Emergent electrodynamics in magnetic systems with a focus on magnetic skyrmions

Transcription

1 C 8 Emergent electrodynamics in magnetic systems with a focus on magnetic skyrmions K. Everschor-Sitte Johannes Gutenberg Universität Mainz Institute for Physics Mainz, Germany Contents 1 Summary of the Lecture 2 2 Introduction 2 3 Free electron in a magnetic texture Berry phase picture Alternative picture: emergent fields Magnetic Skyrmions 6 5 Emergent Electrodynamics for Magnetic Skyrmions Emergent magnetic field Moving skyrmions and emergent electric field Lecture Notes of the48 th IFF Spring School Topological Matter Topological Insulators, Skyrmions and Majoranas (Forschungszentrum Jülich, 2017). All rights reserved.

2 C8.2 K. Everschor-Sitte Fig. 1: An electron traverses a spatially inhomogeneous magnetic texture. This problem can be mapped to one where the electron moves in a ferromagnetic background but instead feels additional emergent magnetic and electric fields. 1 Summary of the Lecture When an electron traverses a spatially or temporally inhomogeneous magnetic texture, it acquires a Berry phase. An intuitive way to understand this Berry phase physics is by mapping it onto a setup where the electron moves in a simple ferromagnet but instead feels additional emergent magnetic and electric fields, see Fig. 1. The corresponding Hall effect that arises, as the electron experiences an effective Lorentz force due to the emergent magnetic field, is often denoted as topological Hall effect. In principle, any spatially inhomogeneous magnetic structure leads locally to an emergent magnetic field. However, most textures do not give rise to a topological Hall effect as the average emergent magnetic field vanishes. As it turns out, there are ideal magnetic textures - skyrmions - which are tailored to investigate these emergent fields and elucidate the term topological Hall effect. 1 2 Introduction The complicated interplay of magnetic structures and currents has been studied for quite some time in history and builds the basis for the modern and application driven field of spintronics. In 1865, James Clerk Maxwell showed that electricity and magnetism are coupled and also interact with electric charges and currents. About 40 years earlier, in 1826, William Sturgeon constructed the first electromagnet which exploits the phenomenon that a current-carrying conductor induces a magnetic field. A modern range of applications for the interplay of magnetism and currents are, e.g., data storage devices based on the giant magnetoresistance (GMR) effect. In 1988, Peter Grünberg and Albert Fert independently discovered this effect which allows to manipulate magnetically the current flow [5, 6]. In 2007, they were awarded the Nobel prize in Physics for their discovery of the GMR effect. John Slonczewski and Luc Berger proposed in 1996 the so-called spin-transfer torque effect [7, 8], where an electric current is used to influence the local magnetization of a material. For example, strong current pulses are used to switch magnetic domains in multilayer devices[9, 10], move ferromagnetic domain walls[11, 12], or induce microwave oscillations in nanomagnets.[13] The control of magnetic structures by currents is important for new types of spintronic applications like non-volatile memory devices.[14] Hence, the understanding of the complicated interplay of electric currents and magnetic structures is of high interest. In this lecture, we will focus on the question what happens to an electron that traverses a spatially and temporally inhomogeneous magnetic texture and how one can intuitively understand 1 Parts of these lecture notes have been published in Refs. [1, 2, 3, 4].

