Curvature-Engineered Spintronic Functionalities

• Curvature-engineered spintronic functionalities are defined by embedding nanoscale curvature to modulate spin interactions, enabling tunable magnetic, electronic, and superconducting responses.
• This approach harnesses curvature to induce effective spin–orbit coupling, Dzyaloshinskii–Moriya interactions, and Berry curvature modifications without altering material parameters.
• Applications include 3D racetrack memories, spin filters, and quantum circuits, demonstrating curvature as a universal design parameter in advanced spintronics.

Curvature-engineered spintronic functionalities encompass a broad set of electronic, magnetic, and superconducting device behaviors that are enabled, programmed, or modulated by the geometric curvature of nanostructures. By directly embedding curvature at the nanoscale—whether through bends in one-dimensional wires, cylindrical or shell geometries, periodic meandering, or three-dimensional nanoarchitectures—one can realize and tune a host of spin-dependent phenomena: effective spin–orbit coupling, emergent Dzyaloshinskii–Moriya interaction (DMI), Berry curvature, nonreciprocal magnonic modes, valley-contrasting spin splitting, and designer chiral spin textures. Applications span quantum spin Hall systems, graphene origami, superconducting circuits, racetrack memories, opto-spintronic devices, and topological logic elements. These functionalities are fundamentally geometric and persist even when material parameters or external fields are held fixed, establishing curvature as a universal design parameter in advanced spintronics.

1. Micromagnetic and Spin-Orbit Mechanisms: Curvature as a Functional Degree of Freedom

Curvature in magnetic and nonmagnetic systems introduces additional terms into the effective Hamiltonians governing spin dynamics. On curved and rolled magnetic multilayers or nanotubes, Heisenberg exchange and DMI acquire geometric corrections that scale as powers of the local curvature $\kappa=1/R$:

• The effective exchange energy becomes $$E_{\rm ex,eff}=A\big[(\nabla_t \mathbf m)^2 + (1/R)^2 m_n^2\big]$$, where $A$ is exchange stiffness, $m_n$ is the normal component, and $\nabla_t$ is the surface-tangent gradient.
• The DMI energy incorporates geometric DMI: $$E_{\rm DMI,eff}=D\big[\mathbf{m} \cdot (\nabla_t \times \mathbf{m}) + (1/R) m_\phi m_z\big]$$, enhancing in-plane chiral torques and enabling the stabilization of Néel domain walls or skyrmionic textures on finite curvature shells (Josten et al., 2021, Raftrey et al., 6 Jun 2025).

In nonmagnetic semiconductors and oxides, curvature–induced spin–orbit coupling (SOC) emerges even in the absence of intrinsic relativistic terms. For a one-dimensional wire, the kinetic Hamiltonian is augmented by a geometric SU(2) gauge field proportional to local curvature $\kappa(s)$:

$$H_{\rm SOC} = \alpha_{\rm curv}(s)\, \sigma_N(s)\, (-i \partial_s)\,, \quad \alpha_{\rm curv}(s) = \frac{\hbar^2}{2m} \kappa(s)\,,$$

where $\sigma_N$ is the spin operator along the local normal (Skarpeid et al., 2023, Siu et al., 2016). Curvature also modifies Rashba interactions in strongly spin–orbit coupled wires, yielding novel geometric spin torques and spin texture modulations (Gentile et al., 2022, Siu et al., 2016).

In two-dimensional electron systems and topological insulators, curvature efficiently rotates the spin-quantization axis, redistributes the spin current components, or even enables nearly perfect spin filtering by breaking the equivalence of opposite edges (Saffarzadeh et al., 2021, Huang et al., 2016).

2. Experimental Demonstrations and Imaging: Direct Probes of Curvature Effects

Direct imaging and fabrication advances demonstrate curvature-driven spin functionalities:

