Lallart M. Ferroelectrics: Characterization and Modeling

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Non-Linear Dielectric Response of Ferroelectrics, Relaxors and Dipolar Glasses

199

The comparison of nonlinear dielectric response of various kinds of polar materials

presented in this chapter gives evidence that such kind of measurement is a very sensitive

tool for determination of the nature of ferroelectrics. Particularly useful for this purpose are

the third-order dielectric susceptibility and the scaled non-linear susceptibility, a

. In our

opinion, the second-order dielectric susceptibility is less significant, since it is more

characteristic of the sample state than of a particular group of ferroelectrics. If anything, this

susceptibility contains information about the distinct polar state of the sample. This

information may be used for checking the presence of a center of inversion. The nonlinear

dielectric response of classic ferroelectric crystals displaying continuous or discontinuous

phase transition stays in a good agreement with predictions of the thermodynamic theory of

ferroelectric phase transition. Predictions of the scaling theory for TGS crystal are also

successfully verified experimentally. The situation is much more complex in disordered

systems like ferroelectric relaxors or dipolar glasses. Respective theories properly explaining

the observed features have still to be developed and tested via dynamic nonlinear dielectric

response.

6. Acknowledgment

The authors are grateful to D. Rytz, A. Tkach, and P.M. Vilarinho for providing samples.

WK thanks the Foundation for Polish Science (FNP), Warsaw, for an Alexander von

Humboldt Honorary research grant.

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Ferroelectrics Study at Microwaves

Yuriy Poplavko, Yuriy Prokopenko,

Vitaliy Molchanov and Victor Kazmirenko

National Technical University “Kiev Polytechnic Institute”

Ukraine

1. Introduction

Dielectric materials are of interest for various fields of microwave engineering. They are

widely investigated for numerous applications in electronic components such as dielectric

resonators, dielectric substrates, decoupling capacitors, absorbent materials, phase shifters,

etc. Electric polarization and loss of dielectric materials are important topics of solid state

physics as well. Understanding their nature requires accurate measurement of main

dielectric characteristics. Ferroelectrics constitute important class of dielectric materials.

Microwave study of ferroelectrics is required not only because of their applications, but also

because important physical properties of theses materials, such as phase transitions, are

observed at microwave frequencies. Furthermore, most of ferroelectrics have polydomain

structure and domain walls resonant (or relaxation) frequency is located in the microwave

range. Lattice dynamics theory also predicts strong anomalies in ferroelectric properties just

at microwaves. That is why microwave study can support the investigation of many

fundamental characteristics of ferroelectrics.

Dielectric properties of materials are observed in their interaction with electromagnetic field.

Fundamental ability of dielectric materials to increase stored charge of the capacitor was

used for years and still used to measure permittivity and loss at relatively low frequencies,

up to about 1 MHz (Gevorgian & Kollberg, 2001). At microwaves studied material is usually

placed inside transmission line, such as coaxial or rectangular waveguide, or resonant cavity

and its influence onto wave propagation conditions is used to estimate specimen’s

properties. Distinct feature of ferroelectric and related materials is their high dielectric

constant (ε = 10

– 10

) and sometimes large dielectric loss (tanδ = 0.01 – 1). High loss could

make resonant curve too fuzzy or dissipate most part incident electromagnetic energy, so

reflected or transmitted part becomes hard to register. Also because of high permittivity

most part of incident energy may just reflect from sample’s surface. So generally

conventional methods of dielectrics study may not work well, and special approaches

required.

Another problem is ferroelectric films investigation. Non-linear ferroelectric films are

perspective for monolithic microwave integrated circuits (MMIC) where they are applied as

linear and nonlinear capacitors (Vendik, 1979), microwave tunable resonant filters (Vendik

et al., 1999), integrated microwave phase shifter (Erker et al., 2000), etc. Proper design of

these devices requires reliable evidence of film microwave dielectric constant and loss

tangent. Ferroelectric solid solution (Ba,Sr)TiO

(BST) is the most studied material for

Ferroelectrics - Characterization and Modeling

204

possible microwave applications. Lucky for microwave applications, BST film dielectric

constant in comparison with bulk ceramics decreases about 10 times (ε

film

~ 400 – 1000) that

is important for device matching. Temperature dependence of ε

film

becomes slick that

provides device thermal stability (Vendik, 1979), and loss remains within reasonable limits:

tanδ ~ 0.01 – 0.05 (Vendik et al., 1999). Accurate and reliable measurement of ferroelectric

films dielectric properties is an actual problem not only of electronic industry but for

material science as well. Film-to-bulk ability comparison is an interesting problem in physics

of ferroelectrics. Properties transformation in thin film could be either favourable or an

adverse factor for electronic devices. Ferroelectric materials are highly sensitive to any

influence. While deposited thin film must adapt itself to the substrate that has quite

different thermal and mechanical properties. Most of widely used techniques require

deposition of electrodes system to form interdigital capacitor or planar waveguide. That

introduces additional influence and natural film’s properties remain unknown.

