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2.1 Introduction
Theoretical aspects of various structural, electrical and magnetic parameters
relevant to the present study are briefly discussed in this chapter. Structural
parameters provide crystallographic information, lattice dimensions, site preferences
in the spinel lattice sites and surface microstructures such as grain size, impurities,
segregations at the grain boundary and composition related data.
The saturation magnetisation of ferrites reflects on the distribution of various
metallic cations over A-and B-sites. Curie temperature tells about the higher limit of
temperature at which the material would work safely in a device. The understanding
of permeability and magnetic losses enables to estimate the performance of the
material at a given frequency of operation. Electrical properties such as resistivity,
dielectric constant and dielectric loss give information on electrical losses, grain
structure, activation energies and distribution of ions in the spinel structure. They
also provide information related to suitability of the materials for work at a given
frequency of operation, in the present case at microwave frequencies.
2.2 Structural aspects
The structural aspects to be dealt in the study of the present materials include
crystal structure of the ferrite, distribution of the metallic cations in the
crystallographic sites of the ferrite, factors influencing the distribution of cations,
determination of crystal structure and its associated parameters like lattice constant,
density, porosity, etc. The microstructures, elemental compositions of the
investigated ferrites and their cation distributions will also be covered in this section
2.2.1 Spinel ferrite structure
The spinel structure was first determined by Bragg and Nishikava [1]. In the
ideal structure of a spinel, the anions form a face centered cubic (fcc) close packing
in which the cations partly occupy the tetrahedral and octahedral interstices. The unit
cell contains 32 anions forming 64 tetrahedral interstices and 32 octahedral
interstices; of these 8 tetrahedral and 16 octahedral sites are occupied by cations.
These are called A- and B- sites, respectively. The general formula of compounds
with spinel structure is AB2O4, and the space group is Fd3m. The unit cell of the
spinel crystal structure, and the tetrahedral (A-site) and the octahedral (B-site)
interstices are shown in Fig.1.
In the A-site, the interstitial is in the centre of a tetrahedron formed by four
lattice atoms. Three anions, touching each other, are in plane; the fourth anion sits in
the symmetrical position on the top at the center of the three anions. The cation is at
the center of the void created by these four anions. In this configuration, four anions
are occupied at the four corners of a cube and the cation occupying the body center
of the cube. However in the B-site, the interstitial is at the center of an octahedron
formed by 6 lattice anions. Four anions touching each other are in plane, the other
two anions sites in the symmetrical position above and below the center of the plane
formed by four anions. Cation occupies the void created by six anions forming an
octahedral structure. This configuration shows that six anions occupy the face
centers of a cube and cation occupies the body center of the cube. Each A-site cation
is surrounded by 4 nearest neighbor oxygen ions and 12 next nearest neighbor B-site
cations. Similarly, each B
neighbor oxygen ions and 6 next nearest neighbor A
Figure 2.1: Unit cell of the cubic spinel alongwith the tetrahedral and o
interstices
The interstices available in an ideal close packed structure of rigid oxygen
anions can incorporate only those metal ions with radius
sites and only those ions with radius
accommodate larger cations such as Co, Cu, Mn, Mg, Ni and Zn with ionic radii in
the range from 0.56 to 0.83 Å the lattice has to be expanded. Thus, the ideal
could not be realized, and the oxygen anions in the spinel structures are generally not
located at the exact positions of the fcc sub lattice. Accordingly, the difference in the
expansion of the tetrahedral and octahedral sites is characterized b
called oxygen parameter
cations. Similarly, each B-site cation in the lattice is surrounded by 6 nearest
neighbor oxygen ions and 6 next nearest neighbor A-site cations.
ell of the cubic spinel alongwith the tetrahedral and o
The interstices available in an ideal close packed structure of rigid oxygen
anions can incorporate only those metal ions with radius r tetra ≤
sites and only those ions with radius rocta ≤ 0.55 Å in octahedral sites. So in order to
accommodate larger cations such as Co, Cu, Mn, Mg, Ni and Zn with ionic radii in
the range from 0.56 to 0.83 Å the lattice has to be expanded. Thus, the ideal
could not be realized, and the oxygen anions in the spinel structures are generally not
located at the exact positions of the fcc sub lattice. Accordingly, the difference in the
expansion of the tetrahedral and octahedral sites is characterized b
called oxygen parameter (u).
