Supplementary Figures

Transcription

1 Supplementary Figure Supplementary Figure S1: Extraction of the SOF. The tandard deviation of meaured V xy at aturated tate (between 2.4 ka/m and 12 ka/m), V 2 d Vxy( H, j, hm ) Vxy( H, j, hm ) 2. The minimum of V d occur at 72.8 A/m in thi cae, which i determined a the cancellation field. The tolerance i choen to be the 3dB point, where the V d i twice higher than the minimum. The inet how emplary curve of V xy after different cancellation field.

2 Supplementary Figure S2: Schematic of field orientation. The magnetization i reoriented by an ternal field H, an aniotropy field H a, an current induced Oerted field demagnetizing field H h, a SOF a / σ d Oerted, a SOT effective field b / σ m, a well a a. The inet how how the coordinate i hifted o that z-axi i along the total in-plane field direction for the convenience of calculation.

3 Supplementary Figure S3: SOT induced ignal. (a) Meaured anomalou Hall effect with an ternal field applied perpendicular to the film plane. (b) Etimated anomalou Hall voltage due to the out-of-plane magnetization reorientation baed on the meaured anomalou Hall reitance. Since our trapolation ue the addition of the voltage between poitive and negative current, the calculated SOT induced AHE voltage i doubled baed on Supplementary Equation (S1). (c) With a 72.8 A/m cancellation field, the econd order PHE voltage curve (red) till ha a 5 V offet, which may be due to the SOT induced AHE voltage. The black curve in the inet i the econd order PHE voltage before the cancellation. The cancellation reduce the voltage by about 5 time. (d) Meaured thermal voltage of Ni 8 Fe 2 (2)/Pt(5). The coercivity i maller than other meaured reult due to the high temperature.

4 Supplementary Note 1. The pin-orbital interaction driven magnetization reorientation The magnetization dynamic driven by pin-orbital interaction follow the generalized Landau- Lifhitz-Gilbert equation. d m/ dt mheff m dm / dt amσ bm σm, (S1) where i the gyromagnetic ratio, H eff i the total effective field including the ternal field H, aniotropy field H and the Oerted field a h Oeted generated by the current, i the damping, σ i the unit vector that i in-plane and orthogonal to the electric current, a and b correpond to the effective field and torque induced from the pin-orbital interaction (SOI), repectively. For a tationary olution, the time varying term in Supplementary Equation (S1) vanihe, leading to m Here H / b/ mσ m eff a. (S2) a /σ i the field-like term and b /σ m i the Slonczewki torque-like term generated by the SOI. A will be hown later, the SOI induced effective field and torque are much weaker than the applied ternal field. The magnetization reorientation due to the SOI can be conidered a a perturbation to the magnetization that i aligned by the total effective field. It i convenient to conider the contribution of the Slonczewki torque-like term eparately a a perturbation. Without the Slonczewki torque-like term, Supplementary Equation (S2) ha a trivial tationary olution that the magnetization m i aligned parallel to the total effective field Heff a /σ. When the Slonczewki torque-like term i conidered, one can write the magnetization reorientation a m m m, where m m under the firt order approximation. There will alo be a change to the total effective field due to the aniotropy and demagnetizing field

5 H a H ax H ay, higher order term m, H az. Supplementary Equation (S2) can be recontructed by neglecting H σ m H b/ m σ m eff a. (S3) / a We can change the coordinate that m and Heff a /σ are in +z direction. After defining σ { co,,in } and H Heff a /σ, we can pand Supplementary Equation (S3) into Hmy H ay b / co. (S4) Hmx H ax The econd part of Supplementary Equation (S4) i independent of the torque term, which ha a trivial olution of m x. Conidering a perpendicular urface aniotropy and demagnetizing field in the y-direction, H ay can be written a H ay 2K M M m y, where K i the effective urface aniotropy energy. Therefore, the olution to the firt part of Supplementary Equation (S4) i m y b / co H M 2K / M. (S5) Thi olution can be undertood in a imple picture, in which the torque term can be viewed a an effective field in the direction of m σ, which i orthogonal to the film plane. Driven by thi field, magnetization will tilt out of plane. The derivation from Supplementary Equation (S2) to Supplementary Equation (S3) i by neglecting the econd order and higher order term. Here we take more precaution in the derivation by keeping the econd order term and neglecting the third order and higher order term. Therefore, Supplementary Equation (S3) can be rewritten a

