1 Introduction
Among various ferroelectric materials, N(CH3)4HSO4 or using the chemical formula TMAHSO4 (tetramethyl ammonium monohydrogen sulfate) and its successive phase transitions have been studied extensively. This compound exhibits commensurate-incommensurate transitions and with those transitions, the ferroelectric phase occurs. Its crystal structure is pseudocubic or it is monoclinic at room temperature with the values of the lattice constant, a≠b≠c=9.7Ǻ and α=β=90°, γ=90.01° [1]. It is known that N(CH3)4HSO4 shows the ferroelectricity in a temperature range between –104 and 40 °C and it has been observed experimentally that this crystal has transition points around –70, –42, 42 and 120 °C [1]. It has also been reported that the hydrated crystal, N(CH3)4HSO4.H2O, exhibits transitions at –104 °C and around 40 °C so that it looses water of crystallization above 60 °C on heating [2]. Crystal growth and the experimental studies on the physical and chemical properties of TMAHSO4 crystals have been reported previously [3]. As the temperature decreases from the room temperature, TMAHSO4 crystals exhibit a second order phase transition at 230 K which is followed by a first order transition at 200 K and between those two temperatures an incommensurate phase occurs [4] as stated above. In the normal phase above Ti=232 K at high temperatures, the crystal structure of TMAHSO4 is orthorhombic. In the incommensurate phase it is also orthorhombic down to Tc=202 K while the crystal structure in the commensurate phase is monoclinic with the volume of the unit cell doubled below Tc at low temperatures. The two different orientations of the SO4 ions in the orthorombic normal phase at high temperatures above 232 K give rise to two opposite orientations in the low temperature phase (below 202 K) and between two phases antiphase rotational displacements can occur in the incommensurate (INC) phase [4].
In the same ferroelectric family, N(CH3)4IO3 and N(CH3)4NO3 undergo the transitions at 207 °C and 10 °C, respectively [5]. Also, in this family, Rb2ZnCl4 [6, 7, 8], Rb2CoCl4, Rb2ZnBr4, (N(CH3)4)2ZnCl4, (N(CH3)4)2CoCl4, K2CoCl4 [9], K2ZnCl4 [8, 9, 10] N(CH3)4HSeO4 [4] and [N(CH3)4]2CuCl4 [11] undergo the successive phase transitions of the normal - incommensurate- commensurate sequence [12].
The transition between ferroelectric and paraelectric phases was detected at Tc=181 °C for K2CoCl4 from the measurements of the dielectric constant in the a direction (εa) at 100 kHz and, also from the measurements of the spontaneous polarization Ps and the coercive field Ec (maximum applied field is 11 kV/cm) at various temperatures [9]. Measurements of the dielectric constant along the c direction (εc) at 100 kHz and the spontaneous polarization Ps on heating for N(CH3)3HSO4.H2O [2], and also measurements of the dielectric constant along the a(εa), bεb and cεc directions at 100 kHz on heating for N(CH3)4IO3 [5] have been reported.
In this study, we analyze the temperature dependence of the dielectric constant εc (along the c direction) which was measured at 100 kHz on cooling for TMAHSO4 and we also analyze the temperature dependence of the spontaneous polarization Ps which was measured along the c direction on heating in the temperature range of –115 °C to –65 °C [1]. For this analysis, we calculate the temperature dependence of the spontaneous polarization Ps (order parameter) and the dielectric constant ε through the dielectric susceptibility χ(=ε−1) using the mean field theory for the ferroelectric- paraelectric transition in TMAHSO4. By expanding the free energy in terms of the order parameter (spontaneous polarization) for the first order transition in TMAHSO4, we derive the expressions for the temperature dependence of the spontaneous polarization Ps and the dielectric susceptibility, which are then fitted to the experimental data [1] for the ferroelectric- paraelectric transition in TMAHSO4. In our recent study [13], we have also used the Landau phenemological theory (mean field theory) to calculate the order parameter (spontaneous polarization) and the dielectric susceptibility as a function of temperature for the ferroelectric- paraelectric transition in BaTiO3 on the basis of observed Raman frequency data [14]. In this work, we use the experimental data for both the spontaneous polarization Ps and the dielectric constant εc [1], as stated above.
