Magnetic skyrmion

Last updated
Fig. 1 The vector field of two-dimensional magnetic skyrmions: a) a hedgehog skyrmion and b) a spiral skyrmion. 2skyrmions.PNG
Fig. 1 The vector field of two-dimensional magnetic skyrmions: a) a hedgehog skyrmion and b) a spiral skyrmion.

In physics, magnetic skyrmions (occasionally described as 'vortices,' [1] or 'vortex-like' [2] configurations) are quasiparticles [3] which have been predicted theoretically [1] [4] [5] and observed experimentally [6] [7] [8] in condensed matter systems. Skyrmions can be formed in magnetic materials in their 'bulk' such as in MnSi, [7] or in magnetic thin films. [1] [2] [9] [10] They can be achiral, or chiral (Fig. 1 a and b are both chiral skyrmions) in nature, and may exist both as dynamic excitations [3] or stable or metastable states. [6] Although the broad lines defining magnetic skyrmions have been established de facto, there exist a variety of interpretations with subtle differences.


Most descriptions include the notion of topology – a categorization of shapes and the way in which an object is laid out in space – using a continuous-field approximation as defined in micromagnetics. Descriptions generally specify a non-zero, integer value of the topological index, [11] (not to be confused with the chemistry meaning of 'topological index'). This value is sometimes also referred to as the winding number, [12] the topological charge [11] (although it is unrelated to 'charge' in the electrical sense), the topological quantum number [13] (although it is unrelated to quantum mechanics or quantum mechanical phenomena, notwithstanding the quantization of the index values), or more loosely as the “skyrmion number.” [11] The topological index of the field can be described mathematically as [11]






where is the topological index, is the unit vector in the direction of the local magnetization within the magnetic thin, ultra-thin or bulk film, and the integral is taken over a two dimensional space. (A generalization to a three-dimensional space is possible).[ citation needed ]. Passing to spherical coordinates for the space ( ) and for the magnetisation ( ), one can understand the meaning of the skyrmion number. In skyrmion configurations the spatial dependence of the magnetisation can be simplified by setting the perpendicular magnetic variable independent of the in-plane angle () and the in-plane magnetic variable independent of the radius ( ). Then the topological skyrmion number reads:






where p describes the magnetisation direction in the origin (p=1 (−1) for ) and W is the winding number. Considering the same uniform magnetisation, i.e. the same p value, the winding number allows to define the skyrmion () with a positive winding number and the antiskyrmion with a negative winding number and thus a topological charge opposite to the one of the skyrmion.

Comparison of skyrmion and antiskyrmion. a, b Neel-like skyrmion and antiskyrmion schematically shown in c and d mapped onto a sphere. The color code represents the out-of-plane component of the spins via the brightness, with bright (dark) spins pointing up (down), and their rotational sense in radial direction going from inside out changing from red (clockwise) via gray (vanishing rotational sense) to green (counter-clockwise). e, f Cross sections of the spin textures along the four highlighted directions shown in c and d Skyrmion and antiskyrmion magnetic configurations.jpg
Comparison of skyrmion and antiskyrmion. a, b Néel-like skyrmion and antiskyrmion schematically shown in c and d mapped onto a sphere. The color code represents the out-of-plane component of the spins via the brightness, with bright (dark) spins pointing up (down), and their rotational sense in radial direction going from inside out changing from red (clockwise) via gray (vanishing rotational sense) to green (counter-clockwise). e, f Cross sections of the spin textures along the four highlighted directions shown in c and d

What this equation describes physically is a configuration in which the spins in a magnetic film are all aligned orthonormal to the plane of the film, with the exception of those in one, specific region, where the spins progressively turn over to an orientation that is perpendicular to the plane of the film but anti-parallel to those in the rest of the plane. Assuming 2D isotropy, the free energy of such a configuration is minimized by relaxation towards a state exhibiting circular symmetry, resulting in the configuration illustrated schematically (for a two dimensional skyrmion) in figure 1. In one dimension, the distinction between the progression of magnetization in a 'skyrmionic' pair of domain walls, and the progression of magnetization in a topologically trivial pair of magnetic domain walls, is illustrated in figure 2. Considering this one dimensional case is equivalent to considering a horizontal cut across the diameter of a 2-dimensional hedgehog skyrmion (fig. 1(a)) and looking at the progression of the local spin orientations.

