Elucidation of the Mechanism for Maintaining Ultrafast Domain Wall Mobility Over a Wide Temperature Range

To realize a data rate of 20 Gbps in the communication standard 5G with a racetrack memory, it is crucial to stably recognize a domain-wall (DW) velocity ( v DW ) of 1200 m/s when the minimum bit length is 60 nm. However, general v DW is as slow as about 100 m/s. Recent reports indeed showed that the fast DW motion occurs using an in-plane external magnetic field however, this mechanism is unsuitable because the rear-edge v DW decelerated, which contrary to the front-edge of DW velocity. Therefore, we designed magnetic wires by bringing the g values of rare-earth and transition-metals close to each other and suppressing the Joule heat generation distribution due to short pulse current, we successfully demonstrated the v DW of 1200 m/s in a wide temperature range without using an external magnetic field. Moreover, the current density ( J ) is low, and the DW mobility ( v DW / J ) is significantly improved 10-times over a wide temperature range compared to other reports.

Spintronics applications are being extensively followed for high-performance logic computing technologies and racetrack memories 3,4,5,1 .The principal difficulties are to reach small bits 6 , high thermal stability, and track them at high speed 7 .In ferromagnetic materials, low coercivity (Hc) is an advantage, however, large critical current density (Jc), bit size, stray field interactions limit, and precessional dynamic limit of the operating speeds are the substantial limitations in ferromagnetic materials 8,9,10 .Moreover, antiferromagnets show a lack of stray fields with a high packing density 11,8,12 .Nevertheless, manipulating and recognizing magnetic state and spin textures of antiferromagnetic are challenging 13,14,15 .To overcome this deficiency an alternative strategy is using amorphous rare earth transition metals (RE-TM) ferrimagnetic systems.Ferrimagnetic materials consist of rare-earth (RE) and transition metal (TM) compounds in which the RE and the TM sublattices are aligned antiparallel with each other and reduce the net angular and magnetic moments.Therefore, ferrimagnets such as GdFeCo alloy can be easily detected using optical and electrical instruments owing to the 3d electrons of the transition metal sublattices 16,17 .Additionally, the perpendicular ferrimagnetic wires show smaller (Jc) than those in-plane wires 18,19 , due to low saturation magnetization Ms 20 .
Theories and experimental results have predicted that the fast current-induced domain wall motion (CIDWM) appears near the angular momentum compensation temperature for ferrimagnetic wires 3,21,22 .To achieve the 20 Gbps for the 5 G application with racetrack memory, domain wall (DW) motion should be realized, with a speed of 1200 m/s and a minimum bit length of 60 nm 6 in magnetic wires.Hence, the most significant issue is the fast and stable DW velocity over 1200 m/s.The idea behind our experiments, illustrated in Fig. 1(a-c), is based on predictions of the DW motion for various RE-TM combinations, which is corresponding to the g values of RE 23 and TM 24 metals.Figure 1(a) shows that the angular momentum compensation temperature (TAMC) of the TbCo is far from the magnetic momentum compensation temperature (TMC), therefore DW velocity is slow at any temperature.In the case of GdCo, both TAMC and TMC become near, the DW velocity increase, however, it is still below 1200 m/s for the 20 Gbps request.On the other hand, the TAMC of the GdFeCo is very near to the TMC, accordingly, DW velocity would be accelerated over 1200 m/s in a wide temperature range.Furthermore, we compared our work with previous reports in Table 1.These results have motivated us to analyze the mechanism of DW motion between TAMC and TMC for compensated ferrimagnet.To verify the prediction of DW motion between TAMC and TMC, we performed systematic experiments for CIDWM in Gdx(Fe88Co12)100-x ferrimagnetic wires.
We exhibit that, by employing the Gd-FeCo ferrimagnet, the DW velocity reaches its maximum (2000 m/s) between magnetic (xMC) and angular momentum (xAMC) compensation composition.Then we analyzed the temperature distribution of the magnetic wire by applying the current (J), and found that the DW velocity increases when the temperature difference between the center and the edge of the magnetic wire is small.We also found that the DW velocity is the fastest for magnetic wires where the TAMC and TMC are close to each other and TAMC is slightly higher than room temperature.As a result, a DW velocity of 1200 m/s can be achieved in a wide temperature range, and DW mobility (vDW/J) is improved more than 10-times compared to other reports.

