J Nanopart Res DOI 10.1007/s11051-011-0303-5
SPECIAL ISSUE: NANOSTRUCTURED MATERIALS 2010
The magnetostatic modes in planar one-dimensional magnonic crystals with nanoscale sizes M. L. Sokolovskyy • M. Krawczyk
Received: 13 September 2010 / Accepted: 22 February 2011 Ó Springer Science+Business Media B.V. 2011
Abstract The plane-wave method is used to determine the spin-wave spectra in nanoscale one-dimensional magnonic crystals formed by a periodic lattice of ferromagnetic stripes. The dynamic demagnetizing field in a new formulation is used in these calculations. The calculated spectra of spin waves are compared with experimental data taken from literature and a good agreement is obtained. The dependence of the spin-wave spectra on magnetic material parameters consisting of magnetic stripes is systematically studied. We proved that the experimentally found magnonic band structure with two magnonic gaps is of magnetostatic nature. Keywords Magnonic crystal Plane-wave method Demagnetizing field Band structure
In recent years, the theory of nanostructured materials has attracted significant attention because of the properties that otherwise do not exist in nature. Materials with periodic modulation of one or more structural parameters are promising for applications. M. L. Sokolovskyy (&) M. Krawczyk Surface Physics Division, Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznan´, Poland e-mail:
[email protected]
Such structures allow us to achieve negative effective permittivity and permeability, (Smith et al. 2000; Ramakrishna 2005; Berrier et al. 2004), zero effective permittivity (Ziolkowski 2004; Silveirinha and Engheta 2006), the phenomenon of giant magnetoresistance (Baibich et al. 1988; Binasch et al. 1989). Artificial electromagnetic dielectric materials with periodicity of the refractive index comparable to the wavelength of light, known as photonic crystals, have already found practical applications in optoelectronics (Joannopoulos et al. 2008). The fundamental feature of periodic structures is the presence of forbidden-frequency gaps (‘band gaps’) in their spectrum, in which no propagation is allowed. Periodically modulated magnetic materials can be regarded as the magnetic counterpart of photonic crystals with spin waves acting as information carriers (Vasseur et al. 1996; Neusser and Grundler 2009; Kruglyak et al. 2010a; Khitun et al. 2010; Gubbiotti et al. 2010). As opposed to magnonics, photonics is a more developed field of research and technology. However, magnonic crystals (MCs) are better candidates for miniaturization, since the wavelength of spin waves is several orders of magnitude shorter than that of electromagnetic waves of the same frequency, in photonic crystals. In very recent publications magnonic band gaps were experimentally observed by Brillouin Light Scattering spectroscopy (BLS) (Tacchi et al. 2010a, b) and by transmission measurements (Nikitov et al. 2006; Chumak et al. 2009). The experimental data provides strong evidence of collective excitations (Kruglyak et al. 2010b) in the
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form of Bloch waves with allowed and forbidden magnonic energy bands but for MCs composed of ferromagnetic and nonmagnetic materials only. It means that band formation is due to weak magnetostatic interactions between localized modes. This results in the formation of narrow bands in the magnonic band structure. Much stronger interaction and behaviour close to photons in photonic crystals are expected when MCs are solely composed of ferromagnetic materials (Puszkarski and Krawczyk 2003; Kruglyak and Kuchko 2003; Kuchko et al. 2005). Our study is motivated by recent research made by Wang et al. (2009, 2010). They designed and fabricated an one-dimensional (1D) MC consisting of a periodic array of alternating cobalt and Permalloy nanostripes. The dispersion relation of spin waves in such crystal was mapped by BLS and was calculated from the Landau–Lifshitz (LL) equation using the finite element method. The main purpose of this article is to explain the experimentally found magnonic band structure and existing magnonic gaps. The results are obtained using the plane-wave method (see e.g., Krawczyk and Puszkarski 2008), with the magnetostatic field in a new formulation described below. Let us consider the MC shown in Fig. 1. The ~ ð~; dynamics of magnetization M r tÞ is described by the LL equation: ~ oM ~ H ~E ; ¼ cl0 M ot
ð1Þ
where c is the gyromagnetic ratio ðc [ 0Þ; l0 is the ~ E is the effective permeability of vacuum and H magnetic field. Considering magnetocrystalline anisotropy negligible within the nanostripes (according to Wang et al. ~ E can be written as follows: 2009), H
x z
y
Co
H0
Py
s
s
2a k
Fig. 1 Geometry of the studied MC. Thickness of the Permalloy and cobalt nanostripes is 2a = 30 nm, whilst its ~ 0 is applied along the width is s = 250 nm. A magnetic field H ~ stripes length, k is the wave vector of the spin waves
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~ 0 ð~; ~ ex ð~; ~ d ð~; ~ E ð~; r tÞ ¼ H r tÞ þ H r tÞ þ H r tÞ: H
ð2Þ
~0. ~ E is the applied magnetic field H The first term of H In our case it is homogeneous in space and directed along the OZ axis. The next component of the ~ ex . In magneteffective field is the exchange field H ically inhomogeneous materials the spatial inhomogeneity of both the exchange stiffness constant Að~ rÞ and the spontaneous (saturation) magnetization MS ð~ rÞ must be taken into account, which leads to the following formula: ~ k2 ð~; ~ m ~ ex ð~; ~ð~; H r tÞ ¼ r r t Þr r tÞ; ð3Þ ex where kex ¼
qffiffiffiffiffiffiffiffi 2A l0 MS2
and kex is the exchange length.
