1. Introduction
The dielectric response of dielectrics with respect to temperature, pressure, frequency and amplitude of the probing electric field is an essential issue in dielectric physics (Jonscher, 1983). This concerns both normal dielectrics and those specified as ferro- or antiferroelectrics (Lines & Glass, 1977). This basic phenomenon of dielectric materials has been extensively studied in the literature both experimentally and theoretically, however, mainly restricting to the linear dielectric response where a linear relationship between polarization,
is fulfilled. Here
At higher electric fields
which contains higher order terms with respect to the external electric field, where
In order to overcome the technological challenge we have constructed a fully automatized
We have applied this new instrument to various basic ferroelectric scenarios, such as the classic first- and second-order ferroelectric transitions of barium titanate (BaTiO3) (Miga & Dec, 2008), triglycine sulphate (TGS) (Miga & Dec, 2008) and lead germanate (Pb5Ge3O11) (Miga & Dec, 2008), the double anomalous second-order transitions of Rochelle salt (Miga et al., 2010a), the smeared transition of the classic relaxor ferroelectrics lead magno-niobate (PbMg1/3Nb2/3O3, PMN) (Dec et al., 2008) and strontium-barium niobate (Sr0
2. Theoretical background of nonlinear dielectric response
Dielectric properties of materials are usually investigated via linear dielectric response. In this case a linear relationship,
Let us consider the Landau-Ginzburg-Devonshire (LGD) theory of ferroelectric phase transitions (PT) (Ginzburg, 1945; Devonshire, 1949). According to this theory the free energy density
where
This relationship between
The second-order susceptibility is proportional to the polarization
Due to positive B, 3 is negative above a continuous PT point. Within the ferroelectric phase P=P
Within the ferroelectric phase 3 has a positive sign. Thus the LDG theory predicts a change of sign of 3 at a continuous PT. The scaling theory (Stanley, 1971) predicts that B scales as B=B0
A similar calculation for a discontinuous PTs, where B < 0 and C > 0, yields 3 > 0 at all temperatures, in particular also above PT (Ikeda et al., 1987). Fortunately from an experimental point of view 3 is given by this same equation (8) within paraelectric phase independent of the ferroelectric PT order. The sign of 3 is a sensitive probe for discrimination between continuous and discontinuous ferroelectric PTs.
In all known ferroelectrics the paraelectric phase is located above the stability range of the ferroelectric one. However, sodium potassium tartrate tetrahydrate (Rochelle salt, RS) (Valasek, 1920, 1921) apart from a classic high temperature paraelectric phase has an additional, unusual one located below the ferroelectric phase. Both PTs, between the paraelectric phases and the ferroelectric one have continuous character. In order to predict the sign of 3 within the low-temperature paraelectric phase one can refer to the theory of Mitsui (Mitsui, 1958). Within this theory the electric equation of state for RS is similar to Eq. (4) with a positive coefficient of the cubic term, P3. Therefore a negative sign of 3 is expected within low-temperature paraelectric phase (Miga et al., 2010a).
Another class of dielectrics are relaxor ferroelectrics. They are usually considered as structurally disordered polar materials, which are characterized by the occurrence of polar nanoregions (PNRs) of variant size below the so-called Burns temperature, T
where J
The measured values of 1 and 3 can be used for calculating the so-called scaled non-linear susceptibility, a3, which is given by (Pirc & Blinc, 1999)
Within the paraelectric phase of classic ferroelectrics a3 is equal to the nonlinearity coefficient B, cf. Eqs. (8) and (11). For ferroelectics displaying a continuous PT, a3 = -8B within ferroelectric phase. The SRBRF model yields negative 3 and positive a3 with two extremes observed at the freezing temperature, T
A schematic comparison of predictions of the above theories is presented in Fig. 1.
