絕緣體 (Si3N4, SiO2)
ne 2τ 1 1 1
m τ τ
τL: collision time for scattering by phonons,
σ ∝ for high temp (T > θ D ), σ ∝ for low temp (T << θ D ),
θ D is Debye temp
τi: collision time for scattering by imperfections.
σ independent of temp
-Eg / 2kT
-Eg / 2kT
σ = nμq = aoT e
1 -Eσ / kT
-E / kT
σ = nμq = a o e
= σoe σ
σo 為常數，此關係稱為Arrhenius Equation，其中k為波茲曼常數
For any material and charge carrier, the conductivity is given by
where ni is the number of charge carriers of species i, ei is their charge and i their mobility.
Migration of ions does not occur to any appreciable extent in most ionic and covalent solids such
as oxides and halides. Rather, the atoms tend to be essentially fixed on their lattice sites and can
only move via crystal defects. Only at high temperatures, where the defect concentrations become
quite large and the atoms have a lot of thermal energy, does the conductivity become appreciable,
e. g. the conductivity of NaCl at ~800oC, just below its melting point, is ~10-3 (Ωcm)-1, whereas at
room temperature, pure NaCl is an insulator.
In contrast, there exists a group of solids called, solid electrolytes, fast ion conductors and
superionic conductors, in which one of the sets of ions can move quite easily. Such materials often
have rather special crystal structures in that there are open tunnels or layers through which the
mobile ions may move. The conductivity values, e. g. ~10-3 (Ωcm)-1, for Na+ ion migration in
β-alumina at 25 oC, are comparable to those observed for strong liquid electrolytes.
σ = σo exp[ -Eσ / kT ] for T >
σ = σo exp[ -(To / T) ] for T <
Typical ionic crystals NaCl
Migration of Na vacancies in NaCl
The magnitude of the ionic conductivity of NaCl
depends on the number of cation vacancies
present, Vacancies are normally created by one of
two methods. On heating a crystal, the number of
vacancies present in thermodynamic equilibrium
increases exponentially and this is the number
that is intrinsic to the pure crystals. Vacancies may
also be created by adding impurities, e. g. addition
of small amount of MnCl2, this will yield solid
solution with fomula,Na1-2xMnxCl.
Schematic ionic conductivity of doped NaCl
Pathway for Na migration in NaCl
The radius of Na+ is rNa+~0.95A . The jumping of ion
Na+ in NaCl is a difficult and complicate process.
First , the ion has to squeeze through a narrow
triangular interstice of radius r’ (=0.45A) ,
whereupon it finds itself in a small tetrahedral
interstitial site of radius 0.59A. The residence time
here is quite short since this site has a hostile
environment with two Na+ ions, 1 and 2, at a distance
of 2.44A as well as four Cl- ions at the same distance.
The ion leaves by squeezing through another gap of
radius 0.45A to occupy the vacant octahedral site on
the other side.
Ionic conductivity of “pure “ NaCl single crystal as
function of temperature
Region I intrinsic conductivity
II extrinsic conductivity
I’ anion vacancy or
III formation of defect complex
cation/anion vacancy pairs or
cation vacancy/aliovalent cation impurity pairs
Schottky pair defect
Conductivity in NaCl crystals
Migration of Na+, Em
Migration of ClFormation of Schottky pair
Dissociation of vacancy
Dissociation of cation
vacancy -Mn+2 pair
Two Frenkel defects
Typical ionic crystals AgCl
The predominant defects in AgCl are Frenkel detects.
Formation energy of Frenkel detects is 1.24eV, migration
energy of cation vacancy is 0.27~0.34eV, migration energy of
interstitial Ag+ is 0.05~0.16eV.
It has been found
experimentally that mechanism 2 is the operative mechanism.
The effect of aliovalent cation impurities on the extrinsic
conductivity region of AgCl is different to that observed in NaCl.
The presence of Cd2+ again increases the number of cation
vacancies, but because the product of the concentrations of cation
vacancies and Ag+ interstitials is constant, the interstitial Ag+
concentration must decrease with increasing Cd+2 concentration.
