Chapter Linear Dielectric Materials | 10 | Electronic, Magnetic, and O

ABSTRACT

A dielectric material is typically a large-bandgap semiconductor (Eg ∼ >4 eV) that exhibits a high resistivity (ρ) The prefix dia means “through” in the Greek language The word dielectric refers to a material that normally does not allow electricity (electrons, ions, and so on) to pass through it There are special situations, for example, exposure to very high electric fields or changes in the composition or microstructure, which may lead to a dielectric material exhibiting semiconducting or metallic behavior However, when the term “dielectric material” is used, it is generally understood that the material is essentially a nonconductor of electricity An electrical insulator is a dielectric material that exhibits a high breakdown field

To better understand the behavior of nonconducting materials, let us first examine the concept of electrostatic induction and what is meant by the terms “free charge” and “bound charge.” First, consider a dielectric such as a typical ceramic or a plastic that has a net positive charge on its surface Now, assume that we bring a conductor B near this charged insulator A (Figure 71a) The electric field associated with the positively charged insulator A “pulls” the electrons toward it from conductor B This is also described as the atoms in conductor B being “polarized,” or affected by the presence of an electric field

The process of the development of a negative charge on conductor B is known as electrostatic induction In this case, the negative charge developed on conductor B is the “bound charge,” because this charge is bound by the electric field caused by the presence of the charged insulator A next to it

The creation of a bound negative charge on conductor B, in turn, creates a net positive charge on the other side of the conductor, because the conductor itself cannot have any net electric field within it If we connect this conductor B with a grounded wire, then electrons will flow from the ground to this conductor and make up for the positive charge (Figure 71b) The flow of electrons from the ground to the conductor can also be described as the flow of positive charge from conductor B to the ground Thus, the positive charge on conductor B is considered the “free charge” The word “free” in this context means that these charges are mobile, that is, they are free to move

If, instead of bringing in conductor B near the charged insulator A, we bring in another dielectric material D, then no induced charge is created on this dielectric material because no free carriers are available (Figure 71c) If we could look into this dielectric material D at an atomic scale, we would see that the electronic clouds are not perfectly symmetric around the nuclei of atoms Instead, the electronic clouds are tilted toward the positive charges in insulator A The presence of an electric field creates dipoles within the atoms of material D, which is polarized by the electric field emanating from dielectric A If we now move dielectric D away from dielectric A, we find that after some time, the dipole moments induced in this material fade away because of fluctuations in thermal energy

A capacitor is a device that stores electrical charge Consider two parallel conductive plates of area (A), separated by a distance d, and carrying charges of +Q and −Q; assume that there is a vacuum between these plates This is the basic structure of a parallel-plate capacitor

The charge (Q) on the plates creates a potential difference V We define the proportionality constant as the capacitance (C) Thus,

Q C V= × (71)

The SI unit of capacitance is a Farad (or coulombs per volt [C/V]) One Farad is a very large capacitance Some supercapacitors have capacitances in this range Most capacitors in microelectronics have a capacitance that is expressed in microfarads (10−6 F), nanofarads (10−9 F), picofarads (10−12 F), or femtofarads (10−15 F)

One of the laws of electrostatics is Gauss’s law, which states that the area integral of the electric field (E) over any closed surface is equal to the net charge (Q) enclosed in the surface divided by the permittivity (ε)

E d A

Q   × =∫ ε0

(72)

Gauss’s law and the fundamentals of many electrical and magnetic properties and phenomena are derived from Maxwell’s equations

We define the surface charge density (σ or σs) as the charge per unit area

σs =

A (73)

The SI unit of charge density is coulombs per square meter (C/m2) Note that the letter “C” represents the capacitance of a capacitor or charge in coulombs The charge density is usually expressed as, for example, microcoulombs per square centimeter (μC/cm2) or picofarads per square nanometer (pF/nm2)

The generation of a voltage (V) between two plates separated by a distance d is also represented using the electric field (E) as follows:

d =

(74)

We can rewrite Equation 71 for the capacitance (C) using Equations 73 and 74 as follows:

E d = = ×

× σs

(75)

This equation tells us that the capacitance of a capacitor, that is, its ability to hold charge, depends on geometric factors, namely, the areas (A) of the plates, and the distance (d) between them

We define the dielectric permittivity (ε) of the material between the plates as

ε σ= s

E (76)

Therefore, from Equations 75 and 76,

d = ε

(77)

We use the special symbol ε0 for permittivity of the vacuum or free space, which is equal to 885 × 10−12 F/m

We can write an expression for the capacitance of a capacitor filled with vacuum (C0) between the conductive plates as

E d

0 0= = =

σ εs,

(78)

The subscript “0” has been added to indicate that this equation refers to a capacitor filled with a vacuum Thus,

ε σ0

= s,0 E

(79)

We define the dielectric flux density (D) or dielectric displacement (D) as the total surface charge density

D = σs (710)

We will see in Section 73 that the dielectric flux density (D) can also be written as a sum of the free charge density and bound charge density (σb) The SI unit for dielectric displacement (D) is the same as that for charge density, that is, C/m2

Consider a capacitor in which the space between the two plates is filled with an ideal dielectric material (Figure 72) The term “ideal dielectric” means that when charge is stored in a capacitor, no energy is lost in the processes that lead to storage of the electrical charge In reality, this is not possible In Section 713, we will define the term “dielectric loss,” which represents the electrical energy that is wasted or used when charge-storage processes occur in a dielectric material These processes are known as polarization mechanisms

In one form of polarization, a tiny dipole is induced within an atom As shown in Figure 72, an applied atom without an electric field has a symmetric electronic cloud around the nucleus The centers of the positive and negative charges coincide, and there is no dipole moment (μ) However, when the same atom is exposed to an electric field, the electronic cloud becomes asymmetrical, that is, the electronic cloud of a polarized atom moves toward the positive end of the electric field The nucleus remains essentially at the same location Therefore, the centers of the positive and negative charges are now separated by a smaller distance (x) This process of creating or inducing a dipole in an atom is known as electronic polarization (Figure 73)

For a dipole with charges +q and −q separated by a distance x, the dipole moment is μ = q × x The international system (SI) unit of dipole moment is C · m We define one Debye (D) as being equal to 33356 × 10−30 C · m Please note that the symbol D is also commonly used for dielectric displacement

In a dielectric material exposed to an electric field, these induced atomic dipoles align in such a way that all the negative ends of the dipole line up near the positively charged plate (Figure 73b) This process means that some of the surface charges on the plates, which were originally free when there was a vacuum between the plates, will now become “bound” to the charges inside the dielectric material When the capacitor is filled with a vacuum, all of the charges on the plates are free The

original free surface charge density (σs) with a vacuum-filled capacitor is now reduced to (σs − σb), where σb is the bound charge density (see Section 73) This reduced charge density means that the voltage (V) across the plates is also reduced Therefore, the new electric field (E) for a capacitor filled with a dielectric is smaller compared to E0 A lower electric field is needed to maintain the same charge, because part of the surface charge is now held or bound by the dielectric material

The ratio E0/E-the electric fields existing in a capacitor filled with a vacuum and a dielectric material, respectively-is defined as the dielectric constant (k) or relative dielectric permittivity (εr)

εr = =k

E 0

(711)

Now, because the total charge on the plates is the same,

Q C V C V= × = ×0 0

The distance between the parallel plates (d) is the same Therefore, Q C E C E E E C C= × = × =0 0 0 0or ( / ) ( / ) We can write Equation 711 as

ε ε

εr = = =k C

C0 0 (712)

Thus, the dielectric constant (k) is also defined as the ratio of the capacitance of a capacitor filled with a dielectric to that of an identical capacitor filled with a vacuum One advantage of defining a dielectric constant (k) or relative dielectric permittivity (εr) as a dimensionless number is

that it becomes easy to compare the abilities of different materials to store charges For example, the dielectric constant of the vacuum becomes 1 The dielectric constant of silicon (Si), alumina (Al2O3), and polyethylene are approximately 11, 99, and 22, respectively (Table 71)

