Time-resolved imaging of Œrsted field induced magnetization dynamics in cylindrical magnetic nanowires
M. Schöbitz, S. Finizio, A. De Riz, J. Hurst, C. Thirion, D. Gusakova, J.-C. Toussaint, J. Bachmann, J. Raabe, O. Fruchart
TTime-resolved imaging of Œrsted field induced magnetization dynamics in cylindrical magneticnanowires
M. Schöbitz,
1, 2, 3, ∗ S. Finizio, A. De Riz, J. Hurst, C. Thirion, D. Gusakova, J.-C. Toussaint, J. Bachmann,
2, 5
J. Raabe, and O. Fruchart † Univ. Grenoble Alpes, CNRS, CEA, Spintec, Grenoble, France Friedrich-Alexander Univ. Erlangen-Nürnberg, Inorganic Chemistry, Erlangen, Germany Univ. Grenoble Alpes, CNRS, Institut Néel, Grenoble, France Swiss Light Source, Paul Scherrer Institut, Villigen PSI, Switzerland Institute of Chemistry, Saint-Petersburg State Univ., St. Petersburg, Russia (Dated: February 9, 2021)Recent studies propose that the Œrsted field plays a significant, previously disregarded role in 3D spintronics.However, there is no direct report of its impact on magnetic textures. Here, we use time-resolved scanning trans-mission X-ray microscopy to image the dynamic response of magnetization in cylindrical Co Ni nanowiressubjected to nanosecond Œrsted field pulses. We observe the tilting of longitudinally magnetized domains to-wards the azimuthal Œrsted field direction and create a robust model to reproduce the magnetic contrasts andextract the angle of tilt. Further, we report the compression and expansion, or breathing, of a Bloch-point domainwall that occurs when weak pulses with opposite sign are applied. We expect that this work lays the foundationfor and provides an incentive to further studying complex and fascinating magnetization dynamics in nanowires,especially the predicted ultrafast DW motion and associated spin wave emissions. Continuous developments in time-resolved magnetic imag-ing techniques have allowed for a shift of interest from sys-tems that lend themselves more readily to imaging, such asflat nanostrips , to more intricate systems such as three di-mensional (3D) nanostructures, with added complexity fromthe volume . Such 3D nanostructures can now be fabricatedwith increasing ease , making the exploration of novel pre-dicted magnetic configurations , such as domain walls (DWs)in Möbius strips or hopfions , feasible. A textbook casefor such an investigation is provided by cylindrical magneticnanowires (NWs), featuring a novel type of DW, the Bloch-point wall (BPW) , which exhibits an azimuthal curling ofmagnetic moments around a Bloch-point on the NW axis .The dynamics of these walls are not yet well understood, butstand out compared to DWs in flat nanostrips due to fascinat-ing theoretical predictions of fast, stable speeds and the con-trolled emission of spin waves . Recent experiments in NWshave shown that the Œrsted field induced by nanosecond cur-rent pulses plays a key role in stabilising walls exclusively ofthe BPW type, and further, imposes an azimuthal circulationparallel to the field . This allows controlling the wall structureand enables fast DW motion with speeds >
600 m / s with anabsence of Walker breakdown. The Œrsted field induced BPWcirculation switching was further studied in a simulation andtheory work, revealing a complex mechanism of the switch-ing process, involving nucleation and annihilation of pairs ofvortex and anti-vortex . Further, magnetic moments in longi-tudinally magnetized domains are predicted to align with theŒrsted field, with the degree of tilt related to a competition be-tween magnetic exchange and Zeeman energy. Previous works ∗ Electronic address: [email protected] † Electronic address: [email protected] have ignored this Œrsted field and it is clear only now thatthe field has a major influence on magnetization dynamics inNWs. However, few of the predictions have been confirmedexperimentally.Here, we make use of time-resolved scanning transmissionX-ray microscopy (STXM) to image magnetization dynam-ics in NWs subjected to nanosecond pulses of Œrsted field.Magnetically-soft Co Ni NWs with diameters of , and
101 nm were electrodeposited in anodized alumina tem-plates and freed by dissolution of the template . Wires weredispersed onto
200 nm thick, × µ m wide X-ray trans-parent Si N windows, suspended in a × intrinsicSi frame. Individual wires were lithographically contactedwith Au pads to allow for the injection of nanosecond pulsesof electric current, in turn creating an Œrsted field aroundthe NW, as in Fig. 1a. Magnetic images were acquired us-ing STXM at the PolLux bend-magnet beamline at the SwissLight Source . The sample was tilted by ◦ with respectto the X-ray beam direction and aligned so that the NW wasoriented to be as parallel as possible to the beam direction.Magnetic contrast was observed due to the X-ray magneticcircular dichroism (XMCD) effect, whereby circularly polar-ized X-ray light is absorbed differently depending on whetherthe magnetization is (anti-)parallel to the photon propagationdirection. Magnetization dynamics were observed with time-resolved STXM as shown in Fig. 1a, making use of the in-trinsically pulsed nature of synchrotron radiation (purple) andphase locking their frequency with the excitation signal ( i.e. current pulses in green) . Time-resolved image series com-prised of 1021 frames, each spaced by
200 ps , were acquiredstroboscopically with a temporal resolution of
70 ps and spa-tial resolution ≈
