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phys. stat. sol. (b) 214, 471 (1999)

Subject classification: 77.80.Bh; 77.84.Dy; S11.1

Manifestation of Incommensurate Phasein the Dielectric Properties of NH4HSeO4 Crystals

B. Andriyevsky (a), O. Myshchyshyn (a), Z. Czapla (b), and S. Dacko (b)

(a) Faculty of Physics, Ivan Franko State University, 8 Kyryla and Mefodiya Str.,290005 Lviv, Ukraine

(b) Institute of Experimental Physics, Wrocøaw University, pl. M. Borna 9,PL-50-204, Poland

(Received January 4, 1999; in revised form May 14, 1999)

Temperature (220 to 280 K) and frequency (0.02 to 1000 kHz) dependences of the real (e0) andimaginary (e00) parts of the complex electric permittivity for the ferroelectric b-cut of NH4HSeO4

crystal have been investigated. The analysis of the experimental results has shown that the effec-tive elastic stiffness coefficient Ceff of the crystal decreases about three to four times in the tem-perature range of 256 to 262 K when compared to the paraelectric phase. Small oscillations of thetemperature dependence of e(T) have been revealed in the temperature range of the incommensu-rate phase (252 to 265 K), which have been interpreted as break off in the continuous change ofthe wave vector of the crystal structure caused by numerous transitions between commensurateand incommensurate subphases.

1. Introduction

Ferroelectric crystals NH4HSeO4 (AHSe) are characterized by an incommensurate (IC)phase existing in the range of 250 to 262 K under normal conditions [1]. The dielectricproperties of these crystals were investigated earlier [1 to 5]. In all these studies a max-imum of e(T) at the phase transition into the ferroelectric (FE) phase (Tc = 250 K) wasreported, but no anomalies were observed at the transition from paraelectric to ICphase (Ti = 262 K). The existence of the IC phase was testified in a rather indirect waywith the aid of dielectric measurements: the approaching of temperature Tc to Ti wasobserved at increasing external electric field [6] or hydrostatic pressure [7] applied tothe crystal. The other anomalies of dielectric properties specific for the IC phase ofAHSe crystals, such as fine periodic structure of the dependences of e(T) and tan d(T)in (N(CH3)4)2ZnCl4 crystals [8], have not been revealed so far.

The aim of the present study is the detailed investigation of the dielectric propertiesof AHSe crystal at different frequencies of the measuring electric field in the tempera-ture range of 220 to 280 K, including paraelectric (PE), IC and FE phases.

2. Experimental

The b-cut sample of AHSe crystal (4.0 � 1.6 � 5.5 mm3) was prepared for measuring itsdielectric properties along the direction of spontaneous polarization. Gold electrodeswere evaporated on the b-faces. To avoid any influence of atmospheric humidity on thecrystal properties it was placed in a vacuum chamber (p = 10±±4 Pa) on a copper platecooled by nitrogen gas. Under these conditions, the sample was mechanically free. The

B. Andriyevsky et al.: Manifestation of Incommensurate Phase in the Dielectric Properties 471

temperature in the chamber and the rate of its lowering in the range of 280 to 220 Kwere controlled automatically with an accuracy not worse than 5 � 10±±3 K. The preci-sion LCR-meter HP 4284A connected with a personal computer was used for dielectricmeasurements. All the discussed results were obtained at frequencies 0.02, 0.2, 0.3, 0.4,0.5, 0.6, 0.8, 1, 3, 5, 8, 10, 12, 15, 20, 50, 80, 100, 120, 150, 200, 250, 300, 400, 500, 600,670, 800, 960, 1000 kHz under a measuring field of 3 V/cm.

3. Results and Discussion

Experimental investigations of AHSe crystals at different frequencies have revealed acomplicated structure of the temperature dependences of the real e0�T� and imaginarye00�T� parts of the complex electric permittivity (Figs. 1, 2). The temperature dependencesof e0�T� for different frequencies in the range of 0.5 to 50 kHz are similar. The maximummagnitude of e0 is practically unchanged under increasing frequency (Fig. 1), but the mag-nitude of e00 increases (Fig. 2). The e0�T� dependences in the temperature range 240 to 260K for frequencies 1 to 100 kHz are characterized predominantly by a single broad maxi-mum. The corresponding temperature dependences of e00 are characterized by a more com-plicated structure in the frequency range of 100 to 670 kHz with two or more extrema.This is caused by the fact that the effects of resonance interaction of the measuring elec-tric field and eigenvibrations of a sample take place just in this frequency range.

