AC electrical properties and dielectric relaxation of [N(C3H7)4]2Cd2Cl6, single crystal

Materials Science and Engineering B 172 (2010) 24–32

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AC electrical properties and dielectric relaxation of [N(C3 H7 )4 ]2 Cd2 Cl6 , single crystal N. Hannachi a,∗ , I. Chaabane a , K. Guidara a , A. Bulou b , F. Hlel a a b

Laboratoire de l’état solide, Faculté des Sciences de Sfax, B.P. 802, 3018 Sfax, Tunisia Laboratoire de Physique de l’Etat Condensé, Faculté des Sciences, Université du Maine, UMR CNRS No. 6087, F-72085 Le Mans Cédex 09, France

a r t i c l e

i n f o

Article history: Received 15 December 2009 Received in revised form 27 March 2010 Accepted 2 April 2010 Keywords: Bis tetrapropylammonium hexachlorado-dicadmate DSC analysis Raman spectroscopy Impedance spectroscopy Equivalent circuit Havriliak–Negami

a b s t r a c t The bis tetrapropylammonium hexachlorado-dicadmate single crystal has been studied by means of a differential scanning calorimetry, Raman measurements and electrical impedance spectroscopy. Differential scanning calorimetry and Raman measurements performed with temperature to disclose a transition at 420 K and prove a key role of the cations in the mechanisms of disorder found on single crystal. The Z and Z  versus frequency plots are well fitted to an equivalent circuit model. The Kohlrausch–Williams–Watts function and the coupling model are used for analyzing electric modulus at various temperatures. The temperature dependence of the electrical conductivity in the different phases follows the Arrhenius law. The frequency dependence of  (ω) follows the Jonscher’s universal dynamic law with the relation (ω) = (0) + Aωn , where ω is the frequency of the AC field, and n is the exponent. The imaginary part of the permittivity constant is analyzed with the Cole–Cole formalism. In the temperature range 377–433 K, the activation energy obtained from the conductivity and the modulus spectra suggests that the ion transport is probably due to a hopping mechanism. © 2010 Elsevier B.V. All rights reserved.

1. Introduction The study of electric transport in disordered crystalline and glassy materials over a wide range of temperature and frequencies has gained considerable interests [1,2]. A variety of models have been proposed to explain the AC conduction mechanisms [3,4]. Dielectric spectroscopy is one of the methods that are able to give valuable information about conduction processes and to understand the nature and the origin of dielectric losses may be useful in the determination of structure and defects in solids [5–7]. In an attempt to study the effects of the size and the geometry of the cation on the electric behavior in this class of compounds, we have successfully synthesized a new compound of formula [N(C3 H7 )4 ]2 Cd2 Cl6 . At room temperature, it crystallizes in the triclinic system (P 1¯ space group) with Z = 2, d = 1.428 g cm−3 and the following unit cell dimensions: a = 9.530(1)Å, b = 11.744(1)Å, c = 17.433(1)Å, ˛ = 79.31(1)◦ , ˇ = 84.00(1)◦ and  = 80.32(1)◦ . The atomic arrangement can be described by an alternation of organic and inorganic layers parallel to the (1 1¯ 0) plan, made up of [N(C3 H7 )4 ] groups and Cd2 Cl6 dimers, respectively (Fig. 1). The organic layers are formed by alternating infinite chains parallel to the c direction. Each chain is made up by one type of cation.

∗ Corresponding author. E-mail address: [email protected] (N. Hannachi). 0921-5107/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.mseb.2010.04.007

The inorganic layer is made up by two different Cd2 Cl6 2− dimers. Each anion is built up by two CdCl4 tetrahedron generated by an inversion center. The material cohesion is assured by van Der Waals interactions [8]. A many works on the doping effects on the improvement of physical properties in this type of compound have been reported [9–12]. In this paper we have investigate the temperature dependence of the electrical conductivity, dielectric constants and Raman spectroscopy of [N(C3 H7 )4 ]2 Cd2 Cl6 .

