Spin waves damping in nanometre-scale magnetic materials
Spin dynamics in magnetic nanostructured materials is a topic of great current interest. To describe spin motions in such magnetic systems, the phenomenological Landau–Lifshitz (LL), or the LL–Gilbert (LLG), equation is widely used. Damping term is one of the dominant features of magnetization dyn...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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Krivoruchko, V.N. 2018-01-05T17:12:44Z 2018-01-05T17:12:44Z 2015 Spin waves damping in nanometre-scale magnetic materials / V.N. Krivoruchko // Физика низких температур. — 2015. — Т. 41, № 9. — С. 864–877. — Бібліогр.: 71 назв. — англ. 0132-6414 PACS: 75.75.–c, 75.78.–n, 75.30.Ds https://nasplib.isofts.kiev.ua/handle/123456789/128069 Spin dynamics in magnetic nanostructured materials is a topic of great current interest. To describe spin motions in such magnetic systems, the phenomenological Landau–Lifshitz (LL), or the LL–Gilbert (LLG), equation is widely used. Damping term is one of the dominant features of magnetization dynamics and plays an essential role in these equations of motion. The form of this term is simple; however, an important question arises whether it provides a proper description of the magnetization coupling to the thermal bath and the related magnetic fluctuations in the real nanometre-scale magnetic materials. It is now generally accepted that for nanostructured systems the damping term in the LL (LLG) equation fails to account for the systematics of the magnetization relaxation, even at the linear response level. In ultrathin films and nanostructured magnets particular relaxation mechanisms arise, extrinsic and intrinsic, which are relevant at nanometre-length scales, yet are not so efficient in bulk materials. These mechanisms of relaxation are crucial for understanding the magnetization dynamics that results in a linewidth dependence on the nanomagnet’s size. We give an overview of recent efforts regarding the description of spin waves damping in nanostructured magnetic materials. Three types of systems are reviewed: ultrathin and exchange-based films, magnetic nanometre-scale samples and patterned magnetic structures. The former is an example of a rare case where consideration can be done analytically on microscopic footing. The latter two are typical samples when analytical approaches hardly have to be developed and numerical calculations are more fruitful. Progress in simulations of magnetization dynamics in nanometre-scale magnets gives hopes that a phenomenological approach can provide us with a realistic description of spin motions in expanding diverse of magnetic nanostructures. Numerous fruitful discussions with S.M. Ryabchenko are gratefully acknowledged. The author would like to thank V.O. Golub for critical reading of the manuscript. This work was partly supported from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 644348 (MagIC). en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур К 80-летию уравнения Ландау–Лифшица Spin waves damping in nanometre-scale magnetic materials Article published earlier |
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Spin waves damping in nanometre-scale magnetic materials |
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Spin waves damping in nanometre-scale magnetic materials Krivoruchko, V.N. К 80-летию уравнения Ландау–Лифшица |
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Spin waves damping in nanometre-scale magnetic materials |
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Spin waves damping in nanometre-scale magnetic materials |
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Spin waves damping in nanometre-scale magnetic materials |
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Spin waves damping in nanometre-scale magnetic materials |
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spin waves damping in nanometre-scale magnetic materials |
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Krivoruchko, V.N. |
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Krivoruchko, V.N. |
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К 80-летию уравнения Ландау–Лифшица |
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Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Article |
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Spin dynamics in magnetic nanostructured materials is a topic of great current interest. To describe spin motions
in such magnetic systems, the phenomenological Landau–Lifshitz (LL), or the LL–Gilbert (LLG), equation
is widely used. Damping term is one of the dominant features of magnetization dynamics and plays an essential
role in these equations of motion. The form of this term is simple; however, an important question arises whether
it provides a proper description of the magnetization coupling to the thermal bath and the related magnetic fluctuations
in the real nanometre-scale magnetic materials. It is now generally accepted that for nanostructured systems
the damping term in the LL (LLG) equation fails to account for the systematics of the magnetization relaxation,
even at the linear response level. In ultrathin films and nanostructured magnets particular relaxation
mechanisms arise, extrinsic and intrinsic, which are relevant at nanometre-length scales, yet are not so efficient
in bulk materials. These mechanisms of relaxation are crucial for understanding the magnetization dynamics that
results in a linewidth dependence on the nanomagnet’s size. We give an overview of recent efforts regarding the
description of spin waves damping in nanostructured magnetic materials. Three types of systems are reviewed:
ultrathin and exchange-based films, magnetic nanometre-scale samples and patterned magnetic structures. The
former is an example of a rare case where consideration can be done analytically on microscopic footing. The
latter two are typical samples when analytical approaches hardly have to be developed and numerical calculations
are more fruitful. Progress in simulations of magnetization dynamics in nanometre-scale magnets gives
hopes that a phenomenological approach can provide us with a realistic description of spin motions in expanding
diverse of magnetic nanostructures.
|
| issn |
0132-6414 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/128069 |
| citation_txt |
Spin waves damping in nanometre-scale magnetic materials / V.N. Krivoruchko // Физика низких температур. — 2015. — Т. 41, № 9. — С. 864–877. — Бібліогр.: 71 назв. — англ. |
| work_keys_str_mv |
AT krivoruchkovn spinwavesdampinginnanometrescalemagneticmaterials |
| first_indexed |
2025-11-26T21:28:31Z |
| last_indexed |
2025-11-26T21:28:31Z |
| _version_ |
1850776773917147136 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9, pp. 864–877
Spin waves damping in nanometre-scale magnetic materials
(Review Article)
V.N. Krivoruchko
Donetsk Institute for Physics and Engineering, the National Academy of Sciences of Ukraine
46 Nauki Ave., Kyiv 03680, Ukraine
E-mail: krivoruc@gmail.com
Received March 23, 2015, published online July 24, 2015
Spin dynamics in magnetic nanostructured materials is a topic of great current interest. To describe spin mo-
tions in such magnetic systems, the phenomenological Landau–Lifshitz (LL), or the LL–Gilbert (LLG), equation
is widely used. Damping term is one of the dominant features of magnetization dynamics and plays an essential
role in these equations of motion. The form of this term is simple; however, an important question arises whether
it provides a proper description of the magnetization coupling to the thermal bath and the related magnetic fluc-
tuations in the real nanometre-scale magnetic materials. It is now generally accepted that for nanostructured sys-
tems the damping term in the LL (LLG) equation fails to account for the systematics of the magnetization relaxa-
tion, even at the linear response level. In ultrathin films and nanostructured magnets particular relaxation
mechanisms arise, extrinsic and intrinsic, which are relevant at nanometre-length scales, yet are not so efficient
in bulk materials. These mechanisms of relaxation are crucial for understanding the magnetization dynamics that
results in a linewidth dependence on the nanomagnet’s size. We give an overview of recent efforts regarding the
description of spin waves damping in nanostructured magnetic materials. Three types of systems are reviewed:
ultrathin and exchange-based films, magnetic nanometre-scale samples and patterned magnetic structures. The
former is an example of a rare case where consideration can be done analytically on microscopic footing. The
latter two are typical samples when analytical approaches hardly have to be developed and numerical calcula-
tions are more fruitful. Progress in simulations of magnetization dynamics in nanometre-scale magnets gives
hopes that a phenomenological approach can provide us with a realistic description of spin motions in expanding
diverse of magnetic nanostructures.
PACS: 75.75.–c Magnetic properties of nanostructures;
75.78.–n Magnetization dynamics;
75.30.Ds Spin waves.
Keywords: magnetic nanostructures and nanoelements, magnetization dynamics, Gilbert damping, spin wave re-
laxation mechanisms.
