
DOI: 10.22661/AAPPSBL.2019.29.4.49
Skyrmions: Emergent Topological Particles In Magnets
NAOTO NAGAOSA
RIKEN CEMS, WAKOSHI, SAITAMA, JAPAN
THE UNIVERSITY OF TOKYO, DEPARTMENT OF PHYSICS, TOKYO, JAPAN
ABSTRACT
A skyrmion is a swirling magnetic texture, characterized by the topological wrapping number of the unit sphere spanned by the directions of the magnetic moments, i.e., the skyrmion number. Theories of magnetic skyrmions are reviewed focusing on their topological aspects and dynamics. The solid angle subtended by the magnetic moments acts as the emergent magnetic field associated with the Berry phase, which governs the quantum dynamics of the conduction electrons coupled to skyrmions as well as the dynamics of skyrmions themselves.
INTRODUCTION
Fig. 1: Schematic picture of a skyrmion.
Skyrmions were originally proposed as a model for hadrons in nuclear physics, and its realization in condensed matter systems has provided rich physics, seen in examples such as Quantum Hall systems, liquid crystals, and BoseEinstein condensates. As for magnets, following an early theoretical proposal [1], skyrmions have been observed in recent years by neutron scattering [2] and Lorentz microscopy [3], which triggered intensive interest. Figure 1 shows the schematic picture of a skyrmion. It is described by the simple Hamiltonian given by [4]
 (1)

where n is the direction of the magnetic moment, J is the ferromagnetic exchange interaction, D is the relativistic DzyaloshinskiiMoriya (DM) interaction allowed by the chiral crystal structure, and B is the external magnetic field. In three dimensions, the skyrmion crystal (SkX) phase is stable only in a narrow region near the magnetic transition temperature and at finite B, while SkX is stable in the much wider region in twodimensions. The solid angle subtended by the magnetic moments leads to the emergent magnetic field for the conduction electrons coupled to them. When a conduction electron hops from site i to site j, the inner product <χ_{i}χ_{j}> of the two spin wavefunctions χ_{i}> and χ_{j}> enters into the transfer integral, whose phase factor acts as the vector potential a_{ij}. This is an example of the Berry phase, and its integral over a closed loop defines the "magnetic flux". This magnetic flux can be expressed by the solid angle spanned by the area on the unit sphere enclosed by the trajectory corresponding to the direction of magnetic moments along the contour. When many spins are involved in the twodimensional magnetic texture, as in the case of a skyrmion, one can consider the magnetic moment structure as the continuous mapping from the real space to the unit sphere (the direction of the magnetic moment) as
 (2)

By n(x,y), the emergent magnetic field is given by
 (3)

Assuming that the direction of the magnetic moments at infinity r → ∞ is unique, Eq.(1) defines the mapping from S^{2} to S^{2}. A skyrmion is a representative magnetic texture where the magnetic moments form a solid angle and wrap the unit sphere as described by the mapping, and the skyrmion number, defined as
 (4)

counts how many times the unit sphere is wrapped, corresponding to the homotopy classification π_{2} (S^{2}) = Z. Note that this skyrmion number cannot change during the continuous deformation of the magnetic structure. In other words, the "singular" configuration of n(x,y) is needed for the change in N_{sk} corresponding to the creation or annihilation of a skyrmion, which usually costs high energy. This fact provides the topological stability of skyrmions. In the context of Eq.(1), the size Î» of a skyrmion is given by Î»~a(_{}), with a being the lattice constant, and Î» is much larger than a when _{} ≫ 1.
When the spin texture is timedependent, the emergent electric field that is produced as given by
 (5)

Through these emergent electromagnetic fields, the conduction electrons and spin textures influence each other leading to a variety of phenomena as discussed in the next section [4].
EMERGENT ELECTROMAGNETIC PHENOMENA
For example, the emergent magnetic field acts on the conduction electrons as the usual magnetic field, which leads to the Hall effect, i.e., the topological Hall effect (THE). This is observed experimentally, where the Hall coefficient of the normal Hall effect and the averaged b_{z} estimated from the density of skyrmions are used to analyze the data. Here, the assumption is that the conduction electron spins adiabatically follow the direction of the magnetic moments. On the other hand, the topological number N_{sk} appears in the Lagrangian describing the dynamics of the skyrmion as the Berry phase term
 (6)

