High-mobility inertial domain walls driven by spin-transfer torque in a ferrimagnetic spinel oxide

Domain: biology
Published: 2026-03-31
Editorial outcome: accept

Abstract
Efficient electrical manipulation of domain walls is key to developing magnetic devices with fast switching capabilities and low energy consumption. Here we demonstrate Bloch-type domain wall velocities exceeding 1 km s-1 in the single-layer ferrimagnetic spinel oxide NiCo O induced by spin-transfer torque at a current density of 2 × 1011 A m-2. This exceptional domain wall mobility is attributed to the combination of giant nonadiabatic spin-transfer torque, low magnetization, and high spin polarization. Additionally, we report a pronounced domain wall inertia effect in this ferrimagnet due to the large nonadiabaticity of the torque. The characteristic time for domain wall acceleration and deceleration is ~ 1 ns, shorter than that reported for typical ferromagnets. Our findings highlight the potential of spinel oxides as a promising platform for engineering high-performance domain wall devices that take advantage of ultrafast ferrimagnetic dynamics.

Introduction
Current-induced domain wall (DW) motion is widely utilized to switch or modulate the magnetic configuration of magnetoelectronic devices1–12. Additionally, DWs can be utilized to store information13–17, perform Boolean logic operations18–21, and emulate neuronal functions22–24 in narrow magnetic strips, so-called DW racetracks. In such systems, DW motion can be driven by spin-transfer torque (STT), arising from an electric current flowing directly through the magnetic layer25–27, by spin-orbit torque (SOT), typically generated by a current in an adjacent heavy metal layer28–32, and by magnon torque, originating from either coherent spin-wave33–35 or incoherent thermal-magnons36,37. SOT has long been widely regarded as the more effective and versatile mechanism for DW manipulation compared to STT38,39. For instance, an ultrafast velocity of 5.7 km-1 has been reported in Pt/CoGd under a current density of 4.2 × 1011 A m-2 and an in-plane magnetic field of 140 mT at angular momentum compensation temperature40. Nevertheless, STT offers several complementary features relative to SOT: it does not require an in-plane bias magnetic field for DW motion; it drives DW in a single-layer magnetic structure, eliminating the need for a heavy-metal spin source; it is effective for both Néel- and Bloch-type DWs; and the direction of DW motion can be easily controlled by the polarity of the applied current irrespective of DW chirality. Notably, recent studies suggest that STT can achieve comparable performances to SOT41–43, renewing interest in STT-based devices. This motivates the search for material properties and systems that lead to large STT-driven DW mobility (defined as velocity per current density) and low energy consumption, the two crucial parameters determining the performance of DW devices.

The STT-induced DW velocity at steady state is 𝜈 = \beta/\alpha u below the Walker breakdown, where \beta is the nonadiabatic torque parameter, \alpha the magnetic damping, and u = g\muB P / 2eMs j the spin-drift velocity with current density j, P the spin polarization, M the saturation magnetization, \mu the Bohr magneton, g the Landé factor, and e the electron charge25–27. A larger nonadiabatic torque simultaneously enhances the DW mobility and eases depinning, thereby reducing the operating current density44. Meanwhile, low magnetization with high spin polarization enables efficient magnetization rotation, thereby promoting DW motion. In light of these considerations, ferrimagnets stand out as attractive materials for DW devices owing to their characteristic low magnetization and significant nonadiabatic torques45,46.

To date, several ferrimagnetic materials have been reported with relatively high performance for STT-driven DW devices. For example, current-driven DW motion was studied in the ferrimagnetic Heusler alloys X Z (X = Mn, Z = Ge, Sn, Sb)47. A DW velocity exceeding 100 m s-1 was obtained with a current density in the order of 1011 A m-2. Moreover, a high DW velocity was observed in Mn N and Mn Ni N thin films with inverted perovskite structure42,43, where DWs can be accelerated up to 2 km s-1. However, reaching such a high velocity requires a current density of 1 × 1012 A m-2. Achieving high DW velocity at low current density thus remains a central objective in the development of energy-efficient STT devices. Spinel oxides, with the general chemical formula AB O , exhibit diverse spin and electronic properties, offering new opportunities for spintronic applications48,49. In particular, the ferrimagnetic compound NiCo O (NCO)50,51, possessing high spin polarization, high electrical conductivity, and low magnetization, holds great promise for high-velocity and low-power DW racetrack devices. Nevertheless, current-driven DW motion remains unexplored in this class of materials.

In this study, we demonstrate efficient current-driven DW motion in a perpendicularly magnetized NCO film combining high DW mobility and low energy dissipation. These features are maintained over a wide range of film thicknesses. To investigate the underlying DW characteristics, we employed scanning nitrogen-vacancy (NV) magnetometry to resolve the DW structure and chirality. Measurements of DW dynamics were carried out using magneto-optical Kerr effect (MOKE) microscopy. We report a DW velocity exceeding 1 km s-1 with a current density of only 2 × 1011 A m-2, corresponding to a DW mobility of 5.7 × 10-9 m3 A-1 s-1. This excellent performance is attributed to the unique intrinsic properties of NCO, including a giant nonadiabatic STT, low magnetization, and high spin polarization. An in-plane magnetic field applied parallel to the wall magnetization produces a moderate decrease of the DW velocity, independently of field or current polarity, consistent with an increase of the DW width and decrease of nonadiabaticity. In contrast, a field applied parallel to the current produces no measurable change of DW velocity, consistently with Bloch DW dynamics below the Walker limit. Furthermore, we observe a pronounced DW inertia effect, also ascribed to the large nonadiabaticity. This effect enables the extraction of key STT parameters, such as the characteristic time for DW acceleration and deceleration, as well as the adiabatic and nonadiabatic torque coefficients. Our study provides guidelines for developing efficient STT-based racetrack memory devices using ferrimagnetic spinel oxides.

