Dr. Alan Jackson,
Central Michigan University Professor of Physics.
INTRODUCTION
Materials research is a very active area of current research. Much of this activity is related to the electronics industry and involves studies of semiconducting materials, but the field is much larger, encompassing a wide range of applications. The scanning probe microscopes are playing an increasingly important role in materials research, giving scientists an unprecedented look at the atomic structure of materials.
TUNNELING
Good descriptions of tunneling can be found in many texts on modern physics or in an introductory quantum mechanics text (c.f. Introduction to Quantum Mechanics, by D. J. Griffiths, Prentice-Hall 1995). The ultimate physics behind tunneling is the wave-like nature of a particle like an electron, represented in quantum mechanics by the wave function. The wave function (actually its square) gives the probability of finding the particle at a particular position. As nature would apparently have it, the wave function cannot go discontinuously to zero, not even at a boundary a classical particle would be forbidden from crossing. The wave function decays quickly with distance into these "forbidden" regions, but the fact that it is non-zero leads to interesting effects like tunneling. A schematic of a particle tunneling through a thin barrier is shown in Figure 1.
Figure 1. A particle tunneling through a barrier. See http://www.Lehigh.EDU/~msf2/stm/stm.htm
POWERS OF TEN PROBLEMS
Competence with scientific notation and making units conversions are key skills for serious students of science. In these lessons, it is assumed that the students have received instruction in the use of scientific notation. They should have had some practice in basic arithmetic operations (+,-,x,/) involving numbers expressed in scientific notation. The exercises given here require the students to make use of scientific notation and units conversions to solve "practical" problems.
STM IMAGE OF GRAPHITE
Most STM images made with atomic scale resolution show atoms as bright spots. Not all surface atoms appear equally bright, however, because the tunneling probability depends not only on the separation between surface and tip, but also on the density of electrons at the surface. The graphite image is a good case in point. Figure 2 is a schematic of the graphite atomic structure, constructed from X-ray diffraction measurements. (X-ray diffraction is a standard tool for determining the structure of crystalline materials.)
Figure 2. Schematic of graphite structure. See http://www.Lehigh.EDU/~msf2/stm/stm.htm
In the diagram, two types of carbon atoms can be seen. The filled circles are carbon atoms that lie directly above atoms in the plane below. The open circles lie above voids in the lower plane. Because the electron density is higher above the filled atoms, they appear as the bright spots in the image given in the student materials. The atoms corresponding to the open circles appear as much fainter spots and are difficult to make out without a prior knowledge of the structure. The students should be able to make measurements to convince themselves that the bright spots in the STM image correspond to the filled circles.
There are many ways to determine the area associated with a given atom in the image. A brute force approach is to count the number of atoms in the image and divide by the total area. Students should be encouraged to find their own technique. They may need to be reminded that the bright spots represent only half the total number of atoms in the plane.
In problem # 5 the students are asked to find the interplanar separation in graphite. The separation is given in Fig. 2 above but cropped from the figure in the students’ materials. Note that the interplanar separation is much larger than the C-C bond length in the plane (about 1.4 Å). This indicates how weakly the planes are bound.
STM IMAGE OF ALUMINUM
The STM image shows an aluminum (Al) surface. The regular arrangement of the atoms is characteristic of a crystalline material, as opposed to an amorphous material, in which there is no long-range order to the atomic positions. The arrangement of atoms on a crystal surface can look different depending on the orientation of the crystal. The hexagonal array shown in the image is characteristic of the (111) surface of Al. In the (100) surface, the atoms would form a square array, with a somewhat larger spacing.
One of the goals of this activity is to encourage the students to think about the relative sizes of atoms. They may be surprised to learn that all atoms are roughly the same size. The largest atom, Cs, as measured by its covalent radius of 2.35 Å, is only a factor of two larger than Al, with a covalent radius of 1.18 Å. Students may wonder about we really mean when we talk about the size of an atom. They may already know that atoms do not have rigid boundaries like billiard balls. In fact, they may have the picture of an atom as an extremely small, positively charged nucleus, surrounded by a cloud of electrons. The edge of the cloud is certainly a hazy concept. The covalent radius is a convenient measure of atom size. By definition, the covalent radius of an atom is equal to half the bond length in the corresponding elemental solid.
