Non Linear Interaction of Microwaves with Ferroelectric

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NON LINEAR INTERACTION OF MICROWAVES WITH FERROELECTRIC

MATERIALS

A Thesis

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulfillment

of the Requirement of the Degree

Master of Science

Nitin Parsa

May, 2016

NON LINEAR INTERACTION OF MICROWAVES WITH FERROELECTRIC

MATERIALS

Nitin Parsa

Thesis

Approved: Accepted:

______Advisor Interim Dean of the College Dr. Ryan C. Toonen Dr. Mario R. Garzia

______Committee Member Dean of the Graduate School Dr. Kye-Shin Lee Dr. Chand Midha

______Committee Member Date Dr. Arjuna Madanayake

______Interim Department Chair Dr. Joan Carletta

ABSTRACT

The main objective of this thesis is to study the ferroelectric behavior of thin film barium strontium titanate and multiferroic behavior of nanoporous polyvinylidene fluoride. The ferroelectric behavior of thin film barium strontium titanate is studied using an experimental set-up based on metal-insulator-metal capacitors and a new approach for microwave power detection is presented. Electric-field-induced, anharmonic dipolar resonances of room-temperature, barium strontium titanate thin films have been used to rectify and detect microwave signals with frequencies ranging from 2 GHz to 3GHz. The resonant frequency was shown to have strong dependence on film thickness with some amount of voltage-controlled tunability. Our experiments involved lock-in detection of a 100% amplitude modulated microwave signal with power levels ranging from -20 dBm to +10 dBm (the maximum power attainable from available instrumentation). An on-resonant sensitivity of 0.6 mV/mW was observed.

This power detection sensitivity was shown to have built-in band-pass filtering corresponding to the resonant line shape. Because the thin films were produced using a relatively inexpensive solution deposition method, we believe that our observed phenomena could be exploited for the purpose of reducing the cost and increasing the availability of engineering applications that relies on microwave power detection.

The multiferroic behavior of nanoporous polyvinylidene fluoride (PVDF) with metallic nanowires is studied on an interdigital capacitor. A simulated model for the interdigital

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capacitor to extract the material properties of multiferroic material (nanoporous PVDF with magnetic nanowires) is presented. The model is accurate and gave the closest results for three-finger and six-finger coplanar waveguide interdigital capacitors (CPWIDC) compared to CPWIDC’s with four and five fingers.

DEDICATION

To my parents

Mr. Radha Kishan Rao

Mrs. Lakshmi

ACKNOWLEDGEMENTS

I would like to express my deepest gratitude to my advisor Dr. Ryan C. Toonen for his encouragement, support and advice during the course of my graduate studies. It would not have been possible to work on my research and write the thesis without his help and support.

I would like to sincerely thank my committee members, Dr. Kye shin Lee and Dr. Arjuna

Madanayake for their support, guidance and motivation.

I would also like to thank my colleagues in Zip Electronic Nanotechnology Lab (ZEN

LAB) Michael Gasper, Sai Prudhvi Kumar Gummadi, Siddardha Mohan Sakhamuri for their help and support.

Last but not least, I would like to thank my parents, my brother for their encouragement and eternal love towards me. My family, friends, Drishti-Indian Student Association and

Akron Cricket Club has the greatest ability to motivate me and help me throughout my graduate studies. I cordially thank all of them.

TABLE OF CONTENTS Page LIST OF FIGURES………………………………………………… x

LIST OF TABLES…………………………………………………. xii

CHAPTER

I INTRODUCTION………………………………………….. 1

1.1 Overview……………………………………………………. 1

1.2 Goal of the Project………………………………………….. 2

1.3 Thesis Organization……………………………………...... 3

II LITERATURE REVIEW……………………………...... 4

2.1 Ferroelectrics………………………………………………… 4

2.1.1 Two Phases of Ferroelectric Materials………………. 5

2.1.2 Crystal Structure……………………………………. 6

2.1.3 BST as a Ferroelectric Material……………………… 7

2.1.4 PVDF as Piezoelectric Polymer…………………….. 8

2.2 Multiferroics………………………………………………… 9

2.3 MIM Capacitor……………………………………………… 9

2.4 Interdigital Capacitor (IDC)………………………………… 9

III ANALYTICAL MODELING OF MIM CAPACITORS AND CPWIDC…………………………………………………….. 12

3.1 Introduction………………………………………………….. 12

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3.2 Analytical Modeling of MIM Capacitor……………………… 12

3.3 Analytical Modeling of Interdigital Capacitor……………….. 13

3.3.1 Conformal Mapping Method ………………………….. 14

3.4 Limitations for Analytical Model …………………...……….. 19

IV DESIGN AND FABRICATION OF MIM CAPACITORS AND CPWIDC……………………………………………………… 22

4.1 Introduction……………………………………………...... 22

4.2 Device Fabrication.……………………………………...... 22

4.2.1 MIM Capacitors Mask Design………………………..... 23

4.2.2 Photolithography Process……..………………………... 24

4.3 Design of Coplanar Waveguide………………………………… 27

4.3.1 Design of CPWIDC……………………………………... 28

4.3.2 Mask Design for CPWIDC……………………………… 30

V SIMULATION AND EXPERIMENTAL RESULTS...... 32

5.1 Introduction…………………………………………………….. 32

5.2 Equivalent Circuit Model of MIM Capacitor..…………………. 32

5.2.1 Microwave Spectroscopy Test Setup……………………. 33

5.3 Microwave Spectroscopy Results………………………………... 34

5.4 MIM Capacitor as Power Detector……………………...... 40

5.4.1 Design of MIM Capacitor Power Detector……………... 40

5.4.2 Small Signal Model and Rectified Output...... 41

5.4.3 Power Detection Results……………………………….... 45

5.5 Simulation of Interdigital Capacitors……………………………. 47

viii

5.5.1 Simulation of One-Port and Two-Port Devices…………. 48

5.5.2 Equivalent Circuit Model for One-Port CPWIDC………. 50

5.5.3 Equivalent Circuit Model for Two-Port CPWIDC……… 50

5.6 CPWIDC Results………………………………………...... 51

VI CONCLUSION AND FUTURE WORK……………………... 53

6.1 MIM Capacitors…………………………………………………. 53

6.2 Coplanar Waveguide Interdigital Capacitors……………………. 53

6.3 Future Work……………………………………………...... 54

BIBLIOGRAPHY…………………………………………………. 55

APPENDIX…………………………………………………………. 58

LIST OF FIGURES

Figure Page

2.1 Relation between Ferroelectric, Pyroelectric and Piezoelectric Materials………………………………………………………. 4

2.2 Phases in Ferroelectric Materials……………………………… 5

2.3 Ferroelectric Crystal Structure a) Paraelectric Phase b) Ferroelectric Phase………………………………………… 7

2.4 Working of Interdigital Capacitor…………………………….. 10

3.1 Gevorigan Model for CPWIDC……………………………….. 14

3.2 Conformal Mapping for Interdigital Capacitors………………. 15

4.1 MIM Capacitors Mask Design……………………………...... 23

4.2 MIM Capacitor Fabrication…………………………………… 24

4.3 Ungrounded Coplanar Waveguide……………………………. 27

4.4 Coplanar Waveguide Interdigital Capacitor…………………… 28

4.5 Layer 1 and 3 Mask Design……………………………………. 30

4.6 Layer 2 Mask Design……….……………………………...... 30

5.1 Equivalent Circuit of MIM Capacitor…………………………. 32

5.2 Test Setup for Microscopic Spectroscopy…………………….. 34

5.3 Microwave Reflection Measurements Schematic…………...... 35

5.4 Frequency vs |푆11| for Sample 1………………………………. 36

5.5 Frequency vs 휀푟푒푎푙 for Sample 1…………………………...... 36

5.6 Frequency vs 휀푖푚푎푔푖푛푎푟푦 for Sample 1……………………….. 37

5.7 Frequency vs |푆11| for Sample 2………………………………. 38

5.8 Frequency vs 휀푟푒푎푙 for Sample 2……………………………… 38

5.9 Frequency vs 휀푖푚푎푔푖푛푎푟푦 for Sample 2……………………….. 39

5.10 Comparing Sample 1 and Sample 2…………………………… 39

5.11 Circuit Diagram for Power Detection………………………… 40

5.12 Small Signal Circuit Model………………………………...... 41

5.13 Rectified Output from Second Order Term……………………. 43

5.14 Output Voltage at Different Power Levels…………………….. 45

5.15 Voltage Output for Different Voltage Bias at 0dBm…………... 45

5.16 Bandpass Filtering for Sample 1 in MIM Capacitor…………... 46

5.17 Maximum Output Voltage at Different Power Levels………… 47

5.18 Structure of One-Port CPWIDC in Ansys HFSS……………… 48

5.19 Structure of Two-Port CPWIDC in Ansys HFSS……………… 49

5.20 Equivalent Circuit for One-Port CPWIDC………………...... 50

5.21 Equivalent Circuit for Two-Port CPWIDC………………..…... 50

5.22 Comparing the Analytical and Simulated Model for CPWIDC……………………………………………………….. 51 5.23 Extracted Permittivity and Loss Tangent…………...………..... 52

LIST OF TABLES

Table Page

1 Curie Temperatures of BST...... 8

2 Design Parameters for Three, Four, and Five-Finger 31 Capacitors…………………………………………………………

3 Design Parameters for Six- Finger Capacitor …………………… 31

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CHAPTER I

INTRODUCTION

1.1 Overview

Ferroelectricity, ferromagnetism and ferroelasticity are the three primary types of ferroic orders that are regarded as well established at this point of time [1]. Ferroelectric materials find applications in agile microwave technology such as frequency agile filters, diplexers, tunable capacitors, resonators, switches, phase shifters, and variable frequency antennas [2]. Ferromagnetic materials find applications in nonreciprocal microwave devices such as circulators, gyrators and isolators. Ferroelastic materials find applications in nanotechnology [3].