6 Hmy H ay b / co H ax mx H b / m in. (S6) y H aymx H axmy b / mx co The firt and third equation in Supplementary Equation (S6) lead to imilar reult a hown in Supplementary Equation (S5). The in-plane magnetization reorientation m x can be derived from the econd equation in Supplementary Equation (S6): m x 2K b / inm y, (S7) / M H ax where 2K m in a uniaxial aniotropy ytem. It can be undertood from ax / M H ax / x Supplementary Equation (S7) that in mot of cae mx my, when b/ H 2Kax / M. When 2 K M H i atified, m x may be larger than m y indicating that the torque term can ax / alo induce a izable magnetization in-plane reorientation in thi pecial ituation. However, we note that the major difference between the effective field induced m x and the Slonczewki torque induced m x a decribed by Supplementary Equation (S7) i that the latter i quadratically proportional to the applied current. Therefore, by taking the econd order tranvere voltage a indicated in Eq. (2), the contribution of Slonczewki torque-like term to the in-plane magnetization reorientation i removed. The above reult are verified by numerically olving Supplementary Equation (S1) in time domain. Moreover, in the periment, we attempt to align the eay axi of the ample along the ternal field direction to minimize the K ax /M term in Supplementary Equation (S7). We alo align the current along the ternal field direction (cept for pin-orbital interaction induced witching periment) and carry out the analyi at aturated regime in order to minimize in in Supplementary Equation (S7). Baed on the dicuion above, we conclude that in our detection method, the firt order contribution of the

7 Slonczewki torque-like term to the in-plane magnetization i to tilt the magnetization out of plane. It effect on the in-plane magnetization reorientation i negligible. The in-plane magnetization reorientation that i linearly depending on the applied current hall be attributed to the current induced effective field term due to the SOI and the Oerted field. The derivation above i baed on the aumption that the pin-orbital effective torque i mall compared with the torque produced by the ternal magnetic field. Here we how the validity of thi aumption. In a ample of Ni 8 Fe 2 (2)/ Pt(5), when applying a current of 5 ma through the 1 mm wide Hall bar, the current denity through Pt i no more than 1 1 A/m 2. Auming the pin Hall angle of Pt i about.7 (ref. 45), the SOT term can be calculated a b jc SH e M d 2 115A/ m. Thi i equivalent to a perpendicular magnetic field with magnitude of 115 A/m applied to the film. The ame current generate an Oerted field of 25 A/m. The maximum current induced effective field detected in our periment (cept for current induced witching) i below 72.8 A/m. In the traction of current induced effective field, we conider the ternal field regime between 2.4 ka/m and 12 ka/m, where the magnetization i well aturated. The out-of-plane magnetization reorientation angle can be calculated to be H b M eff.15, where M eff =.55 T i determined by the anomalou Hall effect meaurement and will be hown later. The in-plane magnetization reorientation angle i h etimated to be I.6 with the current induced effective field h I of 25 A/m and H h I H 1.7 with h I = 72.8 A/m under a 2.4 ka/m ternal field in the z-direction. Therefore, it i reaonable to treat the current induced magnetization reorientation a a mall perturbation.

8 Supplementary Note 2. Planar Hall effect (PHE) voltage profile due to the pin-orbital effective field (SOF) When the in-plane aniotropy i much maller than the ternal magnetic field, the magnetization can be conidered to be aligned approximately in the direction of the ternal magnetic field. The in-plane reorientation due to the field-like term induced by the current hi a / hoerted can be calculated a tan 1 hi h H eff m, where H eff i the uperpoition of the ternal field and aniotropy field and h m i the applied cancellation field. Therefore, the meaured econd order PHE voltage i proportional to in co, which decay with field H eff a oberved perimentally. When the eay axi i aligned along the ternal field direction and the magnetization i nearly aturated, the meaured voltage ignal can be approximately written a V PH 2 hi jw H hm H an, (S8) which plain the typical econd order tranvere voltage profile hown in the inet of Supplementary Figure S1. Here we addre ome potential artifact that may affect our concluion: 1. Mialignment of the ternal field and hielding of the earth field. We take precaution when aligning the ternal magnetic field with repect to the Hall bar ued in the meaurement. However, there may till be a max = 2 in-plane mialignment in the periment. The econd order planar Hall voltage we detect hall be recontructed a V PH V PH ( j, h 2 jwco 2 H m, H ) V PH hi hm co H ( j, h an an m, H ) hi hm H in hi hm H in jwco 2 jwco 2. (S9) H co H H co H an

9 We note that in the derivation, we have made a reaonable approximation that the tranvere field hi hm H in i much maller than the ternal field H co H. an Supplementary Equation (S9) differ from Supplementary Equation (S8) only by a mall amount. Therefore, we conclude that the in-plane mialignment of the ternal field hall not affect our traction. The ame concluion applie to any poible out-of-plane mialignment of the ternal field, which hall be undertood in the imilar fahion. In thi periment, no particular care i taken to hield the earth magnetic field for the imilar reaon. Becaue we are detecting the econd order tranvere voltage, the effect of the mialignment and mall contant tray field can be neglected. 2. Proximity effect induced magnetim in Pt. It ha been recently reported that Pt may become magnetic in adjacent to a magnetic material 46. Thi i one of the reaon why we ue Cu a a pacer layer in between Ni 8 Fe 2 and Pt in mot of our periment, which hall dramatically reduce the proximity effect if not completely eliminate that. However, even if Pt indeed become magnetic, it hall be conidered a a part of Ni 8 Fe 2, which cannot plain why the effective field can be much larger than the maximum poible Oerted field. It may have ome quantitative ramification on the effective thickne of Ni 8 Fe 2 and the interface pin-orbital interaction, which will not affect our main concluion. 3. Surface aniotropy and current induced aniotropy. The urface aniotropy of Ni 8 Fe 2 depend on it thickne and the material in the adjacent layer. However, even in the ample with the larget urface aniotropy, Ni 8 Fe 2 (2)/Pt(5), the effective demagnetizing field i till a large a 44 ka/m, which i much larger than the maximum ternal field 12 ka/m. The perpendicular urface aniotropy doe not affect the in-plane

