## Abstract

Magnetic skyrmions have been subject of growing interest in recent years for their very promising applications in spintronics, quantum computation and future low power information technology devices. In this book chapter, we use the field theory method and coherent spin state ideas to investigate the properties of magnetic solitons in spacetime while focussing on 2D and 3D skyrmions. We also study the case of a rigid skyrmion dissolved in a magnetic background induced by the spin-tronics; and derive the effective rigid skyrmion equation of motion. We examine as well the interaction between electrons and skyrmions; and comment on the modified Landau-Lifshitz-Gilbert equation. Other issues, including emergent electrodynamics and hot applications for next-generation high-density efficient information encoding, are also discussed.

### Keywords

- Geometric phases
- magnetic monopoles and topology
- soliton and holonomy
- skyrmion dynamics and interactions
- med-term future applications

## 1. Introduction

During the last two decades, the magnetic skyrmions and antiskyrmions have been subject to an increasing interest in connection with the topological phase of matter [1, 2, 3, 4], the spin-tronics [5, 6] and quantum computing [7, 8]; as well as in the search for advanced applications such as racetrack memory, microwave oscillators and logic nanodevices making skyrmionic states very promising candidates for future low power information technology devices [9, 10, 11, 12]. Initially proposed by T. Skyrme to describe hadrons in the theory of quantum chromodynamics [13], skyrmions have however been observed in other fields of physics, including quantum Hall systems [14, 15], Bose-Einstein condensates [16] and liquid crystals [17]. In quantum Hall (QH) ferromagnets for example [18, 19], due to the exchange interaction; the electron spins spontaneously form a fully polarized ferromagnet close to the integer filling factor

In order to overcome the lack of a prototype of a skyrmion-based spintronic devices for a possible fabrication of nanodevices of data storage and logic technologies, intense research has been carried out during the last few years [24, 25]. In this regard, several alternative nano-objects have been identified to host stable skyrmions at room temperature. The first experimental observation of crystalline skyrmionic states was in a three-dimensional metallic ferromagnet MnSi with a B20 structure using small angle neutron scattering [26]. Then, real-space imaging of the skyrmion has been reported using Lorentz transmission electron microscopy in non-centrosymmetric magnetic compounds and in thin films with broken inversion symmetry, including monosilicides, monogermanides, and their alloys, like Fe_{1–x}CoSi [27], FeGe [28], and MnGe [29].

_{x}

One of the key parameters in the formation of these topologically protected non-collinear spin textures is the Dzyaloshinskii-Moriya Interaction (DMI) [30, 31, 32]. Originating from the strong spin-orbit coupling (SOC) at the interfaces, the DM exchange between atomic spins controls the size and stability of the induced skyrmions. Depending on the symmetry of the crystal structures and the skyrmion windings number, the internal spins within a single skyrmion envelop a sphere in different arrangements [33]. The in-plane component of the magnetization, in the Néel skyrmion, is always pointed in the radial direction [34], while it is oriented perpendicularly with respect to the position vector in the Bloch skyrmion [26]. Different from these two well-known types of skyrmions are skyrmions with mixed Bloch-Néel topological spin textures observed in Co/Pd multilayers [35]. Magnetic antiskyrmions, having a more complex boundary compared to the chiral magnetic boundaries of skyrmions, exist above room temperature in tetragonal Heusler materials [36]. Higher-order skyrmions should be stabilized in anisotropic frustrated magnet at zero temperature [37] as well as in itinerant magnets with zero magnetic field [38].

In the quest to miniaturize magnetic storage devices, reduction of material’s dimensions as well as preservation of the stability of magnetic nano-scale domains are necessary. One possible route to achieve this goal is the formation of topological protected skyrmions in certain 2D magnetic materials. To induce magnetic order and tune DMIs in 2D crystal structures, their centrosymmetric should first be broken using some efficient ways such a (i) generate one-atom thick hybrids where atoms are mixed in an alternating manner [39, 40, 41], (ii) apply bias voltage or strain [42, 43, 44], (iii) insert adsorbents, impurities and defects [45, 46, 47]. In graphene-like materials, fluorine chemisorption is an exothermic adsorption that gives rise to stable 2D structures [48] and to long-range magnetism [49, 50]. In semi-fluorinated graphene, a strong Dzyaloshinskii-Moriya interaction has been predicted with the presence of ferromagnetic skyrmions [51]. The formation of a nanoskyrmion state in a Sn monolayer on a SiC(0001) surface has been reported on the basis of a generalized Hubbard model [52]. Strong DMI between the first nearest magnetic germanium neighbors in 2D semi-fluorinated germanene results in a potential antiferromagnetic skyrmion [53].

