7 Dielectric Properties Lecture 6

K Asokan

epgp books

   

 

 

Learning Outcomes:

 

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

1.     Detailed study of complex permittivity

2.     The fundamental Dielectric relaxation.

3.     You will learn about Debye equations and its complex plane diagram.

 

The dielectric constant and loss are important properties of interest to physics people because these two parameters, among others, decide the suitability of a material for a given application. The relationships between the dielectric constant and the polarizability under dc fields have been discussed in sufficient detail in the previous lectures. In this chapter we examine the behavior of a polar material in an alternating field, and the discussion begins with the definition of complex permittivity and dielectric loss which are of particular importance in polar materials.

 

6.1 COMPLEX PERMITTIVITY:

 

Consider a capacitor that consists of two plane parallel electrodes in a vacuum having an applied alternating voltage represented by the equation

 

 

Here  is the instantaneous voltage,the maximum value of  and  is the angular frequency in radian per second. The current through the capacitor, is given by

 

 

 

 

 

 

 

In this is the vacuum capacitance , sometimes also called the geometrical capacitance.

In an ideal dielectric the current leads the voltage by 90° and there is no component of the current in phase with the voltage. If a material of dielectric constant  is now placed between the plates the capacitance increases to  and the current is given by

It is noted that the usual symbol for the dielectric constant is , but we omit the subscript for the sake of clarity, noting that &is dimensionless. The current phase or will not now be in phase with the voltage but by an angle (90°-) where  is called the loss angle. The dielectric constant is a complex quantity represented by

The current can be resolved into two components; the component in phase with the applied voltage is and the component leading the applied voltage by 90° is(fig.1). This component is the charging current of the ideal capacitor.

 

The component in phase with the applied voltage gives rise to dielectric loss. is the loss angle and is given by

 

 

is usually referred to as the loss factor and  the dissipation factor. To complete the definitions we note that

Figure1. Real () and imaginary () parts of the complex dielectric constant () in an alternating electric field.

 

The alternating current conductivity is given by

 

6.2 Dielectric relaxation:

 

When a direct voltage applied to a dielectric for a sufficiently long duration is suddenly removed the decay of polarization to zero value is not instantaneous but takes a finite time. This is the time required for the dipoles to revert to a random distribution, in equilibrium with the temperature of the medium, from a field oriented alignment. Similarly the buildup of polarization following the sudden application of a direct voltage takes a finite time interval before the polarization attains its maximum value. This phenomenon is described by the general term dielectric relaxation.

 

When a dc voltage is applied to a polar dielectric let us assume that the polarization builds up from zero to a final value (fig. 2) according to an exponential law

 

Where P(t) is the polarization at time t and  is called the relaxation time, i is a function of temperature and it is independent of the time.

 

The rate of buildup of polarization may be obtained, by differentiating equation (8) as

Substituting equation (8) in (8a) and assuming that the total polarization is due to the dipoles, we get

 

Neglecting  atomic  polarization  the  total  polarization  PT(t)  can  be  expressed  as  the  sum  of  the orientational polarization at that instant, , and electronic polarization,  which is assumed to attain its final value instantaneously because the time required for it to attain saturation value is in the optical frequency range. Further, we assume that the instantaneous polarization of the material in an alternating voltage is equal to that under dc voltage that has the same magnitude as the peak of the alternating voltage at that instant.

 

We can write total polarization as

 

where C is a constant. At time t, sufficiently large when compared with , the first term on the right side of equation (17) becomes so small that it can be neglected and we get the solution for P(t) as

Equation (20) shows that  is a sinusoidal function with the same frequency as the applied voltage. The instantaneous value of flux density D is given by

Equations (25) and (26) are known as Debye equations and they describe the behavior of polar dielectrics at various frequencies. The temperature enters the discussion by way of the parameter  as will be described in the following section. The plot of is known as the relaxation curve and it is characterized by a peak at = 0.5. It is easy to show = 3.46 for this ratio and one can use this as a guide to determine whether Debye relaxation is a possible mechanism.

 

6.3 DEBYE EQUATIONS:

 

One another way to express the Debye equations are

 

    Equations (25)-(27) are shown in fig 2. The conclusion of these equations shows are the following

(1)   For small values of, the real part because of the squared term in the denominator of equation (28) and  is also small for the same reason. Of course, at = 0, we get = 0 as expected because this is dc voltage.

(2)  For very large values of, and is small.

(3)  For intermediate values of frequencies is a maximum at some particular value of.

 

figure2. Schematic representation of Debye equations plotted as a function of log.

 

The values of  and at this value of  are

 

 

The dissipation factor tan also increases with frequency, reaches a maximum, and for further increase in frequency, it decreases. The frequency at which the loss angle is a maximum can also be found by differentiating tanwith respect to  and equating the differential to zero. This leads to

 

These equations shows that

Fig.(2) shows that, at the relaxation frequency defined by equation , decreases sharply over a relatively small band width. This fact may be used to determine whether relaxation occurs in a material at a specified frequency. If we measure  as a function of temperature at constant frequency it will decrease rapidly with temperature at relaxation frequency. Normally in the absence of relaxation s’ should increase with decreasing temperature.

