Local and global energy barriers for chiral domain walls in synthetic antiferromagnet–ferromagnet lateral junctions

Of great promise are synthetic antiferromagnet-based racetrack devices in which chiral composite domain walls can be efficiently moved by current. However, overcoming the trade-off between energy efficiency and thermal stability remains a major challenge. Here we show that chiral domain walls in a synthetic antiferromagnet–ferromagnet lateral junction are highly stable against large magnetic fields, while the domain walls can be efficiently moved across the junction by current. Our approach takes advantage of field-induced global energy barriers in the unique energy landscape of the junction that are added to the local energy barrier. We demonstrate that thermal fluctuations are equivalent to the magnetic field effect, thereby, surprisingly, increasing the energy barrier and further stabilizing the domain wall in the junction at higher temperatures, which is in sharp contrast to ferromagnets or synthetic antiferromagnets. We find that the threshold current density can be further decreased by tilting the junction without affecting the high domain wall stability. Furthermore, we demonstrate that chiral domain walls can be robustly confined within a ferromagnet region sandwiched on both sides by synthetic antiferromagnets and yet can be readily injected into the synthetic antiferromagnet regions by current. Our findings break the aforementioned trade-off, thereby allowing for versatile domain-wall-based memory, and logic, and beyond.

b,c, Comparison of oxidized SAF (60 sec processed) and as-grown reference FM film measured along the magnetic easy-axis (b) and hard-axis (c).

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The plasma oxidation to transform SAF into FM is optimized by monitoring the magnetic hysteresis loop measured on unpatterned films using vibrating sample magnetometry.
The magnetic easy axis hysteresis loops of SAF films that are treated with different oxidation times are shown in Fig. S1a. Here, the plasma oxidation is carried out using reactive ion etching (RIE) at an oxygen pressure of 50 mTorr and an RF bias power of 100 W.   To understand the domain wall (DW) dynamics in the thermally activated regime in which DW motion is dominated by thermal fluctuations, the depinning probabilities for current-and field-driven DW motion are measured inside the FM and SAF regions. Single current or magnetic easy axis field pulses are applied. The DW depinning is monitored by magneto-optical Kerr microscopy. Each depinning process is repeated 10 times. Solid lines represent fits to the depinning probability curves. Here, the depinning threshold current density , and field ( = and ) correspond to = 0.5 at , and , respectively 2,3 .

Supplementary Note 6. Current-driven DW injection (FM  SAF)
probability at FM-SAF junction Figure  respectively. Here we assume that the DW width is much smaller than both and .
When, in the presence of a uniform field along the easy axis, a domain wall with ↑↓ configuration is located at = in the FM region (− < < 0), the energy can be written as On the other hand, when a DW with ↑↓ in the lower layer and ↓↑ in the upper layer is sitting in the SAF region (0 < < ), the energy in the SAF is given by Hence, the total energy landscapes for the cases that the DW is located in the FM and SAF regions are, respectively, Eqs. (S4) show that, when > 0, the slopes of energy landscapes vs. for the cases that the and are the gyromagnetic ratio, the effective magnetic field, the random thermal fluctuating magnetic field, and the Gilbert damping, respectively.
includes the external field, the DMI field, the exchange field, the dipolar field, and the anisotropy field. In contrast, when = 0 (case II. DW is at the FM-SAF boundary), the situation is distinct from the others: there is effectively no DW (see the net magnetic moment landscape in case II in Fig. S9). This is a very special case, thus corresponding to a singular point at which the whole wire is nothing but a single domain due to the fact that the upper moment is larger than the lower moment in the SAF region. This suggests that any direction Let us assume that the DW cross sectional area in the upper layer is = cot where is the distance between the junction endpoint and the DW (see Fig. S10a).
As the DW propagates (i.e. increases) reaching = tan in the tilted junction region, Consequently, we find that = cot thereby showing that the larger is the smaller is . This means that the gradient of DW volume to be nucleated decreases with increasing , as the DW propagates in the junction region, consequently making it easier

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for the DW to propagate at larger . This is clearly shown in experiments as shown in Fig.   S10b. At = 60°, the DW can be initialized by following the shape of junction in the presence of external field (DW initialization). However, the DW relaxation is observed and the shape of

