1

Department of Applied Physics, Department of Applied Physics,

The Hebrew University of Jerusalem, The Hebrew University of Jerusalem,

IsraelIsrael

Yuri FeldmanYuri Feldman

Dr. A. PuzenkoDr. A. PuzenkoDr. Y. HayashiDr. Y. Hayashi

Mrs.EMrs.E. Axelrod. Axelrod

Dr. Dr. Ya.RyabovYa.RyabovDr. A. Dr. A. GutinaGutina

Mr. P. Ben IshaiMr. P. Ben Ishai

Mr. I. Mr. I. BeilinBeilin

Mr. L. Mr. L. LivshitzLivshitz

Dielectric spectroscopy of complex systems in Time and Dielectric spectroscopy of complex systems in Time and

Frequency Domains: Problems and solutionsFrequency Domains: Problems and solutions

2

Dielectric response on mesoscaleDielectric response on mesoscale

Dielectric spectroscopy is sensitive to relaxation processes

in an extremely wide range of characteristic times ( 10 5 - 10 -12 s)

Broadband Dielectric Spectroscopy

Porous materialsPorous materials and colloidsand colloids

Clusters Clusters Single droplets Single droplets and poresand pores

Glass forming Glass forming liquidsliquids

MacromoleculesMacromolecules

1010--221010--44 00 101022 101044 101066 101088 10101010 10101212

Time Domain Dielectric Spectroscopy; Time Domain Reflectometry

f (Hz)f (Hz)

1010--66

WaterWater

iceice

3

Broad-Band Dielectric Spectrometer BDS 80 (NOVOCONTROL) in the frequency

range 10 µµµµHz – 1.8 GHz with automatic temperature control by QUATRO Cryosystem (temperature range -150 0C - +400 0C)

Generator

US

IS Sample

Holder

Sample

Sample

Capacitor

( ) ( ) ( ) )(/)( '''* ωωωωω ss IUiZZZ =+=

( ) 02/)(")(')(* CffZii πωεωεωε −=−=

Measurement SetupMeasurement Setup

4

Time domain dielectric spectroscopy

lumped capacitance approximation in time domain:

Sampling oscilloscope Data Processing

Sampling head Sample sectionPulse Generator

V0(t)R(t)

><><

=Γ≈ε−εΦ−ε

∞ )0()0(

)()0()(

)(

MM

tMMt

t

s

s

Dipole Correlation FunctionSpectrum

ε ω ε*( )[ ( )]

[ ( )]= + ⋅∞

1

0 0Z C

L Q t

L V t

Q t C V t t t V t dt

t

( ) ( ) ( ' ) ( ' ) '= + −

∞

•

∫0

0

ε Φ

Yu. Feldman,et al., (1996)

Rev. Sci. Instrum. 67, 3208.

100KHz 100KHz –– 18 GHz18 GHz

0

0

1I(t) = [V (t)-R(t)] ,

Z

0V(t)=V (t)+R(t)

5

( )( )

( ) ( )( ) ( )tRtV

tRtVZ

tI

tVr

tt −+

==∞→∞→

0

00 limlim

( ) ( ) ( ) ( ) ( ) ( )( ) ( )tRtV

tRtV

Z

tRtV

r

tVtI

tR +

−+==

∞→0

0

0

0 lim

[ ] [ ]

+−

+−−=∞→ )()(

)()()()()()(

1)(

0

000

0 tRtV

tRtVlimtRtVtRtV

ZtI

tD

( ) ( )( ) ( )tRtV

tRtVlim

CZ t +−ε

=σ∞→

0

0

00

0 ( ) ( ) ''0

∫=t

D dttItQ

6

7

8

9

10

)()()( tVtVtI sampleopen −+∝ δ ( ) )()(exp)( ωωωδω sampleopen vvii −=

TDS Problems

Mismatch of signals in time

Solution:

Time Shift in signal using the criteria

0:0

maxmax

2

2

==tt dt

Vd

dt

dI

11

)()()( tVtVtI sampleopen −+∝ δ ( ) )()(exp)( ωωωδω sampleopen vvii −=

TDS Problems

Mismatch of signals in time

12

Solution:

