EPS 1988 Conference,
Budapest
Calculation of Tunnel Current Along Semiclassical
Trajectories for
STM
E. Balázs, Department of Physics, Gy. Papp,
Department of
Atomic Physics, G. Márk, Quantum Theory Group
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
The dependence of tunnel
current on geometric parameters (rotational
hyperboloid shaped tip, plane anode) is studied. The transition
probability through a phenomenological potential barrier is calculated
along semiclassical trajectories in plane wave approximation. The
current is obtained as a sum of suitably weighted current densities
calculated from the above defined transitional probability.
INTRODUCTION
The recent development of the scanning tunneling microscopy (STM) has
stimulated renewed interest in the problem of a particle tunneling
through a potential barrier.
The exact quantummechanical solution of the tunneling problem of STM
involves several principal, and technical difficulties. However, the
main tendencies furnished by experiments may be qualitatively
understood also on the basis of simple models. In the present paper the
authors propose a semiclassical model for the interpretation of the
high lateral resolution of the STM.
MODEL
The geometry of the tipsurface arrangement is approximated by a rotational hyperboloid and a plane.
By this choice the electrostatic equipotential surfaces and lines of
force due to the applied bias and contact potential are well known from
classical electrodynamics.
The electrons are assumed to tunnel along semiclassical trajectories
determined within the frame work of the WKB approximation.
The tunneling of an electron along the trajectory is then considered to
be equivalent to the tunneling between two infinite plane electrodes
being at a distance equal to the arc length of the trajectory measured
from the cathode to the anode. (Thus the 2D problem is approximated by
a 1D one, and the difficulties due to the unknown boundary conditions
referring to the wave function of the electron in the tip may be
avoided.) 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}.
DETERMINATION of the TRAJECTORIES
The model calculation was carried out for W (E_{F} = 8.0 eV, Phi = 5.0 eV)
, and Au (E_{F} = 5.5 eV,
Phi = 4.3 eV) as tip, and sample, respectively. When determining
the WKB trajectories numerically, the starting points were chosen along
the circumference of the hyperboloid (plane) corresponding to the tip
negative (tip positive) case. As it is shown in Fig. l the WKB method
breaks down (as it should do) at the boundary of the forbidden region.
Obviously the particle enters the forbidden region orthogonally in both
cases, since in the classically allowed region it follows the classical
trajectory which near to the electrode coincides with the lines of
forces (see Fig. 2).
Thus, in tip negative case the semiclassical trajectories were
approximated by lines of force of the electrostatic field, as proposed
in [2]. For tip positive biases straight lines perpendicular to the
plane were chosen as trajectories on the basis of Fig.2.
RESULTS
Several calculations were carried out for different biases and
geometrical parameters (radius of curvature of the tip, distance of the
electrodes) for tip negative as well as for tip positive cases.
The overall tendencies of the variation of current with the distance
obtained experimentally (eg. [3]) are in good agreement with our
results. It is also interesting to observe that in all
cases the it tip negative current is higher than the tip positive one,
when all the other parameters are kept constant. The high lateral
resolution of the STM is also well demonstrated in Figs.3, and 4,
where the current density as a function of the radial variable measured
from the symmetry axis of the arrangement is displayed. However, our
model does not support the planeplane approximation for blunt tips (R
> 30A) , i.e. neither in this case the electrodes may be considered
as infinite planes.
When comparing our resolution with the results obtained by Das and
Mahanty^{4}, it turns out that, probably due to the assumed
different potential barriers, our resolution is worth than their, but
it is in good agreement with the results obtained by Garica, et al.^{5}.
The weak distance dependence of the resolution is also manifested by
our results. The prediction of our model concerning the difference of
lateral resolution in tip negative and tip positive cases should be
checked by experiments.
Click on the image to enlarge!

FIG.1. Equipotential lines
of
the tip positive arrangement if the bias
is 2 V, and the distance between the electrodes is 5 A. The data
referring to the electrodes are given in the text. The thick lines
represent the boundary of the classically forbidden region. The WKB
trajectories are also shown. 
Click on the image to enlarge!

Click on the image to enlarge!

FIG.2. Comparison of
different
WKB trajectories.
1: classical
trajectory in the pure electrostatic field of the given geometry.

2:
WKB trajectory for an electron energy higher than the barrier. The
lines of force of the field are shown by dotted lines. 
Click on the image to enlarge!

Click on the image to enlarge!

FIG.3.a. Current density
as a
function of the radial distance measured
from the symmetry axis for 10 A tipsample separation, 30 A tip radius
and 2 V bias.

FIG 3.b. Same as Fig. 3.a.
but
for 5 A tip radius.

REFERENCES
 L. Orosz and E. Balázs, Surface Sci. 177(1986)444.
 J. László and E. Balázs, 10th Werner
Brandt
Workshop, Alicante 1987.
 G. Binnig and H. Rohrer, Helvetica Phys. Acta 55(1982)726.
 B. Das and J. Mahanty, Phys.Rev B 36(1987)898.
 N. Garcia, C. Ocal and F. Flores, Phys.Rev.Letters
50(1983)2002.
Last updated: Jan 31, 2004
by Géza I. Márk
, mark@sunserv.kfki.hu
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