REVIEW ARTICLE PUBLISHED ONLINE: 4 DECEMBER 2013 | DOI: 10.1038/NNANO.2013.243
Topological properties and dynamics of magnetic skyrmions Naoto Nagaosa1,2* and Yoshinori Tokura1,2
Magnetic skyrmions are particle-like nanometre-sized spin textures of topological origin found in several magnetic materials, and are characterized by a long lifetime. Skyrmions have been observed both by means of neutron scattering in momentum space and microscopy techniques in real space, and their properties include novel Hall effects, current-driven motion with ultralow current density and multiferroic behaviour. These properties can be understood from a unified viewpoint, namely the emergent electromagnetism associated with the non-coplanar spin structure of skyrmions. From this description, potential applications of skyrmions as information carriers in magnetic information storage and processing devices are envisaged.
n classical mechanics, particles are represented by point masses or in non-centrosymmetric magnets such as MnSi (refs 10,23–28),
rigid bodies. In field theory, on the other hand, they are represented Fe1–xCoxSi (refs 29–33), FeGe (refs 34–37) and Mn1–xFexGe (ref. 38). Iby wave-like excitations, akin to vibrations on a drumhead. In In this case, an analogous transformation from the helical spiral into other words, they are excitations characterized by higher energy and the triangular-lattice skyrmion crystal under an external magnetic momentum than the ground state (in this case the drumhead at rest), field occurs. (3) Frustrated exchange interactions39 or (4) four-spin as well as a finite lifetime — the sound made by the drum decays after exchange interactions12, which can lead to atomic-sized skyrmion a certain time. In fact, explaining the stability of particles is a highly structures. Case (1) was extensively studied in the 1970s18 aiming at nontrivial issue. One possibility, originally proposed by Skyrme1, is applications for magnetic memory devices, while cases (2), (3) and that the particles are topologically protected, in the sense that they (4) are a focus of intensive research now. Therefore, we remark here are characterized by a topological integer that cannot be changed by on some basic differences among these four cases. First, the size of a continuous deformation of the field configuration. This model — the skyrmions is different. In case (1), skyrmions are typically of the in which the particles are called skyrmions — has been proposed to order of 100 nm to 1 μm, which is comparable to the period λ of the explain hadrons in nuclear physics. Interestingly, however, it has also spiral determined by the ratio of the dipolar and exchange interac- turned out to be relevant in condensed-matter systems such as the tions. In case (2) the size is determined by the DM interaction and is quantum Hall system2, liquid crystals3 and Bose condensate4. Indeed, typically 5–100 nm. In cases (3) and (4), λ and the skyrmion size are it was predicted that topologically protected particles can be stabi- of the order of the lattice constant (~1 nm). As a result, skyrmions lized in chiral magnets5–9 in the form of stable spin textures, referred and skyrmion crystals in cases (1) and (2) are larger than the lattice to as magnetic skyrmions. Following these pioneering theoretical constant, and therefore the continuum approximation is justified. In works, a flurry of recent experiments by neutron scattering10, Lorentz these two cases, the energy density of the skyrmions is much smaller transmission electron microscopy (LTEM)11 and spin-resolved scan- than the atomic exchange energy J, and the topological protectorate ning tunnelling microscopy12,13 have established the existence of holds. In other words, discontinuous spin configurations — called skyrmion phases in magnets. monopoles — with energy of the order of J can create or annihi- Figure 1a shows a schematic of a skyrmion configuration, late the skyrmions. These large-scale skyrmions have many internal
with arrows depicting the direction of spin n(r) at spatial posi- degrees of freedom, and hence are ‘soft’ and highly mobile, and free tion r = (x,y). The spin at the core points down, while spins at the from the commensurability pinning by the crystal lattice. perimeter point up. The topological skyrmion number Nsk is defined Other important issues are γ and m. Depending on the mecha- as a measure of the wrapping of n(r) around a unit sphere14,15 (see nism of stabilization of the skyrmion, the dependence of γ and m Box 1). As shown in Fig. 1g, there are several possible types of skyr- on the energy differs (see Box 1). In case (1), the dipolar interaction 15,16 mion structure corresponding to different values of vorticity m is due to the magnetic charge ρmag ∝ ∇ · n. To suppress ρmag and and helicity γ, as defined in Box 1. the associated energy cost, m = 1 and γ = ±π/2 are required, and Various mechanisms can generate skyrmions in magnetic sys- when m and γ do not take these values the energy cost is higher. tems, and multiple mechanisms often contribute simultaneously. However, the sign of γ = ±π/2 remains undetermined endowing the These are: (1) Long-ranged magnetic dipolar interactions. In mag- skyrmion with helicity degrees of freedom. In case (2), the energy netic thin films with perpendicular easy-axis anisotropy, the dipolar for skyrmion formation is lowest for m = 1 and γ = ±π/2, but the interaction favours an in-plane magnetization, whereas the ani- sign of γ is determined by the sign of the DM interaction, D, which sotropy prefers an out-of-plane magnetization. The competition in turn is determined by the chiral crystal structure. In the cases of between these two interactions results in periodic stripes in which the frustrated exchange interaction (3) or four-spin interaction (4), the magnetization rotates in the plane perpendicular to the thin film. the energies of skyrmions and anti-skyrmions, that is, energies for
An applied magnetic field perpendicular to the film turns the stripe values of m = ±1, are degenerate, and γ can take an arbitrary value. state into a periodic array of magnetic bubbles or skyrmions17–20. Some remarks are in order regarding the topological nature (2) The relativistic Dzyaloshinskii–Moriya (DM) interaction21,22 of magnetic bubbles, which have been extensively studied for
1RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan, 2Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan. *e-mail: [email protected]
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a b c 0.05 80 <111> <110> <111> 0.6 Field polarized Bc2 Ba2 36.3 Bc1 Ba1 6 16.5 0.5 5 7.5 C 0.4 ounts (a.u.) ) 3.41 Conical 1 0.0 1.55 (Å (T)
0.3 y B q <100> 0.70 0.2 A phase 0.32 MnSi 8 7 0.1 0.15 Helical B || <100> B 0.07 0 × 0.05 0.03 20 25 300 3355 0.050.0 0.05 T (K) 1 qx(Å ) d e f
90 nm 90 nm 30 nm
g γ = 0 γ = π γ = π/2 γ = π/2
m = 1 ny
nx
m = 1
Figure 1 | The skyrmion spin structures. a, A schematic of the spin configuration of a single skyrmion. The arrows indicate the direction of the spins, and their colours represent the normal component to the plane, that is, from up direction (red) to the down direction (blue). b, The phase diagram of
