Complex ImpedanceComplex Impedance SpectroscopySpectroscopyComplex ImpedanceComplex Impedance SpectroscopySpectroscopy

S. S. BhogaS. S. Bhoga

Department of PhysicsDepartment of Physics

RTM Nagpur University, NagpurRTM Nagpur University, Nagpur

What is Complex ImpedanceWhat is Complex Impedance Spectroscopy ?Spectroscopy ?

Involves measurement ofInvolves measurement ofReal and imaginary parts of impedance/ Real and imaginary parts of impedance/ admittanceadmittanceOver wide frequency range Over wide frequency range (10(10-3-3 to 10 to 101414 Hz) Hz)

Convey informationConvey information

Microscopic ion dynamics Microscopic ion dynamics Ionic bulk conductivityIonic bulk conductivityElectrode polarisation Electrode polarisation Activation enthalpy for ion migrationActivation enthalpy for ion migrationMicro-heterogeneities Micro-heterogeneities Dielectric constantDielectric constantElectric polarizationElectric polarizationEtc.Etc.

What is Complex ImpedanceWhat is Complex Impedance Spectroscopy ?Spectroscopy ?

(() ) is Fourier transform of autocorrelation function is Fourier transform of autocorrelation function of current density, of current density, ii

It resolves elementary hopping processes It resolves elementary hopping processes It is a very powerful microscope in time.It is a very powerful microscope in time.

0

,)exp()().0(3

)( dttjtiiTk

V

N

tvqV

ti ii

1),(

1)( here

V - volume of the sample qi - Charge on mobile specie

vi - Velocities

- Angular frequency

Why Complex Impedance Why Complex Impedance Spectroscopy ?Spectroscopy ?

Mass transport in solids is vital for many Mass transport in solids is vital for many technological applicationstechnological applications

Mobility of ions under influencesMobility of ions under influences Chemical gradientChemical gradient -- DiffusionDiffusionElectric potentialElectric potential gradientgradient -- Conductivity Conductivity

S

/cm

)

Time (Min)

electronic

Bulk

Fig:- Variation of dc conductivity with time

No instrument is No instrument is capable to capable to determinedetermine

is time/frequency dependentis time/frequency dependent

Disadvantage of dc conductivityDisadvantage of dc conductivityPolarization of ions introduces Polarization of ions introduces errorerror

ac Conductivity ac Conductivity Avoids polarizationAvoids polarization

Single frequencySingle frequency First hand information (screening First hand information (screening test)test)

Bulk conductivity – dcBulk conductivity – dc

Only due to ion migrationOnly due to ion migration

Excludes electrode Excludes electrode polarizationpolarization

Dielectric lossDielectric loss

Grain-boundary contributionGrain-boundary contribution

Why Complex Impedance Why Complex Impedance Spectroscopy ?Spectroscopy ?

Fig:- Variation of ac conductivity with frequency

S

/cm

)

Log f (Hz)

Ele

ctro

de

pol

ariz

atio

n

Bulk

Die

lect

ric

loss

es

Ele

ctro

de

pol

ariz

atio

n

Fig:- Cole-Cole plot (impedance)

Z”

Oh

m)

Z’ (Ohm)RBulk

Die

lect

ric

loss

es

Fig:- Cole-Cole plot (Admittance)

Y”

S)

Y’ (S)GBulk

Ele

ctro

de

pol

ariz

atio

n

Fig:- Variation of ac conductivity with frequency

S

/cm

)

Log f (Hz)

Ele

ctro

de

pol

ariz

atio

n

Bulk

Die

lect

ric

loss

es

Fig:- Variation of ac conductivity with frequency

S

/cm

)

Log f (Hz)

Ele

ctro

de

pol

ariz

atio

n

Bulk

Die

lect

ric

loss

es

Fig:- Variation of ac conductivity with frequency

S

/cm

)

Log f (Hz)

