Anisotropic Inverse Spin Hall Effect

• AISHE is a phenomenon where spin currents are converted into charge currents with direction-dependent efficiency due to broken symmetry in magnetic, crystallographic, and interfacial structures.
• Experimental studies in bilayer systems, organic semiconductors, and engineered ferromagnets reveal that device geometry and interface design critically influence the anisotropic conversion signals.
• Theoretical models incorporating spin–orbit coupling, Berry curvature, and quantum confinement underline AISHE’s potential for tailoring spintronic device functionalities and optimizing spin current quantification.

The anisotropic inverse spin Hall effect (AISHE) describes the conversion of a spin current into a transverse charge current in a manner that depends explicitly on the direction of spin polarization, magnetization, crystallographic orientation, and interfacial symmetry in the materials under consideration. Unlike the conventional inverse spin Hall effect (ISHE), for which the charge conversion efficiency remains invariant under sample rotation in highly symmetric systems, the anisotropic variant emerges when those symmetries are broken—by crystal structure, magnetic ordering, quantum confinement, or engineered interfaces. The phenomenon’s fundamental and applied significance spans the quantification of spin–orbit coupling, device geometry optimization, and the exploitation of new magneto-electronic responses in spintronic systems.

1. Symmetry Breaking and the Origin of Anisotropy

AISHE arises in systems where the symmetry—either at the atomic, crystallographic, or magnetic level—is reduced. In hexagonal close-packed (hcp) metals and antiferromagnets, the intrinsic spin Hall conductivity vector $\boldsymbol{\sigma}(\hat{\mathbf{S}})$ is a function not only of crystal axes but also spin polarization direction (Freimuth et al., 2010). The formal relation

$$\boldsymbol{\sigma}_k(\hat{\mathbf{S}}) = \frac{1}{2} \sum_{i j s} \epsilon_{ijk} \sigma_{ij}^s S_s$$

clarifies that $\boldsymbol{\sigma}(\hat{\mathbf{S}})$—and hence the efficiency of spin-to-charge conversion—depends on both the orientation of the electric field and the measurement direction of the spin current. In particular, in non-cubic systems, even the sign of the spin Hall effect (and thus the inverse effect) may be tuned by varying $\hat{\mathbf{S}}$.

Other sources include antiferromagnetic and altermagnetic ordering, which break cubic symmetry and introduce direction-dependent spin absorption and reflection mechanisms (Leiviskä et al., 31 Jan 2025). In engineered magnetic films, tailored magnetic textures and interfacial anisotropies can produce unconventional spin current polarization components, as in obliquely deposited permalloy, where non-collinear interfacial magnetization provides a platform for robust AISHE signals at room temperature (Aon et al., 6 May 2024).

2. Experimental Realization and Model Systems

The AISHE has been experimentally probed in several distinct geometries:

• Permalloy/normal metal bilayers: Spin pumping through ferromagnetic resonance generates spin currents, which upon entering the normal metal are converted to charge currents with a voltage proportional to the spin Hall angle. Rigorous modeling of the magnetization precession trajectory—including its ellipticity—yields angle and frequency dependent AISHE voltages. The measured signal decomposes into a symmetric ISHE (AISHE) part and an antisymmetric AMR part, with the angular dependence of the charge response confirming the anisotropic nature (Mosendz et al., 2010).
• Organic semiconductors: CoFeB/C₆₀ bilayers exhibit an AISHE voltage whose amplitude and sign strongly depend on the in-plane field direction, arising from $\pi$–$\sigma$ hybridization and curvature-induced enhancement of spin–orbit coupling in the C₆₀ layer. Lorentzian decomposition of the voltage lineshape and angular analysis uniquely separates spin pumping and anisotropic rectification effects, underpinning the assignment of a high spin Hall angle to C₆₀ (Sharangi et al., 2021).
• Altermagnetic insulator/Pt interfaces: Spin Hall magnetoresistance measurements display marked anisotropy with respect to the current direction relative to crystal axes, attributed to the anisotropic altermagnetic order in Ba₂CoGe₂O₇. Phenomenological fits to the data (e.g., $–\cos2\alpha – \cos4\alpha$) are consistent with symmetry-governed modulation of spin–charge conversion via ISHE at the interface (Leiviskä et al., 31 Jan 2025).
• Single-layer ferromagnets: Films with tailored magnetic anisotropy (through oblique deposition) exhibit a distinctive antisymmetric Hall voltage attributed to unconventional spin currents with transverse polarization. The angular shift of the maximum AISHE signal with respect to the deposition flux direction provides unambiguous evidence of interfacial anisotropy’s role (Aon et al., 6 May 2024).