The fascination with the ferroelectric materials for microwave applications arises from the ability to substantially tune the dielectric constant upon the application of a DC electric field. In the earlier days the ferroelectrics were hardly used at microwave frequencies because of the availability of the material in bulk form. The bulk ferroelectric materials require more bias voltage to operate and it is not possible to use the bulk material without substantial losses [4]. However in the present day, thin ferroelectrics have found many applications in microwave engineering as tunable microwave devices

[2]. The dielectric responses of thin dielectric films were found to be better and different than the bulk material. The differences including higher loss tangent and broadening of

1 dielectric response curve have been attributed to lattice defects, charged grain boundaries, oxygen vacancies and local inhomogeneities leading to a distribution of Curie temperatures.

A material that simultaneously exhibits more than one ferroic behavior is a multiferroic material. In this thesis multiferroic refers to ferroelectricity and ferromagnetism. The permittivity of ferroelectric material can be controlled by electric field. Similarly, the permeability of ferromagnetic material can be controlled by magnetic field. The multiferroic material exhibits electric field control of permeability and magnetic field control of permittivity. This magnetoelectric coupling of the multiferroic materials has caught the attention of the microwave engineers [5]. Electric field control of permeability in multiferroics has found to have promising applications such as frequency-agile filters, phase shifters and nonreciprocal components.

1.2 Goal of the Project

The aim of the project is to study the ferroelectric behavior of the barium strontium titanate (BST) thin films for microwave applications. Metal-insulator-metal (MIM) capacitors have been used to study the ferroelectric behavior of thin film BST. Reflection measurements have been performed and electric-field-induced anharmonic dipolar resonances have been observed from the thin film BST. The electric-field-induced anharmonic dipolar resonances are used for microwave power detection.

Coplanar waveguide interdigital capacitors (CPWIDC) have been used to study the multiferroic materials. To make use of the multiferroic behavior of a material its electrical properties such as permittivity and loss tangent are to be known. Nanoporous

PVDF filled with ferromagnetic nanowires may behave like a multiferroic material exhibiting both the ferroelectric and ferromagnetic behavior. Various physical parameters affect the working of CPWIDC. By changing these parameters, the net capacitance can be varied. The net capacitance is used to calculate the permittivity and loss tangent. The capacitors are designed and simulated on a material with known permittivity and loss tangent in Ansys HFSS. The extracted S-parameters from the simulations are used to calculate the capacitance in each case using the lumped parameter model. From the extracted capacitance using the analytical model for interdigital capacitor the permittivity and loss tangent are determined and compared with the known parameters.

1.3 Thesis Organization

In chapter II the literature review for ferroelectrics, multiferroics, CPWIDC and MIM capacitors are discussed.

In chapter III the analytical models of the MIM capacitors and CPWIDC are discussed.

In chapter IV the design and general fabrication process are discussed, and the mask designs for the particular MIM capacitors and CPWIDC designed in this work are presented.

In chapter V the experimental results for the MIM capacitor and the working of the MIM capacitor as a power detector are discussed, along with the simulation results for the

CPWIDC.

In chapter 6 conclusions and future work are discussed. Matlab and Mathematica codes used in this research are presented in the appendix.

CHAPTER II

LITERATURE REVIEW

2.1 Ferroelectrics

Ferroelectrics belong to a class of materials exhibiting a response to physical stimuli.

These are dielectric materials that have unique field dependent permittivity. When an electric field is applied to a ferroelectric material, some of the charges get discharged at the electrode while some of them oscillate. This is called polarization and is expressed as a sum of electric dipoles per unit volume. Ferroelectricity is a subclass of piezoelectricity and pyroelectricity [6] as shown in Figure 2.1.

Piezoelectric Pyroelectric Ferroelectric

Figure 2.1: Relation between Ferroelectric, Pyroelectric and Piezoelectric Materials

When there is a shift of ions or charge in the orientation of the dipoles, there is a slight change in the dimension of the ferroelectric material. This is called electrostrictive effect.

This is similar to piezoelectric effect, but the piezoelectric effect is linear whereas the elecrostrictive effect is quadratic [2].

2.1.1 Two Phases of Ferroelectric Materials

Ferroelectric materials have two phases [2] as shown in Figure 2.2. The first one is called a non-polar phase also called as a ferroelectric phase. The second one is called a polar phase also called as a paraelectric phase. Materials that are in the ferroelectric phase have polarization even when the applied electric field is zero. This is called spontaneous polarization. The materials that are in the paraelectric phase have no spontaneous polarization.

Figure 2.2: Phases in Ferroelectric Materials

Ferroelectric materials tend to become paraelectric beyond a temperature called Curie temperature. When the material is in ferroelectric phase and the temperature is increased we see that up to a particular point the permittivity increases with an increase in temperature. After that point we see that the permittivity decreases with an increase in temperature. This point is called the Curie temperature point. This is based on Curie-

Wiess law which can be given as

퐶 (2.1) 푒푟 = , 푇 − 푇푐

where 푇 is the temperature, 푇푐 is the Curie temperature, 퐶 is a constant and 푒푟 is the relative permittivity. At 푇 = 푇푐 the relative permittivity 푒푟 is maximum. When 푇 > 푇푐 the material is in paraelectric phase and when 푇 < 푇푐 the material is in ferroelectric phase. When a ferroelectric material is in ferroelectric phase a strong hysteresis behavior is observed making it more suitable for non-volatile memory applications. In the paraelectric phase the material loses its spontaneous polarization and has a high dielectric constant. This makes the ferroelectric material more suitable for tunable microwave applications.

2.1.2 Crystal Structure

The ferroelectric crystal has prevoskite structure [2]. In paraelectric phase the crystal has a center-symmetrical structure, while in ferroelectric phase the material has non-center- symmetrical structure [2] as shown in Figure 2.3. Electrostriction is observed in all crystal symmetries whereas piezoelectricity is observed in non-center-symmetrical structure. So the piezoelectric effect is not observed in center-symmetric crystal structure (ferroelectric phase). However the results presented in [7] showed that when an electric field is applied

6 in the ferroelectric phase there is a slight deviation from the center-symmetric-cubic structure for STO (strontium titanate) which is ferroelectric. This was a clear indication that STO exhibits electrostriction and electric-field-induced piezoelectricity. Similar is the case with the thin film BST.

Figure 2.3: Ferroelectric Crystal Structure a) Paraelectric Phase b) Ferroelectric Phase

The crystalline structure of thin BST films in paraelectric phase is often far from centro- symmetric structure due to electrostriction induced strains and lattice mismatches. So we can observe substantial amount of induced piezoelectricity in BST thin films with the induced electric field.

2.1.3 BST as a Ferroelectric Material

Barium strontium titanate (BST) is the most widely used ferroelectric material for high frequency applications. BST based devices have shown to handle high power and very high break down fields [8]. The Curie temperature of BST can be varied by varying the composition of Ba: Sr in the BST mixture as shown in Table (1). The flexibility to vary the Curie temperature is absent in other ferroelectrics. Hence BST finds many 7 applications such as agile microwave devices, infrared imaging, underwater acoustics [9] etc. Above the Curie temperature the material is paraelectric and the hysteresis effect is not predominant.

Table 1: Curie Temperatures of BST

Composition Curie Temperature ( K)

SrTiO3 0

Ba0.2Sr0.8TiO3 105

Ba0.5Sr0.5TiO3 218

Ba0.7Sr0.3TiO3 280

Ba0.8Sr0.2TiO3 324

BaTiO3 390

2.1.4 PVDF as Piezoelectric Polymer

PVDF is a piezoelectric polymer first developed in 1961 [10]. To give the material its piezoelectric properties the material is mechanically stretched to orient the molecular chains and then poled [2] while under tension to align the dipoles. When poled, the material can also be used as a ferroelectric exhibiting both piezoelectric and ferroelectric behavior. PVDF becomes supersensitive to pressure when impregnated with a small quantity of nanotubes [11]. Thus PVDF when combined with nanotechnology provides a

8 new type of piezoelectric material. If the nanowires embedded exhibit ferromagnetic behavior, there is a chance that the material might exhibit multiferroic behavior.

