Transcript 2.5 DEFFECTS IN CRYSTALS1

2.1 Atoms 1
- The atomic model used as a basis for understanding the
properties of matter has its origins in the  particle
scattering of Ernest Rutherford.
- Niels Bohr developed a dynamic model for the simplest of
atoms, the hydrogen atom. Using a blend of classical and
quantum theory
e2
v2
2
 me  me w r
2
4 0 r
r
- The total energy
(2.1)
 of the electron is made up of its kinetic
energy  k and its electrostatic potential energy  p . The value
of  p is taken as zero when the electron is so far removed
from the nucleus, I.e.at ‘infinity’ that interaction is negligible.
Hence
   p   k (2.2)
2.1 Atoms 2
e2
1
 
 m2v 2
4 0 r 2
Substituting from equation (2.1) gives
※  
e2
8 0 r
(2.3)
From the quantum condition for angular momentum,
me wr 2  nh (2.4)
which, substitutes with equation (2.1),leads to
2
2 2
e
n
h
2
w 
 2 4
2
3
4m  0 r
me r
or
me e 2
1

r 4 o n 2 h 2
(2.5)
2.1 Atoms 3
which substitutes into equation (2.3) to give
me e 4
1
 
32 2 02 h 2 n 2
(2.6)
‘de Broglie wavelength’
h

mv
(2.7)
Electron states are described by the solutions of the
following equation which was developed by Erwn
Schrodinger and which bears his name:
2m
   2 (   p )  0
h
2
(2.8)
2.1 Atoms 4
This from of the Schrodinger equation is independent of time and
so is applicable to steady state situation, The symbol  2 denotes
the operator



 2 2
2
x y z
2
2
2
To apply equation (2.8) to the hydrogen atom it is first transformed
into polar coordinates (r , ,  ) and then solved by the method of
separation of the variables, This involves writing the solution in the
form
 (r , ,  )  R(r )( )( )
(2.9)
2.1 Atoms 5
- Pauli Exclusion Principle states that there cannot be more than
one electron in a given state defined by particular set of values
for n, l , m1and . For a given principal quantum number n there
are a total of 2n 2 available electronic.
s
-Electrons occupy states such that following.
1.The S value is maximum allowed by the Pauli Exclusion
Principle,I.e. the number of unpaired spins is a maximum.
2. The L value is the maximum allowed consistent with rule 1.
3. The value of L  S when the shell is less than half-full and
L  S when it is mire than half-full. When the shell is just
half-full, the first rule requires L  0 so that J  S.
 -page 9
 -page 10~11 with reference
2.2 The arrangement of ions in ceramics
Fig 2.1 Packing of ions
(a)simple cubic packing showing an interstice
with eightfold coordination ;
(b)hexagonal close packing;
(c)cubic close packing showing a face-centred
cubic cell

Fig 2.1(a) in simple cubic packing, the centres
of the ions lie at the corners of cubes formed
by eight ions.
it is generally found that anion lattices will
accommodate oversize cations more readly
than undersize cations so that the tolerance to
4
the relatively small Zr ion is exceptional

Fig 2.1(b) and (b)The oxygen ions are more
closely packed together in the close-packed
hexagonal and cubic structures.
2.2 The arrangement of ions in ceramics
MO6

Fig 2.2(a) on this basis the rutile from of TiO2
consists of columns of edge-sharing TiO6
octahedra linked by shared coners of the TiO6
units

Fig2.2(b)A hexagonal from of BaTiO3 ,where the BaO3
lattice is hexagonal close packed,contains layers
of corner-sharing TiO6 groups(Fig2.2(c))
TiO2
BaTiO3
TiO2
Fig 2.2
MO6 octahedra arrangements in
(a)perovskite-type structures,
(b) TiO2
And (c)hexagonal BaTiO3
2.3 Spontaneous polarization







Value of a crystal property depend in the direction of measurement
Crystals having cubic symmetry are optically isotropic
For these reasons, a description of the physical behaviour of a material
has to be based ion a knowledge of crystal structure
Purpose to distinguish polar crystals, spontaneously polarized and
possess a unique polar axis, from the non-polar variety
Of the 32 crystal classes, 11(centrosymmetric) and non-polar variety
Of the remaining 21 non- centrosymmetric classes, 20(piezoelctric, of
these 10 are polar)
An idea of the distinction between polar and non-polar structures can be
gained from Fig.2.3
Fig.2.3 (a) Non-polar array; (b),(c) polar arrays. The arrows indicate the direction of spontaneous
polarization Ps
2.3 Spontaneous polarization




Piezoelectric crystals : when stressed, polarized or change in polarization
When an electric field is applied, become strained
The 10- polar crystal types(pyroelectric, piezoelectric) : because of their
spontaneous polarization
Pyroelectric crystal :
change in temperature => change in spontaneous polarization
additional features cannot be predicted from crystal structure
some material that the direction of the spontaneous polarization can be
changed by an applied electric field or mechanical stress
change due to electric field = said to ‘ferroelectric’
change due to stress = said to ‘ferroelastic’
Poling process: conditions of temperature and time, static electric field
after poling => ferroelectric ceramic makess it capable of poezoelectric,
pyroelectric, electro-optical behaviour
2.3 Spontaneous polarization

The changes in direction of the spontaneous polarization:
-The tetragonal(4mm) structure allows six direction
-The rhombohedral(3m) allows eight direction
-both tetragonal and rhombohedral crystallites are present at a transition
point, the number of alternative crystallographic directions rises to 14 and
the extra alignment attained becomes of practical significance
2.4 Transitions
-Ionic size and the forces are both temperature dependent and may change sufficiently
for a
particular structure to become unstable and to transform to a new one
-If a system is described in terms of the Gibbs function G then, because the molar
S : entropy , V : volume , P: pressure
entropies and
molar volumes of the two
G phase do mot change, the change in G for the system can be
S  ( ) P
written
T
dG =- SdT + Vdp
-It follows that
-‘first-order’ transition: if there is a discontinuity in the derivative of G(T) there is a
G
G
change in