This means that the X and Y coordinates of the skyrmion's centerofmassmotion are a canonical conjugate to each other. More explicitly, the equation of motion of a skyrmion under the conduction electron's current density j is given by
 (7)

where V is the velocity vector of skyrmion, G is the gyrovector given by G = e_{z}N_{sk}, Îº is a constant of the order of unity, and U is the potential acting on the skyrmion. The constant Î± represents the Gilbert damping, while Î² represents the nonadiabatic effect. It contains the spin transfer torque effect due to the conduction electron current density j. This equation of motion is called the Thiele equation, which describes the dynamics of skyrmions. There are several unique features of this equation. One feature is the mass term due to the deformation of the structure during the motion, which is almost 0 when the skyrmion is driven by the current, i.e., in this case, m = 0 [5]. This means that there is no inertia for the currentcontrol of a skyrmion, which offers an advantage in applications. For example, as a backaction of the THE, it gives the Hall effect of the skyrmion, i.e., the transverse motion of skyrmion to the current density j. The pinning potential represented by U induces the velocity V perpendicular to gradU, which helps the skyrmion avoid impurities. This is the reason why the threshold current density j_{c}, for the motion of skyrmions observed experimentally, is so small, and also why the "universal" currentvelocity relation is realized for skyrmions in sharp contrast to that of domain walls [6]. If one measures the THE by increasing the current density j, the change of the Hall signal occurs when j goes beyond j_{c}. This is due to emergent electromagnetic induction, where the emergent electric field e is induced by the motion of skyrmions according to Eq.(5). The direction of e is opposite to that of the voltage due to the THE, and hence the THE signal is reduced above it is j_{c}. This effect has been predicted theoretically [7] and observed experimentally [8]. Figure 2 summarizes these emergent electromagnetic phenomena schematically.
Fig. 2: The emergent electromagnetic phenomena associated with skyrmions. Reproduced from Ref. [4].
VARIOUS TYPES OF SKYRMIONS AND RELATED STRUCTURES
Skyrmion structure is characterized by two quantities, i.e., the helicity and vorticity. In order to illustrate this, let us express n(r) in the polar coordinates as
 (8)

with r = (rcosφ, rsinφ). Θ(r) describes the outofplane magnetization and hence cosΘ(r → ∞) = 1 outside of the skyrmion, while cosΘ(r = 0) = 1 at its core. On the other hand, the winding of the inplane magnetic moment is characterized as
 (9)

where the integer m defines the vorticity, and Î³ is the helicity. Note that the skyrmion number is determined only by m as N_{sk }= m when the core moment is down. Note that different values of Î³ for antiskyrmion simply corresponds to the rotation of the skyrmion. Putting Eqs.(8) and (9) into the DM interaction Hamiltonian in Eq.(1), one obtains
 (10)