The last component of the effective magnetic field is the dipolar field. When the lattice constant in MC is equal to 100 nm or more the dipolar interactions become more important for spin-wave dynamic and should be taken into account. One of the challenges during the computation of this field in magnetic systems is to include boundary effects. In this study, we extend the plane-wave method by taking into account the finite thickness of the considered structures. ~ d can be written as The total demagnetizing field H follows ~ d ð~; ~ ð~ H r tÞ ¼ H rÞ þ ~ hð~; r tÞ:
ð4Þ
In the magnetostatic approximation both the dynamic and the static dipolar fields must fulfil the magnetostatic Maxwell’s equations: ~ ~ r tÞ ¼ 0 ~ ~ rÞ ¼ 0 r hð~; r H ð~ and ; ~ ~ ~ H ~ ð~ ~ð~; r hð~; r tÞ þ m r tÞ ¼ 0 ~z ¼ 0 r rÞ þ Mz ð~ rÞe ð5Þ ~ð~; where m r tÞ is the dynamic magnetization vector ~0 : M ~ ð~; ~z þ component perpendicular to H r tÞ ¼ Mz ð~ rÞe ~ð~; m r tÞ: Using a linear approximation, we can assume Mz MS . Because we are interested in solutions corresponding to monochromatic spin waves, the time dependences of the dynamic magnetization ~ð~; m r tÞ and the dynamic dipolar field ~ hð~; r tÞ take the ixt ~ ~ ~ ~ð~; ~ ~ ð~ form: m r tÞ ¼ m rÞe and hð~; r tÞ ¼ hð~ rÞ eixt correspondingly. According to the ideas presented in (Kaczer and Murtinova 1974) one can solve the equation set 5 analytically. For a structure with periodicity along the OY axis, finite thickness along
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the OX axis and infinitely long along the OZ axis one can obtain the following solution:
first Brillouin zone and forms bands with higher frequency. So our approximation of uniformity of the
~ ð~ H rÞ ¼ 0 3 coshððk þ GÞxÞ mx ðGÞ P6 coshððk þ GÞaÞ þ sinhððk þ GÞaÞ 7 7eiðkþGÞy hx ðx; yÞ ¼ 6 4 5 sinhððk þ GÞxÞ G i my ðGÞ ; 2 coshððk þ GÞaÞ þ sinhððk þ GÞaÞ 3 coshððk þ GÞxÞ my ðGÞ 1 P6 coshððk þ GÞaÞ þ sinhððk þ GÞaÞ 7 6 7 iðkþGÞy hy ðx; yÞ ¼ 6 7e 4 5 sinh ð ð k þ G Þx Þ G i mx ðGÞ coshððk þ GÞaÞ þ sinhððk þ GÞaÞ 2
where G denotes a reciprocal lattice vector of our structure (along the OY axis), k is a Bloch wave vector of spin waves propagating along the OY axis and a is half the thickness of the nanostripes (see Fig. 1). We assume the absence of the static demagnetizing field because of the infinite size of the nanostripes in the direction of MS. The above formulas (6) for the magnetostatic field depend on the position along the OX axis and fulfil standard electromagnetic boundary conditions on the slab surfaces. When the slab is thin enough we can assume that nonuniformity of the magnetostatic field across its thickness has minor influence on the low frequency spin-waves. So we can put its value equal to that in the middle of the slab, i.e., x = 0 into the LL equation 1. However, this assumption is restricted by the wavelength of the spin wave (wavelength should be much greater than thickness of the slab). Those parts of the dispersion with large wave vectors are folded to the
ð6Þ
dynamic magnetization along the OX axis is valid only for the first few bands with low frequency. Because of the Fourier form of our solution (6) we can use it directly in the plane-wave method. After applying the plane-wave method to the LL Eq. 1 and using the solution (6) for the dipolar field, one can obtain the following eigenvalue problem (for details see Krawczyk and Puszkarski 2008): ^ ~k ¼ iXm ~k ; Mm
ð7Þ ~Tk ¼ mx;k with the eigenvector defined as follows: m ðG1 Þ; . . .; mx;k ðGN Þ; . . .; my;k ðG1 Þ; . . .; my;k ðGN Þ; . . .; ^ is the following block matrix: the matrix M ^¼ M
^ xx M ^ yx M
^ xy M ^ yy : M
ð8Þ
The submatrices in Eq. 8 are defined as follows:
Mijxx ¼ Mijyy ¼ 0; X 1 Mijxy ¼ dij þ k þ Gj ðk þ Gl Þ ðGi Gl Þ Gi Gj k2ex Gl Gj H0 l ! 1 1 MS Gi Gj ; MS ðGi Gl Þ þ 1 : H0 cosh k þ Gj a þ sinh k þ Gj a X 1 k þ Gj ðk þ Gl Þ ðGi Gl Þ Gi Gj k2ex Gl Gj Mijyx ¼ dij H0 l 1 1 MS Gi Gj MS ðGi Gl Þ H0 cosh k þ Gj a þ sinh k þ Gj a
ð9Þ
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MS ðGÞ ¼