It is known that the thermodynamical analysis of proper ferroelectrics with the ICphase predicts a local minimum in the temperature dependence of the electric permit-tivity near the temperature of the transition into the FE phase [9]. This conclusion isconfirmed for AHSe crystals, for which a tendency to a minimum near Tc = 250 K isseen on the temperature dependences of e0, and a clear minimum in the range of 238 to250 K is observed in the temperature dependences of e00 (Fig. 1, 2). The position of thee00�T� maximum near T = 250 to 253 K does not depend upon frequency, but the posi-tion of another neighboring one is shifted from the range T = 230 to 240 K at low

472 B. Andriyevsky, O. Myshchyshyn, Z. Czapla, and S. Dacko

Fig. 1. Temperature dependences of the real part of the electric permittivity e0b�T� of the AHSecrystal for the frequencies f = 1 kHz, 10 kHz, 0.1 MHz and 1 MHz of the measuring electric fieldalong the b-axes at cooling run of 0.91 K/min

�f < 100 kHz) and high (f > 200 kHz) frequencies to the vicinity of T = 245 K at thefrequency f = 120 kHz. The latter frequency is close to the resonance one at T = 245 K.In this connection the mentioned shift of the low-temperature maximum of e00�T� dis-plays a nonlinearity of e00 related to the electric and mechanical action on the samplewhen approaching the resonance frequency. This experimental fact can be associatedwith a similar result obtained in [6], in which Tc approaches the almost unchangedvalue of Ti at increasing electric field and disappearing IC phase under fieldsE > 14 kV/cm (the so-called transition into the Lifshitz point).

Manifestation of Incommensurate Phase in the Dielectric Properties of NH4HSeO4 Crystals 473

Fig. 2. Temperature dependences of the imaginary part of the electric permittivity e00b�T� of theAHSe crystal for the frequencies f = 1 kHz, 10 kHz, 0.1 MHz and 1 MHz of measuring electricfield along the b-axes at cooling run of 0.91 K/min

Fig. 3. Temperature dependences of the real (e0) and imaginary (e00� parts of the electric permittiv-ity of the AHSe crystal for the frequency of 1 MHz at cooling run of 0.17 K/min

An interesting peculiarity of the temperature dependences of e0�T� and e00�T� at thefrequency of 1 MHz is its periodic behavior in the temperature range of 255 to 265 K(Fig. 3). At first view, a possible explanation of this fact can be a modulated solitonsuperstructure known in the IC phases [10]. But the detailed analysis shows that thepositions of these extrema are different for different frequencies of the measuring elec-tric field. This testifies the resonance character of the observed periodic structure,which can be classified as the manifestation of resonance sequences at the formation ofstanding waves which fit in the sample of thickness l due to the fast temperaturechange in the elastic stiffness coefficient C of the crystal in the IC phase. For this casethe known relation takes place between the sample thickness l, the wavelength l, thevelocity V of acoustic waves, the resonance frequency fR, the elastic stiffness coefficientC, and the density r,

l � nl

2� n

V

2fR� n

2fR

��C

r

s; �1�

where n � 1; 2; 3 . . . . Neglecting the temperature dependences of the length l and den-sity r of a sample, one can obtain the following condition for the observation ofthe row of vibration resonances with different numbers n at a given frequency(fR � 1 MHz):

C�T� � 4l2r f 2R

n2: �2�

Substituting the known values l, r and fR � 1 MHz into the formula (2), one can derivethe relation between the values C and n. The analysis of the frequency dependences ofe0 and e00 testifies that, at T � 261:5 K, the main resonance frequency (n = 1) of thesample is equal to fR � 250 kHz. Therefore, the maximum of the dependence e00�T� atT � 261.5 K for the frequency 1 MHz in Fig. 3 corresponds to n = 4. In such a case thenumbers of the other maxima on this dependence can be determined uniquely, thuspermitting to calculate the dependence C(n) with the aid of formula (2) and to obtain


Fig. 4. Temperature dependence of the elastic stiffness coefficient C of the AHSe crystal calculatedfrom the data of Fig. 3 and formula (2)

the corresponding dependence of C(T). The result of the calculations is presented inFig. 4. The presence of several extrema on the e0�T� and e00�T� dependences in a nar-row temperature range for 1 MHz is explained by a considerable temperature depen-dence of the elastic stiffness coefficient C(T).

