Graded-index magnonics
The wave solutions of the Landau–Lifshitz equation (spin waves) are characterized by some of the most
 complex and peculiar dispersion relations among all waves. For example, the spin-wave (“magnonic”) dispersion
 can range from the parabolic law (typical for a quantum-mechanical ele...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Cite this: | Graded-index magnonics / C.S. Davies, V.V. Kruglyak // Физика низких температур. — 2015. — Т. 41, № 10. — С. 976–983. — Бібліогр.: 97 назв. — англ. |
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| citation_txt | Graded-index magnonics / C.S. Davies, V.V. Kruglyak // Физика низких температур. — 2015. — Т. 41, № 10. — С. 976–983. — Бібліогр.: 97 назв. — англ. |
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| description | The wave solutions of the Landau–Lifshitz equation (spin waves) are characterized by some of the most
complex and peculiar dispersion relations among all waves. For example, the spin-wave (“magnonic”) dispersion
can range from the parabolic law (typical for a quantum-mechanical electron) at short wavelengths to the
nonanalytical linear type (typical for light and acoustic phonons) at long wavelengths. Moreover, the longwavelength
magnonic dispersion has a gap and is inherently anisotropic, being naturally negative for a range of
relative orientations between the effective field and the spin-wave wave vector. Nonuniformities in the effective
field and magnetization configurations enable the guiding and steering of spin waves in a deliberate manner and
therefore represent landscapes of graded refractive index (graded magnonic index). By analogy to the fields of
graded-index photonics and transformation optics, the studies of spin waves in graded magnonic landscapes can
be united under the umbrella of the graded-index magnonics theme and are reviewed here with focus on the challenges
and opportunities ahead of this exciting research direction.
|
| first_indexed | 2025-12-07T18:04:05Z |
| format | Article |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10, pp. 976–983
Graded-index magnonics
C.S. Davies and V.V. Kruglyak
School of Physics, University of Exeter, Stocker road, Exeter EX4 4QL, United Kingdom
E-mail: V.V.Kruglyak@exeter.ac.uk
Received May 1, 2015, published online August 25, 2015
The wave solutions of the Landau–Lifshitz equation (spin waves) are characterized by some of the most
complex and peculiar dispersion relations among all waves. For example, the spin-wave (“magnonic”) dispersion
can range from the parabolic law (typical for a quantum-mechanical electron) at short wavelengths to the
nonanalytical linear type (typical for light and acoustic phonons) at long wavelengths. Moreover, the long-
wavelength magnonic dispersion has a gap and is inherently anisotropic, being naturally negative for a range of
relative orientations between the effective field and the spin-wave wave vector. Nonuniformities in the effective
field and magnetization configurations enable the guiding and steering of spin waves in a deliberate manner and
therefore represent landscapes of graded refractive index (graded magnonic index). By analogy to the fields of
graded-index photonics and transformation optics, the studies of spin waves in graded magnonic landscapes can
be united under the umbrella of the graded-index magnonics theme and are reviewed here with focus on the chal-
lenges and opportunities ahead of this exciting research direction.
PACS: 75.30.Ds Spin waves;
75.75.–c Magnetic properties of nanostructures;
75.78.–n Magnetization dynamics.
Keywords: magnon, spin waves, graded-index, magnetization.
1. Introduction
It can be somewhat surprising to recognize that the entire
field of magnonics [1] — the study of spin waves [2,3] — is
built upon the foundation of the Landau–Lifshitz equation
[4]. In an ordered ensemble of spins, immediate (and some-
times also somewhat more distant) neighbours are coupled
via the quantum-mechanical exchange interaction, while the
interaction between spins at further distances from each oth-
er is dominated by the magneto-dipole field, described by
the Maxwell equations. By perturbing the static configura-
tion of spins locally, propagating spin waves can be excited.
The Landau–Lifshitz and Maxwell equations operate with
the classical magnetization vector (M) defined as the aver-
age magnetic moment (associated with the spins) per unit
volume. In this approximation, the spin waves take the form
of propagating waves of precessing magnetization, and the
Landau–Lifshitz equation relates the precession of the mag-
netization to the effective magnetic field, which can also be
a function of the magnetization distribution in the sample.
