Abstract
In the MnSi bulk chiral magnet, magnetic skyrmion strings of 17 nm in diameter appear in the form of a lattice, penetrating the sample thickness, 10–1000 μm. Although such a bundle of skyrmion strings may exhibit complex softmatterlike dynamics when starting to move under the influence of a random pinning potential, the details remain highly elusive. Here, we show that a metastable skyrmionstring lattice is subject to topological unwinding under the application of pulsed currents of 3–5 × 10^{6} A m^{–2} rather than being transported, as evidenced by measurements of the topological Hall effect. The critical current density above which the topological unwinding occurs is larger for a shorter pulse width, reminiscent of the viscoelastic characteristics accompanying the pinningcreep transition observed in domainwall motion. Numerical simulations reveal that currentinduced depinning of already segmented skyrmion strings initiates the topological unwinding. Thus, the skyrmionstring length is an element to consider when studying currentinduced motion.
Introduction
An ordered solid exhibits elastic (reversible) or plastic (irreversible) deformations, depending on the strength of the force applied. Conceptually similar phenomena have also been observed in electronic systems, such as ferroelectric/ferromagnetic domain walls^{1, 2}, fluxline lattices in typeII superconductors^{3}, and charge/spin density waves^{4, 5}. Domain walls, for example, are more or less meandering by nature in real materials because of random pinningpotential, and they further deform under an effective force, such as a magnetic/electric field or electric current. When the effective force is relatively weak, the induced deformations are small and return to their original positions if the force is removed; that is, domain walls remain trapped in a potential valley (a reversible/pinning regime). In contrast, as the effective force increases, this reversible/pinning regime eventually collapses, and some domainwall segments begin to exhibit creep, a thermally assisted sluggish motion overcoming potential barriers. Relatively large deformations are thus induced and are no longer reversible (an irreversible/creep regime). This universal behavior is analogous to that in elastic and plastic regimes in ordered solids; therefore, the electronic media described are sometimes viewed as elastic objects.
Magnetic skyrmions^{6,7,8,9,10,11,12}, spinswirling topological objects of 10–100 nm in diameter, constitute a new member of elastic objects in electronic systems. Whereas the skyrmion is a pancakelike entity in an ultrathin film, it forms a cylindrical structure in a bulk sample^{13} (Fig. 1a), similarly to a fluxline in superconductors. In particular, in bulk chiral magnets, such as MnSi and (Fe_{1–x}Co_{x})Si, magnetic skyrmion strings are observed as a thermodynamically stable skyrmionstringlattice (SkSL) phase^{8, 9} in which skyrmion strings pointing along an external magnetic field are arranged in the form of a closepacked triangular lattice. Notably, it has been found for MnSi that the thermodynamically stable SkSL begins to move at extremely low electric current density^{14, 15}, on the order of 10^{6} A m^{–2}, which is 5–6 orders of magnitude smaller than the value required for currentinduced ferromagnetic domainwall motion^{16, 17}. Magnetic skyrmions are therefore attracting considerable research attention as a potential candidate for nextgeneration information carriers^{18,19,20}.
In currentdrive racetracktype applications^{21} based on isolated skyrmions^{22, 23}, one must manipulate a metastable isolated skyrmion because thermodynamically stable skyrmions have a propensity to condense and entirely fill a sample/device^{10, 24}; thus, the behavior of metastable isolated pancakelike skyrmions under current is of practical interest. In this context, ferromagnetic ultrathinfilm stacks, such as polycrystalline Pt/Co/Ta and singlecrystalline Pt/CoFeB/MgO, have recently been investigated^{23}, and it has been observed that, whereas almost all skyrmions move uniformly in the singlecrystalline Pt/CoFeB/MgO, pinned and moving skyrmions coexist in polycrystalline Pt/Co/Ta. Notably, in the latter case, a moving skyrmion sometimes collides with a pinned one and irreversibly merges into one pinned skyrmion^{23}. These observations suggest that under the influence of random pinning potential, currentinduced skyrmion dynamics may be more complex than the uniform flow of a fixed number of skyrmions, particularly near the critical current density at which skyrmions begin to move.