In Section 2, we give the mean field model to derive the Ps and εc as a function of temperature for TMAHSO4. In Section 3, we give our analysis and results. Our results are discussed in Section 4. Conclusions are given in Section 5.
2 Theory
The Landau phenemological theory provides an expansion of the free energy in terms of the order parameter. Using this mean field theory, we can determine the parameters in the free energy expanded in terms of the spontaneous polarization Ps (order parameter) for ferroelectric materials. By expanding the free energy up to the sixth order,
(1)F=a0+a2Ps2+a4Ps4+a6Ps6
Where
(2)a2=αT−Tc,
the coefficients a0,α,a4 and a6 can then be determined. By minimizing the free energy with respect to the order parameter Ps
(3)∂F∂Ps=0
a quadratic equation can be obtained as follows:
(4)a2+2a4Ps2+3a6Ps4=0
We can solve this equation for Ps2 as
(5)Ps2=a43a6∓13a6a42−3a2a6
where a4<0 and a6>0 are taken for a first order ferroelectric - paraelectric transition. For T<Tc, ferroelectric phase can be defined with a positive Ps in eq. (5), whereas for T>Tc, paraelectric phase has Ps=0. Equation (5) gives the temperature dependence of the spontaneous polarization Ps according to eq. (2). The temperature dependence of the inverse dielectric susceptibility can also be obtained by defining
(6)χ−1=∂2F∂Ps2
which gives
(7)χ−1=2a2+12a4Ps2+30a6Ps4
In the paraelectric phase (Ps=0), we have
(8)χ−1=(ε−1)−1=2a2
where ε is the dielectric constant. Thus, the temperature dependence of the reciprocal susceptibility is obtained as
(9)χ−1=2αT−Tc
by using eq. (2). In the ferroelectric phase (Ps≠0), by assuming that
(10)a2a6a42≪1
we get
(11)a42−3a2a6=a4−3a2a62a4
and the spontaneous polarization is then obtained as
(12)Ps2=−2a43a6+a22a4
The reciprocal susceptibility can also be obtained in the ferroelectric phase (T<Tc) as
(13)χ−1=−12a2+16a423a6
Finally, using eq. (2) the temperature dependences of the spontaneous polarization and the reciprocal susceptibility can be expressed in the ferroelectric phase, respectively,
(14)Ps2=αT−Tc2a4−2a43a6
and
(15)χ−1=−12αT−Tc+16a423a6
For the paraelectric phase, in the form of eq. (9) the temperature dependence of the reciprocal susceptibility can be written as a linear relation
(16)χ−1=(εc−1)−1=α0+2αT−Tc
with the slope 2α and the intercept α0 as constants.
3 Analysis and results
In this part, we analyzed the temperature dependences of the dielectric constant and the spontaneous polarization using the experimental data [1] for the ferroelectric- paraelectric transition in TMAHSO4, which were obtained for the dielectric constant εc along the c direction on cooling at 100 kHz and also for the spontaneous polarization Ps was measured along the c direction in this crystal [1], as stated above.
For the analysis of the dielectric constant εc for the ferroelectric and paraelectric phases, eqs. (15) and (16), respectively, were fitted to the experimental data [1]. Also, for the analysis of the spontaneous polarization Ps, we used eq. (14). We plot the temperature dependence of the dielectric constant εc(=χc+1) according to eq. (16) for the paraelectric phase (T>Tc) in TMAHSO4 in Figure 1. Since the slope (=2α) varies considerably at T−Tc≅4 °C, the analysis of εc was conducted for the two temperature intervals using the experimental data [1] for the paraelectric phase of TMAHSO4 (Figure 1). Table 1 gives the α and α0 values which we obtained from our analysis of εa in the paraelectric phase. Similar analysis was conducted for the ferroelectric phase (T<Tc) in TMAHSO4 according to eq. (15) using the observed data for εc. We plot the temperature dependence of the dielectric constant εc in the two temperature intervals because of the scattered experimental data [1], as given in Figure 2. Table 2 gives the values of the parameters in eq. (15) for the ferroelectric phase of TMAHSO4.
