# AC conductivity spectra of KCl: In crystals. Temperature dependent deviations from the Summerfield scaling

**ABSTRACT** The application on the Summerfield scaling law does not lead to a superposition of the conductivity isotherms of KCl:In crystals, but the isotherms are shifted to higher values on the T dc σ ν axis as the temperature deceases. The isotherms, however, do collapse onto a single master curve when an additional scaling factor α T with ] 1 2 . 1 [ ÷ ÷ ÷ ÷ ∈ ∈ ∈ ∈ α is used for frequency axis. The exponent α decreases with temperature. The results of this scaling procedure have been obtained considering the shape of the isotherms being independent of composition.

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Journal of Optoelectronics and Advanced Materials Vol. 6, No. 2, June 2004, p. 465 - 469

AC CONDUCTIVITY SPECTRA OF KCl CRYSTALS. TEMPERATURE

DEPENDENT DEVIATIONS FROM THE SUMMERFIELD SCALING

A. Ioanid*, A. S. Dafinei

Department of Solid State Physics, Faculty of Physics, University of Bucharest,

P.O. Box Mg. 11, 077125 Bucharest - Magurele, Romania

The application on the Summerfield scaling law does not lead to a superposition of the

conductivity isotherms of KCl:In crystals, but the isotherms are shifted to higher values on

the

T

dc

σ

single master curve when an additional scaling factor

frequency axis. The exponent α decreases with temperature. The results of this scaling

procedure have been obtained considering the shape of the isotherms being independent of

composition.

(Received January 7, 2004; accepted June 3, 2004)

Keywords: Conductivity spectra, Scaling law, Master curve, Solid state ionics

ν

axis as the temperature deceases. The isotherms, however, do collapse onto a

α

T with

] 12 . 1 [

∈ ∈ ∈ ∈α ÷ ÷ ÷ ÷

is used for

1. Introduction

Electrical and mechanical spectroscopy are two common methods for the study of the

dynamical processes in solid and liquid materials. In many cases, it has been found that electrical

and mechanical spectra obey the time-temperature superposition principle (TTSP), i.e., the spectral

shape is independent of temperature.The validity of the TTSP suggests that the basic microscopic

mechanisms of the dynamic processes do not depend on temperature, altough the time window of

these processes exhibits generally a strong temperature dependence.

The electrical conductivity spectra reflect thermally activated mouvements of charge carriers

in a solid ionic and electronic conductor matrix, such as inorganic crystals, glasses, polymers,

semiconductors. The TTSP principle is true in the low-frequency regime below a few MHz. The ion

conducting solids may be considered as strong electrolytes where most if not all of the cations are

mobile, or as weak electrolytes where the fraction of mobile ions is some function of temperature

and composition. These observations must receive much attention since they impact strongly on the

assessment of the theoretical approaches to charge transport and to relaxations occuring across the

whole field of solid state ionics [1-4]. On short time and length scales, the mobile ions show a

subdiffusive behaviour, i.e., the mean square displacement (

time, while on longer time and length scales, the subdiffusive behaviour passes over into a diffusive

behaviour with

t~) t (r2

> > > >< < < <

. Indium nanoparticles embedded in KCl crystals have been studied,

recently, by Polosan et al. [5,6].

For a theoretical description of the ion dynamics in a structurally disordered ionic conductor,

one often considers the hopping dynamics of non-interacting or interacting particles in disordered

potential landscape. The short-time subdiffusive motions does not allow specific conclusions on the

shape of the disordered potential landscape or on the role interionic interactions play in the ion

dynamics. In order to obtain information of interionic interactions, detailed analses of the frequency,

temperature and composition dependence of the ac conductivity spectra of ionic conductors are

necessary. The scaling method for the ac conductivity is a way developing the features of interest

> > > >< < < <

) t (r2

) increases sublinearly with

* Corresponding author: anaioanid@solid.fizica.unibuc.ro

Page 2

A. Ioanid, A. S. Dafinei

about these problems. Using of the scaling method consists to choise of a function depending of

frequency, temperature and composition for the frequency axis, so that the isotherms of the real part

of the conductivity

)( ' νσ

do collapse onto a single master curve indicating an universal ionic

relaxation process.

2. Scaling properties of conductivity spectra

In the first comprehensive analyses the validity of the TTSP, particulary in single ion

conducting glasses has been confirmed by the following scaling law

) ( '

σ

466

) ( F

0dc

ν

ννσ

= = = =

(1)

here

dc

σ

denotes the dc conductivity, while

0

ν is a characteristic frequency defined by some authors

∗ ∗ ∗ ∗

ν by

2)( '

σνσ= = = =

that onset frequency of the conductivity dispersion

scaling parameter for the frequency axis.

