3 Dielectric Properties Lecture 2

K Asokan

epgp books

   

 

 

 

Learning Outcomes:

 

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

 

The mechanism and different type of polarization.

●   electronic polarization

●   ionic polarization 

●   orientation polarization

The static dielectric of gases

 

2.   Type of polarizability

 

2.1 Electronic polarizability :

 

In a free atom, the charge distribution is such that the dipole moment in the absence of an external field vanishes; the center of gravity of the electron distribution coincides with the nucleus. Consider now an atom in a static homogeneous external field E. The force exerted on the positive nucleus will then be oppositely directed to the forces exerted on the electrons. As a result, the external field tends to draw the center of gravity of the electrons away from the nucleus. On the other hand, the attractive forces between the electrons and the nucleus tend to preserve a vanishing dipole moment in the atom. Consequently, an equilibrium situation is reached in which the atom bears a finite dipole moment. This has been represented schematically in Fig.(1). The resulting dipole moment is thus induced by the field as a result of an elastic displacement of the electronic charge distribution relative to the nucleus. The induced moment

 

may be represented by

 

????=???                                                                                                      ………(1)

 

Where is called the electronic polarizability of the atom. It should be noted that (1) is actually only the first term of a power series in the field strength. For the usual fields employed in dielectric measurements, however, (1) is a very good approximation.

 

Figure (a) Schematic illustration of the displacement of the electron orbit relative to the nucleus for a hydrogen atom under influence of an external field E.

(b)Simplified model for estimating the magnitude of the electronic polarizability of an atom, as described in the text.

 

To obtain an idea of the magnitude of , consider the following simplified model; Suppose the atom is represented by a nucleus of charge Ze and a homogeneous negative charge distribution inside a sphere of radius r. If the nucleus is displaced over a distance d, under the influence of the field E as shown in Fig.(2). So it can be shown through the laws of electrostatics that when a field E is applied to this atom, the nucleus is displaced from the centre of sphere by a distance

Where r is the radius of the sphere (the atomic radius), and Ze is the nuclear charge. the atom is thus polarized, and the dipole moment

 

?=?? ?                                                                  ………(3)

 

Put the value of d from equation (2) we gets

 

Table 1. Electronic Polarizabilities for some inert gases and closed-Shell Alkali and Halogenic Ions (in units of 10−40 farad. meter2).

 

 

 

 

 

 

 

 

It will be evident that in general atoms with many electrons tend to have a larger polarizability than those with few electrons. Electrons in the outer electronic shells will contribute more to than do electrons in the inner shells, because the former are not so strongly bound to the nucleus as the latter. Positive ions therefore will have relatively small polarizabilities compared with the corresponding neutral atoms; for negative ions the reverse is true.

 

2.2 Ionic Polarizability:

 

So far, we have considered only simple atoms and ions. For molecules one is faced with two more possible influences of an external field:

I. Molecules may have permanent dipole moments which may be aligned in an external field.

2. The distances between ions or atoms may be influenced by an external field.

 

For example, a molecule such as HCI may in first approximation be considered to consist of two ions; the permanent dipole moment is thus equal to the effective charge per ion times the separation of the ions. Symmetric molecules like H2 ,CO2, CCl4, etc. evidently have no permanent dipole moment. An external electric field will tend to orient permanent dipoles along the field direction, and one speaks of orientational polarization.

 

In molecules as well as in atoms an external field will displace the electrons with respect to the corresponding nuclei. Over and above this, however, a displacement of atoms or ions within the molecule may be caused by an external field. For example, in an HCI molecule an external field will change the inter-ionic distance to some extent, leading to a change in the dipole moment. Similarly, in a molecule like CCl4 (which has no permanent dipole moment) a change in the bond angles between the CCl groups will produce a dipole moment because each of these groups by itself does have a dipole moment. This kind of induced polarization is called atomic or ionic polarization because it is a consequence of the displacement of atoms within the molecule. The induced electric dipole moment resulting from elastic displacements of ions within the molecule may again be represented by an expression of the type equation (5) , by replacing by the atomic polarizability .It should be noted that refers to an average over all possible orientations of the molecule with respect to the field. In lecture (1) it will be shown that may be considered a constant up to frequencies in the infrared spectrum. For most molecules, is of the order of 10 per cent of .

