# Transition between radial and toroidal orders in a trimer-based magnetic
metasurface

Vladimir R. Tuz1,2 Andrey B. Evlyukhin3 Volodymyr I. Fesenko4
<EMAIL_ADDRESS>1State Key Laboratory of Integrated
Optoelectronics, College of Electronic Science and Engineering, International
Center of Future Science, Jilin University, 2699 Qianjin St., Changchun,
130012, China  2School of Radiophysics, Biomedical Electronics and Computer
Systems, V. N. Karazin Kharkiv National University, 4, Svobody Sq., Kharkiv
61022, Ukraine  3Institute of Quantum Optics, Leibniz Universität Hannover,
30167 Hannover, Germany  4Institute of Radio Astronomy of National Academy of
Sciences of Ukraine, 4 Mystetstv St., Kharkiv 61002, Ukraine

###### Abstract

Magnetic materials are used in a wide variety of technologies, being key
components of microwave and quantum devices. The properties of such materials
manifest themselves in an external magnetic field and are mainly determined by
the order of intrinsic magnetic moments (spin) associated with their subatomic
particles. Metasurfaces, the two-dimensional counterparts of metamaterials,
can be a platform for studying magnetic orders due to the possibility of
excitation of dynamic magnetic moments in their particles (meta-atoms) by
incoming electromagnetic radiation. In this paper, the change in the
arrangement of magnetic dipole moments in a magnetic metasurface, due to the
influence of an external static magnetic field, is discussed. Each meta-atom
of the metasurface is composed of three identical disk-shaped resonators
(trimer) made of magnetically saturated ferrite. To provide physical insight,
full-wave numerical simulations of the near-fields and transmission
characteristics of the metasurface are complemented by the theoretical
description based on the multipole decomposition method. With these methods,
the study of eigenmodes and scattering conditions of a single magnetic
resonator, trimer, and their array forming the metasurface is performed. It is
found that the magnetic dipole-based collective hybrid mode of the trimer can
be gradually transformed from the radial (pseudomonopole) to azimuthal
(toroidal) order by increasing the bias magnetic field strength. This is
accompanied by various patterns of localization of the electric field inside
the meta-atoms. Due to the unique field configuration of these modes, the
proposed metasurface can be considered for designing magnetic field sensors
and nonreciprocal devices.

###### pacs:

78.20.Ek, 78.20.Fm, 78.67.Pt

††preprint: APS/123-QED

##  I Introduction

The interpretation of magnetic effects in solids has developed from two
concepts [1]. The first is the examination of microscopic electrical currents,
such as orbital currents in atoms, or magnetic dipole moments of subatomic
particles related to their intrinsic angular momentum (spin). Induced magnetic
moments are produced by an externally applied magnetic field, whereas
spontaneous moments are present even in the absence of this field. The second
concept involves the consideration of mutual interactions of the magnetic
moments through ordinary dipole-dipole and quantum mechanical forces. These
so-called exchange forces depend on the separation of the magnetic ions as
well as their internal arrangement, leading to a variety of magnetic orders in
solids. The important aspects of the study of magnetic orders regarding their
nature are (i) the type of arrangement of magnetic moments and (ii) the
characteristics of ordering processes associated with the corresponding phase
transitions and other critical phenomena. These magnetic phase transitions can
occur both from an ordered state to a disordered state and from one magnetic
order to another (e.g., the ordered magnetic moments change and become
disordered at the Curie temperature; on phase transitions and critical
phenomena, see Ref. [2]).

A typical example of the influence of order on the magnetic properties of
solids is the difference between ferromagnets and antiferromagnets (these
materials belong to a class of ferroics; regarding the classification of
materials according to their magnetic properties, see tutorials and reference
books [3, 4, 5, 6, 7]). The order parameter of such materials is related to
magnetic dipole moments and is known as magnetization. Thus, ferromagnets
exhibit parallel alignment of static magnetic moments which results in the
magnetization of the material even in the absence of an external bias magnetic
field. In antiferromagnets, there is the presence of equal magnetic moments
which are aligned in opposite directions resulting in the antiparallel moments
canceling each other. This makes the total magnetic moment of antiferromagnets
equal to zero. Nevertheless, the simple classification into ferromagnetic and
antiferromagnetic orders cannot do justice to the rich variety of other
possible ordered patterns. For instance, there are substances with molecules
in which magnetization vectors have azimuthal arrangement forming closed loops
[8, 9, 10]. They are termed ferrotoroidal materials whose order parameter is
called toroidization or the ferrotoroidal moment (in what follows, for
brevity, the prefix ‘ferro’ is omitted; see recent reviews on the appearance
of the toroidicity in natural materials in Refs. [11, 12]).

Various models are used to describe the possible orders in solids considering
interactions between systems of particles and providing a framework for the
study and classification of critical phenomena including phase transitions
[13]. In particular, the two-dimensional Ising model describes the interaction
of particles that possess a magnetic dipole moment. In this model, it is
supposed that the energy of a system of such particles depends on the
orientation of different magnetic dipole moments relative to each other as
well as to an externally applied magnetic field [14]. Recently, it has been
proposed to study different magnetic orders with metasurfaces [15, 16, 17, 18,
19, 20, 21, 22, 23]. They are arrays of artificially structured subwavelength
resonant particles that facilitate to reach of desirable electromagnetic
functionality. Metasurfaces can maintain artificial magnetism because any
desired configurations of their meta-atoms can be created [24, 25, 26]. In
such meta-atoms, dynamic magnetic moments can arise from macroscopic currents
induced by oscillating electromagnetic fields originating from incoming
radiation.

