2 Dielectric Properties Lecture 1

K Asokan

epgp books

 

 

 

 

Learning Outcomes:

 

From this module students may get to know about the following:

1.     The fundamental Dielectric Constant.

2.     You will learn about electric dipole moment and its derivation.

3.     Detailed study of electric flux density, polarizability and source of polarizability.

 

1. Dielectric Properties

 

1.1 Dielectric constant and Polarizability:

 

Dielectrics are materials which have no free charges; all electrons are bound and associated with the nearest atoms. An external electric field causes a small separation of the centers of the electron cloud and the positive ion core so that each infinitesimal element of volume behaves as an electric dipole. Dielectrics may be subdivided into two groups:

 

Non-Polar: which behave as above.

Polar: in which the molecules or atoms possess a permanent dipole moment which is ordinarily randomly oriented, but become more or less oriented by the application of an external electric field. To start with we will first talk about the electric dipole moment.

 

Electric dipole moment:

 

Electric dipole as an entity composed of the two opposite charges of equal magnitude, q and -q (figure1). The dipole moment of that assembly is given by

 

Figure1. An electric dipole

 

Where d is the distance from negative to the positive charge. The electric dipole moment is therefore equal to one of the charge time’s distance between them. Electric dipole is a vector quantity. The unit of the electric dipole moment is the Debye (1 Debye = 3.33×10-30 coulomb-meter).

 

In 1837, Faraday experimentally found that when an insulating material– also called dielectric (such as mica, glass or polystyrene etc.) – is introduced between the two plates of a capacitor, it is found that the capacitance is increased by a factor which is greater than one. This factor is known as the dielectric constant () of the material. It was also found that this capacitance is independent of the shape and size of the material but it varies from material to material. For glass the dielectric constant is 6, while for water it is 80.

 

Now we will prove that how the presence of the dielectric or insulating material is responsible for the increase in the capacitance.

 

We imagine that two plane parallel plates of area A and separation d are charged with a surface charge density q, one plate being positive and the other negative. If the space between the plates is evacuated and if d is small compared with the dimensions of the plates, there will result a homogeneous electric field between the plates, the field strength being given by

 

Suppose now that the space between the plates is filled with an insulating substance, the charge on the plates being kept constant. It is then observed that the new potential difference  is lower than  and similarly, the capacitance C of the system is increased. The static dielectric constant is then defined by

 

 

 

Thus, as a result of introducing the substance, the field strength is reduced from the value  to the value E, where

From above equations we can say that  is found to be independent of the shape or the dimensions of the conductors and is a solely a characteristic of the particular dielectric medium used.  is called the relative permittivity or the dielectric constant of the medium. In the S.I. systems of units the permittivity of the medium is defined as

 

 

where is an electric constant which represents the permittivity of vacuum while is the relative static  permittivity of the material. The term dielectric constant is sometimes used for both and. The dielectric constant of a material is a macroscopic quantity that measures how effective an electric is in polarizing the material.

 

Derivation of the above equations:

 

Let us apply Gauss theorem to a parallel plate condenser without a dielectric and then with a dielectric. In the first case, when no dielectric is present, the electric field  at any point on the Gaussian surface is given by

 

 

this equcation indicates that the induced charge  tends to weaken the original field (i.e. E<E0) ), which reveals itself as a reduction in the potential difference when dielectric is present and not present respectively, then

 

This shows that the induceed charge  is less than the free charge q and is zero when  or the dielectric is absent.

 

Table 1. List of some materials with dielectric constant

    1.2 Electric flux density and polarisation:

 

Due to the polarization of the medium, charges are induced at the surfaces of the dielectric, the charge due to electric polarisation at the boundary near the positive plate of the capacitor being negative and vice versa. If q is the charge on the plate and q’ is the induceed charge on the boundary of the dielectric, then we can write equcation as

 

1.3 Polarizability and source of polarizability:

 

Because the polarization of a medium –i.e., the alignment of the molecular moment is produced by the field, it is plausible to assume that the molecular moment is proportional to the field. thus we write

where  is a constant called the polarizability of the molecule.

 

In the discussion of dielectric materials, we usually talk about the polarization P of the material, which is defined as the dipole moment per unit volume. if the number of molecules per unit volume is N, and if each has a dipole moment , it follows that the polarization is given by

here we have assumed that all the molecular moments lie in the same direction.

