Scanning tunneling microscopy

Scanning Tunneling Microscopy (STM) is an analytical technique based on the quantum mechanical phenomenon called tunneling. Tunneling is the phenomenon by which a high potential barrier does not eliminate the possibility of finding a particle in a region of high potential or even beyond a region of high potential. The easiest way to describe tunneling is by investigating the example of the one-dimensional harmonic oscillator.

In a classical treatment, a harmonic oscillator defines things such as the forces experienced by a spring or geological compression waves (caused by an earthquake). In a classical harmonic oscillator, the force is preportional to the distance:

1) F = -kx, where k is the force constant

Since the force is related to the potential by the following equation:

An equation for the potential can then be found by equating the non-F parts of equations 1 and 2. If this is done, the variables are isolated, and the equation integrated, the following equation results:

3) V = ½ kx2

The position, x, is classicaly defined by the following equation:

4) x = A sin[(k/m)1/2t + b]

Since the max and min for a sine function are positive and negative one, the particle's x coordinate will oscillate between +A and -A. A is the amplitude of the motion, and represents the absolute maximum displacement from the equilibrium point. There is 0 probability of finding a classically harmonic particle outside of its maximum amplitude, A. The classical picture of the particle's behavior is similar to a pendulum. The particle is displaced from a resting position and released. The particle will speed up as it moves toward its equilibrium position, reaching top speed at exactly the equilibrium position. The momentum, however, will continue to push the particle, even though it slows down as it works against gravity. In an ideal world, at the exact mirror image point of the original displacement, the particle will stop and begin to swing back earthward. In an ideal world, this motion would continue forever. In the real world, at least in the real world for macroscopic objects, the particle will eventually slow down and stop at the equilibrium position. However, the pendulum will never swing beyond it's original maximum displacement from the equilibrium position. This violates the conservation of energy.

For microscopic and especially atomic sized particles, however, the classical treatment of a harmonic oscillator doeas not hold. The formula describing the motion of a quantum mechanical particle is the wave function, Y. To determine the form of the wavefunction, one must solve the schrödinger equation for the harmonic oscillator:

5) -h/4pm (d2Y/dx2) + ½ kx2Y = EY

The well-behaved solutions to this equation are of the form:

6) Y = exp(-ax2/2)

The exact solutions are not important. It is obvious that as x increases or decreases in value, that the value of Y falls off exponentially. The probability, Y2, of finding a particle at very high values of x is not 0 however.

While a classical harmonic oscillator is confined to the region -A < x < A, a quantum mechanical oscillator has some finite probability of being found in the classically forbidden regions x > A and x < -A, where the potential energy is greater than the particle's total energy. This enterance into the classically forbidden regions is called tunneling. This is classically inexplicable. In fact is simply cannot happen, but it does. The tunneling phenomenon is not limited to one-dimensional harmonic oscillators. It is applicable to any large potential barrier. This is where STM comes into play.

The idea behind STM is that there is a certain driving force for an electron to want to move from one surface to another of lower potential. Classicaly, however, this is not possible without a direct connection, say a wire conneting the two surfaces. On an atomic sized scale, however, this is not true. When the distance between two surfaces is small enough, there is a finite probability that an electron will jump from the one surface to the other of lower potential. Experiments have proven this to be true.

An experimenter can determine a certain current that he wants the scanning tunneling microscope to maintain. Since the potential barrier is a function of distance between the two surfaces, so is the current. A computer can thus measure the current flow between a metal tip and a sample which are very close together. If the current increases, the computer can move the tip farther away from the sample, thus increasing the potential barrier, decreasing the probability of an electron jumping from the tip to the sample, and thus decreasing the current. If the current is too low, the computer will do just the opposite, moving the tip closer to the sample, decreasing the potential barrier and increasing the current. By keeping track of the movements of the tip, a realistic picture of the electron density of a surface can be created.

By tunneling current out of a single atom on the tip, the sensitivity of the instrument can be such that single atom layers on a surface can be measured. The STM can resolve local electronic structure at an atomic scale on every kind of conducting solid surface.

A probe tip typically made out of tungsten is attached to a piezodrive, which is a system of very sensitive piezo crystals which will expand or contract in reaction to an applied voltage. By using the piezo to position the tip within a few angstroms of the sample, the electron wavefunctions in the tip and the sample overlap, leading to a tunneling current flow when a bias voltage is applied between the tip and the sample. The tunneling current is amplified and fed into the computer while processing a negative feedback loop to keep the current constant. The computer, by collecting the z distance data, can image a three dimensional plot on-screen. This plot will represent the electron density of the sample surface. This electron density plot can then in turn be interpreted as the general arrangement or positioning of atoms on a conductive surface. Because the distance measurements are so miniscule, vibrations from the environment must be minimized. In the case of most labs, the sample is suspended in a magnetic field on springs, while the entire workbench is levitated off the ground by applying 60 psi N2 to each of the four legs of the work bench.

Even with all of these precautions, the STM is a sensitive instrument. One of the major problems with instrument sensitivity is tip size. The ideal tip would funnel down to a single atom tip which would be the source of tunneling. Tips, however, are seldom so perfect. Instead, researchers must tease and adapt the tip in a variety of ways to ensure that only one atom will tunnel. This can be done by smashing the tip into the sample to rearrange the atoms on the tip, by heating the tip, by applying high voltage differences between the tip and sample to draw the atoms down the tip, etc. Needless to say, STM is an exact and an exacting science.

One major area of STM research currently is to study self-assembled monolayers (SAMs). A self assembled monolayer is a single layer of molecules which aggregates on a surface.

Although no picture is available in this article, in a STM of an alkanethiol on Au(111) there are a series of zig-zag structures. These structures are the herring-bone reconstruction of Au(111). Without further explanation, let it suffice that this is the most thermodynamically stable construction of Au. The much smaller parallel lines crossing between the herring-bones are alkanethiol molecules forming a SAM. the black holes are missing Au atoms in the Au reconstruction. It is thought that these vacancies are caused by the crowding of the SAMs. As the SAMs form, they essentially squeeze the herring-bone structures until a few Au atoms eject out of the surface. The holes then tend to aggregate together.