Thus, the capacitance of a parallel capacitor containing a dielectric material with a dielectric constant (k) is given by modifying Equations 78 and 712 as follows:

C k

d = × ×ε0

(713)

If the dielectric material consists of atoms, ions, or molecules that are more able to be polarized, that is, if they are easily influenced by the applied electric field, then the dielectric constant (k) will be higher In Section 712, we will examine how the dielectric constant (k) changes with electrical frequency ( f ), temperature, composition, and the microstructure of a material Materials with a high dielectric constant are useful for making capacitors Unlike transistors, diodes, solar cells, and so on, capacitors are considered “passive” components One of the goals of capacitor manufacturers is to minimize the overall size of the capacitor while enhancing the total capacitance This is usually achieved by arranging multiple, thin layers of dielectrics in parallel (Figure 74) This device is known as a multilayer capacitor (MLC)

The volumetric efficiency of a single-layer capacitor with area A and thickness d is given by:

ε0 × ×





  





  

k A

d A d

TABLE 7.1 Approximate Room-Temperature Dielectric Constants (or Ranges) for Some Dielectric Materials or Classes of Materials (Frequency ~1 kHz-1 MHz)

volumetric efficiency of a single layer = ×ε0 k

d 2  

 

(714)

Examples 71 and 72 illustrate how the volumetric efficiency is enhanced using multiple layers of a dielectric connected in parallel, instead of using a single, thick layer An analogy to this is that we stay warmer during winter by wearing multiple layers of clothing, rather than wearing one very thick jacket

SOLUTION a. When capacitors are connected in parallel (Figure 7.5), the voltage across each capacitor is

SOLUTION a. From Equation 7.13,

Example 73 illustrates some real-world situations in which there is a decrease in the capacitance because the device structure can transform into capacitors in a series

We define dielectric polarization (P) as the magnitude of the bound charge density (σb) The application or presence of an electric field (the cause) leads to dielectric polarization (the effect) The situation is very similar to that encountered while discussing the mechanical properties of materials

The application of a stress (the cause) leads to development of a strain (the effect) The stress and strain are related by Young’s modulus In this case, the electric field (E) applied and the polarization (P) created are related by the dielectric constant (k)

Assume that the dielectric polarization is caused by N number of small (atomic scale) dipoles, each comprised of two charges (+qd and −qd) separated by a distance x (Figure 72) The dielectric polarization (P) is equal to the total dipole moment per unit volume of the material This is another definition of dielectric polarization The mechanisms by which such dipoles are created in a material are discussed in Section 75

Assume that the concentration of atoms or molecules in a given dielectric material is N; if each atom or molecule is polarized, then the value of dielectric polarization is

P N q x= = × < × >σb d (720)

If σ is the total charge density for a capacitor, then the other portion of the charge density, that is, (σ − σb), remains the free charge This creates a dielectric flux density (D0), as in the case of a capacitor filled with a vacuum From Gauss’s law,

D E0 0= ×ε (721)

Thus, the total dielectric flux density (D) for a capacitor filled with a dielectric material originates from two sources: the first source is the bound charge density (σb) associated with the polarization (P) in the dielectric material; and the other is the free charge density (σ − σb)

Therefore,

D P E= + ×( )ε0 (722)

We can also rewrite the dielectric displacement as D E= ×ε Therefore, we get

P E= = − ×σ ε εb ( )0 (723)

If μ is the average dipole moment of the atomic dipoles created in a dielectric and N is the number of such dipoles per unit volume (ie, the concentration), then we can also write the polarization as follows:

P N= ×( )µ (724)

Polarizability describes the ability of an atom, ion, or molecule to create an induced dipole moment in response to the applied electric field The average dipole moment (μ) of an atom can be written as the product of the polarizability of an atom (α) and the local electric field (E) that an atom within the material experiences

µ α= ×( )E (725)

πε volume volume polarizability (cm= =3

4 )

1 2 2× ( )⋅ ⋅ ⋅−α C V m or F m

(726)

If the volume polarizability is expressed in Å3 (as is often done in the case of atoms or ions), we use the following equation:

πε αvolume in Å C V m or F m3

4 = × ( )⋅ ⋅ ⋅−

(727)

The factors 106 and 1030 are used in these equations because 1 m3 = 106 cm3 and 1 m3 = 1030 Å3 For now, let us assume that the electric field that an atom within a material experiences is the

same as the applied field (E) From Equations 724 and 725, we get

P N E= × ×( )α (728)

From Equations 728 and 723, we get

α ε ε= −( )0

N (729)

We can rewrite this equation as shown below:

ε ε

ε α εr

= = +  

 

  

  

1 N

(730)

Equation 730 is important because it links the dielectric constant (k or εr) of a material to the polarizability of the atoms (α) from which it is made and also to the concentration of dipoles (N)

We also use another parameter called the dielectric susceptibility (χe) to describe the relationship between polarization (the effect) and the electric field (the cause):

P E= χ εe 0 (731)

The subscript e in χe distinguishes the dielectric susceptibility from the magnetic susceptibility (χm), which is defined in Chapter 9 The dielectric susceptibility (χe) describes how susceptible or polarizable a material is, that is, how easily the atoms or molecules in the material are polarized by the presence of an electric field Comparing Equations 723 and 731,

χ εe r= −( ) = −( )k 1 1 (732) Also, from Equations 731 and 732,

P E= −( )ε εr 1 0 (733) Another way to express the dielectric susceptibility is as follows:

χ εe r

bound surface charge density

f = −( ) = =1

D ree surface charge density (734)

The dielectric susceptibility (χe), which is another way to express the dielectric constant (k), depends on the composition of the material We can show from Equations 729 and 731 that

χ α

εe = N

0 (735)

From Equation 735, we can see that the dielectric susceptibility (χe) of the vacuum is zero or the dielectric constant (k) is 1 This is expected because there are no atoms or molecules in a vacuum, so N = 0 The more polarizable the atoms, ions, or molecules in the material, the higher is the bound charge density, and the higher is the dielectric susceptibility (χe) or dielectric constant (k) In Section 75, we will see that several different polarization mechanisms exist for a material (Figure 78)

The dielectric susceptibility values (χe) of silicon, Al2O3, and polyethylene are approximately 10, 89, and 12, respectively, because the dielectric constants are approximately 11, 99, and 22, respectively (Table 71)

When we examine the effect of an externally applied electric field (E) on a dielectric material, we need to account for the electric field that exists inside a polarized material For a solid or liquid exposed to an electric field (E), the actual field experienced by the atoms, molecules, or ions inside the material is different and is referred to as the internal electric field or local electric field (Elocal)

In general, the greater the polar nature of the material, the higher the strength of the dipoles induced, and the larger the local electric field Furthermore, the magnitude of the electric field experienced by the atoms, ions, or molecules inside a material depends on their arrangement

For a cubic-structured isotropic material, a liquid, or an amorphous material, the local electric field is given by

E E Plocal = +

3 0ε (736)

This expression is known as the local field approximation Thus, we can rewrite Equation 728 by substituting Elocal for E as

P N E= × ×( )α local

N P

E P

× = = +

  

ε local

3 0

(737)

Note that E is the applied electric field Therefore,

N E

E E× = −

+ − α ε εε ε

( ) ( )

Eliminating E, dividing by ε0, and rewriting ε/ε0 = εr, we get

ε ε ε

− +

= ×1 2

3 0 ( )N

(738)

Equation 738 is also known as the Clausius-Mossotti equation It describes the relationship between the dielectric constant (k or εr), a macroscopic property, and the concentration of polarizable species (N) and their polarizability (α), which are microscopic properties We have used the local field approximation (Equation 736), which is valid for either amorphous materials or cubic-structured materials Strictly speaking, the Clausius-Mossotti equation should be used only for these types of materials In some materials, molecules have a permanent dipole moment (such as water) Ferroelectrics develop spontaneous polarization because of the rearrangement of ions (Section 711) For such polar materials, the internal electric field is not given by Equation 737 As a result, the Clausius-Mossotti equation (Equation 738) cannot be used for such polar materials

