Probing surface states exposed by crystal terminations at arbitrary orientations of three-dimensional topological insulators

The topological properties of the bulk band structure of a three-dimensional topological insulator (TI) manifest themselves in the form of metallic surface states. In this paper, we propose a probe which directly couples to an exotic property of these surface states, namely the spin-momentum locking. We show that the information regarding the spin textures, so extracted, for different surfaces can be put together to reconstruct the parameters characterizing the bulk band structure of the material, hence acting as a hologram. For specific TI materials like, $\text{Bi}_2\text{Se}_3, \text{Bi}_2\text{Te}_3 \text{and Sb}_2\text{Te}_3$, the planar surface states are distinct from one another with regard to their spectrum and the associated spin texture for each angle ($\theta$), which the normal to the surface makes with the crystal growth axis. We develop a tunnel Hamiltonian between such arbitrary surfaces and a spin polarized STM which provides a unique fingerprint of the dispersion and the associated spin texture corresponding to each $\theta$. Additionally, the theory presented in this article can be used to extract value of $\theta$ for a given arbitrary planar surface from the STM spectra itself hence effectively mimicking X-ray spectroscopy.