MnSi as a function of T and B. Inset: A schematic view of the triangular crystal of skyrmions, corresponding to the A phase. Ba1, Ba2, Bc1 and Bc2 are the critical values of the magnetic field on the phase boundaries.c , The Bragg spots in momentum space corresponding to the skyrmion crystal observed by SANS. d–f, Real-space observation of the helix state (d) and skyrmion crystal state (e) imaged by LTEM. In f, a magnified skyrmion frome is shown. The colour (brightness) corresponds to the direction (magnitude) of the in-plane component of the spins as shown in f. The black colour means the spins are pointing either up or down with a zero in-plane component. g, Skyrmion structures with varying m and γ. The arrows indicate the direction of the in-plane spin component, and the brightness indicates the normal component to the plane, with white denoting the up direction and black the down direction. All the structures of the anti-skyrmions (m = −1) are equivalent on rotation in the x–y plane. The relative stability of each structure is determined by the mechanism or interaction generating the skyrmion. Figure reproduced with permission from: b,c, ref. 10, © 2009 AAAS; d–f, ref. 11, © 2010 NPG. applications in magnetic memories18,40, with a focus on the z terms of the preferred excitation to drive the motion of the magnetic component of the magnetization19. When considering also the in- structures, which is an external magnetic field in the case of bubbles, plane components, some magnetic bubbles are equivalent to skyr- whereas skyrmions are often coupled to the conduction electrons mions, whereas others are topologically trivial. As discussed above, and can be driven by a very low current density43,44 (see section the size of the magnetic bubble due to dipolar interactions (case 1) ‘Dynamics of skyrmions’). is 100 nm to 1 μm, whereas that of a skyrmion with a DM interac- Skyrmions exhibit a variety of unique topological phenom- tion (case 2) is 5–100 nm; the size directly determines the ultimate ena, which can be described by the emergent electromagnetic memory density. Regarding helicity degrees of freedom, magnetic field (EEMF)45–47, associated with the spin configuration (they are bubbles have much richer structures, and often take on complex defined in the section ‘Dynamics of skyrmions’ and in Box 1). An stripy patterns41,42. The case of skyrmions and bubbles differs also in example is the topological Hall effect induced by the emergent
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Box 1 | Topological properties.
The skyrmion number model where the conduction electrons and spins are coupled The swirling structure of a skyrmion is characterized by the ferromagnetically at each site by Hund’s rule coupling. In the topological skyrmion number defined by strong coupling limit, the spin wavefunction |χ(r)> of the con- duction electron at r corresponding to the localized spin n(r) in 1 ∂n ∂n 2 (B1) equation (B2) is given by Nsk = ∫∫d rn · × 4π ∂x ∂y T Θ(r) Θ(r) | χ(r)>= cos , eiΦ(r) sin as the integral of the solid angle, and counts how many times 2 2 n(r) = n(x,y) wraps the unit sphere14,15. Using the symmetry of the skyrmion, one can write where T means the transpose. Therefore, when a conduction elec-
tron hops between two sites r and r + cηα (ηα is the unit vector n(r) = (cosΦ(φ)sinΘ(r), sinΦ(φ)sinΘ(r),cosΘ(r)) (B2) along α(=x,y,z) direction, α is the space index and c a lattice con- stant), the matrix element is given by where we introduce the polar coordinates r = (rcosφ, rsinφ).
Putting this form into equation (B1), we obtain tα(r) = t <χ(r) | χ(r + cηα)>
' 2π 1 dΘ(r) dΦ(φ) r = ' φ = 2π where t is the original transfer integral of the conduction elec- Nsk = ∫dr∫dφ sinΘ(r) =cosΘ(r) r = 0 Φ(φ) φ = 0 0 0 4π dr dφ tron. tα(r) is in general a complex number, and can be writ- icaα(r) icaα(r) ten as tα(r) = |tα(r)|e . This phase factore is analogous Accordingly, one can classify the skyrmion structures as follows. to the Peierls factor in the presence of the external magnetic
Suppose the spins point up at r → ∞ while they point down at r = 0. field and hence we can regard aα(r) as the vector potential of r = ∞ Then, [cosΘ(r)]r = 0 = 2. Now, there are several possibilities for an effective electromagnetic field. Assuming the slowly vary- φ = 2π Φ(φ). The vorticity is defined by the integerm = [Φ(φ)]φ = 0 /2π. ing spin configuration over the lattice constant c, we obtain Therefore, the vorticity determines the skyrmion number as aα(r) = −i<χ(r) | ∂αχ(r)> = ½∂αΦ(1 − cosΘ). From this expres- Nsk = m, once the boundary condition at r → ∞ is fixed. We further sion, one can easily confirm that the emergent magnetic field bz is define the helicity by the phaseγ appearing in related to the solid angle as
Φ(φ) = mφ + γ (B3) ∂ay ∂ax 1 ∂n ∂n bz == – n · × ∂x ∂y 2 ∂x ∂y We show in Fig. 1g various skyrmion structures corresponding to vorticity m = ±1, and γ = 0, ±π/2 and π, Fig. 1a corresponds to Therefore, the total emergent magnetic flux associated with
m = 1 and γ = π/2. a skyrmion is 2πNsk. These considerations can be easily gener- From equations (B2) and (B3), the magnetic charge ρmag = ∇ · n alized to the three-dimensional case and also to the emergent is calculated as electric fielde α as
dΘ m αβγ ρmag = cos (m – 1)φ + γ cosΘ + sinΘ bα = ½ε n · (∂βn × ∂γn) (B4a) dr r
eα = n · (∂αn × ∂tn) (B4b) On the other hand, the DM interaction is given by αβγ where ∂μ = ∂/∂xμ , ε is the totally antisymmetric tensor in three dimensions, and the coupling to the conduction electrons is dΘ m HDM ==Dn · (¢ × n) Dsin (m – 1)φ + γ + sin2Θ described by the Lagrangian dr 2r
or Lint = jμaμ (B5)
dΘ m HDM ==Dn · (ez × ¢)n Dsin (m – 1)φ + γ + sin2Θ with jμ being the density and current density of conduction elec- dr 2r trons as in the case of the Maxwell electromagnetic field Aμ. Here μ is the space–time index. Therefore, the state with vorticity m = +1 and γ = ±π/2 has the We mention here that the well-known spin-transfer torque
lowest energy, where the sign of γ is determined by the sign of arises from equation (B5). Namely, the variation of aμ with respect D, which in turn is determined by the crystal structure. (When to n is given by the DM vector is rotated by ±π/2, γ = 0 or π becomes the stable 109 configuration .) δaμ = –½δn · (n × ∂μn) (B6) In the case of the four-spin interaction (case 3) or the frustrated
exchange interaction (case 4), there is no distinction between the Combined with the variation of the Berry phase term ω as skyrmion and anti-skyrmion, that is, m = ±1 and γ can take an δω = –½δn · (n × ∂tn), equation (B6) leads to the equation of arbitrary value. motion as
Emergent electromagnetic field [∂t + (j · ∇)]n(r,t) = 0 One can describe the interaction between the spin textures and the conduction electrons in terms of the EEMF expressed which describes the current-driven motion of the spin texture by the spin directions. We consider the double-exchange with spatially varying n.
