An Elementary Picture of Dielectric Spectroscopy in Solids: Physical

Research: Science and Education

An Elementary Picture of Dielectric Spectroscopy in Solids: Physical Basis Mario F. García-Sánchez, Jean-Claude M´Peko,*† A. Rabdel Ruiz-Salvador, Geonel Rodríguez-Gattorno,‡ Yuri Echevarría Institute of Materials and Reagents (IMRE), University of Havana, Havana, Cuba 10400; *[email protected] Floiran Fernández-Gutierrez Higher Pedagogical Institute, “E. J. Varona”, Faculty of Sciences, Marianao 14, Havana, Cuba 11400 Adolfo Delgado Institute of Biomedical Engineering, Polytechnical School of Montreal, P.O. Box 6079 Station Centre-ville, Montreal, PQ, Canada H3C 3A7

Dielectric spectroscopy (DS) techniques have become important tools for the characterization of materials of physical, chemical, and biological applications (1–8); for example, electroceramics, semiconductors, biomaterials, ionic conductors, polymers, and ferroelectrics. DS techniques measure and analyze the behavior of the physical properties of dielectric materials as a function of either the time they are exposed to a constant external electric field or the frequency of an external alternating electric field. A simple diagram of an experimental DS system is shown in Figure 1. The electrical data extracted from the experiment can be expressed in terms of polarization, capacitance, permittivity, susceptibility, impedance, admittance, et cetera, since different formalisms of data presentation are used for analyzing the dielectric response. The theoretical basis of DS can be traced to the works of Debye (9, 10) dealing with the evolution of the chargestorage distribution in dielectric materials: in other words, the evolution of the materials’ polarization (P) when subjected to an external electric field or once this electric field has been removed. In his classical model, Debye considered noninteracting dipoles floating in a viscous medium that led to a time dependence of the whole dielectric polarization of the type: P(t) = P0exp(t兾τ), where τ is the relaxation time.1 Consequently the time derivative of the polarization, which characterizes the current flow (I = dP兾dt) across the dielectric medium, is also an exponential function of time: I (t ) =

P (t ) t ∝ exp − τ τ

(1)

This time-domain function of the dielectric response has its equivalent in the frequency domain. For practical reasons most of the dielectric data found in the literature arises from the application of electrical measurements in which the frequency of the alternating electric field applied on the material is varied continuously. † Additional address: University of São Paulo, Institute of Physics at São Carlos, P.O. Box: 369, CEP: 13560-970 São Carlos – SP, Brazil ‡ Additional address: Universidad Nacional Autónoma de México, Facultad de Química, Div. de Estudios de Posgrado, Dpto. de Química Inorgánica Ciudad Universitaria D.F., C.P. 04510 México.

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The analysis of the impedance complex planes applied to nonhomogeneous dielectric materials or systems, such as ceramics or interfaces, has been particularly productive. This data presentation and interpretation procedure has emerged as an independent investigation technique in the framework of DS, often referred to as impedance spectroscopy (IS; ref 2). Although from the measurement point of view there are no differences between DS and IS, one recognizes from the extensive literature that the interpretation of the results is based on different formalisms. Thus, in this article, except in the conclusions, we will use DS and IS to distinguish between the procedures. Since the pioneering work of Bauerle (11), IS has been widely used in electroceramic-type materials to investigate the different contributions to the overall dielectric response aris-

sample thermocouple

multimeter interface

computer

measuring apparatus furnace

Figure 1. A simplified diagram of an experimental system for DS. The electrical data are generated through the measuring apparatus to which the sample is directly connected. Measurements at different temperatures are obtained by placing the sample into a temperature-controllable furnace. A thermocouple is placed near the sample to verify the temperature, read either in terms of induced voltage (millivolts) or temperature. For automated data acquisition, the measuring system can be directly coupled to a computer through the appropriate interface. In the case of time-domain DS experiments, collection of the sample’s electrical data is carried out by apparatus such as picoammeters (Keithley Models), high-resistance meters (Hewlett Packard Models), or any multimeter or measuring method allowing high-precision (direct) current or resistance measurements. In the frequency-domain DS case, the sample’s electrical data are recorded by means of frequency response analyzers (Solartron FRA Models), impedance analyzers (Hewlett Packard and Solartron Models), or high-precision inductance–capacitance–resistance meters (QuadTec LCR Models).

Journal of Chemical Education • Vol. 80 No. 9 September 2003 • JChemEd.chem.wisc.edu


ing from bulk (grains), grain boundaries, and electrodes (2). The study of electrolyte and biological cells, diffusion and corrosion phenomena, and polymers are important examples of the application of IS (and DS in general; ref 2, 12, 13). The parameters (ideal conductance and capacitance) of an equivalent circuit are fitted to reproduce the measured impedance spectra. These circuit parameters are then associated with the physical and chemical processes occurring in the material. An important observation, according to our understanding, is that IS methodology is generally unable to explain the microscopic origin of the relaxation processes on the atomic scale. In fact, the success of DS in studying and interpreting relaxation processes at a microscopic level is perhaps its most important advantage over IS. However, conductance and capacitance contributions from different elements in a nonhomogeneous dielectric medium or different physicochemical processes are hardly ever resolved using DS. This is why IS is more useful for the study of heterogeneous materials, particularly in applied research. In fact, we have employed IS in recent research (14–19) and a typical example will be presented. In practice, the Debye dielectric response shown in eq 1 and its equivalent frequency counterpart deviate from the response encountered in most real dielectric materials. Likewise, the approach of equivalent circuits using, in its original form, ‘ideal’ conductance and capacitance elements, has been criticized owing to the failure of the ‘ideal’ behavior predicted by this formalism (1, 20). Interpretation of dielectric data has included, in both DS and IS cases, the distribution functions of relaxation times, rather than a single (Debye) relaxation time, in the dielectric material (21, 22). A new understanding of the dielectric response and its relationship to microscopic features in solids resulted from the research of Jonscher, Ngai, Dissado, and Hill (1, 20, 23– 28). They introduced the universal dielectric response that is explained in a many-body theory framework (1, 26–28) involving cooperative contribution from interacting electric species (charges or dipoles) during relaxation. Although we will be mainly concerned with the use of DS for characterizing materials, important problems with regard to accurate dielectric data fittings and different proposed models will be briefly considered. This paper is primarily intended for advanced undergraduate students, but is also useful for postgraduate students of chemistry, biochemistry, physics, materials, biology, geology, and engineering fields, who are involved or planing to be involved with materials characterization. This work is the result of several years of collaboration between chemistry and physics groups with strong contributions from students. This article does not present a description of all materials where DS (including IS) techniques are currently or could potentially be applied. However, comments on the widespread use of these techniques and the corresponding references will be given. The purpose of this paper is to provide both an introductory and advanced picture of dielectric relaxation in solids and dielectric data interpretation, as a result of the continuously increasing application of DS in research projects. This involves a description of the principles of DS that are useful for any practical situation, irrespective of the

field where its application is being considered. Treatment of these aspects is given on very simple physical grounds, emphasizing the meaning of the concepts rather than the mathematical background of DS. For those readers who are interested in more comprehensive descriptions we recommend the excellent monographs by Jonscher (1) and MacDonald (2). Basic Elements of Dielectric Spectroscopy In DS, a student or a researcher will be faced with the analysis and correct interpretation of a material’s dielectric responses in terms of the processes or phenomena involved. The dielectric responses might be very different, such as the responses of silicon–oxynitride and a hydrated zeolite illustrate in Figures 2 and 3. Such different dielectric behaviors might even be encountered in the same, or a similar material, if the material is subjected to different conditions of temperature, humidity, and pressure. At this stage, various questions may be asked: •

What kind of process is involved in the dielectric responses presented in Figures 2 and 3?

