Study of Dielectric Properties of
Department of Physics
Magnetism is one of the oldest and an important field in science and technology. By virtue of the different magnetic behaviour exhibited by different materials, they have been put into different applications. Certain materials has also been developed and designed artificially while others are tuned to suit different conditions, applications and needs. Over the years, magnetic materials have been part of the revolutionary development in the field of electronics as well.
On the basis of the response and behaviour of materials in the presence of an applied magnetic field, materials have been broadly classified into the following categories:
Different theories which are based on different assumptions have been developed to explain their distinct response to the applied magnetic field and the exhibition of persistent magnetisation by some materials even in the absence of an applied magnetic field.
Lodestone appears to be the first magnetic material to have been used and studied extensively. It belongs to the class of ferrimagnets. Its chemical formula is given by Fe3O4.
Ferrimagnets exhibits uncompensated antiferromagnetism. This is because the antiparallel magnetic moments in the constituent sub-lattices are not totally compensated. Thus they exhibit spontaneous magnetisation (though lower than ferromagnets) below a characteristic temperature called Neel temperature.
One of the properties of ferrites which make them an important class of materials is the exhibition of a notably strong magnetism and at the same time with a very high electrical resistivity (of the order of 10 10 times that of ferromagnetic iron.1 This property reduces the eddy current loss when they are used as cores of transformers at very high frequencies. The size can also be made so small that the transformer principle can be extended to digital circuits and computer memory arrays. The phenomenon of precession of that an unpaired electron can exhibit in response to an applied magnetic field, forms a basis for the application of ferrites in wave guiding. 2 Ferrofluids production is another application of ferrites with applications in industrial as well as medical fields.3
Fig-1: Structure of cobalt ferrite. 4
A unit cell of a ferrite crystal has 8 formula units. The oxygen atoms make up a f.c.c structural lattice. This arrangement gives rise to two types of site vacancies i.e., the tetrahedral (A) sites and the octahedral (B) sites for the divalent and the trivalent ions to occupy (Fig -1). In normal spinels, all the divalent metal ions occupy the A-sites and the trivalent metal ions occupy the B-sites. Whereas in inverse spinels, half of the divalent metal ions occupy the A-sites and the other half along with the divalent metal ions are distributed among the B-sites. Ferrites in general are very flexible in terms of tuning of their properties. Substitution of iron ions from Fe3O4 with either a transition metal or a rare earth metal changes its electrical as well as its magnetic properties.1
Cobalt ferrite is an example of inverse spinel ferrites. It forms a very stable structure with a very large fraction of volume which is left vacant thus makes it suitable for electron hopping which may be treated as cation migration. It is highly versatile and exhibit remarkable phenomena like moderate saturation magnetization, magnetic anisotropy and high coercivity.5
The electrical and magnetic properties of cobalt ferrite can be suitably tuned by substitutions and doping for cobalt or for iron with either transition or rare earth metal.
Among many researchers who have studied the effect of transition element substitution on electrical properties of cobalt ferrite, Y.D. Kolekar et al5 studied the effect of manganese substitution for iron in cobalt ferrite on its dielectric properties. Likewise A.C. Druc et al6 studied the change in electrical properties of magnesium ferrite due to cobalt substitution for magnesium.
R.C. Kambale et al7 have reported the effect of cobalt substitution for nickel in nickel ferrite on the its structure and electrical properties as well. They reported an increase in grain size with the increase in cobalt ion concentration in the material.
Many researchers have explored the effect of rare earth metal substitution for iron in ferrrites. Md. T. Rahman et al8 reported an increase in lattice parameter due to gadolinium substitution for iron in cobalt ferrite which is attributed to the large ionic radius of Gd3+ compared to those of Co2+ and Fe3+. They also reported the formation of secondary phase and a decrease in electrical conductivity with increase of gadolinium content. A. K. Pradhan et al9 on the other hand reported an increase in crystallite size with no notable change in lattice parameter because of molybdenum doping in cobalt-zinc ferrite. They also reported the domination of the contribution from grain boundary over that from the bulk.
