Abstract
When an electron moves in a smoothly varying noncollinear magnetic structure, its spin orientation adapts constantly, thereby inducing forces that act both on the magnetic structure and on the electron. These forces may be described by electric and magnetic fields of an emergent electrodynamics^{1,2,3,4}. The topologically quantized winding number of socalled skyrmions—a type of magnetic whirl discovered recently in chiral magnets^{5,6,7}—has been predicted to induce exactly one quantum of emergent magnetic flux per skyrmion. A moving skyrmion is therefore expected to induce an emergent electric field following Faraday’s law of induction, which inherits this topological quantization^{8}. Here we report Halleffect measurements that establish quantitatively the predicted emergent electrodynamics. We obtain quantitative evidence for the depinning of skyrmions from impurities (at current densities of only 10^{6} A m^{−2}) and their subsequent motion. The combination of exceptionally small current densities and simple transport measurements offers fundamental insights into the connection between the emergent and real electrodynamics of skyrmions in chiral magnets, and might, in the long term, be important for applications.
Main
Skyrmion lattice phases (SLPs) in chiral magnets such as MnSi and other B20 transition metal compounds are a new form of magnetic order, composed of topologically protected vortex lines (the skyrmions) with a nonzero winding number which are stabilized parallel to a small applied magnetic field. Skyrmion lattices in magnetic materials were discovered only recently by means of small angle neutron scattering (SANS; refs 5, 9). The winding number of the spin structure was thereby first inferred from topological contributions to the Hall effect^{6} and Lorentz force microscopy for thin samples, where the latter even established the existence of individual skyrmions^{7}. So far, SLPs have been identified in all noncentrosymmetric B20 transition metal compounds that order helimagnetically at zero magnetic field, regardless of whether they are pure metals, strongly doped semiconductors or even insulators^{5,10,11}. The excellent theoretical understanding implies that skyrmions are a general phenomenon to be expected in a wide range of bulk materials as well as nanoscale systems^{5,7,12}, with the identification of spontaneous skyrmion lattices in monatomic layers of iron on an iridium substrate being a major new development^{13}.
Exploratory SANS studies^{14} revealed a rotation of the diffraction pattern in the SLP of singlecrystal MnSi when an electric current applied transverse to the skyrmions exceeded an ultralow threshold of j_{c}∼10^{6} A m^{−2}. It is therefore important to emphasize that j_{c} is 10^{5} times smaller than the currents needed to induce a motion in presentday spintorque experiments on ferromagnetic domain walls^{15,16,17}. However, the rotation occurred only in the presence of a small temperature gradient (∼1 K cm^{−1}), inducing gradients in the relevant forces, which in turn caused rotational torques. It was argued that above the critical current density, j_{c}, the skyrmions start to move and that only the sliding skyrmions can be rotated by the tiny torques. Although microscopic probes such as neutron scattering and Lorentz force microscopy may in principle confirm the depinning and motion of the skyrmions, they are not capable of detecting the emergent electrodynamics. Instead, to detect the motion of the skyrmions, measurements of the emergent electric fields are ideally suited, because they are directly proportional to their velocity. This may be readily achieved by means of the Hall effect.
The magnetic properties of MnSi are governed by a combination of ferromagnetic exchange interactions and weak spin–orbit coupling in the absence of inversion symmetry. At zero magnetic field MnSi exhibits a paramagnetictohelimagnetic transition at T_{c}≈28.5 K. In a small range of fields and temperatures below T_{c} the SLP is stabilized^{5}, where the skyrmions (the magnetic whirls) form a hexagonal lattice perpendicular to the applied magnetic field. The lattice constant of the skyrmion lattice, determined from the reciprocal lattice vectors, is set by the wavelength of the helimagnetic state, λ_{helix}∼180 Å.
