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1. The Basics of Scanning Tunneling Microscopy (STM)

A scanning tunneling microscope or STM is a tool to look at individual atoms on the surface of a material.

The basic components of the microscope are:

• the sample you want to study
• a sharp tip to be placed in very close proximity to the sample
• a mechanism to control the location of the tip in the x-y plane parallel to the sample surface (XY scan control)
• a feedback loop to control the height of the tip above the sample (the z-axis)

The basic components of the setup are illustrated and described. Please refer to the figure.

[1] The Sample

A scanning tunneling microscope looks only at the surface of a material. The ideal sample is flat and clean, just a slice through a plane in the crystal structure of a material. It is fixed in place throughout the measurement. Because the STM works by a current to or from the sample, the sample must conduct electricity.

[2] The Tip

The tip is just a very sharp needle, so sharp that it terminates in a single atom. It is not actually touching the sample, rather it is approximately a few Angstroms (~atomic diameters) away. The tip is held at zero voltage, or "ground". Meanwhile, a bias voltage is placed on the sample, on the order of a few millivolts to a few volts. This voltage bias induces a "tunneling" current to flow between the tip and the sample. This current is exponentially dependent upon the distance between the tip and the sample which means that for a small change in the distance between the tip and the sample the current changes by a lot.

In normal electrical conduction the two metallic pieces which are going to conduct the electricity must actually be touching. However in this case the two metallics pieces (the tip and the sample) are not actually touching and yet a current flows by "tunneling". Tunneling is a phenomena that arises from quantum mechanics and it allows electrons to get from the tip to the sample even though the electrons do not really have enough energy to do so.

Here's how tunneling works. Suppose you are driving toward a valley, but there is a small hill between you and the valley. In this case, you are analogous to the electrons and the small hill is analogous to the barrier that the electron sees from the tip when it wants to get into the sample. Of course if you can get over the hill, the car will roll by itself down into the valley, but you must put your foot on the accelerator to give the car some additional energy to get over the hill. Similarly, the electron really wants to get from the tip to the sample (or vice versa), and the net journey is downhill, but the 1 Angstrom gap between the tip and the sample is an energy barrier that acts like a small hill. According to quantum mechanics, the electron doesn't actually need to go up and over the hill, but we say it "tunnels" through the hill, without ever acquiring the full energy that would be needed to go over the hill.

[3] The XY Scan Control

The tip can be scanned across the surface using a crystal called a piezo electric which changes its size by very small amounts when a voltage is applied to it. As the tip is moved in the x or y direction along the surface of the sample, the current will vary according to whether the tip is right on top of an atom (smaller distance), or on top of a space between atoms (larger distance). So an individual atom can be "seen" as an increase in the tunneling current as the XY scan control moves the tip across the surface of the sample.

[4] The Feedback Control

In practice, since current falls off exponentially with distance, the current when the tip is on top of an atom is much much larger than the current when the tip is between atoms. We could just record the current as a function xy position on the surface, but because the current is exponential in the tip-sample distance this would give a distorted image in which the atomic peaks would look much higher than they actually are.

What we are really interested in is the actual height of atoms on the sample surface. We want to measure something which varies linearly with this height. So, we employ a feedback loop to keep the current constant by adjusting the height of the tip. The height of the tip is controlled by a piezo electric crystal. A piezo electric crystal is a material which expands linearly when a voltage is placed across it. The expanding crystal pushes the tip closer to the sample. So the voltage needed to expand the piezo to keep the current constant varies linearly with the actual height of the atoms on the sample. Now by reading the feedback voltage, we can just read directly the height of the tip.

2. Local Density of States: What an STM measures

STM is able to measure the local density of states of a material at it surface as a function of lateral (x-y) position on the sample surface and energy. This page will provide an explanation of what the local density of states is.

Within the sample, each individual electron has a specific energy level. Only a certain number of electrons may occupy any given energy level at any one time. The distribution that gives the number of electrons allowed per energy level as a function of which energy level you consider is called density of states which can be abbreviate as DOS(E) to indicate that this is a quantity dependent upon energy. The DOS is a very useful quantity to be able to measure since it can be used to derive a wealth of information about the crystal's properties.

