Chiral domain wall motion in unit-cell thick perpendicularly magnetized Heusler films prepared by chemical templating

Heusler alloys are a large family of compounds with complex and tunable magnetic properties, intimately connected to the atomic scale ordering of their constituent elements. We show that using a chemical templating technique of atomically ordered X′Z′ (X′ = Co; Z′ = Al, Ga, Ge, Sn) underlayers, we can achieve near bulk-like magnetic properties in tetragonally distorted Heusler films, even at room temperature. Excellent perpendicular magnetic anisotropy is found in ferrimagnetic X3Z (X = Mn; Z = Ge, Sn, Sb) films, just 1 or 2 unit-cells thick. Racetracks formed from these films sustain current-induced domain wall motion with velocities of more than 120 m s−1, at current densities up to six times lower than conventional ferromagnetic materials. We find evidence for a significant bulk chiral Dzyaloshinskii–Moriya exchange interaction, whose field strength can be systematically tuned by an order of magnitude. Our work is an important step towards practical applications of Heusler compounds for spintronic technologies.


Chiral domain wall motion in unit-cell thick perpendicularly magnetized Heusler films prepared by chemical templating
Filippou et al.

Supplementary Note 1. Chemical templating layer (CTL) candidates
The candidate materials for the CTL are listed in Supplementary Table 1. These materials have a B2 structure with a small lattice mismatch (< 7%) compared to the in-plane lattice parameter of tetragonal Mn3Sn found in [1]. Lattice constants are obtained from Pearson's crystal database.
Supplementary Table 1. List of CTL candidates with a B2 structure and a lattice constant between 2.8 and 3Å. *Experimental value obtained from X-ray diffraction (XRD) out of plane lattice constant measurement, for a tetragonal structure.

Supplementary Note 2. Chemical ordering and morphological properties of the (CoZ') CTLs
XRD measurements were carried out at room temperature using a Bruker General Arrea Detector Diffraction System (GADDS) with a Cu Kα X-ray source for the CoAl, CoGa, CoSn and CoGe CTLs that were annealed at various temperatures (TAN). The annealing duration was 30 min in all cases. XRD data for CoAl CTLs shown in Supplementary Fig. 1 show presence of both (001) and (002) peaks in all cases. The existence of the CoAl (001) superlattice peak clearly proves that there is an alternate layering of Co and Al atoms for all annealing temperatures and even without any annealing. The low intensity observed for the MgO (002) substrate peak is due to the step size limitation of the X-ray area detector. High resolution XRD measurements for the CoAl CTL grown at room temperature and without any anneal are shown in Supplementary Fig. 2. The θ-2θ scan, phi scan of MgO (202) and CoAl (101) and reciprocal space mapping of CoAl (101) were measured on a Bruker D8 discover high resolution XRD system. The data shows an excellent epitaxy of the CoAl film with the MgO substrate with in-plane relationship MgO[100]//CoAl [110].
We find no evidence of presence of a second phase in the CoAl CTL.
Another important parameter is the surface roughness of the CTLs. We find that the root-meansquare roughness (rrms), as determined by atomic force microscopy, depends strongly on TAN for certain compositions of the CoGa CTL. The films with the preferred Co53Ga47 composition show the smoothest surface (see Supplementary Fig. 5).

Supplementary Figure 5. Root-mean-square roughness versus annealing temperature for various
CoxGa1-x compositions. For the optimum composition, Co53Ga47, the films remain atomically smooth (~2 Å) even after annealing at high TAN. 8 Electron energy loss spectroscopy (EELS) line scans performed on XTEM micrographs of the CoGa CTLs display the relative concentration of various elements in the film stack with atomic resolution. In Supplementary Fig. 6 the concentration profiles for Co and Ga elements for a CoGa CTL oscillate, such that the maximum intensity of one is at a minimum intensity of the other, clearly indicating the presence of alternating atomic layers of Co and Ga. These EELS spectra were acquired with a 7C probe size (nominal current 70 pA), in 1 Å steps at 0.05-0.005 sec/pixel exposure times.

Supplementary Note 3. Mn3Z Heusler tetragonal structures
The ultra-thin Heusler films grown on the CTL, form alternating layers of Mn-Mn and Mn-Z layers. This is clearly seen in the STEM images included in Fig. 2  and green atoms correspond to Mn and Z atoms, respectively, a and c correspond to the in-plane and out of plane lattice constants, respectively, with c > a.