Our study of the influence of electric current on the SLP was carried out on highpurity single crystals using a standard sixterminal lockin technique (see Methods for details). Shown in Fig. 1a are typical temperature dependences of the Hall resistivity, ρ_{x y}, for small currents (black curves). Its dominant features arise from the temperature dependence of the anomalous Hall effect. The small maximum in ρ_{x y} is therefore a characteristic of the magnetization at the lower boundary of the SLP above a narrow regime of phase coexistence between the conical phase and the SLP (ref. 18). Also shown in Fig. 1a is the Hall resistivity under an applied d.c. current density of j=2.81×10^{6} A m^{−2}, where a suppression (marked in light blue shading) is observed in the temperature range marked by the black arrows. The latter is in excellent agreement with the phase boundaries of the SLP as inferred from the magnetization and susceptibility. The size of the suppression of the Hall signal is similar to the topological Hall contribution Δρ_{x y}≈4 nΩ cm previously inferred^{6} from Hall effect measurements at small currents (see Supplementary Information).
Detailed Hall data for an applied magnetic field of 0.25 T and a wide range of applied d.c. currents, j, are shown in Fig. 1b. From these the evolution of ρ_{x y} with applied electric current density was inferred for selected temperatures, as shown in Fig. 2. At temperatures outside the SLP, shown in Fig. 2a,e, the Hall signal is unchanged as a function of j. Within the SLP, ρ_{x y} is unchanged within the experimental accuracy for small current densities, j<j_{c}, followed by a clear decrease above j_{c} over a finite range of applied currents (light blue shading), which then saturates for even larger currents (the method by which j_{c} was determined is described in Supplementary Information).
Shown in Fig. 3a is the critical current density as a function of temperature, where j_{c}∼10^{6} A m^{−2} agrees within a factor of two with the onset of the rotation of the scattering pattern observed in the SANS study^{14}. As discussed in ref. 14, the exceptionally low value of j_{c} arises from a combination of several factors: first, the very efficient Berryphase coupling between conduction electrons and the spin structure; second, because of the smooth variation of the magnetization the skyrmion lattice couples only weakly to defects and the atomic crystal structure; and third, the long range stiffness and crystalline character of the skyrmion lattice^{9} leads to a partial cancellation of the pinning forces^{19,20}. The value of j_{c} grows by roughly a factor of two when approaching the (first order) transition to the weakly fieldpolarized paramagnetic state. At the same time the suppression of ρ_{x y} given by is greatest in the centre of the skyrmion phase, with a gradual decrease at the lower boundary (see Fig. 3b).
To address our observations on ρ_{x y} from a theoretical point of view we note that the forces driving the skyrmion lattice, which also cause the topological contribution to the Hall signal, originate in quantummechanical phases (Berry phases) picked up by electrons when their spin follows the orientation of the local magnetization M. The Berry phase can be rewritten^{1,8,21} as an effective Aharonov–Bohm phase, associated with ‘emergent’ magnetic and electric fields, B^{e} and E^{e}. As the Berry phase is given by the solid angle covered by , the emergent fields B^{e} and E^{e} measure the solid angle for an infinitesimal loop in space and space–time, respectively
with ∂_{i}=∂/∂ r_{i}, ∂_{t}=∂/∂ t, and ε_{i j k} is the totally antisymmetric tensor. Because the sign of the Berry phase depends on the spin orientation, a majority spin with magnetization parallel to carries the emergent charge q_{↓}^{e}=−1/2 whilst a minority spin carries the emergent charge q_{↑}^{e}=+1/2. For a skyrmion, defined as a magnetic whirl where winds once around the unit sphere in the plane perpendicular to B (while does not vary in the spatial direction longitudinal to B), the total ‘emergent flux’ is given by . It is hence topologically quantized to one flux quantum −2πℏ/q^{e} per skyrmion (the sign accounts for the formation of antiskyrmions in MnSi; ref. 5). Further, according to equation (1) and in complete analogy to Faraday’s law of induction, a skyrmion lattice drifting with velocity v_{d}must induce an electric field E^{e}=−v_{d}×B^{e}, where E^{e}/v_{d}inherits its quantization from B^{e} with E^{e}=E^{e} and B^{e}=B^{e}. This topological quantization of B^{e} and E^{e} makes skyrmion lattices in metals an ideal system to study quantitatively the emergent electrodynamics underlying the coupling of charge and magnetism^{8}. In MnSi, the emergent magnetic field acquires an average strength of 2.5 T, that is B^{e}≈2.5 Te/q^{e}, where e is the electron charge.
