Three-dimensional atomic-arrangement reconstruction from an Auger-electron hologram

Current methods for reconstructing three-dimensional atomic arrangements from photoelectron holograms require data sets recorded using multiple incident photon energies. These techniques are thus difficult to apply to Auger-electron holography, since the kinetic energy of the Auger electron is element specific and independent of excitation energy. We propose a scattering pattern extraction algorithm using a maximum-entropy method for reconstructing the three-dimensional atomic arrangement from a single-energy Auger-electron hologram. The algorithm provides a clear atomic image by taking into account the scattering of the electron by nearby atoms and the non-$s$-wave nature of the Auger electron. We have applied the algorithm to an Auger-electron hologram of Cu(001) recorded at SPring-8’s soft x-ray synchrotron radiation beamline BL25SU and succeeded in determining the positions of 102 atoms of the Cu fcc structure.

Figure 1

(Color online) Schematic of the experimental setup and the two-dimensional display-type analyzer. Circularly polarized synchrotron radiation is irradiated onto the sample. The electrons emitted from the sample focus to the aperture by the electric field in the analyzer. The electron that passed the aperture is projected to the screen. Angular distribution without distortion is observed directly on the screen. When the measured image is divided by the penetration function of the analyzer, the clear angular distribution image is obtained. The example of the observed image is shown in figure.

Figure 2

(Color online) (a) Observed $L_{3}VV$ Auger-electron hologram of Cu(001) with kinetic energy of $914\mathrm{eV}$. [(b)–(f)] Simulated multiple-scattering patterns for emitted waves of $s$, $p$, $d$, $f$, and $g$ angular momenta for a spherical cluster of 241 atoms. Angular resolution is set to 3°.

Figure 3

(Color online) The scattering pattern function $t_{\vec{a}}\left(\vec{k}\right)$ caused by the scatterer located at $(0,0,a)$, and angular momenta dependence for $s$-, $p$-, and $f$-wave functions of emitter. The vertical axis is defined as $\cos \theta =(\vec{a}\bullet \vec{k})∕\mid \vec{a}\mid \mid \vec{k}\mid$. The right models depict the emitter and scatterer positions and the polarity of the emitter wave functions.

Figure 4

(Color online) Reconstructed real-space images. (a) and (b) are the vertical slices [(010) plane] of the reconstructed atomic image by using the $s$-wave and the $f$-wave functions, respectively. The darkness of the images corresponds to $\mid \overrightarrow{a_j}\mid G_j$. The emitter position is indicated by a solid circle. Circles with $1.0\mathrm{\AA }$ radius represent expected positions of Cu atoms, and dashed circles represent atomic positions that are not reconstructed by the calculation. Arrows indicate the position of artifacts. Region (s) reveals a shadow of the first-nearest neighbor atom. (c) The horizontal slices [(001) plane] at various $z$ positions of the reconstructed atomic image by using the $f$-wave function.