Spin-Pumping-Free Determination of Spin-Orbit Torque Efficiency from Spin-Torque Ferromagnetic Resonance

Spin-torque ferromagnetic resonance (ST-FMR) provides a useful tool to investigate various magnetic properties in spintronic systems, where one excites FMR by applying a rf current and detects the dc voltage generated through rectification effects. While this scheme has been used to characterize spin-orbit torques (SOTs) that have attracted much attention recently, it is known that dc voltages can also be generated by spin pumping that overlaps with the signal from the rectification effects. Here, we show a method to determine the SOT generation efficiency by ST-FMR free from spin pumping using two representative material systems, $\mathrm{W}/\mathrm{Co}\text{−}\mathrm{Fe}\text{−}\mathrm{B}/\mathrm{Mg}\mathrm{O}$ and $\mathrm{Pt}/\mathrm{Co}/\mathrm{Mg}\mathrm{O}$. In addition, using the values determined by the method, which are confirmed to agree well with the results of a separately performed extended harmonic Hall measurement, we also quantify the amount of overestimation if the results obtained by a conventional ST-FMR setup are analyzed without considering the spin pumping. The present finding offers a useful insight to obtain a reliable value of SOT efficiency using ST-FMR, in particular for systems that exhibit large SOT, which accompanies large spin pumping.

Figure 1

Schematic illustration of the experimental setup for spin-torque ferromagnetic resonance.

Figure 2

Magnetization angle θ_M or in-plane magnetic field angle ϕ_H dependence of resistance. Applied magnetic field μ₀H = 150 mT and dc current I_dc = 1 mA. (a) $\mathrm{W}/\mathrm{Co}\text{−}\mathrm{Fe}\text{−}\mathrm{B}/\mathrm{Mg}\mathrm{O}$ in the X-Z and Y-Z planes, (b) $\mathrm{W}/\mathrm{Co}\text{−}\mathrm{Fe}\text{−}\mathrm{B}/\mathrm{Mg}\mathrm{O}$ in the X-Y plane, (c) $\mathrm{Pt}/\mathrm{Co}/\mathrm{Mg}\mathrm{O}$ in the X-Z and Y-Z planes, (d) $\mathrm{Pt}/\mathrm{Co}/\mathrm{Mg}\mathrm{O}$ in the X-Y plane.

Figure 3

Magnetic field H dependence of ferromagnetic resonance spectra for the device with $\mathrm{W}/\mathrm{Co}\text{−}\mathrm{Fe}\text{−}\mathrm{B}/\mathrm{Mg}\mathrm{O}$ under magnetic field rotation in the X-Z, Y-Z, and X-Y planes. A rf signal with a frequency of 4 GHz and power of 17 dBm is applied.

Figure 4

ST-FMR spectra in the X-Z plane and symmetric and antisymmetric components of Lorentzian obtained from fitting. Magnetic field angle is (a) θ_H = −40° and (b) θ_H = 40°.

Figure 5

(a)Magnetization angle θ_M dependence of the product of the amplitude of antisymmetric Lorentzian B_asym and (H₂/H₁)^1/2 as a function of current density flowing in $\mathrm{W}$ J_W. Curves denote fitting by B_asym(H₂/H₁)^1/2 =ΔR_x-zIμ₀h_SLsinθ_Mcosθ_M. (b) J_W dependence of Slonczewski-like torque effective field μ₀h_SL. Line shows linear fitting.

Figure 6

In-plane magnetic field angle ϕ_H dependence of first and second harmonic Hall resistances (a) R_ω, (b) R_2ω. Magnetic field μ₀H = 160 mT, ac current I = 2 mA, and frequency f = 10 Hz are applied. The lines in the figures show fitting results with Eqs. (23) and (24), respectively. (c) cosϕ contribution of the second harmonic Hall resistance $R_{2\omega }^{\cos \varphi }$ as a function of $1/({\mu }_0{H}_{\mathrm{ext}}-{\mu }_0{H}_K^{\mathrm{eff}})$. Line denotes linear fit.

Figure 7

Applied current I dependence of experimentally obtained amplitude of symmetric Lorentzian $A_{\mathrm{sym}}^{\exp }$, contribution of rectification effect $A_{\mathrm{sym}}^{\mathrm{rec}}$, and contribution of spin pumping and ISHE $A_{\mathrm{sym}}^{\mathrm{sp}}$ in the Y-Z and X-Y planes for $\mathrm{W}/\mathrm{Co}\text{−}\mathrm{Fe}\text{−}\mathrm{B}/\mathrm{Mg}\mathrm{O}$ sample. $A_{\mathrm{sym}}^{\mathrm{sp}}$ is calculated by subtracting $A_{\mathrm{sym}}^{\mathrm{rec}}$ from $A_{\mathrm{sym}}^{\exp }$.

Figure 8

(a) Input power P₀ versus current flowing to the device I, to $\mathrm{Co}\text{−}\mathrm{Fe}\text{−}\mathrm{B}$ $I_{\mathrm{Co}\text{−}\mathrm{Fe}\text{−}\mathrm{B}}$, and to $\mathrm{W}$ I_W. (b) Input power P₀ versus current flowing to the device I, to $\mathrm{Co}$ $I_{\mathrm{Co}}$, to $\mathrm{Pt}$ $I_{\mathrm{Pt}}$, and to $\mathrm{Ta}$ $I_{\mathrm{Ta}}$.

Figure 9

Magnetic field angle θ_H dependence of resonance field μ₀H_R with microwave frequency and power 4 GHz and 17 dBm in (a)$\mathrm{W}/\mathrm{Co}\text{−}\mathrm{Fe}\text{−}\mathrm{B}/\mathrm{Mg}\mathrm{O}$ and (b) $\mathrm{Pt}/\mathrm{Co}/\mathrm{Mg}\mathrm{O}$. Magnetic field is applied in the X-Z plane. Solid lines in the figures indicate fitting results by the resonance condition described in Ref. [37].

Figure 10

Magnetic layer thickness t dependence of the product of saturation magnetization M_S and t for (a) $\mathrm{W}/\mathrm{Co}\text{−}\mathrm{Fe}\text{−}\mathrm{B}/\mathrm{Mg}\mathrm{O}$ and (b) $\mathrm{Pt}/\mathrm{Co}/\mathrm{Mg}\mathrm{O}$. The lines in the figures are linear fits to the data.