Open access peer-reviewed chapter

Phase Transitions in Layered Semiconductor - Ferroelectrics

By Andrius Dziaugys, Juras Banys, Vytautas Samulionis, Jan Macutkevic, Yulian Vysochanskii, Vladimir Shvartsman and Wolfgang Kleemann

Submitted: November 5th 2010Reviewed: March 14th 2011Published: August 23rd 2011

DOI: 10.5772/20008

Downloaded: 1821

1. Introduction

CuInP2S6 crystals represent an unusual example of an anticollinear uncompensated two-sublattice ferroelectric system (Maisonneuve et al., 1997). They exhibit a first-order phase transition of the order–disorder type from the paraelectric to the ferrielectric phase (Tc = 315 K). The symmetry reduction at the phase transition (C2/c to Cc) occurs due to the ordering in the copper sublattice and the displacement of cations from the centrosymmetric positions in the indium sublattice. X-ray investigations have shown that Cu ion can occupy three types of positions (Maisonneuve et al., 1997). The ordering of the Cu ions (hopping between Cu1u and Cu1d positions) in the double minimum potential is the reason for the phase transition dynamics in CuInP2S6. In (Maisonneuve et al., 1997) it was suggested that a coupling between P2S6 deformation modes and Cu+ vibrations enable the copper ion hopping motions that lead to the onset of ionic conductivity in this material at higher temperatures. At low temperatures a dipolar glass phase appears in CuInP2S6 weakly doped with antiferroelectric CuCrP2S6 or ferroelectric CuInP2Se6 (Macutkevic et al., 2008).

The copper chromium thiophosphate CuCrP2S6 crystallizes in a layered two-dimensional structure of the CuIMIIIP2S6 (M = Cr, In) type described above (Maisonneuve et al., 1995). It is formed by double sheets of sulfur atoms sandwiching the metal cations and P–P groups which occupy the octahedral voids defined by the sulfur atoms. At room temperature the crystal structure has a space group of C2/c (Colombet et al., 1982). At 64 K, the Cu positions are confined to those of an antiferroelectric order where the crystal structure has the space group of Pc (Maisonneuve et al., 1995). Thus, the mechanism of the dielectric transition is likely to involve hopping of the copper ions among two or more positions. Two phase transitions have been observed at 155 K and 190 K by dielectric measurement and differential scanning calorimetry (DSC). The crystal is antiferroelectric below 155 K and paraelectric above 190 K. For the intermediate phase between 155 and 190 K, a quasi-antipolar structure has been proposed. Solid solutions CuCr1-xInxP2S6, x > 0, are therefore expected to reveal disordered dipolar glass phases as a consequence of randomness and frustration as confirmed recently (Maior et al., 2008).

Similar signatures of disorder might also be expected for the magnetic ground state of CuCr1-xInxP2S6, where magnetic Cr3+ ions are randomly replaced by diamagnetic In3+ ions in the antiferromagnetic (AF) compound CuCrP2S6 with a Néel temperature TN ≈ 32 K (Colombet et al., 1982). Owing to its competing ferromagnetic (FM) intralayer and AF interlayer exchange interactions (Colombet et al., 1982), randomness and frustration might eventually give rise to spin glass phases in CuCr1-xInxP2S6, x > 0, similarly as in the related AF compound Fe1-xMgxCl2 (Bertrand et al., 1982; Mattsson et al., 1996). The possible coexistence of this spin glass phase with the dipolar glassy one (Maior et al., 2008) is another timely motivation to study CuCr1-xInxP2S6. Indeed, ‛multiglass’ behavior was recently discovered in the dilute magnetic perovskite Sr0.98Mn0.02TiO3 (Shvartsman et al., 2008), paving the way to a new class of materials, ‛disordered multiferroics’ (Kleemann et al., 2009).

The above mentioned comparison of the two families of dilute antiferromagnets CuCrP2S6:In and FeCl2:Mg is not fortuitous. Originally, a strong structural analogy had been noticed between the lamellar compounds FeX2 (X = Cl or Br) and transition metal (M) thio-phosphate phases, MPS3, such as FePS3 (Colombet et al., 1982). Both families are characterized by van der Waals gaps between their crystalline slabs and their ability to act as intercalation host material. The analogy becomes formally apparent when using the notations Fe2P2S6 or – stressing the occurrence of P2 pairs – [Fe2/3(P2)1/3]S2 (Klingen et al., 1973), and substituting (Fe2+)2 by (Cu+Cr3+). This transcription discloses, however, that in contrast to the FeX2 compounds even the undoped CuCrP2S6 is a ‛dilute magnet’ from the beginning (i.e. in the absence of non-magnetic In3+), since it always hosts two diamagnetic cation sublattices occupied by Cu and P ions. This ‛extra’ dilution must be taken into account for understanding the magnetic and magnetoelectric properties discussed below.

This manuscript includes broad band spectroscopy, SQUID and piezoelectric measurement techniques, which helped to complement the list of already known properties of the investigated crystals and reveal new features such as dipole glass behaviour, magneto-electric coupling, and piezoelectric response. This crystal family is very interesting for various transducers because of the quite broad temperature region (285 to 330 K) for the phase transition.

2. Broad band dielectric spectroscopy of layered crystals

2.1. Ferrielectric phase transition in CuInP2S6, Ag0.1Cu0.9InP2S6 and CuIn1+δP2S6 crystals

Results of the broadband dielectric measurements of Ag0.1Cu0.9InP2S6 are presented in Fig. 1. At low frequencies the dielectric losses increase with increasing temperature mainly due to the high ionic conductivity. The real part of the dielectric permittivity at 1 MHz already corresponds to the static one, because at that frequency ε"is already much smaller than ε'(A. Dziaugys et al., 2010). It was found, that impurity of Ag ions, or addition of extra In ions drastically changes the ferrielectric phase transition temperature (static dielectric permittivity maximum temperature) (Table 1).

Figure 1.

Temperature dependence of the complex dielectric permittivity of Ag0.1Cu0.9InP2S6.

Figure 2.

Frequency dependence of complex dielectric permittivity of Ag0.1Cu0.9InP2S6 crystals measured at different temperatures. Solid lines are best fits according to Eq. (1).

CrystalPhase transition temperature, K
CuInP2S6313 (A. Dziaugys et al. 2010)
CuIn1+δP2S6330 (A. Dziaugys et al. 2011)
Ag0.1Cu0.9InP2S6285 (A. Dziaugys et al. 2010)

Table 1.

