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(revised 4/26/2000)
Scanning Tunneling Microscope
Advanced Laboratory, Physics 407 University of Wisconsin Madison, WI 53706
Abstract
A scanning tunneling microscope is used to demonstrate the principle of quantum mechanical tunneling between the microscope tip and the surface of a conducting sample. Measurements are made on a gold-coated holographic grating and a pyrolytic graphite sample. Since the apparatus is capable of atomic resolution, atomic features of the graphite surface can be directly observed. Mathematical filter algorithms are used to process the sample images and reduce the image noise. The bond angles and bond lengths of the graphite sample are determined.
1 OPERATING PRINCIPLES OF STM
1.1. How the STM Works
There are five scientific and technical processes or ideas that the STM integrates to make atomic resolution images of a surface possible. Each of these processes was used in other areas of science before the invention of the STM.
- The principle of quantum mechanical tunneling.
- Achievement of controlled motion over small distances using piezoelectrics.
- The principle of negative feedback.
- Vibration isolation.
- Electronic data collection.
This Chapter discusses each of these five concepts. The most detail is provided on the process of quantum mechanical tunneling, since this is the fundamental concept that allows the microscope to work. At the end of the discussion of all these concepts, one can see how they integrate to make an STM.
1.2. Ouantum Mechanical Tunneling
Quantum mechanical tunneling is not some obscure process that only occurs under extreme conditions in a crowded basement laboratory of a research university. Quantum mechanical tunneling explains some of the most basic phenomena we observe in nature. One example is the radioactive decay of plutonium. If quantum mechanical tunneling did not occur, plutonium would remain plutonium instead of changing into elements lower on the periodic chart. Plutonium converts to other elements when 2 neutrons and 2 protons are ejected from the nucleus because of tunneling. Even the fundamental force that binds atoms into molecules can be thought of as a manifestation of quantum mechanical tunneling. In this lab, we will look at how tunneling manifests itself in another way. We will attempt to understand how a single electron starts out in one metal and then reappears in another metal, even though they are not touching.
The unusual experimental feature of tunneling is this: when the metals are brought together, but are not quite touching, a small electric current can be measured. The current gets larger the closer the metals are brought together, until it reaches its maximum value when the metals are touching. The concept is analogous to making the dam thinner and thinner by removing cement and noticing that more and more water is leaking through the walls. However, there is a difference between the two analogies. The water physically moves through the pores between the cement, while the electrons do not move in the space between the metals: they just suddenly appear in the other side. The metals must be only 10 angstroms apart to produce detectable tunneling current. Figure 2.2. shows current as a function of the separation between metals [a]. Also plotted in this graph is the measured tunneling current if quantum mechanical tunneling did not occur [b]. The distances involved are so small that special tools are needed to adjust the distances or the small electric currents will not be detected. We will describe these tools in the section on piezoelectrics (see Section 2.4.).
To understand why these small currents occur, the energies involved as the electron moves between the metals must be considered. An electron's energy can be split into two contributions: kinetic energy and potential energy. Kinetic energy (the energy of motion) is large for electrons moving fast and small for electrons moving slowly. Potential energy is the energy available for an electron to convert to kinetic energy if it moves along an electric field. Figure 2.3 plots the potential energy of the electron as it travels from one metal to the other metal. The potential energy shown neglects the complicated aspects of metals, including extra charges from atoms and other electrons on the metals, but does include the general concepts. The potential energy is lower in Metal 2 because this side is connected to the positive terminal of the battery (the terminal to which the electrons are attracted). There is also a large potential energy between the two metals. This is what tends to keep electrons inside their respective metal.