•

What generic electric species are responsible for the response presented in Figures 2 and 3?

•

Why are different dielectric responses found in the same or similar materials?

•

Why can materials with different chemical compositions often exhibit dielectric responses qualitatively similar?

To correctly address these questions (or many other possible questions), one must be familiar with basic concepts of dielectric processes at a microscopic level in the solid state. These concepts can be found in undergraduate textbooks (29–31) and include the elementary definitions such as free charge carriers, direct current (dc), conductivity, dipoles, polarization, capacitance, permittivity, and susceptibility. In this paper we will be concerned with those basic elements of DS not often emphasized in introductory textbooks. In classical literature the concept of the dielectric response of materials is, in general, uniquely associated with the orientation of dipoles composed of tightly bound charges. However, it is now known that hopping electronic or ionic charges may also give rise to polarization processes. The polarization of a material may arise from dipoles created through the chemical interaction of charges within the material (permanent dipoles) or by the application of an external electric field (induced dipoles).2 From the dielectric response point of view, a classification according to their dynamic response is of special relevance. In this context, two kinds of dipoles need to be distinguished: static and dynamic dipoles. The static dipoles can be characterized by strongly bound charge carriers of opposite sign, ±q, with a nearly constant separation distance and with only limited motion around equilibrium positions. In most cases the static dipoles only possess rotational degrees of freedom given by the hopping of at least one of the integrated charges within a double potential well (Figure 4A). Such a static behavior applies to both permanent and induced dipoles leading to a dipolar-type polarization (1, 20, 42). In the case of induced dipoles a different situation arises when dealing with charge carriers that are weakly bonded.

JChemEd.chem.wisc.edu • Vol. 80 No. 9 September 2003 • Journal of Chemical Education

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Research: Science and Education 10ⴚ8

C', C'' / F

10ⴚ9

loss peak 10ⴚ10

dc component

fp = 1.6 × 105 Hz

real part imaginary part 10ⴚ11 102

103

104

105

106

f / Hz Figure 2. Dielectrical response measured on amorphous silicon– oxynitride at 100 C. The imaginary part shows a dielectric peak at a frequency fp =1.6 × 105 Hz. Single dielectric peaks like this or superpositions of successive dielectric peaks are found, under favorable circumstances, in different types of materials: zeolites (32– 34), biological cells (5,35,36), polymers (4,37,38).

real part imaginary part 10ⴚ9

C', C'' / F

In this case, under certain conditions, sufficient energy may become available for their excitation out of the local potential wells and the charges can consequently move over a larger, but still restricted, environment under application of the electric field (Figure 4B). These mobile charges are the dynamic dipoles, and the polarization process associated with them is referred to as charge-carrier polarization (1, 20, 40, 41). It is important to emphasize the main difference between mobile charges in the form of free-charge carriers versus dynamic dipoles. The former species, which are responsible for dc-type processes, are nonlocalized, free charges moving continuously across the material without changing the center of gravity of the charge distribution in the system, therefore, without implying any polarization process. Meanwhile, the charge carriers forming dipoles (dynamic as well as static) are localized, polarizing species that do not contribute to the dctype processes. The appearance of localization or restriction of free motion in a solid is closely related to the nature of the chemical bonds in the system and also to disorder, such as structural or compositional defects. In the course of the polarization process the system undergoes macroscopic changes. The specific term describing this behavior is relaxation—the phenomenon related to the recovery of an equilibrium state within a system after removing an external perturbation. The transit time, termed as the relaxation time, constitutes an important physical parameter determined from the experiment. In general, relaxation phenomena include many properties of interest: dielectric, mechanical, magnetic, and optical. In dielectric studies, DS and other important research techniques, like thermally stimulated depolarization currents (TSDC), have been developed on the basis of the experiences gained from the corresponding (e.g., dielectric) relaxation phenomena. Note that the relaxation is physically associated with the inertia of the system. Here the system inertia is due to the resistance of the dipoles to orientation by the alternating electric field. Therefore, in this context, two competing mechanisms take place: the transfer of energy from the electric field to the material and the orientational motion, which gives rise to a kind of internal friction resulting in dielectric-energy losses. Moreover, as the dc phenomenon does not lead to charge storage, part of the energy transferred by the electric field is lost by the system, a process known as (dc) conductivity losses. The interpretation of the results anticipated from silicon–oxynitride and hydrated zeolite (Figures 2 and 3) requires an understanding of the meaning of the physical magnitudes as complex quantities. It will be shown that the total reliability of DS for carrying out accurate data interpretations often requires the consideration of different physical magnitudes, the most important of which we present below.

10ⴚ10

102

103

104

105

106

f / Hz Figure 3. Dielectrical response measured on a hydrated zeolite (clinoptilolite) at room temperature. Results like these are commonly found in ionic conductors (1, 40, 41) and eventually in most materials with increasing temperature.

Dielectric Magnitudes as Complex Quantities In current undergraduate physics courses or laboratory experiments that are directed at students of physics, chemistry, pharmacy, and biology, the students are asked to describe the electrical response expected from different elementary circuit combinations on application of an alternating current (ac) voltage. According to the basic notions of the physics of dielectrics, the electrical response of a dielectric material that is neither purely resistive nor purely reactive can be approxi1064

A

B

Figure 4. Representation of (A) a static and (B) a dynamic dipole hopping in double and multiple potential wells, respectively.