In this bookchapter, we use the coherent spin states approach and the field theory method (continuous limit of lattice magnetic models with DMI) to revisit some basic aspects and properties of magnetic solitons in spacetime while focusing on 1d kinks, 2d and 3d spatial skyrmions/antiskyrmions. We also study the case of a rigid skyrmion dissolved in a magnetic background induced by the electronic spins of magnetic atoms like Mn; and derive the effective rigid skyrmion equation of motion. In this regard, we describe the similarity between, on one hand, electrons in the electromagnetic background; and, on the other hand, rigid skyrmions bathing in a texture of magnetic moments. We also investigate the interaction between electrons and skyrmions as well as the effect of the spin transfer effect.

This bookchapter is organized as follows: In Section 2, we introduce some basic tools on quantum SU(2) spins and review useful aspects of their dynamics. In Section 3, we investigate the topological properties of kinks and 2d space solitons while describing in detail the underside of the topological structure of these low-dimensional solitons. In Section 4, we extend the construction to approach topological properties to 3d skyrmions. In Section 5, we study the dynamics of rigid skyrmions without and with dissipation; and in Section 6, we use emergent gauge potential fields to describe the effective dynamics of electrons interacting with the skyrmion in the presence of a spin transfer torque. We end this study by making comments and describing perspectives in the study of skyrmions.

## 2. Quantum SU(2) spin dynamics

In this section, we review some useful ingredients on the quantum SU(2) spin operator, its underlying algebra and its time evolution while focussing on the interesting spin 1/2 states, concerning electrons in materials; and on coherent spin states which are at the basis of the study of skyrmions/antiskyrmions. First, we introduce rapidly the SU(2) spin operator

### 2.1 Quantum spin 1/2 operator and beyond

We begin by recalling that in non relativistic 3D quantum mechanics, the spin states ^{1}

where the Pauli matrices ^{2}; then the time evolution of

where appears the Levi-Civita

### 2.2 Coherent spin states and semi-classical analysis

To deal with the semi-classical dynamics of

with

For explicit calculations, this unit 2-sphere equation will be often expressed like

and from which we learn that

and can be identified with the Euler- Lagrange equations following from the variation ** n**|

showing that

## 3. Magnetic solitons in lower dimensions

In previous section, we have considered the time dynamics of coherent spin states with amplitude

### 3.1 One space dimensional solitons

In

The constraint

where

#### 3.1.1 Constrained dynamics

The classical spacetime dynamics of

where

with minimum corresponding to constant field (

where

#### 3.1.2 Topological current and charge

To start, notice that as far as conserved symmetries of (10) are concerned, there exists an exotic invariance generated by a conserved

Because of the

and where

Moreover, seen that

This energy transformation shows that stable solitons with minimal energy correspond to

### 3.2 Skyrmions in 2d space dimensions

In this subsection, we investigate the topological properties of 2d Skyrmions by extending the field theory study we have done above for 1d kinks to two space dimensions. For that, we proceed as follows: First, we turn on the component

#### 3.2.1 Dzyaloshinskii-Moriya potential

The field action

In this expression, the

The interest into this (18) is twice; first it can be put into the equivalent form

but now

where the 1-form

Putting these Eq. (21) back into the expansion of

containing

#### 3.2.2 From kinks to 2d Skyrmions

Here, we study the topological properties of the 2d Skyrmion with dynamics governed by the Lagrangian density (17). From the expression of the

where

Replacing

but this is nothing but the stereographic projection of the 2-sphere

## 4. Three dimensional magnetic skyrmions

In this section, we study the dynamics of the 3d skyrmion and its topological properties both in target space

### 4.1 From 2d skyrmion to 3d homologue

As for the 1d and 2d solitons considered previous section, the spacetime dynamics of the 3d skyrmion in

where

For convenience, we sometimes refer to the three

### 4.2 Conserved topological current

First, we investigate the topological properties of the 3d skyrmion from the target space view; that is without using the spacetine variables

#### 4.2.1 Topological current in target space

The 3d skyrmion field is described by a real four component vector

This 3-form describes precisely the topological current in the target space; this is because on

where we have substituted the 3-form

#### 4.2.2 Topological symmetry in spacetime

In the spacetime

with

describing a compactification of the space

In terms of the angular variables, this current reads like

Because of its spherical symmetry, the space volume

## 5. Effective dynamics of skyrmions

In this section, we investigate the effective dynamics of a point-like skyrmion in a ferromagnetic background field while focussing on the 2d configuration. First, we derive the effective equation of a rigid skyrmion and comment on the underlying effective Lagrangian. We also describe the similarity with the dynamics of an electron in a background electromagnetic field. Then, we study the effect of dissipation on the skyrmion dynamics.