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a to move52. This has been observed recently54, demonstrating the physical reality of the EEMF in solids. In the following sections we 0.4 i will review these recent experimental and theoretical developments from the unified viewpoint of topology and the EEMF.
Skyrmion ii Observation of skyrmions in chiral-lattice magnets
(T) (A phase) B Among many possible skyrmion-hosting materials are the cubic but 0.2 ii i non-centrosymmetric magnets, where non-collinear spin configura- iii tions — including screw spin (single-q) and skyrmion (multiple-q) Helical states — can occur. The antisymmetric DM interaction results in the iii helical spin configuration as described by the Hamiltonian in the 0.0 55,56 0 10 20 30 continuum approximation T (K) b J 0.5 H = ∫dr (¢n)2 + Dn · (¢ × n) – B · n (1) FM t ~15 nm FM SkX Sk 2 0.4 150 yrmion density
SkX + FM where B is the magnetic field. The ground state of the Hamiltonian 0.3 in equation (1) is a helical state with a single wavevector Q whose
(T) 75 magnitude Q = 2π/λ is fixed to be D/J, while its direction is vari- B 0.2 (
H + SkX μ able. When D/J is small, the continuum approximation is justified, m
H+SkX SkX 2 and the spin plane is perpendicular to Q to minimize the energy
0.1 ) SkX 0 in equation (1). MnSi with the B20-type structure is an example of H FM(H) –1 0.0 H this helical magnet with |Q| = 0.043 Å (λ = 18 nm), which shows H a novel temperature (T)–pressure phase diagram23–28,57–65. Neutron FM t ~35 nm scattering experiments on MnSi (ref. 10) and Fe1–xCoxSi (refs 31,32) 0.4 66 H have identified the phase hitherto known as the ‘A phase’ (shown in Fig. 1b) with a two-dimensional skyrmion crystal (SkX) phase. 0.3 H + SkX + FM Theoretical analysis concludes that the SkX is stabilized by thermal (T)
B fluctuations above the conical state in a limited region near the tran- 0.2 SkX sition temperature, while the conical state with the wavevector Q 0.1 parallel to the magnetic field is more stable in the rest of the phase H diagram10. This type of SkX phase can be viewed as the hybridized 0.0 triple-q state, in which n(r) can be expressed as
FM t ~60 nm 3 0.3 n(r) ≈ n + uniform nQi(r + Δri) ∑i=1 0.2
(T) H + FM B Here, nQi(r) = A[ni1cos(Qi · r) + ni2sin(Qi · r), and Qi · Δri is the 0.1 phase of each screw component, A is the magnetization of a single H SkXFM helix with the q-vector Qi, and nuniform is the uniform magnetization 0.0 0.3 induced by the Zeeman effect. The threeq -vectors are all normal to t ~75 nm the applied magnetic field direction, forming an angle of 120° with FM 0.2 each other, and satisfying the relation
(T) 3 B H + FM 0.1 Q = 0 ∑ i H SkX i=1 0.0 0 100 200 T (K) When one skyrmion is singled out from this hexagonal SkX state, it corresponds to the case m = 1, γ = π/2 shown in Fig. 1g. The mag- Figure 2 | Phase diagrams of thin-film samples of chiral magnets as netic helicity of the skyrmion depends on the sign of the DM inter- a function of magnetic field and temperature.a , Phase diagram of a action, as well as that of the screw state. MnSi thin film with a thickness of ~50 nm obtained from the patterns in The SkX can be observed in terms of the magnetic scattering of the diffractographs shown in the insets i, ii and iii, which correspond to neutrons (and possibly of X-rays). Reflecting the large length-scale skyrmion, disordered and helical phases, respectively. The blue dots show of the SkX compared with the lattice spacing, small-angle neutron (T,B) points where diffractographs were actually taken, and the orange dots scattering (SANS) is employed10,31,32,34,37,58,62,63,67–76. The reciprocal- correspond to those whose diffractograms are shown in the inset.b , The space SANS patterns for the SkX on the plane normal to the incident dependence of the SkX phase diagram for FeGe on the sample thickness neutron k-vector shows up as a hexagon with the norm of |Q|, that (t). H, helical state; FM, ferromagnetic state. The colour scale indicates is, Fourier transform of the two-dimensional hexagonal SkX lattice the skyrmion density per square micrometre. Figure reproduced with as shown in Fig.1c (ref. 10), whereas the conical spin phase with the permission from: a, ref. 85, © 2012 ACS; b, ref. 36, © 2011 NPG. wavevector parallel to the magnetic field generates no pattern on this plane. A similar observation of the SkX will be possible with magnetic field of the skyrmions on conduction electrons, which use of resonant soft X-ray scattering with the advantage of being has been experimentally observed48–53. The motion of the skyrmions applicable to crystals of reduced size or thin films; this remains an leads to the temporal change of the emergent magnetic field and ongoing experimental effort. hence to an electromagnetic induction. The induced emergent elec- For the real-space observation of SkX and also individual/ tric field contributes to the Hall effect when the skyrmions begin isolated skyrmions at the nanometre scale, various electron and
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© 2013 Macmillan Publishers Limited. All rights reserved NATURE NANOTECHNOLOGY DOI: 10.1038/NNANO.2013.243 REVIEW ARTICLE scanning probe microscopy techniques may be employed. Examples include the recent application of magnetic force microscopy for the Table 1 | List of transition temperatures (TN) and helical periods observation of a skyrmion crystal on the surface of a bulk crystal77 (λ) of helimagnets. and spin-polarized scanning tunnelling microscopy on ultrathin Material TN (K) λ (nm) Reference magnetic thin films78,79. Furthermore, various spin-polarized elec- MnSi Bulk 30 18 23 tron microscopy techniques80 may be used for the observation of skyrmion crystals and single skyrmions. One important condition Epitaxial thin film 45 8.5 51 for such observations is, however, the necessity of an external mag- Mn1–xFexSi x = 0.06 16.5 12.5 25 netic field, which limits the use of some electron microscopy tools. x = 0.08 10.6 11 25 In this context, LTEM is a powerful method that has nanometric x = 0.10 6.8 10 25 spatial