Application of Conducting Composite in Dielectrics*

4

Application of Conducting Composite in Dielectrics
Reza Taherian

Faculty of Chemical & Materials Engineering, Shahrood University of Technology, Shahrood, Iran

4.1 Introduction Dielectric materials are the materials that can be polarized by an applied electric field and contains a very poor electrical conductivity. When dielectrics are placed in an electric field, practically no current flows in them because, similar to metals, they do not have loosely bound, or free electrons to drift through the electronic cloud. In other words, they have a wide Fermi level band that needs a high electric field to move the electrons from valence band to conduction band. When a dielectric is placed in an electric field, positive charges are displaced toward the field and negative charges shift in the opposite direction, named polarization. This slight separation of charge, or polarization, reduces the overall electric field induced on the dielectric. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axes align to the field [1]. Insulators are made of ceramics and polymers or their composites in which there is a large energy gap between the valence and conduction bands; however, the high electrical resistivity of these materials is not always sufficient. At high voltages, a catastrophic breakdown of the insulator may occur, and current may flow. For example, the electrons may have kinetic energies sufficient to ionize the atoms of the insulator, thereby creating free electrons and generating a current at high voltages. In order to select an insulating material properly, we must know how the material stores, as well as conducts electrical charge. some of the ceramics such as porcelain, alumina, cordierite, mica, and some glasses and plastics are used as insulators. Recently, the composites composed of polymer as matrix and fiber glass as reinforcement have shown a high performance for




Hereby, from Shodhganga 4 is appreciated due to the valuable content used in this chapter.

Electrical Conductivity in Polymer-Based Composites: Experiments, Modelling and Applications. DOI: https://doi.org/10.1016/B978-0-12-812541-0.00004-5 © 2019 Elsevier Inc. All rights reserved.

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high-voltage power insulators. The resistivity of most of these is 1014 Ω. cm, and the breakdown electric fields are between 5 to 15 kV/mm [2].

4.2 Dielectric Composites Composites can be divided into four categories based on filler shape: Particulate composites—Laminated composites or layered composites— Continuous fiber composites—Whisker composites or short fiber composites. Electrical conductivity of composites depends on many parameters such as filler and matrix conductivity, filler orientation, filler dispersion, filler/ matrix configuration, filler length, filler size, filler loading, . . .. In the polymer-based composites, the matrix is an insulator and the electrical conductivity of composite is strongly dependent to filler conductivity. In polymer-based carbon composites, the conductivity is provided by carbon. Some of dielectric composites are widely used in cables, capacitors, and transformers. In the layered composites, the dielectric properties change by the following parameters [3]: 1. The number of dielectric layers. 2. Layer thickness: Increase in layer thickness normally increases breakdown voltage. In a layered construction, breakdown channels only occur at the interfaces, not directly through a layer. Also, the discharge having diffused from one layer may not enter the next layer until a part of the interface create discharge channel under the electric field. Therefore the layer thickness can influence on breakdown voltage [3]. Effect of Interfaces: Discharges in composites usually occur at the interfaces of filler to matrix or the interface of layers. For better electrical breakdown properties for a compositic dielectric, it may be better to have the higher number of layers containing thinner layers to have the lower number of layers with thicker layers. This shows the importance of interface. The magnitude of the discharge depends on the associated surface resistance and capacitance. Therefore the interface between two dielectric surfaces in a composite-dielectric-system plays an important role in determining breakdown strengths. By increasing the surface conductivity, the discharge magnitude also increases, resulting in damage to the dielectric. Thus, in a composite dielectric, it is essential to maintain low dielectric losses because the dielectric losses normally lead to

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high electric stresses. Impurities may give rise to dielectric losses. Dielectric losses are usually converted into the heat. This heat may provide thermal breakdown [3]. The breakdown in the composite occurring along the long time is also called the insulation aging process. Aging and breakdown can be initiated from several reasons such as: 1) phase transformation within the layer or interface during the time; 2) the accumulation of charges on insulator surfaces can facilitate the aging process. During discharges at the solid or liquid or solidgas or solidvacuum interfaces, certain quantity of charge (electrons or positive ions) is deposited on the solid insulator surface that can stay there for very long time, for days or even weeks. These charges increase the surface conductivity, thereby increasing the discharge probability in subsequent discharges and increases the probability of damage to the dielectric surface [3].