mated (at any specific frequency) by either a series or parallel combination of a resistance (Ro) and a capacitance (Co), as shown in Figure 5. Such a representation is called an equivalent circuit. A standard method to treat this problem is by analyzing the behavior of the circuit impedance, Z* = Z′ − jZ″. The impedance is defined as the opposition that the circuit offers to the flow of alternating current at a particular frequency. It is represented as a complex quantity consisting of a real part (resistance or resistive component, Z′) and an imaginary part (reactance or reactive component, jZ″). In some cases, for example the parallel circuit, it is mathematically more practical to analyze the reciprocal of impedance, referred to as admittance, Y * = Y ′ + jY ″, which is comprised of a conductance (real part, Y ′) and a susceptance (imaginary part, Y ″). The relationship between both real and imaginary components and the limiting cases of how close the circuit is to being a pure reactance (susceptance) or a pure resistance (conductance) is treated with particular interest since it remains closely associated with the quality of the circuit. During the DS experiments, the dielectric data of the materials under study are generally extracted in terms of the frequency dependence of Z* or Y *. We summarize how impedance and admittance of the elementary series and parallel circuits can be expressed in terms of the Ro and Co elements in Table 1. It is worth noting that the expressions are built by frequency independent quantities; Co is mostly associated with pure capacitive behavior and the inverse of Ro is associated with the losses. The latter are referred to either pure dielectric losses or conductive losses for either series or parallel circuits respectively. Table 1 includes the complex capacitance [C * = C ′ − jC ″ ≡ Y *兾jω ≡ (jωZ *)1], which is not always taken into consideration in simple circuit analysis. In advanced research on dielectric materials, however, the analysis of this magnitude, especially as a complex quantity, offers an opportunity to gain insight into the peculiarities of the polarization mechanisms in the dielectric media. This results because this magnitude remains directly connected to the property of storage and loss of electric field energy (associated with C′ and C″, respectively) during the polarization processes (29–31). Although possible, deduction of this information from impedance and admittance formalisms is rather indirect and may not be evident when dealing with complex equivalent circuits. Moreover we call attention to the frequency behavior (dependent or not) of each real and imaginary component that can be taken as

A

B Ro

Ro

Co

Co

C

R

Ro

D

Co

R1

R2

C1

C1

Figure 5. Equivalent circuit representation of a dielectric material: (A) a simple series Ro–Co circuit, (B) a simple parallel Ro–Co circuit, (C) a series Ro–Co circuit combined with a parallel resistance, and (D) a series combination of two parallel Ri–Ci circuits.

an identification criterion to relate a given dielectric response to a determined circuit combination. It is worth mentioning that the different expressions of the (di)electrical magnitudes calculated from the series and parallel circuit models, presented in Table 1, still are a ‘point of reference’ for the interpretation of real-electrical data in dielectric studies. On this basis, the equivalent circuits of Figures 5C and 5D will be straightforwardly associated to real dielectric responses. Why Dielectric Studies? When facing the characterization of a given material, one of the most important steps is the choice of the experimental techniques. It is always prudent to incorporate into the research plan techniques that provide direct information on

Table 1. Impedance, Admittance, and Capacitance Calculated for Series and Parallel Ro–Co Circuits Z*(ω) = Z ´ − jZ ˝

Ro–Co Circuits

Series

Parallel

Ro − j

Ro 1+ ω

2

Co2R o2

− j

ω 2 Co2 R o

1 ω Co

1+ ω

ωCo R o2 1+ ω

Y*(ω) = Y ´ − jY ˝

2

Co2R o2

2

Co2R o2

+ j

ω Co 1+ ω

2

Co2R o2

1 + j ω Co Ro

C*(ω) = C ´ − jC ˝

Co 1+ ω

2

Co2R o2

− j

ω Co2 R o 1 + ω2 Co2R o2

Co − j

1 ω Ro

NOTE: General circuit theory giving rise to the derivation of these expressions can be found in refs 43–45.


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the properties that make the material relevant from the application point of view. This consideration explains why electrical measurements are widely applied to the characterization of a variety of materials used in the manufacture of several important devices. The dielectric response of a given material strongly depends on the nature of the chemical bonds, and on the compositional and structural order at short and long range. Therefore, DS provides a tool to study interatomic interaction in materials, not necessarily restricted to electrical applications. In various dielectric studies, the electrical data extracted from a DS experiment are expressed in terms of the frequency dependence of the complex capacitance and thus complex dielectric permittivity.3 Once this complex quantity is obtained, data is presented as in Figures 2 and 3 or in complex diagrams (imaginary part versus real part). Thereafter, one should be able to identify the kind of electric species contributing to the material’s dielectric response and to determine a measure of the strength of interactions present in the material (strongly bonded relaxing charges as in Figure 2 or weakly bonded relaxing charges as in Figure 3 ). It is worth mentioning that, although relegated to a secondary consideration with respect to frequency-domain studies, a complete characterization of a dielectric system often requires an analysis of the behavior of the physical magnitudes in the time domain. Here, the simplest but generally most successful experiment consists of the charge and discharge under a step voltage of a dielectric material inserted into a condensator, from which the time dependence of the polarization or depolarization current is extracted. From this experiment, by applying eq 1, one should be able to get the material polarization as a function of time and the characteristic relaxation time τ and its (generally) thermally-activated dependence if the experiment is conducted at several temperatures, τ = τoexp(∆Ea兾kT), where ∆Ea is the activation energy of the process, k the Boltzmann constant, and T the absolute temperature. A great number of dielectric materials show characteristic relaxation times τ ranging from 105 to 109 seconds. Sampling the current at such short times is often a difficult task and therefore extremely fast processes are not always resolved. This problem can be overcome by using the frequency domain. This is the main reason why most dielectric studies are conducted in the frequency domain where common, available apparatus can reach frequencies as high as megahertz or higher. The equivalence between the time and frequency domain is well understood; in general, the time and frequency responses of materials are linked through Fourier transformation.4 The dynamic evolution of the relaxation in the two domains are in opposite directions—high frequencies are associated with short times and vice versa—as suggested by the simple, well-known relationship connecting the relaxation time and its equivalent relaxation frequency: τ ≡ 1兾ωp. The use of time-domain techniques has allowed us to study interesting (di)electrical features, such as the negative capacitance in humid zeolites (46, 47). Returning to the frequency-domain case for which we obtained the results illustrated in Figures 2 and 3, one recognizes that the dielectric response of materials can be different owing to the nature of the interaction governing the adjustment of the species undergoing polarization by the ap1066

plication of the electric field. Figure 2 shows the typical dielectric response expected from strongly bound charges and, thus, a dipolar-type polarization mechanism. This response is characterized by a dielectric loss peak in the imaginary part of the capacitance or permittivity. This loss peak is classically considered the clear manifestation of a dipolar relaxation process. The frequency of the peak, fp = 1.6 × 105 Hz in this case, corresponds to the relaxation frequency ωp (= 1兾τ) = 2πfp that is a critical point in dielectric spectra and has a special physical meaning as it is associated with the (di)electrical species self-hopping frequency. This frequency is not related to the species’ vibrational frequencies, which are several decades greater in magnitude (29). For simplicity, we present the case of noninteracting dipoles, where the variation of the dielectric permittivity with frequency during the dielectric (dipolar) relaxation may be described in terms of the classical Debye model, ε*(ω) = ε' ( ω) − j ε'' (ω ) ∝