2.2 Multiferroics

In the ferroelectric phase, a ferroelectric crystal has spontaneous ordering of electrically polarized domains, which can be modelled as electric dipole moments. It is the mutual interaction of the domains that gives spontaneous polarization. Similarly, in a ferromagnetic material there is spontaneous magnetization and in ferroelastic materials there is spontaneous deformation. If a material exhibits more than one ferroic behavior in a single phase, the material is called a multiferroic material. Magnetoelectric effect is possible in multiferroic material [1] which makes the multiferroics suitable for microwave applications.

2.3 MIM Capacitor

A metal-insulator-metal (MIM) junction is formed when a dielectric material is sandwiched by two metal layers. MIM capacitor is similar to a parallel plate capacitor.

There are many applications of MIM junctions. They can be used as diodes, capacitors, waveguides and rectifiers. In this research the ferroelectric behavior of BST thin films is studied using MIM capacitors.

2.4 Interdigital Capacitor (IDC)

Microelectromechanical systems (MEMS) were first introduced by Feynman in 1960’s.

Since the time they were introduced they play a major role in science and engineering.

One of the applications of the MEMS is the interdigital capacitor (IDC) [12]. IDC is a

9 periodic comb like structure having parallel electrodes that contribute to a capacitance associated with the electric fields that penetrate into the material (substrate). There are various applications of interdigital capacitors like surface acoustic wave sensors [13], wireless gas dielectric constant sensor [14] and pressure sensor [15]. The major advantages of IDC are that it is very easy to fabricate, very easy to integrate on to other components. In the present study the coplanar waveguide (CPW) transmission technology is used for the study of IDC. Hence the name coplanar waveguide interdigital capacitor (CPWIDC) is used all through the thesis. The working of interdigital capacitor can be best explained from the working of a parallel plate capacitor [16].

Figure 2.4: Working of Interdigital Capacitor

In a parallel plate capacitor the electric field is uniformly distributed as shown in the

Figure 2.4. But in the case of an interdigital capacitor the electric field starts from one group of electrodes that have higher potential coming up and penetrating into the material under test (MUT) and then going in to another group of ground electrodes as shown in the Figure 2.4. So the electric field is not linear as in the case of a parallel plate capacitor.

In this chapter the ferroelectric material properties, multiferroic material properties are discussed along with the working of IDC and MIM capacitors. Parameter extraction is the

10 next step in MIM capacitors and CPWIDC. In MIM capacitors the parameter extraction is necessary to observe the voltage controlled tunability and to observe the anharmonic dipolar resonance in thin film BST. In CPWIDC the parameter extraction is necessary as the electric properties of newly developed multiferroic material are unknown.

In order to determine material parameters, an equivalent circuit model is developed from which equation for capacitance is written in terms of S-parameters. The S-parameters from experimental model or simulated model are used to calculate the capacitance which is further used to calculate permittivity and loss tangent from the analytical analysis. The analytical analysis is discussed in the next chapter.

CHAPTER III

ANALYTICAL MODELING OF MIM CAPACITORS AND CPWIDC

3.1 Introduction

In this chapter the analytical model of a MIM capacitor, the analytical model of the interdigital capacitor and coplanar waveguide are discussed. The analytical model gives expressions for permittivity and loss tangent in terms of capacitance. The MIM capacitor can be approximated to that of a parallel plate capacitor. There are many models available in the literature for interdigital capacitors.

3.2 Analytical Modeling of MIM capacitor

The mathematical equations governing the permittivity of the MIM capacitors can be modeled to that of a parallel plate capacitor. From the basics of a parallel plate capacitor the permittivity is given as εA 퐶 = . (3.1) ℎ푡

휋푑2 Substituting the values for 퐴 = and ε = 휀 휀 (푑 is the diameter of the circular disc, ℎ 4 0 푟 푡 is the thickness of the thin film) in the above equation we get relative permittivity as

4퐶ℎ ′ (3.2) 휀푟 ≈ 2 . 휋휖0푑

The imaginary value of permittivity can be given as

′′ ′ (3.3) 휀푟 ≈ 휖푟 tan 훿 .

It is shown later that the behavior of the MIM capacitor is well in agreement to that of the permittivity equations derived from the Lorentz oscillator model.

By using the concepts of polarization and physical harmonic oscillator [17] we have the equations of real and imaginary permittivity for a Lorentz oscillator model as follows.

Here 휔푝is the plasma frequency [17], 휔0 is the resonant frequency, 훾 is the damping constant and 휔 is the operating frequency.

The real part of permittivity can be calculated as

휔2(휔2 − 휔2) 푝 0 (3.4) 휀푟 = 1 + 2 2 2 2 2. (휔0 − 휔 ) + 휔 훾

The imaginary part of the permittivity can be calculated as

휔2훾휔 푝 (3.5) 휀𝑖 = 2 2 2 2 2. (휔0 − 휔 ) + 휔 훾

3.3 Analytical Modeling of Interdigital Capacitor

The first analytical model for interdigital capacitor was proposed by Gary D Alley on a microstrip transmission line [18] and later a J inverter network model was proposed in

[19]. None of these methods were as accurate as the model presented in [20] by

Gevorigan. Conformal mapping method has been used in Gevorigan’s model that

13 analyzes the interdigital capacitor into a simple parallel plate capacitor. Before studying that we need to look at the interdigital capacitor as per Gevorigan’s model.

Figure 3.1: Gevorigan Model for CPWIDC

As per Gevorigan’s model, the capacitance of the interdigital capacitor can be divided in to three parts as shown in Figure 3.1. Capacitance at the end of the finger sections

(퐶푡푒푟푚𝑖푛푎푙푠), capacitance of three-finger section (퐶푡ℎ푟푒푒푓𝑖푛푔푒푟) and (푁 − 3) periodic sections, where 푁 corresponds to the number of fingers and the net capacitance can be represented as

(3.6) 퐶 = (푁 − 3)퐶푃푒푟𝑖표푑𝑖푐푎푙 + 퐶푡ℎ푟푒푒푓𝑖푛푔푒푟 + 퐶푡푒푟푚𝑖푛푎푙푠.

3.3.1 Conformal Mapping Method

Conformal mapping is the best way to illustrate the analytical model of the interdigital capacitors. Gevorgian’s model in [20] analyzes the interdigital capacitor in terms of a parallel plate capacitor on a multilayered substrate. To illustrate this consider an interdigital capacitor on a single layered substrate. Let ℎ be the height of the substrate, 푊

14 be the width of the fingers, 푆 be the gap between the fingers and 푡 be the metal thickness.

As per the Figure 3.2 we see that because of the symmetry of the electrodes, the capacitance between the half strips of the adjacent electrodes is equal to the capacitance between one half of the strips and a virtual equipotential strip of height h laying at CC.

The first step is mapping the semi-infinite strip 0364 of the Z plane in Figure 3.2b to the upper half of the T plane in Figure 3.2c using

2 휋푧 (3.7) 푇 = cosh ( ) . 2ℎ

Figure 3.2: Conformal Mapping for Interdigital Capacitors

Then the electric field lines are mapped from 01 to 52 in Figure 3.2b to 01 and 52 in

Figure 3.2c. The vertices of the polygon are given as follows

(3.8) 푡0 = 1,

휋푧 2 푡 = cosh ( ) , (3.9) 1 2ℎ

(3.10) 푡3 = 푡4 = ∞,

휋(푊+푆) 2 푡 = − sinh ( ) , (3.11) 5 4ℎ

푡6 = 0. (3.12)

The next step is to map the upper half of the T-plane to the interior of the rectangle in the

W-plane in Figure 3.2d using the following function ( 퐴 and 퐵 are constants)

푡5 푑푡 푤 = 퐴 ∫ + 퐵. (3.13) 푡 √(푡 − 푡0)(푡 − 푡1)(푡 − 푡2)(푡 − 푡5)

This leaves us with

1 퐾(푘) 퐶 = 휀휀 , (3.14) 푛1 2 0 퐾(푘′) where 퐾(푘) and 퐾(푘′) corresponds to elliptic functions and the value of 푘 can be given as

휋(푊 + 푆) 2 휋(푊 + 푆) 2 휋푊 cosh ( ) + sinh ( ) sinh ( ) 4ℎ 4ℎ 푘 = 4ℎ √ , (3.15) 휋(푊 + 푆) 휋푊 2 휋(푊 + 푆) 2 sinh ( ) cosh ( ) + sinh ( ) 4ℎ 4ℎ 4ℎ

′ 2 and 푘 = √1 − 푘 .

Using the above concept for a 3 layered substrate (ℎ1, ℎ2, ℎ3, 휀1, 휀2, 휀3 are the heights and permittivities of the three layers of substrate), we can calculate the capacitance of the (N-

3) periodical sections, capacitance at the ends and capacitance of the three-finger section.