3. Theoretical Framework and Key Equations

Quantitative interpretation of AISHE revolves around generalizations of the ISHE relation, where conversion efficiency is governed by both conventional and anomalous contributions:

$$\mathbf{J}_c = \theta_{SH} \frac{2e}{\hbar} (\hat{\bm{\sigma}} \times \mathbf{J}_s) + \theta_{SH}^A \frac{2e}{\hbar} [\hat{\bm{\sigma}} \times (\hat{\mathbf{M}} \times \mathbf{J}_s)]$$

Here, $\mathbf{J}_c$ is the charge current density, $\mathbf{J}_s$ the spin current density, $\hat{\bm{\sigma}}$ the spin polarization, $\hat{\mathbf{M}}$ the unit vector of magnetization (for anomalous contributions), and $\theta_{SH}$, $\theta_{SH}^A$ are the conventional and anomalous spin Hall angles, respectively (Abrão et al., 2023). The anomalous term is maximized when the injected spin polarization is perpendicular to the converting layer’s magnetization, leading to directionally dependent conversion efficiency.

In systems with quantum confinement and strong spin–orbit coupling, analytical and numerical models predict that AISHE strength is governed by wire orientation, subband occupancy, and the interplay between Rashba and Dresselhaus coefficients (Cummings et al., 2014). The general expressions for energy splitting and the spin current amplitude show explicit angular and density dependence, which is reversed with respect to the bulk (2DEG) case under quantum confinement.

4. Interplay with Magnetoresistive and Rectification Effects

Disentangling AISHE from competing galvanomagnetic effects (e.g., AMR, planar Hall effect, anomalous Hall effect) is crucial for accurate identification and quantification. The total measured voltage under spin pumping or AC current excitation is typically decomposed into symmetric Lorentzian or harmonic components (AISHE) and antisymmetric parts (arising from AMR or rectification), with the angular dependence (e.g., $\sin\alpha$, $\cos(2\phi)$, etc.) serving to isolate the AISHE contribution (Mosendz et al., 2010, Aon et al., 6 May 2024). In some cases, control samples with tailored magnetic anisotropy and deposition conditions (e.g., rotating versus oblique deposition) demonstrate sharply reduced AISHE signals, confirming the necessity of magnetic texture for unconventional spin current generation.

5. Band Structure, Berry Curvature, and Global/Local Anisotropy

The degree of anisotropy in the ISHE depends on whether the underlying physical process averages over the Brillouin zone or is sensitive to local band crossings/anticrossings. In epitaxial Fe₃Si, AMR and PHE show marked orientation-dependent anisotropy, governed by the directionality of carrier velocity and magnetization relative to band crossings. Conversely, the SHE and AHE—being global Berry curvature effects—average out local anisotropies and yield isotropic ISHE signals unless lower crystal symmetry or interface engineering is invoked (Soya et al., 18 Apr 2025).

First-principles approaches (Berry curvature-based Kubo calculations) corroborate large SHC anisotropy in hcp and tetragonal materials, allowing for manipulation of the sign and magnitude of the ISHE by rotation of polarization or current channel (Freimuth et al., 2010). In Dirac ferromagnets, the SHC tensor becomes axially anisotropic as a consequence of modified interband selection rules, with a discontinuity at the paramagnetic limit depending on magnetization orientation (Qu et al., 2023). A plausible implication is that direction-selective ISHE in these systems can be tailored for device-grade functionalities.

6. Quantum, Interfacial, and Nonlinear Regimes

AISHE is further modified by quantum confinement (e.g., in quantum wires, only a single occupied subband can maximize spin separation and hence inverse Hall voltage), by interfacial engineering (e.g., tailoring spinterfaces, curvature-induced $\pi$–$\sigma$ hybridization, or anisotropic spin mixing conductance), and by nonlinear and disorder-free conditions such as those possible in ultracold atomic gases with synthetic spin–orbit coupling. Nonlinear inverse spin galvanic effects are predicted to yield longer spin polarization lifetimes and more tunable anisotropy than in dissipative, disordered solid-state platforms (Miatka et al., 2019).

Interfacial phenomena extend the range of detectable AISHE. For example, spin Hall magnetoresistance in altermagnetic/Pt interfaces displays angular dependence linked not just to magnetic moment orientation but to inherent altermagnetic ordering, providing evidence for a new category of high-symmetry spintronic material responses (Leiviskä et al., 31 Jan 2025).

7. Device Applications and Future Directions

AISHE is a critical tool for quantifying spin Hall angles, spin mixing conductance, spin diffusion lengths, and spin–orbit torque efficiencies. The ability to control the magnitude and directionality of spin-to-charge conversion via engineered anisotropies opens prospects for logic devices, sensors, and reconfigurable memory elements utilizing directional spin current flow and its electrical detection.

Future research aims at quantitative modeling of interfacial anisotropy effects, the role of altermagnetic order and magnon dispersion, device architectures exploiting quantum confinement, THz-frequency probing to separate different spin current dissipation channels, and systematic deployment of control over magnetic and crystallographic texture to maximize or tune AISHE response (Leiviskä et al., 31 Jan 2025, Aon et al., 6 May 2024, Soya et al., 18 Apr 2025).

The collective body of work demonstrates that identification and harnessing of anisotropic inverse spin Hall effects demand rigorous symmetry analysis, comprehensive band structure considerations, and precise measurement techniques capable of distinguishing the complex superposition of spin–orbit coupled responses.