Chirality-induced orbital Edelstein effect in an analytically solvable model

TL;DR

This work addresses how real-space chirality induces an orbital Edelstein effect that can dominate over the spin counterpart. By solving a minimal 3-site helical tight-binding model and computing the orbital angular momentum with the modern orbital magnetization formalism, the authors derive the orbital Edelstein susceptibility $\chi_z^{L_z}$ within Boltzmann transport and show it flips sign with chirality and peaks away from band edges. They demonstrate that the effect stems from nonlocal intersite motion, is robust to three-dimensional coupling and SOC (which mainly renormalizes bandwidth), and can be faithfully captured by an effective single-band description; the spin response is suppressed in the constant-$\tau$ limit and only arises with SOC or momentum-dependent relaxation. The results imply large orbital-driven torques in chiral materials and offer a framework to distinguish CIOS from CISS in experiments, with tellurium as a representative example.

Abstract

Chirality-induced spin selectivity (CISS), a phenomenon wherein chiral structures selectively determine the spin polarization of electron currents flowing through the material, has garnered significant attention due to its potential applications in areas such as spintronics, enantioseparation, and catalysis. The underlying physical effect is the Edelstein effect that converts charge to angular momentum. Besides a spin contribution there exists a contribution based on the orbital angular momentum but the precise mechanism for its generation remains yet to be understood. Here, we introduce the minimal model for explaining the phenomenon based on the orbital Edelstein effect. We consider non-local inter-site contributions to the current-induced orbital angular momentum and reveal the underlying mechanism by analytically calculating the Edelstein susceptibilities in a tight-binding and Boltzmann approach. While the orbital angular momentum is directly generated by the chirality of the crystal, the spin contribution of each spin-split band pair relies on spin-orbit coupling. Using tellurium as an example, we show that the orbital contribution surpasses the spin contribution by orders of magnitude.

Figures (8)

• Figure 1: Chiral helices. (a) Right-handed helix structure characterized by $c>0$. (b) Left-handed structure characterized by $c<0$. The black arrow symbolizes the translational motion of electrons along $z$. It is constrained to a circular motion (red and blue) which generates opposite orbital angular momentum $L_z$(orange and green) for the two opposite crystal chiralities.
• Figure 2: Orbital Edelstein effect (schematic for $c>0$). (a) Band structure of free electrons with orbital angular momentum $L_z$ proportional to the group velocity (color). (b) Upon application of an electric field along $z$, the initial Fermi surfaces (pale) are shifted by $\Delta k_z$ giving rise to a non-equilibrium orbital magnetic moment $\boldsymbol{m}_{L}$.
• Figure 3: Chirality-induced orbital selectivity in a chiral system without spin-orbit coupling. (a) Band structure for which the color encodes the orbital angular momentum $L_z$ in units of $L_0=\frac{1}{\sqrt{3}}\frac{m_e}{g_L\hbar}a^2 t$. (b) Edelstein susceptibility $\chi_z^{L_z}$ in units of $\chi_0=\frac{e^2}{2\pi \sqrt{3}\,\hbar^2}\tau \,|t|ca^2$. The red curve shows the result for the right-handed system ($c>0$) and the blue curve for the left-handed system ($c<0$).
• Figure 4: Band structure of a chiral helix with spin-orbit coupling. (a) The color encodes the orbital angular momentum $L_z$. (b) The color encodes the spin $S_z$. Parameters correspond to tellurium with exaggerated spin-orbit coupling $\lambda=0.05t$.
• Figure 5: Effective 1-band model. (a) Helix structure for $c>0$ with generalized coordinate $q$ (red). (b) Structure along this new effective coordinate. (c) Band structure with effective orbital angular momentum (color). The dashed line indicates the smaller Brillouin zone corresponding to a chain with 3 atoms in the unit cell.