Contents
1. Introduction, background equations ..................................................................................................... 865
2. Ultrathin and exchange-based films ..................................................................................................... 866
2.1. Two-magnon scattering ................................................................................................................ 866
2.2. Nonlocal relaxation ...................................................................................................................... 868
2.3. Unidirectional damping ................................................................................................................ 869
3. Damping in magnetic nanometre-scale samples ................................................................................... 869
3.1. Three-magnon scattering .............................................................................................................. 870
3.2. Longitudinal relaxation ................................................................................................................ 871
3.3. Anisotropic damping in nanoelements ......................................................................................... 872
4. Nanometre-scale patterned magnetic structures ................................................................................... 873
4.1. FMR damping .............................................................................................................................. 873
4.2. Spin waves anisotropic propagation and damping ....................................................................... 874
5. Conclusion ........................................................................................................................................... 875
References ................................................................................................................................................ 876
© V.N. Krivoruchko, 2015
Spin waves damping in nanometre-scale magnetic materials
1. Introduction, background equations
In 1935, Landau and Lifshitz [1] proposed the equation
for the dynamics of the magnetization M of a uniform
ferromagnet in an effective field Heff of the form:
eff eff/ [ ( )]
s
t
M
α
∂ ∂ = −γ × − γ × ×M M H M M H , (1)
where γ is the absolute value of gyromagnetic ratio, t is
time, Ms stands for the saturation magnetization, and α is
the damping constant. The first term in Landau–Lifshitz
(LL) equation describes the precession of the magnetiza-
tion around the micromagnetic effective field Heff, which
is composed of the external magnetic field, the magnetic
anisotropy field, the demagnetization field, etc. In the ab-
sence of damping, this leads to the Larmor precession with
a well-defined frequency, ω. In all real systems, there are
different “friction” processes that cause precession damp-
ing. The second term is phenomenological representation
of one or more (unspecified) energy loss mechanisms. It
breaks the time-inversion symmetry and consequently
leads to the precession damping. This term drives the mag-
netization toward the direction of Heff, whereby angular
momentum is transferred to nonmagnetic degrees of free-
dom (direct damping). Good samples in ferromagnetic
resonance satisfy α << 1.
In the 1955 MMM conference proceedings, Gilbert ar-
gued that the LL damping fails for large enough damping
[2]. Instead, he proposed an expression that, for a small
angle between Heff and M, may be written as follows:
eff/ ( / )G
s
t t
M
λ
∂ ∂ = −γ × + ×∂ ∂M M H M M . (2)
As for Eq. (1), the second term in the r.h.s. of the LL–
Gilbert (LLG) equation is a damping term introduced in
a phenomenological manner. It is instructive to rewrite
Eq. (2) in the form
eff/ /G
G s
t t
M
λ
∂ ∂ = −γ × − ∂ ∂ γ
M M H M .
Clearly, we obtain an attractive interpretation that the damp-
ing introduces an effective magnetic field opposite in di-
rection to Heff and proportional to ∂M/∂t. With λ = α and
2 2 1 ( )/G sMγ = γ + λ , the LL and the Gilbert forms are
mathematically equivalent. Yet, this equivalency (within a
linear approximation) is only for small damping and low
enough microwave frequency f (or ω = 2πf the angular
frequency). Indeed, for example, in the parallel ferromag-
netic resonance (FMR) configuration, with the magnetiza-
tion parallel to the applied field, the time derivative ∂M/∂t
Gilbert term in the equation of motion produces a FMR
linewidth linear with the frequency f, in contrast to the LL
equation.
Damping is measured through the linewidth ΔH of the
absorption peak in the transverse susceptibility spectrum.
In a conventional fixed frequency field-swept FMR exper-
iment, when the magnetization is aligned with either an in-
plane or perpendicular to the plane applied magnetic field,
the absorption linewidth (full width at half maximum) is
given by 4 /sH M f∆ = πλ γ i.e., the linewidth in these
cases is proportional to the frequency with a slope deter-
mined by λ. This is the homogeneous or intrinsic contribu-
tion to the FMR linewidth. However, in many magnetic
systems, while a linear behavior is observed, the linewidth
fails to extrapolate to zero with vanishing frequency. Ex-
periments show an additional frequency-independent con-
tribution to the linewidth:
0 4 /sH H M f∆ = ∆ + πλ γ .
Here zero-frequency contribution to the linewidth ΔH0 is
related to an extrinsic mechanism and reflects the effect of
inhomogeneity on the linewidth. Thus, frequency depend-
ent studies allow distinguishing intrinsic and extrinsic con-
tributions to the relaxation.
Both the LL and the LLG equations were introduced
with suggestion that (i) magnetization motion is small,
(ii) the damping parameter is isotropic, and, in ferromag-
nets, (iii) it is assumed that the length of the magnetization
is conserved locally (|M| = const) as it relaxes to its equi-
librium position. These restrictions are caused by the struc-
ture of the damping term. It is also assumed that α (or λ) is
independent on effective field Heff. Concerning the last
assumption, Callen was the first who pointed out that the
constant α is the result of a small angle expansion θ (θ re-
presents the angle between the magnetization vector and
Heff) [3]. Keeping next-to-leading-order terms, he predict-
ed that [ ] 2~ 1– cos / sinα θ θ , which is sufficient to stabi-
lize a precessing spin-wave mode at larger θ.
A large body of both experimental and theoretical work
confirmed that the magnetization dynamics of a bulk fer-
romagnet is well described by the phenomenological equa-
tions (1) or (2), and the damping mechanism in bulk fer-
romagnets is relatively well understood at present. Yet, we
note here that, even for the bulk systems, as Bar’yakhtar
pointed out [4], the form of the second term in Eq. (1) is
very restrictive. Its structure assumes that the equation of
motion is form invariant under arbitrary rotations of the
coordinate system, whereas in any ferromagnet the equa-
tion only needs to be form invariant under rotations about
the axis along the magnetization direction (in the presence
of anisotropy, the symmetry is lower). Thus, symmetry
allows for a generalized form of the damping term on the
r.h.s. of Eq. (1) (see textbook [5] for more details).
At present, the static and dynamic properties of nano-
structured magnetic materials are among the most active
research topics in magnetism. This is, in particular, due to
the promising prospects for application of such materials
within magnetic data storage [6], magnonic devices [7]
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9 865
V.N. Krivoruchko
(the magnetic counterpart of a photonic crystal), as spin-
wave filters [8], etc. Recently, a multidisciplinary branch
of science called “magnonics” developed as a result of ac-
tivity is this field [9]. It focuses on studies of spin waves,
and more generally on magnetic excitations, in artificial
materials with periodic variations of magnetic properties
— magnonic crystals. Magnonics is strongly coupled to
material science since it requires materials with small
magnetic damping. The main obstacle in the field of
magnonics is still the spin-wave relaxation, which has to
be decreased utilizing new materials or to be compensated
using new energy-efficient means. Magnetoelectronics is
another region of application [10]. As demonstrated by
Slonczewski [11] and Berger [12] the interaction between
localized magnetic moments and conducting electrons
leads to the so-called spin transfer torque, which accounts
for the ability of a spin-polarized current to interact with
the magnetization of a ferromagnetic layer via transfer of
the spin angular momentum. Such a current-induced spin
transfer enables the manipulation of magnetic nanodevices.
Yet, this “pumping” of spins also slows down the magneti-
zation precession corresponding to an enhanced Gilbert
damping constant in the LL equation [13]. The damping
parameter is also a critical figure for the efficient operation
of magnetoelectronic devices.
In device design the LL or the LLG equations are wide-
ly used to describe spin motions in magnetic nanoscale
structures. An important question is whether it provides a
proper description of the damping of the magnetization in
real nanoscale materials or not. Indeed, as follows from
the experiment, the situation is more complex if we deal
with the magnetization dynamics in thin magnetic films
and nanostructured magnets. For example, the damping
constant was found to be 0.04 < α < 0.22 for Cu–Co and
Pt–Co in thin film pillars [14], which is considerably larger
than the bulk value α ≈ 0.005 in Co [15] or α ≈ 0.004 for
thin film Permalloy [16]. Recent works on magnetic nano-
structures demonstrate a strong dependence of the effective
damping on the nanomagnet size and the particular spin-
wave mode [17,18]. It is important to understand how scal-
ing effects damping and linewidth.
