Microwave Processes in Ferrite-ferroelectric Layered Structure

LAPPEENRANTA UNIVERSITY OF TECHNOLOGY FACULTY OF TECHNOLOGY

Master’s Degree Programme in Technomathematics and Technical Physics

Maria Gvozd

MICROWAVE PROCESSES IN FERRITE- FERROELECTRIC LAYERED STRUCTURE

Examiners: Professor Erkki Lähderanta Professor Sergey F. Karmanenko Supervisors: Professor Erkki Lähderanta

ABSTRACT

Lappeenranta University of Technology Faculty of Technology

Master’s Degree Programme in Technomathematics and Technical Physics Maria Gvozd

Microwave Processes in Ferrite-ferroelectric Layered Structure Master’s Thesis

2009

60 pages, 23 figures and 2 appendices.

Examiners: Professor, Erkki Lähderanta Professor, Sergey F. Karmanenko

Keywords: Layered structure, ferroelectric, ferrite, spin wave, electromagnetic wave, dispersing equation

This work is devoted to investigation of wave processes in new hybrid ferrite/ferroelectric structures. Spin wave devices based on ferrite films have disadvanteges. And their applications are limited. Investigated structures allow to overcome disadvantages. This investigation helps to create new class of devices.

Electromagnetic analysis of hybrid spin-electromagnetic waves in ferrite/ferroelectric structures were done. As a result dispersion relation was found. Numerical solution of this dispersion relation gave us follow results. These structures can be effectively tuned by external electric and magnetic field. Methods to increase tuning range were suggested. It was found that such structures have one basic disadvantage which is connected with presence of thick ferroelectric layer. To solve this problem is to use thin ferroelectric films. But this decreases tuning range. It was confirmed by experiment that this structures can be effectively tuned by electric and magnetic fields.

Resonance characteristics of ferrite/ferroelectric resonator were succesfully tuned by magnetic and electric field.

ACKNOWLEDGEMENTS

This master’s thesis was carried out in the laboratory of Physics, Lappeenranta University of Technology in cooperation with laboratory of Microwave and Telecommunication electronics, Saint Petersburg Electrotechnical University "LETI".

I wish to thank Professor Erkki Lähderanta for opportunity to write my master’s thesis here.

I wish to express my deepest gratitude to A. Nikitin for his guidance, support and patience. Without his help this work would not be possible.

Lappeenranta, May 2009 Maria Gvozd.

SYMBOLS Roman letters

H – external magnetic-field E – external electric-field M_o – saturation magnetization f – frequency

k – propagation constant Greek letters



– transverse wave number ω – angular frequency

ωH – ferromagnetic resonance frequency

 ^

–magnetic permeability tensor



– diagonal component of magnetic permeability tensor



a– gyrotropic component of magnetic permeability tensor



– controllability



– permittivity

Abbreviations

MW – microwave

YIG – yttrium iron garnet

GGG – gadolinium gallium garnet BST – barium strontium titanate RES – radio-electronic system SW – spin wave

MSW– magnetostatic wave

SMSW – surface magnetostatic wave FEF – ferroelectric film

TABLE OF CONTENTS:

Introduction...6

1. Modern achievements in radio-electronic devices based on layered structure of active dielectric...8

1.1. Active dielectrics in MW electronic... 8

1.2. Magnetostatic wave in ferromagnetic medium... 13

1.3. Features of hybrid spin-electromagnetic waves in layered structures of ferromagnetic/ferroelectric ... 19

1.4. Surface MSW in tangentially magnetized ferrite films... 21

1.5. Transverse resonance method ... 23

2. Wave processes in layered structures of ferrite/ferroelectric ...27

2.1. Electrodynamic model of layered structure of ferrite/ferroelectric . 27 2.2. Numerical analysis of dispersive relation for layered structures of ferrite/ferroelectric ... 32

2.3. Electrodynamic model and numerical analysis of dispersion relation for layered structures of ferrite/ferroelectric film... 39

3. Experimental investigation of ferrite-ferroelecric layered structure...47

Conclusion ...55

References:...56

Appendice 1 ...59

Appendice 2 ...60

INTRODUCTION

Importance of microwave (MW) communication systems, radar and navigation systems are increasing. Reliability, mobility, energy content requirements now become stronger and stronger. Telecommunication cellular and satellite systems, mobile navigation and radar stations, global and local networks need cheap and electrically- operated devices. This can be obtained if complicated scheme, which use active component, is replaces by tunable MW line based of film materials with nonlinear physical properties. Their important parameters are permittivity, conductivity, permeability change under influence electric and magnetic fields or current.

For example, spin waves (SW) propagating in ferromagnetic films and layered structures based on this film are used successfully many years to construct various processing signal devices in the range of MW frequencies [1 - 7]. Spin waves are used in different MW devices due to properties such as low phase and group velocities, variety of dispersing characteristics, low losses at propagation, ease of excitation and reception, possibility of electrical control for dispersing characteristics with help of variation of static magnetic field. One of the basic advantages of spin waves microwave devices is possibility of their electrical tuning. This tuning is carried out using variation of static magnetic field, but this magnetic tuning has some disadvantages such as huge magnetic systems, low velocity of tuning and big power inputs for control.

