Fast domain walls governed by Œrsted fields in cylindrical magnetic nanowires
M. Schöbitz, A. De Riz, S. Martin, S. Bochmann, C. Thirion, J. Vogel, M. Foerster, L. Aballe, T. O. Menteş, A. Locatelli, F. Genuzio, S. Le Denmat, L. Cagnon, J. C. Toussaint, D. Gusakova, J. Bachmann, O. Fruchart
FFast domain wall motion governed by topology and Œrsted fields in cylindrical magnetic nanowires
M. Schöbitz,
1, 2, 3, ∗ A. De Riz, S. Martin,
1, 3
S. Bochmann, C. Thirion, J. Vogel, M. Foerster, L. Aballe, T. O. Mente¸s, A. Locatelli, F. Genuzio, S. Le-Denmat, L. Cagnon, J. C. Toussaint, D. Gusakova, J. Bachmann,
2, 6 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 Alba Synchrotron Light Facility, CELLS, Barcelona, Spain Elettra-Sincrotrone Trieste S.C.p.A., Basovizza, Trieste, Italy Institute of Chemistry, Saint-Petersburg State Univ., St. Petersburg, Russia (Dated: January 13, 2020)While the usual approach to tailor the behavior of condensed matter and nanosized systems is the choiceof material or finite-size or interfacial effects, topology alone may be the key. In the context of the motionof magnetic domain-walls (DWs), known to suffer from dynamic instabilities with low mobilities, we reportunprecedented velocities >
600 m / s for DWs driven by spin-transfer torques in cylindrical nanowires made ofa standard ferromagnetic material. The reason is the robust stabilization of a DW type with a specific topologyby the Œrsted field associated with the current. This opens the route to the realization of predicted new physics,such as the strong coupling of DWs with spin waves above >
600 m / s . It is well known that specific properties in condensed-matterand nanosized systems can be obtained by either acting on theelectronic structure by selecting an appropriate material com-position and crystalline structure, or by making use of finite-size and interfacial effects, strain, gating with an electric field,etc[1]. These approaches have proven suitable for tailoringcharge transport, optical properties, electric or magnetic polar-ization, etc. However, there are limits regarding what can beachieved with materials, or realized with device fabrication.An alternative strategy entails considering a specific topologyin order to develop the desired properties of a system, yieldingdiverse applications such as the design of wide-band-gap pho-tonic crystals[2] and the control of flow of macromolecules[3],or novel theoretical methods such as for the description ofdefects[4], or intringuing 3D vector-field textures such as hop-fions and torons[5]. As regards magnetism, unusual propertiesresulting from topological features have been predicted, suchas the existence of a domain wall (DW) in the ground state of aMoebius ring[6], or the non-reciprocity of spin waves inducedby curvature and boundary conditions in nanotubes[7].Here, we show that topology plays a critical role in thephysics of DW motion in one-dimensional conduits, a pro-totypical case for magnetization dynamics. For the sake ofsimplicity of fabrication and monitoring, DW motion undermagnetic field or spin-polarized current is usually conductedin planar systems, made of stacked thin films patterned later-ally by lithography. In them, DWs are dynamically unstableabove a given threshold of field or current (Walker limit), un-dergoing transformations of their magnetization texture, asso-ciated with a drastic drop in their mobility. Ways are beinginvestigated to overcome this limitation through the engineer-ing of microscopic properties. Two major routes are the useof the Dzyaloshinskii-Moriya interaction in order to stabilizethe walls[8–10], or of natural or synthetic ferrimagnets withvanishing magnetization to decrease the angular momentum in order to switch and boost the precessional frequency[11–13].The three-dimensional nature of cylindricalnanowires (NWs) gives rise to the existence of a DWwith a specific topology, which respects the rotational in-variance and circular boundary conditions. It is named theBloch-point wall (BPW)[14] and has been experimentallyconfirmed only recently[15, 16]. It was predicted that thiswall can circumvent the Walker limit, but field-driven motionexperiments disappointingly failed to confirm a topologicalprotection[17]. Here, we report experimental results oncurrent-induced DW motion in such NWs. We show thatalthough previously disregarded, the Œrsted field inducedby the current plays instead a crucial and valuable role instabilizing BPWs, contrary to the field-driven case. Thisallows them to retain their specific topology and thus reachvelocities >