site cation in the lattice is surrounded by 6 nearest
site cations.
ell of the cubic spinel alongwith the tetrahedral and octahedral
The interstices available in an ideal close packed structure of rigid oxygen
≤ 0.30 Å in tetrahedral
Å in octahedral sites. So in order to
accommodate larger cations such as Co, Cu, Mn, Mg, Ni and Zn with ionic radii in
the range from 0.56 to 0.83 Å the lattice has to be expanded. Thus, the ideal situation
could not be realized, and the oxygen anions in the spinel structures are generally not
located at the exact positions of the fcc sub lattice. Accordingly, the difference in the
expansion of the tetrahedral and octahedral sites is characterized by a parameter
2.2.2 Distribution of metal ions
An interesting property of ferrites to be extracted with the spinel crystal
structure is the so-called cation distribution. Depending up on the distribution of
metallic cations over the A- and B-sites in the spinel lattice, the ferrites again can
have three different configurations.
1) Normal
2) Inverse
3) Mixed
In normal ferrites, the divalent cations occupy the A-sites and only trivalent
cations occupy the B-sites are. The distribution in the normal ferrites is represented
by the formula (M2+)A [Me3+]B O2-4. Those cations enclosed by paranthesis ( )
occupy A-sites and comprise the tetrahedral sublattice, while the cations enclosed by
square brackets [ ] occupy B-sites and comprise the octahedral sublattice. A typical
example of normal spinel ferrite is bulk ZnFe2O4 [2].
In inverse ferrites, half of the trivalent ions occupy A-sites and the other half
of the trivalent cations and total divalent cations randomly occupy B-sites. These
ferrites are represented by the formula (Me3+)A [M2+Me3+]B O2-4. A typical example
of inverse spinel ferrite is bulk magnetite, Fe3O4, in which all the divalent cations of
Fe occupy only the B- sites [2].
Spinel ferrites with ionic distribution, intermediate between the normal and
the inverse configurations is known as mixed ferrites. The distribution in this case is
represented by the formula (M2+xMe3+
1-x)A [M2+1-xMe3+
1+x]B O2-4. Examples of having
mixed cation distributions are manganese-zinc and nickel-zinc ferrites [3].
In the assignment of cation distributions in ferrites, there is another parameter
of paramount importance in the literature is the degree of inversion. The inversion
parameter, δ, could be defined as the fraction of total number of divalent ions
occupying B-sites [3]. The magnitude of δ depends on several factors including the
method of preparation and nature of the constituents of the ferrites. For normal spinel
ferrites δ = 0 since all the divalent cations for such a configuration reside in A-sites
only, and for inverse spinel ferrites δ = 1 as all the divalent ions occupy B-sites only.
For mixed spinel ferrites, the δ ranges between these two extreme values.
2.2.3 Factors influencing the cation distribution
The useful structural, magnetic and electrical properties of spinels depend not
only on the kinds of cations in the lattice, but also on their distribution over the
available crystal lattice sites. It is thus of major importance to understand the factors
that contribute to the total lattice energy in spinels, i.e., (i) Elastic energy (ii)
Electrostatic (Made lung) energy (iii) Crystal field stabilization energy (iv) d-orbital
splitting and (v) polarization effects.
The elastic energy refers to the degree of distortion of the crystal structure due
to the difference in ionic radii assuming that ions adopt a spherical shape. Smaller
cations, with ionic radii of 0.2-0.4Å should occupy A-sites, while cations of radii
0.4-0.7 Å should enter B-sites. This distribution leads to a minimum in lattice strain.
Since trivalent cations are usually smaller than divalent ones, they can be assumed to
force a situation where the larger sized divalent cations migrate to B-sites, and hence
a tendency towards the inverse arrangement would be expected.