In this regard, the use of metasurfaces for the study of toroidicity has
acquired particularly significant attention. It is due to the fact that
natural materials bearing the toroidal moment are scarce, and the
manifestation of the toroidicity in such materials is usually weak. However, a
strong toroidal response can be created in metasurfaces when their meta-atoms
are properly designed. This implies the excitation of a toroidal dipole moment
from a closed loop of magnetic dipoles inherent in the constituent particles
forming a meta-atom. In this manner, the toroidal response is constructed in
various metasurfaces whose meta-atoms are composed of either metallic [27, 28,
29] or dielectric [30, 31, 32] particles (for a review on the application of
metasurfaces bearing toroidal moments, see Ref. [33]).

In our previous research [34, 35, 36, 37, 38], the general conditions for the
existence of toroidal dipole moments in metasurfaces composed of dielectric
particles are revealed involving the multipole decomposition method and group-
theoretical analysis. These conditions are checked against both numerical
simulations and microwave experiments. Similar to the ferromagnetic and
antiferromagnetic orders in ferroics, the toroidic and antitoroidic orders are
found to exist in such metasurfaces. Further development in this field may
include utilizing magnetic materials in metasurfaces to implement their
tunable configurations [39, 40]. Specifically, these can be structures that
change their dynamic magnetic order under external influence. As a significant
advantage, incorporating magnetic materials into metasurface compositions can
enhance magneto-optical effects and open prospects in developing advanced
nonreciprocal devices [41].

Therefore, in the present paper, our goal is to demonstrate that in a
metasurface composed of magnetic particles, the transition between different
orders of magnetic dipole moments can arise under the action of an external
magnetic field. These magnetic moments belong to the particles that compose
the meta-atoms (here we consider meta-atoms made in the form of trimers) and
their order is associated with the hybridization of the eigenmodes of the
particles. These orders are the azimuthal (toroidal) and radial
(pseudomonopole) arrangements of the magnetic moments (the difference between
radial and azimuthal is that radial is arranged like rays that radiate from or
converge to a common center while azimuthal is of or pertaining to the
azimuth, i.e., in a horizontal closed loop). To describe the transition
between these two orders influenced by an external static magnetic field, we
utilize the multipole decomposition method developed earlier for the
calculation of scattering by magnetic particles [42]. The theoretical
consideration is supplemented by numerical simulations performed with the
commercial COMSOL MULTIPHYSICS solver for the metasurface operating in the
microwave range. The simulation assumes that the particles are made from a
magnetic material (ferrite), which is widely used in practice when
constructing microwave devices.

##  II Problem statement

Figure 1: Schematic view of a trimer-based magnetic metasurface, geometric
parameters of its constituents, and the incident wave conditions. The red
dashed square outlines the metasurface unit cell whose transverse and
longitudinal projections are given in the insets. The blue arrow represents
the direction of the externally applied static magnetic field
$\textbf{H}_{0}$.

In what follows, we consider a metasurface composed of trimer-based meta-atoms
adjusted in a two-dimensional array as shown in Fig. 1. The translation unit
cell of this array is a square with the side size $p$. Each meta-atom consists
of three identical cylindrical resonators (disks) located at the vertices of
an equilateral triangle lying in the $x$-$y$ plane with their axes parallel to
the $z$-axis. The distance between adjacent disks in the trimer is $d$. The
radius and height of the disks are $a$ and $h$, respectively.

We consider that the metasurface is normally irradiated by the field of a
plane wave with the wave vector $\textbf{k}=\\{0,0,-k_{z}\\}$ and wavelength
$\lambda=2\pi c/\omega$, where $\omega$ is the angular frequency and $c$ is
the speed of light in vacuum. We define that the incident plane wave is either
linearly or circularly polarized.

As a material for disks, we choose a widely used ferrite [43]. An external
static magnetic field $\textbf{H}_{0}=\textbf{z}_{0}H_{0}$ is applied along
the $z$-axis parallel to the magnetization $\textbf{M}=\textbf{z}_{0}M_{s}$,
where $\textbf{z}_{0}$ is the unit vector in the $z$-axis direction and
$M_{s}$ is the saturation magnetization. Since the direction of propagation of
the incident wave coincides with the bias of the external static magnetic
field ($\textbf{k}\parallel\textbf{H}_{0}$), the problem is stated in the
Faraday configuration.

In the chosen coordinate frame and with accounting for the time factor
$\exp(-i\omega t)$, ferrite is characterized by the scalar permittivity
$\varepsilon$ and the second-rank permeability tensor written in the form

$\hat{\mu}=\mu_{0}\left({\begin{matrix}{\mu}&{i\mu_{a}}&{0}\cr{-i\mu_{a}}&{\mu}&{0}\cr{0}&{0}&{1}\cr\end{matrix}}\right).$
(1)