Figure 2: (a) A NaCl chain in the NaCl crystal without an applied field. Average or net dipole moment per ion is zero.(b) In the presence of an applied field, the ions become slightly displaced, which leads to a net average dipole moment per ion.

 

3.3 Orientational (Dipolar) Polarization:

 

Certain molecules possess permanent dipole moments. For example, the HC1 molecule shown in Figure 3 (a). a has a permanent dipole moment 0from the Cl- ion to the H+ ion. In the liquid or gas phases, these molecules, in the absence of an electric field, are randomly oriented as a result of thermal agitation, as shown in Figure 3(b). When an electric field E is applied, E ries to align the dipoles parallel to itself, as depicted in Figure 3(c). The Cl- and H+ charges experience forces in opposite directions. But the nearly rigid bond between Cl- and H+ holds them together, which means that the molecule experiences a torque r about its center of mass. This torque acts to rotate the molecule to align 0 with E.

 

However, due to their thermal energy, the molecules move around randomly and collide with each other and with the walls of the container. These collisions destroy the dipole alignments. Thus the thermal energy tries to randomize the orientations of the dipole moments. A snapshot of the dipoles in the material in the presence of a field can be pictured as in Figure 3(d) in which the dipoles have different orientations. There is, nonetheless, a net average dipole moment per molecule that is finite and directed along the field. Thus the material exhibits net polarization, which leads to a dielectric constant that is determined by this orientational polarization.

 

Figure 3. dipoles in the material in the presence of a field

 

To find the induced average dipole moment along E, we need to know the average potential energy Edip of a dipole placed in a field E and how this compares with the average thermal energy 52 per molecule as in the present case of five degrees of freedom. Edip represents the average external work done by the field in aligning the dipoles with the field. If 52 is much greater than Edipthen the average thermal energy of collisions will prevent any dipole alignment with the field. If, however, Edip is much greater than 52 , then the thermal energy is insufficient to destroy the dipole alignments.

 

A dipole at an angle ? to the field experiences a torque ? that tries to rotate it, as shown in Figure 3(c). Work done dW by the field in rotating the dipole by ?? is ? ??.

 

(as in F dx). This work dW represents a small change dE in the potential energy of the dipole. No work is done if the dipole is already aligned with E, when ?=0, which corresponds to the minimum in PE, On the other hand, maximum work is done when the torque has to rotate the dipole from ?=180° to ?=0° (either clockwise or counterclockwise, it doesn’t matter).

 

Let us define the potential energy of a dipole making a 90 angle with the external field as zero. The potential energy corresponding to an angle? between ? and E is then equal to

 

? ?cos?=?.?                             ………(6)

 

According to statistical mechanics, the probability- for a dipole to make an angle between and with the electric field is then proportional to

 

Where is the solid angle between?and ?+??. Hence the average component of the dipole moment along the field direction is equal to

The function L(a) is called the Langevin function, since this formula was first derived by Langevin in 1905 in connection with the theory of paramagnetism. In Fig. 4,L(a) has been

plotted as a function of .Note that for very large values of a, i.e., for high field strengths, the function approaches the saturation value unity. This situation would correspond to complete alignment of the dipoles in the field direction, because then  ?〈cos?〉=?.

 

As long as the field strength is not too high and the temperature is Fig. 6-5. The Langevin function L(a).not too low, the situation may be For a < 1, the slope is 1/3.strongly simplified by making the approximation a1 or .EkT. Under these circumstances the Langevin function L(a) = a/3, so that then

 

   As an example of the condition implied in (10), consider a field of 3000 volts per cm. The dipole moment p of a molecule is of the order of 10-10to esu of charge times 10-8 cm, i.e., about 10-18cgs units, so that .E 10-17 in cgs units. On the other hand, kT at room temperature is of the order of 10-14 erg and for this example the condition is certainly satisfied. In this example saturation would be approached only in the vicinity of 1°K. It may be noted that the quantum mechanical treatment of this problem leads essentially to the same results as obtained here.

 

Figure 4. The Langavin curve L(a) , for a≪ 1 the slope is 1/3.

