Spin Hall Effect (SHE) Overview

• Spin Hall Effect is a spin–orbit coupling phenomenon where an electric current induces a transverse spin current and results in edge spin accumulation.
• It is observed in various materials such as heavy metals, antiferromagnets, and moiré bilayers using optical and transport techniques to measure spin Hall conductivity.
• Material engineering through impurity tuning, structural design, and band engineering enables precise control over the spin-to-charge conversion efficiency for spintronic devices.

The spin Hall effect (SHE) is defined as a collection of relativistic spin–orbit coupling phenomena in which an electrical current generates a transverse spin current, or equivalently, a transverse charge current is generated by a spin current. Unlike the ordinary Hall effect, which yields charge accumulation across a conductor, the SHE produces spin accumulation at the sample edges, enabling interconversion between spin and charge currents without magnetic materials or external magnetic fields. The effect can arise from both intrinsic mechanisms—linked to the electronic band structure and the Berry curvature—and extrinsic mechanisms—arising from spin-dependent scattering off impurities. The SHE is now ubiquitous in the field of spintronics as a standard way to generate, detect, and manipulate spin currents in various materials systems and geometries, from heavy-metal films to moiré-engineered van der Waals bilayers and quantum gases (Sinova et al., 2014).

1. Microscopic Mechanisms: Intrinsic and Extrinsic Spin Hall Effects

The intrinsic SHE arises from the momentum-space Berry curvature of Bloch bands, which provides a transverse velocity to electrons depending on their spin orientation under an applied electric field. The Kubo–Středa formula gives the spin Hall conductivity (SHC) as

$$\sigma_{xy}^s = (e/\hbar)\sum_{n}\int_{\text{BZ}}\frac{d^{3}k}{(2\pi)^3}\,f_n(\epsilon_{n\mathbf{k}})\,\Omega_{n,xy}^s(\mathbf{k}),$$

where $\Omega_{n,xy}^s(\mathbf{k})$ is the spin Berry curvature for band $n$, and $f_n$ is the Fermi–Dirac occupation (Sinova et al., 2014).

Extrinsic mechanisms consist of skew scattering—where spin–orbit coupling in the impurity potential causes spin-dependent asymmetric scattering—and the side-jump effect, in which spin–orbit interaction yields a lateral displacement of the electron wave packet during each scattering event. In systems such as Cu or Ag doped with heavy atoms (Bi, Pb), the skew-scattering mechanism can dominate and result in large spin Hall angles (e.g., $\theta_{\text{SH}} \sim -0.26$ for CuBi), greatly exceeding those in noble metals where the Berry curvature is smaller (Niimi et al., 2014, Fert et al., 2010).

2. Material Platforms and Engineering of the Spin Hall Effect

The SHE is present in a broad range of materials, including:

• Heavy Metals and Alloys: Pt, Au, Ta, and W exhibit large intrinsic SHE due to strong spin–orbit coupling in their d bands. Measured spin Hall conductivities are $$\sigma^s_{xy} \approx 2400\,(\hbar/e)\Omega^{-1}\mathrm{cm}^{-1}$$ for Pt and $-400\,(\hbar/e)\Omega^{-1}\mathrm{cm}^{-1}$ for $\beta$-Ta (Zhang et al., 2017). Doping light metals with heavy impurities (CuBi, AgBi) leverages extrinsic mechanisms to achieve tunable and large $\theta_{\text{SH}}$ (Niimi et al., 2014).
• Antiferromagnets and Fluctuation-Driven Systems: Antiferromagnetic Cr exhibits a fluctuation spin Hall effect (FSHE), where critical spin fluctuations near the Néel temperature ($T_N$) act as scattering sources and enhance $\theta_{\text{SH}}$. Near $T_N$, $\theta_{\text{SH}}$ in Cr reaches $-0.34$, rivaling values found in heavy metals while maintaining much lower resistivity (Fang et al., 2023). Skew-scattering and side-jump mechanisms underpin this spin-fluctuation enhancement.
• Noncollinear and Chiral Magnets Without SOC: Coplanar noncollinear antiferromagnets (e.g., Mn$_3X$ with $X=$ Ga, Ge, Sn) can support an intrinsic SHE by breaking spin rotation symmetry even in the absence of spin–orbit coupling. Calculated SHCs for these systems are on the order of $90$–$613\,(\hbar/e)\Omega^{-1}\mathrm{cm}^{-1}$ (Zhang et al., 2017).
• 5d Transition-Metal Anti-Perovskites: Materials such as Ir$_3$WC and Pt$_3$LaP show giant intrinsic SHEs due to atomic spin–orbit coupling–induced hybridization between $d_{x^2-y^2}$ and $d_{xy}$ orbitals, resulting in sharply peaked Berry curvature hot spots (Jadaun et al., 2020).
• Moiré van der Waals Bilayers: AB-stacked MoTe$_2$/WSe$_2$ moiré bilayers display a giant, gate-tuneable intrinsic SHE with nearly saturated spin accumulation and spin diffusion lengths up to $10\,\mu$m, associated with miniband Berry curvature hotspots near Chern insulating states (Tao et al., 2023).
• Magnetic Interfaces and T-Odd Spin Hall Effects: In Fe$_3$GeTe$_2$/MoTe$_2$ van der Waals heterostructures, an interfacial, time-reversal-odd spin Hall effect driven by a magnetization-dependent spin current dipole is observed, reaching interfacial spin Hall angles up to $\theta_{\text{MSHE}}\sim0.67$, twice that of bulk MoTe$_2$ (Dai et al., 2024).
• Quantum Gas and Cold Atom Realizations: The SHE has also been realized in ultracold atomic gases with synthetic spin–orbit coupling, where spin-dependent Lorentz forces generate spatial spin separation. These systems can be engineered to probe topological quantum spin Hall phases and velocity-insensitive spin transistors (Beeler et al., 2013).