G
G
T
T
entropy at constant temperature,
which implies
latent heat
-‘second-order’ transition: when the first derivative of the Gibbs
is continuous
Latentfunction
heat
but the
second derivative is discontinuous
T
T
T
T
Free-energy changes at transitions : (a)fist-order transition; (b) change in S at constant T and, consequently, late
(c)second-order transition; (d)continuous change in entropy and so no latent heat (discontinuity in
)
2.5 Deffects in crystals1
2.5.1 Non-stoichiometry
-Manganese dioxide is a well-established compound but it always contains
less than the stoichiometric amount of oxygen
:The positive charge deficiency can be balanced by vacant oxygen site
-Iron monoxide always contains an excess of oxygen
:The charge excess can be balanced
-LiNbO3 :The O ions are hexagonally close packed with a third of the octahedral
sites occupied by Nb5+,a, third by Li + and a third empty.
:This can be deficient in lithium down to the level Li0.94Nb1.012O3.
5+
:There is no corresponding creation of oxygen vacancies.instead the Nb
content increases sufficiently to preserve neutrality
-BaTiO3 :there is marked difference in charge and size between the two cations
corresponding to differences between their lattice sites,to complete
solid solutions over the whole possible range where the ions are
identical in charge and close in size and can only occupy one type of
available lattice site such as, Zr4+ and Ti in Pb(ZrxTi1-x)O3
2.5 Deffects in crystals2
2.5.2 Point defects
2.5 Deffects in crystals3
-Point defects
:They occur where atoms are missing(vacancies) or occupy the interstices
between normal sites(interstitials):’foreign’ atom are also point defects
-Line defects,or dislocations
:They are spatially extensive and involve disturbance of the periodicity of the
lattice
-Frenkel and Schottky Defects: paired anions and cations
•
Electronic neutrality must be maintained in crystal.
•
Defects must come in pairs to maintain Q=0.
•
Cation-vacancy + Cation-interstitial = Frenkel Defect
•
In AX-type crystals,
•
Cation-vacancy + Anion-vacancy
= Schottky Defect
(Q=0)
(Q=0)
2.5 Deffects in crystals4
-The equilibrium concentrations of defects in a simple binary oxide MO
nS&nF: Schottky and Frenkel defect concentrations respectively
H S
nS  N exp(
)
ΔHS&ΔHF:enthalpy change accompanying the formation of the associated
2kT
defects.
 HF
(cation vacancy+ anion vacancy and ion vacancy + interstitial
nF  ( NN `)1/ 2 exp(
)
ion)
2kT
-The notation of Kroger
N :concentration of anions or cations
of available interstitial sites
andN`:concentration
Vink
Net positive, negative
.
AgAg  Agi  VAg`
interstitial
vacancy
.
VO  VO  e`
..
VO  VO  e`
.
VM  V `M  h
-In summary,a chemical equation involving defects must balance in 3 respects
1.the total charge must be zero;
2.there must be equal numbers of each chemical species on both sides;
3.the available lattice sites must be filled,if necessary by the introduction of
vacant sites
2.5 Deffects in crystals5
4+
3+
-The introduction of an acceptor Mn on a Ti site in BaTiO3 can be expressed as
Mn 2O3  2 BaO ⇌ 2 Ba Ba  2Mn `Ti  5OO  VO
(2.15)
-which replaces the equilibrium equation for the pure crystal
2TiO2  2 BaO ⇌ 2 Ba Ba  2TiTi  6OO
-Since BaO=BaBa+Oo, equation(2.15) simplifies to
Mn 2O 3 ⇌ 2Mn `OO  VO
(2.16)
-The equilibrium constant for equation(2.16) is
..
..
..
[Mn`Ti]2 [VO]
KA 
[Mn2O3]
-KA is expressed as a function of temperature by
HA
ΔHA : change in enthalpy of the reaction
KA  K `A exp( 
)
K`A : temperature-insensitive constant
kT
3+
-The replacement of Ba in BaTiO3 by the donor La is represented by
1
⇌2LaBa  2OO  2 O2( g )  2e`
La2O3..........
-The equilibrium constant KD is
1/ 2
n: electron concentration
[ LaBa]2 n 2 p O 2
HD
KD  K `D exp(
)
kT
[ La 2O3]
.
2.6 Electrical conductuion1
2.6.1 Charge transport parameters
-The current density j will be given by
j=nQv
-If the drift velocity of the charge is proportional to the force acting on them,
v=uE
u : mobility
 j=nQuE
-For materials for which nQu is constant at constant temperature
j  E
:σ(conductivity) = nQu =(1/ρ)
Ohm’law:
R  G 1  l / A
v
A
Q
v
Flow of charge in a prism E
2.6 Electrical conductuion2
log( / Sm1 )
-The room temperature values of conductivity
characteristic of the broad categories of material
8
Metals
4
0
-4
Semi-conductors
and semi-insulators
insulators
-8
-12
-16
300
1000
T/K
Conductivities of the various classes of material:shad
indicates the range of values at room temperature
Material class
Example
Conductivity
level
dσ/dT
Carrier type
Metals
Ag,Cu
High
Small,negative
Electrons
semiconductors
Si,Ge
Intermediate
Large,positive
Electrons
Semi-insulators
ZrO2
Intermediate
Large,positive
ions or electrons
Insulators
AlO3
Very low
Very large,
positive
Ions or electrons,
Frequently ‘mixed’
Conductivity characteristics of the various classes of material
2.6 Electrical conductuion3
2.6.2 Electronic conduction
(a) Band conduction
Empty conduction
band
EF
Forbidden
zone
Valence band
Energy
(a)Atomic levels having identical energies merging
to a broad band of levels differing slightly in energy
as free atoms condense to form a crystal; (b) band
structure At equilibrium interatomic spacing in a
crystal.
Partly filled
band
Bands arising
from inner
electron
levels
Schematic electron energy band structures for (a) a metallic
crystal and (b) a semiconducting or insulating crystal
2.6 Electrical conductuion4
-The electron density in the conduction
ni  
Etop
Z ( E ) F ( E )dE
Ec
(2.28)
Z(E)dE: total number of states in the energy range
around E per unit volume of the solid
-The Fermi-Dirac function F(E)
E  EF
F ( E )  {exp(
)  1}1
kT
EF : Fermi energy
(2.29)
Conduction band
- -
-
-
-
-
-
-
-
-
-
-
-
Energy
Eg
-Ec
EF
Ev
+
+
+
+
+
+
+ + +
+ Valence band
+
+
+
+
Band structure with electrons promoted from the valence to the conduction band
2.6 Electrical conductuion5
- E-EF>>kT : at room temperature kT≒0.025eV , E-EF: over the 0.2eV
the term +1 can be omitted from equation(2.29)
- The excited electrons and holes occupy states near the bottom of the
conduction band and the top of the valence band respectively
The electrons and holes behave as free particles for state distribution function
- The upper limit of the integration in equation (2.28) is taken as infinity since
since the probability of occupancy of a state by an electron rapidly approaches
zero as the energy increases through the band. Under these assumption,
Ec  EF
EF  Ev
ni  Nc exp( 
), pi  Nv exp( 
)
kT
kT
Nc&Nv:effective state densities for
electrons in the conduction band
and hole in the valence band
-If we put ni=pi,
EF 
Ec  Ev
2
-A more rigorous treatment shows that
Ec  Ev 3kT
me*
EF 