So polarization now can be written as

 

giving the dielectric constant in terms of polarizability. This is a useful result in that it express the macroscopic quantity, in terms of microscopic quantity , thus forming a link between two descriptions of dielectric material.

The electrical susceptibility  of a medium id defined by the relation

 

which relates the polarization to the field. by comparision of this equcation we find that the susceptibility and polarizablity are interrelated by

 

1.4. Sources of polarizability :

 

Bascially, polarizability is a consequence of the fact that the molecules, which are the building blocks of the all substances, which are composed of both positive charge (nuclei) and negative charges (electrons).when a field acts on a molecule, the positive charges are displaced in a direction opposite to that of the field. the effect is therefore to pull the opposite charge apart, i.e., to polarize the molecule.

 

There are different type of polarisation process, depending on the structure of the molecules which constitute the solid. if the molecule have a parmanent moment, i.e., a moment even in the absense of the field, we speaks of a dipolar molecule, and a dipolar substance.

 

 

Figure 4. (a) The water molecule and its parmanent moment p=1.9 debye units (1 debye units = 10-29 coul.meter). (b) CO2 molecule

 

An example of a dipolar molecule is the H2O molecule . The dipole moments of the two OH bonds add vectorially to give a nonvanishimg net dipole moments. Some molecules are nondipolar, possessing no parmanent moments; a common example is the CO2molecule. The moments of the two CO bands cancel each other because of the rectilinear shape of the molecule, resulting in a zero net dipole moment.

 

Despite the fact that the individual molecules in a dipolar substance have permanent moments, the net polarization vanishes in the absence of an external field because the molecular moments are randomely orientated, resulting in a complete cancellation of the polarization.When a field is applied to a substance, however the molecular dipoles tend to align with the field, and this results in a net nonvanishing polarization. This leads to the so-called dipolar Polarizability.

 

If the molecule contains ionic bonds, then the fields tends to stretch the lengths of these bonds. This occur in NaCl, for instance, because the fields tends to displace the positive ion Na+ to the right and the negative ion Cl- to the left, resulting in a stretching in the lengths of the bonds. The effect of this change in length is to produce a net dipole moment in the unit cell where previously there was none. Since the polarization here is due to the relative displacements of oppositely charged ions, we speaks of ionic polarizability.

 

Figure5. Ionic polarization in NaCl. The fields displaces the ions Na+ and Cl- in opposite directions, changing the length of the bond.

 

Ionic polarizability exists whenever the substance is either ionic, as in NaCl, or dipoler, as in H2O, because in each of these classes there are ionic bonds present. But in substances in which such bonds are missing- such as Si and Ge – ionic polarizability is absent.

 

figure 6. Electronic polarization. (a) Unpolarized atom (b) Atom polarized as a result of the field.

 

The third type of polarizability arises because the individual ions or atoms in a molecule are themselves polarized by the field. In the case of NaCl , each of the Na+ and Cl- ions are polarized. Thus the Na+ ion is polarized because the electrons in its various shells are displaced to the left relative to the nucleus, as shown in figure 6, it is called the electronic polarizability. Electronic polarizability arises even in the case of a neutral atom, again because of the relative displacement of the orbital electrons.

 

in general, therefore , we may write for the total polarizability

 

which is sum of the various contributions ;  and  are the electronic, ionic and dipolar polarizabilities, respectiviley. The electronic contribution is present in any type of substance, but the presence of other two terms depends on the material under consideration. Thus the term  is present in ionic substance, while in a dipolar substance all three contributions are present. In covalent crystals such as Si and Ge , which are nonionic and nondipolar , the polarizability is entirely electronics in nature. Dipolar polarizability , for instance exhibits strong dependence on temperature , while the other two contributions are essentially temperature independent.

Figure7. Total polarizability α versus frequency  for a dipolar substance.

 

Another important distinction between the various polarizabilities emerges when one examines the behavior of the ac polarizability that is induced by an alternating field. Figure 7 shows a typical dependence of this polarizability on frequency over a wide range, extending from the static all the way up to the ultraviolet region. It can be seen that in the range from to, where (d for dipolar) is some frequency usually in the microwave region, the polarizability is essentially constant. In the neighborhood of, however, the polarizability decreases by a substantial amount. This amount corresponds precisely, in fact, to the dipolar contribution. The reason for the disappearance of in the frequency range, is that the field now oscillates too rapidly for the dipole to follow, and so the dipoles remain essentially stationary.