Fig. 2 Comparison of a pair of magnetic domain walls with constant angular progression (1D skyrmion), and a pair of magnetic domain walls with two opposite angular progressions (topologically trivial). Magnetic skyrmion in 1D.pdf
Fig. 2 Comparison of a pair of magnetic domain walls with constant angular progression (1D skyrmion), and a pair of magnetic domain walls with two opposite angular progressions (topologically trivial).

It is worth observing that there are two different configurations which satisfy the topological index criterion stated above. The distinction between these can be made clear by considering a horizontal cut across both of the skyrmions illustrated in figure 1, and looking at the progression of the local spin orientations. In the case of fig. 1(a) the progression of magnetization across the diameter is cycloidal. This type of skyrmion is known as a hedgehog skyrmion. In the case of fig. 1(b), the progression of magnetization is helical, giving rise to what is often called a vortex skyrmion.


The skyrmion magnetic configuration is predicted to be stable because the atomic spins which are oriented opposite those of the surrounding thin-film cannot ‘flip around’ to align themselves with the rest of the atoms in the film, without overcoming an energy barrier. This energy barrier is often ambiguously described as arising from ‘topological protection.’ (See Topological stability vs. energy stability).

Depending on the magnetic interactions existing in a given system, the skyrmion topology can be a stable, meta-stable, or unstable solution when one minimizes the system's free energy. [15]

Theoretical solutions exist for both isolated skyrmions and skyrmion lattices. [15] However, since the stability and behavioral attributes of skyrmions can vary significantly based on the type of interactions in a system, the word 'skyrmion' can refer to substantially different magnetic objects. For this reason, some physicists choose to reserve use of the term 'skyrmion' to describe magnetic objects with a specific set of stability properties, and arising from a specific set of magnetic interactions.


In general, definitions of magnetic skyrmions fall into 2 categories. Which category one chooses to refer to depends largely on the emphasis one wishes to place on different qualities. A first category is based strictly on topology. This definition may seem appropriate when considering topology-dependent properties of magnetic objects, such as their dynamical behavior. [3] [16] A second category emphasizes the intrinsic energy stability of certain solitonic magnetic objects. In this case, the energy stability is often (but not necessarily) associated with a form of chiral interaction, which might originate from the Dzyaloshinskii-Moriya Interaction (DMI), [11] [17] [18] or spiral magnetism originating from Double-exchange mechanism (DE) [19] or competing Heisenberg exchange interaction [20] .

  1. When expressed mathematically, definitions in the first category state that magnetic spin-textures with a spin-progression satisfying the condition: where is an integer ≥1, can be qualified as magnetic skyrmions.
  2. Definitions in the second category similarly stipulate that a magnetic skyrmion exhibits a spin-texture with a spin-progression satisfying the condition: where is an integer ≥1, but further suggest that there must exist an energy term that stabilizes the spin-structure into a localized magnetic soliton whose energy is invariant by translation of the soliton's position in space. (The spatial energy invariance condition constitutes a way to rule out structures stabilized by locally-acting factors external to the system, such as confinement arising from the geometry of a specific nanostructure).[ citation needed ]

The first set of definitions for magnetic skyrmions is a superset of the second, in that it places less stringent requirements on the properties of a magnetic spin texture. This definition finds a raison d'être because topology itself determines some properties of magnetic spin textures, such as their dynamical responses to excitations.

The second category of definitions may be preferred to underscore intrinsic stability qualities of some magnetic configurations. These qualities arise from stabilizing interactions which may be described in several mathematical ways, including for example by using higher-order spatial derivative terms [4] such as 2nd or 4th order terms to describe a field, (the mechanism originally proposed in particle physics by Tony Skyrme for a continuous field model), [21] [22] or 1st order derivative functionals known as Lifshitz invariants [23] —energy contributions linear in first spatial derivatives of the magnetization—as later proposed by Alexei Bogdanov. [1] [24] [25] [26] (An example of such a 1st order functional is the Dzyaloshinskii-Moriya Interaction). [27] In all cases the energy term acts to introduce topologically non-trivial solutions to a system of partial differential equations.[ citation needed ] In other words, the energy term acts to render possible the existence of a topologically non-trivial magnetic configuration that is confined to a finite, localized region, and possesses an intrinsic stability or meta-stability relative to a trivial homogeneously magnetized ground-state — i.e. a magnetic soliton. An example hamiltonian containing one set of energy terms that allows for the existence of skyrmions of the second category is the following: [2]






where the first, second, third and fourth sums correspond to the exchange, Dzyaloshinskii-Moriya, Zeeman (responsible for the "usual" torques and forces observed on a magnetic dipole moment in a magnetic field), and magnetic Anisotropy (typically magnetocrystalline anisotropy) interaction energies respectively. Note that equation (2) does not contain a term for the dipolar, or 'demagnetizing' interaction between atoms. As in eq. (2), the dipolar interaction is sometimes omitted in simulations of ultra-thin two-dimensional magnetic films, because it tends to contribute a minor effect in comparison with the others.[ citation needed ]