Apart from the dependence of the properties of ferrites on the nature of the doping species and site occupation of the ions, the dependence of the properties on the methods of preparation and crystallite size have been studied as well. Mathew George et al10 have studied the difference in properties of cobalt ferrite with different crystallite size and reported the contribution of interfacial polarization in the different samples of cobalt ferrites. N. Ponpandian et al11 have studied the effect of grain size effect on the electrical properties of nickel ferrite which resulted in the change of the contribution of the grain boundary and the bulk of the grain relative to each other. This was reported in terms of the increase in the activation energy with grain size reduction.
Different material behaves differently in the presence of an electric field applied across them. Some are more susceptible to change induced by the applied field than the other. Based on this behaviour, materials are broadly classified as:
• Conductors: These materials has some electrons which are free to move and have the capability of producing a steady charge drift across the material when a steady electric field is applied across it thus conduct steady current.
• Insulators: These materials has got electrons which are not free to move but are rather bound to the nucleus and thus are incapable of producing steady current when an electric field is applied across them. These materials may also of charged species which are bound to each other and are immobile.
Dielectrics belong to the group of insulators in which although long range migration of charges may not be possible. However, the centres of positive and negative charges can be separated from each other by some finite distance. Thus if we look at individual species which made up the material we can say that a dipole moment can be induced in such materials.
When such a material is placed in between plates of a parallel capacitor with a steady field applied across the plates , the microscopic separation of charge centres produce a net accumulation of charges on two opposite surfaces of the dielectric such that they produce another electric field opposing the polarizing field.
The dipole moment of a dipole with two opposite charges of magnitude ‘Q’ separated by a distance ‘?’ is expressed as,
p = Q ? ___________________ (1)
Polarization is defined as the net dipole moment per unit volume of the material i.e.,
P = (Q ?)/v ____________________ (2)
Where, ‘v’ is the volume of the dielectric.
If ‘C’ is the be the capacitance of a parallel plate capacitor, then the amount of charge which can be stored in it (with free space between the plates) when a potential difference of ‘V’ is applied across the plates is given by,
Q =C V ____________________ (3)
Where, C = (?0 A) / d ____________________ (4)
Here, A is the cross sectional area of the plates, d is the distance of separation between the two plates and ?0 is the dielectric permittivity of free space.
In case when a dielectric is placed between the plates, then the amount of charge stored in the capacitor under the same condition of potential applied, is given by
Q’ = ? (?0 A /d) ____________________ (5)
Where, ? is the relative permittivity of the medium which is greater than one.
From equation (5) we can say that the amount of charge which can be stored is enhanced by a factor equal to the relative permittivity of the medium.
In terms of energy stored in the capacitor, we have
W = Q2 /2C ____________________ (6)
This indicates that the energy which can be stored in the capacitor also increased.
The main purpose of studying the dielectric properties of a medium is to understand the mechanism of polarization as well as conduction in the dielectric. Since in dielectrics long range charge transport is not possible, only displacement current exists which is the result of polarization. This suggests that polarization and conduction mechanisms are similar in dielectrics.
The following are the four different types of polarization mechanisms observed in dielectrics:
1. Electronic/atomic polarization: This is the result of the displacement of the electron cloud centre relative to the nucleus of an atom. Since all materials are made up of atoms, we can say that this type of polarization mechanism is common to all materials. Since electrons are very light particles, they can respond to the applied unsteady electric field almost instantaneously. This type of polarization persists up to very high frequency region which is of the order of 1015 Hz.