In Equation 738, N is the concentration of dipoles per unit volume (number of molecules/m3) If we assume that each molecule or atom becomes a dipole, then N is related to the NAvogadro (6023 × 1023 molecules/mol), density (ρ in kg/m3), and molecular weight (M in kg/mol) as follows:

N N

M = ×Avagadro

(739)

Substituting for N in Equation 738, we get another form of the Clausius-Mossotti equation:

ε ε ρ

α ε

Avagardro− +

 

 

= ×1

2 3 0

M N

(740)

We can verify that this equation is dimensionally balanced The units are α in F · m2, ε0 in F/m, and Avogadro’s number in number per mole Thus, the unit on the right-hand side of the above expression is m3/mol The unit on the left-hand side is also m3/mol

We can rewrite Equation 740 to get the polarization per mole or molar polarization (Pm), defined as follows:

ε ε

m− +

 

 

= ×1 2

M (741)

The units for molar polarization (Pm) are m3/mol and cm3/mol As we will see in Section 75 and Figure 78, there are five polarization mechanisms-electronic,

ionic, dipolar, interfacial, and spontaneous (ferroelectric) The total polarizability due to ionic and electronic polarization is additive, because these polarizations occur throughout the volume of a material Thus, we can write the total polarizability as

α α α= +e ionic (742)

where, αe and αionic are the electronic and ionic polarizabilities of the atoms, respectively Electronic and ionic polarization are defined in Sections 76 and 77, respectively We can rewrite the Clausius-Mossotti equation (Equation 738) to separate out the ionic and

electronic polarization effects as follows:

ε ε ε

α αr r

= ×( ) + ×( ) 12 1

3 0 N N

(743)

In Equation 743, we cannot incorporate the effects of dipolar (Section 79), interfacial (Section 710), and spontaneous or ferroelectric polarization (Section 711), because the effects that these polarizations have on the local electric field (Elocal) are complex They cannot be described by Equation 736; thus, the Clausius-Mossotti equation cannot generally be used for polar materials, such as water, or ferroelectric compositions of materials, such as the tetragonal form of BaTiO3 An exception to this is a situation where the Clausius-Mossotti equation may be used for polar materials if the electrical frequency ( f ) of the applied field is too high for these polarization mechanisms to exist

We will now examine the different ways in which atoms, ions, and molecules in a material can be polarized by an electric field (Table 72 and Figure 78)

These polarization mechanisms are shown schematically in Figure 78 and are discussed in the following sections

All materials contain atoms (in the form of neutral atoms, ions, or molecules) When subjected to an electric field, each atom is polarized, in that the center of the electronic charge shows a slight shift toward the positively charged electrode

This very slight elastic displacement of the electronic cloud (a few parts per million of the atomic radius), shown as δ or x, occurs very rapidly with reference to the nucleus (∼10−14 seconds; Figure 79) This means that even if the electric field changes its polarity ∼1014 times a second, the electronic polarization process can still follow this rapid change in the direction of the electric field The electric field associated with visible light (which is an electromagnetic wave) oscillates with a frequency of ∼1014 Hz This field interacts with dielectric materials and causes electronic polarization Thus, electronic polarization is related to the optical properties of materials, such as the refractive index This is why electronic polarization is also known as optical polarization

Because the electrons of the outermost shell are the ones most susceptible to the electric field applied, the electronic polarizability (αe) of an atom or an ion depends primarily on its size and the number of electrons in its outermost shell The larger the atom or ion, the farther the electrons are from the nucleus Therefore, the electron clouds surrounding larger atoms or ions are more susceptible to electric fields and are more polarizable Thus, larger atoms or ions have a higher electronic polarizability The extent to which an atom or an ion is polarized via this mechanism is measured by the electronic polarizability (αe)

Consider the nucleus of a monoatomic element (such as argon [Ar]) with radius R When an electric field is applied, the electronic cloud is displaced by a distance δ with respect to the nucleus (Figure 79)

TABLE 7.2 Summary of Polarization Mechanisms in Dielectrics

A dipole is created between the charge q1 of the nucleus (Zq; where Z is the number of electrons surrounding the nucleus) and the charge q2, which is the part of the charge in the electron cloud that no longer surrounds the nucleus because it is displaced This charge q2 is contained in a sphere of radius δ and is given by

R Zq

4 3

4 3 = − = −[( / ) ]

[( / ) ]

πδ π

(744)

Thus, the Coulombic force of attraction between the nuclear charge q1 and the negative charge q2, not shielded by the nucleus and separated by distance δ, is given by Coulomb’s law:

q q= ×1 2 0

24πε δ (745)

We can rewrite this as

Zq Zq R Zq

R =

× −   = −

( ) ( )

πε δ δ

πε

0 34 4

(746)

The magnitude of this attractive Coulombic force is balanced by the force (Fd) that causes the displacement

F Zq Ed = ×( ) (747)

Thus, equating the magnitude of the force that causes the displacement of the electronic cloud and the Coulombic restoring force, we get

( ) ( )

R Zq E

δ πε

= ×

(748)

Solving for displacement (δ), we get

δ πε= 4 0

3R E

Zq (749)

The dipole moment (μ) caused by the electronic polarization is given by

µ δ= ×( )Zq (750)

Substituting for δ from Equation 749,

µ πε= 4 0 3R E

Recalling Equation 725, we can write the dipole moment (μ) as

µ α= ×e E (751)

where αe is the electronic polarizability of an atom or an ion Note that the term “electronic polarizability” does not describe the polarizability of an electron It describes the polarizability of an atom or an ion

Comparing Equations 751 and 754, we get

α πε εe a= =4 30 3 0R V (752)

In Equation 752, Va is the volume of the atom or ion being polarized Because the unit of permittivity is F/m, and the unit of volume is m3, the unit of electronic polarizability (αe) is F · m2

The electronic polarizability values for atoms of different elements are shown in Figure 710 These values are the volume polarizabilities To convert them into SI units, they should be multiplied by 4πε0 × 10−30 (Equation 727)

πε α( )volume polarizability in Å i3

4 = × n C V m or F m⋅ ⋅ ⋅−( )1 2 2

(753)

Example 74 will give us an idea of the magnitudes of the electronic polarizability of atoms and the electron-cloud displacement distances

We can use Bohr’s model to calculate the electronic polarizability (αe) of an atom Using this approach, the electronic polarizability under a static electric field is given by the equation

ωe static, ( )= ×Z q

(754)

where Z is the number of electrons orbiting the nucleus, q is the electronic charge, m is the electron mass, and ω0 is the natural oscillation frequency of the center of the mass of the electron cloud around the nucleus The static electronic polarizability represents the value of electronic polarizability when the electric field causing the polarization is not time-dependent

The electronic polarizability (αe) depends on the frequency of the electric field (ω) as follows:

ω ω βωe =

− + ( )

( )

m j

(755)

where j is the imaginary number and β is a constant that is related to the damping force attempting to pull the electron cloud back toward the nucleus This equation is derived using a classical mechanics approach A quantum mechanical-based approach yields a different equation, but the trend in the change in electronic polarizability as a function of frequency is similar Example 75 illustrates a calculation of the value of static electronic polarizability

SOLUTION For Bohr’s model, using Equation 7.54,

Ions and molecules have electronic polarizability similar to neutral atoms The electronic polarizability of an ion is nearly equal to that of an atom, with the same number of electrons Thus, the electronic polarizability of sodium ions (Na+) (02 × 10−40 F · m2) is comparable to the electronic polarizability of neon (Ne) atoms From Equation 752, we can expect the larger atoms or ions to have a larger electronic polarizability Because cations are typically smaller than anions, the electronic polarizability of anions is generally larger than that of cations Similarly, larger ions such as lead (Pb2+) have a larger electronic polarizability Many real-world technologies make use of these effects, such as the development of lead crystal and optical fibers

Electronic polarization is also linked to optical properties such as the index refractive (see Section 712) Molecules are comprised of several atoms and show electronic polarizability In general, the polarizability of molecules is larger because they contain more electrons