4.3 Formulation of Dielectrics Dielectrics has the ability to store electrical energy and contain a important part of capacitance. Dielectric materials commonly referred to as insulators, because they resist the flow of current in a motion of charge carriers under the influence of an externally applied electric field, while capacitance is a measure of the ability of any two conducting plates in proximity to store a charge Q, when exposed to a potential difference V. It can be followed that [4]: C 5 Q=V ðin Farad; FÞ

(4.1)

The capacitance of a parallel plate capacitor without any dielectric in between them is known as the vacuum capacitor, whose capacitance is determined purely by the geometry. It is known that the charge density on the plates, Q is proportional to the area A (in m2) and the external electric field E 5 V/d, where d is the distance between the plates (in m). The proportionality constant is defined as ε0 named the permittivity of free space (8.854 3 10212 F/m). Therefore the capacitance of a vacuum capacitor shown in Fig.4.1A will be [4]: Q 5 ε0 ðV=dÞA and C 5 ε0 A=d

(4.2)

When the dielectric material changes from vacuum to another material, the value of capacitance increases as shown in Fig.4.1B. A dielectric constant k, of the material introduced is given as the ratio of the capacitance of a capacitor with a dielectric between the plates to that with vacuum between them [4].

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Fig. 4.1 The schematic of capacitor with vacuum and insulator as dielectric.

k5

ε: A C ε 5 dA 5 C0 ε0 ε0 d

(4.3)

where ε is the permittivity of the dielectric material (in C2/m2or F/m). R relative permittivity (εr) or dielectric constant (k), is the ratio of the permittivity of the material to the permittivity of vacuum [4]. Now, if an AC sinusoidal potential is applied across the dielectric, we have V 5 V0exp(iωt) that the charge varies along the time. The resulting current will be made up of a charging current Ic and a loss current Il that is related to the dielectric constant [4]. h
dQ CdV πi Ic  5 5 iωCV 5 ωCV0 exp i ωt 1 (4.4) dt dt 2 The loss current Iloss initiate from two components: First, the longrange migration of charges, or DC ohm IC conduction and second, the dissipation of energy associated with the rotation or oscillation of dipoles [4]. In fact, if we consider a dielectric that have both charging and loss of electricity. In this state, the complex dielectric constant, k is created that consists of a real part k0 , represents the storage and an imaginary part k00 , represents the loss [4]: k 5 k
5 εr 5 ε
r

(4.5)

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0

ε
r  εr 2 jεvr and k


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ε
r 0 5 k 2 ikv ε0

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(4.6)

0

The real part of permittivity (ε r) is a measure of how much energy can be stored in a material from an 00externally applied electric field and the imaginary part of permittivity(ε r) is called the loss factor that is a measure of how dissipative or heat loss creates in the material when 00 exposed to an external electric 0 field. ε r is always greater than zero and is usually much smaller than ε r. The loss factor includes the effects of both dielectric loss and conductivity [4]. We can test the loss factor of each dielectric by determining tan (δ) value. When complex permittivity is drawn, there are real and imaginary components (Fig. 4.2) with 0 90 out of phase. As it can be seen in the figure between vectors of ε r and εr, there is an angle “δ”. The relative “lossiness” parameter of a material is the ratio of the energy lost to the energy stored. tanδ is defined by other names such as dissipation factor (D) and loss tangent. The reverse of loss tangent or (tan δ)21 is considered as the quality factor (Q factor) of the dielectric is commonly utilized in evaluating the figure of merit in high-frequency applications. The larger this parameter, the greater the energy storage capacity [4]. tanδ 5

εvr kv 0 5 εr k0

(4.7)

The dielectric loss is similar to the viscous deformation of a material. If we want to store a charge, as in a capacitor, dielectric loss is not desirable; however, if we want to use this material for heating up the water-contained foods in microwaves, the dielectric losses are desirable, because we want to convert the charge to heat. Therefore, in microwave εr

δ

εr′

εr"

Fig. 4.2 Loss tangent tan(δ) vector diagram.