3 Emergent electrodynamics and skyrmions C8.3 this physics. We will consider this physics on a semiclassical level by Berry phase physics and emergent electrodynamics.[15, 16, 17, 18, 19, 20, 21, 22, 23, 24] Since this is a complicated topic we will mainly only consider the simplest possible model of a free electron moving through a magnetic texture and derive the emergent fields for this case. We will consider why skyrmions are special in this sense and review the first experiments revealing signatures of these emergent fields for magnetic skyrmions. 3 Free electron in a magnetic texture Let s consider a free electron traversing a spatially or temporally inhomogeneous, smooth magnetic texture M(r, t) with constant amplitude M = M(r, t) : [ ] p 2 i t ψ = 2m 11 Jµ M(r,t) ψ. (1) Here, ψ = (ψ,ψ ) T is the two-component wave function, 11 is the 2 2 unit matrix, J > 0 is the strength of the ferromagnetic exchange coupling, and µ is the magnetic moment of the electron. Note that for a particle with mass m, charge q, and Landé factor g the magnetic moment µ and the spin s are related by µ = (gq)/(2m)s. In particular, for a free electron with charge q = e the magnetic moment and the spin are antiparallel: µ = ge 2m s = gµ B σ, (2) 2 where µ B is the Bohr magneton, and σ = (σ x,σ y,σ z ) T denotes the vector of Pauli matrices with ( ) ( ) ( ) i 1 0 σ x =, σ y =, σ z =. (3) 1 0 i In a ferromagnetic material, the finite magnetization allows to define majority (minority) spins whose magnetic moments are parallel (antiparallel) to the magnetization direction. Historically, in the magnetic materials community the majority and minority spins are referred to as spinup and spin-down states, while in the semiconductor community spin-up and spin-down refers to the actual orientation of the electron spin. Here, we follow the latter nomenclature. 3.1 Berry phase picture In the adiabatic limit, the magnetic moment of the electron adapts constantly according to the local magnetization direction, and thereby the electron picks up a Berry phase.[25, 26, 16, 27, 19, 17] Berry Phases Berry phases occur in all parts of physics and arise whenever a system adiabatically evolves along a cyclic process C in some parameter spacex. A simple classical example is the parallel transport of a vector along a closed curve on a sphere. Although the system returns to its initial state it acquires a geometric phase characterized by the enclosed area on the sphere.

4 C8.4 K. Everschor-Sitte A quantum-mechanical example is a system whose states are non-degenerate and which evolves adiabatically in the parameter space X = X(t). In that case, the solution of the time-dependent Schrödinger equationi t ψ(t) = H(X(t)) ψ(t) is given by[26] ψ(t) = n a n (t 0 )e iγn(c) e i t 0 dt ǫ n(x(t )) ψ n (t), (4) where ψ n (t) and ǫ n (X(t)) are the time-evolved eigenstates and energies of the Hamiltonian H(X(t)), ψ(t 0 ) = n a n(t 0 ) ψ n (t 0 ) defines the initial state, and γ n (C) is the Berry phase. Formally, the Berry phaseγ n (C) is given by the line integral of the so-called Berry connectiona n (X) along the closed pathc: γ n (C) = A n (X) dx, (5) C A n (X) = i ψ n X ψ n. (6) Since the latter is a gauge-dependent vector-valued quantity it is also referred to as the Berry vector potential, in analogy to electrodynamics. The Berry phase, however, as a closed contour integral is independent of the gauge choice up to integer multiples of 2π. By means of Stokes theorem one can reformulate the Berry phase in terms of a surface integral. For a three dimensional parameter spacex it is γ n (C) = Ω n (X) ds, (7) S where S is the two-dimensional surface enclosed by the path C. The Berry curvature, which takes a similar role as the magnetic field in electrodynamics, is given by Ω n (X) = X A n (X). (8) Over the last decades physicists have realized that these seemingly harmless Berry phases lead to a lot of interesting physical consequences like quantum,[28] anomalous,[29] topological,[19, 30, 2] and spin[31, 32, 33] Hall effects, electric polarization,[34, 35] orbital magnetism,[36, 37] quantum charge pumping,[38] etc. (For a review of Berry phase effects on electronic properties see Ref. [39].) From the point of view where the electron adapts constantly according to the local magnetization direction, the physical consequences of the accumulated real-space Berry phase are hidden in the spatially and temporally varying spin states which enter the Berry vector potential,a σ = i ψ σ ψ σ withσ =, representing minority and majority spins, respectively. Note that minority and majority spins acquire opposite Berry phases when traversing a magnetic texture. In the following we will show that the real-space Berry curvature can be understood in terms of an emergent magnetic field that acts on the electrons traversing the magnetic texture.