Many ceramic dielectrics exhibit mixed ionic and covalent bonding In ceramics with ionic bonds, each ion undergoes electronic polarization In addition to this, ionic solids exhibit ionic polarization, also known as atomic polarization or vibrational polarization In this mechanism of polarization, the ions themselves are displaced in response to the electric field experienced by the solid, creating a net dipole moment per ion (pav) Consider pairs of ions in an ionic solid such as sodium chloride (NaCl; Figure 711a) Assume that these ions have an equilibrium separation distance of a This is the average separation distance between an anion and a cation Ions in any material are not stationary They vibrate around their mean equilibrium positions; these vibrations of ions or atoms are known as phonons

When an electric field (E) is applied (Figure 711b), the electronic polarization of both cations and anions is established almost instantaneously (in ∼10−14 seconds) This effect, that is, distortion of the electronic clouds for both anions and cations, is not shown in Figure 711 Since anions are bigger than cations because of the extra electrons present, anions typically show higher electronic polarizability than cations

In addition to this electronic polarization, a positively charged cation moves toward the negative end of the electric field Similarly, anions move closer to the positive end of the electric field Figure 711b shows the displacements of ions; cations 1 and 3 move to the right, that is, toward the negative end of the electric field Anions 2 and 4 are displaced toward the positive end of the electric field This means that the separation distance (Δx) between cation 1 and anion 2 is now reduced, compared to their separation (a) without the electric field The separation Δx between cation 3 and anion 2 is now larger than their equilibrium separation distance (a) The extent to which these ions are displaced also depends on the magnitude of the restoring forces imposed by other neighboring ions For example, as cation 3 moves toward anion 4 (Figure 711b), anion 2 tries to pull cation 3 back, and the cation to the right of anion 4 (not shown) repels it

Such asymmetric displacements of ions in response to the presence of an electric field create a dipole moment; this effect is known as ionic polarization The magnitude of ionic displacements encountered in ionic polarization is a fraction of an angstrom (Å) Because ions have a larger inertia than electrons, these movements are a bit sluggish, occurring in about 10−13 seconds When the polarity of the electric field is reversed (Figure 711c), the directions of displacement are also reversed If we have an alternating current (AC) electric field, then the ions move back and forth as long as the electric field does not switch too rapidly, as shown in Figures 711b and c If the frequency ( f ) of the switching field is greater than ∼1013 Hz, that is, if the field switches in less than 10−13 seconds, the ions cannot follow the changes in the electric field direction In other words, the ionic polarization mechanism is seen in ionic solids for frequencies up to ∼1013 Hz (Table 72) If the frequency is greater than ∼1013 Hz, this polarization mechanism “drops out” and does not contribute to the total dielectric polarization (P) induced in the dielectric

Under static electric fields, the magnitude of ionic polarizability (αi) is given by

ωi r (for static electric fields= ×

× ( )Z q

)

(756)

where Z is the valence of the ion (not the number of electrons surrounding the nucleus, which Z represents in electronic polarization equations), q is the electronic charge, Mr is the reduced mass of the ion, and ω0 is the natural frequency of oscillation for a given ion

For AC fields with a frequency ω = 2πf, where f is the electrical frequency in Hz, the magnitude of the ionic polarizability (αi) is frequency-dependent and given by

ω ω βωi r = ×

− + ( )

[ ( ) ]

Z q

M j

(757)

where ω is the frequency of the electric field (in rad/s), j is the imaginary number, and β is a coefficient related to the damping force that tries to bring the displaced ion back to its original position

The details of the treatments for deriving these equations are beyond the scope of this book However, it is important to recognize that the dielectric polarization mechanism is effective only up to a certain frequency (∼1013 Hz)

In general, the magnitude of ionic polarizability (αi) is about 10 or more times larger than the electronic polarizability (αe) This is why most solids with considerable ionic bonding character exhibit much higher dielectric constants (Table 71) Also, note that in addition to ionic polarization, each ion undergoes electronic polarization Thus, the dielectric constant (k) of ionic solids results

from both electronic and ionic polarizations However, the contributions from ionic polarization tend to be dominant, especially at lower frequencies

If μav is the average dipole moment induced by the ionic polarization per ion, then we can write this dipole moment as follows:

µ αi i local= × E

where Elocal is the electric field experienced by the ion As mentioned in Section 74, this is the local electric field (Elocal) and is different from the applied electric field (E) The polarization (P) induced in an ionic solid with Ni ions per unit volume is given by the following equation:

P N N E= × = × ×i av i i localµ α (758)

Recall from the Clausius-Mossotti equation (Equation 743) that the dielectric constant (k or εr) is linked to the different polarizabilities of ions For an ionic solid with no permanent dipoles, we have contributions from both ionic and electronic polarizations Note that this equation applies only to amorphous or cubic structures and nonpolar materials

In Section 78, we will discuss an approach that is useful in predicting the dielectric constants of nonpolar materials from the values of the total (ie, electronic and ionic) polarizability of ions

Shannon measured the dielectric constants of several materials and back-calculated their ionic polarizabilities (Shannon 1993) The frequency range for the dielectric-constant measurements was 1 kHz-10 MHz Thus, both the electronic and ionic polarization mechanisms contributed to the dielectric constant measured

Shannon used experimentally determined values of dielectric constants to first estimate the polarizability of ions or simple compounds Then, he used these values to estimate the total dielectric polarizability αDT( ) of other compounds through the additive nature of ionic and electronic polarizabilities This calculation requires knowledge of the molecular weight and unit cell volume (ie, the theoretical densities) of the compound whose dielectric constant is to be estimated For example, one can measure the dielectric constants of fully dense samples of magnesium oxide (MgO) and Al2O3 and estimate the polarizabilities of Al3+, Mg2+, and O2− ions We can use these polarizability values to calculate the total dielectric polarizability αDT( ) , and hence the dielectric constant of another compound such as magnesium aluminate (MgAl2O4)

The total dielectric polarizability of MgAl2O4 can be expressed as follows:

α α α αMgAl O Mg Al O(2+) (3+) (2-)2 4 2 4= + + (759)

Following this, we can use the value of total polarizability for MgAl2O4 to estimate its dielectric constant by using the following form of the Clausius-Mossotti equation:

ε πα

+ −

3 8

3 4

(761)

where Vm is the molar volume of the compound whose dielectric constant or αDT is being estimated

Shannon also used the following modified forms of the Clausius-Mossotti equation:

π ε εD

T V= −

+  

 

r (762)

In Equations 760 through 762, αDT is the total dielectric polarizability of a material and is commonly expressed as Å3

Shannon’s approach for predicting the dielectric constant is useful, but it has some limitations For many compounds, the calculated values of dielectric constants using Shannon’s approach are very different from the measured values The polarizability of the oxygen ion (O2−), as estimated by Shannon (201 Å3; Figure 712), is lower than that used by other researchers (237 Å3) This leads to the prediction of lower dielectric constants for some oxides, for example, Al2O3 In many other materials exhibiting ferroelectric and piezoelectric behavior or for materials containing compressed or rattling ions, mobile ions, and impurities, the calculated and measured values of the dielectric constants do not match well This can be due to the ionic or electronic conductivity of the material, the presence of interfacial polarization, the presence of polar molecules such as water (H2O) or carbon dioxide (CO2), and the presence of other dipolar impurities Thus, Shannon’s approach cannot be used to calculate the dielectric constants of ferroelectric materials (Section 711)

Despite these limitations, Shannon’s approach serves as a powerful guide for the experimental development of new formulations of materials with high dielectric constants The use of Shannon’s approach is illustrated in Examples 76 and 77

SOLUTION

Molecules known as polar molecules have a permanent dipole moment An example of a wellknown polar material is H2O (water) The polar molecules begin to experience torque when exposed to an external electric field and orient themselves along the electric field This, in turn, causes an increase in the bound charge density (σb) and leads to an increase in polarization (P) The resulting polarization is known as dipolar polarization or orientational polarization. Orientational or dipolar polarization is the mechanism responsible for the relatively high dielectric constant of H2O (k∼78; Figure 713)