2. Ionic polarization: This type of polarization mechanism is exhibited by those materials which are made up of neutral species which in turn are made up of oppositely charged ions. Thus individual species may have some finite dipole moment but by symmetry of their arrangement in a crystal they cancel out each other so that there is no net dipole moment in the absence of an applied electric field. However in the presence of an applied electric field, the displacement of the ions from their equilibrium positions produces a net dipole moment in such material. If we consider the ions to be charged atoms, their masses are equivalent to that of the respective atoms and their displacement requires some finite time. Their response (in terms of fluctuating dipole moment) to the applied unsteady field is much slower than that due to the electrons. Thus this type of polarization mechanism persists at lower frequency region which is of the order of 1013Hz.
3. Dipolar polarization: Dipolar polarization is exhibited by substances which composed of species which individually has got some permanent dipole moment and are free to rotate with respect to each other. However in the absence of an applied electric field, they are distributed and oriented in such a way that their individual dipole moments cancel each other out so that there is no net dipole moment. Even in the presence of an applied electric field, the polarization effect of the field is opposed by the thermal agitation. Thus the effectiveness of the field is reduced and the result is the net contribution from the electric field and the thermal randomness, which is temperature dependent. The individual species are polar molecules which are made up of two or more atoms. The molecules are heavier than atoms or ions and their response time is larger than those of electrons and atoms. This type of polarization persists up to a frequency range of 109Hz.
4. Interface or space charge polarization: this type of polarization can be observed in heterogeneous dielectric medium. Due to its heterogeneity the dielectric permittivity of different parts of the medium may be different. Thus when an electric field is applied, there is charge accumulation at the interface between the different components. This type of polarization requires time (larger than the first three types) to develop as well as to be destroyed. It persists in low frequency range which is of the order of 10 Hz.
Fig-2. gives the pictorial representation of the different polarization mechanisms which can exist in a substance.
At low frequencies, the polarization is the result of the contribution from all the different types of polarization mechanisms. As the frequency of the applied electric field is increased, one by one of the polarization mechanism reaches resonance condition and cease to contribute to the overall polarization of the material (Fig-3). This happens because the entities involve in the mechanism can no longer follow the fast changing electric field.
Fig-2: Different polarization mechanisms. 12
Fig-3: Variation of dielectric constant with frequency 13
Dielectric constant and a.c. conductivity evaluation:
When an electric field is applied across a capacitor, the impedance offered by the capacitor to the applied field is given by,
Z = j/? C
Or, Z = j / (? ? C0) ¬¬_______________(7)
Where, C0 is the capacitance of the capacitor in the absence of a dielectric medium in between the plates.
Or, ? = j / (? C0 Z) ______________ (8)
If we split Z and ? into real and imaginary components, we have
?’ + j?” = j / ? C0 (Z’ + jZ”) ______________ (9)
Comparing the real and imaginary components of the above equation, we have
?’ = Z”/ ? C0 (Z’2 + Z”2) ______________ (10)
and ?” = Z’ / ? C0 (Z’2 + Z”2) ______________ (11)
The above expressions can be used to evaluate the values of the real and imaginary part of the relative dielectric permittivity.
The loss of energy over a complete cycle (charging and discharging) of the applied electric field in the case of an ideal dielectric is zero. However in real cases loss of energy will always accompany the process of charging and discharging. A fraction of the supplied energy is spend
1. For long range displacement of charge which is frequency independent and negligibly small for most dielectrics since long range charge transport is considered impossible.
2. To reduce the randomness in the case of dipolar dielectrics and to cause polarization, this is frequency dependent and involves the relaxation mechanism.
Taking these two factors into consideration, the total current supplied to a capacitor with a dielectric in between the plates becomes,
It = Ic + Il _______________ (12)
Where, It is the total current, I¬c is the charging current and Il is the loss current.