Supplementary Note 4. Kerr microscopy in differential mode for imaging magnetic domains and DW motion
In Supplementary Fig. 9a, we illustrate the domain wall (DW) motion observed using a Kerr microscope in differential mode. The position of the DW is indicated by a dotted line. In Supplementary Fig. 9b, the same images are overlaid with the magnetization direction. The initial magnetic state of the wire is taken at t0 and subtracted from the subsequent images taken at tn.
Application of current pulses t1 through t4 move the DW to the right and this appears as an expansion of the domain in the differential Kerr image. The DW velocity was calculated from measurements of the expansion of the magnetic domain. Figure 9. DW motion for a nanowire imaged with differential Kerr microscopy.

Supplementary
The wire is imaged from top and ↑ magnetization (dark contrast) is indicated by ⊙ while ↓ magnetization (bright contrast) is indicated by ⊗.

Supplementary Note 5. DW velocity versus for different wire widths
The DW velocity for different wire widths was measured. The DW motion is stable and identical for all wire widths. The voltage limits of our power supply were +50V and -45Vso that the width of the nanowire limited the maximum possible current density, . The highest that could be achieved in the narrowest 2 μm wide nanowires is what determines the maximum current density shown in Supplementary Fig. 10 for various nanowires of different widths. These data clearly show that the DW velocity is independent of the nanowire width. (f), in Å. Current densities, , of ~7.5 × 10 7 , ~8.6 × 10 7 and ~9.1 × 10 7 A cm −2 were used for the 7.5, 10 and 15 Å thick Mn3Sn, respectively.

Supplementary Note 9. DW velocity dependence on pulse length for Heusler films with different thicknesses
The DW velocity dependence on the Heusler film thickness for various current pulse lengths was explored. Furthermore, we compare the critical current density for DW motion, c , for current pulse lengths varying from 5 ns for Mn3Sn and 20 ns for Mn3Ge to 100 ns for all thicknesses. The decreasing resistivity with increasing Heusler layer thickness can account for this dependence. For thicker films, the DW motion becomes insensitive to the pulse length.

Supplementary Note 10. SHE dependence on the overlayers: W, Pt
The DW motion in ultra-thin Heusler can be strongly influenced by capping layers that show high Spin Hall Angle (SHA). Fig. 5e showed the DW velocity versus current density and in Supplementary Fig. 15 we present the DW velocity versus Hx and Hy. When we insert a W overlayer (blanket film resistivity ρ=173 μΩ cm) which has a negative sign SHA (see main text and Supplementary Table 3 for SH ), the spin orbit torques are enhanced and this can be seen by the high slope observed in the DW velocity versus Hx field dependence ( Supplementary Fig. 15a).
When a Pt overlayer is introduced, the SHA is of the same sign as the CoGa CTL, thus suppressing the spin orbit torques. As the Pt overlayer thickness is increased from 5 to 10 Å Pt, if we compare Supplementary Fig. 15d and f, we see that the SHA reverses. This can be observed by the change of the asymmetry in the Hy field dependence (see also Supplementary Table 3).

Supplementary Note 11. Hx dependence of DW velocity and 1D model simulations for MnxSb
The fits of the 1D model to the DW velocity data presented in Fig. 5f of the main manuscript are presented below.

Supplementary Note 13. DMI exchange field in the MnxSb nanowires
The Dzyaloshinskii-Moriya interaction (DMI) exchange field determined by 1D model fits to the Hx field dependence data shown in Supplementary Fig. 16 as a function of composition of MnxSb are summarized in Supplementary Fig. 18.