Until recently, the majority of studies in magnonics
dealt with samples either having or assuming uniform con-
figurations (henceforth, we often refer to these as “land-
scapes”) of the magnetization and magnetic field. This
simplicity enabled detailed studies of the spin-wave
(“magnonic”) dispersion and the spectrum of standing spin
waves, which have their wave vector quantized due to con-
finement by the geometrical boundaries of the sample. It
has been established that the magnonic dispersion is intrin-
sically anisotropic in the dipole and dipole-exchange (i.e.
long-wavelength) regimes but is isotropic in the exchange
(short-wavelength) regime. In particular, the anisotropy of
the magnonic dispersion can lead to extremely peculiar
character of spin-wave propagation and scattering from
geometrical boundaries [5–9]. Nonetheless, the direction of
the spin-wave beam remains constant as long as the mag-
netic landscape remains uniform.
However, it was soon realized and increasingly often
exploited that, by deforming the magnetic landscape via
making either the magnetization or the effective magnetic
field or both non-uniform, the propagation path of spin
waves could be deliberately modified. The study of spin
waves in continuously varying magnetic landscapes forms
the scope and definition of the field of graded-index
magnonics [10]. This is similar in spirit to (and indeed, has
been inspired by) graded-index optics [11] (or transfor-
mation optics [12]), which seek to modify the light disper-
sion in photonic and electromagnetic systems using a spa-
tially varying (“graded”) refractive index. However, the
© C.S. Davies and V.V. Kruglyak, 2015
mailto:V.V.Kruglyak@exeter.ac.uk
Graded-index magnonics
magnonic dispersion described by the Landau–Lifshitz
equation is arguably far more complex and peculiar as
compared to that of light, thereby offering extremely rich
opportunities and a bright outlook to the field of graded-
index magnonics. The field of magnonics has received a
tremendous amount of attention in recent years, particular-
ly due to the potential for spin waves to act as information
carriers within the data storage, communication and pro-
cessing technologies [13]. It is therefore perceived that
graded-index magnonics as a theme may well simplify the
construction of, and indeed give rise to, many technologi-
cal applications just as transformation optics has done for
electromagnetic technologies.
Here, we present a nonexhaustive review of the concept
of the graded-index magnonics together with the key re-
search results and directions that are united by this theme.
We begin by briefly reminding the reader of the key equa-
tions governing the dispersion of spin waves in uniform
media and their scattering from magnetic non-uniformities.
The path of spin-wave beams through landscapes with a
continuous variation of parameters that determine mag-
nonic dispersion is then postulated to be a result of multi-
ple scattering events from infinitesimally weak non-uni-
formities [14–16]. This idea is fed into the following
review and discussion of a representative selection of stud-
ies of spin waves in nonuniform magnonic landscapes,
which aim to show the diversity of phenomena falling un-
der the markedly broad umbrella of the graded-index
magnonics concept. The discussion is illustrated using nu-
merical solutions of the Landau–Lifshitz equation (micro-
magnetic simulations [17,18]) obtained using Object-
Oriented Micro-Magnetic Framework [19]. The paper is
concluded with some general remarks on further progress
in the field, with emphasis on opportunities arising from
mapping ideas and methodology from transformation op-
tics onto the exceptionally rich world of the Landau–
Lifshitz equation and its solutions for graded-index mag-
nonic media and devices.
2. Dispersion, propagation and scattering of spin waves
in uniform thin magnetic films
As for any waves, the direction of the spin-wave propa-
gation is given by the group velocity (vg), calculated as the
gradient of the frequency ( 2 )fω = π in the reciprocal
space. The group-velocity vector is therefore orthogonal to
curves of constant frequency, often referred to as iso-
frequency curves (or surfaces in the 3D case) [14]. Hence,
we begin by reviewing the key results concerning the
magnonic dispersion in thin-film magnetic samples.