Time Shift in signal using the criteria

0:0

maxmax

2

2

==tt dt

Vd

dt

dI

δ−⇒ maxmax tt

13

TDS Problems

Vertical Shift in signal

∆+→ )()( tVtV shortshort

False Conductivity

−∆

+≈

∆−−

−∝=

∞→

)()(1

)()(

)()(

)(

)(lim

0tVtV

tVtV

tVtV

tV

tI

shortsample

shortsample

sampleopen

t

σ

σ

14

TDS Problems

Vertical Shift in signal

∆+→ )()( tVtV shortshort

False Conductivity

−∆

+≈

∆−−

−∝=

∞→

)()(1

)()(

)()(

)(

)(lim

0tVtV

tVtV

tVtV

tV

tI

shortsample

shortsample

sampleopen

t

σ

σ

Incorrect Amplitudes

15

∫ ′′=t

D tdtItQ0

)()(

16

TDS Problems

Fourier Transforms – 2 main problems:

Truncation errors

Descrete data

Attenuation factor based on cubic spline with end point correction

[ ] )()()( 21

0

2

∆+∆≈= ∑−

=

−OehHfthF

N

k

Nkn

kn

π

The frequencies of the FFT are, in essence, integral cycles on the interval. Therefore at high frequencies the summation can be

smaller than the second order errors (previously ignored). A correction must be added.

+∆≈= ∑ ∑

= =

M

k endptsk

kkN

nik

k

fai

n Nnheh

NnWeHthF

0

22 )2()2()]([ παπ ππ

∑∑== ∆

−+

∆−

≈endptsk

kkk

M

k

kk

tth

tthth )()()(

0

ϕψ∫∞

∞−

= )()/2(/2

sdseNnWNni ψπ π

∫∞

∞−

−= )()/2( /2 ksdseNn k

Nni

k ϕπα π

Using derivatives - ))(())(( tfFtfFi &=ω Or Samoloon Algorithm

∆≤≤

2

11f

T

17

18

For ε ≈ 1000 ?

Frequency Impedance AnalyzersFrequency Impedance Analyzers

19

Smith ChartSmith Chart

20

TDDSTDDS

21

Sample Holders in BDS and TDS MeasurementsSample Holders in BDS and TDS Measurements

Low Frequency (Low Frequency (<10<107 7 Hz)Hz)

High Frequency High Frequency ((>10>108 8 Hz)Hz)

SolidsSolidsLiquidsLiquids

Electrode polarizationElectrode polarizationContacts Contacts

P, V conditions P, V conditions

επλ

2

240.R ≤

)ab( +> επλ

Small angle spatteringSmall angle spattering

Polish electrodesPolish electrodes

Slide electrodes, open ended coaxial or mesh electrodesSlide electrodes, open ended coaxial or mesh electrodes

22

How to account for leakage from a samples cell during an How to account for leakage from a samples cell during an

extended measurement?extended measurement?

How to maintain a constant pressure during a wide temperature How to maintain a constant pressure during a wide temperature

regime?regime?

How to establish a fixed geometry for a liquid?How to establish a fixed geometry for a liquid?

How to overcome expansion problems for liquids?How to overcome expansion problems for liquids?

How to ensure that there will be no air pockets between How to ensure that there will be no air pockets between

electrode and sample?electrode and sample?

=

a

bC

ln

2 0

0

πε

The 3 electrode sample cellThe 3 electrode sample cell

23

Ground electrode

(Guard)Upper Electrode

Bottom Electrode

=

a

bC

ln

2 0

0

πε

24

25

DS of Glycerol

26

TDDS Sample CellTDDS Sample Cell

Sample

Sample Cell

Coaxial line

Upper electrode

Inner electrode Gold plated Sample

Upper electrode

Inner electrode

Coaxial line

27

Typical Results

0 8 16 24

2000

4000

6000

8000

10000

12000

14000

Responce

t (ns) [GHz]

( )tΦ

Time domain

Dielectric response

function:

Frequency domain

Spectrum of dielectric

permittivity: ( )ωε '

28

Broadband Dielectric Spectroscopy

CellsCells

Time Domain Dielectric Spectroscopy

f (Hz)f (Hz)

Dielectric Response in Biological SystemsDielectric Response in Biological Systems

