LAPPEENRANTA UNIVERSITY OF TECHNOLOGY FACULTY OF TECHNOLOGY
Master’s Degree Programme in Technical Physics
Nikitin Aleksei
Spin-electromagnetic waves spectra in multiferroic heterostructures
Examiners: Professor Erkki Lähderanta Professor Alexey B. Ustinov Supervisors: Professor Erkki Lähderanta
2 ABSTRACT
Lappeenranta University of Technology Faculty of Technology
Master’s Degree Programme in Technical Physics Nikitin Aleksei
Spin-electromagnetic waves spectra in multiferroic heterostructures Master’s Thesis
2015
75 pages, 41 figures and 2 appendices.
Examiners: Professor, Erkki Lähderanta Professor, Alexey B. Ustinov
Keywords: Ferroelectrics, ferrites, spin wave, electromagnetic wave, dispersion relation This work devotes to the theoretical investigations of spin-electromagnetic waves (SEW) propagating in a thin-film multiferroic structures that were composed of a slot-line and structures with several ferrite films. In contrast to earlier works, the spin-electromagnetic waves in the investigated structures are originated from two different electrodynamics coupling. The first one is coupling of the electromagnetic wave localized mainly in the slot-line with the spin wave excited mostly in the ferrite film. The second one is coupling of two spin waves in the different ferrite films separated by a thin ferroelectric film. For theoretical analysis of SEWs propagation in such kind of structures theories of their eigen- wave spectra were developed. Spectra of SEW in the investigated structures were calculated and analyzed. The range of electric and magnetic tunability of dispersion characteristic were investigated. Spectra of SEW in the investigated multiferroic structures are used for investigation of transfer function of periodic structures.
3 ACKNOWLEDGEMENTS
This master’s thesis was carried out in the laboratory of Physics, Lappeenranta University of Technology in cooperation with Department of Physical Electronics and Technology, Saint Petersburg Electrotechnical University "LETI".
I wish to express my gratitude to Professor Erkki Lähderanta, Professor Alexey B. Ustinov for their guidance and support.
I also wish to express my gratitude to my family and friends for their support. Special thanks to Andrey Nikitin for great help.
Lappeenranta, May 2015 Aleksei Nikitin.
4 SYMBOLS
Roman letters
H magnetic field strength E electric field strength M magnetization
f frequency
k propagation constant
Greek letters
permittivity μ permeabilityκ transverse wave number ω angular frequency ω𝐻 = 𝜇0𝛾 𝐻
ω𝑀 = 𝜇0𝛾 𝑀
ω⊥ = ω√ω𝐻(ω𝐻+ ω𝑀)
magnetic permeability tensorAbbreviations
MW microwave
YIG yttrium iron garnet
GGG gadolinium gallium garnet BST barium strontium titanate SW spin wave
EMW electromagnetic wave FF ferrite-ferroelectric
SEW spin-electromagnetic wave SSW surface spin wave
SVSW straight volume spin wave IVSW Inverse volume spin wave MC magnonic crystal
5 TABLE OF CONTENTS:
Introduction ...6
1. Dielectrics in microwave electronics. ...8
1.1. Ferrites ... 8
1.2. Ferroelectrics ... 15
1.3. Multiferroics ... 22
1.4. Periodical structures based on dielectrics in MW electronics ... 25
2. Investigation of spin-electromagnetic waves spectra in multilayered multiferroic structures ... 29
2.1. Theory of spin-electromagnetic waves spectra propagating in the multilayered multiferroic structures. ... 29
2.2. Investigation of spin-electromagnetic waves spectra in thin-film multilayered ferrite-ferroelectric structures. ... 32
3. Investigation of spin-electromagnetic waves spectra in multiferroic structures based on a slot-line. ... 45
3.1. Theory of spin-electromagnetic waves spectra propagating in multiferroic structures based on a slot-line. ... 45
3.2. Investigation of spin-electromagnetic waves in thin-film ferrite-ferroelectric structures based on a slot-line. ... 53
3.3. Investigation of spin-electromagnetic waves in periodic multiferroic structures based on a slot-line with width modulation. ... 58
Conclusion ... 67
References: ... 69
Appendix 1. ... 73
Appendix 2. ... 74
6
INTRODUCTION
A artificial materials that combine properties of several natural materials has attracted interest during last few years. Multiferroics combines properties of ferrites and ferroelectrics. Such materials are promising to develop of microwave (MW) devices as phase shifters and resonators.
Ferroelectric materials are widely used in modern MW devices due to their dependence of dielectric permittivity from bias electric field. This phenomenon allows to control operation characteristic of such device by mean of electric field [1]. The advantages of this method are high speed of operation and low power consumption. At the same time distinguished features of the ferrite material devices are low insertion losses and magnetic field tunability in a wide frequency range. Performance characteristics of such devises are determined by the dispersion of spin waves (SWs) [2, 3].
Owing to development of the thin-film deposition techniques a strong interest for fabrication and investigation of thin-film multilayered structures takes place. Thus, several works [4, 5] reported an analysis of the SW spectrum of a multilayered structure consisting many ferrite layers separated by a gap filled by a nonmagnetic medium. This analysis was carried out following the Damon and Eshbach theory. It should be noted that in this case theoretical results have been obtained in the magnetostatic limit i.e. an electromagnetic retardation was neglected. However in such structures only magnetic field tunability can be realized.
Development of frequency-agile materials for microwave devices has led to appearance of artificial multiferroics [6]. The multiferroics are usually fabricated by a combination of ferrites and ferroelectrics materials so as to obtain structures in the form of multilayers, pillars, spheres, wires and other [7]. A distinctive feature of the multiferroics is dual tunability of their physical properties. The tunability exists basically due to two effects.
The first one is magnetoelectric effect based on mechanical interaction between ferrite and ferroelectric crystal lattices [8]. The second one is effect of electrodynamic coupling of SW and electromagnetic waves (EMWs) in the ferrite-ferroelectric structures [9]. These
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coupled waves are called hybrid spin-electromagnetic waves (SEWs). Well known that necessary condition of this phenomenon is an equality of SWs and EMWs phase velocity for intersection of dispersion characteristics. This condition imposes a thickness limit (200- 500 μm) of ferroelectric layers so as to provide effective hybridization of the spin waves with the electromagnetic waves at the microwave frequencies (C – Q bands). Therefore, relatively high voltage (up to 1000 V) is necessary to apply to the ferroelectric layer for effective electric tuning of the SEW dispersion. Currently, the most urgent task is to find ways to reduce the control voltage. It can greatly expand the possibilities of using ferrite- ferroelectric (FF) structures in microwave devices.