Phase transition temperatures got from the dielectric measurements.

The nature of such phase transition, similar to the pure CuInP2S6, is ferrielectric - ordering in the copper sublattice and displacement of cations from the centrosymmetric positions in the indium sublattice. The ferrielectric dispersion in the vicinity of Tc begins at about 10 MHz for Ag0.1Cu0.9InP2S6 and ranges up to the GHz region (Fig. 2). The characteristic minimum of ε'appears above 100 MHz at T = 285 K for Ag0.1Cu0.9InP2S6 and 500 MHz at T = 330 K for CuIn1+δP2S6 indicating a critical slowing down typical of the order–disorder ferroelectric phase transitions (Grigas, 1996). The frequency plot of the complex permittivity at different temperatures (Fig. 2) was fitted with the Cole–Cole formula:

ε*(ω)=ε+Δε1+(iωτ)αE1
,

where Δεrepresents the dielectric strength of the relaxation, τis the mean Cole–Cole relaxation time, εrepresents the contribution of all polar phonons and electronic polarization to the dielectric permittivity and α is the Cole–Cole relaxation time distribution parameter. Eq. (1) reduces to the Debye formula ifα=0. The obtained parameters are presented in Fig. 3. At higher temperatures the αparameter is very small and indicates Debye type dielectric dispersion.

Figure 3.

Cole–Cole parameters of Ag0.1Cu0.9InP2S6 and CuIn1+δP2S6 crystals.

On cooling the αparameter increases up to 0.133 for Ag0.1Cu0.9InP2S6 and 0.22 for CuIn1+δP2S6 substantially below ferrielectric phase transition temperature Tc. The distribution of the relaxation times is much broader in the ferrielectric phase than in the paraelectric one. The temperature dependence of the dielectric strength Δεwas fitted with the Curie–Weiss law:

Δε=Cp,f|TTCp,Cf|E2
,

where Cp,f is the Curie–Weiss constant and TCp,Cf is the Curie–Weiss temperature obtained from the fitting correspondingly in the paraelectric and ferrielectric phases. The ratios Cp/Cf = 7.8 (6.45 for CuIn1+δP2S6), Cp/TCp ≈10 (for both crystals) and mismatch TCfTCp = 20 K (62 K for CuIn1+δP2S6), show the first-order “order–disorder” ferroelectric phase transition. The temperature dependence of the mean Cole–Cole relaxation time τin the paraelectric phase was fitted with the classical law (Grigas, 1996):

τ=τ0eU/kTCpTTCpE3
,

where τ0is the relaxation time as Tand the exponential factor describes deviations from phenomenological Landau theory close to the phase transition temperature, which appears due to critical fluctuations. The parameters obtained for Ag0.1Cu0.9InP2S6 are τ0=8.1×10-16 s, U/k = 2980 K (0.26 eV).

2.2. Ferrielectric and dipolar glass phase coexistence in CuInP2S6 and Ag0.1Cu0.9InP2S6 crystals

At temperatures below 175 K dielectric dispersion effects can be observed at low frequencies for pure CuInP2S6 (Fig. 4). A similar dielectric dispersion also occurs in Ag0.1Cu0.9InP2S6 and in CuIn1+δP2S6 at low temperatures. Such dielectric dispersion is typical of dipolar glasses (Figs. 4 and 5) (Macutkevic et al. 2008).

Figure 4.

Temperature dependence of the complex dielectric permittivity of CuInP2S6 crystals. Low temperature region.

Figure 5.

Frequency dependence of the complex dielectric permittivity of CuInP2S6 crystals measured at different low temperatures. Solid lines are best fits to Eq. (1).

From dielectric spectra for CuInP2S6 and Ag0.1Cu0.9InP2S6 the Cole–Cole parameters were calculated. Very high and almost temperature independent value of the distribution of relaxation times αindicates a very wide distribution of relaxation times. The mean relaxation time increases on cooling according to the Vogel–Fulcher law:

τ=τ0expEfk(TT0)E4

where Ef is the activation energy and T0 the freezing temperature. Obtained parameters are presented in (Dziaugys et al., 2010; Dziaugys et al., 2011). The freezing temperature is higher for Ag0.1Cu0.9InP2S6 as compared to CuInP2S6 and CuIn1+δP2S6. What is the nature of dipole glass phase in pure CuInP2S6? First of all we must admit that the freezing occurs mainly in the copper sublattice, since the ferroelectric interaction exists only for copper ions in CuInP2S6. Secondly, in the dipole glass phase disorder should exist in the copper sublattice and competitive (ferroelectric and antiferroelectric) interactions occur between copper ions or (and) between copper ions and lattice. Competitive interactions can also occur between copper and indium ions. The static disorder in the copper sublattice was observed by X-ray investigations mainly at higher temperatures (Maisonneuve et al., 1997). This disorder is just random distribution of copper ions between more than the three positions (Cu1, Cu2, Cu3). However, this is only a static picture of disorder in CuInP2S6. The dielectric spectroscopy reveals that the dynamic disorder in CuInP2S6 does not vanish in the ferroelectric phase. The dynamic disorder is hopping of Cu ions between several possible static occupation positions. This hopping freezes at very low temperatures. The CuInP2S6 is a ferroelectric (ferrielectric) for which a dipole glass phase at low temperatures is observed even in the (nominally) pure crystal. Therefore CuInP2S6 with small amount of additions, independent from physical properties of the additions (ferroelectric, antiferroelectric or nonferroelectric) should exhibit the same phase diagram – the ferrielectric phase transition at higher temperatures and the freezing into a dipole glass phase at lower temperatures.

2.3. Phase transitions in antiferroelectric CuCrP2S6 and CuIn0.1Cr0.9P2S6 crystals

The antiferroelectric phase transition in CuCrP2S6 and CuIn0.1Cr0.9P2S6 is accompanied by a step-like dielectric anomaly (Fig. 6). The width of the step is approximately 5K for CuCrP2S6 and 20 K for CuIn0.1Cr0.9P2S6 Taking the temperature, corresponding to the peak point of the step as the temperature of phase transition, it was found that Tc ≈ 170 K for CuCrP2S6 and 167 K for CuIn0.1Cr0.9P2S6.

Figure 6.

Temperature dependence of the real part of the dielectric permittivity of a) CuIn0.1Cr0.9P2S6 and b) CuInCrP2S6. Solid lines are fitted according to the Curie – Weiss law.