 Capacitance of the (N-3) Periodical Sections of a Three Layered Substrate

The capacitance of (N-3) periodical sections can be calculated as

퐾(푘 ) 0 (3.16) 퐶푁 = (푁 − 3)휀3휀푒n ′ , 퐾(푘0)

where

휀 − 1 휀 − 휀 휀 − 1 휀 = 1 + 푞 1 + 푞 2 1 + 푞 3 , 푒n 1n 2 2n 2 3n 2

′ 퐾(푘𝑖n) 퐾(푘0) 푞𝑖n = ′ (푖 = 1,2,3), 퐾(푘𝑖n) 퐾(푘0)

휋푊 휋(푊 + 푆) 2 휋(푊 + 푆) 2 sinh ( ) cosh ( ) + sinh ( ) 4ℎ𝑖 4ℎ𝑖 4ℎ𝑖 푘𝑖n = √ , 휋(푊 + 푆) 휋푊 2 휋(푊 + 푆) 2 sinh ( ) cosh ( ) + sinh ( ) 4ℎ𝑖 4ℎ𝑖 4ℎ𝑖 and ′ 2 푘𝑖n = √1 − 푘𝑖n.

퐿 is the finger length. All the 푞 terms are called filling factors [20].

 Capacitance of Three-Finger Section of a Three Layered Substrate

The capacitance of the three-finger section can be calculated as

′ 퐾(푘03) (3.17) 퐶3 = 4휀0휀푒3 퐿, 퐾(푘03)

17 where

휀 − 1 휀 − 휀 휀 − 1 휀 = 1 + 푞 1 + 푞 2 1 + 푞 3 , 푒3 13 2 23 2 33 2

′ 퐾(푘𝑖3) 퐾(푘03) 푞𝑖3 = ′ , 퐾(푘𝑖3) 퐾(푘03)

(푊 + 2푆)2 1 − 푊 (3푊 + 2푆)2 푘 = √ , 03 푊 + 2푆 푊2 1 − (3푊 + 2푆)2

휋푊 휋(3푊 + 2푆) 2 휋(3푊 + 2푆) 2 sinh ( ) sinh ( ) − sinh ( ) 4ℎ 4ℎ 4ℎ 푘 = 𝑖 √ 𝑖 𝑖 , 𝑖3 휋(푊 + 2푆) 2 2 sinh ( ) 휋(3푊 + 2푆) 휋푊 4ℎ sinh ( ) − sinh ( ) 𝑖 4ℎ𝑖 4ℎ𝑖 and ′ 2 푘𝑖3 = √1 − 푘𝑖3(푖 = 1,2,3).

 Capacitance at the End Finger Section of a Three Layered Substrate

The capacitance of the end finger section of a three layered substrate can be calculated as

퐾(퐾 ) ( ) 0end (3.18) 퐶end = 4푁푆 2 + 휋 휀0휀end ′ , 퐾(푘0end) where

휀 − 1 휀 − 휀 휀 − 1 휀 = 1 + 푞 1 + 푞 2 1 + 푞 3 , end 1end 2 2end 2 3end 2

푊 + 2푔 2 1 − ( end ) 푊 3푊 + 2푔end 푘0end = √ . 푊 + 2푔 푊 2 end 1 − ( ) 3푊 + 2푔end

3.4 Limitations for Analytical Model

In the present research the interdigital capacitor is studied on a single layered substrate.

The expressions for a single layered interdigital capacitor can be obtained just by making a little adjustment to the expressions that are obtained for a three layered capacitor. Just by substituting ℎ2 = ℎ3 = 0 and 휀2 = 휀1 and 휀3 = 1 we get the expressions for a single layered interdigital capacitor. Initially the results presented in [20] are analyzed in

Mathematica using the derived expressions. It appears that there is some error in the end capacitance expression in [20]. Using the inputs given in [20] we cannot reproduce the results if we consider the end capacitance. A different approach for calculating the end capacitance is illustrated in [21]. This method is based on calculating the open end capacitance as per [22] and [23]. Even in this method there seems to be some error. So after a certain number of observations, the end capacitance of the interdigital capacitor is neglected just by setting the interdigital finger gap at the end at a distance where the capacitive effect vanishes. Now the new expression to calculate the net capacitance (퐶s) of the interdigital capacitor can be given as

퐶s = 퐶3 + 퐶푁, (3.19)

where 퐶3 is the capacitance of the three-finger section given as

퐾(푘1) (3.20) 퐶3 = 4휀0휀푒 ′ 퐿, 퐾(푘1)

휀 − 1 휀 = 1 + 푞 푟 , 푒 1 2

′ 퐾(푘1) 퐾(푘2) 푞1 = ′ , 퐾(푘1) 퐾(푘2)

(푊 + 2푆)2 1 − 푊 (3푊 + 2푆)2 푘 = √ , 1 푊 + 2푆 푊2 1 − (3푊 + 2푆)2

2 2 휋푊 휋(3푊 + 2푆) 휋(3푊 + 2푆) sinh ( ) sinh ( ) − sinh ( ) 4ℎ 4ℎ 4ℎ 푘2 = √ , 휋(푊 + 2푆) 휋(3푊 + 2푆) 2 휋푊 2 sinh ( ) sinh ( ) − sinh ( ) 4ℎ 4ℎ 4ℎ

′ 2 퐾(푘1) and 푘𝑖 = √1 − 푘𝑖 . The value of ′ can be evaluated using the Hilberg’s approximate 퐾(푘1) expressions [25].

Since we are also considering the metal thickness the effective width (푊eff) of the strips

[24] can be approximated as

푡 4휋푊 (3.21) 푊 = 푊 + [1 + ln ( )]. eff 휋 푡

퐶푁 is the capacitance of the (푁 − 3) periodic sections and is given as

퐾(푘3) (3.22) 퐶푁 = (푁 − 3)휀0휀푁 ′ 퐿, 퐾(푘3)

20 where

휀 − 1 휀 = 1 + 푞 푟 , 푁 푁 2

′ 퐾(푘3) 퐾(푘4) 푞푁 = ′ , 퐾(푘3) 퐾(푘4)

푊 푘 = , 3 푊 + 푆

2 2 휋푊 휋(푊 + 푆) 휋(푊 + 푆) sinh ( ) cosh ( ) + sinh ( ) 4ℎ 4ℎ 4ℎ 푘4 = √ . 휋(푊 + 푆) 휋푊 2 휋(푊 + 푆) 2 sinh ( ) cosh ( ) + sinh ( ) 4ℎ 4ℎ 4ℎ

Using the above expressions the expression to calculate the dielectric constant is written in terms of IDC’S capacitance. Using the capacitance that is obtained from the lumped parameter model the net dielectric constant is calculated as

퐶 − 4휀0푞03퐿 − (푁 − 3)휀0푞0퐿 (3.23) 휀푟 = 1 + 2 [ ]. 4휀0푞13푞03퐿 + (푁 − 3)휀0푞1푛푞0퐿

The analytical models of MIM capacitor and CPWIDC are studied and the equations to calculate the permittivity are developed. These expressions are later used in chapter 5 to calculate the permittivity and loss tangent from the capacitance obtained using the experimental or simulated S-parameter data. The next step is to fabricate MIM capacitors and CPWIDC from which we get the S-parameter data.

CHAPTER IV

DESIGN AND FABRICATION OF MIM CAPACITORS AND CPWIDC

4.1 Introduction

In this chapter the design of MIM capacitors, the design of coplanar wave guide interdigital capacitors, mask design and fabrication process are discussed. The experiments are performed on the fabricated devices to get the S-parameter data that is used to calculate capacitance which further is used to calculate permittivity and loss tangent of the material.

4.2 Device Fabrication

The device fabrication is done in four steps starting from thin film growth to electrode patterning.

 Thin Film Growth

BST thin films were grown on sapphire substrate with metallic thin film blanket layers

(40 nm titanium adhesion layer /200 nm platinum conduction layer) using the Metal organic solution deposition technique [26].

 Furnace Anneal

In the presence of flowing oxygen the BST films were crystallized using conventional furnace annealing (CFA) at a temperature of 750℃ for duration of 60 minutes. 22

 Thin Film Etch

BST Films were lithographically masked [28] with Microposit SC-1827 photoresist and etched with buffer oxide etchant.

 Electrode Patterning

200 nm thick gold electrodes were sputtered over the etch defined patches of BST and patterned using conventional optical lithography and lift off methods.

4.2.1 MIM Capacitors Mask Design

The mask design for the MIM capacitors is done in AutoCAD for different diameters as shown in Figure 4.1.

Figure 4.1: MIM Capacitors Mask Design

4.2.2 Photolithography Process

The first step in the process of Photolithography is the mask design. The masks are designed from the CAD drawings. Later these drawings are transferred onto the substrate.

