4 <strong>E21</strong> <strong>Annual</strong> <strong>Report</strong> <strong>2011</strong>/<strong>2012</strong>
Chapter 1. Magnetism and Superconductivity 5 Rotating Skyrmion Lattices by Spin Torques & Field or Temperature Gradients Karin Everschor 1, 2 , Markus Garst 1 , Benedikt Binz 1 , Florian Jonietz 2 , Sebastian Mühlbauer 3 , Christian Pfleiderer 2 , Achim Rosch 1 1 Institut für Theoretische Physik, Universität zu Köln, D-50937 Köln, Germany. 2 Physik-Department <strong>E21</strong>, <strong>Technische</strong> Universität München, D-85748 Garching, Germany. 3 Forschungsneutronenquelle Heinz Maier-Leibnitz (FRM II), <strong>Technische</strong> Universität München, D-85748 Garching, Germany. Chiral magnets like MnSi form lattices of skyrmions, i.e. magnetic whirls, which react sensitively to small electric currents j above a critical current density j c . The interplay of these currents with tiny gradients of either the magnetic field or the temperature induce a rotation of the magnetic pattern for j > j c . Either a rotation by a finite angle of up to 15° or – for larger gradients – a continuous rotation with a finite angular velocity is induced. We use Landau- Lifshitz-Gilbert equations extended by extra damping terms in combination with a phenomenological treatment of pinning forces to develop a theory of the relevant rotational torques [1] by extending the method used by Thiele [2]. So far only temperature gradients parallel to the current have been studied experimentally [4]. The observed behaviour for the rotation angle, Fig. 2a), can be modeled within our theory when assuming a weak temperature dependence of Gilbert damping (green curve in Fig. 2b)). In our theory we expect a jump of the rotation angle at j c , depending on the domain size. This appears to be consistent with the observed steep increase of the rotation angle atj c , when taking into account that the experimental results are subject to a distribution of domain sizes. Furthermore we predict that for even larger currents a continuous rotation will occur. The measured distribution of rotation angles shown in Fig. 2 extents up to the maximally possible value of 15 ° for static domains, suggesting that continuously rotating domains are either already present in the system or may be reached by using slightly larger currents or temperature gradients. Figure 1: Schematic plot of forces on skyrmion lattice perpendicular and parallel to the current. In the presence of a temperature or field gradient, these forces change smoothly across a domain, thereby inducing rotational torques. The discovery [3] of the skyrmion lattice in chiral magnets provides new opportunities for investigating the coupling of electric-, thermal- or spin currents to magnetic textures: (i) the coupling by Berry phases to the quantized winding number provides a universal mechanism to create efficiently Magnus forces, (ii) the skyrmion lattice can be manipulated by extremely small forces induced by ultralow currents [4, 5], (iii) the small currents imply that also new types of experiments are possible. Studying the rotational dynamics of skyrmion domains allows to learn in more detail which forces affect the dynamics of the magnetic texture. The basic idea underlying the theoretical analysis for the rotational motion is sketched in Fig. 1. In the presence of an electric current first, dissipative forces try to drag the skyrmion lattice parallel to the (spin-) current. Second, the interplay of dissipationless spin-currents circulating around each skyrmion and the spincurrents induced by the electric current lead to a Magnus force oriented perpendicular to the current for a static skyrmion lattice (for the realistic case of moving skyrmions the situation is more complicated). In the presence of any gradient across the system (e.g. a temperature or field gradient), indicated by the color gradient, these forces vary in strength across a skyrmion domain and lead to rotational torques. Whether the torque arises from the Magnus forces or the dissipative forces depends, however, on the relative orientation of current and gradient and also on the direction in which the skyrmion lattice drifts. Figure 2: a) Average rotation angle ∆φ of skyrmion lattice in MnSi measured by neutron scattering [4]. b) Rotation angle φ as function of effective spin velocity v s. As in experiment the temperature gradient ∇t grows with the square of the applied current. Assumption for thin blue curve: damping constants are independent of t. For thick green curve we assumed a weak temperature dependence of the Gilbert damping constant α to reflect the experimental observation. c) Angular distribution P φ of the intensity for currents of strength j = 0 and j ≈ −2.07·10 6 A/m 2 . We acknowledge discussions with M. Halder, M. Mochizuki, and T. Nattermann and financial support of the DFG through SFB 608, TRR80, and FOR960, as well as the European Research Council through ERC-AdG (Grant No. 291079). K.E. thanks the Deutsche Telekom Stiftung and the Bonn Cologne Graduate School. References [1] K. Everschor et al., Phys. Rev. B 86, 054432 (<strong>2012</strong>). [2] A. A. Thiele Phys. Rev. Lett. 30, 230 (1973). [3] S. Mühlbauer et al., Science 323, 915 (2009). [4] F. Jonietz et al., Science 330, 1648 (2010). [5] T. Schulz et al., Nat. Phys. 8, 301 (<strong>2012</strong>).