The situation may be even more complex in metastable skyrmion strings in a bulk sample. Notably, the diameter of a skyrmion is 10–100 nm, whereas the skyrmionstring length is on the order of the sample thickness (typically, 10–1000 μm). Ordinary bulk crystals at finite temperatures contain a finite density of defects, such as vacancies, as the most thermodynamically stable state, and a defectfree crystal has a higher free energy^{25}. Analogously, such a long skyrmion string is unlikely to be intact at finite temperatures but should contain finite topological defects, that is, pairs of the emergent magnetic monopole and antimonopole^{26,27,28}, as schematically shown in Fig. 1a, b. Although the stability and/or dynamics of the emergent magnetic (anti)monopoles are likely directly linked with the controllability of the metastable skyrmion strings, the details remain unclear because of the lack of experiments targeting metastable skyrmion strings in a bulk sample.
In this article, we describe the behavior of the metastable SkSL in MnSi under pulsed electric current by means of the Hall resistivity, ρ _{yx}, which is a sensitive probe for condensed topological skyrmions^{29, 30}, particularly in MnSi^{31,32,33,34}. We first write a metastable SkSL in a limited area of the barshaped sample by using a local thermal quenching technique^{34,35,36} while keeping the other area in the conical state, which is the thermodynamically most stable at the magnetic field investigated; next, ρ _{yx} is measured at two distant positions before and after inplane current pulse applications to determine how the pulse affects the written metastable SkSL. We find that, whereas the written metastable SkSL is maintained (the reversible/pinning regime) at low current densities (1–2 × 10^{6} A m^{–2}), it undergoes irreversible topological unwinding at higher current densities (>3 × 10^{6} A m^{–2}) rather than being transported along the current direction. Moreover, the critical current density above which the topological unwinding appears is larger for a shorter pulse width, reminiscent of viscoelastic characteristics accompanying a pinningcreep transition in domainwall motion. The smallness of the involved current density and the numerical simulation results suggest that the topological unwinding is initiated by the currentinduced depinning of originally segmented skyrmion strings.
Results
Sample configuration
We targeted the archetypal skyrmionhosting material MnSi and planned to write a metastable SkSL domain in the matrix of a thermodynamically stable conical order to detect the translation of the SkSL, if it occurred, by probing the Hall voltage at a distant position. In planning how to prepare the metastable SkSL in the area of interest, we noted that the SkSL can be created as a metastable state by employing rapid cooling (>100 K s^{−1}) by way of the thermoequilibrium SkSL phase (see Fig. 2a, b)^{34, 37}. If such thermal quenching is locally implemented, then the metastable SkSL domain is also expected to be written locally.
Figure 2c, d displays the sample configuration devised for this purpose. In addition to the standard six contacts c_{1}–c_{6}, two large electrodes e_{1} and e_{2} are attached on the sample surface. The contact resistance of e_{1} and e_{2} is relatively high, ≈10–15 Ω (for comparison, it is less than 0.1 Ω for the current electrodes c_{1} and c_{2}), to facilitate local Joule heating; the heating is followed by rapid cooling after the pulse cessation. When a single electric pulse of an appropriate magnitude and width is applied to the e_{1}–c_{2} pair (or the e_{2}–c_{2} pair), for example, at 15 K and 0.249 T, the SkSLtoconical transition line is quickly crossed during such rapid cooling (Fig. 2a), thus leading to the writing of a metastable SkSL domain around the electrode e_{1} (or e_{2}) only. For convenience, we label the area around the electrode e_{1} and e_{2} as areas 1 and 2, respectively, and refer to the electricheating pulse as a writing pulse.
