Deformation of the moving magnetic skyrmion lattice in MnSi under electric current flow

Topological defects are found ubiquitously in various kinds of matter, such as vortices in type-II superconductors, and magnetic skyrmions in chiral ferromagnets. While knowledge on the static behavior of magnetic skyrmions is accumulating steadily, their dynamics under forced flow is still a widely open issue. Here, we report the deformation of the moving magnetic skyrmion lattice in MnSi under electric current flow observed using small-angle neutron scattering. A spatially inhomogeneous rotation of the skyrmion lattice, with an inverse rotation sense for opposite sample edges, is observed for current densities greater than a threshold value jt ~ 1 MA m−2 (106 A m−2). Our result show that skyrmion lattices under current flow experience significant friction near the sample edges due to pinning, this being a critical effect that must be considered for anticipated skyrmion-based applications at the nanoscale.

A magnetic skyrmion is a swirling spin texture characterized by a discrete topological number, called the skyrmion number.Such topologically nontrivial states of matter were first theoretically predicted in nonlinear field theory [1,2], and subsequently in chiral magnets [3].Experimentally, magnetic skyrmions often condense into triangular-lattice, observed as six-fold magnetic Bragg reflections in small-angle neutron scattering (SANS), first discovered by Mühlbauer et al. in the prototypical chiral magnet MnSi [4].To date, triangular skyrmion-lattice structures are widely confirmed in various magnets ranging from metallic to insulating compounds [5][6][7][8][9][10], and also by various techniques such as Lorentz transmission electron microscopy (TEM) and magnetic force microscopy [11,12].
There are several prominent characteristics of magnetic skyrmions that make them quite intriguing for fundamental as well as technological application viewpoints.One is its topological protection; once created, the skyrmion can hardly be annihilated [5].Hence, there is a possibility for the magnetic skyrmion to be used as a robust information storage/carriage unit in future spintronics devices.In metallic skyrmion compounds, there is another important characteristic, namely its strong cou-pling to electric current flow.The electric current density required to realize the skyrmion-lattice motion is remarkably small at j ∼ 1 MA/m 2 (10 6 A/m 2 ), compared to that required to move magnetic domain boundaries in conventional magnetic materials j ≥ 1 GA/m 2 (10 9 A/m 2 ) [13][14][15][16].The potential for controlling magnetic skyrmions with such a low current density, together with their topologically protected nature, makes spintronics application promising [17].Hence, magnetic skyrmions continue to attract strong attention, and remain under intense scrutiny for elucidating their dynamical behavior under electric current.
A number of reports have appeared to date on the motion of magnetic skyrmions under electric current flow.In the pioneering study of Jonietz et al. using SANS, a rotation of the skyrmion lattice in MnSi was detected for an electric current density greater than the threshold value j t ∼ 1 MA/m 2 [18].There, it was argued that the electric current flow alone introduces translational motion only via spin transfer torques, which cannot be detected by SANS.Hence, a thermal gradient was intentionally applied in their experiment, which provides spatial inhomogeneity of the spin transfer torque.Indeed, the skyrmionlattice rotation was observed only when both the current flow and thermal gradient were present.Nonetheless, by careful examination of the presented data, it can be seen that the skyrmion-lattice peaks are significantly broadened under the electric current flow, the origin of which was not discussed.
Skyrmion-lattice motion under thermally homogeneous conditions was studied in the other works.From Hall-resistivity measurements on MnSi, the skyrmions move in the direction of the electric current above a critical current density j t ∼ 0.5 MA/m 2 , while the emergent electric field is transverse to the current [19,20].From a Lorentz TEM study of FeGe, the skyrmion lattice was observed to 'disappear' as the electric current exceeded a threshold value, indicating the skyrmion lattice to move much faster than the Lorentz TEM time frame [21].These studies show that the electric current flow induces the skyrmion-lattice motion.However, important microscopic information, such as the skyrmionlattice deformation, under thermally homogeneous conditions remains largely unexplored.