The elements of this tensor are

$\begin{split}&\mu=1+\chi^{\prime}+i\chi^{\prime\prime}=\mu^{\prime}+i\mu^{\prime\prime},\\\
&\mu_{a}=\chi_{a}^{\prime}+i\chi_{a}^{\prime\prime}=\mu^{\prime}_{a}+i\mu^{\prime\prime}_{a},\end{split}$
(2)

where
$\chi^{\prime}=\omega_{0}\omega_{m}\left[\omega_{0}^{2}-\omega^{2}(1-b^{2})\right]D^{-1}$,
$\omega_{m}=\mu_{0}\gamma M_{s}$ is the magnetization frequency,
$\omega_{0}=\mu_{0}\gamma H_{0}$ is the Larmor frequency,
$\chi^{\prime\prime}=\omega\omega_{m}b\left[\omega_{0}^{2}-\omega^{2}(1+b^{2})\right]D^{-1}$,
$\chi_{a}^{\prime}=\omega\omega_{m}\left[\omega_{0}^{2}-\omega^{2}(1+b^{2})\right]D^{-1}$,
$\chi_{a}^{\prime\prime}=2b\omega^{2}\omega_{0}\omega_{m}D^{-1}$,
$D=\left[\omega_{0}^{2}-\omega^{2}(1+b^{2})\right]+4b^{2}\omega^{2}\omega_{0}^{2}$,
$\gamma$ is the gyromagnetic ratio, $b=\mu_{0}\gamma\Omega/2\omega$ is a
dimensionless damping constant, $\Omega$ is the linewidth of the ferromagnetic
resonance.

To be specific, commercially available $\mathsf{Al}$-doped polycrystalline
yttrium iron garnet (YIG) G-400 is chosen to be a ferrite material for the
disks [44]. The properties of this material appear in the microwave range
$f\in[6,12]$ GHz, $f=\omega/2\pi$, which we fix for our study. To construct
the metasurface, we consider that the disks are embedded into a homogeneous
medium (host) made from an air-like material with constitutive parameters
$\varepsilon_{h}=\mu_{h}=1$ (although this paper is purely theoretical, see
our experimental microwave samples of dielectric metasurfaces bearing toroidal
resonances and corresponding measurement techniques in Refs. [34, 37, 38]).

##  III Magnetic dipole mode of a single ferrite resonator

As was previously revealed [34, 35, 36, 45], a dielectric (nonmagnetic)
metasurface composed of trimers arranged in square unit cells indeed
demonstrates a dynamic toroidal response, when the structure is excited by a
normally incident wave with a proper polarization. This response arises due to
the hybridization of a natural mode (eigenmode) of individual resonators that
form the meta-atoms. It is the hybrid electromagnetic HE11ℓ mode of a
cylindrical resonator (the $\textrm{HEM}_{11\ell}$ mode in the nomenclature of
Ref. [46]). In the subscripts of this mode abbreviation, the first index
denotes the azimuthal variation of the fields, the second index denotes the
order of variation of the field along the radial direction, and the third
index denotes the order of variation of fields along the $z$-direction. Since
we do not fix the exact number of the field variations along the $z$-axis, we
substitute the third index with the letter $\ell$ (this notation is based on
the mode nomenclature of cylindrical dielectric waveguides introduced in Ref.
[47] where the subscripts $nm$ of the $\textrm{HE}_{nm}$ mode refer to the
$n$th order and $m$th rank, and the rank gives the successive solutions of the
boundary condition equation involving the Bessel function $J_{n}$).

For the HE11ℓ mode, the fields are azimuthally dependent and the $H_{z}$
component is sufficiently smaller than the $E_{z}$ component. From the theory
of dielectric resonators [48], it is known that this mode radiates like a
magnetic dipole $\bf{m}$ oriented in-plane of the disk, and in terms of the
Mie-theory, it is considered as a magnetic dipole (MD) mode [49] (hereinafter
we will use this shorter notation). Within the framework of the present
consideration, it is necessary to find out what changes appear in the
characteristics of the MD mode when the resonator is made of ferrite
influenced by an external static magnetic field. Obviously, these
characteristics are determined in part by the properties of the constitutive
unbounded (bulk) ferrite and the direction of application of the external
static magnetic field with respect to the rotation axis of the disk-shaped
resonator and the primary wave incidence.

###  III.1 Magnetic subsystem conditions

As is well known, the eigenwaves of bulk ferrite that are propagated along the
magnetization direction are the left-handed (LCP) and right-handed (RCP)
circularly polarized waves, which differ in the propagation constants (the
Faraday effect). Near the ferromagnetic resonance, circular dichroism also
takes place, which manifests itself in a significant attenuation of the LCP
eigenwave as it propagates. From a phenomenological point of view, it is
explained by the fact that each of those waves propagates in the medium with a
different effective permeability

$\mu^{\pm}=\mu\pm\mu_{a}=(\mu^{\pm})^{\prime}+i(\mu^{\pm})^{\prime\prime},$
(3)

where $\mu^{+}$ and $\mu^{-}$ are related to the LCP and RCP waves,
respectively.

Moreover, in the ferrite resonator under consideration, the dominating
components of the magnetic field H of the MD mode are transverse to the bias
magnetic field $\textbf{H}_{0}$ ($\textbf{H}\bot\textbf{H}_{0}$, $H_{z}\approx
0$). This feature and the difference in the eigenwaves propagation condition
in the bulk ferrite result in the degeneracy lifting for the MD mode in the
disk-shaped resonator and the corresponding eigenfrequency splitting [42]. In
what follows, we distinguish these split modes as $\textrm{MD}^{+}$ and
$\textrm{MD}^{-}$.