40 nm (full videos in supplementary mate-rial). Magnetic contrast was revealed by a division of eachframe by the average of all frames, the latter essentially beingthe static magnetic state. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b To set ideas, consider the example of a time-resolved se-ries with two frames where dynamics with opposite magneticchanges cause a variation in intensity of ± I D . The intensityin each frame is given as I S ± I D , with I S the static contribu-tion to the transmitted intensity. The average of the two framesis simply the static part, I S , and thus the magnetic contrast ineach frame is given by ( I S ± I D ) / ( I S ) = 1 ± I D /I S (1)While static XMCD contrast is of the order of a few percent,in time-resolved mode, I D /I S (cid:28) , of the order of . . Itshould be noted that if the dynamics do not lead to ± I D that issymmetric about zero, great care must be taken in the analysis.We first investigated the effect of the Œrsted field onuniformly-magnetized domains in Co Ni NWs. Suit-able regions with > µ m domains were detected usingstatic XCMD STXM, after which a < µ m section waschosen within this. A repeating signal of alternat-ing positive (+) and negative ( − ) voltage pulses with am-plitude . × A / m was applied to a
93 nm diameterNW (Fig. 1b). Pulses were spaced by
100 ns to allow for suffi-cient heat dissipation. The frames displayed in Fig. 1c,d showsnapshots of the magnetic contrast observed before and duringthe application of the (+) current pulse. Before the applicationof current in (c), no contrast is observed since magnetizationis at rest along the NW long axis (see the corresponding illus-tration in Fig. 1e) as it is for the majority of the time-resolvedseries. During the application of the current pulse(d), a bipolar contrast is observed across the NW, indicatingthe tilting of magnetization to become more parallel (black)or antiparallel (white) to the X-ray beam direction. The wiremagnetization is thus tilting towards the wire azimuthal direc-tion (see illustration Fig. 1f), consistent with the direction ofthe Œrsted field. This is first direct evidence of the influence ofthe Œrsted field in magnetic NWs. Once the pulse has ended,the magnetization returns to its relaxed state and the differen-tial magnetic contrast is no longer observed. The second pulse,with opposite polarity, gives rise to an inverted contrast.The signal-to-noise ratio is increased by taking the av-erage of the 15 frames acquired during the currentpulse (Fig. 2b). A width-averaged line scan across the wire(blue area in Fig. 2b) reveals the profile of this bipolar con-trast (blue in Fig. 2c), with asymmetric peak amplitudes . and − . . This is expected to be a signature of the tilting ofmagnetization due to the Œrsted field, where appropriate fit-ting may extract quantitative information. In the following, wecreate a model to reproduce this relatively simple physical situ-ation and hence fit the recorded contrast profile to estimate thetilt angle. Our model considers the degree of tilt within a NWcross section, the absorptivity of X-rays in the material and theX-ray beam spot size, which make up the key components ofmagnetic transmission X-ray imaging .We use an Ansatz to describe the tilt by an angle θ of mag-netic moment, m , towards the azimuthal direction when sub- jected to the Œrsted field (Fig. 2a). θ ( r ) = θ sin (cid:16) π rR (cid:17) (2) r is any point within the circular wire cross section, r = | r | , R is the wire radius and θ = θ ( r = R ) . Comparisons withmicromagnetic simulations showed that this Ansatz accuratelydescribes the tilt within the wire .In magnetic material, the absorptivity, µ , or linear rate ofabsorption of X-rays, depends on the chemical composition,the X-ray energy and the magnetization direction versus thepolarity of circularly-polarized X-ray light. For a
100 % cir-cularly (+) or ( − ) polarized X-ray beam parallel to the mag-netization, the absorptivity is given as µ ± and we additionallydefine the average absorptivity, µ av = ( µ − + µ + ) / and thedifference in absorptivity, ∆ µ = µ − − µ + . The amplitudeof the latter is of particular importance, determining I D in (1)and thus the strength of the observed magnetic contrast. Val-ues for µ av and ∆ µ of the studied Co Ni NWs at the CoL3 absorption edge can be extracted from X-ray absorptionspectroscopy (XAS) images acquired with circularly polarizedlight in static STXM. The reduced X-ray intensity behind aNW is described by the Beer-Lambert law, linking the expo-nential decay of light intensity through matter with µ . Inten-sity profiles taken across the NW can thus be fit with the law,however, a non-zero X-ray spot size must be accounted forby a convolution with a Gaussian with width, σ . A detailedanalysis is shown in the supplementary material. The fittingprocedure then provides the only free parameters, absorptivity, µ , and spot size, σ . The analysis relies on the ◦ angle be-tween the NW magnetization and the X-ray beam, induced bythe sample holder orientation. It is also due to this angle thata geometrical adjustment must be made to calculate ∆ µ fromthe extracted µ (see supplementary material).Using this fitting procedure on multiple XAS images, we de-termined µ av = 0 . ± .
002 nm − and also found an aver-age σ = 50 ± , where the uncertainty is the standard de-viation of the data sets. For comparison, the theoretical absorp-tivity for Co Ni at the Co L3 edge is µ av , th = 0 .
019 nm − ,which is a factor 3 larger than our extracted values . This isa common feature in STXM imaging, partly related to back-ground intensity incident on the X-ray detector. We expectthis amounts to ≈
33 % at the PolLux beamline STXM andarises from higher order light from the monochromating mir-ror ( ≈
15 % ) and leakage of the zone plate center stop( ≈ <
18 % ). Accounting for this via a subtraction from thestatic XAS images, we correct µ av = 0 . ± .
005 nm − which is closer to the theoretical value. The reason for theremaining difference is unclear. We similarly correct ∆ µ from . ± .
001 nm − to . ± .
002 nm − (the un-certainty also being the standard deviation). Accounting forthe ≈
50 % degree of circular polarization of X-ray lightin this experiment allows comparing to the theoretical value, ∆ µ th = 0 .