The DOS can vary as a function of position in the crystal which means that one can define a local density of states (LDOS). LDOS is then a quantity which depends on both energy and on position, LDOS(x, y, E). A physically intuitive way to think about local density of states is that it gives the density of electrons of a certain energy at that particular spatial location.

An STM can typically measure and control the current that flows between the tip and the sample (I), the bias voltage between the tip and the sample(V), the xy (in sample plane) position of the tip, and the z (perpendicular to sample plane) distance between the tip and sample.

Using these variables an STM is able to measure the LDOS of a material as function of position on the surface (controlled by where the tip is above the surface) and as a function of energy (controlled by the bias voltage between the tip and sample). The LDOS is proportional to the differential increase in tunneling current given a differential increase in bias voltage or in other words one can measure the LDOS by measuring dI/dV.

3. Measurements

There are several different types of measurements that can be taken with an STM:

One can think of the measurements an STM can make by visualizing a "three dimensional" DOS data set where the three dimensions are x and y position and energy, and the measured quantity is the LDOS. The 3-D data set available to an STM is indicated schematically in the diagram.

As the diagram indicates, the LDOS is the quantity measured. Each plane pictured indicates a different energy at which a measurement of the LDOS can be made. The LDOS is also a function of position on the surface so the surface of the crystal is represented by the faces of the planes.

So the figure schematically represents the 3-D data set that comprises STM measurements by showing the three variables than can be controlled by an STM: x position, y position and energy and also the measurement that is made: LDOS. In practice, subsets of this 3-D data set are taken in any one STM measurement. These subsets are indicated schematically in the figure and are described below.

[1] Topography

A topography shows the total density of electrons on the surface as a function of position. Displaying higher electron density as lighter colors, the atoms then are the white spots in the topograph as the electrons tend to congregate near them and thus prsoduce a higher local density of electrons in that region.

As the tip moves from location to location on the surface, the current gets bigger or smaller depending on the local electron density and by making a map of where the current is big and where it is small, one can image the atomic positions.

In practice, a topograph is measured by fixing the sample at a specific bias V with respect to the tip. All electrons with energy less than V are then free to flow from the tip to the sample (or vice versa depending upon the sign of V). The magnitude of the current depends on the total number of electrons with energy less than V in that spatial location or in other more fancy words, the current depends on the integrated LDOS from zero to the chosen energy V. Feedback is used to keep the current constant and the feedback voltage is recorded as a measure of the height of the sample at each point to produce the topographic image.

[2] dI/dV Map

A dI/dV Map shows the density of electrons at a particular energy as a function of position on the surface. In the figure above, a dI/dV map is equivalent to view a single energy plane. As in the topographies, higher electron density at this chosen energy is displayed as lighter colors.

Impurity atoms on the surface attract electrons of a certain energy so at that energy they are bright in the dI/dV maps. In the figure above, the position of an impurity atom is indicated by the center of the clover leaf pattern in the blue dI/dV map. However, in general at a random energy there will be no especially high electron density at any given position, but rather the electron density will be uniform throughout the surface of the crystal. This means that the dI/dV map will be uniformly black as indicated by the black panel in the figure above.

In short, in a dI/dV map, we fix the energy and measure the LDOS as a function of x-y position.

[3] dI/dV Spectrum

A dI/dV spectrum shows the density of states as a function of energy at a particular spatial position. In the figure above it is shown by a red line which passes through all the energy planes at a particular position. A measurement of the LDOS is made at each energy plane at the point that the red line intersects it.

A superconductor has a very low density of states at low energies. The spectrum shown above is the spectrum for a d-wave superconductor and you can see that the density of states decreases as you move to lower energies (zero energy is in the middle of the graph).

In short, in a dI/dV spectrum you fix the spatial position and measure the LDOS as a function of energy.

[4] Line Cut

A line cut is a series of dI/dV spectra taken at equally spaced spatial points along a line drawn across the surface. In the figure above it is indicated by a series of red lines (each of which intersect each energy plane at a certian spatial position) which are all in a row. The measurement made at the position indicated by each red line is one spectrum in the line cut.