Results
Epitaxial film growth and characterization
Epitaxial NCO films with thicknesses from 10 to 30 unit cells (uc) (1 uc = 0.8 nm), grown on (001)-oriented MgAl O (MAO) substrates, were used for the DW racetrack devices (Methods). The NCO exhibits an inverse spinel structure, as shown in Fig. 1a. In each unit cell, the magnetic moments are carried by 8 Ni ions on octahedral sites and 8 Co ions on tetrahedral sites, which are antiferromagnetically aligned. The resulting uncompensated magnetic moments give rise to ferrimagnetism. The Curie temperature of NCO is approximately 420 K, well above room temperature50. Moreover, NCO is a half-metal with a resistivity of ~ 810 μΩ∙cm. The minority-spin subbands from Ni are dominant at the Fermi surface, resulting in a strong negative spin polarization (P = -0.73)52. The X-ray diffraction (XRD) spectrum in Fig. 1b demonstrates the epitaxial growth of the NCO film with a (001) orientation. The high crystalline quality is further evidenced by the presence of Laue oscillations in the expanded diffraction pattern around the (004) peak (inset). The high-resolution scanning transmission electron microscopy (STEM) image in Fig. 1c reveals the well-ordered interface between NCO film and MAO substrate, confirming the high-quality epitaxial growth of the inverse spinel structure. The magnetic properties were characterized using superconducting quantum interference device (SQUID) magnetometry. The in-plane and out-of-plane magnetic hysteresis loops measured at 300 K are presented in Fig. 1d. The out-of-plane M-H hysteresis loop displays a sharp switching behavior, with a saturation magnetization of 1.1 µ per formula unit (150 kA/m), consistent with a previous report50. In contrast, the in-plane M–H loop does not reach saturation even at magnetic fields exceeding 1 T, indicating strong perpendicular magnetic anisotropy (PMA) of the NCO film. To further characterize the PMA, we measured M-H loops using polar MOKE, as shown in Fig. 1e. We find that strong PMA is sustained across thicknesses from 10 to 30 uc, with comparable coercivities of about 20 mT. This makes NCO highly favorable for DW racetrack memory applications.

DW structure and chirality
Before investigating the current-driven DW dynamics, we examined the DW structure and chirality in a 30 uc NCO using scanning NV magnetometry (Methods)53–55. Near the DW, the spatially nonuniform magnetization has the following projection onto the coordinate system53–55:
cos\psi
M (x) = M ,
x s
cosh(x/\delta)
sin\psi
M (x) = M , (1)
y s
cosh(x/\delta)
M (x) = -M tanh( x ),
z s
\delta
Where x is the direction perpendicular to the wall, \delta is the DW width, and the angle \psi denotes the DW chirality. \psi = \pi/2 corresponds to a Bloch wall, whereas \psi = 0 or \pi corresponds to a right or left Néel wall, respectively (Fig. 2a). The magnetization-induced stray field varies in space depending on DW chirality, and its spatial profile can be probed using a diamond tip containing a single NV center. Figure 2b shows a map of the magnetic stray field over the 30 uc NCO racetrack, with an Up | Down and a Down | Up wall generated by an appropriate magnetic field. To extract the DW width and chirality, we fitted the stray field wall profiles, as shown in Figs. 2c and 2d. We obtained \delta = 21±6 nm and \psi = 1.7±0.3 rad for the Up | Down wall in Fig. 2c. Similarly, \delta = 35±6 nm and \psi = 1.1±0.1 rad was determined for the Down | Up wall in Fig. 2d. Although \psi appears to be close to \pi/2, the uncertainty is too large to conclusively determine the DW chirality. Therefore, we scanned a larger region of the NCO film to obtain an extended data set on DWs with different orientations (Supplementary Note 1). The distribution of chirality with respect to the DW width, as shown in Fig. 2e, indicates a Bloch-type DW configuration for NCO. In films with PMA, the magnetostatic energy inhibits the formation of Néel walls56,57, unless additional energy terms are introduced, for example, via the Dzyaloshinskii–Moriya interaction (DMI)54. The Bloch-wall chirality suggests negligible DMI in NCO. This is also confirmed by the in-plane magnetic field dependence of the DW velocity (Supplementary Note 2).

Efficient current-driven DW motion
The DW velocities in the racetrack were measured by differential MOKE imaging (Methods), as shown in Fig. 3a. After DW nucleation by an external field, a current pulse of duration tp was injected in the racetrack and the resulting DW displacement was captured by MOKE. Here, we define a positive current flowing from left to right of the images and a positive DW displacement as a left-to-right motion. To obtain an accurate measurement of the DW velocity, a series of pulses was applied to drive the DW displacements. An exemplary sequence of DW motion is shown in Fig. 3b. We find that the DWs move in the same direction as the injected current, opposite to the direction of electron flow. This behavior is consistent with the negative spin polarization of NCO derived from theoretical calculations58,59 and magnetic tunnel junction measurements52 (see Supplementary Note 3 for more details).

