Surface Chirality Momentum Locking

• Surface chirality momentum locking is a phenomenon where a material’s inherent handedness deterministically links momentum to intrinsic properties like spin or orbital angular momentum.
• It arises from broken spatial inversion and mirror symmetries, leading to robust, symmetry-protected responses in electronic, superconducting, and photonic systems.
• Experimental evidence from transport measurements, ARPES imaging, and tunable chiral plasmonic effects underscores its potential in applications like spintronics and quantum photonics.

Surface chirality momentum locking refers to the phenomenon where chirality—structural handedness at the atomic, molecular, or lattice scale—enforces a deterministic, symmetry-protected correlation (“locking”) between an electron’s, phonon’s, or photonic mode’s momentum and some intrinsic degree of freedom. This degree of freedom can be spin, orbital angular momentum, charge current, or other vectorial quantities, and the locking takes place primarily at surfaces or interfaces where spatial inversion and/or mirror symmetries are broken. The resulting locked textures are central to the behavior of topological chiral semimetals, chiral superconductors, organic chiral molecules, non-Hermitian photonic interfaces, and tailored metasurfaces. Surface chirality momentum locking supports robust nonreciprocal transport, spin/charge accumulation, orbital currents, and a host of symmetry-enforced responses that underlie many modern phenomena in orbitronics, spintronics, optoelectronics, and quantum photonics.

1. Fundamental Principles and Theoretical Frameworks

Surface chirality momentum locking emerges from the interplay between structural chirality, spin-orbit coupling (SOC), non-Hermitian perturbations, and boundary conditions that break parity and mirror symmetries. In crystalline systems, absence of inversion or mirror symmetry allows for new SOC terms not permitted in centrosymmetric or merely polar materials. For example, in chiral lattices, the effective SOC can contain a $k\cdot\sigma$ component that aligns the spin parallel to momentum (as opposed to the Rashba effect, which locks spin orthogonally) (Sato et al., 28 Jan 2025, Krieger et al., 2022, Polyakov et al., 2020).

A general Hamiltonian incorporating spin-momentum locking in a chiral, non-centrosymmetric system takes the form

$$H_0 = \sum_{k,\alpha} \epsilon_k\,c_{k\alpha}^\dagger c_{k\alpha} + H_{\rm soc} + H_{\rm pair},$$

where $H_{\rm soc}$ includes not only traditional Rashba (orthogonal) terms but also terms allowed by crystal chirality, such as $k\cdot\sigma$. The pairing term in chiral superconductors often exhibits parity mixing, quantified by

$r_t = 2V_u/(V_g + V_u),$

with $V_g$ and $V_u$ the interaction strengths in the singlet and triplet channels, respectively.

In non-Hermitian extensions relevant to the chiral-induced spin selectivity (CISS) effect, the effective Hamiltonian contains an imaginary "twin-pair" exchange term,

$$H = \frac{\hat{p}^2}{2m} + V(x) - i\alpha\,\hat{\sigma} \cdot \hat{p},$$

which breaks parity (P) and time-reversal (T) individually, but preserves combined PT symmetry (Theiler et al., 9 May 2025). Eigenstates of $H$ exhibit one-to-one locking of spin and momentum, enforced by the chiral symmetry.

In topological chiral semimetals, the low-energy bulk Hamiltonian features multifold nodes with monopole-like textures of spin or orbital angular momentum (OAM),

$$H_\Gamma(\mathbf{k}) = v (\mathbf{k} \cdot \mathbf{J}),$$

where $\mathbf{J}$ is a spin-1 or spin-3/2 matrix, yielding $$\langle \mathbf{S}(\mathbf{k}) \rangle \propto \mathbf{k}/|\mathbf{k}|$$ (parallel locking) and similarly for OAM (Hagiwara et al., 2024, Yang et al., 2023, Krieger et al., 2022).

For surface electromagnetic waves (SPPs, chiral surface modes), the spin density $$\mathbf{S} = (\varepsilon_0/4\omega)\,\mathrm{Im}(\mathbf{E}^* \times \mathbf{E})$$ is locked to the surface momentum via the right- or left-hand rule, depending on the material's dispersive properties (Shi et al., 2022, Yu et al., 2022).