600 m / s in the absence of Walker breakdown,which is quantitatively consistent with predictions.DWs with two distinct topologies exist in NWs: thetransverse-vortex wall (TVW) and the BPW (Fig. 1a,b). Theformer has the same topology as all DW types known in 2Dflat strips[18]. The latter is found only in NWs and exhibits az-imuthal curling of magnetic moments around a Bloch point, alocal vanishing of magnetization[19, 20]. This unique topolog-ical feature of NWs is at the origin of the predicted fast speedand stability during magnetic-field or current-driven motion ofBPWs. This is easily explained by considering the time deriva-tive of the magnetization vector . m at any point, described bythe Landau-Lifshitz-Gilbert equation[21]: . m = − γ m × H + α m × . m − ( u · ∇ ) m + β m × [( u · ∇ ) m ] (1)with γ = µ | γ | , γ being the gyromagnetic ratio, α (cid:28) theGilbert damping parameter and β the non-adiabaticity param-eter. H , the total effective field, is comprised of applied fieldsand fields originating from magnetic anisotropy, exchange and a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n a b P ho t on s
16° ElectronsSubstrate XMCD-PEEM imageWire Shadow area c Fig. 1: Schematic of a a TDW and b a BPW. c Schematic of shadowXMCD-PEEM and the contrast resulting from a BPW. dipolar energy. The spin-polarized part of the charge cur-rent induces so-called spin-transfer torques, taken into accountthrough u , with | u | = P ( jµ B /eM s ) [21]. j and P are thecharge current and its spin-polarization ratio, respectively, µ B is the Bohr magneton, e the elementary charge and M s thespontaneous magnetization.In purely field driven cases, the applied field favors theprecession of m around the field direction. In flat strips,for applied fields above a few mT this causes repeated DWtransformations from transverse to vortex walls for in-planemagnetization, and from Néel to Bloch walls for out-of-planemagnetization. This so-called Walker breakdown[22] is fa-cilitated by the fact that all these DW configurations sharethe same topology[23–25]. The mobility is high below theWalker threshold field (scaling with /α ) and low above (scal-ing with α ). The same physics is expected in NWs for theTVW, with the Walker field equal to zero due to the rotationalsymmetry [7, 14]. The phenomenology of current-driven casesis similar: the adiabatic term favors motion, the non-adiabaticterm favors azimuthal precession [third and fourth terms inEq.(1), respectively], and the DW velocity is expected to be ≈ ( β/α ) u below the Walker threshold and ≈ u above it[21, 24],again with a vanishing threshold for TVWs in NWs[26].In contrast to these cases, one expects that magnetizationcannot freely precess azimuthally in a BPW, since it would pe-riodically imply a head-on or tail-on configuration along allthree axes, with an enormous cost in dipolar energy. Instead,the azimuthal rotation should come to a halt and remain in astate essentially similar to the static one (Fig. 1b). This im-plies an absence of Walker breakdown, both under field[7, 14]and current[27, 28], and steady-state motion of the wall. Thesteady circulation is expected to be clockwise (CW) with re-spect to the direction of motion of the DW, while the coun-terclockwise (CCW) circulation may undergo a dynamics-induced irreversible switching event to recover the CW cir-culation and steady state. This picture is valid both for BPWs in wires[14, 27], and vortex walls[7, 28] in thick-walled tubes.Thanks to this locked topology, the mobility of the BPW isexpected to remain high under both field and current. Onlywhen a speed around ≈ / s is attained, the speed is pre-dicted to reach a plateau, with new physics expected to occurvia interactions with spin waves, known as the spin-Cherenkoveffect[7]. However, so far there exists no experimental reportof the mobility of any of these walls under neither magneticfield nor current.Our work is based on magnetically-soft Co Ni wireswith diameter 90 nm, electroplated in anodized aluminatemplates[29]. Following the dissolution of the latter, iso-lated wires lying on a Si substrate are contacted with pads toallow for the injection of electric current. DWs were moni-tored with both magnetic force microscopy (MFM) and X-rayMagnetic Circular Dichroism Photo-Emission Electron Mi-croscopy (XMCD-PEEM) in the shadow mode (Fig. 1c) toreveal the three-dimensional texture of magnetization[16, 30,31]. While in MFM, sharp ns-long pulses could be sent, inXMCD-PEEM the shape of current pulses was distorted toa minimum width of − ns, due to long cabling, UHVfeedthroughs and the sample holder contacts. Micromag-netic simulations were carried