The average DW velocity v avg for positive (+) and negative (-) current is extracted from linear fittings of the displacements in the measured sequence, as shown in Fig. 3c. The method used for determining the DW position is explained in Supplementary Note 4. Figure 3d presents the DW velocity as a function of current density. The DW velocities are identical for both Up | Down and Down | Up configurations. Moreover, they are symmetrically centered with respect to in-plane magnetic bias fields Hx and Hy under both positive and negative current polarities (Supplementary Note 2). This is consistent with the absence of SOT in NCO, indicating STT as the sole current-driven mechanism for DW motion. Moreover, the DW velocity is similar in 10 and 30 uc racetracks (Fig. 3d). Such thickness-independent behavior is consistent with the volumetric character of STT-induced DW motion, further excluding interfacial effects as a cause of DW motion in NCO. The STT-induced DW velocity has odd parity with respect to the current direction, whereas other effects, such as Joule heating (Supplementary Note 5) and thickness nonuniformity along the racetrack, are typically independent of the current direction. Thus, the odd STT contribution to DW motion can be extracted using v avg = (v+ avg - v- avg)/2 and is shown in Fig. 3e. The comparison between Fig. 3d and e shows that STT dominates the DW dynamics. Remarkably, we observe a finite v avg ~ 7 m s-1 with a current density of only 5 × 109 A m-2. The current density required to depin the DW is one to two orders of magnitude smaller than those reported for ferromagnetic60–63, ferrimagnetic42,43,47,64–67, and synthetic antiferromagnetic16,68,69 thin films, and is comparable to that observed in bulk single-crystal antiferromagnetic racetracks70,71. Such a low current density is attributed to the large nonadiabaticity and the high crystalline quality of the NCO film, which minimizes the intrinsic and extrinsic pinning, respectively72. Moreover, we observe a DW velocity exceeding 1 km s-1, with a current density of only 2 × 1011 A m-2 (see Supplementary Note 6 for a discussion on the theoretical velocity limit). According to the one-dimensional model25–27, v avg = \beta/\alpha u below the Walker breakdown and v avg ≈ u above it. In ferromagnets, \beta is typically smaller than \alpha26, and the STT efficiency is therefore limited by u, which represents the rate of spin angular momentum transfer from the electric current to the DW magnetization. However, in NCO, the observed DW velocity is more than one order of magnitude larger than the spin-drift velocity (i.e., v avg ≫ |u|), as shown in Fig. 3e (see Supplementary Note 7 for DW dynamics formulation for NCO). This is possible only in the absence of Walker breakdown and if the nonadiabaticity satisfies \beta/\alpha ≫ 161. Under this circumstance, the nonadiabaticity can be extracted from the linear fit of Fig. 3e, yielding \beta/\alpha ≈ 20. A further evaluation of the \alpha and \beta coefficients is presented in the Discussion Section.

DW inertia
The spin-polarized current interacting with a DW induces two primary effects on its dynamics. The adiabatic STT causes a rotation of the magnetization within the wall plane, resulting in a translational motion. Simultaneously, the nonadiabatic STT tilts the magnetization out of the wall plane, leading to the deformation of the DW structure44. The magnetization tends to relax back to its equilibrium state after turning off the current. This relaxation process gives rise to a DW inertial effect, manifesting as continued DW motion even after the termination of the current pulse73,74. The inertia effect is more pronounced in magnetic systems with large nonadiabaticity75, and can be described using the one-dimensional DW model (Supplementary Note 8)27,73–76. The analytical expression of the instantaneous DW velocity v(t) under a current pulse of duration tp reads76:
v(t) = { \beta/\alpha u(1-e- t/\tau) for 0 ≤ t < tp
\beta/\alpha u(1-e- tp/\tau)e- (t-tp)/\tau for t ≥ tp (2)
Here \tau = (1+\alpha2)/(\alpha\gamma HK) is a characteristic time for DW acceleration and deceleration, HK is the transverse anisotropic field which induces a preferred Bloch-wall chirality, \gamma is the gyromagnetic ratio. According to Eq. (2), when tp ≫ \tau, the DW is accelerated up to a terminal velocity \beta/\alpha u, which is determined by the nonadiabatic STT. In this regime, the DW predominantly moves at the terminal velocity, and the inertial contribution becomes negligible. On the other hand, when tp ~ \tau, the DW does not have sufficient time to reach the terminal velocity and starts to decelerate once the pulse is turned off. Figure 4a illustrates this situation using exemplary parameters of tp = 1 ns and 3 ns, \tau = 1 ns, and \beta/\alpha u = 300 m s-1. Clearly, the DW travels a longer distance during deceleration than acceleration when tp ~ \tau. In the experiment, the DW velocity is measured via the DW displacement, and can be expressed as v avg = 1/tp ∫ 0 tp v(t)dt. Consequently, a shorter tp results in a higher v avg. This phenomenon is evidenced by experimental results presented in Fig. 4b. The DW displacement under pulse injection exhibits a steeper slope for tp = 1 ns compared to tp = 3 ns, indicating a higher measured velocity for a shorter pulse. To further investigate the inertia effect in NCO, we measured v avg as a function of tp under various pulse current densities, as shown in Fig. 4c. A significant increase in v avg is observed as tp decreases. To fit these curves, we compute v avg from Eq. (2):
v avg = \beta/\alpha u(1+ \tau/tp e- tp/\tau - \tau/tp) (3)
From the fitting of v avg with respect to tp, two key parameters can be extracted: \beta/\alpha u and \tau. These values are plotted as a function of |u| in Fig. 4d. The linear dependence of \beta/\alpha u on |u| yields the slope = 17.0 ± 0.3, which is comparable to the estimate of \beta/\alpha obtained from Fig. 3e. On the other hand, \tau shows a weak dependence on |u|, with a magnitude of 0.9 ± 0.2 ns.