resolution and allows the application of a vertical magnetic Fe1–xCoxSi x = 0.10 11 43 29,33 81–86 field . It can be used to observe the in-plane magnetization M x = 0.5 36 90 29,33 of a plate-like specimen of thickness of less than 100 nm, through which the electron beam can transmit. Lorentz transmission elec- x = 0.6 24 174 29,33 tron microscopy makes use of the generated in-plane B field aris- x = 0.7 7 230 29,33 ing from the in-plane M, which generates the Lorentz force on the MnGe T = 20 K 170 3 50 incoming electrons. The deflected electrons from regions with dif- T = 100 K - 3.4 50 ferent in-plane M show interference patterns at under-focused and T = 150 K - 5.5 50 over-focused positions. For example, magnetic domain walls appear Mn1–xFexGe x = 0.35 150 4.7 38 as bright or dark lines in the defocused position80,82, whereas the x = 0.5 185 14.5 38 proper screw state exhibits a striped contrast84. By analysing the two patterns in terms of the transport-of-intensity equation method83, x = 0.7 210 77 38 the local in-plane M distribution can be visualized with a spatial x = 0.84 220 220 38 resolution of a few nanometres. Lorentz transmission electron FeGe Bulk 278 70 34 microscopy can work well for the real-space observation of skyr- Cu2OSeO3 Bulk 59 62 76 mions or SkX with a size of more than a few nanometres, not only Thinned plate - 50 86 in B20-type helimagnets but also in many chiral or achiral magnets, corresponding to cases (1) to (3) described earlier. In a two-dimen- sional model of helical magnets, the skyrmion crystal state is stable over a wide range of T and B compared with the conical state when and collinear (ferromagnetic) spin sates. In thin (<100 nm) films 87–89 the magnetic field is perpendicular to the plane . In the context of MnSi and Fe1–xCoxSi, the SkX phase under an intermediate field of the formation of stable skyrmions in the thin-plate crystal used remains stable even at the lowest temperatures. The increasing in LTEM measurements, it has been theoretically discussed that the stability of the SkX phase compared with the conical phase with uniaxial anisotropy90–93 or the inhomogeneous chiral modulations decreasing film thickness is clearly discerned in Fig. 2b for the SkX 94 — across the layer thickness stabilizes the skyrmion phase over the phase of FeGe (as = (2/√ 3 ) × 70 nm). This is because the spins can- conical state. The importance of the combination of the DM and not complete the spiral along the magnetic field and/or because the dipolar interactions has also been discussed95. magnetic anisotropy is changed when the thickness of the thin film We exemplify the two topological spin textures observed for a becomes the order of, or smaller than, λ (refs 87–95).
(001) thin plate of Fe0.5Co0.5Si with B20-type chiral crystal struc- 11 ture (space group P213) . The LTEM observation of the state below Skyrmions in different magnetic systems the magnetic transition temperature (~40 K) at zero magnetic field Helimagnets with high symmetry but chiral crystal structures reveals the stripy pattern (Fig. 1d) of the lateral component of M such as the cubic B20-type (P213) compounds are likely to host with a period of λ = 90 nm; this corresponds to the proper screw the skyrmion phase either in the presence of an applied magnetic spin structure propagating along [100] or [010]. When a magnetic field or when the material is in a thin-film form. (See Table 1 for field of 50 mT is applied perpendicular to the film, a two-dimen- the list of materials.) The validity of the effective spin Hamiltonian sional SkX as described above is observed in real space (Fig. 1e). A approach (for example, equation (1)) to the chiral-lattice magnet hexagonal lattice is formed by periodic arrays of spin-swirling— struc- has been demonstrated also for the insulating magnet Cu2OSeO3 tures (Fig. 1f) and the lattice spacing (as) is identical to 2/√3 of the with the same space group P213 as the B20 compounds (but with stripe period λ (λ = 90 nm). The helicity in each skyrmion reflects a different atomic arrangement)86. In this material, the unit cell the sign of the DM interaction of this compound, hence indicating is composed of four inequivalent Cu sites; three Cu spins align the uniformity of the chiral domain of the crystal structure. When approximately parallel and one spin antiparallel, forming a locally crystal domains with different chirality are observed, the skyrmion ferrimagnetic state97. In reality, due to the DM interaction aris- helicity can be reversed96. Although LTEM cannot specify the direc- ing from the chiral lattice structure, the coarse-grained local mag- tion of the magnetization perpendicular to the film, it is possible to netization is gradually twisted to form a proper screw state with retrieve such information by knowing the direction of the external a period λ of 50 nm. In spite of the absence of the conduction magnetic field. With the external field applied downward, the black electrons, the insulating helimagnet Cu2OSeO3 shows an analo- background (that is, zero lateral component) can be assigned to the gous magnetic phase diagram in the T–B plane to those of metal- upward direction of spins whereas the black cores can be assigned lic B20-type (for example, MnSi) compounds; the SkX phase in to the downward direction. bulk is also confirmed to exist in a narrow T–B window near the The experimental phase diagrams for the SkX were obtained on phase boundary with the paramagnetic state by SANS experi- the basis of such real-space observations by LTEM. Some results on ments, as well as by magnetization measurements69,76,86. Again, the thin plates of MnSi (ref. 85) and FeGe (ref. 36) are summarized in LTEM observation indicates that the hexagonal-lattice SkX phase Fig. 2a,b. In Fig. 2b, the phase change of the spin texture in FeGe expands to a broad T–B region in the thin (~50 nm) film speci- is represented as contour maps of skyrmion density. As compared men, analogous to the results shown in Fig. 2. with the bulk (Fig. 1b), the SkX phase in thin films extends over a An important point to note here is that the insulating chiral- wider region in the T–B plane, surrounded by the screw (conical) lattice magnet is multiferroic. In multiferroics, the ferroelectric and
