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 such as ${\text{Bi}}_2{\text{Se}}_3,{\text{Bi}}_2{\text{Te}}_3$, and ${\text{Sb}}_2{\text{Te}}_3$, 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 scanning tunneling microscope (STM) which provides a unique fingerprint of the dispersion and the associated spin texture corresponding to each $\theta$. Additionally, the theory presented in this paper can be used to extract the value of $\theta$ for a given arbitrary planar surface from the STM spectra itself, hence effectively mimicking x-ray spectroscopy.

Figure 1

(left) Schematic of the setup involving an arbitrarily cut surface of the TI, STM, and the two contact pads. The surface shown with the contacts has its normal (pointing along ${\widehat{k}}_3)$ at an angle $\theta$ to the crystal-growth axis (along $\widehat{z})$. As a reference, a translucent surface lying in the $x\text{−}y$ plane is shown which is perpendicular to the $z$ axis. For more details refer to the text in Sec. 2. (right) The schematic setup viewed along the $y$ direction clearly shows the surface of interest along with the coordinate system.

Figure 2

(top) The current distribution in the $k_1\text{−}{k}_y$ plane (in units of ${\text{Å}}^{-1})$ for three different surfaces with the STM spin polarized in the $x$ direction. (bottom) The TMR response corresponding to the same situation, shown by plotting $\Delta I/I_0$ as a function of $\gamma$.

Figure 3

Magnetization of the half of the Fermi surface in (top) the $x$ direction and (bottom) the $y$ direction as a function of $\gamma$ for three different surfaces corresponding to $\theta =0,\pi /4$, and $\pi /2.$

Figure 4

(top) The current distribution in the $k_1\text{−}{k}_y$ plane (in units of ${\text{Å}}^{-1})$ for three different surfaces with the STM spin polarized in the $y$ direction. (bottom) The TMR response corresponding to the same situation, shown by plotting $\Delta I/I_0$ as a function of $\gamma$.

Figure 5

The data points show $\Delta I/I_0$ calculated for the STM tip being magnetized in the $x$ direction $\left({\theta }_s=\pi /2,{\varphi }_s=0\right)$ and the ideal case $p=1$ for the contact configuration corresponding to $\gamma =0$. The solid line shows the continuous function $\text{c}cos\theta /v_3$, which is the function followed by the current asymmetry as a function of the surface.

Figure 6

The energy spectrum of the conduction band for three different surfaces. The thick lines show the three constant energy contours corresponding to 0.1, 0.2, and 0.3 eV, respectively. It should be noted that as one moves from the top surface to the side surface, the constant energy contour begins to go from a circle to an ellipse. This happens because the contribution of $k_z$ begins to grow in the in-plane components of the momentum, which magnifies the effect of the difference of the two Fermi velocities $v_z$ and $v_{\parallel }$. The axes of the figures correspond to the in-plane momenta (in units of ${\text{Å}}^{-1})$.

Figure 7

The spin texture for three different surfaces corresponding to $\theta =0,\pi /4$, and $\pi /2$, plotted in the momentum space. It should be noted that the axes of the panels are $k_1$ and $k_y$ (in units of ${\text{Å}}^{-1})$, which are the local in-plane axes; however, the first two columns show the expectation of spin along the global $x$ and $y$ directions. Since $\langle {\sigma }_z\rangle$ is identically zero, it is not plotted; rather, the expectation of the total spin magnitude is plotted to show that it is not constant on the Fermi surface for arbitrary surfaces.

Figure 8

The spin texture for three different surfaces states corresponding to $\theta =0,\pi /4$, and $\pi /2$, plotted in the momentum space. It should be noted that the axes of the panels are $k_1$ and $k_y$ (in units of ${\text{Å}}^{-1})$, which are the local in-plane axes, and the vectors show the in-plane spin components ${\sigma }_1$ and ${\sigma }_y$. The density plot in the background shows the out-of-plane component of the magnetization ${\sigma }_3$.