The exchange interaction makes a negligible contribu-
tion to the dispersion of long-wavelength spin waves,
which are therefore termed “magnetostatic” or “dipolar”
waves. Their dispersion is implicitly defined by [20,21]
2 2 2 2( 1) ( 1)k kz yµ + + µ −ν + +
2 2
2 2 22 cot 0
k kz zk k s ky y
+ µ − − − − = µ µ
(1)
where s is the film thickness, ky and kz are the in-plane
projections of the wave vector with length k, and µ and v
are
2 21 M H
H
ω ω
µ = +
ω −ω
and 2 2
M
H
ω ω
=
ω −ω
v (2)
where S4 ,M Mω = πγ ,H iHω = γ where Hi is the internal
static magnetic field aligned along the z-axis, MS is the
magnetization of saturation and γ is the gyromagnetic ratio.
For each specific frequency value, Eq. (1) describes an
isofrequency curve, every point of which corresponds to
the wave vector of a spin wave that is allowed to propagate
at this frequency. Fig. 1(a) shows two typical isofrequency
curves for dipolar spin waves in a 7.5 μm thick thin film of
yttrium–iron–garnet (YIG), assuming Hi = 1.25 kOe and
MS = 139 G. It is easy to see that generally the direction of
the group velocity (normal to the curve) is not collinear
with that of the wave vector, which is a direct consequence
of the anisotropy of the dispersion relation. One of the two
symmetry axes of the isofrequency curves is parallel to the
magnetization. If ω is greater (smaller) than the frequency
of the uniform ferromagnetic resonance FMRω =
( ),H H M= ω ω +ω the projection of the group velocity
onto the direction of the wave vector is positive (negative).
As the spin-wave wavelength gets shorter, the exchange
interaction cannot be neglected anymore and needs to be
taken into account on equal footing with the magneto-
dipole interaction. The dispersion of such so-called “di-
pole-exchange” spin waves can be written as [22]
2 2
00( )( )H H MDk Dk Fω = ω + γ ω + γ +ω , (3)
where 2 / SD A M= is the exchange stiffness (A is the ex-
change constant) and the zeroth dipole-dipole matrix ele-
ment F00 is defined as
( )
22
00 00 00 21 1 yM z
H
kkF P P
k kDk
ω = + − − ω + γ
and 00
1 exp ( )1 ksP
ks
− −
= − .
Typical dipole-exchange isofrequency curves are shown
in Fig. 1(b) for a 100 nm thick Permalloy [23] thin film,
biased by Hi = 500 Oe. As the frequency of the dipole-
exchange spin wave increases, the anisotropy of the disper-
sion decreases. At very short wavelength (and therefore
very high frequencies), the isotropic exchange interaction
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10 977
C.S. Davies and V.V. Kruglyak
dominates the magnonic dispersion. The dispersion of ex-
change spin waves is isotropic, and so, the corresponding
isofrequency curves have circular shape.
The isofrequency curves presented in Fig. 1 are suffi-
cient on their own to describe the propagation of spin
waves across uniformly-magnetized media, e.g. to explain
the origin of spin wave caustics [9,24]. Spin waves excited
by a nearly point-like magnonic source [9,25,26] have a
broad distribution of wave vectors. However, as can be
seen from Fig. 1, significant sections of the magnetostatic
isofrequency curves are nearly straight. Hence, whilst spin
waves with a range of wave vectors are excited, their group
velocities associated with these wave vectors tend to have
similar, nearly collinear directions, leading to the for-
mation of tightly focused, narrow spin-wave caustic
beams. In context of the perceived magnonic technology, it
is important to note that the noncollinearity of the mag-
nonic phase and group velocities within the beams repre-
sents a complication for designs of magnonic devices ex-
ploiting spin-wave phase [1,27–29].