Ele

ctro

de

pol

ariz

atio

n

Bulk

Die

lect

ric

loss

es

S

/cm

)

Log f (Hz)

S

/cm

)

Log f (Hz)

Ele

ctro

de

pol

ariz

atio

n

Bulk

Die

lect

ric

loss

es

Ele

ctro

de

pol

ariz

atio

n

Fig:- Cole-Cole plot (impedance)

Z”

Oh

m)

Z’ (Ohm)RBulk

Die

lect

ric

loss

es

Fig:- Cole-Cole plot (Admittance)

Y”

S)

Y’ (S)GBulk

Ele

ctro

de

pol

ariz

atio

n

Ele

ctro

de

pol

ariz

atio

n

Fig:- Cole-Cole plot (impedance)

Z”

Oh

m)

Z’ (Ohm)RBulk

Die

lect

ric

loss

es

Ele

ctro

de

pol

ariz

atio

n

Fig:- Cole-Cole plot (impedance)

Z”

Oh

m)

Z’ (Ohm)RBulk

Die

lect

ric

loss

es

Ele

ctro

de

pol

ariz

atio

n

Fig:- Cole-Cole plot (impedance)

Z”

Oh

m)

Z’ (Ohm)RBulk

Die

lect

ric

loss

es

Fig:- Cole-Cole plot (impedance)

Z”

Oh

m)

Z’ (Ohm)RBulk

Die

lect

ric

loss

es

Fig:- Cole-Cole plot (impedance)

Z”

Oh

m)

Z’ (Ohm)RBulk

Die

lect

ric

loss

es

Z”

Oh

m)

Z’ (Ohm)RBulk

Die

lect

ric

loss

es

Z”

Oh

m)

Z’ (Ohm)

Z”

Oh

m)

Z’ (Ohm)RBulk

Die

lect

ric

loss

es

Fig:- Cole-Cole plot (Admittance)

Y”

S)

Y’ (S)GBulk

Ele

ctro

de

pol

ariz

atio

n

Fig:- Cole-Cole plot (Admittance)

Y”

S)

Y’ (S)GBulk

Ele

ctro

de

pol

ariz

atio

n

Fig:- Cole-Cole plot (Admittance)

Y”

S)

Y’ (S)GBulk

Ele

ctro

de

pol

ariz

atio

n

Fig:- Cole-Cole plot (Admittance)

Y”

S)

Y’ (S)GBulk

Ele

ctro

de

pol

ariz

atio

n

Y”

S)

Y’ (S)GBulk

Y”

S)

Y’ (S)

Y”

S)

Y’ (S)GBulk

Ele

ctro

de

pol

ariz

atio

n

Ele

ctro

de

pol

ariz

atio

n

Why Complex Impedance Why Complex Impedance Spectroscopy ?Spectroscopy ?

Provides straight-forward determination of Provides straight-forward determination of electrolyte resistance irrespective of degree of electrolyte resistance irrespective of degree of electrode polarisation.electrode polarisation.

Does not necessarily require use of reversible Does not necessarily require use of reversible electrodes or complicated cell geometryelectrodes or complicated cell geometry

Determine dielectric properties of materialDetermine dielectric properties of material

Ease of studying electrode polarisationEase of studying electrode polarisation

Mathematical Formulation Mathematical Formulation andand

Electrical Equivalent ModelsElectrical Equivalent Models

Mathematical model and Analyses

Mathematical model and Analyses

Plausible Physical modelPlausible Physical model Equivalent circuitEquivalent circuit

System characterizationSystem characterization

ExperimentExperiment TheoryTheory

CIS – CIS – Basics/Electrical EquivalenceBasics/Electrical Equivalence

Sine waves are useSine waves are useVV((tt) = ) = VVmm exp( exp(jjtt ) )

II((tt) = ) = IImm exp( exp(jjt t ))

Input and output waveforms are sameInput and output waveforms are same

ZZ in time domain obeys Ohms law in time domain obeys Ohms law

Z Z (() = |) = |ZZ| exp (-| exp (-jj))

