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Electrical properties of glassy (vitreous) materials
IACS
Prof. A. Ghosh, PhD
Department of Solid State Physics,
Indian Association for the Cultivation of Science,
Jadavpur, Kolkata – 700032
Editor-in-Chief, Indian Journal of Physics (Springer)
Noble Laureates from Kolkata
Sir Ronald Ross
Nobel Prize (Physiology
or Medicine 1902)
Rabindranath Tagore
Nobel Prize (Literature
1913)
Mother Teresa
Nobel Prize (Peace 1979)
C.V. Raman
Nobel Prize (Physics
1930)
Amartya Sen
Nobel Prize (Economics 1998)
OUTLINE OF TALK
 A brief introduction to glass
 Electronic conduction in glass
 Ionic conduction in glass
Crystalline and Non-crystalline (glass or vitreous)
materials
 Crystalline materials:
Materials possessing long range periodicity of
atomic arrangements
 Non-Crystalline materials:
Materials possessing no long range periodicity,
but possessing short range order
Crystalline solids versus Non-crystalline solids
Crystalline solids have of long range
periodicity of atomic arrangements.
Examples
of
some
common
crystalline materials are ice, table
salt, etc.
Disorder solids have no long range
periodicity of atomic arrangements.
Examples
of
some
common
amorphous materials are glass,
polymer, foam etc.
Different types of disorder
(a)Topological
(geometrical or
positional) disorder
Absence of long range
periodicity of atomic
arrangements.
(b) Spin disorder on
regular lattice
(c) Substitutional disorder
on regular lattice
Underlying
perfectly
crystalline
lattice
is
preserved. Atomic sites
possess randomly oriented
spins or magnetic moments.
Underlying perfectly crystalline lattice is
preserved. One type of atoms is randomly
substituted by another type
(d)Vibrational disorder about equilibrium position of a regular lattice
Random motion of atoms of
crystals about their equilibrium
positions at finite temperature
X-ray diffraction and Bragg’s Law
nl = 2 d sin(q)
Where: n is an integer
l is the wavelength of the X-rays
d is distance between adjacent planes in
the lattice
q is the incident angle of the X-ray beam
Bragg’s law tells us the conditions that must be met for the reflected X-ray waves to
be in phase with each other (constructive interference). If these conditions are not
met, destructive interference reduces the reflected intensity to zero!
Li(Mn1/3Ni1/3Co1/3)O2
Crystalline materials: Sharp diffraction
peaks are obtained. Each diffraction peak
corresponds to a particular set of
crystalline planes characterized by Miller
index (hkl).
20
40
(107)
(108)
(110)
(113)
(105)
(006)(102)
(101)
(104)
(003)
Intensity (arb. units)
XRD patterns of crystalline and vitreous materials
60
80
100
120
Intensity (arb. units)
2q (degree)
50Ag2O-0.10B2O3-0.40P2O5
Disorder materials: A broad hump is
obtained instead of sharp diffraction
peaks. The broad hump indicates that the
disorder materials exhibit short range
periodicity.
10
20
30
40
50
2q (degree)
60
Experiment to detect the crystalline and noncrystalline materials: electron diffraction
Electron diffraction pattern of a CdI2 crystal.
Each spot Indicates a set of crystalline plane.
Electron diffraction pattern of Ag2O-V2O5
glass. Diffused circular ring without any spot
indicates the absence of long range
periodicity.
Electron wave function for crystalline and
disorder solids
Crystalline solids
Disordered solids
   0 eik .r
   0 e  r
Crystalline solids: Each electron can be
described by Bloch wave function
ψ=ψ0(r)exp(ik.r) where the function ψ0(r)
denotes the periodicity of the lattice.
Disorder solids: No long range periodicity.
Bloch wave function is not valid to represent
electronic states. The electron wave
functions in disorder solids is represented as
ψ= ψ0exp(-αr), where 1/α is the localization
length.
Wave function and probability density for
crystalline and disorder solids
Type of Solids
Crystalline
Wave Function
  Im( 0 e )
ik .r
Non-localized/
Extended states
Disordered
Localized states
Probability density
   0 e  r
normalizied

