10 December 2001
Physics Letters A 291 (2001) 333–337 www.elsevier.com/locate/pla
Magneto-transport in asymmetric serial loop structures H. Al-Wahsh 1 , A. Mir 2 , A. Akjouj ∗ , B. Djafari-Rouhani, L. Dobrzynski Laboratoire de Dynamique et Structures des Matériaux Moléculaires, ESA CNRS 8024, UFR de Physique, Université de Lille 1, 59655 Villeneuve d’Ascq cedex, France Received 1 October 2001; accepted 27 October 2001 Communicated by L.J. Sham
Abstract In the frame of the long-wavelength Heisenberg model, the magnonic band structures and transmission spectra of asymmetric serial loop structures (ASLS), made of asymmetric loops pasted together with segments of finite length, are examined theoretically. These monomode structures, composed of one-dimensional ferromagnetic materials, may exhibit large stop bands where the propagation of spin waves is forbidden. The width of these band gaps depends on the geometrical parameters of the structure and may be drastically increased in a tandem geometry made of several successive ASLS which differ by their geometrical characteristics. These ASLS’s may have potential applications in spin-injection devices. 2001 Elsevier Science B.V. All rights reserved. PACS: 75.30.Ds; 75.70.Cn; 75.90.+w
In recent years, a great deal of interest has been devoted to the investigation of low-dimensional spin systems—i.e., magnetic structures with a dimensionality less than three [1–3]. This related both to the fundamental interest and to the potential applications of spintronic devices, and is supported by the advanced progress in nanofabrication technology [4]. For example, arrays of very long ferromagnetic nanowires of Ni, permalloy and Co, with diameters in the range of 30 to 500 nm have been created [5,6]. These are very uniform in cross section, with lengths in the range of 20 microns. They thus are realizations of nanowires * Corresponding author.
E-mail address:
[email protected] (A. Akjouj). 1 Permanent address: Faculty of Engineering, Zagazig Univer-
sity, Benha Branch, Cairo, Egypt. 2 Permanent address: Département de Physique, Faculté des Sciences, Université Moulay Ismail, Meknès, Morocco.
one can reasonably view as infinite in length, to excellent approximation. Besides the static and magnetotransport properties of magnetic nanowire arrays, the dynamic properties of magnetic nanostructure are also of considerable interest in both fundamental as well as applied research [6]. These recent developments encouraged us to investigate magnetic excitations in networks composed of one-dimensional (1D) continuous magnetic media. Our choice of 1D magnetic structures is motivated by possible engineer spin-injection devices that render feasible the control of the widths of the pass bands (and hence the stop bands). In previous publications, we proposed [7,8] a model of 1D magnonic crystal exhibiting pass bands separated by large forbidden bands. The geometry of the model presented in Ref. [7] (called a comblike structure CLS) is composed of an infinite 1D monomode waveguide (the backbone) along which N dangling side branches (which play the role of resonators) are
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 7 3 7 - X
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grafted at N equidistant sites, N and N being integers. The presence of defect branches in the comb structure can give rise to localized states within the gaps. It has been shown that these states are very sensitive to the length and number of the side branches, to the periodicity of the system, and to the length of the defect branches. One drawback of the comb structure lies in the difficulty of technical realization of the boundary condition at the free ends of the resonators. On the other hand, the structure presented in Ref. [8] (called a serial loop structure SLS) was made of symmetric loops (rings) pasted together with segments of finite length, the loops play the role of resonators. The above mentioned difficulty was avoided in studying SLS. Interestingly, in studying the transmission rate through a defective geometry, the defect branch in CLS [7] was inserted (fixed) at the middle of the comb, while in the SLS the defect can be located in any cell of the structure, which in turn change dramatically the transmission spectrum. It is worthmentioning also that when the defect wire is located in the middle of the SLS and whatever is the ratio df /d1 (df is the length of the defect wire and d1 is the length of the finite segments) the intensity of the transmitted gap modes remain one. However, if the position of the defected cell is not in the middle, the intensity of the transmitted gap modes is remarkably depressed. Let us direct attention to the point that the quantum size effect (or the sub-band structure) was neglected in the above mentioned networked waveguides. We dealt with a magnetic network where the cross-sections of all wires are considered to be much smaller than the considered wavelength, i.e., a continuum approximation theory was employed in the calculation. In this Letter, we propose a new geometry, called asymmetric serial loop structure (ASLS), of a monomode networked waveguide. The structure is composed of asymmetric loops pasted together with segments of finite length (see Fig. 1(a)). Such structure may exhibit new features, in comparison with the CLS and the SLS waveguides. For example, the existence of larger gaps, the avoidance of the constraint on the boundary condition at the end of the side branches (in the case of CLS), appearance of quasi quantized bands with out inserting a defect, achieving a complete gaps for a small number of loops. Theses new features (which could be of potential interest in waveguide structures) are essentially due the asymmetry of the
Fig. 1. (a) Schematic of the one-dimensional asymmetric serial loop structure. The one-dimensional media constituting the loop and the finite segments are assumed to be of the same material. The lengths of the three wires are denoted d1 , d2 and d3 , respectively. (b) Projected band structure of the ASLS as function of L = d2 − d3 for d1 = 1 and L = d2 + d3 = 2. The shaded areas represent the bulk bands. The dashed curves indicate the frequencies for which the denominator of η (Eq. (1)) vanishes.