Input Output

CIS – CIS – Basics/Electrical EquivalenceBasics/Electrical Equivalence

Real and Imaginary parts Real and Imaginary parts

Z’

Z”

0 (R/2, R/2)

(0,0) (R,0)

Complex impedance Complex impedance responseresponse

Electrical EquivalentElectrical Equivalent

,)(1

r

Z222 RC

R

BG

G

2

2

22 )(1 RC

CR

BG

BiZ

Eliminate Eliminate and rearrange and rearrange

((ZZrr -R -R/2)/2)22 + + ZZii22 = ( = (RR/2)/2)22 . .

Eq. of circle with radius Eq. of circle with radius RR/2/2

CIS – CIS – Basics/Electrical EquivalenceBasics/Electrical Equivalence

Real and Imaginary parts Real and Imaginary parts

Eliminate Eliminate nn andand rearrangerearrange

((ZZrr - -RR/2 )/2 )22 + ( + ( ZZii -{[ -{[RR tan ( tan (/2)]/2)]1/21/2 }) })22 = = r r 22

where, where, r r 22 = ( = (RR/2)/2)22 + {( + {(RR/2)/2)22 + tan( + tan(/ / 2)}2)}22

))()2/sin()(21

)}2/sin()(1{1(2

01

0

1

RZ r

))()2/sin()(21

)}2/cos(){(1(2

01

0

10

RZ i

Z’

Z”

0 (R/2, [R cos(ap/2)]/[2(1+sin(ap/2]

(0,0)

Complex impedance Complex impedance responseresponse

Electrical EquivalentElectrical Equivalent

CPECPE

Depressed by angle Depressed by angle

Centre Centre ((RR/2,[/2,[RR tan{ tan{/2}]/2)/2}]/2)

Top - (Top - (RR/2, [/2, [R R cos(cos(/2)]/[2(1+ /2)]/[2(1+ sin sin . . ( (/2])/2])

Radius Radius rr

CIS – CIS – Basics/Electrical EquivalenceBasics/Electrical Equivalence

Each physical Each physical processes give processes give separate semicircle separate semicircle provided their values of provided their values of are widely different are widely different

As first approximation, As first approximation, each semicircles may each semicircles may be considered as be considered as response of R-C response of R-C combination.combination.

R1R1 + R2

Solid ion conductorSolid ion conductor

Metal electrodesMetal electrodes

Z’Z’

ZZ””

Zw

R2

C2

R1

C1 Cdl

R3

BulkBulk Grain-boundariesGrain-boundaries Metal electrodeMetal electrode

Obtaining equivalent circuit to Obtaining equivalent circuit to simulate phenomenon of simulate phenomenon of electrodeelectrode

Highly convoluted Highly convoluted combination of line and combination of line and semicirclesemicircle

Electrical Equivalence - Electrical Equivalence - ComplicationsComplications

(b)

Z”

Z’

R1C1=1 Z”=(jC2)-1

Z”

Z’

(a)R1

(d)

Z”

Z’

(c)

Z”

Z’

(a’)

C1

R1

C2

Impedance data Impedance data during cooling cycleduring cooling cycle

Cooling 2Cooling 2ooC per minC per min

Dwell time 30 minDwell time 30 min

Measurement at end Measurement at end of dwell timeof dwell time

Experimental - Experimental - MeasurementMeasurement

T6

T5

T4

T3

T2

T1

40 100 160 220 280

Tem

pera

ture

Time

ts Start time te End time

Dwell time, 30 min.