ln( * )
2
4
mh
me * & mh * :the effective electron and hole masses
2.6 Electrical conductuion6
*
*
- under conditions in which me  mh ,EF≒ the center of the band gap.
ni  pi  10 25 exp( 
Ec  EF
Eg
)  10 25 exp( 
)
kT
2kT
- From equation (2.25) the conductivity can be written
  nuee  puhe
ue & uh :electron and hole mobilities
  nie(ue  uh)  1025 e(ue  uh) exp(
Eg
)
2kT
-theory and experiment show a temperature dependence for u lying typically in
the range T 1.5 ~ T 2.5 ,which is so weak compared with that for n (and p) that for
most purpose it can be ignored
Eg
  B exp( 
)
2kT
2.6 Electrical conductuion7
(b) The effect of dopants
- The configuration resembles a hydrogen atom for which the ground state(n=1)
4
(2.39)
Energy E   m2ee 2 2
32  0 h
.
- doping n≠p, the equilibrium relation : e`+ h ⇌ nil
-’nil’ indicates a perfect crystal with all electrons in their lowest energy states
Eg: band gap(at 0K)
from equation(2.39) [e`][h.]  np  K (T )  k `exp(  Eg )
K`: independent of temperature
kT
Conduction band
e-
e-
e- e-
e-
Eq=0.01eV
Donor states
Eg=1.1eV
Acceptor states
Eg=0.01eV
+
+
+
+
+
Valence band
Planar representation of.a silicon crystal doped
With P5+ giving rise to a Psi defect
Effect of n-and p-type doping on the band structure of
a semiconductor (e.g. silicon)
2.6 Electrical conductuion8
-Semiconducting oxide
-
 Several cases of oxide systems in which the conductivity is
controlled by the substitution of aliovalent cations (Chapter 4).
 A limitation to present research is the non-availability of oxides
that approach the parts in 109 purity of available silicon crystals.
 The study of semiconduction in oxides has necessarily been
carried out at high temperatures( >500◦C) because of the
difficulties of making measurements when they have become
highly resistive at room temperature.
1. Oxygen pressure (1)

The features of oxide semiconductors is the effect on their behaviour
of the external oxygen pressure.
Po2 : Oxygen pressure
Pa : Atmosphere pressure
σ : Conductivity
Fig. 2.12 Conductivity of undoped BaTiO3 as a function of pO2 and T.

The general shape of the curves in Fig.2.12 can be explained on the
assumption that the observed conductivity is determined by electron
and hole concentrations.

The electron and hole mobilities depend only on temperature.
1. Oxygen pressure (2)

An estimate of K(T) can be arrived at as follow.
Combining equations (2.35) and (2.40) leads to

uhK (T )
 uen 
e
n
(2.41)
It follows that the value nm of n corresponding to a minimum σm in σ is
given by
uh
nm 2 
K (T )
ue
(2.42)
which, on substituting in equation (2.41), gives
 m 


 e 
2
 4ueuhK (T )
(2.43)
Combining equations (2.35) and (2.43) gives

1 

e
2 
where α=uhp/uen
(2.44)
1. Oxygen pressure (3)

Equation (2.44) enables the relative contributions of electrons and
holes to the conductivity to be estimated from the ratio of the
conductivity to its minimum value, without having to determine K(T).