2
1
normalized

2
 e 2 r
Energy band diagram for crystalline materials
Energy band diagram for crystalline and non-crystalline
semiconductors
In
extrinsic
crystalline
semiconductor Fermi level shifts
from the intrinsic level and moves
near the donor or acceptor level.
There are localized states in the forbidden
energy gap of amorphous solids.
Ec
EF
Ev
The highest energy at which the states are
localized is called the mobility edge, denoted as
Ec.
For crystal, Fermi level EF is devoid of any
states. But for amorphous solids the EF lies within
the region of localized states.
What is Glass ?
Glass is an amorphous solid that exhibits “glass transition”
Experiment: DSC or DTA
Supercooled
liquid
Rapid
quench
(2)
Very
slow
cooling
Glass
Volume
liquid
(1)
Crystal
Glass
transition
(Tg)
Temperature
Freezing
point
(Tf)
Deb and Ghosh,
EPL-Europhys. Lett.
95, 26002 (2011)
Differential scanning calorimetry (DSC)
DSC enables determination of melting,
crystallization, and glass transition
temperatures, and the corresponding
enthalpy and entropy changes, and
characterization of glass transition and
other effects that show either changes in
heat capacity or a latent heat.
A DSC analyzer measures the energy changes that occur as a sample is
heated, cooled or held isothermally, together with the temperature at which
these changes occur. The energy changes enable to find and measure the
transitions that occur in the sample quantitatively, and to note the temperature
where they occur, and so to characterize a material for melting processes,
measurement of glass transitions and a range of more complex events.
Differential scanning calorimetry (DSC)
The
glass
transition
temperature(Tg), is an endothermic
baseline shift.
The temperature Tc indicates the
glass-crystallization transformation
which is an exothermic transition.
The
peak
crystallization
temperature (Tp) denotes maximum
crystallization rate.
Tm indicates the melting of the
sample.
dQ
dT
= Φ = Ch .
= ch . ms . 
dt
dt
where dQ is heat exchanged, dT is the
temperature change, Φ is the heat flow rate
and  is the scan rate, ms is the sample mass
and ch=Ch/ms is the specific heat capacity
The main property that is measured
by DSC is heat flow, the flow of energy
into or out of the sample as a function
of temperature or time with reference
to a reference sample (calibrated
empty pan).
Thermodynamics of glass transition
Gibbs free energy G=U-TS+PV
dG= dU-TdS-SdT+PdV+VdP
= -SdT+VdP (as dU=TdS-PdV)
Now G=G(T, P)
 G 
 G 
dG  
 dT  
 dP
 T  P
 P T
Thus, we can obtain the following relations
 G 
V 

 P T
 G 
S  

 T  p
  2G 
 S 
Cp  
   2 
 T  p
 T  p
First-order phase transitions exhibit a discontinuity in the first derivative of the free
energy with respect to some thermodynamic variable. Thus liquid crystal transition is an
G
example of first-order transition, since the volume V   
changes discontinuously at
 P T
melting temperature.
Second-order phase transitions are continuous in the first derivative of the free energy
but exhibit discontinuity in a second derivative of the free energy. For glass,
  2G 
C p  T  2 
 T  p
is discontinuous at Tg. This apparently indicates that glass transition is a second order phase
transition.
Glass transition as a second order phase transition
To satisfy the criteria of second order phase transition, entropy should be continuous at
the transition i.e. entropy of liquid (S1) should be equal to entropy of glass (S2) at the
transition temperature. Thus S1=S2
or S1  S2
or
 S1 
 S1 
 S2 
 S2 
dT

dP

dT








 dP
 T  p
 P T
 T  p
 P T
Using the following relation
T 
1  V 


V  T  p
kT  
1  V 


V  P T
We can obtain the following relation
dTg  kT 


dP  T 
It has been observed experimentally that the values of
than those of dTg
kT
 T
are appreciably higher
dP
Thus, glass transition is not simple second order phase transition.
Variation of heat capacity with temperature for glass and
crystal
44
The heat capacity for a glass is
comparable to that of a crystal
but considerably smaller than
that of the liquid.
-1
-1
C p( Cal mol deg )
40
36
32
At
glass
transition
temperature the heat capacity
changes discontinuous
28
Crystal
24
glass
Tg
20
350
400
450
500
550
T (K) (log scale)
600
At very low temperature
Cp~T3 (Debye’s T3 law) for
crystal and Cp~T for glasses.
Free volume theory
Free volume = specific volume (volume per
unit mass) - specific volume of the
corresponding crystal. For chalcogenide glasses
10% of the total volume is free at Tg, whereas
for B2O3 glasses 34% of the total volume is free
at Tg.
At the glass transition temperature, Tg, the
free volume increases leading to atomic
mobility and liquid-like behavior. Below the
glass transition temperature atoms (ions) are
not mobile and the material behaves like solid.
Within the free volume theory it is
understood that with large enough free
volume, mobility is high and viscosity is low.
When the temperature is decreased free
volume becomes “critically” small and the
system “jams up”.
*Optical or insulating glass: Network formers
Oxide glasses (SiO2 or P2O5), fluoride glasses (ZrF4 )
Structure of glass: CRN
Modification of glass structure
*Ion conducting glass:
Alkali modified glass (Li2O-SiO2, Ag2O-P2O5)
Structure: MCRN
*Superionic glass:
Alkali halide doped alkali modified glass (LiI-Li2O-SiO2, AgIAg2O-P2O5 )
Structure: ?
*Semiconducting glass: Transition metal ion doped glasses
such as V2O5- SiO2
Determination of ionic conductivity from Complex Impedance
The capacitance (C) and conductance (G = 1/R) of the samples were measured
as a function of frequency 
(a) Dc conductivity
Z*()  Z()  Z() 
1
G(ω)
C()
1  i
G(ω)
G(ω)
G 2 (ω)  ω2 (C(ω)  C0 ) 2
Real part
Z' 
Imaginary part
G(ω) C(ω)  C0 
Z 
G 2 (ω)  ω2 (C(ω)  C0 ) 2
C0 capacitance of the sample cell without sample
From the complex impedance plots (Z-Z), the dc
resistance R was calculated at the intersection of real axis
of impedance (i.e. Z=0)
σ dc =
1  t 
 