Figure 2: (a) Dependencies of the real part (hot color map) and imaginary part
(cold color map) of diagonal ($\mu$) and non-diagonal ($\mu_{a}$) components
of the permeability tensor $\hat{\mu}$ of the ferrite on the frequency $f$ and
bias magnetic field strength $H_{0}$. The colored points at the bottom plane
represent the eigenfrequencies (dispersion curves) of the $\text{MD}^{+}$
(red) and $\text{MD}^{-}$ (blue) modes of a single resonator. (b) Real and
imaginary parts of the scalar effective permeabilities $\mu^{+}$ and $\mu^{-}$
as functions of $H_{0}$ at the fixed frequency $f=8.0$ GHz. Here,
$H_{\textrm{res}}\approx\omega/\mu_{0}\gamma-M_{s}/2$ is the strength of the
bias magnetic field at which the ferromagnetic resonance arises. Schematic
view of the electric E and magnetic H fields flow of the MD mode and
orientation of the magnetic dipole moment m (yellow arrow) are given in the
inset. Parameters of the ferrite are $M_{s}=33.4$ kA/m ($4\pi M_{s}\sim 420$
G), $\Omega\leq 45$ Oe, $\tan\delta=2\times 10^{-4}$, $\gamma=1.759\times
10^{11}$ C/kg, $\varepsilon=15$, whereas the radius and height of the
resonator are $a=4.2$ mm and $h=6.5$ mm, respectively.

The eigenfrequencies of the $\text{MD}^{\pm}$ modes of a ferrite disk-shaped
resonator can be estimated using the following empirical relation [48, 50]

$f^{\pm}\approx\frac{6.324c}{2\pi
a\sqrt{\varepsilon\mu^{\pm}+2}}\biggl{[}0.27+0.36\frac{a}{2h}+0.02\left(\frac{a}{2h}\right)^{2}\biggr{]}.$
(4)

This estimation is valid in the range $a/h\in[0.33,5.5]$ with an accuracy of
about $2\%$ for calculation of the resonant frequency for both $\text{MD}^{+}$
and $\text{MD}^{-}$ modes [48].

To show the discussed characteristics explicitly, we collect all the
parameters related to the bulk ferrite and the MD mode of the ferrite
resonator in Fig. 2. In the chosen frequency range, the ferromagnetic
resonance manifests itself in the bulk ferrite when the strength of the static
magnetic field $H_{0}$ is higher than 2 kOe. From the eigenfrequencies plotted
at the bottom plane in Fig. 2(a), one can readily see that the $\text{MD}^{+}$
mode of the ferrite resonator is significantly perturbed by the ferromagnetic
resonance and its dispersion curve has a discontinuity, whereas the
$\text{MD}^{-}$ mode does not undergo any noticeable change in its dispersion.
Due to circular dichroism, there is a significant attenuation of the
$\text{MD}^{+}$ mode of the ferrite resonator nearby the ferromagnetic
resonance frequency.

One can also notice that the $\text{MD}^{+}$ and $\text{MD}^{-}$ modes do not
degenerate in the ferrite particle even in the absence of the magnetic bias
field ($H_{0}=0$). Previously, this peculiarity has been experimentally
observed for disk-shaped resonators made of YIG as well as barium hexaferrite
[51, 52]. This initial mode splitting, which can reach several GHz, is not
caused by residual magnetization and strongly depends on specific parameters,
such as the saturation magnetization $M_{s}$ and the magnitude of the
anisotropy field (this effect is beyond the scope of our present study; extra
details on this issue can be found in Refs. [53, 44, 51, 52]).

The characteristics of the effective permeabilities $\mu^{\pm}$ are also
plotted in Fig. 2(b) as functions of the external magnetic field $H_{0}$. A
particular strength of this field where the ferromagnetic resonance arises, is
denoted as $H_{\textrm{res}}$. These curves show typical behaviors, where at
the ferromagnetic resonance, the real part of $\mu^{+}$ undergoes changing
from negative to positive values while the imaginary part reaches its maximum.
The real and imaginary parts of $\mu^{-}$ have only a slight monotonous
decrease with increasing $H_{0}$.

For our further consideration, we choose the parameter space of the problem in
such a way that the MD mode is in the region where the real part of $\mu^{+}$
is greater than zero. It is a region above the ferromagnetic resonance
($H_{\text{res}}<H_{0}$), where microwave devices based on low-loss ferrite
materials are usually operated. A schematic representation of the flow of the
electric and magnetic fields as well as the orientation of the magnetic dipole
moment for the MD mode of our interest is shown in Fig. 2(b) just for
reference.

###  III.2 Resonator scattering conditions

To fully reveal the MD mode features of an individual ferrite resonator, we
perform the study of scattered fields involving the multipole decomposition
technique developed earlier [42] to examine the scattering by magnetic
particles. In general, the scattering cross-section in the multipole
decomposition representation is written as [42]

$\displaystyle\sigma_{\rm sca}$ $\displaystyle=$
$\displaystyle\frac{k^{4}}{6\pi\varepsilon_{0}^{2}\varepsilon_{h}^{2}|{\bf
E}_{\textrm{in}}|^{2}}|{\mathbf{p}}|^{2}+\frac{k^{4}}{6\pi\varepsilon_{0}^{2}\varepsilon_{h}^{2}v^{2}|{\bf
E}_{\textrm{in}}|^{2}}|{\mathbf{m}}|^{2}$ (5)
$\displaystyle+\frac{k^{6}}{80\pi\varepsilon_{0}^{2}\varepsilon_{h}^{2}\mu_{h}|{\bf
E}_{\textrm{in}}|^{2}}\sum_{\beta\gamma}|Q_{\beta\gamma}|^{2}+\cdots$

where $k$ is the wave number in the surrounding medium, ${\bf
E}_{\textrm{in}}$ is the electric field amplitude of the incident plane wave.
Note that in Eq. (5) we explicitly indicate only contributions of the
scatterer’s electric dipole $\bf p$, magnetic dipole $\bf m$, and electric
quadrupole $\hat{Q}$ moments. For magnetic particles, all total multipoles
moments are the combinations of two parts originating from the electric
polarization ${\bf P}({\bf
r})=\varepsilon_{0}(\varepsilon-\varepsilon_{h}){\bf E}({\bf r})$ and
magnetization ${\bf M}({\bf r})=(\hat{\mu}-\mu_{h}\hat{1}){\bf H}({\bf r})$
induced by incoming radiation inside the scatterer. Here ${\bf E}({\bf r})$
and ${\bf H}({\bf r})$ are the total oscillating electric and magnetic fields
at the point $\bf r$ inside the scatterer, respectively, and $\hat{1}$ is the
$3\times 3$ unit tensor. In general, the magnetic and electric dipole moments
are given by the sums

