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• # SkyrmionsIn Chiral Magnets

In the original sense of the word, a Skyrmion is a topological soliton solution known to occur in a non-linear field theory for interacting pions originally conceived by the nuclear physicist Tony Skyrme [1]. In a more permissive interpretation of the word Skyrmions, as mathematical objects, have found versatile application in a variety of different areas in physics. In this chapter we briefly review the historic origin of the Skyrmion and define the generalised concept that is nowadays understood as a Skyrmion. We outline previous applications in different fields of physics and then forward to 2009 to give a concise account of the discovery of the Skyrmion lattice phase in the chiral magnet MnSi.

### What is a Skyrmion?

In 1961 before the advent of quantum chromodynamics (QCD) the nuclear physicist T.H.R. Skyrme conjectured that the interior of a nucleus is dominated by a medium formed from three pion fields [1]. He introduced the Skyrme model, a non-linear sigma model, with the intention to describe baryons as the quantised soliton solutions of a field theory which involves only bosonic degrees of freedom. The model is understood as an intermediate between the traditional models which represent the nucleons as point particles interacting through a potential, and a complete description based on quarks and gluons [2]. The pion fields $\mathbf{\pi} = (\pi_1,\pi_2,\pi_3)$ are combined into a $SU(2)$-valued field $$U(\mathbf x) = \sqrt{1 - \mathbf{\pi}(\mathbf x)\cdot\mathbf{\pi}(\mathbf x)}~\mathbb{1} + i \mathbf{\pi}(\mathbf x) \cdot \mathbf{\sigma}~,$$ where $\mathbf{\sigma}$ is the vector of Pauli matrices and we have suppressed a possible time dependence of the fields. For static fields the energy in the Skyrme model is given by $$E = \int d^3\mathbf r \left( -\frac{1}{2} \text{Tr}(R_i R_i) - \frac{1}{16} \text{Tr}([R_i,R_j][R_i,R_j]) \right)~,$$ where we have introduced an associated current $R_i=(\partial_i U)U^\dagger$ and $[\cdot,\cdot]$ denotes the commutator. The vacuum is represented by $U(\mathbf x)=\mathbb{1}$. For the energy to be finite, $U$ must approach a constant at infinity [3]. The energy is invariant under translations and rotations in $\mathbb{R}^3$ and also under the transformation $U \to A U A^\dagger$ with $A \in SU(2)$, one may thus choose $U(\mathbf r \to \infty)=\mathbb{1}$. Effectively, due to this boundary condition space is then topologically (but not metrically) compactified to $\mathbb{S}^3$, and since the group manifold of $SU(2)$ is also $\mathbb{S}^3$, $U$ defines a map from $\mathbb{S}^3$ to $\mathbb{S}^3$. The structure of the homotopically distinct maps $U$ is given by the third homotopy group $\pi_3(\mathbb{S}^3)$ which happens to be isomorphic to $\mathbb{Z}$. The space of all maps $U: \mathbb{S}^3 \to \mathbb{S}^3$ decomposes into distinct subsets characterised by an integer-valued topological charge $B=\int d^3\mathbf r~\mathcal{B}$ with the topological charge density $\mathcal{B}$ given by $$\mathcal{B} = -\frac{1}{24 \pi^2} \epsilon_{ijk} \text{Tr}(R_i R_j R_k)~.$$ The minimal energy solutions for each $B$ are called Skyrmions and their energy is identified with their mass and $B$ with the Baryon number of the nucleus. The $B=1$ Skyrmion has the spherically symmetric hedgehog form [3] $$U(\mathbf x) = \exp(i f(r) \hat{\mathbf{r}} \cdot \mathbf{\sigma})~,$$ where $f(r)$ is a radial profile function obeying an ordinary differential equation with the boundary conditions $f(0)=\pi$ and $f(r\to\infty)=0$.

Skyrmions in their original sense are therefore smooth, topologically stable extremal field configurations which are trivial at spatial infinity and have a finite energy. They are defined by surjective mappings into the order parameter space $\mathbb{S}^3$ and characterised by a non-trivial topological charge $B$. Since the $n$th homotopy group of $\mathbb{S}^n$ is isomorphic to $\mathbb{Z}$ for any $n\ge1$, a more permissive definition of Skyrmion is given by

A Skyrmion is a smooth field configuration defined by a topologically non-trivial, surjective mapping from a base manifold $\mathcal{M}$ into the order parameter space $\mathcal{T}\simeq\mathbb{S}^n$, trivial on the surface of $\mathcal{M}$ and characterised by a finite integer-valued topological charge.

