Annual Report 2011 / 2012 - E21 - Technische Universität München

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 E21, Technische Universität München, D-85748 Garching, Germany.
3 Forschungsneutronenquelle Heinz Maier-Leibnitz (FRM II), Technische 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 (2012).
[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 (2012).