εs 1 + j ωτ

(2)

where ε′(ω) and ε″(ω) are the real and imaginary parts of permittivity, respectively, and εs ≡ ε′(0) is the static— asymptotic, low-frequency—permittivity. This expression, which is the Fourier transform of the time-domain function given in eq 1, coincides with that expressed in terms of the capacitance C* [≡ Co兾(1 + jωτ)] of the series-circuit model in Table 1, with the particularity that the relaxation time, τ, is equal to RoCo. This is the reason why the Debye-type dielectric (permittivity) relaxation phenomenon may be classically represented by such a circuit combination. By contrast, the electrical response of the parallel circuit (Table 1) would correspond to a material showing no frequency dependence or dispersion of the real part of capacitance (or permittivity), while the imaginary part would vary according to ω1, which is well known to be the typical response of the dc processes. The dielectric response (log C * vs log f ) illustrated in Figure 2 corresponds to real data encountered in amorphous silicon–oxynitride (48). The imaginary part C ″ is decomposed into a dc component (dotted line), and the imaginary dielectric component or dielectric losses (dashed line), which are truly connected to the polarization process. Accordingly, the overall dielectric response given in Figure 2 may be classically represented by the combination of series and parallel circuit elements that give the total equivalent circuit and dielectric spectra of the system; that is, a series Ro–Co circuit combined with an additional resistance R in parallel to account for both dipolar polarization and dc conduction (Figure 5C). The typical dielectric response from the charge-carrier polarization of quasi-mobile and weaker-bound charges is shown in Figure 3. Here, no dielectric loss peak is detected and the overall response becomes strongly frequency dependent in the low-frequency region. The reason for this phenomenon, often called low-frequency dispersion (LFD), together with the absence of the loss peak, is the presence of slowly mobile charge carriers dominating the polarization at low frequencies (1, 41, 49–51). In this case the hopping frequency of these mobile species is much higher than the field frequency, thus the appearance of the strong-frequency dispersion in the low-frequency region seems to be caused by



the ‘anomalous’ increase of the separation between the charges forming the dipoles. Although there is not a rigorous theory describing this phenomenon, the occurrence of either a dipolar loss peak or a LFD is determined by the nature and feature of the charge transport in the dielectric system. The LFD, which is commonly found in carrier-dominated systems (fast ionic conductors, surface conduction, and humid insulators as in Figure 3; refs 1, 20, 40, 41) rather than strongly charge-bound dielectric systems, often becomes dominant in most dielectric materials at elevated temperatures when the ionic mobilities become sufficiently high. The most distinctive characteristics of this dielectric response, which is less familiar in classical textbooks, are that both C ′ and C″ become parallel towards the low- and high-frequency limiting regimes. The dielectric response illustrated in Figure 3 corresponds to data measured in a hydrated zeolite at room temperature. However, the identical material measured at high temperature (200 C) revealed no sign of LFD, and the dielectric spectra showed a dipolar-type relaxation process (34) as in Figure 2. Zeolites are microporous materials that are widely used in industry as adsorbents and ionic exchangers, and in purification processes (33), and have been the focus of many dielectric studies (32–34, 46, 47). Many of the physicochemical properties of these materials depend strongly on the nature of extra-framework cations that the zeolites host in their cavities for charge neutrality with respect to the anionic-framework structure. The electrical conduction and polarization mechanisms (which are in close relation with the zeolite structure), chemical nature of extra-framework cations, and hydration are the subject of various investigations, with DS proving to be a powerful technique (32–34, 46, 47). Dipolar processes in zeolites commonly result from the orientation of the dipoles formed by the anionic-framework structure and the cations. The reason for the appearance of LFD in hydrated zeolites probably arises from the effect of water, which may screen the framework-cation dipole interaction, thus allowing the cation to execute restricted but larger hopping motions between equivalent positions. In the context of DS, a much more complete characterization of dielectric materials should often include the analysis of permittivity as well as other dielectric magnitudes of interest, say impedance and admittance. However, purely polarization and the corresponding relaxation processes makes the consideration of permittivity a priority. This is because the permittivity ε* is closely related to the storage and loss of electric field energy (ε′ and ε″, respectively) in the dielectric medium, while impedance and admittance are the result of a more complex function of energy and hence might lead to a misinterpretation of the results. Why Spectroscopy?

Dielectric Spectroscopy Different spectroscopic techniques such as infrared spectroscopy (IR), Raman spectroscopy (RS), nuclear magnetic resonance (NMR), X-ray emission and photoelectron spectroscopies (XES and XPS), and Mössbauer spectroscopy (MS) are well established in the literature as tools for studying the chemical as well as physical properties of materials at the microscopic level. The special feature of any of these techniques

is the possibility of obtaining insights into the peculiarities of molecular interactions, structural properties, and other microscopic properties, such as vibration modes, composition, oxidation state, electronic energy levels, and chemical state of adsorbed species. In contrast to studies performed with these spectroscopic techniques, an important feature of the dielectric relaxation studies is that the experimental observations can be made with a relatively simple apparatus and experimental setup (see Figure 1). Note the great importance of analyzing both the real and imaginary parts of permittivity (or capacitance), since the lack of one can lead to misinterpretations. For example, at low frequencies both silicon–oxynitride and hydrated zeolite show similar behavior of the imaginary parts of the capacitance, however the real parts strongly differ from one another (see Figures 2 and 3). Application of different plotting and representation methods of the physical quantities measured or calculated from the experiment, such as permittivity ε* (or capacitance C *), impedance Z* (or admittance Y *), is helpful for an accurate interpretation of the dielectric spectra, especially when dealing with the superposition of different polarization or conduction mechanisms. This is because each plotting formalism may help to detect the dielectric features that are not observed in other analysis formalisms (2, 18). The plotting methods include the frequency spectra of the dielectric magnitudes as well as their complex planes or diagrams. For a more complete characterization, information should be correlated with the time-domain material response, for example the polarization current’s (I ) behavior, and often with other spectroscopic, structural, or microstructural characterization techniques. Mathematically, the relaxation interval has no limits but one has to know what the frequency limits are, from a practical point of view, identifying the asymptotic behaviors involved. At microscopic level, one of the most important features is the fact that relaxation (hopping) frequencies (ωp) or times (τ) associated with a given species may be identified from the experiment. Additionally, relaxation processes are usually thermally activated: ωp =

1 ∆E ∝ exp − a kT τ

(3)

Hence, the dielectric measurements over a wide range of temperatures will provide an opportunity to obtain the energy barrier or activation energy (∆Ea) associated with each kind of relaxing species. Accordingly, in those cases comprised by a superposition of different dielectric contributions—different electric species with relaxation intervals well delimited— the detailed observation of the evolution of the dielectric spectra with temperature allows discussion of the type of relaxation or conduction processes involved through identification of the different kinds of (di)electrically active species, and their characteristic relaxation frequencies and activation energies. This statement is the basis of the dynamic resolution of this spectroscopy.

