All-magnetic control of skyrmions in nanowire by spin wave

All-magnetic control of skyrmions in nanowire by spin wave Xichao Zhang 1, Motohiko Ezawa 2*, Dun Xiao 3, G. P. Zhao 4, 5, Y. W. Liu 3, Yan Zhou 1 1. Department of Physics, The University of Hong Kong, Hong Kong, China 2. Department of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, Japan 3. Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physical Science and Engineering, Tongji University, Shanghai 200092, China 4. College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China 5. Key Laboratory of Magnetic Materials and Devices, Ningbo Institute of Material Technology & Engineering, Chinese Academy of Sciences, Ningbo 315201, China * E-mail: ezawa@ap.t.u-tokyo.ac.jp E-mail: yanzhou@hku.hk SUPPLEMENTARY INFORMATION Supplementary Figures Supplementary Figure 1: The propagation of a skyrmion driven by the spin wave in the 40-nm-wide nanotrack. The green shaded region in the track corresponds to the region of the spin wave injection. (a) The magnetic field pulse is perpendicular to the track. (b) The magnetic field is parallel to the track. 1

Supplementary Figure 2: The velocity of a skyrmion as functions of time in 800-nm-long nanotracks. The red solid curve denotes the function of the velocity versus time with magnetic pulse perpendicular to the track. The blue dashed curve denotes the function of the velocity versus time with magnetic pulse parallel to the track. 2

Supplementary Figure 3: The propagation of a skyrmion driven by spin waves in the 40-nm-wide nanotrack with the damping coefficient of 0.01 (i.e., the case as shown in Figure 2a of the main text). (a) Top views of the x-component magnetization configuration at selected times. (b) Top views of the y-component magnetization configuration at selected times. (c) Top views of the z-component magnetization configuration at selected times. (d) Closeup of the cross-sectional views of the x-component magnetization configuration at selected times. (e) Closeup of the cross-sectional views of the y-component magnetization configuration at selected times. (f) Closeup of the cross-sectional views of the z-component magnetization configuration at selected times. The color scales present the in-plane and the out-of-plane component of the magnetization. The vertical dash lines in (a-c) denote the closeup region, and the horizontal dash line in (a) denotes the cross-section plane. 3

Supplementary Figure 4: The propagation of a skyrmion driven by spin waves in the Y-junction corresponding to the case shown in Figure 6d of the main text. Upper panel: Top views of the z-component magnetization configuration at selected times. Lower panel: Closeup of the z-component magnetization configuration at selected times. The color scale presents the out-of-plane component of the magnetization. The black line boxes denote the zoomed regions. Supplementary Figure 5: The propagation of a skyrmion driven by strong spin waves in the 40-nm-wide nanotrack with relatively large damping coefficient. The damping coefficient is set to 0.1. The patterned green boxes on the track corresponds to the region of the spin wave injection (135 nm < x < 150 nm), where the amplitude of the spin wave injection field pulse is 1200 mt and the frequency is 100 GHz. The other parameters are the same as that used in Figure 2a of the main text. The color scale presents the out-of-plane component of the magnetization. 4

Supplementary Figure 6: The propagation of a skyrmion driven by the spin wave in the 40-nm-wide nanotrack under an external global oscillating magnetic field, Boc. The patterned green boxes on the track corresponds to the region of the spin wave injection (135 nm < x < 150 nm), of which the profile of the magnetic pulse is shown in Figure 3 of the main text, i.e., the amplitude equals 600 mt and the frequency equals 25 GHz. The spin wave injection scheme and parameters are the same as that used in Figure 2a of the main text. (a) Snapshots of the propagation of the skyrmion driven by spin waves under no Boc. (b) Snapshot of the propagation of the skyrmion driven by spin waves under Boc applied parallel to the nanotrack, i.e., along the x-direction. (c) Snapshot of the propagation of the skyrmion driven by spin waves under Boc applied transverse to the nanotrack, i.e., along the y-direction. (d) Snapshot of the propagation of the skyrmion driven by spin waves under Boc applied perpendicular to the nanotrack, i.e., along the z-direction. (e) Snapshot of the skyrmion under Boc applied parallel to the nanotrack without spin wave injection, i.e., the pulse element is turned off. (f) Snapshot of the skyrmion under Boc applied transverse to the nanotrack without spin wave injection, i.e., the pulse element is turned off. (g) Snapshot of the skyrmion under Boc applied perpendicular to the nanotrack without spin wave injection, i.e., the pulse element is turned off. The frequency and amplitude of the external global oscillating magnetic field, Boc, equal 25 GHz and 50 mt, respectively. The color scale present the out-of-plane component of the magnetization. 5