2. Experimental The synthesis of [N(C3 H7 )4 ]2 Cd2 Cl6 was performed from precursors [N(C3 H7 )4 ]Cl and CdCl2 of high purity. Single crystal of [N(C3 H7 )4 ]2 Cd2 Cl6 was prepared in two steps. At the first, [N(C3 H7 )4 ]Cl was dissolved in aqueous solution. At the second, the obtained solution was added in molar ratio 1:1 to CdCl2 dissolved in HCl (1 M). After 5 days, colorless crystals were obtained by slow evaporation at room temperature. A 2920 MDSC V2.4F differential scanning calorimeter was used to examine phase behavior of [N(C3 H7 )4 ]2 Cd2 Cl6 . The sample (16.52 mg) was weighed into aluminium pans. The spectrum was obtained from temperature sweep experiments by heating the mixed system from 373 to 495 K at the rate of 5 ◦ C/min (Fig. 2). Raman spectra were recorded with a Jobin–Yvon T64000 multichannel spectrometer using a cooled CCD detector. An argon–krypton laser (coherent spectrum) was used for the excita-

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Fig. 1. [0 0 1] view showing alternation of organic and inorganic plans; Cd2 Cl6 represented by grey tetrahedron, empty circles relative to nitrogen atoms and full circles relative to carbon ones. Fig. 2. DSC thermograms of [N(C3 H7 )4 ]2 Cd2 Cl6 .

tion with a wavelength radiation of 514.5 nm and 100 mW power. At the begin, sample was heated up to 358 K; then Raman spectrum was registered in cooling stage from 423 to 323 K. Temperature was controlled by a thermo-regulator with resolution 3◦ . All measurements were conducted under an 50× objective microscope in a backscattering geometry. This was performed on crystal fixed on a goniometer head and suitably oriented for polarization analysis. The instrumental resolution was better than 2 cm−1 . The polarized spectra, in the range 7–4000 cm−1 , were recorded under various polarization configurations (Fig. 3). The electrical measurements were performed using a twoelectrode configuration on a single crystal sample. Several crystallizations tests allowed us to obtain crystals with suitable (1 1¯ 0) faces dimensions to undertake an electric study. The used crystal has a 20.5 mm2 of area and 1.1 mm of thickness. The crystal faces were coated with Ag paste to ensure good electrical contact. Electrical impedances were measured in the frequency range 209 Hz–5 MHz with a TEGAM 3550 ALF automatic bridge monitored by a microcomputer. Measurements were made over the temperature range 377–433 K. 3. Results and discussion 3.1. Differential scanning calorimetry DSC thermograms of [N(C3 H7 )4 ]2 Cd2 Cl6 recorded at a scan rate of 5 ◦ C/min is shown in Fig. 2. Three endothermic events with an onset temperature of 420, 446 and 479 K were observed. The second (b) corresponds to the expansion of the matter; the third (c) presents the melting point. The peak observed at 420 K is assigned to transition of the material.

tions observed in the ranges 3733–3208, 2920–2900, 1480–1453, 1354 and 1325–1139 cm−1 , respectively. The bending vibrations of the cation skeleton occur at 1097 cm−1 . The N–C vibrations appear in the range 1325–1285 cm−1 . The attempt of peaks assignments to vibrational modes observed in the organic cation is referred to frequency and modes reported for similar compound [14–18]. The N–C4 vibrations are habitually observed in the range 900–450 cm−1 . In our material, these modes are obtained in the range 957–455 cm−1 . The band at 1139 cm−1 assigned to rocking of CH3 molecular group shows a considerable broadening accompanied by a decrease in intensity when the temperature increases. Thermal motion can explain this behavior. We can observe the progressive disappearance of the resulting spectral evolution of the Raman peaks at 775, 1102 and 1320 cm−1 from 403 to 423 K corresponding to the (N–C4 ), bending vibration of the cation skeletons and (N–C), respectively, indicating that the cation motions and the N–C vibration are directly involved in the disordering process. Fig. 5a–c describes the variation of I(1480)/I(1453) ratio with temperature for diverse configuration (XY and YY), which where determined approximately by assuming that these bands have a Lorentzian profile and by using a spectral decomposition program (Fig. 6). There is a sudden variation in the relative intensities of some bands. We notice that the ratio decrease with increased tem-

3.2. Raman scattering In order to study the transition observed by differential scanning calorimetry in the vicinity of 420 K, we report the dependence of the frequencies of Raman lines with temperature. In addition, the literature investigation shows that aliphatic tetra-ammonium groups are governed by the reorientational dynamics [13]. Fig. 4a–d shows the evolution of spectra versus temperature at several polarizations (XY and YY) for cooling and heating. The assignments of bands observed in the Raman spectrum of [N(C3 H7 )4 ]2 Cd2 Cl6 at room temperature are listed in Table 1. The methyl group exhibits the antisymmetric stretching, as (CH3 ), symmetric stretching, s (CH3 ), antisymmetric bending, ıas (CH3 ), symmetric bending, ıs (CH3 ) and rocking, r (CH3 ), vibra-

Fig. 3. Raman spectrum of [N(C3 H7 )4 ]2 Cd2 Cl6 at 383 K (shown).