The steps involved in this process are wafer cleaning, photoresist application, soft- baking, mask alignment, exposure and development, and hard-baking [27].

a) b)

Figure 4.2: MIM Capacitor Fabrication

 Wafer Cleaning and Photoresist Application

The wafer cleaning is a process to remove any of the impurities from the surface by cleaning with the chemicals. Soon after cleaning the photoresist is applied onto the surface of the wafer. The wafers are coated on to the surface by a process called as spin coating which produces a uniform thin layer of photoresist on the wafer surface. photoresist is of two types positive and negative.

For positive resists the resist is exposed to UV light where ever the underlying material is to be removed. Such exposure changes the chemical structure of the resist and becomes soluble in the developer. The exposed resist is then washed away by the developer solution, leaving the windows of the bare underlying material. Negative photoresist is the opposite. The negative resist remains on the surface wherever it is exposed and the developer solution only removes the unexposed portions.

 Soft-Baking

Soft-baking is the next step after coating the wafer with photoresist. It is done to remove all the solvents from the resist coating. The photoresist coatings become photosensitive only after soft-baking. Excess soft-baking will degrade the photosensitivity of the resists by destroying the portion of the sensitizer. Under soft-baking will prevent the light from reaching the sensitizer.

 Mask Alignment and Exposure

Mask alignment is the important step in the photolithography process. The mask is a square glass with metal patterns on it. The mask is aligned properly onto the wafer to transfer the patterns on the wafer surface. After the mask has been properly aligned high intensity ultraviolet light is passed on the photoresist. There are three methods to do this.

They are contact, proximity and projection.

In contact printing the resist coated wafer is brought into physical contact with the glass photo mask. The UV light is exposed when the photoresist is in contact with the mask resulting in very high resolution. If there are any scattered fragments in between the

25 photoresist and the mask it creates a problem by creating the defects in the pattern. The other two methods are preferred as the photoresist is not in direct contact with the mask.

 Development

After the wafer is exposed the device can be differentiated in the photoresist as regions of exposed and unexposed resist. The pattern is developed in the resist by the chemical dissolution of the un-polymerized resist regions. Problems resulting from the poor developing process are either the resist will be sticking on the patterned areas or excess resist is removed from the patterned areas. Hence the resist has to be developed for the optimal time.

 Hard-baking

This is the final step in the photolithography process. This step is to harden the photoresist and to improve the adhesion of the photoresist on the surface of the wafer.

 Metal Deposition

The metal deposition is done by DC sputtering or e-beam evaporation. Sputtering is a high energy process. The deposition of the metal will be uniform with sputtering and highly consistent.

 Lift-off Technique

The pattern on the substrate is created with the photoresist using the photolithographic process. Lift off is a technique for that particular defined pattern films on the substrate. A thin film is deposited on the substrate over the photoresist layer and on the defined

26 pattern on the substrate (the area where the photoresist is cleared). This film is later lifted off with a solvent (usually with acetone) leaving behind the film on the defined pattern on the substrate. Good adhesion should be maintained between the film and the substrate.

The entire process should be done at optimal temperature (higher temperatures may damage the photoresist on the substrate). The film should be brittle enough to tear along the adhesion surface. The substrate should be immersed in the solvent until no traces of the film are found otherwise after the substrate is dried, film particles get attached on the surface of the film.

4.3 Design of Coplanar Waveguide

The coplanar waveguide is designed in such a way that the characteristic impedance matches the port impedance so that there are no reflection losses.

푆 푊 푆

푇푚푒푡 퐿

퐻 휀푟 tan 훿

푆 푊 푆 a) Top View b) Side View

Figure 4.3: Ungrounded Coplanar Waveguide

There are many coplanar waveguide impedance calculators that are available online that uses conformal mapping. Just by changing the values of 푆, 푊, 퐿, 푇푚푒푡, 퐻, 휀푟, tan 훿 in

Figure 4.3 we can set the characteristic impedance of the coplanar waveguide to the

27 desired value. The reason we choose coplanar wave guide technology is because of its advantages. In terms of circuit isolation, CPW gives great isolation. Many high frequency

RF switches have used CPW technology to achieve isolation. One more advantage of

CPW is that it offers low dispersion.

4.3.1 Design of CPWIDC

Figure 4.4 : Copl anar Waveguide Interdigital Capacitor

Figure 4.4 shows the structure of a CPWIDC. The white part beneath the fingers is the substrate and the fingers are shown in yellow color. The thickness of the substrate (ℎ) has been chosen to be 60µm. The IDC’S are designed for both one-port and two-port measurements.

The interdigital capacitor’s capacitance can be varied with many factors such as finger length(퐿), gap between fingers(푔), gap between finger ends (푔2), number of fingers (푁),

28 width of the finger(푊), metal thickness(푀푡), and permittivity of the material(휀). So by using all these variables, different combinations were taken in to consideration and 24 different capacitors are designed. Out of 24 different capacitors twelve capacitors are six- finger capacitors with different values of 퐿, 푔, 푊. Among other twelve capacitors, four of each is three, four and five finger capacitors with different values of 퐿, 푔, 푊.

By studying the electric field distribution from the simulations it can be seen that as the number of fingers increases the capacitive effect increases. This can also be observed mathematically from the analytical model [20] (capacitance is directly proportional to the number of fingers). So the total capacitance of the IDC changes linearly with the number of fingers and can be represented as

(4.1) 퐶푡표푡 훼 푁.

The same is the case with the finger width. The finger width has a linear relationship with the total capacitance of the IDC. The other factor that comes next is the length of the fingers. If all other factors are kept constant varying the length of the fingers we see that the length of the fingers is linearly related to the total capacitance as

(4.2) 퐶푡표푡 훼 퐿.

The next factor is the gap between fingers. As the finger gap is increased more separation is created in between the interdigital electrodes which indicate the attenuation in electric field which in turn implies decrease in capacitance. So the gap between fingers is inversely related to the total capacitance as

1 (4.3) 퐶 훼 . 푡표푡 푔

4.3.2 Mask Design for CPWIDC

The mask design for the fabrication of CPWIDC is a three layer process. The masks are designed in AutoCAD. Layer 1 and 3 are designed on the same mask while layer 2 was designed on a different mask set.

Figure 4.5: Layer 1 and 3 Mask Design

Figure 4.6: Layer 2 Mask Design

First the simulations are performed in Ansys HFSS, based on those results the masks are designed in AutoCAD. Two sets of masks for positive photolithography process and the negative photolithography process are designed. The design parameters for CPWIDC are shown in Table (2) and Table (3) respectively.

Table 2: Design Parameters for Three, Four, and Five-Finger Capacitors

푁 푊(µ푚) 푔(µ푚) 푔2(µ푚) 퐿(µ푚) 1 3 100 1.2 15 250 2 3 100 1.6 15 250 3 3 100 1.2 15 500 4 3 100 1.6 15 500 5 4 100 1.2 15 250 6 4 100 1.6 15 250 7 4 100 1.2 15 500 8 4 100 1.6 15 500 9 5 100 1.2 15 250 10 5 100 1.6 15 250 11 5 100 1.2 15 500 12 5 100 1.6 15 500

Table 3: Design Parameters for Six-Finger Capacitors

푁 푊(µ푚) 푔(µ푚) 푔2(µ푚) 퐿(µ푚) 1 6 50 0.8 15 250 2 6 100 0.8 15 250 3 6 50 1.2 15 250 4 6 100 1.2 15 250 5 6 50 1.6 15 250 6 6 100 1.6 15 250 7 6 50 0.8 15 500 8 6 100 0.8 15 500 9 6 50 1.2 15 500 10 6 100 1.2 15 500 11 6 50 1.6 15 500 12 6 100 1.6 15 500

CHAPTER V

SIMULATIONS AND EXPERIMENTAL RESULTS

5.1 Introduction

In the previous chapter the general photolithography process and mask design for interdigital capacitors and MIM capacitors are discussed. In this chapter the equivalent circuit models and experiments on MIM capacitors and simulations for CPWIDC in

Ansys HFSS along with their results are discussed.

5.2 Equivalent Circuit Model of MIM Capacitor

The equivalent circuit model for the MIM capacitors is used to extract the capacitance from the devices using the scattering (S) parameters. One-port measuring techniques have been used to extract the S-parameters. Based on the extracted S-parameters the capacitance is calculated which in turn helps in finding the permittivity and loss tangent of the material.

Figure 5.1: Equivalent Circuit of MIM Capacitor

From the equivalent circuit in Figure 5.1, the reflection parameter 푆11 is calculated as

푍 − 푍 퐿 0 (5.1) 푆11 = . 푍퐿 + 푍0

1 The load impedance is 푍퐿 = , where 푌퐿 = 퐺 + 푗퐵. 퐺 is the inverse of resistance and B 푌퐿 is the inverse of capacitive reactance. 푍0 is the characteristic impedance of the port- one. The load impedance is given as

1 + 푆11 푍퐿 = 푍0 . (5.2) 1 − 푆11

By inverting the above equation we get a complex number. Comparing the real part with

퐺 and imaginary part with 퐵, the capacitance can be given as

퐵 (5.3) 퐶 = , 푀퐼푀 2휋푓 and the loss tangent can be given as

퐺 tan 훿 = . (5.4) 2휋푓퐶푀퐼푀

Substituting these expressions into the equations obtained in chapter 3 for the MIM capacitors complex permittivity we have the real and imaginary permittivity of the material. The equivalent circuit for one-port CPWIDC is also similar to this and it should be noted that the same equations obtained here are used for analyzing the one-port

CPWIDC.

