## 2. Theory

### 2.1. Electron Tunneling

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.
Also
,
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

yields

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 at is given 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.

means that an electron can only tunnel if there is an unoccupied state with the same energy in the other electrode (thus inelastic tunneling is not treated). In case of a negative potential on the sample the occupied states generate the current, whereas in case of a positive bias the unoccupied states of the sample are of importance. Therefore, as shown below, by altering the voltage, a complete different image can be detected as other states contribute to the tunneling current. This is used in tunneling spectroscopy. It should finally be mentioned that the probability of tunneling (expressed by M2) is larger for electrons which are close to the fermi edge due to the lower barrier.
 Imaging the occupied states of SiC(000)3x3 Imaging the unoccupied states of SiC(000)3x3

### 2.3. Lateral resolution

The lateral resolution of STM can not be understood in terms of a Fraunhofer diffraction resolution. The corresponding wave length of the tunneling electron would be > 10 Å.
Therefore the STM works in the near-field regime. The overall geometric curvature of the tip with a radius of curvature of e.g. 1000 Å and = 1 Å-1 would give rise to a resolution of about 50 Å.

 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.