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Fig. 4. Raman spectra in the range 7–4000 cm−1 of [N(C3 H7 )4 ]2 Cd2 Cl6 with temperature.

perature. Its shows an inflection point in the vicinity of 408 K, it is clear that the change of the slope indicates the existent of a disorder. The curve in Figs. 7a and b and 8 describes the variation of half-widths with temperature (1/2 = f(T)) with XY and YY polarization for different frequencies. For most bands, a dramatic modification is observed at vicinity of 408 K. Consequently, all variations of alkyl-ammonium bands characteristics, observed in the temperature range 403–423 K, are probably due to the reorientational dynamics of alkyl chains. It appears clearly that the modes of the cation play an important role in the transition process. This result is an agreement with thermal analysis. 3.3. Impedance analysis and equivalent circuit The impedance spectra obtained at different temperatures are shown in Fig. 9a–c. In the studied temperature range, Z –Z arcs move to a lower value of impedance when the temperature increases. The bulk ohmic resistance values of the sample can be determined from the intercept of the semicircle, at low frequency, with the real axe (Z ) [19]. The equivalent circuit allows the establishment of correlations between electrochemical system parameters and impedance elements characteristics [20–22]. Fig. 9a and b shows the experimental and calculated values in a (−Z –Z ) diagram using the equivalent circuit in the temperature range 377–408 K. The equivalent circuit of [N(C3 H7 )4 ]2 Cd2 Cl6 consists of a resistance Rb (bulk resistance) and CPE1 (capacity of the fractal interface CPE) element [23]. The CPE1 element accounts for the observed depression of semicircles and also the non-ideal electrode geometry. The impedance of CPE

is ZCPE = 1/Q(jω)˛ , where ˛ is related to the deviation from the vertical of the line in the −Z versus Z plot. ˛ = 1 indicates a perfect capacitance and lower ˛ values directly reflect the roughness of the electrode used. For the highest temperature (T > 408 K), the above circuit is inadequate; the measured values disagree with the simulated one. We observe a little tail after the semicircles in the impedance spectra (Fig. 9c). The straight line after the semicircle can be explained with CPE2 corresponding to the double layer capacity of an in-homogeneous electrode surface. The expressions of real and imaginary components of the impedance related to the equivalent circuit were calculated according to the following equations, respectively: Z =

Rp2 Q1 ω˛1 cos(˛1 /2) + Rp 2

(1 + Rp Q1 ω˛1 cos(˛1 /2)) + (Rp Q1 ω˛1 sin(˛1 /2)) +

−Z  =

2

cos(˛2 /2) Q2 ω˛2

(1)

Rp2 Q1 ω˛1 sin(˛1 /2) 2

(1 + Rp Q1 ω˛1 cos(˛1 /2)) + (Rp Q1 ω˛1 sin(˛1 /2)) +

sin(˛2 /2) Q2 ω˛2

2

(2)

The good conformity of the calculated lines with experimental data indicates that the suggested equivalent circuit describes the crystal–electrolyte interface reasonably well. The

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Fig. 5. (a–c) Variation of relative intensities at several temperatures.

extract parameters for the circuit elements are summarized in Table 2. The temperature dependence of the conductivity (ln(T) versus 1000/T) of the single crystal is shown in Fig. 10. It shows an Arrhenius type behavior, T = A exp (−Ea /kˇ T). Two regions indicated as I and II are observed and separated by a discontinuity in the temperature range 398–408 K. This discontinuity is an agreement with the transition observed at 420 K, in the DSC curve, and the variations of alkyl-ammonium bands characteristics. The values of activation energies determined in regions I and II are respectively 1.3(1) and 1.2(1) eV. 3.4. Modulus analysis The complex electrical modulus formalism has been used in the analysis of the electrical properties because it gives the main response of the bulk of sample crystal. It is particularly suitable to extract phenomena such as electrode polarization and conductivity relaxation times. The complex electric modulus can be represented by the following equation [24–27]: M ∗ = 1/ε∗ = M  + jM 