The scattering (i.e. reflection and refraction) of spin
waves from an interface between two uniform magnetic
media [30] can also be described using the isofrequency-
curve method. Let us first consider a magnonic caustic beam
propagating across a uniform 7.5 μm thick YIG film biased
by a magnetic field of non-uniform strength. For simplicity,
we will assume that the field is parallel to the z-axis but its
strength takes different values in and then remains constant
within each of regions A, B and C of the sample (Fig. 2 (a)),
so that Hi,A = Hi,C = 11.7 kOe < Hi,B = 12 kOe. Spin waves
with frequency of 34.8 GHz and a broad distribution of
wave vectors are excited by a dynamic field localized at the
bottom left corner of the sample. As the spin wave beam
propagates from Region A to B, it refracts away from the
normal to the interface and then again refracts towards the
normal as it propagates from Regions B to C (Fig. 2 (a)).The
reflections of the beam from each interface and edge of the
sample are also visible, albeit with a reduced intensity.
To understand the behavior observed in Fig. 2(a), the
isofrequency curves belonging to each region of the sample
are plotted in Fig. 2(b). Upon increasing the effective field
strength (while keeping the frequency and the magnetization
orientation fixed), the magnetostatic isofrequency curves on
either side of the ky-axis are pushed away from each other,
leading to an increased gradient along their quasi-linear sec-
tions. We note that this behavior occurs only for
,FMRω < ω as in the present case, but is reversed otherwise.
The incident spin waves in Region A have a range of wave
Fig. 1. (a) (Color online) Typical isofrequency curves of magne-
tostatic spin waves in a YIG film are plotted using Eqs. (1) and
(2) for frequencies above (5.8 GHz) and below (5 GHz) the FMR
frequency. Examples of group velocities vg corresponding to
wave vectors k are indicated schematically on each curve. (b)
Typical isofrequency curves of dipole-exchange spin waves in a
Permalloy thin film are plotted using Eq. (3) for frequencies of
10 GHz and 17 GHz.
(a)
(b)
400
–400
0
k
vg ky, mm–1
M
–400 400
kz, mµ –1
80
–80
ky, mµ
–1
M
–80 –40 40 80
kz, mm–1
vg
k
5.0 Ghz
5.8 GHz
17 Ghz
10 GHz
5.0 Ghz
5.8 GHz
Fig. 2. (a) (Color online) The calculated distribution of the dynamic
magnetization is shown for a spin wave excited at 34.8 GHz in the
bottom left corner of the YIG sample. The arrows show the group ve-
locity directions of the incident (vg,i), transmitted (vg,t) and reflected
(vg,r) beams. (b) The isofrequency curves calculated for regions A and
C (solid), and B (dashed) are shown for the spin-wave in panel (a).
C20°11.7 kOe
vg r,
Bvg r,41°
vg,t
12 kOe
A
20°
vg,i 11.7 kOe
20°
vg r,
B
41°
vg,t
A, C
vg,i
B
A, C
0.5
–0.5
Hi
min max
(a) (b)0.25 mm
kz, mµ –1
z
y | |m 2
vg r, ′
′ k , my µ –1
0 0.5
978 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10
Graded-index magnonics
vector values and directions, even though the associated
group velocities are nearly collinear (as explained earlier).
Let us fix the wave-vector component that is parallel to the
interface, i.e. ky. The beam directions in the different regions
of the sample will then be given by the normal to the
isofrequency curves at points of their intersection by a verti-
cal line corresponding to the selected ky value.
The concept presented in Fig. 2 is the cornerstone of in-
terpretation of many experimental observations in the field
of graded-index magnonics. Moreover, by tracing the spa-
tial evolution of isofrequency curves in media with con-
tinuous variation of magnetic properties defining the mag-
nonic dispersion, one is able to not only explain but also
predict and design the character of propagation and scatter-
ing of spin-wave beams. In the following, we review some
of the most remarkable effects already observed in graded-
index magnonic media (the properties of which are de-
scribed by the Landau–Lifshitz equation) and furthermore
consider the potential new avenues of research available to
future researchers within this theme.