While analyzing the sample from low temperatures, ε'rises slowly between 30 K and 125 K for CuIn0.1Cr0.9P2S6 and 150 K for CuCrP2S6, after which it increases abruptly and then smoothly, while at 167 K for CuIn0.1Cr0.9P2S6 and 170 K for CuCrP2S6 it starts decreasing. As we can see the ε'maximum is no so well pronounced as in CuInP2S6 (Banys et al., 2004), therefore such property is typical of antiferroelectrics (Kittel, 1951). The T width of this dielectric anomaly and the slope changes just below 167 K (CuIn0.1Cr0.9P2S6) and 170 K (CuCrP2S6) agree with the hypothesis of a slowly evolving short-range dipole order. Knowing that the copper dipole configuration is antipolar at T<150 K, we infer from the relatively continuous decrease of ε'at 125 K (CuIn0.1Cr0.9P2S6) and 150 K (CuInCrP2S6) that the intermediate phase is quasi–antipolar (or incommensurate). It was found that ε'(T)follows a Curie –Weiss law, Eq. (2). The ratio of Cp/Caf >> 2 indicates a first order phase transition to take place in both crystals.

2.4. Inhomogeneous ferrielectrics

The temperature dependence of the real and imaginary parts of the complex dielectric permittivity ε* at various frequencies νof CuCr0.3In0.7P2S6 crystals are presented in Fig. 7. We can see three different regions of dielectric dispersion. At temperatures T > 220 K and frequencies ν< 1 MHz the dielectric dispersion is mainly caused by the high conductivity, similarly as in pure CuCrP2S6 and CuInP2S6 crystals. The dielectric dispersion caused by the relaxation soft mode is observed in the vicinity of the ferroelectric phase transition temperature Tc ≈ 256 K for CuCr0.3In0.7P2S6 and Tc ≈ 247 K for CuCr0.2In0.8P2S6 and at higher frequencies (ν> 1 MHz). The dispersion at low temperatures (T < 170 K) is characteristic of the freezing into the dipole glass state. It is likely that substitutions in the chromium sublattice give rise to a more complex potential relief in which the copper ions move. As a consequence a part of the copper ions does not participate in the cooperative dynamics involved in the ferroelectric ordering. The dependence of the dielectric permittivity on frequency has been fitted with the Cole-Cole formula. The parameter Δε in the vicinity of the ferroelectric phase transition temperature follows the Curie-Weiss law, Eq. 2. To find out its order we have calculated the Curie-Weiss constants in the paraelectric and ferroelectric phases, respectively: Cp ≈ 4940 K and Cf ≈ 2580 K. From the ratio of these constants (1.92) it is clear that crystal with x = 0.7 undergoes a second-order phase transition. The ratio Cp,f /TC is in the order 10, therefore the observed phase transition is mainly of the “order-disorder” type (Grigas, 1996). The Cole-Cole mean relaxation time τ increases with decreasing temperature, according to the Vogel-Fulcher law. No anomaly of the relaxation time is

Figure 7.

Temperature dependence of the complex dielectric permittivity of CuCr0.3 In0.7CrP2S6

observed in the vicinity of the ferroelectric phase transition. In order to get information that is more precise about the relaxation-time distribution function, a special approach has been developed. A detailed description can be found in (Banys et al., 2002). We assume that the complex dielectric spectrum ε*(ν) can be represented as a superposition of independent individual Debye-type relaxation processes (Schafer et al., 1996; Kim et al., 2000; Pelster et al., 1998)

ε*(ν)=ε+Δεf(τ)dlgτ1+iωτE5
.

The distributions of relaxation times of the investigated ferrielectric CuCr0.3In0.7P2S6 are presented in Fig. 8. One can recognize that the relaxation-time distribution function significantly broadens at low temperatures, as it is typical for dipole glasses. Let us consider the copper ions moving in asymmetric double well potentials. The movement consists of fast oscillations in one of the minima with occasional thermally activated jumps between the minima. The jump probability is governed by the Boltzmann probability of overcoming the potential barrier between the minima. The relaxation time in such a system is given by:

τ=τ0exp[Eb/kB(TT0)]2cosh(A/2kBT)E6
.

Figure 8.

Distribution of relaxation times of mixed ferroelectric Cu(In0.7Cr0.3)P2S6 obtained from dielectric spectra (points). The solid lines are best fits according to Eq. (4).

The parameter A accounts for the asymmetry of the local potential produced by the mean-field influence of all the other dipoles. Thus, the local polarization of copper ions is

p=tanh(A/2kBT)E7

and the distribution function ϖ(p)of the local polarizations:

ϖ(p)=2kBT2πσA(1p2)exp[(atanh[p]atanh[p¯])22σA(2kBT)2]E8
.

We further consider that the asymmetry A and the potential barrier Eb of the local potential are randomly distributed around their mean values A0 and Eb0 according to a Gaussian law resulting in the distribution functions:

f(Eb)=12πσEbexp((EbEb0)22σEb2)E9

with

f(A)=12πσAexp((AA0)22σA2)E10
,

where σEb and σA are the standard deviations of Eb and A, respectively, from their mean values. Fits with the experimentally obtained relaxation-time distributions were performed and the results are presented in Fig. 8 as solid lines. Knowing the average asymmetry A and the standard deviation of asymmetry σA we have calculated the distributions of local polarization w(p) (Fig. 9).

Figure 9.

Distribution of local polarizations w(p) of a CuCr0.3In0.7P2S6 crystal: at several temperatures.

A broad distribution of local polarization is observed in both investigated ferrielectrics are typical for inhomogeneous ferroelectrics. It indicates that not all copper ions are ordered in the ferrielectric phase. This fact was confirmed also by X ray investigations of pure CuInP2S6. By further cooling non-ordered copper ions form a glassy phase and finally become frozen. Knowing the distribution function w(p), both the average (macroscopic) polarization

p¯=11pw(p)dpE11

and the Edwards-Anderson glass order parameter

qEA=11p2w(p)dpE12

can be calculated (Fig. 10). The temperature behavior of the average polarization is typical for the second-order ferroelectric phase transition.

Figure 10.

Temperature dependence of the spontaneous polarization P and the Edwards-Anderson parameter qEA of mixed CuCr0.2In0.8P2S6 and CuCr0.3In0.7P2S6 crystals.