Supplementary Figure 7: The propagation of a skyrmion in the nanotrack driven by spin waves injected via the oscillating Oersted field. The oscillating Oersted field is induced by the ac current (6.87 ma, 50 GHz) inside the nanowire upon the nanotrack at x = 142. 5 nm. The amplitude profile of the oscillating Oersted field distribution in the nanotrack is shown in (a), which mainly focuses in the region 130 nm < x < 155 nm and ranges from 0 mt to 550 mt. (b) Snapshot of the skyrmion at t = 0 ns. (c) Snapshot of the propagation of the skyrmion driven by spin waves at t = 5 ns. (d) Snapshot of the propagation of the skyrmion driven by spin waves at t = 9 ns. (e-h) show the control case, in which the oscillating Oersted field is confined in the region of 130 nm < x < 155 nm. The color scales in (b-d) present the out-of-plane component of the magnetization, while the color scale in (a) present the amplitude of the Oersted field distribution. 6

Supplementary Figure 8: Comparison between the motion of skyrmions driven by spin wave excited by square magnetic pulse and sinusoidal magnetic pulse. (a-d) show the snapshots for the cases with dashed line box at t = 0 ns and t = ~20 ns. Supplementary Notes Supplementary Note 1: Perpendicular pulse versus parallel pulse We have carried out a series of simulations with magnetic pulse parallel to the nanotrack, in contrast to the case with magnetic field pulse perpendicular to the nanotrack with the damping coefficient of 0.01 in the main text (see Figure 1 in the main text). As shown in Supplementary Figure 1, with the same amplitude (600 mt), frequency (25 GHz) and antenna contact area (135 nm < x < 150 nm), the perpendicular magnetic field pulse are more efficient to excite spin wave to drive the skyrmion on the 7

nanotrack. At t = 9 ns, the skyrmion driven by spin wave excited by perpendicular magnetic pulse has moved 457 nm, while the one driven by spin wave excited by parallel magnetic field pulse has only moved 293 nm, 64% of former case. The Supplementary Figure 2 shows the velocity function of the skyrmion driven by spin wave excited by parallel magnetic field pulse in comparison to that driven by spin wave excited by perpendicular magnetic field pulse. Obviously, with the same profile and parameter, the spin wave excited by parallel magnetic pulse can only drive the skyrmion to reach a maximum speed of ~ 40 m/s, which is ~ 40% less than the maximum speed (~ 70 m/s) of skyrmion driven by spin wave excited by perpendicular magnetic pulse. However, it should be noted that the spin wave excited by parallel magnetic pulse is also able to realize all the turns and control of skyrmion on L-, T- and Y-junctions, which is in good agreement with the corresponding cases in the main text (also see Supplementary Movies 17 25). Supplementary Note 2: Top views and cross-sectional views of the skyrmion motion driven by spin waves Supplementary Figure 3 shows the detailed top views and cross-sectional views of the skyrmion motion driven by spin waves in the 40-nm-wide nanotrack with damping coefficient of 0.01, which corresponds to the case in Figure 2a of the main text. Supplementary Figure 3a, 3b and 3c show the top views of the x-component, y-component and z-component of the magnetization at selected times, respectively. It can be seen that spin waves excited by the magnetic field pulse at the left side of the nanotrack mainly propagate along the length-direction of the nanotrack, i.e., the x-direction, which have a large in-plane component near the magnetic field pulse source. The spin waves interact with the skyrmion, which has a circled in-plane magnetization distribution, driving the skyrmion to move toward the right side. Supplementary Figure 3d, 3e and 3f show the zoomed cross-sectional views of the x-component, y-component and z-component of the magnetization at selected times, respectively, along the control line of the nanotrack (denoted by the dash line in Supplementary Figure 3a). The arrows present the magnetization. Obviously, in the 8

