Thickness dependence of surface plasmon damping and dispersion in ultrathin Ag films

The thickness dependence of the surface plasmon damping and dispersion of atomically flat and ultrathin silver films deposited on the $\mathrm{Si}\left(111\right)\text{−}(7\times 7)$ surface was investigated by a combined high-resolution electron-energy-loss spectroscopy (HREELS) and scanning tunneling microscopy (STM) system. We found stronger plasmon energy dispersion with the momentum parallel to the surface $\left(q_{\Vert }\right)$ in thicker films, and a significant dependence of the damping edge on the film thickness. Both of them can be associated with the presence of quantum well states (QWS) in the direction perpendicular to the film surface, which influences the interband transitions between the lower and upper $5sp$ bands of silver.

Figure 1

STM images of Ag films with different coverages deposited on $\mathrm{Si}\left(111\right)\text{−}(7\times 7)$ at $120\mathrm{K}$ followed by annealing to RT. (a) 2.5 ML; (b) 5 ML; (c) 7.5 ML; (d) 10 ML; (e) 13 ML. The scale is $200\mathrm{nm}\times 200\mathrm{nm}$ for all images.

Figure 2

(Color) Two sets of HREEL spectra obtained at different scattering angles ${\theta }_s$ for Ag film thickness at 5 ML and 7.5 ML, respectively. An electron primary energy $E_0=20\mathrm{eV}$ and an incident angle ${\theta }_s=60°$ were used. The Gaussian fitting for each curve is also shown in the figure.

Figure 3

(Color) The surface plasmon dispersion versus the parallel momentum $q_{\Vert }$ for different Ag coverages. The best-fitting curves for the dispersion are also displayed. The error bars indicate the finite angular acceptance of the HREELS analyzer, which is typically 1° corresponding to $0.02{\mathrm{\AA }}^{-1}$ in the momentum space. The inset is the LEED pattern where the surface Brillouin zone of Ag(111) is indicated.

Figure 4

(Color) The FWHM of the surface plasmons as a function of the plasmon energy $E_{sp}$ for different film thickness. The critical plasmon energy $E_{sp}^c$ corresponding to the minimal FWHW is indicated by a black arrow. The inset in the figure shows the comparison between the $E_{sp}^c$ and theoretically calculated energy of the final states (in the $\Gamma \text{−}L$ direction) corresponding to the quantum number $n=1$. Note that because the data of dispersion in the $\overline{\Gamma }\text{−}\overline{K}$ direction is not available, only the line shapes of the two curves should be compared. The overall profile of the curve nicely agrees with the experiment.

Figure 5

(a) The band structure of Ag along major symmetry directions. (b) The zoom-in view of the $sp$ bands in the $\Gamma \text{−}L$ direction of the BZ. Two QWS’s with the quantum number $n=1$, at different film thickness, $t_1>t_2$, are marked in the lower $sp$ band as $k_{\perp }\left(t_1\right)$ and $k_{\perp }\left(t_2\right)$. In the surface plasmon damping, electrons jump from these QWS’s to the upper $sp$ band whose final energies are represented by $E_f\left(t_1\right)$ and $E_f\left(t_2\right)$. (c) The in-plane dispersion of the QWS’s and the dispersion of the corresponding final states in the upper $sp$ band along the $\overline{\Gamma }\text{−}\overline{K}$ direction. The ${\omega }_{T1}$ and ${\omega }_{T2}$ represent the onsets of the interband transitions and ${\omega }_{T1}<{\omega }_{T2}$ can be concluded for $t_1>t_2$. (d) is the same as (c) but with $q_{\Vert }\ne 0$.