2.5. Dipole glass state in mixed CuCr1-xInxP2S6 crystal

The temperature dependence of the dielectric properties in the CuCr1-xInxP2S6 mixed crystals with x = 0.5 is presented in Fig. 11. The shoulder-like ε’(T) anomaly shifts toward higher temperatures with increasing frequency. The dielectric relaxation is also expressed in the dielectric losses. The dielectric dispersion at low temperatures (T < 170 K) is typical of dipole glasses. At higher temperatures the dielectric dispersion is clearly symmetric and observed only at higher frequencies. On cooling it strongly passes to lower frequencies and becomes more asymmetric. The dielectric dispersion is described with the Cole-Cole formula. The Cole-Cole mean relaxation time τ increases with decreasing temperature, according to the Vogel-Fulcher law, the parameters of which are noticed in Table 2.

CuIn0.5Cr0.5P2S6CuIn0.4Cr0.6P2S6
T0, K2320
E/k, K (eV)1554 (0.134)1575 (0.136)
τ0,s7.67·10-134.7·10-13

Table 2.

Parameters of the Vogel–Fulcher law.

Figure 11.

Temperature dependence of the complex dielectric permittivity of CuIn0.5Cr0.5CrP2S6

Broad and very asymmetric distributions of relaxation times are observed in both investigated dipolar glasses (Fig. 12). To get more insight into the nature of such distributions, they are fitted by the double well potential model described above.

Figure 12.

Distribution of relaxation times of ferroelectric CuCr0.5In0.5P2S6

Figure 13.

a) Distribution of local polarizations w(p) of CuCr0.5In0.5P2S6 at several temperatures. b) Temperature dependence of the Edwards-Anderson parameter of mixed CuCr0.5In0.5P2S6 and CuCr0.6In0.4P2S6 crystals.

From the double well potential parameters the local polarization distribution has been calculated (Fig. 13). The temperature behavior of the local polarization distribution is very similar to that of other dipole glasses like RADP or BP/BPI (Banys et al., 1994). The order parameter is an almost linear function of the temperature and does not indicate any anomaly.

2.6. Phase diagram of the mixed CuInxCr1-xP2S6 crystals

The phase diagram of CuCr1-xInxP2S6 mixed crystals obtained from our dielectric results is shown in Fig. 14. Ferroelectric ordering coexisting with a dipole glass phase in CuCr1-xInxP2S6 is present for 0.7 ≤ x. On the other side of the phase diagram for x ≤ 0.9 the antiferroelectric phase transition occurs. At decreasing concentration x the antiferroelectric phase transition temperature increases. In the intermediate concentration range for 0.4 ≤ x ≤ 0.6, dipolar glass phases are observed.

Figure 14.

Phase diagram of CuCr1-xInxP2S6 crystals. AF – antiferroelectric phase; G – glass phase; F+G – ferroelectric + glass phase.

3. Magnetic properties of CuCr1-xInxP2S6 single crystals

3.1. Experimental procedure

Single crystals of CuCr1-xInxP2S6, with x = 0, 0.1, 0.2, 0.4, 0.5, and 0.8 were grown by the Bridgman method and investigated as thin as-cleft rectangular platelets with typical dimensions 4×4×0.1 mm3. The long edges define the ab-plane and the short one the c-axis of the monoclinic crystals (Colombet et al., 1982). While the magnetic easy axis of the x = 0 compound lies in the ab-plane (Colombet et al., 1982), the spontaneous electric polarization of the x = 1 compound lies perpendicular to it (Maisonneuve et al., 1997).

Magnetic measurements were performed using a SQUID magnetometer (Quantum Design MPMS-5S) at temperatures from 5 to 300 K and magnetic fields up to 5 T. For magnetoelectric measurements we used a modified SQUID ac susceptometer (Borisov et al., 2007), which measures the first harmonic of the ac magnetic moment induced by an external ac electric field. To address higher order ME effects, additional dc electric and/or magnetic bias fields are applied (Shvartsman et al., 2008).

3.2. Temperature dependence of the magnetization

The temperature (T) dependence of the magnetization (M) measured on CuCr1-xInxP2S6 samples with x = 0, 0.1, 0.2, 0.4, 0.5 and 0.8 in a magnetic field of 0H = 0.1 T applied perpendicularly to the ab-plane are shown in Fig. 15a within 5 T 150 K. Cusp-like AF anomalies are observed for x = 0, 0.1, and 0.2, at TN 32, 29, and 23 K, respectively, as displayed in Fig. 15. While Curie-Weiss-type hyperbolic behavior, M (T – )-1, dominates above the cusp temperatures (Colombet et al., 1982), near constant values of M are found as T → 0. They remind of the susceptibility of a uniaxial antiferromagnet perpendicularly to its easy axis, const., thus confirming its assertion for CuCrP2S6 (Colombet et al., 1982).

Figure 15.

Magnetization M vs. temperature T obtained for CuCr1-xInxP2S6 with x = 0, 0.1, 0.2, 0.4, 0.5, and 0.8 in 0H = 0.1 T applied parallel to the c axis before (a) and after correction for the diamagnetic underground (b; see text).

Figure 16.

Néel and Curie temperatures, TN and , vs. In3+ concentration x, derived from Fig. 15 (M) and Fig. 17 (1/M), and fitted by parabolic and logistic decay curves (solid lines), respectively.

At higher In3+ contents, x 0.4, no AF cusps appear any more and the monotonic increase of M on cooling extends to the lowest temperatures, T 5 K. Obviously the Cr3+ concentration falls short of the percolation threshold of the exchange interaction paths between the Cr3+ spins, which probably occurs at x 0.3.

A peculiarity is observed at the highest In3+ concentration, x = 0.8 (Fig. 15a). The magnetization assumes negative values as T > 60 K. This is probably a consequence of the diamagnetism of the In3+ sublattice, the constant negative magnetization of which becomes dominant at elevated temperatures. For an adequate evaluation of the Cr3+ driven magnetism we correct the total magnetic moments for the diamagnetic background via the function

M=CTΘ+DE13
.