3. Methods of measurement of nonlinear dielectric response
Nonlinear dielectric response can be measured using two different kinds of experimental methods. The first one is based on the investigation of the ac linear dielectric susceptibility as a function of a dc bias field. A schematic presentation of this method is shown in Fig. 2a. One applies to the sample a weak probing ac electric field with fixed amplitude (that warrants a linear response) and a superimposed variable dc bias field, E
for the paraelectric phase and
for the ferroelectric one. (E) and (0) are the susceptibilities for bias, E, and zero electric field, respectively. By use of Eqs. (12) and (13) one can calculate the nonlinearity coefficient B. Its knowledge allows us to calculate the third-order nonlinear susceptibility 3. Unfortunately, the above described method has at least one restriction when investigating the nonlinear dielectric response. Namely, during so-called field heating/cooling runs, unwanted poling and remnant polarization of the investigated sample can evolve under a high bias field.
The second kind of method is free from this restriction. A schematic presentation of this method is shown in Fig. 2b. During the experiment the sample is exposed to an ac probing field with sufficiently large amplitude. Consequently, under this condition the temperature dependences of the linear and nonlinear susceptibilities are determined under zero dc field in heating and cooling runs. That is why the nonlinear susceptibility detected this way can be referred to as a dynamic nonlinear susceptibility related to ac dielectric nonlinearity. Usually the amplitude of this field is much smaller than the bias field strength used in the above described experiment. As a result of the nonlinear P(E) dependence, the output no longer remains harmonic. The distorted signal may be subjected to Fourier analysis revealing all harmonic components in the polarization response. This is the main idea of our nonlinear ac susceptometer (Miga et al., 2007). Using harmonics of displacement current density j
where and E0 are the angular frequency and the amplitude of the applied electric field respectively. In Eq. 14 the terms up to seventh order are involved, hence, susceptibilities up to the fifth order can be regarded as reliable even for strongly nonlinear materials. Neglecting harmonics higher than the order of a considered susceptibility may lead to artificial effects. The next important point is the simultaneous measurement of all displacement current components, which considerably improves the accuracy of the measured susceptibilities (Bobnar et al., 2000). In case that the phase shifts of the displacement current harmonics are known it is possible to calculate the real, χ
4. Experimental results
4.1. Ferroelectrics displaying continuous PTs
Triglycine sulphate, (NH2CH2COOH)3H2SO4 (TGS) is a model ferroelectric displaying a continuous PT. Sodium potassium tartrate tetrahydrate (Rochelle salt, RS), NaKC4H4O64H2O, and lead germanate, Pb5Ge3O11 (LGO) exhibit continuous PTs, as well. Despite the quite different structures of the crystals and different mechanisms of their PTs, a negative sign of the real part of the third-order nonlinear susceptibility is expected in their paraelectric phases. Fig. 3 shows the temperature dependences of the real parts of the linear, the third order nonlinear dielectric susceptibilities, and the scaled susceptibility, a3. The linear susceptibility of a TGS crystal (Fig. 3a) obeys a Curie-Weiss law within the paraelectric phase very well with a critical exponent =1.0000.006. According to predictions of the phenomenological theory of ferroelectric PT, 3’ changes its sign at the PT point (Fig. 3b).
While below T
Fig. 4 shows the temperature dependences of the linear, the second, and the third-order nonlinear susceptibilities of Rochelle salt. The temperature dependences of 1’ and 3’ are, close to the high temperature ferroelectric-paraelectric PT ( 297.4 K), qualitatively similar to those of the TGS crystal. As was mentioned earlier, RS displays an additional low temperature PT. This transition between the ferroelectric and the low temperature paraelectric phase appears at about 254.5 K. At this PT 3’ changes its sign as compared to the high temperature PT (Fig. 4c). As a result of the inverse order of phases, 3’ is negative below and positive above the low temperature PT. In this way the third-order nonlinear susceptibility is negative in both paraelectric phases close to PT points. Fig. 4b shows the temperature dependence of the second-order nonlinear susceptibility. The sign of this susceptibility depends on the net polarization orientation. Therefore it can be changed by polarization of the sample in the opposite direction. The simplest way to change the sign of 2’ is a change of the wires connecting the sample to the measuring setup. So, in contrast to sign of the 3’, the sign of the second order nonlinear susceptibility is not very important. In the case of 2’ most important is its nonzero value and the observed change of sign of this susceptibility at 273 K. This hints at a modification of the domain structure within the ferroelectric phase. The next change of sign at the low temperature PT point is originating from different sources of the net polarization above and below this point. Above 254.5 K this polarization comes from the uncompensated ferroelectric domain structure, whereas within the low-temperature paraelectric phase it originates merely from charges screening the spontaneous polarization within the ferroelectric phase.