Addition of Cd2+ therefore leads to a reduction in the
concentration of the more mobile species. An extrinsic region at
lower temperatures is again observed but it is displaced
downwards to lower values. The degree of downward
displacement increases with increasing Cd2+ concentration.
Typical ionic crystals Alkaline earth fluoride
The most important defect in this group is probably the anion Frenkel defect
in which an interstitial F- ion occupies the center of a cubic that has eight F- ion at the corners.
Conductivity measurements have shown that the anion vacancy is more mobile than the
interstitial F- ion. This contrasts with AgCl in which the interstitial Ag+ is more mobile than the
cation vacancy. In some materials that have the fluorite structure, e. g. PbF 2, the conductivity at
high temperatures becomes quite large .
At room temperature, PbF2 has very low ionic conductivity and as such is a typical ionic solids.
With rising temperature its conductivity increases smoothly and rapidly until at 500 oC a limiting
values of ~5(Ωcm)-1 is reached.
Solid electrolytes as intermediate between normal
crystalline solids and liquids
Ionic conductivity of some solid electrolytes with
concentrated H2SO4 for comparison.
Mobile ions in electrolytes
CaF2 : F-; ZrO2 : O-2; AgI, RbAgI5 , AgCl : Ag+, Na3Zr2PSi2O12, Na:β-alumina : Na+
β-alumina is the name for a family of
compounds of general formula M2O.nX2O3
where n can have various values in the range
5 to 11, M is a monovalent cation, such as
alkali ion, Cu+, Ag+, Tl+, NH4+, H3O+, and X
is a thivalent cation, Al3+, Ga3+, or Fe3+. The
most important member of this family is
sodium β-alumina (M=Na+1, X=Al+3), which
has been known for many years. The high
conductivity of monovalent ions in βalumina is a consequence of its unusual
crystal structure. The Na+ ions reside in the
oxygen-deficient layers and are able to move
very easily because (a) there are more sites
available than there are Na+ ions to occupy
them and (b) the radius of Na+ is less than
that of the O2- ion. There are exists in two
structural modifications, named β and β”,
which differ in the stacking sequence of the
layers. The β” forms with more soda-rich
crystals, n~5-7, whereas β occurs for n~8-11.
The conductivity of β” is several times
larger than that of β.
Conduction of some single crystal β-aluminas. Activation energies,
in electronvolts, are in parentheses.
Crystal structure of β- alumina
Oxide packing arrangement in β and β”-alumina
Oxide layer in β- alumina
sb: spinel block
Conduction plane in β-alumina
C : O-2 in conduction plane
A, B : O-2 in wall plane
m : empty site
br, abr, m : possible Na+ site
Conduction pathway in β-alumina
The conduction pathway of Na+ ion must through the sequence –br-m-abr-m-br-m-. The
activation energy, which is 0.16ev, represents the overall value for migration of a Na + ion
from one br to the next. Both knock-on and interstitialcy mechanism are possible. Na+ ions
cannot move independently to each other in β–alumina bur that, instead, a cooperative
process must occur.
AgI and Ag+ ion solid electrolytes
β-AgI stable below 146oC, has the wurtzite
structure with hexagonal close packed I- ions and Ag+
in tetrahedral sites. The highly conducting α-AgI
phase is body center cubic. In it, iodide ions lie at
corner and body center positions and Ag+ ions appear
to be distributed statistically over a total of thirty-six
sites of tetrahedral and trigonal coordination. The
iodide ions are essentially fixed and Ag+ ions can
readily move from one site to the next in a liquid-like
manner. The disordered Ag+ arrangements and the
easy motion of Ag+ between sites must be related to
the nature of the bonding between silver and iodine.
