In classical physics an electron cannot penetrate into or across a potential
barrier if its energy E is smaller than the potential
within the barrier.
A quantum mechanic treatment predicts an exponential decaying
solution for the electron wave function in the barrier. For a rectangular barrier we get
The probability of finding an electron behind the barrier of the width d is
In scanning tunnneling microscopy a small bias voltage V is applied so that due to the electric field the tunneling of electrons results in a tunneling current I. The height of the barrier can roughly be approximated by the average workfunction of sample and tip.
If the voltage is much smaller than the workfunction eV << , the inverse decay length for all tunneling electrons can be simplified to
The current is proportional to the probability of electrons to tunnel through the barrier:
By using the definition of the local density of states for
the current can be expressed by
With 5eV as typical example for a workfunction value a change of 1Å in distance causes a change of nearly one order of magnitude in current. This facilitates the high vertical resolution.
which means that the current is proportional to the local density of states of the sample at the Fermi energy at a distance d, i.e. the position of the tip.
A more exact calculation of the current density of the square barrier problem
requires the Schrödinger's equation to be solved in the three regions:
before, in and behind the barrier.
The coefficients have to be adapted so that the overall solution is continually
differentiable. Defining the transition probability as
with the approximation .
The current density itself is defined as
For a nonsquare potential the WKB method must be used. This is more adequate
as the potential is changed by the applied voltage and influenced by the
image force on the electron. The WKB method yields a transition probability of
2.2. Bardeen Approach
Another way of describing electron tunneling comes from Bardeen's
approach which makes use of the time dependent perturbation theory.
The probability of an electron in the state
at to tunnel into a state
by Fermi's Golden Rule
The tunneling matrix element is given by an integral over a surface in the barrier region lying between the tip and the sample:
Applying a bias voltage V and approximating the Fermi distribution as a step function (kT Eresolution), the current is
Hence the current is given by a combination of the local densities of states of the sample and the tip, weighted by the tunneling matrix element M.
Schematic of electron tunneling with respect to the density of states of the sample.
Imaging the occupied states of
Imaging the unoccupied states of
The actual atomic resolution can only be understood in a quantum mechanical view:
The most prominent model in this respect is the s-wave-tip model.
The tip is regarded as a protuding piece of Sommerfeld metal with a Radius of
curvature R (see Figure).
It is assumed that only the
s-wave solutions of this quantum mechanical problem (spherical potential
well) are important. Thus, at low bias the tunneling current is proportional to the local
density of states at the center of cuvature of the tip r0:
In this model only the properties of the sample contribute to the STM image which is quite easy to handle. But it cannot explain the atomic resolution.
Interaction which causes a high corrugated tunneling distribution
Calculations and experiments showed that there is often a dz2
like state near the fermi edge present at
the apex atom which also predominantely contributes to the tunneling current.
It is understood that this state (and also the pz like state)
is advantageous for a sharp tip.
Since the tunneling current is a convolution of the tip state and the sample state,
there is a symmetry between both: By interchanging the electronic state of the tip and the sample
state, the image should be the same (reciprocity principle). This can also explain the fact
that the corrugation amplitude of an STM image is often larger than that of
the LDOS of the sample (measured by helium scattering). In this case the tip traces a fictitious
surface with a dz2 like state.
The state of the tip atom is dependent on the material and the orientation. As the tip is quite difficult to handle, it is one of the most difficult problems in a STM experiment.