Therefore, accurate and reliable measuring of dielectric constant and loss factor of bulk and

thin film ferroelectrics and related materials remains an actual problem of material science

as well as electronic industry.

2. Bulk ferroelectrics study

At present time, microwave study of dielectrics with ε of about 2 – 100 and low loss is well

developed. Some of theses techniques can be applied to study materials with higher

permittivity. Approximate classification of most widely used methods for large-ε materials

microwave study is shown in Fig. 1.

Fig. 1. Microwave methods for ferroelectrics study

Because of high dielectric constant, microwave measuring of ferroelectrics is quite

unconventional. The major problem of high-ε dielectric microwave study is a poor

interaction of electromagnetic wave with studied specimen. Because of significant difference

in the wave impedance, most part of electromagnetic energy reflects from air-dielectric

boundary and can not penetrate the specimen. That is why, short-circuited waveguide

method exhibit lack of sensitivity. If the loss of dielectric is also big, the sample of a few

millimetres length looks like “endless”. For the same reason, in the transmission experiment,

only a small part of electromagnetic energy passes through the sample to output that is not

sufficient for network analyzer accurate operation. Opened microwave systems such as

resonators or microstrip line suffer from approximations.

Ferroelectrics Study at Microwaves

205

One of the most used methods utilizes measurement cell in the form of coaxial line section.

Studied specimen is located in the discontinuity of central line. Electric field within the

specimen is almost uniform only for materials with relatively low permittivity. This is

quazistatic approximation that makes calculation formulas simpler. If quazistatic conditions

could not be met, then radial line has to be studied without approximations. For the high ε

materials coaxial method has limitations. Firstly, samples in form of thin disk have to be

machined with high precision in a form of disk or cylinder. Secondly, many ferroelectric

materials have anisotropic properties, so electric field distribution in the coaxial line is not

suitable. This work indicates that a rectangular waveguide can be improved for

ferroelectrics study at microwaves.

2.1 Improved waveguide method of ferroelectrics measurements

The obvious solution to improve accuracy of measurement is to reinforce interaction of

electromagnetic field with the material under study. One of possible ways is to use dielectric

transformer that decreases reflection. For microwave study, high-ε samples are placed in the

cross-section of rectangular waveguide together with dielectric transformers, as shown in

Fig. 2.

Fig. 2. Measurement scheme: a) short-circuit line method, b) transmission/reflection method

A quarter-wave dielectric transformer with

trans sam

εε

= can provide a perfect matching,

but at one certain frequency only. In this case, the simple formulas for dielectric constant

and loss calculations can be drawn. However, mentioned requirement is difficult to

implement. Foremost, studied material dielectric constant is unknown a priori while

transformer with a suitable dielectric constant is also rarely available. Secondly, the critical

limitation is method validity for only one fixed frequency, for which transformer length is

equal precisely to quarter of the wavelength. Moreover, the calculation formulas derived

with the assumption of quarter wave length transformers lose their accuracy, as last

requirement is not perfectly met.

Insertion of dielectric transformers still may improve matching of studied specimen with air

filled part of waveguide, though its length and/or permittivity do not deliver perfectly quarter

wave length at the frequency of measurement. Dielectric transformers with ε

trans

= 2 – 10 of

around quarter-wave thickness are most suitable for this purpose. Influence of transformers

must be accurately accounted in calculations.

2.2 Method description

The air filled section of waveguide, the transformer, and the studied sample are represented

by normalized transmission matrices



, which are the functions of lengths and

Ferroelectrics - Characterization and Modeling

206

electromagnetic properties of neighbour areas. Applying boundary conditions normalized

transmission matrix for the basic mode can be expressed as:

ii ii

γγ

γγ γγ

−−

−

−−

+−













=⋅





−+

















, (1)

where μ

is permeability of i-th medium; γ

is propagation constant in i-th medium; d is the

length of i-th medium. Transmission matrix of whole network can be obtained by the

multiplication of each area transmission matrices:

[]

nn−

=⋅ ⋅⋅TTT T

 

 . (2)

The order of multiplying here is such, that matrix of the first medium on the wave’s way

appears rightmost. Then, for the convenience, the network transmission matrices can be

converted into scattering matrices whose parameters are measured directly.

