Influence of local symmetry on lattice dynamics coupled to topological surface states

We investigate coupled electron-lattice dynamics in the topological insulator ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ with time-resolved photoemission and time-resolved x-ray diffraction. It is well established that coherent phonons can be launched by optical excitation, but selection rules generally restrict these modes to zone-center wave vectors and Raman-active branches. We find that the topological surface state couples to additional modes, including a continuum of surface-projected bulk modes from both Raman and infrared branches, with possible contributions from surface-localized modes when they exist. Our calculations show that this surface vibrational spectrum occurs naturally as a consequence of the translational and inversion symmetries broken at the surface, without requiring the splitting-off of surface-localized phonon modes. The generality of this result suggests that coherent phonon spectra are useful by providing unique fingerprints for identifying surface states in more controversial materials. These effects may also expand the phase space for tailoring surface state wave functions via ultrafast optical excitation.

Figure 1

(a) Schematic of time-resolved ARPES measurements on ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ to measure electronic binding energy oscillations. (b) ARPES spectrum along the $\overline{\mathrm{\Gamma }\text{K}}$ direction before and (c) after excitation at the pump-probe delays indicated. The bulk conduction band and most of the surface state are unoccupied in equilibrium, but become partially filled by the excitation. (d) Time-dependent shift in the binding energy of the bulk conduction band and (e) surface state within the energy-momentum windows indicated by dashed lines in (c). (f) Magnitude of the Fourier transform of the bulk conduction band and (g) surface state dynamics after subtracting a slowly varying background ($6^{\mathrm{th}}$ order polynomial for the bulk and exponential for the surface). Points are from the data and lines are fits (see Sec. pp2 for fitting methodology and Table 1 for fit parameters). The arrows highlight two modes observed in the surface state but not in the bulk.

Figure 2

(a) Time-resolved x-ray diffraction measurement to measure coherent lattice dynamics. [(b)–(d)] Time-dependent relative change in the intensities of the $\left(hkl\right)$ = (4 4 5), (5 5 6), and (3 3 4) Bragg peaks. [(e)–(g)] The corresponding Fourier transforms after exponential background subtraction. Points are from the data, and solid lines are a fit. Two modes are observed, with $(h+k+l)$-dependent amplitudes consistent with modes of ${\text{A}}_{1\text{g}}^1$ and ${\text{A}}_{1\text{g}}^2$ symmetries.

Figure 3

Theoretical calculation of surface and bulk lattice dynamics in ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ and ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ using DFT and a dynamical matrix formalism. (a) Fourier transforms of the Bi atomic motion at the surface and center (“bulk”) of the slab. (b) The calculated response of the surface state compared to the trARPES data for the surface state [reproduced from Fig. 1], showing overall agreement. (c) $\overline{\mathrm{\Gamma }}$-projected density of states of the full slab and for the top QL only. Arrows denote surface modes outside the bulk continuum. (d) Bulk phonon dispersion curves along $\mathrm{\Gamma }$-Z. [(e)–(h)] Same set of analysis repeated for ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, exhibiting similar phenomenology. Experimental data for ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ in (f) reproduced from Ref. [29].

Figure 4

Classical semi-infinite one-dimensional chain of trimers, representing the minimum model containing Raman and infrared phonons. (a) Cartoon of the model. All atoms have identical masses $m$, connected by spring constants $k_1/m={\left(2\pi \right)}^2{\left(1\text{THz}\right)}^2$ and $k_2/m={\left(2\pi \right)}^2{\left(0.8\text{THz}\right)}^2$. At $t=0$ the trimers are uniformly distorted throughout the chain (see $\mathrm{\Delta }x$ arrows) corresponding to excitation of the ${\text{A}}_{1\text{g}}^{}$ optical phonon with $q_z=0$. (b) Phonon dispersion and (c) density of states for the bulk. (d) Surface density of states and (e) FFT of resulting motion for a bulk (red) and surface (blue) atom for the case in which the surface $k_1$ are softened by 2%. A damping time of 20 ps is used. (f) The amplitude of the low-frequency and high-frequency peaks relative to the central peak, plotted versus layer number. Corresponding plots for the unperturbed surface are shown in (g)–(i), and for the 2% hardened surface in (j)–(l).

Figure 5

Complementary sensitivities of multimodal probes. Near the surface (within the top quintuple layer), atomic motion involves the surface-projected bulk phonon spectrum with infrared- and Raman-active branches in addition to surface-localized modes. These oscillations are probed by trARPES measurements of the surface state. Deep within the bulk, only Raman-active modes at $q_z=0$ are excited. These are probed by trARPES measurements of the bulk states, as well as trXRD and transient optical reflectivity. The different frequency of the ${\text{A}}_{1\text{g}}^2$ mode in XRD is due to its different measurement temperature and fluence.

Figure 6

Overview of the Fourier transform fitting methodology. (a) Real and imaginary parts of the Fourier transform of a simulated data set of two oscillators with equal amplitudes and opposite phases. The frequencies are $f_1=1.0$ THz and $f_2=1.5$ THz with lifetimes ${\tau }_1={\tau }_2=1$ ps. (b) Magnitude of the same Fourier transform. The dashed lines depict the magnitude of each oscillator separately. Note that the total magnitude is not simply the sum of the individual magnitudes due to interference. (c) Simultaneous fit to the real and imaginary parts of the FFT for the bulk band dynamics measured by trARPES. (d) The FFT magnitude resulting from this fit, which is also shown in Fig. 1. [(e) and (f)] Same analysis but for the surface band.

Figure 7

Structure factor analysis of trXRD data. [(a)–(f)] Time-resolved x-ray diffraction measurement for six Bragg peaks excited by 800 nm with an incident fluence of $8.2\mathrm{mJ}/{\mathrm{cm}}^2$. [(g)–(h)] Amplitude of both modes extracted by fitting the time-dependent intensities, plotted as a function of $(h+k+l)$ for the six measured peaks. The solid line is a fit from a structure function model assuming modes of ${\text{A}}_{1\text{g}}^1$ and ${\text{A}}_{1\text{g}}^2$ symmetries, respectively. This allows for quantitative extraction of the displacements of all five atoms in the unit cell, sketched as arrows in (i) and (j).

Figure 8

Fluence-dependence of mode frequencies measured by time-resolved XRD at room temperature. (a) Fourier transform of the (5 5 6) Bragg peak dynamics as a function of incident fluence. Solid curves are fits and dashed lines are guides to the eye. (b) Fluence-dependent frequencies extracted from the fits. The ${\text{A}}_{1\text{g}}^1$ mode is fluence-independent at 1.84 THz, while the ${\text{A}}_{1\text{g}}^2$ mode extrapolates to $3.94\pm 0.05$ THz at zero fluence with a slope of $0.040\mathrm{THz}/(\mathrm{mJ}/{\mathrm{cm}}^2)$.

Figure 9

Time-resolved optical reflectivity measurement on ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. (a) Relative time-dependent change in the reflectivity at 800 nm. (b) Fourier transform after background subtraction. Points are from the data, and solid lines are a fit. Two peaks corresponding to the ${\text{A}}_{1\text{g}}^1$ and ${\text{A}}_{1\text{g}}^2$ modes are observed at 1.85 and 3.89 THz.