The total force on an electron with momentum k and spin orientation σ is therefore given^{8} by
where v_{σ k n} is the velocity of quasiparticles in band n, Eis the physical electrical field and F_{H}≪eE is the Hall force from the normal and anomalous Hall effect. Further dissipative drag forces, F_{diss}, arising for are probably much smaller (see Supplementary Information). The extra electric current induced by −q_{σ}^{e}v_{d}×B^{e} transverse to the electrical current has to be cancelled exactly by the change of the electric Hall field , with Δρ_{x y}=ρ_{x y}(j)−ρ_{x y}(0). For a current in the x direction and a magnetic field in z direction we find using equation (2)
where the dimensionless spin polarization can be obtained by calculating the crosscorrelation of the charge current j and the emergent current j^{e} using Kubo formulae. is the ratio of electric currents obtained from E^{e} and E, where a simple approximation for is given in equation (4) in the relaxation time approximation of a multiband system (f_{n σ}^{0}is the Fermi distribution for band n with scattering rate 1/τ_{σ n} and spinorientation σ relative to the local magnetization). Up to the factor , the measurement of the Hall field is therefore a direct measurement of the emergent field E^{e} and of v_{d}=E^{e}/B^{e}, as B^{e} is quantized.
The drift velocity along the current direction, v_{d∥}, in absolute units is obtained from equation (3)
Here we used in the centre of the skyrmion phase and estimated the effective polarization to be (ref. 6). We further used that is approximately independent of the local magnetization and therefore of the temperature. As expected, the corresponding pinning velocities (see labels on the right axes in Fig. 3a) are of the same order of magnitude as the electronic drift velocities, v_{drift}∼j/e n≈0.16 mm s^{−1} for j∼j_{c}∼10^{6} A m^{−2}, where we estimate n≈3.8 ×10^{22} cm^{−3} from the normal Hall constant^{6}. The expression 4πℏM v_{pin} may finally be interpreted as the force per skyrmion and per length needed to depin the SLP (see Supplementary Information).
The critical current and the pinning velocities grow by a factor of two just below the firstorder transition T_{p}^{c}, see Fig. 3a. This cannot be explained by the decrease of the local magnetization (proportional to the small drop in ). Instead, the increase of j_{c} most probably reflects the reduction of the stiffness of the skyrmion lattice when approaching T_{p}^{c}, which is only a very weak firstorder phase transition. With decreasing stiffness the skyrmion lattice adjusts better to the disorder potential, implying that the pinning forces increase substantially^{19,22,23}. Consistent with this picture, there is essentially no temperature dependence of j_{c} at the lowtemperature side of the SLP, where the phase transition is strongly first order. Here shows a gradual temperature dependence in the range from 25.8 K to 26.5 K that may be attributed to the phase coexistence between the conical phase and the skyrmion phase as inferred from highprecision magnetometry^{18}.
Combining the results of our analysis, we show in Fig. 3c a scaling plot of , or equivalently of the parallel drift velocity of equation (5) and the associated emergent electric field , as a function of j/j_{c}. To obtain quantitative values for and in physical units, one may refer to Fig. 3a, equation (6), and with B^{e}=2.5 Te/q^{e}.
For j<j_{c}, the drift velocity vanishes within our experimental precision, as the skyrmion lattice is pinned by disorder. For j>j_{c}, the Magnus forces are sufficiently strong to overcome the pinning forces and the skyrmions start to move. For j≫j_{c} their velocity becomes proportional to the current, as pinning forces can be neglected. This is precisely the picture that has been developed for the depinning transition of charge density waves and vortices^{19,20,22,23}. However, the precise behaviour of skyrmions differs from that of vortices because their dynamics is very different. Because of their strong friction, superconducting vortices flow approximately perpendicular to the current following the Magnus force and the resulting Hall signals are tiny^{22,24}. In contrast, skyrmions drift dominantly parallel to j, thus reducing the relative speed of spin current and giving rise to a large Faraday field in the perpendicular direction. A more technical discussion of the validity of the scaling, the forces on the SLP and its direction of motion is given in Supplementary Information.
