To understand the theory of how STM works, it is vital to know what is tunneling current, and how it relates to all the experimental observations.

Tunneling current is originated from the wavelike properties of particles (electrons, in the case) in quantum mechanics. When an electron is incident upon a vacuum barrier with potential energy larger than the kinetic energy of the electron, there is still a non-zero probability that it may traverse the forbidden region and reappear on the other side of the barrier. It is shown by the leak out electron wavefunction in the following picture.

If two conductors are so close that their leak out electron wavefunctions overlap. The electron wavefunctions at the Femi level have a characteristic exponential inverse decay length K given by

m is mass of electron, is the local tunneling barrier height or the average work function of the tip and sample. When a small voltage, V is applied between the tip and the sample, the overlapped electron wavefunction permits quantum mechanical tunneling and a current, I will flow across the vacuum gap.

At low voltage and temperature

d is the distance between tip and sample. If the distance increased by 1 Angstrom, the current flow decreased by an order of magnitude, so the sensitivity to vertical distance is terribly high. As the tip scans across the surface, it gives atomic resolution image you now see.

This is the STM image of Si(111)-7x7 surface, the white spots represents the position of the atoms.

STM does NOT probe the nuclear position directly, but rather it is a probe of the electron density, so STM images do not always show the position of the atoms, and it depends on the nature of the surface and the magnitude and sign of the tunneling current.

Equation (2) obviously shows the current is exponentially depends on both gap distance and the local barrier height, change of current might be due to corrugation of the surface or to the locally varying local barrier height. The two effects can be separated by the relationship.

The tip is vibrating vertically and the current is measured, in theory, the local barrier height can be calculated. Again, if the tip rasters the surface, map of local barrier height can be obtained. However, the local barrier image also contain topographic features, some questions related to the local barrier have so far remained unexplained. So, extra care needs to be taken in performing such experiments.

Density of States (DOS) represents the amount of electrons exist at specific values of energy. The tunneling conductance, (or I/V ) is proportional to the LDOS.

where r(r, E) is the local density of states of the sample.

Keeping the gap distance constant, measure the current change with respect to the bias voltage can probe the LDOS of the sample. Moreover, changing the polarity of bias voltage can get local occupied and unoccupied states.

When the tip is negatively biased, electrons tunnel from the occupied states of the tip to the unoccupied states of the sample. If the tip is positively biased, electrons tunnel from the occupied states of sample to the unoccupied states of the tip.

Here are the spectra for Si(111)-7 x 7 surface. The bottom spectrum is the area averaged tunneling conductance measured by STM, and the top spectrum is the surface states spectrum measures photoemision and inverse photoemision. Both spectra show similar features.

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