Radial Rashba Spin–Orbit Fields

• Radial Rashba spin–orbit fields are a distinct class of interactions where electron spins align radially with momentum, contrasting the conventional tangential Rashba effect.
• They emerge from twist-engineered, symmetry-protected configurations in layered 2D and van der Waals structures, enabling advanced spin-charge conversion and supercurrent diode effects.
• Both computational models and experimental techniques, including magnetotransport and spin-resolved spectroscopies, validate that tuning the Rashba angle optimizes spintronic and quantum device performance.

Radial Rashba spin–orbit fields are a recently identified class of spin–orbit interactions in crystalline solids and heterostructures in which the induced spin polarization aligns parallel to the electron’s in-plane momentum (radially outward or inward in momentum space), fundamentally contrasting the canonical (tangential) Rashba field that aligns spins perpendicular to momentum. This radial configuration arises from specific symmetry and stacking engineering, including layer twisting in van der Waals (vdW) heterostructures, and is directly relevant for the design of spin–charge conversion devices, unconventional supercurrent diode effects, and the manipulation of quantum phases in 2D materials. The emergence, characterization, and consequences of radial Rashba fields involve both first-principles computations and effective Hamiltonian models, with experimentally accessible signatures chiefly in magnetotransport and tunneling phenomena.

1. Definition and Theoretical Description

Radial Rashba spin–orbit fields (abbreviated as "RR SOF"; Editor's term) are characterized by an effective spin–orbit field in momentum space such that the resulting expectation value of the spin operator for Bloch electrons, $\langle \vec S(\vec k) \rangle$, points along the radial direction of the crystal momentum around high-symmetry points (e.g. $K$, $K'$, or $\Gamma$ points). In contrast, the standard Rashba effect gives a spin texture winding tangentially (i.e. perpendicular to $\vec k$).

The general Rashba Hamiltonian in 2D with $C_3$ symmetry is written as

$$H_R = \lambda_R e^{-i\phi s_z/2} [ \tau \sigma_x \otimes s_y + \sigma_y \otimes s_x ] e^{i\phi s_z/2}$$

where $\lambda_R$ is the Rashba strength, $\tau$ is the valley index, $\sigma$ and $s$ are pseudospin and spin Pauli matrices, and $\phi$ is the "Rashba angle" that determines the orientation of the spin with respect to momentum. For $\phi = 0$, the field is purely tangential; for $\phi = 90^\circ$ ($\pi/2$), it is purely radial (Frank et al., 19 Feb 2024, Kang et al., 20 Feb 2024, Costa et al., 5 Dec 2024).

The effective in-plane spin–orbit field at a 2D Fermi surface can be recast as

$$\hat{\Omega}(k_x, k_y) = \begin{pmatrix} \alpha (\sin \theta_R k_x + \cos \theta_R k_y) - \beta k_y \ \alpha (-\cos \theta_R k_x + \sin \theta_R k_y) - \beta k_x \ 0 \end{pmatrix}$$

with $\alpha$ and $\beta$ corresponding to Rashba and Dresselhaus SOC strengths, and $\theta_R$ the Rashba angle parameterizing the mix between conventional (tangential, $\theta_R=0$) and radial ($\theta_R=\pi/2$) components (Costa et al., 5 Dec 2024).

2. Physical Origin and Symmetry Considerations

The emergence of RR SOFs fundamentally relies on the interplay of crystal symmetry, stacking configuration, and interlayer coupling in vdW layered materials. In untwisted bilayers, each monolayer may host a "hidden" Rashba field with opposite winding due to inversion or mirror symmetry, resulting in global cancellation. Upon introducing a relative twist angle between the layers (as in commensurate twisted transition-metal dichalcogenide bilayers or twisted bilayer graphene), the individual layer Rashba fields undergo a rotation, causing their tangential components to cancel and their radial components to add (Frank et al., 19 Feb 2024, Naimer et al., 12 Sep 2025).

A crucial symmetry protecting the pure radial character is an in-plane 180° rotation axis (a $C_2$ symmetry). For stackings that retain this symmetry, only the radial Rashba component survives; lateral displacements or breakings of $C_2$ symmetry reintroduce tangential winding (Naimer et al., 12 Sep 2025).

The magnitude of the RR SOF is strongly twist-angle dependent, vanishing in untwisted ($\Theta = 0^\circ, 60^\circ$) cases and being maximal near $\Theta = 30^\circ$. The effect displays symmetry not only around the high-symmetry untwisted points but also around $30^\circ$, reflecting the commensurate stacking energy landscape and microscale hybridization (Naimer et al., 12 Sep 2025).

3. Model Hamiltonians and Microscopic Mechanisms

Ab initio calculations and effective impurity models have been used to elucidate RR SOFs. The essential features are captured by model Hamiltonians featuring conventional and radial Rashba components:

$$H_{\text{SOC}} = \lambda_{\mathrm{CR}} (\vec{s} \times \vec{k})_z + \lambda_{\mathrm{RR}} (\vec{s} \cdot \vec{k})$$

with $\lambda_{\mathrm{CR}} = \lambda_R \cos \theta_R$, $\lambda_{\mathrm{RR}} = \lambda_R \sin \theta_R$. The presence of interlayer coupling $w$ is critical for proximity-induced SOC, with $w$ decreasing (typically exponentially) as the size of the commensurate supercell increases (i.e., as the twist angle $\Theta$ departs from special commensurate values that yield small supercells) (Naimer et al., 12 Sep 2025).