Supplementary Note 15. One-dimensional analytical model and DW motion mechanism
The one-dimensional model [2,3] provides not only insights but also a quantitative understanding of the current driven domain wall dynamics. This model has been successful in describing the current induced motion of DWs in straight racetracks formed from single magnetic layers with perpendicular magnetic anisotropy (PMA) via a chiral spin torque [4,5], and in synthetic ferromagnets and antiferromagnets via an additional exchange coupling torque [6]. Here we use this model to investigate the current driven DW motion in nanowires formed from Mn3Z Heusler alloys. We model these ferrimagnetic Heuslers by a ferromagnetic layer whose magnetization is the net magnetization of the two sub-lattices in the ferrimagnet. This is a reasonable approximation when the two sub-lattices are strongly anti-ferromagnetically coupled. Indeed, as we show below, such a model well accounts for many details of the current induced domain wall motion in our Heusler films.
Note that for better illustrating the out of plane axes and torques, we will now refer to ↑ and ↓ as ⊙ and ⊗ respectively. Therefore, a ↑↓ DW configuration will be referred to as ⊙ | ⊗ and ↓↑ as ⊗ | ⊙.
The basic assumption of our model is that the DW has a fixed magnetization profile with the magnetization rotated from the direction perpendicular to the layer, i.e. the z-axis, by a polar angle , and by the azimuthal magnetization angle which is defined in the plane of the wire with respect to the direction ̂. Note that is constant and independent of the lateral position of the DW along the wire when it is manipulated either by current or magnetic field or a combination of the two (see Supplementary Fig. 20). For the case of perpendicularly magnetized nanowires, the DW dynamics can be described within the 1D model by two parameters, namely the position q of the DW along the nanowire and its conjugate momentum 2 / . is the saturation magnetization and is the gyromagnetic ratio. The DW motion in nanowires is governed by , with the domain wall profile located at ⃗ and at time t, given by Here the upper and lower signs correspond to the ⊙ | ⊗ and ⊗ | ⊙ domain magnetic configurations, respectively, and is the domain wall width parameter.
First, we formulate the Lagrangians that include the adiabatic and non-adiabatic spin transfer torques (STTs), external field driven torques, the spin Hall current torque, and the Dzyaloshinskii-Moriya exchange field. The equations of motion are then derived by the Lagrange-Rayleigh equations [7].
With the DW profile function (1), the Lagrangian ℒ in the nanowire that contains the magnetostatic potential energy, including anisotropy, DW kinetic energy and adiabatic spintransfer torque is given by: Here the upper and lower signs correspond to the ⊙ | ⊗ and ⊗ | ⊙ magnetic DW configurations, respectively. ℇ is the magnetostatic energy density of the domain wall per unit area and is given by: ℇ = 2 sin 2 cos 2 − cos − sin cos( − ) − sin cos