The Madelung constant of the spinel structure has been calculated by Verwey
et a1.[4] as a function of oxygen parameter u and the charge distribution among A-
and B- sites. Their results showed that this energy is dependent on the u-parameter.
For u>0.379, the normal distribution is more stable, while for lower u values the
inverse arrangement possesses a higher Madelung constant. It therefore implies the
presence of two kinds of cations in B-sites in inverse spinels leads to an additional
contribution to the Madelung energy. The critical u value then becomes 0.38142.
Madelung energy is higher for the normal spinel if u >0.381 and the inverse ordered
spinel is more stable for u < 0.381.
Crystal field factors used to help account for the site preferences in spinels.
Romeijn [5] was the first one who suggested the application of the crystal field
theory to understand the cation site preference in spinels. Dunitz and Orgel [6] has
calculated the octahedral site preference energies of transition metal ions in oxides
using crystal field theory (CFT) The data show that the systems with d5 and d10
configurations (Fe3+, Mn2+ and Zn2+) have no CFSE and hence no site preference.
The d3 system (Cr3+) has the highest octahedral site preference energy. The d4 and
d9 (Mn3+ and Cu2+) ions can be further stabilized by Jahn-Teller distortion.
Another factor which plays a role in cation distribution is d-orbital splitting
energy. Although, the CFSE contribution to the total bonding energy of a system is
only about 5-10%; it may be the deciding factor when other contributions are
reasonably constant. When the various factors are counter balancing, there can be a
completely random arrangement of metal ions among the 8 tetrahedral and 16
octahedral sites.
2.2.4 Crystal structure determination
Phase identification shall be done by comparing the observed structural
parameters with the standard values reported. The number and position of the peaks
in X-ray diffractin (XRD) pattern confirms whether the materials exhibits single-
phase spinel structure or not. From the experimental values of θ, distance between
the atomic planes 'd' can be calculated using the Bragg relation.
2d sinθ = nλ
where λ is the wavelength of incident radiation (X- rays) and θ is the diffraction
angle.
To find the crystal structure, we need to determine the lattice constant. It is
related to the inter planar spacing for the hkl planes in the cubic structure as
2 2 2hkld h k l a+ + = (1)
where hkl is the plane of reflection. From the above equation, one can get the value
of lattice parameter 'a'.
2.2.5 Lattice constant
A parameter defining the unit cell of a crystal lattice is the length of one of the
edges of the cell or an angle between edges. It is also known as lattice parameters or
lattice constant. Lattice constant refers to the constant distance between the lattice
points. It is calculated using equation (1). Lattice in three dimensions generally has
three lattice constants, referred as a, b, c. However, in cubical structures, all these
three constants are equal and we only refer to “a”.
2.2.6 Density
Density related properties have no connection with an external factor and so
are not mechanical. They must, however, be considered first before any other
property of the materials can be studied.
Measured density is an intrinsic property of materials that denote the
relationship between its mass and unit volume. It is used as an index property or an
independent variable to predict other properties of the materials and is difficult to
characterize because this parameter can be affected by temperature, pressure and the
amount of substitution of different elements. The measured density is determined
using the following formula
2m
m
r hρ
π=
(2)
where m is the mass, r is the radius and h is the height of the sample
2.2.7 X- Ray Density
The X- ray density of the prepared samples of ferrites is calculated by the
relation
3
8X ray
M
Naρ − =
(3)
where M is the molecular weight of the samples, N is the Avogadro’s number and
a is the lattice constant. Each cell has 8 formula units.
2.2.8 Porosity
The storage capacity of any material is referred to as the porosity of a material
depending on the shape, size of grains and on the degree of their storing and packing.
The porosity P of the samples is then determined using the relation
1 m
X ray
Pρ
ρ −
= − (4)
where mρ and X rayρ − are the measured and X-ray densities, respectively
2.2.9 Scanning electron microscopy
Scanning electron microscope (SEM) plays a vital role in the characterization
of materials. SEM can provide internal and surface imaging of all materials with the
magnification, 10,000 x plus. which is not possible in optical microscopes. Atomic
scale resolution can be, achieved at such higher magnifications.
