## 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 ( = 10^{2} – 10^{4}) and sometimes large dielectric loss (tan = 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_{3} (BST) is the most studied material for possible microwave applications. Lucky for microwave applications, BST film dielectric constant in comparison with bulk ceramics decreases about 10 times (_{film} ~ 400 – 1000) that is important for device matching. Temperature dependence of _{film} becomes slick that provides device thermal stability (Vendik, 1979), and loss remains within reasonable limits: tan ~ 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 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- materials microwave study is shown in Fig. 1.

Because of high dielectric constant, microwave measuring of ferroelectrics is quite unconventional. The major problem of high- 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.

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 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- samples are placed in the cross-section of rectangular waveguide together with dielectric transformers, as shown in Fig. 2.

A quarter-wave dielectric transformer with *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 _{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

where _{i} is permeability of *i*-th medium; _{i} 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:

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:

Here _{n} is the weight function; *S*-parameter at frequency *f*_{n};

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.

Measured data and processing curves are illustrated in Fig. 3, 4. In reflection experiment minima of *S*_{11} 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.

BaTiO_{3} is very lossy material with high permittivity. In reflection experiment, Fig. 4, there is fuzzy minimum of *S*_{11}, 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 _{1}, _{2,}
_{3} are diagonal components of permittivity tensor.

Sharp maxima of at _{1}(^{4} – 10^{5} Hz mean piezoelectric resonances that is accompanied by a fluent -decrease near 10^{6} Hz, Fig. 5, *a*. The last is domain relaxation that follows electromechanical resonances. In the microwaves Rochelle Salt _{1} dispersion with _{1} broad maximum characterizes dipole relaxation that can be described by Debye equation

where is relaxation time, () is infrared and optical input to _{1} why (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^{o}C, in the ferroelectric phase between –18^{o} – +24^{o}C, and in the antiferroelectric phase below –18^{o}C, Fig. 5, *b*. To describe *(,T) dependence in all these phases using eq. (1) one need substitute in the paraelectric phase = _{0}/(T *–* ) and (0) *–* () = С*/*(Т *–* θ). Experiment shows that in paraelectric phase C = 1700 K, = 291 K and _{0} =3.210^{-10} s/K. By the analogy this calculations can be done in all phases of Rochelle Salt.

Figure 6 shows main results of microwave study of TGS (another well known order-disorder type ferroelectric). Dipole relaxation in the polar phase demonstrates _{2}(_{2}(*a*. Note, that 1 cm^{1} corresponds to *f* = 30 GHz.

In contrast to Rochelle Salt, TGS is not piezoelectric in the paraelectric phase. In the Curie point _{2}(T) at microwaves demonstrates minimum. The family of *_{2}(

where _{0} = 210^{–10} sec/К.

Microwave properties of the DKDP *_{3}(

### 2.5. Ferroelectrics of displace type at microwaves

In the ferroelectric phase the -dispersion at microwaves depends on domain walls vibration. That is why in the single-domain crystal practically no decrease in at microwaves is observed, as it is shown in Fig. 8, a with the example of LiNbO_{3} crystal. Resonant change in _{3} and _{1} at megahertz frequencies means only piezoelectric resonances while far infrared -maxima are obliged to the lattice vibrations.

However, in the multidomain crystals dielectric dispersion at microwaves results in -decrease that is accompanied by tan maximum near frequency 9 GHz, shown in Fig. 8, b for multidomain LiTaO_{3} crystal (there are also many piezoelectric resonances in the megahertz area).

Polycrystalline ferroelectrics have obviously multidomain structure and, as a result, show microwave -dispersion, as it is shown in Fig. 9 for PbTiO_{3} and BaTiO_{3} ( maximum is observed near frequency of 9 GHz while decreases in two times). More “soft” ceramics Ba(Ti,Sn)O_{3} demonstrate microwave dispersion at lower but microwave frequencies: broad maximum is seen at 1 GHz.

Microwave properties of displace type ferroelectrics in the paraelectric phase depends on soft lattice vibration mode. That is why Lorentz oscillator model is a basic model to describe * frequency dependence:

In this equation let assume

where A is Cochran coefficient, C is Сurie-Weiss constant, is damping coefficient. From and tan temperature dependences at various frequencies, as for instance Fig. 10, a, soft mode temperature dependence can be calculated, Fig. 10, b. Main lattice dynamics parameters of studied ferroelectrics are shown in Table 2.