Eq. (2) can be rewritten as ℒ = + ∫ [̇sin − sin ]
where is the magnetostatic potential energy, = ∫ ℰ . is the magnitude of the in-plane anisotropy field derived from the shape anisotropy of the DW that favors a Bloch wall over a Néel wall, is the out-of-plane field, and are the in-plane magnetic field and its angle with respect to the +̂ direction, respectively. is the Dzyaloshinskii-Moriya interaction exchange field at the DW whose direction is always perpendicular to the DW length direction, thereby favoring Néel type walls but its sign depends on the domain configurations establishing the chirality of the domain walls. The volume spin transfer torque due to the current flowing within the magnetic layer is parameterized by = , where is the Bohr magneton, e is the electron charge, P is the spin polarization of the current and is the current density in the magnetic layer. The dissipative function ℱ that includes damping, non-adiabatic spin-transfer torque, and spin-orbit torque is given by: Here again the upper and lower signs correspond to the ⊙ | ⊗ and ⊗ | ⊙ magnetic DW configurations, respectively, is the non-adiabatic STT coefficient and is the Gilbert damping. When the spin Hall torque is very small, i.e. ≈ 0, from eqs. (6a,b) we obtain: Eq. (6b) tells us about the steady state condition, i.e. ̇= 0, such that: Let us investigate which factors determine the width of the dome. To qualitatively understand this, let us make an aggressive approximation that sin~ for 0 ≤ ≤ 1 when . This shows that the dome width decreases with decreasing and and increasing Δ. As a result, the value of can be uniquely determined as well.
When a longitudinal field = , i.e., = 0, is applied in the presence of and a very small , the dome-like curve is shifted by − thus forming a maximum at = − since (1 + 2 )̇= ± Δ − ( + ) 2 sin , which is observed in our experiments (see Fig. 4 in main text). Note that when ≠ 0, the DW velocity is finite for all values that is different from what we observed from our experimental vs plot. Hence, we conclude that the non-adiabatic STT contribution in Mn3Z is very small.
Let us intuitively understand current-induced DW motion in Heusler alloys (see Supplementary Fig. 22-24). First, we assume that = = 0 and = < 0 or < 0 in ⊙ | ⊗ DW configuration and | | or | | are large ( Supplementary Fig. 21). In the absence of , (or ) stabilizes the Néel wall structure in an anti-clockwise chirality (see Supplementary Fig. 21a). When an electric current is applied along the −̂ direction, i.e, the conduction electrons -with positive spin polarization ( > 0) flow along ̂ direction, the volume spin-transfer torque ⃗ starts to rotate the DW magnetization ⃗⃗⃗ along +̂ direction ( Supplementary Fig. 21b). The volume STT driven motion induces a damping torque ⃗ = ⃗⃗⃗ × ⃗⃗⃗ that rotates ⃗⃗⃗ along the −̂ direction, i.e. clockwise rotation from the top view (see Supplementary Fig. 21c). This leads to a non-collinearity between ⃗⃗⃗ and ⃗ ⃗⃗ (or ⃗ ⃗⃗ ), thus giving rise to ⃗ = − ⃗⃗⃗ × ⃗ ⃗⃗ (or ⃗ = − ⃗⃗⃗ × ⃗ ⃗⃗ ). Note that ⃗ (or ⃗ ) is always along the −̂ direction, thereby compensating ⃗ and slowing down the DW motion ( Supplementary   Fig. 21d) | (see Supplementary Fig. 22 and 25). Note that the DW velocity and precession angular velocity are maximum when the magnetization in the middle of the DW is directed along the ̂direction while they are minimum when the magnetization in the middle of the DW is directed along the ̂-direction. Likewise, we can readily understand the DW motion dependence on ( Supplementary Fig. 23) and = ( Supplementary Fig. 24). The DW width is determined by the fits to the dome in ν vs ( Supplementary Fig. 28).

Supplementary Note 16. Torques to move chiral domain walls and precess domain wall magnetization in wires formed from Heusler alloys
Since out-of-plane field is not applied to perpendicularly magnetized Heusler film in our current driven DW motion experiment, Zeeman field or Zeeman-like effective induced DW motion is not taken into account. In this case, the torques that move or precess the DW are spin-transfer torque and in-plane field driven torques. First, let us consider an adiabatic STT only here. The adiabatic STT originates from total angular momentum conservation in non-uniform magnetic Please note that 7.7 nm is the upper bound for Δ. We note that due to the significant magnetization from the MgO substrates and the tiny magnetization of the Heusler material, it is very difficult to precisely determine the thickness of any dead layer but it is clear from the magnetic measurements that this is less than a fraction of a unit cell equivalent thickness. On the other hand, we can measure the anisotropy of the magnetization reliably and the Heusler films show very high anisotropy values in unit cell thick Heusler layersso large that we cannot rotate their magnetization in plane using available magnetic fields (7T) in a commercial SQUID magnetometer. This is why our paper focuses predominantly on PMA which we show we have been able to produce in single unit cell thick Heusler films for the very first time.

Supplementary Note 18. One-dimensional model simulation of − curves
We reproduce − curve for Mn3Sb (Fig. 3e) with 1D analytical model as shown in Supplementary Fig. 29. Although non-adiabatic spin-transfer-torque parameter is set to be zero in our simulation, the intrinsic pinning is small since and DM are small. In addition, a finite SOT gives rise to = 0 which is apparently far from what we observe from the experiment.
Hence, a finite c is due to the extrinsic pinning that can be induced by inhomogeneous anisotropy or other reasons. Here to emulate the extrinsic pinning, periodic pinning potential is used (see the details in ref. 5 and 8). Note that the fitted pinning potential for 10 Å Mn3Sb is significantly smaller than that for 20 Å Mn3Sb. That could infer that Mn3Sb becomes rougher with the increasing thickness thus increasing the anisotropy inhomogeneity.

Supplementary Table 2.
List of fitting parameters for the fits in Fig. 4 and Fig. 5a-d of the main text and fits to data in Supplementary Fig. 11-13. The corresponding fits of Supplementary Fig.   13, are plotted in Supplementary Fig. 26