Electron Microscopes use a beam of highly energetic electrons to examine
objects on a very fine scale. The electrons emitted from the source are accelerated
and shaped by the condenser lenses and objective lens. This lens-system facilitates
changing the current in the electron beam that bombards the sample and controls the
beam convergence. The interaction of the electrons with the matter in the sample
produces different signals (electrons, photons from the infrared through visible till
the X-ray range, etc.) These signals can be used for imaging different sample
characteristics, like surface morphology or distribution of electrically active crystal
defects or local composition, etc. The measured characteristics can be presented
qualitatively and interpreted intuitively or can be presented quantitatively.
SEM can mainly yield the information related to topography, which include
surface features of an object and how it looks, and morphology representing shape
and size of the grains making up the object.
2.2.10 Energy Dispersive X-ray Spectroscopy
Energy dispersive X-ray spectroscopy (EDS) is a technique used for the
compositional analysis of a material. It characterizes a sample through the
interactions between electromagnetic radiation and matter and analyzing the X-rays
emitted by matter in this particular case. As each element of the periodic table has a
unique electronic structure so it has a specific response to electromagnetic waves. In
this technique, the spectroscopic data are plotted as a graph of counts vs energy.
A beam of electrons or photons is bombarded to the sample which is to be
characterized. When an atom is at rest within the sample, it contains ground state
electrons in discrete energy levels around the nucleus. The incident beam may excite
an electron from the inner shell, and results in the formation of an electron-hole
within the atom’s electronic structure. An electron from an outer shell then fills the
hole, and the excess energy of that electron is released in the form of an X-ray. The
X-rays released in this way, create spectral lines that are highly specific to individual
elements. In this manner, the X-rays emission data are analyzed to characterize the
sample.
2.2.11 Fourier Transform Infrared Spectroscopy
Fourier transform infrared (FTIR) spectroscopy studies have been carried out
for the ferrite samples under present investigation to understand their spinel
structures. The FTIR spectroscopy is also highly relevant for surface analysis
because it allows the investigation of the chemical composition and the nature of the
chemical bonds on the outermost layer.
When the molecule is excited with IR radiation, the frequency of the incident
radiation coincides with the vibrational frequency of some part of the molecule.
Hence, resonance occurs and energy is absorbed. This absorbed energy transforms
the vibrational energy of the molecules due to change in dipole moment. When the
molecule returns to the ground state from its excited state, the absorbed energy is
released resulting in distinct IR peaks in the spectrum.
2.3 Electrical Properties
The electrical properties of the investigated spinel ferrites deal with dc
resistivity as a function of composition and temperature, activations energies for
conduction, dielectric constant and dielectric loss tangent as a function of
composition and frequency. It also gives an estimate of the performance as a
microwave magnetic material.
2.3.1 DC resistivity
Ferrites should have high electrical resistivities to eliminate eddy current
losses and allow full penetration of electromagnetic fields through out the solid.
High resistivity is obtained only when a cation has one valance in one lattice site.
The electronic conduction in ferrites is mainly due to hopping of electrons between
the ions of the same elements present in more than one valence state, distributed
randomly over crystallographically equivalent lattice sites [7]. High sintering
temperatures and reducing atmospheres tend to produce mixed valance cations of an
element in the equivalent lattice sites in an otherwise stoichiometric ferrite, thereby
reducing the resistivity.
Ferrites are ionic solids and their electrical resistivity (ρ) drops exponentially
with rising temperature (T), according to the relation [8]
ρ = ρ0 exp (E / KT)
where ρ0 is the resistivity at 0°C
E is the activation energy, i.e., energy needed for an electron to
jump from one ion to the neighbouring ion of same element
and K is the Boltzmann constant
From the plots of log ρ versus 1/T, activation energies are obtained which
explain the involved conduction mechanisms in ferrite [9, 10].