Local writing of the metastable SkSL
The conjectured operation was substantiated by simultaneously probing the Hall voltages near the electrodes e_{1} and e_{2}. To this end, we measured ρ _{yx}–H profiles at the c_{3}–c_{4} and c_{5}–c_{6} pairs at 15 K; these are shown in Fig. 2e, f, respectively. When no writing pulse was applied (the black curves), the c_{3}–c_{4} and c_{5}–c_{6} pairs exhibited nearly the same profiles, thus reflecting the homogeneous conical order as the initial state. In contrast, after a single writing pulse of 110 mA and 15 ms was applied to the electrode e_{1} at 0.249 T (for the details of the pulsemagnitude dependence, see Supplementary Fig. 1), an enhanced ρ _{yx} was observed for the c_{3}–c_{4} pair (Fig. 2e), whereas such an enhancement was not discerned for the c_{5}–c_{6} pair (Fig. 2f); furthermore, when the magnetic field was subsequently increased (the red curves) or decreased (the green curves) from 0.249 T, ρ _{yx} at the c_{3}–c_{4} pair remained enhanced and then decreased to the values corresponding to the case in which no writing pulse was applied. This additional Hall signal, Δρ _{yx} (Δρ _{yx}≈24–25 nΩ cm), known as the topological Hall effect^{29, 30}, is a hallmark of condensed topological skyrmions^{31,32,33,34}, thus demonstrating successful writing of the metastable SkSL domain in the area 1. We also confirmed that when the writing pulse was applied to the electrode e_{2}, an enhanced Δρ _{yx} of a similar magnitude appeared for the c_{5}–c_{6} pair, whereas Δρ _{yx} ≤ 2 nΩ cm for the c_{3}–c_{4} pair (see Supplementary Fig. 2).
Current application to the metastable SkSL
Having established a method to create metastable SkSL in an areaselective manner, we were able to address the type of dynamics that emerges from the metastable SkSL under current applications. After writing a metastable SkSL in the area 1, we applied positive inplane current pulses (that is, current flowing from c_{1} to c_{2}) of various magnitudes with a fixed pulse width, 25 ms, and then measured ρ _{yx} at the c_{3}–c_{4} and c_{5}–c_{6} pairs simultaneously to observe the effects of the pulse application.
Figure 3a, b displays the topological Hall signal, Δρ _{yx}, at the c_{3}–c_{4} and c_{5}–c_{6} pairs vs. inplane pulse numbers applied, respectively, for several current magnitudes at 15 K. Here, three aspects can be highlighted. First, at the lowest current density, +1.0 × 10^{6} A m^{–2}, the topological Hall signal at the c_{3}–c_{4} pair remains constant even after 3000 pulses are applied (Fig. 3a). This finding suggests that the weak current application introduces no appreciable changes to the written metastable SkSL; that is, at this current density, the metastable SkSL remains in the reversible/pinning regime. Nevertheless, this regime clearly collapses at higher current densities, and the topological Hall signal (equivalently, the density of skyrmion strings) at the c_{3}–c_{4} pair decreases by the repetitive pulse applications. Second, in this irreversible regime, the decrease in the topological Hall signal occurs primarily at the early stage of the repetitive pulse applications; the decrease then becomes more moderate as the pulse number increased (Fig. 3a). Third, no appreciable signal is transferred to the Hall signal at the c_{5}–c_{6} pair (Fig. 3b), even after Δρ _{yx} at the c_{3}–c_{4} pair largely disappears. The second and third aspects are not compatible with the simplest scenario that the metastable SkSL domain may move as a whole along the current direction. In such a case, the Δρ _{yx} signal at the c_{3}–c_{4} pair would remain constant as long as the currentinduced shift of the metastable SkSL were small, and the signal would eventually be transferred to the Hall signal at the c_{5}–c_{6} pair; these are not the case in the experiments. We also performed the experiments with negative inplane current pulses (current flowing from c_{2} to c_{1}), but essentially the same results were obtained (Supplementary Fig. 3).