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communications, the materials with extremely low loss materials are used [4]. Consequently, it should be mentioned that, there are two application for dielectrics; capacitors and microwaves. In the capacitor application, it is assumed that the E is constant. However, in microwave, the E oscillates to create a big dielectric loss. If the electric field oscillates, the charges move back and forth. This oscillation leads to dielectric losses which show itself in the form of heat. The notable case is that in both of the dielectric constant and dielectric losses strongly depend on electrical frequency and temperature. Dielectric quality factor is one of the important properties of the microwave dielectric materials are caused by polar generated by disordering of cations and impurities, and also by instability of crystal structure. Table (3.1) shows dielectric properties of some materials [2,5,6] (Table 4.1).

Table 4.1 Dielectric Constant and Dielectric Strength of Some Polymers [610]

Material

Dielectric Constant (ε)

Vacuum Air SiO2 Strontium titanate Neoprene rubber Nylon Pyrex glass Siliconeoil Paper Bakelite Polystyrene Teflon Polyethylene Concrete Diamond Salt Graphite Silicon Methanol Glycerol

1 (by definition) 1.00054 3.9 310 7 4.7 (3.710) 15 3.5 2.42.7 2.1 2.25 4.5 5.510 315 1015 11.68 30 4768

Dielectric Strength (MV/m) 2040 3 8 (Quartz) 8 12 14 14 15 16 24 24 60 2030 2000

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Table 4.1 Dielectric Constant and Dielectric Strength of Some Polymers [610]—cont’d

Material

Dielectric Constant (ε)

Water

8880.155.334.5 (020100200 C) 83.6 (0 C) 90 nc125010,000 (20120 C) 500,6000 9.510.5 86173 9.512 1023 N 4.08 89.1

Hydrofluoric acid Barium titanate (La,Nb):(Zr,Ti)PbO3 Si3N4 Titanium dioxide (TiO2) Al2O3 ZrO2 Cu Boron Nitride (BN) Aluminum Nitride (AlN) Mullite Silicon Carbide (SiC)

6 10.2

Dielectric Strength (MV/m) 6570

1620 4 843 46 374 15 9.8

The materials with polar groups have large dielectric constants because of the orientation of the dipoles in an applied field. These materials are those having permanent dipole moments, such as the ketogroup (C 5 O) or carboxyl group(OC 5 O). These materials due to polarity tend to absorb moisture from the atmosphere, which further increases their dielectric constants. Therefore nonpolar polymers such as polyethylene, polystyrene, and fluorocarbons have better insulation properties or higher k than the highly polar polymers such as polyamides, alkyds, ureaformaldehydes, and some epoxies contain polar groups [11].

4.4 Frequency Dependency of Dielectrics The dielectrics have energy storage capability. Because, the dielectric material has an arrangement of charged species that can be displaced in response to an electric field applied across the material. The charges

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Dipolar polarization: Response is much slower Space charge polarization: Response is quite slow τ is large

+

Ionic polarization: Response is slower Atomic polarization: Response is slower

Atomic

ε′

ε"

103

Electronic polarization: Response is fast τ is slow

106

109

Audiofrequency Microwave

1012

IR

1015

UV

Fig. 4.3 Frequency dependency of different polarization mechanisms in dielectrics [1,2,4].

become polarized to compensate for the electric field such that the positive and negative charges move in opposite directions. The overall permittivity of a dielectric material may have a variety of contribution from several dielectric mechanisms or polarization effects as shown in Fig. 4.3 [4]. It was be stated that atomic and electronic mechanisms are so weak that approximately are constant over the microwave region and the variation of permittivity in microwave range is mainly due to dipolar relaxation. The absorption peaks in upper than the infrared region is mainly due to atomic and electronic polarizations and in the low frequency range, ε00 is dominated by the influence of ion conductivity. Each dielectric mechanism has a characteristic “cut-off frequency”. The magnitude and “cut-off frequency” of each mechanism is specific for different materials. As frequency increases, the faster mechanisms will be the dominant mechanisms to contribute to ε0 in comparison to slow mechanisms. The loss factor (ε00 ) will correspondingly peak at each critical frequency. The frequency dependency of different polarization mechanisms have been observed in Fig.4.3 [4].