• Magnetic soft X-ray nanotomography on Co/Pd and Pt/Co/Ni multilayers coated onto sub-100 nm fibers reveals local realignment of magnetization with respect to wire axes, domain wall canting, and enhanced frequency of right-handed Néel walls. Quantitative extraction shows curvature-induced DMI reaching $30\%$ to $100\%$ of intrinsic interfacial values for radii $R\sim 25$–$100$ nm, enabling tunable chiral domain wall propagation (Raftrey et al., 6 Jun 2025, Josten et al., 2021).
• Atomic layer deposition (ALD) processes produce conformal Ni$_{80}$Fe$_{20}$ shells on $150$ nm nanowires, with low damping ($\alpha\sim0.01$) and GHz-range spin wave modes. Curvature introduces effective anisotropy $K_{\rm eff} \sim A/R^2$ and is expected to yield DMI-like terms $D_{\rm eff} \sim A/R$ in detailed magnonic spectra (Giordano et al., 2021).
• Periodically folded or curvature-gradient metallic nanochannels display independently tunable spin and charge impedance, facilitating three-dimensional integration and impedance matching in pure spin current devices (Das et al., 2018).
• XMCD and transmission X-ray microscopy elucidate the stabilization of curvature-induced chiral spin configurations at the nanometer scale (Raftrey et al., 6 Jun 2025).

3. Device-Implemented Functionality: All-Geometric Logic, Filtering, Racetrack Memory, and Quantum Transport

Curvature mediation enables and enhances device functionalities:

• 3D Racetrack Memories and Neuromorphic Circuits: Alternating high/low curvature sections create selective pinning/release for homochiral domain walls, allowing for efficient, fast, and structurally robust memory elements and neuron-mimetic synapses (Josten et al., 2021, Raftrey et al., 6 Jun 2025).
• Magnonic Crystals and Filters: Meander-shaped nanowires act as magnonic crystals where periodic curvature determines bandgap widths/centers entirely via geometry. Analytical and numerical results reveal that all higher-order spectral gaps and magnonic transport can be controlled by $\kappa_0 = \ell/R$ without material heterostructuring (Korniienko et al., 2019).
• Curvature-Driven Spin Interferometry: Polygonal or looped Rashba conductors exhibit non-monotonic spin-phase accumulation (dynamic vs geometric/Berry components), with programmable degeneracies and quantized Aharonov–Casher plateaus. Interference-based spin logic is accessible via gate-tuning of Rashba parameters and curvature design (Rodríguez et al., 2021).
• Spin Filters and Valley Contrasts: In curved quantum spin Hall elements and origami graphene, curvature enables two-terminal, high-efficiency spin filtering and valley-contrasting spin splitting by geometric programming of edge orientation and chiral angle (Saffarzadeh et al., 2021, Yamakage et al., 2023, Costa et al., 2013).
• Spin Current Control in AFM and Superconducting Devices: Curvature in antiferromagnets mimics DMI, opening or closing spectral gaps in magnon bands, localizing magnons, and programming geometry-based spin-orbit torques. In diffusive S–F–S Josephson junctions and proximity-coupled curved wires, geometric curvature alone generates long-range triplet supercurrents, enables dynamic $0$–$\pi$ transitions, and mediates chirality-dependent spin polarization—giving direct control of superconducting device phase via curvature (Pylypovskyi et al., 2020, Salamone et al., 2021, Skarpeid et al., 2023).

4. Theoretical Frameworks: Effective Hamiltonians, Geometric Terms, and Symmetry

Curvature-associated spintronic effects are captured by generalized Hamiltonians and energy functionals incorporating geometric invariants:

• Micromagnetism on Curved Surfaces: Curvature introduces geometric DMI and anisotropic exchange, expressed by tensorial energy densities incorporating principal curvatures ($\kappa_1, \kappa_2$), mean curvature $H$, and covariant derivatives on the surface (Raftrey et al., 6 Jun 2025).
• Geometric SOC and Berry Curvature: Effective nonrelativistic SOC fields are geometric gauge fields with spatial profiles controlled by $\kappa(s)$. Symmetry-based frameworks (group theory) systematically classify all curvature-induced SOC allowed at each valley or orbital, with valley-contrasting behavior and scaling $\Delta_{\rm so} \propto \kappa$ (Yamakage et al., 2023).
• Berry Curvature Engineering: Curvature modifies Berry curvature distributions via momentum-space and phase-space mechanisms in both metallic, semiconducting, and superconducting contexts, producing measurable effects in the Hall response and nonlocal transport channels (Lesne et al., 2022, Liao et al., 11 Dec 2024, Du et al., 2019).
• Domain Wall and Spin-Transfer Torque Dynamics: Curvature gradients generate chiral spin-transfer torques (CSTT), giving rise to handed effective fields proportional to curvature gradients, threshold behaviors, and unidirectional domain wall motion—a new geometric handle in racetrack and logic circuits (Bittencourt et al., 2023).

