Coupled Néel domain wall motion in sandwiched perpendicular magnetic anisotropy nanowires

The operating performance of a domain wall-based magnetic device relies on the controlled motion of the domain walls within the ferromagnetic nanowires. Here, we report on the dynamics of coupled Néel domain wall in perpendicular magnetic anisotropy (PMA) nanowires via micromagnetic simulations. The coupled Néel domain wall is obtained in a sandwich structure, where two PMA nanowires that are separated by an insulating layer are stacked vertically. Under the application of high current density, we found that the Walker breakdown phenomenon is suppressed in the sandwich structure. Consequently, the coupled Néel domain wall of the sandwich structure is able to move faster as compared to individual domain walls in a single PMA nanowire.


SI-1) Coupled domain wall (DW) dynamics in two-nanowire sandwich structure as a function of time.
Fig. S1 shows the magnetizations of the upper and bottom nanowires of the sandwich structure with current J = 2.68 ×10 12 A/m 2 applied to the bottom nanowire. The magnetizations of the two DWs are shown to gradually change over time from its initial state to the stable state. Fig. 3 (a) and (c) of the manuscript corresponds to the stable states of the two DWs (t = 25 ns).

SI-2) Coupled domain wall (DW) dynamics for different saturation magnetization
We have performed additional simulations to investigate the dynamics of the DWs in the sandwich structure for different saturation magnetization values.

SI-3) Coupled domain wall (DW) dynamics for different values of the spin transfer torque constant (β)
We have performed additional simulations to investigate the dynamics of the coupled DWs in the sandwich structure with respect to a change in the non-adiabatic constant. Fig. S3 (a) shows the DW position vs time plots of the coupled DWs in the sandwich structure for various values of the non-adiabatic constant. The speed of the coupled DWs is shown to increase with increasing value for the non-adiabatic spin transfer constant. Fig. S3 (b) shows the corresponding dynamics of the DW in the bottom nanowire. The two graphs show that the change in the nonadiabatic constant of the spin-transfer-torque equation results in the increase of the propagation speed and also in the tilting angle of the DW, which both agree with the 1-D equation for DW dynamics below the Walker breakdown regime.

SI-4) DW dynamics in the sandwich structure
The analytical explanation of the coupled DW dynamics is as follow: The 1-D DW dynamics can be written as: , where w is the volume energy density of the system.
The magnetization of the nanowire can be written as: , where q is the position of the DW along the nanowire, and  is the tilt angle of the nanowire. The second term in the tilt angle equation is the additional term that comes from the curling domains. C is a constant that represents the degree of curling (C<1).
The energy of the system can then be calculated as: To obtain the final results, the volume energy density is changed to the areal energy density.
The LLG is then rewritten to include the areal energy density: The dynamics of a current-driven DW in the absence of an applied external field can then be written as: If we consider the Neel state of the DW in the sandwich structure (ψ = π/2 + δ), the dynamics become: It can be seen that the contribution from curling domains (B) results in the suppression of the change in the tilt angle of the DW, which is equal to the suppression of Walker Breakdown. It can also be seen that the presence of the curling domains also add to the forward motion of the DW upon the application of current. For sandwich structure with nanowire thickness of 6 nm and width of 40 nm, the contribution from the B constant is approximately equal to an anisotropy field of μH ≈ 100 mT.

SI-5) Sandwich structure device details
We have performed additional simulations to investigate the effect of the structure width (w) to the dynamics of the coupled DW in the sandwich structure.

Fig. R2 Image of a 46 nm wide nanowire that is made using the above technique
The gold contact pads can be patterned by using lithography twice: first is for the deposition of the insulator and the second is for the gold contact. The insulator and the gold are patterned in different position so that the gold is able to be connected with the upper nanowire without touching the bottom nanowire.  performed additional simulations to investigate how thick the insulator layer at the middle can be made while still maintaining the coupling between the DWs. We found that the DWs can still be coupled with insulator layer of 10 nm, which are thick enough to ensure that there is no current that flows in the lower nanowire.
To generate the coupled DWs, current can be applied to the injection line:  For the three-nanowire sandwich system, the above fabrication method can be extended to add the third nanowire, as shown below:

SI-6) DW pinning in sandwich structure
We have performed additional simulations to investigate the effect of extrinsic pinning to the dynamics of the coupled DW in the sandwich structure. Two pinning sites are created in both nanowires of the sandwich structure as shown below in Fig. S11 (a). Current is only applied to the bottom nanowire. are able to move together with nearly a constant speed and a constant tilting angle up to t = 15 ns where they finally arrive at the pinning sites. The DW in the top nanowire is shown to be stopped by the pinning sites, while the DW in the bottom nanowire is shown to be able to continue its movement. Fig. S11 (c) shows that following the coupling breaking, the DWs start to change their behaviour. Fig S11 (d) shows that the DW in the top nanowire starts to come to equilibrium as it oscillates around the x axis with the natural frequency of the spin precession. At the same time, the bottom nanowire starts to rotate in the xy plane which signals that the DW has experienced the Walker breakdown due to the high applied current density.

SI-7) Threshold current density as a function of nanowire thickness
The current that is needed to drive the coupled DWs in the sandwich structure depends on the local pinning that is induced by the curling magnetic domains. We have performed additional simulations with different PMA nanowire thickness to investigate the effect of the structure dimension to the threshold current. Fig. S12 shows the change in the threshold current in sandwich structure with two nanowires with respect to the nanowire thickness. The results show that it is possible to lower the threshold current by making the nanowire thinner.
For instance, the threshold current is reduced to J = 1.5 ×10 11 A/m 2 when the thickness of both the top and the bottom nanowires of the sandwich structure is reduced to 1 nm. Fig. S12. Threshold current as a function of nanowire thickness. The spacer layer is maintained at 2 nm.

SI-8) Current density distribution within the sandwich structure
We have performed additional simulations using COMSOL to simulate the current density distribution within the sandwich structure. Fig. S13 shows the simulation result. The thickness of each ferromagnetic layer here is 5 nm while the thickness of the middle insulator spacer layer is 2.5 nm. The two smaller boxes at the top of the sandwich structure correspond to the contact pads. The result shows that majority of the flows in the in-plane direction within the top nanowire, with minimal current flowing in the bottom nanowire. Additionally, it is also possible to increase the spacer layer thickness up to 5 nm to ensure even better current isolation while still maintaining the coupling between the DWs in the top and bottom nanowire. At such thickness, it is highly improbable for any current to tunnel through. Therefore we believe that