Spin-wave propagation in cubic anisotropic materials

The information carrier of modern technologies is the electron charge whose transport inevitably generates Joule heating. Spin-waves, the collective precessional motion of electron spins, do not involve moving charges and thus avoid Joule heating. In this respect, magnonic devices in which the information is carried by spin-waves attract interest for low-power computing. However implementation of magnonic devices for practical use suffers from low spin-wave signal and on/off ratio. Here we demonstrate that cubic anisotropic materials can enhance spin-wave signals by improving spin-wave amplitude as well as group velocity and attenuation length. Furthermore, cubic anisotropic material shows an enhanced on/off ratio through a laterally localized edge mode, which closely mimics the gate-controlled conducting channel in traditional field-effect transistors. These attractive features of cubic anisotropic materials will invigorate magnonics research towards wave-based functional devices.


INTRODUCTION
Magnonics is a research field that aims to control and manipulate spin-waves in magnetic materials for information processing [1][2][3]. Spin-waves enable Boolean and Non-Boolean computing with low-power consumption [4][5][6][7][8][9][10]. Their wave properties also allow distinct functionalities [11][12][13][14] such as multi-input/output (non-linear) operations [15,16]. Despite significant progress, however, low signal and on/off ratio of spin-waves have been major obstacles to implementation of magnonic devices for practical use. The on/off ratio of spin wave can be related to the asymmetric ratio of spin wave amplitude. These obstacles are caused by poor excitation efficiency and propagation losses of spin-waves. In this work, we show that cubic anisotropic materials offer a more efficient spin-wave propagation than conventional materials.
Our strategy to improve the spin-wave signal is to modify the dispersion relation. Compared to other waves in solid states, the spin-wave dispersion is highly anisotropic, caused by the long-range magnetostatic interaction. An established way to control the anisotropy of dispersion is to change the relative orientation between the equilibrium magnetization direction m and the direction of wave vector k [i.e., backward volume (m//k), surface (m⊥k for in-plane m), and forward volume modes (m⊥k for out-of-plane m)]. As the dispersion determines all spin-wave properties, a proper modification of the dispersion may allow us to improve spin-wave properties. We note that for improved functionalities of spin-wave devices, not only the spin-wave amplitude but also the spinwave attenuation length and group velocity should be improved.
Here we introduce an epitaxial Fe film as a waveguide for this purpose. The cubic crystalline anisotropy of epitaxial Fe film provides an additional knob to modify the dispersion by changing the relative orientation between the magnetic easy (or hard) axis and the direction of wave vector k.
As we show below, a proper tuning of this relative orientation allows an enhancement of the spinwave amplitude by a factor of 28, as well as enhancements of the spin-wave attenuation length and group velocity by several factors. We also show that the cubic anisotropy provides a giant lateral spin-wave asymmetry due to a laterally localized edge mode at zero external field, which may enable three-terminal nonvolatile spin-wave logic gates.

MATERIALS AND THEORETICAL FRAMEWORK
The spin-wave dispersion in cubic anisotropic materials was first derived by Kalinikos et al. [17].
Instead of starting with this known dispersion directly, here we describe the problem in a rather general way in order to get an insight of how to improve spin-wave properties. We first describe two key requirements in the dispersion relation for improved spin-wave properties. From the spinwave theory for an in-plane magnetization m with an assumption that m varies only along the spin-wave propagation direction [18][19][20][21], the spin-wave amplitude A SW , the group velocity v g , and the attenuation length Λ are respectively given as (see Supplementary Note 1 for details), where are the in-plane and normal effective fields, respectively and F is the free magnetic energy density without magnetostatic interactions, which are treated separately. The subscripts (i.e., θ and φ are the polar and azimuthal angles of m, respectively) on F refer to partial derivatives around equilibrium positions, /|k|d, k is the wavenumber, d is the film thickness, M s is the saturation magnetization, α G is the Gilbert damping, and γ g is the gyromagnetic ratio. In Eq. (1), we refer to the dominant in-plane component only because the normal component is negligible due to strong demagnetization of thin film geometry.
Focusing on the limit of kd ≪ 1 with which P k is small, one finds from Eqs. (1-3) that A SW , v g , and Λ are simultaneously maximized when H 1 ≈ 0 (equivalently F φ φ ≈ 0) and φ ≈ ±π/2 (i.e., surface mode configuration). Therefore, two key requirements for the improved spin-wave properties are vanishingly small in-plane effective field and surface mode. For kd ≪ 1, F consists of the Zeeman and anisotropy energies. The contribution from the Zeeman energy to F φ φ is simply an external in-plane field H(> 0) that should be applied to ensure the surface mode. The central question is thus how to obtain a negative in-plane effective field from the anisotropy energy in order to diminish F φ φ in the surface mode.
