Quantum confinement of electrons at metal surfaces

Confinement of Shockley surface state electrons of Ag(111) to a hexagonal and triangular vacancy resonator. a Pseudo-three-dimensional presentation of an STM image of an atomically resolved hexagonal monatomically deep vacancy island on Ag(111) (bias voltage: $200\,\text{mV}$ , tunneling current: $2\,\text{nA}$ , size: $17.5\,\text{nm}\times 17.5\,\text{nm}$ ). b Experimental $\text{d}I/\text{d}V$ data (dots) and fit (solid line) based on the superposition of Lorentzian line shapes. Copyright 2005, American Physical Society.^[104] The topographic edge length of the underlying hexagonal vacancy is $7.4\,\text{nm}$ . Feedback loop parameters: $300\,\text{mV}$ , $1\,\text{nA}$ . c Gallery of constant-current $\text{d}I/\text{d}V$ maps of a hexagonal vacancy island with an edge length of $5.8\,\text{nm}$ recorded at the indicated sample voltages (white corresponds to a high $\text{d}I/\text{d}V$ signal). Copyright 2005, American Physical Society.^[104] Feedback loop parameters: Sample voltage indicated in the individual map and $0.5\,\text{nA}$ . d As c for a triangular vacancy with an edge length of $22\;\text{nm}$ . Copyright 2005, American Physical Society.^[104] Feedback loop parameters: Sample voltage indicated in the individual map and $2\,\text{nA}$ .

Lifetime analysis of confined Shockley surface electron states at Ag(111). a Calculated level widths for a circular Ag vacancy island using $\Im\Sigma_{\text{i}n}=0$ , $m^\ast=0.42\;m_\text{e}$ and $E_0=-67\;\text{meV}$ , for island radii $7.5\;\text{nm}$ ( $\square$ ), $15\;\text{nm}$ ( $\bigcirc$ ) and $30\;\text{nm}$ ( $\lozenge$ ). Copyright 2005, Elsevier Ltd.^[105] The lines are plots of $-\hbar\sqrt{2m^\ast(E-E_0)}\ln\vert{\cal{R}}\vert/(m^\ast S)$ for the respective island radii. Inset: Energy dependence of reflection coefficient $\vert{\cal{R}}\vert$ used for the calculation (line) along with experimental data.^[103] b Lifetime versus binding energy $E-E_\text{F}$ found for Ag(111) surface state electrons confined to vacancy hexagons ( $\bigcirc$ ), vacancy triangles ( $\triangle$ ), triangular corrals of Ag adatoms ( $\bigtriangledown$ ),^[74] circular and rectangular corrals of adsorbed Mn atoms ( $\square$ );^[73] photoemission data ( $\times$ );^[109] results of theory are depicted as a solid^[109] and a dashed^[110] line; lifetimes extracted from the spatial decay of standing wave patterns near step edges ( $\Diamond$ ).^{[55, 110]} Copyright 2005, Elsevier Ltd.^[105] Lifetime data ( $\bigcirc$ ) were corrected for lossy-boundary scattering and data extracted by the phase coherence length approach ( $\bigtriangledown$ , $\Diamond$ ) divided by a factor of 2.^[56] Inset: Lifetime data in the energy range $-50$ to $50\;\text{meV}$ .

Lateral confinement of quantum well states to a thin Pb layer above buried nanocavities. a Illustration of the preparation of buried resonantors at Pb(111) by Ar $^+$ ion bombardment and subsequent annealing. b Wulff construction of a buried nanocavity using $(111)$ , $(110)$ , $(100)$ Pb surface energies. Facets $\{111\}$ and $\{100\}$ are clearly visible; $\{110\}$ facets appear as the smallest region of the construction. c Pseudo-threedimensional presentation of an STM image of Pb(111) revealing cavities buried beneath two terraces ( $1.2\,\text{V}$ , $0.5\,\text{nA}$ , $100\,\text{nm}\times 100\,\text{nm}$ ). Copyright 2017, American Physical Society.^[143] Inset: Constant-current $\text{d}I/\text{d}V$ map recorded atop the indicated cavity showing electron standing wave patterns ( $0.4\,\text{V}$ , $1\,\text{nA}$ , $7.8\,\text{nm}\times 7.8\,\text{nm}$ ). d Spectrum of $\text{d}I/\text{d}V$ acquired atop the center of a subsurface cavity identified as being four layers below the surface, showing spectroscopic fine structure (arrows) within the band of the highest occupied QWS. Copyright 2016, American Physical Society.^[142] Feedback loop parameters: $-2.5\,\text{V}$ , $1\,\text{nA}$ .

Simulation of lateral QWS confinement. a Model geometry and partitioning of space used in the calculations. The cylindrical cavity with radius $R$ exhibits a surface--vacuum distance $L$ . Electrons are excluded from the cavity and the vacuum above the surface. Surface $A$ 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 (downward) direction. b Density of states (DOS) of the highest occupied QWS for a four-layer Pb film (blue line) integrated over film thickness and atop a $2.7\,\text{nm}$ radius cavity four layers beneath the surface omitting (red line) and including (green line) interband scattering (IBS). Copyright 2016, American Physical Society.^[142] The dashed line shows the DOS in the absence of inelastic lifetime broadening ( $\Sigma_\text{i}=0$ ). The arrows indicate the expected energies of electrons ideally confined to region I. c Comparison of experimental constant-current $\text{d}I/\text{d}V$ maps acquired a four-layer deep cavity ( $1\,\text{nA}$ , $7.8\,\text{nm}\times 7.8\,\text{nm}$ ) with calculated confinement patterns in an ideal hexagon with a $3\,\text{nm}$ edge, a QWS energy of $-1.725\,\text{eV}$ and an effective mass of $1.1\,m_\text{e}$ (area: $7.8\,\text{nm}\times 7.8\,\text{nm}$ ). Copyright 2017, American Physical Society.^[143]