005 nm − . Our calculated value is again lowerthan theory, but still in reasonable agreement considering theerror range. The background subtraction was therefore applied (e) t + 0.8 ns t nsTR 250 nm (c) (d)(f) V o lt a g e ( V ) t (ns) (b) D e t ec t o r (a) + j X -r a y li gh t S i N w i ndo w H œ - -1 1 m z Fig. 1: Time-resolved STXM with electrically contacted magnetic NWs. a) Schematic of the STXM set-up, with incoming circularly polarizedX-ray photon bunches (purple) incident at ◦ to the normal of the SiN window. In time-resolved mode, the frequency of the current pulses(green) inducing the Œrsted field (blue) is phase-locked with the photon bunch frequency. b) Voltage pulse signal measured after the NW acrossa
50 Ω load, with (+) and ( − ) pulses of applied to induce a current density of j = 1 . × A / m in a
93 nm diameter NW. c,d) Framesfrom a time-resolved image series showing the magnetic contrast observed in a
93 nm diameter Co Ni NW with the wire edges indicated byguides to the eye. No current is flowing at time t (c), but at time t + 0 . (d) the current pulse with amplitude . × A / m is beingapplied. e,f) Illustrations of the magnetization in the NW at the time corresponding to frames (c,d).
101 nm97 nm93 nmTheoryTrendline y x z r φ m ẑ φ◌̂ θ H œ ϕ r ̂ θ m (a) - I / I (c) x (nm) 0 400200 500 (b)
250 nm x (d) ϴ ( ° ) j (10 A/m )Raw dataFit300100 3050 Fig. 2: Tilting of magnetization in NW domains as a function of cur-rent density. a) Schematic of a NW with magnetic moment, m , de-scribed with spherical polar angles θ and φ in the cylindrical coordi-nate system ( r , ϕ , z ). An applied Œrsted field tilts m from ˆ z towards ˆ ϕ by an angle, θ . b) Average of all 15 frames acquired during the ap-plication of a current pulse with amplitude . × A / m ina
93 nm diameter NW. The width-averaged line scan is indicated bythe blue arrow. The corresponding intensity profile is shown in bluein (c). The black curve is the fit from our model, with θ = 17 ◦ . d) θ as a function of applied current density, j , in three different NWsamples with diameters , and
101 nm , and with vertical error-bars. Trendlines through each data-set and and analytical solutions(see eq. (4)) for each wire diameter are shown as dashed and solidlines, respectively. to all acquired STXM images. It should be noted that consid-ering the derived values of µ av and ∆ µ as effective, allowsextracting the exact magnetization direction if the values werecalibrated on uniform magnetization at ◦ . To now fit the bipolar contrast profile in Fig. 2c, we re-usethe Beer-Lambert law and include non-uniform magnetization,such as described by our Ansatz in (2): I ( x ) = I exp (cid:26) − Z (cid:18) µ av + 12 ∆ µ ˆ k · m (cid:19) d l (cid:27) (3)This describes the progressive absorption of X-rays througheach elementary segment with length, d l , and with ˆ k · m thecomponent of magnetization along the X-ray beam direction, ˆ k . I depends on m ( r ) which itself depends on θ .The intensity profile is then convoluted with the Gaussianto account for the finite spot size, already determined from thestatic XAS image analysis. By performing the same imagecalculation as in (1) with the dynamic ( θ = 0 ) and the static( θ = 0 ) intensity profile, the bipolar magnetic contrast pro-files can be reproduced. The only free parameter for the fitis θ , while all other variables are fixed as previously deter-mined from the XAS image analysis. We revisit now the mag-netic image in Fig. 2b and the corresponding contrast profilein blue in Fig. 2c, which is fit by the black curve. The agree-ment is excellent ( r = 0 . ), also reproducing the asymmet-ric signal originating from non-linear X-ray absorption due tothe exponential nature of the Beer-Lambert law. From the fitwe determine the tilt of magnetic moments on the surface as θ = 17 . ◦ , caused by the current pulse induced Œrsted field.This value comes with multiple sources of error. First, we de-termined that the uncertainty in the NW diameter measuredfrom scanning electron microscopy images and the uncertaintyin the spot size extracted from the XAS analysis translate to anerror of < ≈
10 % in θ . This remains moderate comparedwith the last source of error, the uncertainty on µ av and ∆ µ ,for which the impact is critical due to the aforementioned ex-ponential in (3). We repeat the fitting of the data in Fig. 2cusing values for µ av of . and .
016 nm − and ∆ µ of . and .
004 nm − which correspond to an uncertainty ofone standard deviation (see supplementary material Fig. S2).This gives θ , max = 42 . ◦ and θ , min = 12 . ◦ which againis strongly asymmetric. Further, for the case of the intensityprofile produced with θ , max , the fit to the data is quite poor( r = 0 . ) indicating that the original fit with θ = 17 . ◦ ismore appropriate.This fitting analysis to determine θ was applied to multi-ple time-resolved image series from three wire diameters andseveral applied current densities, j , with the results plotted inFig. 2d. θ increases linearly with j , as also shown by thedashed trendline (passing through the origin) for each dataset. While the linear dependence of the and
97 nm diam-eter wires is clear, the
101 nm diameter wire exhibits a largerspread. For any j , θ increases from the to to
101 nm diameter wire and using the trendline, we find a tilt rate in θ equivalent to . , . and . ◦ per A / m , respec-tively.To compare the experimental θ we use an analytical modeldeveloped by A. De Riz et al , balancing the competitionbetween exchange and Zeeman Œrsted energies to describe θ ( j ) in longitudinal domains in NWs. To first order, thisreads: θ ≈ . µ jM s R πA (4)with µ = 4 π × − the vacuum magnetic permeabil-ity, M s the spontaneous magnetization of the material and A the exchange stiffness. De Riz et al. found that for j < × A / m , the model is an accurate descriptionand matches well with simulations, meaning that it is appro-priate to compare to the results presented here. Using mag-netic parameters for Co Ni NWs ( M s = 0 .
