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Electrical properties of glassy (vitreous) materials

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1 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 – Editor-in-Chief, Indian Journal of Physics (Springer)

2 Noble Laureates from Kolkata
Sir Ronald Ross  Nobel Prize (Physiology or Medicine 1902) Rabindranath Tagore Nobel Prize (Literature 1913) C.V. Raman Nobel Prize (Physics 1930) Amartya Sen Nobel Prize (Economics 1998) Mother Teresa Nobel Prize (Peace 1979)

3 A brief introduction to glass Electronic conduction in glass
OUTLINE OF TALK A brief introduction to glass Electronic conduction in glass Ionic conduction in glass

4 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

5 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.

6 Different types of disorder
(a)Topological (geometrical or positional) disorder (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. Absence of long range periodicity of atomic arrangements. 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

7 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!

8 XRD patterns of crystalline and vitreous materials
Crystalline materials: Sharp diffraction peaks are obtained. Each diffraction peak corresponds to a particular set of crystalline planes characterized by Miller index (hkl). Disorder materials: A broad hump is obtained instead of sharp diffraction peaks. The broad hump indicates that the disorder materials exhibit short range periodicity.

9 Experiment to detect the crystalline and non-crystalline 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.

10 Electron wave function for crystalline and disorder solids
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. Crystalline solids Disordered solids

11 Non-localized/ Extended states
Wave function and probability density for crystalline and disorder solids Type of Solids Wave Function Probability density Crystalline Non-localized/ Extended states normalizied Disordered Localized states normalized

12 Energy band diagram for crystalline materials

13 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. 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.

14 Glass is an amorphous solid that exhibits “glass transition”
What is Glass ? Glass is an amorphous solid that exhibits “glass transition” Experiment: DSC or DTA Very slow cooling liquid Rapid quench Super- cooled Glass Crystal Freezing point (Tf) transition (Tg) (1) (2) Deb and Ghosh, EPL-Europhys. Lett. 95, (2011) Volume Temperature

15 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.

16 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. 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). 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

17 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) Thus, we can obtain the following relations 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 example of first-order transition, since the volume changes discontinuously at 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, is discontinuous at Tg. This apparently indicates that glass transition is a second order phase transition.

18 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 Using the following relation We can obtain the following relation It has been observed experimentally that the values of are appreciably higher than those of Thus, glass transition is not simple second order phase transition.

19 Variation of heat capacity with temperature for glass and crystal
The heat capacity for a glass is comparable to that of a crystal but considerably smaller than that of the liquid. At glass transition temperature the heat capacity changes discontinuous At very low temperature Cp~T3 (Debye’s T3 law) for crystal and Cp~T for glasses. Crystal glass Tg

20 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”.

21 Modification of glass structure
*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, AgI-Ag2O-P2O5 ) Structure: ? *Semiconducting glass: Transition metal ion doped glasses such as V2O5- SiO2

22 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 Real part Imaginary part 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) (b) Ac conductivity The frequency dependent ac conductivity () at frequency  was determined from the following relation

23 The real part of the permittivity () related to the capacitance by
where 0 is the free space permittivity The imaginary part () of the permittivity was related to the real part of the conductivity as 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 Real part Imaginary part

24 R d


26 Electronic conduction in non-crystalline system

27 e- Electron-conducting glass
Transition metal ion glass (V2O5-GeO2, ZnO-MoO3) 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. V4+ V5+ e- Electrical conduction in Vanadate glass Oxygen Vanadium

28 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. The term polaron comes from the early conjecture that the formation of polaron might occur in a polar (ionic) lattice which is known as the dielectric polaron. However, the formation of polaron is not solely restricted in polar 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.

29 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)

30 The energy of a polaron Wp = e2/2εp (l/rp - 1/R)
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)

31 Models for dc electrical conduction
Mott Model Holstein Model Schnakenberg Model Emin Model Variable range hopping model

32 Mott Model σ = νo(e2C(1-C)/kBTR)exp(-2αR)exp(-W/kBT)
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 for T > θD /2   = WD for T < θD /4 θD is the characteristic Debye temperature J. Phys.: Condens. Matter 19 (2007)

33 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 The conductivity relation σ = σ0 exp[ - (T0 / T) ¼] with To = 19.4 σ3/ kBN(EF) O x=0.8  x=0.9 In general for d-dimensions σ = σ0 exp[ - (T0 / T)1/d+1 ] J. Appl. Phys. 74, 3961 (1993)

34 Holstein Model σ = (8π Ne2R2 vo/3 kBT) exp[- (WH - J)/ kBT]
Based on the molecular crystal model the expression of conductivity in the non-adiabatic 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.

