Tuning the Band Structures of a 1D Width-Modulated Magnonic Crystal by a Transverse Magnetic Field

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1 Tuning the Band Structures of a D Width-Modulated Magnonic Crstal b a Transverse Magnetic Field K. Di, H. S. Lim,,a) V. L. Zhang, S. C. Ng, M. H. Kuok, H. T. Nguen, M. G. Cottam Department of Phsics, National Universit of Singapore, Singapore 754 Universit of Western Ontario, Department of Phsics and Astronom, London, Ontario N6A 3K7, Canada Theoretical studies, based on three independent techniques, of the band structure of a onedimensional width-modulated magnonic crstal under a transverse magnetic field are reported. The band diagram is found to displa distinct behaviors when the transverse field is either larger or smaller than a critical value. The widths and center positions of bandgaps ehibit unusual nonmonotonic and large field-tunabilit through tilting the direction of magnetization. Some bandgaps can be dnamicall switched on and off b simpl tuning the strength of such a static field. Finall, the impact of the lowered smmetr of the magnetic ground state on the spin-wave ecitation efficienc of an oscillating magnetic field is discussed. Our finding reveals that the magnetization direction plas an important role in tailoring magnonic band structures and hence in the design of dnamic spin-wave switches. I. INTRODUCTION Magnonic crstals -7 (MCs), with periodicall modulated magnetic properties on the submicron length scale, behave like semiconductors for magnons and hold great promise for applications in emerging areas like magnonics and spintronics. Such artificial crstals ehibit unique properties not present in natural materials, such as magnonic bandgaps, and offer novel possibilities in controlling the propagation properties of the spin waves (SWs) or magnons. Recentl, enormous theoretical and eperimental efforts have been devoted to studing the fundamental properties and potential applications of MCs. Of all the characteristics, the formation of magnonic band gaps, the most essential feature, is most thoroughl investigated. Studies have shown that the band gaps are controllable b tuning the geometrical dimensions,,3,4,8 lattice smmetr, 6,8 or constituent materials 3,9 of the MCs. The magnetic field dependence of band structures is of particular interest. Both theor and eperiment reveal that magnonic band gaps can be monotonicall shifted and resized in frequenc b simpl varing the strength of an eternal applied magnetic field. 5 The great majorit of studies were restricted to cases where the equilibrium magnetization is nearl uniforml aligned along an eternal field in either the backward-volume or Damon-Eshbach geometries. There is to date, however, ver little knowledge of the band gaps of MCs for intermediate-field cases where the magnetization is arbitraril oriented with respect to the SW wavevector, making the field dependence more comple. a) phlimhs@nus.edu.sg. Here, emploing three independent theoretical methods that give consistent results, we present a detailed stud of the SW band structure of a one-dimensional width-modulated MC with a static magnetic field applied transversel to the long ais of the waveguide sstem. The band structure ehibits different behaviors depending on whether the field is below or above a critical value. The widths and center frequencies of magnonic band gaps are shown to be tunable in a non-monotonic fashion b varing the applied field. Interestingl, some bandgaps attain their maimal width at a field below the critical value, when the equilibrium magnetization is in an intermediate configuration. Finall, results for the ecitation efficienc of the SWs b an applied oscillating magnetic field are also presented and interpreted using group-theor analsis. II. CALCULATION METHODS The [P, P ] = [9 nm, 9 nm] Permallo MC studied (period a = 8 nm, thickness of nm and alternating widths of 4 nm and 3 nm) has the same geometr and material propert as those of Ref. 4. But now we include a variable transverse magnetic field [see Fig. (a)]. To calculate its SW band structure, three independent numerical calculations are performed, based on different assumptions and models and et ielding consistent results, namel, a finite-element method, a microscopic approach, and time-domain simulations, as described below. A proper understanding of the bandgap properties entails a complete description of the SW modes. In the calculations, the saturation magnetization M S, echange constant A and gromagnetic ratio γ of Permallo were set to A/m,