To gain more insight into the decrease in the topological Hall signal, we performed similar experiments but with the alternating application of positive and negative inplane pulses. In this pulse sequence, the sum of the total effective force applied (including its sign) is zero by definition; thus, the net shift of the metastable SkSL is not expected, even if each current pulse were to be accompanied by its finite shift. For comparison, the results are plotted in Fig. 3a, labeled with the current magnitude, +/–8.2 and +/–3.4 (×10^{6} A m^{–2}). Remarkably, these pulse sequences exhibited a similar (or even larger) decrease in the topological Hall signal. Therefore, the observed Δρ _{yx} profiles at the c_{3}–c_{4} pair cannot be a consequence of the translation of the metastable SkSL. Alternatively, its destruction can account for this observation: the metastable SkSL undergoes currentinduced topological unwinding into a nontopological spin texture, which is probably the thermodynamically most stable conical state.
Regarding the origin of the topological unwinding, we can rule out a Jouleheatingassisted mechanism. We estimated the increase of the local temperature in the area 1 by measuring the voltage at the c_{3}–c_{4} pair during the pulse application and found a temperature increase of only 0.2 K (specifically, 15.0 → 15.2 K) under 8.2 × 10^{6} A m^{–2} and 125 ms (the maximum and longest current density applied in this study: for details, see Supplementary Fig. 4). Moreover, the estimated lifetime of the metastable SkSL is beyond 10^{15} years at 15 K^{34}; thus, the slight temperature increase is expected to play a negligible role in the observed topological unwinding. The appreciable difference of the data between +/–8.2 and +8.2 × 10^{6} A m^{–2} (Fig. 3a) also cannot be ascribed to Joule heating, which should be symmetric with respect to the current polarity, at least, for Ohmic contacts (this is the case in the experiments; see Supplementary Fig. 4).
When considering the destruction of skyrmion strings from the perspective of topology, pair creation of the emergent magnetic monopoleantimonopole (Fig. 1a, b) and their subsequent unbinding motion are expected to be involved^{26,27,28}, analogous to the nucleation and subsequent growth in ordinary firstorder phase transitions. To gain insight into which process is more relevant to the currentinduced topological unwinding, it is helpful to consider the magnitude of the involved current density. Both a theoretical paper^{28} and our orderofmagnitude estimates (Supplementary Note 1) predict that the monopoleantimonopole pair creation requires a current density of ~10^{12} A m^{–2}, orders of magnitude larger than the value used in the present experiments, that is, 10^{6}–10^{7} A m^{–2}. Therefore, we conclude that the monopoleantimonopole pairs preexist in the quenched SkSL and hence that their currentinduced unbinding motion plays a key role in the topological unwinding. Meanwhile, 10^{6} A m^{–2} is the order on which the skyrmion strings start to move^{15}, thus suggesting that the topological unwinding is also related with the creep of the skyrmion strings. In fact, the currentinduced topological unwinding at 10 K was found to be less pronounced than that observed at 15 K (Supplementary Fig. 5), in agreement with the notion of creep, which is innately thermally assisted^{38,39,40,41,42,43,44}.
Numerical simulations
To determine how the creep of the skyrmion string and the unbinding motion of the preexisting monopoleantimonopole pairs are correlated with each other in the context of the currentinduced topological unwinding, we performed micromagnetic simulations of the chiral magnetic system including randomly distributed pinning sites (for details, see the Methods section). We found that under certain numerical conditions, a segmented skyrmion string in a ferromagnetic background can be metastable at zero or weak currents (Fig. 4a). Nevertheless, when the applied current exceeds a specific threshold value, the segmented skyrmion string starts to move; remarkably, this process is also accompanied by the motion of (anti)monopole such that the skyrmion string shortens, as shown in Fig. 4b–d (see also Supplementary Movie 1). As a result, the metastable segmented skyrmion string quickly disappears without a longdistance creep (Fig. 4d). This numerical result establishes a clear example of topological unwinding being initiated by the onset of the creep motion of metastable segmented skyrmion string, corroborating the experimental observations.
