Supplementary Notes of spin-wave propagation in cubic anisotropy materials

Transcription

1 Supplementary Notes of spin-wave propagation in cubic anisotropy materials Koji Sekiguchi, 1, 2, Seo-Won Lee, 3, Hiroaki Sukegawa, 4 Nana Sato, 1 Se-Hyeok Oh, 5 R. D. McMichael, 6 and Kyung-Jin Lee3, 5, 7, 1 Department of Physics, Keio University, Hiyoshi , Yokohama , Japan 2 JST-PRESTO, Gobanchon 7, Chiyoda-ku, Tokyo , Japan 3 Department of Materials Science and Engineering, Korea University, Seoul 02841, Korea 4 National Institute for Materials Science (NIMS), Sengen, Tsukuba , Japan 5 Department of Nano-Semiconductor and Engineering, Korea University, Seoul 02841, Korea 6 Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA 7 KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, Korea These two authors contributed equally to this work. Electronic address: kj_lee@korea.ac.kr 1

2 A. NOTE 1: Derivations of theoretical equations for spin-wave amplitude, group velocity, and attenuation length Spin-wave dynamics is described by the Landau-Lifshitz-Gilbert equation, given as m t = µ 0 γ g m H + αm m t, (1) where γ g is the gyromagnetic ratio, m = (cosϕ sinθ,sinϕ sinθ,cosθ) is the unit vector along the magnetization, H is the effective magnetic field, µ 0 is the permeability of vacuum, and α is the Gilbert damping. From Eq. (1), the spin-wave susceptibilty χ at the frequency ω is given by γ g µ 0 M s χ = γ gµ 0 H 2 iαω iω, ωr 2 ω 2 (2) i2ω ω iω γ g µ 0 H 1 iαω where M s is the saturation magnetization, ω r (= γ g µ 0 H1 H 2 ) is the resonance frequency, ω(= γ g µ 0 α(h 1 + H 2 )/2) is the resonance linewidth due to α, and µ 0 H 1 and µ 0 H 2 are two mutually orthogonal effective fields transverse to m. The spin-wave amplitude A SW is proportional to the imaginary part of a diagonal term of χ. At resonance (ω = ω r ), A SW corresponding to Imχ 11 is given as A SW Imχ 11 = H2 H 1 M α(h 1 + H 2 ). (3) For an in-plane m (θ = π/2) and spin-wave propagation in the x direction with an assumption of uniform m across the film thickess, H 1 (=F ϕϕ /µ 0 M s + M s P k sin 2 ϕ) and H 2 (=F θθ /µ 0 M s + M s (1 P k )) are the in-plane and normal effective fields, respectively [1 4]. Here F is the free enegy density including the Zeeman and the anisotropy contributions in the long wavelength limit, the subscripts on F refer to partial derivatives around equilibrium positions, the last terms describe the magnetostatic contribution, P k = 1 (1 e k d )/( k d), k is the wavenumber, and d is the film thickness. In this case, the group velocity v g and attenuation length Λ are given by v g = ω k = γ ( gµ 0 M s P k H2 2 k Λ = v g ω = M s P k α(h 1 + H 2 ) k H1 sin 2 ϕ H 1 ( H2 sin 2 ϕ H 1 ), (4) H 2 H1 H 2 ). (5) 2

3 1.0 M / Ms 0.5 hard axis easy axis magnetic field (mt) FIG. 1: Magnetization curves along the easy and hard axes, measured by a vibrating sample magnetometer. B. NOTE 2: Experimental determination of cubic anisotropy field Figure 1 shows magnetization curves along the easy and hard axes of the epitaxial Fe film. From the magnetization curve along the hard axis, we obtain the cubic anisotropy field µ 0 H A = (66 ±2) mt. C. NOTE 3: Spin-wave logic gates based on laterally localized edge modes of cubic anisotropy materials In this note, we propose reconfigurable spin-wave logic gates, based on laterally localized edge modes in cubic anisotropy materials. Figure 2a shows a schematic illustration of a reconfigurable spin-wave logic device that allows NOT and PASS gates. The source antenna induces spin-waves propagating along the +x-axis (i.e., hard-axis) whereas the detection antenna placed at the topright corner of ferromagnetic waveguide generates a spin-wave-induced voltage. As described in the main text, the edge spin-wave modes are formed at the top or bottom edge, depending on the equilibrium magnetization direction (Fig. 2b and c). We define that the induced voltage measured in the detection antenna corresponds to 1 ( 0 ) for the spin-wave configuration of Fig. 2c (Fig. 2b), because the edge mode is (is not) present at the location of detection antenna. The equilibrium magnetization direction is determined by the Oersted field from a current flowing 3