2 J/m and. 5 Hz m/a, respectivel, with magnetocrstalline anisotrop and surface pinning neglected. The finite-element approach was implemented in COMSOL Multiphsics software with calculations done within one unit cell of the waveguide as illustrated in Fig. (a). First, the sstem was relaed to its energ minimum under a transverse magnetic field b solving the Landau-Lifshitz-Gilbert (LLG) equation. The demagnetizing field Hd () r is obtained b solving the following Poisson s equation () r M() r, () where () r is the scalar potential of the demagnetizing field and Mr () the total magnetization. For the nonmagnetic domain, Mr () is set to zero. Application of the periodic boundar conditions ( a)= ( ) and ( a)= ( ) automaticall guarantees the periodicit of the echange field. The band structure was then obtained b solving the three-dimensional linearized Landau-Lifshitz equation, i dip D T d D, m M h m +m H H M () where m and M are the dnamic and equilibrium magnetizations with three components, h dip () r is the dnamic demag-netizing (a) (b) (c) (d) H V S S g H S nm 4 nm <M / M S >, (3), j, j j j B T j,, j, j,, j z.5 P P odd even odd+even nm T. T.3 T z fields, and D A M S. The Bloch-Floquet boundar conditions ( a) ( )ep( ik a) and m ( a) m ( )ep( ik a) are applied, where k is the SW wavevector in the direction, () and m() are the respective dnamic components of the demagnetizing field and magnetization. Note that in both steps, the same tetrahedral mesh grid was emploed and convergence was obtained for various mesh sizes smaller than the echange length of Permallo l A M 5 nm. In the microscopic approach we emploed an etension to the dipole-echange theor in Ref. that was applied to lateral periodic arras of ferromagnetic stripes coupled via magnetic dipole-dipole interactions across nonmagnetic spacers. In the waveguide MCs considered here, there are interfaces between the Permallo regions in adjacent unit cells [see Fig. (a)], and so that inter-cell as well as intra-cell short-range echange has to be taken into account. This is in addition to the long-range dipole-dipole coupling involving all cells. The waveguide was modeled as an infinitel long spatiall-modulated arra of effective spins that were arranged on a simple cubic lattice, with the effective lattice constant a chosen to be shorter than the Permallo echange length l e mentioned above. The appropriate number of spins is then chosen in an of the phsical dimensions of the waveguide unit cell so the correct size is obtained (e.g., in the thickness dimension a choice of a = nm would correspond to ten cells of spins). The spin Hamiltonian H can be epressed as a sum of two terms: e S FIG.. (a) Schematic view of the magnonic crstal waveguide. The shaded blue block represents the computational unit cell used in the finite-element and microscopic calculations. The green bar indicates the region where the ecitation field is applied in the OOMMF simulations. (b) Instantaneous crosssection profile of the ecitation field for OOMMF. (c) Average value of the normalized equilibrium transverse magnetization M M calculated as a function of the transverse field strength. (d) Ground state magnetization of one unit cell subjected to various. where, V j, j is the total (echange plus dipolar) interaction between the spin components S j and S j, and the summations are over all distinct magnetic sites, labeled b an inde j (or j'), which specifies the position within an unit cell, and (or '), which labels the repeated cells of the MC along the direction. The second term in Eq. (3) represents the Zeeman energ due to the transverse field in the direction, where g and μ B denote the Landé factor and Bohr magneton, respectivel. The interaction (where and denote the Cartesian components, or z) has the form r, j, j, 3rj, j r j, j j, j j, j, B 5 rj, j V J g S, (4) where the first term describes the short-range echange interaction J j, j, assumed to have a constant value J for nearest neighbors and -7

3 zero otherwise. The second term describes the long-range dipoledipole interactions, where r j, j j jt, j j, zj zj is the separation between spins. The parameters of the microscopic and macroscopic models (see Refs.,) are related b M g S a and D SJa gb 3 S B, where a is the effective lattice parameter mentioned above, S is the spin quantum number, and gb in terms of the gromagnetic ratio. The steps in the microscopic theor involve first solving for the equilibrium spin configurations in the arra, using an energ minimization procedure appropriate for low temperatures and treating the spins as classical vectors. Net the Hamiltonian H was reepressed in terms of a set of boson operators, which are defined relative to the local equilibrium coordinates of each