Impedance Spectroscopy The treatment and presentation of dielectrical data in terms of impedance is found to be much more instructive


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Z* =

∑ i

Roi 1 + ω2 Coi 2 Roi2

− j

ω Coi Roi 2 1 + ω 2 Coi 2 Roi 2

(4)

where i denotes the different arcs, i = 1, 2 for the example shown in Figure 6. The impedance response corresponds the equivalent circuit shown in Figure 5D. In practice, all the Roi and Coi parameters involved in the equivalent circuit may be estimated by fitting the experimental data. The large quantity of experimental IS results have established the capacitance (Co) order of magnitude of generic species or processes (Table 2, ref 2, 3). In the example given in Figure 6, capacitance values after fitting the impedance spectra were found to be in the order of 5 × 1010 F (for the highfrequency arc), and 2 × 108 F (for the low-frequency arc), which are identified to arise from the ferroelectric bulk (grains) and the grain boundaries, respectively. The values of the capacitance supply a simple but well-accepted criterion of identification. For example, looking at the static capacitance value Cs = C′(0) of the dielectric response of the amorphous silicon–oxynitride (on the order of 109 F) shown in Figure 2, it is possible to establish that the phenomenon can be associated either with the grain boundaries, bulk ferroelectric, or surface layer. However, based on (a) the fact that silicon– oxynitride is not a ferroelectric-type material and (b) the amorphous nature of the sample as determined by diffraction methods (48), the identification of the response as a surfacelayer dielectric relaxation phenomenon is straightforward. Although the original model by Debye is strictly connected with the dipolar response of dielectric materials (eq 2), we should mention that impedance responses of the type described in Table 1 for the parallel circuit, or in eq 4, are often formally considered in the literature as Debye-like responses since they have the (Debye) form: Z*(ω) ∝ 1兾(1 + jωτ). Excluding this formal consideration, the reader should note that the similarities in both expressions, C * for a series Ro–Co and Z* for a parallel Ro–Co circuits, do not imply the same physical basis with respect to the moving force responsible for the phenomenon, and thus to the expressions being considered and its appropriate equivalent circuit. Care must be taken in the interpretation of the data since the superposition of different conduction processes in hetero1068

4

401 °C

3

Z'' / kΩ

than in terms of permittivity when the materials show high or at least nonnegligible conductive processes. By means of appropriate circuits or combination of circuits, IS formalism has been widely and successfully applied to different materials (2, 3, 14–19, 52, 53) for correlating the electrical impedance response with some internal, structural, or even microstructural properties. The characterization consists of the quantitative determination of the internal resistance and capacitance magnitudes associated with different microregions or local conduction processes in the heterogeneous medium. An example of the impedance data observed in a ferroelectric barium titanate ceramic sample (16, 18, 19) is shown in Figure 6. The diagram consists of two overlapped arcs, each of which can be modelled by a parallel circuit whose impedance response has been given in Table 1. The occurrence of several arcs in Z″ versus Z′ plots may be accounted for by a series combination of different parallel Ro–Co circuits giving a total impedance dispersion of the type,5

2

1

0 0.0

ω

1.5

3.0

4.5

6.0

'

Z / kΩ Figure 6. Impedance response, Z ″ vs Z ′, of a ferroelectric barium titanate (BaTiO3) ceramic sample recorded at a temperature of 401 C. Each point in the graph corresponds to a different measuring frequency ω = 2πf. The impedance spectrum is comprised of two arcs associated with two conductive processes involving different origins and characteristics (see text). Under favorable conditions, such results are also common in electrolyte and biological cells, and polymers, and arise from different local processes contributing to the overall impedance (2, 12).

geneous dielectric media (such as that presented in Figure 6) also leads to changes in both real and imaginary parts of permittivity similar to those shown in Figure 2; that is, with a loss peak of qualitatively identical (Debye) appearance (2, 15, 16, 52, 53). Here, the physical mechanism responsible for such a response is an interfacial polarization process (also known as Maxwell–Wagner polarization) arising from ions trapped at effective blocking interfaces (grain boundaries, electrodes) and should not be associated with a dipolar relaxation phenomenon. Thus, characterization of how electrical conduction processes combine is to some extent better assessed from impedance rather than direct permittivity observations. This is, in part, why the application of impedance spectroscopy has become a special topic within the different DS methods of data presentation and interpretation. Use of DS and IS Techniques in the Characterization of Materials: A Brief Overview Several examples of real DS and IS data (silicon– oxynitride, zeolite and ferroelectric BaTiO3) have been shown. However, as noninvasive techniques, DS and IS are suitable for the characterization of an increasingly important range of materials. Electrochemistry is for instance an important scientific field where DS, especially IS, techniques have broad application. This includes the study of the electrical response of heterogeneous electrochemical cells, which can substantially vary depending on the type of charge present; the microstructure of the electrolyte; and the texture and nature of the electrodes. Chemical reactions at the double layer between the electrode and electrolyte may be studied by monitoring, through equivalent circuits, the variation of the corresponding internal resistance and capacitance. In this way, IS is widely applied to investigate the effect of various electrode materials on the properties of solid electrolyte-based cells (sen-


Research: Science and Education Table 2. Capacitance Values and Their Possible Interpretation Capacitance/F

Phenomenon Responsible

1012

bulk

1011

minor, second phase

1011–108

grain boundary

1010–109

bulk ferroelectric

9

10 –10

7

107–105 104

surface layer sample–electrode interface electrochemical reactions

NOTE: See ref 3.

Table 3. Grain (g) and Grain Boundary (gb) Factors of Impedance Semicircle Depression below the Real Axis, γπ/2, Found in a Barium Titanate Ceramic Sample T/ C

γg

γgb

250

0.046

—

310

0.069

0.190

350

0.081

0.294

401

0.132

0.332

sors and fuel cells) to optimize the device performances. Moreover, IS may also allow the determination of diffusive, electrochemical, and chemical rate constants. In the electrochemical branch of corrosion phenomena, the applications of IS are extended to the study of corrosion rates and to the problem of defining degradation mechanisms. Good reviews in this field of electrical measurements can be found in refs 2 and 54. It is important to consider the scope of the validity and applicability of the Debye model by Cole and Cole; more specifically that the Cole–Cole model (presented later) had its origin as a representation of electrical impedance properties of biological membranes and tissues (35). Applications of DS and IS in the biological field can be found in several literature reports. These include the monitoring of the growth, physiological states, and internal structure of biological cells (5, 55), the study of microbial suspensions (36), the cultivation of suspended and immobilized hybridoma cells (56), and the online monitoring of biological cell concentrations (57). For readers interested in this specific field of electrical measurements we recommend the reviews by Coster et al. (6) and Kell et al. (12). DS has also become a versatile tool for the study of molecular dynamics of polymers, an important class of materials owing to their wide range of applications as biomaterials, as sensors, in electronics, and in constructions. The development of new advanced polymer materials has involved the application of DS. The technique has aided the understanding of the correlation between structure and properties through the identification of the dielectric response from dipoles forming part of either the polymer chain, α-type relaxations, or the side-chain, β-type relaxations (4, 37, 38). In this research field, IS has been used to study charge carrier-transport processes in solid polymeric electrolytes (58, 59), to examine the electrical contribution of both membrane