2.3.2 Dielectric constant
The dielectric properties of polycrystalline materials are generally determined
by a combination of various factors like method of preparation, sintering temperature,
grain size, substitutions used, the ratio of Fe3+/Fe2+ ions, and ac conductivity [11,12]
In the present work the dielectric properties of Ni-Cu and Ni-Mg substituted
Lithium ferrites have been measured by using capacitance bridge method. Air-dried
silver epoxy electrical contacts were deposited on the flat surfaces of sintered pellets,
and the dielectric constant ('ε ) and the dielectric loss tangent (tanδ ) were calculated
using the formula
'
o
Cd
Aε
ε=
Where ‘C ’ is the measured capacitance
‘d’ is the thickness of the sample
‘ A’ is the area of the capacitor’s plate
‘ oε ’ is the permittivity of free space and its value is -128.85×10 F/m
It has been known since the investigations of Blechschmidt [10] in 1938, that
ferrites have high dielectric constants at low frequencies falling to very low values of
ten to twenty at microwave frequencies. The low and high frequency values of
dielectric constant and the dispersion behaviour at frequencies smaller than 1010 This
cannot be explained by atomic polarization mechanism. Koops [13] found that the
dielectric constant of ferrite followed much the same course with changing
frequency as did the electrical resistance. To interpret the frequency response of
dielectric constant in ferrite materials, Koops suggested a theory by assuming that
the grains and grain boundaries of ferrite material can be represented the behaviour
of an in homogeneous dielectric structure, as discussed by Maxwell [14] and Wagner
[15], assuming that the same current flows through both the grain and the grain
boundaries.
As there are evident contradictions between Koops theory and experiments,
Krausse [16] developed a new model on the basis of Schottky barrier layer theory in
semiconductors [17]. According to this model, the ferrite structure can be
represented as a series of barrier layer and undisturbed grain material, i.e.,
electrically a series of combination of RC elements. This is analogous to Koops
model, however the poorly conducting layers are not chemical in nature but
correspond to Schottky barrier layers.
2.4 Magnetic Properties of spinel ferrites
The magnetic properties chosen for a theoretical treatment in this section
include saturation magnetization, coercivity, initial permeability and magnetic loss.
The experimentally obtained and theoretically calculated (as per the Neel’s
molecular field theory) magnetic moments of the materials provide valuable
information related to the distribution of cations among the tetrahedral and
octahedral lattice sites of the ferrites investigated. Followed by the complex
permeability measurements on the toroidal ferrite samples up to a few GHz, an
attempt has been made to characterize the material in terms of the Snoek’s product
for its applicability at microwave frequencies.
2.4.1 Saturation Magnetization
Saturation magnetization (Ms) of a material is the resultant dipole moment
per unit volume when the dipole moments associated with all the molecules are
aligned in the direction of the applied magnetic field, and is given by
Ms = N µ m . . . 2.3
where N is number of dipoles, and µ m is the dipole moment.
A magnetic ferrite crystal has domain structure similar to that of
ferromagnetic metals. The magnetization of each domain is associated to the
magnetic moment from the individual ions present in the ferrite, the free oxygen
atom has a partially unfilled 2p sub-shell which gets filled on acquisition of two
electrons when the free oxygen converts to O2− negative ion. The oxygen ions
therefore have no magnetic moment and so make no direct contribution to the
magnetic moment of the domain. The magnetic moments of the cations arise from
unfilled sub-shells. As in ferromagnetic metals and in paramagnetic solids the
magnetic moment due to the orbital motion of the electrons is quenched by internal
fields. The ionic magnetic moments are therefore due to parallel uncompensated
electron spins in the ions. In case of an atom of iron there are four uncompensated
spins in the 3d sub-shell, so that the iron atom has magnetic moment of 4 µB (µB
being the Bohr magneton) due to electron spins. The divalent iron ion having lost
two electrons from the 4s shell has a magnetic moment of 4 µB . On becoming
trivalent it loses as additional electron from the 3d sub-shell increasing the
uncompensated electron spins to 5 and the resultant magnetic moment becomes 5 µB
.