We next show that spin-wave propagation where the wave vector k is along the hard-axis direction of cubic anisotropic materials can naturally satisfy these two requirements. We consider two cases, easy-easy (Fig. 1a) and hard-hard ( Fig. 1b) cases, in which the first (second) word corresponds to the direction of m (wave vector k). A top view of waveguide for each case is shown in Fig. 1a and 1b where the crystallographic orientations are defined for an epitaxial Fe layer with a cubic crystalline anisotropy [24]. The long axis of the waveguide (i.e., direction of wave vector k) for the easy-easy device is along the easy axis (Fe [100] direction, Fig. 1a) whereas that for the hard-hard device is along the hard axis (Fe [110] direction, Fig. 1b). Our main focus is the hard-hard case, whereas the easy-easy case corresponds to conventional spin-wave propagation and will be used as a reference. In both cases, the external field H is applied in the y-direction and wave vector (k) is in the x-direction. The cubic anisotropy energy density E an is given as where K c is the cubic anisotropy and α(= φ − φ K ), β (= π/2 − φ + φ K ), and γ(= θ = π/2) are direction cosines of the magnetization with respect to the easy axes of cubic anisotropy. Here φ K is the angle between the easy axis and x-axis, and is 0 (π/4) for the easy-easy (hard-hard) case.
On the other hand, the Zeeman energy is −µ 0 M s H sin φ sin θ . With these energy terms, H 1 for the hard-hard and easy-easy cases are readily calculated as where for 0 ≤ h < 1.
For the hard-hard case, therefore, the cubic anisotropy provides a negative effective field and H hard 1 becomes small when H ≈ H A and φ eq = π/2 because M s P k is small. Figure 1c shows that this is indeed the case. For the hard-hard case, the frequency f (= ω/2π) minimizes at H ≈ H A (Fig. 1d) and all of A SW , v g , and Λ are largely enhanced at H ≈ H A , in comparison for the easy-easy case ( Fig. 1e and 1f). We note that the improvements of spin-wave properties discussed in this section are consequences of the known dispersion [17], but there has been no direct experimental proof for spin-wave waveguides made of cubic anisotropic materials.

RESULTS AND DISCUSSION
In order to confirm the theoretical prediction, we measure spin-wave-induced voltages in the time domain for microfabricated devices containing two antennas [i.e., spin-wave excitation and detection antennas (Fig. 2a), see Methods], based on the propagating spin-wave spectroscopy [22,23]. The layer structure of the waveguide is Cr(40)/Fe (25)  It clearly shows that the spin-wave amplitude is much larger for the hard-hard case than for the easy-easy case. Magnetic field H-dependences of A SW , v g , and Λ are summarized in Fig. 3. All the experimental results show enhanced spin-wave properties at H ≈ H A as predicted by the theory.
At H ≈ H A and the antenna distance of 5 µm, the induced voltage of spin-wave packet reaches 3.30 mV for the hard-hard case (Fig. 3a), whereas it is about 0.12 mV for the easy-easy case (Fig.   3c). Therefore, the spin-wave amplitude A SW for the hard-hard case enhances by a factor of 28 in comparison for the easy-easy case. By analyzing the antenna-distance dependence of amplitudes and arrival-times of spin-wave packets [23], we deduce the spin-wave attenuation length Λ and the spin-wave group velocity v g . At the enhancement condition (µ 0 H = 66 mT), Λ for the hard-hard case is about (17.8 ± 0.5) µm whereas Λ for the easy-easy case remains (10.0 ± 4.5) µm (Fig. 3f).
At the enhancement condition, furthermore, v g for the hard-hard case is about (23.4 ± 0.7) km/s whereas v g for the easy-easy case is about (8.9 ± 0.3) km/s (Fig. 3g). We note that uncertainties expressed in this paper are one standard deviation of fit parameters.
Therefore, all these results confirm that spin-wave properties are largely improved at the enhancement condition, qualitatively consistent with the theoretical predictions. For the hard-hard case, we find however that there are interesting quantitative differences between the experimental and theoretical ones. For instance, the one-dimensional theory (Fig. 1) predicts that the ratio of A SW at H = H A to A SW at H = 0 is about 3.5 with considering spin-wave attenuation for the distance of 5 µm. This predicted ratio is much smaller than the experimental one (≈ 9; A H=0 SW ≈ 0.36 mV and A H=H A SW ≈ 3.30 mV, see Fig. 3a).
In order to understand this discrepancy, we perform two-dimensional micromagnetic simulations for the hard-hard case (Fig. 4). When m is aligned along one of easy-axes (i.e., H = 0 and φ = π/4; Fig. 4a), the spin-wave having −k (+k) is localized at the top-left (bottom-right) edge.
The exact opposite trend is obtained when m is aligned along another easy-axis (i.e, H = 0 and φ = 3π/4; Fig. 4d). On the other hand, this laterally localized spin-wave edge mode is absent when H = H A (Fig. 4b). This edge mode originates from the anisotropic dispersion relation de-picted in Fig. 4c. It shows a contour plot of the spin wave dispersion relation of hard-hard case at H = 0. The dispersion relation at the excitation frequency (10.5 GHz) is highlighted. One finds that the wave fronts propagate in the direction of k determined primarily by the antenna geometry whereas the group velocity, v g = dω/dk, is perpendicular to the contour, the oblique direction.
