voltage measurement for spin-orbit torques"

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1 SUPPLEMENTARY for article "Accurate analysis for harmonic Hall voltage measurement for spin-orbit torques" Seok Jin Yun, 1 Eun-Sang Park, 2 Kyung-Jin Lee, 1,2 and Sang Ho Lim 1,* 1 Department of Materials Science and Engineering, Korea University, Seoul 02841, Korea 2 KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, Korea (Date August 31, 2017) A. Derivation of θ M and φ M A detailed derivation of Eqs. (5) and (6) is described in this section. Consider the total energy equation for a system containing the first- and secondorder PMA [Eq. (1)]. By defining H x, H y, and H z as x-, y-, and z-axis components of the external magnetic and SO effective fields (H ext + H), Eq. (1) can be rewritten as follows: E K cos K eff 2 4 tot 1 M 2 M cos MS H x sinm cos M H y sinm sin M H z cos M. (S1) The equilibrium magnetization angles of θ M and φ M can be obtained by solving the following two equations: E M eff 3 tot M S H sin cos H sin cos K,1 M M K,2 M M H x cosm cosm H y cosm sinm H z sinm 0, (S2) 1

2 E tot M M S H sin H cos sin 0. x M y M M (S3) Here, H K,1 eff 2K 1 eff /M S and H K,2 4K 2 /M S. Both θ M and φ M can be obtained by making the derivatives of Eqs. (S2) and (S3) with respect to H x, H y, and H z zero. Etot MS eff HK,1 cos2 M K,2 2 cos2 M cos4 M H i M M H x sinm cosm H y sinm sinm H z cos M H H M H x sinm H y cosm cosm fi 0, H Etot MS M Hx sinm Hy cosm cosm Hi M Hi M H x cosm H y sinm sinm gi 0, H i i i (S4) (S5) f cos M cos M, cos M sin M, sin M, (S6) g sinm sin M, sinm cos M, 0. (S7) From Eq. (S3), φ M = φ H. By using the relation (H x, H y, H z ) = H ext (sinθ H cosφ H, sinθ H sinφ H, cosθ H ) and H K eff H K,1 eff + H K,2, we can simplify Eqs. (S4) and (S5) as follows: eff M H K cos 2 M H K,2 sinm sin 3 M Hext cos M H fi 0, Hi (S8) 2

3 H M sin sin g 0. H ext M H i i (S9) It is possible to obtain θ M / H i and φ M / H i from Eqs. (S8) and (S9) and then to obtain both θ M and φ M by using the following expressions: M M M M H x H y H z, H x H y H z (S10) M M M M H x H y H z. H x H y H z (S11) Detailed expressions for θ M and φ M are as follows: H x cosh H y sinh cos M H z sinm M, eff HK cos 2 M HK,2 sinm sin 3 M Hext cos M H (S12) Hx sinh Hy cosh M. H sin ext H (S13) Given that ( H x, H y, H z ) = ( H DL m z, H FL, H DL m x ), Eqs. (S12) and (S13) are identical to Eqs. (5) and (6), respectively. B. Macrospin simulation A macrospin simulation was performed to mimic the harmonic Hall voltage measurement by numerically solving the Landau Lifshitz Gilbert equation. Since the angular frequency of the AC current (ω) in the harmonic Hall measurement is significantly lower than the Larmor frequency, the quasi-static 3

4 assumption ( m/ t = 0) should be valid, with the resultant equations being as follows: m H m zˆ H m zˆ H H 0 eff 3 K,1 z K,2 z ext H H m yˆ H yˆ DL FL (S14) With the in-plane AC current expressed as I = I 0 sinωt, H in Eq. (S14) can also be replaced with Hsinωt. The following were the input parameters used for the macrospin simulation: H eff K,1 = 5 koe, H K,2 = 0 or 1 koe, H DL = 50 Oe, H FL = 100 Oe, θ H = 86, and φ H = 0 or 90. It was assumed, just for simplicity, that the input values of H DL and H FL are independent of θ M, although they are usually angle-dependent. The values of H ext and ωt were varied from +10 to 10 koe and from 0 to 6π, respectively. Figures S1(a) (c) show the results of a macrospin simulation at φ H = 0 for the three components of m, m, and m, respectively. A similar set of results is shown in Figs. S1(d) (f) for the simulation at φ H = 90. In both cases, i.e., φ H = 0 and 90, H K,2 = 0 koe. Both m and m were obtained using the lock-in amplifier principle, 1 which is also employed in the harmonic Hall voltage measurement. Figures S1(a) and (d) show the macrospin simulation results for the oscillation of m x, m y, and m z under H ext = 4 koe at φ H = 0 and 90, respectively. The components m x, m y, and m z sinusoidally oscillate with respect to their equilibrium positions (refer to Eqs. (7) (9) of the main text). It is a straightforward task to obtain the equilibrium positions (m x, m y, and m z ) of m x, m y, and m z from the oscillating signals, the results of which are shown in Figs. S1(b) and (e) as functions of H ext at φ H = 0 4