This model function accounts for pure Curie-Weiss behavior with the constant C at sufficiently high temperature and for the corresponding diamagnetic background D at all compositions. Table 3 presents the best-fit parameters obtained in individual temperature ranges yielding highest coefficients of determination, R2. As can be seen, all of them exceed 0.999, hence, excellently confirming the suitability of Eq. (12). The monotonically decreasing magnitudes of the negative background values D - 53, -31, and -5 A/m for x = 0.8, 0.5, and 0.4, respectively, reflect the increasing ratio of paramagnetic Cr3+ vs. diamagnetic In3+ ions. We notice that weak negative background contributions, D - 17 A/m, persist also for the lower concentrations, x = 0.2, 0.1 and 0. Presumably the diamagnetism is here dominated by the other diamagnetic unit cell components, viz. S6 and P2.

x[K]C [103A/(mK)]D [A/m]best-fitting rangeR2
025.80.220.720.22-28.72.2T 50 K0.9999
0.125.20.218.160.16-16.40.9T 45 K0.9994
0.223.50.219.530.13-16.61.0T 45 K0.9997
0.412.40.26.560.06-4.50.4T 34 K0.9998
0.59.60.36.990.10-31.40.4T 29 K0.9994
0.84.50.13.190.02-54.60.3T 21 K0.9998

Table 3.

Best-fit parameters of the data in Fig. 15 to Eq. (12).

Remarkably, the positive, i.e. FM Curie-Weiss temperatures, 26 > > 23 K, for 0 x 0.2 decrease only by 8%, while the decrease of TN is about 28% (Fig. 16). This indicates that the two-dimensional (2D) FM interaction within the ab layers remains intact, while the interplanar AF coupling becomes strongly disordered and, hence, weakened such that TN decreases markedly. It is noticed that our careful data treatment revises the previously reported near equality, TN 32 K for x = 0 (Colombet et al., 1982). Indeed, the secondary interplanar exchange constant, Jinter/kB = - 1K, whose magnitude is not small compared to the FM one, Jintra/kB = 2.6 K (Colombet et al., 1982), is expected to drive the crossover from 2D FM to 3D AF ‛critical’ behavior far above the potential FM ordering temperature, .

As can be seen from Table 3 and from the intercepts with the T axis of the corrected 1/M vs. T plots in Fig. 17, the Curie-Weiss temperatures attain positive values, > 0, also for high concentrations, 0.4 x 0.8. This indicates that the prevailing exchange interaction remains FM as in the concentrated antiferromagnet, x = 0 (Colombet et al., 1982). However, severe departures from the straight line behavior at low temperatures, T < 30 K, indicate that competing AF interactions favour disordered magnetism rather than pure paramagnetic behavior. Nevertheless, as will be shown in Fig. 19 for the x = 0.5 compound, glassy freezing with non-ergodic behavior (Mydosh, 1993) is not perceptible, since the magnetization data are virtually indistinguishable in zero-field cooling/field heating (ZFC-FH) and subsequent field cooling (FC) runs, respectively.

The concentration dependences of the characteristic temperatures, TN and , in Fig. 16 confirm that the system CuCr1-xInxP2S6 ceases to become globally AF at low T for dilutions x > 0.3, but continues to show preponderant FM interactions even as x → 1. The tentative percolation limit for the occurrence of AF long-range order as extrapolated in Fig. 16 is reached at xp 0.3. This is much lower than the corresponding value of Fe1-xMgxCl2, xp 0.5 (Bertrand et al., 1984). Also at difference from this classic dilute antiferromagnet we find a stronger than linear decrease of TN with x. This is probably a consequence of the dilute magnetic occupancy of the cation sites in the CuCrP2S6 lattice (Colombet et al., 1982), which breaks intraplanar percolation at lower x than in the densely packed Fe2+ sublattice of FeCl2 (Bertrand et al., 1984).

Figure 17.

Inverse magnetization M-1 corrected for diamagnetic background, Eq. (12), vs. T taken from Fig. 15 (inset). The straight lines are best-fitted to corrected Curie-Weiss behavior, Eq. (12), within individual temperature ranges (Table 3). Their abscissa intercepts denote Curie temperatures, (Table 3).

Figure 18.

Out-of-plane magnetization of CuCr1-xInxP2S with 0 x 0.8 recorded at T = 5 K in magnetic fields |0H| 5 T. The straight solid lines are compatible with x = 0, 0.1, and 0.2, while Langevin-type solid lines, Eq. (14) and Table 4, deliver best-fits for x = 0.4, 0.5, and 0.8.

A sigmoid logistic curve describes the decay of the Curie temperature in Fig. 16,

Θ=Θ01+(x/x0)pE14

with best-fit parameters 0 = 26.1, x0 = 0.405 and p = 2.63. It characterizes the decay of the magnetic long-range order into 2D FM islands, which rapidly accelerates for x > x0 xp 0.3, but sustains the basically FM coupling up to x → 1.

3.3. Field dependence of the magnetization

The magnetic field dependence of the magnetization of the CuCr1-xInxP2S compounds yields additional insight into their magnetic order. Fig. 18 shows FC out-of-plane magnetization curves of samples with 0 x 0.8 taken at T = 5 K in fields -5 T 0H 5 T. Corrections for diamagnetic contributions as discussed above have been employed. For low dilutions, 0 x 0.2, non-hysteretic straight lines are observed as expected for the AF regime (see Fig. 15) below the critical field towards paramagnetic saturation. Powder and single crystal data on the x = 0 compound are corroborated except for any clear signature of a spin-flop anomaly, which was reported to provide a slight change of slope at 0HSF 0.18 T (Colombet et al., 1982). This would, indeed, be typical of the easy c-axis magnetization of near-Heisenberg antiferromagnets like CuCrP2S, where the magnetization components are expected to rotate jump-like into the ab-plane at 0HSF. This phenomenon was thoroughly investigated on the related lamellar MPS3-type antiferromagnet, MnPS3 albeit at fairly high fields, 0HSF 4.8 T (Goossens et al., 2000), which is lowered to 0.07 T for diamagnetically diluted Mn0.55Zn0.45PS3 (Mulders et al., 2002).

In the highly dilute regime, 0.4 x 0.8, the magnetization curves show saturation tendencies, which are most pronounced for x = 0.5, where spin-glass freezing might be expected as reported e.g. for Fe1-xMgxCl2 (Bertrand et al., 1984). However, no indication of hysteresis is visible in the data. They turn out to excellently fit Langevin-type functions,

M(H)=M0[coth(y)1/y]E15

where y=(mμ0H)/(kBT)with the ‛paramagnetic’ moment m and the Boltzmann constant kB. Fig. 18 shows the functions as solid lines, while Table 4 summarizes the best-fit results.

xM0mN = M0/m
0.465.7 kA/m5.6×10-23 Am2 = 6.1(B1.2 nm-3
0.559.6 kA/m8.5×10-23 Am2 = 9.2(B0.7 nm-3
0.824.7 kA/m6.86×10-23 Am2 = 7.4(B0.4 nm-3

Table 4.