3. Experimental Signatures, Probes, and SHE Metrics

Edge spin accumulation is a direct signature of the SHE, observable via optical Kerr or magnetic circular dichroism imaging, transport measurements in Hall-bar and lateral spin valve geometries, and in magnetic switching by spin–orbit torques:

• Optical Probes: Time-resolved Kerr rotation microscopy tracks both the dynamical build-up and decay of spin accumulation, enabling extraction of spin relaxation times, diffusion lengths, and the distinction between bulk and boundary contributions (0806.0019, Sinova et al., 2014).
• Transport and Spin-Torque Measurements: In heavy-metal/ferromagnet heterostructures (e.g., Pt/Co), the SHE induces an antidamping torque sufficient for efficient magnetization switching at current densities as low as $j_{\text{e}}\sim5\times10^7\,\text{A/cm}^2$, competitive with conventional spin-transfer torque devices (Liu et al., 2011).
• Spin Hall Angle ($\theta_{\text{SH}}$) and Spin Hall Conductivity ($\sigma_{\text{SH}}$): The spin Hall angle quantifies the charge-to-spin current conversion efficiency, defined as $$\theta_{\text{SH}} = j_s / j_c = \sigma_{\text{SH}}/\sigma$$, where $j_s$ and $j_c$ are spin and charge current densities (Sinova et al., 2014, Liu et al., 2011).
• Nonlocal Geometry and Inverse SHE: Lateral spin valves facilitate separation of charge and spin transport paths, and the inverse spin Hall effect (ISHE) measurement provides a direct electrical readout of the spin current absorption in candidate SHE materials (Niimi et al., 2014).
• Critical Fluctuation Enhancement: Near magnetic criticality ($T\to T_N$), correlated antiferromagnetic fluctuations can dramatically enhance extrinsic SHE via increased side-jump and skew-scattering rates, as demonstrated in Cr (Fang et al., 2023).

4. Topological and Higher-Rank Generalizations

Recent theoretical work highlights the emergence of higher-rank spin Hall effects in large-spin systems (spin-$F\geq 1$), in which rank-1 (vector) spin currents vanish, leaving universal rank-2 (tensor) spin Hall conductivities. In spin-1 models with intrinsic SOC, a universal rank-2 SHE with $\sigma_{xy}^{(2)} = e/8\pi$ is predicted, and generalizations to spin-3/2 yield higher universal values (Hou et al., 2020). Experimental schemes using pseudospin-1 ultracold fermions are proposed, and such tensor-spin currents may offer new paradigms for multiplexed spintronic logic and dissipationless spin transport.

In magnetic insulators, especially those with strong spin–orbit coupling and broken time-reversal symmetry, the SHE can be studied via the electric-field-induced spin-density polarization rather than conventional spin currents, with distinct bulk and boundary contributions determined by Berry-phase and gauge-dependent terms (Chen et al., 2018).

5. Control, Engineering, and Practical Devices

Material engineering of the SHE includes:

• Impurity Engineering: Tuning the impurity type and concentration in host metals allows the magnitude and sign of the SHE to be systematically controlled, leveraging skew-scattering and side-jump mechanisms as in CuBi and AgBi alloys (Niimi et al., 2014, Fert et al., 2010).
• Surface Roughness and Geometry: Self-affine fractal surface roughness in thin metallic films can enhance the spin Hall angle by an order of magnitude relative to smooth surfaces, particularly by amplifying the side-jump contribution through increased high-momentum scattering (Hajzadeh et al., 2017).
• Band Structure and Orbital Engineering: Moiré superlattice engineering in van der Waals bilayers and tailored inversion-symmetry-breaking in complex oxides can produce large Berry curvature hotspots and enable gate-tunable topological states with giant SHEs (Tao et al., 2023, Mizoguchi et al., 2014).
• Device Applications: Spin–orbit torque MRAM, logic, and neuromorphic computation devices exploit the efficient charge–spin conversion and non-volatility made possible by large-SHE materials and interfacial mechanisms. In-memory computation in interfacial-MSHE devices has been demonstrated for binary convolutional neural networks, achieving energy efficiencies on the order of sub-fJ/MAC (Dai et al., 2024).

6. Extensions: Photonic and Non-Electronic Spin Hall Effects

The photonic spin Hall effect is the optical analogue of the electronic SHE, observed as spin-dependent beam shifts at interfaces due to spin–orbit interaction in light. This effect, though not associated with charge transport, enables precision metrology for thin-film thickness, graphene layer counting, topological insulator axion angles, and magneto-optical constants (Zhou et al., 2014). Weak measurement amplification enhances the detection of nanometric or even picometric spin-dependent displacements, providing all-optical routes to parameter retrieval not easily accessible by electronic means.

7. Outlook and Future Directions

The breadth of material systems supporting the SHE—from transition metals and their alloys, noncollinear antiferromagnets, van der Waals heterostructures, to topological and magnetic insulators—underscores its fundamental and technological significance. Recent progress in fluctuation-driven SHE, higher-rank spin currents, atomtronic devices, and interfacial T-odd spin Hall effects points toward a future of highly tunable, low-power, and multifunctional spin-based information processing platforms. The interplay between symmetry, topology, and local dynamical correlations in dictating the magnitude and functionality of the SHE will continue to be central themes in both fundamental studies and spintronic device engineering (Tao et al., 2023, Zhang et al., 2017, Fang et al., 2023, Dai et al., 2024).