A spin torque meter with magnetic facet domains

Current-induced magnetic domain wall (DW) motion is an important operating principle of spintronic devices. Injected current generates spin torques (STs) on the DWs in two ways. One is the spin transfer from magnetic domains to the walls by the current flowing in the magnet. Current flow in attached heavy metals also generates another ST because of the spin-Hall effect. Both phenomena explain the wall motions well; therefore, their respective contribution is an important issue. Here, we show the simultaneous measurement of both torques by using magnetic facet domains that form mountain-shaped domains with straight walls. When the STs and the external magnetic field push the walls in opposite directions, the walls should have equilibrium angles to create balanced states. Such angles can be modulated by an additional in-plane magnetic field. Angle measurements distinguish the STs because each torque has a distinct mechanism related to the DW structure.


: Facet formation
Supplementary Equation (1) is valid for magnetization dynamics of perfect samples (without pinning). Typical samples have many pinning sites due to sample irregularity that drastically reduces the speed of DW motions known as the DW creep 3,12 . However, non-zero field still produces non-zero speed in the creep regime 3,12 . Thus, all the driving force should be cancelled to stop DWs. Such conditions are easily obtained by inserting ̇= 0 in Supplementary Equation (1). This means that a cross product of total field and m should be zero as follows: Applying the current (I) and perpendicular field (H z ) makes the facet shown in Fig. 2. The DW always has pure in-plane magnetization at the DW centre that produces nonzero torque due to H z . To compensate this effect, ST should produce -H z . The adiabatic SMT and the field-like SOT term do not produce perpendicular field, thus only the non-adiabatic SMT and the damping-like SOT cancel H z . These two terms have clear angular dependences explained in the main manuscript. The DW tilting changes the magnetization gradient along the current direction. Tilted DW magnetization also changes the × value because is transverse (+x or -x in this paper) to the current direction (+y). We can define the strength of SMT-induced perpendicular field (|H SMT |) as ( 0 ∆ 0 ⁄ ), where ∆ 0 is the DW width. The strength of the SOT-induced perpendicular field (|H SOT |) is (π 2 ⁄ ) d . Here, π 2 ⁄ is required due to integration over the DW. Note that, to determine the sign of H SMT and H SOT , the domain polarity (s) and the DMI field (H DMI ) 12 The SMT-only facet has a stabilized angle without the in-plane field: Here, is the DW angle. This equation is split into left and right sides with H x because H x generate opposite at each side. The first order of approximation equation at each side of the facet is: The left side of above two equations should have the same value; thus, sin R ] cos L .
Recent research on the SMT shows that the change of the SMT is similar to the change of the wall. But the variation of SMT is much more than expected from the wall width changes. They explained the origin is that the β depends on the wall width [16][17][18] . Thus, we replace H K with H K * in Supplementary Equation (6) as follows.
Here, H K * is an effective anisotropy representing effects of the wall width variation as well as β variation.
SOT-induced facet tilting has to consider the DW magnetization. Without H x , the DW magnetization angle () is stabilized by H SOT : Due to H DMI ,  should have the same value as . However, applied H x breaks this situation. The projection field of H x and H DMI on the transverse direction of DW magnetization should balance out at each side of the facet as follows: Elimination of  leaves: This is the SMT-only facet-tilting equation.
Note that these tilting equations are quite simplified. The adiabatic SMT and the field-like SOT act as additional in-plane fields that can make an offset in the facet-tilting equation. However, in our experiments, we observed no significant offsets in the facet-tilting experiments (significant offsets are observed in other samples, discuss later). We believe that the induced field by the adiabatic SMT and the field-like SOT is negligibly small Finally, if we ignore the offset of tilting and we measure small change of  -, replacing sin -with Δ -make a more simple equation used in the main text. The difference between sin(x) and x is not larger than 5 % up to x=0.5.

: Facet sharpening
Similar to the facet-tilting equation, we can derive the facet-sharpening equation. The sharpening does not distinguish left and right sides of the facet. Thus, we only consider the left side of the facet. In the case of the SMT-only facet, applied H y expands the DW width and then the facet angle equation is: We use H K * than H K to include the effect of β variation on the wall width [16][17][18] . We divided Supplementary Equations (11) by Supplementary Equation (3) and replaced L ( ) with  ( 0 ). Then, This is the facet-sharpening equation of SMT only case. In addition, assuming | −  0 | ≪ 1 results in following asymptotic equation.
It is notable that facet observation is easy with small H z because larger H z induces more nucleation in the samples that erases the facet domains. Thus, most of observable facet has angle near  ~90˚ As a result, the effect of SMT in sharpening equation is much smaller than that of in the tilting equation.
The facet sharpening equation for SOT only case is also obtained. Projected fields of H y and H DMI on the transverse direction of DW magnetization should balance out that requires, We replaced L ( ) with  ( 0 ) then, Here, ψ 0 is the domain wall magnetization angle at H y =0. C 1 is a constant and is expected within the range of 0~1. Micromagnetic simulations find 0.5 is good for C 1 ( Supplementary Information 7). Combining Supplementary Equations (14) and (16) (16), with C 1 =0.5. Note that we used H DMI , and H K (=2K eff /M S ). We know that the domain wall magnetization is fixed by