5 Emergent electrodynamics and skyrmions C Alternative picture: emergent fields An intuitive way to understand this adiabatic Berry phase physics is to consider a mapping onto a problem, where the electron moves in a uniform Zeeman magnetic field, but instead feels an additional emergent electric field E e and an additional emergent (orbital) magnetic field B e :[15, 22, 20, 16, 19, 30, 17, 18, 23, 24] Emergent magnetic and electric field B e i = 2 ǫ ijk ˆM ( j ˆM k ˆM), E e i = ˆM ( i ˆM t ˆM), (9a) (9b) where i = / r i, and ˆM(r,t) = M(r,t)/M is the local magnetization direction. The emergent magnetic (electric) field measures the solid angle for an infinitesimal loop in space (space-time). Note that the emergent electric fielde e can only be non-zero for a time-dependent magnetic structure. It is important to note that these emergent fields lead to effective forces on the conduction electrons which are, in particular, Lorentz forces. Those effects on the conduction electrons can then be measured in experiments, see Sec. 5.1 and Sec In the following, we derive the above emergent fields following Refs. [23, 1] and show that the emergent magnetic field is indeed the real-space Berry curvature. 2 Derivation of emergent fields The idea is to perform a local transformationψ = U(r,t)ζ, so that the second part of the Hamiltonian becomes trivial: Jµ M(r,t) = Jσ ˆM(r,t) U(r,t) Jσ z, (10) with J = Jgµ B ( /2)M > 0. Hence, one has to rotate the quantization axis from the ẑ-direction to the local magnetization direction ˆM(r,t) for a given pointr in space and timet. Such a unitary transformationu is given by U = e i(θ/2)σ n = cos(θ/2)11 isin(θ/2)σ n, (11) where θ = θ(r,t) = arccos( ˆM ẑ) is the angle of rotation, and n = n(r,t) = ẑ ˆM/ ẑ ˆM is the rotation axis. Note that we have dropped the explicit space and time dependence to simplify the notation. Multiplying Eq. (1) byu from the left and inserting ψ = Uζ leads to a Schrödinger equation forζ of the following form: i t ζ = [q e V e + (p11 qe A e ) 2 + 2m Jσ z ]ζ. (12) By analogy to the Hamiltonian of a free electron under the influence of an electric and orbital magnetic field one can denote the2 2 matrices V e = (i /q e )U t U, A e = (i /q e )U U (13a) (13b) 2 Ref. [23] employs a different sign convention and does not use the Einstein sum convention.

6 C8.6 K. Everschor-Sitte as the emergent scalar and vector potentials. At this level the emergent charge q e is introduced artificially and actually drops out of Eq. (12). Furthermore, note that this transformation is exact. For a magnetization texture M(r,t) which varies smoothly in space and time, one can treat the scalar and vector potentials V e and A e as a perturbation to the unperturbed Hamiltonian H 0 = p 2 /(2m)11+ Jσ z. In the adiabatic approximation, V e and A e act on each band separately, allowing us to introduce electromagnetic potentials for both bands: A e σ = σ A e σ = (i /q e ) ψ σ ψ σ, V e σ = σ V e σ = (i /q e ) ψ σ t ψ σ, (14a) (14b) with σ =, for minority and majority spins, respectively, and ψ σ = U σ. Above potentials have the same form of the Berry vector potential introduced in Eq. (6). Finally, introducing for each band an emergent electric field, Eσ e = Ve σ ta e σ, and an emergent magnetic field, Bσ e = Ae σ, that are felt by the electron, it becomes clear that the real-space Berry curvature acts like an emergent magnetic field, while the mixed space-time Berry curvature acts like an emergent electric field. By an explicit calculation one finds (Bσ) e i = ǫ ijk 2q e 2 ˆM ( j ˆM k ˆM), (E e σ ) i = 2q e ˆM ( i ˆM t ˆM), (15a) (15b) where the upper (lower) sign corresponds to the band for electrons with minority (majority) spin. Let us now assign different emergent charges to the two bands, because the sign of the Berry phase depends on the orientation of the spin.[3] For a minority (majority) spin we define a q e = 1/2 (qe = 1/2), leading to the emergent fields given in Eq. (9). Note that the emergent magnetic and electric fields have been derived under the assumption that the electron follows the magnetic texture adiabatically. Corrections due to nonadiabatic processes are discussed in Refs. [40, 41, 3]. Furthermore, they do not contain fluctuations of the amplitude of the magnetization, and they do not take into account modifications due to the band structure of the system. Moreover, also dissipative drag forces acting on the electrons are not covered by Eqs. (9a) and (9b). a Signs forq e σ are opposite as in Ref. [3]. In principle, any magnetic structure that varies smoothly in position space leads to an emergent magnetic field [Eq. (9a)] and thus to a topological Hall effect. As it turns out, there are ideal magnetic textures skyrmions which are tailored to investigate these emergent fields. 4 Magnetic Skyrmions Like the concept of the Berry phase, a skyrmion is a certain object that can be defined precisely mathematical and which is realized in many areas of physics ranging from nuclear and particle