3. Brief review of effects observed in graded-index
magnonic media
Perhaps, one of the first graded-index effects observed
during the latest boom of magnonics research was the dis-
covery of spin-wave modes confined within so-called
“spin-wave wells”, first in stripes [31,32] and then in
squares and rectangles [33,34]. This confinement is a di-
rect consequence of the existence in the magnonic disper-
sion of a threshold frequency below which spin waves
cannot propagate or be excited. This threshold frequency
approximately scales with the value of the static internal
magnetic field. Hence, spin waves that are allowed in the
regions of a reduced internal magnetic field (typically, cre-
ated by the demagnetising field in magnetic elements of
nonellipsoidal shape due to edge magnetic charges) are not
allowed to propagate into the bulk of the sample, where the
demagnetising field is reduced and the internal field is
therefore increased.
This phenomenon of spin-wave confinement in internal
magnetic field landscapes is analogous to confinement of a
quantum-mechanical electron in a potential well, whereby
the spin wave plays the role of the electron wave function
and the internal magnetic field plays the role of the elect-
ron potential energy. However, the vectorial nature of the
magnetic field makes it challenging to induce confinement
in more than one dimension. Hence, a suitably excited spin
wave could still propagate in the direction orthogonal to
the direction of the field induced confinement, leading to
the idea of spin-wave channeling [35,36]. The lateral ex-
tent of such magnonic channels (typically created close
and parallel to the edge of a thin-film sample) is directly
linked to the spin-wave frequency [37]. Moreover, by con-
tinuously varying the geometry of the magnonic wave-
guide and therefore of the associated non-uniform field
distribution, the spin-wave channels can “cling” to the
edges of a nonrectangular structure, resulting in spin-wave
splitting and potentially magnonic interferometer function-
ality [38].
The anisotropic dispersion of long-wavelength spin
waves yields another (and probably, unique to magnonics)
scheme of wave confinement. Indeed, by fixing the direc-
tion of the wave vector, we see from Fig. 1(a) that there
exists a range of small k-values for which spin wave exci-
tation is allowed for one but is forbidden for the other (or-
thogonal) direction of the magnetization relative to that of
the wave vector. Hence, regions of curved magnetization
can also prohibit propagation and therefore confine magne-
tostatic and dipole-exchange spin waves, as indeed was
observed e.g. in Refs. 39, 40. The same mechanism could
lead to the confinement of spin waves in the in-plane direc-
tion that is orthogonal to that of the static internal magnetic
field, e.g. when it is orthogonal to the edge of a thin-film
magnetic stripe.
The variation of the internal magnetic field has also been
used to continuously tune the wavelength of propagating
spin waves. Let us consider the funnel-shaped Permalloy
element shown in Fig. 3(a). With a transverse bias field ap-
plied (HB = 1.25 kOe), the average demagnetizing field in-
creases in strength as we move along the funnel from left to
right [41]. As a result, the projection of the total internal
field onto the magnetization decreases, thereby decreasing
also the spin-wave frequency at a given wavelength. Con-
sidering instead a fixed frequency, the vertices of the higher-
frequency isofrequency curves in Fig. 1(a) move to higher
Fig. 3. (a) (Color online) The projection of the effective field on the static
magnetization is shown for a 100 nm thick funnel-shaped Permalloy
waveguide. (b) A snapshot of the out-of-plane component of the dynamic
magnetization is shown for a spin wave excited harmonically at 14 GHz
at the far left end of the waveguide.
(a)
(b)
z
25 –25
mx, G
HB =1.25 kOe
2.5 µm
2.5 0
internal field, kOe
y
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10 979
C.S. Davies and V.V. Kruglyak
ky-values, and so, the spin-wave wavelength decreases. This
wavelength reduction is clearly seen in the shown snapshot
of the dynamic magnetization associated with the spin wave
continuously excited at 14 GHz at the far left of the structure
(Fig. 3(b)). It is interesting to note that the demonstration of
this effect for a bulk YIG sample by Schlömann in 1964 was
probably the first-ever manifestation of the graded-index
magnonics principles discussed in literature [42]. The use of
this so-called Schlömann mechanism of spin-wave excita-
tion to couple free-space microwaves to spin waves of many
orders of magnitude shorter wavelength propagating in
Permalloy microstructures was demonstrated in Refs. 43,44.
The formation of spin-wave wells discussed earlier can be
interpreted as a result of the inability of the wavelength vari-
ation (at constant frequency) to compensate for the variation
of the internal magnetic field.