Role of the topology

Topological stability vs. energetic stability

A non-trivial topology does not in itself imply energetic stability. There is in fact no necessary relation between topology and energetic stability. Hence, one must be careful not to confuse ‘topological stability,’ which is a mathematical concept,[ citation needed ] with energy stability in real physical systems. Topological stability refers to the idea that in order for a system described by a continuous field to transition from one topological state to another, a rupture must occur in the continuous field, i.e. a discontinuity must be produced. For example, if one wishes to transform a flexible balloon doughnut (torus) into an ordinary spherical balloon, it is necessary to introduce a rupture on some part of the balloon doughnut's surface. Mathematically, the balloon doughnut would be described as 'topologically stable.' However, in physics, the free energy required to introduce a rupture enabling the transition of a system from one ‘topological’ state to another is always finite. For example, it is possible to turn a rubber ballon into flat piece of rubber by poking it with a needle (and popping it!). Thus, while a physical system can be approximately described using the mathematical concept of topology, attributes such as energetic stability are dependent on the system's parameters—the strength of the rubber in the example above—not the topology per se. In order to draw a meaningful parallel between the concept of topological stability and the energy stability of a system, the analogy must necessarily be accompanied by the introduction of a non-zero phenomenological ‘field rigidity’ to account for the finite energy needed to rupture the field’s topology[ citation needed ]. Modeling and then integrating this field rigidity can be likened to calculating a breakdown energy-density of the field. These considerations suggest that what is often referred to as ‘topological protection,’ or a 'topological barrier,' should more accurately be referred to as a 'topology-related energy barrier,' though this terminology is somewhat cumbersome. A quantitative evaluation of such a topological barrier can be obtained by extracting the critical magnetic configuration when the topological number changes during the dynamical process of a skyrmion creation event. Applying the topological charge defined in a lattice, [28] the barrier height is theoretically shown to be proportional to the exchange stiffness. [29]

Further observations

It is important to be cognizant of the fact that magnetic =1 structures are in fact not stabilized by virtue of their ‘topology,’ but rather by the field rigidity parameters that characterize a given system. However, this does not suggest that topology plays an insignificant role with respect to energetic stability. On the contrary, topology may create the possibility for certain stable magnetic states to exist, which otherwise could not. However, topology in itself does not guarantee the stability of a state. In order for a state to have stability associated with its topology, it must be further accompanied by a non-zero field rigidity. Thus, topology can be considered a necessary but insufficient condition for the existence of certain classes of stable objects. While this distinction may at first seem pedantic, its physical motivation becomes apparent when considering two magnetic spin configurations of identical topology =1, but subject to the influences of only one differing magnetic interaction. For example, we may consider one spin configuration with, and one configuration without the presence of magnetocrystalline anisotropy, oriented perpendicular to the plane of an ultra-thin magnetic film. In this case, the =1 configuration that is influenced by the magnetocrystalline anisotropy will be more energetically stable than the =1 configuration without it, in spite of identical topologies. This is because the magnetocrystalline anisotropy contributes to the field rigidity, and it is the field rigidity, not the topology, that confers the notable energy barrier protecting the topological state.

Finally, it is interesting to observe that in some cases, it is not the topology which helps =1 configurations to be stable, but rather the converse, as it is the stability of the field (which depends on the relevant interactions) which favors the =1 topology. This is to say that the most stable energy configuration of the field constituents, (in this case magnetic atoms), may in fact be to arrange into a topology which can be described as an =1 topology. Such is the case for magnetic skyrmions stabilized by the Dzyaloshinskii–Moriya interaction, which causes adjacent magnetic spins to 'prefer' having a fixed angle between each other (energetically speaking). Note that from a point of view of practical applications this does not alter the usefulness of developing systems with Dzyaloshinskii–Moriya interaction, as such applications depend strictly on the topology [of the skyrmions, or lack thereof], which encodes the information, and not the underlying mechanisms which stabilize the necessary topology.

These examples illustrate why use of the terms 'topological protection' or 'topological stability' interchangeably with the concept of energy stability is misleading, and is liable to lead to fundamental confusion.