${\mathbf{m}}={\mathbf{m}}^{\textrm{E}}+{\mathbf{m}}^{\textrm{H}},\quad{\mathbf{p}}={\mathbf{p}}^{\textrm{E}}+{\mathbf{p}}^{\textrm{H}},$
(6)

where the superscripts E and H are related to the dipole moments determined by
the polarization $\mathbf{P}$ (electric part) and the magnetization
$\mathbf{M}$ (magnetic part), respectively. The exact expressions for the
magnetic and electric dipole moments can be found in Ref. [42]. Here, for the
convenience of our discussion, we present the expressions for these dipole
moments in the long wavelength approximation (LWA) which is well applicable
when the size of scatterers is smaller than the incident wavelength [42, 54].
Thus, for the first-order terms of the LWA, we have

${\mathbf{m}}^{\textrm{E}}\approx-i\omega\frac{1}{2}\int\limits_{V_{p}}[{\mathbf{r}}\times{\mathbf{P}}]d{\mathbf{r}},$
(7)
${\mathbf{m}}^{\textrm{H}}\approx\frac{1}{\mu_{h}}\int\limits_{V_{p}}{\mathbf{M}}d{\mathbf{r}}+\frac{k^{2}}{10\mu_{h}}\int\limits_{V_{p}}[{\mathbf{r}}({\mathbf{r}}\cdot{\mathbf{M}})-2r^{2}{\mathbf{M}}]d{\mathbf{r}},$
(8)
${\mathbf{p}}^{\textrm{E}}\approx\int\limits_{V_{p}}{\mathbf{P}}d{\mathbf{r}}+\frac{\mu_{h}k^{2}}{10}\int\limits_{V_{p}}[({\mathbf{r}}\cdot{\mathbf{P}}){\mathbf{r}}-2r^{2}{\mathbf{P}}]d{\mathbf{r}},$
(9)
${\mathbf{p}}^{\textrm{H}}\approx\frac{ik^{2}}{\omega\mu_{h}}\frac{1}{2}\int\limits_{V_{p}}[{\mathbf{r}}\times{\mathbf{M}}]d{\mathbf{r}}.$
(10)

Note that the second integral terms in Eqs. (8) and (9) are introduced as the
magnetic ${\bf T}^{\rm H}$ and electric ${\bf T}^{\rm E}$ toroidal dipole
moments, respectively [55].

To characterize absorption in a scatterer, the absorption cross-section
$\sigma_{\textrm{abs}}$ is used, which for magnetic particles is determined by
the expression

$\sigma_{\textrm{abs}}=\frac{k}{\varepsilon_{0}\varepsilon_{h}|{\textbf{E}_{\textrm{in}}|^{2}}}{\rm
Im}\int\limits_{V_{p}}\\{({\bf P}\cdot{\bf E}^{*})+\mu_{0}({\bf M}\cdot{\bf
H}^{*})\\}d{\bf r}.$ (11)

Note that for scatterers with real permittivity, as in our case, the
contribution of $\bf P$ into the absorption cross-section is equal to zero.
Therefore, the absorption is associated only with the magnetic subsystem, so
that

$\sigma_{\textrm{abs}}=\frac{k}{\mu_{h}|{\textbf{H}_{\textrm{in}}|^{2}}}{\rm
Im}\int\limits_{V_{p}}({\bf M}\cdot{\bf H}^{*})d{\bf r},$ (12)

where $(*)$ denotes the complex conjugation, $\textbf{H}_{\textrm{in}}$ is the
magnetic field amplitude of the incident plane wave.

Figure 3: (a) Normalized scattering cross-section ($\sigma_{\textrm{sca}}$)
and absorption cross-section ($\sigma_{\textrm{abs}}$) of a single ferrite
resonator frontally irradiated by a linearly polarized plane wave
($\textbf{y}_{0}\parallel\textbf{E}$) as a function of the frequency $f$ and
bias magnetic field strength $H_{0}$. The external static magnetic field
$\textbf{H}_{0}$ is applied in the Faraday configuration, i.e., parallel to
the primary wave propagation ($\textbf{H}_{0}\parallel\textbf{k}$) as shown in
the inset. (b) Electric (${\bf m}^{\rm E}$) and magnetic (${\bf m}^{\rm H}$)
parts of the magnetic dipole moments ($\textbf{m}={\bf m}^{\rm E}+{\bf m}^{\rm
H}$). (c) Manifestation of the magnetic subsystem in the $\textrm{MD}^{\pm}$
mode of the ferrite resonator at its half-height. Here the black and yellow
arrows represent the flow of the magnetic field H and orientation of the
magnetic dipole moment m, respectively. These patterns are plotted for the
points ① and ② in the $f$-$H_{0}$ parameter space depicted in panels (a) and
(b). The problem parameters are the same as in Fig. 2.

