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Unwinding of a Skyrmion Lattice by Magnetic Monopoles

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Science  31 May 2013:
Vol. 340, Issue 6136, pp. 1076-1080
DOI: 10.1126/science.1234657

Hedgehogs, Whirls, and Zippers

Topologically ordered materials at certain ranges of temperature and magnetic field can form a regular lattice of magnetic whirls called skyrmions. Milde et al. (p. 1076) studied the destruction of a skyrmion lattice with varying magnetic fields by imaging the surface magnetic structure. Magnetic force microscopy revealed a pairwise merging of skyrmions on the surface. Furthermore, in the bulk, a hedgehog-like spin structure with the properties of the elusive magnetic monopole was needed to "zip" together the corresponding skyrmion lines.

Abstract

Skyrmion crystals are regular arrangements of magnetic whirls that exist in a wide range of chiral magnets. Because of their topology, they cannot be created or destroyed by smooth rearrangements of the direction of the local magnetization. Using magnetic force microscopy, we tracked the destruction of the skyrmion lattice on the surface of a bulk crystal of Fe1−xCoxSi (x = 0.5). Our study revealed that skyrmions vanish by a coalescence, forming elongated structures. Numerical simulations showed that changes of topology are controlled by singular magnetic point defects. They can be viewed as quantized magnetic monopoles and antimonopoles, which provide sources and sinks of one flux quantum of emergent magnetic flux, respectively.

The notion of topological stability refers to those properties of a system that remain unchanged under continuous (elastic) deformations such as bending or stretching (1, 2). Because topologically stable structures cannot easily be created and destroyed, they play an important role in both fundamental and applied physics. An area in which topological stability is important are spin configurations in magnetic materials. Magnetic domain walls are examples of planar, two-dimensional topological defects. Various types of magnetic whirls form one-dimensional topological structures, and hedgehogs, where the magnetization points in all directions, are examples of pointlike (zero-dimensional) defects. A major challenge in systems exhibiting topological stability is to experimentally observe the unwinding of topologically stable configurations and to identify its mechanism.

The recent discovery of skyrmion lattices in chiral magnets with B20 crystal structure (Fig. 1A) has attracted great interest, as it provides an example of lattice order composed of topologically quantized magnetic whirls (38). A single skyrmion is a linelike structure oriented parallel to a small external magnetic field, where the magnetization winds once around the unit sphere in the plane perpendicular to the field. Skyrmion lattices are regular arrangements of skyrmion lines. They are ideally suited to explore the question of topological phase conversion experimentally. They occur in all B20 compounds with helimagnetic order comprising metallic, semiconducting, and insulating systems such as MnSi, Fe1−xCoxSi, and Cu2OSeO3, respectively. As the distance of skyrmions varies strongly among these systems, one may select compounds amenable to the experimental question of interest. In addition, skyrmion lattices in chiral magnets have been studied microscopically in great detail by means of neutron scattering (3, 4, 8) and transmission electron microscopy (TEM) (5, 6, 9, 10). All properties, including the magnetic phase diagram, are in excellent agreement with theory, providing a sound basis for studies of the topological unwinding. Beyond these fundamental aspects, there is also great interest in skyrmions in chiral magnets as a new route toward spintronics applications (9, 11, 12).

Fig. 1 Phase diagram of Fe1−xCoxSi for x = 0.5 inferred from magnetization, susceptibility, and neutron scattering.

The diagram comprises skyrmion-lattice (s), helimagnetic (h), conical (c), ferromagnetic (fm), and paramagnetic (pm) phases. (A) Typical spin configuration of a skyrmion lattice (from MC data). (B) Sketch of a magnetic configuration describing the merging of two skyrmions. At the merging point the magnetization vanishes at a singular point (arrow). This defect can be interpreted as an emergent magnetic antimonopole, which acts like the slider of a zipper connecting two skyrmion lines. (C) Phase diagram observed under zero-field cooling (zfc). The skyrmion lattice is confined to a small phase pocket (red) just below Tc. The field scale corresponds to the externally applied field for the geometry of the sample studied by MFM. (D) Phase diagram observed under field cooling (fc). For field values in the range of the skyrmion lattice as observed under zfc, the skyrmion lattice phase persists under field cooling as a metastable state down to the lowest T (red shading).