process of the skyrmion moving toward the right, the magnetization around the skyrmion are precessing about the z-axis slightly. The magnetization proceeding the moving skyrmion will first complete a 180-degree rotation from spin-up to spin-down state, and then rotate from spin-down to spin up, developing the motion of the skyrmion. Supplementary Figure 4 shows the propagation of a skyrmion driven by spin waves in the Y-junction corresponding to the case shown in Figure 6d of the main text. It can be seen that the skyrmion presses on the track border at the left side of its motion direction, resulting in the left turn at the junction. Supplementary Note 3: The propagation of a skyrmion driven by high frequency spin waves in nanotrack with relatively large damping coefficient Supplementary Figure 5 shows the propagation of a skyrmion driven by strong spin waves in the 40-nm-wide nanotrack with relatively large damping coefficient. The damping coefficient is set to 0.1 here. The patterned green boxes on the track corresponds to the region of the spin wave injection (135 nm < x < 150 nm), of which the amplitude of the spin wave injection field pulse is increased to 1200 mt from the value used in the main text, and the frequency is also increased to 100 GHz. At t = 5 ns, the skyrmion moves 43 nm along the nanotrack with an average speed of 9 m s -1. At t = 9 ns, it moves 54 nm along the nanotrack with an average speed of 6 m s -1. It can be seen that the skyrmion can still be pushed into motion under relatively large damping coefficient for high frequency spin waves. Supplementary Note 4: The effect of global magnetic pulse on the motion of the skyrmion We have also performed some simulations with magnetic field pulses directly applied to the nanotrack to investigate the effect of magnetic field pulses on the skyrmion motion driven by spin waves. Supplementary Figure 6 shows the propagation of a skyrmion driven by spin waves in the 40-nm-wide nanotrack with/without an applied global oscillating magnetic field, Boc = B0 sin[2πft] ui, i = x, y, z. The 9

patterned green boxes on the track corresponds to the region of the spin wave injection (135 nm < x < 150 nm), of which the profile of the magnetic field pulse is shown in Figure 3 of the main text, i.e., the amplitude equals to 600 mt and the frequency equals to 25 GHz. The spin wave injection scheme at pulse elements and parameters are the same as that used in Figure 2a of the main text. The frequency f and amplitude B0 of the external oscillating magnetic field equal to 25 GHz and 50 mt, respectively. As shown in Supplementary Figure 6a, in absence of global oscillating magnetic field, the skyrmion driven by spin waves injected at the pulse element moves 266 nm within 5 ns. As shown in Supplementary Figure 6b, when the global oscillating magnetic field is applied parallel to the nanotrack, i.e., Boc = B0 sin[2πft] ux, the skyrmion driven by spin waves moves 246 nm within 5 ns. As shown in Supplementary Figure 6c, when the global oscillating magnetic field is applied transverse to the nanotrack, i.e., Boc = B0 sin[2πft] uy, the skyrmion driven by spin waves moves 271 nm within 5 ns. As shown in Supplementary Figure 6d, when the global oscillating magnetic field is applied perpendicular to the nanotrack, i.e., Boc = B0 sin[2πft] uz, the skyrmion driven by spin waves moves 212 nm within 5 ns. Supplementary Figure 6e-g show the effect of the global oscillating magnetic field pulse Boc directly on the whole nanotrack, where the pulse element is turned off, i.e., the spin wave injection at the pulse element is absence. It can be seen that, for three types of the configuration of the global oscillating magnetic field pulse applied on the whole nanotrack, the skyrmion cannot be driven into motion by the global oscillating magnetic field, but the shape of the skyrmion will be influenced. Within the same frames of time and spin wave injection, the magnetic field pulse directly applied on the whole nanotrack with the configuration of perpendicular or parallel to the nanotrack impedes the spin wave-driven motion of skyrmion. However, the spin wave-driven motion of skyrmion is enhanced by the magnetic field pulse directly applied transverse to the whole nanotrack. Obviously, the spin waves propagating along the nanotrack excited by local transverse magnetic field pulse will be enhanced by the global magnetic field pulse applied transverse to the nanotrack and decayed by the global magnetic field pulse applied perpendicular or parallel to the nanotrack, leading to the promotion and reduction of spin wave-driven skyrmion speed. In addition, the global magnetic field pulse applied perpendicular to the whole 10