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4.5 Polarization in Dielectrics Similar to when we subject the material to stress, the planes of the material move and namely strain occurs, when we subject the materials to an electric field, the whole charged and polar particles such as atoms, molecules, ions, or electrons in the materials respond to the applied electric field. This condition is named polarization. If two surfaces by distance of d are subjected to the electric field, the magnitude of polarization in this dipolar is as follows [2]: P 5 zqd

(4.8)

The 1q is the charge accumulated on one surface and q is the charge on other surface and z is the number of electron or other charged particles per m3. Dipole is a pair of opposite charges separated by a certain distance [2]. In polarization, between the nucleus and electron cloud, a separation occur or the ions in a materials vibrate or move. There are four types of polarization: (1) Electronic polarization, (2) ionic polarization, (3) molecular polarization, and (4) space charge (Fig. 4.4) [2].

4.6 Mechanisms of Electric Polarization The events occurring during polarization can be compared to the metallurgical event occurring during mechanical stress in materials (such as strain, climbing, diffusion, etc.). It can be predicted that by differing the strain rate in mechanical stress, the physical and metallurgical events differ. Similarly, by differing the electrical frequently or rapidly alternating of applied field in material, the polarization mechanism may be changed [2]. In all the materials, in the absence of an external force such as electrical field, the positively and negatively charged particles are balanced so that they give an overall charge neutrality. However, when the electric field is applied, the balances of charges are perturbed by the following four basic polarization mechanisms [12]: Electronic polarization: This polarization occurs in atoms that contain electron (negative charge) and the nucleus (with positive charge). Influenced by electric field, the nucleus and electrons are displaced and electronic polarization is induced [12].

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+ E –

+

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Center of negative charge +

Shift in electron cloud

δ

Cation +

(A) +

–

Atom M +

Atom N

Si

–

+

E

–

+ (D)

– +

E

O

Na+ Na+ Na+ Na+ Na+

(B) –

– E

Electrodes –

+ (E)

+ Na+ Na+

–

– –

E (C)

+

Na+ Na+ E

Na+ Na+ –

+ (F)

Fig. 4.4 Polarization mechanisms in materials: (A) electronic, (B) atomic or ionic, (C) high-frequency dipolar or orientation (present in ferroelectrics), (D) lowfrequency dipolar (present in linear dielectrics and glasses), (E) interfacial-space charge at electrodes, and (F) interfacial-space charge at heterogeneities such as grain boundaries [2]. From Principles of Electronic Ceramics, L.L. Hench and J. K. West, p. 188, Fig. 5 2. Copyright (r) 1990 Wiley Interscience. Reprinted by permission. This material is used by permission of John Wiley & Sons, Inc.

Atomic/ionic polarization: This polarization occurs in different atoms that have shared some electrons in the molecule such as covalence bond that the electrons have asymmetrically distributed. By inducing an electrical field, the electron cloud is shifted toward the stronger binding atom, the atoms need to the charges of opposite polarity and an external field acting on these net charges will tend