Fig. 1 shows the construction recipe for $\mathcal{M} = \mathbb{R}^2$ and $\mathcal{T}=\mathbb{S}^2$. We start with the identity map, $\hat{\Omega}(\hat{\mathbf x})=\hat{\mathbf x}$, with $\hat{\mathbf x} \in \mathbb{S}^2$ which can be visualised as a hedgehog configuration (c.f. Fig. 1, top left). The stereographic projection $\mathcal{P}$ maps the sphere onto the two-dimensional plane, $\mathcal{P}(\hat{\mathbf x})\in \mathbb{R}^2$ and thus a Skyrmion configuration is given by the mapping $\hat{M}: \mathbb{R}^2 \to \mathbb{S}^2$, $\hat{M}: \mathbf r \mapsto \hat{\Omega} \circ \mathcal{P}^{-1}(\mathbf r)$. The corresponding topological charge $W$ is given by $\int d^2\mathbf r~\mathcal{W}$ where we have defined the topological charge density $\mathcal{W}$ as $$\mathcal{W} = \frac{1}{8\pi} \hat{M} \cdot (\partial_x \hat{M} \times \partial_y \hat{M})~.$$ $W$ counts the number of times the mapping $\hat{M}$ sweeps out the target manifold $\mathbb{S}^2$. From the construction recipe above it is obvious that for the Skyrmion (Fig. 1, lower left) $W=-1$.

The color code has been chose such that arrows pointing to the north pole are plotted in red and those to the south pole as blue and the equator in green.

A chiral, non-inversion symmetric Skyrmion can be constructed if the hedgehog is additionally combed' by performing a $\pi/2$ rotation $\mathcal{R}$ about the $\hat{z}$-axis in order parameter space (Fig.1, top right), $\hat{N}: \mathbf r \mapsto \mathcal{R} \circ \hat{\Omega} \circ \mathcal{P}^{-1}(\mathbf r)$. $\mathcal{R}$ is a linear map on $\mathcal{S}^2$ and therefore $W=-1$ for this configuration as well. These later, chiral Skyrmions (Fig. 1, lower right) will be the main focus of this thesis. In contrast to a Skyrmion, a vortex does not sweep out the whole sphere. For example a vortex configuration is given by the map $V: \mathbb{R}^2 \to \mathbb{S}^2$, $V: \mathbf r \mapsto \hat e_\phi(\mathbf r)$, where $\hat{e}_\phi = (-\sin(\phi),\cos(\phi),0)^T$ in polar coordinates $(r,\phi)$. The vortex only sweeps out the equator, is singular at $\mathbf r=0$ and has a non-trivial winding for $\mathbf r \to \infty$.

### Skyrmions in other areas of physics

Within this generalised understanding Skyrmions have found versatile application in many different fields of physics. Here we only mention a few. In 1985 Klebanov proposed the possibility of a Skyrmion crystal [4]. A phenomenological application of this kind of a solution could be a neutron crystal, which may exist under high pressure inside neutron stars [5]. The theory might resolve puzzles concerning discrepancies about the maximum mass of stable neutron stars between observations and predictions by more traditional equation of state descriptions [6].

Liquid crystals are states of matter which show characteristics of those of a conventional liquid and those of a solid crystal. Many interesting ordering phenomena have been reported in these systems where the local order parameter is describe by a director field ( a field of headless vectors) rather than a vector field [7]. Among these the blue phases which have a regular three-dimensional cubic structure of defects with lattice periods of several hundred nanometers are particularly interesting. Here so-called $2\pi$ disclinations are singular line defects where the $2\pi$ indicates that the director rotates a full $360^\circ$ as the singular line is encircled. These singular defect configurations are unstable towards a non-singular configuration that differs from its original one only in the immediate neighborhood of the formerly singular line. For the $2\pi$ disclinations these non-singular configurations are given by Skyrmion configurations of directors in $n=2$. Recently it was shown theoretically [8], with the aid of numerical methods, that a highly chiral nematic liquid crystal can accommodate a quasi-two-dimensional Skyrmion lattice as a thermodynamically stable state, when it is confined to a thin film between two parallel surfaces.

