Ceramic Science 4RO3
Lecture 5
October 7, 2013
Tannaz Javadi
Magnetic properties
Susceptability (χ): similar to dielectric materials, is a parameter
which expresses magnetic response of electron in a material to
the applied magnetic field and is a dimensionless quantity.
M: induced magnetization
H: magnetic field
Diamagnetic:
• Materials with small negative susceptibility (χm < 0)
• χm in Superconductors are -1 (Perfect Diamagnet).
• Atoms have no net magnetic moments
• Exposed to a field, a negative magnetization is produced (inherent effect)
• The susceptibility is temperature independent
o quartz (SiO2): -0.62x10-8 m3/kg, Calcite (CaCO3): -0.48x10-8 m3/kg,
water: -0.90x10-8 m3/kg
 Materials with positive susceptibility are either paramagnetic,
ferromagnetic or ferrismagnetic (χm > 1).
Paramagnetism
• Atoms have a permanent non-zero net magnetic moment due
to the sum of orbital and spin magnetic moments.
• The magnetic moments randomly orientated due to thermal
fluctuations when there is no magnetic field.
• The moments align parallel to the field when magnetic field is
applied.
• Susceptibility is positive but very small for paramagnetic
materials
o Montmorillonite (clay) 13x10-8 m3/kg
Ferromagnetism
• The magnetic moments in a ferromagnet aligned parallel to each other
under the influence of a magnetic field.
• These moments will then remain parallel when a magnetic field is not
applied (unlike the moments in a paramagnet)
• Above Tc, the Curie temperature, all ferromagnetic materials become
paramagnetic.
• Because thermal energy is large enough to overcome the cooperative
The Curie temperature is an intrinsic
ordering of the magnetic moments.
property and is a diagnostic
parameter that can be used for
mineral identification
saturation magnetization goes to zero
Saturation Magnetization
• The maximum induced magnetic moment that can be obtained in a
magnetic field (Hsat); beyond this field no further increase in
magnetization occurs.
• An intrinsic property, independent of particle size but dependent on
temperature.
Ferromagnetic hysteresis loop
• Domain that has a direction closest to that of the applied field grows
at the expense of the other domains.
• Such growth occurs by motion of the domain walls.
• Initially domain wall motion is reversible, and if the applied field is
removed the magnetisation will return to the initial demagnetised
state.
• In this region the magnetisation curve is reversible and therefore does
not show hysteresis (OB)
Ferromagnetic hysteresis loop
(Point B): domains aligned in the direction of applied
field and
• saturation magnetization, MS(Point B), domains
completely aligned (- MS in the opposite direction).
Ferromagnetic Vs. Ferroelectric:
• In ferromagnetic materials, favourably
oriented domain growth at the expense
of unfavourably oriented domains.
• In ferroelectric materials, favourably
oriented domains nucleate and grow.
(Point C): the field is reduced to zero,
• the domains do not come back to
their configuration in the virgin state
• a net magnetization in the absence of
field called remnant magnetization, Mr
( -Mr in the opposite direction).
(Point D): magnetization bring back
to zero,
• an extra field in the opposite direction
is applied called coercive field,
- Hc. (+Hc in the opposite direction).
• hard magnet has large coercivity
• soft magnet when coercivity is small.
Antiferromagnetism
• Without an applied field, adjacent magnetic moments (electron
spins associated with magnetic atoms) align anti-parallel to each
other.
• Adjacent magnetic moments are equal in magnitude and opposite
therefore there is no overall magnetisation.
• This occurs below a particular temperature, called
Néel temperature (TN) above which the material behaves as a
paramagnet.
• Ilmenite
Ferrimagnetism
• Antiparallel alignment of moments at particular atomic sites
• Most of these materials consist of cations of two or more types (i.e.
magnetic moment of one crystal sub-lattice is anti-parallel to the other)
• The aligned magnetic moments are not of the same size.
• An overall magnetisation is produced (net magnetization is not equal
to zero) but not all the magnetic moments may give a positive
contribution to the overall magnetisation.
Example for Ferrimagnetic material
• Magnetite (Fe3O4)
• Spinel structure
• The large oxygen ions are close packed in a cubic arrangement
and the smaller Fe ions fill in the gaps.
• The gaps come in two flavors, (two magnetic sublattices)
tetrahedral site (A): Fe ion is surrounded by four oxygens
octahedral site (B): Fe ion is surrounded by six oxygens
• The spins on the A sublattice are antiparallel to those on the B
sublattice
A periodic table showing the elements and the
types of magnetism at room temperature
Diffusion
• Migration of the defects which happens via an
atomistic process called as diffusion.
• Diffusion
causes
changes
in
the
microstructures (sintering, creep deformation,
grain growth)
• Diffusion is also related to transport of
defects or electronic charge carriers (electrical
conductivity and mobility)
Electrical conductivity
•The electrical conductivity in ceramics is a sum of ionic and
electronic conductivity
o Ionic conductors applications:
 chemical and gas sensors,
 solid electrolytes
 fuel cell (optimization the fuel/air ratio in the engines
by using an oxygen sensor made of ceramic ZrO2 in
automobiles)
Atomic
diffusion
rates & electrical
conductivity
Defect types
Defect Concentration (T, PO2, Comp.)