Limitations of applying the concept of topology

One must exercise caution when making inferences based on topology-related energy barriers, as it can be misleading to apply the notion of topology—a description which only rigorously applies to continuous fields— to infer the energetic stability of structures existing in discontinuous systems. Giving way to this temptation is sometimes problematic in physics, where fields which are approximated as continuous become discontinuous below certain size-scales. Such is the case for example when the concept of topology is associated with the micromagnetic model—which approximates the magnetic texture of a system as a continuous field—and then applied indiscriminately without consideration of the model's physical limitations (i.e. that it ceases to be valid at atomic dimensions). In practice, treating the spin textures of magnetic materials as vectors of a continuous field model becomes inaccurate at size-scales on the order of < 2 nm, due to the discretization of the atomic lattice. Thus, it is not meaningful to speak of magnetic skyrmions below these size-scales.

Practical applications

Magnetic skyrmions are anticipated to allow for the existence of discrete magnetic states which are significantly more energetically stable (per unit volume) than their single-domain counterparts. For this reason, it is envisioned that magnetic skyrmions may be used as bits to store information in future memory and logic devices, where the state of the bit is encoded by the existence or non-existence of the magnetic skyrmion. The dynamical magnetic skyrmion exhibits strong breathing which opens the avenue for skyrmion-based microwave applications. [30] Simulations also indicate that the position of magnetic skyrmions within a film/nanotrack may be manipulated using spin currents [9] or spin waves. [31] Thus, magnetic skyrmions also provide promising candidates for future racetrack-type in-memory logic computing technologies. [9] [32] [33] [34]

Related Research Articles

Kondo effect Physical phenomenon due to impurities

In physics, the Kondo effect describes the scattering of conduction electrons in a metal due to magnetic impurities, resulting in a characteristic change in electrical resistivity with temperature. The effect was first described by Jun Kondo, who applied third-order perturbation theory to the problem to account for s-d electron scattering. Kondo's model predicted that the scattering rate of conduction electrons off the magnetic impurity should diverge as the temperature approaches 0 K. Extended to a lattice of magnetic impurities, the Kondo effect likely explains the formation of heavy fermions and Kondo insulators in intermetallic compounds, especially those involving rare earth elements like cerium, praseodymium, and ytterbium, and actinide elements like uranium. The Kondo effect has also been observed in quantum dot systems.

In particle theory, the skyrmion is a topologically stable field configuration of a certain class of non-linear sigma models. It was originally proposed as a model of the nucleon by Tony Skyrme in 1961. As a topological soliton in the pion field, it has the remarkable property of being able to model, with reasonable accuracy, multiple low-energy properties of the nucleon, simply by fixing the nucleon radius. It has since found application in solid-state physics, as well as having ties to certain areas of string theory.

Albert Fert

Albert Fert is a French physicist and one of the discoverers of giant magnetoresistance which brought about a breakthrough in gigabyte hard disks. Currently, he is an emeritus professor at Paris-Saclay University in Orsay, scientific director of a joint laboratory between the Centre national de la recherche scientifique and Thales Group, and adjunct professor at Michigan State University. He was awarded the 2007 Nobel Prize in Physics together with Peter Grünberg.

Montonen–Olive duality

Montonen–Olive duality or electric–magnetic duality is the oldest known example of strong–weak duality or S-duality according to current terminology. It generalizes the electro-magnetic symmetry of Maxwell's equations by stating that magnetic monopoles, which are usually viewed as emergent quasiparticles that are "composite", can in fact be viewed as "elementary" quantized particles with electrons playing the reverse role of "composite" topological solitons; the viewpoints are equivalent and the situation dependent on the duality. It was later proven to hold true when dealing with a N = 4 supersymmetric Yang–Mills theory. It is named after Finnish physicist Claus Montonen and British physicist David Olive after they proposed the idea in their academic paper Magnetic monopoles as gauge particles? where they state:

There should be two "dual equivalent" field formulations of the same theory in which electric (Noether) and magnetic (topological) quantum numbers exchange roles.

Majorana fermion Fermion that is its own antiparticle

A Majorana fermion, also referred to as a Majorana particle, is a fermion that is its own antiparticle. They were hypothesised by Ettore Majorana in 1937. The term is sometimes used in opposition to a Dirac fermion, which describes fermions that are not their own antiparticles.


Helimagnetism is a form of magnetic ordering where spins of neighbouring magnetic moments arrange themselves in a spiral or helical pattern, with a characteristic turn angle of somewhere between 0 and 180 degrees. It results from the competition between ferromagnetic and antiferromagnetic exchange interactions. It is possible to view ferromagnetism and antiferromagnetism as helimagnetic structures with characteristic turn angles of 0 and 180 degrees respectively. Helimagnetic order breaks spatial inversion symmetry, as it can be either left-handed or right-handed in nature.

