Dielectric properties of ultrathin metal films around the percolation threshold

Optical reflection measurements of thin Au films at and around the percolation threshold (film thickness 3–10 nm) are performed in an extremely broad spectral range up to $35000{\text{cm}}^{-1}$. Combining spectroscopic ellipsometry, Fourier-transform infrared spectroscopy, and dc measurements, the dielectric properties of the films can be described over the whole frequency range by Kramers-Kronig consistent effective dielectric functions. The optical conductivity of the films is dominated by two contributions: a Drude component starting at the percolation threshold in the low-frequency range and by plasmons in the near-infrared region, which shift down in frequency with increasing film thickness. The interplay of both components leads to a dielectric anomaly in the infrared region with a maximum of the dielectric constant at the insulator-to-metal transition. The results are compared to predictions from effective-medium approximations and percolation theory.

Figure 1

(Color online) AFM images of 3-, 5-, and 7-nm-thick gold films on $\text{Si}/{\text{SiO}}_2$. All three images are presented at the same scale. With increasing film thickness, the cluster size increases and the films become smoother. For the 7 nm film, the islands are not separated but coalesce to larger clusters.

Figure 2

(Color online) Measured ellipsometric angle $\Psi$ for a 8.1-nm-thick Au film on a $\text{Si}/{\text{SiO}}_2$ substrate for several angles of incidence (dashed lines). Solid lines are the corresponding model fits (for details, see text).

Figure 3

(Color online) Measured ir reflectivity of a 6.5 nm Au film (dashed line) together with the reflectivity calculated from the Kramers-Kronig consistent dielectric functions obtained by a simultaneous analysis of the ir and ellipsometric data. Inset shows the assumed layer structure.

Figure 4

(Color online) (a) Optical conductivity and (b) dielectric constant of gold films obtained from reflection experiments and ellipsometry at room temperature. The included dc values (solid squares) perfectly match to the extrapolated conductivity ${\sigma }_1\left(\omega \to 0\right)$. Dashed arrows indicate the tendency as the film thickness increases

Figure 5

In situ dc-resistivity measurement showing the sharp transition at the percolation threshold. Here the weight thickness is plotted as measured by the quartz microbalance.

Figure 6

(Color online) Divergence of the static dielectric constant (obtained at $1{\text{cm}}^{-1}$) as a function of nominal film thickness $d$ for Au on $\text{Si}/{\text{SiO}}_2$. Solid lines below and above $d_c$ correspond to ${ϵ}_1\left(d\right)\propto {\left(6.4\text{nm}-d\right)}^{-1}$ and ${ϵ}_1\left(d\right)\propto \left(6.7\text{nm}-d\right)$, respectively.

Figure 7

Comparison between the measured maximum ${ϵ}_1\left(\omega ,d_c\right)$ and the frequency dependence calculated by Eq. (3). Over the entire midinfrared range, the measured values are above the predicted ones. At very low frequencies, the experimental ${ϵ}_1\left(\omega \to 0,d_c\right)$ saturates.

Figure 8

(Color online) Frequency-dependent conductivity for various Au films below $d_c$ together with the corresponding quadratic fits. In this regime, the films can be pictured as isolated capacitively coupled metal clusters even in the midinfrared.

Figure 9

(Color online) (a) Decomposition of the measured conductivity of the 6.8 nm film in its Drude and Lorentz components. The symbol at $\omega =0$ indicates the independently measured dc value. (b) Development of the two Lorentz oscillators with film thickness. With increasing thickness, both plasmons shift to lower frequencies and the separation between them increases.

Figure 10

Comparison between observed and calculated plasmon frequencies. With increasing film thickness, the plasmon frequency drops, closely following the behavior given by Eq. (5).

Figure 11

Spectral weight $\int {\sigma }_1\left(\omega \right)d\omega ={\omega }_p^2\pi {ϵ}_0/2$ and electron density $N_e={\omega }_p^2{ϵ}_{0}m/e^2$ of the Drude and the two Lorentz oscillators as function of the film thickness. Dashed line represents the electron density as determined for bulk gold.

Figure 12

(Color online) Thickness-dependent reflectivity of Au films on $\text{Si}/{\text{SiO}}_2$. With increasing film thickness, the reflectivity in the infrared first drops below the value of the bare substrate (dashed line), vanishes for a thickness of 5.3 nm around $6000{\text{cm}}^{-1}$, and then increases again reaching the original value of the bare substrate at about the percolation threshold.

Figure 13

(Color online) Calculated relative transmission $T_{\text{sub}+\text{film}}/T_{\text{sub}}$ for various film thicknesses below the percolation threshold. Over the entire mid-ir range, the transmission through the Au-covered film is higher than through the bare substrate.

Figure 14

(Color online) Comparison between the measured effective ${ϵ}_1$ and the calculated one using Eq. (6) (dashed lines). The BEMA model can reproduce the experimental values only at very low coverage and low frequency.