Quantitative accuracy and microscopic origin of spin-orbit torques extracted from harmonic Hall voltage measurements

The adiabatic harmonic Hall voltage measurement — in which a low-frequency sinusoidal current through a heavy-metal/ferromagnet/oxide stack generates first- and second-harmonic transverse and longitudinal voltages from which the current-induced spin-orbit-torque effective fields are extracted — is one of the standard tools for quantifying spin-orbit torques. The analytical framework of Hayashi et al. [Phys. Rev. B 89, 144425 (2014)] left open a bundle of linked issues: the assumed magnetization-angle independence of the torque amplitudes, a factor-of-two discrepancy between the fieldlike amplitudes extracted by the low-field inverse-$V_{2\omega}$ and large-field-fitting protocols on the same CuIr|CoFeB|MgO devices, the unquantified validity range of the small-modulation linearization, the link between extracted effective fields and microscopic spin transport (underlayer thickness, Pt spin-diffusion length), and the microscopic origin of the fieldlike torque. Later work has substantially advanced each strand — more accurate analytic formulas, full-angle protocols, and the identification of thermoelectric and magnonic artifacts — but no protocol-independent, artifact-corrected determination of the torque amplitudes with quantified validity limits exists, and the mechanism of the fieldlike torque remains unsettled.

Background

When a charge current flows through a heavy metal with strong spin-orbit coupling, such as Pt or the Cu–Ir alloy CuIr, adjacent to an ultrathin ferromagnet (e.g. CoFeB), spin-orbit coupling generates a spin current or an interfacial spin accumulation (via the spin Hall effect — the conversion of a charge current into a transverse spin current in the bulk of a spin-orbit-coupled material — or the Rashba-Edelstein effect, the build-up of a spin accumulation at an interface with structural inversion asymmetry, respectively) that exerts torques on the magnetization. These spin-orbit torques (SOTs) are conventionally decomposed into a dampinglike (DL) component, which behaves like a modified Gilbert damping and is the component that switches magnetization, and a fieldlike (FL) component, which acts like an effective magnetic field transverse to the current. Both are quantified by equivalent effective fields (denoted $\Delta H$ below, or by amplitudes $a_J$ and $b_J$ for the DL and FL terms).

A widely used way to measure these effective fields is the adiabatic (low-frequency) harmonic Hall voltage technique: a sinusoidal current $I=\Delta I\sin\omega t$ slightly tilts the magnetization direction, and the first-harmonic ($V_\omega$) and second-harmonic ($V_{2\omega}$) Hall voltages — arising from the anomalous Hall effect (AHE, the transverse voltage proportional to the out-of-plane magnetization component), the planar Hall effect (PHE, a transverse magnetoresistance varying as $\sin^2\theta\sin 2\varphi$), and the anisotropic magnetoresistance (AMR, the dependence of the longitudinal resistance on the angle between the magnetization and the current) of the longitudinal voltage — encode the tilt amplitude and hence the current-induced effective field. Here $\theta$ and $\varphi$ are the polar and azimuthal angles of the magnetization. Because the tilt is small, the standard analysis linearizes the magnetoresistance in the angular deviations $\Delta\theta,\Delta\varphi$ from the equilibrium direction $(\theta_0,\varphi_0)$; the PHE mixes orthogonal effective-field components into $V_{2\omega}$, so the extraction formulas must invert this mixing (with the ratio $\xi=\Delta R_P/\Delta R_A$ of PHE to AHE resistance measured separately). The measured transverse field also contains an Oersted field (the magnetic field generated by the measurement current itself) that must be subtracted to isolate the SOT contribution.