M ∗ = M∞ 1 −

 dϕ(t)  −jωt

e 0



∞

−

dt

 dt

Fig. 6. Deconvolution of the Raman static spectrum at 358 K for [N(C3 H7 )4 ]2 Cd2 Cl6 .

empirical Kohlrausch–Williams–Watts (KWW) function [28]

(3) ϕ(t) = exp (4)

M∞ = 1/ε∞ is the inverse of the high frequency dielectric constant. The stretched exponential function is defined by the

  −

t KWW

ˇ 

(5)

where the ˇ (0 < ˇ < 1) parameter describes the non-exponential character of the relaxation function, the  KWW is the KWW relaxation time and ϕ(t) is related to the modulus in the angular

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Table 1 Assignments of the observed Raman spectra of [N(C3 H7 )4 ]2 Cd2 Cl6 . Wave number (cm−1 )

Assignments

84 98 118 189 197 246 268 284 309 373 455 611 753 772 783 846 957 1035 1097 1139 1285 1325 1354 1453 1480 1516 2224 2330 2370 2567 2620 2651 2742 2779 2800 2839 2850 2900 2910 2920

4 (Ag ) 12 (B2g ) + 14 (B2u ) 3 (Ag ) 15 (B3g ) 6 (B1g ) 7 (B1g ) 16 (B3u ) 2 (Ag ) + 13 (B2u ) 18 (B2g ) 1 (Ag ) + 4 (NC4 ) 11 (B2g ) 2 (NC4 )+ (CH2 ) (N–C–C–C) 3 (NC4 ) 1 (NC4 ) r (CH2 ) of the cation Bending vibrations of the cation skeleton r (CH3 ) (N–C) + r (CH3 ) of the cation ıs (CH3 ) ıas (CH3 ) ı(CH2 ) Harmonic vibration

s (CH2 ) + s (CH3 )

Fig. 7. (a–c) Half-widths variations of some Raman bands of [N(C3 H7 )4 ]2 Cd2 Cl6 with temperature; (a) ␯ = 246 cm−1 , (b) ␯ = 286 and 309 cm−1 .

CH3 stretch

Table 2 The extract parameters for the circuit elements. T (K)

Rp (×104 )

˛1

Q1 (×10−11 )

˛2

Q2 (10−6 )

377 383 385 386 398 403 408 415 418 423 427 433

152.6 194.5 115.8 111.2 40.4 23.5 30.2 20.7 13.7 10.8 7.5 3.2

0.985 0.966 0.979 0.993 0.979 1.001 0.96 1.05 1.07 1.07 1.12 1.15

0.860 1.29 1.02 0.973 1.04 0.975 1.31 4.30 4.04 5.80 3.35 1.41

0.20 0.36 0.04 0.02 0.161

7.18 2.85 9.10 1.09 0.1

frequency domain by the expression:





∞

M = Ms 1 −

e−jωt

 dϕ(t)  −

0

dt



dt = Ms[1 − ϕ(ω)]

(6)

The Fourier transform of dϕ(t)/dt can be approximated by the Havriliak–Negami (HN) equation given by the following [29–31]:



∞

e 0

−jωt

 dϕ(t)  −

dt

dt =

Fig. 8. Half-widths variations of some Raman bands of [N(C3 H7 )4 ]2 Cd2 Cl6 with temperature;  = 284 and 309 cm−1 .

related as [32–34]

1 ˛ y

[1 + (jωHN ) ]

(7)

where ˛ is a parameter characterizing a symmetrical broadening of relaxation times,  characterizes an asymmetrical broadening (˛ and  positive real smaller than 1) and  HN represents the characteristic relaxation time. These two shape parameters, ˛ and  are

 = 1 − 0.812(1 − ˛)0.387

(8)

Being the corresponding relationship between ˇ and HN parameters given by ˇKWW = (˛)1/1.23

(9)

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Fig. 10. Temperature dependence of conductivity relaxation frequency and conductivity obtained from complex permittivity of the single crystal.