The purpose of current work is the investigation of spin-electromagnetic waves propagating in the thin-film multiferroic structures that were composed of a slot-line and structures with several ferrite films. In contrast to earlier works, the spin-electromagnetic waves in the investigated structures are originated from two different electrodynamic coupling. The first one is coupling of the electromagnetic wave localized mainly in the slot-line with the SW excited mostly in the ferrite film. The second one is coupling of two SWs in different ferrite films separated by a thin ferroelectric film. For theoretical analysis of SEWs propagation in such kind of structures theories of their eigen-wave spectra will be developed. Spectra of SEW in the investigated structures will be calculated and analyzed. The range of electric and magnetic tunability of dispersion characteristic will be investigated. Spectra of SEWs in the investigated multiferroic structures are used for investigation of transfer function of periodic structures. This task is of interest from both fundamental and practical point of view, that is, to create a new tunable MW devices.
8
1. DIELECTRICS IN MICROWAVE ELECTRONICS
1.1. Ferrites
Ferrites are polycrystalline magnetic oxides that can be described by the general chemical formula XO•Fe2O3, where X is a divalent ion such as CO2 or Mn2. Since these oxides have much lower conductivity than metals, MW signals can easily pass through them relating with small insertion loss.
Magnetic films, such as epitaxial yttrium iron garnet (YIG) films, are attractive for development of novel microwave devices. It is due to relatively low SWs propagation losses of less than 30 dB/μs and simple realization of spin-wave phase shift tuning by magnetic field.
It is worth noting another important feature of ferromagnetic materials: the dependence on the magnetic permeability of the applied magnetic field. It means that if the ferrites are placed in varying external magnetic field it becomes possible to control its magnetic characteristics. The advantage of this effect is to control the dispersion of SEWs or SWs in a relatively wide range of wavelengths. However, the magnetic tuning has some shortcomings, for example, low rate of tuning (microseconds), power consumption and the large size of the control systems.
Consider now in more details the principles of MW ferrite devices. At these frequencies, ferrites are used at a relatively constant or slowly changing magnetic field. If an alternating magnetic field and variable magnetization are sufficiently small, complex amplitudes between their constituents should have a linear relation:
𝑚 = 𝜒̂ ∙ ℎ, (1.1)
where m is the alternating magnetization, 𝜒̂ is the tensor of magnetic susceptibility and h is the alternating magnetic field and. The components of this tensor depend on the frequency of the alternating field ɷ and the constant magnetic field H0. A characteristic
9
feature of this tensor is its asymmetry. Therefore it is possible to produce a MW device that do not satisfy the reciprocity principle. The second feature of the magnetic susceptibility tensor of magnetized ferromagnetic is ferromagnetic resonance. The essence of the above phenomenon is that some components of the tensor 𝜒̂ depends on the values of ɷ and H0. There are maxima of the imaginary parts of these components at a certain ɷ (if H0 is constant) or H0 (at constant ɷ), which corresponds to the peak of magnetic losses in the material.
It is worth noting that the influence of tensor’s asymmetry on wave’s propagation was first investigated in Polder’s work [10]. For an infinite medium with a dielectric permittivity wave equation has the following form:
𝛥𝐻 − 𝑔𝑟𝑎𝑑 𝑑𝑖𝑣 𝐻 − Ɛ
𝑐2
𝜕2𝐵
𝜕𝑡2 = 0, (1.2)
where 𝛥𝐻 is Laplace operator of the external magnetic field and B is magnetic induction.
This equation follows from Maxwell's equations. It is known that condition div H = 0 is not satisfied since the magnetic permeability is not isotropic. It is possible to get three homogeneous conditions during the simplification of the equation (1.2). These conditions have a solution only if the determinant is equal to zero. Thus it is possible to obtain the relation between k and ɷ which is called the dispersion relation. This was done in [11] for ferrite layer in which the wave propagates in the direction of the magnetization (kx = ky = 0, kz =k):
𝑘2 =ɷ2Ɛ
𝑐2 (1 + 4𝜋𝑀
𝐻+𝐻𝑎𝑛±ɷ 𝛾⁄ ), (1.3)
where ɷ is the angular frequency, Ɛ is the permittivity, с is the speed of light, M is the magnetization, 𝛾 is the gyromagnetic ratio and 𝐻𝑎𝑛 is the angular momentum created by the static magnetic field.
It was shown in [12] that the solution of the equation (1.3) has two types of waves:
magnetostatic waves (MSWs) (for which the exchange energy can be neglected), and SWs
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(for which electromagnetic energy can be neglected and exchange energy is essential). The dispersion relations of MSWs and SW are shown in figure 1.1. It is worth noting that wave number’s scale for SW is at least 106 times less than in the case of MSW.
Figure 1.1. Dependencies of a frequency on a wave number for SW (a) and MSW (b) [12].
Thus, several types of interactions define the mechanism of wave process in ferromagnetic materials. The first one is due to the exchange interaction of spin magnetic moments (spins), which are related and oriented parallel to each other. If any of them will change their position, then there will be a disturbance, which is transmitted to neighboring spins.
So the above-mentioned SW is implemented. The second one is due to the long-range strengths (in contrast to the exchange interaction) of the magnetic dipole interaction. To describe this effect dipoles should be considered. Attempting to reposition of any of them in the system leads to the disturbance in the system, which begins to spread through the system of dipoles.
It can be concluded that the range of possible wave numbers k can be divided into special areas. Vibration is mixed magnetostatic modes in the case of k » 10-1 cm-1. Exchange interaction does not render influence on vibration spectrum, but it is necessary to consider boundary conditions and spreading effect. Normal vibrations represent magnetic waves in
11
"magnetostatic" approach for the case of 10 < k < 103 cm-1. Dipole and partially exchange interactions play essential role. Normal vibrations are short waves "exchange" SW for the case of 103 < k <105 cm-1. Essential role play dipole and exchange interactions, but SW spreading effect is insignificant.
Two first ranges of wave numbers are most interesting for the purposes of this work.
Therefore, advantages of exchangeless approach can be taken. Solution of magnetostatic equations gives three types of SW [13].
1. Conditions for surface SW (SSW) propagation: magnetic-field vector H0 is in a plane of spatially homogeneous ferrite film and perpendicular to SW propagation direction (H0 k). Propagation of SW is not reciprocal. It means that at different direction of traveling spin waves spins are pressed on different sides of plate surfaces.
2. Straight volume SW (SVSW). Vector H0 is perpendicular to film plane and to wave vector (H0 k). There is no allocated direction in plane plate for SVSW. Direction kand group velocity Vg coincide.
3. Inverse volume spin wave (IVSW). The wave is formed, when vector Н0 lies in the film plane and is parallel to the wave vector (Н0 || k), directions of vectors of the phase and group velocities are opposite.
Each type of the SW has a following dispersion relations:
SSW ɷ2 = ɷ⊥2 +ɷ𝐻2
4 (1 − 𝑒−2𝑘𝐿) (1.4)
SVSW ɷ2 = ɷ𝐻(ɷ𝐻 + ɷ𝑀− ɷ𝑀(1−𝑒−𝑘𝐿
𝑘𝐿 )), (1.5)
IVSW ɷ2 = ɷ𝐻(ɷ𝐻 + ɷ𝑀(1−𝑒−𝑘𝐿
𝑘𝐿 )), (1.6)
12
In the above formulas, L is the ferrite film thickness and ɷ⊥2 = ɷ𝐻(ɷ𝐻+ ɷ𝑀).
Ratios (1.4) - (1.6) allow determining the limits of the existence of the main types of SW:
SSW
,
2
M
H
(1.7)
SVSW and IVSW . (1.8)
Figure 1.2 shows the dispersion relations of fundamental modes SWs for different directions of the constant magnetic field:
Figure 1.2. Dispersion relations of fundamental modes SWs for SSW (a), SVSW (b), and IVSW(c) [14].
SSW has property of non-reciprocity, which is shown that the waves running in opposite directions are pressed to the different sides of the ferromagnetic film. Magnetization decreases exponentially when moving from the film surface. Thin metal conductors are usually used for the SW excitation in the ferromagnetic films. This conductors locate on the surface of the film or near to it. Simple exciting element (antenna [15-17]) is a segment of microstrip line. The k value of exciting SW is defined by cross-section sizes of microstrip antenna and SWs dispersion law. In film structures, in contrast to bulk samples, can be avoided difficulties connected with excitation reproducibility and reception MSW,
13
caused by heterogeneity of internal static magnetic field. The main sources of losses at propagation of magnetostatic wave are relaxation process in YIG and diffraction effects.
The mechanisms of losses will not be consider in this work.
Consider now a design of practical ferrite devices. These may be delay line, circulators, phase shifter, etc. One of the most common example of the spin-wave device on the ferrite film is a phase shifter or a delay line, which are shown in figure 1.3. MW phase shifters are widely used in phased array antennas and other radar and telecommunication systems [18] as tunable elements.
Figure 1.3. The topology of the spin wave phase shifter and delay line [19, 20].
Design of the SW phase shifters are identical to the design of the delay lines. The basic operating characteristic of the phase shifter is the dependence of the phase variation introduced by external magnetic field that determined by SWs dispersion law. The dispersion law of SWs determines both of these characteristics. Indeed, the dispersion law of SWs and distance between the antennas determine the dependence of the phase shift of the frequency.
The SW phase shifter consists of the free (unshielded) ferromagnetic film “2” grown on a dielectric substrate “1” and the input/output antennas “3” fitted by microstrip lines “4”.
Antennas with a supply lines are typically created on a dielectric substrate “5” metallized from the reverse side “6” [19, 20].
An example of another method of using ferrite materials for microwave device is a MSW signal-to-noise enhancer. It is a non-linear frequency selective device, which has a high
14
loss to small signals, or noise, and a reduced loss to large signals above threshold (figure 1.4). It consists of a two-port transmission line capable of launching magnetostatic surface waves into an YIG or ferrite film. Approach to achieving broad band operation is described which relies on surface wave propagation on a metallized YIG surface [21].
Figure 1.4. Measured attenuation versus frequency with different input power levels for a slot- line enhancer and a 75 μm thick YIG film [21].
The calculated variation of frequency with wave number for a surface wave propagating on the metallized surface is shown in figure 1.5 for several different values of the spacing t between the metal layer and the YIG surface. The film thickness was 75 μm, the magnetization was 1800 G and the internal field was 500 Oe. It is seen that even a 10 μm separation of the YIG surface from the metal layer produces a significant reduction in the available bandwidth.
15
Figure 1.5. Calculated frequency versus wavenumber for surface wave propagation on a 75 μm thick YIG film with the spacing (t) of a metal layer from the film as a parameter. The
magnetization was 1800 G and the internal field 500 Oe [21].
It can be concluded that small phase and group velocities, variety of dispersion characteristics, ease excitation and reception and low propagating losses are advantages of SW. The main advantage is possibility of non-mechanical tuning. This tuning is implemented by change of the external magnetic field. Therefore it can be called magnetic tuning. Magnetic tuning is easily implemented in wide frequency range. As a result SW has found wide application in variety of linear and nonlinear devices for example phase shifter, delay lines, filters, resonators, and etc. [22,23] Main disadvantage of such devices, preventing of their development, is connected with magnetic subsystem, because it is slow and power consuming.
1.2. Ferroelectrics
The permanent electric dipole moment possessed by all pyroelectric materials may, in certain cases, be reoriented by the application of an electric field. Such crystals are called ferroelectrics, a term first used by analogy with ferromagnetism. Ferroelectric crystals are divided into domains (a region within a material in which vector of the spontaneous
16
polarization is in a uniform direction) in the absence of an external electric field. The reason for the formation of the domains is exchange forces resulting from the merging of electrons, which belong to neighboring atoms. These forces operates on the interatomic distances.
The existence of domains is due to principle of a minimum internal energy.
It is worth noting that in ferroelectric crystals the domains can be oriented predominantly in one direction by a strong external electric field. Reversing external field reverses the predominant orientation of the ferroelectric domains, though the switching to a new direction lags somewhat behind the change in the external electric field. This lag of electric polarization behind the applied electric field is ferroelectric hysteresis, named by analogy with ferromagnetic hysteresis.
All ferroelectric crystals are necessarily both pyroelectric and piezoelectric. Many of them lose these polar properties at the transition or Curie temperature Tc. A nonpolar phase above Tc is the so-called paraelectric phase. Ferroelectric single crystals grown in the absence of an electric field are inevitably electrically twinned with a domain volume. Ceramics have similar domain structures superimposed on the more general orientation disorder associated with polycrystalline materials. Exposure to an electric field under appropriate conditions can result in complete or partial realignment of the spontaneous polarization. A net dipole moment is not normally detectable in such materials because the surface charges are rapidly neutralized by ambient charged particles.
Ferroelectric materials are widely used in MW electronics. One of the main reasons for this is the strong dependence of their dielectric permittivity Ɛ on the applied external electric field and temperature. These phenomena are shown in figure 1.6 for typical ferroelectrics.
17
Figure 1.6. Dependence of dielectric permittivity Ɛ on the applied external electric field at different temperature for:
a) strontium titanate (SrTi03); b) barium strontium titanate (Ba0.6, Sr0.4)TiO3; c) barium titanate (BaTi03) [24].
Single crystals of barium titanate (BTO) BaTiO3 exhibit typical behavior of a displacive ferroelectric material, which means that the dielectric behavior of BTO close to the ferroelectric phase transition is fully controlled by the soft mode dynamics which slows down close to the transition temperature. In strontium titanate (STO) SrTiO3 a strong soft mode is also observed, which exhibits a frequency decrease (softening) upon cooling;
however, the compound remains paraelectric down to 0 K due to quantum fluctuations.
Barium strontium titanate (BSTO) BaxSr1-xTiO3 is solid state solution of STO and BTO, where x is stoichiometric ratio. Therefore this material change properties from BTO to STO dependending on x. BSTO is an extremely attractive candidate for many ferroelectric applications. It is well known that BSTO has exceptionally high tunability, high breakdown field and relatively low loss tangent at MW frequencies [25].
18
BSTO has perovskite structure, which is shown in the figure 1.7 (the arrows indicate the eigenvector of the soft mode). It derives its high dielectric constant from an ionic displacement and therefore differs from lower dielectric constant materials that only experience an electronic displacement with changing applied voltage. The Ti ions are each surrounded by six oxygen ions. At zero applied voltage, the Ti ions are centered in the oxygen octahedron and the dipole moments cancel. With the application of a voltage, the Ti ions will be displaced and a dipole moment will be induced.
Figure 1.7. Elementary cell of (Ba,Sr)TiO3.
Ferroelectrics in thin-film form are of particular interest in MW tunable applications. For example, this can be seen from BSTO materials. Significant differences are often observed between BSTO bulk single crystals and thin films. The permittivity of the thin films is somewhat lower than in single crystals and the losses are frequently higher. It is worth noting that the dielectric losses are relatively low value at a temperature above the phase transition temperature Tc at which there is the maximum permittivity. Spontaneous polarization is occurred in the ferroelectric at a temperature below Tc. This state of the material is called ferroelectric phase. In addition, at T≫Tc spontaneous polarization is not exist and therefore the state of ferroelectric is in the paraelectric phase. The widespread use of ferroelectric in the MW range is mainly possible in the paraelectric phase, because the dielectric losses in the material are substantially less. Also dependence of dielectric permittivity Ɛ on the applied external electric field is rather high in this temperature range.
19
Based on this phenomenon, it is possible to produce electrically controllable capacitor, waveguide or other transmission line. Phase shifters based on ferroelectric film have received individual mention for its ease of assembly, large working capacity and low loss.
One way to create a phase shifter is to use the structure based on the slot transmission line.
Consider now in more details the wave processes in the slot transmission lines. Main electromagnetic mode in slot-line consists from all six field components and therefore cannot be ascribed to well-known fields E or H type [26]. But if electrodes (that are forming line) absences dielectric layered structure supply surface wave of hybrid type. Therefore fundamental problem about dispersion characteristics of slot-line is necessary to formulate in full wave variant.
In works [27 - 29] were chosen the longitudinal-section magnetic (LSM) and longitudinal- section electric (LSE) basis [30] as basis of expansion slot-line mode. Total of LSE and LSM fields that forming full orthogonal basis allow to describe slot-line wave structure. It is more obvious if width of slot is approaching to width of wave guides. Then they can be a partially filled rectangular wave guides. To describe this wave guide is usually used LSE and LSM basis [30]. In the slot-line LSE and LSM fields connect by currents on electrodes in slot electromagnetic mode.
Well known slot-lines are formed by two external electrodes on the surface of the dielectric plate [31]. Figure 1.8 show one possible way to implement such structures.
20
Figure 1.8. Slot transmission line on a dielectric substrate [31].
The propagation of EMWs in the slot-line on the dielectric substrate were described in [31].
Thus, for example, experimental slot-line wavelength ratios with different gap width are shown in figure 1.9, where λ’
is slot-mode wavelength
.Figure 1.9. Experimental slot-line wavelength ratio with different gap width [31].
21
As shown in figure 1.9, the value of the wavelength in a slot-line decreases with decreasing gap width.
On another hand ferroelectric materials can be used in slot transition lines because of their permittivity, its external field dependence and dielectric losses as it was shown earlier.
Fabrication of this structure is to create a waveguide slot-line with a layer of ferroelectric films of BSTO Ba0.5Sr0.5TiO3. This film may be produced by reactive magnetron RF sputtering. Slot-line electrodes are obtained by vacuum thermal evaporation. Theoretical investigations of such kind of structure (figure 1.10) was done in [32].
Figure 1.10. Cross-section of a slot-line based on ferroelectric layer [32].
As it was shown in figure 1.10 one can mark three regions, which form dielectric layers at a cross-section of a slot-line:
- the first one is a ferroelectric with permittivity Ɛ1;
- the second one is a dielectric with permittivity (Ɛ2) 10;
- the third one is a substrate with permittivity (Ɛ3) 1.
22
Results of numerical solution of dispersion equation is found in [32] and they are shown in figure 1.11.
Figure 1.11. Dependences on propagation constants in a slot-line by permittivity of ferroelectric layer with different gap width for (a) 30 GHz and (b) 60 GHz.
As seen from this figure propagation constant k increases with the increase of permittivity of ferroelectric layer with different gap width 0.01 mm and 0.05 mm.
It can be concluded that the main disadvantage of ferroelectrics is relatively small ratio of tuning in compared with ferrites.
1.3. Multiferroics
As was mentioned in previous sections a ferroelectric crystal exhibits a stable and switchable electrical polarization that is manifested in the form of cooperative atomic displacements. A ferromagnetic crystal exhibits a stable and switchable magnetization that arises through the quantum mechanical phenomenon of exchange. Thus, there are very few 'multiferroic' materials that exhibit both of these properties, but the 'magnetoelectric' coupling of magnetic and electrical properties is a more general and widespread phenomenon (figure 1.12).
23
Figure 1.12. The relationship between ferromagnetic and ferroelectric materials [33].
It is worth noting that the first multiferroics were discovered in the early 1960s [34]. There are two main research directions of the multiferroic structures. The first one is a study of single-phase natural multiferroic materials, in which ferromagnetism and ferroelectricity arise independently. These exist, but thay are rare [35]. The second direction is a development and study of the artificial composite multiferroic structures [36-38]. Such structures can be made by combination of ferromagnetic and ferroelectric (or piezoelectric) materials into the layered or other kind of structures [37]. It became clear in the last few years that artificial multilayered multiferroic structures are attractive for MW applications due to high magnetoelectric performance in comparison with natural multiferroic materials [38].
In the artificial multiferroic structures the interaction between the ferromagnetic and ferroelectric or piezoelectric phases may be due to two effects. The first one is the magnetoelectric effect. This phenomenon is the possibility to induce a polarization by application of a magnetic field (linear effect) or for instance by applying simultaneously an electric and a magnetic field (non-linear effect) [39]. The reverse effect is that a magnetization can be induced by an electric field (linear effect) or by application simultaneously, for instance, of an electric and a magnetic field (non-linear effect).
The second effect is due to the electrodynamic interaction between the microwave electromagnetic and spin waves in the layered ferrite-ferroelectric structures [40]. The
24
interaction leads to a formation of SEWs which is shown in the figure 1.13. Spectrum of the waves is dually controllable by both electric and magnetic fields. Electric tuning of the SEW spectrum is possible due to a dependence of dielectric permittivity from bias electric field whereas magnetic tuning is provided by a dependence of magnetic permeability from bias magnetic field as it was shown in previous sections.
Figure 1.13. Spectrum of the waves in the layered ferrite-ferroelectric structures [40].
In practice, two types of waves are usually used for implementing the spin wave devices:
longitudinal SW propagating along the direction of a constant magnetic field (φ = 0) and the transverse SW propagating at a right angle to the direction of a constant magnetic field (φ = π / 2). The last one is the most promising SW in terms of tuning the dispersion with the dielectric constant of ferroelectric varying.
It can be concluded that thickness of ferroelectric layer must be 200-500 μm to provide electrodynamic interaction in S – Q bands frequencies. Therefore, relatively high voltage (up to 1000 V) is necessary to apply to the ferroelectric layer for effective electric tuning of the SEW dispersion. This disadvantage may be solved by using slot-line based on thin- film ferrite (as it was mentioned in chapter 1.2) and by using multilayered structures consisted of several thin-film ferrites (as it was mentioned in chapter 1.1).
25
1.4. Periodical structures based on dielectrics in MW electronics
Magnonic crystal (MC) is an artificial pattern periodical magnetic structure which has band gaps in the SW spectrum. One of the simplest type of a MC is a periodical structure based on thickness-modulated ferrite film (figure 1.14).
Figure 1.14 Schematic illustration of the MC based on thickness-modulated ferrite film.
It is well known that one of the determining factors for investigation of various linear and nonlinear spin-wave phenomena in ferromagnetic films is the dispersion law [41]. In order to investigate properties of MC calculations are usually made for structures with an infinite number of periods N. The theory predicts presence of allowed bands and band gaps in the SW spectrum. Allowed bands correspond to the frequencies at which the SWs can propagate with small losses. Band gaps correspond to the frequencies at which propagation of SWs is impossible. According to this fact there are gaps on the dispersion characteristics of MC. However instead of band gaps appear sections with a relatively high damping of SWs in the case of MC consisting of a limited number of periods. This phenomenon is shown in the figure 1.15 for different number of periods N [42].
YIG GGG
26
Figure 1.15. Dispersion characteristics of SWs in MC for 1) N=1; 2) N=15; 3) N=1000 [42].
Note that these dispersion characteristics were obtained without losses. It means that attenuation coefficient was equal to zero. Despite the fact that in the band gap the SW transmission coefficient is relatively small, however they propagation is still possible.
Therefore the frequency dependence of the transmission coefficient of SWs (frequency response) for MC with finite length is characterized by alternating bands of transmission with small and large losses (figure 1.16).
Figure 1.16. Dependences of the transmission coefficient (S21) on a frequency [42].
27
It can be concluded from this section that there are band gaps in dispersion characteristics of periodical structures where propagation of wave is impossible. However despite the variety of existing approaches [43-45] to describe the properties of the MC a theoretical study of the propagation of SEWs in thin-film multiferroic periodical structures based on a slot-line was not carried out. Therefore investigation of thin-film multiferroic structures based on a slot-line withfinite number of periods is promising idea for developing of new MW devices.
28 Conclusions from the first chapter:
Nowadays an interest to study composite materials with the novel properties is evident.
Artificial multiferroic structures are attractive for various applications, in particular for frequency agile microwave devices. A distinctive feature of the multiferroic materials is dual electric and magnetic tunability of their physical properties. One way to create structures with the multiferroic properties is to use multilayered structures composed of ferrite and ferroelectric materials. In these structures the multiferroic properties are due to a hybrid nature of their eigen-wave excitations, named spin-electromagnetic waves (SEWs). They are formed as a result of electrodynamic coupling between the spin waves localized mainly in the ferrite layer and the electromagnetic waves propagating mostly in the ferroelectric layer. Electric tuning of the SEW spectrum is possible due to a dependence of dielectric permittivity of ferroelectric layer from bias electric field whereas magnetic tuning is provided by a dependence of magnetic permeability from bias magnetic field.
There are several types of thin-film multiferroic structures for microwave applications in C- band and Q-band. The most effective structures in these frequency ranges are based on the slot transmission line and several ferrite films. Also there is possibility to change spectrum of SEW propagating in investigated structures by using width-modulated configuration. In this case dependences of the transmission coefficient on a frequency have band gaps corresponding to the frequencies at which the wave cannot propagate.
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2. INVESTIGATION OF SPIN-ELECTROMAGNETIC WAVES SPECTRA IN MULTILAYERED MULTIFERROIC STRUCTURES
This chapter is devoted to the investigation of the hybrid SEWs spectra in an arbitrary sequence of the ferrite layers separated by the dielectric (vacuum or ferroelectric) layers.
A theory of SEW spectrum in a multilayered structure consisting of infinite numbers of ferrite and dielectric layers is developed. A numerical modeling of dispersion characteristics of such waves in structures with different parameters is carried out. In contrast to previous works it is shown that using thick ferroelectric layer placed between two ferrites layers makes possible dual tunability of dispersion characteristic.
2.1. Theory of spin-electromagnetic waves spectra propagating in the multilayered multiferroic structures
It is known that SWs propagating perpendicular to the static magnetic field in a tangentially magnetized ferrite layers have effective hybridization only with TE mode of EMWs spectra. More effective interaction of SWs and EMWs is achieved in asymmetrical layered structures. Therefore it is worth to consider the electrodynamic model of wave propagation along the ferrites – ferroelectrics boundary. As it was mentioned earlier the exchangeless approach is actually only for long SW (k
< 10
2cm
-1). This interval of k is the most interesting for the purposes of this work so it will be used further in the exchangeless approach.Let’s derive a dispersion relation of SEWs in multilayered ferrite-ferroelectric structure shown in figure 2.1. This structure consists of N ferrite layers stacked along the z-axis.
Ferrite layers have different thickness a2i, magnetic permeability 2i and different saturation magnetization M2i. Each ferrite is surrounded by the dielectric layers with thickness a2i-1, electric permittivity 2i-1, where i is any integer from 1 to N. Thereby even 2i correspond to ferrite layers and odd 2i-1 correspond to ferroelectric layers. Any single layer or multiple dielectric layers can be ferroelectrics through high dielectric constant.
This corresponds to a ferroelectric in paraelectric phase. Investigated structure is infinite
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in the xz plane and is magnetized to the saturation along z-axis. It can be assumed that EMWs propagate along the x-axis that is perpendicular to the external magnetic field H0 in order to execute conditions for the surface SW propagation.
Figure 2.1. Ferrite-ferroelectric (dielectric) multilayered structure.
The infinity along z axis gives possibility to divide EMWs in this structures into TE and TM modes with the following field structures E(0,0,Ez), H(H Hx, y, 0) and E(E Ex, y, 0), H(0,0,Hz), respectively. It is well known that TE modes of electromagnetic waves and surface SWs have close field structures that lead to the stronger interaction between these waves.
Further, only TE modes of the EMWs will be considered.
The magnetic permeability tensor for the ferrite layer has the following form [46]:
2 2
2 2 2
0 0 ,
0 0 1
i i
i i i
j j
where
2 2
2 2 2
( )
H H M i ,
i
H
2 2i 2 M i2,
H
H g 0H0, M i2 g0M2i,
2.8 1010
g s-1·T-1 is gyromagnetic ratio, 0 4 107H·m-1 is a vacuum permeability, H0
is external magnetic field and M2i is saturation magnetization of ferrite layer with 2i number.
(2.1)
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The Helmholtz equations can be found from the Maxwell equations for each layer. Solutions of these equations are shown below (multiplier ej( t kx) is dropped from consideration, but is taken into account):
– for free space above and below the structure, respectively
{𝐸𝑧 = 𝐴0𝑒𝜅0𝑦, − ∞ < 𝑦 < 0
𝐸𝑧 = 𝐴2𝑁+1𝑒−𝜅0𝑦, 𝑎1+ 𝑎2+ ⋯ + 𝑎2𝑁+1 < 𝑦 < ∞,
(2.2) where 0 k2 2 0 0 .
– for the ferrite layers and for the dielectric layers, respectively
2 2 2 2 2 2
( sin ( ) cos ( )),
z i f i i i f i i
E A y D B y D (2.3)
2 1 2 1 2 1 2 1 2 1 2 1
( sin ( ) cos ( )),
z i i i i i i
E A
y D
B
y D
(2.4)wheref i2 2 0 0 2i 2i k2 , 2i ( 2i2 2i2) /2i, 2 1i 2 0 0 2 1i k2 ,
1 i
i m
m
D a
, Ɛ0=8.854∙10−12 F/m is a vacuum permittivity.
The Maxwell’s equations were used in order to express tangential magnetic field components (Hx) through the tangential electric field components (Ez). Therefore, the relations for the free space and each dielectric and ferrite layer were obtained. Substitution of these relations into the electrodynamic boundary conditions at the layer interfaces gives a system of linear algebraic equations. Therefore, unknown coefficients in the equations (2.2-2.4) for all layers are connected with each other by the boundary conditions. Such connection between the coefficients gives possibility to express the coefficients in the i- layer trough coefficients in the neighboring layers:
,
T ( 1), ( 1) T ,i A Bi i iAi Bi
P Q
(2.5)
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where Pi and Qi are matrixes, that connect unknown coefficients in the neighboring layers with numbers i and i+1. These matrixes were called transfer matrixes.
Finally, continuity of all tangential components of the fields gives possibility to express coefficient A0 in the free space below the structure through An in the free space above:
A A0,
0
T A An,
n . (2.6)
Detailed breakdown of equation (2.6) is given in Appendix 1.
Thus, dispersion relation f( , ) 0 k of SEW can be expressed from the relation (2.6) in the following form:
11 12 21 22
( , ) T +T T T ,
f k
(2.7)Dispersion relation (2.7) describes spectra of the SEWs in the ferrite-dielectric structures with arbitrary filling of the ferrite and dielectric layers. Note that this dispersion relation could be easy modified in the case of presence one or two metal screens below and/or above ferrite-dielectric structures by change form of P0 and Q2N+1 matrixes, respectively.