The complex magnetic texture of skyrmions causes the electron spin to pick up a Berry phase, which allows for an efficient coupling of currents to the magnetic structure (11, 12). This coupling may be elegantly described by associating to each skyrmion an artificial "emergent" magnetic field (1214), which is, because of the topology of the skyrmion, quantized to one flux quantum per skyrmion. The corresponding forces on electrons can be observed directly in the Hall effect (12, 15, 16). As skyrmion lines have a one-to-one association with a quantized magnetic flux, their creation or destruction is naturally associated with quantized sources or sinks of emergent magnetic flux. These can be identified with "emergent magnetic monopoles." Quantized magnetic monopoles were originally introduced as a hypothetical particle by Dirac (17) to explain the quantization of electric charge. Whereas magnetic monopoles have so far not been found experimentally as elementary particles, the concept has been used to describe spin-flip excitations in so-called spin-ice (18, 19).

Combining magnetic force microscopy (MFM), numerical calculations, and topological arguments, we study the transition of a skyrmion lattice in Fe1−xCoxSi (x = 0.5) to conventional magnetic order. Our central result is that the skyrmions unwind by means of hedgehog point defects, which can directly be interpreted as emergent magnetic monopoles and antimonopoles. Figure 1B shows schematically how such a singular point defect merges two skyrmion lines like the slider of a zipper.

For our study, we selected Fe1−xCoxSi (x = 0.5) because the periodicity of the magnetic modulations of ∼90 nm for this composition is large as compared with the resolution of ∼20 nm of the MFM (20). Figure 1, C and D, displays the phase diagram of Fe1−xCoxSi, inferred from magnetization, ac susceptibility, and small-angle neutron scattering in bulk samples (4, 20). Under zero-field cooling (zfc) (Fig. 1C) helimagnetic order (h) appears below the critical temperature, Tc ≈ 45 K, with a modulation vector parallel to 〈100〉. Well below Tc, the helimagnetic order undergoes a spin-flop transition to conical order (c) at Bc1 ∼ 30 mT with the modulation vector parallel to the magnetic field, followed by a transition to a spin-polarized state (fm) at Bc2 ∼ 60 mT. For temperatures T just below Tc, an additional phase (s) stabilizes, the skyrmion lattice.

Under field cooling (fc) (i.e., cooling while keeping the applied field constant), the phase diagram (Fig. 1D) exhibits several important differences with respect to zfc. First, for field values outside the range of the skyrmion lattice, there is only a paramagnetic to conical transition. Second, for field values in the range of the skyrmion lattice, the skyrmion lattice survives as a metastable state down to the lowest T with the same reversible phase boundaries near Tc as for zfc but irreversible phase boundaries well below Tc.

This metastable skyrmion lattice state made it possible to take measurements at T << Tc, which was helpful in two ways. First, as the magnetic moment increases considerably toward low T, the contrast of the MFM data increases substantially, providing unambiguous information. Second, the topological stability of skyrmions relies on the fact that the modulus of the local magnetization is finite everywhere. Close to Tc, strong thermal fluctuations may in principle weaken the topological stability, which is not the case for T << Tc, thereby exposing the generic mechanism of the topological unwinding.

Typical MFM data for decreasing applied fields after initial field cooling in +20 mT to 10 K are summarized in Fig. 2 [see (20) for details]. Each row is composed of the real-space image, an enlarged section of the same image, and a fast Fourier transform (FFT) (see bottom of Fig. 2 for scales). The MFM measurements reveal a hexagonally ordered pattern (Fig. 2, A1 and B1) with one of the reciprocal lattice vectors approximately aligned along 〈100〉. Upon decreasing magnetic field, first at some places on the surface, two neighboring skyrmions, visible as blue dots, coalesce into one elongated pattern (Fig. 2, A2 and B2). Reducing the magnetic field causes the elongated structures to grow in length, reducing the number of skyrmions (Fig. 2, A3/B3 through A5/B5). Eventually a striped pattern forms with numerous defects, which is characteristic of a one-dimensional modulated state parallel to 〈100〉.

Fig. 2 Typical magnetic force microscopy data at the surface of Fe1−xCoxSi (x = 0.5).

Blue (red) colors correspond to a magnetization pointing parallel (antiparallel) to the line of sight into (out of) the surface, respectively. Panels (A1) through (A5): Data recorded as a function of magnetic field after fc at +20 mT down to T = 10 K. Panel (A1) displays data immediately after fc. After the initial cool-down, the field was reduced at a fixed temperature of 10 K (A2 to A5). During this process, the skyrmions, visible as blue spots, merge and form elongated, linelike structures. The left inset shows a Fourier transformation of the real-space signal. Panels (B1) to (B5) enlarge the region marked by the black rectangle in panels (A1) to (A5).

To confirm our observations, we performed the following tests (20). First, the time constant of the metastable state is long as compared with the time needed to record each image (17 min); e.g., Fig. 2, A3 changed little after a wait of 15 hours. Second, using small-angle neutron scattering, we confirmed that the intensity pattern of the magnetic order in the sample volume corresponds to the FFTs of the MFM data recorded at the sample surface [see (20) for details].