4 through a Y-line and an INPUT-line. A current flowing through the Y-line generates a magnetic field in ±y-direction (±H y ) and that flowing through the INPUT-line generates a magnetic field along ±x-direction (±H x ). We define that Y = 1 ( 0 ) corresponds to a magnetic field H y (+H y ) whereas INPUT = 1 ( 0 ) corresponds to a magnetic field +H x ( H x ). Figure 2d shows the truth table of a NOT gate for which Y is set as 0 (corresponding to +H y ). In this condition, if INPUT = 0 (corresponding to H x ), the magnetization points to the top-left corner of the device. As a result, the induced voltage is 1, resulting in the OUTPUT of 1. In the same manner, an INPUT of 1 (corresponding to +H x ) results in the magnetization pointing the top-right corner and thus the OUTPUT is 0. Therefore, the OUTPUT is always the opposite to the INPUT, i.e., NOT gate. The PASS gate is realized by setting Y = 1 (corresponding to H y ) of the NOT gate (Fig. 2e). Here, the OUTPUT is always the same as the INPUT. Figure 3 shows a reconfigurable spin-wave logic gate consisting of three elements (ELEMENT 1, 2, and 3), which performs AND, NAND, OR, and NOR operations. Here we assume the threshold signal to determine 0 or 1 is 0.6. In this spin-wave logic device, the sum of induced voltages detected at antennas 1 and 2 produces another input signal (INPUT 3) to ELEMENT 3, and the final OUTPUT is produced by the detection antenna 3. ELEMENT 3 functions as a NOT/PASS gate depending on the setting of Y3. Based on this device, one can reconfigure the function of the gate by properly setting Y1, Y2, and Y3-lines, where the current flowing in Y-lines generates Oersted field in the ±y-direction on each ELEMENT. For instance, the device becomes an AND gate by setting Y1 = 0, Y2 = 0, and Y3 = 0 (which generates +H y field on all ELEMENTs). In the case of [INPUT 1 = 0 and INPUT 2 = 0 ], the magnetizations of the ELEMENTs 1 and 2 point the top-left corner, resulting in the spin-wave configurations shown in M(1) and M(2). Thus, the induced voltages of the detection antennas 1 and 2 are both 1. Because signals from the detection antennas 1 and 2 are added, the total signal becomes 2, which is larger than the threshold 0.6. Thus it provides INPUT 3 = 1 (corresponding to +H x ). Thus, the magnetization of ELEMENT 3 points the top-right corner and as a result, the OUTPUT is 0. In the cases of [INPUT 1 = 0 and INPUT 2 = 1 ] and [INPUT 1 = 1 and INPUT 2 = 0 ], the total voltage induced by the detection antenna 1 and 2 is 1 (> 0.6). Thus it also provides INPUT 3 = 1 (corresponding to +H x ), resulting in the OUTPUT of 0. In the case of [INPUT 1 = 1 and INPUT 2 = 1 ], the total voltage detected by the antennas 1 and 2 are 0 (< 0.6), which provides INPUT 3 = 0 (corresponding to H x ). As a result, the OUTPUT in this case is 1. From the truth table in Fig. 3b, one finds that the device operated by the aforementioned 4

5 procedure functions as an AND gate. By the same manner, the device functions as NAND, OR, and NOR gates with proper setting parameters of Y1, Y2, and Y3 (Fig. 3c-e). We also propose a reconfigurable logic of XOR/XNOR gate. The schematic illustration and the truth tables are shown in Fig. 4. In this device, the signal at the detection antenna 1 serves as an input of Y2 and the final output is detected at the detection antenna 2. In XOR gate where Y1 is set as 0 (corresponding to +H y ), the spin-wave configuration becomes M(1) shown in Fig. 4b depending on INPUT 1. INPUT 1 = 0 ( 1 ) gives a signal of 1 ( 0 ) to the Y2-line. Y2 of 1 and 0 generate H y and +H y on ELEMENT 2, respectively. With this magnetic field in y- direction, INPUT 2 allows spin-wave configurations shown as M(2) and corresponding OUTPUT (in red). By changing the setting of Y1 from 0 to 1, this gate also performs as an XNOR gate as shown in Fig. 4c. Here, we have demonstrated a spin-wave logic using laterally localized edge modes of cubic anisotropy materials. The reconfigurability and nonvolatility of this device give rise to a large advantage from the applications point of view, allowing eight-types of logic gates including NOT/PASS, AND/NAND/OR/NOR, and XOR/XNOR gates by using only three-types of devices. [1] Damon, R. W. and Eshbach, J. R. Magnetostatic modes of a ferromagnet slab. J. Phys. Chem. Solids 19, 308 (1961). [2] Patton, C. E. Spin-wave instability theory in cubic single crystal magnetic insulators. Phys. Stat. Sol. (b) 92, 211 (1979). [3] Kalinikos, B. A. and Slavin, A. N. Theory of dipole-exchange spin wave spectrum for ferromagnetic films with mixed exchange boundary conditions. J. Phys. C: Solid State Phys. 19, 7013 (1986). [4] McMichael, R. D. and Krivosik, P. Classical model of extrinsic ferromagnetic resonance linewidth in ultrathin films. IEEE Trans. Magn. 40, 2 (2004). 5

6 a d e b m c m FIG. 2: a, Schematic illustration of reconfigurable NOT/PASS gate. b, c, Spin-wave configuration depending on the equilibrium agnetization direciton. Black bars at the top-right corner describes the location of the detection antenna. d, Truth tables of NOT gate and e, NOT gate. 6

7 a b d c AND gate (Set Y1=0, Y2=0, Y3=0) e OR gate (Set Y1=1, Y2=1, Y3=1) NAND gate (Set Y1=0, Y2=0, Y3=1) NOR gate (Set Y1=1, Y2=1, Y3=0) FIG. 3: a, Schematic illustration of reconfigurable AND/NAND/OR/NOR gate. b, Truth tables of AND gate, c, NAND gate, d, OR gate, and e, NOR gate. 7

8 a b XOR gate (Set Y1=0) c XNOR gate (Set Y1=1) FIG. 4: a, Schematic illustration of reconfigurable XOR/XNOR gate. Truth tables of b, XOR gate and c, XNOR gate. 8