Low-energy plasmons in quantum-well and surface states of metallic thin films

We studied low-energy plasmons in ultrathin films of silver in the thickness regimes where the surface states as well as quantum-well states must play significant roles. Realistic band structure was adopted for assessing the quantum-mechanical effect on the low-energy charge dynamics. In addition to the expected quasi-two-dimensional plasmon mode, we find the modes that resemble an acoustic surface plasmon on semi-infinite metal surfaces and an additional plasmon mode related to the interband transitions between the two slab-split surface states. It is found that the dispersion of the latter mode is almost identical to the acoustic surface plasmon except the energy offset at small momenta values. The peaks in the surface response function related to interband transitions between the surfacelike states and bulklike states are identified as well. The present work elucidates the role of quantized electronic states and surface states on the plasmonic excitations in the ultimately thin films potentially used in the future nano-optics devices.

Figure 1

(Right) Calculated band structure for the 3-ML Ag(111) slab. The energy positions at $k=0$ are obtained with the use of the Ag(111) model potential.[44] Dispersion of the slab energy bands with effective masses set to unity (dashed lines) and realistic ones (solid lines) is shown. A pair of even and odd energy bands evolving into the Ag(111) surface state on increase of the slab thickness for realistic (unity) effective masses is presented by thick solid (dashed) lines and denoted as SS${}^{+}$ and SS${}^{-}$, respectively. (Left) Calculated band structure for the jellium slab with thickness equal to three Ag MLs. All energies are relative to the Fermi level, $E_F$.

Figure 2

(Right) Calculated band structure for the 31-ML Ag(111) slab. The energy positions at $k=0$ are obtained with the use of the Ag(111) model potential.[44] Dispersion of the slab energy bands with realistic effective masses is shown. Note that for this slab thickness the splitting of the even SS${}^{+}$ and odd SS${}^{-}$ surface states is negligible and only a doubly degenerate surface state SS exists. (Left) Calculated band structure for the jellium slab with thickness equal to 31 Ag MLs. Note that, in this case, absence of an energy gap in the projected bulk electronic structure does not allow appearance of any surface state. All energies are relative to the Fermi level, $E_F$.

Figure 3

Energy values at the $k=0$ point for quantum-well states in the Ag(111) slabs versus number $n$ of MLs obtained with the use of a model potential of Ref. 44. Dots and squares stand for even and odd quantum states, respectively. Crosses show the experimental values measured in Ag overlayers on Si(111).[47] Solid lines show energies calculated in Ref. 47 according to the phase-shift quantization rule.

Figure 4

Charge-density distribution of the lowest-energy QWSs, $|{\phi }_{nl}{\left(z\right)|}^2$, as a function of $z$ in the 8-ML slab: (a) the Ag(111) model potential[44] and (b) the jellium calculations. The zero position for each curve is located at the corresponding energy $E^{8l}$ shown on the right. The zero energy is at the Fermi level. The QWS number is counted from the bottom. In the Ag(111) case the states corresponding to the surfacelike states are denoted by SS${}^{-}$ and SS${}^{+}$ according to their symmetry respective to the $z=0$ plane.

Figure 5

Normalized surface loss function Im$\left[g\right(q,\omega \left)\right]/q\omega$ for the 1- to 10-, 17-, and 31-ML Ag(111) films evaluated with the use of realistic effective masses in energy band dispersions. The strongly dispersing peak corresponding to a conventional ${\omega }_{\mathrm{sp}}^{-}$ mode of a thin film is denoted by the corresponding symbol. Acoustic surface plasmon dispersion is labeled with “ASP.” Peaks having their origin in the interband transitions between the energy-split SS${}^{+}$ and SS${}^{-}$ quantum states are denoted with “ISP”. The dashed vertical lines for the 3-ML and 8-ML films correspond to cuts along which the induced charge density and the surface loss function are presented in Figs. 7, 8, and 9.

Figure 6

Normalized surface-loss function Im$\left[g\right(q,\omega \left)\right]/q\omega$ for the 1- to 10-, 17-, and 31-ML films calculated within the jellium model with the use of charge-density parameter $r_s=3$ a.u.. Strongly dispersing peak corresponding to a conventional ${\omega }_{\mathrm{sp}}^{-}$ mode of a thin film is denoted by the corresponding symbol. Acoustic surface plasmon dispersion produced by a quantum-well state with slowest Fermi velocity (i.e., that with the smallest binding energy at $k=0$ point) is highlighted by the symbol “ASP”. The dashed vertical lines for the 3-ML and 8-ML slabs correspond to cuts along which the surface loss function is presented in Fig. 9.

Figure 7

Two-dimensional plot of imaginary (top) and real (bottom) parts of induced charge density $n^{\mathrm{ind}}(z,q,\omega )$ produced by the external potential of Eq. (7) versus energy $\omega$ and distance $z$. Calculations are preformed for the 8-ML $m\ne 1$ Ag(111) film at $q=0.02$ a.u.. The energy positions corresponding to the ASP and the ISP and the I and II modes are shown with arrows. The Ag(111) slab is delimited by two horizontal dashed lines.

Figure 8

Imaginary (solid lines) and real (dashed lines) parts of the induced charge density, $n^{\mathrm{ind}}(z,q,{\omega }_{\mathrm{ASP}})$, corresponding to the ASP in the 3-ML (top panel) and 8-ML (bottom panel) $m\ne 1$ Ag(111) films. Calculations are for $q=0.02$ a.u. and ${\omega }_{\mathrm{ASP}}=0.25$ eV and ${\omega }_{\mathrm{ASP}}=0.1$ eV for the 3-ML and 8-ML films, respectively. Vertical solid lines show the atomic plane positions and vertical dashed lines delimit the slabs.

Figure 9

Surface loss function for the 3-ML (top panel) and 8-ML (bottom panel) films evaluated at $q=0.02$ a.u., i.e., along the cuts shown by the vertical dashed lines in Figs. 5 and 6. Solid lines show results obtained for the $m\ne 1$ Ag(111) model, whereas dashed lines correspond to the $m=1$ Ag(111) one. Dotted lines stand for the jellium calculations. The peaks corresponding to the ASP and the ISP and the I and II modes (explained in the text) are highlighted by the corresponding symbols.