5 and 90, respectively. It is also a straightforward task to obtain the oscillation amplitudes ( m x, m y, and m z ) of m x, m y, and m z from the oscillating signals, which are the constant components of the signals obtained by multiplying the oscillating signals with a reference signal sinωt. Figures S1(e) and (f) show m x, m y, and m z as functions of H ext at φ H = 0 and 90, respectively. The following relations, which can be derived from Eq. (10) and the relation m(t) = m + (2 m) sinωt, are then used to obtain V 1ω and V 2ω : V 1ω = V AHE m z + V PHE m x m y and V 2ω = V AHE m z + V PHE (m x m y + m x m y ). There is another way of obtaining V 1ω and V 2ω, which are the constant components of the signals formed by multiplying the oscillating signal of V H = V 1ω sinωt V 2ω cos2ωt [Eq. (10)] with reference signals of sinωt and cos2ωt, respectively. It was confirmed that both procedures give perfectly the same results for V 1ω and V 2ω. 5

6 FIG. S1. Macrospin simulation results for three components of m as a function of ωt [(a) and (d)] and those of m [(b) and (e)] and m [(c) and (f)] as functions of H ext. The left panels [(a), (b), and (c)] show the results for φ H = 0, whereas the right panels [(d), (e), and (f)] show those for φ H = 90. C. Sample preparation The samples were fabricated using an ultrahigh vacuum sputtering system with a base pressure of Torr. All the layers were deposited at a constant Ar pressure of 2 mtorr. No specific substrate cooling or heating was applied 6

7 during the deposition. The deposition rate of the layers, which was used to calculate their thicknesses, was adjusted to approximately 0.07 nm/s by varying the sputtering power. The as-deposited samples were annealed at 350 C for 30 min in a vacuum Torr and then patterned into Hall bars with a width of 5 μm both for the current injection line and the branch. Figure S2 shows the m H loops measured at room temperature under in-plane (squares) and out-of-plane (circles) magnetic fields for an unpatterned Pt/Co/MgO stack. The results were measured with a vibrating sample magnetometer. The PMA of the sample is very strong and hence no saturation occurs in the in-plane loop even at 20 koe. In order to characterize the PMA properties of the Pt/Co/MgO stack, the Generalized Sucksmith Thompson method, which does not require a high magnetic field to saturate a sample along the magnetic hard axis, 2 was used. Details on the method will be described in Section F. Because of this, the secondorder PMA behavior is not clearly visible in the in-plane m H loop. 7

8 FIG. S2. In-plane (squares) and out-of-plane (circles) m H loops of unpatterned Pt/Co/MgO stack. D. Harmonic Hall voltage measurement The harmonic Hall voltages were measured with a homemade apparatus equipped with one current source (Keithley 6221 DC/AC) and two lock-in amplifiers (SRS 830 DSP). The external magnetic field was generated using a superconducting coil. The schematics showing the Hall bar with definition of axes and angles and the measurement circuit are shown in Figs. S3(a) and (b). The magnetization direction was controlled by H ext, which was swept from +90 to 90 koe with two different directions of θ H = 85 and φ H = 0, and θ H = 85 and φ H = 90. When AC currents with a frequency of 401 Hz were applied along the x direction, both V 1ω and V 2ω were induced along the y direction, which were simultaneously measured with two lock-in amplifiers. The measured results are 8

