Orbital Hall Effect: Fundamentals & Applications

• The orbital Hall effect is a transport phenomenon where an electric field induces a transverse flow of orbital angular momentum, governed by orbital hybridization and Berry curvature.
• It emerges from intrinsic band topology, extrinsic impurity scattering, and mesoscopic fluctuations, offering insights into quantum transport in diverse materials.
• Experimental studies in metals, semiconductors, and topological insulators demonstrate robust orbital currents, paving the way for ultra-low-power orbitronics applications.

The orbital Hall effect (OHE) is the phenomenon whereby an applied electric field induces a transverse flow of orbital angular momentum (OAM) in condensed matter, optical, or magnonic systems. OHE is analogous to the spin Hall effect but is fundamentally distinct: it can occur without spin-orbit coupling, in the absence of a net spin transport, and is governed by details of orbital hybridization, Berry curvature, and scattering. OHE has emerged as a central topic in the fields of topological transport, quantum metals, two-dimensional materials, and orbitronics.

1. Microscopic Theory and Operator Formalism

The OHE is typically formalized in the linear-response framework for crystalline solids, where the central quantity is the orbital Hall conductivity, defined via

$$J^L_i = \sigma_{OH}\,\epsilon_{ijk}\,E_j\,\hat{n}_k,$$

with $J^L_i$ the OAM current, $\sigma_{OH}$ the orbital Hall conductivity, $E_j$ the applied electric field, and $\hat{n}_k$ the axis of polarization. For non-magnetic crystals, the OHE is not contingent on sizeable spin–orbit coupling and can manifest even in inversion-symmetric materials (Pezo et al., 2022).

The microscopic OAM current operator is conventionally defined as the symmetrized product

$\mathcal{J}_i^z = \frac{1}{2}\{L_z, v_i\}$

where $L_z$ is the (microscopic) $z$-component of orbital angular momentum and $v_i$ the velocity operator. In the Bloch basis, the Kubo formula for the intrinsic OHE is

$$\sigma^{O}_{xy} = -2\hbar e \int_{BZ} \frac{d^d k}{(2\pi)^d} \sum_n f(\epsilon^n_{\mathbf{k}}) \Im \sum_{m \neq n} \frac{\langle u^n_{\mathbf{k}} | \mathcal{J}^z_x | u^m_{\mathbf{k}} \rangle \langle u^m_{\mathbf{k}} | v_y | u^n_{\mathbf{k}} \rangle}{(\epsilon^n_{\mathbf{k}} - \epsilon^m_{\mathbf{k}})^2}$$

directly analogous to the spin Hall case but with $L_z$ replacing spin (Pezo et al., 2023, Choi et al., 2021).

Two distinct contributions to the OAM operator enter: the intra-atomic part (acting within atomic sites; atomic center approximation, ACA) and the inter-atomic (itinerant) part (arising from the self-rotation of the Bloch wavepacket over the entire unit cell) (Pezo et al., 2022, Pezo et al., 2023). The full modern theory of orbital magnetization establishes that only the total (intra + inter) operator yields gauge-invariant, physically observable OHE.

2. Intrinsic, Extrinsic, and Mesoscopic Regimes

Three major mechanisms drive the OHE:

• Intrinsic OHE: Arises from the Berry curvature associated with the OAM operator, entirely due to the underlying band structure and orbital textures in $\mathbf{k}$ space (Pezo et al., 2022, Choi et al., 2021). The orbital Berry curvature is

$$\Omega^{L_z}_{n,xy}(\mathbf{k}) = -2\hbar^2\,\mathrm{Im}\sum_{m \neq n} \frac{\langle n| J_x^{L_z} | m \rangle \langle m | v_y | n \rangle}{(\epsilon_{n\mathbf{k}} - \epsilon_{m\mathbf{k}})^2}.$$

Chiral orbital textures in $\mathbf{k}$-space (e.g., Dresselhaus- or Rashba-like structures) are crucial for a large intrinsic OHC (Canonico et al., 2019).