## 3. Ferroelectric films investigation

### 3.1. Various methods comparison

Most of existing studies of ferroelectric films (22 published experiments listed in the review by Gevorgian & Kollberg, 2001) are drawn with the use of electrodes. For instance, the opposite-electrodes method is employed to study the system Pt/BST/Pt (Banieki et al., 1998). However, in most cases, ferroelectric film is studied between planar electrodes applied to the opened surface of the film. In that case, film parameters can be extracted from the impedance of interdigital planar capacitor as well as from the coplanar phase shifter study. Nevertheless, in all mentioned methods, the “natural film” microwave and tan remain unknown, because a complex system of “electrode-film-electrode” is investigated. Nevertheless, the data related to the “natural film” as well as to film components properties and substrates properties are important: their frequency and temperature characteristics are shown in Fig. 11.

It is necessary to note that dielectric constant calculation from the planar capacitance is approximate while microwave loss cannot be even estimated. Point is that metallic electrodes strongly affect onto measured _{film} value (and especially onto film’s tan) through the mechanical stress and skin effect in electrodes. Moreover, as a rule, dielectric parameters of film with interdigital electrodes are usually obtained at low frequency (of about 1 MHz); however, next this information is applied to microwave device elaboration. In the mass production small portion of the substrate could be sacrificed for test electrodes area. However, in laboratory study, single film gets unusable after electrodes deposition. So the electrodeless techniques are very important. A comparison of different methods of ferroelectric film study at microwaves is shown in Fig. 12.

Thin ferroelectric film is usually deposited onto dielectric substrate. Practically used films have thickness of 0.1–1 mm. Thermal expansion coefficient and lattice parameter of the substrate are different from those of thin film. Thus, film suffers from mechanical stress. This stress changes films properties comparing to the properties of bulk ferroelectric. Dielectric constant and loss could be decreased by order of magnitude. On the other hand, directional mechanical stress contributes to the anisotropy of film’s parameters. So methods of films study must not only register film’s response, but consider anisotropy as well.

Because of high dielectric constant and loss microwave testing of ferroelectrics is quite complex. In thin film study a question becomes even more complicated by film small thickness. This work presents waveguide method, suitable for thin films study.

### 3.2. Waveguide method description

Common technique for dielectric material measuring in the waveguide usually relies on complex scattering parameters measurement of waveguide section which cross section is filled with studied material. That technique can be easily adapted for measurement of the layered structures where properties of one layer are unknown.

However, this approach faces irresolvable difficulties with thin films. Simple estimation shows that X-band waveguide being entirely baffled with film of 1 m thickness that has = 1000 and tan = 0.05 has phase perturbation of only about 0.4, and brings attenuation of about -0.002 dB. These quantities are obviously out of equipment resolution capabilities. That is why, the goal is to arrange the interaction of film with electromagnetic field in such a way that brings recognizable response.

In proposed method, film-on-substrate specimen is centrally situated along the waveguide (Fig. 13). It is known that electric field intensity is highest in centre of waveguide so highest possible interaction of film with the electric field is provided.

Dielectric constant and loss can be found by solving scattering equations at one certain frequency. However, the accuracy of one-point technique is strongly affected by the accidental error (Baker-Jarvis, 1990). Proposed method accuracy is improved by the recording of complete frequency dependence of scattering parameters using contemporary vector network analyzer. Similarly to the method for bulk samples study, gathered experimental data then processed utilizing nonlinear least squares curve fitting technique (3).

For the S-parameters calculations, electromagnetic field problem can be solved utilizing longitudinal wave representation (Egorov, 1967), (Balanis, 1989). Applying boundary conditions on media boundaries yields a complex nonlinear equation with respect to complex propagation constant:

(8) |

where_{s} is substrate thickness, d_{f} is film thickness, a is width of wide wall of waveguide, is propagation constant, k is free space wave number. In this equation, the position of film-substrate boundary assumed to be exactly at the middle of waveguide, however known displacement can be taken in account.

### 3.3. Experimental result

Described measurement technique was utilized for study of BST thin films. The film of about 1 m thickness was deposited onto 0.5mm MgO substrate in a pulsed laser ablation setup. Special measurement cell was elaborated to provide reliable contact of specimen under test with waveguide walls, Fig. 14. Automatic network analyzer was calibrated with

appropriate X-band calibration kit. Then two reference measurements were performed. First one is a measurement of empty cell. It makes possible to determine cell’s electrical length more precisely. Second one is the measuring of substrate alone. This stage is required to determine the actual loss of the “substrate-in-waveguide” system because only on this background film properties will be recognized.