nanotrack will induce the breathing of the skyrmion, which also resulting in the decrease of the spin wave-driven skyrmion mobility. Supplementary Note 5: The propagation of a skyrmion driven by spin waves generated by spatiotemporal Oersted field Supplementary Figure 7 shows the propagation of a skyrmion in the 800-nm-long, 40-nm-wide and 1-nm-thick nanotrack driven by spin waves injected via the oscillating Oersted field. The oscillating Oersted field is induced by the ac current (6.87 ma, 50 GHz) inside the nanowire with a radius of 2.5 nm upon the nanotrack at x = 142. 5 nm, z = 3.5 nm. The amplitude profile of the oscillating Oersted field distribution in the nanotrack is shown in Supplementary Figure 7a, which ranges from 0 mt to 550 mt at the peak value of the current. It can be seen that the oscillating Oersted field mainly concentrates in the region of 130 nm < x < 155 nm. In this case, both the oscillating Oersted field and the spin waves generated by the oscillating Oersted field will act on the skyrmion simultaneously. As shown in Supplementary Figure 7b-c, the skyrmion moves 303 nm within 5 ns and moves 516 nm within 9 ns. On the other hand, we show the control case (see Supplementary Figure 7e-h), where the oscillating Oersted field generated by the ac current is artificially confined in the space of 130 nm < x < 155 nm, i.e., for x < 130 nm and x > 155 nm, the value of the Oersted field is set to 0. In this case, only the spin waves generated by the oscillating Oersted field will act on the skyrmion. The skyrmion moves 99 nm within 5 ns and moves 176 nm within 9 ns. The speed of the skyrmion is slower than the one shown in Supplementary Figure 7a-d, which may be caused by the decrease in the amplitude of the spin waves generated by the confined oscillating Oersted field. Supplementary Note 6: Radius of a skyrmion Our system is a 2-dimensional magnet. The Hamiltonian reads H = H A + H K + H DM + H Z. (1) HA describes the nonlinear O(3) sigma model, H A = A d 2 x( k n) 2, (2) 11