Dielectric spectroscopy is sensitive to relaxation processes

in an extremely wide range of characteristic times ( 10 5 - 10 -11 s)

HH

HH33NN

++—— C C —— COOCOO--

RR

Ala AspAla Asp ArgArg AsnAsn

CysCys GluGlu GlnGln HisHis

IleIle LeuLeu LysLys MetMet

PhePhe Ser Ser ThrThr TrpTrp

TyrTyr ValVal

10103310102200 101055 101066 101077 101088 101099 101010101010441010111010--11 10101111

PP--NN++ Head group

region

LipidsLipids

ProteinsProteins

WaterWater

γγ -- DispersionDispersion

DNA, RNADNA, RNA

αα--DispersionDispersion β -- DispersionDispersion δ -- DispersionDispersion

TissuesTissues

iceice

Amino acidsAmino acids

29

dc conductivitydc conductivity

• In the cathode

– Na+ + electron → Na

• 2Na + H2O → 2Na+ +

2OH– + H2

• In the anode

– Cl– – electron → Cl

• 2Cl → Cl2

• Metal + Cl → MCl

(e.g., CuCl2)

Problems are:

increase of pH, birth of gases and corrosion of the electrode.These are significant if current is not small.

Na+

Na+

Na+

Cl –

Cl –

Cl –

+ + + + + +

––––––

30

V

+ −−−−

E

Electrical double layer

cation

V

anion

+ −−−−

Eswitch

Frequency

εε εε

Measured characteristic

True characteristic of

material

Electrode polarization effect

What is the electrode polarization?What is the electrode polarization?

31

Electrode Polarization

– List of possibilities of the

corrections

• Substitution method

• Single exponent method (F3)

• Fractal method (F2)

• Blocking electrode

• Electromagnetic induction method

• Combination of plate-capacitance

and fractal method

• Change of the distance (d) of plate-

capacitance

• Others?

Na+

Na+

Cl –

Cl –+ + + + + +

––––––

Na+

Na+

Na+

Na+

Na+

Cl –

Cl –

Cl –

Cl –

Cl –

Cl –

Cl –

Cl –

Cl –

Cl –

Na+

Na+

Na+

Na+

Na+

V

Electric double layer

32

Signals in time domainSignals in time domain

0.1 1 10 100 1000100

150

200

250

300

350

400

Air

Short

SuspensionV (mV)

time (nsec)

Dielectric spectrum in

frequency domain

Dielectric spectrum in

frequency domainFourier transform

0.1 1 10 100 1000

50

100

150

200

250

300

ε'

erythrocyte suspension (4.0%)

frequency (MHz)

Data treatment from time domain to frequency domain

E.g., human erythrocyte suspension

33

100 1000

260

270

280

290

V (mV)

Time (nsec)

Electrode polarizationElectrode polarization

34

Single exponent methodSingle exponent method

2ττττττττ ττττ

R

R p+

Am

plitu

de

time

2ττττττττ ττττ

p

R p+

tmax

Vec(t)

Vp(t)

Z0

R

Cp

Vp

Vo(t)Vx

I

C

( ) ( ) ( )( )( ) Rp

p

sss

sCssVsI

ττττ

+++

+=

11

1

0

0

0

( ) ( ) ( )( )( )

Rp

Rp

sss

ssVsV

τττττ

+++++

=11

1

0

00

Electrode polarization correction function

ττττp=Z0Cp , ττττR=RCp , ττττp=RC , s=jωωωω

( ) ( ) ( )tVx

tV ec

C

p tVp

+=∞→

for low conductive sample

( ) ( )( )

( )max

max

2

/

1 where

,1 2

tV

tV

etV

p

p

t

pR

p

ec

&

−=

−+

= −

τ

τττ τ

35

Example B: Human lymphocyte suspensionExample B: Human lymphocyte suspension

4 5 6 7

101

102

log f / Hz

ε"

0 2000 4000

–1.8

–1.7

t / nsec

log(V

air–Vp)

0.4

0.5

0.60.70.80.91

2

3

4

5

6789

10

Conductivity of membrane (New v.f.)