### 5.1 Equation of a rigid skyrmion

To get the effective equation of motion of a rigid skyrmion, we start by the spin

#### 5.1.1 Rigid skyrmion

We begin by introducing the variables describing the skyrmion in the magnetic background field

In this representation, the velocity

In this relation, the density

with

with

#### 5.1.2 Skyrmion equation without dissipation

To get the effective field equation of motion of the point-like skyrmion without dissipation, we calculate the vanishing condition of the functional variation of the action; that is

from which one can wonder the effective Lagrangian describing the effective dynamics of the skyrmion. It is given by

Notice that the right hand of Eq. (39) looks like the usual Lorentz force (

Concerning the calculation of the

Next, using the relation

This variation is very remarkable because it is contained in the variation of the effective coupling

From this Lagrangian, we learn the equation of the motion of the rigid skyrmion namely

### 5.2 Implementing dissipation

So far we have considered magnetic moment obeying the constraint

In presence of dissipation, we loose energy; and so one expects that

#### 5.2.1 Landau-Lifshitz-Gilbert equation

Due to dissipation, the force

where its both sides have

Notice that in presence of dissipation (

then, substituting

#### 5.2.2 Damped skyrmion equation

To obtain the damped skyrmion equation due to the Gilbert term, we consider the rigid magnetic moment

Substituting

where we have set

with

Implementing the kinetic term of the skyrmion, we obtain the equation with dissipation

## 6. Electron-skyrmion interaction

In this section, we investigate the interacting dynamics between electrons and skyrmions with spin transfer torque (STT) [70]. The electron-skyrmion interaction is given by Hund coupling

### 6.1 Hund coupling

We start by recalling that a magnetic atom (like iron, manganese, …) can be modeled by a localized magnetic moment

where

#### 6.1.1 Emergent gauge potential

Because of the ferromagnetic Hund coupling (

Here, the vector potential matrix

where the factor

#### 6.1.2 Large Hund coupling limit

We start by noticing that the non abelian gauge potential

where we have used the local basis vectors

from which we learn that the two first components combine in a complex gauge field

In the large Hund coupling (

where

### 6.2 Skyrmion with spin transfer torque

Here, we investigate the full dynamics of the electron/skyrmion system

with

#### 6.2.1 Modified Landau-Lifshitz equation

Regarding the spin texture

The variation

The second step deals with the calculation of the space like component

This vector two remarkable properties:

These vectors are respectively interpreted as two spin polarised currents; the

Regarding the factor

To compare this equation with the usual LL equation (

where, due to

#### 6.2.2 Rigid skyrmion under spin transfer torque

Here, we investigate the dynamics of a 2D rigid skyrmion [

To determine

The deviation

where we have set

where we have set

from which we determine the modified equation of motion of the rigid skyrmion in presence of spin transfer torque.

## 7. Comments and perspectives

In this bookchapter, we have studied the basic aspects of the solitons dynamics in various

Regarding these two skyrmionic configurations, it is interesting to notice that, unlike magnetic skyrmions, the missing rotational symmetry of antiskyrmions leads to anisotropic DMI, which is highly relevant for racetrack applications. It follows that antiskyrmions exist in certain Heusler materials having a particular type of DMI, including MnPtPdSn [36] and MnRhIrSn [74]. It is then deduced that stabilized antiskyrmions can be observed in materials exhibiting D

To lift the limitations associated with ferromagnetic skyrmions for low-power spintronic devices, recent trends combine multiple subparticles in different magnetic surroundings. Stable room-temperature antiferromagnetic skyrmions in synthetic Pt/Co/Ru antiferromagnets result from the combination of two FM nano-objects coupled antiferromagnetically [76]. Compared to their ferromagnetic analogs, antiferromagnetic skyrmions exhibit different dynamics and are driven with several kilometers per second by currents. Coupling two subsystems with mutually reversed spins, gives rise to ferrimagnetic skyrmions as detected in GdFeCo films using scanning transmission X-ray microscopy [77]. At ambient temperature, these skyrmions move at a speed of

## Acknowledgments

L. B. Drissi would like to acknowledge “Académie Hassan II des Sciences et Techniques-Morocco”. She also acknowledges the Alexander von Humboldt Foundation for financial support via the George Forster Research Fellowship for experienced scientists (Ref. 3.4 - MAR - 1202992).

## Notes

- For convenience, we often refer to σ→,e→i,σ→.e→i=σi respectively by bold symbols as σ, ei,σ.ei=σi.
- For an electron with Zeeman field Ba, we have ωa=−gqe2meBa with g=2 and qe=−e.