This expression indicates that DM interaction prefers m = 1 and the sign of D determines Î³ = 0 or Î³ = π. Namely, the DM interaction uniquely determines the stable Blochtype skyrmion structure.
The dipoledipole interaction also prefers the Blochtype skyrmion since there appears to be no magnetic charge there, but this interaction does not determine the helicity Î³. Therefore, the helicity degrees of freedom makes the physics of skyrmions richer in dipolar systems. Even the antiskyrmion is shown to be metastable in dipolar systems, where pair annihilation of the skyrmion and antiskyrmion has been demonstrated numerically. Recently, the antiskyrmion crystal state has been reported in ref.[9], where the spin anisotropy within the inplane is strong. Namely, there are four inplane preferred directions, which favor the antiskyrmion.
Another interesting texture related to the skyrmion is the meron, i.e., halfskyrmion. In this structure, the magnetic moments are inplane outside of the structure, i.e., cosΘ(r → ∞) = 0. Therefore, N_{sk} = Â±1/2. Very recently, the crystal of this meron and antimeron has been observe by transmission electron microscopy [10]. It is less stable compared with the skyrmion crystal against the annealing process, and can be regarded as the metastable phase. Namely, the transformation between the meronantimeron crystal and skyrmion crystal occurs with the total N_{sk} being conserved. This fact also indicates the topological stability of the magnetic structure.
CONFINED SKYRMION MOTION
In applications, it often happens that the skyrmions are confined within a limited area or volume. As discussed in Eq.(6), X and Y coordinates are the canonical conjugate to each other. Once the Y coordinates are confined by the potential U(Y) = Y^{2 }/ (2M), this corresponds to the kinetic energy of the skyrmion along the direction of X with M being the mass. Therefore, the mass and inertia appear in the motion of skyrmions along a onedimensional channel. It is shown theoretically that the "universal currentvelocity relation" is modified when the current applied perpendicular to a onedimensional channel, i.e., the slope is enhanced by a factor of 1/Î±, which is usually much larger than 1. However, this enhanced currentvelocity relation is terminated when the skyrmion overcomes the confining potential barrier and goes outside of the sample.
MAGNONSKYRMION INTERACTION
The ferromagnetic state appears between skyrmions, where the ferromagnetic spin waves are defined. Therefore, the interaction between the spin waves and skyrmions is an interesting issue. This problem was studied through the numerical solution of the LandauLifshitzGilbert equation [11]. It was found that the scattering between the spin waves and a skyrmion results in the deflection of the spin waves, i.e., skew scattering and the drift motion of a skyrmion in the opposite direction of the propagating spin waves. This can be interpreted as the spin transfer torque effect due to the spin current carried by the spin waves. The slight deflection of the skyrmion's trajectory, i.e., the skyrmion Hall effect, was also found, which can be analyzed in terms of the conservation of the total momenta of the spin waves and a skyrmion.
Fig. 3: A simulation of magnonskyrmion scattering. The magnons are propagating upward and are scattered with the skew angle 𝜙, while the induced motion of the skyrmion is downward with the angle Î¦. Reproduced from Ref.[11].
BROWNIAN MOTION OF SKYRMIONS
A skyrmion has a definite direction of rotation, as in the case of charged particles under an external magnetic field, and its Brownian motion reflects this fact. Namely, the diffusion coefficient Df of a skyrmion is given by [5]
which indicates that the diffusion is strongly suppressed by the topological nature of the skyrmion, i.e., N_{sk}. This means that the uncertainty due to the Brownian motion is reduced, which is another advantage of the skyrmion as the information carrier.
An interesting observation was reported that focused on the rotation of the microcrystals of skyrmions under the transmission electron microscope [12]. This rotational ratchetlike motion since the direction of the rotation is determined by the skyrmion number N_{sk}. Under an electron beam, the center region is heated up as compared to the rim and the temperature gradient is induced. This causes a flow of spin waves from the center to the rim, which induces the rotation of the skyrmion crystal according to the skyrmion Hall effect.
SKYRMIONS IN THREE DIMENSIONS: THE STRING AND THE MONOPOLE
In three dimensions, the skyrmion forms a string along the external magnetic field, which terminates at the top and bottom surfaces. Naively, this can be regarded as the rodlike object but the internal degrees of freedom are sometimes activated, especially when the disorder potential is present and/or the skyrmion is in motion. The pinning effect by impurities has been discussed extensively for the charge density wave (CDW), the spin density wave (SDW), and the vortex lattice in superconductors. Similar physics can be realized in the skyrmion crystal state, where the distortion of the strings in a glassy state occurs in three dimensions. Replica field theory has been developed for this problem, and the expressions for the various physical properties have been obtained [13]. In particular, the dynamics of the skyrmion glass differs from that of CDW/SDW, and is rather similar to that of the vortex lattice in superconductors. However, the viscous effect is much stronger in the vortex lattice in superconductors compared with that of skyrmions, and the phason excitations can be detected experimentally.
Another unique feature of the skyrmion is its nonreciprocal nature along the magnetic field. Namely, the phason dispersion is asymmetric between q_{z} and q_{z}. This nonreciprocal nature appears as the nonlinear Hall effect of the skyrmion crystal under the current driven motion [14]. The Hall voltage is detected along the skyrmion strings with the current perpendicular to it. This is analyzed by the emergent electric field generated by the distorted string due to the collision with the impurities during the motion.
The endpoints of the skyrmion string correspond to the monopole and antimonopole of the emergent magnetic field. This is because the integral of the flux is quantized for the twodimensional cross section, which jumps by 2π when the cross section sweeps the (anti)monopole point. In MnGe with strong electron correlation and DM interaction, the threedimensional network of skyrmion strings, monopoles and antimonopoles is formed even under the zero magnetic field [15]. This socalled monopole crystal state shows a remarkable change in the topological properties under the magnetic field. Namely, the length of the skyrmion strings changes and eventually disappears by the pairannihilation of monopoles and antimonopoles. This topological transition is well defined in the limit of strong electron correlation, where the magnitude of the magnetic moment is fixed and the topological classification of n(r) works. Experimentally, this topological phase transition leads to anomalous magnetoresistance and also anomalies in the ultrasonic absorption [15].
