Finite size effects in surface states of stepped Cu nanostripes

Cu nanostripes with finite arrays of monatomic steps are self-assembled by Ag-induced faceting of vicinal Cu(111) surfaces. By varying the amount of Ag in the submonolayer range one can tune the internal step spacing $d$ of Cu stripes, while decreasing its total width $w_{\mathrm{Cu}}$. We can observe, by means of angle-resolved photoemission, a progressive transition from two-dimensional surface bands to one-dimensional quantum well states as $w_{\mathrm{Cu}}$ decreases. A direct comparison between surface states of infinite vicinals and nanostripes with the same $d$ indicates a small upwards energy shift in the latter, which is well explained by assuming electron confinement in an infinite quantum well of size $w_{\mathrm{Cu}}$. Nanostripe finite size effects are more straightforwardly observed in Fermi surfaces, which are asymmetrically broadened in the perpendicular direction. This effect is quantitatively analyzed and explained as due to the characteristic spectral broadening observed in photoemission from $w_{\mathrm{Cu}}$-wide one-dimensional quantum wells.

Figure 1

(Color online)(a) Side view of the $\mathrm{Ag}∕\mathrm{Cu}$ periodically (period $L_{\Theta }$) faceted nanostructure as a function of Ag coverage. Up to 0.6 ML, we distinguish an early regime with (112)-oriented facets, whereas a mixture of (335) and (112) facets appears near monolayer completion. In the (112)-facet regime, the density and size of these facets increases, thereby reducing the terrace size $d$ in contiguous Cu nanostripes (width $w_{\mathrm{Cu}}$). The latter display surface states with $w_{\mathrm{Cu}}$-dependent dimensionality, evolving from quasi-two-dimensional bands in large Cu stripes with high density of steps (0.25 ML) to one-dimensional quantum wells in narrow Cu stripes (0.8 ML). This is proved by the dispersing behavior of their Cu surface state in the perpendicular direction of the step array shown in (b). The series of photoemission spectra have been taken with $h\nu =22\mathrm{eV}$ and varying the emission angle in 0.5° steps. They clearly show a dispersing peak with 0.25 ML and a nondispersing QW feature with 0.8 ML.

Figure 2

(Color online) Energy distribution curves for surface states in Cu nanostripes at increasing coverage. The spectra have been taken with $h\nu =22\mathrm{eV}$ and correspond to band minima for dispersing peaks (clean, 0.25 ML and 0. 45 ML) and intensity maxima for nondispersing, QW-like peaks (0.6 ML and 0.8 ML). The solid lines represent line fitting to the data. The peaks display a continuous downward shift towards the peak position of Cu(111) (dotted line), reflecting the size-dependent shift in terraces of increasing size, as previously observed in infinite vicinal surfaces (Refs. 4, 14).

Figure 3

(Color online) (a) Open circles, surface state peak shift in Cu nanostripes with respect to flat Cu(111) versus surface state peak shift in single crystal, infinite vicinals (filled dots and thin lines) with the same step spacing $d$. The upper scale indicates the absolute size of the nanostripe $w_{\mathrm{Cu}}$ at a given $d$. The dotted line represents the expected peak shift in an infinite one-dimensional quantum well of size $w_{\mathrm{Cu}}$. The transition from surface state scattering at weak step barriers in wide stripes to full confinement within narrow Cu nanostripe boundaries is nicely observed. (b) Energy shift versus relative broadening with respect to flat Cu(111) for surface states in Cu nanostripes. Data points lie close to the universal behavior found for scattering in metallic steps [thick line (Ref. 7)], revealing the same nature of the scattering potential in all cases.

Figure 4

(Color online) Fermi surface evolution for $\mathrm{Ag}∕\mathrm{Cu}$ faceted nanostructures as a function of coverage. The photoemission experiments were carried out with $h\nu =22\mathrm{eV}$. The bright spot at $k_x\sim 0.33{\mathrm{\AA }}^{-1}$ identifies the emission from Ag stripes, which is significant only beyond 0.6 ML. The interesting feature is the Cu-related Fermi surface ring. Its radius increases and it becomes broader along $k_x$, respectively, reflecting terrace size increase and stripe width decrease as a function of $\Theta$ (see Fig. 1). At 0.6 ML, the Cu ring collapses into a one-dimensional QW-related vertical streak at $k_y\sim 0.16{\mathrm{\AA }}^{-1}$. The dotted lines indicate the position of the intensity profiles shown in Fig. 5.

Figure 5

(Color online) Momentum distribution curves across the Fermi surfaces perpendicular (left) and parallel (right) to the nanostripes, as indicated with dotted lines in Fig. 4. The $k_x$ scale is referred to the ring center in Fig. 4, nominally found at the zone boundary of the nanostripe step array $\pi ∕d$. Solid lines are fits to the data using Gaussian functions. The increasing asymmetry in peak width for parallel and perpendicular directions is explained as due to stripe size effects.

Figure 6

(Color online) Asymmetric broadening $(\Delta k_{xy}=\Delta k_x-\Delta k_y)$ in momentum distribution curves at the Fermi energy, as determined from the line fitting in Fig. 5. The red dots and the straight lines represent the spectral broadening width in photoemission from infinite quantum wells of size $w_{\mathrm{Cu}}$, as calculated from Eq. (3) and $N>1$. The close relationship between both applies in the whole coverage regime, thereby defining $\Delta k_{xy}$ as the straightforward feature that characterizes finite size effects in nanostripes. The inset shows a schematic description of the electron potential and wave function within the stepped Cu nanostripe. The infinite potential at stripe boundaries leads to the quantum well modulation ($N=2$ in the inset) of the step superlattice electron wave function inside the nanostripe.