5.2.1 Microwave Spectroscopy Test Setup

HP 8714ES RF network analyzer is used to get the reflection data in the frequency range of 2GHz to 3GHz. A Tektronix voltage source PWS4305 is used as a voltage bias. The

33 measurements are taken in a Signatone probe station maintaining a constant temperature of 20℃. A code is written in Matlab graphical user interface (GUI) to collect the S- parameter data at different bias points. Microwaves are sent on to the DUT (device under test) with a probe. The SOL (Short Open Load) calibration was also done at probe level with CS8 substrate. Figure 5.2 shows the test setup used for reflection measurements.

Matlab Voltage GUI Bias

Network Probe Bias-T DUT Analyzer Station

Figure 5.2: Test Setup for Microscopic Spectroscopy

5.3 Microwave Spectroscopy Results

As discussed in chapter ii, in paraelectric films there are no acoustic resonances without the application of the DC electric field. When a DC field is applied to the crystal the symmetry gets disturbed and the crystal behaves like piezoelectric crystal [2]. This can be observed as a resonant absorption of microwave power at particular frequencies.

The dipole behaviour is approximated to that of Lorentz oscillator model as shown in

Figure 5.3b to better understand the resonance behavior. Voltage bias creates the electric field inside the material and sinusoidal voltage makes the diople to oscillate. The plots for complex permittivity are also drawn and are in good agreement with the Lorentz

34 oscillator model as discussed in chapter ii. Figure 5.3a shows the schematic of the reflection measurements.

a) Microwave Reflection Apparatus b) Model for Resonating Dipolar Domains

Figure 5.3: Microwave Reflection Measurements Schematic

 Sample 1

The physical dimensions of this sample are 푑 = 50 휇푚, 퐿푆푃 = 25휇푚, 퐿푂푉 =

15휇푚, ℎ푡 = 190푛푚. This sample was tested in the frequency range of 2-3 GHz

and a resonance at 2.71 GHz is observed. Figure 5.4 shows the anharmonic

dipolar resonance from the reflection data. Figure 5.5 and Figure 5.6 gives the real

and imaginary plots as per the Lorentz oscillator model.

Figure 5.4: Frequency vs |푆 | for Sample 1 11

Figure 5.5: Frequency vs 휀푟푒푎푙 for Sample 1

Figure 5.6: Frequency vs 휀𝑖푚푎푔𝑖푛푎푟푦 for Sample 1

 Sample 2

The physical dimensions of this sample are 푑 = 50 휇푚, 푔 = 25휇푚, 퐿푂푉 = 15휇푚, ℎ푡 =

240푛푚. This sample was tested in the frequency range of 2-3 GHz and with a voltage bias of 0 - 12 volts.

A resonance at nearly 2.55 GHz is observed as shown in Figure 5.7. Similar to the sample

1 the real and imaginary permittivity is also plotted from the equations derived in the chapter iii and the plots in Figure 5.8 and Figure 5.9 are well in agreement with the

Lorentz oscillator model. It should be noted that the microwave small signal power level was held at 0 dBm for both sample 1 and sample 2.

Figure 5.7: Frequency vs |푆11| for Sample 2

Figure 5.8: Frequency vs 휀푟푒푎푙 for Sample 2

Figure 5.9: Frequency vs 휀𝑖푚푎푔𝑖푛푎푟푦 for Sample 2

 Comparing Sample 1 and Sample 2

Figure 5.10: Comparing Sample 1 and Sample 2

The only difference between sample 1 and sample 2 is the thickness of the thin film.

Hence it is evident that the shift in the resonant frequency in Figure 5.10 is due to change in the thickness of the thin film. Also from Figure 5.10 it is clear that as the thickness of the thin film increases more microwave power is absorbed.

5.4 MIM Capacitor as a Power Detector

The electric-field-induced anharmonic dipolar resonances from the BST thin films are used to rectify and detect the microwave signals. Since the resonance has strong dependence on film thickness this power sensor can be made to work in the ISM band.

5.4.1 Design of MIM Capacitor Power Detector

Figure 5.11: Circuit Diagram for Power Detection

For power detection microwaves at different frequencies and at different power levels (-

20dBm to 10dBm, the latter is the maximum power available from the instrumentation) are bounced on the device under test along with the voltage bias of 0-12 volts via a resistor 푅퐿 as shown in Figure 5.11. The value of 푅퐿 is chosen to be around 100k ohms to make sure the DUT is receiving the full voltage bias. The output voltage is measured using a lock in amplifier. Sample 1 was used for power detection.

5.4.2 Small Signal Model and Rectified Output

Figure 5.12: Small Signal Circuit Model

The small signal model for the power detection is shown in Figure 5.12. As mentioned earlier, with the application of the DC electric field, electric dipole moments are induced

(electric polarization is observed) and these dipoles oscillates in response to the input microwave signal. This polarization term can be expressed as a nonlinear function of electric field 퐸 as per [28] and [29] as

3 1 −1 2퐸 (5.5) 푃[퐸] = 휖0퐸2 sinh [ sinh [ ]] , 2 3 퐸2

1 where 퐸2is the electric field scaling factor and 휒푚푎푥 = [28]. ∝1[푇]

Since polarization is in nonlinear terms we can express it locally in terms of Taylor series expansion centered on a quiescent point (퐸 = 퐸푑푐) as

∞ 퐸 푛 푑푛푃[퐸] (5.6) 푃[퐸 , 퐸 ] = 푃[퐸 ] + ∑ 푎푐 . 푑푐 푎푐 푑푐 푛! 푑퐸푛 푛=0 The polarization is often expressed as a nonlinear expansion of susceptibility as

2 3 푃[퐸푑푐, 퐸푎푐] = 휖0{휒1[퐸푑푐]퐸푎푐[푡] + 휒2[퐸푑푐]퐸푎푐 [푡] + 휒2[퐸푑푐]퐸푎푐 [푡] + ⋯ }. (5.7)

Comparing the two expressions for the polarization, the equation for susceptibility at

퐸 = 퐸푑푐 is 1 푑푛푃[퐸] 휒푛[퐸푑푐] = 푛 . (5.8) 휖0푛! 푑퐸

The partial polarization resulting from each order of susceptibility can now be calculated as we have the expression to calculate susceptibility from (5.8).

When it comes to the power detection measurements, the rectified output is observed from the second order polarization term. The partial polarization from the second order polarization term can be given as

(2) 2 (5.9) 푃 [퐸푑푐, 퐸푎푐] = 휖0푥2[퐸푑푐]퐸푎푐 [푡].

where 퐸푎푐[푡] is a sinusoidal input. The expansion leaves a constant term and a second order harmonic. Hence the rectified output is detected from the second order partial

(2) polarization. Assume 퐸푎푐[푡] = 퐸푎푐,0 cos[2휋푓푡] then 푃 [퐸푑푐, 퐸푎푐] can be written as

(1 + cos[2휋(2푓)푡]) 푃(2)[퐸 , 퐸 ] = 휖 휒 [퐸 ] . (5.10) 푑푐 푎푐 0 2 푑푐 2

휖 푥 [퐸 ] 0 2 푑푐 is the DC term and the remaining term is the second harmonic term. 2

a) DC Term b) Second Harmonic

c) Second Order Sinusoidal Term

Figure 5.13: Rectified Output from Second Order Term

It can be derived from Maxwell equations that because of the induced electric field the device under test acts like a voltage source.

Using Maxwell equations we have

∂D (5.11) ∇ × H = , ∂t

∂B (5.12) ∇ × E = − , ∂t

∇ × D = 0, (5.13)

∇ × B = 0. (5.14)

Operating (4.12) with the curl operator ∇ × and using the relations 퐵 = µ0퐻 and

퐷 = 휀0퐸 + 푃 we can get the following equation

1 ∂2퐸 ∂2푃[퐸] (5.15) ∇2퐸 − = µ . 푐2 ∂푡2 0 ∂푡2

Now the value of voltage source can be given by

∂2푃[퐸] (5.16) 푉 = µ 푉. 푠 0 ∂푡2

푉 is the volume of the dielectric. Using (5.5) in (5.16) we have the value of the voltage output as (퐾[푇] is a constant that depends on the temperature)

1 2퐸 2 −1 푑푐 푓 푉푃[퐸]퐸푑푐 cosh [3 sinh ( 퐸 )] 2 (5.17) 푉표 = 퐾[푇] 3 . 2 2 2 2퐸푑푐 퐸2 [1 + ( ) ] 퐸2

The experimentally obtained data did not fit in the equation (5.17). It is observed that the experimentally determined data fitted to an empirical, piecewise model with exponential dependence on voltage bias. The model features two distinct regimes of behavior as shown in Figure 5.14. For values of voltage bias below an anomalous transition point, the trend is characterized by the function

훾(푉퐷퐶−푉퐴) 푉표 = 훼 + 훽(푉퐷퐶 − 푉퐴)푒 , (5.18) where 훼, 훽, 훾 and 푉퐴 are empirical parameters. Above the transition point, the trend is characterized by

휁(푉퐷퐶−푉퐵) 푉표 = 훿 + 휖푒 , (5.19) where 훿, 휖, 휁 and 푉퐵 are also empirical parameters.

