Quantitative characterization of the spin-orbit torque using harmonic Hall voltage measurements

Solid understanding of current induced torques is a key to the development of current and voltage controlled magnetization dynamics in ultrathin magnetic heterostructures. To evaluate the size and direction of such torques, or effective fields, a number of methods have been employed. Here, we examine the adiabatic (low-frequency) harmonic Hall voltage measurement that has been used to study the effective field. We derive an analytical formula for the harmonic Hall voltages to evaluate the effective field for both out of plane and in-plane magnetized systems. The formula agrees with numerical calculations based on a macrospin model. Two different in-plane magnetized films, Pt|CoFeB|MgO and CuIr|CoFeB|MgO are studied using the formula developed. The effective field obtained for the latter system shows relatively good agreement with that estimated using spin torque switching phase diagram measurements reported previously. Our results illustrate the versatile applicability of harmonic Hall voltage measurement for studying current induced torques in magnetic heterostructures.

Figure 1

Schematic illustration of the experimental setup. A Hall bar is patterned from a magnetic heterostructure consisting of a nonmagnetic metal layer (gray), a ferromagnetic metal layer (blue), and an insulating oxide layer (red). The large gray square is the substrate with an insulating oxide surface. Definitions of the coordinate systems are illustrated together. $\vec{M}$ denotes the magnetization and $\vec{H}$ represents the external field.

Figure 2

(a) Magnetization angle with respect to the film normal (${\theta }_0$), (b) first harmonic Hall voltage and (d) and (f) second harmonic Hall voltage plotted against in-plane external field (${\theta }_H$ = 90°). The field is directed along the $x$ axis (${\phi }_H$ = 0°) for (a), (b), and (d) and along the $y$ axis (${\phi }_H$ = 90°) for (f). Open symbols show numerical calculations using the macrospin model. Solid/dashed lines represent the analytical solutions: (a) Eq. (22), (b) Eq. (23), (d) and (f) Eq. (24). (c) and (e) $x$, $y$, and $z$ components of the effective field used for the numerical calculations. Left (right) panel indicates the effective field when the magnetization is pointing along +$z$ (–$z$). Parameters used in the numerical calculations: $H_K$ = 3162 Oe, $H_A$ = −6 Oe, α = 0.01, γ = 17.6 MHz/Oe, $a_J$ = 3 Oe, $b_J$ = 3 Oe, $\widehat{p}=\left(0,1,0\right)$, $\Delta R_A$ = 1 Ω, $\Delta R_P$ = 0.1 Ω, Δ$I$ = 1 A.

Figure 3

(a) Polar (${\theta }_0$) and (b) azimuthal (${\phi }_0$) angles of the magnetization, (c) first and (d) second harmonic Hall voltages ($V_{XY}$), (e) first and (f) second harmonic longitudinal voltages ($V_{XX}$) as a function of a slightly tilted out of plane field (${\theta }_H$ = 5°). The in-plane component of the tilted field is directed along the $x$ axis (${\phi }_H$ = 0°). Open symbols show numerical calculations using the macrospin model. Solid lines represent the analytical solutions: (a) and (b) Eq. (27), (c) Eq. (28), (d) Eq. (30), (e) Eq. (38), (f) Eq. (39). Inset of (b) shows the $x$, $y$, and $z$ components of the effective field used for the numerical calculations. Parameters used in the numerical calculations: $H_K$ = −4657 Oe, $H_A$ = −4 Oe, α = 0.01, γ = 17.6 MHz/Oe, $a_J$ = 0.3 Oe, $b_J$ = 0.3 Oe, $\widehat{p}=\left(0,1,0\right)$, $\Delta R_A$ = 1 Ω, $\Delta R_P$ = 0.1 Ω, $\Delta R_{MR}$ = 1 Ω, $R_0$ = 0 Ω, Δ$I$ = 1 A.

