Layer Orbital Hall Effect in 2D Materials

• Layer Orbital Hall Effect is the generation of a transverse, charge-neutral flow of orbital angular momentum in response to an electric field in layered systems.
• It arises from nontrivial orbital textures and broken crystal symmetries, captured by tight-binding models and effective Hamiltonians near Dirac points.
• The effect offers robust, enhanced orbital currents compared to spin Hall responses, with promising applications in orbitronic devices and information transport.

The Layer Orbital Hall Effect (OHE) denotes the phenomenon in which a transverse, charge-neutral flow of orbital angular momentum (OAM)—rather than spin—is generated in response to a longitudinally applied electric field in crystalline solids. Distinguished from the spin Hall effect (SHE), which relies on spin–orbit coupling (SOC) for efficient charge-to-spin conversion, the OHE can exist independently of SOC and derives fundamentally from the quantum geometry of orbital degrees of freedom and the underlying crystal symmetry. In various two-dimensional (2D) and multi-orbital layered systems, this effect arises due to nontrivial momentum-space "orbital textures," crystal-field-induced orbital splitting, and band structure topology, resulting in orbital Hall currents whose magnitudes can surpass those of the spin Hall current. The OHE's robust manifestation in both metallic and insulating phases—especially in technologically relevant layered van der Waals materials and engineered thin films—has made it an active topic in condensed matter physics, quantum transport, and orbitronics.

1. Microscopic Origin and Model Hamiltonians

The OHE in layered and 2D multi-orbital systems is rooted in the interplay between orbital hybridization, lattice symmetry, and the resulting momentum-dependent distribution of OAM—collectively referred to as "orbital textures." Minimal tight-binding Hamiltonians, particularly on the honeycomb lattice with $p_x$ and $p_y$ orbitals, capture the essential physics. For example, the Hamiltonian incorporates nearest-neighbor hopping ($t_{ij}^{\mu\nu}$), on-site energies (including a possible sublattice potential $V_{AB}$), and atomic SOC: $$\mathcal{H} = \sum_{\langle ij \rangle} \sum_{\mu\nu s} t_{ij}^{(\mu\nu)} p_{i,\mu,s}^\dagger p_{j,\nu,s} + \sum_{i\mu s} ( \epsilon_i + \lambda_I \ell_{z,\mu\mu} \sigma^z_{ss} ) p_{i,\mu,s}^\dagger p_{i,\mu,s}$$ where $\mu, \nu = x, y$ label orbitals and $s$ is the spin. The states $|p_+\rangle \propto p_x + i p_y$ and $|p_-\rangle \propto p_x - i p_y$ define the SU(2) "orbital pseudo-spin" basis.

A key feature is an effective low-energy Hamiltonian near Dirac (K/K′) points: $$\mathcal{H}_\text{eff} = -\hbar v_F (k_x \sigma_x + \tau k_y \sigma_y) + s \lambda_I \ell^z + V_{AB} \sigma^z + H_\ell$$ with $H_\ell$ containing a "Dresselhaus-like" orbital coupling term. The nonzero in-plane orbital textures $\langle \ell_x \rangle, \langle \ell_y \rangle$ in momentum space—arising from these Dresselhaus-type terms—are crucial for generating a robust transverse OAM current even when conventional edge-state physics (QSH or QAH) is absent.

2. Linear Response Theory and Orbital Hall Conductivity

The orbital Hall conductivity (OHC) is calculated via Kubo linear-response formalism: $$\sigma_{OH}^{z} = \frac{e}{\hbar} \sum_{n \neq m} \sum_s \int_{BZ} \frac{d^2k}{(2\pi)^2} (f_{mk} - f_{nk}) \Omega^{(\ell^z)}_{n m k s}$$ where $\Omega^{(\ell^z)}_{n m k s}$ is a Berry curvature-like interband term built from the matrix elements of the orbital current operator $J_y^{(\ell^z)} = \frac{1}{2}\{ \ell^z, v_y \}$ and the velocity operator $v_y$.

Notably, the OHE persists in trivial insulating phases—unlike the spin Hall effect, which typically vanishes unless the system is topological (hosting edge modes). In scenarios with sublattice symmetry breaking or next-nearest-neighbor hopping (breaking particle-hole and electron-hole symmetry), the in-plane orbital textures fail to cancel over the Brillouin zone, yielding a finite OHC even in the absence of edge transport.