T-cells B-cells Cancer

Normal

Transformed

σm 1

0-5 , S/m

B-norm. Magala Farage Raji Bjab Daudi Peer HDMar T-nor.

d

membraneεm, σm

cytoplasmεcp, σcp

nuclear envelopeεne, εne

nucleoplasmεnp, εnp

R

dn

Rn

36

Single exponent method: Limitation

This behavior allows This behavior allows

us to suggest the us to suggest the

frequency (or time) frequency (or time)

selfself--similarity of the similarity of the

electrode polarization electrode polarization

phenomenon, i.e. the phenomenon, i.e. the

fractal nature of this fractal nature of this

effect. effect.

5 µµµµs

300 ns300 ns

100 ns100 nsV t V ep p

t

o

Cp( )/

=− τ

τC << τCp

10-3

10-2

10-1

100

101

102

103

104

10-2

10-1

100

101

102

103

104

ε''

Frequency (MHz)

Without correctionWithout correction

5 µµµµs

300 ns100 ns

Strong electrolyte solution; NaCl (0.1 M) in water

37

Fractal approach Fractal approach

εC0

R

C sν ( ) C sν ( )

Equivalent circuits for a conductive sample with

electrode polarization impedance described by recap Cν

Equivalent circuits for a conductive sample with

electrode polarization impedance described by recap Cν

( ) ( )( ) vjvAjZ

−= ωω0

Electrodes should be polished carefully because the electrode

polarization depends on the fractal nature of the electrodes surfaces.

( ) ( ) iii

iisCRRsCsC iii

ννννν

−−− =⋅≡

i=1,2; 0<ν<1

can be applied to high conductive sample

In Frequency domain:

3810

210

3

–0.94

–0.92

–0.9

–0.88

t / nsec

log V(t)

cylindrical capacitor

102

103

–1.6

–1.55

t / nsec

log V(t)

plate capacitor

( ) cRttBtBtV τνν/21

21 ++= for t/τc <<1

( ) RtBtBtV ++= 21

21

ννfor t/τc >>1

( ) ( ) ( ) ( )[ ]ctRtC

tC

tV τνννν

νννν/exp12211

2211

−−+Γ

+Γ

∝

In Time domainIn Time domain:

( ) cRtBttV τν /+= for t/τc <<1

( ) RBttV += νfor t/τc >>1

a) both electrodes of the sample holder have the same (or

equivalent) electrode polarization (ν1 = ν2; Cν1 = Cν2)b) (ν1 ≠ ν2; Cν1 ≠ Cν2)

There is no dispersion of the conductive solution (sample) in the

time window defined by τc (> tmin = 1/ωmax)

39

Electromagnetic inductive method Electromagnetic inductive method

-- additional remark for near future additional remark for near future --

H. Wakamatsu, 1997. Hewlett-Packard J. 48, 37.

K. Asami et al., 1999. Biophys. J. 76, 3345.

VA

EEEE

SolutionSolutionSolutionSolutionPrimary coil Secondary coil

~

The HP SensorThe HP SensorThe HP SensorThe HP Sensor

There is no electrode polarization!

Problem is amount of sample (400 ml).

40

Data TreatmentData Treatment

What function?What function? dc conductivity correction?dc conductivity correction?Fitting?Fitting?

φ−=

ε−εε−ωε

∞

∞ )(ˆ)(*

tdt

dL

s

Frequency DomainFrequency Domain Time DomainTime Domain

[ ]βαωτεεεωε

)(1)(*

m

s

i+

−+= ∞

∞

1/

exp )(

1

1

>>τ

τ

−

τ

=φνµ−

tfor

ttAt

m

HavriliakHavriliak--NegamiNegami (HN)(HN) Power Power × stretched exponentialstretched exponential

??

1 10 100 1000

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

26.50C

250C

240C

logΨ (t /τ

1)

t /τ1

ε'

ε

41

��Simultaneous fit of both real and imaginary components of the Simultaneous fit of both real and imaginary components of the

complex dielectric permittivity data;complex dielectric permittivity data;

��Problems in selecting the proper fitting function, such as:Problems in selecting the proper fitting function, such as:

The limited frequency and temperature ranges of the experiment;The limited frequency and temperature ranges of the experiment;

Distortion influences of the sample holder;Distortion influences of the sample holder;

Overlapping of several physical processes with different Overlapping of several physical processes with different

amplitudes in the same frequency and temperature ramplitudes in the same frequency and temperature ranges.anges.