Methods
Film growth and characterization
Epitaxial NCO films with thicknesses in the range of 10 uc to 30 uc (8 - 24 nm) were deposited on MAO substrates using off-axis radio frequency magnetron sputtering. The temperature was set to 320 °C and the gas (Ar:O₂ = 1:1) pressure was set to 100 mTorr during the film growth.
XRD spectra were measured using a SmartLab SE diffractometer (Rigaku, Japan) with out-of-plane θ-2θ scanning. The general scanning range was from 35° to 110° at a scan rate of 8°/min, while a confined range from 39° to 47° was scanned at 0.5°/min for higher resolution. A cross-sectional sample was prepared using a focused ion beam for STEM images (JEOL ARM 200CF). The in-plane and out-of-plane M-H hysteresis loops were measured using SQUID magnetometry.

Scanning NV magnetometry
The DW structure and chirality of NCO were characterized using a scanning NV magnetometry (QZabre LLC). Magnetic stray fields near the DW region were mapped using a monolithic diamond tip containing a single NV center. The polar angle of the NV spin was 54° relative to z-axis. The tip stand-off distance was maintained at 227 nm, calibrated using the sample edge near the DW. A microwave antenna (~2.9 GHz) was employed for spin excitation, while spin-state detection was performed via fluorescence microscopy (excitation at 515 nm; detection around 670 nm). Line profiles were fitted to extract DW width and chirality following the method described in the Ref 54. All the measurements were performed in ambient conditions at room temperature.

