Snell’s Law for Surface Electrons: Refraction of an Electron Gas Imaged in Real Space

On $\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}(100)/\mathrm{C}\mathrm{u}(111)$ an interface state band is observed that descends from the surface-state band of the clean copper surface. This band exhibits a Moiré-pattern-induced one-dimensional band gap, which is accompanied by strong standing-wave patterns, as revealed in low-temperature scanning tunneling microscopy images. At NaCl island step edges, one can directly see the refraction of these standing waves, which obey Snell’s refraction law.

Figure 1

STM image of NaCl on Cu(111). The numbers indicate the thicknesses of the NaCl islands in layers.

Figure 2

(a)–(d) $dI/dV$ images of the same region near a NaCl step edge, taken at various voltages. In all cases the wavelength of the $\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}/\mathrm{C}\mathrm{u}(111)$ interface state is larger than that of the Cu(111) surface state. (e) Dispersion of the 2DEG for the clean Cu(111) surface and the $\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}(100)/\mathrm{C}\mathrm{u}(111)$ interface state band (square data points and fit curves). (f) Using the phase-accumulation model, the probability distributions $|\Psi {|}^2$ perpendicular to the surface are determined, yielding only small changes upon the adsorption of NaCl. The origin ($z=0$) was chosen to be the Cu(111) substrate surface. Circles denote the position of Cu atoms.

Figure 3

(a) An atomically resolved STM image shows one of the various possible Moiré patterns that result from incommensurate growth [26]. (b) This pattern gives rise to a strong plane-wave pattern. (c) Illustration of a band gap in the NFE model. Just below and just above the band gap (marked by circles), the electronic wave functions are expected to be standing waves, and the probability distributions $|\Psi {|}^2$ will have a phase shift of $\pi$ with respect to each other. (d) This $dI/dV$ image (acquired in constant-height mode) maps the probability distribution $|\Psi {|}^2$ in real space and energy. It shows the behavior expected from the NFE model.

Figure 4

(a) STM $dI/dV$ image (constant-current mode) of a NaCl island step edge reveals strong standing-wave patterns, not only on the NaCl island but also on the clean Cu(111) surface ($V=55\mathrm{m}\mathrm{V}$). (b) Adding some lines as guides to the eye shows that the standing-wave pattern obeys Snell’s refraction law. Note that the contrast for the area of clean Cu(111) is increased, compared with the rest of the image.