Angular and temperature dependence of current induced spin-orbit effective fields in Ta/CoFeB/MgO nanowires

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1 Supplementary Information Angular and temperature dependence of current induced spin-orbit effective fields in Ta/CoFeB/MgO nanowires Xuepeng Qiu 1, Praveen Deorani 1, Kulothungasagaran Narayanapillai 1, Ki-Seung ee,3, Kyung- Jin ee,3,4, yun-woo ee 5 & yunsoo Yang 1 1 Department of Electrical and Computer Engineering, National University of Singapore, , Singapore Department of Materials Science and Engineering, Korea University, Seoul , Korea 3 Spin Convergence Research Center, Korea Institute of Science and Technology, Seoul , Korea 4 KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul , Korea 5 PCTP and Department of Physics, Pohang University of Science and Technology, Kyungbuk , Korea S1. Second harmonic component V ƒ of the all voltage V With an ac current with a frequency ƒ applied through the nanowire, the magnetization oscillation can be decomposed and characterized by and. Considering both anomalous and planar all effects (AE and PE, respectively), the all voltage V can be written as 1

2 V V V I sintr sin( sin t) AE PE ac AE I tr t t ac sin PE cos sin sin sin where f...., one can get By using Taylor expansion, f x δ f x fx fx sin( sin t) sin cossin t.... Therefore, V AE can be written as V I sintr sin( sin t) I sin tr (sin cossin t) AE ac AE ac AE IacRAE sinsint IacRAE cos sin t IacRAE cos IRAE cos IacRAE sin sint cos t The second harmonic term of V AE is. V f, AE IacRAE cos (S1.1) get When magnetic field is applied in x-z plane ( ), by using Taylor expansion, one can sin t cos sintcos sinsintcos.... sin sint sin sint sin sint cos sint sint Therefore, V PE can be written as V I tr t t PE ac sin PE cos sin sin(sin ) sin t Iac sintrpe cos sinsint cos... sint. I R cos sin t I R I R t ac PE ac PE cos ac PE cos cos In this geometry with, the second harmonic term of V PE is

3 V f, PE IacRPE cos (S1.) When magnetic field is applied in y-z plane ( 9), by using Taylor expansion, one can get sin t cos sintcos sinsintcos... sin sin sin 9 sin cos 9 sin t t t sin sintcos sint sint Therefore, V PE can be written as V I tr t t PE ac sin PE cos sin sin((9sin )) sin t Iac sintrpe cos sinsint cos... sint. I R cos sin t I R I R t ac PE ac PE cos ac PE cos cos In this geometry with 9, the second harmonic term of V PE is V f, PE IacRPE cos (S1.3) Combining equations of S1.1 ~ S1.3, with inclusion of contributions from anomalous and planar all effects, the second harmonic term V of V f can be expressed as V I R cos I R x-z and ac AE f,// ac PE cos ( in plane, ) I R cos V I R y-z ac AE f, ac PE cos ( in plane, 9 ) (S1.4) S. Relation between the magnetization direction and current induced spin-orbit effective fields With the ac current induced effective fields, equations of magnetization direction and current induced effective fields are derived by solving the force balance equation along the xˆ mˆ and yˆ mˆ directions. 3

4 First, with the coordinate system as defined in Fig. 1(b), the magnetization direction can be expressed as mˆ (cos cos,cossin,sin ), xˆ mˆ, and yˆ mˆ, where i j k ˆ ˆ xm 1, sin,cossin coscos cossin sin i j k ˆ ˆ ym 1 sin,, coscos. coscos cossin sin Then, the anisotropy field and current induced effective fields can be expressed as ˆ (,,sin ) an an ˆ yˆ mˆ sin,, coscos y ˆ m ˆ sin cos cos ˆ,1, T T (1) When magnetic field is applied in the x-z plane In this geometry, ˆ (cos,,sin ). The force balance equations along xˆ mˆ and yˆ mˆ are ˆ ˆ ˆ ˆ ˆ ˆ T an xm sin cossin cos sincos sin cos cos T sin sincossin an (S.1) 4

5 ˆ ˆ ˆ ˆ ˆ ˆ T an ym cos sin cos cos sin sin cos cos an cossincos (S.) () When magnetic field is applied in the y-z plane In this geometry, ˆ (,cos,sin ). The force balance equations along xˆ mˆ and yˆ mˆ are ˆ ˆ ˆ ˆ ˆ ˆ T an xm cos sin cossinsin cos sincos sin cos cos T sin sincossin an (S.3) ˆ ˆ ˆ ˆ ˆ ˆ T an ym cos cos sin sin cos cos ancos sin cos. (S.4) From experimental harmonic all voltage measurements, V f,//, V f,, I ac, θ, θ, R AE, and R PE are known. By solving the equations S1.4 and S.1 ~ S.4,, T,, Δθ, and can be obtained. The detailed procedures are listed in the following section S3. S3. Measurements of R AE & R PE and procedures to evaluate & T The value of R PE is obtained by measuring the in-plane angular dependence of all voltage. With a magnetic field of 6 T applied in the film plane to fully saturate the magnetization along the field direction, the all voltage was recorded as a function of θ I-M (the angle between the current and the magnetization) and its angular variation purely arises from the planar all effect. The representative R vs. θ I-M is shown in Fig. S1(a). We measured R PE at different 5

