Quantum Oscillations in the Layer Structure of Thin Metal Films

Understanding the underlying physical principles that determine the internal structure of objects at the atomic scale is critical for the advancement of nanoscale science. We have performed synchrotron x-ray diffraction studies to determine the structural properties of smooth Pb films with varying thicknesses of 6 to 18 monolayers deposited on a Si(111) substrate at 110 K. We observe quasibilayer variations in the atomic interlayer spacings of the films consistent with charge density oscillations due to quantum confinement of conduction electrons and surface-interface interference effects. Quantum oscillations in atomic step height are also deduced.

Figure 1

X-ray reflectivity data (circles) for Pb films with thicknesses 6–18 ML and fits using the model described in the text (solid curves). Nominal coverages are indicated for each curve. The abscissa is the momentum transfer along the surface normal in Si reciprocal lattice units ($1\mathrm{r}\mathrm{.}\mathrm{l}\mathrm{.}\mathrm{u}\mathrm{.}=0.668{\AA }^{-1}$). The tails of the sharp Si(111) and (333) Bragg peaks can be seen at $l=3$ and $l=9$, and the large peaks at $l\approx 3.3$, 6.6, and 9.9 (not shown) correspond to the Pb(111), (222), and (333) Bragg positions, respectively, halfway between which the prominent half-order features can be seen (inverted triangles). As discussed in the text, such features are indicative of a quasibilayer superperiod in the direction normal to the surface.

Figure 2

Because of the confinement of conduction electrons in a quantum well ($N=10$), oscillations in the electron density (a) will be present in the film. The displacement of each atomic plane is proportional to the first derivative (b) of these density variations, from which the changes in individual interlayer spacings (c) can be calculated. The abscissa is the distance into the quantum well, where $z=0$ is the film-substrate well boundary. The ideal positions of the atomic layers, $z_j$, are marked with vertical dotted lines and in (a) and (b) the intersection of these positions with the curves are marked with diamonds. All curves and values are calculated using the model parameters resulting from the fit to the experimental data for $N=10$ shown in Fig. 1.

Figure 3

Step heights that would be observed in an STM or HAS experiment. The points are derived from the difference between the net thicknesses of films differing by one atomic layer and are positioned between the two relevant film thicknesses. The horizontal dotted line is positioned at the bulk Pb(111) interlayer spacing, $t_0=2.84\AA$. Quasibilayer oscillations are evident, consistent with the results seen in STM and HAS studies.