Bandgap and dispersion relation are two important properties of photonic crystals (PhCs) to control light and electromagnetic waves. PhCs are believed to play a key role of future core components of novel electro optical applications. The potential applications extend from simple waveguides or splitters over multiple wavelength demultiplexers and wavelength filters to advanced applications such as single photon sources or laser resonators. Future sensor and data processing industry may profit out of PhCs as well as the telecommunication sector. PhCs with dimensions in the low sub-100-nm region 1 2 , and even in the IR range 3 , have already been successfully fabricated using high-resolution electron beam lithography and nanoimprint lithography. Magnonic crystals (MCs) 4 5 6 7 8 9 10 11 12 , as the magnetic counterpart of PhCs, also exhibit the bandgap and dispersion relation properties that can be exploited to control and manipulate spin waves (SWs) and also to produce effects that are not possible with isolated magnetic nanostructures. The bandgap property can forbid propagation of a certain frequency range of SWs into MCs. Since wavelengths of SWs span several orders of magnitude from tens of microns to below 1 nm, their frequencies may vary from gigahertz to terahertz. Additionally, the frequency for a given wavelength can be shifted by the magnetic field. This broad region in length and time scales is one reason that makes SWs so interesting for high-frequency applications.

The SWs in MCs can be classified according to the magnitude of their wavelength as dipolar- or exchange-dominated SWs 9 . This is essentially because dipole-dipole interaction is long-ranged, while exchange interaction is short-ranged. For wavelengths longer than 1 μm 13 , the dispersion of the SWs is dominated by dipolar interactions. For the in-plane magnetized thin-film magnetic nanostructures, the dipolar-dominated SWs can be classified into magnetostatic surface spin wave (MSSW) and backward volume magnetostatic spin wave (BVMSW) modes depending on their propagation direction with respect to the magnetization 14 . The MSSW modes, whose frequencies lie above the spatially uniform precession (Kittel) mode, are characterized by perpendicular wave propagation relative to the applied in-plane magnetic field. In contrast, the BVMSW modes, with a precession frequency smaller than that of the Kittel mode, are characterized by wave propagation parallel to the applied field. Since BVMSW modes travel ‘backward’ in phase, this leads to a negative dispersion. Therefore, if dipolar interactions dominate, the band structure will be anisotropic with regard to the applied field direction: the positive dispersion for MSSW mode and the negative dispersion for BVMSW mode. For wavelengths below 1 μm 13 , the exchange interaction becomes important so that this contribution has to be taken into account for mixed dipole and exchange spin waves in an intermediate region of length scales. For wavelengths below 100 nm 13 , the magnetostatic contribution to the energy of SWs will be dominated by the exchange interaction. Hence, the dispersion is dominated by the exchange interaction. As a consequence of neglecting the anisotropic dipolar contribution, the dispersion in the exchange limit is positive for the two relative orientations between wave propagation and magnetization direction.

Recently, the dispersion relations of SWs in one-dimensional (1D) MCs, with lattice constants in the order of several hundreds of nanometers or several micrometers, have been investigated. These MCs consist of dipolarly coupled nanostripes 15 16 17 and of contacting alternating stripes of two different materials 10 11 18 19 , and the magnonic band structures of SWs are dominated by dipolar interactions. Among these MCs of lattice constant larger than 200 nm, the largest widths of transmission band and forbidden band (or bandgap) are 2.5 and 2.1 GHz, respectively. Larger values are expected for MCs of lattice constant smaller than 100 nm, in which the magnonic band structures of SWs are dominated by exchange interactions. However, scarce attention has been paid to the magnonic band structures of exchange SWs in MCs with lattice constant in the order of tens of nanometers 9 . Although the fabrication of MCs with nanoscale lattice constants and the detection of exchange SWs are still challenging, the high frequency and short wavelength gave exchange SW an advantage over the dipolar-dominated SW. In this work, we present the results of a micromagnetic study of magnonic band structures for exchange SWs propagating in 1D bi-component magnonic crystal waveguides (MCWs). The waveguides are periodic arrays of alternating stripes of different ferromagnetic materials. The properties of the bi-component bandgaps are studied as a function of the constituent components, the stripe width to lattice constant ratio, and also the applied field orientation.