The AHSe crystal is piezoelectric in the whole temperature range studied, so thatresonance-like regions in the frequency dependences e0 and e00 are expected. Since thesymmetry of the PE phase of AHSe (point symmetry group 2) allows different non-zero components of the piezoelectric tensor (d21, d22, d23 and d25) for the chosenb-direction of the measuring electric field, then several maxima in the e0�f � and e00�f �dependences are expected, corresponding to different vibrational resonances (see for-mula (1)), with different components of the elastic stiffness tensor C(a) and differentgeometric dimensions of the sample (4.0 � 1.6 � 5.5 mm3). The complexity of thesespectra can be also explained by the interaction of vibrational modes caused by theconsiderable nonlinearity of elastic characteristics of AHSe in the temperature range ofits structural instability.

It is revealed that the low-frequency (f < 50 kHz) imaginary part of the electric per-mittivity e00�f � is relatively large only in the temperature range of T < 240 K. This isobviously caused by thermal losses taking place during repolarization motions of rela-tively large ferroelectric domains. In the PE phase (T > 265 K) the maxima of the e00�f �dependence take place at frequencies 120, 200, 300, and 400 kHz, which correspond todifferent resonances during the formation of standing waves.

The real spectrum of e00�f � for the AHSe sample is rather complicated. Then for thecharacterization of its vibrations at a certain temperature we found the effective fre-quency feff which corresponds to a ``weight centreº of the e00�f � spectrum. It was per-formed by calculating the integral

I�f � � �f0

df �e00�f �� 3�

as a function of frequency f in the limits 200 Hz to 1 MHz and determining the fre-quency feff from the condition I(feff) = 0.5I (1 MHz). For the frequency feff, some effec-tive elastic stiffness coefficients Ceff can be calculated using the formula (2), and theeffective velocity of acoustic waves can be determined as

Veff � feff2l : �4�The temperature dependence of feff obtained in such a way is shown in Fig. 5. It is seen thatthe effective frequency feff is changing about twice in the narrow temperature range 256 to259 K. Taking into account the relations (4) and (2), one can expect corresponding changesin the effective velocity of acoustic waves Veff and the effective elastic stiffness coefficientCeff to be about three to four times. This conclusion agrees with the data of Fig. 4.

According to equation (4), the experimentally observed range of resonance frequen-cies fR = 120 to 400 kHz corresponds to the ranges of maximum velocities V of acousticwaves being equal to 1320 to 4400 m/s at l = 5.5 � 10±±3 m, 960 to 3200 m/s atl � 4:0� 10ÿ3 m, and 384 to 1280 m/s at l � 1:6� 10ÿ3 m. These magnitudes agree satis-factorily with the results of ultrasound wave velocity investigations in this crystal at thefrequency 10 MHz: V[010] � 4000 m/s and V[011] � 3300 m/s [11]. By the way, the veloc-ity V[010] � 4000 m/s taken from [11] and the dimension of our sample in the samedirection (l[010] = 1.6 mm) give the first resonance frequency fR = 1.25 MHz (n = 1),


according to formula (4). The elastic stiffness coefficient C22, calculated from the formula

V ��C

r

s�5�

on the basis of the known data for the velocity V�010� � 4000 m/s and the densityr � 2162 kg=m3, is equal to C22 � 3:46� 1010 N/m2. This exceeds approximately twicethe maximum effective magnitudes of Ceff obtained on the basis of the data of Fig. 5and the acoustic wavelength l � 2l � 5:5 mm. The latter value of Ceff is named as theeffective one, because it represents a linear combination of the components Cij

�i; j � 1; 2 . . . 6�, which induce, due to the inverse piezoelectric effect, different strains inthe monoclinic AHSe crystal under electric field directed along the b-axis. Thus, theobtained results show that acoustic waves with velocities twice smaller than V�010�� 4000 m/s can exist in AHSe crystals.