The Ar bubbles causing the disturbances seen in the STM image were created intentionally by a process called ion implantation. Ar ions were fired at the Al surface with enough energy to enter the solid. The Ar atoms can migrate through the solid, preferring to cluster in "bubbles".
EXPONENTIAL DECAY
An important conceptual issue in the operation of the STM is that of exponential decay. It is mentioned in the materials that the tunneling current depends in an exponential way on the tip-surface separation. A comparison of exponential dependence versus linear dependence makes for some interesting discussions. The basics are simple: a linear relationship implies a constant change in y for each step in x. Height on a staircase is a homely example--on a typical staircase, for each step you take forward, you climb a fixed amount higher. In an exponential dependence, each step in x implies a change in y by a constant factor. For example, with each step on an exponential staircase, you could double your altitude with each successive step (2, 4, 8, 16,...). Playing with this concept can make clear to the students the sensitivity implied by the exponential dependence the tunneling current. Some suggested activities:
REFERENCES/ADDITIONAL RESOURCES
The following websites offer a good starting point for further study:
1) http://www-i.almaden.ibm.com/vis/stm/gallery.html
IBM Research Labs: This is a gallery of images made by IBM scientists, including Don Eigler, the father of using the STM to maneuver atoms on a surface. There is not much detail here about how the STM works, but there are lots of interesting pictures.
2) http://inaba.nrim.go.jp/ifes/LinkSTM.html
This site contains links to a large number of web pages belonging to STM groups and other related researchers around the world. There are a large number of links here, offering many opportunities for exploring.
3) http://www.iap.tuwien.ac.at/www/surface/STM_Gallery
This is a nice site managed by the STM group at the Technical University of Vienna in Austria. It has a nice basic description of how the STM works and many high quality images made by members of their research group.
ATOMIC SCALE MICROSCOPY
1) Introduction
Microscopes allow us to investigate objects in much finer detail than we can with the unaided eye. For example, looking through a microscope we discover that a drop of lake water that looks clear and pure to the eye is actually alive with microorganisms like amoebas. But what if we want to see what the amoeba is made of? Can we adjust the resolving power of the microscope to allow us to see the components of, say, the amoeba’s cell wall? There is a simple limit to a traditional microscope’s ability to "see" small things: only objects that are larger than the wavelength of the light (or other wave-like probe) used to view them can be resolved. In the case of visible light, that wavelength is about one tenth of a micron, or about one ten millionth of a meter. Thus, objects much smaller than one micron across cannot be resolved under a conventional light microscope. Individual atoms are about 2 D ( 1 D = 1 ten billionth of a meter = 1 ten thousandth of a micron) across--about 500 times too small to be seen under an optical microscope.
Why would we want to see atoms? The arrangement of atoms at the nanometer (1 nm = 1 billionth of a meter) scale determines macroscopic properties of a material. For example, both diamond and graphite are composed exclusively of carbon atoms, yet because of their different atomic-level architectures (see the figure below), the two materials are very different. Diamond is transparent material and very hard (you may have used a diamond to scratch glass). Each carbon atom is bonded to four others to form a strong network. In graphite, the material in pencil "lead", each carbon atom is strongly bonded to three others (in the graphite figure, only the bonds are shown--the atoms are represented by the points where bonds meet). The atoms are arranged in sheets that are weakly bound to each other. As a result, the sheets can slide easily, making graphite very soft.
Fig. 1 a Atomic structure of diamond
Fig. 1 b Atomic structure of graphite
In many areas of science and technology, significant effort is directed at understanding the connection between atomic arrangements and macroscopic properties, in order to develop materials with more useful properties. Being able to directly "see" atoms makes this process more efficient and reliable.
To overcome the limits of traditional microscopy, Gerd Binnig and Heinrich Rohrer invented an instrument called the scanning tunneling microscope (STM) in the early 1980’s at the IBM research laboratories in Zurich, Switzerland. The STM is capable of imaging the atoms on the surface of a material with a resolution of a fraction of an atomic diameter, allowing the most detailed images of surfaces ever made. The STM revolutionized surface science and earned Binnig and Rohrer the 1986 Nobel Prize in Physics.