Second perspective technology to create electrical tuning devices of MW range is ferroelectric materials. Application of ferroelectrics for tuning devices is based changing of permittivity in electric field [8]. Using ferroelectric materials it is possible to control characteristics of microwave devices by applying a voltage.

Using ferroelectric materials for tuning spin wave microwave devices improve their behaviour and characteristics [9 - 18] and can lead to creation a new category of turning MW devices.

Experimental and theoretical investigations show that ferrite-ferroelectric layered

devices in comparison with spin wave devices are obvious. In addition, using such control mechanism allows to increase tuning velocity and decrease power inputs relating with active losses in electromagnet coils [9 - 18].

The purpose of this workis to: investigate ferrite-ferroelectric layered structure and to create electrodynamic model of layered structure. The purpose is also to find dispersion relation for hybrid electromagnetic spin waves, to analyse obtained dispersing dependences and to make experimental proof of double electric and magnetic tuning.

1. MODERN ACHIEVEMENTS IN RADIO-ELECTRONIC DEVICES BASED ON LAYERED STRUCTURE OF ACTIVE DIELECTRIC.

1.1. Active dielectrics in MW electronic

Ferromagnetics

As a basis of the controlled devices, which are a part of receiving-transferring modules of radio-electronic systems (RES), are applied different nonlinear environments whose physical parameters change with electric or magnetic field.

To create controlled mutual and not mutual MW components is ferromagnetic materials and ferrites. These are dielectric materials which have strong magnetic permeability [19]. MW mechanism of ferrite devices is based on interaction of magnetic field with non compensated magnetic moments of atoms in condensed medium. In this case the electromagnetic wave must penetrate into material and propagate in it. Penetration of magnetic waves into conducting medium is difficult, therefore for our purpose are used compounds of ferromagnetic metals (iron) with other elements. This kind of magnetic dielectrics have big resistivity, 10⁶...10¹¹ Ohm/cm, their specific permittivity is in limits 5..20 and is in almost independent on ferrite composition.

Physically, the waves in magnetic-ordered crystals (in ferromagnetics and antiferromagnetic) are magnetization waves. Such waves can be excite in a wide range of frequencies (from hundreds megahertz to tens hundreds gigahertz). They belong to the class of slow electromagnetic waves as their phase velocities is less than the light velocity [19, 20].

The magnetic properties of ferrite materials are caused by uncompensated antiferromagnetism. They have high magnetic susceptibility which depends on magnetic field strength and temperature.

Main feature of the ferrites is dependence of permeability on external magnetic field. Magnetic characteristics of dielectrics is controlled by change of external magnetic field (for example, monocrystal or ceramic plate, ferromagnetic film), in wide range of length and velocities of electromagnetic waves or of spin waves [20, 21]. But magnetic tuning of radio-electronic components has disadvantages such as low tuning speed (microseconds), significant power consumption, and big size of magnetic system.

In Fig. 1.1. is shown example of a planar ferrite phase shifter based on monocrystal plate of hexaferrite BaFe₁₂O₁₉ [0001] with size a = 7 mm, b = 7 mm, L = 500m, wide of microstrip line is w = 500 m, working frequency is f = 40 GHz [22]. Typical dependence of differential phase shift () on external magnetic field is shown in Fig. 1.2. It is seen that in microstrip line with parameters presented above,  is small for propagation process of electromagnetic waves in the magnetic medium.

Fig. 1.1. Schematic image of the planar ferrite phase shifter.

H

0

0 300 600 900 1200 0

10 20 30

  , d eg re e

H

₀

, Oe

Fig. 1.2. Differential phase shift in the planar structure presented in Fig.1.1. Line is approximation of experimental data (points).

As shown in [23], generally, influence of magnetic field strength on magnetic induction is described by asymmetrical permeability tensor:

0 0

0 0

a a

z

i i

 

  



 

 

   

 

 



, (1.1)

Where

2

2 2

( )

H H M

H

   

  

 

  , _{a 2} ^M ₂

H

 

 

  , _z 1, _H  ₀ H₀, _M  ₀ M₀,  is gyromagnetic ratio, H0 is external magnetic field and M0 is saturation magnetization.

Then dependences of



^and



_a on external magnetic field H₀ can be present as:

0 0

1 ( )( )

M

H H

   

      ^;

) )(

( ₀   ₀ 

 

 H H

M

a .

Properties of ferrite film differ from bulk samples. One differences is various dispersion laws depending on of external field direction in relation to sample and to the propagation wave direction.

Small phase and group velocities, variety of dispersion characteristics, easy excitation and reception and low losses at propagation are advantages of spin waves (SW). Main advantage is possibility of nonmechanical tuning. This tuning is implemented using bias field, therefore it can be called magnetic tuning. Magnetic tuning is easily implemented in wide frequency range. As a result SW has wide application in variety of linear and nonlinear devices. Among such devices are: phase shifter, delay lines, filters, resonators, limiters, convolves and many others [6, 7].

Main disadvantage of such devices is preventing of their development. This is connected with magnetic subsystem, because it is slow and power-consuming.

