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Brillouin light scattering studies of planar metallic
magnonic crystals
G Gubbiotti, S Tacchi, M Madami, G Carlotti, a O Adeyeye, M Kostylev
To cite this version:
G Gubbiotti, S Tacchi, M Madami, G Carlotti, a O Adeyeye, et al.. Brillouin light scattering studies
of planar metallic magnonic crystals. Journal of Physics D: Applied Physics, IOP Publishing, 2010,
43 (26), pp.264003. �10.1088/0022-3727/43/26/264003�. �hal-00569637�
1
Brillouin light scattering studies of planar metallic magnonic
crystals
G. Gubbiotti
1
, S. Tacchi
1
, M. Madami
1
, G. Carlotti
2
, A. O. Adeyeye
3
and M.
Kostylev
4
1
CNISM, Unità di Perugia- Dipartimento di Fisica, Via A. Pascoli, I-06123 Perugia,
Italy
2
CNISM, Unità di Perugia- Dipartimento di Fisica and Università di Perugia, Via A.
Pascoli, I-06123 Perugia, Italy
3
Department of Electrical and Computer Engineering, National University of
Singapore 117576, Singapore
4
School of Physics M013, University of Western Australia, 35 Stirling Hwy, 6009
Western Australia, Australia
E-mail: gubbiotti@fisica.unipg.it
Abstract
. The application of Brillouin light scattering to the study of the spin-wave
spectrum of one- and two-dimensional planar magnonic crystals consisting of arrays of
interacting stripes, dots and antidots is reviewed. It is shown that the discrete set of allowed
frequencies of an isolated nanoelement becomes a finite-width frequency band for an array of
identical interacting elements. It is possible to tune the permitted and forbidden frequency
bands, modifying the geometrical or the material magnetic parameters, as well as the external
magnetic field. From a technological point of view, the accurate fabrication of planar magnonic
crystals and a proper understanding of their magnetic excitation spectrum in the GHz range is
oriented to the design of filters and waveguides for microwave communication systems.
1. Introduction
Similar to photons in photonic crystals,
1
the spectrum of spin excitations in materials with
periodically modulated properties shows bands of allowed magnonic states, alternated with forbidden
band gaps.
2,3,4,5,6,7
This constitutes a new class of artificial crystals, now known as magnonic crystals
(MC), in which collective spin excitations rather than light are used to transmit and process
information.
8,9,10
Since the wavelength of these excitations are shorter than those of light in the GHz
range, MC offer better prospects for miniaturization at these frequencies with the advantage that
frequency position and width of the band gap are tunable by the applied magnetic field.
A MC can be formed starting from uncoupled resonators and making them coupled by some
interaction, such as dipolar or exchange magnetic coupling. Alternatively, one can artificially process
a continuous medium, to make a periodical profile of magnetic properties such as the saturation
magnetisation and the exchange constant. Examples of one-dimensional (1D) MC, where the
collective behaviour is mediated by dynamical dipolar interaction, are multilayered magnetic
structures consisting of alternated ferromagnetic layers.
4
It is also possible fabricate a “lateral”
multilayer in the form of an array of closely spaced parallel magnetic stripes or an array of stripes of
different magnetic materials in direct physical contact with each other (Cobalt and Permalloy, for
Confidential: not for distribution. Submitted to IOP Publishing for peer review 5 February 2010
2
example).
11,12
Since the two materials have different coercivity and saturation magnetisation, this can
be used to open the road to programmable magnonic ground state and to the consequent field-
controlled dynamic response. The advantage of such a kind of “continuous” over discrete MC is that
exchange coupling at boundary regions takes place and the dynamical dipole coupling is maximised.
As a result, spin oscillations can easily transfer from one region to another and therefore spin waves
can propagate across its entire structure with considerable group velocities. Remarkably, 1D MC in the
form of lateral multilayers have also been obtained through the artificial modulation of the magnetic
properties of a continuous film by either laser annealing or ion implantation.
13,14,15
More recently, Ki-
Suk Lee an co-authors performed micromagnetic simulations on a novel 1D waveguide consisting of a
Permalloy nanostripe with periodic modulation of its width.
16
Their predictions about allowed and
forbidden bands of propagating dipole-exchange spin waves in such a system have been just
confirmed by spatially–resolved Brillouin light scattering experiments in microsized modulating
stripes.