��Continuous parameter estimation; Continuous parameter estimation;

�� dc conductivity problems.dc conductivity problems.

DS data treatment; problems of correct fittingDS data treatment; problems of correct fitting

42

New Fitting SoftwareNew Fitting Software--””MatFitMatFit””

43

�Simultaneous fit of both real and imaginary components of the complex dielectric permittivity data;

�Linear and nonlinear fitting methods, from least squares and logarithmic to fitting procedures based on entropy norm;

�Global fit procedure on all selected temperature ranges for continuous parameter estimation;

�Hilbert Transform for computing dc conductivity.

Key features of the programKey features of the program

44

Complex systems involve in the appearance of a new Complex systems involve in the appearance of a new

""mesoscopicmesoscopic" length scale, intermediate between " length scale, intermediate between

molecular and macroscopic.molecular and macroscopic.

Complex systemsComplex systems??

••Porous materials (porous glasses, porous silicon, etc.)Porous materials (porous glasses, porous silicon, etc.)

••FerroelectricsFerroelectrics

••Complex liquids (microemulsions, emulsions, etc.)Complex liquids (microemulsions, emulsions, etc.)

••Glass forming liquids and polymers including compositesGlass forming liquids and polymers including composites

••Biological systems ( protein solutions, membranes and cell Biological systems ( protein solutions, membranes and cell

suspensions)suspensions)

Percolation: The transition associated with the formation of a

continuous path spanning an arbitrarily large ("infinite")

range.

The percolation cluster is a selfThe percolation cluster is a self--similar fractal. similar fractal.

5 10 15 20 25 30 35 40 45

ε s

Temperature ( oC )

0 2 4 6 8 10

10-1

100

101

102

103

σ[µ

S/c

m]

5 10 15 20 25 30 35 40 45

20

40

60

80

100

TpTon

What is the percolation phenomenon?What is the percolation phenomenon?

Three dimensional plots of frequency

and temperature dependence of the

dielectric permittivity εεεε' for the AOT/water/decane microemulsion

Three-dimensional plots of the time and

temperature dependence of the

macroscopic Dipole Correlation

Function for the AOT-water-decane

microemulsion

(t) ψ ≈⟨ ⋅ ⟩⟨ ⋅ ⟩M M t

M M

( ) ( )

( ) ( ),

0

0 0

AOTAOT--waterwater--decanedecane microemulsion (17.5:21.3:61.2 microemulsion (17.5:21.3:61.2 volvol%), %),

W = 26.3, W = 26.3, RRwpwp = 35.6 Angstrom= 35.6 Angstrom

Dielectric response in microemulsions

((Yu.FeldmanYu.Feldman et. al., PRE. et. al., PRE. 1995,19961995,1996))

47

101

103

105

107

Frequency [Hz]

100

101

102

Perm

ittivity' []

-2000

200400

Temperature [°C]

Sample ASample A

10 -110 1

10 3

10 510 7

Frequency [H

z]

10-4

10-2

100

102

Perm

ittivity'' []

-200 0 200 400

Temperature [°C]

Sample CSample C

Dielectric response of the porous glass materials Dielectric response of the porous glass materials

additional treatment in 0.5N KOHadditional treatment in 0.5N KOH

drdryyinging at 200at 20000CC

rinsing in rinsing in deionizeddeionized waterwater

Sample BSample B

Sample DSample D

10 1

10 3

10 5

10 7

Frequency [Hz]

10-4

10-2

100

102

Perm

ittivity'' []

0100

200300

Temperature [°

C]

Sample C after heating Sample C after heating

48

Porous borosilicate glass samplesPorous borosilicate glass samplesAOTAOT--waterwater--decanedecane microemulsionmicroemulsion

,)0()0(

)()0((t)

⟩⋅⟨⟩⋅⟨

≈MM

tMMψ

Ψ(t) = A (t/τ1) -µ exp [- (t/τm) ν]

Ψ(t) ~ exp [- (t/τm) ν]