R A
(b) Ac conductivity
The frequency dependent ac conductivity () at frequency  was determined from
the following relation
 t 
σ(ω) = G(ω)  
A
The real part of the permittivity () related to the capacitance by
 C(ω) 

ε (ω) = 
  t/A 
 ε0 
where 0 is the free space permittivity
The imaginary part () of the permittivity was related to the real part of the
conductivity as
 G(ω) 
ε(ω) = 
  t/A 
 ε 0ω 
The real and imaginary part of the electric modulus [M*() = M() + iM()] were
calculated from the real and imaginary part of the dielectric permittivity by the following
relation
ε(ω)
ε(ω)2 + ε(ω)2
Real part
M(ω) =
Imaginary part
ε(ω)
M(ω) =
ε(ω)2 + ε(ω)2
R
d
I (t )
d  (t )
 Co
Vo
dt
eo S eo R 2
Co 

d
d

d  (t )
I (t )

dt
eo  S  Eo

t
1
 (t ) 
I (t ')dt '

Co  Vo 0
r * (x) =
U*Ref (x)
*
U In (x)
1+ r* (l )
Z (ω) = Z0
1  r* (l )
*
s
r*(l ) = r(0) e[2l /(α+iβ)]
ε* = ε  ε =
σ*(ω) = σ(ω) + iσ(ω) =
1
iωZ*s (ω)C0
1
(t/A)
*
Z s (ω)
Electron-conducting glass
Transition metal ion glass (V2O5-GeO2, ZnO-MoO3)
eV4+
V5+
Electrical conduction in Vanadate glass
Oxygen
Vanadium
Oxide
glasses
containing
transition metal ions show
semiconducting behavior due to
the presence of transition metal
ions in multivalent states.
Electrical conduction in these
glassy semiconductors takes place
by the hopping movement of
electrons or polarons between
transition metal ions of different
valence states.
Electron conduction in glass
The presence of disorder causes localization, which results in the tailing of the bands into the
band gap in disordered solids such as glass. The Fermi level which is situated in the gap in case of
crystalline materials lies within the localized states in case of glasses. The trapping and subsequent
release of charge carriers by the localized states below EF, interrupts the motion of the carriers.
Whenever there is a strong localization the conduction results from thermally activated hopping of
charge carriers between localized states in the mobility gap. In the case of the strong localization the
charge carrier is not simply an electron but rather a polaron.
Ec
The term polaron comes from the early conjecture that the
formation of polaron might occur in a polar (ionic) lattice
EF
which is known as the dielectric polaron. However, the
formation of polaron is not solely restricted in polar
Ev
materials; it can also be formed in covalent materials
The energy levels in an amorphous semiconductor
A polaron has larger effective mass than the free carrier, because polaron carries the induced
distortion caused by it. The extent of induced distortion may be large or small and
accordingly the polaron is termed as large or small.
Temperature dependent dc conductivity of semiconducting glasses
The conduction process at high
temperatures is considered in terms of
optical phonon-assisted hopping of
small polarons between localized
states.
With the lowering of temperature, a
smaller activation energy is preferred
by the electrons and hence the hopping
distance
continuously
changes,
resulting in variable range hopping of
polarons.
J. Phys.: Condens. Matter 19 (2007) 106222
Polaron
the formation of the dielectric polaron in which
an excess electron added to the centre of the
ionic lattice causes ionic readjustment.
Nature and shape of the potential well created due to
polarization of the lattice and formation of polarons
with a polaron radius rp.
The energy of a polaron Wp = e2/2εp (l/rp - 1/R)
Models for dc electrical conduction
Mott Model
Holstein Model
Schnakenberg Model
Emin Model
 Variable range hopping model
Mott Model
Mott proposed the theoretical model for the hopping conductivity of transition
metal oxide glasses, in the light of phonon assisted hopping of small polarons
between localized states. At high temperatures the non-adiabatic nearest neighbor
hopping mechanism gives expression for the conductivity as
σ = νo(e2C(1-C)/kBTR)exp(-2αR)exp(-W/kBT)
νo is the optical phonon frequency, α is the inverse localization length
The activation energy for hopping conduction is given by,
W = WH + WD/2
= WD
for T > θD /2
for T < θD /4
θD is the characteristic Debye temperature
J. Phys.: Condens. Matter 19 (2007) 106222
Variable range hopping model
The variable range hopping (VRH) model is used to investigate the low temperature
behavior of the strongly disordered systems.
The localized states are randomly distributed in energy as well as in space with a
uniform distribution N(EF) , the density of states per unit volume per unit energy at
energies close to the Fermi energy. The hopping probability between two states of
spatial separation R and energy separation W has the form
P ~ exp(-2αRW/kBT)
xV2O5 − (1 − x)SiO2
O x=0.8
 x=0.9
The conductivity relation
σ = σ0 exp[ - (T0 / T) ¼]
with
To = 19.4 σ3/ kBN(EF)
In general for d-dimensions
σ = σ0 exp[ - (T0 / T)1/d+1 ]
J. Appl. Phys. 74, 3961 (1993)
Holstein Model
Based on the molecular crystal model the expression of conductivity in the nonadiabatic hopping case derived by Holstein and Friedman is given by
σ = (3Ne2 R2 J/2kBT)(π/ kBTWH)1/2 exp(-WH / kBT)
For adiabatic hopping the expression of conductivity derived by Emin and Holstein is
given by
σ = (8π Ne2R2 vo/3 kBT) exp[- (WH - J)/ kBT]
N is the site concentration and J is the polaron bandwidth related to the wave function
overlap on adjacent sites. The condition for the nature of hopping in this model
expressed by
J > or < (2kB TWH /π )1/4 (hvo/π)1/2
Where > and < indicate adiabatic and non-adiabatic hopping respectively.
Schnakenberg Model
Schnakenberg proposed a more generalized polaron hopping model where WD ≠ 0.
He considered that hopping is not possible without activation energy WD at low
temperatures, which will be caused by acoustic phonons
Temperature dependence of conductivity in this model is given by
σ ~ T-1[sinh(hvo/kB T)] 1/2exp( -4 WH/hvo)tanh(hvo/kB T)exp(-WD/kBT)
The polaron mass mp is evaluated as
mp = (ħ2/2Ja2) exp(γ)
for non-adiabatic
mp = (ħ2/2ω0a2) exp(γ)
for adiabatic
where, a is the lattice parameter and
γ = Wp/ ħω0.
J. Appl. Phys. 86, 2078 (1999)
Emin Model
Gorham-Bergeron and Emin [108] have extended the calculation to include coupling
of the electron (polaron) to both the acoustical and optical phonon modes. The dc
hopping conductivity is generally given by
ac
1/2
σ = (Ne 2 R 2 /6kT)(J/h) 2[πh 2 /2(E op
+E
)kT]
C
C
Nc is the carrier
concentration R is the
op
ac
ac
×exp[- WD2 /8(E op
+E
)k
T
-W
/kT]×exp[-(E
+E
)/kT]