Best-fit parameters of data in Fig. 18 to Eq. (14).

While the saturation magnetization M0 and the moment density N scale reasonably well with the Cr3+ concentration, 1-x, the ‛paramagnetic’ moments exceed the atomic one, m(Cr3+) = 4.08 B (Colombet et al., 1982) by factors up to 2.5. This is a consequence of the FM interactions between nearest-neighbor moments. They become apparent at low T and are related to the observed deviations from the Curie-Weiss behavior (Fig. 17). However, these small ‛superparamagnetic’ clusters are obviously not subject to blocking down to the lowest temperatures as evidenced from the ergodicity of the susceptibility curves shown in Fig. 15.

3.4. Anisotropy of magnetization and susceptibility

The cluster structure delivers the key to another surprising discovery, namely a strong anisotropy of the magnetization shown for the x = 0.5 compound in Fig. 19. Both the isothermal field dependences M(H) at T = 5 K (Fig. 19a) and the temperature dependences M(T) shown for 0H = 0.1 T (Fig. 19b) split up under different sample orientations. Noticeable enhancements by up to 40% are found when rotating the field from parallel to perpendicular to the c-axis. At T = 5 K we observe M 70 and 2.5 kA/m vs. M 50 and 1.8 kA/m at 0H = 5 and 0.1 T, respectively (Fig. 19a and b).

Figure 19.

Magnetization M of CuCr0.5In0.5P2S6 measured parallel (red circles) and perpendicularly (black squares) to the c axis (a) vs. 0H at T = 5 K (best-fitted by Langevin-type solid lines) and (b) vs. T at 0H = 0.1T (interpolated by solid lines).

At first sight this effect might just be due to different internal fields, Hint = H – NM, where N is the geometrical demagnetization coefficient. Indeed, from our thin sample geometry, 3×4×0.03 mm3, with N 1 and N << 1 one anticipates Hint < Hint, hence, M < M. However, the demagnetizing fields, NM 0 and NM 50 and 1.8 kA/m, are no larger than 2% of the applied fields, H = 4 MA/m and 80 kA/m, respectively. These corrections are, hence, more than one order of magnitude too small as to explain the observed splittings.

Since the anisotropy occurs in a paramagnetic phase, we can also not argue with AF anisotropy, which predicts > at low T (Blundell, 2001). We should rather consider the intrinsic magnetic anisotropy of the above mentioned ‛superparamagnetic’ clusters in the layered CuCrP2S6 structure. Their planar structure stems from large FM in-plane correlation lengths, while the AF out-of-plane correlations are virtually absent. This enables the magnetic dipolar interaction to support in-plane FM and out-of-plane AF alignment in H, while this spontaneous ordering is weakened in H. However, the dipolar anisotropy cannot explain the considerable difference in the magnetizations at saturation, M0=58.5 kA/m and M0 = 84.2 kA/m, as fitted to the curves in Fig. 19a. This strongly hints at a mechanism involving the total moment of the Cr3+ ions, which are subject to orbital momentum transfer to the spin-only 4A2(d3) ground state. Indeed, in the axial crystal field zero-field splitting of the 4A2(d3) ground state of Cr3+ is expected, which admixes the 4T2g excited state via spin-orbit interaction (Carlin, 1985). The magnetic moment then varies under different field directions as the gyrotropic tensor components, g and g, while the susceptibilities follow g2 and g2, respectively. However, since g = 1.991 and g = 1.988 (Colombet et al., 1982) the single-ion anisotropies of both M and are again mere 2% effects, unable to explain the experimentally found anisotropies.

Since single ion properties are not able to solve this puzzle, the way out of must be hidden in the collective nature of the ‛superparamagnetic Cr3+ clusters. In view of their intrinsic exchange coupling we propose them to form ‛molecular magnets with a high spin ground states accompanied by large magnetic anisotropy (Bogani & Wernsdörfer, 2008) such as observed on the AF molecular ring molecule Cr8 (Gatteschi et al., 2006). The moderately enhanced magnetic moments obtained from Langevin-type fits (Table 4) very likely refer to mesoscopic ‛superantiferromagnetic clusters (Néel, 1961) rather than to small ‛superparamagnetic ones. More experiments, in particular on time-dependent relaxation of the magnetization involving quantum tunneling at low T, are needed to verify this hypothesis.

It will be interesting to study the concentration dependence of this anisotropy in more detail, in particular at the percolation threshold to the AF phase. Very probably the observation of the converse behavior in the AF phase, < (Colombet et al., 1982), is crucially related to the onset of AF correlations. In this situation the anisotropy will be modified by the spin-flop reaction of the spins to H, where jumps up to the large and both spin components rotate synchronously into the field direction.

3.5. Magnetoelectric coupling

Magnetic and electric field-induced components of the magnetization, M = m/V,

μ0Mi=F/Hi=μ0μijHj+αijEj+βijkEjHk+γijk2EjEk+δijklHjEkElE16

related to the respective free energy under Einstein summation (Shvartsman et al., 2008)

F(E,H)=F012ε0εijEiEj12μ0μijHiHjαijHiEjβijk2HiEjHkγijk2HiEjEkδijkl2HiHjEkElE17

were measured using an adapted SQUID susceptometry (Borisov et al., 2007). Applying external electric and magnetic ac and dc fields along the monoclinic [001] direction, E = Eaccost + Edc and Hdc, the real part of the first harmonic ac magnetic moment at a frequency f = /2 = 1 Hz,

mME= (α33Eac + 333EacHdc + 333EacEdc + 23333EacEdcHdc)(V/o),

provides all relevant magnetoelectric (ME) coupling coefficients αij, βijk, γijk, and δijkl under suitable measurement strategies.

First of all, we have tested linear ME coupling by measuring mME on the weakly dilute AF compound CuCr0.8In0.2P2S6 (see Fig. 15 and 16) at T < TN as a function of Eac alone. The resulting data (not shown) turned out to oscillate around zero within errors, hence, 0 ( 10-12 s/m). This is disappointing, since the (average) monoclinic space group C2/m (Colombet et al., 1982) is expected to reveal the linear ME effect similarly as in MnPS3 (Ressouche et al., 2010). We did, however, not yet explore non-diagonal couplings, which are probably more favorable than collinear field configurations.