HK represents the easy-axis anisotropy, HDM corresponds to the DMI, H K = K d 2 x(n z ) 2, (3) H DM = d 2 xd [n z ( n x x + n y y ) n n z x x n n y y y ] + D [n z ( n x y n y x ) n n z x y + n n z y x ] = d 2 xd [n z n (n )n z ] D n(x) ( n(x)), (4) where D (D ) is the Neel-type (Bloch-type) DMI and HZ the Zeeman effect, H z = μ 0 H d 2 xn z (x) (5) with H > 0. Here, J is the exchange energy, K is the single-ion easy-axis-anisotropy constant, and n = (n x, n y, n z ) = M/M S is a classical spin field of unit length. We parametrize the spin field n in the polar coordinate as n x = sin[f(r)] cos(φ + α), n y = sin[f(r)] sin(φ + α), n z = cos[f(r)]. (6) We obtain the Hamiltonian H = 2πrdr [A [( r f(r)) 2 + ( 1 + 2K ) r 2 A sin2 [f(r)]] + D [ r f(r) + sin[2f(r)] ] μ 2r 0 H cos f(r)]. (7) with D = cos αd + sin αd. Therefore we obtain the Neel-type Skyrmion (α = π) for D = 0, while the Bloch-type Skyrmion (α = π/2) for D = 0. The strong localization of its core allows us to use a linear ansatz [Zh. Eksp. Teor. Fiz. 95, 178-182 (1989); Phys. Status Solidi B 186, 527-543 (1994)] θ(r) = π(1 r/r) with 0 r R. By substituting this configuration into Eq. (2), we obtain H = ( 4 π μ 0H + Kπ) (R R Sk ) 2 D2 π 4 16 π μ 0H+4Kπ + Aπ(π2 + γ Ci(2π) + log2π), (8) where γ = 0.58 is the Euler s gamma constant and Ci is the cosine integral defined by C i (x) = cost x t (4) in the main text. dt with Ci(2π)= 0.023. Thus the skyrmion radius is given in Eq. Supplementary Note 7: Square pulse versus sinusoidal pulse We have also studied the skyrmion motion driven by spin wave excited by magnetic field pulse with different profiles, say, square magnetic field pulse and sinusoidal magnetic field pulse. The results are shown in Supplementary Figure 8. We put the antenna at the center of a nanotrack (1600 nm 40 nm 1 nm) with the same 12

set of parameters employed in the main text. The pulse direction is perpendicular to the nanotrack. It is found that the spin wave excited by square magnetic field pulse can drive the skyrmion backwards the antenna. When the frequency of the pulse is smaller than 10 GHz, the skyrmion driven by the spin wave excited by sinusoidal magnetic field pulse is basically motionless. However, in this case (blue dot region), we noted that the skyrmion on the left side of the skyrmion has a tiny displacement towards the antenna, while the one on the right side has a tint displacement backwards the antenna, within the simulation time of 20 ns. When the frequency is larger than 20 GHz and the amplitude is larger than 300 mt, the sinusoidal magnetic field pulse is also effective to generate spin wave to drive the skyrmion to move towards the end of the nanotrack with an appreciable speed. It is worth mentioning that when the frequency and amplitude of the pulse respectively equal to 40 GHz and 500 mt (in contrast to Supplementary Figure 8(a) and 8(d)), the spin wave excited by sinusoidal profile can push the skyrmion further away from the antenna. Supplementary Movie Legends Supplementary Movie 1: Spin wave-driven motion of a skyrmion on the nanotrack (800 nm 40 nm 1 nm) with the damping coefficient of 0.01 within 10 ns. The amplitude of the square magnetic field pulse is 600 mt and the pulse width and spacing are equal to 0.02 ns. The DMI constant is set to be 4 mj m -2. The out-of-plane component of magnetization is color coded, i.e., teal blue: into the plane, red: out of the plane, white: in-plane. Supplementary Movie 2: The spin wave driven motion and annihilation of a skyrmion in the nanotrack (800 nm 40 nm 1 nm) within 10 ns. The amplitude of the square magnetic field pulse is 600 mt and the pulse width and spacing are equal to 0.02 ns. The DMI constant is set to be 4.3 mj m -2. 13

