Distinguishing Bulk and Surface Electron-Phonon Coupling in the Topological Insulator ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ Using Time-Resolved Photoemission Spectroscopy

We report time- and angle-resolved photoemission spectroscopy measurements on the topological insulator ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. We observe oscillatory modulations of the electronic structure of both the bulk and surface states at a frequency of 2.23 THz due to coherent excitation of an $A_{1\mathrm{g}}$ phonon mode. A distinct, additional frequency of 2.05 THz is observed in the surface state only. The lower phonon frequency at the surface is attributed to the termination of the crystal and thus reduction of interlayer van der Waals forces, which serve as restorative forces for out-of-plane lattice distortions. Density functional theory calculations quantitatively reproduce the magnitude of the surface phonon softening. These results represent the first band-resolved evidence of the $A_{1\mathrm{g}}$ phonon mode coupling to the surface state in a topological insulator.

Figure 1

(a) Schematic of the experimental setup including 1.5-eV pump and 6-eV probe pulses. (b) ARPES spectrum of the sample at equilibrium (unpumped) at 40 K. (c) trARPES spectra after optical excitation. The dotted lines represent the SS dispersion extracted by momentum-distribution curve (MDC) fitting. (d) Zoom in on the right-hand branch of the extracted dispersions, showing a time-dependent renormalization of the band. (e) The time-dependent scaling and (f) shift of the SS dispersion.

Figure 2

Oscillations in the SS (top) and CB (bottom). (a) The time-dependent shift of the SS and a smooth curve fit (tenth-order polynomial). (b) Residuals from the fit. The oscillations can be fit as a beating pattern due to the sum of two cosine functions (see text for fit details). (c) Photoemission intensity at the CB edge obtained by integrating within the window indicated in the inset. The color scale in the inset is oversaturated to emphasize the CB. The smooth curve fit is a tenth-order polynomial. (d) Residuals from the fit. The oscillations can be fit with a single undamped cosine.

Figure 3

(a) Magnitude of the Fourier transforms of the surface and bulk oscillations. The vertical dashed lines are at 2.05 and 2.23 THz (8.48 and 9.22 meV). (b) Cartoon depiction of the optical excitation of coherent phonons in the bulk and surface bands. For clarity, the magnitude of the oscillations is greatly exaggerated with respect to that observed in the experiment.

Figure 4

(a) The lattice distortions employed to compute the surface phonon softening [47]. The arrows represent the $A_{1\mathrm{g}}$ distortions. The structure on the left represents the upper half of the 6-QL slab used to model surface-only distortions. A vacuum layer separates each 6-QL slab. On the right is the 3-QL structure used to model bulk distortions. There is no vacuum layer and hence no surface. (b) The band structure obtained for the 6 QL slab, which has the Dirac SS clearly visible. (c) Total energy of the frozen phonon configurations (per distorted QL) for bulk and surface, showing the softening of the surface phonon. Solid lines are parabolic fits (without linear terms).