C
D
A
C
A
hopping distance

The expression for hopping rate is
Г =(Jij / ħ)2 [ħ2 π /2(Ecop+Ecac)kB T]l/2exp[ -WD2 / 8(Ecop+Ecac )kBT]
exp[ - WD /2kBT].exp[ - ( exp[ - (EAop+EAac )/kBT]
Jij is an electron transfer integral between sites i and j,
and Ecop, Ecac, EAop and EAac are defined as
Ecop = ħ2 / 4kBT .1/N Σg [2Ebop / ħ ωg,op]cosech(ħ ωg,op /2kBT)ωg,op2
Ecac = ħ2 / 4kBT .1/N Σg [2Ebac / ħ ωg,ac]cosech(ħ ωg,ac /2kBT) ωg,ac 2
EAop = [2kBT/ ħ ω0]Eboptanh(ħ ω0 /2kBT)
Phys. Rev. B 66, 132203 (2002)
EAac = 1/N Σg [2kBT / ħ ωg,ac]Ebac tanh(ħ ωg,ac /2kBT)
ω0 is the maximum longitudinal optical frequency, ωg,op is the optical frequency, ωg,ac is
the acoustic phonon frequency at wave vector g and N is the number of phonon modes
Ac electrical conduction
The study of ac conductivity in several amorphous semiconductors, insulators, polymers
shows that the ac conductivity in all the cases can be expressed by
σ1 (ω) = Aωs
The conductivity can be split into two parts, as

0.6V2O5 -0.4Ag2O
σtot(ω) = σdc+ σ1 (ω)
1 ()    p Y()2  / [1  2  2 ]d
0
αp is the polarizability of a pair of sites,
Y(τ) is the distribution function
Models for ac electrical conduction
Electron tunneling or Quantum mechanical tunneling
Small polaron tunneling model
Large polaron tunneling model
Hopping over barrier model
Correlated barrier hopping model
0.6V2O5 -0.4Ag2O
Phys. Rev. B 68, 224202 (2003)
Electron tunneling
In this model the random variable ξ = 2αR. Here, α and R has the same meaning as discussed
before. It is assumed that the R is constant for all sites. The relation (α is relaxation time) gives
that
  ot exp(2R) / cosh( / 2k BT)
Δ is the difference of energy of the two energetically favorable sites for tunneling. The real part of
ac conductivity can be expressed (when Δ →0 ) as
1 ()  (Ne2  / 6k BT)
max

