Hot-electron-assisted femtochemistry at surfaces: A time-dependent density functional theory approach

Using time-evolution time-dependent density functional theory (TDDFT) within the adiabatic local-density approximation, we study the interactions between single electrons and molecular resonances at surfaces. Our system is a nitrogen molecule adsorbed on a ruthenium surface. The surface is modeled at two levels of approximation, first as a simple external potential and later as a 20-atom cluster. We perform a number of calculations on an electron hitting the adsorbed molecule from inside the surface and establish a picture, where the resonance is being probed by the hot electron. This enables us to extract the position of the resonance energy through a fitting procedure. It is demonstrated that with the model we can extract several properties of the system, such as the presence of resonance peaks, the time electrons stay on the molecule before returning to the surface when hitting a molecular resonance and the lowering of the resonance energy due to an image charge effect. Finally we apply the TDDFT procedure to only consider the decay of molecular excitations and find that it agrees quite well with the width of the projected density of Kohn-Sham states.

Figure 1

A fit of the Kohn-Sham potential at a ruthenium surface. The dotted curve shows the self-consistent ground-state Kohn-Sham potential of a four-layer ruthenium (0001) slab averaged over the directions parallel to the surface. The solid curve shows our fit from Eq. (1). The vertical lines indicate the positions of the layers in the slab. The DFT calculation was made with the GPAW code (Refs. 24, 25).

Figure 2

(Color online) An example of the evolution over time of a hot electron with a momentum directed toward the molecule. The hot-electron orbital is shown at times: 0, 0.15, 0.30, 0.45, 0.60, 0.75, 0.90, and 1.05 fs. The gray scale (online: color grading) indicates the phase of the orbital. The two dots, which are visible at $t=0\text{fs}$, indicate the positions of the nitrogen atoms and the gray line indicates the surface. The unit cell is cylindrical with a radius of $4\text{Å}$ and a length of $40\text{Å}$ and is exactly contained in the shown boxes.

Figure 3

The difference between an electron that hits resonance and one that does not. The $y$ axes are the orbital of the hot electron, $\varphi$, projected to the subspace spanned by the two $2{\pi }^{\ast }$ orbitals of the molecule squared. The $x$ axes show the time. Top panel is an example of an electron hitting resonance (Wf. No. 1 from Table with $p_0=0.8$). Bottom panel is an example of an electron not hitting resonance (Wf. No. 1 from Table with $p_0=0.4$). Please notice the more than 2 orders of magnitude difference in the $y$-axis scales.

Figure 4

The amount of electron that gets into the $2{\pi }^{\ast }$ orbitals of the molecule within the first two femtoseconds plotted as a function of the average momentum of the hot electron, $p_0$. The five different curves are for the five different orbitals given in Table .

Figure 5

The Fourier transform of wave functions 4 and 5 from Table integrated over the momenta parallel to the surface. The horizontal axis indicates the momentum in the direction perpendicular to the surface, $p_z$.

Figure 6

The gray peaks show $W\left(ϵ\right)$ from Eq. (3), where the delta functions have been replaced by Gaussians with a spread of 0.1 eV, for Wf. No. 1 in Table . The different subplots are for different average momenta, $p_0$. The black curves show the fitted Lorentzian, $R\left(ϵ\right)$, from Eq. (4) with the values $\Gamma =1.4\text{eV}$ and ${ϵ}_{\text{res}}=9.8\text{eV}$ above the Fermi level and ${\alpha }_{\text{res}}=5.4\times {10}^{-3}$. The left $y$ axes indicate $W\left(ϵ\right)$ and the right $R\left(ϵ\right)$.

Figure 7

The black lines show the same maximum overlaps depicted on Fig. 4. The gray lines are obtained by inserting Eqs. (3, 4) in Eq. (2) and varying $\Gamma$, ${ϵ}_{\text{res}}$, and ${\alpha }_{\text{res}}$ until the best least-squares fit is obtained. We find this to be at $\Gamma =1.4\text{eV}$ and ${ϵ}_{\text{res}}=9.8\text{eV}$ above the Fermi level and ${\alpha }_{\text{res}}=5.4\times {10}^{-3}$.

Figure 8

(Color online) The ruthenium cluster with adsorbed nitrogen, on which we perform calculations in Sec. 3B. The gray atoms are ruthenium and the dark (online: blue) nitrogen. The cluster has 20 ruthenium atoms.

Figure 9

The momentum space representation of the hot-electron wave function squared and integrated over the directions parallel to the surface.

Figure 10

The black curve shows the amount of electron that gets into the $2{\pi }^{\ast }$ orbital of the molecule within the first three fs plotted as a function of the average momentum of the hot electron, $p_0$. The gray lines are obtained by inserting Eqs. (3, 4) in Eq. (2) and varying $\Gamma$, ${ϵ}_{\text{res}}$, and ${\alpha }_{\text{res}}$ until the best least-squares fit is obtained. We find this to be at $\Gamma =0.36\text{eV}$, ${ϵ}_{\text{res}}=4.9\text{eV}$, and ${\alpha }_{\text{res}}=1.9\times {10}^{-3}$.

Figure 11

The orbital of the hot electron, $\varphi$, projected to the plane spanned by the two $2{\pi }^{\ast }$ orbitals of the molecule squared as a function of time. The black curve is at a momentum of $p_0=0.8\text{a}\text{.u}\text{.}$ and the gray curve is at $p_0=0.2$.

Figure 12

The orbital of the excited electron projected plane spanned by the $2{\pi }^{\ast }$ orbitals of the nitrogen molecule squared as a function of time for four different calculations. The only difference between upper and lower panels is that they use a normal and a logarithmic scale, respectively, on the $y$ axes. Two of the calculations are performed on the 20-atom cluster from Fig. 8 and the other two on the 10-atom cluster from Fig. 13. The difference between the two calculations on each cluster is the state of the excited electron. Either the Kohn-Sham orbital with the largest overlap with the $2{\pi }^{\ast }$ orbitals of the molecule is used or simply one of the $2{\pi }^{\ast }$ orbitals found from a gas phase calculation is used.

Figure 13

(Color online) One of the ruthenium clusters used in Sec. 4. The gray atoms are ruthenium and the dark (online: blue) are an adsorbed nitrogen molecule. The cluster has 10 ruthenium atoms.

Figure 14

The density of states for the 20 atoms cluster of Fig. 8 and the density of states projected onto the plane spanned by the $2{\pi }^{\ast }$ orbitals of the nitrogen molecule. The inserted bar has a length of 1.1 eV, which corresponds to the expected uncertainty in energy of the $2{\pi }^{\ast }$ resonance (see text). The light gray filling indicates the occupied states. Left axis is the density of states and the right the projected density of states.