Supplementary Movie 3: Annihilation of a skyrmion and destroy of the background magnetization of the nanotrack (800 nm 40 nm 1 nm) within 10 ns. The amplitude of the square magnetic field pulse is 700 mt and the pulse width and spacing are equal to 0.02 ns. The DMI constant is set to be 4 mj m -2. Supplementary Movie 4: Spin wave-driven motion and annihilation of a skyrmion on the 60-nm-wide L-corner within 5 ns. The amplitude of the square magnetic field pulse is 600 mt and the pulse width and spacing are equal to 0.02 ns. The DMI constant is set to be 3.5 mj m -2. Supplementary Movie 5: Spin wave-driven motion of a skyrmion on the corner-shaped 60-nm-wide L-corner within 5 ns. The amplitude of the square magnetic field pulse is 600 mt and the pulse width and spacing are equal to 0.02 ns. The DMI constant is set to be 3.5 mj m -2. Supplementary Movie 6: Spin wave-driven motion of a skyrmion for another case of corner-shaped 60-nm-wide L-corner within 5 ns. The amplitude of the square magnetic field pulse is 600 mt and the pulse width and spacing are equal to 0.02 ns. The DMI constant is set to be 3.5 mj m -2. Supplementary Movie 7: Spin wave-driven motion of a skyrmion from the central branch to the left branch in the corner-shaped 60-nm-wide T-junction within 5 ns. The amplitude of the square magnetic field pulse is 600 mt and the pulse width and spacing are equal to 0.02 ns. The DMI constant is set to be 3.5 mj m -2. Supplementary Movie 8: Spin wave-driven motion of a skyrmion from the left branch to the right branch in the corner-shaped 60-nm-wide T-junction within 5 ns. The amplitude of the square magnetic field pulse is 600 mt and the pulse width and spacing are equal to 0.02 ns. The DMI constant is set to be 3.5 mj m -2. Supplementary Movie 9: Spin wave-driven motion of a skyrmion from the right 14

branch to the central branch in the corner-shaped 60-nm-wide T-junction within 5 ns. The amplitude of the square magnetic field pulse is 600 mt and the pulse width and spacing are equal to 0.02 ns. The DMI constant is set to be 3.5 mj m -2. Supplementary Movie 10: Spin wave-driven motion of a skyrmion from the central branch to the left branch on the 60-nm-wide Y-junction within 5 ns. The amplitude of the square magnetic field pulse is 600 mt and the pulse width and spacing are equal to 0.02 ns. The DMI constant is set to be 3.5 mj m -2. Supplementary Movie 11: Spin wave-driven motion of a skyrmion from the central branch to the left branch on the 60-nm-wide T-junction within 10 ns. The amplitude of the square magnetic field pulse is 600 mt and the pulse width and spacing are equal to 0.02 ns. The DMI constant is set to be 3.5 mj m -2. Supplementary Movie 12: Spin wave-driven motion of a skyrmion from the central branch to the right branch in the 60-nm-wide T-junction within 10 ns. The amplitudes of the square magnetic field pulses of working pulse element are 600 mt before t = 2.4 ns and then increase to 700 mt. The pulse width and spacing are equal to 0.02 ns. The DMI constant is set to be 3.5 mj m -2. Supplementary Movie 13: Spin wave-driven motion of two skyrmions toward two opposite directions of the nanotrack (800 nm 40 nm 1 nm) within 5 ns. The amplitude of the square magnetic field pulses is 600 mt and the pulse width and spacing are equal to 0.02 ns. The DMI constant is set to be 4 mj m -2. Supplementary Movie 14: The case corresponding to Supplementary Movie 4, where Supplementary Movie 15: The case corresponding to Supplementary Movie 5, where 15

Supplementary Movie 16: The case corresponding to Supplementary Movie 6, where Supplementary Movie 17: The case corresponding to Supplementary Movie 7, where Supplementary Movie 18: The case corresponding to Supplementary Movie 8, where Supplementary Movie 19: The case corresponding to Supplementary Movie 9, where Supplementary Movie 20: The case corresponding to Supplementary Movie 10, where Supplementary Movie 21: The case corresponding to Supplementary Movie 11, where Supplementary Movie 22: The case corresponding to Supplementary Movie 12, where 16