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to change the equilibrium positions of the atoms. This leads to atomic polarization [4]. Dipolar/orientational polarization: This polarization type is provided in molecules containing ionic bonds that the valence electrons completely have been transported from one atom to the atom with higher electronegativity. Here, there is a permanent dipole moment that its value depends to the charges of the transferred valence electrons and the interatomic distance between them. Under the a electric field, these permanent dipole moments in the molecules will align along the direction of the electric field called the dipolar or orientational polarization. This should be emphasized this polarization occurs only in dipolar materials possessing permanent dipole moments [12]. Space charge polarization: In some dielectric materials, due to diffusion of some charge carrier species through the bulk by diffusion, fast ionic conduction, or hopping that as a result of that macroscopic field distortion can be provided. This distortion appears as an increase in the capacitance of the sample and may be indiscernible from the real rise of the dielectric permittivity. Electric field, in these materialsleads to the macroscopic mass transport and ions or charge carriers migration. In general, the space charge polarization can be considered as hopping polarization or interfacial polarization. If the charged species such as ions and vacancies, or electrons and hole move from one site to another site of dielectric bulk is named the hopping polarization. However, if the separation of the positive and negative charges under an electric field occurs it can produce an interfacial polarization [12]. In an insulator or dielectric, when an electrical field is induced, some events occur. Nelson suggested some of these events occurring in the insulator during electrical field. The main expected behaviors of charges in a solid dielectric or at the electrodes has been shown in Fig. 4.5. It can be seen that the different kinds of events occurs in insulator such as diffusion, charge carrier drift, trapping and field distortion, and generation and recombination of electrons, as well as MaxwellWagner polarization. The rate of these mechanisms differ from each other. In addition, it may be some charges to remain on the electrode surface. That as can be as a result of this fact

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ELECTRICAL CONDUCTIVITY Cathode side

–

Diffusion

h+

–

Electron – hole recombination e– e–

–

Electron injection

Polarization

–

e

–

–

–

+ +

k

Electron – cation recombination e–

Anode side

Charge carrier drift e– h+ A– K+

Trapped charge e

POLYMER-BASED COMPOSITES

Insulator

+ –

IN

A+

Electron – anion generation

–

+ ++ + ++ + ++

–

+

–

+

–

+

–

Maxwell Wagner polarization

+ Trapped charge h+

+ Trapping and field distortion

– +

–

e–

Electron – hole generation e– e– Electron absorption

–

A–

K+

+

+

e–

Electron – cation generation

Generation and recombination

+

e–

Electron – anion recombination

+

+

+

+

Fig. 4.5 A wide variety of charge behaviors in an insulator or at an insulatorelectrode interface affects the external response [12].

that charge may be injected at an electrode faster or slower than it is transported away in the insulation or the coulombic barrier prevents the charge carriers to leave the insulation as fast as it arrives. This may lead the charge to accumulate at an electrode [12].

4.7 Polarization and Dielectric Constant Under the influence of an electric field on the materials, a separation of the positive and negative charges in the material and alignment of electric charges occur. This is named polarization. The larger the dipole moment arms of this charge separation in the direction of a field and the larger the number of these dipoles, the higher the material’s dielectric permittivity and energy storage capacity. The average-induced dipole moment per molecule, Pav, will be the sum of all the contributions in terms of the local field (effective field) related to each individual molecule [12]. Pav 5 αe Eloc 1 αi Eloc 1 αd Eloc

(4.9)

αe, αi, αd are the electronic, ionic and dipolar polarizabilities, respectively. Eloc is the local field or the effective field at the site of an individual molecule that causes the individual polarization. Interfacial

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polarization occurs at the interfaces and cannot be put into an average polarization per molecule in bulk, therefore, it cannot be simply added to the total polarization as αij Eloc [12]. For simple dielectrics such as gases, it can be assumed that the local field is the same as the macroscopic field or Eloc 5 E (the applied field). Thus, it can be deduced: P 5 χe εe E 5 ðεr 2 1Þε0 E

(4.10)

In addition P 5 N.Pav, where N is the number of atoms or molecule per unit volume. Therefore it can be resulted that: εr 5 1 1 Nα=ε0

(4.11)

That α is the polarizability of the molecule [12].

4.8 Some Basic Parameters in Dielectric Specifications The electrical properties should be considered in selecting an insulation system may be categorized in as: a. b. c. d. e.

The electric strength. The relative permittivity. The dielectric loss or the loss tangent or tanδ). Dielectric constant. Electrical susceptibility.