In two dimensions, the spatial variation of the internal
magnetic field and magnetization enables the steering of spin
waves both in continuous films and networks of magnonic
waveguides. The possibility of steering spin-wave caustics in
thin magnetic films arises directly from the strict relationship
between the directions of the static magnetization and caus-
tics at a given value of the internal magnetic field. However,
the continuous variation of the internal magnetic field and
magnetization can lead to even more striking consequences,
such as a complete disappearance of one of the spin-wave
beams launched into a T-junction of magnonic wave guides
[10]. Fig. 4(a) shows the configuration of the internal mag-
netic field and magnetization in an asymmetrically magne-
tized T-junction of 5 µm wide/100 nm thick Permalloy
waveguides. Fig. 4(b) and (c) show snapshots of the dynamic
magnetization due to a spin-wave beam propagating from the
vertical “leg” into the right “arm” of the junction from time-
resolved scanning Kerr microscopy (TRSKM) and OOMMF
simulations, respectively. The spin-wave beam that was sup-
posed to propagate to the left arm of the junction is absent,
because the non-uniform field and magnetization (i.e. the
“graded magnonic index”) steer it into the lower edge of the
left arm, from which it is then backward-reflected into the
right arm of the junction. The curves and arrows in Fig. 4 (c)
show the direction of the group velocity of the incident and
reflected spin waves calculated using the approach presented
in Fig. 2 (b). Of a special note is the curving of the spin-wave
beam towards the normal to the rear edge of the arm by the
graded magnonic index. This beam curving is a spin-wave
analogue of the so-called "mirage effect", which was also
observed in simulations from Ref. 45.
Even relatively small regions of graded magnonic index
could either present obstacles for or find application in
magnonic data and signal processing devices [46]. Indeed,
any bending of magnonic waveguides necessarily leads
either to spatial variation of the angle between the static
magnetization and the wave vector, or to curvature of the
static magnetization. In the former case, the graded
magnonic index can lead to scattering and even transfor-
mation of the propagating spin wave [47,48]. In the latter
case, the curved magnetization can lead to such exotic ef-
fects as the curvature induced (“geometrical”) magnetic
anisotropy [49,50], which could be described in terms of
magnetic energy contributions characteristic of the
Dzyaloshinskii–Moriya interaction [51,52].
At the same time, transverse Bloch-type magnetic do-
main walls represent a natural reflectionless potential for
propagating spin waves. Indeed, spin waves propagating
through such domain walls preserve their amplitude but
acquire a phase shift. This phase shift is directly related to
the magnetization rotation within the domain wall - specif-
ically, the spin-wave phase shifts are 90° and 180° for 180°
and 360° domain walls, respectively. In contrast to Bloch
walls, the amplitude reflectivity of spin waves from Néel-
type magnetic domain walls can vary significantly (ranging
between the extremes of zero and unity) depending on the
Fig. 4. (Color online) Spin waves in an asymmetrically magnetized
Permalloy T-junction (after Ref. 10). (a) The calculated distributions
of the static magnetisation (arrows) and the projection of the internal
magnetic field onto the magnetization (color scale) are shown for the
magnetic field of HB = 500 Oe applied at 15° to the vertical sym-
metry axis. Each arrow represents the average of 5×5 mesh cells.
(b) A TRSKM snapshot of the spin-wave beam propagating into the
arm of the Permalloy T-junction is shown for the bias magnetic field of
HB = 500 Oe applied at 15° relative to the leg of the junction. The fre-
quency of the cw magnetic "pump" field was 8.24 GHz. (c) The numer-
ically simulated out-of-plane component of the dynamic magnetization
corresponding to the experimental snapshot from panel (b) is shown
together with the directional unit vectors of the group velocities v̂ and
wave vectors k̂ extracted for the incident (index “i”) and reflected
(index “r”) spin-wave beams at kx = 0.94 µm–1. The pumping frequen-
cy in the simulations was 7.52 GHz. The difference in the frequency
values in the experiments and simulations was due to inevitable differ-
ences between the measured and simulated samples.
