Quantum Theory of Optical Spin Texture in Chiral Tellurium Lattice

Abstract

The absence of inversion symmetry in chiral tellurium (Te) creates exotic spin textures within its electron waves. However, understanding textured optical waves within Te remains a challenge due to the semi-classical limitations of long-wavelength approximation. To unveil these textured optical waves, we develop a spin-resolved deep-microscopic optical bandstructure for Te analogous to its electronic counterpart. We demonstrate that the degeneracies in this optical bandstructure is lifted by the twisted lattice of Te, which induces optical gyrotropy. Our theory shows excellent agreement with experimental optical gyrotropy measurements. At the lattice level, we reveal that the chirality of Te manifests as deep-microscopic optical spin texture within the optical wave. Our framework uncovers the finite-momentum origin of optical activity and provides a microscopic basis for light-matter interactions in chiral crystalline materials.

Figures (4)

• Figure 1: Spin-resolved deep-microscopic optical bandstructure of Te. (a) Schematic of the Te lattice, $\hat{n}_\text{OA}$ represents the screw axis. Here, $a=4.457 \mathring{A}$, $c=5.929 \mathring{A}$. (b) Top view of the Te crystal lattice. The unit cell is highlighted in yellow. Here, $2R=2.346 \mathring{A}$. (c) Screw orientation of the two enantiomers of chiral Te. (d) First Brillouin zone of Te crystal along with the high-symmetry points. (e) Electronic band-structure of Te calculated using Purdue-PicoMax (f) Spin-resolved deep-microscopic optical bandstructure at $60$THz (=$5\mu$m). The classical limit of the bandstructure is highlighted in yellow near the $\mathbf{\Gamma}$ points. The dominant bands ($m=1,2$) determine the macroscopically measurable quantities such as the optical gyrotropy.
• Figure 2: Super-dispersive optical gyrotropy and chirality density for undoped Te. (a) Experimental plot of the projection of the longitudinal gyration tensor component $\bm g_{33}$ on the photon momentum, vs. (b) Theoretical prediction using Purdue-PicoMax at $\bm{q}=(0,0,0.1)2\pi/a$ . The Feynman diagram in the inset reflects the transverse SILC of the induced transverse current including the momentum-exchange processes. Here, $c$ and $v$ represent the conduction and valence bands respectively. (c,d) The spin-resolved chirality density projection and total spin on the isofrequency surface of RCP ($+$) at $\omega=60$THz (c) and LCP ($-$) (d). The concentration near the pole of the sphere and the difference between RCP and LCP illustrates the suitable axis (optic axis) for observing high optical activity.
• Figure 3: Hidden optical waves reveal polarization texture at the lattice-level. Vectorial distribution of the optical waves for the first transverse optical band. The plots are shown for three different planes of the Te crystal: (a) $x=0$, (b) $z=0$, and (c) $y=0$.
• Figure 4: Deep-microscopic optical spin texture reveals orientation of polarization of the optical waves in Figure 3. Vectorial distribution of the optical spin texture of the optical waves in the first transverse optical band. The colormap shows the projection of the spin density along the screw axis. The plots are shown for three different planes of the Te crystal: (a) $x=0$, (b) $z=0$, and (c) $y=0$.