🔬 mesoscale physics

Engineering nonlinear spin-orbit torque driven by intra-band transport in MoSe $_2$ /CrI $_3$ and WSe $_2$ /CrI $_3$ van der Waals heterostructures

This study reveals that in MoSe $_2$ /CrI $_3$ and WSe $_2$ /CrI $_3$ van der Waals heterostructures, symmetry-forbidden linear spin polarization necessitates nonlinear intraband transport as the dominant mechanism for generating a tunable, purely field-like spin-orbit torque that exhibits strong material dependence and can be controlled via doping, twist angle, and electric fields.

Original authors: Leyla Majidi, Azadeh Faridi, Reza Asgari

Published 2026-09-28

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CC BY 4.0

Original authors: Leyla Majidi, Azadeh Faridi, Reza Asgari

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of modern electronics, the ability to control magnetic information with electricity is the key to building faster, smaller, and more efficient computers. For decades, scientists have relied on a mechanism called spin-orbit torque to flip magnetic bits, a process that usually requires heavy metals or exotic materials with strong internal magnetic interactions. The goal has always been to find a way to do this with less energy and in thinner layers, pushing the boundaries of what is possible in ultra-thin devices. Recently, researchers have turned their attention to a new class of materials known as van der Waals heterostructures. These are stacks of atomically thin sheets, like layers of paper, that can be peeled apart and reassembled in different combinations. By pairing a magnetic sheet with a semiconductor sheet, scientists hope to create a system where an electric current can manipulate magnetism with extreme precision. However, a major puzzle remained: in certain combinations of these materials, the standard rules of physics seemed to forbid the usual way these magnetic flips happen, leaving scientists unsure of what mechanism was actually driving the process.

A team of researchers has now solved this puzzle by investigating a specific pairing of materials: a magnetic sheet made of chromium iodide stacked on top of a semiconductor sheet made of either tungsten diselenide or molybdenum diselenide. In their study, they focused on a scenario where a specific type of magnetic interaction, known as Rashba spin-orbit coupling, is effectively absent. Under these conditions, the standard, linear response that typically generates magnetic torque is completely blocked by the symmetry of the crystal structure. Instead of a dead end, the researchers discovered that the system relies on a more complex, nonlinear process. When an electric current flows through the semiconductor layer, it does not simply push electrons in a straight line to create a magnetic effect. Instead, the electric field modifies the distribution of the electrons themselves within their energy bands. This subtle reshaping of the electron population creates a spin polarization that would not exist under normal, linear conditions. This nonlinear effect becomes the dominant force, generating a magnetic torque that is purely field-like, meaning it acts to push the magnetic direction sideways rather than damping its motion.

The findings reveal that this nonlinear torque is not a minor correction but a powerful phenomenon that can be precisely tuned. The researchers found that the strength and even the direction of this magnetic push depend heavily on how the material is doped, or whether it has an excess of negative or positive charge carriers. In the case of the molybdenum diselenide stack, the torque generated in the negatively doped state was nearly ten times stronger than in the tungsten diselenide stack. Furthermore, as the researchers adjusted the chemical potential, which controls the density of these charge carriers, the direction of the torque flipped back and forth multiple times. This behavior is driven by the specific internal properties of the materials, particularly the way the electron spins interact with their movement within the crystal lattice. The study shows that the molybdenum-based material has a unique negative interaction parameter that causes these sharp peaks and reversals, a feature that the tungsten-based material lacks.

Beyond the choice of material, the researchers demonstrated that the magnetic torque can be controlled by physically twisting the two layers relative to one another and by applying an external electric field. By rotating the top layer by angles between zero and thirty degrees, they could either amplify the torque or suppress it entirely, and in some cases, induce a reversal in its direction. This twisting changes the magnetic environment at the interface, altering how the layers talk to each other. Similarly, applying a gate electric field, which acts like a voltage knob, allowed them to modulate the strength of the torque by nearly an order of magnitude. In some configurations, simply turning up the voltage was enough to flip the sign of the magnetic torque. These results suggest that by combining the right material choice with structural twisting and electrical gating, engineers can create highly flexible, ultra-thin devices where magnetic states are controlled with remarkable efficiency and low power consumption. The work establishes that in these specific atomic stacks, the nonlinear movement of electrons is the central engine for magnetic control, offering a new pathway for the next generation of spintronic technology.