: Temperature and the Oersted field problem
The electric current needed to generate sufficient spin-torque effects requires current density larger than 10 9~1 0 10 A m -2 . Thus, flowing such current density in the film structure (~1 mm in width, ~10 nm in thickness) requires several hundred mA of total current. This current heats up the sample as well as the sample stage. The stabilized temperature should be higher than room temperature. For example, film that is 3 mm in width and 0.6 mm in length shows clear facet domains with 0.29 A of total current, but with a sample temperature of ~370 K.
This current is sufficient for changing the material parameters.
To remove this heating effect, it is good to reduce the sample size to decrease the total current; however, the small size should generate significant Oersted field distribution. Typically, current density ~10 10 A m -2 , sample width ~1 mm, and total thickness ~10 nm induces a perpendicular field gradient ~1 Oe mm -1 along the sample width near the sample centre. If the Oersted field gradient is comparable to the spin-torque fields, we can see gradual angle variation of the facets. To reduce this problem, it is advantageous to increase the sample width.
Therefore, we have to tune the sample size. In this paper, we select a sample size of 1 mm in width and 0.2 mm in length. This sample shows facet formation with 0.15 A of current. The total current becomes almost half that of film 3 mm in width and 0.6 mm in length, so the temperature increase is reduced by ~1/4. As a result, we performed all experiments under the sample temperature of ~320 K. We think that this temperature increase does not induce a significant difference from room temperature (see the next section) Also, we only observe a small area (<100 m×100 m) near the centre of the film to minimize the Oersted field effect (<0.1 Oe).
Note that the Oersted field also has a transverse component (x-directional field by y-directional current) which does not exceed several tens of Oersted by ~10 10 A m -2 current density. However, the magnetic layer is placed in almost the centre of the film stack that cancels the transverse component of the Oersted field. Moreover, tilting and sharpening are observed with the ~kOe in-plane field. Therefore, the in-plane component of the Oersted field is negligible in our experiments.

: Comparison with other methods
We check our results through a comparison with other methods. In the wire structure, magnetic field-induced domain wall motions with small electric current are a well-known method to extract the pure current effect 16 .
The domain wall speed is mainly determined by the magnetic field, but the current makes a small speed deviation. We fabricate a wire 30 m in width and select the current density to be 7.1×10 9 A m -2 . The total current is sufficiently small and does not induce a meaningful temperature increase (<2 K) 16 . Supplementary   Figure 5a shows the wire structure and measured domain wall speeds with +I  /H K )). We plot these dependences by converting each spin-torque field to each ε in Supplementary Fig. 5b. Then, ε ST (=ε SOT +ε SMT ) has a similar slope on H y with measured ε in the wire.

: Other sample 1: Sub/Ta(3 nm)/Pt(3 nm)/CoFeB(0.9 nm)/Pt(0.6 nm)/MgO(1.5 nm)/Ta(2 nm)/Pt(1.5 nm)
The sample has a thicker insertion Pt layer (0.6 nm) than that of the sample in the main script (0.4 nm). The film is patterned to 1 mm in width and 0.2 mm in length. The sample has an additional capping layer of Pt 1.5 nm to improve electrode contacts, which drastically reduces the total resistance. As a result, 0.3 A of current shows clear facet domains at a sample temperature of ~320 K.
The film shows perpendicular magnetization and stripe growth behaviour. Such stripe growth shows strong asymmetry in domain growth. Supplementary Figure 6a-c shows the typical stripe growth. We saturate the magnetization in the -z direction and apply 10 Oe of H z . Next, the stripe domains grow from the nucleation site up to 20 s. After growth, -10 Oe of H z are applied to compress the expanded domain. However, the boundary of the stripe domain does not go back to the initial nucleation site, and only the relative proportion of +z and -z magnetization are changed in the striped domain (insets of Supplementary Fig. 6b,c). Therefore, we can expect that there is a magnetic field on the boundary of the striped domain for converting the uniform magnetization to the striped domain state. Here, we call this field a stripe field, H stripe . The insets of Supplementary Fig. 6a schematically show where H stripe acts as a force and the direction of the field. We think that the main cause of  Supplementary Fig. 7d).

