Harmonic Hall Voltage Measurements

• Harmonic Hall voltage measurements is a contact-based technique that quantifies current‐induced torques and nonlinear magnetotransport in spintronic materials.
• It involves detecting first and second harmonic voltage responses under oscillating currents and applied magnetic fields to extract parameters like spin Hall angles and effective fields.
• Advanced analysis using lock-in amplification and fitting strategies enables separation of damping-like, field-like, and thermoelectric contributions in multilayer devices.

Harmonic Hall voltage measurements represent a precise, contact-based methodology for quantifying current-induced torques—especially spin-orbit and orbital Hall torques—as well as nonlinear magnetotransport effects in thin-film and multilayer spintronic structures. Applying an oscillating current to a patterned Hall-bar device generates voltage responses not only at the applied frequency (first harmonic) but also at higher harmonics, with the second harmonic capturing the interplay of effective magnetic fields, thermal gradients, and symmetry-breaking electronic phenomena. This technique is central to extracting key physical parameters such as spin and orbital Hall torque efficiencies, field-like and damping-like effective fields, spin Hall angles, nonlinear Hall coefficients, and unidirectional magnetoresistance contributions across a wide range of material systems, including heavy-metal/ferromagnet, light-metal/ferromagnet, antiferromagnet, magnetic topological insulator, and quantum materials (Mahapatra et al., 2024, Lau et al., 7 Jan 2026, Min et al., 2023).

1. Experimental Methodology and Instrumentation

Measurements are performed on Hall-bar or cross-shaped microdevices (typical channels: 2–30 μm wide, contact separation: 2–100 μm) patterned from multilayer stacks (e.g., LM/FM or HM/FM bilayers, AFM/HM, or quantum conductors). The excitation protocol involves driving a sinusoidal (or cosinusoidal) ac current, $I(t) = I_0\sin\omega t$, with $I_0$ chosen to realize current densities $J \sim 10^6$–$10^7$ A/cm² in the active conducting layer. The devices are typically situated under a static magnetic field $B_{\mathrm{ext}}$ (in-plane or rotated), whose magnitude ($20$ mT – $10$ T) and direction ($\phi$: $0^\circ$–$360^\circ$) are controlled via vector magnet or rotary probe stage (Mahapatra et al., 2024, Neumann et al., 2018, Skowroński et al., 2021).

Lock-in amplifiers are configured to simultaneously detect the first ($V^{1\omega}$) and second ($V^{2\omega}$) harmonic voltage, referenced to $\omega$ and $2\omega$, respectively, in both longitudinal ($V_{xx}$) and transverse (Hall, $V_{xy}$) channels. Precise phase calibration and filtering procedures are necessary to suppress crosstalk and environmental interference (Min et al., 2023).

2. Theoretical Framework for Harmonic Voltages

The first harmonic resistance is predominantly governed by the equilibrium orientation of the magnetization or symmetry axis: $$R_{xx}^{1\omega} = R_0 + \Delta R_{xx}^{1\omega} \cos^2\phi$$

$R_{xy}^{1\omega} = R_{AHE} m_z + R_{PHE}\sin 2\phi$

where $R_{AHE}$ and $R_{PHE}$ are the anomalous and planar Hall resistances, respectively, and $m=(\cos\phi, \sin\phi, 0)$ for in-plane magnetization (Mahapatra et al., 2024, Hayashi, 2013).

The second harmonic arises from current-driven modulations in the magnetization induced by effective fields; in the LM/FM context,

$$R_{xy}^{2\omega}(\phi) = [R_{AHE}\frac{B_{SL}}{B_{ext}+B^{k}_{eff}} + \alpha B_{ext} + R^{0}_{\Delta T}]\cos\phi + 2R_{PHE}\frac{B_{FL}+B_{Oe}}{B_{ext}}\cos2\phi\cos\phi$$

and

$$R_{xx}^{2\omega}(\phi) = R^*\sin\phi - 2\Delta R_{xx}^{1\omega}\frac{B_{FL}+B_{Oe}}{B_{ext}} \cos^2\phi\sin\phi$$

where $B_{SL}$ ($=$ damping-like), $B_{FL}$ ($=$ field-like), and $B_{Oe}$ (Oersted) fields are separated based on their distinct angular and field dependencies; $R^* = gR_{\Delta T} + R_{UMR}$ with $g$ the Hall-bar aspect ratio and $R_{UMR}$ the unidirectional MR amplitude (Mahapatra et al., 2024, Neumann et al., 2018, Lin et al., 2024).

Small-angle assumptions require $B_{ext} \gg B_{SL}, B_{FL}$ for linear response validity; accurate extraction of torque amplitudes depends on simultaneous fits of $R^{2\omega}$ versus $\phi$ and/or $B_{ext}$.

3. Data Analysis: Fitting Strategies and Parameter Extraction

Standard protocols for extracting field-like and damping-like torque amplitudes involve decomposing $R_{xy}^{2\omega}(\phi)$ into

$A_{cos}\cos\phi + A_{PHE}\cos2\phi\cos\phi$

with $A_{cos}/R_{AHE}$ linearly dependent on $1/(B_{ext}+B^{k}_{eff})$ (slope yields $B_{SL}$), and $A_{PHE}/2R_{PHE}$ linear in $1/B_{ext}$ (slope yields $B_{FL}+B_{Oe}$). $B_{Oe}$ is deduced from the layer geometry and current partition; subtraction isolates the intrinsic orbital/spin Hall field (Mahapatra et al., 2024, Hayashi, 2013, Lau et al., 7 Jan 2026).

