PHYSICAL REVIEW B 87, 064424 (2013)

Resonant frequencies of a binary magnetic nanowire

K. L. Livesey,1 J. Ding,2 N. R. Anderson,1 R. E. Camley,1 A. O. Adeyeye,2 M. P. Kostylev,3 and S. Samarin3

1Department of Physics and Energy Science, University of Colorado at Colorado Springs, 1420 Austin Bluffs Parkway,Colorado Springs, Colorado 80918, USA

2Information Storage Materials Laboratory, Department of Electrical and Computer Engineering, National University of Singapore,117576, Singapore

3School of Physics M013, University of Western Australia, 35 Stirling Hwy, Crawley WA 6009, Australia(Received 11 December 2012; revised manuscript received 4 February 2013; published 28 February 2013)

Binary nanowires (Co50Fe50/Ni80Fe20) that are 260-nm wide, 25-nm thick, and 4-mm long are fabricatedusing “self-aligned shadow deposition” lithography. Each wire is separated from the next by a gap of 115 nm.Ferromagnetic resonance experiments performed on the nanowires indicate resonant frequencies associated withthe Co50Fe50 and the Ni80Fe20 parts of the nanowires. These frequencies are shifted from those measured innanowires made from just one material or the other due to exchange and dipolar coupling between the twomaterials. A semianalytic theory is outlined that correctly predicts the resonant frequencies in the binary stripesat various values of static applied field. This calculation is checked against a micromagnetic simulation, and theagreement between both methods and the experiment is good.

DOI: 10.1103/PhysRevB.87.064424 PACS number(s): 75.30.Ds, 75.75.Cd, 75.78.Cd

I. INTRODUCTION

During the last 15 years or so, there has been an increased in-terest in microwave dynamics of nanoscale magnetic elementsmade by nanopatterning magnetic films and multilayers.1 Thisinterest has been largely driven by the recent progress in thefield of magnetic storage and magnonics.2 Once the dynamicsof single elements had been understood,3–9 the interest shiftedto the collective dynamics of sets of nanoelements separatedby gaps.10 The collective dynamics originates from long-ranging dipole (dynamic) stray fields of individual elements,which can reach nearest neighbors and renormalize resonancefrequencies of individual elements.10 It has been demonstratedthat on periodical arrays of nanoelements collective excitationstake the form of propagating Bloch waves.11 As in any periodicmedium, the spectrum of these excitations is characterized bya set of transmission bands (here called “magnonic bands”)and prohibited frequency zones (here called “magnonicgaps”).12 Later, it was demonstrated both theoretically13 andexperimentally14,15 that the frequency width of the magnonicbands can be significantly increased by filling in the gapsbetween the nanoelements with some other magnetic material.This type of magnonic crystals was termed “bicomponentcrystals.”

The building block of such a bicomponent periodic structureis a unit cell comprising one element made of each materialin exchange contact to each other. In particular, the buildingblock of the one-dimensional (1D) magnonic crystal14 is abicomponent magnetic stripe, also known as a nanowire (NW).The magnetic and microwave properties of individual bicom-ponent stripes have not been investigated yet, although it maybe important for a number of spintronic applications since thestripes are in exchange contact to each other and are conduct-ing. Thus injection of spin-polarized electron currents throughthe interface is possible. This geometry may have also somerelevance for microwave assisted magnetization switching.16

In other words, although bicomponent magnonic crystals thatform a continuous film [see Fig. 1(a)] have been extensively

studied both experimentally and theoretically,11–15,17 thisstudy looks at individual bicomponent or binary stripesthat are separated by an air gap from one another [seeFig. 1(b)].

Importantly, the fabrication of bicomponent planar arrays,where two materials are in the same plane and are in anexchange contact, is a big technological challenge. Thereare various limitations with the quality of 1D magnoniccrystals produced with the multilevel electron beam litho-graphic approach described in Ref. 14, including the issueof alignment of the two contrasting nanowires. Recently, anadvanced fabrication technique based on “self-aligned shadowdeposition” has been developed, which specifically addressesthis problem18 and this technique has been used to make thematerials studied in this work.

In this work, we study the ferromagnetic resonance (FMR)response of bicomponent stripes, namely Ni80Fe20/Co50Fe50,experimentally and theoretically. The dynamic response ofsuch structures is important to understand if they are to beused for any microwave applications, such as microwave-assisted switching or microwave signal filtering. We performthis study for two possible ground states for the material:“parallel” and “antiparallel.” In the parallel ground state, themagnetization vectors in the two materials are co-alignedand are along the stripe. In the antiparallel ground state, themagnetization in one material is antiparallel to that in the othermaterial.

The paper is organized as follows. In the next section,we describe details of the sample fabrication and of thebroadband vector-network analyzer measurements of the FMRresponse of the sample. Section III contains details of thequasianalytical theory we propose to describe the dynamicresponse of the material. This theory provides an extensionto earlier theoretical works that describe nanowires made ofonly a single material. Section IV contains a discussion of theexperimental and theoretical results as well as comparison ofour original theory with the results of direct micromagneticsimulations.

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K. L. LIVESEY et al. PHYSICAL REVIEW B 87, 064424 (2013)

FIG. 1. (Color online) Schematic of the geometry for (a) the usualcontinuous 1D binary magnonic crystal comprising many binarywires all exchange coupled together and (b) an array of binarynanowires separated by gaps, as is investigated in this work.

