Electron quantization in arbitrarily shaped gold islands on MgO thin films

Low-temperature scanning tunneling microscopy has been employed to analyze the formation of quantum well states (QWS) in two-dimensional gold islands, containing between 50 and 200 atoms, on MgO thin films. The energy position and symmetry of the eigenstates are revealed from conductance spectroscopy and imaging. The majority of the QWS originates from overlapping Au 6$p$ orbitals in the individual atoms and is unoccupied. Their characteristic is already reproduced with simple particle-in-a-box models that account for the symmetry of the islands (rectangular, triangular, or linear). However, better agreement is achieved when considering the true atomic structure of the aggregates via a density functional tight-binding approach. Based on a statistically relevant number of single-island data, we have established a correlation between the island geometry and the gap between the highest-occupied and the lowest-unoccupied molecular orbital in the finite-sized islands. The linear eccentricity is identified as a suitable descriptor for this relationship, as it combines information on both island size and island shape. Finally, the depth of the confinement potential is determined from the spatial extension of QWS beyond the physical boundaries of the Au islands. Our paper demonstrates how electron quantization effects can be analyzed in detail in metal nanostructures. The results may help elucidating the interplay between electronic and chemical properties of oxide-supported clusters as used in heterogeneous catalysis.

Figure 1

(a) STM topographic image and (b) corresponding $dI$/$dV$ map of 2 ML MgO/Ag (001) covered with ML Au islands ($50\times 50$ nm${}^2$, 1.7 V, 100 pA). The inset in (b) shows a $dI$/$dV$ spectrum of the bare oxide. Whereas the sharp rise at 2.5 V marks the onset of the MgO conduction band, the small shoulder at 1.7 V is due to an MgO/Ag(001) interface state (arrow).

Figure 2

(a) Differential conductance spectra taken in the center (blue) and in the lower part (orange) of the rectangular Au island shown in the inset (setpoints: −2.0 and $+$2.5 V for negative and positive polarity, respectively). The insets (from left to right) show a topographic image, a structure model, and the quasirectangular outline of the island ($7.5\times 7.5$ nm${}^2$, −0.1 V, 250 pA). (b) Filled-state $dI$/$dV$ maps of the islands in (a). The selected bias voltages are indicated by arrows directly in the $dI$/$dV$ spectra. Note the pronounced localization of the state density at the cluster perimeter.

Figure 3

(a) Empty-state $dI$/$dV$ spectrum taken in the center of the rectangular island shown in Fig. 2. The spectral course is deconvoluted into Gaussians of constant width (gray), representing the individual QWS of the cluster. (b) Corresponding $dI$/$dV$ maps measured at the bias positions marked by arrows in (a) ($7.5\times 7.5$ nm${}^2$). The symmetry of the QWS can be rationalized with a simple 2D particle-in-a-box model. (c) DFTB calculations showing the local-state density in the respective Au island. The second and third maps result from a superposition of two closely spaced eigenstates of the system.

Figure 4

Dispersion relations for the (a) rectangular, (b) triangular, and (c) curved-linear cluster as deduced from the corresponding $dI$/$dV$ spectra. The measured bias values are plotted against the quantum numbers of the particle-in-a-box models with corresponding geometry. A linear fit through the data yields the effective electron mass and the onset energy of the underlying Au electronic states.

Figure 5

(a) Empty-state $dI$/$dV$ spectra taken in the upper (orange) and lower (blue) parts of the triangular island shown in the inset. Intensity maxima have been obtained by deconvoluting the curve into a set of Gaussians. (b) Corresponding $dI/dV$ maps taken at the peak positions marked by arrows in (a) ($8\times 8$ nm${}^2$). The symmetry of the QWS can be explained either with the eigenstates of an isosceles, right-angled triangular potential or with DFTB calculations performed on a structure model of the Au island. Here, the symmetry of the third, fourth, and fifth $dI$/$dV$ maps result from a superposition of two adjacent eigenstates.

Figure 6

(a) Empty- and filled-state $dI$/$dV$ spectra taken in the center of the curved-linear island shown in the inset ($9\times 9$ nm${}^2$). The first five maxima, obtained from a Gaussian deconvolution, are marked by arrows. (b) $dI$/$dV$ maps taken at the determined bias positions ($9\times 9$ nm${}^2$). The symmetry of the QWS is compatible with those of a 1D particle-in-a-box model. The respective LDOS maps calculated with the DFTB method are shown below. The third and fourth maps are superpositions of two adjacent states.

Figure 7

Size and shape dependence of the HOMO-LUMO gap Δ plotted versus (a) the total number of atoms, (b) the ratio of semimajor and semiminor axis $L_x$/$L_y$, (c) the linear eccentricity $e$, and (d) the mean coordination number $c_{\lambda }$ $=$ 0.001. To emphasize the shape effect on the gap size, four sample islands have been highlighted by letters (A–D). The corresponding islands are shown in Fig. 8. Note the good linear relationship between gap size and linear eccentricity in (c).

Figure 8

Pair correlation functions of four sample islands with similar atom counts (A $=$ 134, B $=$ 128, C $=$ 130, D $=$ 135) but increasing eccentricity. Whereas the symmetric island shows a rapid decay, asymmetric clusters are characterized by pronounced tails in the pair-correlation function. The corresponding gap sizes can be deduced from the diagrams in Fig. 7.