5.4.3 Power Detection Results

10 dBm

0.001 V

o 0 dBm V 10 4

-10 dBm

-20 dBm 10 5 0 2 4 6 8 10 12 VDC V Figure 5.14: Output Voltage at Different Power Levels

Figure 5.15: Voltage Output for Different Voltage Bias at 0dBm

The output voltage of sample 1 was measured at different frequencies using a lock in amplifier. Figure 5.15 clearly indicates that there is a resonance at 2.71 GHz and the corresponding output voltage detected for the input power of 0 dBm was around 0.45 mV. At 12 V bias the output voltage vs frequency was plotted and it is observed that the device has some inbuilt bandpass filtering capability as shown in Figure 5.16.

Figure 5.16: Bandpass Filtering for Sample 1 in MIM Capacitor

The output voltage was collected as a function of voltage bias for different values of power level. The data points corresponding to 푉퐷퐶=12V were fitted to a straight line as shown in Figure 5.17 and a sensitivity of 0.6 mV/mW was extracted from its slope.

Figure 5.17: Maximum Output Voltage at Different Power Levels

5.5 Simulation of Interdigital Capacitors

The simulations for the study of coplanar waveguide interdigital capacitors were performed in Ansys HFSS. Ansys HFSS uses finite element analysis (FEA) for the electromagnetic simulations. 24 different simulations have been performed for 24 different capacitors. For the simulation purposes a random material has been created with permittivity of 3.9 and loss tangent of 0.001.

The main purpose of this simulation is to build a lumped parameter model that uses the extracted S-parameters from the simulations to find the values of the admittance in the lumped parameter model. Since a lumped parameter model is used, the total size of the device in all the 24 cases should be much smaller (at least 1/10th) than the wavelength.

The capacitance value obtained from this admittance is used in the analytical model to extract the material permittivity and the loss tangent of the material. So to summarize, the purpose of this simulation is to build a generalized extraction code that uses S-parameters from the coplanar waveguide interdigital capacitor simulations to extract the material permittivity and loss tangent. Matlab has been used here for programming purpose.

The coplanar waveguide interdigital capacitor (CPWIDC) couples the two coplanar waveguides by the electromagnetic field in its region. The electrical coupling can be represented in terms of a capacitor whereas the magnetic coupling can be represented using an inductor. In the present case only the electrical coupling is considered. For this purpose, we are operating the CPWIDC at the frequency (푓) very much smaller

푓 compared to 푓푅푒푠, the resonant frequency ( ⁄ =0.2). 푓푅푒푠

5.5.1 Simulation of One-Port and Two-Port Devices

Figure 5.18: Structure of One-Port CPWIDC in Ansys HFSS

For simulation of a one-port device in HFSS, the first step is to design a coplanar wave guide tramsmission line. Then a subsrate with its material properties (permittivity and loss tangent) is defined. After designing it the CPWIDC structures are drawn in HFSS as shown in Figure 5.18 and the port excitation is given . This process is done for all the capacitors and the capacitors are simulated in the frequency range of 1 GHz -8 GHz

(frequency sweep). The same process is repeated for the two-port device except the port excitation is given from both the ends as shown in Figure 5.19.

The S-parameter data is collected, analyzed in Matlab with the help of CPWIDC circuit model for both one-port and two-port devices to find the capacitance from which permittivity and loss tangent are determined.

Figure 5.19: Structure of Two-Port CPWIDC in Ansys HFSS

5.5.2 Equivalent Circuit Model for One-Port CPWIDC

Figure 5.20: Equivalent Circuit for One-Port CPWIDC

The equivalent circuit model for one-port CPWIDC is shown in Figure 5.20. The capacitance is the net capacitance of the interdigital capacitor and the resistor represents the losses in the material. This model is similar to the circuit model of a MIM capacitor.

So using the equations in section 4.2 we can calculate the capacitance and loss tangent of a one-port CPWIDC.

5.5.3 Equivalent Circuit Model for Two-Port CPWIDC

Figure 5.21: Equivalent Circuit for Two-Port CPWIDC

The equivalent circuit model for one-port CPWIDC is shown in Figure 5.21. As mentioned earlier the resistor represents losses in the interdigital capacitor. This model can be assumed as a simple Y (admittance) model. The capacitance can be determined

50 from the admittance of the interdigital capacitors in terms of S-parameters using network analysis and the resulting equation for admittance is given as

((1 − 푆11)(1 − 푆22)) − (푆12푆21) (5.20) 푌𝑖푑푐 = . 푍0(2푆21)

The S-parameter values for the CPWIDC’S are obtained from the simulations in HFSS.

5.6 CPWIDC Results

 Analytical Model vs Simulated Model

There is a slight difference between the analytical model and the simulated results. For

푁 = 3 and 푁 = 6 the capacitance value from the simulations and the capacitance value from the analytical model are found to be nearly the same.

Figure 5.22: Comparing the Analytical and Simulated Model for CPWIDC

 Permittivity and Loss Tangent

For 푁 = 3 and 푁 = 6 the extracted permittivity is close to that of the real permittivity.

For 푁 = 4 and 푁 = 5 the permittivity takes a deviation from the actual value. Similar is the case with the loss tangent.

Figure 5.23: Extracted Permittivity and Loss Tangent

CHAPTER VI

CONCLUSION AND FUTURE WORK

6.1 MIM Capacitors

Electric-field-induced anharmonic dipolar resonances (from reflection measurements) are observed from the MIM capacitors on the thin film BST and a new approach for microwave power detection is proposed using the dipolar resonances. A sensitivity of

0.6 mV/mW was observed from the MIM devices. The resonance of sample 1 was around 2.71 GHz and for sample 2 it was around 2.55 GHz. The only difference between sample 1 and sample 2 is the thickness of the ferroelectric thin film (BST). So it is clear that the shift in the resonance is because of the thickness of the thin film. Hence we can control the resonant frequency by changing the thickness of the thin film. Thus the device can be made to work in the ISM band (2.4-2.5 GHz) making it suitable for engineering applications. The MIM capacitors on BST thin films which are used as microwave power detector have a built in band pass filtering. This characteristic can be well studied and practically implemented eliminating the need for external filters for microwave power detection using the traditional circuit components.

6.2 Coplanar Waveguide Interdigital Capacitors

To use a multiferroic material for microwave applications its material properties are to be determined. For the CPWIDC the programming codes for permittivity and loss tangent

53 extraction are written. In other words, the interdigital capacitor is used as a permittivity sensor. The simulations are performed in Ansys HFSS. The simulation data is used to extract the permittivity and loss tangent of the material. The extracted permittivity and loss tangent for three-finger and six-finger capacitors gave good results compared to four and five-finger capacitors.

6.3 Future Work

MIM capacitors on BST thin films were used for microwave power detection. The rectified output was detected experimentally. The analytical expression for calculating the rectified output was derived. However, the experimentally obtained data is not in agreement with the analytically derived expression. The experimentally determined data fitted to an empirical, piecewise model with exponential dependence on voltage bias with two distinct regions of behavior. Efforts are being made to relate the empirical model to the physical theory.

For CPWIDC a simulated model was developed and extraction codes were written. To implement this model practically, the growth of magnetic nanowires in nanoporous

PVDF (multiferroic material) is needed. The growth of multiferroic material is currently under progress. The method employed for the growth is the electrodeposition technique.

After the material is grown, the devices are fabricated on top of the material. The fabricated devices are tested to determine the electric properties of the multiferroic material experimentally.