DW racetrack preparation and MOKE microscopy imaging
The DW racetrack devices were prepared by standard photolithography using a laser writer (Heidelberg DWL66+), followed by reactive ion etching with Ar plasma (Plasmalab 80Plus, Oxford Instruments). The contact electrodes were connected using wire bonding. Microwave SMA cables with a bandwidth of DC – 3 GHz were used for pulse injection. The pulses were generated using a voltage pulser (UTV50P, Kentech Instruments Ltd).
The magnetic domain images were captured using a home-built wide-field MOKE microscope. A white LED (Thorlabs, MCWHLP1, wavelength: 400-750 nm) was mounted as the light source. A high-resolution digital CMOS camera (Hamamatsu, C13440-20CU) and an optical objective (× 100, Mitutoyo) were used for imaging. The pixel size was calibrated to be 0.078 μm for the captured images under this setup. The determination of the DW position was explained in Supplementary Note 4. All measurements were performed in air at room temperature.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors studied STT dirven DW motion in ferrimagnet NCO, and reported a DW velocity exceeding 1 km/s with a current density of only 2e11 A/m2, corresponding to a DW mobility of 5.7e-9 m3/(As). They attributed it to the giant nonadiabatic STT, low magnetization, and high spin polarization in NCO. They also observed a pronounced DW inertia effect, which is used to determine material parameters. However, several critical issues have to be addressed before the work can be further considered for publication in Nat. Commun.\n1. The authors use the ferromagnetic theory to interpret their key results, including the DW profile, and STT driven DW motion. They even use the same theory to extract material parameters. Given the ferrimagnetic nature and the complex crystalline structure as shown in Fig. 1(a), it is doubtful that the FM theory would give an accurate interpretation.\n2. As the authors mentioned, typically the nonadiabatic STT is small in FM and can be large in FiM due to spatial variation of magnetization. This indicates the antiferromagnetic coupled sublattice plays an important role, which has been neglected in the theoretical analysis.\n3. Moreover, the nonadiabatic STT coefficient beta in this work is 1.4, much larger than GdFeCo (0.5). The authors should justify this and provide theoretical limit of such coefficients in different material systems.\n4. In terms of the DW speed or DW mobility, 5 km/s under Jc=4.2e11 A/m2 has been reported in FiM (GdFeCo) driven by SOT [Nat. Elec. 3, 37 (2020)]. The authors should justify why one would still choose NCO.\nOther comments\n1. Could the author give a theoretical limit of DW motion in NCO, which would be helpful for assessment of its potential.", "reviewer_2": "Key results: Race Track Memory has been studied for some time as a potential next-generation magnetic storage technology. Key factors for optimizing materials for this application include domain wall mobility and the associated energy consumption. In the submitted paper, the authors investigate domain walls in the ferrimagnetic spinel oxide NCO. They demonstrate high domain wall mobility, low energy consumption, and propose the broader class of ferrimagnetic spinel oxides as promising candidates for future applications.\nData & methodology: To my understanding, the measurements are carefully performed and the theoretical framework for data analysis is sound. The supplementary material provides ample additional information. I have no doubts about the validity of the data or the conclusions drawn.\nClarity of paper: Overall, the paper is well written and accessible, with previous literature mostly cited appropriately. There is some analogy to domain wall motion in antiferromagnets driven by SOT (Gomonay et al., Phys. Rev. Lett. 117, 017202 (2016)) and domain wall motion in ferrimagnets by magnonic currents (Donges et al., Phys. Rev. Research 2, 013293 (2020)) that the authors may wish to consider. The figures are clear and informative. However, in Fig. 4 (figure and caption), I would suggest avoiding the term “simulation” for the result shown in Fig. 4a. In my understanding, this is the outcome of an analytical calculation presented in the Supplementary Information. The term “computer simulation” by contrast, usually refers to large-scale numerical modeling. Additionally, in the supplement (line 116), “procession” should be corrected to “precession.”\nOriginality and significance: Optimizing domain wall mobility while minimizing energy consumption is of broad relevance in spintronics. Fig. 5 convincingly demonstrates exceptionally high domain wall mobility despite the material’s finite magnetization — to date, only an antiferromagnet has shown greater mobility. This is a noteworthy advance and will be of immediate interest to the spintronics community.\nConclusion: Nature Communications is dedicated to publishing high-quality research representing important advances in specific scientific disciplines. In this paper, the advance lies in the discovery of high domain wall mobility in a material with finite magnetization. I consider these findings sufficiently significant for the spintronics and materials science communities to warrant publication in Nature Communications.", "reviewer_3": "In this manuscript, the authors demonstrate Bloch-type domain wall (DW) velocities exceeding 1 km/s in the single-layer ferrimagnetic spinel oxide NiCo O induced by spin-transfer torque (STT) at a current density of 2 × 1011 A/m2. Their 2 4 exceptional DW mobility is attributed to the combination of giant nonadiabatic STT, low magnetization, and high spin polarization. Additionally, they report a pronounced DW inertia effect in this ferrimagnet due to the large nonadiabaticity of the torque. The characteristic time for DW acceleration and deceleration is ~1 ns, shorter than that reported for typical ferromagnets. The authors’ findings are sound, and the manuscript is well written. Therefore, I recommend for publication after revision of several minor points. My suggestions regarding this manuscript are as follows;\n(1) The origin of large nonadiabaticity: The authors systematically elucidate the large nonadiabaticity of STT in the NiCo O 2 4 sample. However, they did not mention the physical origin of this large nonadiabaticity. Why does the NiCo O sample 2 4 exhibit such a large nonadiabaticity? In my opinion, the discussion on the origin of this large nonadiabaticity would be helpful for the reader. Please provide additional discussion on this point in revised manuscript.\n(2) The relativistic effect: Recent paper [Science 370, 1438 (2020)] have demonstrated the relativistic kinematics of domain wall motion at high speed. The authors’ observations also show a very high domain wall velocity up to 1 km/s. However, in Figure 3d. it appears that the domain wall velocity diverges at large applied current. What is the origin of this divergence? Is there any relativistic effect in domain wall motion at such high velocity?