Technical Summary: Engineering Nonlinear Spin-Orbit Torque in TMDC/CrI3 Heterostructures

Problem Statement
The role of nonlinear carrier dynamics in current-driven spin phenomena within van der Waals (vdW) magnetic heterostructures remains poorly understood. While Spin-Orbit Torque (SOT) is a established mechanism for electrical magnetization control, conventional understanding relies heavily on linear-response regimes and materials with significant Rashba spin-orbit coupling (SOC). In TMDC/CrI3 heterostructures, specifically WSe2/CrI3 and MoSe2/CrI3, the interplay between intrinsic SOC, proximity-induced exchange fields, and carrier dynamics in the zero-Rashba limit presents a theoretical gap. It is unclear whether nonlinear intraband transitions become the dominant mechanism when linear responses are symmetry-forbidden, and how external parameters like twist angle and electrostatic gating can tune these effects.

Methodology
The authors develop a microscopic theory of current-induced SOT using a single-band steady-state Boltzmann approach. The system is modeled as a TMDC monolayer (WSe2 or MoSe2) coupled to a ferromagnetic CrI3 monolayer via vdW interactions.

Hamiltonian: An effective low-energy Hamiltonian is constructed incorporating the valley index, intrinsic SOC, and proximity-induced exchange fields. The model accounts for the lifting of valley degeneracy and spin-splitting in both conduction and valence bands.
Transport Formalism: The non-equilibrium distribution function is derived using the relaxation time approximation, systematically constructing higher-order corrections (up to second order) to the Fermi-Dirac distribution.
Spin Polarization Calculation: The total spin polarization is decomposed into intraband ( $S_{oc}$ ) and interband ( $S_{in}$ ) contributions. The authors explicitly calculate the spin expectation values using eigenstates of the Hamiltonian and evaluate the contributions from both linear and nonlinear terms.
Parameters: Numerical simulations utilize parameters from previous literature (Ref. 36, 38) for twisted bilayers, considering room temperature ( $k_BT = 25.7$ meV) and varying chemical potentials ( $\mu$ ) for both n-type and p-type doping. The study focuses on the limit of negligible Rashba SOC.

Key Contributions and Results

Symmetry-Forbidden Linear Response: The study demonstrates that in the zero-Rashba SOC limit, linear spin polarization is strictly forbidden by symmetry. Neither intrinsic interband nor extrinsic intraband transitions contribute to the spin polarization in the linear-response regime. Consequently, the leading-order physical mechanism for SOT is the nonlinear intraband transition, driven by electric-field-induced modifications of the carrier distribution.
Nature of the Torque: The resulting nonlinear spin polarization generates a purely in-plane, field-like torque. This contrasts with conventional SOT devices where damping-like torques are often dominant or where mixed inter/intraband mechanisms exist. The torque magnitude and sign are highly sensitive to the chemical potential and doping type.
Material Dependence and Magnitude:
- MoSe2 vs. WSe2: A strong material dependence is observed. The n-doped MoSe2/CrI3 bilayer exhibits a torque nearly an order of magnitude larger than that of WSe2/CrI3.
- Sign Reversals: The MoSe2/CrI3 system displays two sign reversals in the torque as a function of chemical potential, attributed to the negative sign and smaller magnitude of the conduction-band SOC parameter ( $\lambda_c$ ) in MoSe2 compared to the positive $\lambda_c$ in WSe2.
- Doping Asymmetry: Pronounced asymmetries exist between n- and p-doped systems, with the n-doped regime generally yielding larger torque magnitudes.
Tunability via Twist Angle and Gating:
- Twist Angle: Rotating the TMDC layer relative to CrI3 (twist angles 0°–30°) significantly modulates the proximity-induced exchange parameters ( $B_c$ and $B_v$ ). This can enhance or suppress the torque and induce or eliminate sign reversals. Notably, a twist-induced sign reversal of the valence-band exchange field occurs near 8° for WSe2 and 16° for MoSe2.
- Electrostatic Gating: Applying a transverse gate electric field (−2 to 2 V/nm) modifies the proximity exchange parameters, enabling nearly an order-of-magnitude modulation of the torque in n-doped WSe2/CrI3 and sign reversals in p-doped structures.
- Combined Control: The combination of doping, twist angle, and gate field provides a versatile platform for simultaneous control of the torque's sign and magnitude.

Significance
The paper establishes that TMDC/CrI3 bilayers serve as a versatile platform for engineering nonlinear spin-orbit phenomena. By identifying the nonlinear intraband mechanism as the dominant source of SOT in the zero-Rashba limit, the work challenges conventional linear-response predictions. The findings highlight that structural (twist angle) and electrostatic (gate field) controls offer powerful, complementary means to tune the SOT, including substantial modulation and controllable sign reversals. This suggests a pathway toward ultrathin, low-power spintronic devices where torque efficiency and direction can be programmatically engineered through material selection (MoSe2 vs. WSe2) and external parameters, distinct from the mechanisms governing heavy metals or topological insulators.

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