Lateral Electron Confinement with Open Boundaries: Quantum Well States above Nanocavities at Pb(111)

We have studied electron states present at the Pb(111) surface above Ar-filled nanocavities created by ion beam irradiation and annealing. Vertical confinement between the parallel crystal and nanocavity surfaces creates a series of quantum well state subbands. Differential conductance data measured by scanning tunneling spectroscopy contain a characteristic spectroscopic fine structure within the highest occupied subband, revealing additional quantization. Unexpectedly, reflection at the open boundary where the thin Pb film recovers its bulk thickness gives rise to the lateral confinement of electrons.

Figure 1

(a) STM image of Pb(111) showing buried cavities on a lower (dark gray) and an upper (light gray) terrace (0.5 nA, 1.08 V, $94\mathrm{nm}\times 94\mathrm{nm}$). Inset: Schematic illustration of a subsurface nanocavity, with geometry based upon Wulff construction using (111), (110), and (001) Pb surface energies. (b) Constant-current (0.5 nA) $dI/dV$ spectra (vertically offset) acquired atop cavities located at different depths, exhibiting signatures of unoccupied QWS. Inset: Calculated QWS energies as a function of the Pb layer thickness. (c) Left: Calculated energy band structure of a four-layer Pb(111) thin film showing the dispersion of the highest occupied QWS (blue line) with a 0.2 eV energy shift applied to align band minimum. Right: Experimental constant-height $dI/dV$ spectrum acquired atop the center of a subsurface cavity identified as being four layers below the surface, showing spectroscopic fine structure (red arrows) within the band of the highest occupied QWS (feedback loop parameters: 1 nA and $-2.5\mathrm{V}$).

Figure 2

Density of states (DOS) of the HOQWS for a four-layer Pb film (blue line) integrated over film thickness and atop a 2.7 nm radius bubble four layers beneath the surface omitting (red line) and including (green line) interband scattering. The dashed line shows the DOS in the absence of inelastic lifetime broadening (${\mathrm{\Sigma }}_i=0$). The arrows in the main figure indicate the expected energies of electrons ideally confined to region $I$. Inset: Model geometry and partitioning of space used in calculations. The cavity with radius $R$ exhibits a surface-vacuum distance $L$. Electrons are excluded from the cavity and the vacuum above the surface. Surface $S$ separates region $I$, the space between the cavity and the crystal surface, from region $II$, where the electron is no longer confined in the vertical (downwards) direction.

Figure 3

(a),(c),(e) Experimental $dI/dV$ spectra [feedback loop parameters, 1 nA and $-2.5\text{V}$ (a) or $-2\mathrm{V}$ (c),(e)]. The data were normalized [47] to ensure an improved approximation to the DOS. Arrows in (a) and (c) indicate additional spectroscopic features caused by the reduced symmetry of the (111) facets. (b),(d),(f) Calculated DOS of the HOQWS atop cavities of indicated radii and depths. Shaded areas in (b),(d),(f) show the range of energies with parabolic HOQWS dispersion.

Figure 4

Experimentally determined (dots) and calculated (squares) linewidths $\mathrm{\Gamma }$ of laterally confined QWS electrons associated with a six-layer deep cavity as a function of energy. The energy is referenced to the onset energy ${ϵ}_9$ of the HOQWS. Calculations were performed for the indicated cavity radii and represent the sum of inelastic (${\mathrm{\Gamma }}_{\mathrm{e}\text{-ph}}+{\mathrm{\Gamma }}_{\mathrm{e}\text{−}\mathrm{e}}$, dashed line) and elastic contributions to $\mathrm{\Gamma }$. Experimental data were obtained for cavity radii ranging between 1.5 and 2.2 nm. The data encircled by a dotted line are linewidths of confined states with energies close to the flat region of the HOQWS band. Uncertainty margins represent the statistical uncertainty of the fit procedure. Inset: Calculated energy dependence of the reflection coefficient $|\mathcal{R}|$ at the boundary above a cavity with 2.5 nm radius.