Spin-Wave Band Structure in 2 DMagnonic Crystals with Elliptically Shaped Scattering Centres

Spin waves in 2D periodic magnetic nanocomposites are studied by means of the plane wave method. The effect of the ellipticity and in-plane rotation of the scattering centers on the band structure is investigated, to indicate new possibilities of fine tuning of spin-wave filter passbands.


Introduction
Magnetic composites with a structure modulated periodically on the nanoscale are the subject of a very intensive research activity, and the interest in their properties is increasing [1][2][3][4][5][6][7].By analogy to photonic crystals (PCs), they are referred to as magnonic crystals (MCs) [8], since the role of information carriers in these materials is played by magnons, or spin-wave quanta.Magnonic crystals have properties that cannot be reduced to those of their constituent materials, as manifested, among others, by the band structure of their spin-wave spectrum.The periodic structure given to a magnetic composite strictly determines the possible occurrence of magnonic gaps, or energy ranges forbidden to propagating spin waves [9][10][11].The anticipated full control over the spin waves propagating in MCs, similar to that of the electromagnetic waves in PCs, combined with a long-term stability of the programmable magnetic state makes MCs excellent for both research and application purposes [12,13].Moreover, the wavelength of spin waves is much shorter than that of electromagnetic waves of the same frequency.This provides additional possibilities in the miniaturization of MC-based devices [14][15][16].For a broad survey of the current state of both the experimental and theoretical research in MCs and their potential applications, please refer to review papers [17,18].
The most frequently mentioned of the numerous potential applications of MCs include microwave resonators, magnonic waveguides, spin-wave emitters, and filters [19].These potentia applications, along with the possibility of modeling the energy spectrum of spin-wave excitations propagating in MCs, are the very reason of the intensification of the research on new magnonic materials with so far unknown properties and functionalities [14,16].In particular, two-dimensional (2D) MCs with band gaps in the spin-wave spectrum have potential applications in diverse magnonic devices, such as spin-wave filters or switches, or current-controlled delay lines [20].For example, the latest results of micromagnetic simulations [21] indicate the occurrence of wide magnonic gaps, implying a possible application in spin-wave filters, in 2D Fe/YIG MCs, twocomponent magnetic composites with iron scattering centers embedded in a matrix of yttrium iron garnet.A particular role in the modeling of magnonic gaps is played by the deformation of the scattering centers in the plane of spinwave propagation [22].
In this study we examine the possibilities of modeling the spin-wave spectrum of 2D MCs that could be used for fine tuning of spin-wave filter passbands [23].We present the results of calculations of the magnonic band structure of a Co/Fe composite with scattering centers in the shape of elliptic cylinders.We find that for different filling fraction values there are specific in-plane rotation angles for which modifying the rod ellipticity can alter the position of the allowed band without changing its width or Advances in Condensed Matter Physics cause a substantial shrinking of two adjacent bands without changing the width of the gap between them.

The Model
Figure 1(a) depicts schematically a section of the system under consideration in the plane perpendicular to the rod axis (the x-y plane), which is the plane of spin-wave propagation.The system includes cobalt rods (scattering centers), which are assumed to be parallel to each other and have an infinite length.The rods are arranged in sites of a 2D square lattice and embedded in an iron matrix.The system is infinite in the plane of periodicity.The filling fraction ff, describing the proportion of the rod material in the whole volume of the system, in the case of elliptic cylinders is given as f f = πR a R b /a 2 , where R a and R b are the semiaxes of the rod cross section, and a is the lattice constant.Let us define the ellipticity RR of the rods as the semiaxis ratio: RR = R a /R b .The angle α between the major semiaxis R a and the x-axis of the crystallographic system is the angle of rotation of the rods (in the plane of periodicity).Applied to the system, an external magnetic field perpendicular to the plane of periodicity is assumed to be strong enough to enforce a uniform magnetization throughout the system.
We shall consider in-plane propagation of spin waves, that is, their propagation in the plane of periodicity, in 2D MCs as described above.Thus, the wave vectors to be considered are limited to the Brillouin zone of the 2D lattice.Figure 1(b) shows the high-symmetry line in such a 2D Brillouin zone.In the case of square lattice the line starts at the zone center (point Γ) to pass through X and M and return to Γ (segments C, D, and E in Figure 1(b)).The introduction of rods with elliptical cross section breaks the symmetry of the square lattice.If the semiaxes of the ellipse follow the axes of the 2D crystallographic system, the structure has the symmetry of a rectangular lattice.Points X and X are not equivalent anymore, and the high-symmetry line in this case leads from M to X , Γ, X, and back to M (segments A, B, C, and D).For an ellipse rotated by 45 • points, X and X are equivalent symmetry points, whereas M and M are not.Thus, the high-symmetry line leads from Γ to X, M, Γ, M , and X (segments C, D, E, F, G).In the general case, that is, for any angle of rotation of the ellipse, the full line shown in Figure 1(b) must be considered.
Our theoretical approach is based on a set of equations including the linearized Landau-Lifshitz equation and Maxwell's magnetostatic equations [24].When the applied magnetic field is parallel to the rods, the internal static magnetic field is uniform, which allows us to only take into account the exchange and dynamic dipolar interactions.The ferromagnetic materials of the rods and the matrix are characterized by two quantities: the spontaneous magnetization M s and the exchange stiffness constant A. For the materials considered in this study the specific values of these two magnetic parameters are for iron M s = 1.752e6A/m and A = 2.1e − 11 J/m, and for cobalt M s = 1.390e6A/m and A = 2.8e − 11 J/m [25].Crucial for the magnonic nature of the structure under consideration is the assumption that these two material parameters are periodic functions of position, with the same periodicity as the 2D lattice on which the magnonic crystal is built.With this assumption our equations can be solved by the plane wave method.The main point of this method is the Fourier expansion of the material parameters.Bloch theorem is applied to the dynamic functions, such as the demagnetizing field potential and the dynamic component of magnetization.Thus, the equations are transformed to the reciprocal space, where their solution is equivalent to the diagonalization of a 2N × 2N matrix, N being the number of plane waves used in the Fourier expansion (for more details, see [24] and references therein).