loops structure which is quit different from the case of CLS or SLS. We report on results of calculated band structures and transmission coefficients. We also show that the width of the band gaps may be enlarged by coupling several ASLS of different physical characteristics. The 1D infinite ASLS can be modeled as an infinite number of unit cells pasted together. In each unit cell, the two arms of the ring have different lengths d2 (of medium 2) and d3 (of medium 3). This results in asymmetric loop of length d2 + d3 (see Fig. 1(a)) which is pasted to a segment of length d1 (of medium 1). We focus in this letter on homogeneous ASLS where the media 1, 2 and 3 are made of the same material. The dispersion relation of the infinite ASLS, that relates the pulsation of the spin wave ω to the Bloch wave vector k, can be derived using
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the Green function method [9]. It can be written as cos(kd) = η(ω) where d is the period of the structure and η=
1 αL 2 sin( αL 2 ) cos( 2 )
1 5 × sin(αd1 ) cos(αL) − cos(αL) − 1 4 4 (1) + cos(αd1 ) sin(αL) .
√ Here L = d2 + d3 , L 2= d2 − 2 d3 , α = (ω − γ H0 )/D, and D = (2J a M)/(γ h¯ ). M, H0 , J , and γ stand, respectively, for the spontaneous magnetization, the static external field [7], the exchange interaction between neighboring magnetic sites in the simple cubic lattice of lattice parameter a constituting the ferromagnetic medium, and the gyromagnetic ratio. Fig. 1(b) displays the projected band structure (the plot is given as the reduced frequency Ω = = γ H0 /D versus L) of + α 2 , with H ω/D = H an infinite ASLS for given values of L, d1 and H such that L = 2, d1 = 1, and H = 1, respectively. There is a complete absolute gap below the lowest band due to the presence of the external field H0 . The shaded areas, corresponding to frequencies for which |η| < 1, represent bulk bands where spin waves are allowed to propagate in the structure. These areas are separated by minigaps where the wave propagation is prohibited. Inside these gaps, the dashed lines show the frequencies for which the denominator of η (Eq. (1)) vanishes: the dashed horizontal and curved lines, that correspond to the vanishing of sin(αL/2) and cos(αL/2), respectively, define the frequencies at which the transmission through a single asymmetric loop becomes exactly equal to zero. In Fig. 1(b), one can distinguish between two types of minigaps: those of lozenge pattern that originate from the crossings of the zero transmission lines, and the gaps around Ω = 25 or 60 (occurring for any value of L) that are related to the periodicity of the structure. One interesting point to notice in the band structure of Fig. 1(b), namely, at certain values of L (for instance, L ∼ 0.44), one can obtain a series of narrow minibands separated by large gaps; this is because the points at which the minibands close, align
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more or less vertically in such a way that a few successive bands may become very narrow. We now turn to the study of the transmission coefficient. We start with a study of a simple example, namely a wave guide consisting of a unique asymmetric loop. The transmission factor T can be written as 2 2(S2 + S3 )S2 S3 , T = (2) 2 2 (C2 S3 + C3 S2 + S2 S3 ) − (S2 + S3 ) where Ci = cosh(j αdi ), Si = sinh(j αdi ), and j = √ −1. The transmission is equal to zero only when S2 + S3 = 2 sin(αL/2) cos(αL/2) = 0. The zero frequencies, corresponding to the eigenmodes of a single loop, are given by 2 + (2m + 1)π , Ωg = H (3) L and
4nπ 2 Ωg = H + , L
(4)
where m and n are integers, and Ωg = ωg /D. The variations of T versus the reduced frequency, Ω = ω/D, are reported in Fig. 2(b) for d2 = 1, d3 = 0.5, i.e., L = 0.5. In the case where the number of the asymmetric loops greater than one, the zeros of the transmission coefficient enlarge into gaps. In the particular case of a symmetric loop (d2 = d3 , i.e., L = 0), the transmission coefficient becomes T=
16 . 25 − 9 cos2 (αd2 )
(5)