II. FABRICATION AND EXPERIMENT

Large area arrays of binary (Co50Fe50/Ni80Fe20) nanowires(NWs) were fabricated using a “self-aligned shadow depo-sition” technique as described in Ref. 18. Periodic arraysof NW of length 4 mm, width w = 260 nm, and edge-to-edge separation g = 115 nm were defined in a 240-nmthick deep ultraviolet lithography resist on top of a60-nm bottom antireflection coating on silicon substrates.Details of the resist template fabrication process are de-scribed elsewhere.18 To create Co50Fe50/Ni80Fe20 binary NWs,25-nm Co50Fe50 and Ni80Fe20 films were deposited in anelectron beam deposition system, without breaking the vac-uum, with the sample holder tilted at an angle of ∼+23◦ and−23◦, respectively. A shadow area is formed on the left-handside of the pattern when the Ni80Fe20 material is depositedfrom the left of the sample due to the thick resist sidewalls,while the shadow area appears on the right-hand side of thepatterns when Co50Fe50 material is deposited from the rightof the sample. A lift-off process followed the deposition ofthe binary materials. Thus the width of each of the individualmagnetic materials is 130 nm and the total width of the stripeis 260 nm. Shown in Fig. 2(a) is the SEM image of thebinary NW array. For control experiments, nanowire arraysconsisting of only Ni80Fe20 NWs and only Co50Fe50 NWswere also fabricated from the same resist template using asimilar deposition technique, as shown in Figs. 2(b) and 2(c),respectively.

The collective magnetic hysteresis loops were characterizedusing a longitudinal magneto-optical Kerr effect (MOKE)setup with a laser spot size of about 50 μm. A clear two-stepswitching process corresponding to the switching of theNi80Fe20 elements at low field (Hsw1 = ±350 Oe) and theCo50Fe50 elements at high applied field (Hsw2 = ±800 Oe)can be observed in the MOKE loop of the binary NW arrayshown in Fig. 2(d). The corresponding M-H loops for theCo50Fe50 and Ni80Fe20 NW arrays are shown in Figs. 2(e)and 2(f), respectively. As expected, both loops display single-step switching with the coercive field of the Co50Fe50 NWarray distinctly higher than that of the Ni80Fe20 NW array dueto the higher saturation magnetization of Co50Fe50.

For the ferromagnetic resonance spectroscopy (FMR)measurements, ground-signal-ground (G-S-G) type co-planarwaveguides (CPWs) having 20 μm-wide signal lines werefabricated on top of the nanostructures to excite and detect theFMR signal using standard photolithography, deposition, andlift off of Al2O3 (50 nm)/Ti (5 nm)/Au (150 nm). The FMRresponses were measured using a microwave vector networkanalyzer (VNA) with the two ports connected to the CPW

FIG. 2. (Color online) Scanning electron microscope (SEM)images of the (a) binary, (d) Co50Fe50, and (c) Ni80Fe20 NWs. Theinsets show color-coded schematic diagrams of the three types ofNWs. To the right are the corresponding magnetic hysteresis loops in(d)–(f).

using G-S-G-type microwave co-planar probes. The staticfield Happ is applied along the easy (longitudinal) axis of thewires, while the microwave magnetic field produced by thesignal line of CPWs is in-plane and perpendicular to the Happ.FMR measurements were performed at room temperature bysweeping the frequency for fixed Happ. This measurementwas repeated for a number of different Happ values startingfrom the negative saturation field −Hsat = −1400 Oe, passingthrough zero, and then gradually increasing to the positivesaturation field Hsat (forward half of a loop). The field is thensubsequently decreased to −Hsat (backward half of a loop).

Shown in Fig. 3 are the FMR absorption spectra for thebinary nanowire array [see Fig. 3(a)] as well as for the referencesingle-element Co50Fe50 [see Fig. 3(b)] and Ni80Fe20 [seeFig. 3(c)] wire arrays. For clarity, only the ascending half-loop(i.e., from the negative to the positive saturation) is shown.In these figures, the vertical and the horizontal axes are thefrequency and applied field, respectively. The bright regions inthe field of the graph indicate low microwave absorption, whilethe dark regions represent strong absorption. The dots indicatepositions of the maxima of the resonance peaks. Two modesare seen for the binary nanowire structure when the sample isin the saturated state (for −1500 Oe < Happ < 250 Oe) [seeFig. 3(a)]. The frequency of both modes decreases when theamplitude of Happ is decreased. Multiple absorption peaksare visible for Happ in the range from 250 to 1000 Oe. Thismay be attributed to the antiparallel magnetization alignmentin the two materials composing the bicomponent wire assuggested by the hysteresis loop shown in Fig. 2(d). Thetwo-mode response reappears for the fields above 1000 Oe.This suggests completion of magnetization reversal for the

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RESONANT FREQUENCIES OF A BINARY MAGNETIC . . . PHYSICAL REVIEW B 87, 064424 (2013)

FIG. 3. (Color online) The FMR absorption as a function of staticapplied field and frequency. Larger FMR absorption is indicated bythe darker shading. The points indicate where the FMR absorptionpeak has its maximum for a given value of the applied field. The resultsfor (a) binary NWs, (b) isolated Co50Fe50 NWs of width 130 nm, and(c) Ni80Fe20 NWs of width 130 nm are shown.

binary array. For the single-component Co50Fe50 and Ni80Fe20

NW arrays, only one mode is observed for the saturated state,while two modes are visible for the transition field range asshown in Figs. 3(b) and 3(c). As reported in Ref. 19, thelower-frequency mode is the response of unswitched wires,while the higher-frequency mode is the response of switchednanoelements. In the whole-field range shown in the graphs,the resonance frequency of the Co50Fe50 NWs is much higherthan that of the Ni80Fe20 NWs due to the significantly highervalue of the saturation magnetization for the Co50Fe50. We willdiscuss the difference between the results for the binary stripesand those for the single-material stripes in detail in Sec. IV.

FIG. 4. (Color online) The geometry of a single stripe that isrepeated to make up the 1D magnonic crystal. Each NW is separatedfrom its neighbors by a gap of 115 nm.

III. THEORY

The geometry of a single Co50Fe50/Ni80Fe20 stripe withtotal width 260 nm and height 25 nm is illustrated in Fig. 4.This NW is repeated (each stripe is separated from the next by agap of 115 nm) along the y direction to form the 1D magnoniccrystal. The NWs are each 4-mm long, which is so long asto assume they are infinite in the theory. The magnetizationof both materials ( �M1 and �M2, respectively) lies in plane andpoints along the stripe in the z direction in Fig. 4, whichcorresponds to the parallel ground state.