B = (ω)˛ sin

 ˛  2



+

1 ( − 1) ln(C)(ω)˛ sin 2

−( − 1) 1 + (ω)˛ cos

C = 1 + (ω)2˛ + 2(ω)˛ cos



= arctan

Fig. 9. Experimental and calculated values using the equivalent circuit (—). (a–c) Semicircle plots of −Z  versus Z at different temperature.

In the electric modulus formalism the Havriliak–Negami equation have the following form:



M  = M∞ 1 − M  = M∞

A2

A + B2



(10)

B A2 + B2

(11)

where A, B, C and are given below: A = 1 + (ω)˛ cos

 ˛  2

+

+( − 1)(ω)˛ sin



1 ( − 1) ln(C) 1 + (ω)˛ cos 2

 ˛  2

 ˛ 

2 (11)

 ˛ 

2

 ˛  2



2 (11)

and



(ω)˛ sin ˛/2



 ˛ 

1 + (ω)˛ cos ˛/2

(11)

M and M data for all temperatures are fitted simultaneously to the values obtained from the modulus formalism (Eqs. (10) and (11)). Best fits for M and M at different temperatures of the sample are shown in Figs. 11 and 12, respectively. In each temperature,  ≈ 1/ε at the values of the real part reache a constant value M∞ ∞ high frequencies. At low frequencies, it approaches to zero that the electrode effect can be neglected in the modulus representation [35]. The peaks in the M plots shift toward higher frequencies and their height increase with increasing temperature. This reveals that when the frequency is high, the temperature for which the measuring frequency is equal to fp is also high, while the peaks are broader and asymmetric on both sides of the maxima than predicted by ideal Debye behavior. The conductivity relaxation frequency fp is given by the relation fp = f0 exp(−Em /(kˇ T)); where f0 is the characteristic phonon frequency, Em is the activation energy for conductivity relaxation, kˇ is the Boltzmann’s constant and T is the temperature. The temperature dependence of the conductivity relaxation frequency of the compound is plotted in Fig. 10. This is well described by the Arrhenius relation. The activation energy Em obtained from the modulus spectra (Fig. 10) in regions I and II are respectively 1.0(5) and 1.0(2) eV. Values of E calculated from conductivity and Em obtained from the modulus spectra are close, suggesting that the mobility of the charge carrier is probably due to a hopping mechanism [36]. The asymmetric M plot is suggestive of stretched exponential character of relaxation times of the material. The stretched exponential function is defined by the empirical Kohlrausch–Williams–Watts (KWW) function (Eq. (5)). The value of ˇ was determined from the full width-at-half-maximum of the M spectrum (ˇ = 1.14/FWHM). The obtained FWHM from the imaginary part of the complex modulus diagrams M are different. The ˇ value is temperature

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Fig. 13. Temperature dependence of ˇ value.

Figs. 11 and 12. Frequency dependencies of M and M  at different temperatures (show), respectively. Solid curves are the best fits to the modulus formalism.

dependent (Fig. 13). Two plates indicated as I and II are observed. The values of ˇ determined in plates I and II are respectively 0.65 (low temperature) and 0.37 (high temperature), the change in the slope is detected around 397 K. This result is in agreement with that determined from the conductivity. We can conclude that the interactions between the charge carriers vary with the temperature. 3.5. Conductivity analysis The frequency dependence of AC conductivity at various temperatures for the sample is shown in Fig. 14. The conductivity results are fitted by the following equation referred as Jonscher’s law [37]: (ω) = (0) + Aωn

(12)

where (0) is the frequency independent component, A is a temperature dependent constant, and n is the power law exponent. It has been applied to many materials to analyze the AC conductivity behavior in glasses and amorphous semiconductors [38–41]. The exponent n represents the degree of interaction between mobile ions and the environments surrounding them. Many manifestations of the hopping models and experiments to give value of n in the range of 0.6–1 have been given in Refs. [42,43]. The transport mechanism is explained by the thermally activated hopping pro-

Fig. 14. Frequency dependencies of the real part of AC complex conductivity with temperatures. Solid curves are the best fits to the modulus formalism.