9 shown in Fig. S4. It is seen from Fig. S4 that the results for V 1ω and V 2ω do not show irreversible jumps, an indication of incoherent magnetization behavior during the harmonic Hall voltage measurement (refer to Ref. [20] of the main text). Incoherent magnetization behavior was reported to occur during the Hall voltage measurement only at θ H > 85. 3,4 From these, it is reasonable to consider that the magnetization behavior is coherent during the harmonic Hall voltage measurement. The linearly of the Generalized Sucksmith Thompson plot in Fig. S6 is another indication of showing coherent magnetization behavior. Note that the Generalized Sucksmith Thompson plot is derived from the total energy equation using the assumption of coherent magnetization (refer to Ref. [35] of the main text). Details on the Generalized Sucksmith Thompson plot will be described in Section F. FIG S3. Schematics showing (a) Hall bar with definition of axes and angles and (b) measurement circuit. 9

10 FIG S4. Results for V 1ω and V 2ω as functions of H ext measured at three different I 0 values of 1.0 ma (black squares), 1.5 ma (red circles), and 2.0 ma (blue triangles). E. Planar Hall voltage measurement To separate the total second harmonics signal into AHE and PHE, it is necessary to measure one or both of the constituents. In this study, V PHE was obtained by measuring the V 1ω value with φ H swept from 0 to 360. Under the conditions of θ H = 90 and H ext H eff K, a rough estimate of V PHE can be obtained using the approximate relation: V 1ω (V PHE /2)sin2φ H. Because the nominal field angles (θ H nom and φ H nom ), which are usually adjusted mechanically in the beginning of the measurement, are quite different from the actual angles (θ H and φ H ) in the analytical equations, it is necessary to find the equations relating these 10

11 two sets of angles, which are given as follows: θ H = c θ θ nom H + θ and φ H = c φ φ nom H + φ. Here, the superscripts nom and denote the nominal and offset angles, respectively and the coefficients c θ and c φ denote the correction factors for θ H and φ H, respectively. With the corrections considered, the relation for V 1ω can be derived as follows: 1 nom V VAHE w H VPHE w nom sincos( ) ( / 2)cos sin2( H ) (S15) The parameters of θ, φ, c φ, and V PHE can be extracted by fitting the experimental results with Eq. (S15). The PHE measurement results are shown in Fig. S5, together with the fitting curves. FIG S5. Results for V 1ω as a function of φ H nom at three different I 0 values of 1.0 ma (black squares), 1.5 ma (red circles), and 2.0 ma (blue triangles). The lines are fits to the experimental results using Eq. (S15). 11

12 F. Extracting first- and second-order PMA parameters The Generalized Sucksmith Thompson method is a well-known technique for the accurate determination of H eff K,1 and H K,2. The key to the method is the use of the following equation, which can be derived from the total energy equation: 2 eff 2 1mz cosh mz sinh ext K,1 K,2 z ; 2 mz 1 mz H H H m (S16) The experimental results measured at various conditions collapse into a line in the plot of αh ext versus m z 2 (Fig. S6), from which H K,1 eff and H K,2 can be extracted from the intercept and slope, respectively. 12

13 FIG S6. Plot of αh ext versus m z 2 measured at three different I 0 values of 1, 1.5, and 2.0 ma and two different φ H values 0 and 90. REFERENCES 1. Meade, M. L. Advances in lock-in amplifiers. J. Phys. E 15, (1982). 2. Okamoto, S., Kikuchi, N., Kitakami, O., Miyazaki, T., Shimada, Y. & Fukamichi, K. Chemical-order-dependent magnetic anisotropy and exchange stiffness constant of FePt (001) epitaxial films. Phys. Rev. B 66, (2002) 3. Rosenblatt, D. P., Karpovski, M. & Gerber, A. Monitoring magnetization reversal and perpendicular anisotropy by the extraordinary Hall effect and anisotropic magnetoresistance. J. Appl. Phys. 108, (2010). 4. Franken, J. H., Hoeijmakers, M., Lavrijsen, R. & Swagten, H. J. M. Domain- 13

14 wall pinning by local control of anisotropy in Pt/Co/Pt strips. J. Phys.: Condens. Matter 24, (2012). 14