Fig. 2 Schematic of an STM. See the Website at the Technical University of Vienna http://www.iap.tuwien.ac.at/www/surface/STM_Gallery/
In the mid-80’s a related device known as the atomic force microscope or AFM was invented by Binnig and Christopher Gerber of IBM’s Almaden Research Center and Calvin Quate of Stanford University. The AFM can also image surfaces at the atomic level, but can be used to study to a wider variety of materials than the STM, including biological materials like DNA molecules.
2) The scanning tunneling microscope (STM): How it works
At a conceptual level, there are three elements that make the STM work: tunneling current, piezoelectric positioners and feedback (see Fig. 2). The STM has a narrow tip that is positioned over the surface of the sample to be studied. A small voltage difference is applied between the tip and the sample, causing a small electric current (the tunneling current--more on tunneling below) to flow between the tip and the sample surface. The strength of the current depends very sensitively on the separation between the tip and the nearest surface atom. Changing the tip-surface separation by 0.5 D can change the tunneling current by 1000 times or more. The amount of tunneling current can therefore be used to monitor the distance between the tip and the surface.
Piezoelectric positioners are used to move the STM tip across a surface in a highly controlled way. A piezoelectric material has the property that its length changes slightly when a voltage is applied across its ends. The STM tip is attached to a system of piezoelectric rods that can move the tip in any direction across the surface to atomic-scale accuracy by applying the proper voltages.
To obtain a topological map of the surface, a reference current is established with the tip at a given starting position over the surface. The tip is then carefully moved a small distance across the surface by the piezo-positioners. As the tip moves across the surface the tunneling current changes, reflecting a change in the tip-surface distance. This change in current is fed-back to an electrical circuit that controls the up and down motion of the tip over the surface. If the tunneling current increases (indicating that the tip-surface separation has decreased), the feedback circuit causes the tip to be raised up until the current returns to the reference value. Similarly, if the current decreases, the tip is lowered to return the current to the reference level. In this way, the feedback circuit works to keep the tunneling current, and therefore the tip-surface separation, constant as the tip is scanned across the surface. (A piezo-positioner is also used to control the up-down motion, by the way.) A computer tracks the up and down and side to side motion of the STM tip simultaneously, creating a map showing the height of the tip over the sample. Since the tip-surface separation is kept constant by the up-down positioning, this map effectively charts the height of the surface atoms.
The sensitive dependence between the current and the tip-surface separation is due to the quantum mechanical phenomenon called "tunneling". As you may know, in an electrical circuit, electric charge flows naturally from a higher voltage to a lower voltage. The situation is akin to a marble placed on a smooth incline: the marble "prefers" to be at the lowest possible position on the incline and spontaneously rolls downhill to get there. Of course, if you place the marble behind an obstacle on the incline, a wad of gum, maybe, it gets stuck and can not roll down. The marble does not have the energy to get over the gum and continue downhill. In the case of the STM, the obstacle to the flow of tunneling current is the absence of a conducting path between the tip and the surface. According to the laws of classical physics, without a conducting path, the electrons do not have enough energy to "jump" from the tip to the surface. No current can flow unless the tip and the surface are actually in contact. However, according to quantum mechanics (the theory that describes the motion of atomic-scale particles like electrons) there is a small probability that an electron can make the jump, even if the tip and surface are not in contact. The probability decreases very quickly (exponentially, mathematically speaking) as the tip-surface separation increases, accounting for the sensitivity of the current to the distance.
It is difficult to fully understand the peculiar nature of tunneling without considering the macroscopic analogy. Imagine placing your marble on the incline behind the wad of gum. Without a push, the marble cannot get over the gum and roll down the incline. Tunneling in this case would mean that the marble would have some probability of getting over the obstacle without any push and if you watched long enough, it finally would. If you are wondering how the marble actually manages to get past the gum, quantum mechanics does not offer much help. The "tunneling" concept does not provide the details of how it happens (at least not in a way that helps our imagination), only that it can happen. It’s interesting to note that the laws of quantum mechanics apply even to macroscopic objects like the marble; however, tunneling probabilities are such that they are significant only for atomic-scale objects. The probability for your marble to tunnel past the gum is so small that you would have to wait an astronomically long time for it to happen.