The main feature that distinguishes the orientational polarization mechanism from other mechanisms is the presence of permanent dipoles This mechanism of dipolar polarization is seen only in materials that have molecules with a permanent dipole moment Materials in which molecules develop a net polarization or have a permanent or built-in dipole moment are known as polar materials or polar dielectrics Polar materials in which polarization appears spontaneously, even without an electric field, are known as ferroelectrics (Chapter 8) They do not contain molecules with a permanent dipole moment

If each permanent dipole had a dipole moment of μ and if the concentration of such dipoles was N, then the maximum polarization that can be caused by this mechanism alone is N × μ Not all dipoles can remain aligned with the applied electric field, because thermal energy tries to randomize their orientations Thus, this polarization mechanism begins to fade away with increasing temperature At substantially high temperatures, the orientations of dipoles with respect to the applied electric field become completely randomized The net polarization begins to decrease, and this polarization mechanism stops The dipolar polarizability (αd) associated with this mechanism is given by

α µd

= 1 3

K (763)

where μ is the permanent dipole moment of the molecules, KB is the Boltzmann’s constant, and T is the temperature

Dipole moments associated with polar molecules are typically very large compared to those induced by the polarization of atoms or the displacements of ions Dipolar or orientational polarizability (αd) values and the resultant dielectric constants are therefore large for polar materials

Dipolar polarization is typically seen in polar liquids, gases, or vapors (eg, H2O, alcohol, and hydrochloric acid [HCl]) Unlike in their vapor form, molecules with permanent dipoles are not free to rotate in polar solids (eg, ice, instead of water vapor), even if they are present This means that the effect of orientational polarization in polar solids is smaller compared to that in liquids and gases or vapors

For example, nitrobenzene (C6H5CO3) is a polar liquid with k∼35 (Figure 714) The dielectric constant of liquid C6H5CO3 is expected to decrease with increasing temperatures (Equation 763) When liquid C6H5CO3 freezes into a solid, the dipoles are present but are frozen and unable to rotate This is why the dielectric constant of C6H5CO3 decreases to about 3 (Figure 714a) This lower value for the dielectric constant reflects the smaller extent of electronic and ionic polarization, compared to the extent of orientational polarization and ignoring the effects of the differences in the molar volumes of the solid and liquid phases

In some materials (such as hydrogen sulfide [H2S]), the dielectric constant continues to increase with decreasing temperatures, even below the freezing or melting temperature (Tm) This continues up to the critical temperature T0, below which the dipoles cannot rotate and the orientational polarization mechanism ceases (Figure 714b)

Some dielectrics contain relatively mobile ions (eg, H+, Li+, and K+) At high temperatures, these ions can drift under the influence of an electric field The movement of such charge carriers is eventually impeded by the existence of interfaces (such as grain boundaries) in a material or a device This can create a buildup of double-layer-like capacitors at interfaces, such as grain boundaries, in a polycrystalline material or material-electrode interfaces The increase in polarization due to such movements of mobile ions in a material and the creation of polarization at the interfaces is known as space-charge polarization, interfacial polarization, or Maxwell-Wagner (M-W) or MaxwellWagner-Sillars (M-W-S) polarization Trapping electrically charged ions, electrons, and holes at the interfaces is the essential process behind interfacial polarization (Figure 715)

This polarization mechanism differs from other mechanisms we have discussed (Figure 78) First, the polarization mechanism is usually more prominent at higher temperatures This is because the rate of the diffusion process, which often occurs through atoms or ions jumping or hopping from one location to another, increases exponentially with increasing temperature (Chapter 2) Second, the diffusion of atoms or ions is relatively slow compared to that of electrons and holes Thus, this polarization mechanism is usually operated under static (DC) or AC fields, in which the electrical frequency ( f ) is small (a few mHz to several Hz) If the electrical frequency ( f ) is too high, the otherwise mobile ions cannot rapidly follow this frequency The interfacial polarization mechanism then ceases to exist Many silicate glasses and crystalline ceramic materials containing mobile ions

(eg, lithium niobate [LiNbO3], lithium tantalate [LiTaO3], and lithium cobalt oxide, [LiCoO2]) exhibit this polarization mechanism This polarization mechanism is also quite common at the liquid electrolyte-electrode interfaces because diffusion in liquids occurs rather readily This polarization mechanism is commonly used in many electrochemical reactions encountered during the operation of supercapacitors, batteries, fuel cells, and similar devices

The existence of interfacial polarization is often considered a strong possibility whenever unusually high dielectric constants are seen, especially at high temperatures and low frequencies For example, the data for the dielectric constant for a calcium-copper-titanium oxide (CCTO) ceramic are shown in Figure 716

A ferroelectric material is defined as a material that exhibits spontaneous and reversible polarization (Chapter 8) This polarization is typically very large in magnitude compared to other polarization mechanisms, such as electronic and ionic polarizations

A prototypical example of a ferroelectric material is the tetragonal polymorph of BaTiO3 The term “polymorph” means the particular crystal structure of a material Consider the two polymorphs of BaTiO3-one is cubic, and the other is tetragonal The tetragonal form of BaTiO3 is also

known as the pseudocubic form The cation/anion ratio for the tetragonal polymorph of BaTiO3 at room temperature (300 K) is ∼101 For the cubic polymorph, which is stable at higher temperatures, the c/a ratio is 100 The lattice constant is ∼4 Å Thus, the physical difference in the dimensions of the unit cells is actually very small However, the differences in the electrical properties between the cubic and tetragonal forms of BaTiO3 are significant

For the typical single-crystal or polycrystalline BaTiO3, the centrosymmetric cubic phase is stable at temperatures above ∼130°C The temperature at which a ferroelectric material transforms into a centrosymmetric paraelectric form is known as the Curie temperature. In cubic BaTiO3, the titanium (Ti) ion “rattles” very rapidly around several equivalent but off-center positions present around the cube center For each of these off-center positions of the titanium ion, the unit cell structure has a dipole moment Thus, at any given time, the time-averaged position of the titanium ion appears to be exactly at the cube center As a result, the cubic phase of BaTiO3 has no net dipole moment from the viewpoint of electrical properties From a structural viewpoint (eg, while using x-ray diffraction) the crystal structure appears cubic and is centrosymmetric In cubic BaTiO3, all the dipole moments that are associated with the barium (Ba2+), titanium (Ti4+), and oxygen (O2−) ions cancel one another out This high-temperature phase, derived from an originally ferroelectric parent phase that now has no dipole moment per unit cell, is known as the paraelectric phase

When the temperature approaches the Curie temperature (Tc∼130°C for BaTiO3), the titanium ions begin to undergo other very small displacements At temperatures below Tc, barium ions are displaced by a distance of ∼6 pm (1 pm = 10−12 m) Titanium ions are displaced in the same direction by ∼11 pm Oxygen ions (O2−) are displaced by ∼3 pm After these displacements occur, the unit cell becomes tetragonal The tetragonal structure is not centrosymmetric It is a polar structure, that is, the tetragonal unit cell of BaTiO3 has a net polarization (Figure 717)

The polarization mechanisms require displacements of ions, electronic clouds, dipoles, and so on (Table 72) These displacements are small; nevertheless, they require a small but finite amount of time Consider a covalently bonded material such as silicon, where the electronic polarization is established quickly (∼10−14 seconds) Now, consider changing the polarity of the applied electric field The electronic clouds shift, and the induced dipoles realign with the new field direction within another ∼10−14 seconds The induced dipoles can align rapidly back and forth in an alternating electric field even if the electrical field switches at a frequency of, for example, 1 MHz A frequency

( f ) of 1 MHz means that the field switches back and forth 106 times per second, or in one microsecond (μs; 1 μs = 10−6 seconds) Under a static (ie, f = 0) field or an electric field oscillating with a frequency of 106 Hz, we expect the electronic polarization process to contribute to the dielectric constant (k) of silicon This continues to very high frequencies, ranging up to 1014 Hz, because electronic polarization is the only mechanism of polarization that survives up to such high frequencies Thus, the dielectric constants of silicon and other covalently bonded solids (such as germanium [Ge] and diamond) are expected to remain constant with frequencies up to the range of ∼1014 Hz