Ferroelectrics

Second method of nonmechanical control of radio-electronic components (RES) is using of electric field. «Electrical» control method differs in high speed (part of less than nanoseconds), small energy consumption (tuning is done without leak current through control circuit) and also small sizes of electrical system. However, turndown RES parameters with electrical tuning, as a rule, are less wide. Dielectrics having nonlinear dependence of physical characteristics on electrical field belong to ferroelectric group.

Ferroelectrics are materials which have a spontaneous polarization in absence of external electric field in a certain interval of temperatures and mechanical stress.

Change of permittivity by applied electrical field is basic property of ferroelectrics,

electrical field, therefore it can be called electrical tuning. Electrical field changes impedance of the component and leads to change of resonance characteristics, or it changes electromagnetic wave propagation speed (phase velocity) in transmission line.

Applicability of ferroelectric as a tuning component of radio-electronic circuits is defined by permittivity value  of material and value of control coefficient [24, 25]:

max

( 0)

( )

n E

E

 

 

^{, (1.2)}

or ^{max max min}

max

( 0) ( )

( 0)

E E

m E

     

 

  

^(%),

which is defined as the permittivity ratio, corresponding different values of applied electrical field or percentage wise. Electric field influence on permittivity takes place much faster in comparison with the "magnetic" control. Moreover "electric" influence has small power consumption.

Ferroelectric films represent the greatest interest for MW electronics. Thin films allow to integrate micro- and nano- technology for creating MW devices, to solve problem of matching MW circuits and provides significant turndown of material permittivity from applied voltage. At higher power level it is important to have good thermal contact of film with a substrate to provide effective heat removal from active area. MW properties of thin ferroelectric films essentially depend on production technology and differ from properties of bulk samples. For MW electronics requirements, the most suitable ferroelectric materials are perovskite structures, having the general chemical formula АВО3, where А and В are cations, О is anions (oxygen). Generally are used films of barium-strontium titanate BaхSr1–xTiO3 (BST), in which relation Ba and Sr is x change between 0 – 0.7. For devices working at a room temperature is used composition х = 0.5 – 0.6.

Film and bulk sampels have different electrophysical parameters and even if they have similar component composition. Characteristics of films also change in wide range that depends on the oxygen content, structural quality of films, their thickness, voltage and

crystal structure deformation. Results of some ferroelectric films, show that, BST film represent a real basis components for MW devices and for their practical introduction in modern RES. Moreover frequency dispersion of permittivity is absent up to 100 GHz in BST [26].

Relative permittivity in ferroelectric BST is order of 10³ and weakly depends on frequency up to 10¹¹ Hz. Electrical field with intensity (10-30) Vm^-1 changes initial values of permittivity in several times. Dielectric losses in MW range can be estimated as dielectric loss tangent that equal (3-5)10^-2.

Due to

 ( ) E

dependence ferroelectrics are applied in devices such as [27]: frequency multipliers (or harmonic generators), mixer, i.e. signal converters with frequency f_Signal and f_Generator in sum frequency f_Generator + f_Signal and in intermediate frequency f_IF

=1/2(f_Generator + f_Signal), parametric generators, stabilizers (dependence of the effective permittivity which is averaged dynamic conductivity over a period, is used for stabilization of MW power that passes through waveguide), commentators and switches (for fast commutation in radar and antenna engineering are used ferroelectric, including in single-tuned network (resonator, resonance stub, waveguide window), phase sifters, i.e. devices with controlled phase of transmission coefficient.

1.2. Magnetostatic wave in ferromagnetic medium

Magnetostatic (MSW) or spin (SW) waves are slow MW waves, exciting in ferromagnetic materials. For the first time MSW were described theoretically for homogeneously magnetized ferrite layer in 1961. Most usual material used for of MSW propagation is epitaxial films of yttrium-iron garnet Y2Fe5O12 (YIG). YIG has a garnet structure with lattice constant is а0 =12.376 Å. Structural transitions at low temperature in YIG are absent, but а0 decrease with 12.376 Å to 12.359 Å when temperature decreases from 295 K to 4.7 K. This material has low MW losses and weak anisotropy in a wide frequency range in comparison to ferrites with hexagonal structure or spinel type [28]. YIG monocrystals have dielectric constant



^{ 16,}

dielectric loss tangent

tg

less than 5·10^-4 at frequency 15 GHz, and saturation magnetization 1780 Gs at room temperature.

An important simplification [21, 29] is applied to consider MSW. Phase velocity of MSW, V_f = /k, as a rule is much less than light velocity (c

/ 

).This mean that in wide range of wave numbers phase velocity of electromagnetic waves exceeds by several orders phase velocity of spin waves.Therefore it is possibile to neglect electromagnetic part of a spectrum, i.e. use infinity light velocity in Maxwell equations (to neglect electromagnetic delay). Similar operation simplify the Maxwell equations.

Instead of full system of six equations in the given approach it is enought to consider only two equations, called magnetostatics. The solution of the magnetostatics equations, give the spin waves spectrum to consist only dispersing characteristics, corresponding to slow SW. The dispersing characteristics corresponding to fast electromagnetic waves in magnetostatics approach are lost.