Our MFM data suggest a mechanism for the reduction in the number of skyrmions and therefore the topological winding number at the surface: Skyrmions coalesce and form linelike structures. Two immediate (and closely connected) questions arise: (i) How and to what extent does the change in the magnetic structure on the surface reflect similar processes in the bulk? (ii) How can the merging of skyrmions be understood from a microscopic and topological point of view? To answer these questions, we performed a classical Monte Carlo (MC) simulation of 42 × 42 × 30 spins coupled to their nearest neighbors by ferromagnetic exchange and Dzyaloshinksy-Moriya interactions. To track metastable behavior, we used a Metropolis algorithm based on local updates and also used micromagnetic simulations, including the effects of thermal fluctuations (20). The MC calculation reproduces the equilibrium phase diagram and captures the metastable behavior, consistent with the micromagnetic simulations and our experiments.

In our simulations, we followed a temperature and field protocol similar to that used in the experiments. First, we cooled the system at a fixed field crossing the skyrmion phase. Similar to experiment, the skyrmion phase survives down to the lowest temperatures. Second, at low temperature the field is reduced, permitting transverse thermal fluctuations of the skyrmion lines to grow until the skyrmions touch and merge. Snapshots of typical field configurations in the bulk are shown in Fig. 3A, where the pattern at the front of the simulated box shows marked similarities with the MFM data.

Fig. 3 Monte Carlo simulation for a system first field cooled at B = 0.16 J (B||[110]) down to T = 0.6 J.

[See (20) for other temperatures.] After the cool-down, the field is reduced at constant temperature. Below a critical field, skyrmions start to touch and merge. (A) Typical magnetic configurations shown by contour surfaces of equal magnetization in [110] direction for B = 0.16 J, B = 0.036 J, and B = 0. The arrows labeled MP and AMP point to a monopole and antimonopole, respectively. (B) Winding number W (or, equivalently, number of skyrmions) per area in units of the helical wavelength Embedded Image on the front and back surface of the simulated box computed while B is reduced. (C) Number of monopoles (MP) and antimonopoles (AMP) per volume in units of Embedded Image. The plots show averages over 15 cooling cycles; error bars denote standard deviations of the mean.

Figure 3B shows how, on average, the density of skyrmions, measured by the winding number W, is reduced. For T = 0.6 J ≈ 0.65 Tc [see (20) for other temperatures], a small density of skyrmions survives even at B = 0. During these sweeps, we did not obtain any field configurations typical for the conical phase (i.e., a phase with ordering vector parallel to the B field). Instead, we always obtained a strongly hysteretic transition to a helical phase driven by the merging of skyrmions, reminiscent of our experiments (other types of transition are, e.g., observed when increasing the field). Taken together, our numerical results therefore suggest that the structures seen in our MFM data reflect a similar transition in the bulk of the sample. However, a quantitative comparison of theory with experiment is beyond the scope of the work reported here, because the effects of disorder in Fe1−xCoxSi cannot be recovered from our numerical simulation of a clean system.

To appreciate the physics of the merging of the skyrmions observed experimentally and numerically, we note that both the topological nature of the skyrmions and their interaction with electrons are best described (12) in terms of the (fictitious) emergent electromagnetic fields (13, 14, 21)

Bie=2εijkn^(jn^×kn^),Eie=n^(in^×tn^) (1)

where n^(r,t)=M/|M| is the local orientation of the magnetization, ∂i = ∂/∂ri and εijk is the totally antisymmetric tensor. Taken together Bie and Eie account for the Berry phase that the spin of a conduction electron accumulates when following the magnetic texture adiabatically. The experimental consequences have been detected directly in terms of an additional (topological) contribution to the Hall signal and an emergent electric field, providing evidence of the motion of the skyrmions (12, 15, 16).