Here, we investigate the deformation of the magnetic skyrmion lattice under an electric current flow using the SANS technique, paying careful attention towards keeping any thermal gradient as small as experimentally achievable.We observe clearly a broadening of the skyrmion-lattice-reflection peaks for j > j t , indicating the skyrmion lattice to deform considerably when the lattice starts to flow translationally.We further find that the peak broadening is due to a spatially inhomogeneous rotation of the skyrmion lattice; by measuring opposing-sample edges, we observed a counter-rotating behavior of the skyrmion lattice.While the rotation direction does not change under inversion of the external magnetic field, it does revert when the electric current direction is reversed.Finally, the peak broadening and corresponding lattice rotation remain finite even after the current is turned off, indicating a plastic deformation of the skyrmion lattice under the force exerted by the electric current.
SANS experiments were carried out at NG7 (National Institute of Standards and Technology) and SANS-II (Paul Scherrer Institut).The single-crystal-MnSi samples were cut in a rectangular shape of 1.4 mm (width) × 7.5 mm (height) × 0.4 mm (thickness) for NG7.The incident neutron wavelength was λ i = 6 Å with ∆λ/λ i = 14 %.To further limit any temperature inhomogeneity in the measured region, and also to check the sample position dependence, only a small part of the sample was illuminated by using a very narrow beam of 2.0 mm (width) × 1.0 mm (height).Full details of our experiments can be found in the Supplemental Material [22].
A photograph of the sample mount for the NG7 experiment and its schematic illustration are shown in Figs.1(a) and 1(b), respectively.An electric current up to 2 A (j = 3.6 MA/m 2 ) was applied along the [0 0 1] direction [23].The temperature gradient along the current-flow direction was estimated by measuring the position dependence of the ordering temperature, and was confirmed to be less than 0.035 K/mm at the sample region investigated in the present study under j = 2.7 MA/m 2 [22].This is at least one order of magnitude smaller than in the previous experiment [18].The sample mount was installed into a horizontal-field magnet with the magnetic field applied along [1 -1 0] parallel to the incident neutron beam.
Representative SANS patterns measured at NG7 are shown in Figs.2(a)-2(d).They were measured from imaging an entire horizontal cross-section of the sample, as shown in Fig. 1(c).Figures 2(a) and 2(b) show the skyrmion reflections observed at the two temperatures, T = 28.3K and 28.6 K, respectively, under B ext = 0.2 T and j = 0 [24].The six-fold magnetic reflections characteristic of the skyrmion lattice were observed in a temperature range of 28 K < T < 29.2 K at B ext = 0.2 T [22].These observations are identical to those in the earlier works [4,18].
Figures 2(c) and 2(d) show the SANS patterns obtained at j = 2.7 MA/m 2 and B ext = 0.2 T. By comparing with the zero-current-flow counterparts in Figs.2(a) and 2(b), a considerable azimuthal broadening of the skyrmion-lattice peaks was observed.The peak broadening is temperature dependent at j = 2.7 MA/m 2 ; the width is considerably larger at T = 28.3K compared to that at T = 28.6 K.This result clearly indicates the significant deformation of the skyrmion lattice under large electric current.
To examine quantitatively the current dependence of the skyrmion-lattice peaks, in Figs.3(a) and 3(b), we show respectively the total scattered SANS intensity and azimuthal spot widths as functions of current density at three selected temperatures and B ext = 0.2 T. The spot width remains almost constant in the low current density region (j < j t ∼ 1 MA/m 2 ) for all measured temperatures.At T = 28.3K, the width shows a steep increase for j > j t .This clearly confirms the existence of a threshold current density for skyrmion-lattice deformation.It may be noted that the value of j t is in good agreement with the current density above which skyrmion-lattice motion was detected in the earlier studies [18,19].Hence, the peak broadening detected above j t in the present study is a direct consequence of skyrmion-lattice motion driven by the electric current.In stark contrast to the skyrmion-lattice peaks, we observed no peak broadening for the reflections in the helical phase at B ext = 0 up to 1.67 MA/m 2 , as shown by the open triangular symbol in Fig. 3(b).