s s DMS;Py þ MS;Co 1 D 2 s GD MS;Py MS;Co sin G 2
for G ¼ 0 ; for G ¼ 6 0
(a) 20
Frequency (GHz)
In order to solve this problem numerically, we need to truncate the Fourier series and keep a finite number N of its terms. Also we must calculate the Fourier coefficients of the material parameter: the static magnetization and the exchange length squared. These coefficients can be calculated by integration over the volume Vc of the unit cell. In our case of the 1D MC (Fig. 1) it is possible to perform this integration analytically and obtain the following formulas for the static magnetization:
0
π 2s
π s
3π 2 s
2π s
15
(b) 10
5
ð10Þ where D is a period of the MC, MS;Py and MS;Co are the static magnetization of the Permalloy and cobalt nanostripes, respectively. The eigenvalues of Eq. 9 are obtained using standard numerical procedures designed for solving matrix eigen problems. In addition, we perform a convergence test to check the obtained results. We find satisfactory convergence of the numerical solutions of (9) when we use N = 43 reciprocal lattice vectors, so all calculations are performed with 43 plane waves. The spin-wave spectra of the 1D MC studied in (Wang et al. 2009) are shown in Fig. 2b, a, without external magnetic field and with an applied static magnetic field along the stripes, respectively. In our calculations we assume values of material parameter (spontaneous magnetization and exchange constant for cobalt and Permalloy) equal to those in the experimental paper (Wang et al. 2009). It is for cobalt: MS ¼ 1:15 106 A=m, exchange constant A ¼ 2:88 1011 J=m and for Permalloy: MS ¼ 0:658 106 A=m; A ¼ 1:11 1011 J=m. For the gyromagnetic ratio we assume average value, i.e., the same for cobalt and Permalloy, c ¼ 194:6 GHz=T. The green lines in Fig. 2 represent the results of our calculation with the plane-wave method, whilst the open circles are the experimental data. The evident feature of the spectrum is the presence of the two energy band gaps. It is found that applying an external magnetic field the frequencies of the entire band structure shift up and both band gaps become narrower. As one can see, a good agreement is achieved for all frequencies except the highest ones
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0 0.000
0.008
0.016
0.024
Wavevector (nm-1) Fig. 2 Spin-wave spectra of the studied MC. a Spin-wave spectrum in dependence on the wave vector of the MC shown in Fig. 1 for l0H0 = 0.2 T. The green lines represent results of our calculations, whilst the open circles are the experimental data taken from Wang et al. 2009. b Spin-wave spectrum of the MC in the same configuration as in a but for l0H0 = 0 T. Experimental points are taken also from Wang et al. 2009. (Color figure online)
and those that are close to the centre of the Brilluoin zones. The discrepancy for the third band increases when we apply a nonzero external magnetic field (Fig. 2a). An explanation could be the fact that we assume the gyromagnetic ratio to be uniform in our MC and moreover we take in our calculations the magnetostatic field in the middle of the plane of the MC (in x = 0). Including into the calculations inhomogeneity of the gyromagnetic ratio allow the authors of (Wang et al. 2009) to obtain a better agreement for all energy bands of the studied structure (Fig. 3 in Wang et al. 2009). Higher frequencies of the first mode observed in the experiment as compared to those obtained from our calculations can have two sources: it can be connected with limitations of the experimental setup or/and with real structure of the MC different from that used in the calculations. In the BLS light is collected under limited range of angles, i.e., the signal is coming to Fabry–Perot interferometer from some range of the transversal components of the light’s wave vectors. Assuming that the experimental setup has the resolution taken from Sandweg
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(b) 16
12
12
4 0 0.4
0.8
1.2
Static magnetization, MS [106 A/m] in Py