The Role of Magnetostatic Interactions
Figure 2 shows examples of the so-called magnonic spectra, represented by the spin-wave spectra of 2D Co/Fe magnetic composites, calculated along the high-symmetry line in the 2D Brillouin zone.The spectra shown in Figures 2(a) and 2(b) have been obtained for lattice constants a = 50 nm and a = 100 nm, respectively.A circular cross section of the rods (RR = 1) and a filling fraction f f = 0.5 are assumed in both cases.For small lattice constants the exchange interactions play a dominant role; however, as the lattice constant grows, their importance diminishes to the advantage of the magnetostatic interactions [26,27].As a result, the spin-wave frequency range lowers; for the lattice constant of 50 nm the frequency of the ten lowest modes is below 80 GHz (Figure 2(a)), but only ranges from 10 GHz to 35 GHz for a = 100 nm (Figure 2(b)).The flattening of successive bands results in the opening of a magnonic gap: for a = 50 nm all the bands overlap, while for a = 100 nm a gap occurs between the lowest band and the rest of the spectrum.

The Ellipticity and Rotation of the Rods
Figure 3 presents the effect of the cross-sectional ellipticity of the scattering centers and their rotation in the plane of spin-wave propagation on the magnonic spectrum in a Co/Fe composite with a lattice constant of 300 nm and a filling fraction of 0.3.For unrotated rods (Figure 3(a)) the maximum ellipticity (corresponding to touching rods) is RR = 2.6, which means the major and minor semiaxes R a and R b can range from 92.7 nm up to 149.8 nm and down to 57.4 nm, respectively.In the whole range of rod ellipticity the bottom of the lowest band is nearly constant, remaining between 10.84 GHz and 10.89 GHz.Also the top of the second band varies very slightly, only ranging from 12.04 GHz to 12.15 GHz.However, the width of both bands grows rapidly, as the top of the first band and the bottom of the second one converge.Consequently, the gap between the bands shrinks to vanish completely for RR = 2.1 (R a = 134.3nm, R b = 64.0nm).The two bands merge to form a single wide band, separated from the rest of the spectrum by a third gap, the width of which varies from 1.26 GHz for right circular cylinders (RR = 1.0) to 0.79 GHz for RR = 2.1 (the closing of the second gap) to 0.37 GHz for RR = 2.6 (the maximum ellipticity).Figure 3(b) shows the band and gap widths plotted versus the rod ellipticity.Also for rods rotated by 45 • (Figure 3(c)) the second gap is seen to shrink, though not as rapidly as in the case of unrotated rods.The gap closes for RR = 3.45, which corresponds to R a = 212.0nm and R b = 40.5 nm.A significant difference with respect to the composite with unrotated rods is seen in the behavior of the second band, which moves down the frequency scale with nearly constant width as RR grows from 1.0 to 2.8 (cf. Figure 3(d)).
In a Co/Fe composite with a filling fraction of 0.5 and rods unrotated in the plane of periodicity (the major semiaxis following the x direction), the rod ellipticity can range from 1.0 (circular cross section) to 1.57, which for the assumed lattice constant a = 300 nm corresponds to the major semiaxis ranging from 119.7 nm up to 150.0 nm, and the minor semiaxis from 119.7 nm down to 95.5 nm.As the ellipticity of the cylinders grows, the gaps are seen to shrink and the bands to widen (Figure 4(a)).Although the bottom of the lowest band at first moves towards higher frequencies, the change is to slight to compensate the concurrent rising of the top.In the second band the top is seen to descend slightly, while the bottom moves much faster in the same direction.As a consequence, the second gap, between the first and second bands, shrinks rapidly with growing ellipticity to vanish completely for RR = 1.2 (R a = 131.1 nm, R b = 109.3nm).The first and second bands merge to form one relatively wide band (of width ranging from 1.17 GHz for RR = 1.2 to 1.32 GHz for RR = 1.57), separated from the rest of the spectrum by a third