Design and sample characterization
We initially discern the magnetic properties of Gdx(Fe88Co12)100-x films.Figure 2(a) exhibits the hysteresis loops were measured using the Polar-Magneto-Optical Kerr Effect (PMOKE) at room temperature.The polarity of the Kerr rotation (θK) signals changes between Gd24(Fe88Co12)26 and Gd25(Fe88Co12)75 samples are consistent with a transition from being FeCo dominated to being Gd dominated in the magnetic moment.Figure 2(b) shows the Ms and Hc of deposited Gdx(Fe88Co12)100-x films at different compositions.The saturation magnetization (Ms) of Gdx(Fe88Co12)100-x reaches its magnetization compensation composition at (xMC) ~24 at.%.While the Ms is minimum and the coercive fields reach their maximum at magnetic compensation composition 25,26 .The angular momentum compensation composition (xAMC) was determined using the following equations: where Ai=Mi/ , also Mi (i=Gd or FeCo) is the magnetic moment and  =giµB/ħ is the corresponding g factor, µB and ħ is the Bohr magnetron and Planck constant, respectively.
The g factor is gGd ≈ 2.0 and gFeCo ≈ 2.16 27,24 , hence, the xAMC is estimated to be xAMC~23, respectively.In our study, the DW moves along the direction of pulse current, which can be supported by the spin-orbit torque (SOT) in the heavy metal/ferrimagnet system 28,29 .Figure 3(a),(b) shows that the DW velocity increases by raising the J for all Gdx(Fe88Co12)100-x compositions.The maximum DW velocity vDW= 750 m/s and 2000 m/s are observed with a sample of Gd24(Fe88Co12)76 for 30 and 3 nsec pulse duration, respectively.As a result, without applying an in-plane external magnetic field, the fastest DW velocity of 2000 m/s has been obtained as far as we know in the reports.In addition, the domain wall velocity of both the up-down and down-up domain walls was the same 2000 m/s.In racetrack memory, it is necessary to shift recorded data patterns with the current without destroying them.Therefore, the results show that high-speed data transfer is possible as racetrack memory.On the other hand, the result 22 of achieving 5700 m / sec by accelerating only one side DW of the recorded magnetic domain using the in-plane external magnetic field is attracting attention, however, the DW on the other side decelerates significantly, in consequence, the method using the in-plane external magnetic field cannot be used practically, because it causes data destruction 22,3,2,30 .Except for Gd24(Fe88Co12)76, we observed a linear relationship between DW velocity with current density for all GdFeCo compositions (Fig. 3(a-b)).To clarify the J dependence of DW velocity, we first determine the magnetic features of the Gdx(Fe88Co12)100-x wires using the hysteresis loops at various temperatures (as shown in the supplementary Fig. 2).From those results, the magnetic compensation temperature (TMC) for each composition is obtained.
Next, using the TMC, the angular momentum compensation temperature TAMC was evaluated from Equations 1-4 of the supplement.As can be seen from Equation 1of the supplement, the DW velocity becomes faster near the TMC.On the other hand, it is considered that the DW velocity increases even near the angular momentum compensation temperature (TAMC).Since an electric current is applied to the magnetic wire at room temperature, it is desirable that TAMC and TMC are present near room temperature to achieve a high DW velocity.It can be considered that the reason why the DW velocity of Gd24(Fe88Co12)76 is the fastest in Fig. 3(c) is that it meets this condition.Joule heat generation 31,32 can be taken into account that, by applying a pulse current, nevertheless the longer the pulse width, the higher the Joule heat generation.When a pulse width is 3nsec, the Joule heat generation is small, and the temperature of the magnetic wire is close to TAMC as shown by the blue cross mark in Fig.