7 Emergent electrodynamics and skyrmions C8.7 Fig. 2: (Color online) (a) A (non-chiral) skyrmion configuration (bottom) is obtained by unfolding a hedgehog (top). Arrows correspond to the local magnetization direction ˆM = M/ M. The color code is chosen according to the out-of-plane component of the arrows: from red ( up ) over green (in-plane) to blue ( down ). (b) Schematic plot of a chiral skyrmion lattice with an additional in-plane winding of the magnetization. physics over high-energy physics to condensed matter physics. It is named after the nuclear physicist Tony Skyrme who in the early 1960 s studied a certain nonlinear field theory for interacting pions, showing that quantized and topologically stable field configurations nowadays called skyrmions do occur as solutions of such field theories.[42, 43] In the original work, Skyrme considered three-dimensional versions of skyrmions, but later the notion of a skyrmion was generalized to arbitrary dimensions: One can define a skyrmion as a topologically stable, smooth field configuration describing a non-trivial surjective mapping from coordinate space to an order parameter space with a non-trivial topology. In the following, we restrict the discussion to the two-dimensional unit sphere,s 2, describing the magnetization direction ˆM. An intuitive picture of a skyrmion and the mapping to a sphere is shown in Fig. 2 (a), where infinity (the boundary of the skyrmion) is mapped onto the north pole. Note that a skyrmion is everywhere non-singular and finite. In contrast to vortices, skyrmions are trivial at infinity, i.e., all arrows at the boundary point in the same direction out of plane. Skyrmions can be classified according to their integer winding number W counting the number of times the field configuration wraps around the whole sphere. To obtain the winding number of a magnetization configuration which varies in the xy plane as the example shown in the bottom panel of Fig. 2 (a), one has to integrate the solid angle swept out by ˆM: Winding number W = 1 4π ˆM ( x ˆM y ˆM)dxdy. (16) At the end of the 1980 s skyrmion structures were shown to be the mean-field ground states

8 C8.8 K. Everschor-Sitte of certain models for anisotropic, non-centrosymmetric magnetic materials with chiral spinorbit interactions subjected to a magnetic field.[44, 45, 46] Although some further theoretical works appeared on magnetic skyrmions and similar textures,[47, 48, 49, 50, 51, 52, 18] the real breakthrough was in 2009 when a hexagonal lattice of skyrmion-tubes perpendicular to a finite, external magnetic field [as sketched in Fig. 2 (b)] was experimentally discovered in the cubic helimagnet manganese silicide (MnSi).[53] Since 2009 skyrmions have been detected in several magnetic bulk materials[54, 55, 2, 56] and thin films,[57, 58, 59, 60] including metals, semiconductors, and an insulating, multiferroic material.[61, 62] Experimentally they have been observed by means of neutron scattering in momentum space[53, 56] as well as directly by microscopy techniques in real space.[57, 58, 63, 59, 60] Furthermore, artificial skyrmions have been realized at room temperature by patterning magnetic nanodots.[64, 65, 66, 67] Skyrmions are a rather generic phase and can occur due to various mechanisms,[68] mainly based on the interplay of two interactions favoring different alignments of the magnetization. More than 40 years ago skyrmion-like textures were also discussed extensively in thin films with perpendicular easy-axis anisotropy in the form of magnetic bubble domains,[69] where long-ranged magnetic dipolar interactions favoring an in-plane magnetization compete against the anisotropy preferring an out-of-plane magnetization. In chiral magnets the interplay of the competing energy scales is controlled by the strength of spin-orbit coupling and the ferromagnetic exchange coupling, which gives rise to the relativistic Dzyaloshinskii-Moriya interaction.[70, 71] Other mechanisms that may lead to skyrmions via a canting of the spin structure are frustrated exchange interactions[72] or higher order-spin exchange interactions.[59] In a recent works, it has also been shown that skyrmions might be stabilized dynamically.[73, 74] Magnetic skyrmions are very promising candidates for spintronics applications, such as ultrahigh density information storage and information processing due to their special properties: i) they are topologically protected (if supported by the boundary conditions) and thus more stable than other types of magnetic configurations; ii) skyrmions react to ultra-low electric currents,[75, 3] much smaller than the currents needed to move domain walls in magnetic wires. 5 Emergent Electrodynamics for Magnetic Skyrmions The emergent fields are particularly interesting for magnetic skyrmions, because, for example, as we will see in the following, the topology of the skyrmions causes the emergent magnetic field to be quantized. However, studying these emergent fields for skyrmions makes only sense if the requirements under which they have been derived are fulfilled. This includes in particular the adiabaticity condition that the conduction electrons pass the topologically stable knots while their spins adjust to the orientation of the local magnetization M(r, t), as shown in Fig. 3. Furthermore the emergent fields have been derived for a simple model, and therefore only a qualitative understanding of experimental signatures is possible at this level. In skyrmion systems, different length scales appear: the size of the atomic unit cell corresponding to the wavelength of the electrons, the diameter of the skyrmions, the non-spinflip mean-free path, and the spinflip scattering length. In the adiabatic limit, where the size of the skyrmions is much larger than the non-spinflip scattering length, band structure effects are negligible, and one can concentrate on real-space Berry phases. This is for example the case in the B20 structures like MnSi where the hierarchy of energy scales