Once induced by the electric current, the broadening of the skyrmion-lattice peaks persists, even after the electric current is removed.In Fig. 3(c), starting from the pristine state, the width increases monotonically up to j = 2.7 MA/m 2 above j t , whereas when lowering the current, the width stays almost the same down to j = 0.This clearly indicates that the skyrmion lattice remem-   bers the deformation generated in the driven state.In other words, the skyrmion lattice locally deforms like plastic matter when driven under electric current.This plastic deformation may be due to impurity pinning [25].Figures 3(d) and 3(e) show the temperature dependences of respectively the total scattered SANS intensity and azimuthal spot widths at fixed current density.At j = 2.7 MA/m 2 , the width shows a significant temperature dependence; it monotonically increases as temperature is decreased.Below j t , the width shows negligible temperature dependence.Figure 3(f) shows the temperature dependence of the width at j = 0, measured with decreasing temperature, after having applied the current density 2.7 MA/m 2 at T = 28.6 K. Independent of the temperature, the spot widths remain broadened compared with the pristine state at j = 0 after the current ramping, confirming the robust memory effect for the skyrmion-lattice deformation.
To investigate the origin of the broadening of the skyrmion reflections, we performed SANS measurements on the left-edge (+) or right-edge (-) of the sample, as shown schematically in Fig. 1(d).The size of the neutron illumination area is approximately 0.2 mm (width) × 1.0 mm (height).The SANS patterns obtained at T = 28.6K and B ext = -0.2T without electric current are shown in Figs.4(a) and 4(e) for the left-and right-edges, respectively.The observed patterns are identical for the two edges, and in good agreement with the data shown in Fig. 2(b).In marked contrast to the zero-current condition, under j = 2.7 MA/m 2 the reflection patterns taken from the left-and right-edge parts exhibit counterclockwise and clockwise rotations, respectively, as shown in Figs.4(b) and 4(f).
Next, the effect of inverting the current and magneticfield directions was investigated.Figures 4(c) and 4(g) show data taken under an inverted magnetic field B ext = 0.2 T. Apparently, the rotation direction does not change by inverting B ext .On the other hand, the inversion of the current direction (from j = +2.7 to −2.7 MA/m 2 ) results in a sign change of the rotation angle, as clearly seen by comparing Figs.4(d) and 4(h) (j = −2.7 MA/m 2 ) with Figs.4(b) and 4(f) (j = +2.7 MA/m 2 ).This indicates that the skyrmion-lattice rotation observed in the thermally homogeneous condition depends not on the magnetization direction, but only on the current direction.It should be noted that the transverse translational motion is due to the scalar spin chirality, which depends on the third order of the magnetization density M ( R), whereas the longitudinal translational mode along the electric current is due to the dissipative tensor which is in the second order in M ( R) [26][27][28].The independence of the observed rotation angle to the magnetization direction thus indicates that the rotation effect should be attributed to the longitudinal translational motion of the skyrmions under the electric current.The averaged rotation angles for the six skyrmion reflections are plotted in Figs.4(i) and 4(j) for the left-and right-edge parts, respectively.A counter-rotating behavior that is clearly antisymmetric around j = 0 is observed for the two edge positions with a rotation angle of approximately ±5 degrees for j = 2.7 MA/m 2 at T = 28.6K and B ext = ±0.2T. Hence, we conclude that the peak broadening is mainly due to a spatially inhomogeneous rotation of the skyrmion lattice, with counter-rotating behavior for the left-and right-edges.It should be further noted that the sum of absolute rotation angle at each edge is in semiquantitative agreement with the overall broadening observed from imaging across the whole sample [Fig.1(c)], supporting our conclusion.
Plausibly related to the present observation is a recent magneto-optical Kerr study of the motion of single skyrmions under current flow in a CoFeB-based trilayer, where the velocity of the skyrmion near the edge of the sample is reported to be 40 % less than that at the center of the sample [29].It has also been found that the pinning effect is stronger when the confining potential is acting on the skyrmions near the edge [30].Therefore, we expect that a shear flow of the skyrmion-lattice domains may appear in our sample where the velocity of the skyrmion-lattice domains is higher in the middle region and slower by the pinned motion near the edges.Such a shear flow generates a rotational torque on the skyrmionlattice domains as shown in Fig. 4(k), which may be the origin of the counter-rotating behavior observed in the present study.In other words, the skyrmion lattice flows with a shear component plausibly due to significant friction near the edges.From the rotation direction of the skyrmion lattice relative to the direction of the electric current as shown in Figs.4(b)-4(g), we can deduce that the skyrmion motion in MnSi is along the charge current not the electron current.This is in agreement with the conclusion based on the topological Hall resistivity measurement that macroscopically the skyrmions in MnSi drift in the same direction as the electric current flows [19,31].