77 MA / m and A = 1 . × − J / m ), (4) is plotted as solid line for eachNW diameter in Fig. 2d. The tilt rate is θ = 23 . , . and . ◦ per A / m for , and
101 nm diameter wires,respectively. Direct comparison with our experimental resultshence show good agreement for the
101 nm diameter wire, buta lower tilt angle than expected for the other two samples. Fur-ther, within its range of uncertainty, it is difficult to deduce a /R dependence from this data set.We now turn to the imaging of BPWs under the influenceof the Œrsted field in the same NWs. DW positions were de-termined using static XMCD, after which a current pulse ofsufficient amplitude ( j ≈ × A / m ) was sent throughthe NW to ensure a) the BPW DW type by transforming wallsof the transverse-vortex kind to BPWs and b) that the BPWis sufficiently pinned on an extrinsic pinning site to allow fora reproducible magnetization process over millions of pulses.Multiple time-resolved series were acquired while applyinga similar voltage pulse signal as in Fig. 1b, with different j around j c , the critical current density expected for BPW circu-lation switching .We imaged a tail-to-tail BPW in a
101 nm diameter NW,first in the regime when j < j c by applying a current den-sity of ± . × A / m . Fig. 3a-d show the frame averageof the frames acquired during (a) the application of the current pulse, (b) the first
100 ns rest period, (c) the
200 nm (a)(b)(c) (f)(e) H œ (g) H œ (d) (h) - I / I x (nm) 0 400200 0.000-0.0040.004-0.008 (k) (l) LeftRightRaw dataFit
LeftRightRaw dataFit
100 300 500 x (nm) 0 400200100 300 500 - I / I -1 1 m z
200 nm (i) (j)
Fig. 3: BPW in a
101 nm diameter NW subjected to an Œrsted field.Average of all wavelet filtered time-resolved frames acquired dur-ing (a) the application of a current pulse with amplitude . × A / m , (b) the
100 ns rest period, (c) the − ) currentpulse and (d) the
100 ns rest period. The BPW position is indicatedby orange circles. The micromagnetic simulations in (e-h) correspondto the images in (a-d) and show a tail-to-tail BPW in its compressed(a), expanded (g) and relaxed (f,h) state. The direction of the appliedŒrsted field is indicated where applicable. Simulated time-resolvedmagnetic contrast of a compressed (i) and expanded (j) BPW in a
100 nm diameter NW. Each frame was blurred with a Gaussian. Con-trast profiles for the case of a compressed (k) and expanded (l) BPW,corresponding to line scans left (solid lines) and right (dashed lines)of the wall center. The true line scans from the image in (a) and (c)are shown in blue, while the fit from our model is shown in black. ( − ) current pulse and (d) the second
100 ns rest period. Allframes were filtered with a hat wavelet filter to improve thesignal-to-noise ratio. In addition to the Œrsted field tilting ob-served within the domains during the pulse application, all fourimages contain a strong contrast at the BPW location (orangecircle).As expected for this regime, in Fig. 3a,c no specific con-trast is visible at the wall center, indicating that the sign ofazimuthal curling directly around the Bloch-point remains un-changed for the duration of the image series. The appliedŒrsted field pulses (see black arrows in Fig. 3e and f) arethus once parallel and once antiparallel to the wall circula-tion, so that different dynamics are expected during each ofthe two pulses. Indeed, in (a) and (c) we observe two differ-ent stronger contrasts around the wall center. In (a) there arefour small symmetric lobes of contrast with a stronger inten-sity than seen in the longitudinal domains far from the wall.The bipolar contrast to either side of the wall center is indica-tive of a change of circulation that is now opposed to the staticcirculation ( i.e. state I S ), which can be understood as follows:the bipolar contrast, albeit stronger, matches that within thedomains, suggesting a tilt towards the Œrsted field direction.This tilt direction opposes the intrinsic BPW circulation and asthe BPW does not reverse its sign of circulation, the observedchange therefore reflects rather a compression of the wall.Such a compression was also evidenced with micromagneticsimulations (Fig. 3e) of a tail-to-tail BPW in a
100 nm diame-ter Co Ni NW, subjected to an antiparallel Œrsted field in-duced by a current with amplitude × A / m . These sim-ulations were obtained with our homemade finite element free-ware F EE LLG
OOD , based on the Landau-Lifshitz-Gilbertequation. The simulated distribution of magnetization wasfurther used to simulate XAS images expected for an ab-sorption according to eq. (3) and governed by our experimen-tally determined values of µ av and ∆ µ . Importantly, the ◦ sample holder alignment was accounted for in the simulatedimaging. Magnetic contrast images were calculated by apply-ing eq. (1) to simulated XAS images of a dynamic and staticBPW. A Gaussian filter was applied to reproduce the effect ofa finite width spot size as in the experiment (unfiltered imagesare shown in the supplementary material). The magnetic con-trast simulated for a compressed tail-to-tail BPW is shown inFig. 