980 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10
Graded-index magnonics
thickness of the film and the wave vector. Given that mag-
netic domain walls themselves have been proposed as in-
formation carriers, there is also scope for creating hybrid
spin-wave/domain-wall devices.
In addition to the interaction of plane spin waves with
confined regions of graded magnonic index, a traditionally
active and discovery-rich research area of magnonics has
dealt with spin waves in extended, strongly nonuniform
micromagnetic states and textures, featuring magnetic vor-
tices and antivortices [56–59], complex domain structures
and skyrmions [62,63]. The relevant research results are
reviewed in other articles of this Special Issue [64–66].
At last but not least, the graded magnonic index can and
has in fact been used in design of magnonic crystals [67] in
several ways. The nonuniform internal magnetic field can
either serve to modulate the magnonic properties in the di-
rection of spin wave propagation [68,69], or to channel spin
waves through e.g. topographically defined landscapes
[70,71]. Alternatively, as discussed earlier, the same can be
achieved as a result of the non-uniform configuration of the
magnetization in patterned magnetic films [72–74].
4. Conclusions and outlook
Over the past decade, magnonics has emerged as one of
the most rapidly growing research fields in magnetism and
a potential rival of semiconductor technology in the field
of data communication and processing. However, together
with loss reduction, the control of the spin-wave trajectory,
the shortening of the wavelength of studied spin waves and
the associated miniaturization of realized functional mag-
nonic devices remain major challenges in both experi-
mental research and technological development in mag-
nonics. Here, we have reviewed the concept of graded-
index magnonics, which could help meet these challenges.
Indeed, the propagation of spin waves in graded magnonic
media can be controlled using sub-wavelength, continuous-
ly varying magnetic non-uniformities rather than physical
patterning. This should minimize scaling of the magnonic
device size with the spin-wave wavelength, in contrast to
e.g. magnonic crystal based approaches. This sort of cross-
over from studying magnonic phenomena associated with
ubiquitous nonuniformity of micromagnetic configurations
in geometrically patterned magnetic systems to the exploi-
tation of the graded magnonic index is predicted to drive
the magnonics research in the nearest future.
From the point of view of the Landau–Lifshitz equa-
tion, this trend will lead to two major directions of theoret-
ical development. Within the first of them, the concepts,
ideas and sometimes whole classes of solutions and theo-
retical methods developed in transformation optics and
quantum mechanics will continue to be mapped onto
magnonic systems [42,49,50]. In particular, the study of
exchange spin waves, which have an isotropic dispersion
described by a parabolic law in the continuous medium
approximation, will benefit from this approach. Within the
second direction, researchers will face the challenge of
developing a completely new theoretical formalism that
will fully account for the rich and exciting complexities
inherent to the Landau–Lifshitz equation. In addition to the
already discussed anisotropic dispersion of magnetostatic
and dipole-exchange spin waves, the challenges include
the non-linearity of the Landau–Lifshitz equation [75,76]
and exotic contributions to the magnetic energy, such as
the magneto-elastic coupling [77,78] and (nonreciprocal)
Dzyaloshinskii–Moriya [51,52] interaction, and magnetic
dissipative function [79,80].
We have limited the discussion above to the case of pat-
terned thin-film magnetic structures, which are in the focus
of current experimental studies and in which the graded
magnonic index is created by virtue of their patterning.
The key advantage of such samples is that, due to the mag-
netic hysteresis, their graded magnonic landscapes could
potentially be programmed e.g. by the external magnetic
field [81]. However, it is clear that the scope of the concept
is far broader. Indeed, the graded magnonic index can be
created through application of external non-uniform stimu-
li, ranging from the magnetic field due to the electrical
currents [82] or magnetic charges [83] through to electric
field, spin currents [85] and thermal gradients, including
those created optically [25,87,88]. An exciting extension of
the concept is that of non-stationary, dynamically con-
trolled graded-index landscapes [89,90]. Alternatively, the
means of nano- and micro-scale materials engineering al-
low one to create essentially arbitrary magnonic landscapes
[91–94], provided that the spin-wave damping could be
controlled at a reasonably low level. Finally, one should
not forget that the world of magnetic materials is not lim-
ited to transition metal ferromagnet and YIG samples. In-
deed, the spin dynamics and therefore spin waves in multi-
sublattice magnetic materials are generally faster and argu-
ably richer than one might think [95–97], and are still gov-
erned by the generalization of the Landau–Lifshitz equa-
tion. Extension of the graded-index magnonics concept to
such systems is certainly possible but is beyond the scope
of this paper.