Quantum vortex Quantized flux circulation of some physical quantity

In physics, a quantum vortex represents a quantized flux circulation of some physical quantity. In most cases, quantum vortices are a type of topological defect exhibited in superfluids and superconductors. The existence of quantum vortices was first predicted by Lars Onsager in 1949 in connection with superfluid helium. Onsager reasoned that quantisation of vorticity is a direct consequence of the existence of a superfluid order parameter as a spatially continuous wavefunction. Onsager also pointed out that quantum vortices describe the circulation of superfluid and conjectured that their excitations are responsible for superfluid phase transitions. These ideas of Onsager were further developed by Richard Feynman in 1955 and in 1957 were applied to describe the magnetic phase diagram of type-II superconductors by Alexei Alexeyevich Abrikosov. In 1935 Fritz London published a very closely related work on magnetic flux quantization in superconductors. London's fluxoid can also be viewed as a quantum vortex.

The quantum spin Hall state is a state of matter proposed to exist in special, two-dimensional, semiconductors that have a quantized spin-Hall conductance and a vanishing charge-Hall conductance. The quantum spin Hall state of matter is the cousin of the integer quantum Hall state, and that does not require the application of a large magnetic field. The quantum spin Hall state does not break charge conservation symmetry and spin- conservation symmetry.

Topological insulator State of matter with insulating bulk but conductive boundary

A topological insulator is a material that behaves as an insulator in its interior but whose surface contains conducting states, meaning that electrons can only move along the surface of the material. Topological insulators have non-trivial symmetry-protected topological order; however, having a conducting surface is not unique to topological insulators, since ordinary band insulators can also support conductive surface states. What is special about topological insulators is that their surface states are symmetry-protected Dirac fermions by particle number conservation and time-reversal symmetry. In two-dimensional (2D) systems, this ordering is analogous to a conventional electron gas subject to a strong external magnetic field causing electronic excitation gap in the sample bulk and metallic conduction at the boundaries or surfaces.

In condensed matter physics, a quantum spin liquid is a phase of matter that can be formed by interacting quantum spins in certain magnetic materials. Quantum spin liquids (QSL) are generally characterized by their long-range quantum entanglement, fractionalized excitations, and absence of ordinary magnetic order.

Antisymmetric exchange

Antisymmetric exchange, also known as the Dzyaloshinskii–Moriya interaction (DMI), is a contribution to the total magnetic exchange interaction between two neighboring magnetic spins, and . Quantitatively, it is a term in the Hamiltonian which can be written as

Weyl semimetal

Weyl fermions are massless chiral fermions embodying the mathematical concept of a Weyl spinor. Weyl spinors in turn play an important role in quantum field theory and the Standard Model, where they are a building block for fermions in quantum field theory. Weyl spinors are a solution to the Dirac equation derived by Hermann Weyl, called the Weyl equation. For example, one-half of a charged Dirac fermion of a definite chirality is a Weyl fermion.

Chiral magnetic effect

Chiral magnetic effect (CME) is the generation of electric current along an external magnetic field induced by chirality imbalance. Fermions are said to be chiral if they keep a definite projection of spin quantum number on momentum. The CME is a macroscopic quantum phenomenon present in systems with charged chiral fermions, such as the quark-gluon plasma, or Dirac and Weyl semimetals. The CME is a consequence of chiral anomaly in quantum field theory; unlike conventional superconductivity or superfluidity, it does not require a spontaneous symmetry breaking. The chiral magnetic current is non-dissipative, because it is topologically protected: the imbalance between the densities of left-handed and right-handed chiral fermions is linked to the topology of fields in gauge theory by the Atiyah-Singer index theorem.

M. Zahid Hasan is an endowed chair Eugene Higgins Professor of Physics at Princeton University. He is known for his pioneering research on quantum matter exhibiting topological and emergent properties. He is the Principal Investigator of Laboratory for Topological Quantum Matter and Advanced Spectroscopy at Princeton University and a Visiting Faculty Scientist at Lawrence Berkeley National Laboratory in California. Since 2014 he has been an EPiQS-Moore Investigator awarded by the Betty and Gordon Moore foundation in Palo Alto (California) for his research on emergent quantum phenomena in topological matter. He has been a Vanguard Fellow of the Aspen Institute since 2014. Hasan is an elected fellow of the American Academy of Arts and Sciences. His father is 5 time Member of parliament Md. Rahamat Ali

Dual photon A hypothetical elementary particle that is a dual of the photon under electric–magnetic duality

In theoretical physics, the dual photon is a hypothetical elementary particle that is a dual of the photon under electric–magnetic duality which is predicted by some theoretical models, including M-theory.

