Visualizing spin-dependent bulk scattering and breakdown of the linear dispersion relation in Bi${}_2$Te${}_3$

We performed a scanning tunneling spectroscopy investigation of the electronic properties of the topological insulator Bi${}_2$Te${}_3$ in the energy range between $-200$ and +700 meV with respect to the Fermi level. For unoccupied states, tunneling into topological surface states dominates. Analysis of Fourier-transformed (FT) $dI/dU$ maps allowed us to obtain their energy dispersion relation and to visualize the breakdown of the linear dispersion relation typical of massless particles. For occupied states, no signature of scattering events involving surface states can be detected. FT $dI/dU$ maps reveal in this case two scattering vectors pointing along the $\overline{\Gamma }\overline{M}$ and $\overline{\Gamma }\overline{K}$ directions. Both are identified with scattering events within bulk states. Interestingly, vectors pointing along $\overline{\Gamma }\overline{K}$ which correspond to bulk backscattering events have a length longer than the minimum distance necessary to match opposite pockets on a constant energy cut. Comparison with calculated spin-resolved constant-energy cuts shows that this is a direct consequence of the chiral spin texture present in bulk states when their energy and momentum are close to the resonances of the spin-polarized topological surface states.

Figure 1

(a) Bi${}_2$Te${}_3$ crystal structure: alternating planes of Bi and Te are stacked on top of each other up to the formation of a quintuple layer. Quintuple layers are separated by van der Waals gaps which define the natural cleavage plane. (b) Constant-current image showing the defects present on the surface. The inset reports an atomically resolved image. (c) STS spectra and (d) calculated band structure of a 30-layer Bi${}_2$Te${}_3$ slab (thick blue lines) and the projected bulk band structure (thin gray lines); (b) and (c) set point: $I=50$ pA, $U=-0.3$ eV.

Figure 2

(a)–(c) $dI/dU$ maps and (d)–(f) corresponding FT $dI/dU$ maps obtained within the empty electronic states. A scattering vector with sixfold symmetry pointing along the $\overline{\Gamma }\overline{M}$ direction can be seen. It is related to scattering of surface states and becomes visible for energies where the hexagonal warping already developed. (g) Energy dispersion relation of the topological surfaces state. For energies higher than $+500$ meV with respect to the Fermi level the linear dispersion relation typical of Dirac fermions breaks down, in agreement with the ab initio calculations reported in the inset. Tunneling current: 50 pA for all images.

Figure 3

(a)–(d) $dI/dU$ maps and (e)–(h) corresponding FT $dI/dU$ maps obtained at $-50$, $-150$, $-200$, and $-300$ meV with respect to the Fermi level. Two scattering vectors (q${}_1$ and q${}_2$) are visible. Both correspond to scattering processes involving bulk states. q${}_3$ is ascribed to the FT of the scattering potential caused by the defects themselves. Tunneling current: 50 pA for all images.

Figure 4

(a)–(c) Calculated constant energy cut at $-50$ meV with spin resolution represented by projections of the spin vector $\vec{S}$ on Cartesian axes. The spin vector components shown are sums of those within the Wigner-Seitz spheres around each atom in the surface quintuple layer block. Positive (negative) values of the spin vector components are shown in red (blue), while the size of the circle reflects the component absolute value. (d) and (e) Geometrical construction of scattering events: the filled area obtained by the intersection of two constant-energy cuts translated by vectors q${}_1$ and q${}_2$ represents possible scattering events and explains the anisotropic intensity observed in FT $dI/dU$ maps.