R 4 d() / [1  () 2 ]
min
The characteristic tunneling distance Rω at ωτ = 1
R  (1/ 2 )ln(1/  0 )
The ac conductivity
1 ()  (Ce 2 k BT / )N(E F ) 2 R 4
The frequency exponent
s = 1-4 / ln(1/ωτ0)
C is constant, N(EF) is the density of
states at the Fermi level and
N=kBTN(EF)
Small polaron tunneling model
The relaxation time for small polaron tunneling at high temperatures
  0 exp(WH / k BT)exp(2R)
At low temperatures the relaxation time is not thermally activated
  0 exp(4WH / 0 )exp(2R)
WH is the hopping energy α is the spatial decay parameters for the s-like wave function, R is the
intersite separation and ω0 is the vibrational frequency corresponding to the lattice distortion
The characteristic tunneling distance Rω
Rω = (1/ 2α)[ln(1/ωτ0 ) − (WH / kBT)]
at high temperatures
Obviously at a characteristic frequency frequencies higher than ω= ωc where
ωc = (1/ τ0 )exp(−WH / kBT)
The frequency exponent s
s = 1- 4/[ln( 1/ ωτ0) -WH/kBT]
Large polaron tunneling model
When the polaron energy is derived from the polarization changes in the deformed lattice, the
resultant excitation is called large or dielectric polaron
The Polaron hopping energy is given by,
WH = WHO(1-rp/R)
where rp is the polaron radius, WHO is given by
WHO = e2/4εprp
εp is the effective dielectric constant
Long has obtained the expression for ac conductivity as
σ1(ω) = [(π2ekBT )2 /12][N(EF)]2ωRω4/[2αkBT+WHOrp/Rω2]
xV2O5 − (1 − x)ZnO
The frequency exponent s has been calculated as
s=1-(1/Rω')(4+6βWHOrp'/ Rω' 2 )/(1+(βWHOrp'/ Rω'))2
R'ω and r'p are related by the dimensionless equations
Rω' 2 +[βWHO+ln(ωτ)] Rω' -βWHOrp' = 0
Where
Rω' = 2αRω , rp' = 2αrp, β = 1/kBT
For large values of rp', s decreases from unity with increasing temperature and gives values
predicted by quantum mechanical tunneling model. For small values of rp', s decreases with
the increasing temperature becomes minimum at a certain temperature and ultimately again
increases with the increase of temperature.
J. Phys.: Condens. Matter 21 (2009) 145802
Correlated Barrier Hopping (CBH) Model
The case of a thermally activated single electron transition relaxation variable W is given by
W = WM - (e2/πε'ε0R)
The relaxation time is given by
τ = τ0exp(W/kBT)/cosh(Δ/kBT)
In this CBH model the ac conductivity in the narrow band limit (Δ0<<kBT) is expressed by
σ1(ω) = (π2/24) N2'0Rω6
N is the concentration of pair sites
Rω is the hopping distance at frequency ω given by
Rω = e2/[πε'ε0 /{WM-kBTln(l/ωτ0)}]
xV2O5 − (1 − x)ZnO
The frequency exponent s
s = l-kBT / [WM-kBT ln(l/ωτ0)]
s ~ 1-6kBT/WM
J. Phys.: Condens. Matter 21 (2009) 145802
Alkali modified SiO2
Ion jump in defect structure
Energy
E
Eeff
Lattice Distance
No
electric
field
With electric
field
• Diffusion Eq.:
 Eσ 
D=Doexp - 
 kT 
σ = qN μ
Eeff<E
Dq

k T
Anderson-Stuart model for alkali oxide glass
E= ∆EB+∆ES
∆EB is binding energy, ∆Es is
strain energy .
∆EB is required to overcome
Coulomb attraction with the
nonbridgng oxygen ions.
∆Es is required to dilate the
glass network when moving from
one site to another.
ZZ0e2  1
1 
ΔE B =



γ  r+r0 λ/2 
Schematic potential energy
landscape of an alkali ion
ΔES =4πGrD (r  rD ) 2
Ionic conduction in glass
Moving ions carry charge, and thus produce an electrical response. The total conductivity of
an ion conducting glass is the result of two factors: conduction current and molecular dipole
relaxation Suppose, ions with charge q, are subjected to an electric field Є
σdc = n c (Ze)μ
 ΔE 
n c (T) = N0exp   crn 
 k BT 
 ΔE mig 
D(T)= 2 υ0exp  

k
T

B

 (T) =
(Ze).D (T)
k BT
 is jump between two sites of distance , Σ is the
degree of freedom, υ0 is the frequency of jumps attempt
by the ions, and Emig is the energy that must be
overcome for the jump process to take place
 ΔE mig 
 2 υ0 (Ze)
 (T)=
exp  

k BT
 k BT 
  ΔE mig  ΔE crn  
N0  2 υ0 (Ze)2

 (T)=
exp  


k BT
k BT


σdc =
 ΔE 
σ0
exp   act 
T
 k BT 
 Ionic conduction model for dc conduction
 Weak Electrolyte Theory
 Random site model
 Anderson-Stuart model
 Diffusion path model
 Volume expansion and conduction pathway
 Cluster-Bypass model
 Percolation model
Different models on ion conduction
Cluster model
Alkali halide entered
the host glass to form
connected pathways of
Alkali halide clusters
Weak electrolyte
model
A small number of
mobile
ions
dissociated
from
Alkali
halide
contribute to the
transport process.
Potential energy diagram according to diffusion path model
Diffusion path model
Mobile ions ions mainly
correlated with halide
ions in the connected
pathways move
Clustering in glass at Tg
shows the pathways for
ion migration
A 2–D representation of
modified random network
The arrows indicate the
location of the presumed
pathways for ion migration.
Ac conductivity
Polarization
Four types of polarization in ionic electrolytes
The different types of polarization depending on time in an electric field
Ac conductivity spectra
Ion jump in a disordered landscape in different time scale
The low frequency data were
consisted with power law model
(Jonscher & Almomd-West):
/() = dc[1 + (  /  c ) n ] , n < 1
The high frequency data were
explained using the site relaxation
model (Funke):
/() = () [1 + c / ]-q , q > 1
Almond-West (A -W) formalism
Jonscher proposed the following empirical relationship for the dispersion in the imaginary part
of the ac complex dielectric constant (dielectric loss)
 