out with the home-made finite-element code FeeLLGood[32], based on the Landau-Lifshitz-Gilbert equation including spin-transfer torques. See supple-mentary information for additional details on the methods[50].Domain wall velocities were experimentally investigatedprimarily with MFM imaging. Fig. 2b shows an atomic forcemicroscopy (AFM) image of the left hand side of the con-tacted NW from Fig. 2a. The corresponding magnetic forcemicroscopy (MFM) image in Fig. 2c shows the initial magneticconfiguration, with two DWs located at . and 7.2 µm fromthe edge of the left contact. By applying a current pulse ofduration . and amplitude . × A / m , the left handDW moved over a distance of ≈ µ m (Fig. 2d), correspondingto an average velocity of ≈
350 m / s . However, the right handDW remains pinned, highlighting a common and key issue forinferring DW velocities from motion distances: pinning on ge-ometrical or microstructural defects hampers DW motion[33].Depinning not only requires a current density above a criticalvalue j dp , but re-pinning can also occur at another locationwith a deeper energy well, while the current pulse is still be-ing applied. This results in DW propagation with an effectivetime span possibly much shorter than the nominal pulse du-ration. Consequently, the values for DW velocity convertedfrom motion distance and nominal pulse length are a lowerbound of an unknown higher velocity (see supplementary ma-terial for a quantitative discussion). Furthermore, with suchlarge current densities the effect of Joule heating may not beneglected. However, measurements of the NW resistance dur-ing the pulse showed that the samples never exceeded the Curietemperature (see supplementary material) and that the resultsdescribed herein are not caused by thermal activation.Fig. 2e (open circles) shows the discussed lower bound for a μ m bcd j
862 4 μ m j j (10 A/m ) 3.00 0.5 1.0 1.5 2.0 2.5 ≤ t < 10 ns10 ≤ t < 15 ns15 ≤ t < 25 ns25 ≤ t < 35 ns35 ≤ t ns e v D W ( m / s ) / = / = / = Fig. 2: a SEM, b AFM and corresponding c , d MFM images ofa
90 nm diameter Co Ni NW with Ti/Au electrical contacts. c Initial state, with two DWs d Same wire, after a current pulse with . × A / m magnitude and . duration. e Domain wall ve-locity as a function of applied current density, duration (see inner cap-tion), monitored with MFM (open circles) and XMCD PEEM (filledcircles) from 4 individual NWs. The dashed lines are expectationsfrom the one-dimensional model below the Walker breakdown, for v = ( β/α ) u with β/α = 1 , , . DW velocity, as a function of applied current density, inferredfrom a multitude of MFM images before and after pulseswith durations ranging from to
15 ns . Consistent with theexpected occurrence of re-pinning, lower velocities are in-ferred from longer pulse durations. Still, DW velocities upto >
600 m / s were observed for applied current densities ≈ . × A / m . This sets a five-fold record for purelyspin-transfer torque motion of DWs in a standard ferromag-netic material, i.e. , with large magnetization, with reportedvalues hardly exceeding m/s[34]. Similar or higher speedshave been measured recently, however in low-magnetizationferrimagnets, thereby enhancing the efficiency of spin-transfertorque[35]. Here, it is the topology of the wall that enhancesthe DW speed, not a special material. Similarly, these DWvelocity measurements are not enhanced by DW inertia, sincesimulations showed that this effect will only come into playin sub-nanosecond pulse experiments (see supplementary ma-terial). The black dotted lines in Fig. 2e act as a guide to theeye through the speed predicted by the one-dimensional modelbelow the Walker breakdown v = ( β/α ) u , for three differentratios of β/α : 1, 2 and 3 (for Co Ni M s = 0 . M A / m , P ≈ . , resulting in u ≈ . / s per A / m ). This isnot intended as precise modelling, but rather to show that the experiments are clearly not compatible with v = u , support-ing the absence of Walker breakdown for the BPW. Instead avalue of β/α (cid:39) is inferred. Note, however, that the adverseeffects of DW pinning reappear in the form of a threshold cur-rent density j dp ≈ . × A / m required to set any DWin motion. Even above this value, DW motion was not fullyreproducible, with some pinning sites associated with a larger j dp .To link unambiguously the measured velocity with theory,the DW type must be identified. For this purpose, we em-ployed shadow XMCD-PEEM and imaged NWs before andafter injecting a given current pulse (Fig. 3a,b, and full sym-bols in Fig. 2e). Note that the values for speed are lower thanthose measured with MFM, as