Hayashi, Kim, Yamanouchi and Ohno [Phys. Rev. B 89, 144425 (2014)] derived compact analytical formulas for $V_\omega$ and $V_{2\omega}$ for both out-of-plane and in-plane magnetized samples including the PHE mixing, verified them against numerical macrospin simulations of the Landau-Lifshitz-Gilbert equation (the standard equation of motion for magnetization dynamics, solved for a single-domain magnetization), and applied two analysis protocols to in-plane magnetized Pt|CoFeB|MgO and CuIr|CoFeB|MgO devices: (i) a low-field procedure using the inverse of $V_{2\omega}$ versus a weakly tilted field, and (ii) a large-field fit in which the magnetization is rotated from in-plane to out-of-plane and $V_{2\omega}(H)$ is fitted with $a_J$ and $b_J$ as constants. The DL amplitude from the large-field fit agreed with independent spin-torque switching phase-diagram measurements on CuIr devices, but the FL amplitudes from the two protocols differed by roughly a factor of two (about 22 Oe vs about 42 Oe at $1\times10^8$ A/cm$^2$), a discrepancy the paper stated "requires further investigation for clarification." The derivations also assume $a_J$ and $b_J$ are independent of the magnetization angles $(\theta_0,\varphi_0)$, keep the second-order terms $\Delta\theta^2,\Delta\varphi^2$ only as a warning that they eventually dominate, and interpret the sign of the FL term under the assumption that the spin Hall effect is its source.

Subsequent work has shown that the raw second-harmonic signal is contaminated by several non-torque contributions: anomalous and ordinary Nernst voltages from Joule-heating-induced thermal gradients, and a magnon-creation-annihilation contribution to the planar Hall resistance. These artifacts, together with fitting degeneracies, are now known to bias extracted SOT amplitudes in many material systems.

Problem Statement

The common objective is to make the harmonic Hall voltage measurement a quantitatively trustworthy and physically interpretable probe of spin-orbit torques: extracted effective fields should be independent of the analysis protocol, valid within stated operational limits, and traceable to a microscopic mechanism. Building on the analytical framework of Hayashi et al., Phys. Rev. B 89, 144425 (2014), the following linked questions remain open:

1. Angular dependence of the torque amplitudes. The standard analytic $V_{2\omega}$ formulas treat the dampinglike and fieldlike amplitudes $a_J$ and $b_J$ as independent of the equilibrium polar and azimuthal magnetization angles $(\theta_0,\varphi_0)$, even though the fieldlike term is known to exhibit strong magnetization-angle dependence. How are the analytic expressions and extraction protocols to be modified to incorporate explicit $(\theta_0,\varphi_0)$ dependence of $a_J$ and $b_J$, and how large is the resulting correction to previously extracted torque amplitudes?

2. Origin of the protocol-dependent fieldlike amplitude. For the same CuIr|CoFeB|MgO devices, the low-field inverse-$V_{2\omega}$ procedure and the large-field fitting procedure give fieldlike amplitudes differing by roughly a factor of two (about 22 Oe versus about 42 Oe at $1\times10^8$ A/cm$^2$). What is the origin of this analysis-dependent spread — magnetization-angle dependence of the torque, unremoved thermoelectric (anomalous or ordinary Nernst) signals, magnonic magnetoresistance contributions, nonlinear modulation terms, fitting degeneracy, or a combination — and under what corrected protocol do the two approaches agree?

3. Validity limits of the small-modulation linearization. The linear harmonic formulas hold only while the current-induced angular deviations satisfy $|\Delta\theta|,|\Delta\varphi|\ll1$; second-order terms eventually dominate, but no systematic quantification exists of the operational ranges — maximum effective field, maximum current density or excitation amplitude — within which the linear formulas remain accurate to a stated tolerance for each measurement geometry.

4. Link to microscopic spin transport. The dampinglike amplitude extracted for the Pt|CoFeB|MgO stack of the source study is substantially smaller than values reported for comparable Pt-based stacks, and the connection between the extracted effective fields and microscopic transport — the underlayer thickness dependence, the Pt spin-diffusion length (the characteristic distance over which a spin current decays in Pt), current shunting (the division of the applied current between the metallic layers of the stack), and the Oersted-field subtraction — is to be established quantitatively for such stacks.

5. Microscopic origin of the fieldlike torque. The fieldlike term is interface-sensitive, but the relative contributions of the interfacial Rashba-Edelstein effect, coherent spin rotation during transmission across the interface, and bulk (spin Hall, and more recently orbital) mechanisms remain to be discriminated theoretically and experimentally.

The measurement-and-analysis protocols should come with quantified validity ranges and artifact corrections whose extracted dampinglike and fieldlike amplitudes are protocol-independent and consistent with independent techniques, and the microscopic account of the fieldlike torque should explain its sign, magnitude, and interface sensitivity across heavy-metal/ferromagnet/oxide stacks.

References

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