If σ= σm, α=1 then uhpm=uenm
When n is large
e
e
 uen
(2.46)
When p is large

h

 u hp
(2.47)
ue = 0.808T-3/2exp(-εu/kT)m2V-1s-1 , Eu= 2.02kJmol-1 (0.021eV)
This give ue= 15X10-6m2V-1s-1 at 1000°C and 24X10-6m2V-1s-1 at
600°C. uh is likely to be about 0.5ue.
e
1. Oxygen pressure (4)

The further analysis of the dependence of σ on po for BaTiO3 is
mainly based on recent work by Smyth[3].
Controlled factor for Conductivity : po , n, p, VÖ, VT“i“(VB“a)
2
2

The various po regions are now considered separately for the 1000°C
isotherm of acceptor-doped or nominally pure BaTiO3.
A - B in Fig.2.13(a), the equilibrium reduction equation is
OO ⇔ ½ O2(g) + Vö + 2e’
(2.48)
which, by the law of mass action, leads to
Kn = n2[Vö]pO21/2
(2.49)
where Kn is the equilibrium constant.
The oxygen vacancy concentration is regarded as insignificant
compared with that arising through loss of oxygen.
Therefore, since
n  (2Kn)1/ 3 pO 2 1/ 6
(2.50)
2
1. Oxygen pressure (5)
Fig. 2.13 Schematic representation of the dependence of n, p, [VÖ] and [VT“i“]
on pO2 for (a) acceptor-doped and (b) donor-doped BaTiO3.(After Smyth)
1. Oxygen pressure (6)
B – D in Fig.2.13(a), the oxygen vacancy concentration, now determined by
the acceptor impurity concentration [A’], is little affected by changes in pO2
and remains sensibly constant.
1/ 2
 Kn 

n 
..
 [VO ] 



pO 2 1/ 4
(2.51)
The p-type contribution to semiconductivity arises through the oxidation
reaction involving take up of atmospheric oxygen by the oxygen vacancies
according to
Vö + ½ O2(g) ⇔ OO + 2h’
(2.52)
leading to
p = [Vö]1/2 Kp1/2pO21/4
(2.53)
At pO2 ≒ 100Pa, n = p, σ = σm and the material behaves as an intrinsic D-E
Over this pO2 regime the discussion is more speculative since measurement
against which the model can be checked have not been made.
1. Oxygen pressure (7)

In the region D-E the dominating defect changes from VÖ to VT“i“ since
the oxygen vacancies due to the acceptors are now filled.
The conductivity is largely governed by acceptor concentration and may
be independent of pO2 over a small pressure range.
In the E-F region the equilibrium is
O2(g) ⇔ VT“i“ + 2OO + 4h’
(2.54)
so that
K’p = p4[VT“i“]pO2-1
(2.55)
which, because p ≒ 4[VT“i“] leads to
p = (4K’p)1/5pO21/5

Measurements in the region 10-17Pa<pO2<105Pa as shown in Fig.2.12
show good agreement between the σ-pO2 slopes and the calculated
n-pO2 and p-p02 relations given above. Increased acceptor doping
moves the minimum in the σ-pO2 towards lower pressures.
2. Donor-doped BaTiO3(1)

The effect of pO2 on the conductivity of a
donor-doped system has been studied for
lanthanum-substituted BaTiO3 as shown in
Fig.2.14 for 1200°C.
The behaviour differs from that for acceptor
-doped material.
(i) There is a shift of the curves towards
higher oxygen pressures.
(ii) When at higher lanthanum the conductivity
is independent of pO2.
At low pressures the curves coincide with
those of the ‘pure’ ceramic.
 At the lowest pO2 values (AB) loss of oxygen from the crystal is
accompanied by the formation of Vö and electrons according to
equations (2.48) and (2.50).
2. Donor-doped BaTiO3(2)

As pO2 is increased, n falls to the level controlled by the donor
concentration so that n≒[La·Ba] as shown in the following equation:
La2O3 ⇔ 2 La·Ba + 2OO + ½ O2(g) + 2e’
(2.57)
When n is constant over B-C, corresponding to the plateau in the curves
of Fig.2.14, there are changes in the energetically favoured Schottky
disorder so that [Vö]∝pO2-½ , according to equation (2.49), and
[VT“i“]∝pO2½ . At C the condition
4[VT“i“] = [La·Ba]
(2.58)
is established from the equilibrium
2La2O3 + 4TiO2 ⇔ 4 La·Ba + 3TiTi + VT“i“ + 12OO + ‘TiO2’
(2.59)
where ‘TiO2’ indicates incorporation in a separate phase.
Both [VT“i“] and [Vö] remain sensibly constant over the range C-E so
that, according to equation (2.55),
p = K”ppO2¼ and n= K”ppO2-¼
(2.60)
At still higher values of pO2 (E-F), the dependence of p on pO2 is same
with equation (2.56).
3. Properties of doped BaTiO3(1)

The n- and p-type substituents, at low concentrations, have important
effects on the room temperature behaviour of BaTiO3.
Acceptor-doped material : Iow oxygen pressure without losing its high
resistivity at room temperature, Piezoelectric properties under high
compressive stress. Oxygen vacancies are also associated with the fall in
resistance that occurs at temperatures above 85℃ under high DC fields.
Donor-doped material : The basis of positive temperature coefficient
(PCT) resistors, The insulating dielectrics formed with high donor
concentrations have a low oxygen vacancy content and are therefore less
prone to ageing and degradation.

The effects of aliovalent substituents in PbTiO3 and Pb(Ti, Zr)O3 are,
broadly speaking, similar to those in BaTiO3.
4. Band model



There is less confidence then elemental semiconductors in band models
for the oxide semiconductors because sufficiently precise physical and
chemical characterization of the materials is often extremely difficult.
In addition, measurements are necessarily made at high temperatures
where knowledge of stoichiometry, impurity levels, dislocation content,
defect association and other characteristics is poor.
Fig.2.15 shows a tentative band model for doped barium titanate.
Fig.2.15 Tentative band model for doped BaTiO3(energes in electronvolt)
(d) Polaron conduction

The band model is not always appropriate for some oxides and the
electron or hole is regarded as ‘hopping’ from site to site.

‘Hopping’ conduction occurs when ions of the same type but with
oxidation states differing by unity occur on equivalent lattice sites and
is therefore likely to be observed in transition metal oxides.