Numerous experiments with the same samples show reliable repeatability of experiments. Fig. 15 illustrates an example of measured data fitting for MgO substrate and 0.84m thick BST film.

Average value of substrate permittivity is = 9.9, tan = 310^{-4}. As to the film fitting to the calculation exhibits good agreement and yields = 450, tan = 0.05. Both reflection of shorted waveguide with sample and transmission in the 2-port system can be used, however the first method is preferable because of higher sensitivity.

Presented technique of thin ferroelectric film examination can be applied also for study of relatively thick (10m and more) films that have non-ferroelectric nature. The method can be successfully used, for example, to study semiconductor films deposited onto dielectric subtract when traditional metering technique brakes down.

### 3.4. Uncertainty of non-resonant waveguide method for thin films measurement

Ideal contact of studied specimen with waveguide walls is hard to achieve and there small air gaps on the interface. These gaps may degrade accuracy substantially. Though some estimations consider 2.5-7 m gap acceptable (Champlin & Glover, 1966), this value is too general and hard to verify. So it is desirable to enforce interface contact as much as possible.

Though films permittivity and loss are estimated using least squares curve fitting technique (3), let’s begin uncertainty estimation with single point accuracy. Parameters of studied film are derived from indirect measurements. They contain uncertainties of dimensions measurements, scattering parameters magnitude and phase uncertainties, and rounding errors of processing procedure. In waveguide experiment magnitude and phase of reflection coefficient are measured directly (real and imaginary part to be precise, but that does not change further explanations). Their simulation values depend on sample’s physical dimensions, permittivity and loss:

where

Due to low loss substrate effective loss of measurement cell is low. In such conditions permittivity is mainly found by phase measurement, whereas loss is found from magnitude measurements (Janezic & Jargon, 1999). Large difference in sensitivities allows separate analysis of permittivity and loss uncertainties.

Sensitivities of calculated values of film’s permittivity and loss to uncertainties of directly measured values can be estimated using rules of implicit function differentiation. Then for every given frequency permittivity and loss uncertainty can be expressed as:

(10) |

where ^{–4}, their values presented in Table 3. Table 4 presents summary on instrument uncertainties.

Parameter | Value | Note |

L | 10 m | Micrometre screw |

d_{f} | 10 nm | Reflectometer |

_{s} | 0.1 | 1% |

tan_{s} | 10^{-4} | 10% |

|S_{11}| | 0.02 | HP 8510C |

S_{11} | 2 | HP 8510C |

For the film under consideration uncertainty of permittivity measurement / is about 14%, while uncertainty of loss measurement

If method applied for film study in production process, i.e. the same substrate used in all measurements, then uncertainty of substrate permittivity and loss could be eliminated and permittivity uncertainty improves to about 10%.

These values present worst case estimation of single point measurement. Uncertainty of final parameters is reduced by least squares processing. With 50 point equally distributed along measurement frequency range sensitivity to uncertainty of scattering parameter determination can be reduced by order of magnitude to

Then averaged uncertainty of film’s permittivity reduces to 2% and loss to 10%.

### 3.5. Coplanar line method

In actual devices a system of electrodes is deposited on the surface of ferroelectric film. Geometry of the electrodes depends on film permittivity. At the same time metal electrodes can modify film permittivity and loss. Therefore it would be desirable to perform measurement directly in the device with deposited electrodes.

One of the most usable electrode system forms a coplanar line, Fig.16. Measurement of permittivity and loss of ferroelectric film integrated in coplanar line is discussed below.

Electromagnetic field problem for the structure given in Fig.16 was solved by finite element method. Electromagnetic field of quasi ТЕМ mode was described in terms of vector and scalar potential applying the Lorenz gauge. As a result the problem was reduced to partial differential equation for scalar potential

where

(13) |

where

Dependences of characteristic impedance and effective permittivity of coplanar line deposited on the ferroelectric film and low permittivity dielectric wafer versus permittivity of ferroelectric film and its thickness are shown in Fig.17.

Measurement cell for study of permittivity and loss tangent of ferroelectric film is shown in Fig.18. Two-port measurement of frequency dependences of scattering parameters was performed by vector network analyzer.