spin. Finall, keeping onl the terms up to quadratic order in an operator epansion, we solve for the dipole-echange SWs of the waveguide MC. In general, the procedure requires numericall diagonalizing a N N matri, where N is the total number of effective spins in an unit cell (one period) of the MC. For most of our numerical calculations we emploed values for a in the approimate range. to.4 nm, which implies N ~ 8 or larger. The coupled modes will depend, in general, on the Bloch wavevector component k associated with the periodicit of the MC in the direction. B means of a Green s function approach, the microscopic theor can also be used to calculate a mean-square amplitude for the SW modes. This can be done for each of the discrete SW bands as a function of wavevector k and the position anwhere in the unit cell for one period of the MC. In the time-domain calculation of the spin dnamics using OOMMF, 3 the waveguide etending 4-μm in the direction was discretized to individual ΔΔΔz =.5.5 nm 3 cuboid cells. The equilibrium magnetization of the waveguide subject to a transverse field was obtained through solving the LLG equation with a relativel large Gilbert damping coefficient α =.5. Net a pulsed magnetic field H() t H sinc( f t) in the z direction, chosen to ecite all SW branches, was applied to a.5 5 nm 3 central section of the waveguide [see the odd+even field of Fig. (b)]. This contrasts with the previous two methods where no ecitation field is needed. The time evolution of the magnetization was obtained with the damping coefficient α set to.5. Dispersion relations are then acquired b performing a Fourier transform of the out-of-plane component m z in time and space, with contributions from all the discretized magnetizations considered. from the to the direction due to the competing demagnetizing and applied field. The simulated average M M versus plot presented in Fig. (c), indicates that the waveguide is almost completel magnetized in the direction above a critical field of about H C =.9 T. This value accords well with the analticall estimated average demagnetizing field of.3 T, based on the assumption that the infinitel-long waveguide has a uniform average width of 7 nm and is uniforml magnetized in the transverse () direction. For all values of considered, the magnetization is nearl uniform, since the echange interaction is dominant due to the relativel small size of the waveguide. The calculated dispersions of magnons in the MC waveguide under transverse magnetic field =,. and.3 T are shown in Fig.. It is clear that, for all values, the MC waveguide ehibits complete magnonic band gaps arising from Bragg reflection and anticrossing between counter-propagating modes. The frequencies of each branch, at the Γ and X points, as functions of field strength are presented in Figs. 3 (a) and (b), respectivel. At the Γ point, instead of being a monotonic function, the frequencies of all five branches first decrease for H C and then increase for H C, where H.9 T. Such a behavior under a transverse magnetic field has III. MAGNONIC BAND STRUCTURES UNDER instead of the tunable widths of the gap openings. We illustrate this TRANSVERSE FIELDS using the first band gap (see Fig. 4). In Ref. 7, it is reported that Under an increasing transverse field applied in the direction, without a static magnetic field, onl even-smmetr branches, 3, the equilibrium magnetization in the waveguide is graduall oriented and 5 (odd-smmetr branches and 4) will be ecited b an even (odd) ecitation field, meaning that there is no coupling between the 3-7 C been eperimentall observed in ferromagnetic nanowires 4,5 of rectangular or circular cross-section, for which the SW wavevector component along the wire ais is zero. Fig. 3(b) shows that at the X point, the curves feature an obvious dip onl for branches labeled and 4, and the fields at which a frequenc minimum occurs are lower than H C. Of special interest is how the complete magnonic band gaps change with magnetization as is varied. Interestingl, Figs. 3(c) and (d) reveal that the overall variation of the bandgap parameters with increasing field is non-monotonic, which contrasts with previous reports of MCs.,5-7 With increasing field, the width of the first band gap first increases and then decreases. It remains constant after the equilibrium magnetization is totall aligned along the direction. We note that a maimum value of ~ 7. GHz is attained at about =. T, when the average angle between M and the ais is about π 4. B comparison, the second band gap decreases monotonicall and vanishes above =.5 T. The center frequenc of the first (second) band gap monotonicall increases (decreases) as the field increases. It should be noted that such band gap tunabilit is mainl a result of the field-tunable separation between center frequencies of gap openings induced b coupling between counter-propagating modes, 4 S

4 even and odd modes. Therefore, gap openings appears between bands with the same smmetr, e.g., between and 3 or and 4. When = T, the gap opening is ~ 9.7 GHz between branches and 3, and ~.5 GHz between branches and 4, the frequenc overlap of which gives the first complete band gap of ~.8 GHz. It is clear from Fig. 4 (b) that the relativel small change in the widths of the gap openings cannot eplain the large variation of the first complete band gap width for <. T. Fig. 4 (c) indicates that for <. T, the center frequenc of the first (second) gap opening shifts up (down) sharpl, leading to a decreasing center-to-center distance of the gap openings, which should be the main reason for the observed band gap tunabilit. A similar discussion shows that when increases, the two gap openings responsible for the second complete band gap shift awa from each other, causing the band gap to decrease and finall to close full. FIG.. (a c) Magnonic dispersion curves under various transverse magnetic fields. The blue solid, red dashed and green bold lines are data obtained from COMSOL, microscopic and OOMMF calculations, respectivel. The wavevector at the Δ point is.3 nm -. (d f) COMSOL-simulated mode profiles of m z for =,. and.3 T, respectivel. IV. MODE SYMMETRY AND EXCITATION EFFICIENCY > H C, the ground state has a higher smmetr as the magnetization The magnon modes, calculated using the three methods with is completel saturated along either the or direction, respectivel. consistent results, can be classified based on their mode profile The smmetr group of the wavevector at the Γ and X points is the smmetr of the dnamic magnetization, as indicated in Fig.. The same as that of the ground state, while that at a general point Δ is respective smmetries of the magnetic ground state [see Fig. (d)] usuall a subgroup, leading to a lower smmetr. For =,.,.3 for =,., and.3 T correspond to the respective C h, C i, C h T, the smmetr groups corresponding to the wavevector at Δ are C, groups, whose character tables 6 are presented in Table. The lower C, and C h, respectivel. In all the cases considered, the ais of smmetr associated with the ground state for < < H C precludes rotational smmetr, if one eists, is coincident with the direction of the labeling of the branches as even or odd smmetr. For = or the transverse field. 4-7

5 (a) Frequenc (GHz) (c) Gap width (GHz) H T..4 H T FIG. 3. The transverse field dependences of (a) mode frequencies at the Γ point, (b) mode frequencies at the X point, (c) complete band gap widths, and (d) center frequencies of band gaps. (a) Frequenc (GHz) nd gap st gap = T π/a π/a k (b) Frequenc (GHz) (d) Gap center (GHz) (b) 3 Frequenc (GHz) (c) Frequenc (GHz) nd gap st gap TABLE. Character tables of smmetr groups (after [6]) C h E C σ h i A g A u - - B g - - B u - - C E C A B - C i E i A g A u - C h E σ h A' A" - C E A Using time-domain OOMMF simulations, we have earlier established that 7, for the same MC in the absence of an eternal field, onl A (B) smmetr branches can be ecited b an A (B) smmetr magnetic field [corresponding to odd (even) field in Fig. (b)] for time-domain OOMMF simulations. However, for a transverse field below H C, the above conclusion becomes inapplicable as no point smmetr operation (besides the identit operation) eists for modes at a general point Δ in the Brillouin zone. This is illustrated b the absence of smmetr for the corresponding mode profiles. Fig. 5 indicates that, although the amplitude of m z at the Δ point has inversion smmetr, its phase, on the other hand, has no such smmetr. In the linear ecitation regime, the ecitation efficienc for a particular mode m(r, t) b an ecitation field h(r, t) is proportional to the overlap integral 8 of m(r, t) and the torque eerted b h(r, t): mm, h m * M h dv M h m * dv, (5) FIG. 4. (a) Formation of the first complete band gap as the frequenc overlap between the first (green hatched area) and second (red hatched area) gap openings. (b) Widths and (c) center frequencies of the first (green circle line) and second (red triangle line) gap openings. FIG. 5. The (a) absolute values and (b) phases of the dnamic magnetization m z for the respective modes, at the Γ, Δ and X points, of the five dispersion branches in Fig.. The mode profiles were calculated for =. T. 5-7