For the calculation of the scattering and absorption cross-sections as well as
the introduced dipole moments, the corresponding procedures are implemented in
the RF module of the commercial COMSOL MULTIPHYSICS software. The realization
of the computational model in this solver for both single particle and
metasurface has been described earlier (see Refs. [42, 56]), therefore here we
omit all the details so as not to overburden the presentation. To reveal the
scattering characteristics caused by the excitation of both $\text{MD}^{+}$
and $\text{MD}^{-}$ modes, we consider the frontal irradiation of the disk-
shaped resonator by a wave with linear polarization. For definiteness, we fix
that the vector E of the incident wave is oriented along the $y$-axis. After
simulation, we retrieve from the COMSOL MULTIPHYSICS data both scattering and
absorption cross-sections as functions of the frequency $f$ and the external
static magnetic field strength $H_{0}$. The implementation of expressions
(7)-(10) for the dipole contributions in the COMSOL MULTIPHYSICS also makes it
possible to visualize the distribution of fields and the orientation of the
dipole moments inside the resonator and, thus, to reveal the influence of the
bias magnetic field on the MD mode behaviors. The obtained results are
summarized in Fig. 3.

In Fig. 3(a), one can readily identify the manifestation of both the magnetic
subsystem in its essence and the split MD resonances of the disk-shaped
ferrite particle. In particular, the maximal magnitudes in the scattering and
absorption cross-sections can be uniquely matched with the MD± dispersion
curves and ferromagnetic resonance position in the $f$-$H_{0}$ space,
respectively, shown above in Fig. 2(a). However, since the primary wave has a
linear polarization, here the areas of existence for both split MD modes
experience a discontinuity in the region of ferromagnetic resonance. In fact,
the MD± resonances disappear completely in the region where the real part of
$\mu^{+}$ has a negative value, within which the magnetic losses are
significant and the field cannot be localized inside the particle [44]. We
should note that such an effect for axially magnetized ferrite disks has been
theoretically predicted [53] and experimentally confirmed for
$\mathsf{Al}$-doped YIG disks in the microwave frequency range (see also Fig.
3 in Ref. [44]). This additionally verifies our results obtained.

Figure 3(b) demonstrates the separate contributions of the electric
$\textbf{m}^{\textrm{E}}$ and magnetic $\textbf{m}^{\textrm{H}}$ parts of the
magnetic dipole moment m to the scattering. One can conclude that the
$\textbf{m}^{\textrm{H}}$ part makes a substantial contribution near the
frequency of ferromagnetic resonance, where it affects the behaviors of the
MD+ mode and is responsible for absorption. In the rest of the range of
parameters $f$-$H_{0}$, the contribution of this part of the magnetic dipole
moment is negligible, and the resonant scattering properties of the ferrite
disk are determined mainly by the $\textbf{m}^{\textrm{E}}$ contribution. The
apparent resonant behavior of the scattering cross-section in the range $f>10$
GHz and $H_{0}>3$ kOe arises due to the excitation of an electric dipole
resonance, but it is not important for our present consideration.

The specific influence of the magnetic subsystem is that the overall magnetic
moment m of the MD+ mode undergoes rotation as the magnetic field strength
$H_{0}$ increases, whereas no such changes are observed for the MD- mode. The
representative patterns of the magnetic field magnitude and flow as well as
the orientation of the vector m for these split modes are shown in Fig. 3(c).
In particular, the presented pictures suggest that the rotation of the
magnetic dipole moment m for the MD+ mode is clockwise, with increasing
$H_{0}$. Further, it is of interest to find out how this rotation manifests
itself in a system of several electromagnetically coupled ferrite particles,
particularly, in trimers that form a metasurface.

##  IV Magnetic dipole moments hybridization in trimer

Let us now turn to the study of the spectral behaviors of our metasurface
based on trimers. It is known that the electromagnetic response of the
metasurface with composite meta-atoms (clusters) containing three or more
resonant particles is determined both by the properties of individual
particles and, to a greater extent, by the near-field coupling between these
particles within the cluster [57]. This electromagnetic coupling results in
existing collective hybrid modes of the cluster. Among a large variety of such
collective modes, of particular interest are the so-called radial
(pseudomonopole) and azimuthal (toroidal) modes [31, 58, 36] that arise from
the MD mode hybridization (hereinafter we denote it as the collective MD
mode). These modes are not radiative, which is important from the practical
viewpoint making it possible to realize strong field localization inside the
system when the metasurface is operated on these modes (nonetheless, for the
trimers, the pseudomonopole exhibits a nonzero magnetic octupole moment
$Q^{(m)}_{yxx}$, that in some cases may unmask it [31]). Although both these
modes of the trimer arise from the MD mode of individual disk-shaped
resonators considered above, there is some difference in the arrangement of
three magnetic dipole moments, each of which belongs to the corresponding
particle (recall that these magnetic moments still lie in the horizontal plane
of the metasurface). In particular, the toroidal dipole mode appears from a
head-to-tail arrangement of the in-plane magnetic dipole moments providing the
toroidal dipole moment of this system is oriented out-of-plane. In contrast,
for the pseudomonopole, there is their tail-to-tail (or head-to-head)
arrangement (see Table A in Appendix A).

We should note that the choice of a trimer-based meta-atom for the formation
of our metasurface is conditioned by the fact that such a cluster provides the
most straightforward rotational arrangement of particles able to support the
toroidal mode [34]. Moreover, such a cluster configuration allows one to
efficiently separate the toroidal dipole from other higher-order electric and
magnetic multipoles that exist in the meta-atom, compared to clusters composed
of the even number of particles like a four-rod configuration [31].