3i. The similarity with the image in (a) is striking, thusconfirming the qualitative explanation of a BPW compression.Minor differences, e.g. the size of certain features comparedto in (a), are likely related to the Gaussian filter applied to thesimulation.Conversely, in (c) there are only two large lobes of oppo-site contrast, suggesting that the circulation of the static state isnow being enhanced. This should be the result of an expansionof the BPW (see simulation in Fig. 3g) as the applied Œrstedfield is parallel to the wall circulation. The magnetic con-trast from the simulated expanded wall (g) is shown in Fig. 3j,with key contrast features again matching with those observedin (c). This combination between time-resolved imaging andsimulated imaging provides a powerful tool to explain ob-served contrasts.We return to our model for a quantitative verification of thequalitative explanation. We plot in blue in Fig. 3k,l contrastprofiles from line scans taken across the wire, through the re-gions of contrast left (solid curve) and right (dashed curve) ofthe BPW center. Fig. 3k and (l) correspond to the images in(a) and (c), respectively. In this case, fitting is achieved be-cause the static ( I S ) and the dynamic, either compressed or ex-panded, state intensity profiles are defined by separate θ = 0 .It must be noted that the convolution with the Gaussian spotsize along x only is now slightly less valid because the mag- netization is no longer homogeneous along z , and a finer anal-ysis should convolute along both the x and z direction. Still,the black curves in Fig. 3k,l are fits to the contrast profiles andshow an excellent agreement on either side of the wall center.The BPW compression proposed to explain (a) is confirmednumerically with the fits in (k): left and right of the wall cen-ter, θ tilts from to − ◦ and from to − ◦ , respectively.The Œrsted field reverses the sign of circulation, compressingthe wall and giving rise to the bipolar contrast and hence fourcontrast lobes around the wall center. Similarly, the fits for (l)reveal θ tilts from to ◦ and from to ◦ , left and rightof the wall center, respectively, confirming the enhancementof the static circulation, or an expansion of the BPW. (a) and(b) together show the breathing of the BPW, predicted only bysimulations until now . The contrast patterns were observedin multiple image series and are inverted in the case of a BPWwith opposite static circulation (see supplementary material).For the interpulse periods displayed in Fig. 3b,d a strongwhite or black contrast, respectively, is observed at the DWlocation. This should be a result of small scale BPW motion,however, intricacies of this contrast, such as the direction ofmotion and a time evolution of the contrast are not yet under-stood (see supplementary material for further discussion).We finally mention the case when j > j c , for which theBPW switches its sense of circulation . This could not beobserved experimentally with time-resolved STXM becauseDWs disappeared for j ≥ j c , which we attribute to heat as-sisted DW depinning. Future measurements, possibly with en-gineered DW pinning sites are required to observe this effectwith temporal resolution.In conclusion, we have used time-resolved STXM to imagedynamic changes of magnetic textures in cylindrical NWs. Weobserve the effect of the Œrsted field on longitudinally mag-netized domains and evidence the breathing of a BPW whensubjected to pulses with opposite sign. A quantitative analysisof the magnetic contrast is provided by a robust model basedon the absorptivity of X-rays and a description of the magne-tization in a NW cross section. This highlights the depth ofinformation obtainable with time-resolved magnetic imagingand that a direct comparison of the observed dynamics to i.e. simulations and theory is possible. This type of direct collab-oration will significantly grow the understanding of magnetic3D nanosized systems and enable better control over them.See supplemental material for the following: full time-resolved image series shown in Fig. 1, Fig. 2 (D3_43.tif) andFig. 3 (C4_86.tif); calculation of the projection of magnetiza-tion in a tilted NW; detailed explanation of the XAS analysis,impact of the uncertainty in absorptivity on the asymmetricerror bars on θ ; unfiltered simulated images of a breathingBPW; images of breathing of a BPW with switched circula-tion; and an explanation of magnetic contrast of BPW motion.M. S. acknowledges a grant from the Laboratoired’excellence LANEF in Grenoble (ANR-10-LABX-51-01).The project received financial support from the French Na-tional Research Agency (Grant No. JCJC MATEMAC-3D).This work was partly supported by the French RENATECHnetwork, and by the Nanofab platform (Institut Néel), whoseteam is greatly acknowledged for technical support. Partof the work was performed at the PolLux STXM endsta-tion of the Swiss Light Source, Paul Scherrer Institut, Villi-gen PSI, Switzerland, financed by the German Bundesminis- terium für Bildung und Forschung (BMBF) through contracts05K16WED and 05K19WE2.The data that support the findings of this study are availablefrom the corresponding author upon reasonable request. K. Litzius, I. Lemesh, B. Krüger, P. Bassirian, L. Caretta,K. Richter, F. Büttner, K. Sato, O. A. Tretiakov, J. Förster, R. M.Reeve, M. Weigand, I. Bykova, H. Stoll, G. Schütz, G. S. D. Beach,and M. Kläui, Nat. Phys. , 170 (2016). R. Juge, S.