The addition of Li2O to NiO leads to an increase in conductivity, as
illustrated in fig.2.16.
1
 EA 
u  exp 

 kT 
-1
-2
0
5
10
15
[Li+]/at%
Fig. 2. 16 Resistivity of NiO as a function of lithium content
(2.61)
2.6.3 Ionic conduction(1)
 Crystals
Ionic conduction depends on the presence of vacant sites into which
ions can move. In the absence of a field, thermal vibrations proportional
to kT cause ions and vacancies to exchange sites.
 The Nernst – Einstein equation links this process of self-diffusion with
the ion drift σi caused by an electric field:
i
N iQ i
(2.62)
Di
kT
where Di is the self- or trace-diffusion coefficient for an ion species i,
Qi, is the charge if carries and Ni is its concentration.



Features that contribute to ionic mobility are small charge, small size and
lattice geometry.
A highly charged ion will polarize, and be polarized by, the ions of
opposite charge as it moves past them, and this will increase the height
of the energy barrier that inhibits a change of site.
2.6.3 Ionic conduction(2)

Some structures may provide channels which give ions space for
movement.

The presence of vacant sites
assists conduction since it
offers the possibility of ions
moving from neighbouring
sites into a vacancy which,
in consequence, moves in the
opposite direction to the ions
(Fig.2.17).
Fig.2.17 Energy barriers to ionic transport in a crystal
(a) in the absence of a field and (b) with applied field E

The cations usually have to pass through the relatively small gap between
three O2- ions to reach any neighbouring cation vacancy.
2.6.3 Ionic conduction(3)



The crystal is highly ionic in character the barrier is electrostatic in origin,
and ion in its normal lattice position is in an electrostatic potential energy
‘well’ (Fig.2.17).
When an electric field E is imposed barrier heights are no longer equal,
and the jump probability is higher for the jump across the lower barrier (in
the illustrated case, to the right) of height Ej – ΔEj where
ΔEj = e E a/2
(2.63)
Since we know bias in jump probability in one direction, it is not difficult
to arrive at the following expression for the current density:
j
nv A
 Ej 
E exp  
NT
 kT 
(2.64)
in which nv/N is the fraction of Na+ sites that are vacant and A is a
constant describing the vibrational state of the crystal. Since it is assumed
that the vacancy is part of the Schottky defect, then nv = ns and hence,
using equation (2.12), we obtain
(2.66)
 1 
A
Hs 
 
T
exp
 Ej 
kT



2 
2.6.3 Ionic conduction(4)

Because the temperature dependence of σ is dominated by the
exponential term, the expression for conductivity is frequently written


   0 exp 
Ei 

kT 
(2.67)
in which Ei = Ej + ΔHs/2 is an activation energy and σ0 is regarded as
temperature independent.
 Glasses



1. Conductivity σ depends upon temperature.
2. σ decreases as the size of the mobile ion increase.
3. σ decreases as the concentration of blocking ions increases.
 Mixed-phase materials


In practice ceramics are usually multiphase, comprising crystalline
phase, glasses and porosity.
The overall behaviour depends on the distribution as well as properties
of these constituents.
2.6.5 Schottky barriers
to conduction

Fermi-Dirac fuction

 E  EF  
F ( E )  exp 
  1
 kT  

1
(2.69)
where EF is the Fermi energy.

The energy øm required to remove an electron with the Fermi energy to a
point outside the metal with zero kinetic energy is called the ‘work
function’ of the metal.

Excited semiconductor, the effective work function øs of the semiconductor is the energy difference between the Fermi energy and the
vacuum level.
Schottky barriers
Fig.2.19 Metal - n - type semiconductor junction(øm>øs) (a)before contact and (b) after contact.

In the vicinity of the junction, typically within 10-6 - 10-8m depending on
the concentration of n dopant, the donors are ionized.

Electrons moving up to the junction from the semiconductor then
encounter an energy barrier – a Schottky barrier – of height |e|Ub.
Metal-semiconductor
 At metal-semiconductor current-voltage characteristic is following.
I
Reverse bias
Forward bias
U
Fig.2.20 Current-voltage characteristic for a metal-semiconductor rectifying junction
N-type semiconductor sandwiched



A sandwich comprising a semiconductor between two metallic
electrodes presents the same effective barrier irrespective of the sense
of an applied voltage.
The situation is illustrated in Fig.2.21.
Current – voltage characteristic in Fig.2.22.
Fig.2.21 n-type semiconductor sandwiched
between two metal electrodes.
Fig.2.22 Current-voltage characteristic for back to
back Schottky barriers.
Junction n-type and p-type


Junction is that between n and p types of the same semiconductor.
The situation before and after contact is illustrated in Fig.2.23.
Fig.2.23 Junction between an n-type and a p-type semiconductor
(a) before contact, (b) after contact.
Junction n-type and p-type

About n-p junction reverse bias, Forward bias in Fig.2.24.
Fig.2.24 n – p junction. (a) reverse bias, (b) forward bias.
2.7 Charge displacement processes
2.7.1 dielectric in static electric fields
(a)Macroscopic parameters
1.Atomic polarization
:Atomic polarization occurs in all materials,it is a small
displacement of the electrons in an atom relative to
the nucleus
2.ionic polarization
:ionic polarization involving the relative displacement
of cation and anion sublattices
3.dipolar polarization
:dipolar materials, such as water, can become polarized
because the applied electric field orients the molecules
4.space charge polarization
:space charge polarization involves a limited transport of
charge barriers until they are stopped at a potential barrier,
Fig. 2.25 Various polarization processes
possibly a grain boundary or phase boundary
2.7.1 dielectric in static electric fields