8 4 0 0.8
1.2
Fig. 3 Confirmation of magnetostatic nature of the observed spin waves. In this figure, we show the spin-wave frequencies versus static magnetization (a, b) and exchange constant (c) of the Permalloy and cobalt nanostripes of the MC. As one can Fig. 4 Profiles of spin waves. Profiles of the spin waves with frequencies taken from the 1st, 2nd and 3rd bands of the magnonic band structure are shown. As one can see, the spin waves are localized in the Permalloy nanostripes
1.4
16 12 8 4 0
Static magnetization, MS [106 A/m] in Co
0.6
1
1.4
Exchange constant A [10-11 J/m] in Py
see, the low frequency spectrum of the MC almost do not depend on the exchange constant in Permalloy and has a week influence from the static magnetization of cobalt
3rd band
2nd band
1st band
Frequency (GHz)
8
(c) Frequency, f [GHz]
16
Frequency, f [GHz]
Frequency, f [GHz]
(a)
10
5
0 0.000
0.008
0.016
0.024
Wavevector (nm-1)
et al. (2010): Dk ¼ 0:45 103 nm1 ; from our calculations we obtain frequency around 2 GHz for k = 0 and H0 = 0. In the case of the uniform thin film very small magnetic field will shift up frequency of the Damon–Eshbach mode to a higher frequency, e.g., putting l0 H0 ¼ 0:005 T shifts up the frequency to 2.0 GHz for k = 0. This is very small value and it is reasonable, that there could be comparable anisotropy field in real Co/Py stripes, at least in Co stripes. The role of the interfaces was skipped in article (Wang et al. 2009). We perform additional calculations with small empty spaces between neighbour stripes, i.e., between Co and Py. We find that 4-nm thick air spaces result in 3 GHz frequency of the first mode in the centre of the Brilluoin zone with H0 = 0.
By increasing the empty spaces increase of the frequencies of the modes will be observed. In order to understand the nature of the observed spin-wave spectra we calculate the dependences of the spin-wave frequency on the static magnetizations and the exchange constants in the nanostripes. The results are presented in Fig. 3. As one can see there is almost no effect of the exchange constant of Permalloy for the first three bands with lower frequency and there is minor effect of the static magnetization of cobalt on the magnonic band structure of the studied MC (the dependence of the spectrum on the exchange constant in cobalt is found to be the same as that for Permalloy). At the same time, the spectrum of our system is sensitive to the magnitude of the static
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magnetization of Permalloy. The independence of the magnonic band structure on the exchange constants points out that the first three bands observed in the experiment have magnetostatic nature whilst in the bands with higher frequency exchange interaction starts to play important role. To have a deeper insight into the behaviour of the spin waves in our MC we calculate and plot the amplitude profiles of the wave with arbitrary chosen frequencies from the 1st, 2nd and 3rd bands—Fig. 4. According to these profiles the amplitude of the spin waves from all three bands are concentrated in the Permalloy nanostripes. This result explains insensitivity of the studied material with respect to the static magnetization of the cobalt nanostripes. In summary, we calculate the spin-wave spectra of a nanoscale periodic magnetic structure with a finite thickness. The plane-wave method is employed for this purpose. The dynamic demagnetizing field is obtained in the form suitable for the plane-wave method. We obtain a good agreement with the published Brillouin light scattering measurements. Localization of the spin waves in the Permalloy nanostripes is shown and the magnetostatic nature of the observed spin waves is justified. Acknowledgements The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under Grant Agreement nr228673 for the MAGNONICS project. The calculations presented in this article were performed in Poznan Supercomputing and Networking Center.
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