 

2.4 The static dielectric constant of gases:

 

We are now in a position to give an atomic interpretation of the static dielectric constant of a gas. It will be assumed that the number of molecules per unit volume is small enough so that the interaction between them may be neglected. In that case, the field acting at the location of a particular molecule is to a good approximation equal to the applied field E. Suppose the gas contains N molecules per unit volume; the properties of the molecules will be characterized by an electronic polarizability an ionic polarizability and a permanent dipole moment . From the discussions in the preceding two sections it follows. that, as a result of the external field E, there will exist a resulting dipole moment per unit volume or polarization is:

Note that only the permanent dipole moment gives a temperature dependent contribution, because and are essentially independent of T. If the gas fills the space between two capacitor plates of area A and separation d, the total dipole moment between the plates will be equal to

 

??????=? ? ?                                                              ………(12)

 

This simple relation shows immediately that the same total dipole moment would be obtained by assuming that the dielectric acquires an induced surface charge density P at the boundaries facing the capacitor plates. It represented the induced surface charge density at the dielectric-plate interface. Therefore, combination of (   − 1) = 0 = and (1l) leads immediately to the Debye formula for the static dielectric constant of a gas

 

  As an example of an application of this formula, we show in Fig.(5) the temperature dependence of some organic substances in the gaseous state. Note that (??−1) has been plotted versus the reciprocal of the absolute temperature, leading to straight lines, in agreement with formula (13). From the slope of the lines and knowledge of the number of moleculesper unit volume, the dipole moment ?may be obtained. Also, from the extrapolated intercept of the lines with the ordinate, one can calculate(??+??). The determination of dipole moments has contributed a great deal to our knowledge of molecular structure. For example, CCl4 and CH4, according to Fig.(5), do not possess permanent dipole moments, in agreement with the symmetric structure of these molecules. Similarly, the fact that H2O has a dipole moment of 1.84 Debye units, whereas CO2 has no dipole moment, indicates that the CO2 molecule has a linear structure, whereas in H2O the two OH bonds must make an angle different from 180o with each other.

 

Figure5. Temperature variation of the static dielectric constant of some vapors

 

    Summary:
  1. We considered mechanisms of polarization and determined polarizability
  • electronic polarization (which induces dipoles at all),
  • ionic polarization (which shifts existing ions),
  • orientation polarization (which rotates existing dipoles),
  1. The static dielectric constant of gases has been discussed in detail.
you can view video on Dielectric Properties Lecture 2

   References:

  1. J. Daintith (1994). Biographical Encyclopedia of Scientists. CRC Press. p. 943. ISBN 0–7503-0287-9.
  2. James, Frank A.J.L., editor. The Correspondence of Michael Faraday, Volume 3, 1841– 1848, “Letter 1798, William Whewell to Faraday, p. 442.”. The Institution of Electrical Engineers, London, United Kingdom, 1996. ISBN 0-86341-250-5
  3. Microwave Engineering – R. S. Rao (Prof.). Retrieved2013-11-08.
  4. P. Debye (1913), Ver. Deut. Phys. Gesell. 15, 777; reprinted 1954 in collected papers of Peter J.W. Debye Interscience, New York
  5. Chiang, Y. et al.: Physical Ceramics, John Wiley & Sons1997, New York
  6. Giere, A.; Zheng, Y.; Maune, H.; Sazegar, M.; Paul, F.; Zhou, X.; Binder, J. R.; Muller, S.; Jakoby, R. (2008). “Tunable dielectrics for microwave applications”. 2008 17th IEEE International Symposium on the Applications of Ferroelectrics. p. 1. doi:10.1109/ISAF.2008.4693753.ISBN 978-1-4244-2744-4
  7. Bunget, I., & Popescu, M. (1984). Physics of solid dielectrics. Amsterdam: Elsevier.
  8. Kao, K. (2004). Dielectric phenomena in solids with emphasis on physical concepts of electronic processes. Amsterdam: Academic Press.
  9. Kasap, S. (2006). Principles of electronic materials and devices (3rd ed.). Boston: McGraw-Hi