2. Manifestations in Electronic, Photonic, and Composite Systems

Electronic and Superconducting Systems

Chiral organic superconductors such as $\kappa$-(BEDT-TTF)$_2$Cu(NCS)$_2$ (space group $P2_1$) display record-high electrical magnetochiral anisotropy (EMChA) and nonreciprocal superconducting diode effects, traced to a sturdy chirality-driven spin–momentum locking. The EMChA coefficient $\gamma'$ can exceed that in heavy-element Rashba systems by orders of magnitude despite nominally weak atomic SOC, indicating an effective SOC in the chiral phase that is 100–1000$\times$ the bare value (Sato et al., 28 Jan 2025). In these materials, the extracted spin-momentum rigidity tensor $\rho_{ij}$ far exceeds expectations from atomic-scale SOC, highlighting the role of chirality and possibly exchange-driven mechanisms.

Topological chiral semimetals (e.g., CoSi, RhSi, PtGa, PdGa; space group $P2_13$) house bulk multifold fermions (spin-1, spin-3/2) whose spin or OAM exhibits pure radial (monopole-like) locking to momentum, reflected at the surface as Fermi arc states with the OAM or spin vector strictly parallel to (or, for specific orbitals, orthogonal to) the arc momentum (Hagiwara et al., 2024, Krieger et al., 2022, Yang et al., 2023). This is contrasted with the orthogonal locking in conventional Rashba or topological insulator surface states.

At interfaces, the CISS effect is a direct manifestation of non-Hermitian chirality-induced exchange, producing robust spin accumulation (“spin voltage”) and enantiomer-selective spin filtering, even in the absence of significant atomic SOC or magnetic order (Theiler et al., 9 May 2025).

Photonic and Plasmonic Systems

Surface plasmon polaritons (SPPs) at metal–dielectric interfaces naturally possess spin–momentum locking: the transverse spin of the evanescent field is deterministically tied to the in-plane propagation vector (Kalhor et al., 2015, Yu et al., 2022). In chiral photonic interfaces—such as chiral-gain media—the interplay between chiral non-Hermiticity and SPP spin–momentum locking leads to gain-momentum locking: one propagation direction is selectively amplified, while the opposite is attenuated, creating robust unidirectional edge-wave propagation and selective orbital angular momentum emission (Serra et al., 2024).

Metasurfaces engineered with L-shaped unit cells and tensorial polarizabilities can exhibit both in-plane and out-of-plane transverse spin components, both locked to the propagation vector via strong $x$–$y$ coupling and broken rotational symmetry. This results in anomalous spin–momentum locking: flipping the input polarization reverses both transverse spins and the allowed propagation direction (Kandil et al., 2021).

3. Topological and Symmetry Constraints

Surface chirality momentum locking is fundamentally enforced by the breaking of all mirror and inversion symmetries. In the bulk, Chern numbers and monopole charges of multifold nodes enforce the existence and connectivity of Fermi arcs with locked spin or OAM textures at the surface (Hagiwara et al., 2024, Krieger et al., 2022, Yang et al., 2023). The sign of the locking (radial or tangential) and the associated physical responses (e.g., longitudinal magnetoelectric effect, orbital Hall effect) are tied to the handedness of the crystalline enantiomer and are robust under symmetry-allowed perturbations, including weak disorder and moderate interactions.

For photonic systems, the locking of spin to momentum direction follows from Maxwell’s equations in the presence of broken mirror symmetry at interfaces; in dispersive or non-Hermitian materials, the handedness of the locking (right- or left-hand rule) is determined by sign conventions for permittivity and permeability (Shi et al., 2022, Yu et al., 2022, Serra et al., 2024).

Within superconducting and CISS devices, PT symmetry (unbroken phase) ensures real energy spectra and maintains robust spin-filter behavior, while open boundaries or engineered interfaces enable the emergence of spin skin modes localized by the non-Hermitian term (Theiler et al., 9 May 2025).

