Warping effects in the band and angular-momentum structures of the topological insulator Bi${}_2$Te${}_3$

We performed angle-resolved photoemission (ARPES) experiments on Bi${}_2$Te${}_3$ with circularly polarized light. ARPES data show very strong circular dichroism, indicating the existence of orbital angular momentum (OAM). Moreover, the alignment of OAM is found to have a strong binding energy dependence. Such energy dependence comes from a relatively strong band warping effect in Bi${}_2$Te${}_3$ compared to Bi${}_2$Se${}_3$. OAM close to the Dirac point has an ideal chiral structure ($\sin \theta$) without an out-of-plane component. The warping effect comes in as the binding energy decreases, and circular dichroism along a constant energy contour can no longer be explained by a simple $\sin \theta$ function but requires a $\sin 3\theta$ term. When the warping effect becomes even stronger near the Fermi energy, circular dichroism gains an additional $\sin 6\theta$ term. Such behavior is found to be compatible with the theoretically predicted OAM structure.

Figure 1

(a) Experimental geometry used in the study and Brillouin zone of Bi${}_2$Te${}_3$. Sample surface is parallel to the $xy$ plane. Circularly polarized photons (spiral arrow) come in in the $xz$ plane at an angle of 40${}^{\circ }$ to the sample surface. (b) $\Gamma$-K cut taken with LCP. (c) Same cut taken with RCP. (d) Difference data taken with LCP and RCP.

Figure 2

(a) Illustration of Dirac cone of Bi${}_2$Te${}_3$. (b) Dichroism in the momentum space at various binding energies. (c) Constant dichroism map at selected binding energies of 0, 100, and 250 meV.

Figure 3

Circular dichroism $I_{\mathrm{CD}}$ as a function of $\theta$ at binding energies of (a) 250 meV, (b) 200 meV, (c) 150 meV, (d) 100 meV, (e) 50 meV, and (f) 0 meV. The definition for the azimuthal angle $\theta$ is shown in Fig. 2c.

Figure 4

CD along the constant energy contour at (a) 0 meV, (b) 100 meV, and (c) 250 meV. Dots and lines represent the data and the fit, respectively. Also drawn are schematics of the in-plane (black arrow) and out-of-plane components of the OAM. For the out-of-plane components, the red segment of the contour has in-to-page OAM while blue does out-of-page, as shown by the symbols.