The dipole moment P of the dipole

polarized material can be regarded as made up of elementary dipolar prisms
the dipole moment per unit volume of material is termed the polarization P and
can vary from region to region From Fig. 2.26 the magnitudes of the vectors
are given by
p  Qx
p   pAx   pV
or
p
 p p
V
in general  p  n  p where n is the unit vector normal to the surface enclosing
the polarized material and directed outwards from the material
Fig 2.26 Elementary prism of polarized material
2.7.1 dielectric in static electric fields
Fig 2.27 The role of the dielectric in a capacitor


From Gauss’s theorem the electric field E between and normal to two parallel plates
carrying surface charge density  and separated by a vacuum is
E   / 0
In (b) the polarization charge density  T appearing thus the effective charge density giving
rise to E is reduced to  T   p so that
E

T  p
0
Total charge density T is equivalent to the magnitude of the dielectric displacement
vector D
D  0E  p
2.7.1 dielectric in static electric fields


If the dielectric ‘linear’, so that polarization is proportional to the electric field within the material
which commonly case
P   e 0 E
It follows from that
D   0 E  e 0 E  (1  e ) 0 E

And since
D  T
QT
U
 (1   e ) 0
A
h

QT is the total charge on the capacitor plate therefore the capacitance is
QT
A
 (1   e ) 0
U
h
Since vacuum has zero susceptibility the capacitance C
0
A
C0   0
h
C


The permittivity
 of the dielectric is defined by
   0 (1  e )

is

 1  e   r
0
and r is the relative permittivity(or dielectric constant)of the dielectric
2.7.1 dielectric in static electric fields
(b)From induced elementary dipoles to macroscopic properties
Ea :applied external field
Edp :depolarizing field
Em :internal macroscopic field ( Ea  Edp)
Fig 2.28 The ‘local’ field in a dielectric
• In reality the atomic nature of matter dictates that the local field which is known as the Lorenz field
EL  Em  E p  Ed
• E p is the contribution from the charges at the surface of the spherical cavity.
•
Ed is due to the dipoles within the boundary, E pcan be shown to be p / 3 0
•However certain crystals of high symmetry and glasses it can be Ed  0 and for these cases.
EL  Em 
P
P
 Ea  Edp 
3 0
3 0
•In the more general case it is assumed that
EL  Em  P
•  is the ‘internal field constant’
2.7.1 dielectric in static electric fields

The dipole moment P induced in the entity can now be written
P  EL

If it is assumed that all entities are of same type and have a density N then,
P  NP  N ( Em  P)


or
N /  0
p
 e 
 0 Em
1  N
In the particular case for which   1 / 3 0 rearrangement of equation
leads of the Clausius-Mosotti relationship
 r  1 N

 r  2 3 0
Ferroelectrics possess very high permittivity values which vary considerably with both applied
field strength and temperature. The permittivity reaches a peak at the Curie-Weiss law
A
r 
T  c
2.7.2 Dielectrics in alternating
electric fields

2.7.2 Dielectrics in alternating electric
fields
(a)Power dissipation in a dielectric

Fig 2.29 Sinusoidal voltage applied to a perfect capacitor

Fig 2.30 phasor diagram for a perfect capacitor
If voltage describedU 0 sin(wt ) then the current
is U 0 cos(wt ) and leads U by 90 Since the
instantnaneous power drawn from the
voltage source is I cU ,the time average power
dissipated is P where,
1 T
P   I cUdt or
T 0
1 T
P   U 0 I c sin( wt ) cos( wt )dt  0
T 0
The applied voltage at a given point in
time is represented by a horizontal line
and the instantnaeous current by a
vertical line since it leads the voltage

by 90
2.7.2 Dielectrics in alternating
electric fields

Time average power loss is
P


1 T
1
UIdt

T 0
T

T
0
U 0 sin( wt ) I 0 cos( wt   )dt
Intergrating equation gives
1
P  U 0 I 0 sin 
2
Since I 0  I c / cos and I c  wU 0C
1
1 2
P  U 0 I c tan   U 0 wC tan 
2
2

For disspated power density in the dielectric
P 1 2
 E0 w 0 r tan
V 2
  tan is termed the ‘loss factor’ of the dielectric
r
Fig 2.31 phasor diagram for a real capacitor
and w 0 r tan  is the ‘dielectric conductivity’
 AC  w 0 r tan
2.7.2 Dielectrics in alternating
electric fields

U  U0 exp(jwt ) can represent a complex sinusoidal
voltage The time differential of U is given

U  jwU 0 exp( jwt )  jwU
(b) The complex permittivity

The instantaneous charge on a ‘lossless’ vacuum
capacitor C0 is
Q  UC0


0
And I  Q  U C  jwC U
0
0
Complex relative permittivity  
   j  It follows that

'
''
I  jw  C0U  jw  C0U  w  r C0U
'
''
r

As indicated in Fig 2.32 It can be seen from the figure
''
that
 r '  tan 

The current I 1 in phase with U can be written

I1  w  r
''

Fig 2.32 Capacitative and ‘loss’
components of total current I

0 A
h
U
So that the current density is given by
I1
''
''
 w r  0 E  w E
A
Average dissipated power density is given by
P 1 2
1 2
1 2
''
'
 E0  AC  E0 w 0  r  E0 w 0  r tan
V 2
2
2
2.7.2 Dielectrics in alternating
electric fields
(c) Frequency and temperature dependence of dielectric properties
(1) Resonance effects


If the damping (resistive) force
is assumed to be

proportional to the velocity x of the moving charged
particle the equation becomes


2
m x  m x m w0 x  QE0 exp( jwt )
Solving and ignoring the transient term yields
x(t )  

eE0 exp(jwt )
2
m{( w0  w2 )  jw}
Since -ex(t) is the induced dipole moment per atom,
the complex polarization P  is
P  N (e) x(t )

'
''