In magnetostatic approach gives spectrum of SW enough exact in a wide range of wave numbers. Magnetostatic approach gives different results only in long-wave part of spectrum, where phase velocities of spin and electromagnetic waves are comparable. Thus a range of wave numbers in which magnetostatic approach is unsuitable, depends on particular range of SW and values of permittivity of ferromagnetic layer and dielectric layers surrounding it. Magnetostatic approach gives considerable error in millimetric range of electromagnetic wavelengths and when the ferromagnetic layer contacts the dielectric layers with big values of permittivity (for example, with ferroelectric layers). For correct description of SW properties in these cases it is necessary to apply full electrodynamic theoretical model based on full Maxwell system.

The result of the full electrodynamic problem in the layered structures including ferromagnetic layers is called a spectrum of hybrid electromagnetic SW. Such name point out that the spectrum of SW is obtained taking into account effects of electromagnetic delay.

Due to such approach the term MSW has been expanded, earlier it was consider not so successful [19, 29] in comparison with the term "spin waves". In western literature the term MSW is used [30]. In the this work we will use "magnetostatic" and "spin" waves as synonyms.

Adequate description of MSW processes is given by the dipole-exchange theory [23, 31]. This theory considers heterogeneous exchange interaction in ferromagnetic spin system. This theory considers features of surface spin on a film (exchange boundary conditions). In dipole-exchange theory a number of results for anisotropic ferromagnetic environments are obtained, spectrum of MSW in free and shielded films is investigated,and a theory of linear and parametrical excitation is constructed. The theory allows calculating work characteristics of SW devices.

One of the basic problems is excitation of propagating magnetization waves. To construct strict theory of excitation it is necessary to account of all possible interaction energy of ferromagnetic: dipole-dipole, exchange and anisotropic. To solve this problem often is used some approximation.

One approximation is exchangeless approach which assumes that exchange interaction (a spatial dispersion) makes only small influence to SW spectrum. This approximation is physically justified only in case of the long spin waves propagating in thick ferromagnetic layers. Applicability criterion is a small wave length (1 m) in comparison with wave length of SW and thickness of a ferromagnetic layer.

Possible wave numbers k can be divided into three range.

1. Small k values (k  10^-1 cm^-1). In this case vibration is mixing magnetostatic modes. Exchange interaction does not replace influence on a spectrum of vibration, but it is necessary to consider boundary conditions and spreading effect.

2. Average k values (10< k <10³ см^-1). Normal vibrations represent magnetic waves in "magnetostatic" approach. Dipole and partially exchange interactions play main role.

3. Big value k (10³ < k <10⁵ см^-1). Normal vibrations are short waves "exchange"

SW. Essential role play dipole and exchange interactions, but SW spreading effect is insignificant.

Two first ranges of wave numbers are most interesting; therefore we use exchange approach. The magnetostatic equations gives three types of SW [31].

1. Conditions for propagation of surface SW: Magnetic-field vector Hi is in a plane of YIG film and perpendicular to a SW direction (Hi  k). Propagation of SW is nonreciprocal; They are pressed to different sides of the plate surfaces at the different direction of traveling wave.

2. Straight volume SW (SVSW). Vector H_i is perpendicular to film plane and to a wave vector (H_i  k). For SVSW there is no allocated direction in a plate plane.

Direction of k and group velocity V_g coincide.

3. Inverse volume spin wave (IVSW). The wave is formed, when vector Н_i lies in the film plane and is parallel to a wave vector (Нi || k). Directions of phase vector and group velocities are opposite.

SMSW is most convenient for practical applications. These are transverse waves in tangential magnetized ferromagnetic film. Dispersion equations SMSW are presented in Eq. (1.3) - (1.5). Wave vector k is perpendicular to a magnetic field direction (k  H), and frequency range of MSW is in limits from  to (H + m/2).

These three types of SW:

SMSW

( 1 ) 4

2 2

2

2 М k L

М H

H

     е

^{  }

 









^(1.3)

SVSW

[ ]

2 2

2 2

n Y

Y M

H

H k X

k









   

 ^(1.4)

IVSW

]

[ 2 2

2 2

Y n

n M

H

H Y k

Y









  

 ^(1.5).

where X_n is root of transcendental equation, L is thickness of the film.

z Z y

k L k

tg k )

( 2 ,

z Y

Z k

k L ctg k )

( 2 for symmetric and antisymmetric waves.

Yn is root of transcendental equation

y x

x k

k L tg k )

( 2 ,

y x

x k

k L tg k )

( 2 .

Equations (1.3) - (1.5) allow to define limits of basic types for SW:

SMSW

1

( )

2

2

М

H H М H

          

SVSW and IVSW ²

1

)]

( [



_H

   

_H

 

_H

 

_М ^.

Fig.1.3 Dispersing characteristics a) straight volume (SVSW) b) inverse volume (IVSW) and c) surface magnetostatic waves (SMSW).