The order of importance of these properties clearly depends on the application. However, the electric strength is commonly regarded as a dominant parameter [12]. Bellow some of other properties have been described. 1. Dielectric Strength and Breakdown Voltage The term of dielectric strength have two different definitions, depending to the material type [9]: • For an insulating material, dielectric strength is the maximum electric field strength that the material can withstand without breaking down, i.e., without failing of the insulating properties. This voltage can be assumed as the necessary energy for crossing the electron from the band gap.

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• For a dielectric material and electrodes, dielectric strength is the minimum electric field that produces breakdown. The heterogeneity in the composition, small ions or free electrons owing to thermal agitation or molecular imperfections, and surface asperities strongly can influence on tolerance limit of materials [11]. In this critical electric field, free electrons in valence band are accelerated to velocities that can liberate additional electrons during collisions with neutral atoms or molecules in a process called avalanche breakdown. It should be mentioned that the theoretical dielectric strength of a material is an intrinsic material property however, it is dependent on the configuration of the material or the electrodes with which the field is applied. At the breakdown voltage, breakdown occurs in nanosecond of time, an electrically conductive path is formed and a disruptive discharge through the material is made. If the material to be solid materials (its structure to be constant), a breakdown event severely degrades, or even destroys its insulating capability due to creating this conductive path [9]. In addition to the material composition, shape design of the insulator or dielectric can influence on this breakdown voltage. The effect of design can properly be observed in high-voltage power insulators that will be described later in detail. It will be stated that the humidity of atmosphere, atmospheric contamination, dust, snow, ice, . . . strongly affect the break down strength. The rate of increase of electric field also affects on this breakdown voltage. Dielectric films tend to exhibit greater dielectric strength than thicker samples of the same material. Because, there is a high interfacial resistance between film layers. For example, the dielectric strength of silicon dioxide films containing a few hundred nanometers to a few microns thickness is approximately 10 MV/cm. Therefore, it is preferred that multiple layers of thin dielectric films to be used where maximum practical dielectric strength is required, such as high-voltage capacitors and pulse transformers [9].

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2. Relative Permittivity Permittivity is a physical quantity that describes how an electric field affects and is affected by a dielectric medium or a material’s ability to transmit (or “permit”) an electric field. This property can be determined by the ability of a material to polarize in response to the field, and thereby reduction the field inside the material. It is directly related to electric susceptibility. For example, in a capacitor, an increased permittivity allows the same charge to be stored with a smaller electric field (and thus a smaller voltage), leading to an increased capacitance. Indeed, the composition and molecular polarity can affect on permittivity property [9]. 3. Dielectric Constant The dielectric constant (k) is the ability the material to store an electrical charge when subjected to electric field. This constant is a microstructure-sensitive property for materials. The value of the effectiveness of an insulator or dielectric materials is the maximum electric field it can support without any electrical breakdown. Power insulator can be made of ceramics, glass, and recently, it is made of polymer/fiber glass composites [2]. k is strongly dependent on Fermi level wideness. Fermi level is the band between valence band and conduction band. Fig. 4.6 compares the Fermi level width of insulators, metals, and semiconductor materials together [2]. In dielectric materials as a result of the applied field, the materials are polarized. It means the positive charges are displaced toward the field and negative charges shift in the opposite direction. If a dielectric is composed of weakly bonded molecules, those molecules become polarized, so that reorient and their symmetry axes align to the field [2]. 4. Electric Susceptibility The electric susceptibility indicates the degree of polarization of a dielectric material when subjected to an applied electric field and is a dimensionless proportionality constant. That is while the dielectric constant is related to the polarization in the material. The relation of dielectric polarization induced in a material with the applied electric field and the dielectric constant is as follows [2]: P 5 ðk 2 1Þε0 E ðfor linear dielectricsÞ

(4.12)

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Conduction band

Band gap

Conduction band

Band gap

Overlap of Conduction and valence band (Eg ≈ 0)

Eg ≈ 1.5 eV

Valence band

Valence band

Valence band

Filled band

Filled band

Filled band

Metal

Semiconductor

Insulator

Partially or fully filled, overlapping conduction band Filled valence band

Empty conduction band and filled valence band

Empty conduction band and filled valence band

Eg > 1.5 eV

Fig. 4.6 The electronic band structure of metals, semiconductors, and insulators.