Lead germanate (LGO) displays all peculiarities of a continuous ferroelectric PT (see Fig. 5 a, c, e) (Miga et al., 2006). However, a small amount of barium dopant changes this scenario (Miga et al., 2008). Ba2+ ions replacing the host Pb2+ influence the dielectric properties. 2% of barium dopant causes a decrease of the linear susceptibility, broadening of the temperature dependence of 1’, and a decrease of the PT temperature. Despite all these changes the temperature dependences of 1’ for pure and barium doped LGO are qualitatively similar. A completely different situation occurs, when one inspects the nonlinear dielectric response. Small amounts of barium dopants radically change the temperature dependence of the third-order nonlinear susceptibility (Fig.5 b). Similarly to linear one the anomaly of the third-order nonlinear susceptibility shifts towards lower temperature and decreases. The most important difference is the lacking change of sign of 3’. In contrast to pure LGO for barium doped LGO the third-order nonlinear susceptibility is positive in the whole temperature range. Therefore one of the main signatures of classic continuous ferroelectric PTs is absent. Due to the positive value of 3’ the scaled nonlinear susceptibility is negative in the whole temperature range (Fig. 5d). No change of a3 is observed. This example shows the high sensitivity of the nonlinear dielectric response to the character of the PT. In the discussed case a change of character of the PT is due to the presence of barium induced polar nanoregions (PNRs). The occurrence of PNRs results in weak relaxor properties of barium doped LGO.
4.2. Ferroelectrics displaying discontinuous PT
Barium titanate, BaTiO3 (BT), is a model ferroelectric displaying three discontinuous ferroelectric PTs (von Hippel, 1950). Two of them appear – at rising temperatures - between rhombohedral, orthorhombic and tetragonal ferroelectric phases at about 200 K and 280 K respectively. The final discontinuous PT appears between the ferroelectric tetragonal and the cubic paraelectric phase at about 400 K. Fig. 6 shows the temperature dependences of the real parts of the linear (a) and third-order nonlinear (b) susceptibilities, and the a3 coefficient (c) of a BT crystal in the vicinity of the ferroelectric-paraelectric PT. The amplitude of a probing ac electric field was equal to 7.5 kVm-1.
Temperature hysteresis is one of the typical features of discontinuous PT. Therefore for viewing this phenomenon Fig. 6 presents results for cooling and heating runs. The third-order nonlinear susceptibility is positive, both within the ferroelectric and the paraelectric phases (Fig. 6b). Just below T
4.3. Relaxor ferroelectrics
The classic relaxor ferroelectric lead magno-niobate (PbMg1/3Nb2/3O3, PMN) (Smolenskii et al., 1960) displays an average cubic structure in the whole temperature range (Bonneau et al., 1991). No spontaneous macroscopic symmetry breaking is observed in this relaxor. On the other hand and in contrast to PMN, the relaxor strontium-barium niobate Sr0
Fig. 7 presents the temperature dependences of the real parts of the linear, 1’ (a), second-order, 2’ (b), and third-order, 3’ (c), dielectric susceptibilities and of the scaled nonlinear susceptibility a3 (d) of the PMN single crystal recorded along [100] direction (Dec et al., 2008). The linear susceptibility displays features of a relaxor i.e. a large, broad and frequency-dependent peak in the temperature dependence. Nonzero 2’ as presented in Fig. 7b is an indicator of net polarization of the crystal. The presence of this polarization was independently confirmed by measurements of thermo-stimulated pyroelectric current of an unpoled sample. Integration of this current indicates an approximate value of the average polarization as low as 310-5 C/m2. This polarization is much smaller than the spontaneous polarization of ferroelectrics, but is well detectable. The observed pyroelectric response and nonzero 2’ hints at an incomplete averaging to zero of the total polarization of the PNR subsystem. Fig. 7c shows the temperature dependence of the third-order nonlinear susceptibility. In contrast to the predictions