SSW has property of nonreciprocity, meaning waves running in opposite directions are

pressed to different sides of a ferromagnetic film. Magnetization decreases exponentially when moving from a film surface. In ferromagnetic films thin metal conductor are usually used for excitation SW located on the surface of film or near to it. Usual exciting element is called antenna [30, 31, 32] which is segment of microstrip. The k value in the case of a microstrip segment can change in wide limits and is defined by cross-section sizes according to the dispersing law. Using film structures, as opposed to volume samples, are avoid the difficulties connected with excitation reproducibility and reception MSW, caused by heterogeneity of an internal static magnetic field.

Relaxation process is the main sources of losses at propagation of magnetostatic wave in YIG and diffraction effects. In this work we will not consider loss mechanisms.

1.3. Features of hybrid spin-electromagnetic waves in layered structures of ferromagnetic/ferroelectric

Use of ferroelectric materials for tuning of microwave spin-wave devices improves work characteristics and lead to creation new class of tuning MW devices. Advantage of such devices is based on layered structures, containing ferromagnetic and ferroelectric layers. Tuning mechanism is based on change of ferroelectric layer permittivity, instead of usual mechanism of changing static magnetic field allowing substantially to decrease weight and sizes of SW devices (due to exclusion from the construction of electromagnet coil). Besides, using such control mechanism increases tuning velocity and decreases power inputs, which are connected with the active losses in electromagnetic coil.

In [13 - 18] is presented electrodynamic theory of arbitrarily magnetized layered structures, including ferromagnetic and dielectric layers. Also this theory takes into account dipole-dipole and exchange interactions in spin system of ferromagnetic and electrodynamic and exchange boundary conditions on the layers interface.

These works are based on the theory and are constructed using Green's functions [33].

In this case find integral representation for variable dipole magnetic field of plane SW using Maxwell equations. Substituting integral formula in linearized equation of moment motion leads to integro-differential equation for magnetization vector of SW.

We can solve obtained equation using Bubnov-Galerkin method. We expansion into a series distribution of magnetization variable along orthogonal system of vector functions. It satisfies to boundary conditions on ferromagnetic layer surfaces.

In this method a set of basic functions is made by eigenfunctions of the differentially- matrix operator. This describes magnetization distributions in a ferromagnetic layer at spin-wave resonance (spin-wave modes). This method gives [13 - 18] the dispersion relation and the analysis is made for layered structure.

For spin-wave devices usually is used two types of spin waves: longitudinal spin waves, propagating parallel with the direction static magnetic field, and transverse spin waves propagating perpendicular with the direction of static magnetic field.

Longitudinal spin waves interact with electromagnetic waves of lowest type TM0, without cutoff [13 - 18]. The control mechanism of dispersing is permittivity change in symmetric structure. This mechanism is ineffective if we have asymmetry. It is connected with attenuation of interaction between longitudinal SW and electromagnetic waves in the case of asymmetry [13-18].

Spin waves propagating across a magnetization direction in magnetized structures have an electromagnetic field. Its electric component lies in a plane, perpendicular to a propagation direction. The spin waves propagating in across a direction of a static magnetic field in magnetized layered structures are hybridized only with electromagnetic waves of TE type [13 - 18]. Cutoff frequency changes at change of permittivity ferroelectric layers.

Thus, the most perspective are transverse SW, if we control dispersion using permittivity changes. Transverse spin waves have nonreciprocal character, i.e. their

dispersion characteristics depend on sign of longitudinal wave number. The analysis of the dispersion equation in work [17] shows that the interaction of transverse spin and electromagnetic waves is more effective in asymmetrical layered structures, than in the symmetric.

1.4. Surface MSW in tangentially magnetized ferrite films

Let's consider in detail propagation process of SMSW. The case when the plate is metallized from both sides and frequency is not dependent from k, i.e.



_gr(group velocity) = 0 (Fig. 1.4, a) is not interesting. We consider distribution of a surface wave in a free plate (Fig. 1.4, b). This problem in magnetostatic approach (see 1.3) has been investigated by Damon and Eshbach [34] and represents big practical interest.

Fig. 1.4. Dispersion dependences and group velocities of surface waves in different structures. Continuous lines s = 1, dotted lines s = -1. In calculation is used

2 , 5

t  t  d t  d

. S is characteristic of propagation direction of magnetostatic

wave relative to magnetization vector, t, t₁, t₂ are distances from ferrite film to metal shield and d is thickness of ferrite film. Ferrite film is metallized from both sides

(a), ferrite film without metal shield (b), metal shield over ferrite film (c), metal shield on the distance from ferrite film (d), metal shield on the distance from both

sides ferrite film (e).

In this case dispersing equation looks like:

2 2 2 2

(

_{H M}

/ 2) (

_M

/ 2) e

^kd

      

^ . (1.6) where _H  ₀ H₀, _M  ₀ M₀,  is gyromagnetic ratio, H0 is external magnetic field and M0 is saturation magnetization.

This equation is called Damon and Eshbach equation. This dispersion dependence has frequency limits:

1

H

2

M



_

     

, (1.7)

where



_

 

_H

( 

_H

 

_M

)

.