Scattering matrix of coplanar line section with the length equal to

where

Measured frequency dependences of scattering matrix parameters were approximated by formulas (13) and (14) by least square method:

where _{n} assuming tested film to have permittivity _{f} and loss tangent

Fig.19 demonstrates measured and calculated after solving of approximation problem (15) the frequency dependences of scattering matrix parameters for the structure presented in Fig.18.

Relative uncertainty of film permittivity measurement can be defined as uncertainty of implicit measurement:

where

Analysis of the formula (16) predicts that uncertainty of film thickness measurement makes the main contribution in measurement uncertainty of film permittivity. Estimation prediction of film permittivity measurement uncertainty is about 10% if uncertainty of film thickness measurement is about 10 nm for the film thickness about 500 nm and its permittivity around 200. The uncertainty rises up while either film thickness or its permittivity decreases.

Uncertainty of film loss tangent measurement is larger than uncertainty of permittivity measurement because of smaller value of the sensitivity of scattering matrix parameters to alteration of film loss tangent

Described technique was verified during measurement of ferroelectric films deposited by sol-gel method on semi-insulated silicon substrate. Some results of the verification are presented in table 5.

### 3.6. Resonator method description

Thin ferroelectric films can also be studied using composite dielectric resonator (CDR) method. Simple equations for effective permittivity and loss, based on parallel layers model, in case of significant difference in layers thickness, which is a case for thin films, give inadequate results. Thus, electromagnetic problem for “film-on-substrate” composite dielectric resonator should be solved without approximations.

To calculate resonant frequencies of the CDR one may solve electromagnetic problem for the configuration, depicted in Fig. 20. Square shaped CDR of length L and thickness d_{s} is made from material with _{s} and placed inside waveguide with ab cross-section parallel to its narrow wall. Waveguide is filled with _{a}. Dielectric film of thickness d_{f} and permittivity _{f} is deposited onto resonator surface. The problem is solved using partial domains method.

In every domain 1 and 2 electromagnetic field may be expressed using x-components of electrical Г^{e} = (Г^{e},0,0) and magnetic Г^{m} = (Г^{m},0,0) Hertz vectors. To do that electromagnetic field in every domain is presented as series of eigen functions:

where i is number of partial domain,

Using equality requirement for tangential components of the field in

where S is waveguide cross-section. Integral equations kernels

Basic mode oscillations can be excited as in waveguide, so in resonator itself. However, in matched waveguide section there are no parasitic oscillations, which are natural to standalone resonator.

CDR’s made of Al_{2}O_{3} (ε =9.6), BaTi_{4}O_{9}, and DyScO_{3}, SmScO_{3}, LSAT with ε = 26.3, 25.1, 22.7 respectively were simulated and studied experimentally. CDR dimensions ratio was in the range d/L = 0.2…0.01. Results summary is presented in Fig. 21.

Because all measurements are held relative resonant frequency of resonator without film, uncertainty depends mainly on dimensions uncertainty. Resonant frequency shift for CDR with film depends on film’s thickness, Fig. 22. Frequency range, where this method is applicable, depends on CDR’s resonant frequencies, which in turn depend on dimensions and substrate permittivity.

To reduce measurement uncertainty the resonator thickness d_{s} should be rather small. However it leads to increasing of resonator length L which would be greater than narrow wall width. In this case the resonator should be placed by such a way that ferroelectric film surface would be almost parallel to wide wall of waveguide with some angle shift to satisfy resonator excitation condition.

## 4. Conclusion

Electrodeless methods for bulk and thin film ferroelectric materials study presented. In non-resonant methods partial filling of waveguide cross-section may be applied for rare and expensive materials study. Permittivity and loss of studied specimen are determined by solving scattering equations for waveguide section with studied specimen against measured scattering parameters. The choice for transmission or reflection method depends on sensitivity of scattering matrix to change of and tan of studied specimen. Study of scattering parameters frequency dependency and subsequent their approximation with appropriate measurement cell models allows to increase measurement accuracy and reduce susceptibility to random error.

To measure and tan of thin films studied specimen may be located lengthwise inside of rectangular waveguide. This allows better interaction of thin film with electromagnetic field.

For the system with metal electrodes deposited on the surface of ferroelectric film the coplanar line technique may be applied. This method can be extended for other electrode layouts.

Another option for thin films study might be use of resonant techniques. To increase sensitivity to film’s properties thin composite resonator required. However, to tune it to required frequency composite resonator should have larger dimensions, thus irregular waveguide might be required. Otherwise sample should be oriented with large tilt angle, almost normal to waveguide’s wide wall.