The direct observation of the emergent electric field of skyrmions reported in this paper allowed us to measure their depinning transition and subsequent motion quantitatively. This opens the possibility to address fundamental questions of the coupling of magnetism, electric currents and defects, respectively. The control and detection of the motion of magnetic whirls (skyrmions) by the interplay of emergent and real electrodynamics therefore promises to become an important route towards spintronic applications.
Methods
Sample preparation.
Single crystals of MnSi were grown by optical floatzoning under ultrahighvacuumcompatible conditions^{25}. The specific heat, susceptibility, and resistivity of small pieces taken from this single crystal were in excellent agreement with the literature, where the residual resistivity ratio was of the order 100. This indicates good, although not excellent sample purity. The sample quality was the same as for the samples studied in the SANS experiments reported in ref. 14. Samples for the measurements reported here were oriented by Laue Xray diffraction, cut with a wire saw, and carefully polished to size. Current leads were soldered to the small faces of the sample and Au wires for the voltage pickup were spotwelded onto the surface of the sample. All data reported in this paper were recorded for a magnetic field parallel to the 〈100〉 axis and electric current parallel to the 〈110〉 axis.
Spintorque transport.
For our measurements of the Hall and longitudinal resistivity we modified a standard sixterminal phasesensitive detection system such that large d.c. currents could be superimposed on a small a.c. excitation. The setup is based on a method used for measurements of superconducting tunnel junctions^{26}. It was tested on highpurity Cu to ensure proper operation. In all experiments the a.c. excitation amplitudes were not larger than a few per cent of the applied d.c. currents. The samples were carefully anchored to the cryogenic system to minimize ohmic heating and temperature gradients. In particular, compared with the SANS experiments reported in ref. 14, all thermal gradients were minimized. The Hall signal and the longitudinal resistivity, ρ_{x y} and ρ_{x x}, respectively, were measured simultaneously at a low excitation frequency of 22.5 Hz. To correct for the remaining tiny amount of uniform ohmic heating, which generated a small systematic temperature difference between the sample and thermometer of less than a few tenths of a kelvin for the largest currents applied, we calculated from the longitudinal resistivity ρ_{x x} the true sample temperature. However, we have tested carefully that our results do not depend on the precise way in which these small ohmic heating effects are corrected.
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Acknowledgements
We wish to thank P. Böni, H. Hagn, N. Nagaosa, T. Nattermann, S. Mayr, M. Opel, B. Russ, B. Spivak, G. Stölzl and V. M. Vinokur for helpful discussions and support. R.R., A.B., M.W. and C.F. acknowledge financial support through the TUM Graduate School. K.E. acknowledges financial support through the Deutsche Telekom Stiftung and the Bonn Cologne Graduate School. Financial support through Deutsche Forschungsgemeinschaft TRR80, SFB608 and FOR960 is gratefully acknowledged.
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T.S., R.R. M.H., M.W. and C.P. developed the experimental setup; T.S. and R.R. performed the experiments; T.S., R.R. and C.P. analysed the experimental data; C.F. wrote the software for analysing the data; A.B. grew the singlecrystal samples and characterized them; K.E., M.G. and A.R. developed the theoretical interpretation; C.P. supervised the experimental work; C.P. and A.R. proposed this study and wrote the manuscript; all authors discussed the data and commented on the manuscript.
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Schulz, T., Ritz, R., Bauer, A. et al. Emergent electrodynamics of skyrmions in a chiral magnet. Nature Phys 8, 301–304 (2012). https://doi.org/10.1038/nphys2231
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DOI: https://doi.org/10.1038/nphys2231
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