Skyrmions were predicted to occur in quantum Hall systems close to the Landau level filling fraction $\nu =1$ for sufficiently small Zeeman splitting $g \mu_B B$ (compared to the the cyclotron gap $\omega_c = e B / mc$) [9]. The incompressible ground state of a two-dimensional electron gas at this filling fraction is ferromagnetic. For sufficiently small $g < g_c$ the charged excitations of the system were argued to be Skyrmions where their winding number is related to the charge $\nu e$ of the Skyrmion. The equivalence of physical charge and topological charge in the system is a consquence of the quantum Hall effect and is responsible for the dominating role of Skyrmions in determining many physical properties [10]. Brey and collaborators proposed that ground state close to $\nu=1$ is a crystal of charged Skyrmions [11]. Nuclear magnetic resonance measurements in $\mathrm{GaAs}$ provided only indirect evidence [12,13].

Topologically, skyrmions are equivalent to certain magnetic bubbles (cylindrical domains) in ferromagnetic thin films, which were extensively explored in the 1970s for data storage applications [14]. In ferromagnets where long-range order is frustrated due to long-range dipole-dipole interactions a wealth of different magnetic patterns can be seen, such as domain walls, vortices and periodic stripes. In Ref. [15] Lorentz transmission electron microscopy (LTEM) was used to show that a magnetic field applied perpendicular to a thin film of hexaferrite turns the periodic stripe domain state into a periodic, hexagonal lattice of chiral Skyrmion bubbles (c.f. Fig 1, lower right). In contrast to other materials however where the inversion symmetry of the atomic unit cell is broken, in hexaferrite the helicity of the Skyrmion is not fixed by crystal structure, but represent a $\mathbb{Z}_2$ degree of freedom and a random distribution of different helicities in the lattice can be observed. Here even helicity reversals within a single Skyrmion where observed. Note that the helicity is independent of the winding number which can be seen from the fact that one may smoothly deform helicities into one another.

Bogdanov and collaborators studied the mean-field theory of easy-axis ferromagnets with chiral spin-orbit interactions. They argued that in certain parameter regimes a mixed state with a finite density of Skyrmions much like the vortex lattice in type II superconductors becomes the thermodynamically stable phase [16]. Although the stability analysis was carried out in the circular unit cell approximation the Skyrmion lattice was predicted to be hexagonal [16]. Here the presence of easy-axis anisotropy turned out to be a necessary ingredient for the stabilisation of the mixed phase within the mean-field treatment. Also they assumed the magnetization vector to be homogeneous along the z-axis [17].

### Discovery of the Skyrmion lattice in MnSi

In 2009 Mühlbauer et al. reported the discovery of a Skyrmion lattice phase in the chiral magnet $\mathrm{MnSi}$ by a small angle neutron scattering study (SANS). Although in this section we concentrate on $\mathrm{MnSi}$ as the first chiral magnet the spontaneous formation of this phase of whirling magnetisation has been observed in, the Skyrmion lattice phase has since then been discovered in many other compounds as well. In 2010 the same group discovered a Skyrmion lattice phase in the doped semiconductor $\mathrm{Fe_{1-x}Co_x Si}$ [18,19]. The Skyrmion lattice phase in this material was also later confirmed by real-space images using Lorentz transmission electron microscopy (Lorentz TEM) [19]. Since then the Skyrmion lattice has been observed in a variety of different materials both as a bulk phase as well as in thin films. Here we only want to mention that the electronic properties of this set of compounds is very diverse: Among these are metals, insulators, semi-conductors and also a multi-ferroic material. This show that the Skyrmion lattice is not a peculiarity of $\mathrm{MnSi}$ but rather a general phenomenon in this class of materials.