Fig 2.33 Variation in rand rwith


e
frequency close to a resonance frequency w0 
and
so that
Ne 2
1
Ne 2
1


{ 2
}


1

m 0 (w0  w2 )  jw r
m 0 (w0 2  w2 )  jw
it follow

''
r
Ne 2
w
 1
{ 2
}
2 2
2 2
m 0 (w0  w )   w
2.7.2 Dielectrics in alternating
electric fields

(2) Relaxation effects

Polarization processes occurs in ceramics for
which the damped, forced harmonic motion
approach is inappropriate.
for example, because of the random structure of
glass the potential energy of a cation moving
through a glass can be shown schematicalliy as
in Fig. 2.34
Pd
Pds
Fig 2.34 schematic one-dimensional representation
of the electrostatic potential in a glass


Figure 2.35 illustrates how on the application of
a field and following the initial instantaneous
atomic and ionic polarization
The slow diffusional polarization Pd approaches
its final static value Pds
Fig 2.35 Development of polarization by slow diffusional
processes.
2.7.2 Dielectrics in alternating
electric fields

It assumed that time t the polarization Pd (t ) develops at a rate proportional to Pds  Pd (t )

1
P d  {Pds  Pd (t )}





In which 1 /  is a proportionality constant. Integrating with initial condition Pd  0
when t=0 gives Pd  Pds {1  exp(  t )}

Where is a relaxation time

If assumed that the polarizing field is E  E0 exp( jwt ) it can be shown that
1 '
'
P d  { rs   r) 0 E   Pd (t )}

In which is the value of the permittivity measured at low frequencies or with a
static field applied ' It can
' be integrated to give

t
Pd  C exp( )   rs  r  0 E 

1  jw

If the transient C exp(t /  ) is
neglected it leads to
 rs'   r' 
  1
1  jw

r
and
'
'



'
rs
r


1

r
or
1  w2 2
w
''
'
'
 r  ( rs   r )
1  w2 2
Fig 2.36 variation in permittivity with for a
dielectric showing ‘Debye’ relaxation
2.7.2 Dielectrics in alternating
electric fields

Fig 2.36 show graphically Debye
equations

The relaxation frequency is wr  1 / 
because the polarization occurs by
the same temperature-activated
diffusional processes which give rise
to DC conductivity,
 depends on temperature through
an exponential factor
   0 exp(
Fig 2.37 permittivity dispersion and
dielectric loss for a glass
A
kT
)
2.7.2 Dielectrics in alternating
electric fields
 r'
 r''
Fig 2.38 variation of  r and  r with frequency. Space charge and
dipolar polarizations are relaxation processes and are strongly
temperature dependent; ionic and electronic polarizations are
resonance processes and sensibly temperature independent.
Over critical frequency ranges energy dissipation is a maximum
''
as shown by peaks in  r (w)
'
''
• various polarization processes which lead to dielectric dispersion and attendant energy dissipation
are summarized in fig 2.38
•In conclusion, it is opportune to mention the relationship between the refractive index n and the
relative permittivity
2
r
 n
2.7.3 Barium titanate
- the prototype ferroelectric ceramic
-Fig.2.42 Variation in the potential energy of
Ti4+ aling the c axis
-Tetragonal BaTiO3 the energy if the Ti4+ion in
terms of its position along the c axis takes the form
of two wells
-Applied field in the opposite direction to the
polarization may enable a Ti4+ion to pass over the
energy barrier between the two states and so
reverse the direction of the polarity at that point
-When this happens the energy barriers for
neighboring ions are reduces and the entire
region affected by the field will eventually switch
into the new direction.
Fig.2.43 (a) surface charge associated with spontaneous
polarizaion;
(b) formation of 180° domains to minimize electrostatic
energy
2.7.3 Barium titanate
- the prototype ferroelectric ceramic
Fig.2.43(a),(b) : These regions are called domains and the whole configuration
shown comprises 180° domains.
•
Surface mosaic : carrying apparent charges of opposite sign, resulting in a
reduction in Ed and in energy
•
The presence of mechanical stress in a crystal results in the development of
90° domains configured so as to minimize the strain.
•
The configurations ca be modified by imposing either an electric or a
mechanical stress.
•
A polycrystalline ceramic that has not been subjected to a static field
behaves as a non-polar material even though the crystals comprising it are
polar.
•
One of the most valuable features of ferroelectric behaviour is that
ferroelectric ceramics can be transformed into polar materials by applying a
static field.
•
Poling and depoling processes are illustrated schematically in Fig.2.44
2.7.3 Barium titanate
- the prototype ferroelectric ceramic
-Fig.2.44 Poling in a two-dimensional ceramic: (a) unoriented material; (b)oriented by 180° domain changes;
(c) oriented by 180° and 90° domain changes; (d) disoriented by stress
-The random directions of the crystallographic axes of the crystallites of a
ceramic limit the extent to which spontaneous polarization can be developed.
-The fractions of the single-crystal polarization value, polar axes, alignments
0.83(tetragonal), 0.91(orthorhombic), 0.87(rhombohedral) perovskites structure
-The orientations occurring in a simple domain structure are shown schematically
in Fig.2.45(b)
2.7.3 Barium titanate
- the prototype ferroelectric ceramic
•The thickness of the layer separating the
domains is of the order of 10nm but varies with
temperature and crystal purity.
•The wall energy is of the order 10mJm
2
Fig.2.45 (b) schematic diagram of 180°
and 90° domains in barium titanate
•The almost vertical portions of the loop are due to
the reversal of the spontaneous polarization as
reverse 180 °domains nucleate and grow.
•The horizontal portions represent saturated states :
crystal is single domain with a permittivity of 160
obtainable in the polar direction
Fig.2.46 (a) Hysteresis loops for a
single-domain single crystal of
BaTiO3
•The coercive field at room temperature - 50Hz,
0.1MVm  1, saturation polarization(0.27Cm )
2.7.3 Barium titanate
- the prototype ferroelectric ceramic
-The coercive field is higher and the remanent
polarization is lower than for a single crystal
-Both 180 °and 90 °changes take place during a cycle
and are impeded by the defects and internal strains
within the crystallites
-The hysteresis loss, single crystal : 0.1MJm
1
3
Fig.2.46 (b) Hysteresis loops
for BaTiO3 ceramic
-Rapid rise in temperature, at 1000Hz power : 100MWm
•The dissipation factor in ceramic at high fields : very high
•But at the 100Vmm
1
level tanδ : less than 0.1 for undoped material
•A unusually characteristic of ferroelectric materials: properties change with
Time
t0 : arbitrary zero for the time
p a '