To summarize the crucial points that make the STM work: the tunneling current is fantastically sensitive to the tip-surface separation; piezo-electric positioners can control the motion of the STM tip across the surface to sub-angstrom distances; and feedback circuitry positions the tip up and down to maintain a constant current and hence a constant tip-surface separation as the tip is scanned across the surface.
POWERS OF TEN LESSONS
GOAL: To develop competence in using Scientific Notation; to provide practice in unit conversion; to stimulate student interest in matter at the atomic scale.
ACTIVITY ONE: Units and Scientific Notation
In science we often deal with numbers that are very large or very small. Scientific notation is designed to make working with large and small numbers easier. By now you have studied scientific notation in your class and you have learned the basic rules for adding, subtracting, multiplying and dividing numbers expressed in scientific notation. The following exercises will give you additional practice both with scientific notation and with doing conversions between different sets of units. Let’s start first with a brief example.
In the metric system, three of the fundamental units by which all things are measured are:
| Measurement | Fundamental Unit | Symbol |
| Length | Meter | m |
| Mass | Kilogram | kg |
| Time | Second | s |
To express large and small quantities of these fundamental units, prefixes are used. Each prefix represents a "power of ten" since our numbering system is based upon ten.
| Power of Ten | Prefix | Abbreviation |
| 10-18 | atto- | a |
| 10-15 | femto- | f |
| 10-12 | pico- | p |
| 10-9 | nano- | n |
| 10-6 | micro- | µ |
| 10-3 | milli- | m |
| 10-2 | centi- | c |
| 10-1 | deci- | d |
| 100 | unit 1 | |
| 101 | deca- | da |
| 103 | kilo- | k |
| 106 | mega- | M |
| 109 | giga- | G |
| 1012 | tera- | T |
| 1015 | peta- | P |
| 1018 | exa- | E |
Often it is necessary to change from one unit to another: "How many meters are in 1000 km?"; or "How many grams are in 365 mg?". Remember that in solving problems of this type, we are just expressing the same measurement in a different unit that differs by a certain multiple (power) of ten. For example:
1000 km x 103 m / km = 1.0 X 106 m.
Or:
365 mg x 1 g / 10-3 mg = 0.365 g.
Sometimes it is necessary to change from one system of measurement to another. For example, "How many centimeters are in 12.8 feet?"
Solution:
12.8 ft x 12 inches/ft x 2.54 cm/inch = 390 cm
While this latter example does not require power of ten conversions as the previous examples did, the method of solution is virtually the same: multiply by the correct conversion factor (or factors), canceling units until the desired unit is reached. This method should be used throughout your study of physics.
Problems involving units and powers of ten
The annual budget of Central Michigan University is about 250 million dollars.
2) A typical woman is 1.6 m tall. Assume that she is made mostly of carbon atoms. (Humans also contain lots of water, i.e. oxygen and hydrogen, as well as many other types of atoms, but let’s keep things simple.)
3) Human eyes can resolve a spot roughly 1.0 X 10-1 mm across. Let’s assume that the spot is in the shape of a cube. Let’s also think of atoms as tiny cubes with sides of approximately 2.0 Å. The scanning rate for a certain Scanning Tunneling Microscope (STM) is about 100 Å/minute. Each scan covers a strip 2.0 Å wide.
ACTIVITY TWO: STM image of graphite
A basic application of the STM is to investigate the arrangement of atoms on surfaces. How atoms are arranged, e.g. their spacing and the angles between bonds is of great practical interest to researchers. Figure 1 shows a picture of the surface of graphite with a length scale clearly marked for reference. Study the figure and then answer the following questions.
Figure 1. STM image of graphite. See http://www.Lehigh.EDU/~msf2/stm/stmhopg.htm
Figure 2. Schematic of the graphite structure. The lengths indicated are in Å units. See http://www.Lehigh.EDU/~msf2/stm
ACTIVITY THREE: STM image of aluminum
Figure 3 is an image of an aluminum (Al) surface. The bright dots are the Al atoms. The large hexagonal features represent disturbances in the electron "cloud" at the surface, caused by impurities below the surface layer. Study the diagram and then answer the following questions.
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Figure 3. An aluminum surface. The large hexagonal features are due to subsurface impurities.
(See http://www.iap.tuwien.ac.at/www/surface/STM_Gallery/electronwaves.html)
Questions? Comments??
James Gormley