Table 1:
Values of the coefficients for the dielectric constant εc at various temperatures according to eq. (16) in the paraelectric phase (T>Tc) of TMAHSO4 using the experimental data [1] within the temperature intervals indicated.
| (εc-1)−1 | α×10−3 (∘C−1) | α0×10−3 | Temperature interval (°C) |
|---|
| Eq. (16) T>Tc | 0.36 | 7.0 | 0.01<T−Tc<4.0 |
| 1.11 | 2.8 | 4.0<T−Tc<8.31 |
Table 2:
Values of the coefficients for the dielectric constant εc at various temperatures according to eq. (15) in the ferroelectric phase (T<Tc) of TMAHSO4 using the experimental data [1] within the temperature intervals indicated.
| (εc-1)−1 | α×10−5 (∘C−1) | (a42/a6)×10−3 | Temperature interval (∘C) |
|---|
| Eq. (15) T<Tc | 2.16 | 7.6 | −126.89<T−Tc<−47.44 |
| 7.0 | 2.32 | −33.53<T−Tc<−0.77 |
In the second part of our analysis, we obtained the temperature dependence of the spontaneous polarization according to eq. (14) for the ferroelectric- paraelectric transition in TMAHSO4 using the experimental data [1]. Since the spontaneous polarization was measured at various temperatures under the electric field (100 kHz) along the cdirection of the crystal axes for TMAHSO4 [1], a non-zero spontaneous polarization occurs above Tc. Below about –104 °C, the spontaneous polarization suddenly drops in the ferroelectric phase. At this transition temperature, the dielectric constant increases abruptly, as observed experimentally [1].
In order to analyze the spontaneous polarization according to eq. (14), we assumed that a continuous increase in the spontaneous polarization occurs with the increasing temperature above Tc because of the scattered experimental data [1]. This then provides a linear dependence of the spontaneous polarization (squared) on the temperature (eq. (14)). Figure 3 gives a linear plot of Ps2 against T−Tc of TMAHSO4 along the cdirection using the experimental data [1] according to eq. (14) in the two temperature intervals, as in the dielectric constant εc. The two temperature intervals were chosen for our analysis of Ps vs T−Tc because of the scattered experimental data [1], as stated above. From the two linear plots (Figure 3), we extracted the values of the parameters (eq. (14)) for the two temperature intervals indicated, as given in Table 3.
Table 3:
Values of the coefficients for the spontaneous polarization Ps at various temperatures according to eq. (14) in TMAHSO4 within the temperature intervals indicated.
| Ps (μC/cm2) | (α/a4)×10−6(μC/cm2)2/∘C | (a4/a6)×10−6(μC/cm2)2 | a4(μC/cm2))−2 | −a6×105(μC/cm2))−4 | Temperature interval (∘C) |
|---|
| Eq. (14) T>Tc | 73.0 | −1.8 | 4.93 | 27.38 | 0<T−Tc<5 |
| 1.8 | 276.0 | – | – | 5<T−Tc<45 |
4 Discussion
The temperature dependence of the dielectric constant was analyzed in the paraelectric (T>Tc) and ferroelectric (T<Tc) phases according to eqs (16) and (15), respectively, using observed dielectric constant [1] for TMAHSO4. As shown in Figure 1 (T>Tc) and Figure 2 (T<Tc), the experimental data [1] are scattered, in particular in the ferroelectric phase (Figure 2). So, the rounding data were analyzed in the two temperature intervals and the two straight lines provided the values of the coefficients of eqs (15) and (16). As given in Table 1, the slope value α decreases eq. (16) as the transition temperature is approached from the paraelectric phase (T>Tc) (Figure 1), whereas it increases as T approaches Tc (Table 2) from the ferroelectric phase (T<Tc) (Figure 2) in TMAHSO4, as expected.
From our second analysis of the measured spontaneous polarization Ps [1] as a function of temperature (Figure 3) according to eq. (14), we also extracted the coefficients in the two temperature intervals, as given in Table 3. Nearly in the same temperature interval by comparing the values of the coefficients from the analysis of the dielectric constant εc (0.01 °C <T−Tc<4 °C) (Table 1) and from the analysis of the spontaneous polarization Ps(0<T−Tc<5 °C) (Table 3) in the paraelectric phase (T>Tc), we obtained the values of the coefficients a4 and a6, as given in Table 3.