We calculate the frequencies of an isolated stripe like thatillustrated in Fig. 4 ignoring the dipolar coupling betweenstripes. It is known that this dipolar coupling can lead tocollective magnetostatic modes with Bloch wave vector inthe y direction, perpendicular to the stripes, with wavelengthsof several stripes.11 However, in this instance, the distancebetween stripes is 115 nm and therefore the interstripe dipolarcoupling is weak compared to that within the stripe. Thisapproximation is supported by the good fit between the theoryand experiment, which is discussed in Sec. IV.

The magnetization is assumed uniform through the stripethickness (x direction). The stripe is only 25-nm thick (seeFig. 4) and this is sufficiently thin that the stripe can effectivelybe considered as a 2D structure. (For example, see Ref. 22where it is shown that for Ni80Fe20 films thinner than 30 nm,the first dipole-exchange mode in the thickness direction is farhigher in frequency than the uniform mode such that it canbe ignored.) Also, the magnetization is assumed uniform inthe stripe (z) direction and we look for standing modes withvariation perpendicular to the stripe’s edges in the y direction.

The semianalytical theory used to match calculated fre-quencies with those seen in Fig. 3 is based on that in Refs. 20and 21. However, in those works, only a single magneticmaterial is considered and here an extension must be madeto two materials. This leads to some interesting complicationsdue to the material interface. The theory proceeds as follows.(1) We find the magnetization equations of motion in bothmaterials and solve for the exchange modes using boundaryconditions at the edges (y = ±w = ±130 nm) and at theinterface (y = 0). In this first step, the dipolar energy isignored. (2) We write the dipolar field of the stripe interms of a magnetostatic Green’s function20 and expand themagnetization in terms of the exchange modes. (3) We solvefor the dipole-exchange modes and their frequencies using aperturbative expansion in terms of the exchange modes.

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K. L. LIVESEY et al. PHYSICAL REVIEW B 87, 064424 (2013)

The effective magnetic field felt at position y across thestripe’s width is given by

�Heff(y) = −Happz + 2A(y)

M(y)2

∂2 �M(y)

∂y2

+ �hdip(y) + 2K(y)

M(y)Mz(y)z, (1)

where the first term is due to the applied magnetic field with z

representing a unit vector along the stripe direction, the secondterm is due to the magnetic exchange interaction, and the thirdterm is the magnetic dipolar field of the stripe at position y.The fourth, and final, term in Eq. (1) is an effective field ofmagnetocrystalline uniaxial anisotropy. It is known that poly-crystalline cobalt films develop uniaxial growth anisotropywith the anisotropy axis oriented perpendicular to the filmplane (“normal uniaxial anisotropy”) and the strength of thisanisotropy may vary a lot from sample to sample.23 Thereforethe Co50Fe50 is assumed to have some anisotropy, which wefind by fitting the theory to experiment, while the Ni80Fe20

is assumed to have K ∼ 0. The exchange constant A(y),anisotropy constant K(y), and the saturation magnetizationM(y) are all functions of y that are discontinuous at theinterface and are given by

A(y) ={

A1, −w < y < 0,

A2, 0 < y < w,(2)

K(y) ={

K1 ∼ 0, −w < y < 0,

K2, 0 < y < w,(3)

M(y) ={

M1, −w < y < 0,

M2, 0 < y < w,(4)

where w = 130 nm is half the width of the stripe (see Fig. 4).The Landau-Lifschitz equation of motion for the magneti-

zation is

iω

γ�m(y) = [M(y)z + �m(y)] × �Heff, (5)

where �m = (mx,my,0) represents the dynamic magnetizationand oscillatory solutions with angular frequency ω have beenassumed. γ is the gyromagnetic ratio and is assumed constantin both materials for simplicity.

The effective magnetic field in Eq. (1) is substituted intothe equation of motion (5) and then the equation of motionmust be linearized about the ground state. As mentioned inIntroduction and Sec. II, there are two ground states that mustbe considered, and these give very different results for thefrequencies. We therefore split the rest of the calculation intotwo parts. In Sec. III A, the calculation for parallel alignmentof �M1 and �M2 will be discussed. In Sec. III B, the calculationwill be described when there is antiparallel alignment in thebinary stripe. This occurs when the static applied field Happ islarge enough to reverse the Ni80Fe20 magnetization but is notyet large enough to reverse the Co50Fe50 part of the NW.

A. Parallel ground state

This ground state is illustrated in Fig. 4. Substituting theeffective magnetic field in Eq. (1) into the equation of motionand linearizing about the aligned ground state, Eq. (5) becomes

in matrix form

iω

γ

(0 −11 0

) �m(y)

M(y)=

[2A(y)

M(y)

∂2

∂y2+ Happ − 2K(y)

M(y)

]

×(

1 00 1

) �m(y)

M(y)+

(hx

dip(y)h

y

dip(y)

)

= N (y)�m(y)

M(y)+

(hx

dip(y)h

y

dip(y)

), (6)

where the dimensionless angle of the dynamic magnetizationis given by

�m(y)

M(y)= 1

M(y)

(mx(y)my(y)

)≡ �a(y). (7)

We will look to find the eigenfunctions of the linear operatorN (y), which correspond to the exchange modes of the stripealong the y direction. Then the dimensionless angle �a(y) andthe dipolar fields in Eq. (6) will be expanded in terms of theexchange mode basis set in order to find the dipole-exchangemodes and their frequencies.