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a b c B = 0 mT B B= =100 100 mT mT B = 180 mT
500 nm 500 nm 500 nm
d e f B = 100 mT Mz 3 2 1 4
5 1 2 3 4 5 6 7 8 9
My
Mx
6 9 100 nm 100 nm 8 7
Figure 3 | Various spin textures in dipolar-driven skyrmion systems. a–c, TEM images of a (001) Sc-doped barium ferrite thin film.a , The striped domains at zero field.b , The triangular skyrmion lattice in a 100-mT field. Bright and dark colours correspond to the two helicitiesγ = ±π/2 of skyrmions, which are randomly distributed even though the positions of skyrmions are ordered. c, By increasing the magnetic field to 180 mT, the size of skyrmions is reduced. d, Lorentz microscope image of the triangular lattice of skyrmions with the triple-ring structure and random helicities. Colour wheel and white arrows indicate the magnitude and direction of the in-plane magnetization. e, Close up of a skyrmion. f, Mapping of spin direction in the skyrmion to the Mx –My
–Mz sphere. Numbers 1–9 correspond to the distance from the core shown in e. The direction of the spin rotation goes back and forth corresponding to the helicity degrees of freedom. Figure reproduced from ref. 41, © 2012 NAS. magnetic orders coexist98–102. In chiral-lattice magnets, in general, We show some examples of the helimagnetic (including the chemical bond, for example, the hybridization between mag- skyrmion) phases as observed by LTEM for the centrosymmetric netic d-orbital and ligand p-orbital states, is modulated by the magnet, Ba(Fe1–x−0.05ScxMg0.05)12O19 (x = 0.16), hereafter referred to spin–orbit interaction when the compound exhibits ferromagnetic as BFSO (ref. 41). The parent compound BaFe12O19 (BFO) is a well- or related orders, thereby producing a macroscopic electric polari- known hexaferrite, which is commercially produced in large quan- 102 zation (P) . In the case of Cu2OSeO3 with space group P213, for tities, with a magnetic transition temperature above 700 K and a example, symmetry considerations predict that P = 0 for M || [001], magnetic easy-axis along the c axis. In a thin (~50 nm) c-plane film while non-zero P || [001] for M || [110], and non-zero P || [111] for of BFSO, in which the magnetic anisotropy is controlled and slightly M || [111] for a uniform M (refs 86,103,104). In accordance with the weakened by Sc doping, a highly disordered helimagnetic texture vortex-like M distribution of the skyrmion, a corresponding local with frequent reversals of magnetic helicity is observed at room P distribution is also expected, producing the polarization-density- temperature and in zero magnetic field (Fig. 3a). When a magnetic wave-like feature in the SkX phase103,104. This localP distribution field is applied normal to the film, the SkX phase emerges as shown may also enable the manipulation of skyrmions in terms of electric by under-focused LTEM images (Fig. 3b,c); each skyrmion shows field gradients105,106. as either a black or a white circle-disk in a random manner, indicat- As described in the section ‘Observation of skyrmions in chiral- ing that the skyrmion helicity (for example, clockwise or anticlock- lattice magnets’, the emergence of skyrmions is not restricted to the wise curl of the in-plane M) is not correlated, but disordered in this DM-interaction helimagnets but also occurs in centrosymmetric triangular-lattice of SkX. On increasing the field strength, the skyr- magnets in a thin-film form. In this case, the dipolar interaction mion size tends to decrease, while the SkX lattice constant increases plays a key role, together with the uniaxial magnetic anisotropy. (Fig. 3c). This is in contrast to the case of the nearly magnetic-field- Similar to the case of helimagnets, the striped phase is observed at independent shape of the SkX in the chiral-lattice DM magnet. low fields when the out-of-plane magnetic anisotropy exceeds some To scrutinize each skyrmion, its magnified picture based on the critical value to suppress the uniform in-plane M. On the other transport-of-intensity analysis is shown in Fig. 3d,e. The in-planeM hand, the bubble lattice phase in such an anisotropic Heisenberg curl (magnetic helicity) changes its direction (sign) twice in going system may show a rich variety of bubble shapes due to the higher- from the peripheral to core regions, forming a triple-ring feature. order terms of the magnetic anisotropy as well as to the magnetic When mapped on to the unit sphere, the local M shows a pendu- helicity degree of freedom. lum-like oscillation, as shown in Fig. 3f. Nevertheless, in total, the
904 NATURE NANOTECHNOLOGY | VOL 8 | DECEMBER 2013 | www.nature.com/naturenanotechnology
© 2013 Macmillan Publishers Limited. All rights reserved NATURE NANOTECHNOLOGY DOI: 10.1038/NNANO.2013.243 REVIEW ARTICLE change in M wraps around the total sphere only once, indicating field, k is the wavevector, ν, τ , –e and f0 are the velocity, mean free Nsk = 1. In this way, the helicity degree of freedom allowed in the time, charge and equilibrium distribution function of electrons, centrosymmetric magnet may greatly enrich the skyrmion features, respectively. In equation (3), j is the current density of conduction 41 as predicted also by simulations . Note that the equation of motion electrons, HS is the spin Hamiltonian, αG the Gilbert damping con- for a skyrmion does not depend on the helicity but on the skyrmion stant and β represents the non-adiabatic effect107–109. Because of the number as described in the section ‘Dynamics of skyrmions’. Lorentz force −e[(E + e) + ν × (B + b)] in equation (2), a Hall effect Recently, a spontaneous atomic-scale magnetic skyrmion lat- occurs due to b, called the topological Hall effect (THE). In the SkX tice has been discovered by spin-polarized scanning tunnelling phase, the periodic array of the unit flux (ϕ0 = h/e, where h is the microscopy in a hexagonal Fe film of one atomic layer on an Ir(111) Planck constant) associated with each skyrmion leads to the almost surface12. The skyrmion lattice is incommensurate with the underly- uniform emergent magnetic field as ing atomic lattice. Theoretical analysis has concluded that the four- spin interaction, combined with the DM interaction, stabilizes the 3ϕ0
