Magnonic band structure investigation of one-dimensional bi-component magnonic crystal waveguides
© Ma et al.; licensee Springer. 2012
Received: 13 July 2012
Accepted: 27 August 2012
Published: 4 September 2012
The magnonic band structures for exchange spin waves propagating in one-dimensional magnonic crystal waveguides of different material combinations are investigated using micromagnetic simulations. The waveguides are periodic arrays of alternating nanostripes of different ferromagnetic materials. Our results show that the widths and center frequencies of the bandgaps are controllable by the component materials, the stripe widths, and the orientation of the applied magnetic field. One salient feature of the bandgap frequency plot against stripe width is that there are n-1 zero-width gaps for the n th bandgap for both transversely and longitudinally magnetized waveguides. Additionally, the largest bandgap widths are primarily dependent on the exchange constant contrast between the component materials of the nanostructured waveguides.
KeywordsMagnonic crystal Magnonics Spin wave Bandgap Micromagnetic simulations 75.40.Gb 75.30.Ds 75.78.Cd 75.78.-n
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 , have already been successfully fabricated using high-resolution electron beam lithography and nanoimprint lithography. Magnonic crystals (MCs) [4–12], 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 . This is essentially because dipole-dipole interaction is long-ranged, while exchange interaction is short-ranged. For wavelengths longer than 1 μm , 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 . 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 , 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 , 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 [15–17] 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 . 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.
Magnetic parameters of ferromagnetic metals Co, Fe, Py, and Ni
M s (106 A/m)
Results and discussion
Transversely magnetized MCWs
Longitudinally magnetized MCWs
Unlike the dispersion curves of transversely magnetized waveguides, a negative dispersion is observed near the BZ center (q = 0) where SWs with small wave vectors are dominated by long-range dipolar interaction. The dispersion curves of dipole-dominated SWs are positive or negative depending on whether the orientation of wave propagation is transverse or parallel to the external field. In contrast, the dispersion curves of SWs with a large wave vector whose properties are dominated by short-range exchange interaction are positive and independent of the orientation of the external field. Therefore, the SW modes near the first BZ boundary are exchange spin waves. It is interesting to note that the widths of the first three bandgaps in the longitudinally magnetized 16Co/4Fe MCW (see Figure 4f) are narrower than the corresponding ones of the transversely magnetized case (cf. Figure 2f). However, the bandgaps of the other five MCWs (see Figure 4a,b,c,d,e) are wider than those of the corresponding transversely magnetized ones (see Figure 2a,b,c,d,e). For instance, the width of the first bandgap for longitudinally (transversely) magnetized 16Co/4Fe MCW is 2.5 GHz (7.0 GHz), while the width of the first bandgap for longitudinally (transversely) magnetized 16Fe/4Ni MCW is 10.5 GHz (6.0 GHz).
It can be seen from Figure 6b that the largest first, second, and third bandgaps among the MCWs studied are all observed in the M Co/N Ni MCWs, which have the largest exchange constant ratio. The smallest second and third bandgaps are observed in the M Co/N Fe MCWs (7.5 and 4.0 GHz), which has the smallest exchange constant ratio. Interestingly, however, the smallest first bandgap is observed in the M Fe/N Py MCWs (2.5 GHz).
For the longitudinally magnetized MCWs (Figure 6c), the maximum widths of the first three bandgaps monotonically decrease with decreasing exchange constant ratio. Unlike the transversely magnetized waveguides, the smallest first, second, and third bandgaps are all observed in the M Co/N Fe MCWs (3.0, 2.0, and 0.0 GHz). In general, the larger the exchange constant contrast between the material components of a nanoscale bi-component MCW, the wider would be its bandgap.
A closer inspection of Figure 6b,c reveals two unusual behaviors in M Co/N Fe MCW in comparison with the other five MCWs. Firstly, its higher-order bandgaps has a narrower width than those of the lower-order bandgaps, in sharp contrast with the other five MCWs. Secondly, the bandgaps of transversely magnetized M Co/N Fe MCWs are wider than the corresponding ones in the longitudinally magnetized case, and the other five MCWs, on the other hand, exhibit a completely opposite behavior.
The contrasting behaviors between the M Co/N Fe MCWs and the other five types of MCWs may be attributed to the magnetic parameter contrasts between the component materials of the MCWs. From Figure 6a, it can be seen that the M Co/N Fe MCWs have an exchange length contrast larger than 1, but its magnetization contrast is smaller than 1. The other five types of MCWs, on the other hand, have static magnetization contrast larger than 1, but the exchange length contrast smaller than one.
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 n th bandgap for both the transverse and longitudinal cases. The largest bandgap widths were observed in the M Co/N Ni 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 M Co/N Fe 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.
This project was supported by the Ministry of Education, Singapore under grant no. R144-000-282-112.
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