Current-driven fast magnetic octupole domain-wall motion in noncollinear antiferromagnets

Antiferromagnets (AFMs) have the natural advantages of terahertz spin dynamics and negligible stray fields, thus appealing for use in domain-wall applications. However, their insensitive magneto-electric responses make controlling them in domain-wall devices challenging. Recent research on noncollinear chiral AFMs Mn3X (X = Sn, Ge) enabled us to detect and manipulate their magnetic octupole domain states. Here, we demonstrate a current-driven fast magnetic octupole domain-wall (MODW) motion in Mn3X. The magneto-optical Kerr observation reveals the Néel-like MODW of Mn3Ge can be accelerated up to 750 m s-1 with a current density of only 7.56 × 1010 A m-2 without external magnetic fields. The MODWs show extremely high mobility with a small critical current density. We theoretically extend the spin-torque phenomenology for domain-wall dynamics from collinear to noncollinear magnetic systems. Our study opens a new route for antiferromagnetic domain-wall-based applications.


Section 3: Determination of MODW position and displacement.
In the experiment, the method of determining the MODW position and displacement is shown in Supplementary Fig. 2. Firstly, we converted the pixel contrast image into the horizontal line profiles and calculated an averaged profile (black line).A smooth profile (red line) was obtained using the adjacent-averaging method.We compared the contrast profile with the raw MOKE image to determine the exact MODW position.The midpoints of the rise and fall lines (The positions of green and blue dashed lines) were defined as the MODW positions.The MODW displacement was calculated by comparing the MODW positions before and after pulse injection.We averaged the displacements of both left and right domain walls for the velocity calculation.In Supplementary Section 11, we discuss and conclude that the nonadiabatic torque should play a crucial role in MODW motion in noncollinear AFMs.Under this circumstance, the extrinsic pinning effect may affect threshold current density for domain-wall motion and cause an underestimation in velocity 1,2 .We found a threshold pulse duration is required for the MODW motion, similar to the previous report 3 .Therefore, we study the role of Joule heating by comparing the vMODW under different pulse durations in a Mn3Ge Néel-wall device.

Supplementary
The current density is 4.36 × 10 10 A m -2 .The pulse durations in Supplementary Fig. Apart from spin torques on the magnetic octupoles, the longitudinal pulse current may also create Joule heating in the sample.We intentionally reduced the pulse duration when increasing the amplitude of current density to diminish the heating effect in the experiment.
Joule's Law can estimate the Joule heating generated by the injected pulse current: where Q g , I, R, and t denote the amount of induced heating, applied current, sample resistance, and pulse duration time.On the other hand, the temperature increase can be expressed by the where n, C v,m , ΔT denote the molar amount, molar heat capacity at constant volume, and temperature increase, respectively.The Qa is smaller than the Qg due to the heating dissipation.We can generally neglect the heat dissipation for a nanosecond pulse while not for a microsecond pulse since the temperature gets saturated 5 .
Therefore, we ignore the heat dissipation for a nanosecond pulse and have Qa ≈ Qg.
Supplementary Figure 5b shows the Qg obtained from Supplementary Fig. 5a concerning the current density for the nanosecond pulses.Here we use n = 6.8×10 -12 mol, Cv,m = 102.2J mol - 1 K -1 and  ≈ 150 Ω •  for the estimation.Ultimately, the evaluated temperature increase is about 43 K, comparable with the previous estimation [6][7][8] .For microsecond pulses where the heating dissipation cannot be neglected, it reads Qa ≪ Qg.Therefore, we estimated the temperature increase during the pulse injection by monitoring the longitudinal resistance using an oscilloscope.Firstly, we measured the temperature dependence of longitudinal resistance under a small DC current of 50 μA as the reference by heating the device (Supplementary Fig. 5c).When we estimated the temperature increase, the initial resistance R0 = 5.51 Ω before the pulse injection corresponds to the environmental temperature of 275 K in the atmosphere.We note this environmental temperature is anomalously low which may be due to the temperature drift in a cryostat when we measured the R-T curve.However, this temperature drift does not affect the estimation because it will be cancelled when calculating the temperature change before and after pulse injection.Afterward, we applied a pulse (9.2×10 9 A m -2 , 50 μs), much longer than the experimental value (9.2×10 9 A m -2 , 4.5 μs) to get a saturated temperature.We then monitored the resistance change during the pulse injection, as shown in Supplementary Fig. 5d.The spiking peaks originate from the overcompensation of the probe due to the impedance mismatching 9 .The resistance Rt during the pulse injection is not more than 6.14 Ω, corresponding to the temperature of 305 K.The temperature increase due to Joule heating under the microsecond pulse duration is not more than 30 K. The temperature thus remains lower than the Néel temperature (370 K for Mn3Ge) 10 when the pulse current is applied in our experiment.

Fig. 2 |Section 4 :Supplementary Fig. 3 |
Determination of MODW position and displacement.The MODW position was extracted from the horizontal contrast profile.The midpoints of the rise and fall lines show the MODW positions.The MODW displacement was determined by two H corresponding MODW positions.The j in this representative MOKW image is set to 3.44 × Representative MOKE images of flow motion.Representative MOKE images of flow motion.The contrast profiles in the right panel show the MODW positions and displacements.The j is set to 3.44 × 10 10 A m -2 , 100 ns; 4.81 × 10 10 A m -2 , 50 ns; and 6.64 × 10 10 A m -2 , 26 ns, respectively.We checked that the contrast drop around 27 μm remains unchanged under the sweeping magnetic field (Supplementary Fig. 6 in Section 7) and thus should not be the magnetic contrast.

Supplementary Fig. 4 |Section 6 :
4 (b)-(d) are set to 40 ns, 50 ns, and 60 ns, separately.We estimate the vMODW to be 375 ± 16 m/s from (b), 354 ± 58 m/s from (c), and 296 ± 21 m/s from (d).The vMODW is summarized in Fig 4(e).The average velocity from different pulse durations is 342 ± 41 m/s and the error is about 12%.There are two main origins for the velocity variation under different pulses.The domain-wall velocity shows intrinsic stochastic behavior due to the pinning effect.The depinning probability oscillates with the pulse width and depends on the pinning landscape 2 .In addition, the broad domain-wall width of Mn3X gives rise to another source of velocity uncertainty when determining the domain boundary (e.g., the ~1.4 μm domain-wall width of Mn3X 4 can cause a velocity difference as large as 56 m/s under a pulse width of 50 ns).Both origins contribute to large error bar in velocities.Estimation of vMODW with different power injection.The pulse durations are 40 ns in (b), 50 ns in (c), and 60 ns in (d).The vMODW is extracted from MODW profiles compared with (a).The right domain wall in (c) tilts after pulse injection.We thus chose the left, middle, and right positions for the velocity estimation, which induces more significant errors in (c).The current density is 4.36 × 10 10 A m -2 .Estimation of temperature increase due to Joule heating.