Now, consider a material in which the bonding has some ionic character, such as SiO2 In this material, we expect both the ionic and electronic polarization mechanisms to contribute to its dielectric constant (k) This will be true as long as the ionic and electronic polarization mechanisms can follow the alternating electric fields The ionic polarization mechanism is slower compared to that of electronic polarization because it involves the displacement of ions Thus, up to a frequency of ∼1012-1013 Hz, both electronic and ionic polarization mechanisms contribute to the dielectric constant of SiO2 At higher frequencies, the ions cannot follow the back-and-forth switching of the polarity of the electric field As a result, the ionic polarization mechanism stops, lowering the dielectric constant The electronic polarization mechanism survives up to ∼1014 Hz and continues to contribute to the dielectric constant Thus, we expect the dielectric constant of SiO2 to become smaller (or for it to relax) as we proceed from low to high frequencies We can expect this trend for any material with more than one polarization mechanism (Figure 718)

The lowering of the dielectric constant with increasing frequency is known as dielectric relaxation or frequency dispersion This low-frequency dielectric constant is also known as the static dielectric constant (ks), although the value is not necessarily measured under DC fields The highfrequency dielectric constant is designated as k∞ This is also known as the optical frequency dielectric constant, and it is the value of dielectric constant when only the electronic (optical) polarization mechanism remains

In practice, dielectric materials with multiple polarization mechanisms show a decrease in the dielectric constant (k) However, the decrease is rather steady and not as abrupt as that shown in Figure 718, because with multiple types of ions or atoms, the polarization mechanisms do not

stop at a particular frequency but rather over a range of frequencies, even for the same type of polarization

As an example, Figure 719 shows the lowering of the dielectric constant for lithium-iron-nickelvanadium oxide ceramics The data are shown for temperatures ranging from 23°C to 300°C

At any given temperature, the dielectric constant decreases with increasing frequency The lower-frequency dielectric constants increase with increasing temperature This is very much an indication of interfacial polarization, or Maxwell-Wagner polarization Its presence is not surprising because this material contains relatively mobile lithium ions

The only polarization mechanism that does not cease to exist at high frequencies is electronic polarization The high frequencies at which only the electronic polarization mechanism survives correspond to a wavelength (λ) of light (Figure 720)

Therefore, electronic polarization is also known as optical polarization (see Section 76) We can show that the high-frequency dielectric constant (k∞) of a material is equal to the square of its refractive index (n):

k n∞ = 2 (764)

Recall the Clausius-Mossotti equation (Equation 743) that correlates the dielectric constant with polarization

Applying this equation for high-frequency conditions, that is, by replacing the term εr = ε∞ and removing the ionic polarization term, we get

ε ε ε

− +

= ×( ) 12 1

(765)

Combining Equations 764 and 765, we get the so-called Lorentz-Lorenz equation:

n N

− +

= ×( ) ε αe e (766)

This can be rewritten by replacing the concentration of atoms (Ne) as follows:

M N2

2 3

− +

 

  =

× ρ

α ε

(767)

A note of caution is in order regarding the use of the Lorentz-Lorenz equation Recall that the Clausius-Mossotti equation was derived using the local internal electric field (Elocal) calculation (Equation 736) Because the Lorentz-Lorenz equation is derived using the Clausius-Mossotti equation, it applies to nonpolar materials with a cubic symmetry or to amorphous materials It cannot be applied to ferroelectric or other polar dielectric materials

Materials that contain ions or atoms with a large electronic polarizability (αe) have a higher refractive index (n) A common example of such a material is called the lead crystal (Figure 721) This is actually an amorphous silicate glass that contains substantial (up to 30-40 wt% PbO) concentrations of lead ions (Pb2+) Because lead ions have a large electronic polarizability, the refractive index of the lead crystal is much higher (n up to 17) than that of a common soda-lime glass (n ∼ 15) Note that the polarizability values shown in Figure 710 include both electronic and ionic polarizabilities

Another important example of the use of higher electronic polarizability (αe) to achieve a higher refractive index (n) is its application in optical fibers (Figure 722) In optical fibers, a small yet significant mismatch (∼1%) is created between the refractive indices of the core and the cladding By doping the core of the fibers with dopants such as germanium, the refractive index of the core region is maintained higher than that of the cladding Optical fibers are usually made from ultra-highpurity SiO2 The core is doped with germanium oxide (GeO2), which enhances the refractive index of the core region This increases the total internal reflection at the core-cladding interface, thereby restricting the light waves (ie, information) to within the optical-fiber core Doping the fibers with fluorine (F) causes the refractive index of SiO2 to decrease

The following example illustrates the extent of contribution of the electronic and ionic polarization mechanisms to the dielectric constant of ionic materials

Another feature associated with polarization mechanisms is the notion of dielectric loss When ions, electron clouds, dipoles, and so on are displaced in response to the electric field (Figure 78), these displacements do not occur without resistance This resistance is similar to the effect of friction on mechanical movement The electrical energy lost during the displacements of ions, the electronic cloud, or any other entity that causes dielectric polarization, is known as the dielectric loss One way to represent dielectric losses is to consider the dielectric constant as a complex number Thus, we define the complex dielectric constant εr*( ) as

ε ε εr r r * = −′ ′′j (768)

In Equation 768, j is the imaginary number −1

εr′ , known as the real part of the dielectric constant, represents the charge-storage process, which is the same quantity that we have referred to so far as εr (or k) The imaginary part of the complex dielectric constant εr′′( ) is a measure of the dielectric losses that occur during the charge-storage process

An ideal dielectric is a hypothetical material with zero dielectric losses (ie, εr′′ = 0) This means that all the applied electrical energy is used to cause the polarization that leads to charge storage only A real dielectric is a material that does have some dielectric losses All dielectric materials have some level of dielectric loss because the displacements of ions, electron clouds, and so on, cannot occur without resistance from neighboring atoms or ions The dielectric losses increase if the applied field switches in such a way that the polarization mechanisms can follow these changes in the applied electric field It is usually desirable to minimize or lower the dielectric losses for microelectronic devices However, dielectric losses can be useful for applications in which heat must be generated A common example is that of a microwave oven The water molecules in food, which are permanent dipoles, tumble around during the polarization caused by the microwave’s electric field The resultant dielectric losses cause the generation of heat (Vollmer 2004)

While developing materials for capacitors to store charge (Section 722), we prefer to use a low-loss dielectric; in other words, we want a small εr′′ The need for increasing the dielectric constant (k, or now what we refer as εr′) while maintaining the dielectric losses at small levels poses a problem because polarization processes are required to achieve higher dielectric constants When these polarization processes occur, they cause dielectric losses Polarization and dielectric losses originate from the same basic processes (some type of displacement of ions and electron clouds, in addition to the reorientation or rotation of dipoles, etc; Figure 78)

Because the dielectric constant depends on frequency ( f or ω), it is reasonable to assume that dielectric losses are also frequency-dependent

Consider a hypothetical dielectric material with only the dipolar polarization mechanism If the electrical frequency ( f or ω) is too high, then the dipole cannot rotate or flip back and forth in response to the oscillating electric field We may also change the magnitude of the electric field from some initial value E0 to another value E without changing its direction If this happens, the induced dipole moment (ionic or electronic polarization) adjusts from some starting value of μ0 (at E0) to another final value μ (at E) Such changes in the dipole moment require a small but finite time The relaxation time (τ) is the time required for a polarization mechanism to revert and realign a dipole or change its value when the electrical field switches or changes in magnitude

In a hypothetical dielectric material with a single polarization mechanism and a relaxation time τ, the dielectric losses are very small if the frequency of the electrical field switches too rapidly When ω >> 1/τ, the polarization mechanism is unable to follow the change in E On the contrary, if ω << 1/τ, then the polarization process can follow the changes in the electric field However, the induced dipole moment does not switch or readjust that often, so the dielectric losses are again small When ω = 1/τ, similar to a resonance condition, the dielectric losses are maximized because when the polarization switches or adjusts itself in a synchronized fashion as the electric field switches again Figure 723 shows this variation of dielectric losses with frequency for a hypothetical dielectric with a single polarization mechanism with relaxation time τ