More encouraging results were found in testing higher order ME coupling as found, e. g., in the disordered multiferroics Sr0.98Mn0.02TiO3 (Shvartsman et al., 2008) and PbFe0.5Nb0.5O3 (Kleemann et al., 2010). Fig. 20 shows the magnetic moment mME resulting from the weakly dilute AF compound CuCr80In20P2S6 after ME cooling to below TN in three applied fields, Eac, Edc, and (a) at variant Hdc with constant T = 10 K, or (b) at variant T and constant 0Hdc = 2 T.

Figure 20.

Magnetoelectric moment mME of CuCr0.8In0.2P2S6 excited by Eac= 200 kV/m at f = 1 Hz in constant fields Edc and Hdc and measured parallel to the c axis (a) vs. 0H at T = 5 K and (b) vs. T at 0H = 2 T.

We notice that very small, but always positive signals appear, although their large error limits oscillate around mME= 0. That is why we dismiss a finite value of the second-order magneto-bielectric coefficient 333, which should give rise to a finite ordinate intercept at H = 0 in Fig. 20a according to Eq. (17). However, the clear upward trend of <mME> with increasing magnetic field makes us believe in a finite biquadratic coupling coefficient. The average slope in Fig. 20a suggests 3333 =omME/(2VHdcEacEdc) ≈ 4.410-25 sm/VA. This value is more than one to two orders of magnitude smaller than those measured in Sr0.98Mn0.02TiO3 (Shvartsman et al., 2008) and PbFe0.5Nb0.5O3 (Kleemann et al., 2010), 3333 ≈ -9.010-24 and 2.210-22 sm/VA, respectively. Even smaller, virtually vanishing values are found for the more dilute paramagnetic compounds such as CuCr0.5In0.5P2S6 (not shown). The temperature dependence of mME in Fig. 20b shows an abrupt increase of noise above TN = 23 K. This hints at disorder and loss of ME response in the paramagnetic phase.

3.6. Summary

The dilute antiferromagnets CuCr1-xInxP2S6 reflect the lamellar structure of the parent compositions in many respects. First, the distribution of the magnetic Cr3+ ions is dilute from the beginning because of their site sharing with Cu and (P2) ions in the basal ab planes. This explains the relatively low Néel temperatures (< 30 K) and the rapid loss of magnetic percolation when diluting with In3+ ions (xc 0.3). Second, at x > xc the AF transition is destroyed and local clusters of exchange-coupled Cr3+ ions mirror the layered structure by their nearly compensated total moments. Deviations of the magnetization from Curie-Weiss behavior at low T and strong anisotropy remind of super-AF clusters with quasi-molecular magnetic properties. Third, only weak third order ME activity was observed, despite favorable symmetry conditions and occurrence of two kinds of ferroic ordering for x < xc, ferrielectric at T < 100 K and AF at T < 30 K. Presumably inappropriate experimental conditions have been met and call for repetition. In particular, careful preparation of ME single domains by orthogonal field-cooling and measurements under non-diagonal coupling conditions should be pursued.

4. Piezoelectric and ultrasonic investigations of phase transitions in layered ferroelectrics of CuInP2S6 family

Ultrasonic investigations were performed by automatic computer controlled pulse-echo method and the main results are presented in papers (Samulionis et al., 2007; Samulionis et al., 2009a; Samulionis et al., 2009b). Usually in CuInP2S6 family crystals ultrasonic measurements were carried out using longitudinal mode in direction of polar c-axis across layers. The pulse-echo ultrasonic method allows investigating piezoelectric and ferroelectric properties of layered crystals (Samulionis et al., 2009a). This method can be used for the indication of ferroelectric phase transitions. The main feature of ultrasonic method is to detect piezoelectric signal by a thin plate of material under investigation. We present two examples of piezoelectric and ultrasonic behavior in the CuInP2S6 family crystals, viz. Ag0.1Cu0.9InP2S6 and the nonstoichiometric compound CuIn1+δ P2S6. The first crystal is interesting, because it shows tricritical behavior, the other is interesting for applications, because when changing the stoichiometry the phase transition temperature can be increased. For the layered crystal Ag0.1Cu0.9InP2S6, which is not far from pure CuInP2S6 in the phase diagram, we present the temperature dependence of the piezoelectric signal when a short ultrasonic pulse of 10 MHz frequency is applied (Fig. 20). At room temperature no signal is detected, showing that the crystal is not piezoelectric. When cooling down a signal of 10 MHz is observed at about 285 K. It increases with decreasing temperature. Obviously piezoelectricity is emerging.

Figure 21.

Temperature dependences of ultrasonically detected piezoelectric signal in an Ag0.1Cu0.9InP2S6 crystal. Temperature variations are shown by arrows.

Figure 22.

Temperature dependences of the amplitude of piezoelectric signal and the least squares fit to Eq. (18), showing that the phase transition is close to the tricritical one.

The absence of temperature hysteresis shows that the phase transition near Tc = 283 K is close to second-order. In order to describe the temperature dependence of the amplitude of the ultrasonically detected signal we applied a least squares fit using the equation:

Up=A(TcT)βE18

In our case the piezoelectric coefficient g33 appears in the piezoelectric equations. The tensor relation of the piezoelectric coefficients implies that g = d εt-1. According to (Strukov & Levanyuk, 1995) the piezoelectric coefficient d in a piezoelectric crystal varies as d η0/(Tc-T). Assuming that the dielectric permittivity εt can be approximated by a Curie law it turns out that the amplitude of our ultrasonically detected signal varies with temperature in the same manner as the order parameter η0. Hence, according to the fit in Fig. 22 the critical exponent of the order parameter (polarization) is close to the tricritical value of 0.25.

Figure 23.

dc field dependence of the piezoelectric signal amplitude in a Ag0.1Cu0.9InP2S6 crystal

Figure 24.

