Direct observation of electron thermalization and electron-phonon coupling in photoexcited bismuth

We investigate the ultrafast response of the bismuth (111) surface by means of time-resolved photoemission spectroscopy. The direct visualization of the electronic structure allows us to gain insights on electron-electron and electron-phonon interaction. Concerning electron-electron interaction, it is found that electron thermalization is fluence dependent and can take as much as several hundreds of femtoseconds at low fluences. This behavior is in qualitative agreement with Landau's theory of Fermi liquids, but the data show deviations from the behavior of a common three-dimensional degenerate electron gas. Concerning electron-phonon interaction, our data allows us to directly observe the coupling of Bloch states to the coherent $A_{1g}$ mode. It is found that surface states are much less coupled to this mode when compared to bulk states. This is confirmed by ab initio calculations of surface and bulk bismuth.

Figure 1

(a) Angle-resolved photoemission data taken on Bi (111) surface at 130 K and along the ${\Gamma }_s$-${\mathrm{M}}_s$ direction of the surface Brillouin zone. $b$ is a bulk band whereas $ss$ is a surface band and $sr$ a surface resonance. Solid line: DFT-LDA calculation of bulk band $b$; dashed line: DFT-GGA calculation of surface states $ss$ and $s{s}^{\prime }$. (b) Fermi surface of the Bi (111) surface.

Figure 2

DFT-GGA simulations of the electronic structure of bulk Bi, taken from Ref. 27. The dashed red arrows stand for direct transitions caused by the 1.6-eV pump laser pulse.

Figure 3

Photoelectron intensity map in the ${\Gamma }_s$-${\mathrm{M}}_s$ direction. (a) Before the arrival of the pump pulse, (b) after the arrival of the pump pulse, fluence $F=0.22\mathrm{mJ}/{\mathrm{cm}}^2$. The polarization of the probe pulse is close to $s$ polarization, which reveals the surface states above and below the Fermi level.

Figure 4

Photoelectron spectra at $k_{\parallel }=0.38{\AA }^{-1}$ as a function of pump probe delay. Panels (a) and (b) were taken with a pump fluence of $0.22\mathrm{mJ}/{\mathrm{cm}}^2$ and $0.04\mathrm{mJ}/{\mathrm{cm}}^2$, respectively.

Figure 5

(a) Number of excited electrons in state $s{s}^{\prime }$ as a function of pump-probe delay for a long scan (fluence was $0.84\mathrm{mJ}/{\mathrm{cm}}^2$). (b) Number of excited electrons in state $s{s}^{\prime }$ for different fluences and just after photoexcitation. From pink line to blue line, fluences are $0.12-0.2-0.36-0.52-0.68-0.84\mathrm{mJ}/{\mathrm{cm}}^2$. (c) The shift of the surface band as a function of pump-probe delay for the same fluences as in (b). (d) The characteristic time associated with the number of excited states and shift of the surface state $ss$ as a function of pumping power.

Figure 6

(a) Evolution of the electronic temperature as a function of pump-probe delay for $F=0.16\mathrm{mJ}/{\mathrm{cm}}^2$. (b) The thermalized electronic temperature as a function of laser pump fluence. The red curve is a fit $\propto F^{\alpha }$ with $\alpha =0.6$. (c) Electron thermalization time as a function of the thermalized electronic temperature $T_e^{\mathrm{eq}}$. The full red curve is a fit $\propto 1/T^{\beta }$ with $\beta =0.7$. The dashed black curve represents ${\tau }_l$ computed using Eq. (2).

Figure 7

(a) Ultrafast dynamics of band $b$ following photoexcitation at $0.6\mathrm{mJ}/{\mathrm{cm}}^2$. (b) Ultrafast dynamics for the surface band $ss$.

Figure 8

Binding energy of the bulk state $b$ (full red line) and surface state $s$ (black dashed line) as a function of pump-probe delay. The error bar represents the typical energy error of the fitting procedure.

Figure 9

DFT simulations of band dispersion in bismuth. The bands are calculated along the ${\Gamma }_s{\text{-M}}_s$ direction and for various atomic displacements $u/c$ in the $A_{1g}$ mode. (a) Dispersion of the surface state $ss$ and (b) bulk state $b$.