• Extrinsic OHE: Dominates in the presence of significant disorder and strong doping, driven by Fermi-surface processes—skew scattering and side-jump terms mediated by short-range impurity potentials. Quantum-kinetic analyses show that, at typical carrier densities, up to 95% of the OHE in systems such as doped graphene or TMDs is extrinsic—the sum of skew scattering and side-jump terms—while the intrinsic contribution is subdominant (Liu et al., 2023).
• Mesoscopic Fluctuations: In chaotic or diffusive finite devices, the OHE exhibits universal sample-to-sample fluctuations determined by random matrix theory (RMT). In four-terminal Landauer–Büttiker geometries, the OHE amplitude (root-mean-square current) displays universal values: $\mathrm{rms}\,I^o = (e^2 V/h)\times 0.36$ (weak SOC, COE symmetry), and $0.18$ (strong SOC, CSE symmetry) (Fonseca et al., 2023). These translate into two universal relations between the orbital Hall angle and conductivity:

$$\mathrm{rms}\,\Theta_{\rm OH}\,\sigma = \begin{cases} 0.36 & \text{(weak SOC)} \ 0.18 & \text{(strong SOC)} \end{cases}$$

verified both in tight-binding numerics and experiment.

3. Topological and Geometric Aspects

The OHE connects deeply to the topology and symmetry of the Bloch bands, though this link is more nuanced than for the quantized spin or charge Hall effects.

• Projected Orbital Angular Momentum (POAM) Topology: In 2D group-IV monolayers (e.g., Ge, Sn, Pb), the topology of the projected orbital angular momentum spectrum defines Chern numbers from the windings of Wannier charge centers, yielding quantized plateaux for the OHC inside band gaps:

$$\sigma_{xy}^{L_z} \approx \frac{e}{h}\sum_i \lambda_i C_i$$

where $\lambda_i$ are POAM eigenvalues and $C_i$ their sector Chern numbers (Wang et al., 2024). This engenders a bulk-boundary correspondence with chiral edge states in the POAM spectrum and orbital texture at sample edges.

• Triangular and Kagome Lattices, s-Electron OHE: Even in minimal models with only s orbitals (l=0), geometric current loops on, e.g., the kagome lattice, can produce a robust OHE via inter-atomic (offsite) circulations, demonstrated by cycloid trajectories and chiral edge states transporting OAM (Busch et al., 2023). On the triangular lattice, orbital insulator phases with quantized OHCs and Lz-carrying edge states (but no spin-Hall response) are realizable (Barbosa et al., 2023).
• Quantum Hall Edge Currents and OHE: In quantum Hall systems, each chiral edge state carries a definite orbital polarization, leading to a transverse OAM current whose Hall resistivity scales as $B^2$ and dominates the linear charge response at high fields (Göbel et al., 2024).

4. Experimental Observation and Device Implications

The OHE has been directly confirmed in multiple materials via state-of-the-art experiments:

• Magneto-optical Detection: In Cr, current-induced in-plane orbital accumulation transverse to the current was detected by longitudinal MOKE, with the OHC and the orbital diffusion length $l_o=6.6 \pm 0.6$ nm extracted from thickness-dependent measurements and ab initio comparisons (Lyalin et al., 2023). Comparable techniques yielded semiquantitative agreement in Ti (Choi et al., 2021).
• Spin Pumping and Orbital-to-Spin Conversion: In transition metal heterostructures, spin- and orbital-current-to-charge conversion efficiencies were contrasted using spin-pumping-FMR. The orbital response dominates over the spin response across 19 transition metals, with the OHC exceeding the SHC by typically factors of 2–5. Even in weak-SOC metals such as Ti, Cr, and Mo, OHE-driven currents can greatly surpass SHE-driven signals (Costa et al., 10 Jun 2025). The drift-diffusion model incorporating coupled OAM and spin relaxation, as well as 4f spacer-mediated conversion, quantitatively accounts for the OHE/SHE interplay and the observed spin-orbit torques (Sala et al., 2022).
• Heterostructures and Topological Insulators: In 3D topological insulators (e.g., Bi$_2$Se$_3$, Bi$_2$Te$_3$), the bulk OHE exceeds the bulk SHE by up to three orders of magnitude, owing to the large orbital moments in the Bloch states. This substantially enhances the efficiency of OHE-driven orbital torques in adjacent ferromagnets, supporting experimentally observed large switching efficiencies (Cullen et al., 29 Jan 2025).
• Ultrafast Regime: Ultrafast OAM currents and edge accumulations consistent with OHE can be driven by femtosecond laser pulses in metallic nanoribbons, with dynamic signatures distinct from steady-state charge currents (Busch et al., 2023).