It is now widely accepted that the mechanisms usually
applied to describe the spin waves damping in bulk mag-
netic systems cannot explain the experimental data ob-
tained on ultrathin films and nanostructured magnetic ma-
terials. Possible novel mechanisms of spin waves damping
in such systems were predicted and are under extensive
discussion at present. In this report, the questions concern-
ing the magnetization relaxation in nanostructured magnet-
ic materials have been reviewed. Naturally, in this inten-
sively expanding research field, we are forced to restrict
ourselves to some narrow area. The mechanisms for mag-
netization damping in heterostructures of ferromagnet and
normal conductors are beyond the scope of this report, for
a review on this topic see, e.g., Ref. 10. In this paper, we
do not touch upon the question of the LL (LLG) equations
and the magnetization damping generalization for fast
(on the time scale of about 100 fs) magnetization dynam-
ics. A recent review concerning ultrafast dynamics and the
optical manipulation of magnetic order can be found in
Refs. 19–21. We will concentrate here mainly on three
types of interesting systems: (i) ultrathin and exchange-
based films, (ii) magnetic nanometre-scale samples, and
(iii) patterned magnetic structures. Ultrathin films is a rare
example when theoretical consideration can be done ana-
lytically on microscopic footing. The nano-scale magnets
and patterned magnetic structures are systems modelling of
which can be hardly developed analytically, and numerical
calculations based on the generalized LL (LLG) equations
are typically used. Specific of nanometre-scale samples is
originated by their finite format and the fact that, due to the
highly inhomogeneous internal magnetic field within the
nanoelement, their magnetic resonances may exhibit strong
spatial localization, so that the resonance spectra are dis-
crete. Concerning patterned magnetic systems, their prop-
erties are a nontrivial combination of those of thin film and
nanoelement. Some features here we illustrate on the ex-
ample of thin film with mesh of holes (antidots).
2. Ultrathin and exchange-based films
It is now generally accepted that for ultrathin and ex-
change-based films specific relaxation processes arise,
which operate at small-length scales, and which are not
relevant for bulk materials. These processes conventionally
are parted into two classes, intrinsic and extrinsic, and both
are crucial for understanding the magnetization dynamics.
The intrinsic processes have been well known and (con-
ventionally) summarized as Gilbert damping [22]; the key
role of the extrinsic relaxation processes has been under-
stood and analyzed in detail most recently. Their signifi-
cance arises because of the fact that a large fraction of the
magnetic moments are in or near surfaces and interfaces.
We illustrate the specific role of the surface in nanostruc-
tured magnets on the example of ultrathin and exchange-
based films.
2.1. Two-magnon scattering
Two-magnon damping in ultrathin films and different
layered structures was proposed in Ref. 23 and is based on
two magnon scattering processes induced by defects or, in
general, magnetic inhomogeneity. The inhomogeneity con-
tribution is associated with disorder and their effect is at
least twofold. First, fluctuations in the materials magnetic
properties such as its anisotropy or magnetization lead to a
linewidth that is frequency independent. In a simple pic-
ture, independent parts of the sample come into resonance
at different applied magnetic fields. Second, disorder can
couple the uniform precessional mode with k equals zero,
excited in an FMR experiment, to degenerate finite-k spin-
866 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9
Spin waves damping in nanometre-scale magnetic materials
wave modes. This mechanism of relaxation of the uniform
mode, known as two-magnon scattering [22], requires a
spin-wave dispersion with finite-k modes that are degener-
ate with zero-k mode, which only occurs for certain mag-
netization orientations. Let us briefly remind the main ide-
as of this field (for more details see, e.g., textbook [22]).
Following [24], the scattering of a magnon with wave
vector k1 into another magnon k2 by a process that does
not conserve momentum is described by the Hamiltonian:
21
1 2
1 2
,
( , )( h.c.)sc
k k
H V k k a a+= +∑ kk , (3)
where a+k and ak are the creation and annihilation magnon
operators and V(k1,k2) represents the scattering perturba-
tion. For an ideal (strictly periodic) crystal V(k1,k2) = 0 if
k1 ≠ k2 due to impulse conservation law. Yet, if the crystal
(film) is imperfect V(k1,k2) ≠ 0.
Consider relaxation of a spin wave with wave vector k1.
Within simple perturbation-theory calculation, the magnon
number
1
nk rate of change is
1
1 1 22
2
2
1 2
2 | ( , ) | ( ) ( )k
k k
k
dn
V k k n n
dt
π
= − δ ω − ω∑ kk
. (4)
Thus, standard transition probability calculation yields for
the two-magnon contribution to the linewidth
1 2
2
(2) 2
1 2| ( , ) | ( )k k
k
H V k kπ
∆ = δ ω − ω∑
. (5)
This expression implies that incoming magnon k1 relaxes
by scattering into all degenerate modes with wave vector
k2. If k1 = 0, i.e., we deal with homogeneous magnetiza-
tion precession, these processes called a 0-k processes of
relaxation. The remaining task is for a given sample to
model the surface and/or interface defects, and to find their
contribution to the matrix elements V(k1,k2).
Usually, the two-magnon scattering is less important in
bulk ferromagnet, but is of particular importance in mag-
netic nanostructures because real magnetic superlattices
and ultrathin films have steps and defects on a length scale
of some hundreds of nanometres. This is the order of mag-
nitude for long wavelength magnons, into which the uni-
form motion of magnetization can scatter. Following Mills
et al. [23], consider an in-plane magnetized film. The FMR
frequency ωFMR of an ideal ultrathin film equals that ωk of
a spin wave with wave vector k equals zero. As is known,
in a thin ferromagnet film and at long wavelengths, dipolar
couplings produce terms in ωk linear in |k|. The initial
slope of ωk depends on the angle φ between k and the
magnetization MS, assumed in the plane. If 0( )c Hϕ < ϕ ,
or if 0( )c Hπ−ϕ < ϕ , then this initial slope is negative.
Here ( ) 1/2
0 0 0sin /( ) ( )[ ]c SH H B Hϕ = + , where H0 is
magnetic field applied parallel to MS, 0 0 4 SB H M= + π ,
and HS is the surface anisotropy field taken positive when
the surface normal is a hard axis. Exchange contributes to
ωk term proportional to Dk2, with D being the exchange
stiffness.
Thus, for 0( )c Hϕ < ϕ and also 0( )c Hπ−ϕ < ϕ we
have spin waves of wave vector kc(φ) degenerate with the
FMR mode, as illustrated in Fig. 1. Static defects such as
surface imperfections scatter energy from the FMR mode
to the finite wave vector modes of the same frequency, to
relax the FMR mode by a dephasing process. This is, for
the ultrathin film, a direct analog of the relaxation mecha-
nism discussed many years ago [24]. Associated with such
scatterings is necessarily a shift in frequency of the FMR
mode.
The authors [23] modelled surface defects as rectangu-
lar islands and argued that the dominant contribution to the
two-magnon matrix element has its origin, for typical sam-
ples, in the perturbation of the surface anisotropy field near
the islands. Under the conditions specified in Ref. 23 and
usually realized in experiments, this theory provides that
the extrinsic linewidth can be written as follows:
1/21/21/2 2
0 0(2)
1/21/2 2
0 0
( / 2) / 2
arcsin
( / 2) / 2
FMR
FMR
H
ω +ω −ω ∆ ∝
ω +ω +ω
, (6)
where, ω0 = γ(4πMS + HS). If ω0 >> ωFMR, then Eq. (6)
predicts that the linewidth should vary linearly with fre-
quency, very much as the prediction of the LLG equation.
However, under typical conditions ωFMR ~ ω0 and in this
regime Eq. (6) yields strong deviations from linear behav-
ior. For experimentally accessible FMR frequencies, anal-
ysis of Eq. (6) shows that the linewidth increases with fre-
quency much slower than expected from the linear law.
This means, in particular, that the damping term is non-
local in time. A general consequence is that the linear fre-
quency dependence predicted from the LLG phenomeno-
logy is qualitatively incorrect for the real materials.
Fig. 1. A schematic illustration of the spin wave dispersion with
two angles of propagation, one is less than the critical angle φc
and another is larger, for in-plane magnetized film.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9 867
V.N. Krivoruchko
Resonance experiments [25] on magnetic multilayer
nanostructures in a frequency range of more than two or-
ders of magnitude show an unambiguous nonlinear fre-
quency dependence of the linewidth (2) ( )H∆ ω . It was
found that two-magnon scattering is not effective in FMR
when applied field is perpendicular to the film plane, but
may be very important for various directions of the exter-
nal field applied in the film plane. The viscous Gilbert
damping and two-magnon scattering were clearly separat-
ed. Analysis of angular and frequency dependent meas-
urements has shown that a transverse scattering within the
magnetic subsystem is a main mechanism of relaxation.