and electrolyte solution in polymeric membrane–electrolyte systems (60), and to discuss structure–property relationships in such relevant materials (61). For readers interested in dielectric responses of polymers we recommend the review by Williams et al. on dielectric relaxation processes in polymers (13) and the recent, comprehensive paper by Zhang et al. on polymeric membrane selectivity (62). We have given some relevant examples among various fields of current interest in materials characterization, where DS and IS have found strong applications and success. The growing ability and creativity of DS and IS users are expected to increase even further the spectrum of application of these spectroscopic techniques, in particular for adsorption studies and monitoring host–guest interactions where there are already some initial successes (63). The Dielectric Response of Most Real Dielectric Materials In terms of permittivity and impedance complex plane plots (Figure 6), the classical Debye model remains consistent with the occurrence of simple, symmetric (permittivity or impedance) semicircles that ideally have their centers on the real axis. While this model is a good approximation to the behavior of liquids or solutions containing simple polar molecules, examples of near-Debye behavior in solids are relatively few. Departures from the ‘ideal’ Debye response often consist in (a) a depression of the semicircles center below the real axis or (b) the observation of asymmetric semicircles. A closer fitting of the data given in Figure 6, for instance, revealed a depression of the two impedance semicircles below the real axis by an angle π兾2 with respect to the high-frequency intercept (axis origin). These results are presented in Table 3 for various measuring temperatures. To achieve better agreement with the experimental data, the Debye model was improved by considering processes with more than one relaxation time (1, 2, 21, 22, 35). The mathematical, empirical expressions of some of the frequently used models, Cole–Cole, Davidson–Cole, and Havriliak–Negami, are displayed in Table 4. The expressions are given in terms of the complex permittivity ε*(ω) and may be rewritten, in the case of impedance relaxation, in terms of the impedance Z*(ω), the static permittivity εs being replaced by Ro. These models are derived from descriptions of the dielectric relaxation in terms of distributions of Debye-type relaxation times. The characteristic parameters α and β define the form of the distribution and represent degrees of divergence from the ‘ideal’ Debye model.6 Jonscher has pointed out that these acceptable, near-Debye empirical approaches are just a good proposition in most cases of practical interests since they do not always lead to physically well-accepted interpretations and also do not cover the wide variety of dielectric features experimentally observed (1), even with the more general biparametric Havriliak–Negami function.7 It can be proved that the selection and use of the appropriate distribution functions almost always ensure a good, albeit not always satisfactory, fitting of the data, in spite of the physicochemical meaning of the parameters. Modern interpretations of dielectric phenomena became available after the relevant works of Jonscher, Ngai, Dissado, and Hill (1, 20, 23–28). The new concept of dielectric re-


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laxation in solids by these authors is based on several experimental results from which the empirical power-law relations were proposed, initially by Jonscher, for both ε*(ω) and its Fourier transform I(t) of the type (1, 20, 25–27, 42):

ε*(ω) ∝ ( j ω τ )S ;

I (t ) ∝

t τ

−(1 + S )

(5)

Such dependencies, which strongly contrast with the previous Debye and near-Debye approaches, are actually explained and justified in a many-body theory framework (1, 26–28): that is, the power-law behaviors for both ε*(ω) and I(t) are proposed to arise as a consequence of the cooperative interaction of all constituent electric species (charges or dipoles) during relaxation, as it is expected to occur in most real dielectric materials. The power-law exponent (S) is considered to represent a measure of the degree of correlation between interacting species in the system. A detailed presentation of some specific physical models associated with this dielectric response, including an analytical derivation of the power-law behavior, can be found in the relevant works of Jonscher and collaborators (1, 20, 26, 27). The empirical dielectric response defined in eq 5 and currently found in most real dielectric materials has been proposed as the universal dielectric response (UDR). For the dielectric permittivity case, the overall behavior of the ‘universal’ response over a wide frequency range according to Jonscher, Disado and Hill’s observations8 is also included in Table 4. In the present interpretation, it is important to note that the dielectric loss peak is considered to consist of three regions. These are the low- and high-frequency branches having power-law dependencies and arising from the dominance of cooperative many-body processes, while at frequencies around the peak (ω near ωp, within a frequency region below and above the relaxation frequency), the behavior is dominated by a Debye-like response that arises from one-particle transitions over a potential barrier.9 Although there is at present a lack of a rigorous theory on the LFD phenomenon observed in carrier-dominated sys-

tems (Figure 3), this response may be easily accommodated in the framework of the ‘universal’ theory as a particular case in which the familiar dipolar loss peak is replaced by a second low-frequency power law.10 However, a satisfactory unified theory on this topic of dielectric relaxation is still needed and needs to be capable of explaining all the above general features in terms of recognizable physical processes independently of the particular detailed physical and chemical nature of the materials involved. Any contribution to this open question is encouraged and would be welcome. We would finally like to emphasize that from the practical point of view—dielectric permittivity and impedance data fittings—the models of Cole–Cole, Davidson–Cole, and Havriliak–Negami are used in current research. Application of the UDR philosophy would require the consideration of an ‘universal’ capacitance, commonly suggested as Cn(ω) = Co(jωτ)n−1, instead of an ‘ideal’ frequency independent capacitance Co in the equivalent circuits. In fact, during fitting and interpretation of impedance spectra, Cole and Cole recognized that better representative equivalent circuits require that the classical Debye capacitor element Co be substituted by a constant-phase element (CPE), which has a frequencydependent empirical admittance of the type Y *CPE = A(jω)ψ (2), where ψ ≡ 1 − α for the Cole–Cole model. The similarities of the Jonscher and near-Debye formalisms from the circuit representations point of view have been pointed out elsewhere (1, 2). In terms of impedance relaxation, for instance, a parallel ‘Jonscher-type’ R −Co(j ωτ)n−1 circuit is shown to give rise to an impedance semicircle whose center is depressed by an angle of (1 − n)π兾2 below the real axis, the equivalent depression in the Cole–Cole model being by an angle of απ兾2 ≡ (1 − ψ)π兾2. Differences in these models have to be found in their theoretical conceptions on relaxation phenomena, and thus interpretation of the behavior of some final (di)electrical parameters resulting from the experimental data processing. With respect to Table 3, for instance, if assuming the Cole– Cole model with γ ≡ α, the fact that the grain boundary semicircles are much more depressed than the grain semicircles

Table 4. Models for the Interpretation and Fitting of Dielectric Responses, and the Asymptotic Behaviors of the Corresponding Dielectric Losses Models

ε*(ω)

ε’’(ω)

Debye

= εs兾[1 + jωτ]