In any dielectric material, there are different relaxation times for different mechanisms of polarization For example, the displacement of the electronic cloud around a nucleus occurs very rapidly,

and the relaxation time τ is ∼10−14 seconds On the contrary, for interfacial polarization, the relaxation time (τ) can be several seconds because this mechanism involves a relatively longer-range motion of ions Thus, for materials with multiple polarization mechanisms, the real part of the dielectric constant εr′( ) shows changes similar to a set of cascades The dielectric loss component εr′′( ) shows a series of maxima that follow the different polarization mechanisms (Figure 724)

For a dielectric material with a single relaxation time (τ), the frequency dependence of εr′ and εr′′ can be written quantitatively as follows:

ω τr s′ ′= = + − +∞

∞k k k k

1 2 2 (769)

ε ωτ

ω τr s ′′ ′′= = −

+  

 ∞k k k( ) 1 2 2

(770)

Equations 769 and 770 describe changes in the real and imaginary parts of the dielectric constant and are also known as Debye equations

We can see from Equation 770 that the maximum in k″, known as the Debye loss peak, occurs when ω = 1/τ (see the problems at the end of this chapter and Example 712)

In many materials, there can be different relaxation times for the same polarization mechanism This is because different atoms and ions may be involved There is then a distribution of relaxation times due to the different polarization mechanisms and the distribution of relaxation times applicable for each mechanism for a given dielectric

The following example illustrates an application of the Debye equations

SOLUTION We first convert the frequency (f) into angular frequency (ω) by using

The changes in the dielectric properties of dielectrics encountered in applications of microelectronics are more complex For example, CCTO (CaCu3Ti4O12) was reported as a giant dielectric constant material The changes in the real and imaginary parts of the dielectric constant for CCTO are shown in Figure 725

Note several details from the data in Figure 725:

1 First, the dielectric constant is very large We should therefore immediately suspect that this material may be a ferroelectric or that a space-charge polarization (see Section 710) may be present Crystal structure analysis and other measurements have shown that this material is not a ferroelectric The measured dielectric constant must be an apparent dielectric constant

These experimental data show that the apparent or measured dielectric constant is a microstructure-sensitive property If we were to predict the dielectric constant using the Clausius-Mossotti equation, then the dielectric constant, although expected to be a function of frequency, will not be expected to show microstructure-sensitive characteristics This is because the Clausius-Mossotti equation does not account for interfacial or ferroelectric (spontaneous) polarization

2 Overall, as predicted by the Debye equations, the dielectric constant decreases with increasing frequency (Figure 725)

3 In this case, the increase in the dielectric constant with the sintering times may be because when the sintering times are low (eg, 25 hours), the samples may not be dense These materials are basically a composite of a dielectric and air Because air has a low dielectric constant (k′ ∼ 1), the overall measured values of the dielectric constant are lower

4 As the measured density increases with increasing sintering times, the overall dielectric constant also increases The data in Figure 725 (marked as the first step) also show that the increase in the dielectric constant with sintering times is higher for the higher-frequency region (∼5 × 105 to 3 × 106 Hz) In these materials, the interfacial or Maxwell-Wagner polarization at the grain boundaries plays an important role

5 The relatively moderate increase in the dielectric constant in step 2 (frequency range ∼104 to 5 × 105 Hz; Figure 725) is related to the formation of another phase, which involves a process known as liquid-phase sintering In this process, a liquid phase is formed that can assist densification However, it can also lead to the formation of grain boundaries or surface phases that have a chemical composition different from that of the original ceramic material

6 The relaxation of the low-frequency dielectric constant (in the range of 102-104 Hz-the region to the left of that marked as the second step) may be due to interfacial polarization at the ceramic-electrode interface via a Schottky barrier effect If this is the case, it may be possible to change the electrode materials to see if the apparent dielectric constant changes

In summary, the interpretation of changes in the apparent or measured dielectric constant or losses with frequency requires a considerably detailed analysis We can attempt to correlate these changes with the different polarization mechanisms and microstructures To determine the controlling polarization mechanisms, the dielectric properties are measured as functions of the temperature and frequency while changing the processing conditions systematically The change in the dielectric constant of CCTO ceramics as a function of the temperature is shown in Figure 716 The technique of measuring k′ and k″ as a function of the frequency ( f ) is known as impedance spectroscopy The measurements of k′ and k″ can be accomplished using an instrument called an impedance analyzer The variations of k′ (x-axis) and k″ (y-axis) plotted as a function of the frequency appear as sets of arcs or semicircles and are known as Cole-Cole plots

For a parallel capacitor, the capacitance is given by

d = × ×ε ε ω0 r′ ( )

(773)

This is similar to Equation 713; the difference is that now we explicitly show the dependence of the dielectric constant on electrical frequency by incorporating the variation of the dielectric constant with frequency (Figures 718 and 723)

We can model a real dielectric material as a pure, lossless capacitor with capacitance C, connected in parallel to a resistor with a resistance (Rp) The subscript p tells us that in this equivalent circuit, the resistor is connected to the capacitor in parallel This is an equivalent circuit of a real dielectric material (Figure 726) A circuit comprised of a capacitor and a resistor (in series or parallel) is also known as a resistor-capacitor (RC) circuit For an ideal dielectric, there is no loss, that is, Rp = 0

The conductance (Gp = 1/Rp) of a dielectric is given by

G R

r= =1 0ω ε ε ω ′′ ( )

(774)

Thus, for an ideal capacitor, if Rp = 0, then Gp = ∞ This means that as the capacitor charges and discharges, the current appears to flow “through” the capacitor We have, of course, seen before that a dielectric material is a nonconductor, and very little, if any, current can actually flow through it Equation 774 can be used to calculate the equivalent resistance or conductance of a dielectric material with specific capacitor geometry The relatively simple appearance of Equations 773 and 774 is a bit misleading If we want to calculate these values as a function of the frequency (ω), then we must keep in mind that both εr′ and εr′′ change with frequency

For a real dielectric, we define impedance (Z) as a measure of the resistance offered by a circuit under AC fields The impedance of a real dielectric (Figure 726) can be written as follows:

Z R jX= +p C(775)

where XC is the capacitive reactance It is the equivalent of a capacitor’s resistance to the flow of current through it The magnitude of the capacitive reactance (XC) is given by

C f Cc p = =1 1

2ω π (776)

Thus,

Z R j

f C = +p

2π (777)

The subscript “p” is used to indicate a parallel arrangement of the resistor and capacitor (Figure 726)

The admittance (Y) is defined as the inverse of the impedance (Z) Note that the admittance (Y) and impedance (Z) are both complex numbers The SI unit for admittance is Siemens, also often reported as mho, the inverse of Ohm The admittance of a capacitor made using a real dielectric

material can be written as a combination of a conductor, with conductance Gp, and a capacitor, with capacitance Cp (Figure 726) The total admittance is written as a complex number:

Y G j C= +p pω (778)

Y R

j C

Y R

j f C

= +

1 2

π (779)

The admittance (Y) can be rewritten as follows using Equations 773 and 774:

d j

d = +ω ε ε ω ω ε ε ω0 0r r

′′ ′( ) ( )

(780)

Examples 710 and 711 illustrate the calculations of capacitive reactance and its frequency dependence

We know from Joule’s law for a resistor that the power dissipated in a resistor is given as

Power dissipated = × =V I V

(781)

The equivalent of this for a capacitor is

Power input = × = ×V I V Y2 (782)

We substitute for Y from Equation 779 to get

Power input

= +j CV V R

(783)

The second term of this expression gives the power dissipated in a capacitor with equivalent resistance Rp (Figure 726)

Thus, the power lost in a real dielectric is given by

Power loss in a real dielectric

= V R

(784)

The dissipated power appears as heat in the dielectric material It is sometimes important to note how much power is dissipated per unit volume If we consider a capacitor (Figure 73) with volume = A × d, the power loss per unit volume is given by substituting for 1/Rp from Equation 774 as shown below:

Power loss in a real dielectric per unit volume p

r= × ×

= ×

×V R A d

A d

2 2 0ω ε ε ω′′ ( )