 

Variation of  as a function of frequency is referred to as dispersion in the literature on dielectrics. Variation of as a function of frequency is called absorption though the two terms are often used interchangeably, possibly because dispersion and absorption are associated phenomena.

 

6.4 COMPLEX PLANE DIAGRAM:

 

Cole and Cole showed that, in a material exhibiting Debye relaxation a plot of against , each point corresponding to a particular frequency yields a semi-circle. This can easily be demonstrated by rearranging equations (25) and (26) to give

 

The right side of equation (35) may be simplified using equation (25) resulting in

practice very few materials completely agree with Debye equations, the discrepancy being attributed to what is generally referred to as distribution of relaxation times.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Summary:

 

In this present chapter Debye equation was derived and its complex plane diagram was discussed. Complex permittivity was defined and derived. Dielectric relaxation were discussed and described the behavior of polar dielectrics at various frequencies.

you can view video on Dielectric Properties Lecture 6

   References:

  1. Peter Y. Yu, Manuel Cardona (2001). Fundamentals of Semiconductors: Physics and Materials Properties. Berlin: Springer. p. 261. ISBN 3-540-25470-6.
  2. José García Solé, Jose Solé, Luisa Bausa, (2001). An introduction to the optical spectroscopy of inorganic solids. Wiley. Appendix A1, pp, 263. ISBN 0-470-86885-6.
  3. John H. Moore, Nicholas D. Spencer (2001). Encyclopedia of chemical physics and physical chemistry. Taylor and Francis. p. 105. ISBN 0-7503-0798-6.
  4. Chiang, Y. et al.: Physical Ceramics, John Wiley & Sons 1997, New York
  5. 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 Mikroelektronik

    References and Suggestive Readings

  1. C.J.F. Bottcher, O.C. von Belle & Paul Bordewijk (1973) Theory of Electric Polarization: Dielectric Polarization, volume 1, (1978) volume 2, Elsevier ISBN 0-444-41579-3.
  2. Arthur R. von Hippel (1954) Dielectrics and Waves ISBN 0-89006-803-8
  3. Arthur von Hippel editor (1966) Dielectric Materials and Applications: papers by 22 contributors ISBN 0-89006-805-4.
  4. Jackson, John David (August 10, 1998). Classical Electrodynamics (3 rd ed.). John Wiley & Sons. ISBN 978-0-471-30932-1. 808 or 832 pages.
  5. Scaife, Brendan (September 3, 1998). Principles of Dielectrics (Monographs on the Physics & Chemistry of Materials) (2 nd ed.).Oxford University Press. ISBN 978-0198565574.

    Web Links

 

1.       http://www.lightandmatter.com/html_books/0sn/ch11/ch11.html

2.       http://wiki.4hv.org/index.php/Dielectric_Sphere_in_Electric_Field

3.       http://www.doitpoms.ac.uk/tlplib/dielectrics/index.php

4.       http://falsecolour.com/aw/eps_plots/eps_plots.pdf

    Additional Topics to be studied

 

1)   Dielectric relaxation time

2)  Complex permittivity of metals

 

Debye relaxation

 

Debye relaxation is the dielectric relaxation response of an ideal, non interacting population of dipoles to an alternating external electric field. It is usually expressed in the complex permittivity ε of a medium as a function of the field’s frequency ω:

 

where ε∞ is the permittivity at the high frequency limit, Δε = εs − ε∞ where εs is the static, low frequency permittivity, and τ is the characteristic relaxation time of the medium.

 

This relaxation model was introduced by and named after the physicist Peter Debye (1913).

 

Variants of the Debye equation

 

1)  Cole–Cole equation:

 

This equation is used when the dielectric loss peak shows symmetric broadening The Cole–Cole equation is a relaxation model that is often used to describe dielectric relaxation in polymers.

 

It is given by the equation

 

where  is the complex dielectric constant,  and  are the “static” and “infinite frequency” dielectric constants,  is the angular frequency and  is a time constant.

 

The exponent parameter , which takes a value between 0 and 1, allows to describe different spectral shapes. When , the Cole-Cole model reduces to the Debye model. When , the relaxation is stretched, i.e. it extends over a wider range on a logarithmic scale than Debye relaxation.

 

Cole-Cole relaxation constitutes a special case of Havriliak-Negami relaxation when the symmetry parameter (β) is equal to 1 – that is, when the relaxation peaks are symmetric. Another special case of Havriliak-Negami relaxation (β<1, α=0) is known as Cole-Davidson relaxation, for an abridged and updated review of anomalous dielectric relaxation in dissored systems see Kalmykov.

 

2)   Cole–Davidson equation

 

This equation is used when the dielectric loss peak shows asymmetric broadening.

 

3)Havriliak–Negami relaxation

 

This equation considers both symmetric and asymmetric broadening.

 

Havriliak–Negami relaxation is an empirical modification of the Debye relaxation model, accounting for the asymmetry and broadness of the dielectric dispersion curve. The model was first used to describe the dielectric relaxation of some polymers, by adding two exponential parameters to the Debye equation: