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PHYSICAL REVIEW B 69, 012405 ~2004!

Stop bands and defect modes in a magnonic chain of cells showing single-cell spectral gaps

Housni Al-WahshFaculty of Engineering, Zagazig University, Benha Branch, 11241 Cairo, Egypt and Equipe des cristaux photoniques et phonon

UMR CNRS 8024, UFR de Physique, Universite´ de Lille1, 59655 Villeneuve d’Ascq, France~Received 24 May 2003; revised manuscript received 7 August 2003; published 15 January 2004!

A magnonic chain of cells consisting of a serial loop structure with dangling resonators~SLS-DRs! isdesigned to obtain possibly large stop bands. Contrary to all known systems of this kind, a spectral gap ofnonzero width occurs here even with a single cell. General expressions for the dispersion relation and for thetransmission coefficient are given for all serial systems of this kind within the framework of the long-wavelength Heisenberg ferromagnetic model. The width of the stop bands is very sensitive to the number ofresonators and may be drastically increased in a tandem geometry made of several successive SLS-DRs whichdiffer by their geometrical characteristics. These SLS-DRs may have potential applications in spin injectioninto devices.

DOI: 10.1103/PhysRevB.69.012405 PACS number~s!: 75.30.Ds, 75.70.Cn, 75.90.1w

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Low-dimensional spin systems—i.e., magnetic configutions with a dimensionality less than three, have attracenthusiastic attention in the past few years.1–3 This was re-lated both to the fundamental interest and to the potenapplications of spin-transport devices, and is supportedthe advanced progress in nanofabrication technology.4 Arraysof very long ferromagnetic nanowires of, for example, Npermalloy and Co, with diameters in the range of 30 to 5nm have been created.8,9 These are very uniform in crossection, with lengths in the range of 20 microns. They thare realizations of nanowires which one can reasonably vas infinite in length, to an excellent approximation. In adtion to the static and spin-transport properties of magnnanowire arrays, the dynamical properties of a magnnanostructure are also of considerable interest in both funmental as well as applied research.9 The study of spin wavesis a powerful tool for probing the dynamic propertiesmagnetic media in general and those of laterally pattermagnetic structures in particular.5 On the other hand, due tthe possible use of electron spin for storage and informatransfer in quantum computers,6 there have been many recestudies on spin transport in semiconductor nanostructur7

These recent developments actuated us to investigatenetic excitations in networks composed of one-dimensio~1D! continuous magnetic media. Our choice of 1D magnestructures is motivated by possible engineered spin-injecdevices that render the control of the widths of the pbands~and hence the stop bands! feasible.

In previous publications, we proposed10,11 a model of a1D magnonic crystal exhibiting pass bands separatedlarge forbidden bands. The geometry of the model presein Ref. 10@calleda comblike structure~CLS!# is composedof an infinite 1D monomode waveguide~the backbone!along whichn8 dangling side branches~which play the roleof resonators! are grafted atn equidistant sites,n and n8being integers. The stop bands originate from the periodiof the system determined by the distance between two neboring sites and from the eigenmodes of the dangling sbranches. The presence of defect branches in the comb sture can give rise to localized states within the stop bandhas been shown that these states are very sensitive to

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length and number of resonators, to the periodicity ofsystem, and to the length of the defect branches. One dback of the comb structure lies in the difficulty of a technicrealization of the boundary condition at the free ends ofresonators. On the other hand, the structure presented in11 @calleda serial loop structure~SLS!# was made of sym-metric loops~rings! pasted together with segments of finilength, the loops playing the role of resonators. The abmentioned difficulty was avoided in studying SLSs. Intereingly, in studying the transmission rate through a defectgeometry, the defect branch in CLSs~Ref. 10! was inserted~fixed! at the middle of the comb, while in the SLSs thdefect can be located in any cell of the structure, whichturn change dramatically the transmission spectrum. Iworth mentioning also that when the defect wire is locatedthe middle of the SLSs and whatever is the ratiodf /d1 (df isthe length of the defect wire andd1 is the length of the finitesegments! the intensity of the transmitted gap modes rem1. However, if the position of the defected cell is not in tmiddle, the intensity of the transmitted gap modes is remaably depressed.

In the above mentioned networked waveguides the qutum size effect~or the subband structure! was neglected. Wedealt with a magnetic network where the cross sections owires are considered to be much smaller than the considwavelength, i.e., a continuum approximation theory was eployed in the calculation. Such an approximation is vaprovided that the relevant wavelengths are large compawith the lattice spacing, i.e., we dealt only with longwavelength excitations. Therefore, in using the Heisenbmodel of a ferromagnet we are neglecting the dipole-dipinteractions compared with the exchange contribution toHamiltonian.10,12This macroscopic approach is analogousthat used by Cottam13 in magnetostatic calculations. Let ualso mention also that the continuum theory has the advtages that explicit analytic expressions of different magneproperties~e.g., dispersion curves, transmission coefficien!can be calculated.

In this work, we introduce a geometry, calledserial loopstructure with dangling resonators~SLS-DR!, of a mono-mode networked waveguide. The structure is composed

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BRIEF REPORTS PHYSICAL REVIEW B69, 012405 ~2004!

symmetric loops~rings! pasted together with segmentsfinite length and connected with dangling resonators; see1~a!. Such a structure may exhibit interesting features,comparison with the CLS and SLS waveguides. Forample, the existence of larger gaps, the appearance of qquantized bands without inserting a defect,achieving a com-

FIG. 2. ~a! The spin wave band structure of the infinite serloop structure with out resonators (N850). We have chosend1

51, d25d1 , H51, and N→`. One observes an absolute stoband below the first band due to the presence of the externalH0. ~b! The same as in~a! but for a SLS-DR withN851, d3

50.5d1. Note the stop band observed in~b! @in comparison with~a!# due to the existence of the resonators connected to the loo

FIG. 1. ~a! Schematic of the one-dimensional serial loop struture with dangling resonators studied in the present work. Thedia are designated by an indexi, with i equal to 1 for the finitebranch, 2 for the loop, and 3 for the resonators~the side branches!.Each loop has a length 2d2 and is distant byd1 from neighboringloops. Each cell is composed of a finite branch and the loop cnected to its right extreme. The resonators are connected apoints of connection of the loop with the finite segments.~b! Waveguide with one loop, with dangling side resonators, connected aextremities to two semi-infinite leading lines.

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ig.n-si-plete stop band for a single loop. These features~whichcould be of potential interest in waveguide structures! areessentially due to the existence of the resonators connewith the loops~which is quite different from the case oCLSs or SLSs!. We report on results of calculated banstructures and transmission coefficients. We also showthe width of the stop bands may be enlarged by couplseveral loop structures with dangling resonaters~SLS-DRs!of different physical parameters.

The 1D infinite SLS-DR can be modeled as an infinnumber of unit cells pasted together@see Fig. 1~a!#. Each unitcell is composed of a finite wire~medium 1! of lengthd1 inthe direction of propagation, connected to a loop ‘ring’~me-dium 2! of length 2d2 ~each loop is constructed of two wirewith the same lengthd2), and the loop is connected~at thepoints of connection to the finite wire of medium 1! to afinite number of dangling side resonators~medium 3! oflength d3. The period of the SLS-DR structure isD5d11d2.

We focus in this letter onhomogeneous SLS-DRs whemedia 1, 2, and 3 are made of the same material. The dis-persion relation of the infinite SLS-DRs, that relates the psation of the spin wavev to the Bloch wave vectork, can bederived using the Green function method.14 It can be writtenas cos(kD)5h(v), where

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FIG. 3. Transmission rate versus reduced frequency for a wguide composed of one loop withN851. ~b! The same as in~a! but

for N852. For convenienceH is considered to be 1,d251 andd350.5d2.

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BRIEF REPORTS PHYSICAL REVIEW B69, 012405 ~2004!

H0 , J, andg stand, respectively, for the spontaneous mnetization, the static external field,10 the exchange interactiobetween neighboring magnetic sites in the simple cubictice of lattice parametera constituting the ferromagnetic medium, and the gyromagnetic ratio.Interestingly, if N850 ord350 we recover the results of Ref.11.

Figures 2~a! and 2~b! display the projected band structuof an infinite SLS-DR for given values ofd1 , d2 , d3 , H,and N8, such thatd151, d25d1 , d350.5d1 , H51, andN850 and 1 respectively. The plot is given as the reducfrequencyV5v/D85H1a2, with H5gH0 /D8 versus thedimensionless wave vectorkD (2p<kD<1p). There is acomplete absolute stop band~where the propagation of spiwaves is prohibited! below the lowest band due to the preence of the external fieldH0. There are also other stop banwith different widths ~besides the asymmetry in the pabands! in the spectrum shown in the figure. Introducing tresonator into the wave guide@Fig. 2~b!# the degeneratepoints between the second and third bands and as weltween the sixth and seventh bands which appear atkD50@in Fig. 2~a!# are removed and wide stop bands are observThe width of these stop bands is increased with an increanumber N8 of the resonators. The above observation is alsreflected in the plot of the transmission factor.

Now we turn to a study of the transmission rate throuthe one-dimensional SLS-DR wave guides . We begin wit

TABLE I. Width of the stop band of the individual cell for thsystems studied in Refs. 10 and 11~CLSs and SLSs! and in thepresent paper.

System

D V ~The secondlowest zero

of transmission!

CLS with one resonator~Ref. 10! .0.5SLS—one loop~Ref. 11! ZeroSLS-DR—one loop with one resonator .5SLS-DR—one loop with two resonators .12

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study of a simple example, namely, a structure consistinga single loop with a finite number of resonators@see Fig.2~b!#. The transmission factorT can be written as

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whereT35S3 /C3. Again if N850 or d350 we recover theresults of Ref. 11. The transmission factor isequal to zerowhenS250 or C350. The zero frequencies, correspondito the eigenmodes of a single loop, are given by

Vg5H1@mp/d2#2 ~3!

and

Vg5H1@~2m811!p/2d3#2, ~4!

wherem and m8 are integers, andVg5vg /D8. The varia-tions of T versus the reduced frequency,V5v/D8, are re-ported in Fig. 3~a! for d251, d350.5d2, andN851. In thecase where the number of the resonatorsN8 is greater thanone, and the number of loops is still one, the zeros oftransmission coefficient enlarge into stop bands. This poinshown in Fig. 3~b! for d251, d350.5d2, andN852.

In the particular case of a single loop with zero numberresonators~i.e., N850), the transmission coefficient becomes T215@2529 cos2(ad2)#/16. In contrast with thetransmission coefficient of a single loop with dangling resnators, the transmission power of a single loop with out renators never reaches zero values.11 That’s why, in SLSs, thestop bands originate only from the periodicity. On the cotrary, in SLS-DRs, the stop bands are due to the conjugeffect of the periodicity and the zero transmission associawith the side branches which play the role of a resonaTable I presents the width of the region whereT!0.001~i.e.,of practically stop bands! in the case of systems studiedRefs. 10 and 11 together with the present system. The cruinfluence of the numberN8 of the resonators on the stoband width can be easily appreciated.

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FIG. 4. Variations of the transmission a powthrough a SLS-DR forN54 loops,N853 reso-nators, d151, d25d1 and d350.5d1. ~b! Thesame as in~a!, but for d350.75d1. ~c! Transmis-sion power through a tandem structure buildthe above SLS-DRs~a! and~b! ~the plot is givenfor N58). The superposition of the forbiddebands in~a! and ~b! is well seen in~c!.

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BRIEF REPORTS PHYSICAL REVIEW B69, 012405 ~2004!

The transmission rate through a finite-size SLS-DR ctaining N54 loops with N853, d151, d25d1, and d3

50.5d1 is reported in Fig. 4~a!. The existence of wide stopbands separated by narrow pass bands shows up cleathis figure. Despite the finite number of loops in Fig. 4~a!,the transmission factor approaches zero in regions cosponding to the observed gaps in the magnonic band sture. Figure 4~b! shows the transmission power for anothdifferent SLS-DR withN54, N853, d151, d25d1 andd350.75d1. By associating in tandem the above SLS-DRone obtains@Fig. 4~c!# an ultrawide stop band where thtransmission is canceled over a large range of frequengoing fromV.63–130. In this structure, the huge stop baresults from the superposition of the forbidden bands ofindividual SLS-DR@Figs. 4~a! and 4~b!#.

Now, if a defect is included in the structure, a localizstate can be observed in the stop band. A defect in SLS-can be realized by replacing a finite wire of lengthd1 by asegment of lengthdfÞd1 in one cell of the structure. Thetransmission power versus the reduced frequency for a sture with ten loops, three resonators, and a defect segmelength df50.5d1 located in the middle of the structure,reported in Fig. 5. The frequency of the defect mode insthe stop band depends on the length of the defect segmwhereas the intensity of the peak in the transmission sptrum depends on the numbersN andN8 in the SLS-DR. It isworth pointing out again that, in all of our calculations, tcross sections of all media are considered to be much smthan the considered wavelength of the magnons, so aneglect the quantum-size effect~or the subband structure!.

In conclusion, we have presented a 1D monomode sttures exhibiting very large magnonic stop bands. A theorcal investigation of the magnonic band structure of aSLS-DR using a Green’s function method is presented.solute stop bands exist in the spin wave band structure oinfinite SLS-DR. Compared to other 1D networks suchCLS waveguides, the observed stop bands in SLS-DRsignificantly larger. The calculated transmission powermagnons in finite loop structures parallels the band spectof the infinite periodic SLS-DRs. The existence of the st

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bands in the spectrum is attributed to the conjugate effecthe periodicity and the zero transmission associated withdangling side branches which play the role of resonatorsthese systems, the stop band width is controlled by the gmetrical parameters. Numerical results on localized modeperturbed waveguides were also reported. Since it is geally the case that magnetic periodic networks have wtechnical applications, it is anticipated that this class of mterials, which can be referred to as ‘‘magnonic crystals,’’ wturn out to be of significant value for prospective applictions. One would expect such applications to be feasiblespintronic devices, since magnon excitation energies alsoin the microwave range.

ACKNOWLEDGMENT

H.Al-W. gratefully acknowledge the hospitality of thLaboratoire de Dynamique et Structure des Mate´riaux Mo-leculaires, Universite´de Lille 1.

FIG. 5. Transmission spectrum versus the reduced frequenca ten-loop SLS-DR with one defect segment of lengthdf50.5d1

located in the middle of the waveguide. The other parameters c

sidered areH51, d151, d250.5d1 , d350.5d1, andN853. In thefrequency range displayed in this figure one can see that the pfalling in the gaps are very narrow, and present a strong amplitu

U.

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v.

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