Diffractive-wave guiding of surface electrons on Au(111) by the herringbone reconstruction potential

The surface potential of the herringbone reconstruction on Au(111) is known to guide surface-state electrons along the potential channels. Surprisingly, we find by scanning tunneling spectroscopy that hot electrons with kinetic energies twenty times larger than the potential amplitude (38 meV) are still guided. The efficiency even increases with kinetic energy, which is reproduced by a tight-binding calculation taking the known reconstruction potential and strain into account. The guiding is explained by diffraction at the inhomogeneous electrostatic potential and strain distribution provided by the reconstruction.

Figure 1

(a) STM image of Au(111), $I=1\mathrm{nA},V=-480\mathrm{mV}$; herringbone reconstruction and a few impurities (bright and dark spots) are visible. (Inset) Spatially averaged $dI/dV$ spectra taken on hcp and fcc areas as indicated by arrows, $I_{\mathrm{stab}}=1\mathrm{nA},V_{\mathrm{stab}}=-540$ mV, $V_{\mathrm{mod}}=10\mathrm{mV}$. (b)–(d) $dI/dV$ maps of the same surface area as in (a) recorded in constant-current mode at voltages as marked in the insets, $I=1\mathrm{nA},V_{\mathrm{mod}}=10\mathrm{mV}$. Arrow in (c) indicates a guided standing wave. (Insets) Fourier transformations (FT) of the $dI/dV$ images; arrows in (b) mark spots corresponding to the herringbone periodicity; arrows in (d) mark the enhanced intensity within the circle in the direction parallel to the reconstruction lines.

Figure 2

Calculated LDOS maps at $E-E_{\mathrm{F}}$ as marked in the subfigures; reconstruction lines run vertically; insets in upper right corner are FTs of the real-space images; arrows in LDOS images mark standing waves running along the reconstruction lines; pairs of white rectangles in (c) and (d) show a region close to a point scatterer and the corresponding zoom-in: wave guiding is visible at 250 mV, but not at 0 mV; arrows in FT point to the increased intensity along the reconstruction; the TB calculation includes the electrostatic potential and the strain of the reconstruction lines as well as impurities modelled as Gaussian potentials with amplitude 2.5 eV and width 0.3 nm.

Figure 3

Color plot of the angular distribution of the FT averaged around the dominant $\left|\mathbf{k}\right|$ (ring structure in FTs of Figs. 1 and 2, respectively, Figs. 6 and 7) (a) FT from ideal theoretical LDOS including reconstruction potential and strain but no disorder; (b) angular and energy dependence of backscattering probability according to a semiclassical Gutzwiller path calculation of the LDOS including reconstruction potential, strain and impurities (random phase shift of wave at each impurity $|{\beta }_R|<0.05\pi$); (c) same as (a) but with impurities as in Fig. 2; (d) FT from experimental LDOS for one domain of the herringbone reconstruction after angular smoothing (second-order polynomial fit) and setting $eV=E-E_{\mathrm{F}}$. The color scale in all images covers the range [0.52, 1.00]; arrows at the dark lines in (a) and (c) mark the dominant diffractive minimum [Eq. (2)]; all theoretical FTs are averaged over 50 realizations of disorder and different edge configurations for each energy.

Figure 4

Angular dependence of the FT of theoretical LDOS maps, but in contrast to Fig. 3 without considering any strain effects, i.e., only considering the reconstruction potential Eq. (1) (same color scale and contrast as in Fig. 3). (a) with and (b) without disorder scattering.

Figure 5

(a) FT of $dI/dV$ map at $V=250\mathrm{mV}$ recorded within a single domain of the reconstruction lines; dashed rings border the area used for the angular plot in (b); angles of maximum intensity $\alpha =0{}^{\circ }$ and $\alpha =180{}^{\circ }$ are marked. (b) Angular distribution of FT amplitude radially averaged over the ring marked in (a); minimum $\mathrm{FT}{}_{\min }$ and maximum $\mathrm{FT}{}_{\max }$ are marked. (c) Anisotropy $A$ according to Eq. (4) from experimental data [circles] and from TB simulations ($▿$: without, $▾$: with impurities). Experimental error bars decrease with energy as indicated.

Figure 6

$dI/dV$ maps of the identical surface area at voltages $V$ as marked below the images, $I=1\mathrm{nA},V_{\mathrm{mod}}=10\mathrm{mV}$. (Insets) Fourier transformations of the $dI/dV$ images; size of $dI/dV$ map and Fourier transformation are the same for all $V$, contrast is adapted to improve visibility.

Figure 7

Theoretical LDOS maps at voltages $V$ as marked below the images for a particular configuration of impurities, compare with the experimental LDOS in Fig. 6. The reconstruction lines run from top to bottom. (Insets) Fourier transformations of the $dI/dV$ images; size of $dI/dV$ map and Fourier transformation are the same for all $V$, contrast is adapted to improve visibility.

Figure 8

(a) Power spectrum of Fourier transformation of $dI/dV$ map recorded at $V=-200$ mV; regions for back-transformation in (c) and (d) are marked by red circles; (b) five backfolded constant-energy circles in $\mathbf{k}$ space for Au(111) with gaps originating from miniband formation and marked preferential scattering vectors parallel to the reconstruction lines (arrows). (c) Image resulting from inverse Fourier transformation of (a) after applying an ideal band-pass filter selecting the ring structure, which is bordered by the two red circles in (a); white circles mark areas of wave guiding. (d) Image resulting from inverse Fourier transformation of (a) after applying an ideal low-pass filter selecting the area bordered by the inner red circle; (e) and (f) same as (c), (d) but for a $dI/dV$ image recorded at $V=100$ mV and back-transformed from (g). (g) Power spectrum of Fourier transformation of $dI/dV$ map recorded at $V=100$ mV, regions for back-transformation in (e) and (f) are marked by red circles.

Figure 9

(a) STM image of Au(111), $I=1$ nA, $V=-0.48$ V with marked lines where the corrugation $C$ is measured. (b) $dI/dV$ intensity along line 2 in (a) recorded at $V=0.25$ V; minima (Min) and maxima (Max) as used for the evaluation of $C$ are marked. (c) Corrugation $C$ as determined by Eq. (B1) as a function of electron energy with respect to $E_{\mathrm{F}}$ for the lines marked in (a), labels include the type of region (Dis = dislocation line) as well as the number marking the lines in (a). Insets show $dI/dV$ images corresponding to the marked points of relatively large corrugation; grey area marks the channeling energies; $I=1$ nA, $V_{\mathrm{mod}}=10$ mV for all data points and the insets.