\n(3) Minor comments: There are several minor mistakes and suggestions.\n(3-1) for Figure 1: On line 97th, Figure 1c presents transmission electron microscopy (TEM) data, but the manuscript explains it as a magnetic hysteresis loop. Please provide an appropriate explanation of the TEM data.\n(3-2) for Figure 3: On line 165th, Figure 4e should be corrected to Figure 3e.\n(3-3) Suggestion I: In Figure 2b, please provide the scanning line of the NV center measurement results shown in Figures 2c and 2d. This will be helpful for readers.\n(3-3) Suggestion II: Please include a linear fitting line in Figure 3e.", "reviewer_4": "The authors of this manuscript report fast domain wall (DW) motion at velocity up to 1 km/s in NiCo₂O₄ ferrimagnetic spinel oxide under relatively low current density via spin transfer torque (STT). They attribute this rapid motion to the high spin polarization and large nonadiabaticity arising from the staggered spin configuration. In addition, they report inertial effects enabling DW motion for ~1 ns after turning off the pulse. While the observation of rapid domain wall motion at low current density is indeed advantageous from the viewpoint of racetrack memory, it is hard for me to thick that work represent a major technological advance. Fast domain wall motion has already been reported multiple times in the literature, not only in the STT-driven cases as cited by the authors, but also in spin-orbit torque systems. It is not clear what specific advantage arises from achieving domain wall motion purely by STT, and the explanation for the fast motion does not show new physics. Moreover, the significant inertia could in fact be an obstacle for precisely controlling domain wall position using the current pulse. Overall, although the data are interesting, I am not convinced that this work meets the standards required for the publication in Nature Communications. I therefore do not recommend publication.\nBelow I provide several minor comments:\n1. The manuscript claims that the characteristic time of ~1 ns is shorter than in typical ferromagnets, but a massless domain wall motion has previously been reported [J. Vogel Phys. Rev. Lett. 108, 247202 (2012)].\n2. The material used here appears to have high resistivity, which is not favorable in terms of power-consumption. The authors suggested volume-normalized energy consumption, but given the relatively large film thickness used in this study, I question whether this is an appropriate parameter. A comparison in terms of energy per device length might be more relevant.\n3. The domain walls appear to be quite wide (tens of nanometers), which does not seem advantageous in terms of domain density (bit-density).\n4. Appropriate citations are needed for the experiments on the in-plane magnetic field dependence of domain wall velocity.\n5. The fact that the domain wall moves along the current direction by STT is intriguing but is treated rather superficially. The authors should discuss whether previous studies on the directionality of STT exist, and whether alternative explanations beyond STT are possible for the opposite STT.\n6. The resistance of the sample should be specified. Given the high resistivity of the material, the resistance is likely much larger than 50 Ω, raising concerns about impedance mismatch when injecting 1 ns current pulses. The inset in Fig. 4b shows a current pulse profile; it should be clarified whether this waveform was measured after transmission through the sample using an oscilloscope, or whether it simply obtained without the sample."}, "author_response": "1\nReviewer #1\n(Remarks to the Author)\nMy main concern is the validity of the theoretical model. The authors justified that they use the 1D\nmodel with antiferromagnetically coupled sublattices, which is an unsatisfied answer. The same\nmodel has been extensively used in the previous studies of ferrimagnets such as CoTb, GdFeCo,\nor antiferromagnets. Therefore, nothing is new if the authors rely on such simplified model to\nexplain the experimental results, and there is no reason to believe NCO has new physics compared\nto other ferrimagnets or antiferromagnets. If the authors believe there is new physics in NCO, I\nsuggest the author to seriously consider more dedicated theories that can capture the unique\ncharacteristics of NCO, such as the complicated lattice structure and exchange interactions beyond\nthe nearest neighbor coupling. At this stage, I do not recommend its publication in nature\ncommunications.\nResponse: We thank the reviewer for taking additional time to review our manuscript.\nWe understand the reasoning by the referee: if a well-established model is used to interpret\nthe data, then nothing new emerges from the data. However, we do not agree with the implicit logic\nof this comment. Established models do not prevent new discoveries. Rather, they provide a\nbenchmark that makes new discoveries visible. Novelty not only arises from model violations, but\nalso from revealing parameters whose values become the discovery, such as the extremely high\ndomain wall (DW) mobility and nonadiabaticity in our case, and from extending models to different\nmaterials classes, such as spinel oxides rather than ferromagnetic metals or ferrimagnetic alloys.\nIn fact, a majority of novel findings reported across different fields make use of established\nmodels for their interpretation. Famous examples include the “discovery” of massless electron\ndispersions in a variety of 2D materials, which follows a rather simple tight-binding model first\nestablished in the 1940s. The 1D model is widely used to formulate DW dynamics in racetracks\nunder various driving forces (magnetic field, STT or SOT) across a broad range of materials.\nAlthough different material systems employ the same 1D model, this does not imply a lack of new\nphysics. For example, the 1D model was applied to CoTb to interpret the increase of DW velocity\n2\nnear the compensation point1, to GdFeCo to support evidence for STT-driven DW motion in a\nferrimagnetic alloy2, to Bi-YIG to explain the saturation of DW velocity3, and to Fe GeTe and\n3 2\nFe GaTe to report current-induced motion in 2D materials4,5.\n3 2\nIn regard to this manuscript, we report an extremely large STT-driven DW velocity\nexceeding 1 km/s under a moderate current density of 2 × 1011 A m-2 in a single-layer spinel oxide\nmaterial without the need for an external magnetic field. The novelty lies in the following\nexperimental observations:\n1) First demonstration of current-driven DW motion in the ferrimagnetic spinel oxides;\n2) Demonstration of the giant nonadiabatic character of the STT and its role in high DW\nmobility in this class of materials;\n3) First observation of DW inertia in a ferrimagnet with a characteristic time ~ 1 ns, smaller\nthan typical ferromagnets;\n4) Lowest depinning threshold, high DW mobility and lowest energy consumption for DWs\ndriven by STT compared to most ferromagnets and ferrimagnets.