The unifying property for all of these materials is that they crystallise in the so-called B20 structure. The symmetry transformations are described by the space group $P2_13$ with a cubic Bravais lattice [20]. With only 12 symmetry operations this space group is among the smallest compatible with the cubic lattice crystal system. The point symmetry at the component sites is $C_3$, the cyclic group of 3-fold $2\pi/3$ rotations about an appropriate $[111]$ axis. The nonsymmorphic group $P2_13$ contains in addition 3 screw rotations which involve 2-fold rotations about one of the three $[100]$ axis followed by an appropriate non-primitive translation $(0,\frac{1}{2},\frac{1}{2})$. Most notably the list of symmetry transformations does not include the inversion. The lack of inversion symmetry has profound consequences for the Ginzburg-Landau free energy description of these materials and for the symmetry constraints on the magnetic configuration that the materials can show. Materials with non-inversion symmetric atomic unit-cells can support non-inversion symmetric magnetic structures. There are other mechanisms by which Skyrmion lattice phase can be stabilised. We will return to this point at the end of this chapter. Although we concentrate prodominantly on $\mathrm{MnSi}$ in this chapter, the magnetic phases of $\mathrm{MnSi}$ are generic for chiral magnets. Particularly the phase diagram Fig. 2 can be seen as a generic phase diagram for $B20$ compounds that order helimagnetically.

The primitive cell of manganese silicide ($\mathrm{MnSi}$) contains four pairs of the 2 component formula units $\mathrm{Mn}$ and $\mathrm{Si}$ located at $(u,u,u)$, $(\frac{1}{2}+u,\frac{1}{2}-u,\bar u)$, $(\frac{1}{2}-u,\bar u,\frac{1}{2}+u)$, and $(\bar u,\frac{1}{2}+u,\frac{1}{2}-u)$ with $u_{\rm Mn}=0.138$ and $u_{\rm Si}=0.845$. $\mathrm{MnSi}$ is an itinerant ferromagnet with a fluctuating magnetic moment of 0.4 $\mu_{\rm B}$ and a saturated moment of 2.2 $\mu_{\rm B}$ per manganese atom. Before the discovery of the skyrmion lattice phase, it already attracted attention due to a high pressure anomaly: Although described very well by Fermi-liquid theory at ambient pressure, $\mathrm{MnSi}$ shows a non-Fermi liquid phase above a critical pressure of $p_c \sim 14.6~\rm kbar$ with the temperature dependence of the resistivity given by $T^{3/2}$ [22,23]. In addition in the pressure region $12~\mathrm{kbar}-20~\mathrm{kbar}$ a state of partial magnetic order was encountered in neutron scattering experiments [24].

At ambient pressure and zero applied magnetic field, $\mathrm{MnSi}$ develops magnetic order below a transition temperature $T_c = 29.5 K$ that is the result of three hierarchical energy scales. The strongest scale is the ferromagnetic exchange favoring a uniform spin polarisation (spin alignment). The lack of inversion symmetry of the cubic B20 crystal structure results in chiral spin-orbit interactions, which may be described by the rotationally invariant Dzyaloshinsky Moriya (DM) interaction favoring canted spin configurations [25,26]. The DM interaction originates from relativistic effects, i.e. spin orbit coupling $\lambda_{\rm SO} \sim 10^{-2}$, and is the lowest order chiral spin-orbit interaction [27,28,29]. In addition there are very weak crystalline field interactions which break the rotational symmetry and align the ordering wave vector of the magnetic structures along the $[111]$ axes [29].

### Magnetic phases of MnSi

The magnetic phase diagram of MnSi, Fig. 2, shows four distinct magnetic phases: a helical phase, a conical phase, a field-polarized phase and the previously mentioned skyrmion lattice phase (for historical reasons referred to as the `A-phase'' in the diagram). In the following we briefly describe the magnetic order in each of these.