-G. Je, D. de Souza Chaves, L. D. Buda-Prejbeanu,J. Peña-Garcia, J. Nath, I. M. Miron, K. G. Rana, L. Aballe, M. Fo-erster, F. Genuzio, T. O. Mente¸s, A. Locatelli, F. Maccherozzi,S. S. Dhesi, M. Belmeguenai, Y. Roussigné, S. Auffret, S. Pizzini,G. Gaudin, J. Vogel, and O. Boulle, Phys. Rev. A , 044007(2019). S. Finizio, S. Wintz, S. Mayr, A. J. Huxtable, M. Langer, J. Bailey,G. Burnell, C. H. Marrows, and J. Raabe, Appl. Phys. Lett. ,212404 (2020). C. Donnelly and V. Scagnoli, J. Phys.: Condens. Matter ,213001 (2020), http://arxiv.org/abs/1909.08956v2 . C. Donnelly, S. Finizio, S. Gliga, M. Holler, A. Hrabec, M. Odstr-cil, S. Mayr, V. Scagnoli, L. J. Heyderman, M. Guizar-Sicairos,and J. Raabe, Nat. Nanotech. , 356 (2020). S. Bochmann, A. Fernandez-Pacheco, M. Maˇckovi`c, A. Neff,K. R. Siefermann, E. Spiecker, R. P. Cowburn, and J. Bachmann,RCS Adv. , 37627 (2017). G. Williams, M. Hunt, B. Boehm, A. May, M. Taverne, D. Ho,S. Giblin, D. Read, J. Rarity, R. Allenspach, and S. Ladak, NanoRes. (2017), 10.1007/s12274-017-1694-0. L. Skoric, D. Sanz-Hernández, F. Meng, C. Donnelly, S. Merino-Aceituno, and A. Fernández-Pacheco, Nano Lett. , 184 (2020). A. Fernandez-Pacheco, R. Streubel, O. Fruchart, R. Hertel, P. Fis-cher, and R. P. Cowburn, Nat. Commun. , 15756 (2017). S. Grytsiuk, J.-P. Hanke, M. Hoffmann, J. Bouaziz, O. Gomonay,G. Bihlmayer, S. Lounis, Y. Mokrousov, and S. Blügel, Nat. Com-mun. (2020), 10.1038/s41467-019-14030-3. S. Da Col, S. Jamet, N. Rougemaille, A. Locatelli, T. O. Mente¸s,B. S. Burgos, R. Afid, M. Darques, L. Cagnon, J. C. Toussaint,and O. Fruchart, Phys. Rev. B , 180405 (2014). R. Feldtkeller, Z. Angew. Physik , 530 (1965). W. Döring, J. Appl. Phys. , 1006 (1968). R. Wieser, E. Y. Vedmedenko, P. Weinberger, and R. Wiesendan-ger, Phys. Rev. B , 144430 (2010). R. Hertel, J. Phys.: Condens. Matter , 483002 (2016). M. Schöbitz, A. De Riz, S. Martin, S. Bochmann, C. Thirion, J. Vo-gel, M. Foerster, L. Aballe, T. O. Mente¸s, A. Locatelli, F. Genuzio,S. Le Denmat, L. Cagnon, J. Toussaint, D. Gusakova, J. Bach-mann, and O. Fruchart, Phys. Rev. Lett. , 217201 (2019). A. D. Riz, J. Hurst, M. Schöbitz, C. Thirion, J. Bachmann, J. Tou-ssaint, O. Fruchart, and D. Gusakova, “Mechanism of current-assisted bloch-point wall stabilization for ultra fast dynamics,”(2021). J. Raabe, G. Tzvetkov, U. Flechsig, M. Böge, A. Jaggi, B. Sarafi-mov, M. G. C. Vernooij, T. Huthwelker, H. Ade, D. Kilcoyne,T. Tyliszczak, R. H. Fink, and C. Quitmann, Rev. Sci. Instr. ,113704 (2008). S. Finizio, S. Wintz, B. Watts, and J. Raabe, Microsc. Microanal. , 452 (2018). R. Nakajima, J. Stöhr, and Y. U. Idzerda, Phys. Rev. B , 6421(1999). U. Flechsig, C. Quitmann, J. Raabe, M. Böge, R. Fink, andH. Ade, AIP Conf. Proc. , 505 (2007). D. Jiles, T. Chang, D. Hougen, and R. Ranjan, J. Phys. , 1937(1988). P. Talagala, P. S. Fodor, D. Haddad, R. Naik, L. E. Wenger, P. P.Vaishnava, and V. M. Naik, Phys. Rev. B , 144426 (2002). http://feellgood.neel.cnrs.fr. S. Jamet, S. D. Col, N. Rougemaille, A. Wartelle, A. Locatelli,T. O. Mente¸s, B. S. Burgos, R.Afid, L. Cagnon, J. Bachmann,S. Bochmann, O. Fruchart, , and J. C. Toussaint, Phys. Rev. B , 144428 (2015). upplementary material: Time-resolved imaging of Œrsted field induced magnetization dynamics incylindrical magnetic nanowires M. Schöbitz,
1, 2, 3, ∗ S. Finizio, A. De Riz, J. Hurst, C. Thirion, D. Gusakova, J.-C. Toussaint, J. Bachmann,
2, 5
J. Raabe, and O. Fruchart Univ. Grenoble Alpes, CNRS, CEA, Spintec, Grenoble, France Friedrich-Alexander Univ. Erlangen-Nürnberg, Inorganic Chemistry, Erlangen, Germany Univ. Grenoble Alpes, CNRS, Institut Néel, Grenoble, France Swiss Light Source, Paul Scherrer Institut, Villigen PSI, Switzerland Institute of Chemistry, Saint-Petersburg State Univ., St. Petersburg, Russia (Dated: February 9, 2021)
Full time resolved image series
The first image series (D3_43.tif) attached to this manuscript shows a full time-resolved image series from which still frameshave been shown in the main manuscript (see Fig. 1c,d and Fig. 2a). The image series is comprised of 1021 frames, each taken atan interval of
200 ps and a square pixel size of
25 nm . A hat wavelet filter has been applied in order to reduce noise and improvethe visibility of the magnetic contrast. In the image series, we observe the tilting of magnetization away from the nanowire(NW) long axis and towards the azimuthal Œrsted field direction when a (+) and ( − ) polarity, current pulse with amplitude . × A / m is applied to a
93 nm diameter Co Ni NW. We clearly see the effect of both of the pulses, as well as a veryfaint ringing effect that appears after each pulse and lasts for a duration of tens of ns.The second image series (C4_86.tif) shows the full time-resolved series, from which still frames have been shown in the mainmanuscript in Fig. 3a-d. There are 1021 frames, taken at
200 ps intervals and the same hat wavelet filtering was applied. In theimage series, we observe the "breathing" of a Bloch-point domain wall (BPW) as explained in the main manuscript (see Fig. 3of main manuscript), when a (+) and ( − ) polarity, current pulse with amplitude . × A / m is applied to a
101 nm diameter Co Ni NW. In addition, in the interpulse time we can now visualize the slow growing and shrinking of a black andwhite contrast, respectively, indicating the slight motion of a BPW. This is further discussed later in this supplementary material.