The operator N (y) acts in the same way on the x and y

components of the dynamic magnetization so both may beexpanded using the same basis set of modes. We thereforedrop for now the x and y indices. The eigenequation for theoperator on either side of the interface is

N (1)�(1)n =

(2A1

M1

∂2

∂y2+ Happ

)�(1)

n = N (1)n �(1)

n , (8)

N (2)�(2)n =

(2A2

M2

∂2

∂y2+ Happ − 2K2

M2

)�(2)

n

= N (2)n �(2)

n , (9)

where �(1)n is the nth eigenfunction in material 1 (Ni80Fe20) and

N (1)n is the corresponding eigenvalue. In the coupled system,

the eigenvalues in both materials must be equal, i.e., N (1)n =

N (2)n , and we assume that at the boundaries (y = ±w), the

magnetization is completely unpinned for this exchange-onlyoperator. Therefore the general solutions to Eqs. (8) and (9)are

�(1)n = C(1)

n cos(κ (1)

n (y + w)), − w < y < 0, (10)

�(2)n = C(2)

n cos(κ (2)

n (y − w)), 0 < y < w, (11)

with the wave vectors in each material, respectively, given byκ (1)

n ≡ M12A1

(N (1)n − Happ) and κ (2)

n ≡ M22A2

(N (2)n − Happ + 2K2

M2)

coupled according to

κ (1)n =

√A2

A1

M1

M2κ

(2)n

2 + K2

A1

M1

M2. (12)

The wave vectors may be real or imaginary. When there isno shape anisotropy (K2 = 0), then they are both real andexchange modes propagate through both materials, albeit withdifferent amplitudes. However, when anisotropy (a positiveconstant) is present, then the lowest order solutions to theeigenproblem (n = 1,2, . . .) have real κ (1)

n but imaginary κ (2)n .

This corresponds to a spin wave that is resonant in material1 but not in material 2 and so has an exponential decayin amplitude in material 2. Equation (11) becomes a cosh

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RESONANT FREQUENCIES OF A BINARY MAGNETIC . . . PHYSICAL REVIEW B 87, 064424 (2013)

function. For eigenmodes indexed by n above a critical numbern > n′, both the wave vectors are real.

It remains to use the boundary condition for the magneti-zation at the interface in order to find the relative strengths ofthe constants C(1)

n and C(2)n in Eqs. (10) and (11) and also to

find the discrete values that κ (1)n can take on. Then the set of

eigenfunctions will be known.For the exchange boundary conditions, Rado-Weertman’s

expressions24 are used,

∂�(1)n

∂y+ A12

A1

[�(1)

n − �(2)n

]∣∣∣∣y=0

= 0, (13)

∂�(2)n

∂y− A12

A2

[�(2)

n − �(1)n

]∣∣∣∣y=0

= 0, (14)

where A12 is the interface exchange energy per unit area withunits of erg cm−2.

There has been recent discussion about the effect that dipo-lar fields have in changing this boundary condition and pinningthe magnetization at interfaces, especially in nanometer-sizedelements.25 However, we are yet to include dipolar fields inour calculation so there is no issue with Eqs. (13) and (14).When we later add in dipolar interactions, no assumptionsneed to be made about the pinning at boundaries. This is onedistinct advantage of this theoretical approach as the dipolarpinning does not need to be fitted from experiment but fallsout naturally in the solution of the dipole-exchange modes. Ofcourse, the exchange boundary conditions have been assumed,but we will see that these are less important than the dipolarcontributions for the fundamental spin wave frequencies.

Substituting the general solutions (10) and (11) into theboundary conditions (13) and (14) results in the equations

−C(1)n κ (1)

n sin(κ (1)

n w) + A12

A1

[C(1)

n cos(κ (1)

n w)

−C2 cos(κ (2)

n w)] = 0, (15)

−C(2)n κ (2)

n sin(−κ (2)

n w) − A12

A2

[C(2)

n cos(−κ (2)

n w)

−C(1)n cos

(κ (1)

n w)] = 0. (16)

C(1)n can be eliminated in terms of C(2)

n using either Eq. (16) or(17), i.e.,

C(1)n = C(2)

n

A12 cos(κ (2)

n w)

−A1κ(1)n sin

(κ

(1)n w

) + A12 cos(κ

(1)n w

)≡ C(2)

n An. (17)

Requiring that there must be nonzero solutions to Eqs. (16)and (17) means that only discrete wave numbers are allowed,κ (2)

n (n = 1,2, . . .), which are solutions to the following equa-tion:

−κ (1)n κ (2)

n sin(κ (1)

n w)

sin(κ (2)

n w) + A12

A1κ (2)

n cos(κ (1)

n w)

× sin(κ (2)

n w) + A12

A2κ (1)

n sin(κ (1)

n w)

cos(κ (2)

n w) = 0,

(18)

remembering that κ (1)n is dependent on κ (2)

n according toEq. (12). Therefore the final eigenmodes of the binary stripe

are given up to a constant by

�n(y) = Cn

{An cos

(κ (1)

n (y + w)), −w < y < 0,

cos(κ (2)

n (y − w)), 0 < y < w.

(19)

The exchange eigenmodes must be correctly normalized inorder to form a suitable basis for our expansion of thedynamic magnetization when calculating the dipole-exchangefrequencies. The normalization (or choice of constants Cn) isdefined in terms of an inner product in function space andrequires that ∫ w

−w

dyW (y)�n(y)�n′ (y) = δnn′ , (20)

where δnn′ is the Kronecker δ function and W (y) is a suitablychosen weighting function. In previous calculations that haveused this method to find dipole-exchange modes,20,21 theweighting function is 1 since the exchange modes lie injust one material. With two materials, this is no longerthe case. It turns out that the weighting function is themagnetization, W (y) = M(y), and so is discontinuous at theinterface y = 0. The problem is analogous to that of findingoscillations on a string with inhomogeneous mass26 and therethe weighting function is the mass density. If we were findingthe exchange modes numerically using a discrete number ofsites in the y direction, then this inhomogeneity in the magneticproperties would cause the operator N to be non-Hermitian andboth left and right eigenvectors would exist.27 Multiplyingthe corresponding left and right discrete eigenvectors andnormalizing recovers the same weighting function as is usedhere for continuous eigenvectors.

Now that the exchange modes (19) are known, the dynamicmagnetization can be expanded:

mx(y) =∞∑

n=0

M(y)Fxn �n(y), (21)

my(y) =∞∑

n=0

M(y)Fyn �n(y), (22)

where Fxn and F

yn are the coefficients that will be found in

order to define the dipole-exchange modes.Equations (21) and (22) are substituted into the equation of

motion (6) and the dipolar fields are defined in terms of theGreen’s function for a magnetic stripe:8

hxdip(y) =

∫ w

−w

Gxx(y − y ′)mx(y ′)dy ′, (23)

hy

dip(y) =∫ w

−w

Gyy(y − y ′)my(y ′)dy ′ (24)

with the only nonzero components of the magnetostaticGreen’s function given by

Gxx(y − y ′) = 2

dln

[(y − y ′)2

(y − y ′)2 + d2

], (25)

Gyy(y − y ′) = −Gxx(y − y ′) − 4πδ(y − y ′), (26)

where d = 25 nm is the thickness of the stripe and CGS unitsare used.

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K. L. LIVESEY et al. PHYSICAL REVIEW B 87, 064424 (2013)

Both the left- and right-hand sides of the equation ofmotion (6) are multiplied by M(y)�n′(y) and then integratedbetween −w < y < w. Making use of the orthogonalitycondition of the exchange modes, Eq. (20), the result is thatthe equation of motion becomes an infinite set of equationsrelating the coefficients Fx

n and Fyn :

iω

γF x

n′ = (Nn′ + G

yy

n′n′)F

y

n′ +∑n�=n′

Fyn G

yy

nn′ , (27)

− iω

γF

y

n′ = (Nn′ + Gxx

n′n′)Fx

n′ +∑n�=n′

Fxn Gxx

nn′ , (28)

where the constants Nn are the eigenvalues correspondingto each exchange mode �n(y) and the constants Gii

nn′ areintegrations over the magnetostatic Green’s functions [seeEqs. (25) and (26)] given by

Giinn′ =

∫ w

−w

dy

∫ w

−w

dy ′M(y)M(y ′)�n′(y)�n(y ′)Gii(y − y ′).

(29)

Writing Eqs. (27) and (28) in matrix form results in

0 =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

−iωγ

N1 + Gyy

11 0 Gyy

12 . . .

N1 + Gxxzz

iωγ

Gxx12 0 . . .

0 Gyy

21−iωγ

N2 + Gyy

22 . . .

Gxx21 0 N2 + Gxx

22iωγ

. . .

......

......

. . .

⎞⎟⎟⎟⎟⎟⎟⎟⎠

×

⎛⎜⎜⎜⎜⎜⎜⎝

Fx1

Fy

1

Fx2

Fy

2

...

⎞⎟⎟⎟⎟⎟⎟⎠

. (30)

Although this is an infinite-sized matrix, it can be truncated tosize j × j with j sufficiently large that the solutions do notchange from size j − 1 to j to within a required accuracy.Certain frequencies ω will cause the matrix equation to havenonzero solutions for Fx

n and Fyn , and these coefficients in turn

provide the dipole-exchange modes of the stripe. Notice againthat no assumptions have been made about the dipole pinningat interfaces in order to find these modes.

B. Antiparallel ground state

When the magnetization in the Ni80Fe20 part of the stripereverses to align with the applied magnetic field but themagnetization in the Co50Fe50 part remains aligned with the z

direction, the procedure outlined in Sec. III A is altered slightly.The main difference is that the exchange modes are differentfor this magnetic ground state. When both magnetizationswere aligned, the exchange modes corresponded mostly tosinusoidal oscillations in both materials [see Eq. (19)]. Here,however, all the exchange modes are sinusoidal in one materialand decay in the other material.

The equation of motion for magnetization in the Ni80Fe20

side of the stripe [see Eq. (6)] is slightly changed because the

linearization is about a different ground state. It becomes

iω

γ

(0 −11 0

) �m(y)

M1

=[−2A1

M1

∂2

∂y2+ Happ

] (1 0

0 1

) �m(y)

M1+

(hx

dip(y)

hy

dip(y)

)

= N (1) �m(y)

M1+

(hx

dip(y)

hy

dip(y)

). (31)

The equation of motion for the unreversed Co50Fe50 side ofthe stripe remains the same.

Again making the eigenvalues in both materials match(N (1)

n = N (2)n ) and guessing sinusoidal solutions [see Eqs. (10)

and (11)] leads to a different equation relating the wavevectors in each material as compared to Eq. (12) for parallelmagnetization, namely,

κ (1)n =

√−K2

A1

M1

M2− A2

A1

M1

M2κ

(2)n

2. (32)

Solution of this equation requires that either κ (1)n or κ (2)

n bereal and the other be imaginary. In other words, all possibleexchange modes correspond to an off-resonance driving in oneof the materials for this case of antiparallel magnetizations.Solutions that have κ (1)

n imaginary correspond to a decay inamplitude in material 1 and those that have κ (2)

n imaginarycorrespond to decay in material 2. From now on, we onlyconsider the magnitudes of the wave vectors when writing κ (1)

n

and κ (2)n and decaying modes are written as cosh(κy) rather

than cos(iκy).By repeating the procedure of finding exchange modes with

unpinned surfaces and interface exchange density A12 at y =0, two types of exchange modes are found. One correspondsto decay in material 1 (Ni80Fe20):

�1n(y) = Dn

{Bn cosh

(κ

(1)n,1(y + w)

), −w < y < 0,

cos(κ

(2)n,1(y − w)

), 0 < y < w,

(33)

with the coefficient

Bn =[

−A12 cos(κ

(2)n,1w

)A1κ

(1)n,1 sinh

(κ

(1)n,1w

) + A12 cosh(κ

(1)n,1w

)]

, (34)

and the other exchange mode corresponds to decay in material2 (Co50Fe50)

�2n(y) = En

{Cn cos

(κ

(1)n,2(y + w)

), −w < y < 0,

cosh(κ

(2)n,2(y − w)

), 0 < y < w,

(35)

with the coefficient

Cn =[

−A12 cosh(κ

(2)n,2w

)−A1κ

(1)n,2 sin

(κ

(1)n,2w

) + A12 cos(κ

(1)n,2w

)]

. (36)

The allowed wave numbers are again found by consideringthe boundary conditions at the material interface y = 0 [seeEqs. (13) and (14)]. For modes that decay in material 1, the

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RESONANT FREQUENCIES OF A BINARY MAGNETIC . . . PHYSICAL REVIEW B 87, 064424 (2013)

wave numbers are solutions to the equation

−κ(1)n,1κ

(2)n,1 sinh

(κ

(1)n,1w

)sin

(κ

(2)n,1w

)+ A12

A1κ

(2)n,1 cosh

(κ

(1)n,1w

)sin

(κ

(2)n,1w

)− A12

A2κ

(1)n,1 sinh

(κ

(1)n,1w

)cos

(κ

(2)n,1w

) = 0, (37)

together with Eq. (32). For modes that decay in material 2, thewave numbers are solutions to the equation

−κ(1)n,2κ

(2)n,2 sin

(κ

(1)n,2w

)sinh

(κ

(2)n,2w

)− A12

A1κ

(2)n,2 cos

(κ

(1)n,2w

)sinh

(κ

(2)n,2w

)+ A12

A2κ

(1)n,2 sin

(κ

(1)n,2w

)cosh

(κ

(2)n,2w

) = 0, (38)

together with Eq. (32).The constants Dn and En in Eqs. (33) and (35) are

determined by the orthogonality condition which for this con-figuration is given with respect to slightly different weightingfunctions:

−∫ 0

−w

dyM1�in(y)�j

n′ (y) +∫ w

0dyM2�

in(y)�j

n′(y)

= δnnij , (39)

where δnn′ is only equal to 1 if n = n′ and is zero otherwise,and the function ij is given by

ij =⎧⎨⎩

1, i = j = 1,

−1, i = j = 2,

0, i �= j.

(40)

Since there are two types of exchange modes, the expansionof the dynamic magnetization involves two sums:

mx(y) =∞∑

n=0

M(y)[Bx

n,1�1n(y) + Bx

n,2�2n(y)

], (41)

my(y) =∞∑

n=0

M(y)[B

y

n,1�1n(y) + B

y

n,2�2n(y)

]. (42)

Again, the equation of motion (6) can be rewritten in termsof the coefficients Bx

n,1, Bxn,2, By

n,1, and By

n,2. There is a differentequation of motion for the coefficients of exchange modes thatdecay in material 1 compared to those for material 2. Theequations for each of the four types of coefficients are

iω

γBx

n′,1 = (Nn′,1 + G

yy

n′n′,11

)B

y

n′ +∑

(n,m)�=(n′,1)

Byn,mG

yy

nn′,m1,

(43)

− iω

γB

y

n′,1 = (Nn′,1 + Gxx

n′n′,11

)Bx

n′ +∑

(n,m)�=(n′,1)

Bxn,mGxx

nn′,m1,

(44)

− iω

γBx

n′,2 = (−Nn′,2 + Gyy

n′n′,22

)B

y

n′ +∑

(n,m)�=(n′,2)

Byn,mG

yy

nn′,m2,

(45)

iω

γB

y

n′,2 = (−Nn′,2 + Gxxn′n′,22

)Bx

n′ +∑

(n,m)�=(n′,1)

Bxn,mGxx

nn′,m2,

(46)

where the constants are given by

Nn,1 = H − 2A1

M1

[κ (1)

n

]2, (47)

Nn,2 = H + 2A2

M2

[κ (2)

n

]2 + 2K2

M2, (48)

Giinn′,mm′ =

∫ w

−w

dy

∫ w

−w

dy ′M(y)M(y ′)�m′n′ (y)�m

n (y ′)

×Gii(y − y ′). (49)

The dummy variables m and m′ can correspond to 1 or 2(decay modes in material 1 or 2) and i can be either x or y.The magnetostatic Green’s functions Gxx and Gyy were givenin Eqs. (25) and (26).

As for the case of parallel magnetizations, the infinite setof equations formed by Eqs. (43)–(46) can be truncated andlinear algebra methods are employed to find the eigenvalues(frequencies) and eigenvectors (dipole-exchange modes) of thebinary stripe.

IV. RESULTS AND DISCUSSION

Frequencies calculated using the method described inSec. III show a good fit to the experimental results if the fol-lowing parameters are used. For Ni80Fe20, 4πM1 = 8357 Oeand A1 = 10−6 erg/cm. For Co50Fe50, 4πM2 = 21360 Oe,K2 = 0.8 × 106 erg/cm3, and A2 = 0.5 × 10−6 erg/cm. Forboth materials, a gyromagnetic ratio γ = 2.87 GHz/kOe isassumed. All of these parameters are within the range oftypically measured values for the two alloys. This value of γ

corresponds to a g factor of 2.05. This is not an unreasonablevalue as it is slightly lower than the value of 2.08 reportedfor 25-nm thick NiFe films in Ref. 28. For stripes rather thanfilms, the increased number of interfaces is accompanied bya reduction in g. Finally, for the interface exchange a valueof A12 = 1 erg/cm2 is used, which corresponds to weakcoupling as we will show below. These parameters producethe solid lines shown below in Fig. 5, while the dots showthe experimentally measured frequencies, the same maximumabsorption frequencies shown in Fig. 3(a).

The fit is good in Fig. 5, although the theoretical valuesare below the experimental values especially at large valuesof the applied field for the lower frequency. The slope of thefrequency-versus-field line is obviously slightly lower for thetheoretical values. The slope can be increased by increasingeither the gyromagnetic ratio or the magnetization in theNi80Fe20. However, increasing either of these values not onlyincreases the slope but also increases the y-axis intercept andtherefore causes a worse fit at low values of the applied field.The only other failing of the theory is the inability to predictthe mode with frequency near 12.5 GHz between Happ = 400and 600 Oe. In this region, the Ni80Fe20 stripe has reversedto align with the applied field and there is an “antiparallel”ground state. The lowest frequency mode is predicted well butthe second mode is not. Perhaps this mode has an importantcontribution due to the dipolar coupling between neighboring

064424-7

K. L. LIVESEY et al. PHYSICAL REVIEW B 87, 064424 (2013)

FIG. 5. (Color online) The theoretically predicted resonant fre-quencies (solid lines) of the binary magnonic crystal as a functionof static magnetic field applied along the stripe. Experimentallymeasured frequencies are shown by dots and are the same as thoseindicated in Fig. 3(a). The parameters used in the theory are given inthe main text.

NWs, which the current theory ignores. From the absorptionmap shown in Fig. 3(a), it can be seen that this mode has lowerabsorption than the other mode in this field range.

The most encouraging thing about these fitting parametersis that they also provide a good fit to the experimentalresults when isolated 130-nm wide Ni80Fe20 and Co50Fe50

stripes, separated by air gaps, are respectively measured. Theexperimental (dots) and theoretical (solid line) results areshown together in Fig. 6. The experimental values are the sameas the maximum microwave absorption shown in Figs. 3(b)and 3(c). The frequencies look rather similar to the resultsfor the binary stripe in Fig. 5. The major difference is theappearance of an addition mode when the binary NW is in theantiferromagnetic ground state, as discussed in Sec. II, whichis not possible for the isolated single stripes.

FIG. 6. (Color online) The theoretically predicted resonant fre-quencies (solid lines) of the isolated stripes as a function of staticmagnetic field applied along the stripe. Experimentally measuredfrequencies are shown by dots and correspond to the maximumabsorption points in Figs. 3(b) and 3(c). The results for Ni80Fe20 andCo50Fe50 are shown together although the measurements are doneseparately. The parameters used in the theory are the same as thoseused for the binary NWs in Fig. 5.

TABLE I. Measured frequencies corresponding to microwaveabsorption maxima in the FMR experiment with Happ = 0 for threedifferent NWs.

NW 130 nm 130 nm 130 nm/130 nmcomposition Ni80Fe20 Co50Fe50 Ni80Fe20/Co50Fe50

Frequency associated 9.10 . . . 8.50with Ni80Fe20 (GHz)Frequency associated . . . 26.32 26.38with Co50Fe50 (GHz)

Although the plots look similar, there are small shiftsin the frequencies of the stripes when comparing those inisolation to those coupled to another material in a binarystripe. These small frequency shifts are too small to be seenwhen comparing Figs. 5 and 6 on the scale drawn, but aresummed up in Table I. For zero applied field, the resonantfrequency in a 130-nm-wide Ni80Fe20 NW is measured tobe 9.10 GHz in a single stripe and 8.50 GHz in a binarystripe. There is a decrease in the frequency of 0.60 GHz whenthe stripe is coupled to an adjoining Co50Fe50 stripe. For thesingle 130-nm-wide Co50Fe50 stripe, the resonant frequencyis measured to be 26.32 GHz, while when it is coupled to theNi80Fe20 NW it is 26.38 GHz (very little change).

To understand the frequency shifts between isolated andcoupled stripes, we first plot the theoretically predicted dipole-exchange modes with Happ = 0 for the binary NW. The twocomponents of the dynamic magnetization are shown in Fig. 7for the seven modes with the lowest frequencies. The appliedfield is taken to be zero. The panels on the left [Figs. 7(a)–7(g)]show the out-of-plane component of the magnetization mx as afunction of distance through the stripe width. The correspond-ing panels on the right [Figs. 7(h)–7(n)] show the in-planecomponent my . The Ni80Fe20 occupies −130 nm < y < 0 andthe Co50Fe50 occupies 0 < y < 130 nm. The frequency ofeach mode is written on each panel.

The frequencies that are measured in the FMR experimentand which are shown in Figs. 5 and 6 correspond to thefundamental frequencies in each material: the first and theseventh (bottom) modes in Fig. 7. Both of these modes havethe longest wavelengths and so dipolar energy dominatesover exchange giving them a different shape compared to theother modes. In particular, these two modes do not have freeboundaries whereas the exchange-dominated modes do due tothe exchange boundary conditions we chose in the calculation.We can estimate the exchange contribution to the frequencyof the lowest mode in the NiFe [see Figs. 7(a) and 7(h)]. Thewavelength is at least λ = 300 nm (giving an upper limit tothe frequency contribution). This corresponds to a wave vectork = 2π/λ, an exchange energy Eex = Ak2, and an effectivefield Hex = Ak2/M ∼ 66 Oe. Using the parameters for NiFeand the simple Kittel formula for the resonant frequency ofa finite object, this corresponds to a frequency shift of about0.24 GHz at Happ = 0. This number is only about 3% of thefrequency value, 8.54 GHz at Happ = 0.

The reason that the two fundamental modes are the onlytwo that can be measured using FMR is that these two modesabsorb by far the most power from the rf microwave field,with amplitude hrf , which is applied along the y direction.

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RESONANT FREQUENCIES OF A BINARY MAGNETIC . . . PHYSICAL REVIEW B 87, 064424 (2013)

FIG. 7. (Color online) The out-of-plane (mx , left) and in-plane(my , right) dynamic magnetization as a function of distance alongthe stripe width for the lowest seven modes, as calculated using thesemianalytic theory described in Sec. III. The static applied field iszero in this diagram. NiFe occupies the left of the stipe (y < 0) andCoFe occupies the right (y > 0).

We can estimate the power density absorbed per unit areaperpendicular to the stripe’s width (y direction) using

P =∫ +w

−w

hrfdmy

dtdy

∝ ωhrf

∫ +w

−w

mydy. (50)

In other words, the power absorbed by each mode is pro-portional to the area under the curves in Figs. 7(h)–7(n). Byinspection, it is obvious that the first and seventh modes willabsorb the most power as they have a large net moment. Belowin Fig. 8, the power absorbed by each mode is plotted as afunction of mode frequencies. The two fundamental modes

0 5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

1.0

1.2

Frequency (GHz)

Pow

erab

sorb

ed(a

rb.u

nits

)

FIG. 8. (Color online) The power absorbed by each mode,calculated using Eq. (50), as a function of mode frequency for the 12lowest frequency modes in the binary stripe. The fundamental modesin each material have by far the largest power absorption.

absorb roughly five times the energy per unit time of the nextmost-absorbing mode.

Notice that the fundamental mode in Ni80Fe20 [seeFigs. 7(h)] has a large, positive amplitude of in-plane mag-netization in the Co50Fe50 (y > 0). This means that there isa strong dipolar field generated in the Co50Fe50 that affectsthe fundamental mode frequency of Ni80Fe20 and contributesto the 0.60-GHz drop in the frequency as compared with anisolated 130-nm-wide Ni80Fe20 stripe. The frequency drop isbecause the dynamic in-plane magnetization in both materialspoints in the same direction, which softens the mode associatedwith the 130-nm-wide Ni80Fe20 stripe as compared to whenit is in isolation. The interface exchange coupling, as we willdiscuss next, is weak and therefore the dipolar energy is thedominant contribution in changing the Ni80Fe20’s fundamentalfrequency when it is coupled to the Co50Fe50. In contrast, thefundamental mode in the Co50Fe50 [see Figs. 7(g) and 7(n)] hasa small oscillatory amplitude in the Ni80Fe20 (y < 0). The netcontribution of the dipolar field in Ni80Fe20 to the fundamentalfrequency of Co50Fe50 is therefore close to zero. This matchesthe experimental observation that the frequency is relativelyunchanged from that of an isolated 130-nm-wide Co50Fe50

stripe (see Table I).It is not perhaps so obvious from Fig. 7 that the interface

exchange coupling is weak. By plotting the dimensionlessangles ax = mx/M(y) and ay = my/M(y), it is more obvious.When the interface exchange coupling is strong and ferro-magnetic, these angles are continuous at the interface and themagnetization directions on either side are locked to the samedirection. For the value A12 = 1 erg/cm3, chosen becauseit causes the theory and experiment for both the isolatedstripes and binary stripes to match well, these angles are notcontinuous and, in fact, have large jumps. As an example, thein-plane angle of the lowest frequency mode is shown as afunction of distance through the stripe’s width in Fig. 9. Alarge discontinuity at the interface (y = 0) can be seen and istypical of the other modes. The interface exchange is so weakthat the frequencies do not vary markedly if the exchange isremoved completely (A12 → 0). However, a small interfaceexchange is required to give the best match to experiment.

The frequencies calculated using the semianalytical methodhave been checked using a micromagnetics program written

064424-9

K. L. LIVESEY et al. PHYSICAL REVIEW B 87, 064424 (2013)

FIG. 9. (Color online) The in-plane angle of the magnetizationprecession as a function of distance along the stripe width for thelowest frequency mode. The applied field is zero. The discontinuity atthe interface can be seen indicating weak interface exchange coupling.

by the authors. The micromagnetics calculation of eigenmodesand frequencies is similar to that detailed in Ref. 29. The binaryNW is split into long rectangular cells that are 25-nm-thick (thetotal thickness of the nanowire). Then the dynamics of eachcell is calculated as the magnetization relaxes from an initial“kicked” state via the Landau-Lifshitz equation with dampingα:

∂ �M∂t

= −|γ | �M × �Heff − α

Ms

|γ | �M × ( �M × �Heff). (51)

A Fourier transform in time reveals the resonant frequenciesof the system and a Fourier spatial transform enables thecorresponding eigenmodes of the system to be found.

All the frequencies calculated with the micromagneticsimulation and semianalytic theory agree within 0.1 GHz.Moreover, the mode profiles match very well. Therefore thesemianalytical method is shown to be robust. The questionthen is what advantage this method has over micromagnetics.Importantly, this method is less computationally demanding.

By far the most computationally demanding step in thissemianalytic method is the calculation of the dipolar matrixelements Gii

nn′ [see Eq. (29)], which involves integrating overthe magnetostatic Green’s functions and exchange modesindexed by n and n′. This step is done using built-in numericalintegration packages within MATHEMATICA. However, the max-imum matrix size used to calculate the frequencies and modesis 28 × 28. Increasing the matrix size further does not changethe calculated frequencies in the first three digits and doesnot change the mode profiles noticeably. The computationaltime taken to calculate the 28 × 28 dipolar matrix elementsis less than the time needed to calculate the dipolar fields atevery time step of the micromagnetics calculation. Also, themicromagnetics simulation must be analyzed using Fouriertransforms to find the eigenfrequencies and eigenmodes

whereas the semianalytic method produces them in the firstinstance. Moreover, a given initial magnetization configurationfor the stripe used in the micromagnetics calculation may nothave a projection onto all the different eigenstates and Fourieranalysis may not provide all of the frequencies and modes thatwe are looking for. Therefore the micromagnetics calculationoften needs to be repeated for a variety of different startingconfigurations or “initial kicks” to find the complete set ofmodes. The semianalytic method described here does not havethis issue and only needs to be run once.

V. CONCLUSION

In this paper, the resonant frequencies in a binary nanowire(NW) or stripe have been studied. Arrays of these binaryNWs were fabricated using the “self-aligned shadow deposi-tion” technique and were characterized using vector-network-analyzer FMR. A comparison between the experimentallymeasured frequencies and those calculated using a semi-analytical theory and via micromagnetics has been made,and all three sets of values agree quite well. In particular,by matching theory and experiment for single-material andfor binary stripes, the exchange coupling between the twomaterials comprising the NW is estimated to be weak, withA12 ∼ 1 erg/cm2.

The semianalytic theory is an extension of previous studiesthat calculated the standing dipole-exchange modes in singlematerials. The interface between two different materialsand the discontinuous material properties that occurs therecomplicates the method, in particular the orthogonal expansionin terms of exchange-only modes. The frequencies calculatedwith this method have been checked with micromagnetics,proving the method robust.

The FMR experiment only measures the fundamentalfrequencies associated with the two materials comprisingthe stripes. This is because this method is only sensitiveto modes that have a large net magnetic moment. However,other techniques such as Brillouin light scattering (BLS),3,10

micro-BLS,30 and microwave photovoltage techniques31 canmeasure magnetization oscillations that have small or zero netmoment. These methods may be employed in the future totest the match between our theory and experiment for morefrequencies.

ACKNOWLEDGMENTS

M.P.K and S.S. acknowledge support from the AustralianResearch Council and the Australian-Indian Strategic Re-search Fund. J.D. and A.O.A. acknowledge the Ministry ofEducation, Singapore. The work of R.E.C. and N.R.A. wassupported by US ARO Grant No. 666 W911NF-10-1-0255.

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