And Il = G0 + G (?)V _______________ (13)
Equation (12) may be rewritten as,
Ic = It – G0 + G (?)V _______________ (14)
When ? =0, Ic = (dQ/dt)
Ic = It – G (? = 0) V _______________ (15)
Using V = V0 exp (j?t)
And Q = C V,
We may rewrite equation (15) as
It – G (? = 0) V = C (dV/dt)
Or, It – G (? = 0) V =C0 (j??) V
Or, It – G (? = 0) V = C0 (?’ -j?”) V
Or, It = j??’C0V +??”C0 V + G0V
Comparing the above equation with the expression
It = Ic + G0 + G (?)V,
We get, G(?) = ??”C0
This is the expression for the frequency dependent conductance of the dielectric. The frequency dependent conductivity may be expressed as
?(?) = ??”?0 ______________ (16)
And the total a.c. conductivity of the dielectric medium may be expressed as,
? = ?0 + ?(?) ______________ (17)
The equivalent circuit modelling representation of the impedance data was first developed by Randles and Warburg. This technique has been applied to spectroscopy analysis of dielectrics and other materials to study their conduction mechanisms. Fig-4 gives an example of an equivalent circuit for a dielectric with its respective graphical representation (Cole-Cole plot) of the behaviour of the dielectric.
Fig-4: A. Equivalent circuit; B. Nyquist plot; C. Impedance vs. frequency plot.14
Objective of the present project:
In this project work, we aimed at understanding the basics of dielectric behaviour of materials especially that of the substituted cobalt ferrites. This includes:
1. Preparation of Mnx Co(1-x) Fe2O4 samples. ( x = 0.0, 0.10, 0.20, 0.30, 0.4 and 0.5 ) Where, M is a divalent transition metal.
2. Study the crystal structure of the prepared samples through X-ray diffractometer.
3. Study the electrical behaviour of the different samples by using LCR meter.
We have prepared poly-crystalline samples of Mx Co(1-x) Fe2O4 ( x=0.0, 0.10, 0.20, 0.30, 0.40 and 0.50).
Sol-gel method: The starting materials for this method are nitrates of high purity cobalt nitrate hexa-hydrate (Co (NO3)2.6H2O), iron nitrate nona-hydrate (Fe (NO3)3.9H2O), manganese acetate (Mn(C2H3¬O2)2) and Citric acid monohydrate (C6H8O7.H2O). The different nitrates and the acetate were weighted first in the required proportions as per required stoichiometric ratio using an electronic balance supplied by METTLER TOLEDO with an accuracy of 0.0001gm.The starting materials were dissolved either in distilled water. The materials were mixed. Excess amount of citric acid dissolved in distilled water was added to the mixture to convert them into citrates. Required amount of ethylene glycol was added to the mixture and heated at about 60oC under constant stirring by using a magnetic stirrer until a wet gel is obtained. The gel was dried at 1200c so as to obtain a fine powder. The resultant mixture was pressed into cylindrical disc shaped pellets of 13mm diameter by using a hydraulic hand operated press with a maximum load of 1.5ton/cm2. The compacts were sintered at 1050oC for 24h.
The X-ray diffractometer was used to study crystal structure of the sample and to detect the presence of secondary phase if any. It was carried out at room temperature. For X-ray generation, a setting of 100mA and 50KV was maintained. The rotating anode diffractometer (RIGAKU TTRAX) with a radiation of wavelength of 1.54056 Ao (Cu-K?) was used. The data were collected in a usual ?-? scan with an angular velocity of 30 per minute and a step size of 0.01o. (Fig-5 represents the ray diagram for studying the X-ray diffraction of a crystal.)
Fig-5: Ray diagram for X-ray spectroscopy 15
This measurement involves placing of the dielectric sample between two plates of a parallel plate capacitor. The variation of the dielectric constants and the loss tangent for the samples with different compositions was carried out by the use of Wayne Kerr Electronic model 4300 with two terminal sample holders and the LCR meter with working frequency range 20 Hz to 1MHz.
Impedance analysis involved the modelling of the impedance offered by the samples into an equivalent electrical circuit which was done with the help of the Zsimpwin software.