    References and Suggestive Readings

  1.  Mussig & Hans-Joachim, Semiconductor capacitor with praseodymium oxide as dielectric, U.S. Patent 7,113,388published 2003-11-06, issued 2004-10-18, assigned to IHP GmbH- Innovations for High Performance Microelectronics/Institute Fur Innovative Mikroelektroni
  2. Kuhn, U.; Lüty, F. (1965). “Paraelectric heating and cooling with OH–dipoles in alkali halides”. Solid State Communications 3 (2): 31. doi:10.1016/0038-1098(65)90060-8.
  3. “Self-correcting crystal may lead to the next generation of advanced communications”. KurzweilAI.doi:10.1038/nature12582. Retrieved 2013-11-08.
  4. Lee, C. H.; Orloff, N. D.; Birol, T.; Zhu, Y.; Goian, V.; Rocas, E.; Haislmaier, R.; Vlahos, E.; Mundy, J. A.; Kourkoutis, L. F.; Nie, Y.; Biegalski, M. D.; Zhang, J.; Bernhagen, M.; Benedek, N. A.; Kim, Y.; Brock, J. D.; Uecker, R.; Xi, X. X.; Gopalan, V.; Nuzhnyy, D.; Kamba, S.; Muller, D. A.; Takeuchi, I.; Booth, J. C.; Fennie, C. J.; Schlom, D. G. (2013). “Exploiting dimensionality and defect mitigation to create tunable microwave dielectrics”. Nature502 (7472): 532–536. doi:10.1038/nature12582.PMID 24132232.
  5. “Electrically tunable dielectric materials and strategies to improve their performances”. Progress in Materials Science55: 840–893. 2010-11-30.

    Web Links

 

1.      http://chemwiki.ucdavis.edu/Physical…Polarizability

2.      http://chemwiki.ucdavis.edu/u_Materi…ezoelectricity

3.      http://www.qmul.ac.uk/~ugez644/index.html#microwave 

4.      http://www-users.aston.ac.uk/~pearcecg/Teaching/PDF/LEC2.PDF

 

Additional Topics to be studied

 

1.  Interface and space charge polarization

2.Lorenz model

 

Interface and space charge polarization

 

Surfaces, grain boundaries, interphase boundaries may be charged, i.e. they contain dipoles which may become oriented in an external field and thus contribute to the polarization of the material. Conducting granules in insulated matrix may play a role of induced dipoles and cause the space charge polarization.

In the absence of a field, there is no separation between the positive charges and the negative charges. In the presence of an applied field, the mobile positive ions migrate toward the negative electrode but remains in the dielectric (electrode is blocking). At the another electrode developes a net negative charge. The dielectric therefore exhibits space charge polarization

 

 

The charge layer at electrode interface is called double layer due to its structure structure: it consists itself of two layers: Stern layer, where no mobile carriers are present and diffuse layer, where a balance between diffusive and drift currents is developed. If both electrodes are blocking two double layers of different charge polarities develops at their interfaces.

 

The conducting grains in a dielectric matrix become dipoles due to electrostatic induction. Grain boundaries and interfaces between different materials frequently give rise to interfacial polarization even if both the materials are dielectrics

Polarization vector

 

In order to describe the bulk material the sum of all the particles we sum up all individual dipole moments contained in the volume unit of the material. This gives us the polarization vector P

 

 

If dipolar moments of all the particles are equal and have the same direction

then

where n0 is concentration of the particles

 

Polarization vector

 

If we want to know the charge density ρ inside a small probing volume, it is zero in the volume of the material, because there are just as many positive as negative charges. At the surfaces there is indeed some charge. At one surface, the charges have effectively moved out a distance l, at the other surface they moved in by the same amount. We thus have a surface polarization charge:

 

Local field

 

The equation P = χ ∙ E refers to the external field, i.e. to the field that would be present in our capacitor without a material inside. On the other hand, the induced dipole moment (or electric force which acts at the molecular dipole) depends on the local field at the place of the molecule. The factor which connects the induced dipole moment and the local electric field is the molecular polarizability α (basically a microscopic parameter)

All electrical fields can (at least in principle) by solving the Poisson equation. It couples the charge distribution and the potential V(x, y, z):

Doing this is pretty tricky, however. We can obtain usable results in a good approximation in a much simpler way by using Lorentz model. Let’s remove a small sphere (containing a few 10 atoms) from the material. We want to know the local field in the center of this sphere while it is still in the material.

 

Lorenz model

 

Our local field consists of three components:

 

●  external E

●  field of polarized continuous dielectric outside the sphere Ec

●  field of near atoms located inside the sphere Eb

The calculation of Ec is a standard problem from electrostatics. Electric field caused by a charge at the surface of the sphere: