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Theory of dual-tunable thin-film multiferroic magnonic crystal
Aleksei A. Nikitin, Andrey A. Nikitin, Alexander V. Kondrashov, Alexey B. Ustinov, Boris A. Kalinikos, and Erkki
Citation: Journal of Applied Physics 122, 153903 (2017);
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Published by the American Institute of Physics
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Theory of dual-tunable thin-film multiferroic magnonic crystal
Aleksei A. Nikitin,1,2 Andrey A. Nikitin,1,2 Alexander V. Kondrashov,1,2 Alexey B. Ustinov,1,2
Boris A. Kalinikos,1 and Erkki La
Department of Physical Electronics and Technology, St. Petersburg Electrotechnical University,
St. Petersburg 197376, Russia
Laboratory of Physics, Lappeenranta University of Technology, Lappeenranta 53850, Finland
(Received 18 August 2017; accepted 5 October 2017; published online 19 October 2017)
A theory has been developed for the waveguiding characteristics of dual-tunable multiferroic magnonic crystals (MCs). The crystals are constructed as periodically width-modulated microwave
transmission slot-lines placed in between thin ferrite and ferroelectric films. Dispersion characteristics of the spin-electromagnetic waves (SEWs) in the investigated periodic waveguiding structure
were derived using the method of approximate boundary conditions and the coupled-mode
approach. The transmission-loss characteristics (TLCs) were calculated by the transfer-matrix
method. The results show that the TLCs of the structures consist of pass-bands and stop-bands. The
stop-bands are due to Bragg reflections in the structure. The magnetic and electric fields control the
stop-band frequencies. The ferroelectric film polarization produced with the application of control
voltage to the slot-line electrodes reduces its dielectric permittivity and provides up-shift of the
stop-band frequencies. The most effective electric tuning is achieved in the area of the maximum
hybridization of SEWs. As a result, the investigated multiferroic MCs combine the advantages of
thin-film planar topology and dual tunability of magnonic band-gaps. Published by AIP Publishing.
Nowadays, increasing interest to study composite materials for their possible microwave applications is evident.1
Artificial multiferroic structures are promising candidates for
microwave devices due to a combination of advantages of
ferrite and ferroelectric materials.
The development of frequency-agile materials for
microwave devices has led to the appearance of multilayered
multiferroic structures.2–6 An interaction between the ferromagnetic and ferroelectric phases is realized through electrodynamic coupling of spin waves (SWs) and electromagnetic
waves (EMWs). This interaction leads to a formation of
spin-electromagnetic waves (SEWs).2 The frequency spectrum of these waves is dually controllable by both electric
and magnetic fields. Electric tuning of the SEW spectrum is
due to a dependence of ferroelectric dielectric permittivity
on the bias electric field, whereas magnetic tuning is provided by a dependence of magnetic permeability on the bias
magnetic field. In addition, devices based on multilayered
multiferroic structures demonstrate small insertion losses3
and small power consumption.6
Owing to the dual tunability, multiferroic materials have
been used to develop microwave devices. Such devices
added the advantages of electric tuning to spin-wave devices.7–15 Among such devices, there are a delay line7 based
on the yttrium iron garnet film and single-crystal lead magnesium niobate-lead titanate bilayer, the tunable microwave
resonators with wide frequency tuning range8 and changeable quality factor,9 the ferromagnetic resonance phase
shifters,10–12 and a multiband filter.13 Besides, an increased
interest to investigate one-dimensional and two-dimensional
magnonic crystals (MCs) was observed (see, e.g., Refs.
16–19 and literature therein).
Typical MCs are made from magnetic materials with a
spatial periodic modulation of their physical properties or
geometry. The periodicity of the magnetic film structures
results in the appearance of the band-gaps in the spin-wave
spectrum. It modifies the dispersion of spin waves in the
vicinity of the band-gaps. Similar phenomena were observed
for EMWs in photonic crystals20,21 and for matter waves of a
Bose-Einstein condensate.22,23
Various linear and nonlinear microwave phenomena in
the MCs are caused by peculiarities of the SW dispersion
near the magnonic band-gaps.24–31 The magnonic crystals
were used to construct various microwave devices such as
phase shifters,32 spin-wave logic devices,33 magnetic field
sensors,34 microwave oscillators,35 and others. Note that the
epitaxial yttrium-iron garnet (YIG) films were successfully
used in MCs due to small magnetic losses.36
It is clear that a combination of multiferroic and MC
features is promising for the development of a new class of
microwave devices. Recent advances in this field include the
development of periodic ferrite-ferroelectric structures. In
particular, the periodic structure composed of a thin-film MC
and a ferroelectric slab was fabricated and studied.37 After
that, a number of theoretical and experimental works were
carried out.38–44 In these works, rather thick ferroelectric
layers (of thickness more than 100 lm) were used in order to
provide an effective hybridization of microwave SWs with
EMWs. As a result, a relatively high control voltage (up to
1000 V) was needed for an effective electric tuning of SEW
122, 153903-1
Published by AIP Publishing.
Nikitin et al.
In order to reduce the control voltage, the so-called allthin-film multiferroic structures were suggested.5,11,12,45,46
However, the problem of relatively high control voltage for
multiferroic MCs still remained. To solve this problem, in
this work, we suggest a thin-film MC based on a slot transmission line. As it was shown, the thin-film regular structures based on a slot transmission line demonstrate an
effective electric tunability under voltage less than 100 V.5
Therefore, these structures could be useful for thin-film multiferroic MCs with a relatively low control voltage.
The purpose of the present work is twofold: (i) to
develop a general theory for waveguiding characteristics of
thin-film MCs based on a slot transmission line and (ii) using
this theory to find ways to enhance the electric tuning range
for reduction of the control voltage.
This paper is organized as follows. Section II describes
the topology of the thin-film MCs under investigation.
Section III is devoted to the theoretical model. The dispersion characteristics of SEWs and the transmission-loss characteristics (TLCs) of the periodical structure are investigated
in Sec. IV. Section V provides a summary and conclusions.
The thin-film multiferroic MC based on a slot-line having a periodic modulation of the slot width is shown in Fig.
1. It is composed of several layers enumerated with index j:
a sapphire substrate (j ¼ 1), a polycrystalline ferroelectric
film of the barium strontium titanate (j ¼ 2), an epitaxial YIG
film with saturation magnetization M0 (j ¼ 3), and a gadolinium gallium garnet substrate (j ¼ 4). We denote the thickness
of the layers as dj and their relative permittivities as ej . We
assume that a SEW propagates along the slot-line in the multiferroic structure, i.e., along the x-axis. The structure is magnetized to saturation tangentially along the z-axis.
The transmission slot-line is assumed to be formed as a
narrow slot between two infinitely thin and perfectly conducting metal electrodes placed between the ferrite (j ¼ 3)
and ferroelectric (j ¼ 2) films, as is shown in Fig. 1. These
electrodes provide two functions, namely, waveguiding of
J. Appl. Phys. 122, 153903 (2017)
SEWs and electric biasing of the ferroelectric film. The thinfilm multiferroic MC is formed by segments of the slot-line.
The segments of narrow and wide slots (w1 and w2 ) are hereinafter referred to as segment I and segment II. The lengths
of each segments are l1 and l2 , respectively. Thus, the period
of the MC is K ¼ l1 þ l2 . The ferrite and ferroelectric layers
are assumed to be relatively thin (on the order of unity of
micrometers). Other dielectric layers are assumed to be relatively thick (on the order of hundreds of micrometers).
Running ahead, we note that during our simulations, the
following parameters were varied: the YIG film thickness d3 ,
the BST film thickness d2 , the barium strontium titanate film
permittivity e2 , the external magnetic field H as well as geometrical parameters of the MC such as the slot widths w1 and
w2, the period K, and the number of the periods N. All the
rest parameters were fixed. These were d1 ¼ d4 ¼500 lm,
e1 ¼ 10, e4 ¼ 12, e3 ¼ 14, and M0 ¼ 1750 G.
The development of the theoretical model was carried
out in several stages. At the first stage, the dispersion relations for the SEWs propagating in the multiferroic MCs
were found according to the coupled-mode approach.47 This
approach supposes that forward and backward waves are
propagating independently and that a waveguide parameter
variation (in our case, it is the slot width) provides a coupling
between them. In this case, a dispersion equation has the following form:
cos ðK KÞ ¼ cosðk1 l1 Þ cosðk2 l2 Þ
k12 þ k22
sin ðk1 l1 Þ sin ðk2 l2 Þ;
2 k1 k2
where K is the Bloch wave vector, and k1 and k2 are the
wave numbers of the SEWs in the segments I and II. The
wave numbers in Eq. (1) were numerically calculated
according to the SEW dispersion relation based on the
approximate boundary conditions method described in detail
in Ref. 5.
Note that Eq. (1) takes into account two aspects of the
wave process in the investigated MCs. First, the SEWs are
damped waves for the frequencies where the Bloch wave
vector has a complex value. In the microwave range, this
phenomenon causes the Bragg gaps. Second, the roots of the
Eq. (1) are real at the frequencies outside the band-gaps.
Such solutions are responsible for propagating waves.
At the second stage, TLCs of the periodic multiferroic
structure were obtained according to the transfer-matrix
method.48 This method allows one to calculate transmission
characteristics of a finite-length periodic structure taking into
account SEW insertion losses.
FIG. 1. The thin-film multiferroic MC based on a slot transmission line with
modulation of the slot width.
The dispersion characteristics of the spin-electromagnetic
waves and transmission characteristics of the thin-film
multiferroic magnonic crystals were calculated and analyzed by using the theoretical model discussed in Sec. III.
Nikitin et al.
The corresponding calculations were carried out for the typical parameters of the experimental multiferroic structures
based on ferrite and ferroelectric films, as was outlined in
Sec. II.
Figure 2 shows the dispersion characteristics for the regular thin-film multiferroic structures with geometry presented in Fig. 2, i.e., without width modulation. The
calculations were carried out for the different slot widths
w ¼ 25 lm (solid lines) and w ¼ 90 lm (dashed lines) with
the following parameters: d3 ¼ 13.6 lm, H ¼ 1350 Oe,
d2 ¼ 2 lm, and e2 ¼ 1500. For a comparison, Fig. 2 also
shows the dispersion branches for the fundamental electromagnetic mode of the individual slot with the ferroelectric
film on the dielectric substrate and for the surface spin wave
mode of the free-standing magnetic film (dotted lines). One
can clearly see a hybridization of the two fundamental modes
appearing due to their electrodynamic interaction. It
becomes more pronounced near the point where the dispersion branches of the pure EMW and the pure SW cross each
As is clear from Fig. 2, a reduction of the slot width
w shifts the SEW dispersion characteristic to the higher
wave numbers. Consequently, the SEWs at a fixed frequency
accumulate the different phase shifts in different segments of
the periodic structure. The band-gaps appear at the frequencies where this phase shift is a multiple of p.
The SEW dispersion characteristic and the TLC of the
thin-film multiferroic MC of Fig. 1 are shown in Figs. 3(a)
and 3(b), respectively. The MC was formed by a series
connection of two slot-line segments investigated before.
The period and the number of the periods were taken to be
K ¼ 1.7 mm and N ¼ 10. The lengths of the segments I
and II were equal, i.e., l1 ¼ l2 ¼ K/2. This set of parameters
was chosen in order to achieve the second purpose of this
work. Indeed, as it was shown in Ref. 5, the effective electric and magnetic tuning ranges for multiferroic structures
with slot transmission lines are possible only around the
point of effective hybridization of electromagnetic and
spin waves. For the considered set of the parameters, the
frequency of hybridization is located near 5.74 GHz (see
Fig. 2). Therefore, the periodic modulation of the slot
FIG. 2. Dispersion characteristics of SEWs in the slot-lines for widths of
w ¼ 25 lm (solid lines) and w ¼ 90 lm (dashed lines). Pure EMWs and SW
are presented with dotted lines.
J. Appl. Phys. 122, 153903 (2017)
FIG. 3. Dispersion (a) and transmission-loss (b) characteristics of SEWs of
the thin-film multiferroic magnonic crystal.
width of the MC was chosen in such a way that the first
magnonic band-gap appears near 5.74 GHz. Thus, effective
electric tuning of the TLC becomes possible.
As can be seen from Fig. 3, the modulation of the slot
width leads to an appearance of the band-gaps in the spectrum of the SEWs [see Fig. 3(a)]. These band-gaps cause the
dips in the TLC of the investigated structure [see Fig. 3(b)].
The dips have finite depths due to a limited number of the
periods. In the considered case, the depth of the first bandgap (denoted by I in Fig. 3) is about -34 dB. The width of the
band-gaps reduces with the frequency increasing. This
behavior is determined by an efficiency of hybridization
between the EMWs and the SWs.
Let us now discuss transmission-loss characteristics.
Figure 4 demonstrates an influence of the slot-line parameters
such as the width of the segments I and II [see Figs. 4(a)] and
the period K [see Fig. 4(b)] on the TLCs for thin-film multiferroic MCs. As can be seen from Fig. 4(a), a change in the
modulation of the slot-line width affects the first bandgap of
the thin-film multiferroic MC. For the first band-gap, an
increase in the slot width difference (w1 w2 ) leads to a
reduction in the band-gap frequency and to an increase in the
band-gap width. These two effects are due to an increase in
the wavenumber difference for SEWs propagating in segments I and II (see Fig. 2). Note that the band-gap widths
were calculated at a level of 3 dB from the maximum loss.
Similar effects can be observed in the case of different
periods K [see Fig. 4(b)]. According to Bragg’s diffraction
law, the maxima of reflection occur at k ¼ np=K. Therefore,
an increase in the length of the period decreases the wavenumbers, which correspond to the band-gaps. Due to this
Nikitin et al.
FIG. 4. Influence of the slot line width w (a) and period length K (b) on a
MC band structure.
phenomenon, the band-gaps are shifted to the lower frequencies. In addition, this down-frequency shift determines
increasing in the width of the first band-gap.
Finally, an influence of the number of periods N on the
behavior of the transmission-loss characteristic is analyzed.
As follows from the transfer-matrix method, an increase in
this parameter increases the losses for a TLC. For the considered structure, the transmission coefficient of the first bandgap is 34 dB for 10 periods, while for 20 periods, this value
reaches 68 dB.
Also, numerical simulations were carried out for the different thicknesses of the ferroelectric and ferrite films, i.e.,
d2 and d3 . In particular, Fig. 5(a) shows that a decrease in the
ferroelectric film thickness d2 leads to a shift of the band-gap
position to higher frequency and brings a change in a width
of the first band-gap. This behavior is determined by the
decrease in the difference between the SEW dispersion in
different segments of the MC. In addition, the higher frequency band-gaps depend weakly from the ferroelectric film
thickness. Such a behavior is determined by their frequency
positions that are higher than 5.74 GHz. Therefore, the
higher frequency band-gaps are located far from the point of
effective hybridization between the EMW and the SW (see
Fig. 2). In this case, properties of the ferroelectric film play a
dominant role in the wave process whereas the influence of
the ferroelectric subsystem is negligible.
Figure 5(b) demonstrates an influence of the ferrite film
thickness d3 on the transmission-loss characteristics. As one
can see, the frequencies of the first band-gap depend weakly
J. Appl. Phys. 122, 153903 (2017)
FIG. 5. Transmission-loss characteristics for different thicknesses d2 of the
ferroelectric film (a) and d3 of the ferrite film (b).
on the ferrite film thickness. At the same time, the width of this
band-gap is increased. It is because of an increase in the SEW
hybridization for thick ferrite films.2 Sufficient reduction in a
group velocity with decreasing the ferrite thickness provides
higher losses and down-shift in the frequency of the bandgaps. Simultaneously, the narrow band-gaps appeared for the
MC based on the thin ferrite film. As shown in Fig. 5(b), the
widths of the first band-gap are 38.6 and 19.8 MHz at 20 dB
for 20- and 9–lm thick of the ferrite films, respectively.
Let us consider now the electric and magnetic tunability
of the transmission-loss characteristics. The results of modeling of the electric tuning are shown in Fig. 6(a). The calculations were carried out for the following parameters of the
MC: d3 ¼ 13.6 lm, H ¼ 1350 Oe, d2 ¼ 2 lm, K ¼ 1.7 mm,
N ¼ 10, w1 ¼ 90 lm, and w2 ¼ 25 lm. An influence of control
voltage U applied to the slot-line electrodes was simulated as
a reduction of the ferroelectric film permittivity e2 . Note that
due to the different widths of the slot-line gaps of the segments I and II (see Fig. 1), the electric field is not the same
in different segments and was calculated as E1;2 ¼ U=w1;2 .
The relative permittivity of the BST film as a function of the
electric field E was calculated by the following formula:
e2 ðE1;2 Þ ¼ e2 ð0Þ k E21;2 :
The following typical parameters of the barium strontium titanate film were used: e2 ð0Þ ¼ 1500 and
k ¼ 0.194 cm2/kV2.5
As can be seen from Fig. 6(a), an increase in the control
voltage shifts the band-gaps to the higher frequency. Such
Nikitin et al.
J. Appl. Phys. 122, 153903 (2017)
influence of different geometrical parameters on the waveguiding characteristics and band-gap frequency positions
was analyzed. It was found that the investigated structures
can provide an excellent signal rejection of more than 30 dB.
The optimal rejection efficiency and the required band-gap
bandwidth can be obtained by adjusting geometry of a thinfilm magnonic crystal.
Furthermore, an enhancement of the electric tuning
range and a reduction of the control voltage for the multiferroic MC were achieved due to its unique geometry and physical properties. Thus, application of the bias voltage of 200 V
to 2-lm-thick ferroelectric film leads to the shift of the bandgap position by 8.35 MHz. At the same time, the change of
the external magnetic field by 10 Oe shifts the band-gaps by
30 MHz. Therefore, the proposed structures are perspective
for the development of new microwave devices and investigation of new physical phenomena.
FIG. 6. Electric (a) and magnetic (b) tuning of the transmission-loss
The work at SPbETU on numerical modeling was
supported by the Russian Science Foundation (Grant No.
14–12-01296-P). The work at SPbETU on Development of
Computer Program for Numerical Simulation was supported
by the Ministry of Education and Science of the Russian
Federation (Project “Goszadanie”) and the Russian Foundation
for Basic Research (project No. 16-32-000715 mol_a). The
work at LUT was supported in part by the Academy of Finland.
behavior of the TLCs can be explained as follows. It is
known that a decrease in a ferroelectric permittivity shifts a
SEW dispersion characteristic to the lower wavenumbers. In
a case of the MC, the values of wavenumbers that correspond to the maxima of reflection (k ¼ np=K) remain constant. Therefore, in the case of e2 decreasing (i.e., increase in
the control voltage), this condition will be satisfied in the
area of the higher frequency compared to the zero voltage
case. In particular, the electric tuning of the TLC for the first
band-gap reaches values of 8.35 MHz for the electric voltage
of 200 V. Note that the electric tuning is decreased for higher
stop-bands due to weak interaction of spin and electromagnetic waves at frequencies higher than 5.8 GHz for the investigated slot-line structure.
Turn now to the magnetic tuning. Transmission-loss
characteristics were simulated for different values of the
external magnetic field H [see Fig. 6(b)]. The following magnetic fields were used: 1340 Oe (solid line), 1345 Oe (dashed
line), and 1350 Oe (dotted line). Figure 6 shows that an
increase in the external magnetic field leads to the shift of
the SW spectrum toward the higher frequencies.
Waveguiding characteristics of the thin-film multiferroic
magnonic crystals based on a slot transmission line were theoretically investigated. In particular, dispersion and
transmission-loss characteristics were calculated according
to the coupled-mode approach and transfer-matrix method,
respectively. According to these general theories, an
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