Results and discussions:
1. Variation of Z’ with frequency: The variation of Z’ with frequency is as shown in Fig-6. The plot shows a decreasing trend in the impedance value with increase in frequency for all the prepared samples. This gives the implication that the conductance increase with the increasing frequency. The plateau in the low frequency region is observed. This can be attributed to the lesser activeness of the conductive grains in the region and the main contribution is made by the space charge or the short range hopping mechanism. The curve then enters the negative slope region which marks the transition from lower frequency mechanism to the higher frequency mechanism of conduction involving the more conductive grain or the localised hopping mechanism.
Fig-6: Variation of Z’ with frequency
2. Variation of Z” with frequency: The variation of Z” with frequency indicates the presence of relaxation peaks which marks the transition of mechanism from lower frequency side to higher frequency side (Fig-7). The peaks are broad; this may be interpreted to be caused by the involvement of different mechanisms of polarization which involves entities of different masses. Thus they possess different relaxation time and resonate at different frequencies. Or in other words there is a distribution of hopping parameters. 16
Fig-7: Variation of Z” with frequency
3. Variation of ?’ with frequency: The variation of ?’ with frequency is as shown in Fig-8. The value of ?’ decreases with the increase in frequency for all the prepared samples and they finally becomes almost frequency independent. This can be attributed to the fact that at lower frequencies, all the possible mechanisms of polarization makes contribution towards the overall effect thus higher of ?’. As the frequency value increase, those polarization mechanisms which involve entities of higher masses cease to contribute because of the inability of the respective entities to follow the fast changing field. In the higher frequency region, only electron hopping can persists.
Fig-8: Variation of ?’ with frequency
4. Variation of ?” with frequency: The values of ?” with frequency shows a rapid decreasing trend with increasing frequency and finally the curves merge at high frequencies (Fig-9). The high ?” at lower frequency can be attributed to the involvement of the polarization mechanisms which have appreciable relaxation times. Thus more energy is spent in orientation or oscillation by virtue of their higher inertia.
Fig-9: Variation of ?” with frequency
5. Variation of a.c. conductivity with frequency: The a.c. conductivity of the different samples shows an increasing trend with increasing frequency (Fig-10). In ferrites, the electron hopping Fe3+ + e- ? Fe2+ is considered to be the principal mechanism of conduction. This is also considered as a mechanism of dipolar polarization. At lower frequency, this hopping between the two localised states is not very much active. As frequency increase, the increase in conductivity is also due to the higher activeness of the conductive grains.
The a.c. conductivity vs. frequency curves were fitted with the Jonscher’s universal power law16.
? = ?0 +A(?)n
where, ?0 is the frequency independent or the d.c. conductivity and A is the pre-exponential factor.
The exponential ‘n’ can have values from 0 to 1 for conduction mechanism which involves hopping of electron or ions between states which are either neutral when occupied or neutral when empty. The values of ‘n’ is greater than one for hopping mechanism which are equivalent to the dipole rotation.17
Fig-10: Fitted a.c. conductivity vs. frequency curves.
6. Impedance spectroscopy: The impedance plots are obtained for different samples at room temperature. For each sample, there is only a single semi-circular arc in the higher frequency side which indicates the main contribution from the grains (Fig-11). The smaller arc diameter indicates the smaller Z’ and Z” values even at lower frequency region (in which the grain boundary is supposed to make major contribution) indicates the lesser grain boundary contribution in the respective samples.
Fig-11: Nyquist plot for different samples (lines represents the calculated values and symbols represents the observed values).
Polycrystalline samples of manganese substituted cobalt ferrite (Mnx Co(1-x)Fe2O4) with x ranging from 0 to 50 per cent of the total cobalt content have been prepared by the sol-gel method. The dielectric properties of the prepared samples have been studied with a range of frequency 100Hz to 1MHz. The Koops model of dielectrics 18 according to which a dielectric is made up of highly conducting grains separated from each other by less conducting boundaries is valid for the variation of the different quantities with frequency. At low frequency, the boundary made the major contribution while at high frequency the grains made the major contribution. The impedance spectroscopy indicates the presence of single arcs (on higher frequency side) for each sample indicating the domination of the grain over the boundaries.
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