71 (1997) 1852.