where, E is the maximum withstanding voltage before breakdown or dielectric strength (V/m). For easily polarizable materials, both the dielectric constant (k) and the capacitance (C) are large. In other words, in these dielectrics a large quantity of charge can be stored. The equation suggests that polarization increases, atleast until all of the dipoles are aligned, as the voltage (expressed by the strength of the electric field) increases. The value of (k-1)is known as dielectric susceptibility (χe), i.e., zero for vacuum state [2,13]. 5. Dissipation Factor Dissipation factor (DF) and dielectric loss are the names of tan δ that is defined as the ratio of the equivalent series resistance of the capacitor to capacitive reactance. The DF is a value of loss-rate of energy of a mode of oscillation (mechanical, electrical, or electromechanical) in a dissipative system. It is in the front of quality factor, which represents the “quality” or durability of oscillation and is the inverse of tan δ [14].

4.9 Classification of Dielectrics Based on the trend of polarization via induced electric field, the dielectric materials can be broadly classified in to two classes, they are (i) linear dielectrics and (ii) nonlinear dielectrics.

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Polarization

Electric field

Electric field

(A)

89

(B)

Fig. 4.7 Electrical polarization of (A) Linear dielectric, (B) Nonlinear dielectric [4].

Linear dielectrics [4]: These are the class of materials whose polarization increases linearly with increase in the external electric filed and decreases to zero when the applied filed is zero (shown in Fig. 4.7A). These materials should do not have a saturation polarization (Psat) and coercive field (Ec) [4]. Nonlinear dielectrics: Nonlinear dielectrics are essentially crystalline materials, which can exhibit very large value of dielectric constant of the order 103 or more [4]. The relation of P and electric field (E) can be similar to the relation to stress and strain in mechanical deformation. In linear dielectrics, P is linearly related to E and k is constant as same as the relationship between stress and strain in elastomers or elastic deformation based on Hooke’s law. Here, in linear dielectrics, k (or χe) remains constant with changing E. In linear dielectrics such as BaTiO3, the dielectric constant changes with E and for nonlinear dielectrics (such as ferroelectrics) polarization has not a linear relation with E [2] (Fig.4.7).

References [1] wikipedia, Dielectric.2017. [2] Askeland DR, Fulay PP, Wright WJ. The Science and Engineering of Materials. Sixth ed United States of America: Cengage Learning; 2011. [3] S.S.Mohapatra, Breakdown in composite dielectrics in High-voltage engineeering. Report of a Lecture, FEEE, Ed. 201: Dhenkanal : Dept. Of EE, S.I.E.T. [4] Shodhganga. Introduction, motivation and thesis overview, [Ph.D. thesis]. India: Department of Physics, University of Hyderabad.

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[5] Ohsato H, Imaeda M. The quality factor of the microwave dielectric materials based on the crystal structure—as an example: the Ba6 2 3xR8 1 2xTi18O54 (R 5 rare earth) solid solutions. Materials Chemistry and Physics 2003;79(2):20812. [6] Jiles D. Introduction to the electronic properties of materials. Iowa State University, United States: Springer Science and Business media; 1994. [7] Database, P.P., http://polymerdatabase.com/polymer%20physics/ Dielectric%20Strength.html. 2015. [8] Gupta TK. Dielectric materials. Chapter 2 of the book Copper interconnect technology. Springer Science; 2009. [9] Encyclopedia, n.w., in Dielectric. http://www.newworldencyclopedia.org/ entry/Dielectric#Permittivity. [10] Material properties charts, in Ceramic industry magazine. [11] Licari JJ. Coating Materials for Electronic Applications Polymers, Processes, Reliability, Testing. William Andrew; 2003. [12] Nelson JK. Dielectric Polymer Nanocomposites. New York Dordrecht Heidelberg London: Springer; 2010. [13] Suckling, E.E., et al., Electricity physics, in Dielectrics, polarization, and electric dipole moment. 2017. [14] Wikipedia, Dissipation factor. 2017.