cess between two sites separated by an energy barrier. Recently, the frequency dependence of electrical conductivity for the ionic single crystal has been reported [44,45]. The Jonscher’s universality of AC conductivity is not limited to the case of glasses but extends to cover the ionic single crystal. The n greater than 1 is excluded out in the theory, but it has been found in the experimental analysis of the crystals of K2 SO4 [45], Rb3 H(SO4 )2 [46] and Rb3 H(SeO4 )2 [46]. In our case the values of n are greater than 1. It suggests that the interaction between the mobile ions and defect sites is involved in the conduction mechanism. In fact the Raman scattering studies indicate that the disorder in the material increase with increased temperature. This behavior is related to the reorientational dynamics of the tetra-ammonium alkyl chains. A plot of −ln A against n (Fig. 15) indicates a linear temperature-independent and structureinsensitive correlation between the values of these two parameters [47–50]. 3.6. Dielectric properties Study of the dielectric properties is an important source for valuable information about conduction processes [51]. Fig. 16a shows the frequency dependences of the dielectric constant ε (ω) at each temperatures.

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where εs is the static permittivity, ε∞ is the permittivity at high frequency, and ω1 is the relaxation angular frequency of the Debye process. The dielectric spectra show a quasi-Debye-type behavior in high frequency range. A considerable deviation from the Debyetype was observed when frequencies shifts to lower values. Theses comportment involves a more complicate modelization. Consequently, the complex permittivity can be described as: ε∗ (ω) = ε∞ +

εs − ε∞



0

1−˛ + jε ω 0 1 + jω/ω1



(14)

The imaginary part of the complex permittivity is: 

ε (ω) =

Fig. 15. Correlations between −ln A and n. The line was best fitted to the experimental data points.

(εs − ε∞ )(ω/ω1 ) 1 + 2(ω/ω1 )

1−˛



1−˛



sin ((1 − ˛))/2

cos ((1 − ˛))/2 + (ω/ω1 )

2(1−˛)

+

0 ε0 ω

(15)

where the parameter ˛ represents the tilting angle (˛/2) of the circular arc from the real axis in the complex permittivity plane, ω is the frequency,  0 the specific conductivity and ε0 the dielectric permittivity of vacuum [53]. As the temperature increases, the dielectric constants at low frequency show a dispersive behavior [54]. Since in the low frequency region of the ε spectrum a clear tail connected to the conductivity was observed. Best fits using the function (15) give a suitable fitting of the curves resulting from the experimental data. Fig. 16b shows the tangent losses, tan ı = ε (ω)/ε (ω), as a function of frequencies at different temperatures. All the curves show a similar behavior at frequencies smaller than 2 kHz. However, at frequencies below 2 kHz, an intense increase of dielectric loss was observed below 403 K. This type and frequency dependence of Z and Z suggests the presence of dielectric relaxation in the compound [55,56]. 4. Conclusions This study shows that the bis tetrapropylammonium hexachlorado-dicadmate crystal undergoes one transition as a function of temperature as determinated by differential scanning calorimetry and Raman spectroscopy. This can be explained by a dynamical disordering process of the tetra-ammonium groups. The Cole–Cole plots of impedance complex measurements have been performed in the electrical analysis. The AC conductivity and dielectric relaxation behavior of [N(C3 H7 )4 ]2 Cd2 Cl6 single crystal compound have been studied as a function of temperature and frequency. The dielectric data have been analyzed in modulus formalism with a distribution of relaxation times using KWW stretched exponential function, indicates a non-Debye relaxation behavior of the conducting ions. The analysis of the temperature variation of M peak indicates that the relaxation process is thermally activated. The activation energy obtained from modulus spectra and complex impedance suggests that the ionic transport in the investigated material can be described by a hopping mechanism. The AC conductivity is analyzed by Jonscher’s law, suggests that the interaction between the mobile ions and defect sites is involved in the conduction mechanism. The n exponent values was found greater than 1 indicating the increases of disorder in the title compound.

Fig. 16. (a) Frequency dependence of imaginary part of complex permittivity at several temperatures. Solid and doted curves are the best fits and the experimental data, respectively. (b) Variation of the tangent losses (tan ı) versus frequency at several temperatures for [N(C3 H7 )4 ]2 Cd2 Cl6 sample.

It is proposed that the Debye model is the conventional model used in the description of the dielectric relaxation and which may be taken in from [52]: ε∗ = ε = jε − ε∞ +

εs − ε∞ 1 + (jω/ω1 )

(13)

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