SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHYS3883 Direct observation of the skyrmion Hall effect Wanjun Jiang 1,2,3, *,, Xichao Zhang 4,*, Guoqiang Yu 5, Wei Zhang 1, Xiao Wang 6, M. Benjamin Jungfleisch 1, John E. Pearson 1, Xuemei Cheng 6, Olle Heinonen 1, 7, Kang L. Wang 5, Yan Zhou 4, Axel Hoffmann 1,, Suzanne G. E. te Velthuis 1, 1 Materials Science Division, Argonne National Laboratory, Lemont, Illinois, USA, 60439 2 State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China 3 Collaborative Innovation Center of Quantum Matter, Beijing 100084, China 4 School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen 518172, China 5 Department of Electric Engineering, University of California, Los Angeles, California, USA, 90095 6 Department of Physics, Bryn Mawr College, Bryn Mawr, PA, USA, 19010 7 Northwestern-Argonne Institute of Science and Engineering, Northwestern University, Evanston, IL, USA, 60208 * These authors contributed equally. To whom correspondence should be addressed. E-mail: jiangw@anl.gov, hoffmann@anl.gov, tevelthuis@anl.gov NATURE PHYSICS www.nature.com/naturephysics 1 2016 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
The followings parts are to be discussed in the supplementary information. Part 1: Theoretical description of the topological charge Part 2: Dissipative force tensor and its dependence on skyrmion size Part 3: Evolution of topological charge on the profile of skyrmion spin texture. Part 4: Micromagnetic simulation formulism Part 5: Magneto-optical Kerr effect imaging for the whole device Part 6: MOKE movie for the skyrmion motion in the interior and edge of the device Part 7: MOKE movie for the skyrmion motion upon the reversal of current direction. Part 1: Theoretical description of the topological charge A magnetic skyrmion in a planar geometry is characterized by a topological charge, i.e., the skyrmion number. The topological charge is given by "# (), = (S-1) where the topological charge density "# () is given by "# = (), (S-2) Thus, the topological charge of the magnetic skyrmion is = (), (S-3) which is also referred to as the Pontryagin number. It counts how many times is wrapped around the unit sphere as the coordinate (, ) spans the whole planar space. A point of the planar space is parameterized as = cos, = sin. By applying the mapping = 0 as 1, = 1 as = 0, as 1, and lim (, ) = lim (, ), we consider the compactification of the planar space to a sphere in the xyz-space parameterized by = 1 cos = 1 cos cos, = 1 sin = 1 cos sin, and = cos. Here z is defined by "# = = "#. (S-4) Hence, we can write = (, ) = sin cos, sin sin, cos, (S-5) 2
By substituting equation (S-5) into equation (S-3), we obtain sin " " = = cos = cos, (S-6) Thus, when the spins at point towards the +z direction while the spin at = 0 points towards the z direction, cos = 2, resulting in the negative skyrmion number Q = -1. In the same way, when the spins at point towards the -z direction while the spin at = 0 points towards the +z direction, cos = 2, resulting in the positive skyrmion number Q = +1. Note that the direction of the spins at is given by the direction of the applied magnetic field, and thus the skyrmion number reverses sign when the field is reversed and the sign of the topological charge is opposite to the sign of the applied magnetic field in the +z direction. Furthermore note that this definition is independent of the size of the skyrmion. Part 2: Dissipative force tensor and its dependence on skyrmion size In the modified Thiele equation, equation (1) of the main text, the dissipative force term is given as. Here = 4 " " and " = " " "#, is the dissipative tensor describing the effect of the dissipative force on the moving magnetic skyrmion. For a single isolated magnetic skyrmion, = = and " = " = 0. Considering the position vector in cylindrical coordinates, = (,, ), the spin texture of an isolated Néel skyrmion can be described by: = sin cos + sin sin + cos (S-7) where = 0 for Néel skyrmion. This simplifies the equation to () = sin + cos. (the angle with respect to axis) can be approximately written as follows:, = " 0, < " " > " ", (S-8) 3
where P and " represent the position and width of the domain wall, respectively. is the radial distance from the skyrmion core to the point r. Solving the integral of " = " " = = = "# results in: (S-9) " where d is the diameter of the Néel skyrmion that is defined as the diameter of the circle where mz = 0. The domain wall width " = /"" 21 nm is estimated based on the exchange stiffness = 10 10-12 J m-1, while the effective magnetic perpendicular anisotropy "" = 2.3 104 J m-3 (µ0heff = 70 mt) is determined by measuring the inplane hard axis hysteresis loop. Fig. S1. Dependence of the dissipative tensor () on the diameter (d) of skyrmion. Calculation of by considering a fixed domain wall width (" 21 nm) and P = d/2 shown is the dependence of on the size of the skyrmion. The inset shows the exemplary spin profile ( ) when the diameter d of skyrmion equals 100 nm. Part 3: Evolution of topological charge on the profile of skyrmion spin texture. In the following, we will discuss two different types of spin profile to discuss the independence of topological charge on the detailed skyrmion spin textures. The first one is the so-called skyrmion bubble stabilized by the chiral interfacial Dzyaloshinskii- 4