#### Helical phase

Cooling the system at zero or only small applied magnetic field below the critical temperature $T_c \sim 29~\mathrm{K}$ a phase transition to the helical phase is encountered. In this phase the magnetization winds around an axis parallel to the spiral propagation vector $\mathbf q$ as shown in Fig. 4 with the local magnetic moment $\mathbf M$ perpendicular to $\mathbf q$. The period of the helix, $\lambda_{\rm h} = 2 \pi / |\mathbf q|$ is controlled by the competition of the ferromagnetic exchange with the chiral spin-orbit coupling. The weakness of the spin-orbit interaction leads to a wavelength $\lambda_{\rm h} \sim 190~\overset{\circ}{A}$ which is large as compared with the lattice constant, $a \sim 4.56~\overset{\circ}{A}$. This large separation of length scales results in an efficient decoupling of the magnetic and atomic structures. The direction of propagation $\hat q = \mathbf q / |\mathbf q|$ is determined by tiny crystal field anisotropies. Therefore, the alignment of the helical spin spiral along the cubic space diagonal $[111]$ is weak and is only fourth power in the small spin-orbit coupling, $\lambda_{\rm SO}^4$. The decoupling from the underlying atomic structure results in an extremely coherent helical phase with a huge correlation length of $10^4~\overset{\circ}{A}$ as reported in this neutron scattering study [30]. While the paramagnetic to helical transition is expected to be second order on a mean-field level, interactions between the helimagnetic fluctuations were theoretically predicted to give rise to important corrections. Indeed it was recently shown that a Brazovskii-type scenario is realized where an abundance of strongly interacting fluctuation distributed uniformly over a sphere in momentum space drives the transition first order [31].

#### Conical phase

Setting out in the helical phase one finds a phase change upon increasing the applied magnetic field above $B_{c_1} \sim 0.1~\mathrm{T}$. The stronger magnetic field allows for a net reduction in free energy by building up a uniform magnetic moment in the direction of the applied field. While for high magnetic fields above $B_{c_2} \sim 0.6~\mathrm{T}$ the DM interaction can be completely neglected and the magnetic configuration completely polarizes, there is an intermediate field range where the magnetization winds both around a spiral propagation vector $\mathbf q$ parallel to $\mathbf B$ and in addition possesses a uniform magnetization in the direction of $\mathbf B$ as the the magnetisation vectors tilt towards $\hat q = \hat B$.

The phase is referred to as the concial phase and is despicted in Fig. 5. On general grounds a crossover between the helical and the conical phase is expected where the ordering wave vector $\mathbf q$ rotates continously from the helical $[111]$ direction towards the direction of the applied field. If applied along special high symmetry axis one may encounter a second order phase transition however. The angle between the propagation vector $\mathbf q$ and the local magnetization $\mathbf M$ is smooth function of the applied magnetic field $\mathbf B$ and decreases to zero for $\mathbf B > \mathbf B_{c_2}$.

#### Skyrmion lattice phase

A first order phase transition separates a tiny pocket in the magnetic phase diagram close to $T_c$ at finite magnetic field from the surrounding conical phase. This region, termed for historical reasons "A-phase", has a hexagonal lattice of anti-skyrmions perpendicular to the applied magnetic field as its ground state. An illustration of the skyrmion lattice is despicted in Fig. 6. The configuration possesses a translational invariance along the direction of the applied magnetic field. The magnetisation configuration should therefore be imagined as an ordered arrangement of whirling tubes similar to the flux lattice in a type II superconductor. Fig. 6 shows only a single layer. The magnetic configuration can be approximated by a superposition of three helices with their propagation vectors lying in a plane perpendicular to the applied magnetic field and relative angles of $120^\circ$ plus a uniform magnetic moment antiparallel to the applied field. The relative phases are aligned such that the magnetization in the center of the skyrmion points antiparallel to $\mathbf{B}$. The lattice constant is therefore given by $2 \lambda_{\rm h}/\sqrt{3}$. The large lattice constant ensures an efficient decoupling of the magnetic structure from the underlying atomic lattice and allows for the orientation towards the applied field. The orientation of the hexagonal lattice within the plane however is determined by crystal field anisotropies. For a magnetic field in the $[001]$ direction, for instance, one of the three $\mathbf q$ vectors pins weakly in the $[110]$ direction of the atomic crystal. The building blocks of the lattice are referred to as \textit{anti}-skyrmions as their their winding number per magnetic unit cell $$W =\frac{1}{4 \pi} \int_{\rm UC} \hat M \left( \partial_x \hat M \times \partial_y \hat M \right) \label{eq:2WindingNumberDensity}$$ is quantized to -1. Here $\hat M = \mathbf M / |\mathbf M|$ and the integration is taken over the two-dimensional magnetic unit cell, which contains exactly one "knot".