6 temperatures for the temperature dependence of current induced effective fields as shown in Fig. S1(b). By measuring out-of-plane anomalous all voltage loops, R AE is obtained at different temperatures. The temperature dependence of R AE is shown in Fig. S1(c). The ratio between R PE and R AE shows non-monotonic relationship with temperature. a.5 b.7 R (). -.5 R PE R PE () I-M (deg) Temperature (K) c 1 d 6.1 R AE () Temperature (K) R PE /R AE (%) Temperature (K) Fig. S1. (a) Example of R vs. θ I-M for extracting R PE. Temperature dependence of R PE (b), R AE (c), and R AE /R AE (d). With R PE and R AE obtained above and the equations developed in sections S1 ~ S, we are able to evaluate and T. Two approaches are performed. The first one is by assuming constant effective field values as shown in Fig. (e, f). The second one is calculating the 6

7 effective field values at each magnetic field. The detailed procedures of these two approaches are as follows. (1) Fitting the V and f,// V f, curves by assuming constant values of and T (a) Obtain θ at each applied magnetic field by using the first harmonic all voltage data. (The first harmonic all voltage loop does not depend on the measurement scheme, either longitudinal or transverse scheme, as concluded from the comparison at different temperatures) (b) With an external magnetic field applied at and the fitting parameters of, T, and an, we can obtain //, I, //, I,, I,, I, //, I, //, I,, I,and, I by solving equations (S.1 ~ S.4) (maximal positive and negative ac current need to be considered for each equation). The subscript of other parameters indicates the measurement scheme and the current direction, for example, //, I is the value in the longitudinal scheme with maximal positive ac current applied. (c) Obtain R PE and R AE as describe earlier. (d) Calculate V and f,// V f, by using equation (S1.4): IacRAE cos ( //, I //, I ) ( //, I //, I ) V f,//( ) IacRPE cos 4 and IacRAE cos (, I, I ) (, I, I ) V f, ( ) IacRPE cos 4 Thus we can obtain the fitting curves of V and f,// V f,. (e) The fitting parameters of, T, and an are chosen for the best fitting to the amplitude and position of the peak in the experimental V and f,// V f, curves. 7

8 () Calculation of angular dependent and T (a) Convert the field dependence of V f,// and V f, curves into a dependence using the first harmonic experimental data. (b) At each θ, one has the values of, V, f,// V f,, an,, I ac, R AE, and R PE. an is, obtained from the above assuming constant effective field values fitting. (c) By solving equations of (S.1 ~ S.4) (maximal positive and negative ac current need to be considered for each equation) and (S1.4), one can obtain //, I, //, I,, I,, I, //, I, //, I,, I,, I as well as and T. Thus angular ( ) dependences of and T are obtained. S4. inear correlation between current induced effective fields and I ac armonic loops are measured at 3 K to evaluate the current induced effective fields with different I ac. and T are obtained by using the method as in Fig. (e, f). As shown in Fig. S, both and T show a linear relationship with I ac. The current (8.3 μa) used for the measurements in the main text is within the linear region where Joule heating is not playing a role. a b (Oe) I ac (A) T (Oe) I ac (A) Fig. S. Current induced effective fields vs. I ac. (a) and T (b) vs. I ac at 3 K at θ = 1 from another device. 8