 ()  

 p
''
a
  
   
  p 
b 1
' ()  ()
' ()  d.c  An
  a    b 1 
 '()        
 p   p  


Equating the dielectric loss frequency ωp to the ion hopping frequency, ωH = 2πνH, and letting
a = -1 and b = n
   1    n 1 
 '()  K 
 
 
 H 
 H  
 KH  KH1 n n
dc  KH
dc  AH n
   n 
()  dc 1  
 
  H  
Unified Site Relaxation (USR) model
Jonscher power law model provides a good fit to the spectra in the low frequency
regime, it fails as frequencies increases above few MHz (>10MHz). Nice fits are however,
obtained when the exponent n is replaced by another exponent m larger than unity. The high
frequency conductivity spectra then can be written as*
m

1 

σ (high freq.)  σ hf 1+

 ωt1 
1<m2 and t1 (=1/1) denotes the crossover time to high frequency dc regime and hf is
the high frequency plateau value of conductivity
The basic assumption of USR model is that when the ion has performed a jump,
two different processes of site relaxation are important:
1. Relaxation by shifting of the Coulomb cage.
2. Site-identity relaxation.
(a) and (b) are the typical single
particle potentials encountered by
hopping ions (at t=0). (c) The
relaxation processes along two
possible competing routes.
* K. Funke, Prog. Solid State Chem., 22 (1993) 111
Linear response theory
=

kB. T. d 0
< j(0) j(t) >
e -i  t dt
2
*()
V

Nernst-Einstein equation with a frequency
dependent diffusion coefficient
D*()
kB T HR
nc q 22  2
 '() 
0  r (t )  sin(t )dt
6k B TH R
1
c
t
2

 dc
Frequency
 Short time : < r 2 (t) > is small (< r 2 (t) > ~ t 1-n ) the ion transport is
characterized by non-random backward - forward hops and '() ~  n
Long time : a crossover to diffusive dynamics occurs at  = c with
< r 2 (t) > ~ t , the ions execute an unrestricted random work of step size
ξ at the rate h
 dc = q2 ξ 2 nc c / 12 k T HR
1-n
time
'()
*() =
q2 nc
t
< r (t) >
Fluctuation-dissipation theorem
n
Dc conductivity and hopping frequency
-2
-5
l o g 1 0 [  d c (W
-6
-7
-8
-9
-1
-4
l o g 10 [ ( h ) (r a d s ) ]
x = 0.0
x = 0.1
x = 0.2
x = 0.3
x = 0.4
x = 0.5
x = 0.6
-1
-1
cm )]
9
x AgI-(1-x) AgPO3
-3
x AgI- (1-x) (Ag2O - P2O5 )
8
x=0.2
x=0.3
x=0.4
x=0.5
x=0.6
7
6
5
4
3
-10
-11
2
3
4
5
6
7
1000/T(K
8
-1
9
10
)
4
5
6
7
8
9
-1
1000/T(K )
10
Dc conductivity and hopping frequency show thermally activated nature.
dc= q nc  = Cnch
Ag+ ion concentrations:
Ag+ Total
xAgI-(1-x)(Ag2O-2B2O3)
Ag+ from AgI
-3
10
-3
)
22
nc(cm
22
nc(cm )
10
x=0.0
x=0.1
x=0.2
x=0.3
x=0.4
x=0.5
21
10
10
Mobile Ag+ Ions
uncorrelated (HR=1)
21
Mobile Ag+ Ions
correlated (HR=0.2)
20
10
3
4
5
6
1000/T(K
7
-1
8
)
0.0 0.1 0.2 0.3 0.4 0.5 0.6
x(AgI)mol fraction
Frequency exponents (n):
0.8
0.8
x AgI - ( 1 - x ) AgPO3
x AgI - ( 1 - x ) ( Ag2O - 2B2O3 )
x AgI - ( 1 - x ) ( Ag2O - V2O5 )
x AgI - ( 1 - x ) AgPO3
0.7
0.7
Sidebottom, Phy. Rev.
Lett.82,3653(1999):
n
n
n= 2/3 for threedimensional motion
0.6
n=2/5 for twodimensional motion
0.6
x = 0.0
x = 0.2
x = 0.4
x = 0.6
(b)
(a)
0.5
120 150 180 210 240 270 300
T(K)
n=1/3, for onedimensional motion
0.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
x
n is almost independent of temperature & composition
Correlation of ion dynamics with microscopic lengths
The time dependent mean-square displacement <r2(t)> of the mobile ions
can be obtained from conductivity spectra by
 r 2 (t ) 
12k BTHR t
  '() sin(t ')d 
dt
'
0 0
2