Power loss in a real dielectric per unit volume r= × ×ω ε ε ωE 2

0 ′′ ( ) (785)

7.16.1 CONCEPT OF TAN δ Consider a resistor connected to voltage (V) Assure that the voltage changes with time in a sinusoidal fashion as follows:

V V j t= 0exp( )ω (786)

You may recall that the definition of the function exp ( jx) is

exp( ) cos( ) sin( )jx x j x= + (787)

Thus,

exp( ) cos( ) sin( )j t t j tω ω ω= + (788)

In a pure resistor, the current (I) and the voltage are said to be “in phase” This means that as the voltage changes over time, the current instantly follows the change in voltage This is shown in a phasor diagram in Figure 727 A phasor is a rotating vector that shows the phase angle for a particular type of current or voltage in a circuit component A phasor diagram shows the relative positions of the phasors for currents and/or voltages corresponding to a circuit The lengths of the phasor arrows are generally in scale with the magnitude of the voltage or current they represent A phasor diagram is also known as a vector diagram

At ωt = 0, V = V0; at ωt = π/2 or 90°, Vj = V0 As shown in Figure 727, as the voltage phasor rotates for a pure resistor from ωt = 0 to ωt = π/2, the phasor representing resistance also rotates from ωt = 0 to ωt = π/2

When an AC voltage V (Equation 786) is applied to this capacitor, a charge Q appears on this capacitor and is given by

Q C V C V jwt= × = × 0 exp( ) (789)

The buildup of charge on the capacitor can be described by defining a charging current (Ic) This current is given by

dtc =

(790)

Rewriting the expression for the charging current using Equation 789, we get

I C

dtc =

(791)

Because V V j t dV dt V j j t= =0 0exp( ), exp( )ω ω ω/ Thus,

I jCV j t j CVc = =0ω ω ωexp( ) (792)

Compare Equations 786 and 792 (for voltage V and Ic,respectively) when applied to an ideal capacitor Note that the charging current Ic always leads the applied voltage V by exactly 90° This is shown in the phasor diagram for an ideal capacitor, Figure 728a The current and voltage waveforms for an ideal capacitor are shown in Figure 728b

Referring to Figure 728, if the phase angle ωt = 0, the charging current (Ic) will be along the axis of the imaginary and the voltage (V) will be along the axis of the real As the phasor for Ic rotates through any angle, the phasor for the voltage also rotates by the same amount, thus maintaining a steady phase difference of 90° at all times (Figure 728b)

One way to represent a real dielectric is through a parallel combination of a resistor and a lossless capacitor (Figure 726) We then describe the total current (Itotal) as the vector sum of a charging current (Ic) and a loss current (Iloss) The charging current associated with the ideal capacitor leads the applied voltage by 90° The loss current originates from two sources First, the dielectric material has a resistance under DC conditions that is usually very large However, a very small amount of current still flows through a dielectric material Second, for an AC voltage, as the polarization processes continue to occur, the back-and-forth displacements of electronic clouds, ions, molecular dipoles, and so on also cause a dissipation of energy and are thus part of the impedance The loss current (Iloss) in a real dielectric due to both of these contributions is always in phase with the voltage, as these processes always follow the applied voltage (V)

The loss current (Iloss) in a dielectric can be written as follows:

I G Vloss = × (793)

where G is the conductance and V is the voltage The conductance can be expressed as a sum of the DC and AC components

The loss current can be written as

I G G Vloss dc ac= +( ) × (794) The total current (I) is given by the vector sum of these two currents, namely, the charging current (Ic) and the loss current (Iloss) This total current (Itotal) always leads the voltage, but not by 90° As shown in Figure 729, the total current (Itotal) therefore leads the voltage at an angle (90 − δ)° The angle δ is known as the loss angle Recall that the charging current (Ic) is jωCV (from Equation 792)The total current (Itotal) in a capacitor made using a real dielectric can be written as

I I I j C G G Vtotal c loss dc ac= + = + +( )ω (795)

From Figure 729, we can also note that

tan δ = I I

(796)

Thus, tan δ (pronounced tan delta) is also a measure of the dielectric losses There are no dielectric losses in an ideal dielectric This means that IL = 0, tan δ = 0, and the total current is equal to the charging current

The concept of tan δ is very important for describing properties of dielectric materials This parameter can be measured using an impedance analyzer similar to measuring the dielectric constant We can compare how “lossy” different dielectric materials are relative to each other The real part of the complex dielectric constant is a measure of a material’s chargestoring ability The magnitude of tan δ indicates the inefficiency of the material in terms of the charge-storing ability

In an ideal dielectric, which does not exist, polarization processes are assumed to occur without any electrical energy waste, with no energy wasted during charge storage, that is, tan δ = 0 In many real dielectrics, the tan δ values range from ∼10−5 to 10−2 Some special applications, such as ceramic materials used as dielectric resonators for microwave communications, require very low tan δ values In this case, the values of a parameter defined as the quality factor (Q) are reported The quality factor of a dielectric (Qd) is defined as

Qd ≈ 1

tanδ (797)

We can also show that tan δ is the ratio of the imaginary and real parts of the complex dielectric constant (k*)

The dielectric constant is defined as εr /* = C C0, and because Q CV= , Q C V= εr* 0 The current I dQ dt C dV dttotal / /= = ( ), or

I C

dttotal r = 

 ε

(798)

If V V j t= 0 exp( )ω , then dV dt j V j t j V/ = =ω ω ω0 exp( ) ; therefore, Itotal is

I C dV

dt j C j Vtotal r r r= 

  = −( )ε ε ε ω* 0 0′ ′′ (799)

To derive Equation 799, we substituted for the complex dielectric constant in terms of its real and imaginary parts (see Equation 768)

I j C V C Vtotal r r= +ωε ωε′ ′′0 0 (7100)

Note that to derive this equation, we used the value j 2 = −1 Now compare Equations 795 and 7100 The first term of Equation 7100 represents the charge-

storage in the dielectric, that is, the charging current The second term is the magnitude of the loss current (Iloss) Thus, the ratio of the magnitude of these two parts is tan δ:

tan δ ωε ωε

ε ε

= =r r

′ C V

C V 0

Thus,

tan δ εε = =r

′ k

k (7101)

Another interpretation for tan δ or loss tangent is that it is the ratio of the imaginary and real parts of the complex dielectric constant or the complex relative dielectric permittivity One can think of tan δ as the ratio of the price we pay for storing the charge in a capacitor (in terms of the energy that is wasted and that appears as heat) Heating due to dielectric losses is useful in some applications However, in many applications, the generation of heat changes the temperature of the dielectric material, and the dielectric properties may change with changes in temperature In addition, heating a dielectric material can change its dimensions; the geometrical changes also lead to changes in capacitance In general, we prefer to use temperature-stable and low-loss dielectric materials

Because both the real and imaginary parts of the dielectric constant are frequency-dependent (Figure 724), tan δ also depends on the frequency

Recall that for a hypothetical material with a single polarization mechanism with relaxation time (τ), the frequency dependences of the real and imaginary parts of the dielectric constants are given by the Debye equations (Equations 769 and 770) Therefore, the frequency dependence of tan δ for such a hypothetical material is given by

tanδ ε ε

ωτ ω τ

= = −( ) +

k k

k k 2 2 (7102)

where ks and k∞ are the values of the low-frequency (static) and high-frequency dielectric constants

While the value of εr′′ reaches a maximum at ωt = 1/τ (Figure 723), tan δ reaches a maximum at a slightly higher frequency, which is given by

ω δ τmax

(for tan ) =

k∞

 

 

(7103)

Recall that the power loss in a real dielectric per unit volume is given by Equation 785 We can write this in terms of tan δ and εr′ (dielectric constant) as follows:

Power loss in a real dielectric per unit volume r= × × × ×ω ε ε δE2 0 ′ tan( ) (7104)

Example 712 shows how to calculate the frequency at which the value of tan δ reaches a maximum

SOLUTION We start with Equation 7.102, which describes the dependence of tan δ on frequency ω.