5. Quantitative Design Rules and Parameter Regimes

Device optimization is dictated by analytic relationships between curvature and spin-functional parameters:

Effect	Key Scaling Law	Device Implication
Curvature-induced DMI	$D_{\rm curv} \sim A \kappa$ or $A H$	Chiral wall stability, skyrmion nucleation
Geometric SOC	$\alpha_{\rm curv} \sim \hbar^2 \kappa / 2m$	Programmable spin textures, Rashba-like effects
Magnon gap (AFM)	$\omega_{-}^{\rm gap} \simeq c|\kappa|$	Local frequency gating in antiferromagnetic guides
CSTT velocity	$$v_{\rm CSTT} \sim (u \lambda^2 / \alpha) \partial_s \kappa$$	Chiral rectifiers, unidirectional DW motion
Pure spin current (QSH)	$I^{S_y}(K)=\sin(KW/2)(e/2\pi)V$	Tunable spin injectors by geometry
Berry curvature dipole (oxide)	$D_x \sim$ (geometry and band parameters)	Nonlinear Hall and photogalvanic conversion
Spin precession length	$$\ell_{\rm SP}^{-1} = \sqrt{\kappa^2 + l_\alpha^{-2}}$$	Spin control via curvature in Rashba channels

Maximum geometric control is typically realized for $R \sim 25$–$100$ nm (magnetic shells/nanotubes), $K \sim 10^{-2}\,\rm nm^{-1}$ (QSH devices), or $K \sim 0.04\,\rm nm^{-1}$ for DMI modulation in multilayers.

6. Emerging Directions: Multifunctionality, Integration, and Outlook

Curvature as a design parameter supports:

• Multifunctional logic: Integration of 3D spin current control, geometric memory, and phase logic in a single substrate via curvature patterning (Das et al., 2018, Skarpeid et al., 2023).
• Strain-driven, reconfigurable architectures: Nanoscale curvature is tunable via piezoelectric, mechanical, or differential thermal/chemical routes, enabling dynamic reprogramming of spintronic states without relying on magnetic fields or gates (Huang et al., 2016, Salamone et al., 2021, Gentile et al., 2022).
• Berry curvature engineering: Purely geometric control of nonlinear and topological Hall effects, spin–photon conversion, and dissipationless trajectories in antiferromagnetic and superconducting platforms (Du et al., 2019, Liao et al., 11 Dec 2024).
• Material generality: Mechanisms are operable in metallic, semiconducting, oxide, and superconducting systems—provided characteristic length scales (mean free path, exchange length, coherence length) are commensurate with the imposed curvature.

Device classes envisioned include domain wall diodes, chirality-sensitive racetrack memory, opto-spintronic transducers, and geometric phase-locked superconducting qubits—each leveraging nanoscale curvature to enable or enhance spin functionality (Gentile et al., 2022, Khan et al., 27 May 2024).

7. Representative Materials, Fabrication Strategies, and Integration

Materials and methods span:

• Ferromagnetic multilayers and alloys: Pt/Co/Ni, Co/Pd, NiFe-based nanotubes (Josten et al., 2021, Giordano et al., 2021, Raftrey et al., 6 Jun 2025).
• 2D materials and rolled architectures: Graphene origami, monolayer graphullerene, oxide microtubes (Costa et al., 2013, Khan et al., 27 May 2024).
• Topological insulators: Bismuthene domes, quantum spin Hall systems (Saffarzadeh et al., 2021, Huang et al., 2016).
• Semiconductors and oxides: LaAlO₃/SrTiO₃ interfaces, InGaAs loops (Lesne et al., 2022, Gentile et al., 2022).
• Direct-write and self-assembly: Conformal ALD, strain-driven roll-up, focused-ion-beam patterning, piezoelectric actuation.

Design rules recommend matching curvature scales to the relevant magnetic, electronic, or superconducting length, ensuring conformal and homogeneous shell deposition or membrane rolling for fidelity to the prescribed geometry.

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In total, curvature-engineered spintronic functionalities constitute a rigorous, material-independent framework for programming spin currents, spin textures, and quantum geometric responses in nanoscale devices. They unify structural engineering and spintronics, opening an extensive phase space for innovation in memory, logic, and sensing hardware by exploiting geometry alone as a tunable parameter (Josten et al., 2021, Raftrey et al., 6 Jun 2025, Skarpeid et al., 2023, Das et al., 2018, Gentile et al., 2022, Saffarzadeh et al., 2021, Giordano et al., 2021, Costa et al., 2013, Khan et al., 27 May 2024).