ω+1; ω1;

Cole–Cole

= εs/[1 + (jωτ)1−α] ; 0 ωp

ω ωp ω >> ωp



should be accounted for by the fact that grain boundaries are characterized by a larger quantity of (microscopic) defects, thus implying a much more broad distribution of relaxation times than inside the grains. Also, as briefly mentioned in a previous section with respect to the interfacial polarization mechanism appearing in heterogeneous materials, grain boundaries commonly act as effective blocking barriers at which a great quantity of charge carriers become accumulated (2, 15, 16, 46, 47). This fact naturally involves a great probability of finding larger distributions of relaxation times at these microregions. Assuming the Jonscher formalism for which γ ≡ 1− n,11 the reason for the behavior of the results reported in Table 3 is that, as involving several blocked, potentially-interacting charge carriers together with a larger (micro)structural disorder throughout the material, the charge interaction at the intergranular microregions leads to a less correlated charge motion than inside the grains. When considering the effect of thermal energy delivered to the dielectric system, Table 3 shows an increase of the semicircles depression (i.e., increase of the Cole–Cole parameter α ≡ γ, to which corresponds a decrease of the Jonscher parameter n) for both grain and grain boundary microregions with increasing temperature. Such a behavior, which is also commonly reported in the literature (1), should arise from the expected excitation of a larger number of charge carriers by the thermal energy—an increase of the potentially-interacting charges participating in the conduction process, giving rise to a larger distribution of relaxation times, if considering the Cole–Cole model, or an (most probably) increased but less correlated charges interaction, if considering the Jonscher model. The reader must actually note the importance of finding in the future the correct answer to these remaining questions or details of data interpretation, an answer involving perhaps the finding of an unified theory. The applications of DS and IS techniques are successfully being expanded from the practical point of view to many fields of interest, where internal (resistance and capacitance) parameters extracted from the analysis of experimental data are correlated to the phenomena or processes involved in each

case. Of special interest, we have shown that for materials characterized by heterogeneity, the dielectric response along the whole material is mainly and grosso modo determined by the capacitive (and resistive) behaviors arising from the interfacial phenomena between the microregions rather than from features at atomic-scale structure (although these latter are involved). In such cases IS has been successfully used and has allowed us some interesting studies (14–19). However, when the heterogeneity does not control the dielectric response, DS allows gain insights of the atomic world. Conclusions We have presented the main features and potential applications of DS: (a) strong connection of the dielectric permittivity with the gain and loss of energy in the dielectric material, (b) distinction between ‘dynamic’ and ‘static’ dipoles from their dielectric responses, (c) determination of important dynamical parameters, such as hopping frequencies, and (d) capacitance based-identification criterion as presented in Table 2. Although we have omitted the mathematical details that support this technique, we have shown why DS helps to identify the nature of the electrically active species as well as their dynamics. It is important to remark that combining the information obtained by DS (relaxation frequency or time, activation energy, type of dielectric process: dipolar relaxation, interfacial polarization, LFD, or DC, etc.), it is possible to gain knowledge about interaction features at the microscopic level, which can be correlated to structural, compositional, and dynamic properties of materials studied by other physical or chemical techniques. In summary, DS is a powerful technique for material science research. With a simple experimental setup, information can be obtained about the electrical and transport properties related to the structural features of the material under study. The physics lying behind every response can be treated in a very simple fashion, making both preliminary, qualitative, but also high-level investigations feasible. More-

List of Symbols and Abbreviations DS: dielectric spectroscopy

C: capacitance

IS: impedance spectroscopy

ε: dielectric permittivity

dc: direct current

C*; ε*: complex capacitance and dielectric permittivity

ac: alternating current

Z *: complex impedance

FT: Fourier transform

Y*: complex admittance

CPE: constant-phase element

χ: dielectric susceptibility

E: electric field

P: polarization

Ea: activation energy

I: (polarization or depolarization) current

L (≡ A兾h): geometrical factor of the material under study, considered to be inserted into a planar capacitor consisting of two parallel metallic electrodes of area A and at a distance h apart.

f : linear frequency ω (≡ 2πf ): angular frequency ωp; fp: relaxation angular and linear frequency τ (≡ 1兾ωp): relaxation time

R: resistance

α; β: Cole–Cole and Davidson–Cole’s exponents

G: conductance

S;m;n: Jonscher’s universal exponents


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over, the utilization of DS with the complementary use of other characterization techniques can help to bring about solid conclusions about the properties of materials of a very different nature and chemical composition. Therefore we strongly recommend the use of DS to complement or correlate the studies of material characterization. Acknowledgments The authors kindly acknowledge the referees’ comments and suggestions during the revision of the manuscript. We are also very grateful to D. Georganopoulou, A. Grey, D. Carruana and D.W. Lewis for the critical reading of the manuscript, suggestions, and assistance with English language. Notes 1. In the Debye model the relaxation time τ is a thermally activated parameter (τ = τ0exp(∆Ea兾kT), where ∆Ea is the activation energy of the process, k the Boltzmann constant, and T the absolute temperature). The viscosity of the fluid in which the dipoles are ‘floating’ is also a thermally activated parameter. 2. In general, the application of an electric field is sufficient, but is not necessary, to induce an electric polarization. In piezoelectric materials, for example, polarization can also arise from the application of a mechanical stress (39). 3. For comparative studies conducted in different materials, the dielectric permittivity is a quantity that is physically much more instructive since it does not depend on the material geometry and dimensions (i.e., a dimensionless magnitude) while the capacitance does: ε* ∝ C*兾L. L (≡ A兾h) is the geometrical factor of the material inserted into a planar capacitor consisting of two parallel metallic electrodes of area A and at a distance h apart. 4. As a consequence of basic definitions of dielectric magnitudes, the polarization (P ) of a material can be expressed as (1, 29– 31),

P(ω) = εoχ(ω) E(ω)

(a)

where εo (= 8.854 pF兾m) is the vacuum permittivity, χ (≡ ε − 1) is the dielectric susceptibility (ε is the dielectric permittivity), and E is the electric field applied to the material. Moreover, according to eq 1, the time dependence of polarization satisfies a relation of the type: P(t) = Po f (t). By analogy to eq a, P (t) can be rewritten as,

P(t) ≡ εoφ(t)Eo

(b)

where φ(t) is now the dielectric response function in the time domain and should be equivalent to χ(ω) in the frequency domain. The mathematical operation generally used to do this is the Fourier transform (FT), which allows conversion from one domain into another domain: χ(ω) = FT[φ(t)] and φ(t) = FT[χ(ω)]; for mathematical details see refs 1 and 2. 5. From this expression, it is expected that the diameter of each impedance arc in Figure 6-type graphs coincides with Ro, while at the maximum of the arc, Z ″ (ω = ωo), the condition ωoCoRo = 1 is satisfied. This condition can be used to calculate the value of Co once that of Ro has been estimated. 6. In the limit of a continuous distribution of relaxation times, the ideal Debye expression becomes replaced by an integration, ε*(ω) = εs兰[g(τ)兾(1 + jωτ)]dτ; Z*(ω) = Ro兰[h(τ)兾(1 + jωτ)]dτ where g(τ) and h(τ) are the distribution functions of the relaxation