\nWe appreciate the reviewer’s thoughtful suggestion to consider a more dedicated theory. A\ncomplex and more complete theory may reveal interesting physics, but goes beyond the scope of\nthis paper and our own capabilities. Regarding the exchange interactions, the nearest-neighbor\ncoupling is typically much stronger than the next-nearest-neighbor coupling, and is therefore\nwidely considered as the dominant contributor to magnetic order. This is also applicable to NiCo O\n2 4\n(NCO) which exhibits an inverse spinel structure. The nearest-neighbor exchange coupling J ,\nA-B\noriginating from Ni ions on octahedral sites (A sites) and Co ions on tetrahedral sites (B sites), is\n(2018). DOI: https://doi.org/10.1103/PhysRevLett.121.057701.\nElectron 2, 389–393 (2019). DOI: https://doi.org/10.1038/s41928-019-0303-5.\n10.1126/science.aba5555.\nFe3GeTe2. Nat Commun 15, 4851 (2024). DOI : https://doi.org/10.1038/s41467-024-48893-y.\nWaals magnet. Nat Commun 16, 10790 (2025).DOI: https:// doi.org/10.1038/s41467-025-66637-4.\n3\nlarger than the next-nearest-neighbor couplings (J and J )6 , resulting in ferrimagnetism7 .\nA-A B-B\nMoreover, considering a complicated lattice structure or including exchange interactions beyond\nthe nearest neighbors may modify the exchange energy by 𝐸 = (𝐽 +𝐽 +𝐽 )∑ 𝒎 ∙\nex A−B A−A B−B 𝑖𝑗 𝑖\n1 𝜕𝐸\n𝒎 , and consequently influences the effective field term (H = − ) of the LLG equation.\n𝑗 eff\n𝑀𝑠𝜕𝒎\n𝐴\nThis influence appears, for example, in the DW width 𝛿 = 𝜋√ . However, it does not affect the\n𝐾𝑢\nanalytical expression of 1D model, as an effective exchange coupling 𝐽 = 𝐽 +𝐽 +𝐽\neff A−B A−A B−B\ncan always be introduced to account for the modification from the exchange interactions.\nIn summary, our experimental results demonstrate that spinel oxide ferrimagnets represent\na novel material platform for STT-based DW devices, offering unexpected and exceptionally high\nDW mobility combined with low pinning and minimal energy consumption. Using NV microscopy,\nwe provided the first insights into the type and width of DWs in NCO. By incorporating the 1D\nmodel, we further revealed the giant nonadiabatic STT and the short timescale of DW inertia in\nthis ferrimagnetic compound. We therefore believe that our findings provide meaningful advances\ntoward the understanding and development of efficient current-driven DW motion in different\nmaterials and racetrack systems.\n(1979). DOI: https://doi.org/10.1103/PhysRevB.19.499.\nPhys. 132, 020901 (2022). DOI: https://doi.org/10.1063/5.0095326.\n(Remarks to the Author)\nKey point of the submitted manuscript is the exceptionally high domain wall mobility driven by\nspin-transfer torque (STT). My previous report was overall positive, and the authors’ replies to my\nremarks as well as the corresponding changes in the revised manuscript are satisfactory. After\nreviewing the other referees’ reports, I noted that Referee 4 raised serious concerns regarding the\nnovelty of the work, claiming that high domain wall mobilities have been reported before —\nhowever, without providing any specific evidence. Referee 1 (in remark #4) compares the present\nresults seriously to a previous study that demonstrated similarly high domain wall mobilities, but\nin a very different system and driven by spin–orbit torque (SOT), a distinct mechanism. I find the\nauthors’ response to this remark convincing. In conclusion, I recommend publication of the\nmanuscript in Nature Communications.\nResponse: We thank the reviewer for evaluating the revised version of our manuscript, including\narguments presented in the rebuttal letter, and for recommending publication.\nReviewer #3\n(Remarks to the Author)\nI would like to express my gratitude to the authors for their tremendous efforts in alleviating my\nconcerns. Therefore, I am pleased to recommend the publication of this manuscript.\nResponse: We thank the reviewer for the clear appreciation of our work and for recommending\npublication of this manuscript.\n(Remarks to the Author):\nThe authors have responded well to the reviewers’ comments and have provided persuadable\nanswers in several aspects. While I believe that the NCO material presents meaningful results in\nterms of the development of domain-wall-motion-based devices, I still question whether this is a\nsignificant advance. Although spin-transfer torque (STT) in ferrimagnets has not been extensively\nstudied, I do not think the results present a significant advance compared to previous ferrimagnet\nstudies, which required an external magnetic field to achieve fast domain-wall motion. The need\nfor an external field is not a major issue, as it can be easily expected if the Dzyaloshinskii–Moriya\ninteraction (DMI) is tuned to stabilize a Néel wall.\nWhile the introduction of a new material platform is certainly of interest and could stimulate further\nstudies, however, because the current findings can still be explained within the scope of existing\nmodels, it is hard for me to agree that this work ‘improve the fundamental understanding of\nphenomena leading to high DW mobilities.’\nBy the way, I think presenting the longitudinal field Hx dependence would constitute a genuine\nnew finding. Since non-adiabaticity is inversely proportional to the domain-wall width (in terms of\nmistracking), verifying whether the domain-wall mobility decreases as Hx modifies the wall width\nwould significantly enhance the novelty of the work.\nResponse: We appreciate the reviewer’s positive remarks on the last-round response. We further\nelaborate below on the significance of this work in order to address the reviewer’s concerns. We\nalso performed additional analysis of the DW velocity as a function of the applied magnetic field.\nThe concern that fast field-free STT-driven DW motion may not represent a significant\nadvance compared to in-field SOT-driven DW motion is debatable. In fact, a multitude of studies\nhave been dedicated to developing field-free SOT switching precisely because in-field operation is\nnot always possible or practical in integrated memories. Additionally, we believe that STT-driven\nDW motion does not compete with SOT, but rather complements it, as these two distinct\nmechanisms have different advantages and disadvantages. Currently, the main drawbacks of STT\ncompared with SOT are its lower efficiency and slower operation speed. Recent work has identified\n6\nthe ferrimagnetic Heusler alloy Mn Ge as a promising material system, in which STT-MTJs8 can\n3\nachieve performance comparable to SOT-MTJs in terms of switching speed, reliability, and energy\nconsumption. Our work demonstrates that STT can likewise achieve comparable performance to\nSOT-driven DW motion in a single-layer ferrimagnet like NCO, owing to its large nonadiabaticity\nand high spin polarization. These advances will motivate researchers to explore material properties\nand systems with even better STT performance in the future.