2.2. Investigation of spin-electromagnetic waves spectra in thin-film multilayered ferrite- ferroelectric structures.
The theoretical model described above can be used to determine the spectra of the
SEWs in structures that consist of two ferrite films separated by relatively thin
ferroelectric layer. Let’s begin the investigation from the simplest case. Consider
now two ferrite layers separated by free space. It will give an opportunity to explain
complex cases.
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Investigated structures that are consisted of two ferrite films on the dielectric substrates are shown in the figure 2.2.
Figure 2.2. Two layered ferrite structure.
As it was discussed in the first chapter one of the most popular thin film ferrite structures for the MW application is a single crystal YIG film on a gadolinium gallium garnet (GGG) substrate. Therefore, usual parameters for such materials were used in the calculations.
Dispersion characteristics of surface SW were considered in the two cases. The first one is structure composed of the ferrite films with equal parameters (a2=a4= 20 m, M2=M4= 1750 G). The second one is the structure based on the films with different thicknesses and saturation magnetizations. The most interesting case is intersection of dispersion characteristics of the surface SW in different films. In order to satisfy this condition following parameters a2= 20 m, M2 = 1750 G, a4=6 m, M4= 1790 G were chosen.
Dispersion characteristics of the SW in the first case at different distance between ferrite films (a3) are shown in the figure 2.3.
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Figure 2.3. Dispersion characteristics of the surface SWs in the two ferrite films with equal parameters separated by free space a3.
As it can be seen from this figure if the distance has devoted to an infinity then SWs propagate independently and dispersion curves coincide (black solid line). If the distance has decreased then waves interact and dispersion curves repulse (red dash line). As would be expected, interaction attenuates with the decrease of wavelength. Interaction between SWs and repulsion of the dispersion curves increases at the decrease of the distance (blue dash dot line). Finally, dispersion curve corresponds with dispersion characteristic of the SWs in the ferrite film with double thickness (green dot line).
The same calculations were repeated for the second case of the different ferrite parameters (figure 2.4). Like in the previous case, SWs are not interacting and propagating independently at the infinity apart from each other. Therefore, dispersion curves intersect each other (black solid line). And again if the distance between the films has decreased then waves interact near intersection point (red dash line). As it was mentioned above, interaction increases with approach of the films (blue dash dot line). Dispersion curve at a3=0 is shown in the figure 2.4 by green dot line.
100 200 300 400
6.2 6.3 6.4 6.5 6.6
Frequency, GHz
Wave numbers (cm-1 )
a3=
a3= 100 m a3= 25 m a3= 1 m a2=a4= 20 m;
M2=M4=1750 G;
35
Figure 2.4. Dispersion characteristics of the surface SWs in the two ferrite films with different parameters separated by free space.
It can be concluded that that two ferrite layers placement leads to the splitting of the fundamental mode into two branches. Besides interaction between SWs in different ferrite layers repulses these branches. Insertion of the ferroelectric layer between ferrites leads to complication of the spectra, because it is formed by several hybridizations. Due to this fact the spectra have three dispersion branches. In order to investigate the SEWs spectra features in such structure, following layer parameters were chosen. Two ferrite layers with thickness a2= 20 m, permittivity 2= 14, magnetization M2 = 1750 G, thickness a4=6 m, permittivity4= 14, magnetization M4= 1790 G separated by ferroelectric with thickness a3= 25 m and permeability 3= 1500. These are shown in the figure 2.5. The external field H0 was 1500 Oe.
30 60 90 120
6.1 6.2 6.3 6.4
a2= 20 m; M2= 1750 G;
a4= 6 m; M4= 1790 G;
a3=
a3= 200 m a3= 25m a3= 1 m
Frequency, GHz
Wave numbers (cm-1 )
36
Figure 2.5. Two ferrite layers separated by ferroelectric.
Dispersion characteristics of SWs in two ferrite layers with distance a3= 25 m are shown in figure 2.6 by red dash lines. It is well known that surface SWs are not reciprocal. It means that SWs field distributions depend on the propagating direction at fixed external magnetic field direction. If the wave propagates along x axis, then the maximum of the field distribution is on the top surfaces of the ferrites. In the opposite direction it corresponds to the bottom ferrite surfaces.
20 40 60 80 100 120 -120 -100 -80 -60 -40 -20
6.1 6.2 6.3 6.4
SEW
SW SEW
EMW EMW
SW
Frequency, GHz
Wavenumber, cm-1
Figure 2.6. Spectra of SEWs in the multilayered ferrite-ferroelectric structures with thin ferroelectric layer between ferrites. Black dot line is EMW, red dash line is SW and blue
solid line is SEW spectra.
37
Thus it was shown that SWs interact in the intersection point of dispersion characteristics. It leads to the changing of the SWs spectra. Dispersion characteristics of the SWs in the independent ferrite layers (a3=, 3=1) and in the two ferrite layers at the distance a3= 200 m and a3= 25 m are shown in the figure 2.7 (a, b) by black short dot lines and red dash lines, respectively.
As shown in figure 2.7, the dispersion characteristic of EMW (black dot line) intersects both SW dispersion characteristics. If the EMW propagates along x axis then due to field configuration SW in the thick ferrite (a2) interacts with the EMW that is located in the
Figure 2.7. Spectra of SEWs in the multilayered ferrite-ferroelectric structures with (a) thick a3= 200 m and (b) thin a3= 25 m ferroelectric layers between ferrites
(Calculation parameters are shown in the figure 2.5)
b) a)
10 100
-100 -10
6.1 6.2 6.3 6.4
Frequency, GHz
Wavenumber, cm-1
-120 -90 -60 -30
6.1 6.2 6.3 6.4
30 60 90 120
Frequency, GHz
Wavenumber, cm-1
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ferroelectric layer (a3). It leads to the formation of the SEW dispersion characteristics. This is shown by the green dashed lines in the figure 2.7. Note that such spectrum was calculated for the structure that contains only one thick ferrite. It is seen that the lower dispersion SEW branch intersects dispersion characteristic of the SW in the thin ferrite layer. It leads to wave interaction in two ferrite layers separated by the ferroelectrics layer and formation of a new type of hybrid SEW spectrum that consists of the three dispersion branches (blue solid lines). In case of the opposite propagation direction interaction between the EMW and SW in the thin ferrite layer (a4) is stronger than in the previous case. Therefore, SEW spectra in the structure that contains only thin ferrite layer were calculated (green dashed lines). In this case the lower SEW dispersion branch approaches to the SW dispersion characteristic in thin ferrite layer. Thus spectrum of the SEW in the two ferrite layers separated by ferroelectric layer in all propagation directions consists of three dispersion branches.