EEMF aμ originating from the coupling of the spin textures to the cedure is demonstrated in Fig. 4c. The topological Hall resistivity T conduction electrons (see Box 1). One remarkable consequence of ρH at 25 K (Fig. 4c) is about 8 nΩ cm, comparable with or slightly T this coupling is the spin-transfer torque effect as described in Box 1, larger than the bulk value. As seen in the contour map of ρH, the and hence the combined equations of motion for the conduction expanded region of the SkX phase in the MnSi thin film can be electrons and the localized spins52 deduced (Fig. 4d). As an even more conspicuous example of the THE in the SkX ∂f 1 phase, we show results of the Hall resistivity for a bulk polycrystal of + ν · ¢r f – e (E + e) + ν × (B + b)· ¢k f = – ( f – f0) (2) ∂t τ B20-type cubic MnGe, which has a short helical pitch λ that changes from 3 nm to 6 nm with temperature50. The temperature-depend- ent change of λ indicates that there should be another important ∂n δHS ∂n factor, which is likely to be the magnetic anisotropy, in addition + (j · ¢)n = –n × + n × αG + β(j · ¢)n (3) ∂t δn ∂t to the competition between the ferromagnetic exchange and DM interactions. Reflecting this nature, the SkX feature seems to be where equation (2) is the Boltzmann equation for the electron dis- qualitatively different from the conventional one, that is, the vertical tribution function f(r,k,t) at time t, and equation (3) is the Landau– magnetic field-induced hexagonal skyrmion lattice observed typi- Lifshitz–Gilbert equation for the spin. In equation (2), E(B) is the cally in chiral-lattice magnets. Recent SANS measurements indicate electric(magnetic) field, e(b) is the emergent electric(magnetic) that the three-dimensional skyrmion lattice as shown schematically
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6 –2 a h j = 0 j = 2.81 × 10 A m b 7 B (mT) Electron flow 350 Topological m) 6 300 Hall e ect Ω 270 10
– 250 5 220 200 (×10 180 xy 4 150 ρ b 130 Spin-transfer Skyrmion i 3 torque Hall e ect j (MA m–2) 6 0 0.168 m) 0.281 Ω 0.393 10
– 0.562 5 0.674 e 0.786 (×10 Emergent 0.899 xy 1.124 electromagnetic induction ρ B = 250 mT 1.686 4 2.241 2.810 24 25 26 27 28 29 30 31 T (K) Δ
j ρ b MnSi thin film (50 nm) e MnGe bulk (polycrystal) 1 xy m) 5.84 20 0.4 (×10 30 K Ω 5.74 0 10 – – 0 11 0.2 5.64 –1 Ω
20 K (×10 cm) m)
cm)
20 xy
Ω 5.54 T = 28.42 K –2 10 K ρ μΩ ( (n 2K Δ
yx 0.0
yx 40 10K ρ ρ ρ 15K 0 xy
m) 5.56 20K 30K 5 K (×10
60 Ω 25K 35K 0.2 5.46 j –1 10 c – c f 160 K – R0H 200 K 11 20 T=25 K 5.36 –2 Ω T 0.2 100 K (×10 m)
10 ρyx 70 K xy 5.26 T = 28.21 K –3 0 50 K ρ cm) cm) Δ Ω 10 0.0
μΩ ρ ( (n Observed 20 0 xy yx 5.12 T m) (×10
ρ 30 K
Hall 30 Ω ρ 40 A 0.2 20 K 5.02 –1 ρyx 10 K 10 j –
c – 50 5 K 11
4.92 –2 Ω
0.0 0.5 1.001.5 51015 (×10 m)
μ H (T) μ H (T) xy 4.82 T = 27.02 K –3 0 0 ρ
T Δ
ρyx (nΩ cm) ρ 0 xy
m) 4.47 10 5 0 5 10 (×10 Ω 4.37 –1 d g 10 j
– c 5.0 – M – H 11 15 4.27 –2 Ω ρyx – H 0.3 (×10 m)
ρ T yx xy 4.17 T = 26.00 K –3 1.0 ρ 10 ( μΩ
Δ
(T) (T) H ρ H cm) 0 0 0 1 xy μ μ
m) 4.5 0.5 5 (×10 Ω 4.4 0 10 – 0.2 – 0 11 0.0 0 100 200 4.3 –1 Ω (×10 m) 5101520253035 40 45 T (K)
xy 4.2 T = 25.83 K –2 T (K) ρ 0123 j (106 A m–2)
Figure 4 | Topological phenomena of skyrmions. a, Schematic picture of skyrmion motion and associated physical phenomena under the flow of electrons. Electron current drives the flow of a skyrmion by means of the spin-transfer torque mechanism. (Note that the direction of the charge current is the opposite to the electron flow.) Electrons are deflected by the Lorentz force due to the emergent magnetic fieldb of the skyrmion, which results in the THE. The velocity of the skyrmion has its transverse component proportional to αG–β, that is, the skyrmion Hall effect. The motion of the skyrmion is accompanied by the time-dependent emergent magnetic field b (pink), and hence the emergent electric fielde , that is, emergent electromagnetic induction. b–g, Experimental observation of the THE. b, Hall resistivity ρH = ρyx in a MnSi thin film as a function of the magnetic field N A T 2 T at various temperatures. c, The decomposition of the Hall resistivity to various contributions as ρH = ρH + ρH + ρH = R0H + SAρxxM + ρH . d, Colour contour T T map of the magnitude of the topological contribution ρH + ρyx. e–g, Same as for b–d, except for a MnGe bulk sample. The inset of g is a model calculation of the three-dimensional skyrmion crystal. h–j, Experimental data for the current-driven emergent electromagnetic induction in bulk MnSi. h, Hall resistivity ρyx as a function of temperature for several magnetic field strengths. The black curves are for the limit of zero d.c. current density j, and the red are for j = 2.81 × 106 A m–2. Black arrows indicate the phase boundaries of the A phase, that is, skyrmion crystal state. It is noted that the Hall signal is suppressed in the skyrmion crystal phase due to the topological contribution (blue shaded region). i, Current density j dependence of the Hall resistivity in the skyrmion lattice phase of MnSi under a large applied d.c. electric current. Black arrows indicate the phase boundaries of the skyrmion crystal state. j, The change in the Hall resistivity ρyx as a function of j at B = 250 mT for several temperatures. The suppression of the signal above the threshold current density jc, which corresponds to the onset of the skyrmion crystal motion, is observed. This can be understood by the emergent electric field that is generated by the temporal change of the emergent magnetic field associated with skyrmions, that is, emergent electromagnetic induction. The error bars represent the statistical error of the data after subtracting a spline fit54. Figure reproduced with permission from: b–d, ref. 51, © 2013 APS; e–g, ref. 50, © 2011 APS; h–j, ref. 54, © 2012 NPG.