Projection of magnetization in a tilted nanowire
In the manuscript we use an Ansatz to describe the tilt by an angle θ of magnetic moment, m , towards the azimuthal directionwhen subjected to the Œrsted field, θ ( r ) = θ sin (cid:16) π rR (cid:17) (1) r is any point within the circular wire cross section and r = | r | . Further, R is the wire radius and θ = θ ( r = R ) .We now reformulate m , which in spherical coordinates is (1 , θ, φ ) on axes that are (ˆ r , ˆ ϕ , ˆ z ) (see schematic in Fig. 2ain the manuscript). This can be projected into a cartesian coordinate system using a standard conversion, giving m =(sin ( θ ) cos ( ϕ ) , sin ( θ ) sin ( ϕ ) , cos ( θ )) with ϕ = arctan ( y/x ) the angle between the x -axis and r (see schematic in Fig. 2ain the manuscript), where ( x, y ) is the cartesian coordinate of r . Further, we need to exhibit the component of m along the X-raybeam direction, ˆ k , given by the dot product ˆ k · m . This is well defined by the plane of the sample holder, which is tilted by α = 30 ◦ from the z -axis and passing through the x -axis. It follows that ˆ k · m = 0 + sin ( θ ) sin ( ϕ ) cos ( α ) + cos ( θ ) sin ( α ) (2)As stated in the manuscript, (2) is directly related to the magnetic contribution to the absorption of X-rays and an exact solutionrelated exclusively to θ is found by substituting θ , ϕ and α . ∗ Electronic address: [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b (a) x (nm) I / I Raw dataFitSpot (x 20)Theory -x x (c)(b)
Fig. S1: Analysis of static XMCD STXM images to extract X-ray absorptivity and spot size. a) XAS image of a section of
93 nm diameterCo Ni NW taken with static STXM using negative circularly polarized X-ray light at the Co L3 edge. In (b) the corresponding XMCDimage of the same NW section shows two longitudinal magnetic domains, with their magnetization direction indicated (green). The intensityprofile on the XAS image in (a) is plotted in (c) (blue). Also plotted are the fitted convoluted intensity profile (black), the theoretical transmittedintensity profile before convolution (red) and the Gaussian X-ray spot size (green, increased by factor 20).
X-ray absorption spectroscopy analysis
The absorptivity of the studied Co Ni NWs at the Co L3 absorption edge is extracted from intensity profiles taken per-pendicular to the wire long axis in X-ray absorption spectroscopy (XAS) images taken in static scanning transmission X-raymicroscopy (STXM). A typical XAS image is shown in Fig. S1a. The line in blue indicates the line scan with which the intensityprofile in blue in Fig. S1c is determined. The observed profile is related to the absorptivity of X-rays and can be fitted with theBeer-Lambert law, according to which the theoretical transmitted intensity of X-ray light, I ( x ) , for a profile along x , is I ( x ) = I exp [ − µt ( x )] (3)For the case of NWs, the thickness is t ( x ) = 2 p F ( R − x ) , where the F = 4 / term is a geometrical adjustment made forthe ◦ sample holder and R is the NW radius measured in SEM. However, the intensity profile that can be calculated with (3)assumes an infinitely small X-ray spot size. We model the realistic spot size with a Gaussian, normalized to have an integral equalto unity, I spot ( x ) = exp (cid:0) − x / σ (cid:1) (4)where σ is the standard deviation of the Gaussian, which is a measure of the spot width. We fit the observed XAS intensity profilewith the convolution of (3) and (4) to accurately extract the only free parameters, µ and σ . The effect of the convolution is tospread the absorption profile laterally. This at the same time blurs the image and also decreases the apparent absorption, however,the total absorption remains fixed. This step is crucial to extract realistic values of µ with such an analysis. The intensity profilesfrom the XAS images are nearly perfectly reproduced with the convoluted Beer-Lambert law (black dashed curve in Fig. S1c).Minor mismatches arise from the asymmetry in the true spot shape, however, these have little impact on the extracted values.The red and green curve in Fig. S1c display the transmitted X-ray intensity profile before convolution and the Gaussian spot size(scaled by a factor 20) used for said convolution, respectively.However, due to the NW and sample holder orientation, the µ extracted by this analysis does not equal exactly the theoretical µ ± ,th and we extract instead µ ± ≈ µ av ± ∆ µ (cid:16) ˆ k · m (cid:17) , which critically depends on the projection of magnetization along theX-ray beam direction. It is thus critical that we know the value of ˆ k · m for our analysis. To this end, the intensity profiles acrossthe NWs (see blue line in Fig. S1a) were chosen in an area of uniform magnetization, i.e. inside a longitudinally magnetizeddomain. This was known using the corresponding static XMCD images (see Fig. S1b) where homogeneous black or whitecontrasts indicated domains with uniform magnetization parallel or antiparallel to the wire long axis, respectively. Thus, dueto the ◦ sample holder, ˆ k · m = ± sin (30 ◦ ) = ± . and as a result, the µ from our fit ≈ µ av ± ∆ µ . Nonetheless, µ av and ∆ µ can be correctly calculated by carrying out this analysis on intensity profiles on multiple XAS images taken with bothpolarizations of light for the different NW samples and accounting for a factor 2 in ∆ µ . Asymmetric errors on θ As discussed in the manuscript, the uncertainty in the values of µ extracted in our XAS analysis leads to asymmetric errorbars in θ , due to the exponential nature of the Beer-Lambert law. However, this asymmetry is also reflected in the goodness x (nm) 0 400200 500300100 - I / I x (nm) 0 400200 500300100 x (nm) 0 400200 500300100 - I / I - I / I ϴ = 12.7° ϴ = 42.7° ϴ = 17.2° (c)(b)(a) Fig. S2: Fitting magnetic contrast with different values of µ . The raw data (blue) is fit by the model (black) presented in the main manuscript.In (a) the ideal parameters of ∆ µ = 0 . − and µ av = 0 .