The longitudinal relaxation into the thermal bath, when the
length of the magnetization vector does not conserve
|M| ≠ const (see subsection 3.2, below), is up to two orders
of magnitude smaller.
In the presence of a periodic scattering potential the
spin relaxation in ultrathin ferromagnets is not a monoto-
nous function of the frequency, as has been usually as-
sumed. In report [26], a phenomenological theory of the
two-magnon scattering in periodically structured thin-film
systems was developed. The authors demonstrate that the
frequency dependence of the overall spin relaxation in a
large class of ferromagnetic systems is not monotonous
and depends on the defect structure. In particular, the spin
relaxation rate is found to increase substantially at charac-
teristic frequencies related to the periodicity of the magnon
scattering potential. Consequently, the usual practice of
separation of intrinsic and extrinsic spin relaxation pro-
cesses by means of their frequency dependence, strictly
speaking, must be reconsidered. The exact structure of
defects in a material, being periodic in the majority of fer-
romagnetic systems, needs to be cleared up first.
Damping of spin waves in magnonic crystals was ana-
lyzed in reports [27]. Authors obtained an analytical ex-
pression for the effective damping coefficient of propagat-
ing spin waves as function of system parameters and
shown, in particular, that it strongly depends on the spin
wave frequency and bias magnetic field. This dependence
is more pronounces when damping is localized in the vi-
cinity of interface. Thus, the effective damping coefficient
can be designed by means of a proper choice of the depth
modulation of magnetic parameters.
These results are important for future developments,
since they could explain the anomalous spin relaxation in
magnonic crystals and help to tailor spin relaxation in
spintronic devices by artificially inducing a defect structure
to activate a desired spin relaxation channel in a specific
frequency range.
2.2. Nonlocal relaxation
The FMR linewidth is related to the relaxation rate of
the magnon with zero wave number k. In order to obtain
more information on the magnon relaxation, in Ref. 28
investigated the behavior of magnons with k ≠ 0, which
can be measured with Brillouin light scattering (BLS). It
was found that the damping increases strongly with the
wave vector. This means that damping term is nonlocal in
space. That is another example of a damping mechanism
not present in bulk materials yet operating in ultrathin
ferromagnets.
Both the FMR and BLS linewidth data are consistently
explained by a relaxation mechanism based on two-mag-
non scattering processes due to the local fluctuation of
the exchange coupling caused by interface roughness. As
already mentioned, standard calculation of the transition
probability yields for the magnon’s energy relaxation ex-
pression (5). This expression suggests that the incoming
magnon k1 relaxes by scattering into all degenerate modes
with wave vector k2. Authors [28] considered a simple
model of exchange-based FM/AFM films where the
roughness gives rise to a large fluctuating field because the
FM magnetization interacts alternatively with one or the
other AFM (antiferromagnetic) sublattice via the atomic
exchange coupling. Defects on the FM–AMF interface
result in a local perturbation of the exchange coupling with
energy 2Jint cos(θ) per unit area, where Jint represents the
local interfacial exchange energy and θ is the angle be-
tween the FM and AFM moments. Then it can be shown
that the strength of the scattering perturbation is [28]:
2 int
1 2 1 1 1 2( , ) ( ) ( ) ( ) cos .
2
d
x y
k
H A
V k k H k H k S k k
A
= γ + − θ ω
(7)
Here Hint = Jint/Msd stands for the local exchange coupling
field (d is the film thickness), Ad and A are the defect and
the film area, respectively, S(k1 – k2) is a structure factor
for the defects,
2 2( ) cos ( ) 2 sin ( ) cos ( ),x H s k aH k H M kd Dk H= θ− θ + π θ + + θ
2( ) cos ( ) 4 (1 /2) cos ( ),y H s aH k H M kd Dk H= θ− θ + π − + + θ
and the frequency of the k magnon ωk is
1/2[ ( ) ( )]x yH k H kω = γk . Here H is the external field ap-
plied in the film plane at an angle θH with the direction of
the unidirectional anisotropy field Ha, θk is the angle of the
magnon wave vector in the plane, and D is the exchange
stiffness of the FM film. Under the conditions usually real-
ized in experiments, the model [28] predicts that both FMR
and BLS linewidths vary with the interface energy as 2
intJ
and with film thickness as 1/d2. In addition, the BLS line-
width is inversely proportional to the magnon frequency.
The thickness dependence of the two-magnon model,
with an explicit ~ 1/d2 dependence in the scattering
strength and a ~ 1/d thickness dependence in the mode
counting, does not agree well with the published experi-
mental results [29], which show the linewidth increasing
linearly with 1/d for d > 10 nm and decreasing for a thinner
sample. The 1/d2 dependence of the scattering strength can
be eliminated if the perturbing field is assumed to act on
868 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9
Spin waves damping in nanometre-scale magnetic materials
spins throughout the thickness of the sample, rather than
on the surface spins.
A further generalization of this aspect of magnetization
dynamics is the implementation of nonlocal effects in both
space and time [30]. This can be achieved by introducing a
retardation kernel, which takes into account temporal re-
tardation within a characteristic time τ and spatial one with
a characteristic scale ζ. The latter simulates an additional
mutual interaction of the magnetic moments in different
areas of the film within the retardation length ζ. For esti-
mation, the size of inhomogeneity or defect on the film
surface may be considered as the length ζ. As for the feed-
back time, the authors [30] presented arguments why it is
reasonable to consider the τ in the interval 0 fs < τ < 100 fs.
This time-scale is relevant for ultrafast dynamics.
The nonlocal damping becomes significant when the
wavelength of magnons approaches the exchange length of
the material, i.e., for high-order exchange dominated spin
waves. In report [31], exchange attenuation of standing
spin waves is calculated for an ultrathin ferromagnetic film
of the order of exchange length thick. It is shown that the
wave vectors of standing spin waves in such film achieve
values that are proportional to the inverse film thickness.
The nonlocal exchange attenuation at such wave vectors
becomes dominant and can result in smearing of the stand-
ing spin wave spectrum.
Thus, the nonlocal feedback in ferromagnetic resonance
leads to linewidth broadening and consequently to spin
damping. Whether the magnitude of retardation is able to
exceed the Gilbert damping depends strongly on the fre-
quency. In particular, the calculations [30] suggest that, for
sufficiently high frequencies, retardation effects dominate
the intrinsic damping.
2.3. Unidirectional damping
Recently it was demonstrated experimentally that damp-
ing in exchange-coupled systems is not only anisotropic
but also unidirectional [32]. This was evidenced by an asym-
metry in the damping by inversion of the magnetic field
polarity. The study reveals that this asymmetry in the damp-
ing is enhanced by the increase in the exchange bias field.
It was shown unambiguously that the anisotropy in the
intrinsic Gilbert damping originates from the pinned spins
of the bilayers and is proportional to the exchange-coupl-
ing field Hex between bilayers.
To reproduce the experimental features of the resonance
absorption, a modified relaxation term in the LLG equation
of motion (2) was proposed [32]:
( )exeff/ 1 cos ( ) /Gt t∂ ∂ = −γ × + λ − ξ θ ×∂ ∂H MM M H M M .
(8)
The term ~ ξ represents the anisotropic damping that de-
pends on the angle
ex
θH M between the direction of the
magnetization M holed by an antiferromagnetic layer and
the coupling field Hex. For the samples NiO(67.5nm)/
Ni81Fe19 the fit to the experimental data gives for ξ = 0.18.
It was also demonstrated that the value of ξ depends on the
thickness of the antiferromagnetic layer. Below the critical
thickness ξ is zero, i.e., the relaxation parameter remains
unchanged by inversion of the polarity. However, ξ in-
creases with Hex.
Note that this relation Steiauf and Fähnle [33] used ear-
lier in a phenomenological extension of the ab initio densi-
ty-functional electron theory to derive an equation of mo-
tion for the spin dynamics in magnets. The most important
result of their calculations is that the magnetization dynam-
ics with a scalar damping term of the Gilbert’s form does
not depend on the orientation of M and is valid only for a
few specific situations. Even for the case of a homogene-
ous magnetization, the damping term has to be replaced by
a more general term of the form ˆ1/ ( ( ) / )SM t×α ∂ ∂M M M
with a damping matrix ˆ ( )α M , which depends on the orien-
tation of the magnetization. The orientation dependence of
ˆ ( )α M is already substantial in some bulk materials (e.g., in
hexagonal Co), and it is very strong for systems with re-
duced dimensionality like monatomic layers or monatomic
wires. The authors demonstrate that in the systems with
reduced dimensionality there are orientations for which the
damping is identically zero and other orientations for
which the damping is very large. It was also argued that in
the presence of spin-orbit interaction the damping parame-
ter would depend on the direction of the magnetization,
too. Thus, the spin-orbit coupling makes the spin degree of
freedom respond to its orbital environment.
Safonov [34] has also introduced ad hoc a matrix form
of magnetization damping for small magnetization mo-
tions. The damping tensor is scaled by only one phenome-
nological damping parameter, which can be obtained from
the experiment.
3. Damping in magnetic nanometre-scale samples
An important specific of nanosized samples stems from
their finite size and the fact that, even in an ellipsoidal
element, their magnetization distribution can be nonuni-
form and, thus, the internal magnetic field can be inhomo-
geneous. The magnetic structure shown in Fig. 2 illustrates
a possible metastable remanent state — two magnetic vor-
texes of opposite chirality — that can be realized in na-
nometer samples subjected to rapidly pulsed magnetic field
[35]. Inhomogeneous internal magnetic field Heff plays a
dominant role in the effective pinning of the dynamic
magnetization and induces inhomogeneous magnetization
configurations that form a potential wall for spin waves
[36,37]. Due to the highly inhomogeneous internal magnet-
ic field within the nanometre-scale sample, its magnetic
resonances may exhibit strong spatial localization so that
the resonance spectra are discrete [37]. As a result, intrin-
sic nonlocal effects, moderated by spin-wave mode con-
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9 869
V.N. Krivoruchko
finement, are important contributors to damping in mag-
netic nanostructures.
Time-resolved scanning Kerr microscopy investigation
of magnetization dynamics of arrays of nonellipsoidal nano-
magnets of different sizes directly confirms these conclu-
sions [38]. In the experimental spectra acquired from an
element of length of 236 nm and thickness of 13.6 nm, two
branches of excited modes were observed to coexist above
a particular bias field. Micromagnetic simulations and Fou-
rier imaging revealed that modes from the higher frequen-
cy branch had large amplitude at the center of the element
where the effective field was parallel to the bias field and
the static magnetization. Modes from the lower frequency
branch had large amplitude near the edges of the element
perpendicular to the bias field. The simulations revealed
significant canting of the static magnetization and effective
field away from the direction of the bias field in the edge
regions. For the smallest element sizes and/or at low bias
field values, the effective field was found to become anti-
parallel to the static magnetization. The simulations re-
vealed that the majority of the modes were delocalized
with finite amplitude throughout the element while the spa-
tial character of a mode was found to correlate with the
spatial variation in the total effective field and the static
magnetization state. The frequencies of the edge modes are
strongly affected by the spatial distribution of the static
magnetization state both within the element and within its
nearest neighbors.
Note also that, in the absence of translational invariance
caused by the finite size of the sample, there is no require-
ment for conservation of momentum. However, similar to
bulk samples, the suggestion of momentum conservation is
conceptually useful for the nanostructured samples, as
well. The fact that all spin-wave modes are standing waves
implies that all modes have a total momentum of zero, i.e.,
each state now has both +k and −k components, so that the
sum of the wave vectors for a given mode is zero [39]. As
a result, the standard Gilbert damping is insufficient to
describe the spin dynamics in nanomagnets.
At present, several different theoretical models attempt
to explain the dependence of the resonance frequencies and
damping on nanomagnet size [17,18,33,40]. On the other
hand, a comparison of experimental data with theory is not
so simple because an extraction of the damping parameter
from measurements of ensemble nanomagnets is not a triv-
ial task. Indeed, (i) the resonance frequencies might differ
from nanomagnet to nanomagnet; (ii) shape distortions can
give rise to mode splitting; (iii) the dipolar interaction be-
tween nanomagnets is nonzero, etc. Therefore, only sys-
tematic comparison of data obtained on individual nanomag-
nets with theory gives possibility to examine the dependence
of damping on various spin-wave modes in nanomagnets
of different size.
3.1. Three-magnon scattering
As already mentioned, in small elements, the wave vec-
tor becomes a set of discrete values. This limits the availa-
bility of states at the frequency of ω. As a result, localized
modes become more important for nanoscale samples than
usual magnetostatic waves in extended films. These modes
often have low frequencies and provide a mechanism for a
decay that does not involve usual magnetostatic waves. In
general, in nanomagnets different nonlinear mechanisms
can exist, which results in an anomalous increase in the
linewidth at low frequencies. As a rule, the measured
linewidth for the nanoscale magnets does not exhibit a lin-
ear dependence on frequency even at the lowest frequen-
cies [36]. Micromagnetic simulations confirmed this be-
havior.
Let us illustrate some specifics in the field on an exam-
ple of three-magnon damping in thin film recently discuss-
ed by Camley [39]. (For a bulk ferromagnet, such three-
magnon scattering is important for the saturation of ferro-
magnetic resonance [22].)
Following Ref. 39, let us assume that the quantized
wave vectors in the xoy-plane are given by /z zq n L= ± π ,
where Lz is the length of the nanoelement in the oz direc-
tion and n is an integer. In addition, in finite systems, all
the modes are essentially standing ones and contain both
positive +qz and negative –qz wave-vector contributions.
As a result, the conservation of energy and momentum
rules are modified so that now we can have a three-
magnon process where an initial spatially uniform mode
(k = 0) with a frequency ω0 decays to two modes, one at
frequency 0 / 2ω + ∆ω and the other at 0 / 2 –ω ∆ω. For this
process to occur, there must be two states spaced equally
(in frequency space) about ω0/2. Because the dispersion
relation is no longer continuous, this can only happen at a
few special points. The process also conserves momentum:
the initial wave vector has k zero and the total final wave
vector is zero, as each state now has both +k and −k com-
ponents. This set of decay possibilities is illustrated in
Fig. 3.
It is important to note that the existence of the three-
magnon process depends critically on the dimensions of
the nanoelement. In particular, as was shown by Camley
[39], the three-magnon process, which leads in thin films
to waves at ω/2, produces a different result in rectangular
Fig. 2. (Color online) A possible magnetic structure, two magnetic
vortex of opposite chirality, which can be realized in nanometer
samples under the effect of pulsed field. (Borrowed from [35]).
870 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9
Spin waves damping in nanometre-scale magnetic materials
magnetic nanoelements. This nonlinear relaxation mecha-
nism leads to responses at several frequencies that are
symmetrically spaced around ω/2. When the driving field
is large and near the resonance frequency for the uniform
mode, the excitation spectrum can be quite complicated.
However, many of the peaks seen in that spectrum result
from nonlinear combinations of the original wave.
3.2. Longitudinal relaxation
Both the Landau–Lifshitz and the Gilbert phenomeno-
logical damping terms suggested that the magnetic losses
could be characterized by a single intrinsic damping con-
stant of relativistic nature. As stressed in Introduction,
within the LL (LLG) damping term’s structure the length
of |M| stays constant while the projection of the magnetiza-
tion onto the z-axis, Mz, increases as it relaxes to its equi-
librium position and the angle between vectors Heff and M
decreases. In magnetic nanoelements the internal magnetic
field is strongly inhomogeneous and one can suggest a re-
laxation mechanism when the projection of Mz stays con-
stant but the precessional energy is dissipated into the
transverse magnetization components, i.e., a local reduc-
tion of the magnetization length and longitudinal relaxa-
tion. Naturally, the question arises whether there is any
experimental evidence supporting this suggestion. The
answer is positive. Recent experiments [25,41] and theo-
ries [39,42] point out that even at low temperature there is
no reason to assume a fixed magnetization length for mag-
netic nanoelements. In particular, the uniform motion of
the magnetization may scatter into excited states of the mag-
netic subsystem — magnetic thermal bath (spin waves,
Stoner excitations, etc.). Respectively, the projection of M
onto the z-axis stays constant but |M| is decreased since the
precessional energy is scattered into the transverse compo-
nents Mx and My. This is another example proving that the
phenomenological Gilbert (as well as the Landau–Lifshitz)
damping parameter cannot correctly describe the magnetic
relaxation in magnetic nanostructures.