 

The polarizability remains similarly unchanged in the frequency range from, to, and then drops down at the higher frequency. The frequency lies in the infrared region, and corresponds to the frequency of the transverse optical phonon in the crystal. For the frequency range the ions with their heavy masses are no longer able to follow the very rapidly oscillating field, and consequently the ionic polarizibility , vanishes, as shown in Fig. 7.

 

Thus in the frequency range above the infrared, only the electronic polarizability remains effective, because the electrons, being very light, are still able to follow the field even at the high frequency. This range includes both the visible and ultraviolet regions. At still higher frequencies (above the electronic frequency we, however, the electronic contribution vanishes because even the electrons are too heavy to follow the field with its very rapid oscillations.

 

The frequenciesand, characterizing the dipolar and ionic polarizabilities, respectively, depend on the substance considered, and vary from one substance to another. However, their orders of magnitude remain in the regions indicated above, i.e., in the microwave and infrared, respectively. The various polarizabilities may thus be determined by measuring the substance at various appropriate frequencies.

 

Summary:

  1. Electric dipole moment with its derivation have been discussed in detailed.
  2. We have discussed what polarization is and how it is described by the polarization vector P
  3. susceptibility and permittivity have been studied in brief.
  4. Electric flux density along with polarization and source of polarization have been discussed.
  5. In the last frequency dependence on the polarizability was explained.
you can view video on Dielectric Properties Lecture 1

    References:

  1. Bunget, I., & Popescu, M. (1984). Physics of solid dielectrics. Amsterdam: Elsevier.
  2. Kao, K. (2004). Dielectric phenomena in solids with emphasis on physical concepts of electronic processes. Amsterdam: Academic Press.
  3. Kasap, S. (2006). Principles of electronic materials and devices (3rd ed.). Boston: McGraw-Hill
  4. A.R. West – “Solid State Chemistry and it’s Applications”, Wiley (1984)
  5. R.H. Mitchell – “Perovskites: Modern & Ancient ”, Almaz Press, (www.almazpress.com) (2002)
  6. P. Shiv Halasyamani & K.R. Poeppelmeier – “Non-centrosymmetric Oxides”, Chem. Mater. 10, 2753-2769 (1998).
  7. M. Kunz & I.D. Brown – “Out-of-center Distortions around Octahedrally Coordinated d0 Transition Metals”, J. Solid State Chem. 115, 395-406 (1995).

    References and Suggestive Readings

  1. R.D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993).
  2. R. Resta, Rev. Mod. Phys. 66, 899 (1994)
  3. Souza, J. Iniguez and D. Vanderbilt, Phys. Rev. B 69, 085106 (2004) R.W. Nunes and X. Gonze, Phys. Rev. B 63, 155107 (2001).P. Umari and A. Pasquarello, Phys. Rev. Lett. 89, 157602 (2002).
  4. X. Wang and D. Vanderbilt, in preparation.
  5. N. Sai, K.M. Rabe, and D. Vanderbilt, Phys. Rev. B 66, 104108 (2002).
  6. X. Wu, D. Vanderbilt, and D.R. Hamann, submitted to Physical Review B.
  7. D.R. Hamann, X. Wu, K.M. Rabe, and D. Vanderbilt, and, Phys. Rev. B. 71, 035117 (2005).

    Web Links

  1. http://chemwiki.ucdavis.edu/u_Materials/Optics/Dielectric_Polarization
  2. http://www.tf.uni-kiel.de/matwis/amat/elmat_en/kap_3/backbone/r3_2_1.html
  3. http://www.tf.uni-kiel.de/matwis/amat/elmat_en/kap_3/backbone/r3_1_1.html#_dum_8
  4. http://www.doitpoms.ac.uk/tlplib/dielectrics/summary.php
  5. http://physics.stackexchange.com/questions/134542/how-do-we-conclude-that-polarized-dielectric-in-electric-field-reduces-overall-f

    Additional Topics to be studied

 

 I. Dielectric Loss.

II. Impurities in dielectrics

 

Theory of dielectrics

 

This section presents a brief description of the atomic interpretation of the dielectric and optical properties of insulator materials on the basis of classical theory. This section is essentially concerned with the static dielectric constant, the frequency dependence of dielectric constant and dielectric losses.