As mentioned above, surface SWs are not reciprocal. It means that SEWs propagating in different directions have different behavior (figure 2.7). Namely, influence of the ferroelectric layer permittivity on the intermediate SEW dispersion branch in the case of wave propagation opposite to the x-axis direction is much weaker than in the direction along the x-axis. Further increase in the number of ferrite layers makes SEW spectrum more complicated but gives additional features. These features can be used in the novel MW devices.
In order to illustrate features of these spectra in both propagation directions, ferroelectric layer permittivity 3 was changed. Influence of the ferroelectric layer permittivity on dispersion characteristics is shown in figure 2.8. It is worth noting that ferroelectric permittivity 3 was change from 1500 to 1000.
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As it was shown above, decrease of ferroelectric layer thickness leads to decrease of interaction between the EMW and SW. Finally these waves have a weak interaction at the thickness on the order of several micrometers. It can be considered that these waves propagate independently. On the other hand, in the case of ferrite-free space-ferrite structure interaction between SW in the different ferrites depends on the distance between them. Thus it can be concluded that presence of a thin ferroelectric film between the ferrite layers gives possibility to electric tuning of dispersion characteristics, because the changing of the ferroelectric layer permittivity influences on spin wave interaction (figure 2.8.).
To sum up, the dispersion relation strongly depends on the parameters of the structure. In order to investigate this influence to the wave’s interaction the values of wave number
Figure 2.8. Electric tunability of dispersion characteristic of SEW in the multilayered ferrite-ferroelectric structure thick a3= 200 m and (b) thin a3= 25 m ferroelectric
layers between ferrites (Calculation parameters are shown in the figure 2.5).
a)
b)
-100 -10
6.1 6.2 6.3 6.4
10 100
Frequency, GHz
Wavenumber, cm-1
-120 -90 -60 -30
6.1 6.2 6.3 6.4
30 60 90 120
Frequency, GHz
Wavenumber, cm-1
40
variation k with changing of the ferroelectric layer permittivity from 1500 to 1000 were calculated. The wave number variations for the investigated structure with different ferroelectric layer thicknesses from 10 to 30 m are shown in figure 2.9. It should be noted that wave number variations were calculated only for the middle and bottom SEW dispersion branches since number variations are much bigger for them.
It is seen from the figure that decrease of ferroelectric layer thickness in the FF structure leads to decrease of interaction between the EMW and the SW. Finally, at the thickness on the order of several micrometers this interaction can be relatively small especially for middle dispersion branch. One can see the wave number variation for the bottom branch amounts to higher values than for the middle branch. Therefore influence of the ferroelectric permittivity is stronger and electric tunability of SEW on the bottom dispersion branch is higher. Besides the electric tuning interval increases with the increase of the ferroelectric layer thickness.
Figure 2.9. Dependencies of the wave number variation on a frequency of (a) middle and (b) bottom SEW dispersion branches in the FF structure with different thicknesses of the
ferroelectric layer thicknesses a3= 10 m … 30 m.
6.12 6.15 6.18 6.21 6.24 6.27 6.30
0 2 4 6 8 10 12 14
6.23 6.24 6.25 6.26 6.27 6.28 6.29 6.30 0.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5
b)
Wave number variation , cm
-1Frequency, GHz
10 m a) 15 m 20 m 25 m 30 m
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The influences of the thicknesses of both ferrite layers also were investigated. Thicknesses of the bottom ferrite layer were changing in the range from 6 to 20 m. Dependencies of the wave number variations on a frequency are shown in Figure 2.10. Thicknesses of the upper ferrite layer were change in the same ranging and dependencies are shown in Figure 2.11.
It is seen from the figure that increasing of the thickness of the bottom ferrite layer leads to decreasing of interaction between SWs. Therefore the value of the wave number variation decreases for the middle dispersion branch and increases for the bottom one. In this case the frequency of the wave number variation maximum shifts to the lower values. It should be underlined that wave number variation for the middle branch has different behavior then for the bottom branch. The wave number variation has a peak near the frequency of wave interaction for the bottom branch and growth with the frequency increasing for the middle branch.
Figure 2.10. Dependencies of the wave number variation on a frequency of (a) middle and (b) bottom SEW dispersion branches in the FF structure with different thicknesses of the
bottom ferrite layer a2= 6 m … 20 m.
6.10 6.15 6.20 6.25 6.30
0 3 6 9 12
150.06.22 6.24 6.26 6.28 6.30
0.5 1.0 1.5 2.0 2.5
b)
Frequency, GHz
a)
Wave number variation , cm
-16 m 8 m 10 m 12 m 14 m 16 m 18 m 20 m
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It can be seen from this figure that increasing of thickness of the upper ferrite layer leads to stronger interaction between waves. Frequency of the wave number variation maximum shifts to the higher frequencies values. It should be noted that including of thick ferrite layers leads to stronger interaction between SWs and EMW. Therefore the electrical tuning of the SEW in this case can be much higher.
As soon as SWs could be easily tuned be the external magnetic field, magnetic tunability of SEWs spectra were also investigated (figure 2.12).
Figure 2.11. Dependencies of the wave number variation on a frequency of (a) middle and (b) bottom SEW dispersion branches in structure with different thicknesses of the
upper ferrite layer a4= 6 m … 20 m.
6.10 6.15 6.20 6.25 6.30
0 3 6 9 12 15 18
6.22 6.24 6.26 6.28 6.30
0.0 0.5 1.0 1.5 2.0
2.5 a
)
Frequency, GHz
b
)
Wave number variation , k, cm
-16 m 8 m 10 m 12 m 14 m 16 m 18 m 20 m