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By taking advantage of the large difference in coercivity and spin-flop field between the FM and SAF regions, DWs can be readily and reliably created by applying an external field.
The field | | is chosen to be | |(≈ 0.1 kOe) < | | < | |(≈ 3 kOe) such that a DW is nucleated in the FM region. The created DW is injected into the SAF region by current pulses. By repeating the DW nucleation and injection cycle, we demonstrated multi-DW injection, as shown in Fig. S11a. Each cycle for DW injection consists of a field pulse of ± 20 mT for 2 sec and a current pulse of 0.76 × 10 A cm for 10 ns (see Fig. S11b). An external field is set to zero when the current pulse is applied. Consequently, eighteen DWs can be a. When the DW is observed to be clearly injected across the interface from Kerr microscope, we are sure that the DW is injected since the DW has been displaced by more than the Kerr microscope resolution.
b. When the DW is observed not to be injected by current pulse from Kerr microscope, an easy axis magnetic field is applied to check if the DW is actually injected across the interface. First, let us consider a DW that is initially in the FM region at the FM-SAF junction. If the DW was successfully injected across the interface so that it moves to the SAF region, the application of a negative (− < 0.4 kOe: propagation field in the SAF region) would move the DW from the left to the right since the total DW configuration in the SAF region is ↓↑ (Note that the upper layer moment is larger than the lower layer). On the other hand, if the DW fails to be injected so that it still sits in the FM region, a negative (− < 30 Oe: propagation field in the FM region) would move the DW from the right to the left. Consequently, the DW moving direction by a negative is opposite depending on whether the DW is successfully injected across the interface or not, from which we can rigorously judge the DW injection. Since the propagation field in the SAF region (~0.4 kOe) is significantly larger than the FM region (~30 Oe), the application of in-between two propagation fields, i.e., 30 Oe < − < 0.4 kOe can also tell whether the DW is injected across the interface. Fig. S12b describes this method to judge the DW injection across the interface.
c. The same protocol as above can be applied to the case that the DW is injected from the SAF into the FM region.

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Supplementary Note 13. The energy barriers for field and current-driven DW motion Figure  Let us clarify the field-, the STT-and SOT-driven DW motions. The STT and SOT are generated by current. The DW angle corresponds to the azimuthal angle since the polar angle = in the middle of DW. Importantly, note that the energy landscape and barrier is a function of DW position solely by definition as shown in Fig. S13 above.

a. Field-driven DW motion
The reviewer is right that in the steady state the DW angle does not change but the DW position defined by changes by an easy axis component of field. For the FM or SAF case, an easy axis field that is larger than a threshold value can displace DWs by tilting the DW energy landscape and thereby overcoming the local energy barrier (see Fig. S13a below, and note that field components perpendicular to the easy axis do not displace the DWs since they do not affect the energy landscapes). The key point here is that the global energy landscape is uniformly tilted over the whole FM or SAF, which just allows the DW to get out of the local energy barrier. This means that the global energy landscape does not give rise to an additional energy barrier in addition to the local barrier.
In sharp contrast, for the FM-SAF junction the easy axis field fundamentally modifies the global energy landscape, thus adding a global energy barrier to the local barrier, as shown in To evaluate the shape of the magnetic field pulse, a Hall probe sensor combined with a digital oscilloscope were employed to measure the pulse shape resulting from the Hall voltage in response to the magnetic field changes. As shown in Fig. S14a, the shape of the field pulses with different pulse lengths exhibit a rise and fall time, each of ~ 1 ms, and a pulse duration corresponding to the nominal input voltage pulse length of from 10 to 100 ms (see arrows).
Contrary to the nominal input pulse length, the actual magnetic field pulse length is slightly longer. For instance, the 10 ms nominal pulse length has a measured length of ~16 ms, as shown in Fig. S14b. We assume here that the current density randomly fluctuates with a magnitude of with respect to the applied current density , and independently of each other over a time scale of . In typical metallic systems, the Drude model shows that a temporal fluctuation of decays within ~10 − 10 sec that corresponds to the electron scattering time. The averaged magnitude is derived from the spectral density function that is given by where ( ) is the electrical conductivity at frequency , is the Boltzmann constant, is the temperature, and is the volume of sample. If a DW is depinned within the time ∆ that is much larger than , we obtain the average magnitude by integrating Eq. S1 with respect to , for the interval 0 < < /∆ as follows: where is the d.c. conductivity ( = 0). Using = 300 K, ~1.28 × 10 Ω m , DW volume ~2.5 × 10 nm and ∆ = 1 ns, we obtain ≅ 4.5 × 10 A cm ⁄ . These values are much smaller than ~10 A/cm 2 by more than two orders of magnitude (i = FM, SAF, and FM  SAF), thus showing that the DW depinning induced by thermally driven current density fluctuations is negligible.