10 magnetization reorientation, and therefore ha no influence on the planar Hall voltage. However, the urface aniotropy doe affect the anomalou Hall voltage, which depend on the out-of-plane magnetization reorientation. Thi i a mall effect, which will be addreed in the Supplementary Note 3. The current may induce heat and change the temperature, leading to a change of the aniotropy field. However, the periment i done at a relatively low current denity (< 1 1 A/m 2 ), where no apparent change of temperature i oberved. It i alo reported lately that a large electric field may modify the aniotropy of an ultra-thin magnetic film. However, it i not clear whether electric current can directly modify the aniotropy of the magnetic film. We would like to point out that any type of aniotropy term due to current will appear in the denominator in Supplementary Equation (S8). The effect of the current induced aniotropy change cannot account for the ternal field dependence of econd order planar Hall voltage a oberved in the inet of Supplementary Figure S1. 4. Domain tructure and edge tate. The derivation of magnetization reorientation due to Slonczewki torque-like term and current induced effective field i carried out uing macro pin model. However, when domain tructure form, it i poible the Slonczewki torque-like term may alo contribute to in-plane magnetization reorientation. Thi i the reaon the traction i carried out at an ternal field higher than 2.4 ka/m, where Ni 8 Fe 2 i well aturated. Since the citation of current i alo uniform, we pect macro-pin model to be ufficient to decribe thi mall perturbation to a aturated tate. In addition, due to the demagnetization at edge and corner, certain edge domain tate may perit at field higher than 2.4 ka/m. However, ince the ample ued for the SOF characterization i a large a 1mm x 1 cm, we pect the contribution of edge tate to be

11 negligibly mall. In fact, the tracted current induced field in NiFe(2)/Cu(x) a hown in Fig. 2(b) aymptotically approache 25 A/m a the Cu thickne increae. Thi agree well with the theoretical prediction baed on Ampere Law and therefore upport the accuracy of our propoed perimental technique. Supplementary Note 3. Anomalou Hall effect (AHE) voltage profile due to the pin-orbital torque (SOT) A decribed in Supplementary Note 1, the Slonczewki torque-like term (for generality, we hereafter call it pin-orbital torque, SOT) can tilt magnetization out of plane and give rie to an anomalou Hall effect. In order to quantify the magnitude of thi AHE induced voltage, we meaured the anomalou Hall reitance of a repreentative ample Ni 8 Fe 2 (2)/Pt(5) (Supplementary Fig. S3 (a)). The magnetization i aturated out-of-plane in an ternal field of about 44 ka/m. The value i ignificantly lower than M (1T) of Ni 8 Fe 2 due to the itence of the urface aniotropy, which i etimated to be K J / m. The anomalou Hall reitance i about R AH Uing the ame parameter etimated in Supplementary Note b 1, a 5 ma bia current can generate a SOT with magnitude of ~ 115A/ m, which reult in an anomalou Hall voltage due to the out-of-plane magnetization reorientation V AH H IRAHbco M 2K / M, (S1) with it magnitude imulated in Supplementary Figure S3 (b). Although thi value i much maller than the PHE voltage due to the field-like term, uch Hall-like voltage curve can till be

12 oberved in Supplementary Figure S3 (c) after the cancellation of the current induced effective field. Another poible voltage ignal i the thermal voltage from the anomalou Nernt effect 47 and the pin Seebeck effect 48, which have the ame profile a the anomalou Hall effect. Thi thermal voltage i due to the perpendicular temperature gradient generated by paing current through the metallic film. However, we performed a calibration of thermal voltage by placing a.5 W heater on top of a continuou film with dimenion of 4 mm x 2 mm and a thermal voltage of 1.5 µv i obtained along the long axi a hown in Supplementary Figure S3 (d). In our econd order tranvere voltage meaurement, the Hall bar ha a dimenion of 1 cm x 1 mm. A 5 ma current through the film can generate a power of.75 W. Since we meaure the voltage along the hort axi, the thermal voltage can be etimated to be.45 µv, which i negligible in our calculation. Supplementary Reference 45. Liu, L., Buhrman R.A., and Ralph D. C., Review and Analyi of Meaurement of the Spin Hall Effect in Platinum. Preprint at (211). 46. Huang, S. Y. et al., Tranport Magnetic Proximity Effect in Platinum. Phyical Review Letter 19, 1724 (212). 47. Huang, S. Y. et al., Intrinic Spin-Dependent Thermal Tranport. Phyical Review Letter 17 (21), (211). 48. Uchida, K. et al., Spin Seebeck inulator. Nature Material 9 (11), (21).