The experimental technique used by Mühlbauer et al. was small angle neutron scattering (SANS). Neutron scattering is an ideal tool for the study of magnetic order in bulk phases as neutrons predominantly scatter from the magnetic structure in a solid-state system due to their magnetic moment. The lack of an electric charge allows them to penetrate deep into the system under investigation. The neutrons scatter elastically due to the interaction of their spin with the nuclei and unpaired electrons of the magnetic atoms in the sample and the scattered neutrons are recorded by detectors placed behind the sample. The Fourier modes in the magnetic order are recorded as Bragg peaks in reciprocal space. The Skyrmion lattice can be approximated by three helices with their ordering wavevectors in a plane normal to the applied magnetic field and relative angles of $120^\circ$. In a typical neutron scattering experiment the incoming neutron beam is perpendicular to the applied magnetic field. In such a setup not all of the 6 reflection spots (two per helix at $+\mathbf q$ and $-\mathbf q$) can be seen simultaneously. The setup chosen by Mühlbauer et al. aligned the incoming beam parallel to the applied field. This setup is much more advantageous and allows to record all 6 spots at the same time, c.f. Fig. 7.
Other experimental techniques were also able to prove the existence of the Skyrmion lattice. In recent years powerful real-space imaging techniques have been modified and applied to chiral magnetic systems which allow for a direct visualization of the spatial magnetization configuration. The advantage of such methods is that not only a single spin texture, but also the crystallization and melting process during phase conversions can be observed. Fig. 8 shows images of the Skyrmion lattice phase in a thin film of $\mathrm{Fe_{0.5}Co_{0.5}Si}$ recorded by Lorentz transmission elctron microscopy (LTEM). LTEM is a modification of traditional electron microscopy in which the Lorentz forces between the electrons in a beam and the sample are utilised to generate images which allow for the real-space observation of the magnetic structure of materials. The drawback of LTEM is that samples have to be electron transparent and therefore the technique can only be applied to thin films. Also LTEM images only the in-plane component of the magnetisation. Real-space images of the surface of bulk materials can be recorded using the magnetic field microscopy (MFM). MFM images forces between the surface of a sample and the magnetic stray field of a cantilever tip coated with a ferromagnetic film. The total force acting on the cantilever is inferred from small changes in its resonance frequency. It is complementary to LTEM in the sense that it is only sensitive to the out-of-plane component of the magnetisation. Fig. 9 shows MFM images of Skyrmions from the surface of bulk $\mathrm{Fe_{0.5}Co_{0.5}Si}$. Red (blue) color indicates an out-of-plane component of the magnetisation that is anti-parallel (parallel) to the line of sight. For more information about real-space imaging techniques. % - mention susceptibilty Other physical quantities also show signatures in the Skyrmion lattice phase. For instance the magnetic AC suscpetibility $\chi$ shows a sudden drop to a lower when entering the Skyrmion phase from the conical phase by increasing the applied field. It then rises exceeding the value in the conical phase before entering the conical phase once again for higher magnetic fields [32]. A more dramatic effect can be seen in measurements of the Hall effect in $\mathrm{MnSi}$. Here due to the unique topology of the Skyrmion lattice an additional top hat contribution to the Hall signal can be seen in the Skyrmion lattice phase [33]. % alle Zitate unterbringen Ever since the original discovery of Skyrmions in chiral magnets in 2009, many exciting developments have deepened our unstanding of these fascinating structures. Here we mention only a few. Neubauer et al. showed that the topological properties of the Skyrmion lattice lead to additional contribution to the Hall signal, called the topological Hall effect [34]. Everschor et al. analyzed the spin-transfer effects resulting from an electric current driven through a Skyrmion lattice, and, in particular, focussed on the current-induced rotation of the magnetic texture by an angle in such a setup [35]. Schulz and collaborators have shown that the forces acting on conduction electrons moving through a Skyrmion lattice can be accounted for by the introduction of emergent (fictious) electromagnetic fields. This offered fundamental insights into the connection between the emergent and real electrodynamics of skyrmions in chiral magnets [36]. Iwasaki et al. showed in a numerical study that a single skyrmion can be created by an electric current in a simple constricted geometry comprising a plate-shaped specimen of suitable size and geometry [37]. In experimental realisation of Skyrmion creation however with a different mechanism was reported by Romming and collaborators in 2013. They showed that on an ultrathin magnetic film in which individual skyrmions can be written and deleted in a controlled fashion with local spin-polarized currents from a scanning tunneling microscope [38]. There have been many more interesting and noteworthy publications which we cannot mention here and without question there will be many more.