3(d)
. Therefore, it is seen that the DW velocity of the Gd24(Fe88Co12)76 shows the highest in the GdFeCo in Fig. 3(c).On the other hand, when the pulse width is 30nsec, it is considered that the temperature of the magnetic wire has risen to the vicinity of the orange cross mark in Fig. 3(d).The result of the temperature simulation is described in Fig. 4(b).Therefore, as a result of the current density dependence of the DW velocity with a pulse width of 30nsec in Fig. 3(a), the temperature of the magnetic wire is near TAMC, so the DW velocity becomes faster.However, when the current is further increased, the temperature exceeds the TAMC, so the DW velocity slows down.Thus, only this 30nsec pulse injection result of Gd24(Fe88Co12)76 does not show a straight line due to this effect.
By applying excessive current density, multi-domains can be created easily for the samples which are far from the magnetic compensation composition point xMC.For these samples, the perpendicular anisotropy Keff becomes smaller than that of samples near the xMC 33,34 .Thus, it is taken into account that a short pulse current is required to avoid multidomains at high current density J.As a result, to apply larger current densities (J) into the Pt/ Gdx(Fe88Co12)100-x wires and determine the faster DW motion in magnetic wires, we compared the CIDWM using single voltage pulses with a pulse duration of 30nsec and 3nsec for all Gdx(Fe88Co12)100-x composition, respectively.A wide range of current densities can be injected into the samples among xMC and xAMC using a short pulse current.Accordingly, the fast DW motion at low current density can be realized in Gdx(Fe88Co12)100-x wires without using an external magnetic field.

Thermal stability and fast domain wall motion without external magnetic field
Generally, to attain even fast DW velocity, larger current densities are needed.Figure 4(a) shows the DW velocity as a function of pulse duration for the Gd25(Fe88Co12)75 sample.
Despite applying the same current density (J =1.7×10 11 A/m 2 ), the DW velocity becomes faster as the pulse duration becomes shorter.As discussed previously, in ferrimagnetic materials the temperature can affect their magnetic properties 31 .
Increasing the input current density also increases the domain wall velocity, as shown in Fig. 3(a),(b).When the pulse width is fixed at 3nsec, and the number of pulses is increased, the DW displacement distance is ideally proportional to the number of pulses, as shown in the inset of Fig. 4(a).Similarly, fixing the current density and increasing the pulse width should increase the domain wall velocity.However, when the current density was fixed (J =1.7×10 11 A/m 2 ) and the pulse width was increased as shown in Fig. 4(a), the domain wall velocity became slower, contrary to the expectation.Hence, this expectation can be explained due to the Joule heating effect in the GdFeCo wires.As shown in Fig. 4(b) the effect of Joule heating on the DW velocity can be clarified as follows: When the applied current flows through the magnetic wire, the temperature in the GdFeCo layer rises, which depends on pulse duration.However, an increase in the sample temperature with higher current density will drive the sample away from the compensation region, then the DW velocity decreases.
The Joule heating effect decreases the DW velocity for the duration of the large pulses current, which is consistent with the previous reports 21,3 .
Our experimental data shows that fast and stable DW motion occurs in short pulse current.
Correlated with these large current densities is a substantial amount of Joule heating that enhances the temperature of the samples.Therefore, it is a crucial issue to recognize that how strongly the temperature can influence the physical properties of ferrimagnetic wires.To determine the temperature change induced by the pulse duration, we performed a numerical simulation based on the heat diffusion in wires (as shown in Supplementary Figs.4-5). Figure 4(b) exhibits the temperature profile after the application of a current density of J =1.7×10 11 A/m 2 at the different pulse duration as a function of position x. Figure 4(b) presents that the temperature of the samples increased at the center of the wire for both 30nsec and 3nsec pulse duration.It is seen that the high-temperature gradient occurs between the edge part and center part of the GdFeCo wire when the 30nsec pulse current is injected into the sample.Contrarily, the 3nsec pulse duration shows a broad peak (small temperature gradient) at the center of the wire.Accordingly, the temperature gradient increases linearly and causes decreases in the DW velocity with a negative linear relationship with the sample temperature (as shown in the supplementary Fig. 6).Therefore, it is clearly observed that the DW propagation is more uniform and stable when a short pulse current of 3nsec is injected into the sample, which agrees with experimental results in Figure 4(a).
Figure 4(c) shows the DW velocity of the Gd24(Fe88Co12)76 as a function of operating temperature for pulse current of 30nsec and 3nsec, respectively.DW speed obtained stable between the temperature range of R.T. <Top<70°C for Gd24FeCo76 when pulse current of 30nsec and 3nsec is injected into the wire.Especially for a short pulse current of 3nsec, the DW velocity remains relatively constant, over 1200 m/s between room temperature and 70 °C, which is suitable for the racetrack memories (see supplementary Fig. 3 for details).
As a result, a wide temperature range with a DW velocity of 1200 m/s has been obtained in Gd24(Fe88Co12)76 for a short pulse current which is more stable in comparison with the previous reports 21,3,35 .

