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Multiply Periodic States and Isolated Skyrmions in an Anisotropic Frustrated Magnet

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Multiply Periodic States and Isolated Skyrmions in an Anisotropic Frustrated Magnet

A O Leonov et al. Nat Commun.

Abstract

Multiply periodic states appear in a wide variety of physical contexts, such as the Rayleigh-Bénard convection, Faraday waves, liquid crystals and skyrmion crystals recently observed in chiral magnets. Here we study the phase diagram of an anisotropic frustrated magnet which contains five different multiply periodic states including the skyrmion crystal. We clarify the mechanism for stabilization of these states and discuss how they can be observed in magnetic resonance and electric polarization measurements. We also find stable isolated skyrmions with topological charge 1 and 2. Their spin structure, interactions and dynamics are more complex than those in chiral magnets. In particular, magnetic resonance in the skyrmion crystal should be accompanied by oscillations of the electric polarization with a frequency depending on the amplitude of the a.c. magnetic field. These results show that skyrmion materials with rich physical properties can be found among frustrated magnets. We formulate rules to help the search.

Figures

Figure 1
Figure 1. Zero-temperature phase diagram of the frustrated triangular antiferromagnet.
The eight phases are as follows: fully polarized ferromagnetic state (FM), conical spiral (CS), 2q-state, 2q′-state, vertical spiral (VS), VS plus two in-plane sinusoidal modulations (M), flop state (FL) and skyrmion crystal (SkX). Dashed line is the upper critical field, above which isolated skyrmions are unstable. The NNN exchange constant, J2=0.5, magnetic anisotropy, K, and magnetic field, h, are measured in units of the NN exchange constant, J1=1.
Figure 2
Figure 2. Multi-q states.
In-plane components (arrows) and out-of-plane components (colour) of spins in four selected states from the phase diagram. (a) 2q-state, (b) 2q′-state, (c) flop state and (d) skyrmion crystal. The inset shows the wave vectors of the three fundamental modulations in the multi-q states.
Figure 3
Figure 3. Helicity reversals.
(a) False colour plot of the toroidal moment density, tz, in the skyrmion with the helicity angle χ=π/2. (b) Fan-like oscillations of the out-of-plane spin component, Sz. The skyrmion centre is located at x=0.
Figure 4
Figure 4. Unusual features of skyrmions in frustrated magnets.
(a) The interaction energy, U12, versus the distance r12 between two skyrmions (black lines with rectangular markers), and skyrmion and antiskyrmion (red lines with circular markers). Solid (dashed) lines show the interaction for equal (opposite) helicities of the topological defects. (b) Metastable skyrmion with the topological charge Q=2. (c) Metastable skyrmion–antiskyrmion crystal with a rectangular lattice. In b and c, colour indicates the topological charge density, ρQ, while arrows show the in-plane components of spins.
Figure 5
Figure 5. Multiferroic and dynamical properties of multi-q states.
Magnetic field dependence at K=0.04 of (a) the average spin vector 〈S〉, (b) the electric polarization vector P and (c) the imaginary part of the in-plane magnetic susceptibility, χ′′(ω) (arbitrary units). Black and red lines in a and b are the out-of-plane and in-plane components of the vectors. The electric polarization was calculated using equation (12) for g1=g2 and g3=0. The inset in c shows magnetic modes in the 2q-state near the transition to the flop state.
Figure 6
Figure 6. Coupled dynamics of skyrmion helicity and centre of mass.
(a) Circular trajectories spanned by the centres of rotating skyrmions in the in-plane a.c. magnetic field for various amplitudes of the a.c. magnetic field hω. (b) Frequency of the helicity rotation, ωhel, measured in units of the frequency of the a.c. field ω=0.224J1−1 versus hω. (c) Time evolution of the x-component of spin at a chosen site, marked by a circle in the series of snapshots (d), which also shows current positions of the skyrmion centre (red point) and its trajectory for hω=0.01. This calculation was performed for h=0.2 and K=0.04.

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