1072

times for a permittivity or impedance relaxation process, respectively. The proposed, acceptable reason of assuming distribution of relaxation times rests on the fact that, at the microscopic level, a dielectric contains a large number of defects such as local charge inhomogeneities, adsorbed species or impurities, variation in composition and stoichiometry, leading to a dependence of the relaxation time on the environment of the relaxing species, while only their average effects over the entire material can be observed or measured. 7. As can be seen in Table 4, the Cole–Cole and Davidson– Cole models can be considered particular cases of the biparametric Havriliak–Negami model for β = 1 and α = 0, respectively. 8. According to the theory developed by Dissado and Hill, and confirming Jonscher’s empirical equation, eq 5, the value of unity for both m and n exponents (Table 4) corresponds to full correlation, while the value of zero corresponds to the complete lack of correlation (1, 26, 27). Moreover, these exponents are shown to be independent of one another, suggesting that they represent two separate physical mechanisms. 9. In the ‘universal’ Jonscher–Dissado–Hill model, the Debye response holding around the dielectric loss peak is considered to result from the (commonly) thermally-activated character of the relaxation processes, as initially recognized in the development of the classical Debye model. In fact, it can be proved that the dynamic response of a charge hopping between two potential wells involves a Debye-type frequency dependence (1). 10. None of the basic circuit models given in Table 1 is able to reproduce the LFD dielectric response presented in Figure 3. Such a material response could be accounted for by considering a single ‘universal’ capacitance, generally proposed as of the type Cn = Co(jωτ)n−1, which shows an identical power law for both real and imaginary components; that is, parallel components. The overall response of Figure 3 would thus mean the assumption of two independent mechanisms leading to a variation of an initial n1 (powerlaw) exponent (n1 > 1/2) yielding C ′ > C ″ at high frequencies, to another n2 exponent (n2 < 1/2) yielding C ″ > C ′ at low frequencies. 11. In the formalism of impedance data fitting using an ‘universal’ capacitance, Cn = Co(jωτ)n−1, the parameter n has the meaning of how much the impedance response departs from the Debye model, as do also the parameters α and β in the near-Debye models. (Note, for instance, that a parallel Ro−Cn equivalent circuit would represent a Debye-type response for n = 1 when the ‘universal’ capacitance becomes frequency independent, Cn ≡ Co). This is a purely formal data treatment, without any attempt to represent a physical model. However, if trying some congruent data interpretation, differences have to be found from the fact that the parameters α and β in the near-Debye models are associated to distribution function of relaxation times, while n is actually associated to the correlation of the charges interaction in the dielectric system.

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35. Cole, K. S. J. Gen. Physiol. 1928, 12, 29. 36. Harris, C. M.; Todd, R. W.; Bungard, S. J.; Lovitt, R. W.; Morris, J. K.; Kell, D. B. Enzyme Microb. Technol. 1987, 9, 181. 37. Watts, D. C. Biomaterials 1998, 19, 551. 38. Bedekar, B. A.; Tsujii, Y.; Ide, N.; Kita, Y.; Fukuda, T.; Miyamoto, T. Polymer 1995, 36, 4735. 39. Jaffe, B.; William, R. C.; Jaffe, H. Piezoelectric Ceramics; Academic Press: London, 1971. 40. Jonscher, A. K.; Deori, K. L.; Reau, J.-M.; Moali, J. J. Mater. Sci. 1979, 14, 1308. 41. Jonscher, A. K. J. Mater. Sci. 1991, 26, 1618. 42. Jonscher, A. K. Nature 1975, 256, 566. 43. Benson, F. A.; Harrison, D. Electric-Circuit Theory; Edward Arnold Publishers: London, 1975. 44. Daniel, V. V. Dielectric Relaxation; Academic Press: London, 1967. 45. Edminister, J. A. Theory and Problems of Electric Circuits; McGraw-Hill: New York, 1965. 46. Fernández Gutiérrez, F.; González, E.; Berazaín, A.; Henández Vélez, M. In Proceedings of the 5th IEEE International Conference on Conduction and Breakdonw in Solids Dielectrics, Leicester, UK; IEEE Dielectrics and Insulation Society: New York, 1995; p 78. 47. Fernández Gutiérrez, F.; Ruiz-Salvador, A. R.; M‘Peko, J.-C.; Henández Vélez, M. Europhysics Lett. 1998, 44, 211. 48. Fernández Gutiérrez, F.; Sanchez Garrido, O.; Henández Vélez, M.; Bueno, R. M.; Martínez-Duart, J. M. Solid State Electronics 1998, 42, 925. 49. Bak, G. W.; Jonscher, A. K. J. Mater. Sci. 1999, 34, 5505. 50. Jonscher, A. K. J. Mater. Sci. 1995, 30, 2491. 51. Bisquert, J.; García-Belmonte, G.; Fabregat-Santiago, F.; Bueno, P. R. J. Electroanal. Chem. 1999, 475, 152. 52. Hirose, N.; West, A. R. J. Am. Ceram. Soc. 1996, 79, 1633. 53. Sarkar, P.; Nicholson, P. S. J. Phys. Chem. Solids 1989, 50, 197. 54. Brett, C. M. A.; Brett, A. M. O. Electrochemistry, Principles, Methods and Applications; Oxford University Press: Oxford, 1993. 55. Matanguihan, R. M.; Konstantinov, K. B.; Yoshida, T. Bioprocess Eng. 1994, 11, 213. 56. Noll, T.; Biselli, M. J. Biotechnol. 1998, 63, 187. 57. Mishima, K.; Mimura, A.; Takahara, Y.; Asami, K.; Hanai, T. J. Ferment. Bioeng. 1991, 72, 291. 58. Matveeva, E. S.; Calleja, R. D.; Martinez, E. S. Synthetic Metals 1994, 67, 207R. 59. Borkowska, M.; Siekierski, M.; Przyluski, J. Appl. Surface Sci. 1996, 92, 447. 60. Asaka, K. J. Membrane Sci. 1990, 50, 71. 61. Benavente, J.; Garcia, J. M.; Riley, R.; Lozano, A. E.; de Abajo, J. J. Membrane Sci. 2000, 175, 43. 62. Zhang, W.; Spichieger, U. E. Electrochim. Acta 2000, 45, 2259. 63. Wubbenhorst, M.; van Turnhout, J.; Klap, G.; Jansen, J. C.; Quintel, A.; Hulliger, J. IEEE Trans. Dielec. Insul. 2000, 7, 523.


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An Elementary Picture of Dielectric Spectroscopy in Solids: Physical

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