9 Emergent electrodynamics and skyrmions C8.9 Fig. 3: The magnetic moment of the electron adjusts adiabatically to the direction of the local magnetization of the skyrmion. This picture is taken from Ref. [76]. (ferromagnetic exchange coupling, Dzyaloshinskii-Moriya interaction, and crystalline field interactions) determines the size of the skyrmion structures. Since the ratio of the Dzyaloshinskii- Moriya and the ferromagnetic exchange interaction is small in B20 structures, the diameter of a skyrmion is rather large and the skyrmion textures can be considered as smooth such that the adiabaticity assumption, that was used while deriving the emergent fields, is fulfilled. In the following we will explicitly consider the emergent magnetic and electric field for skyrmions and review the first experimental results. 5.1 Emergent magnetic field Comparing the winding number with the emergent fields one observes that the emergent magnetic field is of particular interest for magnetic skyrmions: The topology of skyrmions ensures the emergent magnetic flux per skyrmion to be quantized B e dσ = 4π W. (17) skyrmion Note that for topological trivial magnetic phases the skyrmion density vanishes [18], and therefore no emergent magnetic field is expected. This observation allows to detect the topological winding number via the emergent magnetic field using Hall measurements.[77, 18, 78, 2, 47, 48] In the following we will now review parts of the first studies for the emergent magnetic field performed in MnSi,[2, 79] which were first evidence of the non-trivial topological nature of the skyrmion lattice phase. Since only in the skyrmion lattice phase the emergent magnetic field occurs, the statement is, that only in this phase a further contribution to the Hall signal exists. Here, the emergent magnetic field integrated over a magnetic unit cell is quantized. As the emergent magnetic field arises due to the non-trivial topology of the skyrmion lattice, this effect is denoted as the topological Hall effect. Its size is proportional to the emergent magnetic field, thus to the skyrmion winding number and to the skyrmion density [78]. Since in MnSi the

10 C8.10 K. Everschor-Sitte magnetic lattice is a lattice of skyrmions withw = 1, [53, 2, 57] also the emergent flux has a negative sign, implying that the emergent magnetic fieldb e is oriented opposite to the external applied magnetic field B. The topological Hall effect occurs besides the normal and the anomalous Hall effect, and allows for a direct measurement of the topological properties (chirality and winding number) of the spin structure in the skyrmion lattice phase. Hall effect and anomalous Hall effect When an electric current flows through a conductor which is subject to a perpendicular magnetic field, a Lorentz force acts on the electrons. Typically, the current is applied in thexdirection and the magnetic field in thez direction. Since the Lorentz force is perpendicular to both the direction of the current flow and the magnetic field, a potential difference builds up in the y direction. It can be measured as a transverse voltage by applying contacts. This effect is called Hall effect, and the transverse voltage is usually denoted as Hall voltage. In non-magnetic materials and for small fields, the Hall resistivity ρ xy, which is defined as the ratio between the Hall voltage and the applied current, increases linearly with the applied magnetic field. In many ferromagnetic materials, an additional signal occurs due to the finite ferromagnetic magnetization M. This is known as the anomalous Hall effect, and it is proportional to the magnetizationm [80, 81, 82, 29], so that the Hall resistivity for ferromagnetic materials is usually given by ρ xy = R 0 B +µ 0 R s M, (18) where R 0 (R S ) is the normal (anomalous) Hall coefficient, and µ 0 is the vacuum permeability. The prefactor of the normal Hall effect, R 0, depends on details of the band structure and on the relative sizes of the scattering rates. In the adiabatic limit, where the spin of the charge carriers (with infinite lifetime) adjusts constantly to the smoothly varying magnetic structure, the size of the topological Hall signal may be estimated as [17, 19] ρ B xy P R q e 0 e Be z, (19) where e is the electron charge, q e = 1/2 is the emergent charge and P is the local spinpolarization of the conduction electrons. Here, the external magnetic field is applied inz direction, so that the skyrmion lattice lies in thexy plane and induces an emergent magnetic field inz direction, too. To avoid confusion with Sec. 5.2, where we consider the change of the resistivity as a function of the emergent electric field instead of the emergent magnetic field, we use the labels B and E for the resistivity change. Results of Hall measurements for MnSi are shown in Fig. 4. Basically, the Hall signal rises linearly with increasing magnetic field, but for magnetic fields in the range of the skyrmion phase, a small additional contribution can be observed. To make this additional contribution better vis-

