Orbital-related gyrotropic responses in Cu$_2$WSe$_4$ and chirality indicator

In recent years, counterparts of phenomena studied in spintronics have been actively explored in the orbital sector. The relationship between orbital degrees of freedom and crystal chirality has also been intensively investigated, although the distinction from gyrotropic properties has not been fully clarified. In this work, we investigate spin and orbital Edelstein effects as well as the nonlinear responses in the ternary transition-metal chalcogenide Cu$_2$WSe$_4$, which has a gyrotropic but achiral crystal structure. We find that in the Edelstein effect, magnetization is dominated by the orbital contribution rather than the spin contribution. On the other hand, both the nonlinear chiral thermoelectric (NCTE) Hall effect–a response to the cross product of the electric field and the temperature gradient–and the nonlinear Hall effect–conventional second-order response to the electric field–are found to be dominated by the Berry curvature dipole. We further find that spin-orbit coupling plays only a minor role in these effects, whereas the orbital degrees of freedom are essential. Finally, we demonstrate that the orbital magnetic-moment contributions to both the Edelstein effect and the NCTE Hall effect are closely linked to chirality, and we discuss the possibility of using them as a chirality indicator.

💡 Research Summary

This paper investigates the interplay between orbital degrees of freedom, gyrotropic crystal symmetry, and nonlinear transport phenomena in the non‑centrosymmetric but achiral semiconductor Cu₂WSe₄ (space group I‾4₂m, point group D₂d). Using density‑functional theory (GGA‑PBE) together with maximally‑localized Wannier functions, the authors construct an accurate tight‑binding model that reproduces the DFT band structure (indirect gap ≈ 1.2 eV) and captures the orbital character of the valence (Se‑p, Cu‑d) and conduction (W‑d, Se‑p) bands.

The Edelstein effect—electric‑field‑induced magnetization—is decomposed into spin (m_S) and orbital (m_O) contributions. For an electric field along x, only the α_xx component survives due to D₂d symmetry (α_xx = −α_yy). Calculations (τ ≈ 10 fs, T = 30 K, E = 10⁴ V/m) show that in the valence band the orbital term dominates by a factor of 3–5 over the spin term, while in the conduction band the strong spin‑orbit coupling (SOC) of W enhances the spin contribution but the orbital part remains comparable. Importantly, the orbital Edelstein response is essentially unchanged when SOC is switched off, demonstrating that SOC is not essential for the effect in this material.

Next, the authors study two second‑order responses that require gyrotropic symmetry: (i) the nonlinear chiral thermoelectric (NCTE) Hall effect, where a transverse current appears under mutually perpendicular electric field E and temperature gradient ∇T, and (ii) the conventional nonlinear Hall effect (NLH) driven solely by E². Both responses are expressed through a second‑rank tensor χ_ij^(ℓ) (ℓ = 1 for charge, ℓ = 2 for heat) and a third‑rank tensor σ_ijl, respectively. Each tensor can be split into a Berry‑curvature‑dipole (BCD) term and an orbital‑magnetic‑moment (OM) term. Numerical integration over dense k‑meshes reveals that the BCD contribution accounts for 70–90 % of the total signal, while the OM term is comparatively small. This dominance of BCD persists regardless of whether SOC is included, indicating that the underlying band‑crossing physics is primarily orbital in nature. The S₄ roto‑reflection symmetry forces the z‑components of the effective velocity‑weighted Berry curvature (Ω′_z) and the perpendicular component of the orbital moment (m_O,z) to vanish, leaving only Ω′_x and Ω′_y (with opposite signs) to contribute, which explains the observed relations σ_xyz = σ_yzx ≈ −σ_zxy/2 and χ_xx = −χ_yy.

Having quantified these responses, the authors propose a “chirality indicator” based on the symmetry properties of α_ij and χ_ij^(ℓ). In truly chiral crystals (lacking any roto‑reflection axis), the Berry curvature and orbital moment acquire z‑components that break the relations α_xx = −α_yy and χ_xx = −χ_yy, providing a measurable signature of handedness. By contrast, Cu₂WSe₄, being achiral, obeys the symmetric constraints exactly, making it an ideal reference system to separate pure gyrotropic effects from chirality‑related ones.

In summary, the study demonstrates that (1) the orbital Edelstein effect can dominate over its spin counterpart even in a material with moderate SOC, (2) nonlinear Hall and NCTE Hall effects are governed by the Berry‑curvature dipole rather than orbital magnetization, (3) spin‑orbit coupling plays only a secondary role, and (4) the orbital magnetic‑moment contributions to Edelstein and NCTE responses are directly linked to crystal chirality, offering a practical experimental probe for handedness. These insights advance the emerging field of “orbitronics” and provide a clear pathway for identifying and exploiting gyrotropic but achiral materials in future spin‑orbit‑free electronic technologies.