17
Similar to the case of 1D structure, also two-dimensional (2D) artificial magnonic crystals can be
fabricated in the form of ordered arrays of either closely packed magnetic dots (coupled by dipolar
interaction) or ferromagnetic antidots, i.e. a periodic array of holes drilled into a continuous magnetic
film. In these systems the properties of collective spin wave can be controlled by changing the dots
(holes) shape, dimension and symmetry arrangement in the array. The remarkable difference in the
case of antidot samples is due to the presence of continuous portion of the magnetic film which makes
possible the propagation of guided waves along certain particular directions.
The purpose of the present paper is to review the recent experimental observation of collective
modes in 1D and 2D MC by the Brillouin light scattering (BLS) technique, which has proved to be a
very powerful tool for the investigation of magnetisation dynamics in such structures. In this respect,
BLS has some advantages with respect to other experimental methods, such as ferromagnetic
resonance (FMR) with vector network analyzer (VNA)
18
and time resolved scanning Kerr microscopy
(TRSKM),
19
usually employed to probe magnetisation dynamics in nanostructures. First of all, thanks
to the wave vector conservation in the magnon-photon interaction, one has the possibility to measure
the dispersion relation (frequency vs. wave vector) of the collective spin excitations, provided that the
periodicity of the MC is such that the Brillouin zone (BZ) boundary lies in the wave vector range
accessible in conventional BLS experiment. Furthermore, it has been recently shown that using a
large-aperture objective, BLS can be used as scanning probe technique, permitting the map-out of the
spatial distribution of magnetic normal modes with a lateral resolution of a few hundreds of
nanometers.
20,21,22,23
This paper has the following organization. Section 2 is dedicated to deep ultraviolet lithography
technique and addresses its capability to obtain well-controlled nanostructures. In Section 3, the
analytical theory used to calculate the spectrum of collective excitations in quasi-1D plane MC
consisting of array of longitudinally magnetised stripes is described. In Sections 4.1 and 4.2 we review
the experimental observation by BLS of collective spin excitations in arrays of non-contacting and
contacting stripes, respectively, while Sections 5.1 and 5.2 contain results for 2D MC made by
antidots and dots arrays. Eventually, Section 6 presents the conclusions and outlines some perspective
in the field of MC.
2.
Fabrication and experiment 1D dense stripes array
The fabrication of 1D and 2D magnonic crystals is based on recent abilities to produce dense arrays
of magnetic elements arranged with sub-micrometer precision and with a very narrow distribution of
shapes, sizes and distances. The statistical variations of these parameters can often hide, at least
partially, the effect of magnetostatic and magnetodynamic interactions in arrays of thousands
elements. Conventional nanofabrication techniques that are used in the microelectronic industry are
not always compatible with magnetism because the process involves high temperature which will
degrade the quality of the ferromagnetic films. It is also very difficult to use reactive ion beam etching
to pattern magnetic films as it is not easy for the reactive gases to form volatile compounds when in
3
contact with magnetic materials. Some of the key issues to be considered in the development of
fabrication techniques for magnetic nanostructures are critical dimension control, resolution, size and
shape homogeneity, patterned area and alignment accuracy. The nanofabrication methods used for
synthesizing nanomagnets in recent years include electron beam lithography and lift-off processes,
focused ion beam (FIB) etching, interferometric lithography, nanoimprint lithography, and anodic
aluminum oxide membranes. For a general review of the various techniques for fabricating ordered
magnetic nanostructures, the reader is referred to Refs. 24,25,26 and 27. Here we shortly describe the
fabrication technique used at the University of Singapore to prepare most of the samples reviewed in
this article, i.e. deep ultraviolet (DUV) lithography at 248 nm exposure wavelength. Alternating phase
shift mask (PSM) was used to pattern large areas (typically 4x4mm
2
) with either closely packed
magnetic stripes or antidot arrays on commercially available silicon substrates,
28
with lateral
dimensions much below the conventional resolution limit of optical lithography. One unique
advantage of this technique is the fact that unlike e-beam lithography, thicker resists can be used to
make high aspect ratio nanostructures. DUV lithography also has the ability to tune side wall profile
by employing focus offset and resist processing temperatures. In addition, this nanofabrication
technique is also compatible with conventional charge based complementary metal oxide
semiconductor (CMOS) platform, thus enabling the integration of novel magneto-electronic devices.
To create patterns in the resist, the substrate is coated with a 60 nm thick anti-reflective layer followed
by 480 nm of positive DUV photoresist. A Nikon lithographic scanner with KrF excimer laser
radiation is used for exposing the resist. To convert the resist patterns into ferromagnetic
nanostructures, the Permalloy (Ni
8
0
Fe
2
0
) layer is deposited using physical vapour deposition
techniques such as e-beam evaporation and sputtering on the resist patterns. The layer sitting on resist
was lifted-off by dissolving the resist in solvent OK73 (trade name of the resist solvent). In the lift-off
process it is crucial to have a clean break-off of the film at the pattern edges of the resist. To reduce
step coverage, a collimating sample holder was designed. This holder restricts the incidence angle of
the incoming material, thereby allowing only material in the path normal to the surface to reach the
sample.
29
Lift-off was determined by the colour change of the patterned film and confirmed by
examination under a SEM. With the special sample holder, the lift-off process was much easier and
high aspect ratio patterned nanostructures with film thicknesses (> 120nm) were achieved. Note that
the electron beam lithography and lift-off can also used to produce a lateral multilayer consisting of
two families of stripes in direct physical contact with each-other, as made in Ref. 12. The first family
of stripes (for example Permalloy stripes) is defined on polymethyl methacrylate (PMMA) resist
followed by electron beam deposition and lift-off. For the fabrication of the second nanostripe array,
another layer of PMMA resist is then deposited. At this point, high-resolution electron beam
lithography with precise alignment can be used to define the position of the second nanostripe array.
Subsequently, a second magnetic film (for example, Cobalt film) is deposited. The final structure thus
consists of adjacent stripes in direct physical contact. (see Fig. 7)
3. Theoretical model for the quasi-1D plane magnonic crystals
Two main theoretical approaches have been used in the literature to describe the band structure of
magnonic crystals, i.e. the plane wave method expansion and the dynamical matrix method. The
former can be applied to periodical arrays in any dimension including periodic arrays of magnetic
particles embedded in a non-magnetic medium or in a matrix made of a different ferromagnetic
material.
30,31,32,33
The latter
.
is an hybrid method
34,35
combining micromagnetic simulations for
calculating the ground state and an eigenvalue/eigenvector approach, which requires the computation
of a dynamical matrix, whose elements are related to the torque acting on the magnetisation in each
cell. The eigenvalues are the frequencies of the magnonic modes of the system and the eigenvectors
give spatial profiles of the modes. The elements of the dynamical matrix can be calculated analytically
for any contribution to the total energy (magnetostatic, exchange, Zeeman, magnetic anisotropy)
without introducing numerical approximations, while the final diagonalization of the dynamical matrix
is done numerically. So far, this method has been applied mainly to single planar dots of different
4
shapes.
23,35,36
One also has to note that different micromagnetic simulation packages, such as
OOMMF,
3
7
can be adopted for studying coupled dynamics of nanoelements, following the space
resolved dynamics of magnetisation in time domain and operating a Fourier transform to the frequency
domain.
3
8,39,40
However, this latter method does not provide information on the frequency dispersion
over the whole artificial Brillouin zone. If one restricts to the case of collective spin excitations in 1D-
MC, it is possible to formulate analytical models which yield direct physical insight to the
characteristics of collective excitations.
2-6,41
One of these analytical approaches is presented in the
following to analyse the case of 1D plane MC consisting of dynamically coupled nanostripes of
(quasi)-infinite length, magnetised along their length. Within this framework, it is quite
straigthforward to derive a number of results which can easily illustrate the effect of dynamical
coupling in systems of interacting stripes, as we do in the following. The starting point to solve this
problem is the linearized Landau-Lifschitz equation of motion for magnetic moment
(
)
(
)
( , , ) ( ) ( , , ) ( , )
i x y z x x y z x z
ω γ
− = − + × +
m M m H h (1)
In this equation γ is the gyromagnetic coefficient, M is the equilibrium magnetisation, with |M|
equal to the film saturation magnetisation M
s,
. The equilibrium magnetisation is directed along the
magnetic field H which is applied in the film plane. To be specific we will consider the situation
which is usually implemented in Brillouin light scattering experiments: M and H are directed in the
plane array along the stripe longitudinal axis y. In this magnetic ground state the stripes are completely
saturated and there is no static demagnetising field (the stripes are of infinite length along y). The
equilibrium magnetisation depends on the co-ordinate x which lies in the array plane and is
perpendicular to the longitudinal axis y. This is the direction of array periodicity and thus |M| changes
periodically. In particular, if an array of stripes separated by air nanogaps is considered |M| varies
between M
s
in the stripes and zero in the gaps. As M and H are aligned along y the small linear
dynamic magnetisation component m has one component in the array plane (m
x
) and one
perpendicular to the array plane (m
z
). Its precession frequency is ω. This spatially inhomogeneous
precessing magnetisation gives rise to an effective field h which in our case of free oscillations
consists of an effective exchange field
2
exc
α
= ∇
h m
(2)
where
α
the exchange constant, and of a dynamic dipole field
d
h
which is described by the
magnetostatic equations
( , ) 0
( , ) 4 ( , )
d
d
x z
x z x z
π
× =
⋅ = − ⋅
h
h m
∇
∇ ∇
(3)
Three different methods were used to solve the system (1)-(3) for 1D plane magnonic crystals.
12,14,41
They mainly differ by the way how the magnetostatic equations (3) are solved. In Ref. 12 this system
of equations is transformed into a 2D Poisson equation for the magnetostatic potential. In Ref. 14 a
known simple solution in the Fourier space
42
is used and the system of equations (2-3) is transformed
into an infinite system for spatial Fourier amplitudes of dynamic magnetisation. In this work we will
follow the approach of the Green’s function in the direct space described in Refs. 43 and 44.
Importantly, all free approaches require numerical treatment of the problem in the last stage of
solution, however the Green’s function approach allows drawing some important qualitative
conclusions based on approximate analytical formulas.
![Fig. 4 Experimental and calculated spin-wave frequency dispersion for an easy axis applied magnetic field H=500 Oe for stripes arrays #1, #2 and #3 described in the text. Vertical lines indicate the edges of the Brillouin zone (k=nπ/a) while colored areas indicate the frequency band gap. Reprinted with permission from [47]. G. Gubbiotti, G. Carlotti, P. Vavassori, M. Kostylev, N. Singh, S. Goolaup, and A. O. Adeyeye 2005 Phys. Rev. B 72 224413.© 2005, American Institute of Physics. Reprinted with permission from [3]. G. Gubbiotti, S. Tacchi, G. Carlotti, N. Singh, S. Goolaup, A. O. Adeyeye, and M. Kostylev 2007 Appl. Phys. Lett. 90 092503. © 2007, American Institute of Physics.](/figures/fig-4-experimental-and-calculated-spin-wave-frequency-35pk2np9.webp)
![Fig. 8 (Left panel) Frequency dispersion of collective modes measured by BLS measure in zero external applied field. Dashed lines indicate the BZ boundaries. (Right panel) Magnetic field dependence of the frequency centre and width of the first band gap on applied magnetic field. Reprinted with permission from [12]. Z. K. Wang, V. L. Zhang, H. S. Lim, S. C. Ng, M. H. Kuok, S. Jain, and A. O. Adeyeye, Appl. Phys. Lett. 2009 94 083112. © 2009, American Institute of Physics.](/figures/fig-8-left-panel-frequency-dispersion-of-collective-modes-wf0096dq.webp)
![Fig. 7 SEM image of a magnonic crystal in the form of a 1D periodic array of 30-nm-thick Permalloy and Cobalt nanostripes, each of width 250 nm. Reprinted with permission from [12].Z. K. Wang, V. L. Zhang, H. S. Lim, S. C. Ng, M. H. Kuok, S. Jain, and A. O. Adeyeye, Appl. Phys. Lett. 94, 083112 (2009). © 2009, American Institute of Physics.](/figures/fig-7-sem-image-of-a-magnonic-crystal-in-the-form-of-a-1d-3tljwon0.webp)
![Fig. 1 Comparison between the calculated profile of dynamic magnetisation for the lowest frequency mode (left) and the next one (right panel) for an array of stripes of widths w=175 nm, separation ∆=35 nm, thicknes 20 nm, period a=210 nm. Solid line (dashed line) curves refer to the case of k=0 (k=kBZ=π/a). The dimensionless coordinate in abscissa is normalized to the stripes width (w). Reprinted with permission from [47]. G. Gubbiotti, G. Carlotti, P. Vavassori, M. Kostylev, N. Singh, S. Goolaup, and A. O. Adeyeye 2005 Phys. Rev. B 72 224413. © 2005, American Institute of Physics.](/figures/fig-1-comparison-between-the-calculated-profile-of-dynamic-1ke0ratb.webp)