Kinetic nature of the regime transition
We then focused on the kinetic nature of the transition between the reversible/pinning and topological unwinding regimes. In the case of domain walls, an irreversible creep motion occurs more frequently under a larger d.c. force^{38,39,40,41,42,43,44}. When a pulsed or a.c. external force is applied, however, the pulse width or frequency dictates the onset of creep motion as well: at a given effectiveforce magnitude, domainwall segments remain pinned and behave like an elastic solid on short timescales (or at high frequencies), whereas they begin to creep like a viscous fluid on longer time scales (or at low frequencies)^{1, 45,46,47,48}. Such a dual nature depending on timescale or frequency may be described as a viscoelastic characteristic and can be qualitatively accounted for by considering that deformations exceeding the reversible/pinning regime cannot be induced instantaneously. Although this feature appears to be involved in a wide class of pinned electronic elastic objects, such a viscoelastic nature has not been explored in the skyrmion strings to date.
To address this issue, we investigated the decrease in the topological Hall signal for various pulse widths, t _{p}, with a sufficiently large pulse magnitude, +8.2 × 10^{6} A m^{–2} (Fig. 5a). As expected, relative to the case of long pulse widths, such as t _{p} ≥ 25 ms, much less topological unwinding was observed for the shortest pulse width, 5 ms (for the apparent convergence of the Δρ _{yx} profiles for t _{p} ≥ 25 ms, see Supplementary Note 2). However, it should be emphasized that the low degree of topological unwinding for t _{p} = 5 ms occurred not because the integrated pulse width (that is, the pulse number multiplied by each t _{p}) was the shortest: as illustrated in Fig. 5b, even if the Δρ _{yx} profiles are compared at a given integrated pulse width, the data for t _{p} = 5 ms still exhibits the least topological unwinding. This finding suggests that the metastable SkSL tends to remain in the reversible/pinning regime for a shortpulse application, whereas it is more likely to enter the topological unwinding regime for a longerpulse application. This timescaledependent dynamic transition is analogous to the viscoelastic characteristics accompanying the pinningcreep transition of domainwall motion^{1, 45,46,47,48}, again highlighting the key role of the pinningcreep transition of the segmented skyrmion strings at the onset of topological unwinding.
Pulse width vs. critical current density
The timescaledependent regimetransition of the metastable SkSL suggests that the critical current density above which the topological unwinding manifests itself is a function of the pulse width. In estimating the critical current density for various pulse widths, we introduce the quantity F (:0 ≤ F ≤ 1):
where Δρ _{yx}(n) is a value measured after the positive inplane current pulse is applied n times and Δρ _{yx}(0) denotes a value after applying the writing pulse; thus, F(n) represents the unwound fraction of the metastable SkSL after npulse applications, and F(n) = 0 and 1 correspond to the aswritten SkSL and fully unwounded nontopological state, respectively.
Figure 6a summarizes the results at 15 K after 200 pulse applications, F(n = 200), as a function of the current density, j, for various pulse widths, t _{p}, showing the clear dependence of F on j and t _{p}. In all cases, the F(n = 200)–j profiles (and also the F(n = 100)–j profiles: see Supplementary Fig. 6) are well described by a power law with a fixed exponent, F(n = 200)~j ^{α} with α = 3.4 (for F(n = 100), α = 3.5: see also Supplementary Fig. 6). This powerlaw behavior suggests that, in a strict sense, the critical current density is infinitesimally small; however, this consequence is consistent with the current understanding of creep motion, in which an infinitesimal force allows for creep motion with a finite probability as long as the temperature is finite^{38,39,40,41,42,43,44}.