The critical behavior of the dielectric constant ε depends in general on the frequencies measured. In the case of (NH4)2SO4 crystal, measurements on the dielectric behavior were conducted at microwave frequencies [15]. A normal type of dielectric behavior of the crystal occurs at the high frequencies whereas at very high frequencies the clamped dielectric constant is measured, however, at very low frequencies the measured ε value yields the free dielectric constant [16]. Thus, for the (NH4)2SO4 the shape of the ε against T curves varies at different frequencies. Also, for the hydrated crystal N(CH3)4HSO4.H2O containing one molecule of water of crystallization, the dielectric constant was measured at 100 kHz on heating which exhibits a sudden increase at Tc=–104 °C (it was Tc=–102 °C in TMAHSO4) and it varies slightly with increasing temperature [2] as the TMAHSO4 crystal studied here. It has been pointed out that the frequency dependence of the dielectric constant was measured in a frequency range from 10 kHz to 1 MHz with the dielectric dispersion discerned (applied electric field was 10 kV/cm) [2].
It has been reported that a dielectric dispersion due to dipolar reorientation occurs below 3.3 kMcs in (NH4)2BeF4 while the dispersion in (NH4)2SO4 takes place at about 10 kMcs [17]. It has also been pointed out that the dielectric behavior of the (NH4)2SO4 crystal undergoes a radical change at some frequency of 10 kc/sec and 24,000 Mc/sec [16]. Additionally, the actual value of the frequency depends on the thickness of the crystal and the rate of cooling or warming of the crystal near the transition temperature affects the shape of the dielectric constant versus temperature curve considerably [16]. By disregarding the dielectric dispersion, all those studies indicate that when the dielectric constant ε exhibits an anomalous change at Tc as the temperature increases from the ferroelectric to the paraelectric phase at high frequencies (100 kHz), it can be analyzed in the case of the TMAHSO4 crystal as we analyzed in this study and also the measured dielectric constant of N(CH3)4HSO4.H2O [2] can be analyzed similarly using our mean field model introduced here. At very high frequencies or very low frequencies due to the dielectric dispersion as stated above, the dielectric constant may not exhibit any sudden increase at Tc even there occurs a phase transition from the ferroelectric to the paraelectric phase so that our analysis becomes inadequate. In other words, our mean field model is invalid for the analysis of the dielectric constant at very high and very low frequencies for those crystals which belong to the (NH4)2SO4 family.
In regard to the temperature dependence of the spontaneous polarization (Ps) it also increases suddenly as the dielectric constant (ε) at Tc with increasing temperature from the ferroelectric to the paraelectric phase for the TMAHSO4 crystal at 100 kHz as observed experimentally [1]. This anomalous behavior of the spontaneous polarizationwas also observed for a hydrated crystal N(CH3)4HSO4.H2O at 100 kHz (on heating) [2]. According to the anomalous behavior of the Ps at Tc, we analyzed the temperature dependence of the spontaneous polarization in the paraelectric phase (at 100 kHz) of TMAHSO4 using the mean field model as stated above. As in the dielectric constant, a sudden increase in the spontaneous polarization at Tc, which was observed experimentally at 100 kHz restricts the validity of our mean field model. At very high frequencies and also at very low frequencies, a sudden increase in the spontaneous polarization may not occur as the dielectric constant ε, for which the mean field model is no longer valid for TMAHSO4 or some related materials in the same family. As we analyzed the temperature dependence of Ps for the TMAHSO4 crystal, the temperature range was \~ 50 °C with respect to Tc (Table 3). In the proximity of Tc, for our first analysis of Ps (0<T−Tc< 5 °C) the critical exponent for the order parameter (spontaneous polarization) can also be extracted just above the Tc in the paraelectric phase. However, this requires accurate experimental data for the spontaneous polarization within a narrow temperature range. Regarding the temperature range of 0<T−Tc< 5 °C and 5 °C<T−Tc< 45 °C for our analysis of the spontaneous polarization Ps (Table 3), the mean field model introduced here describes satisfactorily the observed behavior of Ps [1].