Figure 4: Transmission spectra ($|T|^{2}$) of the trimer-based magnetic
metasurface illuminated by a normally incident plane wave with either left-
handed (LCP, upper panel) or right-handed (RCP, bottom panel) circular
polarization at four specific strengths of the bias magnetic field $H_{0}$.
Here $p=35$ mm and $d=1$ mm, while other parameters of the problem are the
same as in Fig. 2.

Next, we study the manifestation of these modes in the transmission spectra of
our metasurface and fix changes in the position of the corresponding resonant
frequencies with an increase in the strength of the bias magnetic field
$H_{0}$. For this study, we consider that the metasurface is illuminated by a
normally incident plane wave with either left-handed (LCP) or right-handed
(RCP) circular polarization. Here, we have chosen circularly polarized waves
to irradiate the structure, since they are eigenwaves of the longitudinally
magnetized ferrite, providing independent excitation of the MD+ and MD- modes
for the purity of our experiment.

In the framework of the RF module of the COMSOL MULTIPHYSICS, a single unit
cell of the metasurface containing a trimer is implemented, whereas their two-
dimensional arrangement is imitated by applying the periodic (Floquet)
boundary conditions repeating the unit cell infinitely in both transverse
directions. We dispose of the radiating and receiving ports above and below
the metasurface, respectively. Just like before, after simulation, we retrieve
values of the transmission coefficient $|T|^{2}=|S_{21}|^{2}$ as a function of
the frequency $f$ and the strength of the external magnetic field $H_{0}$ for
the LCP and RCP incident waves.

Figure 5: (a-f) Contributions of the toroidal dipole, electric dipole, and
magnetic dipole moments (presented on the logarithmic scale) to the effective
scattering cross-section calculated for the single unit cell of the trimer-
based magnetic metasurface and rearrangement of the magnetic dipole moments in
the trimer meta-atom with increasing the bias magnetic field strength $H_{0}$.
The metasurface is irradiated with the LCP wave. Here, yellow, black, and red
arrows represent the orientation of the individual magnetic moments of disk-
shaped particles and the flow of the magnetic and electric fields within the
unit cell, respectively. They are calculated at the frequencies marked by
vertical grey lines ending with arrows. Here $p=35$ mm and $d=1$ mm, while
other parameters of the problem are the same as in Fig. 2.

The results of these calculations are summarized in Fig. 4. The collective MD
mode of the trimer is effectively excited in the metasurface when it is
irradiated with an LCP wave, while when it is irradiated with an RCP wave,
only a very slight manifestation of this mode is observed in the transmission
spectra. When $H_{0}$ increases, the corresponding resonant frequency is
shifted towards higher frequencies, and the quality factor of the resonance
changes, whereas the manifestation of the resonance in the spectra of the RCP
wave becomes to be smoothed out (just for convenience, in the figure, we have
marked the positions of the collective MD resonance on the frequency scale
with grey vertical lines ending with arrows and also indicated its particular
resonant frequencies in rectangular frames). The change in the quality factor
is explained by the approaching of the mode frequency to the ferromagnetic
resonance frequency and, thus, an increase in the level of magnetic losses.
Moreover, this effect is also caused by a change in the characteristic of the
collective MD mode itself. Thus, the appearance of the collective MD mode in
the transmission spectra of the RCP wave at low $H_{0}$ indicates that the
field distribution of this mode is different from that at the higher $H_{0}$,
where this resonance completely disappears. It means that in the particular
$H_{0}$ range, the components may appear in the field of the collective MD
mode which can interact with the magnetic subsystem even when the metasurface
is irradiated with the RCP wave. All these features can be revealed from the
dipole contributions and electromagnetic field patterns calculated for the
metasurface unit cell.

Therefore, to elucidate all conditions of the corresponding resonance, the
magnitude of the toroidal dipole T, electric dipole p, and magnetic dipole m
moments of the trimer included in the metasurface unit cell are calculated in
the COMSOL MULTIPHYSICS for different values of $H_{0}$. The results are
depicted in Fig. 5. We consider only the case when the metasurface is
irradiated with the LCP wave (i.e., we are interested in the hybridization of
the MD+ mode only). These are the contributions to the effective scattering
cross-section, which would be in the case of scattering by the unit cell into
free space (they are calculated with respect to the center of mass of the
trimer using Eqs. (7)-(10), where the volume $V_{s}$ is replaced by the
volumes of all disks forming the trimer; in our case, the centers of mass of
the trimer and the unit cell coincide). The obtained spectral curves are
supplemented by patterns of the magnitude distribution and flow of the
magnetic and electric fields within the unit cell. These patterns are plotted
in the plane fixed at the half-height of the disks (at $z=0$).

The obtained spectral curves suggest that in the selected frequency range,
only the electric parts of the electric dipole ($\textbf{p}^{\textrm{E}}$) and
toroidal dipole ($\textbf{T}^{\textrm{E}}$) moments make a significant
contribution to the studied resonant state (we have not presented here the
contribution of the magnetic part of the toroidal dipole moment
($\textbf{T}^{\textrm{H}}$) because of its negligible value). The small
magnitude of the overall magnetic dipole moment m is explained by the fact
that, due to the specific symmetry of the trimer, the magnetic dipole moments
of individual resonators are mutually compensated in the cluster. As $H_{0}$
increases, the resonant frequency of the collective MD mode experiences a
redshift, and the magnitude of the toroidal dipole moment noticeably
increases. The most remarkable feature is that with the change in $H_{0}$,
there is also a rearrangement in the order of the magnetic dipole moments
belonging to the collective MD mode.