This anisotropic propagation leads to the edge localization of spin-waves, which in turn makes an additional difference of A SW between H = H A and H = 0 because our experimental set-up detects an induced voltage integrated over the full width of waveguide.
This localized edge mode at zero field allows us to significantly improve spin-wave asymmetry using cubic anisotropic materials. It can be compared to the conventional spin-wave nonreciprocity in the aspect of "asymmetric propagation of spin-wave". Conventional spin-wave nonreciprocity refers to an asymmetric spin-wave amplitude depending on the spin-wave propagation direction when spin-waves are excited by a magnetic field generated by microwave antennas [25][26][27][28]. This amplitude asymmetry results from a nonreciprocal antenna-spin-wave coupling, caused by the spatially nonuniform distribution of the antenna field. The asymmetry factor (≡ A k>0 SW /A k<0 SW ) for the conventional spin-wave nonreciprocity is about 2 [27,28]. In contrast, the lateral spin-wave asymmetry in cubic anisotropic materials can be very large when the detection antenna is properly designed. Figure 4e shows that the spin-wave profile with respect to the location of the excitation antenna highly is symmetric at the top and bottom edges. By placing a detection antenna at one of the edges (see Fig. 4d), therefore, one can obtain a very large difference in the spin-wave amplitude between the cases for φ = π/4 and for φ = 3π/4 (the asymmetry factor ≈ 40 for Fig.   4f).
In order to confirm the lateral spin-wave asymmetry, we experimentally measure the spatial distribution of the magnon density by micro-focused Brillouin Light Scattering (BLS) spectroscopy [29] (see Methods) for the hard-hard device. We inject an RF current at 10.4 GHz in the excitation antenna and measure the BLS spectra at H = 0 as a function of the distance x from the signal line (Fig. 5a). The BLS spectra at an edge (i.e., bottom edge, y = 115 µm) show a clear dependence on x (Fig. 5b). The BLS intensity shows a highly asymmetric distribution with respect to the signal line (Fig. 5c-f), in agreement with modeling results. We note (BLS intensity for x < 0)/(BLS intensity for x > 0) ≫ 1 in Fig. 5c, corresponding to giant lateral spin-wave asymmetry. The change in the asymmetry depending on the equilibrium magnetization direction is also consistent with modeling results. These results thus confirm the formation of laterally localized edge mode in cubic anisotropic materials, which is tunable by changing the equilibrium magnetization direction.
It is worthwhile comparing the laterally localized edge mode of cubic-anisotropy-based spinwave devices with gate-voltage-controlled conducting channel of traditional three-terminal fieldeffect transistors (FETs). A similarity is that in the spin-wave devices, the edge channel is controlled by the equilibrium magnetization direction (which can be set by an external field), whereas in FETs, the conducting channel is controlled by a gate voltage. Therefore, the equilibrium magnetization direction of cubic-anisotropy-based spin-wave devices serves as a gate voltage of FETs.
This similarity makes it possible to mimic the functionalities of three-terminal FETs with threeterminal cubic-anisotropy-based spin-wave devices, by controlling the equilibrium magnetization direction and measuring the induced voltage with an antenna placed at an edge (see Fig. 4d).
There is also an important difference. Traditional FETs are volatile because the conducting channel is closed when the gate voltage is turned off. In contrast, the cubic-anisotropy-based spinwave devices are nonvolatile because the spin-wave edge channel is maintained by the easy-axis anisotropy, even after removing an external field. In this respect, the cubic-anisotropy-based spinwave devices can serve as three-terminal nonvolatile logic gates. In supplementary note 3, we propose proof-of-principle spin-wave logic gates performing Boolean functions of NOT, PASS, AND, NAND, OR, NOR, XOR, and XNOR. We note however that the proposed gates have inputs (spin-waves with a magnetic field) and outputs (only spin-waves) that are different so that they require additional spin-wave-to-current converters, which are definitely detrimental for practical use.
One may combine spin-wave logic with spin-transfer torque [30] or electric-field magnetization switching technique [31] to remove or at least simplify this additional part. CONCLUSION We have demonstrated that the cubic anisotropic material is a promising candidate for coherent magnonic devices by virtue of largely enhanced spin-wave properties and laterally localized edge modes. The enhanced spin-wave properties will greatly improve the signal-to-noise ratio of magnonic devices. Up to now, conventional spin-wave nonreciprocity has been employed to generate a π phase-shifted wave [27] and different spin-wave amplitude [26][27][28], providing plenty of magnonic functionalities. The lateral asymmetry reported here will add a new functionality, three-terminal nonvolatile spin-wave logic function.