Nonreciprocal collective magnetostatic wave modes in geometrically asymmetric bilayer structure with nonmagnetic spacer P. I. Gerevenkov1V. D. Bessonov2V. S. Teplov2A. V. Telegin2A. M. Kalashnikova1and N. E. Khokhlov1

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Nonreciprocal collective magnetostatic wave modes in geometrically asymmetric

bilayer structure with nonmagnetic spacer

P. I. Gerevenkov,1, ∗V. D. Bessonov,2V. S. Teplov,2A. V. Telegin,2A. M. Kalashnikova,1and N. E. Khokhlov1

1Ioﬀe Institute, 194021 St. Petersburg, Russia

2M.N. Mikheev Institute of Metal Physics, UB of RAS, 620108 Ekaterinburg, Russia

(Dated: March 21, 2023)

Nonreciprocity, i.e. inequivalence in amplitudes and frequencies of spin waves propagating in op-

posite directions, is a key property underlying functionality in prospective magnonic devices. Here

we demonstrate experimentally and theoretically a simple approach to induce frequency nonreciproc-

ity in a magnetostatically coupled ferromagnetic bilayer structure with a nonmagnetic spacer by its

geometrical asymmetry. Using Brillouin light scattering, we show the formation of two collective

spin wave modes in Fe81Ga19/Cu/Fe81 Ga19 structure with diﬀerent thicknesses of ferromagnetic

layers. Experimental reconstruction and theoretical modeling of the dispersions of acoustic and op-

tical collective spin wave modes reveal that both possess nonreciprocity reaching several percent at

the wavenumber of 22 ·104rad/cm. The analysis demonstrates that the shift of the amplitudes of

counter-propagating coupled modes towards either of the layers is responsible for the nonreciprocity

because of the pronounced dependence of spin wave frequency on the layers thickness. The pro-

posed approach enables the design of multilayered ferromagnetic structures with a given spin wave

dispersion for magnonic logic gates.

INTRODUCTION

Nonreciprocity, i.e. inequivalence of a medium prop-

erties sensed by a (quasi)particle traversing it in posi-

tive and negative directions, underlies the operation of

indispensable elements of modern electronic, microwave

techniques, and optics. Limitations of transistor den-

sity and power consumption associated with conventional

electron-charge technologies promote other quesiparticles

as information carriers for next generation of computing.

An emerging competitor of electronics, magnonics [1, 2]

employs spin waves and their quasi-particles, magnons,

to encode, carry, and process the information. Thus,

magnonics is eager for the development of basic nonre-

ciprocal elements, such as spin wave diode [3, 4], cir-

culator [3], half-adder [5], etc. The clear advantage of

magnonics is that nonreciprocity of spin waves’ ampli-

tudes is intrinsic for magnetic thin ﬁlms, as was recog-

nized already in the 1960s [6]. However, the functionality

and tunability of nonreciprocal magnonic elements [7, 8]

broaden signiﬁcantly if they support frequency nonre-

ciprocity as well, when waves of the same wavenumber

traveling in opposite directions possess diﬀerent frequen-

cies [9]. These require ﬁnding media or designing artiﬁ-

cial structures possessing strong frequency nonreciprocity

for spin waves while being compatible with miniaturiza-

tion and maintaining low losses. Intrinsic frequency non-

reciprocity of the spin waves requires that the medium

possess chirality [10], which is lacking in most materi-

als relevant for magnonics applications. Therefore, sym-

metry breaking of various origins in thin ﬁlms and het-

erostructures is now seen as the most promising way to

realize non-reciprocity of spin waves [11–16].

∗petr.gerevenkov@mail.ioﬀe.ru; http://www.ioﬀe.ru/ferrolab/

It has recently been recognized that the collective be-

havior of spin waves in coupled waveguides, magnonic

crystals, etc. oﬀers extended possibility to design and

tune dispersion properties. This owns to the fact that

even small change in properties of one of the element of

such a structure may result in drastic changes in behavior

of coupled spin wave modes [17, 18]. In this respect, het-

erostructures with two coupled magnetic layers have re-

cently attracted great interest due to their ability to sup-

port collective magnetization dynamics and to the possi-

bility of controlling it by independently adjusting charac-

teristics of individual constituents [19] and coupling be-

tween them [20, 21]. In contrast to the widely considered

nonreciprocity associated with interfacial anisotropy [22]