First of all, one can see from the patterns of the electromagnetic field
distributions plotted within the unit cell that the appearance of the fields
in each ferrite disk closely resembles that of the MD mode of the single
resonator, and for all selected values of the magnetic bias field $H_{0}$,
there are just different manifestations of the same collective MD mode. The
evolution of this collective mode consists in the fact that as $H_{0}$
increases, the individual magnetic moment in each disk undergoes a sequential
clockwise rotation. At the extreme fixed values of $H_{0}=3.4$ kOe [Fig. 5(a)]
and $H_{0}=5.0$ kOe [Fig. 5(f)], the arrangement of magnetic dipole moments in
the trimer corresponds to the radial and toroidal order, respectively, while
in-between [Figs. 5(b-e)], this arrangement has an intermediate order.

This transformation from the radial to toroidal order resembles a phase
transition in solids and is accompanied by a change in the localization of the
electric field in the central part of the cluster. Remarkably, at the stage of
existence of the radial order, the electric field intensity has a local
minimum in the center of the cluster and six bright hotspots on the periphery
of the disks. Contrariwise, in the toroidal order, it has a bright hotspot in
the center of the cluster and three less bright hotspots on the periphery of
the disks. The observed feature makes it possible to realize control over the
operating regimes of the trimer-based magnetic metasurface by tuning the
strength of the external magnetic field. More precisely, the near-field
coupling between the particles forming meta-atoms depends on the strength of
the bias magnetic field, which changes the character of electromagnetic
excitation of the metasurface from the radial order to the toroidal one. Due
to the distinctive field configurations of these orders, the proposed
metasurface can be considered for designing magnetic field sensors and
magnetic field-controlled and nonreciprocal devices (e.g., see Refs. [59,
60]).

##  V Conclusions

In conclusion, we have presented a thorough investigation of the transition of
the radial order to the toroidal one in a magnetic metasurface conditioned by
the influence of an external static magnetic field. The metasurface with
trimer-based clusters arranged in a two-dimensional array with the translation
unit cell based on identical disk-shaped ferrite resonators (made of
$\mathsf{Al}$-doped YIG material) has been considered. The study has been
carried out with the use of the COMSOL MULTIPHYSICS finite-element
electromagnetic solver. To provide physical insight, the simulations have been
supplemented by analytical and numerical modeling of the eigenfrequencies and
scattering characteristics of the individual ferrite resonator. The evolution
of the collective mode ordering with changing the bias magnetic field has been
considered. It was found that the MD-based collective hybrid mode of the
trimer can be gradually transformed from the radial to toroidal order with
increasing the bias magnetic field strength.

We believe that the proposed design of the metasurface can be useful in
practical applications due to the possibility of easily switching the nature
of the excitation between different magnetic orders and conditions of the
electric field localization inside the system. Also, the results obtained here
can be extended to metasurfaces containing particles made of optically active
materials with tensor-valued permittivity.

## Acknowledgments

V.R.T. is grateful for the hospitality and support from Jilin University,
China. A.B.E. thanks funding support from the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy within
the Cluster of Excellence PhoenixD (EXC 2122, Project ID 390833453).

##  Appendix A Symmetry-adapted linear combination (SALC) approach

Figure 6: Symmetry adapted basis for the trimer with the $C_{3v}$ symmetry.

The chosen basis for dipole magnetic moments is shown in Fig. 6. The unit
vectors ${\bf m}_{i}$ with odd and even index $i$ are oriented parallel and
perpendicular to the planes of symmetry $\sigma$, respectively. One can
calculate SALCs using the projection operators [61]:

${\bf m}_{kj}^{\Gamma}\\!=\\!\dfrac{1}{g}\sum_{R}\chi_{kj}^{\Gamma}(R)R{\bf
m}_{i},$ (13)

where the summation is performed with respect to the elements $R$ of the
$C_{3v}$ group. These elements are $e$, $C_{3}$, $C_{3}^{-1}$, $\sigma_{1}$,
$\sigma_{2}$, and $\sigma_{3}$ (see Ref. [36]). $\Gamma$ is a representation
$A$ or $B$ of the $C_{3v}$ group, $\chi_{kj}^{\Gamma}(R)$ is the $kj$-th
component of the irreducible representation (IRREP) $\Gamma$ of the element
$R$, $R{\bf m}_{i}$ is a basis element ${\bf m}_{i}$ transformed by the
operator $R$, and $g$ is the order of the group (in our case, $g=6$).

The normalized eigenvectors of the trimer are presented in the third column of
Table A. Corresponding eigenmodes are also identified performing the full-wave
simulation in Ref. [31].

Table A: Azimuthal (toroidal) and radial (pseudomonopole) eigenmodes of a trimer with the $C_{3v}$ symmetry in terms of vectors ${\bf m}_{i}$ IRREP | Eigenmode image | Description
---|---|---
$A$ |  |  $\begin{array}[]{c}\textrm{Azimuthal mode}\\\ \dfrac{{\bf m}_{2}+{\bf m}_{4}+{\bf m}_{6}}{\sqrt{3}}\end{array}$
$B$ |  |  $\begin{array}[]{c}\textrm{Radial mode}\\\ \dfrac{{\bf m}_{1}+{\bf m}_{3}+{\bf m}_{5}}{\sqrt{3}}\end{array}$

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