Seshadri [35] has considered case when tangentially magnetized plate has contact with metal from one side and with air from other. In this case the dispersion law looks like:

(1 2

^H

2 )

^{H M} 2^kd

M M H M

s s e

s

    

    

 



  

 

. (1.8) Into this expression s enters unlike (1.6). From (1.8) follows, that when s = 1,

frequency has limits

H M



_ 











(1.9) and at

k  

frequency turn into wave frequency







_H 



_M propagating in half-infinite ferromagnet with boundary with metal. Then s = -1, frequency in limits (1.7) turn into wave frequency

1

H

2

M

    

, propagating in half-infinite ferromagnet with boundary with dielectric.

Dispersing characteristics of surface waves were calculated for more complicate structures, they are shown in Fig. 1.4 (d, e). For waves in structure Fig. 1.4 (d), which were investigated by Bongianni [36], dispersion equation is:

( )(1 )

2

(1 2 2 )

( )(1 )

H M H kd

M M M H

s thkt

s e

s thkt

    

    

  



  

  

. (1.10)

Interesting feature of this structure, as the similar to structure Fig. 1.4, (e) [37], is nonmonotonic dependence

 ( ) k

^(at s = 1): Wave is straight at small k and inverse (



_gr

 0

) at big k.

1.5. Transverse resonance method

Transverse resonance method allows fast and simple observation of dispersion equation for TE and TM waves in partially filled wave guide.

For example consider empty rectangular regular wave guide. Width of this wave guide is a, and permittivity is  Assume that TE wave propagates along axis z with field structure E(0,E_y, 0)

, H H( _x, 0,H_z)

(Fig. 1.5).

Fig. 1.5 Transverse resonance method for rectangular wave guide.

For TE wave the ratio ^y

x

Z E

 H (for unit

Vm V

A A

m

 ) gives input resistance of

transmission line (for free space ₀ ⁰

0 y 120

x

Z E H

 

    ). Wave resistance to the right and left of x plane must be equal with sign opposite. Transverse resonance condition is

Z  Z

in cross-sectionxconst.

From a condition of transformation resistance [38] is known that full equation for wave resistance is:

1 01 1 1

01

01 1 1 1

( )

( )

Z jZ tg d Z Z

Z jZ tg d





 



 

 . (1.11) where ₁ is transverse wave number, d1 is width, Z01, Z1 - wave impedance.

Let's make so that the plane coincides with wall x0 in the regular wave guide. Then, as resistance of a metal wall is zero, then equation (1.11) gives Z jZ tg₀ (a)

, resistance to the left of a plane x is also 0, and we obtain the dispersion relation

0 ( ) 0

jZ tg a  for a rectangular regular wave guide. This coincides with the dispersion relation described in the literature [39].

Now we will consider a case of a regular wave guide with dielectric filling Fig. (1.6).

Fig. 1.6 Transverse resonance method for rectangular wave guide with dielectric filling. d1, d2, d3, d4 are thickness of dielectric layers with permittivity    ₁, ₂, ₃, ₄.

Let's choose a resonance plane at xd₁d₂. From equation (1.11) in area xd₁ wave resistance will be Z₁ jZ tg₀₁ (_{1 1}d )

, where ₀₁ ⁰

1

Z 

  , as resistance of free space bounded in area i is _0i ⁰

i

Z 

  , where _i²    ² _{0 0 i} k². Wave resistance in area x (1.11) will be ₀₂ ^{1 02 2 2}

01 1 2 2

( )

( )

Z jZ tg d Z Z

Z jZ tg d





 



 

 . Repeating calculation on the other side of a

plane xd₁d₂, we will obtain similarly ₀₃ ^{4 03 3 3}

03 4 3 3

( )

( )

Z jZ tg d Z Z

Z jZ tg d





 



 

 , where

4 04 ( 4 4)

Z  jZ tg  d

. Using condition of transverse resonance we will obtain dispersion relation in the form of Z Z

.

Conclusions from the first chapter:

In this part we considered electrophysical properties of ferroelectrics and ferromagnets.

Possibility to control the properties of these materials by magnetic and electric field is shown. Due to this possibility these materials have received wide application in RES.

However, devices created on these materials have several disadvantages. This disadvantages can be avoided by using ferrite-ferroelectric layered structures.

In the previous works was shown, that we can tune SW devices using permittivity of ferroelectric layer.

2. WAVE PROCESSES IN LAYERED STRUCTURES OF FERRITE/FERROELECTRIC

The purpose of this part is the electrodynamic analysis of wave process in layered structures of dielectric (ferroelectric) – ferrite – dielectric – ideal metal ( ) and investigation of the dispersion equation.

As follow from previous part [18], spin wave propagating to perpendicular direction of static magnetic field in tangential magnetized layered structures, hybridizes only with electromagnetic waves type TE. Transverse spin waves and electromagnetic waves have more effective interaction in asymmetrically layered structures than in symmetrical structures. Therefore in this part we investigate electrodynamic model of wave propagation process along ferrite/ferroelectric interface. It is well known from previous part that exchangeless approach is physically justified only for long MSW (k

< 10²cm^1). Dispersion dependence on that k interval is not the purpose of this part, and therefore we use exchangeless approach.