3. Orbital Textures, Valley Selectivity, and Berry Curvature

The concept of "orbital textures" refers to the momentum-resolved expectation values of OAM within the Brillouin zone, denoted as $\langle \ell_{\alpha} \rangle$ ($\alpha=x,y$). In simple models with only nearest-neighbor hopping and particle-hole symmetry, these in-plane components cancel out when integrated over all occupied states. However, symmetry-breaking terms lift this cancellation.

The orbital Hall effect in this context is characterized by:

• Strong valley selectivity: The OHC is sharply localized near $K$ and $K'$ points in the Brillouin zone, and the sign of the response is tied to valley and orbital indices.
• Intrinsic contribution: The "orbital Berry curvature" is generally associated with differences in the occupation between bands derived from different orbitals—akin to the anomalous Hall effect but for the orbital degree of freedom rather than charge or spin.

4. Comparison with the Spin Hall Effect and Robustness

Unlike the SHE, which requires significant spin–orbit coupling (SOC) to couple spin and orbital degrees of freedom and is quantized only in topological phases, the OHE can be:

• Larger in magnitude than the SHE in many parameter regimes; in certain models the OHC plateau exceeds the quantized spin-Hall conductance found in quantum spin Hall states.
• Nonquantized: The OHC, being determined by the nontrivial orbital texture, is nonuniversal and not protected by a topological invariant, even when the SHE is quantized.
• Robust against disorder: Anderson-type disorder (modeled via numerical Chebyshev expansion of the Kubo–Bastin formula) does not destroy the OHE plateaux, even for strong scattering. While edge states associated with the SHE get washed out, the bulk OHC remains nearly unchanged.

A direct comparison puts this in context: | Property | Spin Hall Effect (SHE) | Orbital Hall Effect (OHE) | |-------------------------|-------------------------------|-------------------------------------------| | Requires strong SOC | Yes | No (Dresselhaus-like OAM textures suffice)| | Quantized in insulator | Only in QSH topological regime| No, OHC is generically nonquantized | | Magnitude | $\sigma_{SH} \sim e^2/h$ | Often $\sigma_{OH} > \sigma_{SH}$ | | Robustness to disorder | Depends on edge state nature | Robust (bulk-driven) |

The OHE thus offers an independent and potentially more efficient information carrier than spin, applicable also in trivial insulators.

5. Layer Geometry, Candidate Materials, and Device Implications

The OHE is particularly salient in multi-orbital 2D materials (such as honeycomb-lattice monolayers) and has been discussed for flat bismuthene on SiC—a platform where only $p_x$ and $p_y$ orbitals dominate the low-energy physics. Identifying and engineering materials with large Dresselhaus-type orbital couplings and broken particle-hole symmetry (via sublattice potential or next-nearest-neighbor hoppings) is a promising route to maximize OHC for device applications.

The theoretical prediction is that such OHE currents can be harnessed for:

• Nonvolatile information transport and switching in “spin–orbitronic” or “orbitronic” devices.
• Pure orbital current injectors: Devices capable of injecting a charge-neutral transverse flow of OAM, even in conventional (non-topological) insulators.
• Robust orbital-torque transducers that exploit the OHE where the SHE is weak or absent.

Moreover, the identification of robust OHC in trivial and disordered phases broadens the scope of possible device architectures beyond those limited to topological insulators or large intrinsic SOC.

6. Theoretical Extensions and Open Directions

The findings provide a basis for several extensions:

• Derivation of criteria for the existence of robust orbital textures—and corresponding nonvanishing OHE—in general classes of symmetry-broken layered materials.
• Analytical construction of effective Hamiltonians at Dirac and other high-symmetry points incorporating both intrinsic SOC and crystal-field effects, including their Dresselhaus-like orbital analogs.
• Systematic paper of disorder, edge roughness, and multilayer effects on the persistence and control of orbital currents.
• Exploration of possible experimental probes, such as Kerr rotation microscopy or ARPES with OAM sensitivity, to observe current-induced OAM accumulation at edges.

In summary, the Layer Orbital Hall Effect in multi-orbital 2D systems arises from Dresselhaus-like in-plane orbital textures and persists across both metal and insulator regimes, providing a robust, tunable, and in many cases dominant angular momentum transport channel beyond the conventional spin degree of freedom. The phenomenon offers profound implications for fundamental transport theory and the design of future quantum and orbitronic information technologies.