Investigation of longitudinal relaxation mechanisms is a
very challenging problem, which has recently received
new impulse in connection with ultrafast remagnetization
(see reviews [19,20]). Within the goal of this report, the
purpose is to search the longitudinal relaxation channels in
small magnetic elements in relation with their geometric
characteristics and micromagnetic configuration. Note that
this interesting scientific direction is just emerging.
As both relaxation mechanisms (longitudinal and trans-
verse) can be active in nanostructures, the questions arise
how to include longitudinal relaxation into description and
how to separate experimentally the scattering to nonmag-
netic degrees of freedom (direct damping) from the dissi-
pative relaxation to the magnetic thermal bath (longitudinal
relaxation). One of the theoretical backgrounds to study
this question was been known for a long time. We mean
the Bloch–Bloembergen equation written as follows
eff
2 1
/ x x y y z s
G z
M M M M
t
T T
+ −
∂ ∂ = −γ × − −
e e
M M H e . (9)
Indeed, in this case two different relaxation times are in-
troduced into the magnetization dynamics: the longitudinal
relaxation time T1 for the component of the magnetization
parallel to the field Heff, i.e., the direct path into the ther-
mal bath, and the so-called transverse time, T2, by which
energy is scattered into the transverse magnetization com-
ponents Mx and My. Analysis of the ferromagnetic reso-
nance in nanostructures within the framework of Eq. (9) is
performed in Ref. 41.
Here we briefly outline a possible alternative formal-
ism, which is based on generalization of the damping term
proposed by Bar’yakhtar [4].
The general form of the LL–Bar’yakhtar equation
(LLBar) reads:
( ) eff
eff eff
ˆ ˆ/ ( ) ( )e
ij
i j
t
x x
∂
∂ ∂ = −γ × + λ −λ
∂ ∂
H
M M H M H M . (10)
As usual, the first term in the r.h.s. defines the precession
of the vector M in effective field Heff, while the second
and the third terms describe the local and nonlocal relaxa-
tions, respectively. Here ˆ ( )λ M and ( )ˆ ( )e
ijλ M are the relaxa-
tion tensors of relativistic and exchange nature, respective-
ly, and in general are functions of the magnetization
vector. These tensors describe how crystallographic and
magnetic symmetries of the system manifest themselves
into the relaxation mechanisms of magnetic subsystem.
It is important to note that, even in a local approxima-
tion, i.e., keeping only the second term in the r.h.s. of
Eq. (10), the Bar’yakhtar equation, in contrast to the Lan-
Fig. 3. (Color online) The lowest-frequency dipole-exchange
mode in a nanoelement for propagation parallel to the applied
field (typical Permalloy parameters used). The finite length of
the sample leads to a discretization of the allowed wave vectors.
The dashed lines indicate possible decays from the uniform mode
(k = 0) to two modes equally spaced about the half-frequency
position. (Borrowed from [39]).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9 871
V.N. Krivoruchko
dau–Lifshitz formalism, does not conserve the length of
the magnetization vector and |M| ≠ const. For simplicity,
we focus here on the intermediate range of temperature,
when it is reasonable to expect that relativistic effect can-
not compete with the exchange force that tries to keep the
magnetization vector length constant and variations of the
magnetization vector length cannot significantly contribute
to the magnon-magnon scattering and, thereby, to the FMR
linewidth broadening. The dynamic contribution of the
nonlocal damping becomes significant only when the
wavelength of magnons approaches the exchange length of
the material. Therefore, both static and dynamic contribu-
tions of the nonlocal damping are vanishing, and Eq. (10)
can be written as:
eff effˆ/ st M∂ ∂ = −γ × − γ αM M H H . (11)
Here, again for simplicity, expanding the relaxation tensor,
we keep only the coefficients of the expansion that take
into account crystallographic symmetry and write
ˆ ˆ ˆ( ) (0) sMλ ≈ λ = −γ αM . Note that, in spite of its relatively
simple form, Eq. (11) still includes a contribution due to
the exchange energy arising from changes of the magneti-
zation vector length, i.e., does not conserve the length of
the magnetization vector. Indeed, by representing M as
M=M m , where m is the unit vector, and substituting this
in Eq. (11), after simple algebra one can obtained a system
of two coupled equations [42]:
eff effˆ/ [ ( )]sM
t
M
∂ ∂ = −γ × − γ ×α ×m m H m m H ,
effˆ/ sM t M∂ ∂ = γ αm H .
Thus, Eq. (11) describes nonconservative magnetization
dynamics.
Concerning the experiment, it was demonstrated [25,41],
that both angular and frequency dependent measurements
of the linewidth give a direct possibility to distinguish the
transverse scattering rate within the magnetic subsystem
and the longitudinal relaxation into the thermal bath. For
the particular system investigated in [25], Fe/V multilayers,
the longitudinal relaxation scattering was about two orders
of magnitude faster than transverse (Gilbert) damping.
A number of magnetic resonance experiments have
been performed by various authors on assemblies of ran-
domly oriented nanoparticles (see, e.g., [43] and references
therein). The agreement between experimental data and
theoretical predictions is, however, rather poor and does
not allow for accurate quantitative analysis of the experi-
mental results. The only exception is the high-temperature
limit where, according to the conventional theory, the spec-
trum progressively collapses into a single nearly Lorentzian
line. At lower temperatures, significant broadening of
the single line was found together with its progressive
shifting to lower fields with cooling. We note here the re-
port of Noginova et al. [44], where the magnetization dy-
namics of γ-Fe2O3 nanoparticles has been studied by elec-
tron magnetic resonance (EMR) at 77–380 K and an origi-
nal model of the longitudinal spin relaxation was proposed.
In this report, slightly asymmetric spectra observed at
room temperature become much broader and symmetric,
and shift to lower fields upon cooling. The longitudinal
relaxation time, T1 ≈ 10 ns, was determined by a specially
developed modulation method. The shift and broadening of
the spectrum upon cooling were assigned to the effect of
the surface-related anisotropy. To describe the overall
spectral shape, the original “quantization” model was used
which includes summation of resonance transitions over
the whole energy spectrum of a nanoparticle considered as
a giant exchange cluster. This approach, supplemented
with some phenomenological assumptions, provides satis-
factory agreement with the experimental data.
Yet, as already noted, such interesting scientific area as
longitudinal magnetization relaxation in small-scale mag-
netic objects is just at its beginning. Further work in theory
and experiment is needed to overcome our very little un-
derstanding in this field.
3.3. Anisotropic damping in nanoelements
We briefly discuss anisotropic damping in nanoscale
samples based on recent theoretical analysis made in
Ref. 42. This analysis utilizes the Bar’yakhtar generaliza-
tion of the damping term, namely, Eq. (11). The simula-
tions were carried out for a thin magnetic disk and an el-
lipse.
The dominant peaks in the spectra (attributed to the
magnonic resonances of different spatial characters) are
fitted to Lorentzian curves in order to extract their ampli-
tudes, frequencies ω, and full width at half-maximum Δω.
The latter two parameters are used to estimate the relative
net relaxation rates given by the LLBar model as
/LLBar LLBar LLBarΓ = ∆ω ω . The angular dependence of
LLBarΓ is calculated for the edge mode for an isotropic
disk and an ellipse. For the disk, the symmetry of LLBarΓ
was found to be either isotropic or twofold for an out-of-
plane and in-plane reduction of the relaxation tensor, re-
spectively. In contrast, for the ellipse the symmetry is al-
ways twofold, even for the case when the tensor is altered
out of plane and, thereby, homogeneous in plane. There-
fore, the shape anisotropy also contributes to the symmetry
of the relative relaxation rate. This is also supported by the
fact that the relative relaxation rates are different for in-
plane and out-of-plane reductions of the relaxation tensor,
i.e., along easy and hard axes, respectively.