5. Material Systems: Roles of Band Structure, Topology, and Disorder

The nature and magnitude of the OHE depend sensitively on material class and microscopic parameters:

• Wide-gap Semiconductors: OHE is predominantly intra-atomic (atomic center) in origin, with ACA sufficing for calculation; 2D TMDs (e.g., MoS$_2$) exemplify this regime (Pezo et al., 2022, Pezo et al., 2023). The clean limit yields robust OHC plateaux in both trivial and quantum spin Hall phases (Canonico et al., 2019, Barbosa et al., 2023).
• Narrow-gap Semiconductors and Metals: The OHE is dominated by inter-atomic contributions arising from delocalized Wannier functions and band-mixing; ACA overestimates the OHC and may even get the sign or magnitude wrong for transition metals (Pt, V) and narrow-gap IV–VI compounds (SnTe, PbTe) (Pezo et al., 2022).
• Role of Disorder: In gapped 2D Dirac materials, the intra-atomic (single-band) OHC is robust against moderate disorder, while the inter-atomic (itinerant) component is fragile and decays rapidly (Pezo et al., 2023). In metallic systems, extrinsic mechanisms (skew-scattering and side jump) dominate the OHE at experimentally relevant dopings, accounting for $\sim$95% of the total OHC (Liu et al., 2023).
• Gap and Topology Dependence: The OHC scales inversely with the band gap, $\sigma_{OH} \propto 1/\Delta$, regardless of whether the gap is topological or trivial. Contrary to the charge Hall effect, OHE is not quantized and is only weakly influenced by the topological class of the band structure (Pezo et al., 2023).

6. Extensions to Magnons and Structured Light

• Magnon OHE: In honeycomb antiferromagnets, magnons exhibit an intrinsic OHE arising from the magnon orbital Berry curvature. This effect is rooted in the lattice geometry and exchange interaction—independent of spin–orbit coupling—and is several orders of magnitude larger than the magnon spin-Nernst effect. Magnetoelectric coupling permits detection via edge voltages (Go et al., 2023).
• Orbital Hall Effect of Light: In optical materials, the OHE describes the transverse shift of light possessing orbital angular momentum $\ell$ and/or spin $\sigma$. The deflection angle follows a universal law $\theta_{\sigma,\ell} \propto 2\sigma + \ell$—highlighting the fundamentally different roles of spin and OAM in light propagation and their disparate couplings to refractive-index gradients (Qiu et al., 2024).

7. Applications, Open Problems, and Perspective

Emergent orbitronics exploits OHE-derived currents and accumulations for ultra-low-power information processing, magnetic switching, and non-dissipative interconnects. OHE-driven torques in heterostructures offer high efficiency, especially when augmented via orbital-to-spin conversion in proximity to strong SOC materials. Robustness in light-element systems, high-temperature antiferromagnets, and nanostructured circuits is under continuous exploration (Costa et al., 10 Jun 2025, Canonico et al., 2019, Sala et al., 2022). Disorder-engineering, interface chemistry, and topological band design remain active areas for increasing OHE observables and enabling new device functionalities.

In summary, the orbital Hall effect constitutes a universal transport phenomenon originating from the interplay of orbital textures, band topology, scattering, and geometry. Its signatures span mesoscopic fluctuation regimes, robust bulk and edge currents, topological and trivial insulators, and extend to bosonic and photonic quasiparticles, positioning OHE at the core of contemporary condensed matter physics, quantum materials research, and orbitronics (Costa et al., 10 Jun 2025, Fonseca et al., 2023, Wang et al., 2024, Liu et al., 2023, Pezo et al., 2023).