Ncq
<r2(t)> can be rewritten as
 r 2 (t )  R2 (t )  H R
<R2(t)> is the mean-square displacement of the center of charge of the mobile
ions.
Correlation of ion dynamics with microscopic lengths
Behavior of <R2(t)> at crossover point [<R2(tp)> ]:
 Long time behavior of <R2(t)> [√<R2()>]:
<R2()> can be obtained using <R2(t)> from the dielectric permittivity spectra,
6 k B T
2
0  ( o )   (  ) 
 R () 
Nc q 2
The characteristic length, <R2(tp)> signifies the distance the mobile ions have
to travel in order to overcome the forces, causing correlated forward –
backward motion
The length √<R2()> signifies the spatial extent of non-random diffusive
motion of the mobile ions where ions perform localized hops between two
neighboring sites.
Physical interpretation of microscopic length scales <R2(tp)> and <R2()>
The characteristic transition time, tp,
signifies the transition from subdiffusive
to diffusive behavior.
The characteristic length, <R2(tp)>
would signify the distance the mobile
ions have to travel in order to overcome
the forces, which cause
correlated
forward – backward motion
The length scale, √<R2()> signifies the
spatial extent of non-random diffusive
motion of the mobile ions where ions
perform localized hops between two
neighboring sites.
Correlation of Li+ Ion Dynamics with Microscopic length scales
x LiI-(1-x) LiPO3
The value of √<R2(tp)> decreases as
the LiI content increases. This means
that the Li+ ions have to cover smaller
distances in order to overcome the
forces
causing
the
backward
correlations with the increase of LiI
content .
The value of √<R′2(∞)> decreases
with the increase of the LiI content.
These results suggest that √<R2(tp)>
and √<R′2(∞)> show almost similar
trend with the variation of LiI content
in the glass matrix.
A. Shaw and A. Ghosh, J. Phys. Chem. C
116, 24255 (2012)
Compositional dependence of <R2(tp)> in silver borophosphate glasses
0.5 Ag2O-0.5[xB2O3-(1-x) P2O5]
x=0.8
The characteristic length scales
√<R2(tp)> shows a mixed glass
former effect similar to that of the
dc conductivity indicating that the
ion dynamics is strongly correlated
to the structural details at
microscopic level.
S. Kabi and A. Ghosh, Europhys. Lett. 100, 26007 (2012)
Correlation between <R2()> and RAg-Ag in Ag2O-(B2O3-P2O5) glasses
Spatial extent of subdiffusive ionic
motion shows one (for y=0.50) or two
(for y=0.40) minimum depending on
composition. The variation of this
length scale is just opposite to that of
the conductivity.
The mean ion-ion distance shows
similar compositional dependency as
that of microscopic length scale,
√<R2()>.
The decrease of ion-ion separation
may be correlated to the increase of
conductivity which indicates that the
ion dynamics is governed by the
structural modification.
FTIR spectra : silver borophosphatete glasses
0.5 Ag2O-0.5[xB2O3-(1-x) P2O5 ]
x=0.5
The FTIR spectra of silver borophosphate glasses show
vibrational bands due to borate and phosphate units.
The vibrational band due to BO4 units is observed at
800-1000 cm-1. The vibrational band due to BO3 units is
observed at 1200-1500 cm-1 .
The vibrational band corresponding to phosphate unit
coordinated to three bridging oxygen (~1090 cm-1)
gradually shifts to lower wave number with increase of
B2O3 content in the glasses. This indicates the formation
of P-O-B linkages in these units.
the
compositional
dependence
of
relative
concentration of BO4 unit is consistent with that
estimated from NMR spectra [ Elbersa et al Solid State
Nucl. Mag. Res. 27, 65 (2005)]
Kabi and Ghosh, Europhys .Lett. 100, 26007(2012)
y Li2O-(1-y)[xBi2O3-(1-x) B2O3 ]
Characteristic length scales of mobile Li+ ions
The values of <R2(tp)> for a particular
composition are independent of temperature.
The composition dependence of <R2(tp)> is
similar to that of σdc. This result is similar to that
of silver and sodium borophosphate glasses.
The microscopic parameters primarily depend
on the two factors: Coulomb interaction between
the ions and the modification of the glass
network.
For mixed former glasses structural
modification is the main reason for the
compositional variation of these parameters.
A. Shaw and A. Ghosh, J. Chem. Phys. 139, 114503 (2013)
Correlation of glass network structure with Microscopic length scales
y Li2O-(1-y)[xBi2O3-(1-x) B2O3 ]
The composition dependence of relative
area of the BO4 unit is almost similar to that of
the dc conductivity as well as the
characteristic mean displacement and spatial
extent of localized motion of mobile ions.
Li+ ions can stay in different sites, such as
non-bridging sites of BO3 trigonal units, BiO6
octahedral units or in the vicinity of BO4anionic sites. Among these the BO4- anionic
sites are favorable for ion transport as the
negative charge is spread over the four boron
oxygen bonds.
Thus the potential wells around BO4 sites
become broader and shallower. Thus, with the
increase of BO4 units the Coulomb energy
minimizes which in turn increases of the value
of and also the cage width increases.
Scaling models for the conductivity spectra
Scaling is an important feature in any data evaluation. The ability to scale
different conductivity isotherms so as to collapse on a common curve
indicates that the process can be separated into a common physical
mechanism modified only by temperature scale.
A. Ghosh and A. Pan, Phys. Rev. Lett. 84, 2188(2000)
A. Ghosh and M. Sural, Euorphys. Lett. 47, 688(1999)
()/dc = F(/c)
This model is being referred to as “Ghosh’s Scaling Model” in the literature:
Phys. Rev. B, 72, 174304(2005); Phys. Rev. B, 84, 174306 (2011); J. Chem.
Phys. 127, 124507 (2007) ; Mater. Sci. & Eng. B, 149, 18(2008); Solid State
Ionics, 176, 1311(2005); Physica B, 452,142(2014); Ionics 20,399(2014); RSC
Advances 5, 21614 (2015); Appl. Phys. A (2015)
Structure dependent scaling of conductivity spectra
in mixed former glasses
(a) For Ag2O-(MoO3–P2O5) glasses the relaxation
dynamics is independent of structural modification
of the glass network, since the extent of modification
is less as observed in FTIR spectra.
(a)
(b) For Ag2O-(B2O3–P2O5) glasses the relaxation
dynamics is dependent on structural modification of
the glass network. The extent of modification is
significant in this case.
0.5 Ag2O-0.5 (x B2O3 -(1-x) P2O5)
(  /dc )
10
10
2
X=0
X = 0.2
X = 0.4
X = 0.5
X = 0.6
X = 0.8
X = 1.0
1
(b)
10
0
10
-6
10
-5
10
-4
-3
10
10
-2
10
-1
10
0
(/c)
10
1
10
2
10
3
10
4
Scaling of conductivity spectra for silver borophosphate
glasses
0.50 Ag2O - 0.50 - ( x B2O3- (1 - x ) P2O5)
(  /dc )
100
10
x=0
0.20
0.40
0.50
1
1E-6 1E-5 1E-4 1E-3 0.01
0.1
1
10
100 1000 10000
(/c)
(  /dc )
0.50 Ag2O - 0.50 - ( x B2O3- (1 - x ) P2O5)
10
x=0.60
x=0.80
x=1
1
1E-5
1E-4
1E-3
0.01
0.1
1
10
100
(/c)
S. Kabi and A. Ghosh, Europhys .Lett. 100, 26007 (2012)
1000
Scaling of mean square displacement curves: Choice of the
scaling parameters
We have chosen the scaling parameters of the length scale and time axis as
√<R2(tc)> and tc respectively.
 The reason behind the choice is that, these two parameters are correlated with the
scaling parameters σdc and ωc used for conductivity scaling.
Scaling of mean square displacement of mobile ions
Fig. (a) shows mean square displacement
of mobile ions for Ag ion conducting glass
systems.
This scaling is also consistent with the
scaling of the conductivity spectra [Ghosh
and Pan, PRL, 84, 2188 (2000)]
S. Kabi and A. Ghosh, Europhys. Lett.,
108, 36002 (2014)
Dielectric Relaxation
Debye Relaxation: Debye relaxation is the dielectric
relaxation response of an ideal, noninteracting population of
dipoles to an alternating external electric field

