Coherent Dirac plasmons in topological insulators

We explore the ultrafast reflectivity response from photo-generated coupled phonon-surface Dirac plasmons in ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ topological insulators several quintuple layers thick. The transient coherent phonon spectra obtained at different time frames exhibit a Fano-like asymmetric line shape of the $A_{1g}^2$ mode, which is attributed to quantum interference between continuumlike coherent Dirac plasmons and phonons. By analyzing the time-dependent asymmetric line shape using the two-temperature model (TTM), it was determined that a Fano-like resonance persisted up to $\approx 1$ ps after photo excitation with a relaxation profile dominated by Gaussian decay at $\le 200$ fs. The asymmetry parameter could be well described by the TTM for $\ge 200$ fs, therefore suggesting the coherence time of the Dirac plasmon is $\approx 200$ fs.

Figure 1

(a) Time-resolved reflectivity signal observed for a ${\mathrm{Sb}}_2{\mathrm{Te}}_{3}8$-QL-thick film. The dashed line represents the background. The inset represents a schematic of the experimental layout and the coupling dynamics in the surface region. (b) The time evolution of the coherent phonon oscillation signals obtained after background subtraction.

Figure 2

(a) DWT chronogram obtained from the oscillatory signal shown in Fig. 1. The color bar represents the DWT intensity. A time lag between the $A_{1g}^1$ and $A_{1g}^2$ modes can be clearly observed as indicated by $\mathrm{\Delta }\tau$. (b) Static FFT spectrum obtained from the oscillatory signal in Fig. 1.

Figure 3

(a) Time evolution of the DWT intensity for the $A_{1g}^1$ and $A_{1g}^2$ modes as a function of the time delay. The vertical dashed lines represent the peak positions. (b) Time-sliced DWT phonon spectra obtained for the different time delays at 0.0, 0.5, and 1.0 ps. The dashed curves indicate the fit to the spectra using Eq. (1). For clarity, the spectra have been vertically shifted.

Figure 4

(a) The time evolution of the coupling strength (1/$q)$. The solid line represents a fit using the TTM result using the $e-ph$ coupling constant of $G=2.0\times {10}^{16}{\mathrm{Wm}}^{-3}{\mathrm{K}}^{-1}$, whereas the dashed line represents a Gaussian fit. (b) The time evolution of the electron and lattice temperatures for ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ calculated by the TTM.

Figure 5

Schematic band structures for photoexcitation and subsequent relaxation processes in ${\mathrm{Sb}}_2{\mathrm{Te}}_3$. (a) A 1.5 eV photon excites electrons from the occupied bulk valence band $\left({\mathrm{BVB}}_1\right)$ below the Fermi level to the ${\mathrm{SS}}_2$ band as well as the ${\mathrm{BVB}}_2$ band, generating a nonequilibrium electron population, whose energy distribution $f(E)$ can be approximated using the Fermi-Dirac distribution function. The nonequilibrium electron distribution with a maximum electron temperature of 1800 K $(\approx 164$ meV) is well overlapped with the optical phonon energy $({\omega }_2$ as the $A_{1g}^2$ mode), enabling the Fano resonance. The thick dashed arrow represents a possible direct optical transition from deeper-lying bulk states into the ${\mathrm{SS}}_1$ band, whereas the thin solid and dashed arrows represent the electronic transitions within the SS bands at the energy of the $A_{1g}^2$ as a probe action. (b) Ultrafast dephasing of the electrons near the Dirac cone and the coherent emission of the coupled optical phonons $({\omega }_1$ as the $A_{1g}^1$ mode) within the ${\mathrm{SS}}_2$ bands. (c) Relaxation of electrons from the higher-lying ${\mathrm{BVB}}_2$ band into the lower-lying bulk conduction band $\left({\mathrm{BCB}}_1\right)$ and into the ${\mathrm{SS}}_1$ band is indicated by the arrows.