: Other sample 2: Sub/Ta(3 nm)/Pt(3 nm)/CoFeB(0.9 nm)/MgO(1.5 nm)/Ta(2 nm)
We also perform facet experiments with Sub/Ta (3 nm Nevertheless, the results are interesting because the facet tilting is not explained by the SOT effect. Supplementary Figure 9c shows the tilting results of the facets. The applied +I and + H z should make a positive slope from the SOT effect, but the results exhibit a negative slope, which implies that the amount of (π/2)H SMT /H K * overcomes H SOT /H DMI and that H SMT is meaningful. If H DMI has a value between 3 kOe and 5 kOe,    The application of a nonzero in-plane field (H x and H y ) complicates the problem because the in-plane fields tilt the DW magnetization angles. In addition, both the field and the thermal activation should change the relative portion of φ + χ and φχ states because of the relative change in potential energy depths; however, these problems are beyond our scope. Here we show a crude model including magnetization tilting and relative portion change to recommend directions for future work. The roughness of domain walls should also be taken into account, but here we ignore this effect.
First, we think about the SOT effect. Supplementary Figure 11 shows an example of domain wall magnetization Next, a projection of H y on the domain wall produces H y cosψ, which increases the energy depth of the φ + χ magnetization state. A simple Arrhenius assumption leads to a relative portion of the φ ± χ state as e (Eφ±χ/kT) /{e (Eφ+χ/kT) + e (Eφ-χ/kT) }, where E φ±χ is the energy depth of the φ ± χ state and kT is the thermal energy.
Under the zero in-plane field, E φ+χ and E φ-χ are equal (=E 0 ) and the relative portion is 1/2 for each magnetization state. If the applied in-field strength is small enough, the relative portion goes to (1+ζ φ±χ )/(2+ζ φ+χ +ζ φ-χ ) with E φ±χ /kT=E 0 /kT +ζ φ±χ . As a result, the averaged total SOT field is H SOT (=0)×[{cos(φ+χ)(1+ζ φ+χ )+cos(φ-χ)(1+ζ φ- where ζ is proportional to the parallel component of the in-plane field on the domain wall magnetization; however, ζ is a non-dimensional parameter, meaning that an additional factor (r T ) is required to convert the field to ζ. Moreover, r T should be determined by M S , wall width, kT, magnetic layer thickness, and an independent segment length of the domain wall. If we obtain a variation of the SOT field as a function of the in-plane magnetic field, we obtain the effect of the relative portion change of the φ ± χ states on the facet equations.
After some calculations and simplifications, we get the first order facet tilting and sharpening equations for the SOT-only case. Here, we ignore the domain tilting. On the right side of the above equations, the first coefficient of H x and H y comes from magnetization tilting and the second term from the change of the relative portion of φ + χ and φχ states.
A similar description is possible for the φ=±90° states (Supplementary Note 5). Supplementary Equation (18a) will be changed as follows. Here, we ignore the domain tilting. For the SMT-only facet, we think about the domain wall type (Bloch or Néel) and their wall width differences.
The wall width ∆ w depends on the θ w (θ w =0 or 180° for a Bloch wall and θ w =90 or 270° for a Néel wall), such as ∆ w (θ w )=∆ B +∆ BN cos 2 θ w with a Bloch wall width (∆ B ) and a difference in widths of the Bloch and Néel walls (∆ BN ). At a zero in-plane field, the wall width (∆ 0 ) is ∆ w (χ). The application of small in-plane fields produces small tilting of θ w , which results in wall width change. If we average the wall width variation, including the ψ = φχ situation, we obtain the following equations for the SMT-only case. In this case, the changes in relative portions of the φ + χ and φχ states owing to thermal activation do not produce a first-order contribution. This SMT-only facet also has the same forms of equations as the large DMI case if we introduce an effective anisotropy field ̃ as follows.
̃ = π∆ 0 S 2∆ sin 2 sin (25) It should be noted that we know SMT strength depends on wall width, but so does ; thus, ̃ should be replaced with ̃ * .
It is well known that the Sub/Ta (5 nm Fig. 12b). In Supplementary   Fig. 12c, H  Employing our crude model, we get SOT,0 / ̃D MI =-0.007±0.0003. Supplementary Figure 12d and e shows the tilting data with no meaningful angle variations, but sharpening shows a clear angle variation. Therefore, we expect that the SOT on the facet tilting is almost compensated by the SMT effect ( SOT,0 / ̃D MI + SMT,0 /