2.1. Electrodynamic model of layered structure of ferrite/ferroelectric

Investigated structure is presented in Fig. 2.1. Wave propagates along x axis and magnetization is perpendicular direction to it, along axis z. Thickness of ferroelectric layer is d; ferromagnetic film (YIG) is b and linear dielectric (substrate for ferromagnetic film) is a.

Fig. 2.1. Layered structure (ferroelectric–ferritedielectric–ideal metal) and its electrodynamic model.

For description of the ferromagnetic environment we use magnetic permeability tensor (1.1), which take into account the direction of magnetic field and electromagnetic field dependence on coordinate in the form

e

^ikx:

b d a

z y

x



B



k

0 0

0 0

a a

z

i i

 

  



  

 

  

 

 



. (2.1)

Solution can be received from first two Maxwell equations:

H i

E

rot   

₀

J E i

H

rot  

₀



. (2.2)

In our problem is dielectric layer; it means that current density is ^{J 0}.

Let's write the Maxwell equations as projections to coordinate axes. There are no dependence of field on coordinate z in the selected coordinate system, so 0





z , and solution can be obtained as two independent modes: TE modes with structure



E (0,0,Ez),



H (Hx,Hy,0) and TM modes



E(Ex,Ey,0),



H(0,0,Hz). Let’s consider transverse electric (TE) wave. This wave corresponds with surface MSW. Maxwell equations projected to axis in appropriate cross-section areas can be written as:

in free space

0

0

0

y x

z

z

x

z

y

H H

i E

x y

E i H y a b d

y

E i H

x







 

 

  



     

 



 

 



, (2.3)

in ferroelectric a b y a b d

H x i

E

H y i

E

E y i

H x H

y z

x z

z y x





























 





 



 







0 0

1 0









, (2.4)

in ferrite film





















 







 



 







) (

) (

0 0

2 0

y x a z

y a x z

z y x

H H i x i

E

H i H y i

E

E y i

H x H

















a yab, (2.5)

in linear dielectric y a

H x i

E

H y i

E

E y i

H x

H

y z

x z

z y x























 





 



 

 





0

0 0

3 0









. (2.6)

Let’s find magnetic field components Hx and Hy in terms of Ez from relation (2.3, 2.4, 2.6) and we obtain wave equation for each environments:

0

0

0 2 2

2 2

2



 

 





z j z

z E

x E y

E

   

_{, (2.7)}

where j = 0,1,3 for each layer, respectively.

Next derive wave equation for ferromagnetic film and obtain magnetic field components H_x and H_y in terms of E_z, obtain:

z x y

a z

z E

y H x

H x

E y

E ₂ ₀ ²₀₀₂

2 2 2

)

( 







 

 





 . (2.8)

Now let’s find

y H x

H_{x y}







 . For this purpose we need to differentiate second equation in (2.5) by x and third equation by y and after that sum them. Result will be:

z y a

x

E

y H x

H



 



_{0 2}



 

 





. We substitute this expression into equation (2.8), then receive:

2

0

0 0 2 2

2 2

2



 

 





 z z

z

E

x E y

E     

_{, (2.9)}

where _{2 2}

2 2 2

2

( )















 





 

 





M H

a .



_

 

_H

( 

_H

 

_M

)

[24].

Equations (2.7) and (2.9) can be solved by variable separation method.

In different volumes Ez can be presented in the form:

In free space ya b d and take into account boundary condition for infinity

ikx y

z

A e e

E 

₀ ^{0 } _{, (2.10)}

where₀  k² ²₀₀ .

Inside the ferroelectric layer ab yabd

ikx

z

A y B y e

E  (

₁

sin 

₁



₁

cos 

₁

)

^ , (2.11)

where₁  ²₀₀₁k² .

In the ferromagnetic film ayab

ikx

z

A y B y e

E  (

₂

sin 

₂



₂

cos 

₂

)

^ , (2.12) where₂  ²₀₀₂_k² .

Finally, inside linear dielectric

ikx

z

A y e

E 

₃

sin( 

₃

)

^ , (2.13)

where₃  ²₀₀₃k² .

The partial solution for linear insulator can be written taking into account boundary condition on ideal metal (E_z|_y00B₃ 0).

For solving we use usual electrodynamic boundary conditions: tangential components of vectors E

and H

are equal on boundary of phase sections:

d b a b a a y j z d b a b a a y j

z

E

E |

_{ ,  ,  }



^1

|

_{ ,  ,   ,}

d b a b a a y j x d

b a b a a y j

x

H

H |

_{ ,  ,  }



^1

|

_{ ,  ,   . (2.14)}

Field component E_z is defined by equations (2.10 - 2.13) and H_x is defined from system equations (2.3 - 2.6).

In insulator and in free space:

y E H

_x

i

^z



 



0 ^{, (2.15)}

and in ferromagnetic film

)

(

^{2 2}

0 a

z a z

x

x E y

i E

H   









 





 . (2.16)

To determine all unknown constants we use 6 boundary conditions (2.14). Imposing these conditions we obtain homogeneous system of six equations. From condition that determinant of the combined equations is zero we obtain dispersion equation. Because transcendental equation has complicated form, it is not possible to present dependence

 from k in explicit form. Full dispersion equation is given in the Appendix 1.