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Anticlockwise rotation mode Breathing mode
a b T = 57.5 K, H H c T = 57.5 K, H || H Cu2OSeO3, H || [110] dc ac dc ac 800 F F erri 15 1,000 Oe errimagnetic 8 1,000 Oe Ferrimagnetic 400 Oe 800 Oe C onical 600 Oe 600 180 Oe
400 Oe C 6 onical 350 Oe 10 150 Oe Oe) 340 Oe Sk B) B) yrmion crystal
320 Oe (d (d
Sk 400 120 Oe 11 11 yrmion crystal S S 280 Oe 4 ∆ Helical ∆ (single-q domain) 240 Oe 80 Oe Magnetic field ( 200 Oe 5 Skyrmion crystal 160 Oe 50 Oe 200
2 Helical/
140 Oe Helical/ 100 Oe 40 Oe conical Helical 50 Oe conical (multi-q domain) 0 Oe 0 Oe 0 0 0 40 50 0 1 2 3 4 0 1 2 3 4 5 Temperature (K) Frequency (GHz) Frequency (GHz)
Figure 5 | Experimental data for skyrmion magnetic resonances in Cu2OSeO3. a, Magnetic phase diagram determined by the magnetic susceptibility. The vertical arrow indicates the magnetic field scan atT = 57.5 K. b,c, Microwave absorption spectra ΔS11 for the static magnetic fieldH dc perpendicular to the a.c. magnetic fieldH ac at various magnetic fields atT = 57.5 K. c, Similar plot to b for Hdc parallel to Hac. The schematics at the top are of the corresponding collective modes. The colours indicate the normal component of the spins from positive (red) to negative (blue). The arrowheads show the time evolution profile of a skyrmion in its crystal phase. Figure reproduced with permission from ref. 125, © 2012 APS. in the inset to Fig. 4g may be realized in the whole T–B region of the ferromagnets. Recently, the current-driven motion of the skyrmion helimagnet72. At zero field, the compound shows no net magnetiza- crystal in a thin film of FeGe has been studied by LTEM44. By com- tion and the SkX may be alternately composed of skyrmions and paring the motion of the helical and the skyrmion crystal phases, anti-skyrmions; this state may be viewed as the multiple-q state of it was observed that a current density less than 106 A m−2 can drive the three <100> helices orthogonal to each other. An external mag- the motion only in the latter case. This small critical current den- netic field may induce a net magnetization and cause a non-zero sity enables the manipulation of information with low-power con- scalar net spin chirality (skyrmion number), giving rise to the THE. sumption per bit in the slow-speed regime. However, high-speed The results of Hall resistivity for MnGe are shown in Fig. 4e and the manipulation requires a similar current density to that for domain extracted topological component in Fig. 4f (ref. 50). In the mag- wall motion. T netization process, the topological Hall resistivity ρH reaches a maxi- Equations (B4a) and (B4b) in Box 1 indicate that once the skyr- mum value of –200 nΩ cm, which appears nearly constant as long as mions move, an electric field is induced by the relation λ (~3 nm) stays nearly constant below 70 K. This is a feature of the ∂b real-space EEMF b acting on the conduction electron, as in the case ¢ × e = − of the Lorentz-force Hall effect that is independent of temperature ∂t T or electron scattering rate. The magnetic-field variations for ρH can be well accounted for by the cubic lattice of the SkX. The magnitude or more explicitly e = νd × b, where νd is the drift velocity of the of pb is estimated to be ≈40 T. Thus, the THE of the high-density skyrmion crystal. As a function of the current density j, νd remains SkX can greatly enhance the action of the external magnetic field. zero below a critical value jc, and begins to increase for j > jc. There are several theoretical issues studied besides the THE dis- Correspondingly, the induced emergent electric field e gives a cussed above, such as the dynamics of the conduction electrons and Hall effect in the opposite direction to the topological Hall effect52 111–113 spin system , and possible three-dimensional skyrmion crys- (Fig. 4h). Therefore, Δρyx as a function of j shows a characteristic tals114,115. Also some experiments suggest the existence of the three- behaviour, which is observed experimentally in MnSi (Fig. 4i,j)54. dimensional skyrmionic states116,117. Theoretically, it is important to derive the equation for the motion of the centre of mass of a skyrmion. This can be achieved Dynamics of skyrmions by the method originally proposed by Thiele and its generaliza- The real advantage of skyrmions compared with other magnetic tion107–109. The results read71 nanostructures lies in their peculiar dynamics especially when combined with the conduction electrons or electric polarization. dνd MS + G × (j – νd) + κ (αGνd – βj)= –¢U (4) Intensive interest has been triggered by the experimental discov- dt ery43 of current-driven skyrmion crystal motion by an ultralow cur- 6 −2 rent density of the order of 10 A m , which is five or six orders where νd = dR/dt = (X˙ ,Y˙ ), R = ((X,Y) are the centre of mass coordi- of magnitude smaller than that needed for domain wall motion in nates), MS is the mass of the skyrmion, κ is a dimensionless constant
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a b HL, = 2 , pure 30 β αG HL, β = 2αG, with impurities 25 HL, β = αG, pure
20 SkX, β = 0, 0.5αG, αG, 2αG, ) pure, with impurities −1
15 HL, β = αG,
(m s with impurities v 10 HL, β = 0.5αG, pure HL, β = 0.5αG, with impurities 5 HL, β = 0, pure, with impurities 0 0510 15 20 c j (A m−2) 1010 d T = 0 ns T = 2.8 ns 5 MA cm–2 v = 55.7 m s–1 T = 0 ns T = 2.8 ns mz 1 MA cm–2 v = 12.1 m s–1 +1 0 T = 0 ns –1 5 MA cm–2 T = 2.8 ns T = 0 ns 1 MA cm–2 Final position 20 nm
Figure 6 | Simulations of the skyrmion motion driven by current. a, Current–velocity (j–v) relation for the helix structure (HL) and skyrmion crystal (SkX) calculated by solving the Landau–Lifshitz–Gilbert equation numerically. The j–v characteristics for a skyrmion crystal (blue) are universal, and independent of αG, β and the impurity pinning effect, while thej –v characteristics for the helical phase are sensitive to these factors (red and pink). b, Snapshot of the skyrmion crystal in motion and its Fourier transformation (inset). The green dots specify the position of impurities. c, Zoom-in view showing the deformation of an individual skyrmion as well as of the skyrmion lattice at motion. d, Motion of a skyrmion under the application of current in a stripe- shaped sample without (upper panels) and with (lower panels) a notch structure. Figure reproduced with permission from: a–c, ref. 122, © 2013 NPG; d, ref. 109, © 2013 NPG.