In total, the results obtained in an isotropic disk and an
ellipse are threefold [42]. First, the highest reduction of
relative relaxation rate is observed when the shortest ei-
genvector of the relaxation tensor is parallel to the hardest
degree of freedom. Second, the reduction (enhancement) of
characteristic frequency leads to the reduction (enhance-
ment) of the contribution of the corresponding relaxation
872 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9
Spin waves damping in nanometre-scale magnetic materials
channel. Third, the authors [42] showed that the symmetry
of the magnonic relative relaxation rate with respect to the
direction of saturation is determined by the superposition
of the symmetries of the relaxation tensor and the ellipse of
motion. The latter is mode-specific due to the competition
between the shape anisotropy and the exchange energy
(that eventually becomes dominant for high-order magnons),
thereby making the absolute value of the relative relaxation
rate frequency dependent.
These conclusions could be used to design a test exper-
iment, e.g., measurements of the magnonic linewidths in
nanoelements with in-plane and out-of-plane uniaxial ani-
sotropies. Then by estimating the characteristic frequencies
from the micromagnetic simulations, the corresponding
components of the relaxation tensor could be extracted. An
angular dependence of magnetic losses (or effects associat-
ed with the same physics) has already been observed ex-
perimentally [45–49].
4. Nanometre-scale patterned magnetic structures
Patterned magnetic media have attracted a lot of atten-
tion recently due to both fundamental and technological
interest. High-quality nanopatterned magnetic structures
can be a 2D magnonic crystal [50–52], the basis for mag-
netic metamaterials [53], and offers new ways for the de-
velopment of data processing without moving electrical
charges with working frequencies in the GHz range
[7,54,55]. Fundamentally, the question of how nanosized
non-magnetic inclusions affect macroscopic and micro-
scopic magnetic properties of continuous thin film is of
great interest. Particularly, the switching mechanism of
patterned magnets during the magnetization reversal pro-
cess is among the important issues that are not well under-
stood in the nanoscale regime yet. When the desired re-
sponse time is in the GHz range, the switching behavior of
magnetic elements is determined by the FMR frequencies
and the corresponding damping times. However, the pat-
terning leads to less uniform and more complex magnetiza-
tion dynamics that has not yet received exhaustive expla-
nation even on a phenomenological level.
We illustrate some specifics in this field on an example
of a thin film with mesh of holes (antidots). The effect of
holes size and lattice parameters on magnetization dynam-
ics has been studied in Refs. 56–61. It was shown that
magnetic antidots enable to induce a configurational ani-
sotropy. Its value is determined by the symmetry of the
lattice, size and shape of the holes, and can be controlled
during sample preparation [62]. The extra broadening of
resonance line is also detected. Let us imagine the antidot
lattice as a two-dimensional network of interconnected
magnetic nanowires. Then it is reasonable to expect anisot-
ropy effects of spin wave propagation, as well. Indeed, it
was observed [63] that, depending on the orientation of the
external field, spin waves propagate at velocities being
comparable with or a factor of about two smaller than the
velocities found for a continuous film. The damping char-
acteristics were also highly anisotropic suggesting that
extrinsic damping, e.g., due to edge-roughness mediated
scattering, is relevant in an antidot lattice. A deep under-
standing of the respective parameters is important for fun-
damental research. Naturally, all these features are very
critical for applications.
4.1. FMR damping
In a film with nonmagnetic inclusions (e.g., holes), in
addition to the intrinsic damping inherent to the particular
material, one can also expect specific relaxation channels
due to the film’s patterned structure. However, the investi-
gation of relaxation mechanisms in relation to the geomet-
ric parameters and micromagnetic configuration of a pat-
terned structure is a challenging problem that has received
a little attention up to date. We note here the report by
Martyanov et al. [64], where measurements were per-
formed on geometrically scaled antidot arrays with a rec-
tangular unit cell. (In Ref. 64 two samples with different
periods, but with a fixed ratio of the hole’s radius, r, to the
lattice period, a, were studied.) The authors found a strong
attenuation of the uniform FMR mode compared to the
resonance mode of a continuous film and attributed the
broadening to a two-magnon process, which scatters ener-
gy from the uniform mode to other degenerate modes with
higher wave numbers.
The authors [60,61,65] used micromagnetic simulations
and a semianalytical approach to model the FMR depend-
ence on geometrical parameters of the antidot lattice. The
approach used directly accounts for the effects of the mag-
netic state nonuniformity, allows avoiding a numerical
calculation on the step of restoring a system’s linear dy-
namic properties, and offers physical transparency for what
is usually hidden behind numerical micromagnetic simula-
tions. An example of micromagnetic modeling of static
magnetic configuration of the hexagonal antidot lattice in
external magnetic field normal to the film plane is shown
in Fig. 4(a). The domain structure obtained consists of three
regions with different magnetization orientation in the mag-
netic unit cell. In two of them, regions A and B in Fig. 4(a),
a direction of magnetic moments differs from the direction
of spontaneous magnetization. In the third area, region C in
Fig. 4(a), the magnetic moments are parallel to the unit cell
total magnetization. Since the external field is oriented per-
pendicular to film plane, i.e., symmetry of the magnetic
system is not broken, the projection of the spontaneous
magnetization on the film plane is directed along one of
the “easy” axes, which coincides with the short diagonal of
the unit cell of a hexagonal lattice of antidots. (For the case
shown in Fig. 4(a) this is a direction along the arrow in
the upper left corner, angle α = 60° to the ox-axis.) While
the “hard” axis is the direction along the long diagonal
of the cell.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9 873
V.N. Krivoruchko
The calculation of local dynamic susceptibility tensor
yields that the resonance spectra consist of three different
quasi-uniform modes of magnetization precession related
to the precession of the magnetic moment in the regions A,
B and C. An example of resonance frequencies distribution
within a unit cell is shown in Fig. 4(b). Each of the reso-
nant modes follows a two-fold variation with respect to the
in-plane orientation of the applied magnetic field. The easy
axes of the modes are mutually rotated by 60 degrees and
combine to yield an apparent six-fold configurational ani-
sotropy.
Figure 5 summarizes the revealed theoretical depend-
ences of the total FMR linewidth, ΔH, in the hexagonal
magnetic antidot lattice as a function of the geometrical
factor (r/a) for structures with different lattice periods a
(the damping parameter α is fixed). It is apparent that the
width of the FMR line increases with an increase of mag-
netic inhomogeneity (i.e., with an increase of the density of
holes). The dependence for the FMR linewidth on the pa-
rameter (r/a) as compared to a continuous film could be
reasonably fitted by a scaling form
/ (0) / (0) ~ ( / )[ ( ) ]H r a H H r a η∆ ∆ ∆− .
The critical exponent η is function of the lattice period
(η = 2.22, 2.30, and 3.33 for structures with lattice period
a = 50,75, and 100 nm, respectively).
Another important parameter here is a film thickness.
The variation of the film thickness leads to changes in ani-
sotropic properties of the system, too. With increasing film
thickness, there is not only an increase of the induced ani-
sotropy (a magnitude of resonance frequency angular de-
pendence), but also a shift of the spectrum minimum to
lower frequencies. In addition, with increasing film thick-
ness, a multiple resonance structure becomes more pro-
nounced in spite of larger resonance line-width. In accord-
ance with theoretical predictions, the experimental data of
FMR spectra [65] of the antidot lattices for film thicknesses
10 nm and 25 nm indeed contain only three modes, which
are well fitted by three quasi-uniform modes of magnetiza-
tion precession. For thicker films, three resonance modes
are clearly observed for the in-plane field geometry and
additional (spin-wave) modes are detected in the normal-
to-plane geometry. As it expected, the inhomogeneous dis-
tribution of magnetization and the effective field in antidot
lattices lead to substantial differences in the forms of spin-
wave lines and the resonant fields for the film with antidot
lattice and homogeneous film [65].
4.2. Spin waves anisotropic propagation and damping
The antidot lattice, if considered as a two-dimensional
network of interconnected magnetic nanowires, is a rele-
vant model system to study the possible anisotropy effects
of spin wave propagation. Such effects were recently studi-
ed by time-resolved magneto-optical Kerr microscopy [63].