and / or the Dzyaloshinskii-Moriya interaction [23–25],

the diﬀerence in the dispersion of collective magneto-

static spin wave (MSW) modes propagating in opposite

directions in multilayered structures also relies on the

interaction between layers. In [26–29], nonreciprocity

is demonstrated in synthetic antiferromagnets, systems

with antiferromagnetic ordering of ferromagnetic layers

due to indirect exchange interaction. Nonreciprocity has

been demonstrated in systems of exchange-coupled layers

with a parallel orientation of magnetization [4, 30, 31].

On the other hand, in [30, 32], it was shown that multilay-

ered systems characterized by diﬀerences in the magnetic

parameters of their constituents, e.g. saturation magne-

tization and anisotropy, also support the nonreciprocity

of MSW, which is based on asymmetry in magnetic pa-

rameters. The antiparallel orientation of magnetizations

in dipole-coupled layers also serves as a source of nonre-

ciprocity [24, 28].

In this article, we propose to realize nonreciprocal

MSW propagation in a ferromagnetic bilayer structure,

where the two dipole-coupled ferromagnetic layers pos-

sess similar magnetic properties, but have diﬀerent thick-

nesses. The structure exhibits a parallel orientation

arXiv:2210.14882v2 [cond-mat.mtrl-sci] 19 Mar 2023

of the magnetizations of the layers. Using Brillioun

light scattering, we examine the dispersion of thermal

MSWs in the bilayer structure and reveal the forma-

tion of collective acoustic (in-phase) and optical (out-

of-phase) modes. Both modes demonstrate considerable

nonreciprocity reaching several percent at a wavenum-

ber of 22·104rad cm−1. Using a theoretical model which

takes into account dipolar, exchange, and anisotropy con-

tributions to spin wave dispersion, we show that, along

with the diﬀerence in layers’ magnetic parameters, the

leading contribution to the nonreciprocity comes from

the diﬀerence in the spin wave frequencies of the layers

due to their diﬀerent thicknesses. The thickness eﬀect

is pronounced in ﬁlms with a thickness of the order of

10 nm and can be additionally enhanced due to interfa-

cial spin pinning. We further suggest the conditions for

maximizing the nonreciprocity, which gives an optimal

geometrical asymmetry deﬁned by the relation between

the layer thicknesses and the pinning conditions at the

interfaces. Such artiﬁcially induced nonreciprocity gives

the opportunity to design magnonic logic gates based on

their geometrical parameters rather than on the proper-

ties of the materials.

SAMPLE AND METHODS

To study the possibility of noreciprocal MSWs propa-

gation in a magnetic multilayered structure with asym-

metry in the geometrical characteristics of the magnetic

constituents, we have chosen a bilayer structure con-

sisting of two dipolarly coupled layers of the ferromag-

netic alloy galfenol separated by a copper layer, speciﬁ-

cally Fe81Ga19(7 nm)/Cu(5 nm)/Fe81Ga19(4 nm), grown

by sputter deposition on a (100)-GaAs substrate. In

the following, the 7-nm galfenol layer is referred to as

the top layer, and the 4-nm layer adjacent to the sub-

strate – as the bottom layer (inset in Fig. 1 a). The top

layer is protected by Al (3 nm) and SiO2(120 nm) cap-

ping layers. The preparation and characterization of the

structure are described in details elsewhere [33]. Further-

more, a single 20 nm ﬁlm of galfenol on a (100)-GaAs

substrate was deposited with the same capped layers as

in the case of bilayer structure [34]. The single ﬁlm has

a thickness comparable to the total thickness of the lay-

ered structure, and thus it is used as a reference to de-

termine the MSW dispersion in the case of a single ﬁlm.

We choose galfenol-based structures because they exhibit

low Gilbert damping [35], high values of magnetization

precession lifetime [36, 37] and long MSW propagation

length [38], which makes galfenol a prospective material

for magnonic applications. According to ferromagnetic

resonance measurements, the saturation magnetization

of the ferromagnetic layers is estimated as µ0MS= 1.7 T

at layer thicknesses of 7 and 20 nm [37, 38]. The satura-

tion magnetization of the 4 nm layer is slightly reduced

due to the interface with the GaAs substrate [39] and is

µ0MS= 1.6 T. All galfenol layers demonstrate eﬀective

cubic magnetocrystalline anisotropy with the parameter

KC= 2.7·104J m−3and an additional uniaxial growth-

induced anisotropy in the ﬁlm’s plane [40]. The parame-

ters of the uniaxial anisotropy are KU t =−0.3·104J m−3

for the top layer and KU b =−1.1·104J m−3for the bot-

tom layer in the bilayer structure. KU=−1·104J/m3

for the 20-nm single ﬁlm [38].

The dispersion of thermal MSWs was studied by

the Brillouin light scattering (BLS) technique in the

backscattering geometry [41–43] (inset in Fig. 1 a). A sin-

gle mode laser with a wavelength of 532 nm is focused on

the sample surface into a spot with a diameter of 50 µm

using an objective lens. Frequency resolution is achieved

using a six-pass Fabry-Perot interferometer. The resolu-

tion in the wave vectors is achieved by tilting the optical

axis of the detection beam from the normal to the ﬁlm’s

plane in the range of angles θ= 0−70 ◦. The correspond-

ing range of wavenumbers is ky= 0 −22 ·104rad cm−1.

All experiments are carried out at room temperature in

an external magnetic ﬁeld of µ0Hext = 100 mT applied

in the sample plane and perpendicularly to the regis-

tered wavevector. The orientation of Hext is chosen ei-

ther along the easy (EA) or hard (HA) magnetization

axes in the ﬁlm’s plane (see details in Sec. II Suppl. Ma-

terials [44]).

RESULTS AND DISCUSSION

Figure 1 a presents a typical BLS spectrum

of the bilayer structure obtained at θ= 35 ◦,

ky= 13.5·104rad cm−1. Two peaks are visi-

ble in the Stokes and anti-Stokes parts of the spectrum

with central frequencies fof 11 and 18 GHz. The

experimental dependencies of fon the wavenumber ky

are obtained by approximating the peaks in the BLS

spectra using Lorentz functions (shown by symbols in

Fig. 1 b and c). fof the high frequency peak increases

with wavenumber, while the lower one demonstrates

only a weak dependence of its position on ky. This

behavior is observed for both directions of Hext, parallel

to HA and EA. To reveal the origin of the two peaks in

the BLS spectra, we compare the dispersions obtained

with those in the 20-nm single galfenol ﬁlm shown in

Fig. 1 d. For the single ﬁlm, the BLS spectra also contain

two peaks. The frequency of one peak increases with ky,

and thus can be readily assigned to the magnetostatic

surface spin wave (MSSW). The frequency of the second

peak is weakly dependent on the wavenumber, and

corresponds to the spin wave resonance (SWR). In

the studied range of ky, the frequencies of the SWR

are higher than those of the MSSW, according to the

typical properties of these waves in single magnetic

layers [45]. In striking contrast, in the bilayer structure,

the high frequency peak shows the stronger dependence

on ky. Furthermore, at ky= 0, the frequencies of both

dispersion branches coincide, while they diﬀer in the

single layer. Therefore, the two dispersion branches in

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NonreciprocalcollectivemagnetostaticwavemodesingeometricallyasymmetricbilayerstructurewithnonmagneticspacerP.I.Gerevenkov,1,V.D.Bessonov,2V.S.Teplov,2A.V.Telegin,2A.M.Kalashnikova,1andN.E.Khokhlov11IoeInstitute,194021St.Petersburg,Russia2M.N.MikheevInstituteofMetalPhysics,UBofRAS,620108Ekaterinbur...

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Nonreciprocal collective magnetostatic wave modes in geometrically asymmetric bilayer structure with nonmagnetic spacer P. I. Gerevenkov1V. D. Bessonov2V. S. Teplov2A. V. Telegin2A. M. Kalashnikova1and N. E. Khokhlov1.pdf

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Nonreciprocal collective magnetostatic wave modes in geometrically asymmetric bilayer structure with nonmagnetic spacer P. I. Gerevenkov1V. D. Bessonov2V. S. Teplov2A. V. Telegin2A. M. Kalashnikova1and N. E. Khokhlov1

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