of the SRBRF model, 3’ is positive in the whole temperature range. The positive sign of 3’ may result from a term 180BP21 exceeding unity in Eq. 7. Positive sign of 3’ results in a negative sign of a3. Consequently the sign of a3 differs from that predicted by the SRBRF model. Having in mind that corrections due to the fifth harmonic contribution produce large noise at temperatures below 210 K and above 310 K (Fig. 7d, main panel), less noisy a3 data calculated only from first and third harmonics are presented in the inset to Fig. 7d. Since both 1’ and 3’ do not display any anomalies in the vicinity of the freezing temperature T
Fig. 8 shows results of measurements of the linear and nonlinear dielectric response of SBN61 crystal (Miga & Dec, 2008). The probing electric ac field was applied along [001], which is the direction of the polar axis below T
The results obtained for both relaxor ferroelectrics are qualitatively similar. Therefore, they are independent of the presence or absence of a structural phase transition and macroscopic symmetry breaking. As discussed in Section 2 the dielectric properties of relaxors are mainly determined by PNRs, which were detected in both of the above relaxors. Unfortunately, the early version of the SRBRF model (Pirc et al., 1994) predicts a negative sign of the third-order nonlinear dielectric susceptibility, which is not confirmed in experiments. Consequently the sign of the scaled nonlinear susceptibility a3 is incorrect as well. Very probably this unexpected result is due to the fact that the PNRs primary do not flip under the ac electric field, but merely change their shape and shift their centers of gravity in the sense of a breathing mode (Kleemann et al., 2011).
4.4. Dipolar glasses
4.4.1. Orientational glasses
The formation of dipolar glasses in incipient ferroelectrics with perovskite structure, ABO3, such as SrTiO3 and KTaO3, by A-site substitution with small cations at low concentrations has been a fruitful topic since more than 20 years (Vugmeister & Glinchuk, 1990). For a long time probably the best-known example has been the impurity system K1-xLixTaO3 (KLT for short) with x<< 1, whose complex polar behavior is known to be due to the interaction of the (nearly) softened transverse-optic mode of the host-lattice and the impurity dynamics (Höchli et al., 1990). Fig. 9 shows the structure model of A-site substituted Li+ viewing the nearest neighbor environment in the KTaO3 lattice from two different perspectives. At very low concentrations, x ≈ 0.01, it reveals signatures of glasslike behavior (Höchli, 1982; Wickenhöfer et al., 1991), while a ferroelectric ground state with inherent domain structure is encountered at higher concentrations, x ≥ 0.022 (Kleemann et al., 1987).
Only recently a similar system has been discovered with qualitatively new properties. The impurity system Sr1-
In addition to conventional tests of the glass transition, e. g. by verifying the divergence of the polar relaxation times, the behavior of the nonlinear susceptibility is believed to similarly decisive. As was first acknowledged in spin glass physics (Binder & Young, 1986), but later on also in the field of orientational glasses (Binder & Reger, 1992), criticality at the glass temperature, Tg, is expected to give rise to a divergence of the third-order nonlinear susceptibility,
4.4.2. Orientational glass K0.989Li0.011TaO3
The experiments on KLT were performed on a Czochralski grown single crystal sample with x = 0.011 with dimensions 3×2×0.5 mm3 and (100) surfaces (Kleemann et al., 1987). Dipolar relaxation was studied as a function of temperature T via measurements of the complex dielectric susceptibility, = ′ - iχ″ vs. T, by use of different experimental methods adapted to different frequency ranges, 10-3 ≤ f ≤ 106 Hz. They included a Solartron 1260 impedance analyzer with 1296 dielectric interface (Fig. 10a and b) and a digital lock-in analyzer (Wickenhöfer et al., 1991) (Fig. 10c) for linear, and a homemade computer-controlled digital susceptometer (Miga et al., 2007) for non-linear dielectric susceptibility data (Fig. 11) at high precision under relatively low excitation voltages.