011 nm − as noted in the main manuscript were used to giving an angle, θ = 17 . ◦ . In (b) ∆ µ and µ av were increased by their one sigma uncertainty and the resulting angle is . ◦ . In (c) ∆ µ and µ av weredecreased by their one sigma uncertainty, giving θ = 12 . ◦ . (c) (d)
200 nm (a) (b)
Fig. S3: Simulated time-resolved magnetic contrast of a compressed (a,c) and expanded (b,d) BPW in a
100 nm diameter NW. The frames (a,b)have been filtered with a Gaussian in order to simulate a finite width spot size. The unfiltered frames are shown in (c,d). of fit. In Fig. S2a, the raw data (blue) is fit by the model presented in the main manuscript (black) using the parameters of ∆ µ = 0 . − and µ av = 0 .
011 nm − extracted from the XAS analysis, giving an angle, θ = 17 . ◦ . Further, the r forthis fit is . . In (b) the fit is repeated on the same raw data using instead ∆ µ = 0 .
001 nm − and µ av = 0 .
006 nm − , which arethe values extracted by the XAS analysis, but increased by one standard deviation. The resulting angle is . ◦ and r = 0 . .Finally in (c) the fit is repeated with ∆ µ = 0 .
004 nm − and µ av = 0 .
016 nm − , i.e. the values increased by one standarddeviation, giving θ = 12 . ◦ and r = 0 . . We see a significant difference in the goodness of fit between the upper (Fig. S2b)and lower error bound (Fig. S2c). It should therefore be noted that while the uncertainty on the values of µ is appropriate, theupper bound for θ is normally an overestimation of the error. This is true for all of the results presented in Fig. 2d in the mainmanuscript. Unfiltered simulated images of a breathing BPW
The simulated magnetic contrast shown in Fig. 3i,j in the main manuscript are produced with a zero width spot size. In orderto reproduce the magnetic contrast observed in the experimental images in Fig. 3a,c of the main manuscript, a Gaussian blur wasapplied to simulate a finite width spot size in the simulation. The filtered (a,b) and unfiltered (c,d) simulated magnetic contrastimages are shown in Fig. S3.
Breathing of a BPW before and after intrinsic circulation switching
As mentioned in the main manuscript, the BPW breathing was observed in multiple time-resolved image series. In particular,the breathing of a wall was imaged before and after the wall unexpectedly reversed the sign of its intrinsic circulation in betweentwo image series. As a result the contrast observed for the compression (and expansion) occurs during the opposite polarity pulsein these two image series. Frame averages of these two different time-resolved image series are shown in Fig. S4. The average ofall frames acquired during the application of the duration (+) current pulse is given in (a,c) while the − ) current pulse
200 nm (a)(b) (c)(d)
Fig. S4: Frame averages of two different time-resolved image series. The average of all frames acquired during the application of the duration (+) current pulse is given in (a,c) while the average of all frames acquired during the application of the duration ( − ) current pulseis given in (c,d). Panels (a,b) are reproduced from Fig. 4a,c of the main manuscript. Panels (c,d) are from a time-resolved image series acquiredafter an unexpected switching of circulation of the same BPW. The BPW center is indicated by an orange circle. is given in (c,d). (a,b) are reproduced from Fig. 4a,c of the main manuscript. A compression of the BPW occurs in (a) while, anexpansion occurs in (b). The wall then unexpectedly reversed its circulation before the next image acquisition and as a result, thecontrast expected for a BPW compression is observed in (d), during the pulse which earlier caused a wall expansion. Similarly,the expansion is observed in (c). Importantly, the Œrsted field induced domain tilting remains the same, thus the swapped contrastcannot be a result of a reversed order of pulse polarity. This is yet another confirmation that this breathing effect is related to theintrinsic azimuthal circulation of the BPW. Magnetic contrast of a BPW motion event
We note in the main manuscript that the strong white or black magnetic contrast observed in Fig. 4b,d (see also Fig. S5a,b)is related to BPW motion. As the wall moves, the magnetization along the wire axis is entirely reversed compared to the state I S and as such, a strong contrast is seen depending on the direction of motion. Particular, however, are the two dots of black(white) contrast spread diagonally around the strong white (black) contrast in Fig. S5a,b. These are related to the intrinsicazimuthal circulation of the BPW and are expected where the wall motion begins and ends ( i.e. the left and right extremity ofthe strong contrast). As the wall moves, magnetization either tilts from the azimuthal to the longitudinal direction if the wallmoves farther away, or conversely tilts from the longitudinal to the azimuthal direction if the wall moves closer. This tilting effectis phenomenologically similar to that due to an applied Œrsted field, and as such a bipolar contrast should be expected acrossthe wire. The bipolar contrast should be reversed to the left and right extremity of the motion event, because the tilt direction isopposite. We once more use our model to explain the contrast. We first assume that the schematic in Fig. S5c explains our wallmotion event, where a 1D BPW initially located to the right is shifted to the left to become BPW*. Assuming the initial positionprovides the state I S , then the dynamic magnetic contrast profile expected at the four indicated locations (
200 nm (a)(b) x - I / I
12 34BPW* x BPW (c) (d)
Fig. S5: Analysis of the magnetic contrast of BPW motion in a
101 nm diameter NW. The average of all frames acquired during the interpulseperiods after (a) the duration (+) current pulse and (b) the duration ( − ) current pulse, both with amplitude . × A / m2