35 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)

36 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 Nc is the carrier concentration R is the 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) EAac = 1/N Σg [2kBT / ħ ωg,ac]Ebac tanh(ħ ωg,ac /2kBT) Phys. Rev. B 66, (2002) 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

37 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 0.6V2O5 -0.4Ag2O The conductivity can be split into two parts, as σtot(ω) = σdc+ σ1 (ω) 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, (2003)

38 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 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 The characteristic tunneling distance Rω at ωτ = 1 The ac conductivity C is constant, N(EF) is the density of states at the Fermi level and N=kBTN(EF) The frequency exponent s = 1-4 / ln(1/ωτ0)

39 Small polaron tunneling model
The relaxation time for small polaron tunneling at high temperatures At low temperatures the relaxation time is not thermally activated 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]

40 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] 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. xV2O5 − (1 − x)ZnO J. Phys.: Condens. Matter 21 (2009)

41 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Rω 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)}] The frequency exponent s s = l-kBT / [WM-kBT ln(l/ωτ0)] s ~ 1-6kBT/WM xV2O5 − (1 − x)ZnO J. Phys.: Condens. Matter 21 (2009)

42 Ionic conduction in glass systems

43 Alkali modified SiO2

44 Ion jump in defect structure
Energy Lattice Distance No electric field E Eeff Eeff<E With electric field Diffusion Eq.:

45 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. Schematic potential energy landscape of an alkali ion

46 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  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

47 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

48 Different models on ion conduction
Cluster model Weak electrolyte model Diffusion path model Alkali halide entered the host glass to form connected pathways of Alkali halide clusters A small number of mobile ions dissociated from Alkali halide contribute to the transport process. Mobile ions ions mainly correlated with halide ions in the connected pathways move Enter speaker notes here. Potential energy diagram according to diffusion path model A 2–D representation of modified random network The arrows indicate the location of the presumed pathways for ion migration. Clustering in glass at Tg shows the pathways for ion migration

49 Ac conductivity Polarization
Four types of polarization in ionic electrolytes The different types of polarization depending on time in an electric field

50 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

51 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) Equating the dielectric loss frequency ωp to the ion hopping frequency, ωH = 2πνH, and letting a = -1 and b = n

52 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* 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. 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

53 Linear response theory
Fluctuation-dissipation theorem V *() = < j(0) j(t) > e -i  t dt kB. T. d Nernst-Einstein equation with a frequency dependent diffusion coefficient q2 nc *() = kB T HR D*() 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

54 Dc conductivity and hopping frequency show thermally activated nature.
dc= q nc  = Cnch

55 Ag+ ion concentrations:
Ag+ Total Ag+ from AgI Mobile Ag+ Ions uncorrelated (HR=1) Mobile Ag+ Ions correlated (HR=0.2)

56 Frequency exponents (n):
Sidebottom, Phy. Rev. Lett.82,3653(1999): n= 2/3 for three-dimensional motion n=2/5 for two-dimensional motion n=1/3, for one-dimensional motion n is almost independent of temperature & composition

57 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 <r2(t)> can be rewritten as <R2(t)> is the mean-square displacement of the center of charge of the mobile ions.

58 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, 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.

59 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.

60 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, (2012)

61 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, (2012)

62 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.

63 FTIR spectra : silver borophosphatete glasses
0.5 Ag2O-0.5[xB2O3-(1-x) P2O5 ] 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 cm-1. The vibrational band due to BO3 units is observed at 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)] x=0.5 Kabi and Ghosh, Europhys .Lett. 100, 26007(2012)

64 Characteristic length scales of mobile Li+ ions
y Li2O-(1-y)[xBi2O3-(1-x) B2O3 ] 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, (2013)

65 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 BO4- anionic 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.

66 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, (2005); Phys. Rev. B, 84, (2011); J. Chem. Phys. 127, (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, (2015); Appl. Phys. A (2015)




70 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. (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. (a) (b)

71 Scaling of conductivity spectra for silver borophosphate
glasses S. Kabi and A. Ghosh, Europhys .Lett. 100, (2012)

72 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.

73 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, (2014)

74 Dielectric Relaxation
Debye Relaxation: Debye relaxation is the dielectric relaxation response of an ideal, noninteracting population of dipoles to an alternating external electric field Non-Debye Relaxation Cole-Cole Relaxation: This equation is used when dilectric loss peak shows symmetric broadening Cole-Davison Relaxation: This equation is used when the dilectric loss peak shows asymmetric broadening Havriliak-Negami (HN) equation: This equation considers both symmetric and asymmetric broadening 0<αhn≤1 and 0<αhnγhn≤1

75 Electric modulus formalism
Havriliak - Negami (HN) equation

76 Thanks Enter speaker notes here.

77 Dielectric Relaxation
Havriliak-Negami (HN) equation

78 Electric modulus formalism
Havriliak-Negami (HN) equation

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