gap, which has a maximum width of 0.65 GHz for RR = 1.2, and a minimum width of 0.25 GHz for RR = 1.57(cf. Figure 4(b)).A completely different behavior of the spin-wave spectrum is seen for the same filling fraction ( f f = 0.5), but with rods rotated by 45 • in the plane of periodicity (Figure 4(c)).Two ranges of RR can be distinguished in this case.In the first range, from 1.0 to 2.0 (the major semiaxis growing from 119.7 nm to 169.3 nm, and the minor semiaxis shrinking from 119.7 nm to 84.6 nm), the width of both lowest bands decreases substantially, the first one shrinking from 0.52 GHz to 0.28 GHz, and the second from 0.34 GHz to 0.22 GHz (Figure 4(d)).At the same time, the midlevels of both bands move, from 11.0 GHz to 11.1 GHz in the case of the first band, and from 11.7 GHz to 11.6 GHz for the second.Interestingly, in this range of RR the second gap remains nearly unchanged, its width only ranging from 0.29 GHz to 0.30 GHz.In the other part of the RR dependence the spectrum behaves as in the cases considered previously: the bands widen and the gaps shrink rapidly.However, in this case even for the maximum ellipticity of the rods (RR = 3.14) the second gap will not close completely, though its width falls as low as 0.02 GHz.

Conclusions
In the 2D magnetic composites considered in this paper, with Co rods embedded in an Fe matrix, the increase in importance of the magnetostatic interactions results in the formation of band gaps in the spin-wave spectrum.These magnonic gaps are destroyed as the exchange interactions begin to play a dominant role.On the other hand, the increase in importance of the exchange interactions not only results in a widening of the bands, but also, consequently, causes the possible gaps to move towards higher frequencies.In contrast, when the magnetostatic interactions gain in importance at the cost of the exchange interactions, the frequency range in which magnonic gaps will occur lowers substantially.This finding is consistent with the results obtained for 2D composites with an EuO matrix [28].However, an entirely different behavior is observed in 3D magnonic crystals, in which absolute magnonic gaps are destroyed by the magnetostatic interactions [29].The use of rods in the shape of elliptic cylinders as scattering centers in 2D magnetic composites implies the introduction of two addition structural parameters: the cross-sectional ellipticity of the rods and the angle of their rotation in the plane perpendicular to the rod axis (the plane of spin-wave propagation).In contrast to the lattice constant, a change of which will strongly modify the spinwave spectrum, these new parameters allow fine tuning of the width and position of the bands and gaps.For specific in-plane rotation angles, changing the rod ellipticity will modify the position of a band without changing its width or cause two adjacent bands to shrink substantially without affecting the width of the gap between them.Thus, an appropriate use of rods of elliptical cross section offers additional possibilities in the design of spin-wave filters with precisely adjusted passband.

Figure 1 :Figure 2 :
Figure 1: (a) Schematic view of the 2D MC under consideration, section in the plane of periodicity.(b) High-symmetry line over the 2D Brillouin zone for ellipses arranged in a square lattice.

Figure 3 :
Figure 3: (a), (c) Three lowest magnonic gaps (shaded) versus rod ellipticity RR for 2D Co/Fe MCs with lattice constant 300 nm, filling fraction 0.3, and two angles of in-plane rotation of the rods: (a) α = 0 • and (c) α = 45 • .(b), (d) The width of the lowest gaps and bands versus RR in (a) and (c), respectively.The width of the second (2g) and third (3g) gaps is plotted with dashed line.Solid line represents the width of the lowest bands: the first (1b), the second (2b), and, in the range of RR values in which the gap between them does not occur, the band resulting from their merging (1 + 2).

Figure 4 :
Figure4: The same as in Figure3, but for filling fraction 0.5.