Electronic properties of graphene

Graphene is a semimetal whose conduction and valence bands meet at the Dirac points, which are six locations in momentum space, the vertices of its hexagonal Brillouin zone, divided into two non-equivalent sets of three points. The two sets are labeled K and K'. The sets give graphene a valley degeneracy of gv = 2. By contrast, for traditional semiconductors the primary point of interest is generally Γ, where momentum is zero. Four electronic properties separate it from other condensed matter systems.

The term Dirac matter refers to a class of condensed matter systems which can be effectively described by the Dirac equation. Even though the Dirac equation itself was formulated for fermions, the quasi-particles present within Dirac matter can be of any statistics. As a consequence, Dirac matter can be distinguished in fermionic, bosonic or anyonic Dirac matter. Prominent examples of Dirac matter are Graphene, topological insulators, Dirac semimetals, Weyl semimetals, various high-temperature superconductors with -wave pairing and liquid Helium-3. The effective theory of such systems is classified by a specific choice of the Dirac mass, the Dirac velocity, the Dirac matrices and the space-time curvature. The universal treatment of the class of Dirac matter in terms of an effective theory leads to a common features with respect to the density of states, the heat capacity and impurity scattering.

Magnetic topological insulators are three dimensional magnetic materials with a non-trivial topological index protected by a symmetry other than time-reversal. In contrast with a non-magnetic topological insulator, a magnetic topological insulator can have naturally gapped surface states as long as the quantizing symmetry is broken at the surface. These gapped surfaces exhibit a topologically protected half-quantized surface anomalous Hall conductivity perpendicular to the surface. The sign of the half-quantized surface anomalous Hall conductivity depends on the specific surface termination.

Iron germanide Chemical compound

Iron germanide (FeGe) is an intermetallic compound, a germanide of iron. At ambient conditions it crystallizes in three polymorphs with monoclinic, hexagonal and cubic structures. The cubic polymorph has no inversion center, it is therefore helical, with right-hand and left-handed chiralities.


A hopfion is a topological soliton. It is a stable three-dimensional localised configuration of a three-component field with a knotted topological structure. They are the three-dimensional counterparts of skyrmions, which exhibit similar topoligical properties in 2D.


