Birth and decay of coherent optical phonons in femtosecond-laser-excited bismuth

The transient reflectivity of bismuth crystal excited by a 45 fs laser pulse in the near-infrared range has been recovered with an accuracy of ${10}^{-5}$, at initial sample temperatures ranging from 50 to 510 K, and at pump fluences from $2\text{mJ}/{\text{cm}}^2$ to $21\text{mJ}/{\text{cm}}^2$. The coherent phonon excitation and decay processes were imprinted into the time-dependent reflectivity and this allows us to uncover the temporal phonon history preceding the structural transformation of solid Bi. Analysis showed that the first coherent atomic displacement was produced by the polarization force and the electron pressure force during the laser pulse, and that manifests itself by a negative change in the reflectivity. The frequency of the subsequent reflectivity oscillations was chirped, redshifted from the initial value due to the lattice heating. The amplitude decreased gradually while electrons transferred their energy to the lattice. Heating and thermal expansion of the lattice transformed the initially coherent harmonic vibrations of atoms into strongly nonlinear chaotic motion that signifies the onset of disordering of the solid. This process was identified through measurement of the damping rate of the reflectivity oscillations and interpretation of this rate as the decay rate of an optical phonon into two acoustic phonons. The analysis of the reflectivity oscillations provides evidence that the overheated solid experiences only the onset of the solid-liquid phase transition but did not proceed into the liquid phase. General relations between the laser-exerted forces, the atomic motion, and the optical parameters were established. The proposed theory reproduces well the measured transient reflectivity across a wide range of crystal temperatures and laser excitation fluences.

Figure 1

Transient reflectivity signals $\Delta R/R_0$ for different crystal temperatures at a constant pump fluence of $6.9\text{mJ}/{\text{cm}}^2$. The curves at temperatures higher than 50 K are offset for better readability; the horizontal lines indicate the zero level for each depicted time-domain signal.

Figure 2

Initial frequency of the $A_{1g}$—phonon mode as a function of the crystal temperature at a constant pump fluence of $6.9\text{mJ}/{\text{cm}}^2$; the line corresponds to a fit performed with Eq. (39)—see Sec. 4 in the text below.

Figure 3

Damping constant of the $A_{1g}$ mode as a function of crystal temperature at a constant fluency of $6.9\text{mJ}/{\text{cm}}^2$. The dashed line corresponds to a fit with Eq. (38), and the solid line to a fit with Eq. (38) where the exponent was a fitting parameter—see Sec. 4 in the text below.

Figure 4

Transient reflectivity signals $\Delta R/R_0$ for various pump fluences at room temperature. The horizontal lines correspond to the zero level for each depicted time-domain signal.

Figure 5

Initial frequency of the $A_{1g}$ mode as a function of the pump fluence at room temperature; the dashed line corresponds to frequencies calculated with Eq. (40)—see Sec. 4 in the text below.

Figure 6

Damping constant of the $A_{1g}$ mode as a function of pump fluence at room temperature; the line corresponds to a linear fit to the data points.