\nAs argued in response to Referee 1, interpreting results using an existing model does not\nimply a lack of novelty. Applying this statement retroactively would wipe off novelty from the\nlarge number of experimental studies of DWs performed in recent years, which use this model for\nanalysis and data simulations. Besides the demonstration of STT-driven DWs in a new class of\nmaterials, we believe that the novelty of our study lies in the exceptionally high nonadiabaticity,\nSTT mobility, and inertia characteristic of this type of ferrimagnetic system. These findings are\nnew and cannot be derived a priori from the 1D model.\nAs the reviewer noted, STT in ferrimagnets has rarely been experimentally examined,\ndespite extensive theoretical works suggesting that it dominates DW motion in ferrimagnets and\nantiferromagnets with staggered magnetic moments9,10,11,12,13. The 1D model helps us parametrize\nthe origin of the exceptional STT performance in NCO, providing a useful phenomenological\nframework to optimize and compare materials enabling high DW mobility, but does not predict\nwhich material performs better than others or which material exhibits giant nonadiabatic STT, or\nhigh spin polarization combined with low magnetization.\nenabled by low magnetization. Nat. Nanotechnol. 20, 360–365 (2025). https://doi.org/10.1038/s41565-024-01827-7.\nL180405 (2021). DOI: https://doi.org/10.1103/PhysRevB.103.L180405.\n144431 (2020). DOI: https://doi.org/10.1103/PhysRevB.101.144431.\n054414 (2024). DOI: https://doi.org/10.1103/PhysRevB.110.054414.\nRev. Lett. 106, 107206 (2011). DOI: https://doi.org/10.1103/PhysRevLett.106.107206.\nB 83, 054428 (2011). DOI: https://doi.org/10.1103/PhysRevB.83.054428.\nmodifications of the DW width and its effect on the DW mobility. We present the in-plane field\ndependence of DW velocity below and in the revised Fig. S2 of Supplementary Note 2.\nFig. S2 DW velocities as functions of in-plane magnetic fields. a, b, H and H\nx y\ndependence of DW velocities for Down | Up and Up | Down domain configurations. The current\ndensity and pulse duration are set to 8.1 × 1010 A m-2 and 2 ns, respectively. The error bars are the\nstandard deviations of the fits. c, d, Calculated DW width change ∆/∆ as a function of H and H .\n0 x y\nmagnetization along y (i.e., perpendicular to the current direction x). Therefore, the DW width ∆ is\nmostly sensitive to a field applied along y rather than x. In the revised Supplementary Information,\nwe theoretically calculate the influence of an in-plane field H on ∆ by minimizing the DW energy\nP\nbased on Ref. 14 (again the 1D model, sorry!). We obtain\n𝐻 𝜋𝐻\n∆= ∆ [1− K cos2𝜓+ P cos(𝜓−𝜓 )] (𝑅1)\n0 2𝐻 2𝐻 H\nU U\n𝐴\nwhere ∆ = √ , 𝜓 and 𝜓 are the angle of wall magnetization and applied field relative to the x-\n𝐾U\ndirection, respectively, 𝜇 𝐻 =\n2𝐾U\nis the effective uniaxial anisotropy field, and 𝜇 𝐻 =\n𝑀s\nln (2)\nt𝑀 = 80±20 mT is the demagnetizing field due to magnetic charges on the walls, which\ns\n𝛿\nleads to a preferred Bloch-wall chirality. We estimate 𝜇 𝐻 = 2.8 T by taking 𝐾 = 0.2 MJ/m3 and\nM = 150 kA/m, which is significantly larger than the demagnetizing field 𝜇 𝐻 (~ 80 mT) and the\ns 0 K\nmaximum applied field 𝜇 𝐻 (≤ 40 mT).\nTherefore, if a magnetic field 𝐻 = 𝐻 (Figs. S2a and 2c) is applied perpendicular to the\nP x\nwall magnetization, we have\n𝜋2\n∆ ≈ ∆ [1+ 𝐻2], (𝑅2)\n0 8𝐻 𝐻 x\nU K\nwhereas if 𝐻 = 𝐻 (Figs. S2b and 2d) we get\nP y\n𝜋\n∆ ≈ ∆ [1+ 𝐻 ]. (𝑅3)\n0 2𝐻 y\nU\nFigures S2c and 2d show the calculated ∆/∆ as a function of the applied magnetic fields H and\n0 x\nH . The behavior of ∆/∆ differs under these two field directions. For H , the change in DW width\ny 0 x\ndepends on the anisotropy field 𝐻 , because H tilts the wall magnetization and thereby modifies\nK x\nthe demagnetizing energy. According to Eq. (R2) and Fig. S2c, the change ∆/∆ becomes weaker\n0\nas 𝐻 increases. In the limit 𝐻 ≫ H , the wall magnetization cannot be tilted, and the DW width\nK K x\nis not affected by H . In contrast, applying H does not tilt the wall magnetization; instead, it aligns\nx y\nresearch (Vol. 1). Academic press. chapter IV, 77-121 (1979). https://doi.org/10.1016/B978-0-12-002951-8.50008-X.\n9\nthe wall magnetization to the field direction to lower the Zeeman energy. Consequently, the DW\nexpands under H , see Eq. (R3).\ny\nQuantitatively, the maximum change in DW width under a 40-mT H is estimated to be\nx\nonly 0.8%, and becomes even smaller for larger 𝐻 . This small variation may explain why the DW\nK\nvelocity remains almost constant under H in Fig. S2a. However, a H of 40 mT expands the DW\nx y\nwidth by up to 2.3%. As the reviewer pointed out, the increase in DW width can reduce the\nnonadiabaticity, accounting for the observed decrease in DW velocity under H in Fig. S2b.\ny\nPrevious theoretical studies demonstrated that nonadiabaticity inversely scales with DW width in\nterms of mistracking15,16,17. Other potential mechanisms, such as in-plane-field–induced Walker\nbreakdown, only affect the DW velocity in the presence of H , but not H . Specifically, H lowers\nx y x\nthe Walker breakdown threshold, while H suppresses it. In this case, the DW velocity should\ny\nchange under H but not under H . This is in contradiction to our experimental observations and\nx y\nconsistent with DW motion remaining below the Walker limit. Additionally, we observed that DW\nvelocity 𝜐 ≫ 𝑢, which further demonstrates the DW motion remains below the Walker limit, as\n𝜐 = 𝑢 would be expected once it enters the Walker regime. Thus, Walker-breakdown–induced\nvelocity changes can be ruled out.\nWe have included a summary of this analysis in the main text and reported Fig. S2a-d and\nthe full derivation of Eqs. R2 and R3 in the revised Supplementary Note 2.\nRev. B 91, 094411 (2015). DOI: https://doi.org/10.1103/PhysRevB.91.094411.\n054428 (2006). DOI: https://doi.org/10.1103/PhysRevB.92.054434.\n054434 (2015). DOI: https://doi.org/10.1103/PhysRevB.92.054434.\nAll changes to the text have been highlighted in blue in the revised manuscript and supplementary\ninformation files. Additionally, we have included the following changes:\nRevised manuscript:\n1. In lines #78-82 of Introduction Section and lines #247-253 of Discussion Section, we added\ndiscussions on the in-plane magnetic field dependence of the DW velocities.\nSupplementary material:\n1. In Supplementary Note 2, we added the full derivation of the influence of in-plane magnetic\nfield on the DW width and nonadiabaticity, which explains the observed dependence of DW\nvelocities on the in-plane magnetic fields."}]

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