 

Electric susceptibility and permittivity

 

It was Michael Faraday who first noticed that when a capacitor of value C0 under vacuum is filled with a dielectric material, its charge storage capacity (capacitance) increases to a value of C. The ratio χ’ of the increase of capacitance C =C-C0 to its initial capacitance- C0,

 

χ’ is called the electrical susceptibility of the dielectric. The most often used terminology is the dielectric permittivity or dielectric constant instead of susceptibility, which is defined as the ratio of the capacitance C of the capacitor filled with a dielectric to the value C0 of the same capacitor under vacuum. From the above equations the relationship between the electric susceptibility and the dielectric permittivity is given as:

 

   Thus, by definition, the electric susceptibility and permittivity are non-dimensional real quantities. The dielectric constant or permittivity of a material is a measure of the extent to which the electric charge distribution in the material can be distorted or polarized by the application of an electric field.
    Clausius and Mossotti relation for dielectric permittivity
   Consider a molecule of a dielectric medium situated in a uniform electric field E. The total electric field acting on this molecule Eloc will have three main components- E1, E2, and E3. Here E1 is the applied electric field E, E2 is the field from the free ends of the dipole chain and E3 is the near field arising from the individual molecular interactions. In solids we have to consider the actual effective field acting on a molecule in order to estimate the dielectric permittivity. For electronic and ionic polarization, the local field for cubic crystals and isotropic liquids can be given by the Lorenz field, given by

By assuming the near field E3 is zero, Clausius and Mossotti derived a relation for the dielectric constant of a material under electronic and ionic polarization.

 

Here, εr is the relative permittivity at low frequencies, αi is the effective ionic polarizability per ion pair, Ni is the number of ions pair per unit volume, αe is the electronic polarizability and Ne is the number of ions (or atoms) per unit volume exhibiting electronic polarization. The atomic/ionic polarizability αi and the electronic polarizability αe cannot be separated at low frequencies and hence they are together represented as the induced polarizability αind

 

Hence equation can be written as:

This is known as the clausius –Mossotti equation for non polar dielectrics. Above the frequencies of ionic polarization relaxation, only electronic polarization will contribute to the relative permittivity, which will be lowered to εr∞ (relative permittivity at optical frequencies).

 

In this form, it is known as Lorentz-Lorenz equation. It can be used to approximate the static dielectric constant εr of non polar and non magnetic materials from their optical properties. In the case of dipolar materials we cannot use the simple Lorentz field approximation and hence the Clausius–Mossotti equation cannot be used in the case of dipolar materials.

 

Dielectric loss

 

The permittivity of a dielectric material has both real and imaginary mathematical representations. The imaginary part of permittivity is represented in mathematical equations as ε׀׀. This imaginary part of permittivity describes the energy loss from an AC signal as it passes through the dielectric. The real part of permittivity, εr is also called the dielectric constant and relative permittivity. The permittivity of a material describes the relationship between an AC signal’s transmission speed and the dielectric material’s capacitance. When the word “relative” is used in front of permittivity, the implication is that the number is reported relative to the dielectric properties of a vacuum. The imaginary part of the dielectric permittivity which is a measure of how much field is lost as heat during the polarization of a material by an applied alternating electric field is also termed as dielectric loss. The characteristic orientation of the dipoles in an electric field results in a frequency variation of dielectric constant and loss over a broad band of frequencies. The typical behavior of real and imaginary part of the permittivity as a function of frequency is show in Figure.

 

Figure: Frequency dependence of dielectric permittivity for an ideal dielectric material.

 

The relative permittivity of material is related to a variety of physical phenomena that contribute to the polarization of the dielectric material. In the low frequency range the ε11 is dominated by the influence of ion conductivity. The variation of permittivity in the microwave range is mainly caused by dipolar relaxation, and the absorption peaks in the infrared region and above, are mainly due to atomic and electronic polarizations. The dielectric properties of solid dielectrics at microwave and radio frequencies are highly influenced by the ionic positions and changes caused by the lattice vibrations.

 

Two types of dielectric losses are identified in crystalline solids at high frequencies, namely intrinsic losses and extrinsic losses. The dielectric dispersion in solids depends on the factors such as ionic masses, electric charge/valence state of the ions, spring constant of the bond, lattice imperfections etc. The dielectric losses close to the lattice vibration frequencies are generally estimated in terms of the anharmonicity of lattice vibrations. The low frequency phonons are responsible for the intrinsic dielectric losses in solid dielectrics. The intrinsic loss mechanism occurs due to the interaction between the phonons and the microwave field or due to the relaxation of the phonon distribution function. The lattice phonon modes will determine intrinsic limits of the high frequency dielectric losses in crystalline solids. The extrinsic losses are occurred due to the interaction between the charged defects and the micro.