In the simple particle model, where skyrmions are dense and move randomly, this edge effect works only near the edge (∼ 1 nm).To explain the generation of a shear stress deeper inside the sample, a numerical simulation including inter-skyrmion interactions with taking into account friction amongst the skyrmions as well as the robustness of their formed triangular lattice, is required.
Lastly, we comment on the effect of the thermal gradient.Despite our effort to minimize the thermal gradient, a small but finite gradient remained along the electriccurrent direction.This thermal gradient was pointed out as the origin of the skyrmion lattice rotation in Ref. [18] and [32].However, since the rotational motion under the thermal gradient is due to the spin transfer torque, which should depend on the macroscopic magnetization direction, one expects that the sign of the rotation angle is switched when the external magnetic field is reversed.We clearly found that the rotation angle at each edge does not change upon inversion of the magnetic-field direction, which excludes the possibility of the remaining thermal gradient giving rise to the skyrmion-lattice rotation in the present setup.
In summary, we have used SANS to study skyrmionlattice motion in bulk MnSi under electric current flow in a thermally homogenous condition.The azimuthal width of the skyrmion-lattice peaks shows significant broadening above a threshold current density j t ∼ 1 MA/m 2 .We show this peak broadening originates from a spatially inhomogeneous rotation of the skyrmion lattice, that shows opposite senses of rotation at the sample edges.An intriguing memory effect was also observed for the peak broadening and the corresponding rotation of the skyrmion lattice, indicating that the driven skyrmion lattice considerably deforms due to the friction near the sample edges.

FIG. 1 :
FIG. 1: Photograph (a) and schematic drawing (b) of the MnSi sample and sample mount for the NG7 experiment.(c, d) Schematic illustration of the beam illumination area for investigating the peak broadening (c) and counter-rotating behavior at the sample edges (d) of the skyrmion-lattice peaks.

FIG. 2 :
FIG. 2: (a)-(d) SANS measured at Bext = 0.2 T. The skyrmion reflections were measured at T = 28.3K (28.6 K) under j = 0 (2.7 MA/m 2 ), respectively.All data were measured for 10 minutes.When the intensities of the six skyrmion reflections are not equivalent, the sample is slightly misoriented away from the Bragg position.The azimuthal width direction is indicated by the arrow in panel (a).

FIG. 3 :
FIG. 3: Current-density dependences of (a) the total scattered SANS intensity and (b), (c), the azimuthal spot widths of the skyrmion reflections measured at three temperatures.The temperature used in (a)-(c) is the sensor temperature, and therefore not the true temperature estimated from the ordering temperature of the helical phase.Thus, the increase/decrease of the intensity at T = 28.3K/29.0K by applying the electric current is due to sample heating.In (b), the azimuthal width of the skyrmion (Bext = 0.2 T) and helical (Bext = 0) reflections at T = 28.6K measured by the thicker sample are also shown.Temperature dependences of (d) the total scattered SANS intensity and (e, f) azimuthal widths of the skyrmion reflections at each current density.In (c) and (f), additionally the dependence of the width after applying j = 2.7 MA/m 2 is shown.

FIG. 4 :
FIG. 4: SANS measured from the left-edge (+) or right-edge (-) of the sample as schematically illustrated in Fig. 1(d).(a)-(h) The skyrmion reflections at T = 28.6K respectively measured at Bext = -0.2(0.2 T), j = 0 (2.7 MA/m 2 ), and at the left-edge (right-edge).All data were measured for 20 minutes.Dashed lines are guides to eye for the peak positions of the skyrmion reflections in pristine condition.The currentdensity and magnetic-field dependences of the averaged rotation angles of the skyrmion reflections at T = 28.6K at (i) left-and (j) right-edges.The circle and triangle denote the data at Bext = -0.2 and 0.2 T, respectively.The dashed lines are guides to eye.(k) Schematic illustration of the skyrmionlattice rotation above jt.Blue and red circles stand for the pristine and moved skyrmions, respectively.Black arrows and v dom explain the velocity of the skyrmion-lattice domain near the sample edges.