DDff = 3= 3νν, , where where DDff is a fractal dimension is a fractal dimension

2

1+=

νµγ ( ) α

ααα

ττ

α

+−+⋅

=

1

1

1

11

mS1

1−=

να

Morphology parameters:

sm : cut-off cluster size

γγγγ : polydispersity index

η η η η : cut-off rate index

Dynamic parameters:

ττττ 1 : minimal time

α : α : α : α : scaling parameter

( ) ( )

*

Ψ−∆+=

Λ

∞ ttd

dFεεωε

(Chem. Phys. (2002) V. 284 , 139(Chem. Phys. (2002) V. 284 , 139--168)168)m

d

ds

sD1

DD

lg

lgΘ−

=

49

Percolation:Percolation: Transfer of electric excitation through the developed system of open pores

( ) ( ) ( )

*

Ψ−−+=

Λ

∞∞ ttd

dFs εεεωε

-100 -50 0 50 100 150 200

Temperature [°C]

020

4060

Permittivity' []

Freq. [Hz]=5.10e+03 Freq. [Hz]=1.13e+04 Freq. [Hz]=3.24e+04

Dielectric relaxation in percolationDielectric relaxation in percolation

10-3 10-2 10-1 100

0.0

0.2

0.4

0.6

0.8

Sample A

Sample B

Sample C

Correl

atio

n funct

ion

time ( s )

Df= 3ν, where Df is a fractal dimension

Ψ(t) ~ exp [- (t/τm) ν]

50

The Fractal Dimension of Percolation PassThe Fractal Dimension of Percolation Pass

Sample AA BB CC DD

Fractal dimension Df 00..9999 11..8899 11..3311 22..55

51

φλ

=

−

Λ

d D

( ) ( )w A aξ ξ ξα β= −exp

w w : size distribution function

αααααααα, , ββββββββ, A, A: empirical parameters

Λ

λφ =

V

V

pφφφφ : porosity of two phase solid-pore system

Vp : volume of the whole empty space

V : whole volume of the sample

ΛΛΛΛΛΛΛΛ , , λλλλλλλλ : upper and

lower limits of self-

similarity

DD : regular fractal

dimension of the

system

( ) ( )Φ = ∫φ ξ ξ ξµ

,D w d

1

µµµµ = λλλλ / ΛΛΛΛ

ξξξξ : scale parameter ξξξξ ∈∈∈∈ [µµµµ,1]

Porous medium in terms of regular and random fractalsPorous medium in terms of regular and random fractals

52

1<< 1,<<a hen w

,

µµ

µα

αφ α

α

+

−++

−−

⋅−++

+≈

1

1

1

1

1

1 Dd

Dd

D−≈

4

1φ

Sample

Fractal

dimension

Df

Porosity φφφφ (%)

( obtained from relative

mass

decrement measurements )

Porosity φφφφ (%)

( obtained from

dielectric

measurements )

A 0.99 38 33

B 1.89 48 47

C 1.31 38 37

D 2.5 50 68

Porosity DeterminationPorosity Determination ((A.Puzenko,etA.Puzenko,et al., Phys. Rev. (al., Phys. Rev. (BB), ), 6060, 14348, 1999), 14348, 1999)

µµµµ = λλλλ / ΛΛΛΛ1 << 3,=d

1 << ,0 1

µ

αα >+

53

10 1

10 2

10 3

10 4

10 5

10 6

10 7

Frequency [Hz]

100

101

102

103

104

105

Perm

ittivity' []

100

200

300

400

500

Temper

ature [K

]

ε′

III

II

I

PS HR-TEM micrographs and dielectric response

-Disordered network of crystallitesDisordered network of crystallites

and poresand pores

- Fractal structureFractal structure

- Preserve the crystalline naturePreserve the crystalline nature

54

-6 -5 -4 -3 -2 -1 0 1

0

0.2

0.4

0.6

0.8

1

lg (time)

correlation function

ΨΨΨΨ (t/τ τ τ τ ) ~ e -(t/τ)τ)τ)τ)νννν

Df is the fractal dimension of

the pore channel lengths D f = 3 ν

f

D4

f D4

1

1

1

D4

1 f

−≈

−−

−=

−

µµφ

Dielectric response

at the percolation

Corr

elat

ion funct

ion

lg (time)