Regarding the dielectric strengths, the temperature dependence of the mode strengths can determine the critical behavior of the dielectric constant close to Tc. By expressing the frequency dependence of the dielectric constant as a sum of damped simple harmonic oscillators of frequencies (transverse-optic frequencies) ωTOi, dielectric strengths Si and damping γi [18, 19, 20],
(17)ε(ω)=ε∞[1+∑iSiωTOi2ωTOi2−ω2+iωγi]
where i is summed over the number of oscillators and ε∞ is the square of the index of refraction of light, the low-frequency clamped dielectric constant can be written as
(18)ε(0)=ε∞1+∑iSi
where Si is independent of ε∞. Thus, the measurements of the mode dielectric strengths become important as they determine the clamped zero-frequency dielectric constant ε0. According to eqs (17) and (18), the dielectric strengths Si of the modes can be determined from the transverse and longitudinal frequencies at which εωLO=0. It has been pointed out that the lowest A1TO and E modes exhibit largest dielectric strength at all temperatures for PbTiO3 and that the lowest E mode of BaTiO3 has all the dielectric strength at room temperature [20]. Thus, those modes with large Si values (A1 and E modes in PbTiO3, and E mode in BaTiO3) dominate the dielectric behavior along the ferroelectric c axis at all temperatures and they are associated with the transition below Tc [20]. In analogy with the ferroelectrics PbTiO3 and BaTiO3, in the case of polycrystalline and crystalline forms of TMAHSO4 as we studied here, the lowest frequency modes with the larger values of the dielectric strengths Si are expected to associate with the ferroelectric-paraelectric phase transition and they dominate the critical behavior of the anomalous behavior of the dielectric constant near Tc. As known in practice, multiple layers of thin dielectric films with maximum dielectric strength are used in various devices.
Our analysis for the dielectric constant (ε) and the spontaneous polarization (Ps) can be performed for crystals of (CsxRb1-x)2 ZnCl4 (0<x<1) which has a β-K2SO4 type crystal structure. By analyzing the experimental data for the dielectric constant εa (along the a direction) of CsRbZnCl4 and Cs0.2Rb1.8ZnCl4 on heating at 100 kHz [1] using the mean field model presented here, its ferroelectric properties can be investigated. Also, ferroelectric properties of N(CH3)4HSO4.H2O can be studied by analyzing the experimental data for the dielectric constant εc (along the c direction) and the spontaneous polarization Ps in the case of hydrated and anhydrous crystal on heating at 100 kHz [2] using the mean field model given here. Additionally, from the analysis of experimental data for the dielectric constant of N(CH3)4IO3 and N(CH3)4NO3 on heating [5] on the basis of the mean field model studied here, the mechanism of the phase transitions in those crystal structures can be investigated. Using this mean field model, from the analysis of the dielectric constant for the crystal of (CsxRb1-x)2 ZnCl4, N(CH3)4IO3 and N(CH3)4NO3, their spontaneous polarization can be predicted as a function of temperature.
5 Conclusions
The analysis of the dielectric constant and the spontaneous polarization was performed at various temperatures to investigate the ferroelectric properties of N(CH3)4HSO4 using the mean field model introduced here. Increasing and decreasing behavior of the inverse susceptibility (dielectric constant) in the paraelectric (T>Tc) and ferroelectric (T<Tc) phases, respectively, with increasing T−Tc was predicted from the mean field model, as observed experimentally for N(CH3)4HSO4. The spontaneous polarization increases in the paraelectric phase with increasing T−Tc in the presence of the applied field (100 kHz), as also observed experimentally for this crystal structure.
Similar analysis can be carried out for new ferroelectrics in the (NH4)2SO4 family such N(CH3)4HSeO4 using our mean model studied here.