9 S5. Comparison of current induced effective fields in Ta/CoFeB/MgO and Pt/CoFeB/MgO Current induced effective fields are compared between Ta/CoFeB/MgO and Pt/CoFeB/MgO nanowires (both of 6 nm width) by measuring current induced switching and harmonic anomalous all loops. Figure S3(a, b) show the current induced switching in Ta and Pt nanowires at 3 K, respectively. With a 4 Oe external magnetic field applied along the current direction, R can be changed between a high and a low resistance state by an in-plane current, indicating the magnetization switching of the CoFeB layer between the M z > and M z <. For the Pt nanowire, the switching occurs at ~ 1.3 ma, corresponding to a current density of A/cm assuming the current is flowing uniformly throughout the Pt/CoFeB layer. For the Ta nanowire, the switching current is ~.11 ma which corresponds to a current density of A/cm. In addition to the different switching current density for a Pt and Ta nanowire, the switching sequences are in the opposite sense for the Pt and Ta case indicated by the arrows in Fig. S3(a) and S3(b). The above observation suggests that the current induced effective fields responsible for the switching are opposite in Pt/CoFeB/MgO and Ta/CoFeB/MgO, as consistent with the opposite sign of the spin all angles in Pt and Ta. To further confirm the opposite directions of effective field of and T in Ta and Pt nanowires, we have performed the second harmonic anomalous all loop measurements in two geometries, as illustrated in Fig. (a) and (b) with = 1. The V ƒ loops measured in the longitudinal geometry with currents collinear with fields for Pt and Ta nanowires are shown in Fig. S3(c) and S3(d), respectively. With field sweeps along the current flow direction, V ƒ shows a positive (negative) peak at a positive (negative) magnetic field for a Pt nanowire, while it shows a negative (positive) peak at a positive (negative) magnetic field for a Ta nanowire, which 9

10 indicates the sign of in the Pt case is opposite to that of a Ta nanowire. For the transverse component measurements, the magnetic field is applied perpendicular to the current flow, as shown in Fig. (b). In this case V ƒ shows a positive peak at both polarities of magnetic fields from a Pt nanowire, while it shows negative peaks for a Ta nanowire, as shown in Fig. S3(e) and (f), indicating that the direction of T is opposite in the Pt and Ta case. a 1 6 Ta b 4 Pt R () c V f, // (V) Ta I I (ma) R () d V f, // (V) Pt I I (ma) e V f, (V) (Oe) 3 Ta I (Oe) V f, (V) f (Oe). Pt I (Oe) Fig. S3. Comparison of current induced switching and V f AE loops in Ta and Pt nanowires. R vs. I for Ta (a) and Pt (b) nanowires. V ƒ AE loops with I ac = 8.3 μa for Ta (c, e) and with I ac = 7.7 μa for Pt (d, f) nanowires. 1

11 S6. Transport property Ta/CoFeB/MgO The transport property of the device was studied by measuring the channel resistance (R channel ) and all resistance (R ) with a 1 T magnetic field to saturate the magnetization out-offilm plane at different temperatures. Figure S4 shows R channel and R versus temperature. Both R and R channel show nonlinear behaviors with temperature. It should be noted that R channel mostly represents the property of the nm Ta layer, since another metal layer CoFeB is only.8 nm thick R channel () Temperature (K) R () Fig. S4. Temperature dependence of R channel and R. S7. Influence of PE on evaluation of current induced effective fields The influence of PE on current induced effective fields at 3 K has been evaluated. In both approaches, either assuming the magnetization direction independent effective fields (Fig. S5(a)) or solving the effective fields at each magnetization direction (Fig. S5(b, c)), and T are underestimated without consideration of PE. 11

12 a With PE correction Without PE correction T 7.4 Oe ( 44 Oe per 1 8 A/cm ) 3.7 Oe ( 194 Oeper 1 8 A/cm ) 6.4 Oe ( 381 Oe per 1 8 A/cm ) 3.1 Oe ( 197 Oeper 1 8 A/cm ) b c (deg) -5-1 With PE correction Without PE correction (deg) T (deg) - -4 With PE correction Without PE correction (deg) Fig. S5. Influence of PE on evaluation of current induced effective fields at 3 K. (a) and T by assuming independent effective fields on the magnetization direction, without and with consideration of PE. (b) Angular dependence of and T without and with consideration of PE. S8. Anomalous Nernst effect Anomalous Nernst effect (ANE) also has contribution to the second harmonic voltage: R sin V V V ANE f f ANE ANE RAE We have measured the ANE contribution at different temperatures with field sweeping along the out-of-plane direction as shown in Fig. S6(a-d). The difference in V f at positive and negative saturated magnetic field is originated from the ANE, as current induced effective fields and Oersted field induced V f are symmetric to the out-of-plane field and thus their 1

13 contributions are excluded. By comparing V ANE, V f,//, and V f,, one can see that ANE contribution is very small. a. V ANE = -.8 V b 1..5 V f (V) V f (V) V K K c (Oe) d (Oe). -.1 V V V f (V) -.5 V f (V) -.5 e 1 K (Oe) V ANE / (μv) 1 K Minimal V ƒ, ǁ (μv) (Oe) Minimal V ƒ, (μv) 3 K K K K Fig. S6. ANE at different temperatures. (a, b, c, d) V f loops at out-of-plane direction at different temperatures. A 8.3 μa ac current is used for the measurements. (e) Comparison of V ANE, V f,//, and V f, at different temperatures. 13

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