  1. 1 2 3 4 Bogdanov AN, Rössler UK (July 2001). "Chiral symmetry breaking in magnetic thin films and multilayers". Physical Review Letters. 87 (3): 037203. Bibcode:2001PhRvL..87c7203B. doi:10.1103/physrevlett.87.037203. PMID   11461587.
  2. 1 2 3 Iwasaki J, Mochizuki M, Nagaosa N (October 2013). "Current-induced skyrmion dynamics in constricted geometries". Nature Nanotechnology. 8 (10): 742–7. arXiv: 1310.1655 . Bibcode:2013NatNa...8..742I. doi:10.1038/nnano.2013.176. PMID   24013132. S2CID   780496.
  3. 1 2 3 Sondhi SL, Karlhede A, Kivelson SA, Rezayi EH (June 1993). "Skyrmions and the crossover from the integer to fractional quantum Hall effect at small Zeeman energies". Physical Review B. 47 (24): 16419–16426. Bibcode:1993PhRvB..4716419S. doi:10.1103/physrevb.47.16419. PMID   10006073.
  4. 1 2 Rössler UK, Bogdanov AN, Pfleiderer C (August 2006). "Spontaneous skyrmion ground states in magnetic metals". Nature. 442 (7104): 797–801. arXiv: cond-mat/0603103 . Bibcode:2006Natur.442..797R. doi:10.1038/nature05056. PMID   16915285. S2CID   4389730.
  5. Dupé B, Hoffmann M, Paillard C, Heinze S (June 2014). "Tailoring magnetic skyrmions in ultra-thin transition metal films". Nature Communications. 5: 4030. Bibcode:2014NatCo...5.4030D. doi: 10.1038/ncomms5030 . PMID   24893652.
  6. 1 2 Romming N, Hanneken C, Menzel M, Bickel JE, Wolter B, von Bergmann K, et al. (August 2013). "Writing and deleting single magnetic skyrmions". Science. 341 (6146): 636–9. Bibcode:2013Sci...341..636R. doi:10.1126/science.1240573. PMID   23929977. S2CID   27222755.
  7. 1 2 Mühlbauer S, Binz B, Jonietz F, Pfleiderer C, Rosch A, Neubauer A, et al. (February 2009). "Skyrmion lattice in a chiral magnet". Science. 323 (5916): 915–9. arXiv: 0902.1968 . Bibcode:2009Sci...323..915M. doi:10.1126/science.1166767. PMID   19213914. S2CID   53513118.
  8. Hsu PJ, Kubetzka A, Finco A, Romming N, von Bergmann K, Wiesendanger R (February 2017). "Electric-field-driven switching of individual magnetic skyrmions". Nature Nanotechnology. 12 (2): 123–126. arXiv: 1601.02935 . Bibcode:2017NatNa..12..123H. doi:10.1038/nnano.2016.234. PMID   27819694. S2CID   5921700.
  9. 1 2 3 Fert A, Cros V, Sampaio J (March 2013). "Skyrmions on the track". Nature Nanotechnology. 8 (3): 152–6. Bibcode:2013NatNa...8..152F. doi:10.1038/nnano.2013.29. PMID   23459548.
  10. Husain S, Sisodia N, Chaurasiya AK, Kumar A, Singh JP, Yadav BS, et al. (January 2019). "Co2FeAl Heusler Alloy Ultrathin Film Heterostructures". Scientific Reports. 9 (1): 1085. doi:10.1038/s41598-018-35832-3. PMC   6355792 . PMID   30705297.
  11. 1 2 3 4 5 Heinze S, Bergmann K, Menzel M, Brede J, Kubetzka A, Wiesendanger R, Bihlmayer G, Blügel S (2011). "Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions". Nature Physics. 7 (9): 713–718. Bibcode:2011NatPh...7..713H. doi:10.1038/nphys2045.
  12. von Bergmann K, Kubetzka A, Pietzsch O, Wiesendanger R (October 2014). "Interface-induced chiral domain walls, spin spirals and skyrmions revealed by spin-polarized scanning tunneling microscopy". Journal of Physics. Condensed Matter. 26 (39): 394002. Bibcode:2014JPCM...26M4002V. doi:10.1088/0953-8984/26/39/394002. PMID   25214495.
  13. Finazzi M, Savoini M, Khorsand AR, Tsukamoto A, Itoh A, Duò L, et al. (April 2013). "Laser-induced magnetic nanostructures with tunable topological properties". Physical Review Letters. 110 (17): 177205. arXiv: 1304.1754 . Bibcode:2013PhRvL.110q7205F. doi:10.1103/physrevlett.110.177205. PMID   23679767. S2CID   21660154.
  14. Hoffmann M, Zimmermann B, Müller GP, Schürhoff D, Kiselev NS, Melcher C, Blügel S (August 2017). "Antiskyrmions stabilized at interfaces by anisotropic Dzyaloshinskii-Moriya interactions". Nature Communications. 8 (1): 308. arXiv: 1702.07573 . Bibcode:2017NatCo...8..308H. doi:10.1038/s41467-017-00313-0. PMC   5566362 . PMID   28827700.
  15. 1 2 Göbel, Börge; Mertig, Ingrid; Tretiakov, Oleg A. (2021). "Beyond skyrmions: Review and perspectives of alternative magnetic quasiparticles". Physics Reports. 895: 1–28. doi: 10.1016/j.physrep.2020.10.001 . ISSN   0370-1573.