The Summerfield scaling law [5], expressed by

dc

∗ ∗ ∗ ∗

, and have used

∗ ∗ ∗ ∗

ν as a

)

T

( F

)( '

σ

dcdc

σ

ννσ

= = = =

(2)

has the main advantaje to utilize directly available quantities as scaling parameters for the frequency

axix instead of an arbitrarly determined parameter

) t (r

>=

>=>=

>= < < < <

diffusion of the mobile ions which is related to the dc conductivity via the Nernst-Einstein relation,

D

NqT

B

respectively, while

B

k is Boltzmann’s constant. This assumption is valid at low number density of

mobile ions,

10N

≤ ≤ ≤ ≤

, when the mobile particles does not interact via long-range Coulomb

forces. The scaling function F is independent of temperature. Only the single alkali glasses

(Na2O.B2O3, Li2O.B2O3, Na2O.GeO2) seem to obey equation (1), while the conductivity spectra of

several mixed alkali glasses systems (Li2O.Na2O.B2O3, Na2O.Rb2O. B2O3) and alkali tellurite

glasses (Na2O.TeO2) [6] cannot be superimposed by applying the Summerfield scaling law. The

application of the Summerfield scaling does not lead to a superposition of the isotherms, but

ν

∗ ∗ ∗ ∗

ν . In the diffusive dynamics ,

. Here

0

D denotes the coefficient of self-

> > > >< < < <

) t (r2

increases linearly with time, so that

tD

0

2

N,

k

0

2

dc

= = = =σ

and q being the number density and the charge of the mobile ions,

320cm

− − − −

isotherms are shifted to higher values on the

T

dc

σ

axis as the temperature decreases. The

isotherms, however, do collapse onto a single master curve when an additional scaling factor

α

T~f

with

3 . 1

− − − −= = = =α

is used for frequency axis [7]. This scaling law is founded in the random

barrier model (RBM) without Coulomb interactions between the mobile ions. In this system, the

average interaction strength between the particle is the ratio

aeff0

2

max

E

1

a4

q

E

V

επε

= = = =

. Here,

a4

q

V

eff0

2

επε

= = = =

denotes the Coulomb’s interactions between the particles,

0 ε and

eff

ε

are the

permittivity of free space and an effective permittivity, respectively. Distance “a” depends on the

Page 3

AC conductivity spectra of KCl crystals. Temperature dependent deviations from the Summerfield…

concentration of the particle, defined by the fraction of site occupied by particles,

E is the energy barriers between occupied sites. The scaling law with the expression

( '

σ

467

3/ 1

x~c

− − − −

[8].

a

)T

T

( f

)

dcdc

α

σ

ννσ

= = = =

(3)

is a theoretical result, the scaling procedure according (3) holds for alkali tellurite glasses, but with

α positive and strongly depending on the type of the alkali ion and on the alkali oxide content [8].

Experimentally α decreases with increasing alkali oxide content.

On the other hand, at high concentration of the mobile ions (e.g.

Coulomb interactions between the mobile alkali ions. The strength of these interactions

characterized by V, increases with the alkali oxide content. Therefore, with increasing strength of the

3 22cm 10~N

− − − −), act the

interactions in the RBM model (i.e., with increasing ratio

a

E

V), α increases. This is a major

contrast between theoretical and experimental results.

A positive value of α implies that the

passes over into the diffusive dynamics increases with temperature. This fact suggests that the

interionic Coulomb interactions are not the decisive factor determining the scaling properties of the

ac conductivity spectra and that likely the number of available diffusion pathways decreases with

increasing temperature.

3. Results and discussions

This paper shows the scaling properties of the conductivity spectra in KCl crystals. The

isotherms of the real part of the conductivity

Hz)10x4 . 110(

frequency range and

500300(

÷ ÷ ÷ ÷

are based on the log-log dependence of the conductivity on frequency, in Fig. 1 these isotherms are

plotted in the onset frequency of the conductivity dispersion range.

> > > > < < < <

) t (r2

where the subdiffusive ion dynamics

)( ' νσ

have been obtained in the

62÷ ÷ ÷ ÷

K)

temperature range. Because the discussions

Fig. 1. Conductivity isotherms of KCl crystals.