11 Emergent electrodynamics and skyrmions C8.11 Fig. 4: The left panel shows the total Hall resistivity ρ xy of MnSi in the skyrmion phase. The right panel shows only the additional Hall contribution in the skyrmion phase. For a better visibility, curves for different temperatures are shifted vertically. The Figures are taken from Ref. [2]. Note that the sign of the Hall effect reported in Ref. [2] is not correct and we have therefore added a minus sign toρ xy compared to the original Figure. ible, the authors approximated the Hall signal by a straight line for temperatures slightly above and below the skyrmion phase and subtracted this linear part from the whole curves. The result of this data analysis is plotted in the right panel of Fig. 4. For better visibility, the curves for different temperatures have been shifted vertically. To exclude the fact that the observed additional signal is not just due to an enhanced anomalous Hall effect, which could in principle explain such a feature, Neubauer et al. [2] considered the behaviour of M(B). Inside the skyrmion phase, the slope of the magnetization as a function of magnetic field is reduced, while it is enlarged at both boundaries of the skyrmion phase. Therefore, the additional contribution to the Hall signal does not track the evolution of the magnetization. Consequently, it has to be mainly attributed to the topological winding of the magnetic structure. However, due to the variations of the slope of M(B) the additional contribution to the Hall signal, which is plotted in the right panel of Fig. 4, does not have the shape of a simple step function, i.e. constant inside the skyrmion lattice phase and zero outside, as does ρ B xy. Further experimental details can be found in Refs. [2]. Another main result of the experiments of Ref. [2] is that the sign of the topological signal is opposite to that of the normal Hall contribution. Furthermore, the Hall signal was observed to be basically the same for different orientations of the magnetic fields implying that also the topological contribution is mainly independent of the direction of the magnetic field. To conclude, the study of the Hall effect allows to observe experimentally the emergent magnetic field felt by the charge carriers while constantly adapting their spin-orientation with respect to the smoothly varying magnetic texture. 5.2 Moving skyrmions and emergent electric field To observe the emergent electric field a time-dependent magnetic texture is needed. One way to obtain a moving magnetic texture is to exploit the spin-torque effect[7, 8] by send-