Diffusion Kinetics
• Fick's First Law of Diffusion J:
diffusion flux (moles/m2-s), and basically
means the amount of material passing
through a unit area per unit time;
D: diffusion coefficient or diffusivity in m2/s;
 Diffusivity is a
temperature dependent
parameter
x: the position in m.
C: the concentration in m3 .
Q: activation energy,
D = D0 exp (-Q/kT)
K: Boltzmann's constant
D0: pre-exponential factor in m2/s.
Diffusion Kinetics
• Fick's Second Law of Diffusion
o It predicts how the concentration changes as a function of time
under non-steady state conditions
t: is the time in seconds
Driving force: the chemical potential which drives the
migration of species from regions of higher chemical
potential to lower chemical potential
Temperature Dependence of Diffusivity
In general, diffusivity can be expressed as
γ: is governed by the
possible number of jumps at
an instant
λ: is the jump distance and is
governed by the atomic
configuration and crystal
structure.
Schematic of the planes of
atoms with arrows showing
the cross-movement of species
Γ: the jump frequency (# of atoms jump/ s)
ν: the vibration frequency (s-1)
ΔG*: the activation energy of migration (J)
k: Boltzmann Constant (J/K).
 exponential temperature dependence
resulting in significant increase in the
diffusivity upon increasing the
temperature
Examples of Diffusion in Ceramics
Diffusion in lightly doped NaCl (NaCl containing small amounts of CdCl2)
CdCl2 → CdNa• + 2ClClx + VNa′
• In addition, NaCl will also have certain intrinsic sodium and chlorine vacancy
concentration (VNa′ and VCl•) due to Schottky dissociation, depending on the
temperature.
ΔGNa* is the migration free energy for sodium vacancies
[VNa‘] is the sodium vacancy concentration
the diffusivity of sodium ions is
governed by vacancy diffusion;
Vacancy concentrations depends on
dopant concentrations
The diffusivity dependence on temperature shows two regimes
Defect migration & Creation
the vacancy
concentration is
governed by
thermally intrinsic
defect creation
mechanism (Schottky
defect formation)
 high temperature intrinsic
diffusivity exhibits a steeper
slope with higher activation
energy which include not only
the
energy
for
defect
migration but also for defect
creation
Defect migration
Extrinsic region is
dominant where
vacancy
concentration is
constant as it is
determined by
the solution
concentration i.e.
[VNa'] = [CdNa•] .
 low temperature extrinsic
regime
where
vacancy
concentration is independent
of temperature and is
determined
by
solute
concentration
Mobility
Diffusivity
Conduction in
ionic compounds Mobility: velocity (ν) of an entity per unit driving
force (F);
F can be defined as either of chemical potential gradient or electrical potential gradient.
 The most general driving force for atomic transport: the virtual
force that acts on a diffusing atom or species and is due to negative
gradient of the chemical potential or partial molar free energy.
Where μi is the chemical potential of i and NA is the Avogadro's Number
Relation between mobility and diffusion
Absolute mobility, Bi is given by
 To obtain the relation between mobility and diffusivity of species, i, we need to
write the flux in a general form as a product of concentration, ci, and velocity, vi, i.e.
substituting for Fi
for an ideal solution with unit activity of species i
R is the gas constant. So, the change in the
chemical potential can be written as
Relation between mobility and diffusion
compare the above equation with Fick's first law, diffusivity of species
i can be written as
The above equation is called Nernst Einstein Equation
Analogue to the Electrical Properties
Electrical force is given as
φ is potential, E is the electric field, Zi is the atomic number, e is the electronic
charge and Zie is the total charge on the particle.
Using the above relations, one can write the flux, Ji , as
Now, since Ji = ci .vi , the velocity can be written as
So, defining electrical mobility, μi as velocity per unit electric field
Electronic conductivity
 electrons are free particles with a drift velocity, vd, under an applied
electric field.
F=ZeE
F: the force on an electron
• various scattering phenomenon control drift velocity.
• under an applied E, drift velocity increases as expressed by Newton's
law of motion
m is the mass of carrier,
v is velocity and τ is the relaxation time.
m* is the effective mass of the carrier
(the mass that it seems to have when
responding to forces)
Under steady state conditions
The relaxation time, τ, in metals and
semiconductors shows a temperature
dependence
And mobility, μ, is
o τ ̴T-3/2 due to thermal
scattering
o τ ̴ T3/5 due to impurity
scattering.
Ionic Conduction: Basic Facts
Conduction in ionic solids is often governed by concentration of
impurities, dopants and point defects.
Conduction happens through hopping type which is migration of
charges between either two dissimilarly charged ions or counter
migration of ions and vacancies.
