Coherent phonons in a ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ film generated by an intense single-cycle THz pulse

We report an observation of coherent phonons of $E_g^1, E_u^1, A_{1g}^1$, and $E_g^2$ symmetry generated in a single-crystal film of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ by an intense single-cycle THz pulse. The atomic vibrations reveal themselves through periodic modulation of the refractive index of the film. The largest signal is detected at the frequency of 4.05 THz that corresponds to the $E_g^2$ mode. The generation of $E_g^2$ phonons is interpreted as resonant excitation of the Raman mode by the second harmonic of THz-driven nonlinear $E_u^1$ oscillator, the fundamental frequency of which (2.05 THz) is approximately half that of $E_g^2$. The origin of nonlinearity in this case is cubic lattice anharmonicity, while generation of $E_g^1$ (1.1 THz) and $A_{1g}^1$ (2.25 THz) phonons is a manifestation of quartic anharmonicity enhanced by the occasional combination relations between phonon frequencies in ${\mathrm{Bi}}_2{\mathrm{Se}}_3$.

Figure 1

Time profile of the pump THz pulse (in the inset) and its spectrum obtained using FFT. Arrows show the position of relevant phonon modes of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ on the frequency scale.

Figure 2

Response of the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ film to the pump THz pulse measured using isotropic (Δ$T/T$ denotes relative change of transmittance; the curve is shifted upward by ${10}^{-3}$) and anisotropic detection.

Figure 3

Spectra of oscillations in the anisotropic and isotropic THz-induced response of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. The anisotropic spectrum was shifted upward to facilitate comparison. Arrows illustrate attribution of the observed spectral lines to phonon modes of the crystal.

Figure 4

Dependence of the oscillation amplitudes $A\left(E_u^1\right)$ (multiplied by 4, circles) and $A\left(E_g^2\right)$ (squares) on the sample orientation. The angle between THz pump and 800-nm probe polarizations was kept constant (45°). The solid line is the $r=a\left|sin3\phi \right|$ function with an arbitrarily chosen amplitude $a$.

Figure 5

(a), (b) The dependence of oscillation amplitudes $A\left(E_u^1\right)$ and $A\left(E_g^2\right)$ on the power of the laser beam fed into the THz generation stage ($P_{800\mathrm{nm}}$). (c) The dependence of $A\left(E_g^2\right)$ on $A\left(E_u^1\right)$. Dashed lines represent fits of the experimental data: square root in panel (a), linear in (b), and parabolic in (c).

Figure 6

(a) Electric field of the pump THz pulse substituted as a driving force into Eq. (8). (b) Trajectory of the IR-active oscillator obtained solving Eq. (8). (c) Trajectory of the Raman-active oscillator driven by $Q_I^2$, the solution of Eq. (9) (dash-dotted curve). The dotted curve represents the squared electric field of the THz pulse as a reference.

Figure 7

(a) Convolution of the experimental anisotropic response of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ with the wavelet at 2 THz having a duration of $\sim 600$ fs. (b) Convolution of the same anisotropic signal with the wavelet at 4 THz having a duration of $\sim 400$ fs. The dashed line is the squared trace from panel (a). The wavelets are shown as dotted curves.

Figure 8

(a) Electric field of the pump THz pulse reflected from a gold mirror (dashed line) and from the $\mathrm{PrF}{\mathrm{e}}_3{\left(\mathrm{B}{\mathrm{O}}_3\right)}_4$ crystal (solid). (b) Spectra of the THz waveforms shown in (a). (c) Reflectance spectrum of $\mathrm{PrF}{\mathrm{e}}_3{\left(\mathrm{B}{\mathrm{O}}_3\right)}_4$ [54]. Measurements were made under ambient conditions.

Figure 9

THz response of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ measured using THz pulses reflected from a gold mirror (dashed line) and from the $\mathrm{PrF}{\mathrm{e}}_3{\left(\mathrm{B}{\mathrm{O}}_3\right)}_4$ crystal (solid line). The inset shows spectra of these signals. Measurements were made under ambient conditions.