Nevertheless, we can conditionally derive a critical current density of a practical sense, above which a finite F(n) becomes experimentally appreciable: in view of the present experimental accuracy, we set this criterion as F(n) = 0.03 and label the (practical) critical current density with j _{c,n }. Figure 6b displays j _{c,n = 100} and j _{c,n = 200} at 15 K as a function of pulse width; although j _{c,n = 200} is reasonably lower than j _{c,n = 100} at a given pulse width, a clear correlation between pulse width and critical current density can be seen for the both profiles, thus highlighting the viscoelastic characteristics at the onset of topological unwinding. Although under the present definition, the value of j _{c} depends on the pulse number and pulse width, it can be safely concluded that j _{c} in the limit of long pulse (t _{p} → ∞) is on the order of 10^{6} A m^{–2}, in good agreement with the value reported for the thermodynamically stable SkSL under d.c. currents^{14, 15}.
Powerlaw analysis of the pulsewidth dependence
The j _{c}–t _{p} profiles at 15 K shown in Fig. 6b are both described by a power law with nearly the same exponent: j _{c} − j _{c}(t _{p} → ∞)~t _{p} ^{–1/δ} with δ = 0.43 ± 0.05. The F(n)–j profiles at a lower temperature, 10 K, were also analyzed using power laws, but at relatively large n (n = 200 and 1000) because of the less pronounced topological unwinding (Supplementary Fig. 6); in this manner, we obtained α = 2.4–2.5 and δ = 0.50 ± 0.06 at 10 K (Supplementary Fig. 7), to be compared with α = 3.4–3.5 and δ = 0.43 ± 0.05 at 15 K. Thus, unlike the exponent α, the exponent δ appears to vary only weakly in the temperature range of 10–15 K. Although there are currently no numerical results to be compared, the value of δ would be a potential basis to test the microscopic model for the currentinduced dynamics of the metastable SkSL. At a minimum, the microscopic models for the depinning kinetics have been discussed in charge density waves by comparing the experimentally and theoretically derived δ values^{49,50,51,52}.
Discussion
We expect that the innately segmented skyrmion strings are likely to be relevant also to the currentinduced dynamics of the thermodynamically stable SkSL. Nevertheless, the consequence of the creep motion is expected to be crucially different between the metastable and thermodynamically stable cases: in a metastable SkSL, the monopoles and antimonopoles move to eliminate the skyrmion string unless they are pinned, eventually lowering the system’s total free energy. In a thermodynamically stable SkSL, by contrast, the topological unwinding is not triggered because the breakdown of the thermodynamically stable state inevitably increases the free energy. The currentinduced dynamics of the thermodynamically stable SkSL are therefore expected to involve the steady flow of the monopoles and antimonopoles, which may be accompanied by fluctuations in the emergent electric field^{27, 28}.
We have also shown that the existing emergent magnetic (anti)monopoles are harmful when one tries to drive metastable skyrmions with an electric current; therefore, in the context of the skyrmion application, the device thickness must be chosen so that skyrmion strings (or cylinders) do not contain topological defects. In the thermoequilibrium SkSL phase, the order of the expected segment length is given as a × exp(Δ_{MPAMP}/k _{B} T), where a is the lattice constant of the considered material and Δ_{MPAMP} is the paircreation energy of the monopole and antimonopole. According to the numerical results^{27}, Δ_{MPAMP} at the lowest temperature is ≈6 J for a simple cubic lattice, where J is the magnetic exchange energy and approximately equals to the magnetic transition temperature^{53}; hence, for MnSi, Δ_{MPAMP}/k _{B} ≈200 K at low temperatures. Thus, in the thermoequilibrium SkSL phase (≈27 K), the expected segment length is ~700 nm (or shorter, given that Δ_{MPAMP} may decrease at high temperatures). Because our sample thickness is ≈100 μm, one can expect that each quenched skyrmion string contains monopoleantimonopole pairs on the order of 100, provided that the topological defect density is frozen at the value of the thermoequilibrium phase. The direct observation of a segmented skyrmion string remains a challenging issue, which may nevertheless provide useful information when designing a topologicaldefectfree skyrmion device.