References
[1] S. Sawada, T. Yamaguchi and H. Suzuki, Ferroelectrics, 63 (1985) 3.10.1080/00150198508221379Search in Google Scholar
[2] H. Suzuki, F. Shimizu, T. Yamaguchi and S. Sawada, Jpn. J. App. Phys., 24 (1985) Supplement 24–2 353–355.10.7567/JJAPS.24S2.353Search in Google Scholar
[3] H. Arend, R. Perret, H. Wuest and P. Kerkoc, J. Cryst. Growth, 74 (1986) 321.10.1016/0022-0248(86)90120-XSearch in Google Scholar
[4] N.L. Speziali and G. Chapuis, Acta Cryst., B47 (1991) 757.Search in Google Scholar
[5] T. Yamaguchi, H. Suzuki, F. Shimizu and S. Sawada, Jpn. J. App. Phys., 24 (1985) Supplement 24–2 359–360.10.7567/JJAPS.24S2.359Search in Google Scholar
[6] S. Sawada, Y. Shiroishi and A. Yamamoto, Ferroelectrics, 21 (1978) 413.10.1080/00150197808237281Search in Google Scholar
[7] S. Sawada and K. Butsuri, Solid state physics, 14 (1979) 51.Search in Google Scholar
[8] K. Gesi, J. phys. soc. jpn., 45 (1978) 1436.10.1143/JPSJ.45.1777Search in Google Scholar
[9] T. Yamaguchi, F. Shimizu, M. Morita, H. Suzuki and S. Sawada, J. Phys. Soc. Jpn., 57 (1988) 1898.10.1143/JPSJ.57.1898Search in Google Scholar
[10] H. Haga, A. Onodera, M. Tokunaga and Y. Shiozaki, J. Phys. Soc. Jpn., 62 (1993) 1597.10.1143/JPSJ.62.1597Search in Google Scholar
[11] H. Yurtseven and I. Yardimci, High Temp. Mater. Proc., 27 (2008) 135.10.1515/HTMP.2008.27.3.135Search in Google Scholar
[12] H. Kasano, M. Mashiyama, K. Gesi and K. Hasebe, J. Phys. Soc. Jpn., 56 (1987) 831.10.1143/JPSJ.56.831Search in Google Scholar
[13] H. Yurtseven and A. Kiraci, J. Mol. Model., 19 (2013) 3925.Search in Google Scholar
[14] S. Gupta, J. Raman Spectrosc., 33 (2002) 42.Search in Google Scholar
[15] Couture, Le Montagner, Le Blot and Le Traon, Comt. Rend., 242 (1956) 1804.Search in Google Scholar
[16] S. Hoshino, K. Wedun, Y. Okaya and R. Pepinsky, Phys.rev., 112 (1958) 405.10.1103/PhysRev.112.405Search in Google Scholar
[17] H. Ohshima and E. Nakamura, J. Solids., 27 (1966) 481.Search in Google Scholar
[18] F. Jona and G. Shirane, Ferroelectric crystals, MacMillan, New York (1962).Search in Google Scholar
[19] E. Fattuzzo and W.J. Merz, Ferroelectricity, Wiley, New York (1967).Search in Google Scholar
[20] G. Burns and B.A. Scott, Phys. Rev., B7 (1973) 3088.10.1103/PhysRevB.7.3088Search in Google Scholar
,
M. Celik
![Figure 1: The reciprocal dielectric constant εc$${\varepsilon _c}$$ as a function of T−Tc$$T - {T_c}$$ in the paraelectric (T>Tc$$T \gt {T_c}$$) phase of TMAHSO4 in the two temperature intervals indicated. Solid line represents the fit of eq. (16) to the observed data [1] (See Table 1).](/document/doi/10.1515/htmp-2016-0016/asset/graphic/j_htmp-2016-0016_fig_001.jpg)
![Figure 2: The reciprocal dielectric constant εc$${\varepsilon _c}$$ as a function of T−Tc$$T - {T_c}$$ in the ferroelectric (T<Tc$$T \lt {T_c}$$) phase of TMAHSO4 in the two temperature intervals indicated. Solid line represents the fit of eq. (15) to the observed data [1] (See Table 2).](/document/doi/10.1515/htmp-2016-0016/asset/graphic/j_htmp-2016-0016_fig_002.jpg)
![Figure 3: Spontaneous polarization squared (Ps2)$$P_s^2)$$ as a function of T−Tc$$T - {T_c}$$ using the experimental data along the c$$c$$ direction at 100 kHz [1]. Solid lines represent the best fit to the experimental data [1] according to eq. (14) for the temperature intervals indicated for TMAHSO4 (See Table 3).](/document/doi/10.1515/htmp-2016-0016/asset/graphic/j_htmp-2016-0016_fig_003.jpg)