t t
t
p  a log10 ( )
t0
Usually a << 0.05
a  a' loge 10
2.7.3 Barium titanate
- the prototype ferroelectric ceramic
•The property : negative for permittivity, positive for Young’s modulus
•After sufficient lapse of time : negligible => mechanical, electrical stresses,
exceeding the Curie point
•Very advantage of ceramic ferroelectrics : properties can be modified by adjusting
the composition and ceramic microstructure
•Additions and the substitution of alternative cations : effects =>
1.
2.
3.
4.
5.
Shift the Curie point and other transition temperatures;
Restrict domain wall motion;
Introduce second phases or compositional heterogeneity;
Control crystallite size;
Control the oxygen content and the valency of the Ti ion
•The effects are important for the following reasons =>
1. Changing the Curie point enables the peak permittivity to be put in a
temperature range in which it can be exploited.
Ba2+ of BaTiO3 : substitution of Sr2+ = lowers Tc, substitution of Pb+
increase Tc
2.7.3 Barium titanate
- the prototype ferroelectric ceramic
Fig.2.47 The effect in the Curie point of the
substitutiom of either strontium ot lead for barium in
BaTiO3
Fig.2.48 The effect of grain size on the
permittivityof a BaTiO3 ceramic
2. Dissipation factor due to domain wall motion : Fe3+, Ni2+, Co3+ can
occupy Ti4+ sites reduce
3. CaZrO3 additions : broadening of the permittivity-temperature peak.
Materials contain regions of variable composition that contribute a range if
Curie points so that the high permittivity is spread over a wider temperature
range
2.7.3 Barium titanate
- the prototype ferroelectric ceramic
4. Cations that have a higher valency than those they replace: (>0.5 cat.%)
Ti4+ =>Nb5+, Ba2+=>La3+; inhibit crystal growth. Rising the permittivity
level below the Curie point<Fig.2.48>. Crystal size is also controlled by
sintering conditions
5. Higher-valency substituents = lead to low resistivity; lower-valency
substituents (Ti3+=>Mn3+) = act as acceptors, enable high-redielectrics to
be sinterd in atmospheres with low oxygen contents
2.7.4 Mixtures of dielectrics
•‘Connectivity’ : classifying different types of mixture
•Any phase in a mixture : self-connected in zero, one, two, three dimensions
•Connectivity of 0 : dispersed particles, Connectivity of 3: medium surrounding
particles
•A disc containing a rod-shaped phase extending between its major surfaces
connectivity : 1 with respect to the rods, 3 with respect to the intervening phase
•Mixture stack of plates of two different phases : connectivity of 2-2
•In all, 10 different connectivities for mixtures of two phases(0-0, 1-0, 2-0, 3-0, 1-1, 2-1,
3-1, 2-2, 3-2, 3-3), for mixtures of three phases(20),
for mixtures of four phases(35)

1
 2 deduced that permittivity( ) of random dispersion of spheres
•3-0 system, Maxwell
in matrix of relative
is
For  2  1 ,V f  0.1 equation
(2.12
reduces to
3V f (1   2 )
1)
m
 m   2{1 
1  2 2  V f (1   2 )
}
m
1  3V f
 2(
)
2
2.7.4 Mixtures of dielectrics
-Convert the values of permittivity found for porous
bodies to the value expected for fully dense bodies.
-Eq.(2.121) : V f
is less than about 0.1V (agreement)
-Resistance-volume concentration relations for
dispersions of conductive particle in insulating
media(Fig.2.49)
-The resistivity remains high until a critical
concentration
in the neighbourhood of 0.05-02 is reached when it
Fig.2.49 log resistivity versus volume fractiondrops by several orders of magnitude
of conductive particles in an insulating matrix
-Transition from a dispersion of separated particles to
one of-Fig.2.50
connected
: Aaggregates.
capacitor containing a two-phase 13 dielctric consisting of rods of
extending
1
from one elctrode to the other in 
a 2medium of
is equivalent in behaviour
Fig.2.50 Equivalent structures for dielectrics
with (a) 1-3 and (b) 2-2 connectivity
-The structure consists of two capacitors in
parallel so that;
2.7.4 Mixtures of dielectrics
m A
h

 1 A1
h

 2 A2
 m  (1  V f ) 2  V f 1
h
-Fig.2.50(b) : 2-2 connectivity dielectric with the main planes of the phases parallel to
the electrodes is equivalent to the structure
-Effectively two capacitors in series
h
h
h
 1  2
A m A1 1 A2  2
 m  (1  V f ) 2  V f 1
n
n
1
1
 m  (1  V f ) 2  V f 1
n
1
 m n  V fi i n
i
Where n=+-1, or for a multiplicity of phases of partial volumes Vf1, Vf2,….Vfi
2.7.4 Mixtures of dielectrics
-The approximationx
nevertheless
n
 1 n ln x
is only valid for small values of x and, n , but,
ln  m  V fi i
i
-Differentiation of equation of above eq. With respect to temperature gives
1
m
 m

T
V fi  i

 i T
i
-Which gives the temperature coefficient of permittivity for a mixture of phases and,
although not in exact agreement with observation, is a useful approximation.