2.2. Numerical analysis of dispersive relation for layered structures of ferrite/ferroelectric

First we assume that a = b = 0 and make all algebraic calculation with Appendix 1. We obtain following dispersion relation for surface TE wave propagating in metallize dielectric layer

0 1 1

)

( 

 d   

tg

_{. (2.17)}

It is similar to equation which we obtain by transverse resonance method for structure of free space – dielectric – metal. This method was described in chapter 1.5.

Results are shown in Fig. 2.2. From this figure we can estimate possible electric field.

Fig. 2.2 show two modes, zero and first, of dielectric wave guide and tuning by external electric field (change of permittivity).

0 25 50 75 100 7

14 21

f, GHz

TE₀ TE₁

k, cm^-1

Fig.2.2. Dispersion characteristics of ТЕ₀ and ТЕ₁ wave in the structure of free space–

dielectric–ideal metal. Here  1000and d = 500 m is shown with continuous line, and  2000and d = 500 m with dotted and dashed line.

0 200 400 600

6 7 8

f , G Hz

H = 1500 Oe H = 1550 Oe H = 2000 Oe

k, cm

^-1

Fig. 2.3. 1, 2, 3 are dispersion characteristics for SMSW with different external field (dotted line is_ and

2

M H

  respectively).

Now let’s consider metallized ferromagnetic film (assumed a0), so make all algebraic calculations on dispersion relation and obtain

1 2 3

0 2

2 2

( )

^a

k

ctn b  

 

 

^

 

_{. (2.18)}

In limiting case of dielectric (i.e. _a 0,  ₀) this gives 2.17.

Calculation results of dispersion relation for surface MSW with different bias fields is shown in Fig. 2.3. From figure one can see magnetic tuning in broad frequency range which is one of the basic advantages of ferrite films. It is significant, that influence of metal shield distance on dispersing characteristics is insignificant if thickness is more than 300 m and  10.

Equation 2.18 agrees with theory presented in chapter 1.4, as you can see Fig. 2.3 is similar with Fig. 1.4, (c) (dotted line).

Solution of dispersion relation (Appendix 1) was calculated using numerical methods in Maple 11 program. Graphical dependences that were obtained using these numerical methods are shown in Fig. 2.4. Here we can see advantage of this layered structure; it is possible to control dispersion characteristics by changing the control electric field (i.e.  of ferroelectric) with fix magnetic field. Near frequency

1

H

2

M

  

dispersion characteristics are strongly displaced in area of large k, i.e. hybrid waves are slowed down. In Fig. 2.4 when frequency is 6.2 GHz then phase velocity of hybrid wave is half of the velocity of wave in open dielectric wave guide.

Let’s consider in detail how thickness of ferromagnetic and ferroelectric layers influences on dispersion characteristics of hybrid wave (Fig. 2.5 and 2.6). The cutoff frequency moves downwards if thickness of the ferroelectric layer increases. This is presented in Fig. 2.5. Additionally hybridization range of SMSW and TE wave are increasing.

0 40 80 2

4 6

f, GHz

k, cm^-1

500

1000

1500

2000

Fig. 2.4. Dispersion characteristics of hybrid electromagnetic spin wave with different

 of ferroelectric film. In calculation is used a500m, b20m, d 500m, H = 1500 Oe; straight dotted line is surface MSW.

Fig. 2.5. Dispersion characteristics of slow hybrid waves with different thickness of ferroelectric layer. In calculation is used b20m, a500m, H = 1500 Oe,

500

  .

Such behavior of dispersion characteristics points out the basic disadvantages of ferrite-ferroelectric layered structure. It is necessary to give big bias voltage to obtain visible tuning of ferroelectric layer. Switching time will be an order 1 ms because big capacitance of this structure is big.

Decreasing of ferromagnetic film thickness (Fig. 2.6) decreases hybridization (repulsion between fast and slow waves) between SMSW and TE wave of open dielectric wave guide, because SMSW has more "smooth" dispersing characteristics in case of thin film. This behavior of dispersion characteristics agrees with theory [23]. It is necessary to use thick ferromagnetic films for increasing operating frequency interval near frequency

1

H

2

M

  

, where we receive operating tuning.

Analysis of more thin ferrite film layers would require to take into account spatial dispersion, but it is beyond exchangeless approximation used by us.

In Fig. 2.7 is shown result of full solution of dispersion equation with fast and slow hybrid waves at different. From this figure one can see when  is big then degree of hybridization increases (increase of repulsion between fast and slow hybrid waves).

Fig. 2.6 Dispersion characteristics of slow hybrid waves and SMSW with different thickness of ferromagnetic film. Here a500m, d 500m, H = 1500 Oe,

1000

  .

20 40 60

4 8 12

f, GHz

k, cm^-1

"slow" (= 1000) SMSW

"fast" (= 1000) "fast" (= 500) "slow" (= 500)

Fig. 2.7. Dispersion characteristics of slow and fast hybrid waves. In calculation is used a500m, b20m, d500m, H = 1500 Oe,  1000.