ATOMIC SCALE TUNNELING AND IMAGING
The physical interaction
used in obtaining the surface information for a scanning tunneling microscope
is quantum mechanical tunneling. A brief and simplified discussion of tunneling
through a vacuum barrier was given above. Now a more detailed examination
of the tunneling mechanism follows. The simplest explanation of the scanning
tunneling microscope as a very sensitive profilometer depends on the belief
that at the atomic scale the STM is actually taking a surface profile of
the sample. At an atomic scale the notion of surface topography is unclear.
A simple assumption would be that surface topography at the atomic scale
is a contour of the charge density of the surface material. Only electrons
near the Fermi level contribute to the tunneling and all electrons below
the Fermi level contribute to the charge density, so, assuming that the
topography produced by changes in the tunneling current is a contour of
the charge density may not be entirely correct.3
In addition to the uncertainty
regarding the surface topography, the electron transport processes in an
STM are different from the standard mechanism described as tunneling. Usually,
the tunneling process has a barrier width of 20-30 angstroms; but in an
STM, the barrier is only a few angstroms wide. Also, overlap between the
surface potentials of the probe and sample may place the top of the potential
barrier lower than that of the vacuum level.4
Sometimes the potential barrier level is even lower than the Fermi level.
The local atomic structure between the probe and the sample also play an
important role in determining the current. Finally, many forces exist between
the probe and the sample affecting the tunneling current.
Ballistic Transport and Tunneling
Atomic scale tunneling differs
from conventional tunneling in one important aspect. The width of the potential
barrier is small compared to that in usual discussions of tunneling. In
fact, with a small enough barrier tunneling might not be an accurate description
of the process occurring. Arguments invoking the uncertainty principle
show that the location of the electron is unknown with respect to the barrier.5
Thus, regarding atomic scale potential barriers the difference between
the ballistic transport and tunneling is nonexistent. Two approaches with
the uncertainty principle show this nondistinction.
The first approach shows
the uncertainty of the electron energy, in the region of the barrier, is
greater than its kinetic energy. Inside the region of the barrier, the
electron has a velocity determined by its kinetic energy.
With this velocity, the time for an electron to pass through the barrier
is:
Using the energy-time uncertainty relation:
Assuming E-UB = 3eV, and the barrier thickness is approximately
2 Å, the energy uncertainty, D E, is about
3.4 eV. Thus, the uncertainty of the energy is larger than the absolute
value of the electron kinetic energy.
The second approach shows
the uncertainty in position of an electron may be greater than the barrier
thickness. The de Broglie wavelength of the electron in the classically
allowed region of the barrier is given by equation 12.
Given an electron with a kinetic energy of 3eV, the de Broglie wavelength
of the electron is approximately 7.1 Å, which is the same order of
magnitude as the barrier thickness.
Modeling the Tunneling Current
Regardless of the uncertainty
relations given above, a tunneling Hamiltonian approach using first order
perturbation theory may be used to calculate the tunneling current. Based
on a method developed by Bardeen6
for metal-insulator-metal tunneling junctions, this approach begins by
considering two subsystems instead of attempting to analyze the Schrödinger
equation of the combined system. For each subsystem, solving the stationary
Schrödinger equation determines the electronic states. Then, using
time dependent perturbation theory, the electron transfer rate between
the electrodes may be found. The overlap of the surface wavefunctions of
the two subsystems at a separation surface determines the amplitude of
the electron transfer, also known as the tunneling matrix M. By modifying
the wavefunctions of one surface due to the presence of the other, the
Bardeen approach to tunneling may be applied to the tunneling in an STM.
This method is also known as the modified Bardeen approach (MBA).7
Using this approach, the
tunneling current is,
.
In this expression f(E) is the Fermi function, V is the applied
voltage, and Mmn is the tunneling
matrix element between the state of the probe and the sample. Evaluating
the tunneling matrix Mmn is
usually the most difficult step in determining the tunneling current. This
difficulty stems from lack of knowledge of the probe and sample wavefunctions.
If the probe and sample wavefunctions are known Mmn
can be evaluated using the expression developed by Bardeen
.
Equation 14 determines the tunneling matrix Mmn
. In this expression, cn is the
modified wavefunction for the probe and ym
is the wavefunction for the sample. The integral is performed over the
surface area defined by S.
Local Density of States (LDOS) Topography
Although the MBA model works
reasonably well for a variety of conditions, this model requires knowledge
of the wavefunction of the surface and the probe. As mentioned previously,
these wavefunctions are usually not known, especially for the tunneling
probe. Assuming a simple model for the probe, and using the first order
perturbation theory for determining the tunneling current, leads to the
conclusion that the tunneling current is proportional to the local density
of states.
The simplest model of the
probe is that it is a point source of current. Assuming the probe is a
point source of current allows the analysis to determine properties of
the sample only. In actual STM measurements, the probe and sample wavefunctions
are responsible for the tunneling current. With the probe modeled as a
mathematical point source of current, equation 13 for the current at small
voltages reduces to8
.
In this equation, EF is the Fermi
level of the sample surface, ym is
the wavefunction for the surface at position r. The quantity r(r,EF)
is the local density of states of the sample.
Lateral Resolution Models
Initial attempts to estimate
the resolution of a scanning tunneling microscope showed that a very high
resolution was possible with even a moderately sharp probe. The initial
spherical probe model by Binnig, the s-wave model, and probe-sample interactions
provide insight to the origin of atomic resolution in the scanning tunneling
microscope.
One of the first estimates
of the lateral resolution of the STM came from Binnig in 1978.9
Assuming the tunneling probe was spherical in shape with a radius R, Binnig
determined the tunneling current would be related to the probe radius by
equation 16.
Figure
3
This equation is based on the idea that the probe radius is much larger
than the distance between the probe and the surface. In this case the current
lines are almost perpendicular to the sample surface (Figure 3).
With k
» 1 Å-1 and R»
100 Å, the current concentrates in a small radius of approximately
14 Å; thus, the expected resolution should not exceed this value.
Modern scanning tunneling microscopes routinely exceed this resolution.
Thus, a more accurate model of the probe-sample interaction is necessary.
Soon after achieving atomic
resolution with the STM, Tersoff and Hamann10
proposed the s-wave model of probe sample interaction to account for the
experimental results of Binnig and Rohrer. They modeled the probe as a
protruded piece of Sommerfeld metal with a radius of curvature of R. The
probe wavefunctions were then assumed to be solutions of the Schrödinger
equation for a spherical potential well. Assuming only the s-wave solution
was important, Tersoff and Hamann were led to a simple solution for the
tunneling current. Under a low bias the s-wave model shows the tunneling
current to be proportional to the Fermi level local density of states at
the center of the curvature of the probe ro (equation 17).
Figure
4
The s-wave probe model removes the probe interactions from the analysis
in a way similar to the point source current model described earlier. Thus,
the s-wave model reflects properties of the surface only.