STM 1991 Conference, Interlaken
(Switzerland)
Calculation of Effects of Tunnel Junction Geometry in
Scanning Tunneling Microscopy
G. I.
Márk^{a}
Institut fur Physik , Universitat Basel, Klingelbergstrasse
82, CH4056
Basel, Switzerland
E.
Balázs
Institute of Physics, Technical University, H1521 Budapest,
Hungary
Corresponding author: G. I.
Márk , mark@sunserv.kfki.hu
, Nanostructures
Laboratory , MFA Research
Institute for
Technical Physics and Materials Science
H1525 Budapest, P.O.Box.49., Hungary
Our other publications on STM
simulation are available here.
ABSTRACT
A theoretical study of current
density for the case of tunneling through phenomenological^{1}
and variationally^{2} calculated metalvacuummetal (MVM)
potential barriers is presented. The twodimensional time dependent
Schrödinger equation is solved numerically. Spectroscopic effects
(e.g. the effects of low work function tip patches) as well as electron
focusing effects are investigated within the framework of this model.
Predictions of semiclassical calculations (e.g. geometrical asymmetries
in I(V) curves) are verified.
This work was partially supported by the Swiss National Science
Foundation.
^{1} N. Garcia, C. Ocal and F. Flores, Phys. Rev. Lett.
50, 2002 (1983).
^{2} E. Balazs & L.Orosz, Surf. Science 238, 341
(1990).
^{a} Permanent address: Institute
of Physics, Technical University,
H1521 Budapest,
Hungary
METHOD
The time dependent Schödinger equation is solved by Fourier
transform method. Given an initial state Psi_{
0 }(r) the method allows us to calculate
the time evolution of the wave function. The potential seen by an
electron when tunneling from one electrode to the other is approximated
as
V(s) = V_{elst}(s) + V_{xc}(s)
+ V_{im}(s) + V_{oc}(s)
where s is the arch length of
a semiclassical WKB trajectory measured from the cathode to the actual
position of the electron, V_{elst}(s)
is the electrostatic potential due to the electron density and for the
exchange correlation potential the usual Wignerformula is used. The
image potential is calculated as an infinite sum. The potential V_{oc}(s) arises from the
external bias and the contact potential. The potential barrier
approximated in this way aligns with that obtained in a quasiSCF way^{1}.
In tip negative case the semiclassical trajectories were approximated
by the lines of force of
the electrostatic field, for tip positive biases straight lines perpendicular to the
sample plane were chosen.
ASSUMPTIONS
The tip and the surface are of the same material, (Au, E_{F} = 5.5 eV,
Phi = 5.2 eV). The incident wave function is assumed to be a
Gaussian wave packet
RESULTS
 Tip negative current is smaller than tip positive one.
 Large spatial oscillations occur in the current in the
interface region
 Influence of the tip shape seems to be important
ACKNOWLEDGEMENT
The authors express their gratitude to the IBM, Hungary for supporting
their participation at this Conference.
GIM gratefully acknowledges the hospitality at the Institute fur
Physik, Universitat Basel, Switzerland
REFERENCES
 N. Garcia, C. Ocal and F. Flores, Phys.Rev.Letters
50(1983)2002.
 E. Balázs and L. Orosz, Surf. Sci. 238(1990)341.

FIG.1. STM model geometry.
Surface of the plane sample and the hyperbolic tip is shown together
with an isodensity line of the incoming wave packet.


FIG.2. Contours of
constant V_{eff}(z,r).
Contour spacing is 0.02 Hartree.
Thick line: V_{eff} = E_{F}.


FIG.3. Current density for
tip negative bias (wave packet coming from the tip). Upper figure is
for the whole model, lower figure is
for the tunneling channel.


FIG.4. Current density for
tip positive bias (wave packet coming from the sample). Upper figure is
for the whole model, lower figure is
for the tunneling channel. 


FIG.5.a. Radial current
density for tip negative bias (wave packet coming from the sample).
Thick line: measured at tip apex, thin line: measured at sample
surface. Oscillations are numerical artefacts. 
FIG.5.a. Radial current
density for tip positive bias (wave packet
coming from the tip). Thick line: measured at tip apex, thin line:
measured at sample surface. Oscillations are numerical artefacts.


FIG 6. Radial tunnel
current density measured at tip apex for tip positive case (wave packet
coming from the tip) and at sample surface for tip negative case (wave packet
coming from the sample). Oscillations at the top of the tip negative curve are
due to numerical error.

Last updated: Feb 10, 2004
by Géza I. Márk
, mark@sunserv.kfki.hu
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