12 C8.12 K. Everschor-Sitte ing a sufficiently large spin-polarized current through the sample which pushes the magnetic structure forward. Usually a spin polarized current is obtained by sending an electric current through a sample with a finite magnetic moment, but there are also other options to create spin currents like thermal or magnetic field gradients or via the intrinsic spin Hall effect.[83, 84, 85, 86, 31, 32, 33] The spin torque effect allows, for example, to move ferromagnetic domain walls[11, 12] with the potentially application of racetrack memories.[14] However, the threshold current density, above which the domain walls get unpinned from disorder, is very high,j c A/m 2. A key feature of the smooth skyrmion lattice in chiral magnets is the very efficient coupling to electric currents. This is reflected in an ultra-low threshold current density above which the skyrmion lattice starts moving. For example, in MnSi,[75, 3] the threshold current density is five orders of magnitude smaller compared to the one for traditional spin-torque effects, j c 10 6 A/m 2. For current densities well above the threshold current density, a rigid skyrmion lattice moves mainly along the current direction with a drift velocity v d, and the relative speed between the spin current and the magnetic structure is reduced.[3, 4, 87, 88] For a rigid drifting magnetic texture the direction of the magnetization ˆM(r,t) depends only on the differencer v d t, i.e. ˆM(r,t) = ˆM(r v d t). Substituting this ansatz into the expression for the emergent electric field and using t ˆM = (vd ) ˆM this leads to t ˆM = (vd ) ˆM: E e = v d B e. (20) which provides a general connection between the emergent magnetic and the emergent electric field for a drifting magnetic structure. Eq. (20) reflects Faraday s law of induction, indicating that a change of the magnetic flux causes an electric field. Since the emergent magnetic field B e is non-zero only in the skyrmion phase and furthermore quantized, this also carries over to the emergent electric field due to Eq. (20) for fixed v d. Note that E e is perpendicular to the applied magnetic field and the drift velocity, and induces extra forces on the charge carriers. Since v d is, for current densities well above the threshold current density, mainly oriented parallel to the applied current, the drift of the skyrmion lattice reduces the relative speed between the spin currents and the magnetic structure. Furthermore, the force exerted by the emergent Faraday field E e on the charge carriers is in perpendicular direction to the applied current, for current densities well above the threshold current density. The crucial point is that the emergent electric field originating from the motion of the skyrmion lattice leads to a reduction of the Hall signal[24, 3] compared to the Hall field of the non-moving skyrmion lattice. To be more precise, in a Galilean invariant one-band system, the extra electric current induced by q e σ v d B e has to be exactly canceled by the change of the electric Hall field, i.e. E q e σv d B e, wherev d is the part of the drift velocity parallel to the applied current that differs only slightly from v d for current densities well above j c. The reason for this is that a Galilean transformation of a coordinate system, where skyrmions and electrons are at rest (i.e. no current is applied and no current induced forces are present), corresponds to a coordinate system with a uniform motion with a finite electric current and a moving skyrmion lattice. In this moving reference frame, the drift velocities of the electrons and the skyrmions are the same, and the force from the emergent magnetic field cancels the force from the emergent electric field. Hence, in a Galilean invariant one-band system, the relative sizes of the emergent Hall- and the Faraday effect are equal. In general, i.e. in a system without Galilean invariance, the contributions from both emergent fields to the Hall signal do not cancel each other. However, also in a more complex system with

13 Emergent electrodynamics and skyrmions C8.13 multiple bands, one nevertheless expects a finite contribution from the emergent electric field to the Hall effect. This insight paves the way for the experimental detection of the emergent electric field by measuring the change of the Hall field. Using basically the same geometries as in Sec. 5.1, where the magnetic field was applied in z direction and the electric current in x direction, and defining the difference in the resistivity due to the applied current ρ E xy by ρ E xy ρ xy (j) ρ xy (0) = ρ E yx, (21) we obtain E = j ρ E yx σe yxe σ xx = je σ xx = P q e e Ee y = P q e e (v d B e ) y, (22) where σ ij denotes the conductivity tensor and σyx E the corresponding conductivity difference. As mentioned previously, to avoid confusion we label the resistivity change due to the emergent magnetic (electric) field by ρ B xy ( ρ E xy). The difference in the transverse current j E is accordingly given by j E = σe yxe, and P is the dimensionless spin polarization. The relation between the difference in the Hall field E and the emergent electric field E e, given by E = P q e /e Ey e from Eq. (22), confirms that up to the factor of the spin polarization P it is possible to measure the emergent electric field E e via a Hall measurement. Another interesting aspect is that one can determine the parallel component of the drift velocity v d from the same Hall measurement by virtue of Eq. (20) since the emergent magnetic field of the skyrmion lattice is quantized: v d = Ee B e E e q e B e P e j ρ E xy q e B e P = v j ρ E xy pin, (23) j c ρ xy where in the last step we introduced ρ xy = ρe xy (j j c) ρ E xy (j j c), and the pinning velocityv pin is given by e ρ v pin j c xy q e. (24) Be P We used that ρ xy e /(Be P q e ) v s /j = q e /(em) P is approximately independent of the local magnetization and therefore of the temperature. Here, v s = j e /M is the velocity of the current associated to the chargesqσ e. With this expression for the pinning velocity (Eq. (24)), the term4π Mv pin can be interpreted as the force per skyrmion and per length needed to unpin the skyrmion lattice. Fig. 5 shows the Hall resistivity ρ xy as a function of temperature in the skyrmion lattice phase of MnSi. The curves plotted in black represent Hall resistivities in the absence of an electric DC current. Different curves correspond to different applied magnetic fields. As described in Sec. 5.1, the electrons in MnSi are subject to the normal, the anomalous and in the skyrmion phase also to the topological Hall effect. In Fig. 5, the pronounced features are due to the temperature dependence of the anomalous Hall effect. Close to the lower boundary of the skyrmion phase, one can observe a small maximum in the Hall resistivity. This is due to a characteristic change in the magnetization, while going from the conical phase to the skyrmion phase and in-between passing a small regime of phase coexistence [89]. The red curves of the upper panel are measured with an applied electric DC current of j = A/m 2, and each curve belongs to a different magnetic field strength. With the applied current the Hall signal is suppressed in the skyrmion lattice phase compared to the zero-current Hall signal. The