The physical mechanism of fast domain wall motion in GdFeCo
The experimental realization of this proposal is shown in Fig. 5(a-d) which the DW velocity is a function of Tb and Gd concentration.So far, we have reported many papers on the current driven DW motion of TbCo magnetic wires, but the DW velocity was as slow as 100 m/sec or less even if the composition and structure were changed 1,2,36 .Previous studies clarified that the fast DW motion appears at the angular momentum compensation point, which is consistent with the results given by Saima et al 2   We, therefore, propose that bringing the g values of RE and TM near to each other is essential to design a fast DW motion.
In order to realize a fast data rate of 20Gbps, both low current density J and fast DW motion are required.Therefore, large DW mobility vDW/J is an important parameter.To compare our work with previous reports, it is clear that the DW mobility in the GdFeCo sample is much higher than that of other FM and FIM materials, which is shown in Fig. 5(e) 28,29,2,8,21 .The DW mobility for 30 and 3nsec pulse width is 33× 10 -10 m 3 /As and 88× 10 -10 m 3 /As.These results open a new window for technology to design a new type of racetrack memory with low power consumption and high-velocity thermal stability.
In summary, we achieve a wide temperature range and fast domain wall motion (vDW>1200 m/s) between magnetic and angular momentum compensation points without an external magnetic field.We have experimentally demonstrated and theoretically explained that a short pulse current of 3nsec is more stable and faster than a long pulse width of 30nsec.
Our work implies that bringing the g values of RE and TM close to each other is a fundamental parameter for designing fast and stable racetrack memory for 5G applications.
These results could be a clue to realize the physical mechanism of DW motion in amorphous ferrimagnetic alloys, and therefore DW mobility is remarkably enhanced more than 10-times, which will invigorate research towards the development of memory applications.
the position of the domain wall (as viewed by PMOKE) by the duration of the current pulse (Fig. 2(c)).Note that all measurements are performed at room temperature.

Supplementary Files
This is a list of supplementary les associated with this preprint.Click to download. supl.pdf

Figure 2 (
Figure 3(a),(b) show the dependence of DW velocity as a function of current density for Pt/ Gdx(Fe88Co12)100-x /SiN wires (20<x<28) measured with a pulse duration of 30 and 3 nsec,

Figure 3 (
Figure 3(c) exhibits the DW velocity as a function of Gd concentration under the condition with a constant current density of J =1.7×10 11 A/m 2 .It is seen that the DW velocity increases by increasing Gd content up to x=24, and then vDW decreases by increasing Gd content over x=25.The maximum DW velocity vDW = 1500 m/s (>20 Gbps) appears between xMC and xAMC point for Gd24FeCo76 with short pulse current 3nsec.

Figure 3 (
Figure 3(d) shows that the TMC (green dots) and TAMC (violet dote) are different for each Gdx(Fe88Co12)100-x composition.By increasing the Gd concentration the TMC (green dots) and TAMC (violet dote) were shifted towards the higher temperature.
. We first extracted the TbCo DW velocity as a function of Tb concentration in Fig. 5(c).Saima et al. have shown that the highest DW speed attains a maximum at around x ≈ 0.21-26.Fig. 5 (c) presents that the xAMC is far from the xMC due to the different g values for Tb~1.5 and Co~2.22 atoms.It is seen that to reach the maximum DW velocity of TbCo sample has a narrow margin of Tb composition between xAMC and xMC.Thus, it is difficult to accelerate the DW velocity of the TbCo wires.

Figure 5 (
Figure 5(d) shows the DW velocity dependence of Gd composition and presents that DW velocity reaches its maximum between xAMC and xMC.Hence, The GdFeCo sample has a wide

Figure 1 .
Figure 1.(a-c) Schematic illustration of the concept of DW motion in ferrimagnetic compared to our results.

Figure 2 .
Figure 2. (a) MOKE measurements as a function of perpendicular magnetic field for GdxFeCo100-x films at room temperature.(b) saturation magnetization and coercivity as a function of Gd composition.(c) configuration of domain wall motion in GdxFeCo100-x wire with consecutive current pulses.

Figure 3 .Figure 4 .
Figure 3. (a-b) Domain wall velocity as a function of current density with pulse duration 30 ns for GdxFeCo100-x /SiN wires (20<x<28), (c) Domain wall velocity as a function of current density with pulse duration 30, 3 nsec for GdxFeCo100-x /SiN wires (20<x<28) (d) summary of TM and TAM as a function of Gd concentration.For Gd24FeCo76 sample, the TAMC is 320 K