In contrast with the transmission coefficient of an asymmetric single loop, the transmission of a symmetric one never reaches zero values (see Fig. 2(a)). That is why, in symmetric SLS, the gaps originate only from the periodicity. On the contrary, in ASLS, the gaps are due to the conjugate effect of the periodicity and the zero transmission associated to a single asymmetric loop which plays the role of a resonator. The transmission rate through a finite-size of ASLS containing N = 5 loops with L = 0.44 , L = 2 and d1 = 1 is reported in the middle panel of Fig. 2(c). Clearly, the existence of wide gaps separated by narrow bands show up. Despite the finite number of loops in Fig. 2(c), the transmission approaches zero in regions corresponding to the observed gaps in the magnonic band structure of Fig. 1(b). It is worth
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Fig. 3. (a) Transmission spectrum versus the reduced frequency for a 8 loops ASLS with one defect segment of length df = 0.1d1 located in the middle of the waveguide. The other parameters is = 1, d1 = 1, L = 2, L = 1. In the frequency considered to be H range displayed one can see that the peaks falling in the gaps are very narrow and present a strong amplitude.
Fig. 2. (a) Transmission factor versus reduced frequency for a wave-guide with one loop in the case of SLS (d2 = d3 ). (b) The same as in (a) but for ASLS with L = 0.5 and L = 1.5. For convenience is considered to be 1. (c) Variations of the transmission power H through ASLS for N = 5 loops, d1 = 1, L = 2 and L = 0.44. (d) The same as in (c), but for d1 = 1.5, L = 2 and L = 1.1. (e) Transmission power through a tandem structure build of the above ASLS’s (c) and (d) (the plot is given for N = 20). The superposition of the forbidden bands in (c) and (d) is well-seen in (e).
noticing that the general features discussed in Fig. 1(b) are still valid for any values of d1 and L and various L. However, the shape of the band structure changes drastically for fixed values of d1 and L and various L. Fig. 2(d) shows the transmission power for another different ASLS with d1 = 1.5, L = 2 and L = 1.1. Now, by associating in tandem the above ASLS’s, one obtains (Fig. 2(e)) an ultrawide gap where the transmission is cancelled over a large range of frequencies going from Ω 35 to Ω 105. In this
structure, the huge gap results from the superposition of the forbidden bands of the individual ASLS (Fig. 2(c),(d)). If a defect is included in the structure, a state can be created in the gap. A defect in ASLS’s can be realized by replacing a finite wire of length d1 by a segment of length df = d1 in one cell of the waveguide. The transmission spectrum versus the reduced frequency for a structure with 8 asymmetric loops and a defect segment of length df = 0.1d1 located in the middle of the structure, is depicted in Fig. 3. The frequency of the defect mode inside the gap depends on the length of the defect segment whereas the intensity of the peak in the transmission spectrum depends on the number N of loops in the ASLS. The band structure as well as the defect modes also gives rise to well define peaks in the delay time we shall present in a more detailed paper. In this Letter, we have considered new 1D monomode structures exhibiting very large magnonic band gaps. A theoretical investigation of the magnonic band structure of 1D ASLS using a Green’s function method is presented. Absolute band gaps exist in the spin wave band structure of an infinite ASLS. Compared to other 1D networks such as CLS-waveguides, the observed gaps in ASLS are significantly larger. The calculated transmission rate of magnons in finite loop structures parallels the band spectrum of the infinite periodic ASLS. The existence of the gaps in the spectrum is attributed to the conjugate effect of the
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periodicity and the zero transmission associated to a single asymmetric loop which plays the role of a resonator. In these systems, the gap width is controlled by the geometrical parameters. Numerical results on localized modes in perturbed waveguides were also reported. Since it is generally the case that magnetic periodic networks have wide technical applications, it is anticipated that this new class of materials, which can be referred to as “magnonic crystals”, will turn out to be of significant value for prospective applications. One would expect such applications to be feasible in spintronic devices, since magnon excitation energies also fall in the microwave range.
Acknowledgements H. Al-Wahsh and A. Mir gratefully acknowledge the hospitality of the Laboratoire de Dynamique et Structure des Matériaux Moléculaires, Université de Lille 1.
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