ts2 te2

ts5 te5

ts3 te3

ts1 te1

ts4 te4

Experimental – Experimental – Set upSet up

Sample 1Sample 2Sample 3Sample 4Sample 5Sample 6

Mass flow meters

Ar gas cylinder O2 gas cylinder CO2/SO2 gas

Key padDisplay

Low High

parameters

HP4192A

HP 16048 test leads

IBM compatible PC(Pentium processor

G P I B

Keithley 7001

R.F. Switch

TemperatureProgrammerFurnace

Six samplesSix samples5 to 13mm5 to 13mm diadia

Frequency Frequency 5 to 13x105 to 13x1066 Hz Hz

Gas partial pressureGas partial pressure100ppm to 10%100ppm to 10%

Temperature - Temperature - R.T. to 700R.T. to 700ooCC

Experimental – Experimental – Set upSet up

G P I B

Solartron 1287Mass flow meters

Ar gas cylinder O2 gas cylinder CO2/SO2 gas Solartron 1255B FRA

IBM compatible PC(Pentium processor

Furnace

Sample Holder

Temperature programmer

FrequencyFrequency

10x1010x10-6-6 to 1x10 to 1x1066 Hz Hz

TemperatureTemperature

R.T. to 700R.T. to 700ooCC

Scribner Z-plot and Z-Scribner Z-plot and Z-view view

Data Fitting/Data Fitting/Circuit SimulationCircuit Simulation

0

10

20

0 20 30 40 50 60

”(k

10

Z’ (k)

4 8 10 12

0

2

4

0 6 2

Fitted (CNLSF) data toFitted (CNLSF) data to

Sum of squares is minimized Sum of squares is minimized by unity weightingby unity weighting

j

ZZZZ

1

0 )()()(

2/122iii IRS

Data Fitting/Data Fitting/Circuit SimulationCircuit Simulation

0

10

20

0 20 30 40 50 60

”(k

10

Z’ (k)

4 8 10 12

0

2

4

0 6 2

Parallel to interface Cg

||

C || R | | R || R ||

C i

C i

C i

R i R I R i

Bulk (interior)

( a) Perpendicular to interface ( b)

Grain

( c)

C ||

R ||

CPE

CPE

CPE

R

C

R i

C i

R

C

R

C

R

C R

C

Conduction Mechanism Conduction Mechanism

0.9

1

1.1

1.2

1.3

1.4

0Ea

(eV

)

0.9

1

1.1

1.2

1.3

0 5 10 15

Vacancy %

La

Dy

Y

Intragrain

Intergrain

0.9

1

1.1

1.2

1.3

1.4

0Ea

(eV

)

0.9

1

1.1

1.2

1.3

0 5 10 15

Vacancy %

La

Dy

Y

0.9

1

1.1

1.2

1.3

1.4

0Ea

(eV

) 0.9

1

1.1

1.2

1.3

1.4

0Ea

(eV

)

0.9

1

1.1

1.2

1.3

0 5 10 15

Vacancy %

La

Dy

Y

IntragrainIntragrain

IntergrainIntergrain

0.9

1

1.1

1.2

1.3

1.4

0Ea

(eV

)

0.9

1

1.1

1.2

1.3

0 5 10 15

Vacancy %

La

Dy

Y

0.9

1

1.1

1.2

1.3

1.4

0Ea

(eV

) 0.9

1

1.1

1.2

1.3

1.4

0Ea

(eV

)

0.9

1

1.1

1.2

1.3

0 5 10 15

Vacancy %

La

Dy

Y

IntragrainIntragrain

IntergrainIntergrain

0.9

1

1.1

1.2

1.3

1.4

0Ea

(eV

) 0.9

1

1.1

1.2

1.3

1.4

0Ea

(eV

)

0.9

1

1.1

1.2

1.3

0 5 10 15

Vacancy %

La

Dy

Y

0.9

1

1.1

1.2

1.3

1.4

0Ea

(eV

) 0.9

1

1.1

1.2

1.3

1.4

0Ea

(eV

)

0.9

1

1.1

1.2

1.3

0 5 10 15

Vacancy %

La

Dy

Y

IntragrainIntragrain

IntergrainIntergrain

kT

Eff aop exp

Intragrain conduction modifies on doping

But not grain-boundary