The range of spin-wave phenomena covered by this
brief review challenges both the expertise of the authors
and the journal page limits for this Special Issue. Yet, we
hope that our contribution will help to inspire and guide
future researchers though the exciting world of graded-
index magnonics, which is both governed and created by
the Landau–Lifshitz equation.
Acknowledgements
The research leading to these results has received
funding from the Engineering and Physical Sciences Re-
search Council of the United Kingdom under projects
EP/L019876/1, EP/L020696/1 and EP/P505526/1. Sup-
porting research data may be accessed at
https://ore.exeter.ac.uk/repository/handle/10871/17998.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10 981
https://ore.exeter.ac.uk/repository/handle/10871/17998
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10 983
1. Introduction
2. Dispersion, propagation and scattering of spin waves in uniform thin magnetic films
3. Brief review of effects observed in graded-index magnonic media
4. Conclusions and outlook
Acknowledgements
|
| id | nasplib_isofts_kiev_ua-123456789-128079 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T18:04:05Z |
| publishDate | 2015 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Davies, C.S. Kruglyak, V.V. 2018-01-05T17:41:54Z 2018-01-05T17:41:54Z 2015 Graded-index magnonics / C.S. Davies, V.V. Kruglyak // Физика низких температур. — 2015. — Т. 41, № 10. — С. 976–983. — Бібліогр.: 97 назв. — англ. 0132-6414 https://nasplib.isofts.kiev.ua/handle/123456789/128079 The wave solutions of the Landau–Lifshitz equation (spin waves) are characterized by some of the most
 complex and peculiar dispersion relations among all waves. For example, the spin-wave (“magnonic”) dispersion
 can range from the parabolic law (typical for a quantum-mechanical electron) at short wavelengths to the
 nonanalytical linear type (typical for light and acoustic phonons) at long wavelengths. Moreover, the longwavelength
 magnonic dispersion has a gap and is inherently anisotropic, being naturally negative for a range of
 relative orientations between the effective field and the spin-wave wave vector. Nonuniformities in the effective
 field and magnetization configurations enable the guiding and steering of spin waves in a deliberate manner and
 therefore represent landscapes of graded refractive index (graded magnonic index). By analogy to the fields of
 graded-index photonics and transformation optics, the studies of spin waves in graded magnonic landscapes can
 be united under the umbrella of the graded-index magnonics theme and are reviewed here with focus on the challenges
 and opportunities ahead of this exciting research direction. The research leading to these results has received
 funding from the Engineering and Physical Sciences Research
 Council of the United Kingdom under projects
 EP/L019876/1, EP/L020696/1 and EP/P505526/1. Supporting
 research data may be accessed at
 https://ore.exeter.ac.uk/repository/handle/10871/17998. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур К 80-летию уравнения Ландау–Лифшица Graded-index magnonics Article published earlier |
| spellingShingle | Graded-index magnonics Davies, C.S. Kruglyak, V.V. К 80-летию уравнения Ландау–Лифшица |
| title | Graded-index magnonics |
| title_full | Graded-index magnonics |
| title_fullStr | Graded-index magnonics |
| title_full_unstemmed | Graded-index magnonics |
| title_short | Graded-index magnonics |
| title_sort | graded-index magnonics |
| topic | К 80-летию уравнения Ландау–Лифшица |
| topic_facet | К 80-летию уравнения Ландау–Лифшица |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/128079 |
| work_keys_str_mv | AT daviescs gradedindexmagnonics AT kruglyakvv gradedindexmagnonics |