 *      
1  i0 
Non-Debye Relaxation
Cole-Cole Relaxation: This equation is used when
dilectric loss peak shows symmetric broadening
    

[1  (icc )cc ]
Cole-Davison Relaxation: This equation is used when
the dilectric loss peak shows asymmetric broadening
    

[1  (icd )]cd
Havriliak-Negami (HN) equation: This equation
considers both symmetric and asymmetric broadening

[1  (ihn )hn ] hn
0<αhn≤1 and 0<αhnγhn≤1
    
Electric modulus formalism
 
d  

M*  M 1   exp(it )   dt 
 dt  
 0
2  M''
φ(t)  
cos(ωt)dω
π 0 ωM

 

t
 t   exp  
  ; 0<<1
  m  


Havriliak - Negami (HN) equation
   

1








s

 [1  (ihn ) hn ] hn



Thanks
Dielectric Relaxation
ε(ω0 )  ε  
ε(ω0 ) 
Havriliak-Negami (HN) equation
    

[1  (ihn )hn ] hn
2


2


 ε(ω)
0
 ε(ω)
0
ω
dω
ω 2  ω0 2
ω0
dω
ω 2  ω0 2


    Re 
[1  (ihn ) hn ] hn



  Ιm 
 hn ] hn
[1

(
i

)


hn








Electric modulus formalism
1




M ()  *
 M  iM  2
i 2
2
 ()
  
  2
*
 
d  

M*  M 1   exp(it )   dt 
 dt  
 0

 

t
 t   exp  
  ; 0<<1
  m  


2  M''
φ(t)  
cos(ωt)dω
π 0 ωM
Havriliak-Negami (HN) equation
   

1





  s

 [1  (ihn ) hn ] hn