Silver is a polarizing cat ion since its outer 4d
electrons are relatively ineffective in shielding
nuclear charge. Iodide is a 1arge and polarizable
anion and so covalent bonds readily form between
Ag+ and I- that are characterized by structures with
low coordination numbers. During conduction, silver
can readily move from one tetrahedral site to the next
via an intermediate, three-coordinate site; covalent
bonding at the intermediate site helps to stabilize it
and reduce the activation energy for conduction.
The I- anions (shown blue) form a b.c.c. unit,
with the Ag cations rapidly diffusing across
distorted tetrahedral interstitial sites (shown
red) through the trigonal interstitial sites
(shown green). The black circles show
distorted octahedral sites.
The phase diagram for the system Rb-Agl
Two binary compounds are present, Rb2AgI3 and RbAg4I5. The latter melts incongruently to AgI
and liquid at ~230oC and has a lower limit of stability at 27oC. Decomposition of RbAg4I5 to AgI
and Rb2AgI3 below 27oC takes place only slowly although it is hastened by the presence of
moisture or iodine vapor. With care, RbAg4I5 may be readily cooled to below room temperature
without decomposition. In order to prepare RbAg4I5, a 1:4 molar mixture of RbI and AgI is melted
in vacuum at ~500oC and then quenched to room temperature. On subsequent annealing at 165oc for
10hr this reacts further to give RbAg4I5. The crystal structure of RbAg4I5 is rather different to that
of α-AgI, but it also contains a random arrangement of silver atoms distributed over a network of
face-sharing tetrahedral sites. Rubidium atoms are immobilized in sites that have a distorted
octahedral environment of I- ions.
Sodium superionic conductor (NASICON, Na3Zr2PSi2O12)
The framework is built of corner –
sharing ZrO6 octahedra and (P, Si)O4
tetrahedra and a three-dimension
network of tunnels in which Na+ ions
Oxide ion conductors
The high temperature, cubic polymorph of zirconia
has the fluorite crystal structure, and may be stabilized
to room temperature by formation of solid solution
with CaO, Y2O3, etc. Such stabilized zirconia are good
O2- ion conductors at high temperatures, mainly
because the mechanism of so1id solution formation
involves the creation of vacant O2- sites in order to
preserve electric neutrality, e.g. lime-stabilized
zirconia has the formula (CaxZr1-x)O2-x: 0.1≦x≦0.2.
With each Ca2+ ion that is introduced one O2- ion
vacancy is created. Typical conductivities
stabilized zirconias (e. g. 85mol％ZrO2, 15mol%CaO)
are 5×10-2 (Ω-cm)-1 at 1000oC With an activation
At lower temperatures stabilized
zirconias have conductivities that are many order of
magnitude 1ess than those of the Na+ and Ag+ ion
solid electrolytes. Their usefulness stems from the fact
that they are refractory materials, can be used up to
very high temperatures (e. g. 1500oC) and have good
oxide ion conductivity, which is an unusual property.
Thoria (ThO2) and hafnia (HfO2) may be doped and
are good O2- ion conductors at high temperatures.
Glasses ion conductor
Alkali borate glasses
Alkali silicate glasses
The best we can do at present is to have a set of guidelines which show us the
likely structural characteristics that are a prerequisite for high ionic conductivity.
(a) A large number of the ions of one species should be mobile (i.e. σ=neμ)
(b) There should be a large number of empty sites available for the mobile ions
to jump into. This is essentially a corollary of (a) because ions can be mobile only
if there are empty sites available for them to occupy.
(c) The empty and occupied sites should have similar potential energies with a
low activation energy barrier for jumping between neighboring sites. It is no use
having a large number of available interstitial sites if the moving ion cannot get
(d) The structure should have a framework, preferably three dimensional,
permeated by open channels through which mobile ions may migrate.
(e) The anion framework should be highly polarizable.
導電性質與機制: 交流阻抗分析 (RLi2O-B2O3 glasses)
(ΔH1/2)2=(2/ π) (ΔH(0)1/2)2tan-1(α τnmrΔH1/2)
A : WP= β T2 (phono interaction)
B : WD= α exp(-ET1/kT) (diffusion)