Moriya interaction (DMI) term in the presence of additional dipole interaction. It exhibits a large inner core of uniform negative perpendicular magnetization. Inside the chiral Néel-type domain wall, spin rotates linearly along the radial direction, shown in Fig. S2 (a). The second type of skyrmion spin textures does not contain a large inner core, but instead the spin rotates continuously across the spin textures in a sinusoidal fashion, shown in Fig. S2 (b). This is more typical for skyrmions in bulk systems with negligible dipolar interactions. It is however, interesting to realize that the topological charge = 1 4 "# (by wrapping each individual spin into the unit sphere/or by integrating over the entire plane) are actually identical as -1 in both cases. As pointed out in Part 1 that topological charge can be calculated as: = cos. When the spins at point the +z direction while the spin at = 0 points the z direction, cos = 2, this gives rise to the equivalent negative topological charge of Q = -1 that is independent on the detailed spin profile. Thus, while the governing energetics are different, these two different skyrmion spin textures are topologically equivalent. = 1 = 1 Fig. S2. Two different type of spin textures that are topological equivalent. (a) skyrmion bubbles with a large inner core of uniform perpendicular magnetization. The spin rotates linearly inside the chiral Néel-type domain wall. (b) A skyrmion dominated by the 5
competition between symmetric (Heisenberg) and chiral antisymmetric (DzyaloshinskiiMoriya) exchange interactions contains no inner core, but instead the spin rotates continuously across the spin textures. These two skyrmion spin textures show the same topological charge of -1. Part 4: Micromagnetic simulation formulism The dimensionless Landau-Lifshitz-Gilbert (LLG) equation including the spin torque from the spin Hall effect (SHE) can be written as " = "" + " " (S-10) where = is the reduced magnetization, is the saturation magnetization, t is the time, α is the damping coefficient, is the gyromagnetic ratio, = ℏ ", ℏ is the reduced Planck constant, " is the spin-hall angle, e is the electron charge, is the thickness of the ferromagnetic layer. is the unit vector of the out-of-plane direction, " =, 0 is the in-plane current injected via the heavy metal. The reduced effective " field is defined as "" =. Where the average energy density E is a function of M reads as: = +"# " " + " 2 (S-11) " where and K are the exchange and anisotropy energy constants, respectively. H and Hd(M) are the external applied and magneto-static self-interaction fields. "# is the DMI constant. The five terms at the right-hand side correspond to the exchange energy, the anisotropy energy, the applied field (Zeeman) energy, the magneto-static (demagnetization) energy and the DMI energy, respectively. The geometry of the device used in the simulation are the following: Length = 1000 nm, Width = 500 nm, Thickness = 1.1 nm. The simulation parameters used for LLG equation are given below: Exchange Stiffness, = 10 10-12 J/m, Dzyaloshinskii-Moriya Interaction "# = 0.5 mj/m2, 6
Effective perpendicular Magnetic Anisotropy K eff = 2.3 10 4 J/m 3 (µ 0 H eff = 70 mt), Saturation Magnetization, = 6.5 10 5 A/m, Damping Coefficient α = 0.02, Driving Current Density j = 3 10 9 A/m 2 with the polarization ratio of P = 0.4. By alternating the sign of electron current direction and sign of topological charge, the following micromagnetic simulation studies have been carried out, shown in Fig S3. These simulation results are consistent with experiment results presented in the Fig. 3 (c) of the main text. Fig. S3. Micromagnetic simulation study of the evolution of skyrmion motion on the sign of electron current/topological charge. 7
Part 5: Magneto-optical Kerr effect imaging for the whole device 20#μm Fig S4. Magneto-optical Kerr effect imaging of current driven skyrmion motion. The dimension of the Hall-bar like device is 100 (width) 500 (length) m (devices discussed in Figs. 2, 3(a), and 3(b)). The applied magnetic field is = -5.2 Oe. Upon applying currents, a few skyrmions formed underneath the electrodes were driven into the area of interests, where the current driven skyrmion motion was studied. For skyrmions moving in the interior of the Hall-bar like devices (typically chosen from the orange block that is far away from the edge), the trajectory follows approximately a straight line, from which the skyrmion Hall angle and the average skyrmion velocity can be determined. Part 6: MOKE movie for the skyrmion motion in the interior and edge of the device. The associated differential MOKE movie acquired by applying pulsed currents of amplitude = 2.8 10 A/cm, 50-µs duration, and 1-Hz frequency. The applied 8
magnetic field is = -5.2 Oe. This MOKE movie shows the = +1skyrmions moving in the interior of the device following a straight and diagonal slope with a well-define skyrmion Hall angle and average velocity. The skyrmions move along the device edge showing oscillatory trajectory due to the competition between the edge repulsive force and the topological Magnus force. Part 7: MOKE movie for the skyrmion motion upon the reversal of current direction This movie shows the motion of a = 1skyrmion driven by a negative electron current of 50-µs duration, and 1-Hz frequency. The applied magnetic field is = +5.2 Oe. It is clear to see that the direction of stochastic motion is reversed by reversing the current direction, as compared to Figs. 2 (a) (e) in the main text. This is done at a current density of = 1.4 10 A/cm. 9