We consider a MCW with a length 1,000 nm (x direction) in the form of laterally patterned periodic arrays of alternating stripes of different ferromagnetic materials (Figure 1). Each stripe in the MCW has a length of 140 nm (y direction) and a thickness of 10 nm (z direction). We investigated the magnetization dynamics of bi-component MCWs each composed of two of the following ferromagnetic metals, namely Co, Fe, Permalloy (Py), and Ni, in a total of six possible combinations. For brevity, the MCWs with respective stripe widths of M and N nm of different ferromagnetic materials are referred to as MCo/NNi, MCo/NPy, MCo/NFe, MFe/NNi, MFe/NPy, and MPy/NNi. Two kinds of SW modes, which depend on the orientation of the applied magnetic field H, in transversely and longitudinally magnetized waveguides were investigated as shown in Figure 1a,b, respectively. The magnonic band structures of exchange SWs with wavelengths down to several nanometers and frequencies up to hundreds of gigahertz are numerically investigated.

The Object Oriented Micromagnetic Framework program 19 was used to numerically calculate the dynamics of the magnetizations by solving the Landau-Lifshitz-Gilbert equation 20 21 . The simulation cell size used is 2 × 2 × 10 nm³, the damping constant α = 0.01, and the gyromagnetic ratio γ = 2.21 × 10⁵ m/As. The magnetic parameters of the four ferromagnetic metals (Co, Fe, Py, and Ni) used in the simulations are specified in Table 1. A static in-plane magnetic field was applied along the y and x directions (see Figure 1) corresponding to the transverse and longitudinal geometries, respectively. In order to excite SWs, a ‘sinc’ function 22 with H ₀ = 1.0 T and field frequency f _H  = 250 GHz was applied locally to a volume element ΔxΔyΔz (=10 × 140 × 10 nm³) in the middle of the MCWs (x = 500 nm). SWs, with frequencies ranging from 0 to 250 GHz, were thus excited and propagated along the x direction. The dispersion curves of SWs propagating along the length (x direction) of the MCWs were calculated using the following procedures 12 . First, the time dependence of the magnetization of each cell is recorded. The dispersion curve for the propagation of spin waves is then obtained by performing a 2D Fourier transform of the out-of-plane component of the magnetization in space and time.

M _s, saturation magnetization; A, exchange constant; l _ex, exchange length 6 .

The magnonic band structures of exchange SWs in 1D bi-component magnonic crystal waveguides were investigated using the micromagnetic methods. Two kinds of SWs were studied according to the relative orientation between the applied field and the waveguides: the transverse and the longitudinal cases. From the calculated dispersion curves of SWs, we found that the widths and center frequencies of the bandgaps are controllable by the component materials, the stripe width to lattice constant ratio as well as the orientation between the applied field and the waveguide. A striking feature of the dispersion curve is that there are n-1 zero-gaps for the nth bandgap for both the transverse and longitudinal cases. The largest bandgap widths were observed in the MCo/NNi MCWs, which have the largest exchange constant ratio. By comparing the band structures of exchange SWs in both the transverse and the longitudinal cases, we found that for the same MCW, the widths of the bandgaps in the longitudinal case are wider than those in the transverse case except for the MCo/NFe MCWs. The investigation of MCs with nanoscale periods, in which SW frequencies can reach values up to the terahertz range and with wavelengths of just a few nanometers, opens the way to practical applications of the dynamic properties of such MCs in much faster devices of nanometer size.

The authors declare that they have no competing interests.

FSM carried out the calculations and analyses. HSL designed the project and participated in the analyses. VLZ, SCN, and MHK helped in the interpretation of the results and drafting of the manuscript. All authors read and approved the final manuscript.