APPLICATIONS OF SKYRMIONS: SKYRMIONICS
Topological stability and high mobility, i.e., easy manipulation by currents, are the advantages of skyrmions in their application to memory devices [16,17]. In particular, the skyrmions at interfaces are considered to be the most useful because (i) in twodimensions the skyrmion is regarded as an isolated particle, (ii) the strength of the antisymmetric spinorbit interaction can be tuned by the potential gradient at the interface, and (iii) they are advantageous for the integration by e.g. employing the multilayer structures.
There are three fundamental manipulations of skyrmions for memory device applications: (i) writing and erasing, (ii) driving and (iii) reading, each of which will be discussed below.
(i) Writing and erasing:
The processes of writing and erasing are associated with the change in the skyrmion number, and hence are hindered by topological protection. Therefore, one needs to introduce the singular configuration of the moments, which can occur, for example, at the boundary of the system. An early theoretical study used a notch structure to create a skyrmion by the inplane current, where the singularity at the boundary or the edge could be the core of the skyrmion. Skyrmion creation by current injection into a plane has also been proposed. Another possible method to create skyrmions is to shed laser light to heat up the local region to reverse the moments and create the core of a skyrmion. The creation and annihilation of skyrmions by an electric field, which changes the DM interaction, is also possible. Experimentally, it has been shown that skyrmions are created when a current flows through a narrow region [18].
(ii) Driving:
Driving a skyrmion by current has been already discussed above, especially in the confined region. This method of driving has advantages over that by a magnetic field because switching on and off can be done more quickly and also in the nanoscale region by the electric current.
(iii) Reading:
There are several possible methods to detect the existence of the skyrmions. One way is to see the THE induced by the skyrmion. When the skyrmion is confined in the narrow channel, the voltage drop perpendicular to the channel associated with the current along the channel is enhanced by the existence of a skyrmion. Another promising way is to employ tunneling magnetoresistance. The magnetization at the core of the skyrmion structure is then reversed, and hence the antiparallel configuration enhances the tunneling resistance. This phenomenon has been widely used in conventional spintronics and the technique is already established. Based on these fundamental manipulations, the skyrmion circuits and logic gates have been designed theoretically and partly realized experimentally.
SKYRMIONS IN HYBRID SYSTEMS
The physics of skyrmions is enriched when they appear in magnets combined with other systems such as superconductors and topological insulators. Skyrmions are topological particles, and it is interesting to consider whether or not this topological nature could be printed onto superconductivity or not. Concerning this question, it has been theoretically proposed that when the swave superconductor is coupled to the skyrmion lattice with the exchange interaction, the chiral p+ip superconductivity and associated Majorana edge channel will appear [19]. This superconductivity is made from the ingap states, i.e., Shiba states, forming the minibands. This can be regarded as the twodimensional generalization of the idea proposed earlier to create the Kitaev chain superconductors in the Shiba states of onedimensional array of magnetic ions on the superconductor. Also, a zerodimensional Majorana bound state is expected in the case where a skyrmion is put on a superconductor [20].
A topological insulator (TI) is a new state of matter with a gap in the bulk states while the surface metallic state is protected by the topology of the bulk states. In a threedimensional TI, the surface state is described by the Weyl fermion, which is gapped when coupled to the magnetic moments and then turns to be the quantized anomalous Hall system. The texture of this surface magnet is an interesting issue since the momentumspin locking in the surface state can be regarded as the strong coupling limit of the Rashba interaction and hence the strong DM interaction is expected for the magnetic moments [21]. Furthermore, the anomalous behavior of the Hall effect during the magnetization process, which is identified as the THE combined with the theoretical calculation of the model describing the magnetically doped TI [22], has been observed experimentally in the heterostructure of
CONCLUSION
In modern physics, a particle is the excitation of a field and can be created and annihilated. Therefore, the long lifetime of a particle requires some unique mechanism, and the skyrmion employs topological protection for this purpose. This field theoretical picture of particle allows for many internal degrees of freedom and highly nontrivial natures such as the spin and novel statistics, in sharp contrast to the point mass in classical dynamics. A skyrmion is a composite particle made from many magnetic moments and embodies this picture of particles in the field. It has several species characterized by helicity and vorticity. In a threedimensional crystal, the skyrmion becomes a string with the endpoints being a monopole and antimonopole. This is exactly the classical analogue of string theory in high energy physics. Therefore, skyrmionic systems offer a laboratory where tabletop experiments can be used to study the basic concepts of field theory. The quantum nature of skyrmions may become important, although the large number of magnetic moments hinders the manifestation of the quantum effect at present.
Acknowledgements: The author acknowledges the collaborations and discussions with J. Iwasaki, W. Koshibae, M. Mochizuki, Z. Jang, A. Beekman, C. SchÃ¼tte, A. Rosch, X.Z. Yu, N. Kanazawa, Y. Taguchi, Y. Tokura, M. Kawasaki for the collaborations. The author was supported by the Ministry of Education, Culture, Sports, Science, and Technology Nos. JP24224009 and JP26103006, and by Japan Science and Technology Agency (JST) Core Research for Evolutional Science and Technology (CREST) Grant Numbers JPMJCR1874 and JPMJCR16F1, Japan.
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Naoto Nagaosa is the deputy director of the RIKEN Center for Emergent Matter Science (CEMS) and a professor of applied physics in The University of Tokyo (U.Tokyo). After receiving a D.Sci from U.Tokyo, he worked at the Institute for Solid State Physics in U.Tokyo, the Department of Physics at the Massachusetts Institute of Technology, before joining the Department of Applied Physics in U.Tokyo. His research field is theoretical condensed matter physics. 