Methods
Sample preparation and setup
A single crystal of MnSi was grown via the Czochralski method. The sample was cut and polished to a size of 2.5 × 1.1 × 0.1 mm^{3}, with the largest surface normal to the <100> axis. The resistivity ratio ρ(300 K)/ρ(4.2 K) was ≈52. Gold current leads of 0.3 mm ϕ were attached to the sample and fixed with indium. The electrodes, e_{1} and e_{2,} were constructed using carbon paste to achieve a high contact resistance, ≈10–15 Ω. The sample was mounted on a sapphire substrate in contact with a heat bath and fixed with varnish.
Transport measurements
The Hall resistivity, ρ _{yx}, was measured at 33 Hz with a low a.c. current excitation (≈7.97 × 10^{4} A m^{–2}) under a magnetic field parallel to the <100> axis using lockin amplifiers (Stanford Research Systems, SR830) equipped with a transformer preamplifier (Stanford Research Systems, SR554). The Hall resistivity values presented here are antisymmetrized between the positive and negative magnetic fields. The pulse currents that were used to write the metastable SkSL or to exert an effective force on the written metastable SkSL were generated by a function generator (NF Corporation, WF1947) connected to a bipolar amplifier (NF Corporation, 4502A). Before applying the writing pulse to the electrode e_{1} at +0.249 T (or –0.249 T), we applied a field of 1T (or –1T) to enter the ferromagnetic state and erase any residue of the metastable SkSL in the previous measurements. When measuring Δρ _{yx} as a function of the inplane pulse number, we waited 1 min after the last pulse was applied to ensure that the sample temperature was sufficiently equilibrated with the sampleholder temperature. When multiple pulses were applied between measurement points, we set the interval between successive two pulses to be 6 s.
Micromagnetic simulation
The simulation was performed for a simple cubic lattice consisting of 60 × 30 × 100 magnetic moments with an open boundary condition in the z direction and periodic boundary conditions in the x and y directions. We considered the following model Hamiltonian:
where J is the exchange interaction, D is the DzyaloshinskiiMoriya interaction energy, h _{z} is the magnetic field along the z direction, e _{x} (or e _{y}, e _{z}) is the unit vector that connects with the nearest neighbor site along the x (or y, z) direction, n _{ r } is the unit vector of the local magnetic moment at site r, and n _{{z,r}} is the z component of n _{ r }. The last term represents the impurity in the model: the easy axis anisotropy K _{imp} was introduced at randomly selected sites, and Λ is the set of random numbers. In simulating the currentinduced dynamics of the skyrmion string at zero temperature, we inserted the Hamiltonian into the following Landau–Lifshitz–Gilbert equation:
where j represents the (spinpolarized) electric current density. The units of nondimensional time t and current j = j are 1/(γJ) and 2eγJ/(pa ^{2}) (p: polarization of the magnet), respectively. We chose the following parameter set {J = 1.0, D = 0.2, K _{imp} = 0.2, h = 0.018, α = β = 0.04, j = 0.04}. The density of the random impurity was set as 10%.
Data availability
The data that support the findings of this study are available from the corresponding author on request.
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Acknowledgements
We thank S. Hoshino, K. Shibata, G. Tatara, N. Ogawa and M. Ikeda for fruitful discussions. This work was partially supported by CREST, JST (grant no. JPMJCR16F1) and KAKENHI (grant nos 25220709, 26103006, 15H03553, 15K05192 and 15H05459). NN is supported by ImPACT Programof Council for Science, Technology and Innovation (Cabinet Office, Government of Japan).
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F.K. performed all the experiments and analyzed the data. A.K. grew the single crystals used for the study. Y.O. cut and polished the sample. F.K. and H.O. planned the project. F.K. wrote the manuscript. W.K. and N.N. performed numerical simulations. All authors discussed the results and commented on the manuscript.
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Kagawa, F., Oike, H., Koshibae, W. et al. Currentinduced viscoelastic topological unwinding of metastable skyrmion strings. Nat Commun 8, 1332 (2017). https://doi.org/10.1038/s41467017013532
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DOI: https://doi.org/10.1038/s41467017013532
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