Time-resolved spectroscopy at surfaces and adsorbate dynamics: Insights from a model-system approach

We introduce a model description of femtosecond laser induced desorption at surfaces. The substrate part of the system is taken into account as a (possibly semi-infinite) linear chain. Here, being especially interested in the early stages of dissociation, we consider a finite-size implementation of the model (i.e., a finite substrate), for which an exact numerical solution is possible. By time-evolving the many-body wave function, and also using results from a time-dependent density functional theory description for electron-nuclear systems, we analyze the competition between several surface-response mechanisms and electronic correlations in the transient and longer time dynamics under the influence of dipole-coupled fields. Our model allows us to explore how coherent multiple-pulse protocols can impact desorption in a variety of prototypical experiments.

Figure 1

The model of Eq. (1), for a mobile adsorbate (with one core and two interacting valence levels) on a 5-site substrate. The adsorbate charge fluctuations are coupled to a local/surface plasmon. The substrate-adsorbate effective potential and the laser perturbation are also schematically shown.

Figure 2

Adiabatic potential energy surfaces (PES) for the model system of Eq. (1) with one core and two valence levels on the adsorbate. The heat map is for a system with $L=5, U=4, {\epsilon }_1=-2$, and ${\epsilon }_2=1$, with colors related to the density of PES ranging from high (red) to low (blue). The light green curves show the eight lowest PES, with nuclear energy levels for the ground state potential surface and a parabolic fit (black curve) superimposed. For comparison, the lowest PES for $L=1$ are also shown, for adsorbate interaction $U=1$ (black dashed curves) and $U=4$ (black solid curves). For $U=1$, the valence levels are at ${\epsilon }_1=-0.75$ and ${\epsilon }_2=2$ to provide adsorbate level fillings similar to $U=4$.

Figure 3

The imaginary part of the local Green's function $G_{vv}$ and density-density response function ${\chi }_{vv}$ is shown for the two valence levels $v_1$ (lower) and $v_2$ (upper). All panels show the same quantity without (shaded regions) and with (lines) a local plasmon mode of frequency ${\omega }_p=4$ and coupling $\gamma =1$.

Figure 4

The imaginary part of the local Green's function $G_{vv}$ is shown for the two valence levels $v_1$ (lower) and $v_2$ (upper). The shaded regions show the case of an infinitely heavy adsorbate and the lines correspond to a mass $M=352$, for a plasmon mode of frequency ${\omega }_p=4$ and coupling $\gamma =2$.

Figure 5

Dependence of the adsorbate wave packet dynamics on the length $L$ of the substrate and the interaction strength $U$. Panel (a) shows snapshots of the wave packet for $L=1$ (red) and $L=3$ (orange), as well as for $L=5$ with an electron-plasmon coupling $\gamma =0$ (green) and $\gamma =1$ (blue). In (b) we show the evolution of the nuclear wave packet for $U=1$ (green) and $U=4$ (blue). In both panels ${\mathrm{\Lambda }}_{v{v}^{\prime }}\left(t\right)=A{e}^{-{(t-t_0)}^2/\tau }cos\left(\omega t\right)$, with $t_0=10, \tau =13$, and $\omega =6\pi /8$, of amplitude $A=2$.

Figure 6

Heat map of the occupation in the hundred first natural orbitals of the reduced nuclear density matrix, as a function of time. Superimposed is the entanglement entropy $S_n$ for the nuclear subsystem.

Figure 7

Time evolution (in units of fs) of the nuclear density (red/blue), average position (orange/green), and plasmon density (back), for different pulse protocols (front). The parameters are as discussed in Sec. 2a with $L=5$ and $U=4$. In (a) two pulses of identical integrated intensity and duration 6 fs (orange) and 35 fs (green) are shown; in (b) two pulses of identical amplitude $A=2$ and duration 6 fs (orange) and 35 fs (green) are shown. The panels at the bottom show the respective bond kinetic energies $K_{ad}$.

Figure 8

Time evolution of the nuclear density (red/blue), average position (orange/green), and plasmon density (back), for two different pulse protocols (front). Parameters are as in Fig. 7. In (a) two pulses are applied consecutively with delays of $14\mathrm{fs}$ or $18\mathrm{fs}$, and ${\mathrm{\Lambda }}_{v{v}^{\prime }}\left(t\right)=A{e}^{-{(t-t_0)}^2/\tau }cos\left(\omega t\right)+A{e}^{-{(t-t_1)}^2/\tau }cos(\omega t+\phi )$. Here $t_0=10, \tau =13, \omega =6\pi /8$, and $A=2$, with $t_1=24$ (red/green) or $t_1=28$ (blue/orange), and $\phi$ chosen to give the second pulse the same envelope-carrier relation as the first. In (b) a core electron is removed during the action of a 6 fs pulse, at ${\tau }_c=10$ (red/green) or ${\tau }_c=11.3$ (blue/orange) with $w=6$.

Figure 9

Panel (a) gives the exact nuclear potential ${\epsilon }_{\mathrm{KS}}$ for a dimer with one valence level at ${\epsilon }_1=-U/2$, while (b) shows the electron density at the adsorbate (red) and the exact electronic KS potential $|T_{\mathrm{KS}}|$ (blue) and $arg\left(T_{\mathrm{KS}}\right)$ (green) for the same system. The insets are snapshots at $t=0,t=17$, and $t=34$ of the nuclear wave packet and corresponding ${\epsilon }_{\mathrm{KS}}$. In all cases ${\mathrm{\Lambda }}_{v{v}^{\prime }}\left(t\right)=A{e}^{-{(t-t_0)}^2/\tau }cos\left(\omega t\right)$, with $t_0=10, \tau =13$, and $\omega =6\pi /8$, of amplitude $A=10$ in (a)–(b).

Figure 10

In the three leftmost panels we show the electron density, bond kinetic energy, and mean internuclear distance, after the application of a square potential with amplitude $A=2$ (blue), $A=6$ (green), or $A=10$ (orange). The three central panels show the argument and modulus of the exact electronic Kohn-Sham potential $T^{\mathrm{KS}}\left(t\right)$, as well as the time structure of the external laser field. In the panels farthest to the right we show 3D plots of the exact nuclear potential for pulses of amplitude $A=2$ (top) and $A=10$ (bottom).

Figure 11

In the three leftmost panels we show the electron density, bond kinetic energy, and mean internuclear distance, after the application of a square potential with amplitude $A=2$ (blue), $A=6$ (green), or $A=10$ (orange). The three central panels show the argument and modulus of the exact electronic Kohn-Sham potential $T^{\mathrm{KS}}\left(t\right)$, as well as the time structure of the external laser field. In the panels farthest to the right we show 3D plots of the exact nuclear potential for pulses of amplitude $A=2$ (top) and $A=10$ (bottom).

Figure 12

In the three leftmost panels we show the electron density, bond kinetic energy, and mean internuclear distance, after the application of a square potential with amplitude $A=2$ (blue), $A=6$ (green), or $A=10$ (orange). The three central panels show the argument and modulus of the exact electronic Kohn-Sham potential $T^{\mathrm{KS}}\left(t\right)$, as well as the time structure of the external laser field. In the panels farthest to the right we show 3D plots of the exact nuclear potential for pulses of amplitude $A=2$ (top) and $A=10$ (bottom).