Sample

Stretching

parameter

νννν

Fractal

dimension

Df

Porosity φφφφ (%)

( obtained from

dielectric

measurements)

20 µµµµ 0.88±0.02 2.64±0.06 74±1

30 µµµµ 0.87±0.02 2.61±0.06 72±1

20 µ

30 µ

Hopping conductivity in a percolating model

PS MidPS Mid--T process II (300T process II (300--400 K) 400 K)

Porosity DeterminationPorosity Determination

E. Axelrod, et al., JNCS. (2002)

55

DC conductivityDC conductivity

Hilbert transform - we applied the Kramers-Kronig relations to the real part of the dielectric function, and subtracted this term from the

imaginary part of the measured dielectric data.

101

102

103

104

101

102

103

104

105

106

100

101

102

103

104

ε'

438 K

488 K

102

103

104

105

100

101

102

103

H[ε

'(ω)]

Frequency ( Hz )

ε''

Frequency ( Hz )

102

103

104

105

10-1

100

101

102

103

ε''-ε'(H

ilb) ~ σ

dc/ω

Frequency ( Hz )

438 K

488 K

> 100 > 100 °°°°°°°°CC

56

KTN CrystalsKTN Crystals

High temperature → Low temperature

cubic → tetragonal → othorthorhombic → rombohedralic

( )0TT

Cs −=ε

CuONbTaK :335.065.01

••What does Cu do to the dielectric relaxation?What does Cu do to the dielectric relaxation?

••Does Cu interact with Does Cu interact with NbNb ??

••What effects do the differing ionic radii of Cu What effects do the differing ionic radii of Cu

and K cause ?and K cause ?

Cu IonNb Ion

PerovskitePerovskite Structure ABOStructure ABO33

Cu doped at 2% by weight Cu doped at 2% by weight –– 1:1000 unit 1:1000 unit

cell volumescell volumes

QuestionsQuestions

57

Crystal 120 KTNCrystal 120 KTN –– Dielectric Losses through the Ferroelectric phase Dielectric Losses through the Ferroelectric phase

transitiontransition

Ferroelectric PTFerroelectric PT

22ndnd PTPT

33rdrd PTPT

DC ConductivityDC Conductivity

Arrhenius Process

∆−=

Tk

E

B

exp0ττ

Activated Process

−

=kTT

Dexp0ττ

∆−=

Tk

E

B

exp0ττ

−+=

kT

EC

kT

E da expexp0ττ

has been linked has been linked

to dynamic to dynamic

relaxation of the relaxation of the

offoff--centre centre NbNb ions ions

in their in their

octrahedraloctrahedral site. site.

58

Maximum Peak FrequencyMaximum Peak Frequency

0.0025 0.0030 0.0035 0.0040 0.0045 0.0050

-15.5

-15.0

-14.5

-14.0

-13.5

-13.0

-12.5

ln(1/fmax)

1000/T [K-1]

Crystal 120

T=354 oK

T=295 oK

T=290 oK

T=230 oK

3*

02.0

228

354

≈=∆

=

=

z

eV

KT

KT

o

k

o

x

µ

ParaelectricParaelectric PhasePhase

Ferroelectric PhaseFerroelectric Phase

5105.51

12.0

×==

=

ηC

eVEa

Where Where ηη is the maximum is the maximum

defect concentrationdefect concentration

eVE 37.0=∆

eVEd 32.0=

59

Free Volume Model

••Ferroelectric Phase Ferroelectric Phase –– Structural Shift constrains Structural Shift constrains CuCu relaxationrelaxation

KK++ 1.37 pm1.37 pm

OO--22 1.40 pm1.40 pm

CuCu+2+2 0.73 pm0.73 pm

CuCu+1+1 0.77 pm0.77 pm

••CuCu ions perturbate the Lattice structureions perturbate the Lattice structure

••Some of these Some of these ““DefectsDefects”” provide the space needed for the Oxygen provide the space needed for the Oxygen

OctahedraOctahedra to to ““tilttilt””..

••Alternatively the Defects allow the migration of Oxygen Alternatively the Defects allow the migration of Oxygen

VacanciesVacancies

CuCu ionion