Although in the crystals, as e.g. alkali halides, the ionic conductivity is controlled by

vacancy dynamics, the spectral shapes of the conductivity of the KCl crystals and of other materials

2.0 2.5 3.03.5 4.0

-3.0

T=317 K

T=337 K

T=358 K

T=391 K

T=430 K

T=454 K

T=486 K

log(σ')

log(ν)

Page 4

A. Ioanid, A. S. Dafinei

(single alkali glasses, other inorganic crystals, polymers, semiconductors) are similar. In Fig. 2. are

presented the isotherms of the

)( ' νσ

of the KCl crystals scaled according the Summerfield scaling

law (2). Obviously, the application of the Summerfield scaling does not lead to a superposition of

468

the isotherms, but the isotherms are shifted to higher values on the

T

dc

σ

ν

axis as the temperature

decreases. The isotherms, however, do collapse onto a single master curve when an additional

scaling factor

T with

] 1 2 . 1 [

÷ ÷ ÷ ÷∈ ∈ ∈ ∈α

, is used for the frequency axis, Fig. 3. The exponent α

decreases with temperature. It is important to note that the shape of the resulting master curve do not

differs on those of the original isotherms and that the frequency range for the master curve is

diminished to the vicinity of the onset frequency of the conductivity dispersion. Consequently, it has

been concluded that the ion and defect dynamics are important only below ~0.5MHz, while the

electron dynamics are prevalent for higher frequencies. Therefore, the scaling procedure (3) remove

the deviations from the Summerfield scaling with positive values for α and that decrease with

increasing temperature.

α

Fig. 2. Conductivity isotherms scaled according to the Summerfield law.

The electrical and optical properties of the alkali halides crystals are governed by structural

defects: cation or anion vacancies, impurities and interstitial atoms. In the system of the KCl crystal

with stabilized defects one may consider that the mobile particles [stabilized defects] are localized

on well-defined sites with equal energies (because the matrix is crystalline) and that the heights of

the barriers between these sites are spatially uncorrelated random variables (RBM model). Recently,

Sidebottom [9] demonstrated that, in ion conducting crystals, the shape of the conductivity spectra

depends strongly on the local dimensionality of the diffusion pathways. Thus, the scaling procedure

results for KCl crystals can be analysed with the arguments from Section 2, i.e., α is positive and

decreases with increasing temperature via decreasing the number of available diffusion pathways.

These results also agree with the temperature dependence of the concentration of donor defects

obtained from the isothermal frequency dependence of the ac-conductivity by extending the theory

of the ac-conductivity for the disordered compounds [10].

3456789

0

1

2

3

4

Summerfield Scaling

T

T=292 K

T=317 K

T=337 K

T=358 K

T=391 K

T=430 K

T=454 K

T=486 K

log(σ/σdc)

log(ν/σdcT)

Page 5

AC conductivity spectra of KCl crystals. Temperature dependent deviations from the Summerfield…

469

Fig. 3. Isotherms of the conductivity of KCl crystals scaled according to Eq. (3). The

reference temperature T0 is 292 K.

crystals are a new exemple of ion conducting materials displaying deviations from the Summerfield

scaling. Although has crystalline (ordered) matrix, the analysis identifies the same structural feature

being responsable for these deviations that for ionically conducting glasses (e.g., for tellurite glasses

[6]). The isotherms

)( ' νσ

collapse onto a single master curve when an additional scaling factor

with α positive and decreasing with increasing temperature via decreasing the number of available

diffusions pathways.

References

[1] A. Hunt, Solid State Commun. 80, 151 (1991).

[2] K. Funke, Prog. Solid State Chem. 22, 1119 (1993).

[3] A. Bunde, M. D. Ingram, P. Maass, J. Non-Cryst. Solids 172-174, 1222 (1994).

[4] P. Maass, M. Meyer, A. Bunde, Phys. Rev. B 51, 8164 (1995).

[5] S. Polosan, E. Apostol, E. Vasile, V. Topa, J. Optoelectron. Adv. Mater. 5(3), 699 (2003).

[6] S. Summerfield, Philos. Mag. B 52, 9 (1985).

[7] S. Murugavel, B. Roling, Phys. Rev. Lett. 89, No. 19, 195902 (2002).

[8] B. Roling, Phys. Chem. Chem. Phys. 3, 5093 (2001).

[9] D. L. Sidebottom, Phys. Rev. Lett. 83, 983 (1999).

[10] S. R. Elliott, Adv. Phys. 36, 135 (1987).

4. Conclusions

The results obtained from the scaling properties of the ac-conductivity spectra of KCl

α

T

34567

0

1

2

3

4

292 K α=1.2

317 K α=1.2

337 K α=1.2

358 K α=1.15

391 K α=1.1

430 K α=1.05

454 K α=1

486 K α=1

log(σ/σdc)

log((ν/σdcT)(T/T0)α)