The detailed analysis has shown that another quasi-periodic structure is characteristicof the temperature dependences of e0, which becomes visible after differentiating thee0�T� curve (Fig. 6). Since the character of any periodic function F(x) is repeated in itssecond-order derivative (d2F�x�=dx2) taken with opposite sign, we have twice differen-tiated the function e0�T� for deriving the data presented in Fig. 6.

The positions of extrema of the function ÿ�d2�e0�=dT2�=e0 do not practically dependon the frequency of the electric field. About five maxima (or minima) of the functionÿ�d2�e0�=dT2�=e0 occur per each 2 K (see Fig. 6). The above-mentioned peculiarities ofthis weak periodic temperature dependence of the AHSe crystal are similar to charac-teristic manifestations of the modulated soliton superstructure taking place in othercrystals with the IC phase [12]. We can emphasize that the form of the periodic tem-perature dependence of ÿ�d2�e0�=dT2�=e0 (a similar form will be peculiar for the e0�T�dependence, too) resembles the temperature dependence of the intensity of X-raypeaks and the mean-square dynamic deviations of atoms in the IC phase of a ZnP2


Fig. 5. Temperature dependence of the effective frequency feff of the AHSe crystal calculated withthe aid of formula (3)

crystal from their equilibrium positions [10]. In this connection the temperature rangeTi ÿ Tc in AHSe can be divided into narrow, approximately equal in length, ranges ofcommensurate subphases, which are separated by incommensurate subphases. It wasalready mentioned that the period of this structure is approximately equal to 2/5 K. Natu-rally, the transitions between such narrow subphases will be accompanied by the corre-sponding maxima of the dielectric permittivity e0�T�, which can be visualized by sub-tracting the e0�T� dependence averaged on the temperature range larger than 2/5 K.The same result can be reached by taking derivatives of the dependence e0�T� (it is


Fig. 6. Temperature dependences of the relative second-order derivative of the electric permittivityÿ�d2�e0b�=dT2�=e0b of the AHSe crystal in the IC phase for the frequencies f = 10 and 100 kHz ofthe measuring electric field at cooling run of 0.17 K/min

Fig. 7. Temperature dependences of the parameters (1) Y1�T� � ÿ�d2�e0b�=dT2�=e0b and (2) Y2�T�� T

265:5

Y1�T� dT in the IC phase of the AHSe crystal for the frequency f � 10 kHz

enough to restrict ourselves to the second-order derivative) (Fig. 7). It is reasonable toassume that the magnitudes of the corresponding maxima of this weak periodic struc-ture of e0�T� (or Y1�T� � ÿ�d2�e0b�=dT2�=e0b have to be proportional to the changes inthe wave vector dk of the crystal structure which causes these maxima. Taking intoaccount this circumstance, we can represent the relative change of the wave vectorDk(T) in the IC phase at cooling in the temperature range of Tc < T < Ti as being inthe form of the integral

Dk�T� � �TTi

dk � c�TTi

Y1�T� dT � cY2�T� ; �6�

where c is a constant. The dependence Y2(T) obtained in such a way is shown in Fig. 7.Finally, from the discussion of our experimental results we can summarize that thewave vector of the crystal structure of AHSe associated with the IC phase dependsalmost continuously upon temperature. This agrees satisfactorily with the results of di-rect measurements of the temperature dependence of the wave vector in AHSe bymeans of neutron diffraction study [13].

4. Conclusions

1. Minimum (or tendency to minimum) in the temperature dependences of real andimaginary parts of the complex electric permittivities e0 and e00 of AHSe crystal is foundin the FE phase (T < Tc). It does not contradict the analogous conclusion of theoreticalinvestigations of crystals with IC phases.

2. The largest changes in electric permittivity of the AHSe crystal are observed in themiddle of the IC phase, 256 to 262 K. In this region the effective elastic stiffness coeffi-cient Ceff decreases about three to four times when compared with its magnitudes inthe PE phase.

3. A fine periodic-like temperature dependence of the electric permittivity has beendetected for the first time in the IC phase of the AHSe crystal. It is characteristic of themodulated soliton-like superstructure. The analysis of this dependence shows that thewave vector in the IC phase of the AHSe crystal changes with temperature almostcontinuously.

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