Fig. 10a and b show dielectric susceptibility data, χ′(T) and χ″(T), for frequencies 10-3 < f < 106 Hz, which reveal various signatures of glassy behaviour. The peak of χ′(T) in Fig. 10a converges toward a finite glass temperature, T
A complementary test of glassy criticality refers to the third-order nonlinear susceptibility,
4.4.3. Multiglass Sr0.98Mn0.02TiO3
The experiments on Sr0.98Mn0.02TiO3 were performed on a ceramic sample prepared by a mixed oxide technology (Tkach et al., 2005). Preponderant incorporation of Mn2+ onto A-sites of the perovskite structure (Fig. 9) was confirmed by energy dispersive X-ray spectra (Tkach et al., 2006), Mn2+ ESR analysis (Laguta et al., 2007), and EXAFS spectroscopy (Lebedev et al., 2009; Levin et al., 2010). Fig. 12 shows the components of the complex dielectric susceptibility
Another striking indicator of the dipolar glass state is the memory effect, which arises after isothermally annealing the sample below T
The glass transition may also be judged from spectra
5. Conclusion
In subsections 4.1 – 4.4 we presented different groups of polar materials separately and compared qualitatively their dielectric properties with predictions of suitable theories. It is likewise interesting to compare quantitatively different materials. The scaled nonlinear susceptibility, a3, is defined independently of the kind of material and its symmetry. This quantity is a measure of the nonlinearity of the investigated object. For an adequate comparison we have chosen values of a3 within centrosymmetric phases of different materials close to their temperatures of phase transition or of peak positions of the linear susceptibility, respectively. Fig. 14 shows thus collected values of a3. In view of the very large differences between the different a3 values a logarithmic scale is used. Consequently, only the magnitude values, |a3|, are presented. The highest |a3| was found for Rochelle salt, RS. Lower nonlinearity appears in sequence in the ferroelectric crystals TGS, LGO, BT, barium doped LGO, multiglass Sr0.98Mn0.02TiO3, orientational glass K0.989Li0.011TaO3 and relaxor ferroelectrics PMN and SBN 61. Obviously, classic ferroelectrics undergoing continuous PT are characterized by high nonlinearity, while structural disorder and presence of PNRs diminish the nonlinearity. Consequently relaxor ferroeletrics are characterized by the smallest values of a3. In other words, displacive ferroelectrics exhibiting soft-mode softening are most affected by nonlinearity, while typical order-disorder systems do not obtain their ferroelectricity primarily from the nonlinear interionic potential. The mechanism of their phase transition rather reflects the statistics of local hopping modes, in particular when being accompanied by quenched random fields as in relaxor ferroelectrics (Westphal et al., 1992; Kleemann et al., 2002), but probably also in partial order-disorder systems like BaTiO3 (Zalar et al., 2003).
The comparison of nonlinear dielectric response of various kinds of polar materials presented in this chapter gives evidence that such kind of measurement is a very sensitive tool for determination of the nature of ferroelectrics. Particularly useful for this purpose are the third-order dielectric susceptibility and the scaled non-linear susceptibility, a3. In our opinion, the second-order dielectric susceptibility is less significant, since it is more characteristic of the sample state than of a particular group of ferroelectrics. If anything, this susceptibility contains information about the distinct polar state of the sample. This information may be used for checking the presence of a center of inversion. The nonlinear dielectric response of classic ferroelectric crystals displaying continuous or discontinuous phase transition stays in a good agreement with predictions of the thermodynamic theory of ferroelectric phase transition. Predictions of the scaling theory for TGS crystal are also successfully verified experimentally. The situation is much more complex in disordered systems like ferroelectric relaxors or dipolar glasses. Respective theories properly explaining the observed features have still to be developed and tested via dynamic nonlinear dielectric response.
Acknowledgments
The authors are grateful to D. Rytz, A. Tkach, and P.M. Vilarinho for providing samples. WK thanks the Foundation for Polish Science (FNP), Warsaw, for an Alexander von Humboldt Honorary research grant.
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