UMR is isolated via

$$(R_{UMR}/J) = \frac{\textrm{coefficient of } \sin\phi \textrm{ in } R_{xx}^{2\omega} - gR_{\Delta T}}{J}$$

Two-field-scan harmonic Hall voltage analysis accelerates measurements by sweeping $V_{2\omega}(H)$ at fixed $\phi=45^\circ$ and $\phi=0^\circ$ only, robustly separating $H_{DL}$, $H_{FL}$, and Nernst artifacts without ambiguities associated with heavily correlated fits (Lin et al., 2024).

4. Corrections, Assumptions, and Device Geometry Effects

Quantitative extraction of torque-related parameters mandates corrections for thermoelectric artifacts (ordinary and anomalous Nernst effects) via antisymmetrization under $B_{ext}\rightarrow -B_{ext}$ and explicit fitting of $V_{2\omega}$ offsets (Mahapatra et al., 2024, Lin et al., 2024).

Current inhomogeneity near voltage lead junctions substantially reduces the apparent magnitude of the measured torques and spin/charge Hall angle. Finite-element simulations establish a geometry-dependent reduction factor $f(ar)$,

$$f(ar) = \langle j_x^2\rangle_D / \langle j_x^2\rangle_{D,ar\to 0}$$

with $ar = w_V/w_I$ (voltage lead width over channel width). $ar = 1$ yields $f \approx 0.7$, demanding correction for intrinsic quantifications; aspect ratios $ar < 0.2$ are required for $>90\%$ accuracy (Neumann et al., 2018).

5. Extensions: Nonlinear Hall Effects and Advanced Material Systems

Second harmonic detection extends beyond linear spin/orbital Hall torque quantification: it is the primary means of probing nonlinear Hall effects (NLHE), nonreciprocal Hall phenomena, and frequency mixing in quantum materials. In noncentrosymmetric conductors,

$V_{y,2\omega} = R_{NL} I_\omega^2$

where $R_{NL}$ quantifies the quadratic, second-order Hall response. Broadband mixing, including sum and difference frequency generation, can be directly identified via lock-in or FFT analysis and relates to underlying disorder and symmetry mechanisms (e.g., extrinsic NLHE in FIB-deposited Pt due to grain-boundary-induced skew scattering) (Min et al., 2023).

For gated 2D systems, spurious second harmonic responses may derive from gate-voltage oscillations and can be eliminated by individually grounding voltage probes during measurements, verifying field parity and artifact symmetry to isolate intrinsic quantum-metric-dipole NHE contributions (Xu et al., 2024).

6. Applications, Scope, and Limitations

Harmonic Hall voltage measurements are now standard for:

• Quantifying spin and orbital Hall torque efficiencies (e.g., $$\theta_{OHT} = \zeta_{SL} = (2e/\hbar)(M_s t_F)B_{SL}/J_{LM}$$) (Mahapatra et al., 2024, Ziętek et al., 2022).
• Assessing unidirectional magnetoresistance and nonreciprocal transport effects.
• Rapid calibration of spin Hall conductivity $\sigma_{SH}$, field-like/damping-like torque ratios, and interface transparency via combined static and dynamic protocols (Skowroński et al., 2021).

These methods are validated by simulation (macrospin LLGS/LLG models, finite-element current maps), essential for benchmarking extraction routines and correcting for parameter degeneracies (e.g., the dominant role of $M_s$ in $V_{1\omega}$ and $V_{2\omega}$ responses) (Ziętek et al., 2022, Yun et al., 2017).

Current limitations include: the breakdown of small-angle approximations for large current/field drive, nonuniform or multi-domain magnetization, and strong thermal gradients that may challenge artifact correction schemes. Analytical expressions require refinement for tilted fields, second-order perpendicular magnetic anisotropy, and advanced multilayer architectures (Yun et al., 2017, Hayashi et al., 2013).

7. Summary Table: Key Experimental Components

Parameter	Typical Value/Range	Significance
Channel width	2–30 μm	Device sensitivity, mitigation of geometric artifacts
AC current density $J$	$10^6$–$10^7$ A/cm²	Ensures detectable harmonic voltage
Field magnitude $B_{ext}$	20 mT–10 T	Controls magnetization orientation, linear response regime
Lock-in frequency	10–3000 Hz	Adiabatic regime, filtering of noise/artifacts
Aspect ratio $ar$	0.07–2.67	Determines current homogeneity correction

Harmonic Hall voltage measurements continue to provide a rigorous, reproducible platform for extracting spin, orbital, and nonlinear Hall transport phenomena, with ongoing refinement in experimental protocol, artifact suppression, and simulation-guided data analysis (Mahapatra et al., 2024, Neumann et al., 2018, Lin et al., 2024, Min et al., 2023, Skowroński et al., 2021).