Temperature dependences of the longitudinal ultrasonic attenuation and velocity in a Ag0.1Cu0.9InP2S6 crystal along the c-axis

In the low temperature phase hysteresis-like dependencies of the piezoelectric signal amplitude on dc electric field with a coercive field of about 12 kV/cm were obtained (Fig. 23). Thus the existence of the ferroelectric phase transition was established for Ag0.1Cu0.9InP2S6 crystal. The existence of the phase transition was confirmed by both ultrasonic attenuation and velocity measurements. Since the layered samples were thin, for reliable ultrasonic measurements the samples were prepared as stacks from 8-10 plates glued in such way that the longitudinal ultrasound can propagate across layers. At the phase transition clear ultrasonic anomalies were observed (Fig. 24). The anomalies were similar to those which were described in pure CuInP2S6 crystals and explained by the interaction of the elastic wave with polarization (Valevicius et al., 1994a; Valevicius et al., 1994b). In this case the relaxation time increases upon approaching Tc according to Landau theory (Landau & Khalatnikov, 1954) and an ultrasonic attenuation peak with downwards velocity step can be observed. The increase of velocity in the ferroelectric phase can be attributed to the contribution of the fourth order term in the Landau free energy expansion. In this case the velocity changes are proportional to the squared order parameter. Also the influence of polarization fluctuations must be considered especially in the paraelectric phase.

Obviously the increase of the phase transition temperature is a desirable trend for applications. Therefore, it is interesting to compare the temperature dependences of ultrasonically detected electric signals arising in thin pure CuInP2S6, Ag0.1Cu0.9InP2S6 and indium rich CuInP2S6, where c-cut plates are employed as detecting ultrasonic transducers. Exciting 10 MHz lithium niobate transducers were attached to one end of a quartz buffer, while the plates under investigation were glued to other end. Fig. 24 shows the temperature dependences of ultrasonically detected piezoelectric signals in thin plates of these layered crystals. For better comparison the amplitudes of ultrasonically detected piezoelectric signals are shown in arbitrary units. It can be seen, that the phase transition temperatures strongly differ for these three crystals. The highest phase transition temperature was observed in nonstoichiometric CuInP2S6 crystals grown with slight addition of In i.e. CuIn1+δ P2S6 compound, where δ = 0.1 - 0.15. The phase transition temperature for an indium rich crystal is about 330 K. At this temperature also the critical ultrasonic attenuation and velocity anomalies were observed similar to those of pure CuInP2S crystals.

Figure 25.

The temperature dependences of ultrasonic signals detected by c-cut plates of CuInP2S6, Ag0.1Cu0.9InP2S6 and indium rich CuInP2S6 crystals.

Absence of piezoelectric signals above the phase transition shows that the paraelecric phases are centrosymetric. But at higher temperature piezoelectricity induced by an external dc field due to electrostriction was observed in CuIn1+δ P2S6 crystalline plates. In this case a large electromechanical coupling (K = 20 – 30 %) was observed in dc fields of order 30 kV/cm. It is necessary to note that the polarisation of the sample in a dc field in the field cooling regime strongly increases the piezosensitivity. In these CuIn1+δP2S6 crystals at room temperature an electromechanical coupling constant as high as > 50 % was obtained after appropriate poling, what is important for applications.

5. Conclusions

It was determined from dielectric permittivity measurements of layered CuInP2S6, Ag0.1Cu0.9InP2S6 and CuIn1+δP2S6 crystals in a wide frequency range (20 Hz to 3 GHz) that:

  1. A first-order phase transition of order – disorder type is observed in a CuInP2S6 crystal doped with Ag (10%) or In (10%) at the temperatures 330 K and 285 K respectively. The type of phase transition is the same as in pure CuInP2S6 crystal.

  2. The frequency dependence of dielectric permittivity at low temperatures is similar to that of a dipole glass phase. Coexistence of ferroelectric and dipole glass phases or of nonergodic relaxor and dipole glass phase can be observed because of the disorder in the copper sublattice created by dopants.

Low frequency (20 Hz – 1 MHz) and temperature (25 K and 300 K) dielectric permittivity measurements of CuCrP2S6 and CuIn0.1Cr0.9P2S6 crystals have shown that:

  1. The phase transition temperature shifts to lower temperatures doping CuCrP2S6 with 10 % of indium and the phase transition type is of first-order as in pure CuCrP2S6.

Layered CuInxCr1−xP2S6 mixed crystals have been studied by measuring the complex dielectric permittivity along the polar axis at frequencies 10-5 Hz - 3 GHz and temperatures 25 K – 350 K Dielectric studies of mixed layered CuInxCr1−xP2S6 crystals with competing ferroelectric and antiferroelectric interaction reveal the following results:

  1. A dipole glass state is observed in the intermediate concentration range 0.4 ≤ x ≤ 0.5 and ferroelectric or antiferroelectric phase transition disappear.

  2. Long range ferroelectric order coexists with the glassy state at 0.7 ≤ x ≤ 1.

  3. A phase transition into the antiferroelectric phase occurs at 0 ≤ x ≤ 0.1, but here no glass-like relaxation behavior is observed.

  4. The distribution functions of relaxation times of the mixed crystals calculated from the experimental dielectric spectra at different temperatures have been fitted with the asymmetric double potential well model. We calculated the local polarization distributions and temperature dependence of macroscopic polarization and Edwards – Anderson order parameter, which shows a second-order phase transition.

Solid solutions of CuCr1−xInxP2S6 reveal interesting magnetic properties, which are strongly related to their layered crystal structure:

  1. Diamagnetic dilution with In3+ of the antiferromagnetic x = 0 compound experiences a low percolation threshold, xp 0.3, toward ‛superparamagnetic’ disorder without tendencies of blocking or forming spin glass.

  2. At low temperatures the ‛superparamagnetic’ clusters in x > 0.3 compounds reveal strong magnetic anisotropy, which suggests them to behave like ‛molecular magnets’.

Crystals of the layered CuInP2S6 family have large piezoelectric sensitivity in their low temperature phases. They can be used as ultrasonic transducers for medical diagnostic applications, because the PT temperature for indium rich CuInP2S6 crystals can be elevated up to 330 K.

Acknowledgments

Thanks are due to P. Borisov, University of Liverpool, for help with the magnetic and magneto-electric measurements.

© 2011 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike-3.0 License, which permits use, distribution and reproduction for non-commercial purposes, provided the original is properly cited and derivative works building on this content are distributed under the same license.

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Andrius Dziaugys, Juras Banys, Vytautas Samulionis, Jan Macutkevic, Yulian Vysochanskii, Vladimir Shvartsman and Wolfgang Kleemann (August 23rd 2011). Phase Transitions in Layered Semiconductor - Ferroelectrics, Ferroelectrics - Characterization and Modeling, Mickaël Lallart, IntechOpen, DOI: 10.5772/20008. Available from:

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