The permalloy film was periodically patterned into a
square lattice. The holes (diameter of 120 nm) are arranged
on a square lattice with a period of 800 nm. For the contin-
uous film, one obtains a relaxation time of 0.6 ns, which
does not depend on propagation angle φ. For the antidot
lattice the authors [63] observed angle-dependent relaxa-
tion characteristics. At regime A (φ = 0°, f = 8 GHz, μ0H =
= 68 mT) the effective relaxation time is 0.70 ns, and was
comparable with the plain film. A minimum relaxation
time of 0.37 ns is found at regime B (φ = 22°, f = 5.2 GHz,
μ0H = 20 mT). If compared to regime A, the effective re-
laxation increases by a factor of almost two. A larger re-
Fig. 4. (Color online) Domain structure of hexagonal antidote
lattice r/a = 0.3 in Permalloy film. The grayscale shows the local
magnetization deviation from the direction of unit cell total mag-
netization (a). Resonance frequencies distribution on antidot lat-
tice cell H = 0.7T, r/a = 0.35, h = 10 nm. Angle between anisot-
ropy axes and field direction is 78° (b). (Borrowed from [65]).
Fig. 5. (Color online) The dependence of the FMR linewidth ΔH,
as function of the geometrical factor r/a for hexagonal antidot
lattice with different lattice period a (Borrowed from [60]).
874 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9
Spin waves damping in nanometre-scale magnetic materials
laxation time, 0.49 ns, is regained at regime C (φ = 30°,
f = 8 GHz, μ0H = 75 mT) but still the value is smaller than
that of at φ = 0° and of the continuous film. The relaxation
times vary with the characteristic mode profiles illustrated
in Fig. 6(a)–(c). In regime B the propagating mode has a
higher oscillation amplitude near the edges of the holes if
compared to A and C regimes. Near the hole edges, the in-
ternal field can vary due to edge roughness and possible
reduced saturation magnetization. Via magnon-magnon scat-
tering, such irregularities can open further relaxation chan-
nels. This might explain the anisotropy of damping found
in the angular dependent experiments. However, a theory
linking mode profiles and edge roughness with relaxation
is missing.
These results show that by removing only a small frac-
tion of material from a film to form an antidot lattice, the
properties of propagating spin waves are significantly al-
tered. Depending on the orientation of the external field,
spin waves can propagate at velocities being comparable
with or a factor of about two smaller than for a plain film.
In contrast to the film, the damping characteristics turn out
to be highly anisotropic suggesting that extrinsic damping
due to, e.g., edge-roughness mediated scattering is relevant
in an antidot lattice. The dependence of the damping on the
orientation of M represents an additional option to opti-
mize a magnetization reversal process in a nanostructured
system by choosing a magnetization trajectory which is
most appropriate from the viewpoint of damping.
Spin-wave modes in thin-film antidot lattices are inves-
tigated theoretically in Ref. 66 using micromagnetic simu-
lations and a semianalytical theoretical approach. The si-
mulations reveal a rich eigenmode spectrum consisting of
edge and center modes. Both spatially localized and spin
waves were found to be extending over many unit cells.
For lattices of unit-cell lengths ranging from 200 to 1100 nm,
the authors found that the characteristic mode eigenfre-
quencies could be correlated with both local inhomogenei-
ties of the demagnetization field and specific wave vectors
caused by geometry-imposed mode quantization condi-
tions.
In conclusion of this section, note that the recent pro-
gress in simulations of the excitation and propagation, and
other wave characteristics of spin waves, in nanometre-
scale patterned magnetic elements can be found in review
paper [67]. Many examples of micromagnetic modelling
for numerical calculations, employing various dimensions
and shapes of patterned magnetic elements, are given here,
too. The current limitations of continuum micromagnetic
modelling and of simulations based on the LLG equation
of motion of magnetization are also discussed, along with
further research directions for spin-wave studies.
5. Conclusion
The high-frequency excitations in confined magnetic
structures exhibiting nonuniform magnetization distribu-
tions are the subject of a growing interest at present. In
addition to interest from the fundamental physics point of
view, there are practical reasons for this. The purpose is to
investigate and then to control (to reduce/enhance) the re-
laxation channels in such systems in relation with their
geometric characteristics and magnetic structure. Natural-
ly, in this connection, the question was raised whether the
LL (LLG) phenomenological damping parameter is appli-
cable to magnetic nanomagnetic elements and nanostruc-
tured samples. Based on the results of recent advanced
experiments and theoretical analysis the answer is “No”.
Analysis within the Bar’yakhtar generalization of the damp-
ing term [42] inspires hopes that this approach can be
a reasonable phenomenology, which can provide us with
a realistic description of spin motions in ultrathin ferro-
magnet films and in large number of magnetic nanostruc-
tures. Yet, further work is definitely needed to prove this
expectation. In addition, it should be taken into account
that all results based on a phenomenological approximation
are valid for time scales longer than the picosecond and
dimensions greater than the nanometre. Phenomena associ-
ated with faster-dynamics and atomic spatial resolution
require microscopic consideration.
The absence of fruitful microscopic theory, which is
able to give true recommendation to control the relaxation,
is one of the reasons why nanomagnonic devices realized
so far suffer from the relatively short damping length.
However, there are reasons to expect that situation can be
drastically change in the nearest future. Very recent exper-
imental publications demonstrate that the ultra-thin-film
ferrimagnet insulator YIG (yttrium iron garnet) allows
Fig. 6. (Color online). (a)–(c) Spatial spin-wave mode profiles
obtained by time-resolved magneto-optical Kerr microscopy (up-
per row). Shown are modes in the three different regimes:
(a) – 0A ϕ = ° , f = 8 GHz, μ0H = 68 mT, (b) 25B − ϕ = °,
5 : 2 GHzf = , μ0H = 20 mT, and (c) 30C − ϕ = °, f = 8 GHz,
μ0H = 75 mT. Simulation data (lower row) are obtained at
μ0H = 20 mT and, in particular, k = 0. The eigenfrequencies are
(a) 3.8 GHz, (b) 4.3 GHz, and (c) 3.9 GHz. Bright (dark) contrast
reflects high (low) spin-precession amplitude. (Borrowed from
[63]).
(a) (b) (c)
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 9 875
V.N. Krivoruchko
overcoming the damping problem for nanomagnetism and
to boost further developments in nanomagnonic devices
technology for processing at GHz operational frequencies.
Liu et al. [68] studied ferromagnetic resonance of YIG
films with a thickness of 7 to 26 nm finding a promising
damping constant of 0.001. This value is already a factor
of about three better than the metallic ferromagnets used in
magnonics. Hahn et al. showed that the damping constant
is unaffected when nanostructuring ultra-thin YIG films
[69]. Pirro et al. [70] studied a 100 nm thick YIG thin film
and observed a spin-wave decay length of 31 μm. Unprec-
edented results for now were obtained in Ref. 71. Authors
demonstrate that in nm-thick YIG films propagating spin
waves can exhibit a damping parameter of 2.3 10–4. Esti-
mated macroscopic of damping length value was of about
one mm allowing for magnonics-based nanotechnology
with ultra-low damping. The spin wave group velocities
are found to vary between 0.6 km/s and 1.2 km/s depend-
ing on the wave vector and applied magnetic field. This
experimental work demonstrates that the thin-film ferri-
magnet insulator YIG allows overcoming the damping prob-
lem for nanomagnetism and boosting further developments
in nanomagnonic devices technology for processing at
GHz operational frequencies. These results definitely mo-
tivate further fundamental investigation in the field.
Numerous fruitful discussions with S.M. Ryabchenko are
gratefully acknowledged. The author would like to thank
V.O. Golub for critical reading of the manuscript. This work
was partly supported from the European Union’s Horizon
2020 research and innovation programme under the Marie
Skłodowska-Curie grant agreement No 644348 (MagIC).
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1. Introduction, background equations
2. Ultrathin and exchange-based films
2.1. Two-magnon scattering
2.2. Nonlocal relaxation
2.3. Unidirectional damping
3. Damping in magnetic nanometre-scale samples
3.1. Three-magnon scattering
3.2. Longitudinal relaxation
3.3. Anisotropic damping in nanoelements
4. Nanometre-scale patterned magnetic structures
4.1. FMR damping
4.2. Spin waves anisotropic propagation and damping
5. Conclusion
References
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