6 where M is regarded as effectivel uniform since the lateral size of the MC lies within the echange-dominated regime, m * the comple conjugate of m, and the integration over volume V is done within regions where the ecitation field is nonzero. Intensit (arb. unit) Τ.5 Τ. Τ.5 Τ Frequenc (GHz) FIG. 6. Magnon spectra, at the Δ point, ecited b the same even-smmetr field and calculated for various values. Groups of peaks corresponding to the five branches are labeled,, 3, 4 and 5 respectivel. For the regimes of = and > H C, onl modes with the same smmetr as that of the ecitation field have non-vanishing ecitation efficienc. In contrast, in the < < H C range, there is no simple correspondence between the smmetr of a mode and that of the ecitation field. In this last case, an ecitation field with an even distribution [middle panel of Fig. (b)] in the -z plane can ecite modes of all branches (i.e. of both even and odd smmetries). Figure 6 presents the OOMMF-simulated ecitation spectra ecited b the same even ecitation field for ranging from to.3 T. The modes of branches and 4 can onl be ecited for < < H C, with the maimum efficienc occurring at HT. T. It is to be noted that Lee et al. 9 claimed that the A + B field [corresponding to the odd + even field in Fig. (b)] in Ref. 7 is not sufficientl general to generate complete magnonic band structure. However, based on Eq. (5) and micromagnetic simulations, we have shown that the A + B field can indeed ecite all the modes. Additionall, this is true even for the single-width nanostripe considered in Ref. 9, because the demagnetizing-field-induced effective pinning of the dnamic magnetization along the ais results in a relativel small et nonzero ecitation efficienc b the A + B field. Although the relative ecitation efficienc generall decreases with increasing node number in the -direction, our simulations show that the high-order m = 3 mode can be ecited b the A + B field, which can be identified under logarithmic scale (not shown). Finall, for the OOMMF simulations, using just one laer of cells across the thickness provides sufficient accurac for the frequenc range considered. For higher frequencies, 4. Τ.5 Τ.3 Τ 5 the above discussion can be triviall etended to include perpendicular standing spin waves (PSSW). V. DISCUSSION AND CONCLUSION Our dnamic magnetic-field tunabilit of the bandgap has advantages over that based on structural dimensions or material composition. For instance, it is not feasible to reshape band structures in the latter case b tuning corresponding parameters once the MCs are fabricated. In contrast, we have demonstrated here that bandgaps can be dnamicall tuned b appling a transverse field. More importantl, b changing the direction of magnetization, some bandgaps can be reversibl switched on and off. Chumak et al. showed that a periodicall applied magnetic field opens bandgaps in uniform spin-wave waveguides termed dnamic magnonic crstals. We have shown, conversel, the possibilit to dnamicall close a bandgap b emploing a simple uniform magnetic field, which ma be utilized for nanoscale SW switches. In conclusion, the band structure of a D width-modulated nanostripe MC under a transverse magnetic field has been studied using three independent theoretical approaches, the size and center frequenc of magnonic bandgaps are found to be highl tunable b the transverse field. Furthermore, some bandgaps can be dnamicall switched on and off b simpl varing the field intensit, providing novel functionalities in magnonics. Further analsis shows that the bandgap tunabilit arises from the tunable separation between gap openings instead of the width of the gap openings. The breaking and recovering of the ground state smmetr due to the transverse magnetic fields are shown to have important implications for the mode classification and ecitation efficienc of an ecitation field. We have analzed the role of the ecitation field, which is inherent in the OOMMF simulation method. A full description of the magnonic band structure is obtained, consistent with the other two methods which we emphasize do not involve an choice of ecitation field. Also, we have shown that, contrar to the assertion of Lee et al., 9 the A + B field is able to ecite all modes of the MC. Possible applications of our transverse-field results are the ecitation of magnonic modes having an odd number of nodes across the stripe width, and dnamicall tunable SW switches and filters. ACKNOWLEDGEMENTS This project is supported b the Ministr of Education, Singapore, under Grant No. R44--8-, and the Natural Sciences and Engineering Research Council (Canada). References Z. K. Wang, V. L. Zhang, H. S. Lim, S. C. Ng, M. H. Kuok, S. Jain, and A. O. Adeee, ACS Nano 4, 643 (). 6-7

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