  16. Brey L, Fertig HA, Côté R, MacDonald AH (1997). "Skyrmions in the quantum hall effect". Lecture Notes in Physics. 478: 275–283. Bibcode:1997LNP...478..275B. doi:10.1007/bfb0104643. ISBN   978-3-540-62476-9.
  17. Kiselev NS, Bogdanov AN, Schafer R, Rossler UK (2011). "Chiral skyrmions in thin magnetic films: new objects for magnetic storage technologies?". Journal of Physics D: Applied Physics. 44 (39): 392001. arXiv: 1102.2726 . Bibcode:2011JPhD...44M2001K. doi:10.1088/0022-3727/44/39/392001. S2CID   118433956.
  18. Nagaosa N, Tokura Y (December 2013). "Topological properties and dynamics of magnetic skyrmions". Nature Nanotechnology. 8 (12): 899–911. Bibcode:2013NatNa...8..899N. doi:10.1038/nnano.2013.243. PMID   24302027.
  19. Azhar M, Mostovoy M (January 2017). "Incommensurate Spiral Order from Double-Exchange Interactions". Physical Review Letters. 118 (2): 027203. arXiv: 1611.03689 . Bibcode:2017PhRvL.118b7203A. doi:10.1103/PhysRevLett.118.027203. PMID   28128593. S2CID   13478577.
  20. Leonov AO, Mostovoy M (September 2015). "Multiply periodic states and isolated skyrmions in an anisotropic frustrated magnet". Nature Communications. 6: 8275. arXiv: 1501.02757 . Bibcode:2015NatCo...6.8275L. doi:10.1038/ncomms9275. PMC   4667438 . PMID   26394924.
  21. Skyrme TH (1961). "A non-linear field theory". Proc. R. Soc. Lond. A. 260 (1300): 127–138. Bibcode:1961RSPSA.260..127S. doi:10.1098/rspa.1961.0018. S2CID   122604321.
  22. Rossler UK, Leonov AA, Bogdanov AN (2010). "Skyrmionic textures in chiral magnets". Journal of Physics: Conference Series. 200 (2): 022029. arXiv: 0907.3651 . Bibcode:2010JPhCS.200b2029R. doi:10.1088/1742-6596/200/2/022029. S2CID   14383159.
  23. Landau LD, Lifshitz EM (1997). Statistical Physics. Course of Theoretical Physics. 5.
  24. Bogdanov AN, Yablonskii DA (1989). "Thermodynamically stable 'vortices' in magnetically ordered crystals. The mixed state of magnets". Sov. Phys. JETP. 68: 101–103.
  25. Bogdanov A, Hubert A (1994). "Thermodynamically stable magnetic vortex states in magnetic crystals". J. Magn. Magn. Mater. 138 (3): 255–269. Bibcode:1994JMMM..138..255B. doi:10.1016/0304-8853(94)90046-9.
  26. Bogdanov A (1995). "New localized solutions of the nonlinear field-equations". JETP Lett. 62: 247–251.
  27. Dzyaloshinskii IE (1964). "Theory of Helicoidal Structures in Antiferromagnets. I. Nonmetals". Sov. Phys. JETP. 19: 960.
  28. Berg B, Lüscher M (1981-08-24). "Definition and statistical distributions of a topological number in the lattice O(3) σ-model". Nuclear Physics B. 190 (2): 412–424. Bibcode:1981NuPhB.190..412B. doi:10.1016/0550-3213(81)90568-X.
  29. Yin G (2016). "Topological charge analysis of ultrafast single skyrmion creation". Physical Review B. 93 (17): 174403. arXiv: 1411.7762 . Bibcode:2016PhRvB..93q4403Y. doi:10.1103/PhysRevB.93.174403. S2CID   118493067.
  30. Zhou Y, Iacocca E, Awad AA, Dumas RK, Zhang FC, Braun HB, Åkerman J (September 2015). "Dynamically stabilized magnetic skyrmions". Nature Communications. 6: 8193. Bibcode:2015NatCo...6.8193Z. doi:10.1038/ncomms9193. PMC   4579603 . PMID   26351104.
  31. Zhang X, Ezawa M, Xiao D, Zhao GP, Liu Y, Zhou Y (June 2015). "All-magnetic control of skyrmions in nanowires by a spin wave". Nanotechnology. 26 (22): 225701. arXiv: 1504.00409 . Bibcode:2015Nanot..26v5701Z. doi:10.1088/0957-4484/26/22/225701. PMID   25965121. S2CID   2449410.
  32. 1 2 3 Zhang X, Ezawa M, Zhou Y (March 2015). "Magnetic skyrmion logic gates: conversion, duplication and merging of skyrmions". Scientific Reports. 5: 9400. arXiv: 1410.3086 . Bibcode:2015NatSR...5E9400Z. doi:10.1038/srep09400. PMC   4371840 . PMID   25802991.
  33. Zhou Y, Ezawa M (August 2014). "A reversible conversion between a skyrmion and a domain-wall pair in a junction geometry". Nature Communications. 5: 4652. arXiv: 1404.3350 . Bibcode:2014NatCo...5.4652Z. doi:10.1038/ncomms5652. PMID   25115977. S2CID   205328864.
  34. Zhang X, Zhou Y, Ezawa M, Zhao GP, Zhao W (June 2015). "Magnetic skyrmion transistor: skyrmion motion in a voltage-gated nanotrack". Scientific Reports. 5: 11369. Bibcode:2015NatSR...511369Z. doi:10.1038/srep11369. PMC   4471904 . PMID   26087287.