Anomalous behavior in the phonon dispersion of the (001) surface of Bi${}_2$Te${}_3$ determined from helium atom-surface scattering measurements

We employ inelastic helium atom-surface scattering to measure the low-energy phonon dispersion along high-symmetry directions on the surface of the topological insulator Bi${}_2$Te${}_3$. Results indicate that one particular low-frequency branch experiences noticeable mode softening attributable to the interaction between Dirac fermion quasiparticles and phonons on the surface. This mode softening constitutes a renormalization of the real part of the phonon self-energy. We obtain the imaginary part, and hence lifetime information, via a Hilbert transform. In doing so, we are able to calculate an average branch-specific electron-phonon coupling constant $\langle {\lambda }_{\nu }\rangle$ $=$ 1.44.

Figure 1

Schematic diagram of the helium beam detection mechanism. The incident beam energy is controlled by the nozzle temperature and the incident and final angles may be adjusted via the polar controls of the sample manipulator and detector, respectively.

Figure 2

Scan curves showing the kinematically allowed scattering events for $E_i$ $=$ $43.09$ meV and ${\theta }_i$ $=$ 45.62${}^{\circ }$. Each blue parabola corresponds to a unique ${\theta }_f$, which are plotted here in 1${}^{\circ }$ increments from 35${}^{\circ }$ to 50${}^{\circ }$. The red parabola and data points correspond to the phonon events shown in the third panel of Fig. 8.

Figure 3

Hexagonal unit cell of the Bi${}_2$Te${}_3$ crystal comprised of three QLs and belonging to the space group $R\overline{3}m$. Note that the Te2 layer within each QL is a center of inversion symmetry.

Figure 4

Bulk and surface reciprocal space structure of Bi${}_2$Te${}_3$. Panel (a) shows the bulk Brillouin zone and high-symmetry points. The surface Brillouin zone is represented as a projection along ${\mathbf{q}}_{\mathbf{z}}$. The extended surface reciprocal lattice is shown in (b) depicting high-symmetry points $\overline{\Gamma },\overline{M}$, and $\overline{K}$ as well as reciprocal-lattice vectors ${\mathbf{G}}_1$ and ${\mathbf{G}}_2$.

Figure 5

Schematic diagram of ion and PC locations in the PCM. The figure indicates that the surface PC are more spread and easier to deform than their bulk counterparts. The figures to the right show the dipolar PC deformation associated with lattice distortions along the $z$ direction.

Figure 6

Calculated bulk dispersion curves along the high-symmetry directions $\Lambda$ [(a) and (b)], $\Delta$ (c), and $\Sigma$ (d). The $C_{3v}$ symmetry of the $\Lambda$ direction allows one to project out purely longitudinal $A$ modes and doubly degenerate, transverse $E$ modes. Modes along the $\Delta$ and $\Sigma$ directions have mixed polarization. The calculated dispersions were fit to available Raman (red triangles), IR (green squares), and inelastic neutron scattering (blue circles) data.

Figure 7

Diffraction patterns indicating the two high-symmetry directions $\overline{\Gamma }\overline{M}$ and $\overline{\Gamma }\overline{K}\overline{M}$ on the surface of Bi${}_2$Te${}_3$. The horizontal axis depicts momentum transfer as a fraction of the pertinent lattice vector.

Figure 8

Time-of-flight scans for the inelastic HASS measurements. The abscissa has been converted to energy difference for clarity. Phonon creation and annihilation events are manifest as blue inelastic peaks.

Figure 9

Phonon dispersion along high-symmetry directions $\overline{\Gamma }\overline{K}\overline{M}$ and $\overline{\Gamma }\overline{M}$ for Bi${}_2$Te${}_3$. Gray areas represent the projection of bulk phonon bands onto the surface Brillouin zone, whereas the dotted lines indicate surface phonon modes with at least 30$\%$ of the oscillator strength concentrated within the first three layers of the 30 QL slab surface regions. Green and blue dots indicate modes polarized perpendicular and parallel to the surface plane, respectively. The TOF measurements are depicted as orange dots with error bars.

Figure 10

Diagram showing the dispersion of the surface DFQ excitations. The spin chirality of the Fermi surface is depicted by the red arrows. The screening effects of the DFQs can be understood in terms of the scattering of electrons in the Dirac cone with the momentum transfer supplied by a phonon. We identify two distinct types of transitions: intraband and interband, although the latter are suppressed by energetic considerations. Note that low-energy intraband transitions are confined to a circle of diameter $2k_F$.

Figure 11

(a) Plot depicting the renormalization of the $\nu$-branch energy in the range $0<q<2k_F$. The dotted black lines indicate the renormalized phonon frequencies in the RPA calculation given by Eq. (33), which are fit to experimental data (orange points). (b) Plot of the $\nu$-branch-specific coupling function. The steady increase of ${\lambda }_{\nu }\left(\mathbf{q}\right)$ with $\mathbf{q}$ is consistent with the softening of the $\nu$-branch phonon frequencies. The attenuation just before $q\approx 2k_F$ is due to the fact that scattering of DFQs across the Fermi surface by phonons is not possible due to time-reversal invariance.