As the integral of Be/ over a surface describes the solid angle covered by n^, the emergent magnetic flux of each skyrmion is exactly given by one (negative) flux quantum Bedσ=2π|qe|=Φ0 [we use a convention, where the conduction electrons of the majority (minority) bands carry the charge 1/2 (−1/2), respectively (12)]. Therefore, the topologically quantized winding number or, equivalently, the quantized magnetic flux has to change when two skyrmions merge. Because of the topological nature of the winding number, this is in fact only possible by a singular field configuration for which the local magnetization vanishes at a point in space. The implications for the topological properties may thereby be seen by integrating Be over a closed surface ∂Ω of the volume Ω

ΩBedσ=ΩBedr=2π|qe|(NoutsNins)=Φ0 (2)

Here Nouts (Nins) is the number of ingoing (outgoing) skyrmion lines, respectively. Hence, when two ingoing skyrmions merge, there must be a singular field configuration, a hedgehog defect with winding number +1, which creates one quantum of emergent magnetic flux. The point of coalescence therefore carries a quantized emergent magnetic charge; i.e., it is an "emergent magnetic monopole." Similarly, when an ingoing single skyrmion line splits into two, an antimonopole with winding number −1 is located at the point of separation. Figure 1B shows schematically such an antimonopole; the location of a monopole (MP) is marked by an arrow in Fig. 3A. Monopoles and antimonopoles are related by a time-reversal symmetry transformation, M → −M, followed by a rotation by π around an axis perpendicular to the magnetic field.

The merging of skyrmions at the surface of the crystal observed in our MFM experiment implies that one of two processes has taken place. Either the merging of the skyrmion lines has started in the bulk and the antimonopole, which zipped two skyrmion lines together, has moved through the surface. Or, when the merging of two skyrmion lines started at the surface, a monopole has entered the surface. As the line segments can be interpreted as elongated skyrmions, also the merging of the segments implies that an antimonopole (monopole) has moved out of (into) the surface, respectively.

Both our experiments and our numerical calculations suggest that the merging of skyrmions underlies the conversion of the skyrmion phase into the helical phase. The change of topology is thereby governed by the creation and motion of topological point defects, which we identify as emergent magnetic monopoles and antimonopoles. Figure 3C shows the density of separate (20) monopoles and antimonopoles during the field sweep. A comparison of Fig. 3, B and C, shows that the destruction of skyrmions at the surface is directly associated with the proliferation of monopoles and antimonopoles [see also fig. S8]. We expect that the energy and dynamics of the monopoles govern the metastability (4, 22) of the skyrmion phase. Because of their singular core, monopoles are expected to pin much stronger to impurities in the sample, an effect not taken into account in our simulations.

It is instructive to compare the emergent magnetic monopoles discussed here with the magnetic monopoles considered in spin ice (18, 19). Monopoles in spin ice are sources of the "real" magnetic H-field, but their magnetic charge is not quantized and depends on microscopic details. By contrast, the emergent monopoles that we identify here are sources of the emergent magnetic field that follows Dirac's quantization condition for monopoles (17); i.e., they carry one quantum of emergent flux. Furthermore, in spin-ice at zero magnetic field, the monopoles are "deconfined"; i.e., it requires only a finite amount of energy to separate monopole and antimonopole. In the skyrmion phase, the situation is different (similar arguments apply to the helical phase): Deep in the skyrmion phase, it requires a finite amount of energy per length to zip two skyrmions together. Consequently, there is a linear potential (i.e., a finite string tension) holding monopole and antimonopole together. Only during the conversion from one phase to the other, the string tension vanishes or becomes negative. In disordered materials, the string tension may be a random function that competes with potentials pinning the monopoles.

An interesting open question is whether phases of deconfined emergent monopoles in chiral magnets exist, where monopoles and antimonopoles proliferate as independent entities. A candidate for such a phase is the metallic state of MnSi at high pressure. Its properties differ markedly from those of conventional metals [the resistivity is proportional to T3/2 over almost three decades in T (23)], with highly unconventional "partial" magnetic order on intermediate time and length scales (24) and an unconventional Hall signature (15). Further experiments and theoretical studies are needed to study the connection of the partial order in MnSi with the emergent monopoles and the electronic properties in the non–Fermi liquid phase of MnSi.

Supplementary Materials

www.sciencemag.org/cgi/content/full/340/6136/1076/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S8

References (2535)

References and Notes

  1. See supplementary materials on Science Online.
  2. Acknowledgments: We thank R. Bamler, P. Böni, K. Everschor, L. Fritz, M. Garst, R. Georgii, S. Mayr, and the team of FRM II, and, especially, D. Meier. We are grateful to B. Pedersen for checking the orientation of our sample on the diffractometer HEIDI at FRM II. Financial support through European Research Council AdG 291079 (TOPFIT) and through SFB 608, SFB TR 12, and SFB TR 80 of the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged. A.B. A.C., and J.K. acknowledge support through the TUM Graduate School, S.B. and C.S. through the Bonn-Cologne Graduate School of Physics and Astronomy (BCGS), D.K. through RTG 1401/2 (DFG), J.S. by the Australian Research Council through a Future Fellowship (FT110100523), and S.B. through the Emmy-Noether group (DFG) of L. Fritz.
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