expected for less sharp pulseshapes with consequentially larger width. Returning to theDW type, the first striking fact is the following: from hun-dreds of DWs imaged after current injection, all were of theBPW type. These unambiguously appear as a symmetric bipo-lar contrast in the shadow[16], corresponding to azimuthal ro-tation of magnetization as on Fig. 3a-b. This sharply con-trasts with all our previous observations of NWs, imaged inthe as-prepared state or following a pulse of magnetic field,for which both TVWs and BPWs had been found in sizeableamounts[16, 17]. The second striking fact is that the sign ofthe BPW circulation is deterministically linked to the sign ofthe latest current pulse, provided that its magnitude is abovea rather well-defined threshold which, as shown in Fig. 3c,lies around . × A / m . In contrast with a one-timeWalker event discussed previously, this holds true irrespectiveof whether or not the wall has moved under the stimulus ofthe current pulse, and is independent of the pulse duration atthe probed timescales. We hypothesize that these two factsare related to the Œrsted field associated with the longitudi-nal electric current, its azimuthal direction favoring the BPWwith a given circulation. Indeed, for a uniform current den-sity j , the Œrsted field is H = jr/ at distance r from theNW axis. For the present NWs with radius R = 45 nm and j = 1 × A / m this translates to
28 mT at the NW sur-face, which is a significant value.In order to support this claim, we conducted micromagneticsimulations including the Œrsted field, which had not beenconsidered in previous works. Starting from a DW at rest with R = 45 nm , we used α = 1 to avoid ringing effects and ob-tain a quasistatic picture, suitable to describe the PEEM ex-periments, for which the pulse rise time is several nanosec-onds. We evidenced that while the added effect of spin-transfer torques may alter the transformation mechanisms, itis of second-order compared to the Œrsted field and consider-ing or disregarding these torques does not quantitatively im-pact switching. Accordingly, below we present only resultsdisregarding these torques. Within the domains the peripheralmagnetization tends to curl around the axis, while it remainslongitudinal on the NW axis. We first consider TVWs as theinitial state and find that these transform into BPWs with CW c ab μ m-2 210-1 j (10 A/m )01 P s w i t c h ( j ) Wire 1Wire 210 ≤ t < 20 ns20 ≤ t < 30 ns30 ≤ t < 40 ns40 ≤ t ns Fig. 3: a , b Consecutive XMCD-PEEM images of a NW with a tiltedx-ray beam (orange arrow). The azimuthal circulation of the fourBPWs seen in the NW shadow is indicated by the white arrows, con-sistent with the Œrsted field of the previously applied current (blueand red arrows in the right hand schematic, respectively). From a to b , a
15 ns and . × A / m current pulse switches % ofBPWs. DW displacement from a to b cannot be discussed as directlyresulting from spin-transfer torque, and the density of current lies be-low the threshold for free motion c BPW switching probability as afunction of j for two different wire samples (squares and triangles).Pulse durations are categorized and color coded, see included labels.The grey region indicates the current density required for switchingin simulations. circulations with respect to the current direction, if the cur-rent density exceeds . × A / m . The underlying pro-cess is illustrated on Fig. 4a, displaying maps of the radial andazimuthal magnetization components, m r and m ϕ , respec-tively, on the unrolled surface of a NW as a function of time.These highlight the locations of the inward and outward fluxof magnetization through the surface, signature of a TVW[18].While these local configurations are initially diametrically op-posite, they approach each other until they eventually merge,expelling the transverse core of the wall from the NW. Thisis associated with the nucleation of a Bloch point at the NWsurface, which later on drifts towards the NW axis, ending upin a BPW. This process is similar to the dynamical transfor-mation of a TVW into a BPW upon motion under a longitu-dinal magnetic field[17], and explains the absence of TVWsin our measurements, for which the applied current densitieswere always larger than . × A / m . In order to under-stand the unique circulation observed, we now consider a BPWas the initial state. BPWs with a circulation matching that ofthe Œrsted field do not change qualitatively, only their widthincreases during the pulse. On the contrary, BPWs shrink iftheir initial circulation is CCW, i.e. opposite to the Œrsted mr m φ φφφφπ - ππ - ππ - ππ - π a φφφ μ m 1 π - ππ - ππ - π μ m 1z z b φπ - π Fig. 4: DW transformations by the Œrsted field in micromagneticsimulations for a TVW to BPW, with j = 0 . × A / m , and b BPW circulation reversal, with j = − . × A / m . Left andright are color maps of the radial and azimuthal magnetization com-ponents, m r and m ϕ , respectively, over time on the unrolled surfaceof a n m diameter, µ m -long NW with α = 1 . field. For j ≤ . × A / m the CCW BPW reaches anarrow yet stable state, and recovers its initial state after thepulse. Beyond this value the circulation switches through atransient radial orientation of magnetization (Fig. 4b). Afterthe switching of circulation, the BPW expands and reaches astable CW state. The value of the critical current density re-quired for circulation switching is in quantitative agreementwith the experimental one (Fig. 3c, ≈ . × A / m ), al-though the simulation does not incorporate thermal activationand considers α = 1 . This suggests that the switching processis robust and intrinsic, in agreement with the narrow experi-mental distribution of critical current. In our simulations thetime required for switching is < ns, though switching timesan order of magnitude faster are expected for realistic valuesof α < . , which explains why no dependence on the pulsewidth was observed in the experiments, where all pulse widthswere above ns.In experiments where the DW type was visible, DW mo-tion events were observed for applied current densities largerthan the critical current density required for the circulationswitching event. Thus, in these the circulation is always CCWwith respect to the propagation direction, i.e. CW with re-spect to the current direction, because the charge of electronsis negative. Remarkably, this sense of circulation is oppositeto the situation expected when neglecting the Œrsted field,which would select the CW circulation with respect to thepropagation direction, as dictated by the chirality of the LLGequation[14, 27, 28]. There must therefore be a competitionfor the circulation sense and for the case of n m diameterNWs, the Œrsted field dominates. Despite this, we find thatthe BPW motion still follows v ≈ ( β/α ) u whether or not theŒrsted field is considered. Notice that the β parameter is ex-pected to depend on the DW width, however for widths muchsmaller than the ones studied here[36]. The predictions of highmobility and possibly spin-Cherenkov effect are thus probablynot put into question.Surprisingly, the Œrsted field was previously only consid-ered in a single report for NWs of square cross-section[37].No qualitative impact was found, likely because a NW side ofat most nm was considered, and a simple analytical modeldescribing magnetization in the domain and balancing ZeemanŒrsted energy with exchange energy shows that the impact ofthe Œrsted field scales very rapidly as R . This is also ac-curately confirmed by simulations. The situation closest tothe present case is the report of flat strips made of spin-valveasymmetric stacks[38]. Such strips can be viewed as the un-rolled surface of a wire, the curling of the BPW translating intoa transverse wall, which tends to be stabilized during motiondue to the Œrsted field.To conclude, we have shown experimentally and by sim-ulation that the Œrsted field generated by the spin-polarizedcurrent flowing through a cylindrical NW has a crucial impacton DW dynamics, while it had been disregarded so far. ThisŒrsted field robustly stabilizes BPWs, in contrast with thefield-driven case[17]. This stabilization allows for the key fea-tures predicted for their specific topology to apply[14, 27, 28]:we evidenced DW velocities in excess of
600 m / s confirmingthe absence of Walker breakdown[7, 39] and setting a five-fold record for spin-transfer-torque-driven DW motion in largemagnetization ferromagnets[34]. This suggests that the exper-imental realization of further novel physics is at hand, such asthe predicted spin-Cherenkov effect with strong coupling ofDWs with spin waves. ACKNOWLEDGMENTS
MS acknowledges a grant from the Laboratoire d’excellenceLANEF in Grenoble (ANR-10-LABX-51-01). The project re-ceived financial support from the French National ResearchAgency (Grant No. JCJC MATEMAC-3D). This work was partly supported by the French RENATECH network, and bythe Nanofab platform (Institut Néel), whose team is greatlyacknowledged for technical support. We thank Jordi Prat forhis technical support at the ALBA Circe beamline and OlivierBoulle for useful discussions. ∗ Electronic address: [email protected][1] R. E. Newnham,
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