NiO doped with Li2O under oxidizing conditions gives rise to oxidation
of Ni2+ ions to Ni3+ ions. Mixed presence of Ni ions in +2 and +3 states
leads to hopping type conduction of electrons between two states.
Thermally activated phenomenon (Mobility of charge carriers and hopping)
The higher the dopant concentration (within appropriate limits), the
higher the conductivity
Carrier concentration is independent of temperature (within extrinsic
region) and mobility is strongly affected by temperature
Ionic and Electronic Conductivity
Electrical conductivity (σi) is
defined as charge flux per unit
electric field with units (Ω-1cm-1) or
S/m. It can be expressed as
• For ionic species, we can apply
Nernst-Einstein equation
Ji = ci .vi
Ji is the flux of species i
Similar to diffusivity, temperature dependence of ionic
conductivity also exhibit extrinsic and intrinsic regions at low
and high temperature, respectively.
Total conductivity and Transference Number
Since all charged species contribute to the electrical conductivity, we
can write total conductivity as
Fraction of total conductivity carried by each charged species is called
as transference number, ti and is expressed as
and it is straightforward to see that
electronic conductor
ionic conductor
mixed conduction
Characteristics of Ionic Conduction
Long range migration of ionic charge carriers, the most
mobile species, through the lattice under application of an
electric field (e.g. migration of Na+ ions in soda-silicate glasses)
Dependent on the presence of vacant sites in
neighbourhood of mobile defects/ions.
Can occur through grain boundaries such as in
polycrystalline ceramics or through the lattice as in fast ion
conductors.
Characteristics of Ionic Conduction
When external field is absent, the thermal energy, kT, is required for
counter migration of ions and vacancies overcoming the migration
energy Ea, which is nothing but process of self diffusion.
In the presence of electric field, the potential energy is tilted to one
side leading to higher driving force for migration towards one side than
to another side.
Characteristics of Ionic Conduction
Ionic conductivity is promoted by
•Small ionic size
•Small charge i.e. less Coulomb interaction between ions
•Favourable lattice geometry
•Cations are usually smaller than anions and hence, they
diffuse faster. For example, in case of NaCl, smaller size
of Na+ ion (102 pm) as compared to Cl- ions (181 pm)
makes them diffuse faster
n: ionic density,
α: irreversibility of the jump
Theory of Ionic Conduction
• β' and β''-alumina: very high conductive ceramics (10 - 10-1 (Ω.cm)-1,
@300 K to 675 K) Typical activation energies: ~3.5-4.5 kCal/mole.
• Spinel oxides (Fe3O4):
0.5 (Ω.cm)-1 and have very low activation energies (0.35 kCal/mole),
representing almost temperature independent behaviour.
• Y2O3, HfO2, SiO2, Al2O3: insulating ceramics (10-5 to 10-14 (Ω.cm)-1,
@ 400 K to 1000 K
 Metals
• Ce ~ constant
• µe decreases as temp increases
• σ decreases as temp increases
 Semiconductors & insulators
• Ce increases as temp increases (dominates)
• σ increases as temp increases
 Ionic conduction
• Ci is either constant (extrinsic) or increases as temp
increases (intrinsic)
• µi increases as temp increases (diffusion)
• σ increases as temp increases
Examples of Ionic Conductors in Engineering
Applications
Conducting ceramics are used in a variety of applications
such as:
SiC and MoSi2 as heating elements and electrodes
ZnO and SiC as varistors (A semiconductor diode with
resistance dependent on the applied voltage) for circuit
protection
YSZ, β -Alumina as electrolytes in fuel cells and
batteries
Materials like YSZ in gas sensing applications
Superconductors
• Exhibit a sudden drop in
electrical resistance to exactly
zero when cooled below room
temperature.
• Happens at a specific
temperature called critical
temperature, TC.
• The phenomenon was
discovered by Heike Kamerlingh
Onnes in 1911 when he was
studying properties of mercury
at liquid helium temperatures.
Inset shows a magnified view near a
superconducting transition region.
Scaling behavior of mixed-state hall effect in MgB2 thin films, Physica C: Superconductivity, Soon-Gill Jung et. al., v 450, 2006
Superconductors
 elemental metals & metallic alloys: (Sn, Al, niobium nitride,
niobium-titanium, and niobium-germanium alloys), most of
these are superconducting at temperatures below 30 K.
 In 1986 Bednorz & Müller demonstrated
superconductivity in a perovskite structured lanthanum based
cuprate oxide (La2Cu2O4 ) which showed a TC of 35 K.
 chemical substitution in perovskite cuprates increases the
transition temperatures to 77 K and beyond.
o YBa2Cu3O7-x(YBCO):TC ~92 K
highest TC when they are slightly oxygen deficient (x = 0.15)
Superconductivity disappears at x ≈ 0.6, (structure of YBCO changes from
orthorhombic to tetragonal)
Meissner Effect
The magnetic field is completely expelled from the
interior of the superconductor, when it is placed within a
magnetic field.
 Application
magnetically levitated trains
Maglevs