Between the spinning electrons in the neighbouring metal ions strong
quantum mechanical forces of interaction occur. In ferrites, according to Neel [18],
there exist three kinds of exchange interactions: the interaction between the various
magnetic ions located at A-site (A-A interaction), the interaction between the various
magnetic ions located at B-site (b-B interaction), and the interaction of magnetic ions
at A-sites with those at B sites (A-B interaction); of those A-B interaction
predominates in strength over A-A and B-B interactions. These interactions align all
the magnetic spins at A-site in one direction comprising A-sub lattice and those at B-
site in the opposite direction comprising B-sub lattice. The net magnetic moment of
the lattice is therefore the difference between the magnetic moments of B and A sub
lattices, i.e., M = M MB A− .
An interesting aspect is exhibited by the mixed zinc ferrites in which the ionic
distribution is given by [19].
( ) [ ]Zn Fe M Fe Ox x x x
2
1
3
1
2
1
3
4
2+−+
−+
++ −
assuming A-B interaction to be the predominant and that M2+ ions have a magnetic
moment of m units and Fe3+ ions have a theoretical magnetic moment of 5 µB , the
resultant moment per formula unit would be
M = M MB A−
= m (1 - x) + 5(1 + x) - 5(1 - x)
= 10x + m (1 - x)
this relation is shown by the broken lines in fig. 2.1. Experimentally observed
decrease in magnetic moment for x values greater than 0.5 is due to the fact that for
larger amount of zinc, the A-sublattice gets so much diluted by the diamagnetic zinc
that A-B interaction weakens and B-B interaction becomes important. The
magnetization measurements help us in knowing the cation distribution over A-and
B-sites and the role of exchange interactions.
2.4.2 Curie temperature
Magnetic properties of materials arise form unpaired electron spins, that result
in a net magnetic moment. Perfect alignment of magnetic spins of all molecules is
possible only at zero Kelvin. Randomizing effects of thermal motion cause net
magnetic moment to decrease as temperature increases from zero Kelvin; this
decrease becomes more rapid as the temperature nears the Curie point. The Curie
point is the temperature at which the aligning effect of the exchange interaction is
cancelled by the disordering effect of random thermal motion.
E j = E thermal
that is, - 2l j Z S2 = - 9S µB H eff = 3K Tc
where H eff = effective magnetic field,
E j = exchange energy,
Z = number of near neighbours, and
S = spin quantum number.
From the above equation it is obvious that the Curie temperature is
proportional to the exchange energy and the number of nearest magnetic neighbours.
The Curie temperature increases as the number of spins per atom increases, and this
increased spin momentum in turn results in a stronger interaction between the
neighbouring magnetic ions. The idea of Curie temperature gives an estimate of the
temperature at which the device can be operated safely.
2.4.3 Initial permeability
Like all ferromagnetic substances ferrites exhibit hysteresis. The ratio of
magnetic induction (B) to applied magnetic field (H) at a point on the hysteresis
loop, that represents the state of magnetisation of the material, is called the
permeability of the material and is written as
µ = B/H
The value of permeability for an initially unmagnetized ferrite specimen in the
presence of an infinitesimally small magnetic field is called as initial permeability (
µ j ).
Since ferrites are used at high frequencies, it is interesting to see how athe
permeability varies with frequency. According to Snoek [20], ferromagnetic
resonance frequency, fres is connected to the permeability, µ rot and the saturation
magnetization, Ms by the following relation
f res(µ rot -1) = 4/3 γ Ms
Where γ is a constant, called gyromagnetic ratio. For a certain value of
maximum application frequency (fres ), the highest permeability is obtained for the
material with the highest saturation magnetization. Another parameter that
determines µ rot and f res is magnetic anisotropy, K1 ; µ rot increases in inverse
proportion and fres proportionally to the anisotropy.
Polder in his exhaustive work [21] had subsequently separated the complex
permeability into real and imaginary components and established the importance of
imaginary permeability in determining the magnetic losses in ferrite materials.
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