of the order of unity, and G = 2πNskez (where ez is the unit vector the three k = 0 skyrmion ‘optical’ modes; namely, anticlockwise along the z direction) is the gyrovector. U is the potential for the (ACW) rotation, clockwise (CW) rotation and breathing modes skyrmion due to boundary effects, magnetic fields and impurities. of the skyrmion core, as schematically shown in the top panels of
The mass term in equation (4) comes from the deformation of the Fig. 5b,c. When an alternating current (a.c) magnetic field (Hac) moving skyrmion compared with the static solution, and can be of the microwaves is applied parallel to the skyrmion plane (that neglected in the low-frequency limit. Then, equation (4) indicates a is, Hac ⊥ Hdc), the former two rotational modes should be excited. canonical conjugate relation between X and Y. The velocity Rd /dt is In the SkX phase region denoted by the red curves in Fig. 5b, a perpendicular to the force or along the equal potential contour sim- skyrmion-related magnetic resonance at 1.0 GHz is observed ilar to the case of a charged particle in a magnetic field. It is straight- (indicated by closed triangles), distinct from the magnetic reso- forward to generalize this equation of motion to many skyrmions. nances of the conical and ferromagnetic (spin collinear) states; Because the skyrmion crystal breaks translational symmetry, this mode is assigned to the ACW motion mode of the skyr- there is a Goldstone mode, that is, a phonon of the skyrmion crys- mion based on simulation results, although the higher-lying CW tal. Note that rotational symmetry in spin space is broken by the mode is not clearly distinguishable from the conical state mag- external magnetic field, and hence the spin wave modes are gapped, netic resonance mode. The well-defined breathing mode of the and this phonon mode is the only gapless low-lying mode. It is pre- skyrmion appears at 1.5 GHz in the configuration ofH ac || Hdc as 2 dicted that there is only one phonon branch with dispersion ωk ∝ k shown in Fig. 5c, where no magnetic resonance is allowed for the (refs 52,118,119). It is noted here that the mass term has been sug- other magnetic phases. The SkX phase also shows a ferroelectric gested to play an important role in the dynamics of a skyrmion polarization as described in the section ‘Skyrmions in different bounded in a circular disk as observed in numerical simulations120,121. magnetic systems’, therefore all the modes are magnetoelectric Another prediction from equation (4) is the skyrmion Hall effect. in nature; they should be active also for a.c. electric-field exci-
In the presence of the Gilbert damping αG and non-adiabatic effect tation and hence are termed electromagnons. Such magnetically β in the Landau–Lifshtz–Gilbert equation, there occurs a velocity and electrically active skyrmion excitation modes are expected to
ν⊥ = (αG − β)j transverse to the current j when higher-order terms in show the optical magnetoelectric effect, such as the directional 52,71,122–124 αG and β are neglected . dichroism, that is, a distinct response depending on the direction The collective modes of the skyrmion crystal were experimen- of light propagation (microwave)126,127. tally studied by the magnetic resonance in insulating Cu2OSeO3 (ref. 125). In its bulk form, the SkX phase is confined within a Perspectives on ‘skyrmionics’ rather narrow window of magnetic field H( dc) and T as shown in From the viewpoint of applications skyrmions have several poten- Fig. 5a, contrary to the extended SkX region of its thin-film form. tial advantages compared with the domain walls in ferromag- The SkX hexagonal lattice plane (referred to as thexy plane) is nets107,108. Both are subject to the spin-transfer torque effect, and can formed perpendicular to the Hdc direction (z). Then, we can define be driven by a spin-polarized current. However, even though the
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© 2013 Macmillan Publishers Limited. All rights reserved NATURE NANOTECHNOLOGY DOI: 10.1038/NNANO.2013.243 REVIEW ARTICLE equations of motion seem to be similar in the two cases, the physics formation of composite structures including merons, reversal of are essentially different122,123,124. Namely, the equation of motion for helicity, possible three-dimensional skyrmion structures140, and the the magnetic domain wall reads with constant current density non-equilibrium phenomena under a current141. From an applica- tions perspective, the nanoscale fabrication of samples is an impor- • ∂U tant step, along with the design and demonstration of skyrmion αGX = βjx – (5) ∂X logic circuits enabled by fundamental understanding of the basic physics of the skyrmions. which should be compared with equation (4). (Here MS has been neglected assuming steady motion.) The essential difference comes Received 27 June 2013; accepted 17 October 2013; from the fact that the emergent magnetic field bz is finite for skyrmi- published online 4 December 2013 ons, which gives the canonical conjugate relation between X and Y, whereas for a magnetic domain wall bz is zero, and the tilting angle References from the easy plane is the conjugate variable to X. Equation (5) 1. Skyrme, T. H. R. A unified field theory of mesons and baryons.Nucl. Phys. 31, 556–569 (1962). yields the relation X˙ = (β/αG)jx without the pinning potential. Especially, when β = 0, there is no motion induced by the current j This is the original paper that proposed the skyrmion as a model for hadrons. x 2. Sondhi, S. L., Karlhede, A., Kivelson, S. A. & Rezayi, E. H. Skyrmions and the as long as it is smaller than a critical value. This is called intrinsic crossover from the integer to fractional quantum Hall effect at small Zeeman pinning. Considering that αG,β << 1, the pinning effect is enhanced energies. Phys. Rev. B 47, 16419–16426 (1993). compared with the case of skyrmion by a factor of 1/ β (>>1) 3. Wright, D. C. & Mermin, N. D. Crystalline liquids: the blue phases. (refs 71,122). Rev. Mod. Phys. 61, 385–432 (1989). Rosch et al.71 introduced a phenomenological expression for 4. Ho, T. L. Spinor Bose condensates in optical traps. Phys. Rev. Lett. the pinning force as F = −4πM f(ν /ν )(ν /ν ) where f is a scal- 81, 742–745 (1998). pin S d pin d d 5. Bogdanov, N. & Yablonskii, D. A. Thermodynamically stable “vortices” in ing function and νpin is a velocity characterizing the strength of the magnetically ordered crystals. The mixed state of magnets.Sov. Phys. JETP 122,123 pinning force. Iwasaki et al. studied the current-driven motion 68, 101–103 (1989). of both the helical state and skyrmion crystal in a model with DM 6. Bogdanov, N. & Hubert, A. Thermodynamically stable magnetic vortex states interaction taking into account the impurity pinning effect due to in magnetic crystals. J. Magn. Magn. Mater. 138, 255–269 (1994). the easy-axis anisotropy. They observed a universal current–veloc- 7. Röβler, U. K., Bogdanov, N. & Pfleiderer, C. Spontaneous skyrmion ground ity relation for the skyrmion crystal almost independent of α , β states in magnetic metals. Nature 442, 797–801 (2006). G This is an early theoretical work predicting the formation of skyrmions in and impurity pinning, in sharp contrast to the motion of the heli- a magnetic system. cal state that shows current–velocity relations similar to those of a 8. Binz, B., Vishwanath, A. & Aji, V. Theory of the helical spin crystal: magnetic domain wall122,123 (Fig. 6a). The reduced critical current A candidate for the partially ordered state of MnSi. Phys. Rev. Lett. density for the skyrmion crystal was attributed to the deformation 96, 207202 (2006). of the crystal and of individual skyrmions to avoid the impurity 9. Tewari, S., Belitz, D. & Kirkpatrick, T. R. Blue quantum fog: Chiral potential122,128,129 (Fig. 6b,c). Note that this result comes from equa- condensation in quantum helimagnets. Phys. Rev. Lett. 96, 047207 (2006). 10. Mühlbauer, S. et al. Skyrmion lattice in a chiral magnet. Science tion (4), which gives the motion of skyrmions transverse to the 323, 915–919 (2009). potential gradient. The pinning of skyrmions has some resemblance This is the first experimental report on neutron scattering that identifies to that of vortices in superconductors130. The reduced critical cur- the A phase in a chiral magnet as the skyrmion crystal phase. rent density could enable low-energy operation for future memory 11. Yu, X. Z. et al. Real-space observation of a two-dimensional skyrmion crystal. devices. 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