Figure 4

(a) Polar (${\theta }_0$) angle of the magnetization, (b) first and (c) and (d) second harmonic Hall voltages as a function of a slightly tilted out of plane field (${\theta }_H$ = 5°). The in-plane component of the tilted field is directed along the $x$ axis (${\phi }_H$ = 0°). Solid symbols show numerical calculations using the macrospin model. Solid lines represent the analytical solutions: (a) Eq. (42), (c) and (d) Eq. (43). (e) and (f) $x$, $y$, and $z$ components of the effective field used for the numerical calculations. Parameters used in the numerical calculations: $H_K$ = −4561 Oe, $H_A$ = −4 Oe, α = 0.01, γ = 17.6 MHz/Oe, $\widehat{p}=\left(0,1,0\right)$, $\Delta R_A$ = 1 Ω, $\Delta R_P$ = 0.1 Ω, $\Delta R_{MR}$ = 1 Ω, $R_0$ = 0 Ω, Δ$I$ = 1 A. (c,e) $a_J$ = 3 Oe, $b_J$ = 3 Oe, (d,f) $a_J$ = 3 Oe, $b_J$ = −3 Oe.

Figure 5

Experimental results for Sub|3 Pt|1 CoFeB|2 MgO|1 Ta (in nanometers). (a) Magnetization hysteresis loops: open and solid symbols represent out of plane ($H_z$) and in-plane ($H_X$) field sweeps, respectively. (b) First harmonic voltage as a function of external field with different tilt angles (${\theta }_H$). $V_{IN}$ = 0.5 V. (c)–(e) External field dependence of the second harmonic voltage ($V_{2\omega }$) for three different ${\theta }_H$. $V_{IN}$ = 3.5 V. The solid lines represent fitting results using Eq. (43). (f) and (g) The dampinglike term $\Delta H_Z$ and the fieldlike term $\Delta H_Y$ obtained from the fitting of $V_{2\omega }$ plotted against the excitation voltage amplitude ($V_{IN}$) for ${\theta }_H$∼9°. (h) and (i) ${\theta }_H$ dependence of the effective field per unit excitation voltage amplitude for $\Delta H_Z$ and $\Delta H_Y$. The right axis shows the corresponding effective field if a current density of 1 × 10${}^8$ A/cm${}^2$ were applied to the underlayer (3-nm Pt). Squares and circles in (f)–(i) represent the corresponding effective field for magnetization pointing along +$x$ and $-x$ axes, respectively. The green dashed line in (i) show the calculated Oersted field if all the current flows into the underlayer. The field is calculated for 1 nm above the top surface of the Pt underlayer.

Figure 6

Experimental results for Sub|10 CuIr|∼1.3 CoFeB|2 MgO|1 Ta (in nanometers). (a) First harmonic voltage as a function of external field with different tilt angles (${\theta }_H$). $V_{IN}$ = 0.5 V. (b) and (c) External field dependence of the second harmonic voltage ($V_{2\omega }$) for ${\theta }_H$∼−9° (b) and 9° (c). $V_{IN}$ = 3.5 V. The solid lines represent fitting results using Eq. (43). (d) and (e) ${\theta }_H$ dependence of the effective field per unit excitation voltage amplitude for the dampinglike term $\Delta H_Z$ and the fieldlike term $\Delta H_Y$. The right axis shows the corresponding effective field if a current density of 1 × 10${}^8$ A/cm${}^2$ were applied to the underlayer (10-nm CuIr). Squares and circles represent the corresponding effective field for magnetization pointing along +$x$ and $-x$ axes, respectively. The green dashed line in (e) show the calculated Oersted field if all the current flows into the underlayer. The field is calculated for 1 nm above the top surface of the CuIr underlayer.

Figure 7

Experimental results for sub|10 CuIr|∼1.3 CoFeB|2 MgO|1 Ta (in nanometers). (a) and (b) External field dependence of the inverse of the second harmonic voltage (1/$V_{2\omega }$) for ${\theta }_H$∼3° (a) and 9° (b). The solid lines represent linear fitting to the data. $V_{IN}$ = 3.5 V. (c) $\Delta H_Y$, obtained from the fitting and Eq. (33), plotted against the excitation voltage amplitude ($V_{IN}$) for ${\theta }_H$∼3°. (d) ${\theta }_H$ dependence of the effective field per unit excitation voltage amplitude for the fieldlike term, $\Delta H_Y$. The right axis shows the corresponding effective field if a current density of 1 × 10${}^8$ A/cm${}^2$ were applied to the under-layer (10-nm CuIr). Squares and circles represent the corresponding effective field for magnetization pointing along +$x$ and $-x$ axes, respectively. The green dashed line in (d) show the calculated Oersted field if all the current flows into the underlayer [same as in Fig. 6].

Figure 8

Schematic illustration of the setup used to calculate the Oersted field. Current is assumed to flow in the bottom layer (gray slab). The profile of the Oersted field is calculated in the upper layer (blue slab).