14 C8.14 K. Everschor-Sitte Fig. 5: Hall resistivity as function of temperature T in the skyrmion phase of MnSi. In the upper panel, the black curves represent the Hall resistivity for various magnetic fields without an applied electric DC current, and the red curves are the corresponding ones with an applied DC current of j = A/m 2. With the applied current the Hall signal is suppressed in the skyrmion phase (colored blue). In the lower panel, the magnetic field is fixed,b = 250 mt, and the different curves represent different current densities. The Figure is taken from Ref. [3]. difference between two corresponding (black and red) curves is colored in blue. The boundaries of the skyrmion phase are indicated by black arrows marking the suppression region of the Hall signal. Note that they agree very well with the boundaries of the skyrmion phase obtained from other results like susceptibility measurements. In the lower panel, the magnetic field is fixed, B = 250 mt, and the different curves represent different applied DC current densities. Figure 6 summarizes the main features of the skyrmion lattice phase in MnSi subject to an electric current. In panel (a) the critical current density is plotted as a function of temperature. j c increases when raising the temperature to the boundary of the skyrmion lattice phase. Note that this cannot be explained by the fact that the local magnetization amplitudem decreases as the temperature increases. Instead, the explanation for the increase inj c is most likely that close to the phase transition the stiffness of the skyrmion lattice is reduced, so that the local magnetic structure can adjust much better to the disorder potential. In total, this leads to much higher pinning forces [90, 91, 92] that have to be overcome by the current. In panel (b) of Fig. 6, the change of the Hall resistivity due to the emergent magnetic field for large current densities ρ xy is plotted as a function of the temperature. It has an extremum in the center of the skyrmion lattice phase. The size of the reduction of the Hall resistivity is about ρ xy 4 nω cm. This is similar to the value obtained from the emergent magnetic field [2].

15 Emergent electrodynamics and skyrmions C8.15 Fig. 6: Summary of features of the skyrmion lattice phase in MnSi subject to an electric current. Panel (a) displays the temperature dependence of the critical current density and the pinning velocity (see Eq. (24)) given in absolute units on the left-hand-side and right-hand-side axes, respectively. Panel (b) illustrates the change of Hall resistivity due to the emergent electric field for large currents, i.e. plotted is ρ xy = ρe xy (j j c) ρ E xy (j j c) as a function of temperature. The scaling plot shown in panel (c) is produced by plotting the transverse electric field, E = j ρ E xy, in units of j c ρ xy as a function of j/j c. This is equivalent to plotting the drift velocityv d in units ofv pin as indicated by the right-hand-side axis. The Figure is taken from Ref. [3] Panel (c) of Fig. 6 is a scaling plot showing the transverse electric field, E = j ρ E xy, in units of j c ρ xy as a function of j/j c. Since E E e and E e v d, this corresponds also to measurements of the emergent electric field E e and the parallel drift velocity v d in units of v pin B e and v pin respectively. For current densities smaller than the critical current density, the drift velocity of the skyrmion lattice is zero (within the experimental precision and up to a very small creep, i.e. a tiny motion due to thermal (or quantum) fluctuations [91]). In this case, the current-induced forces on the skyrmion lattice are not strong enough to overcome the pinning forces due to disorder and the atomic lattice.

16 C8.16 K. Everschor-Sitte Abovej c the skyrmion lattice gets unpinned by the current and starts to move. Note that the size of the critical current density needed to unpin the skyrmion lattice is aboutj c 10 6 A/m 2. It is by a factor of about 10 5 smaller than the current densities needed to unpin, e.g. ferromagnetic domain walls in present-day spin-torque experiments [11, 12, 93]. For j j c, the drift velocity of the skyrmion lattice increases linearly with the applied current density, because in this regime the pinning forces are much smaller than the current-induced forces and can therefore be ignored. For more details and experimental results see, e.g., Ref. [3]. To conclude, the experimental data of Ref. [3] clearly establish the predicted emergent electrodynamics, implying the evidence of the motion of the skyrmion lattice.

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