4. Surface Chirality Momentum Locking: Experimental Evidence

A wide range of experimental modalities confirm surface chirality momentum locking:

• Transport in chiral superconductors: Giant nonreciprocal magnetochiral signals and diode effects in $\kappa$-CuNCS, with $\gamma' \approx 10^6$ T$^{-1}$A$^{-1}$ near $T_c$ (Sato et al., 28 Jan 2025).
• Spin-ARPES: Direct imaging of parallel (radial) spin and orbital textures on Fermi arcs in chiral semimetals like PtGa, with the surface spin lying orthogonal to the arc contour, matching bulk theory (Krieger et al., 2022).
• CD-ARPES and NanoESCA: Observation of chiral OAM textures in Bi$_2$Se$_3$ and CoSi, mapping OAM directly to surface momentum (Park et al., 2011, Hagiwara et al., 2024).
• STM/STS: Real-space visualization of charge density waves on CoSi(001) surfaces with momentum vector $Q_{\rm cdw}$ locked by the crystal chirality; Q flips under mirror symmetry, confirming momentum-chirality locking (Li et al., 2021).
• Chiral photonics: Unidirectionality and gain-momentum locking in chiral-gain SPP interfaces, with gain selectively supplied based on propagation direction and polarization (Serra et al., 2024).
• Optical and noncontact measurements: Near-field microscopy and lateral force detection for chiral nanoparticles above planar surfaces, demonstrating enantiomer-dependent propulsion due to locked spin-momentum evanescent fields (Wang et al., 2013, Kalhor et al., 2015).

5. Applications and Functional Implications

Surface chirality momentum locking is central to several emerging technologies:

• Chiral Spintronics and CISS Devices: Enabling spin filtering, nonreciprocal charge transport, and enantiomer-tunable magnetoresistance without the need for heavy elements or magnetic layers (Theiler et al., 9 May 2025, Sato et al., 28 Jan 2025).
• Topological Orbitronics: Harnessing OAM-momentum locking at surfaces for pure orbital current generation, orbital–spin conversion, or orbitally polarized photocurrents in chiral semimetals (Hagiwara et al., 2024, Yang et al., 2023).
• Quantum and Non-Hermitian Photonics: Implementation of lossless one-way SPP amplifiers, chiral edge-cavity lasers, and OAM routers via gain-momentum locking (Serra et al., 2024, Kandil et al., 2021).
• Metasurface Engineering: Spin- and chirality-controlled routing, diode-like wave propagation, and enhancement of light–matter interactions in chiral plasmonic metasurfaces (Kandil et al., 2021, Shi et al., 2022).
• Sensing and Enantiomer Recognition: Enantiomer-selective mechanical propulsion, orbital signature detection, and chirality-specific surface responses.
• Topology-driven Quantum Phases: Realization of axion insulators and topological superconductivity via chirality-locked order parameters and CDWs on surfaces (Li et al., 2021).

6. Distinctions from Conventional Spin-Orbit and Rashba Effects

Surface chirality momentum locking fundamentally differs from Rashba-type orthogonal locking:

• Chirality Imposed Locking: Originates solely from crystal/lattice handedness and absence of mirror/inversion symmetries, not from built-in electric fields.
• Parallel (Radial) Locking: Spin or OAM is locked parallel (radially) to momentum (or to the Fermi arc), contrasting with perpendicular locking in conventional Rashba or topological insulator surfaces (Krieger et al., 2022, Polyakov et al., 2020).
• Enhanced Effective SOC: The effective SOC and spin–momentum stiffness can dramatically exceed atomic SOC by orders of magnitude due to structural chirality (Sato et al., 28 Jan 2025).
• Non-Hermitian Exchange Mechanisms: In CISS and related phenomena, locking arises entirely from symmetry-enforced non-Hermitian exchange, independent of velocity fields or classical spin–orbit mechanisms (Theiler et al., 9 May 2025).

7. Universal Features and Future Directions

Surface chirality momentum locking provides a unifying framework for understanding nonreciprocal and directionally selective phenomena across electronic, photonic, and composite systems. The essential ingredient is the symmetry-enforced, deterministic locking of a vectorial degree of freedom to surface momentum, protected against moderate material imperfections.

Future directions include:

• Exploiting giant orbital responses and surface chirality for room-temperature quantum and classical spin-orbital devices.
• Engineering hybrid heterostructures that combine Rashba, topological, and chiral locking for enhanced and controllable responses.
• Exploring chiral quantum photonics, including spin–momentum–locked photon sources, orbital pumps, and chiral nanodevices with built-in nonreciprocity.
• Expanding the domain of chirality-momentum locking into phononic, magnonic, and hybrid platforms, leveraging universality in Maxwell and Schrödinger-type wave equations (Shi et al., 2022, Yu et al., 2022).

The rapid emergence of these phenomena across disparate platforms signals a new regime of symmetry-driven engineering grounded in chirality-based momentum locking, with far-reaching implications for condensed matter physics, quantum technology, and materials science.