The Hamiltonian approach can be generalized to include valley-Zeeman terms, higher-order SOC, and Dresselhaus-type interactions, especially in the context of twisted multilayers (Costa et al., 5 Dec 2024, Kang et al., 20 Feb 2024). The interplay of geometric design (twist, stacking, interlayer distance) and electronic hybridization (modulated by strain and atomic relaxation) tunes both the magnitude and "directionality" (Rashba angle) of the SOC.

4. Experimental Manifestations and Detection Protocols

RR SOFs leave unique signatures in a variety of experimental observables, especially those sensitive to the orientation of spin textures in momentum space:

A. Magnetotransport

• Transverse magnetic focusing (TMF): The RR SOF shifts Fermi contours parallel (rather than perpendicular) to the in-plane field, resulting in magneto-transport quantities (e.g., TMF peak amplitudes) displaying characteristic symmetries under $B_\parallel \to -B_\parallel$ only at a "sweet spot" angle between the transport and field axes given by $\varphi = -\theta_R$ (Kang et al., 20 Feb 2024).
• Dyakonov–Perel spin relaxation: The rate exhibits swap-symmetry as a function of field and direction at the sweet spot.

B. Tunneling Anisotropies

• The tunneling conductance (TAMR/MARR) and tunneling anomalous Hall effect (TAHE) have angular dependences that are shifted by an amount directly proportional to $\theta_R$. In the presence of a secondary SOC channel (e.g., Dresselhaus), maximal conductance shifts are $\Delta\Phi = \theta_R/2$ (TAMR) and $\Delta\Phi = \theta_R$ (TAHE), enabling direct experimental extraction of the Rashba angle (Costa et al., 5 Dec 2024).

C. Superconducting and Josephson Devices

• In Josephson junctions and superconducting diodes, the interplay of radial and tangential Rashba fields generates a supercurrent diode effect with directionality determined by the Rashba angle. A pure RR component leads to a diode effect for collinear barrier magnetizations—a feature unattainable with tangential SOC alone. The critical currents in opposite polarity become unequal, with the magnitude and sign of the asymmetry governed by $\theta_R$ (Costa et al., 18 Nov 2024, Bhowmik et al., 17 Jul 2024).

D. Optical and Spin-Resolved Spectroscopies

• Spin-resolved ARPES, time-resolved Faraday rotation, and related pump–probe schemes are capable of mapping the associated spin textures and measuring the Rashba angle, particularly in twisted or engineered vdW heterostructures (Frank et al., 19 Feb 2024, Costa et al., 5 Dec 2024).

5. Microscopic Influences: Interlayer Coupling, Atomic Relaxation, and Moiré Effects

The interlayer hybridization parameter $w$—which quantifies mixing between $K/K'$ points in different layers—controls the strength of proximity-induced SOC. Smaller commensurate supercells (possible at special twist angles) exhibit enhanced $w$, thus stronger RR SOF. As the supercell size increases (more incommensurate twist), $w$ decreases, and the effect vanishes (Naimer et al., 12 Sep 2025).

Atomic relaxation, both as an overall reduction in interlayer distance and as local nonuniform modulations, further influences the proximity-induced SOC and thus the RR field's amplitude. Enhanced interlayer proximity increases the RR SOF, whereas larger separations or relaxation-driven inhomogeneity can weaken or spatially modulate the effect (Naimer et al., 12 Sep 2025). The Moiré potential and geometry can, therefore, play a controlling role in spatially varying the orientation and strength of the SOC.

6. Applications and Theoretical Implications

The ability to realize a pure radial Rashba SOF enables several advanced functionalities:

• Spin–charge conversion: An RR field provides omnidirectional spin accumulation symmetry, promising for Edelstein effect applications where spin orientation can be aligned with any in-plane axis, maximizing spin–charge interconversion efficiency (Frank et al., 19 Feb 2024, Naimer et al., 12 Sep 2025).
• Spin–orbit torque: For spin–orbit torque switches, an RR SOF affords new geometries in which current-induced spin accumulation acts parallel to the electron flow.
• Superconductivity and quantum phases: The Rashba angle $\theta_R$ modulates the pairing symmetry and superconducting current rectification, with RR SOFs stabilizing unconventional supercurrent diode effects robust to the field and interface orientation (Costa et al., 18 Nov 2024, Bhowmik et al., 17 Jul 2024).
• Topological states and correlated phases: Control of the RR SOF is a tool to manipulate topological states in moiré superlattices and twisted heterostructures by controlling the symmetry and magnitude of spin–orbit fields (Frank et al., 19 Feb 2024, Naimer et al., 12 Sep 2025).
• Spintronic architectures: Gate-tunable RR fields in proximitized graphene or TMD devices offer a new paradigm for low-power, directional, and highly efficient spintronic elements (Frank et al., 19 Feb 2024, Costa et al., 5 Dec 2024).

7. Outlook and Future Directions

The observation and engineering of RR SOFs in twisted vdW structures prompt several avenues for research:

• Systematic exploration of stacking and twist configurations to maximize the radial component, aided by first-principles and symmetry-based model calculations.
• Extension to materials with intrinsically large SOC, e.g., Janus dichalcogenides or topological insulators, to augment the amplitude and tunability of RR SOFs.
• Experimental verification in devices via magnetotransport, spin-resolved spectroscopies, and superconducting transport, leveraging predicted angular shifts and diode efficiencies as robust diagnostic signatures.
• Investigation of relaxation effects, Moiré band engineering, and local strain, to spatially modulate spin–orbit fields and engineer designer spin textures beneficial for optospintronic and quantum information applications.

A plausible implication is that future platforms combining gate-tunable, twist-engineered RR SOFs (with symmetry protection from in-plane $C_2$ axes) and proximity-induced superconductivity offer the potential for robust, directional, and nonreciprocal quantum devices with applications spanning spin–charge conversion, nonreciprocal transport, and topological quantum computation.