In case of non-magnetic materials scattering equations, derived from (2), can be solved for

every given frequency. However, this point-by-point technique is strongly affected by

accidental errors and individual initiations of high-order modes. To reduce influence of

these errors in modern techniques vector network analyzer is used to record frequency

dependence of scattering parameters (Baker-Jarvis, 1990). Special data processing procedure,

which is resistive to the individual errors, such as nonlinear least-squares curve fitting

should be used:

()

min , ,

meas

nn n

SSf

εε

σεε

′′′

−



. (3)

Here σ

is the weight function;

meas

S is measured S-parameter at frequency f

;

()

εε

′′′

calculated value of scattering parameter at the same frequency, assuming tested material to

have parameters ε′ and ε″. Real and imaginary parts of scattering parameters are separated

numerically and treated as an independent, i.e. the fitting is applied to both real and

imaginary parts.

Proper choice of weight is important for correct data processing. Among possible ways,

there are weighted derivatives, and the modulus of reflection or transmission coefficients.

These methods emphasize the influence of points near the minimum values of the reflection

or transmission, which just exactly have the highest sensitivity to properties of studied

material.

The choice between short-circuited line or transmission/ reflection methods depends on

which method has better sensitivity, and should be applied individually.

2.3 Examples of measurements

Three common and easily available materials were used for experimental study. Samples

were prepared in the rectangular shape that is adjusted to X-band waveguide cross section.

Side edges of samples for all experiments were covered by silver paste. Summary on

measured values is presented in Table 1.

Ferroelectrics Study at Microwaves

207

Material

Reflection Transmission

tanδ

TiO

96 0.01 95 0.01

SrTiO

290 0.02 270 0.017

BaTiO

590 0.3

Table 1. Summary on several studied ferroelectric materials

Measured data and processing curves are illustrated in Fig. 3, 4. In reflection experiment

minima of S

are deep enough to perform their reliable measurement, so numerical model

coincides well with experimentally acquired points. For transmission experiment total

amount of energy passed trough sample is relatively low, but there are distinct maxima of

transmission, which also are registered reliably.

Fig. 3. Measured data and processing for reflection experiments: TiO

, ε = 96, thickness

2.03 mm (a); SrTiO

of 3.89 mm thickness with 6.56 mm teflon transformer (b)

Fig. 4. Measured data and processing for: 1.51mm BaTiO

with 6.56 mm teflon transformer (a),

reflection experiment; 3.89 mm SrTiO

, transmission experiment (b)

Ferroelectrics - Characterization and Modeling

208

BaTiO

is very lossy material with high permittivity. In reflection experiment, Fig. 4, there is

fuzzy minimum of S

, so calculation of permittivity with resonant techniques is inaccurate.

Change in reflection coefficient across whole X-band is about 0.5 dB, so loss determination

by resonant technique might be inaccurate too. Our calculations using fitting procedure (3)

show good agreement with other studies in literature.

2.4 Order-disorder type ferroelectrics at microwaves

There are two main frequency intervals of dielectric permittivity dispersion: domain walls

relaxation in the polar phase and dipole relations in all phases. Rochelle Salt is typical example

of this behaviour, Fig. 5. Here and after ε

, ε

are diagonal components of permittivity tensor.

Fig. 5. Rochelle Salt microwave study: ε′

and ε″

frequency dependence at 18

С (a);

ε′

temperature dependence at frequencies (in GHz): 1 – 0.8; 2 – 5.1; 3 – 8.4; 4 – 10.2; 5 – 20.5;

6 – 27; 7 – 250 (b)

Sharp maxima of at ε′

(

) in the frequency interval of 10

– 10

Hz mean piezoelectric

resonances that is accompanied by a fluent ε′-decrease near 10

Hz, Fig. 5, a. The last is

domain relaxation that follows electromechanical resonances. In the microwaves Rochelle

Salt ε′

dispersion with ε″

broad maximum characterizes dipole relaxation that can be

described by Debye equation

()

() ( )

1 i

εε

εω ε

ωτ

∞

−∞

, (4)

where τ is relaxation time, ε(∞) is infrared and optical input to ε

why ε(0) is dielectric

permittivity before microwave dispersion started.

Microwave dispersion in the Rochelle Salt is observed in all phases (in the paraelectric

phase above 24

C, in the ferroelectric phase between –18

– +24

C, and in the

antiferroelectric phase below –18

C, Fig. 5, b. To describe ε*(ω,T) dependence in all these

phases using eq. (1) one need substitute in the paraelectric phase τ = τ

/(T – θ) and

ε(0) – ε(∞) = С/(Т – θ). Experiment shows that in paraelectric phase C = 1700 K, θ = 291 K

and τ

=3.2⋅10

-10

s/K. By the analogy this calculations can be done in all phases of Rochelle

Salt.