Electron Cooling and Debye-Waller Effect in Photoexcited Bismuth

By means of first principles calculations, we compute the effective electron-phonon coupling constant $G_0$ governing the electron cooling in photoexcited bismuth. $G_0$ strongly increases as a function of electron temperature, which can be traced back to the semimetallic nature of bismuth. We also use a thermodynamical model to compute the time evolution of both electron and lattice temperatures following laser excitation. Thereby, we simulate the time evolution of (1 $-1$ 0), ($-2$ 1 1) and (2 $-2$ 0) Bragg peak intensities measured by Sciaini et al. [Nature (London) 458, 56 (2009)] in femtosecond electron diffraction experiments. The effect of the electron temperature on the Debye-Waller factors through the softening of all optical modes across the whole Brillouin zone turns out to be crucial to reproduce the time evolution of these Bragg peak intensities.

Figure 1

(a) Calculated phonon density of states in ${\mathrm{meV}}^{-1}$ per cell for $T_e=0\mathrm{K}$ (thin line) and $T_e=2100\mathrm{K}$ (thick line). The inset shows the fraction of acoustic ($\mathrm{TA}+\mathrm{LA}$) and optical (TO and LO) phonons contributing to the effective electron-phonon coupling $G_0$ for electron temperature $T_e$ ranging from 294 K to 4000 K. (b) Calculated effective electron-phonon coupling constant $G_0$ (in $\mathrm{W}·{\mathrm{m}}^{-3}·{\mathrm{K}}^{-1}$) as a function of $T_e$ (in K) compared to the values fitted in Ref. 15 to reproduce the time evolution of (111) Bragg peak intensities measured by Fritz et al. [7]. The inset shows the calculated band structure of Bi along high symmetry lines for $T_e=0\mathrm{K}$.

Figure 2

(a) Mean square displacements (in ${\AA }^2$) of one Bi atom as a function of lattice temperature $T_l$ (in K) from ab initio calculations for $T_e=294\mathrm{K}$ and $T_e=2100\mathrm{K}$ (thin and thick solid lines). The vertical line indicates the melting temperature of Bi. (b) Calculated normalized intensities defined by $I_{hkl}(T_l,T_e=T_l)/I_{hkl}(T_l=294\mathrm{K},T_e=294\mathrm{K})$ for (1 $-1$ 0), ($-2$ 1 1) and (2 $-2$ 0) Bragg peaks as a function of lattice temperature $T_l$ in K. The horizontal lines are the experimental intensities obtained in FED experiments [11] 14 ps after the arrival of the 200-fs pump pulse of fluence $0.84\mathrm{mJ}·{\mathrm{cm}}^{-2}$. The experimental data are shown in Fig. 3c.

Figure 3

(a),(b) Lattice $T_l$ and electron temperature $T_e$ in K, as a function of time delay $t$ in ps obtained from the model proposed in Ref. 15 for a 30-nm-thick Bi film excited by a 200-fs laser pulse of incident fluence $F_{\mathrm{inc}}=1.11\mathrm{mJ}·{\mathrm{cm}}^{-2}$ at a wavelength of 775 nm. The inset shows the function $g$ in ${\mathrm{nm}}^{-1}$ as a function of $z$ in nm. (c) Normalized intensity decays of (1 $-1$ 0), ($-2$ 1 1), and (2 $-2$ 0) Bragg peaks measured by Sciaini et al. [11] as a function of $t$ compared to the theoretical calculations. The thin lines represent $I_{h,k,l}(T_l,T_e=T_l)/I_{h,k,l}(T_l=294\mathrm{K},T_e=294\mathrm{K})$ while the thick lines represent $I_{h,k,l}(T_l,T_e)/I_{h,k,l}(T_l=294\mathrm{K},T_e=294\mathrm{K})$.