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APPENDIX

 Mathematica Code to calculate IDC Capacitance as per Conformal Mapping

W=100*10-6;(*effective width including thickness of the metal,W being the actual finger Width Weff=W+t/(1+Log[(4**W)/t])*) S=0.8*10-6;(*Spacing between the Fingers*) h=60*10-6;(*Height of the Substrate*) L=500*10-6;(*Length of the Fingers*) r=3.9;(*Permitivitty of the Medium*) 0=8.85*10-12;(*Permitivitty of the Free Space*) Nf=6;(*Number of Fingers*) Sg=15*10-6;(*Distance from CPW to Ground Plane*) G=15*10-6;(*end Finger Gap Width*) mt=3*10-6;(*Thickness of the metal*)

2 1 W 2 S 3 W 2 S 2 1 W 3 W 2 S k1[W_,S_]=W/(W+(2*S))* ; 2 k11[W_,S_]= 1 k1 W, S ; k2[W_,S_,h_]=Sinh[(*W)/(4*h)]/Sinh[(*(W+(2*S)))/(4*h)]* W 2 S 2 Sinh 4 h 1 3 W 2 S 2 Sinh 4 h W 2 Sinh 4 h 1 3 W 2 S 2 Sinh 4 h ; 2 k22[W_,S_,h_]= 1 k2 W, S, h ; A[W_,S_]=EllipticK[k11[W,S]]; B[W_,S_]=EllipticK[k1[W,S]]; P[W_,S_,h_]=EllipticK[k2[W,S,h]]; Q[W_,S_,h_]=EllipticK[k22[W,S,h]]; q1[W_,S_,h_]=A[W,S]/B[W,S]*P[W,S,h]/Q[W,S,h];

e=1+q1[W,S,h]*((r-1)/2); C3=4*0*e*(B[W,S]/A[W,S])*L (*Capacitance of 3 finger Section *)

k3[W_,S_]=(W/(W+S)); 2 k33[W_,S_]= 1 k3 W, S ; k4[W_,S_,h_]=Sinh[(*W)/(4*h)]/Sinh[(*(W+S))/(4*h)]*

2 2 Cosh W S Sinh W S 4 h 4 h 2 Cosh W 2 Sinh W S 4 h 4 h ; 2 k44[W_,S_,h_]= 1 k4 W, S, h ; A1[W_,S_]=EllipticK[k33[W,S]]; B1[W_,S_]=EllipticK[k3[W,S]]; P1[W_,S_,h_]=EllipticK[k4[W,S,h]]; Q1[W_,S_,h_]=EllipticK[k44[W,S,h]]; qN[W_,S_,h_]=A1[W,S]/B1[W,S]*P1[W,S,h]/Q1[W,S,h]; N=1+qN[W,S,h]*((r-1)/2); CN=(Nf-3)*0*N*(B1[W,S]/A1[W,S])*L (*Capacitance of a periodic Section *) 8.05693×10-14 7.72125×10-14 CT=C3+CN (*Total Capacitance *) Sg

k5[W_,Sg_]= W Sg ; 2 k55[W_,Sg_]= 1 k5 W, Sg ;

W h 1 W Sg k6[W_,Sg_,h_]= h 1 ; 2 k66[W_,Sg_,h_]= 1 k6 W, Sg , h ; A2[W_,Sg_]=EllipticK[k55[W,Sg]]; B2[W_,Sg_]=EllipticK[k5[W,Sg]]; P2[W_,Sg_,h_]=EllipticK[k6[W,Sg,h]]; Q2[W_,Sg_,h_]=EllipticK[k66[W,Sg,h]]; Ccpw1=2*0*A2[W,Sg]/B2[W,Sg]+0*(r-1)*P2[W,Sg,h]/Q2[W,Sg,h];

 Matlab Code for Extracting Data from IDC close all clear all %Location of the Touch Stone file pname = 'C:\Users\np42\Desktop\'; %Touch Stone file that contains Information about S parameters fname = 'hfssn3.s2p';

%Defining the Parameters of IDC N = 3; h = 60*(10^(-6)); t = 3*(10^(-6)); W = 106.76*(10^(-6)); S= 1.2*(10^(-6)); L = 500*(10^(-6));

% Permittivity of Free Space, units = F / m EPS0 = 8.85418781762E-12;

name= [pname, fname]; [~,x,y] = readColData(name,9,17,0); labels = [];

D = [x,y];

% Copy data into frequency, real and imag vectors f = D(:,1) * 10^9; ReS11 = D(:,2); ImS11 = D(:,3); ReS12 = D(:,4); ImS12 = D(:,5); ReS21 = D(:,6); ImS21 = D(:,7); ReS22 = D(:,8); ImS22 = D(:,9);

%Creating the S Parameter Matrix

S11=ReS11+1i*(ImS11); S12=ReS12+1i*(ImS12); S21=ReS21+1i*(ImS21); S22=ReS22+1i*(ImS22);

% Calculate measured Resistance and Capacitance vectors Y3=((1-S11).*(1-S22)-(S12.*S21))./((50).*2*S21);

B3=imag(Y3); C3=B3./(2*pi.*f); G3=real(Y3);

% Parallel Resistance R3=(1./G3);

q0 = Q0(W,S); q1n = Qn(W,S,h);

%Q0(W,S) and Qn(W,S,h) are Predefined Functions

% Calculating filling factors for C3 q03 = Q03(W,S); q13 = Q3(W,S,h);

EPSr=2.*(((C3)-(4.*EPS0.*q03.*L)-((N- 3).*EPS0.*q0.*L))/(4.*EPS0.*q03.*q13.*L+((N-3).*EPS0.*q0.*L.*q1n)))+1;

% Capacitance of IDC in pF (10^(-12)) is the Scaling. Cidc=C3./(10^(-12));

%Frequency in Ghz F=f./(10^(9));

%Loss Tangent Calculation

LT=G3./(10.*B3);

%Plotting the Obtained capacitance from S parameters vs Frequency plot(F,Cidc); Figureure(1) title('Capacitance vs Frequency'); xlabel('Frequency in GHz'); ylabel('IDC Capacitance in pF');

%Plotting the extracted Permitttivity vs Frequency Figureure(2) plot(F,EPSr); title('permittivity vs Frequency'); xlabel('Frequency in GHz'); ylabel('Extracted permittivity');

%Plotting Loss Tangent Vs Frequency Figureure(3) plot(F,LT); title('Frequency vs Loss Tangent'); xlabel('Frequency in GHz'); ylabel('Loss Tangent') function q = Q0(W, S) k = W / (W + S); kp = (1 - k^2)^0.5; q = ellipke(k) / ellipke(kp); k = []; kp = []; % clear local memory end function q = Qn(W,S,h) if ((W/h) < 10) k = (sinh(pi*W/(4*h)) / sinh(pi*(W+S)/(4*h))) * sqrt( ((cosh(pi*(W+S)/(4*h)))^2 + (sinh(pi*(W+S)/(4*h)))^2) / ((cosh(pi*W/(4*h)))^2 + (sinh(pi*(W+S)/(4*h)))^2) ); else k = sqrt(2) * exp(-pi*S/(2*h)); end; kp = (1 - k^2)^0.5; q = (ellipke(k) / ellipke(kp)) / Q0(W,S); k = []; kp = []; % clear local memory end

function q = Q3(W,S,h)

61 k = ( sinh(pi*W/(4*h)) / sinh(pi*(W+2*S)/(4*h)) ) * sqrt( (1 - ((sinh(pi*(W + 2*S)/(4*h)))^2 / (sinh(pi*(3*W+ 2*S)/(4*h)))^2)) / (1 - (((sinh(pi*W/(4*h)))^2) / ((sinh(pi*(3*W+ 2*S)/(4*h)))^2)) )); kp = (1 - k^2)^0.5; q = (ellipke(k) / ellipke(kp)) / Q03(W,S); k = []; kp = []; % clear local memory end function q = Q03(W,S) k = (W / (W + 2*S)) * sqrt((1-((W+2*S)/(3*W+2*S))^2) / (1- (W/(3*W+2*S))^2)); kp = (1 - k^2)^0.5; q = ellipke(k) / ellipke(kp); k = []; kp = []; % clear local memory end function [labels,x,y] = readColData(fname,ncols,nhead,nlrows) % process optional arguments if nargin < 4 nlrows = 1; % default if nargin < 3 nhead = 0; % default if nargin < 2 ncols = 2; % default end end end

% open file for input, include error handling fin = fopen(fname,'r'); if fin < 0 error(['Could not open ',fname,' for input']); end for i=1:nhead, buffer = fgetl(fin); end maxlen = 0; for i=1:nlrows buffer = fgetl(fin); % get next line as a string for j=1:ncols [next,buffer] = strtok(buffer); % parse next column label maxlen = max(maxlen,length(next)); % find the longest so far end end labels = blanks(maxlen); frewind(fin); % rewind in preparation for actual reading of labels and data % Read and discard header text on line at a time 62 for i=1:nhead, buffer = fgetl(fin); end

% Read titles for keeps this time for i=1:nlrows

buffer = fgetl(fin); % get next line as a string for j=1:ncols [next,buffer] = strtok(buffer); % parse next column label n = j + (i-1)*ncols; % pointer into the label array for next label labels(n,1:length(next)) = next; % append to the labels matrix end end

data = fscanf(fin,'%f'); % Load the numerical values into one long vector nd = length(data); % total number of data points nr = nd/ncols; % number of rows; check (next statement) to make sure if nr ~= round(nd/ncols) fprintf(1,'\ndata: nrow = %f\tncol = %d\n',nr,ncols); fprintf(1,'number of data points = %d does not equal nrow*ncol\n',nd); error('data is not rectangular') end data = reshape(data,ncols,nr)'; % notice the transpose operator x = data(:,1); y = data(:,2:ncols);

 Matlab GUI to get required Data from MIM Capacitors