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The physics of electron scattering states is illustrated by the escape depth,
the perpendicular wave vector composition, the charge density, and the
optical potential. The calculation of low-energy electron diffraction (LEED)
states in the range from 6 to 30 eV is performed by a direct solution of
the Schrödinger equation for the relaxed GaAs(110) surface.
In the lower left image, blue iso-surfaces are drawn from the resulting
charge density. In the vacuum region at the top of the image the
charge density is roughly equal to unity since there the wave function is
dominated
by the incoming electron beam, which is normalized. The
scattering and penetration into the surface has a strong spatial modulation,
as can be seen in the presented part, containing one unit cell parallel to
the surface and some of the topmost atomic layers perpendicular to it.
The atomic positions are indicated by green spheres.
For the perpendicularly incident electron
calculated here the charge tends to accumulate at the zig-zag bonding
chains perpendicular to the surface.
The overall surface sensitivity can be quantified by the 'penetration' (or
'escape') depth. To calculate it, the charge distribution is averaged parallel
to the surface and an exponential function
is fitted to the resulting perpendicular
dependence. As shown in the top left this gives an escape depth of
several Å with a minimum around 20 to 30 eV.
Please note that the minima in the escape depth below the plasma
frequency (at about 15.8 eV) are related to gaps, especially in that band of
the bulk band structure which contributes most to the final state, see
the top right figure. The resulting increased surface sensitivity is
not obvious from the charge iso-surfaces, but is an average property of the
entire charge density.
Since the translational symmetry perpendicular to the surface
is broken, there does not rigorously exist a perpendicular wave vector.
Nevertheless, an approximate momentum composition of the wave function
inside the
solid can be given by calculating the Fourier spectrum
of the phase of the wave function. The result is shown together with
the corresponding part of the bulk conduction band structure. The spectral
weight is encoded in a red color scale with high intensities shown
in dark red.
When the direct solution of the Schrödinger equation is Fourier
analyzed one sees that it consists primarily of the harmonic function due
to one particular wave vector. This differs from the solutions obtained from
matching procedures, where often the wave function has a number of
non-neglegible Fourier coefficients.
This difference has two reasons:
The direct solution allows only an approximate analysis, and
the matching wave function has limited quality.
The strong coincidence of one band of
the calculated bulk band structure with a prominent plane wave
from the Fourier analysis of the
scattering state reflects not the general situation. It is valid for the
low energies shown here and could be a speciality
of III-V semiconductors, because larger differences are observed e.g. for
layered crystals.
To give an impression of the geometry, the first image on the lower right
shows the ground state charge density of the GaAs(110) surface.
With this and the energy as input, a local density function allows to
calculate the optical potential, which describes the inelastic losses of
the excited electron. The following images of iso-surfaces from this
optical potential illustrate the spatial modulation of the attenuation.
The damping is stronger around the As atoms than around
the Ga atoms. This reflects
the structure in the charge density as well as the nonlinear action
of the local density function.
All images are animated in dependence of the energy.
See also:
- Calculated optical potential
- Calculation and physics of scattering
states
- Scattering states for realistic surfaces with
multi-grid acceleration
- The combined analysis of valence
photoelectron spectra and diffraction patterns
- Paper: C. Solterbeck, O. Tiedje, T. Strasser, S. Brodersen, W. Schattke.
I. Bartos,
Optical potential and escape depth
for electron scattering at very low energies,
J. Electron Spectrosc., accepted.
- Paper: C. Solterbeck, O. Tiedje, F. Starrost, and W. Schattke,
Scattering States for Very Low Energy
Electron Spectroscopy,
J. Electron Spectrosc. 88-91, 563 (1998)
- Paper: S. Lorenz, C. Solterbeck, W. Schattke, J. Burmeister, and
W. Hackbusch,
Electron scattering states at solid surfaces calculated with realistic
potentials, Phys. Rev. B 55, 13432 (1997), also available as
cond-mat/9702118
- Poster: C. Solterbeck, O. Tiedje, T. Strasser, S. Brodersen, A. Kistner,
W. Schattke, I. Bartos,
Determination of the Optical Potential
for Electron Scattering at very low Energies (660KB gzipped Postscript,
8.5MB gunzipped), poster at the 12th International Conference on
Vacuum Ultraviolet Radiation Physics, August 3 - 7, 1998, San Francisco,
USA
Acknowledgment:
This work was supported by the Bundesministerium für Bildung,
Wissenschaft, Forschung und Technologie
(Project No. 05 605 FKA).
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