In the scanning tunneling microscope, one of the metals is the sample being imaged (sample) and the other metal is the probe (tip). The sample is usually flatter than the probe, as shown in Figure 2.6. If the probe is sharpened into a tip it will most likely have one atom at the end. All of the tunneling electrons will pass through this atom. As we will discuss later, this feature leads to the atomic resolution capabilities of the microscope.
continuously flowing from one metal to the other. It is then necessary to solve only the one-dimensional steady-state Schroedinger equation, given by:
( ) ( ) ( ) ( )
2 2
2m^2
x U x x E x x
h (1.5)
where E is the kinetic energy of the electron. Note that U(x) is the potential energy of the electron as a function of position, as shown in Figure 2.3. U(x) is smaller than the electron energy in the metals and larger than the electron energy in the barrier. For simplicity we can assume U(x) = U 0 a constant in the barrier.
In the metal, the general solution to the above equation is given by:
(Metal 1) (^) ( )
( 0 ) 2
ikx ikx (^) , m E U x Ae −^ Be +^ k
h
(Metal 2) Ψ ( x ) = Ee − ikx^^ + Fe + ikx (1.7)
and in the barrier (the classically forbidden region) the solution is:
(barrier) ( )
( 0 ) 2
x x (^) , m U E x Ce − μ^^ De +^ μ^ μ
h
Equations 1.6 and 1.7 show that the phase of the electron wavefunction varies uniformly in the metals. The wavelength is λ = 1/k. Higher energy electrons have a smaller wavelength. When a high energy electron wave encounters the boundary of the metal, it "leaks out" a small amount, as discussed in the previous section. The "intensity" of the electron wave decays as a function of distance from the boundary. Mathematically, the argument of the exponential function becomes real and the electron wavefunction decays. (For imaginary arguments, the wave function would have oscillatory behavior.)
To gain a quantitative insight into the electron tunneling phenomena, it is necessary to derive an expression for the transmission coefficient, i.e. the transmitted flux from the sample to the tip through the barrier of width L. The barrier is considered wide but finite, such that the electron wavefunction exponential decay in the barrier is significant. Furthermore, the electron wavefunction and its first derivative must be continuous (join smoothly) at the sample-barrier and tip-barrier boundaries to conserve energy and mass. If we set up a coordinate system in which the surface of the sample (Metal 1) is at x = 0 and the tip (Metal 2) is at x = L, and apply the boundary conditions for continuity:
( )
A B C
ik A B μC
(at the sample surface, x = 0) where D, the amplitude of the reflected wavefunction at the tip-barrier boundary, is neglected, since D << A, B, C. However, D is not insignificant at the tip-barrier boundary. At the tip-barrier boundary, x = L, continuity would require:
L L ikL L L ikL
Ce De Fe Ce De ikFe
μ μ
μ μ^ μ μ
− −
The probability of finding an electron in the barrier region at x = 0, due to quantum tunneling, is given by:
2 2 2
C^ k A (^) k
δ δ
To find the effective tunneling transmission coefficient,
2 F A
i.e. the relative
probability or frequency of occurrence of an electron tunneling out of the sample surface, across the sample-tip-barrier region, and into the tip, combine the tip-barrier boundary equations (at x L) and Equation 1.16 to get:
( ) 2
F ik (^) e L ik A (^) ik
δ μ δ
≈ −^ −^ +
which produces the desired quantitative result:
( (^ ) )
2 (^2 ) 2 / 2
L T E F^ k e e L^ m^ h A (^) k
δ δ δ
− Φ
≈ ≈ ^ ∝
where:
2 2 2 0
2 mE k
E k δ E U E
h
Substituting typical numbers of Φ = 5x10-19joules, m = 9.11x10-31kilograms, and
h = 1.05x10^ -34joule-seconds, results in: T E ( ) = e −^2 L , L in Angstroms.
This formula shows that for each angstrom change in separation, the probability that an electron tunnels decreases by an order of magnitude. This demonstrates mathematically that tunneling current is indeed a sensitive measure of the distance between the tip and sample.
In the STM, one of the metals is the sample being looked at and the other metal is the probe. The sample is usually flatter than the probe, as shown in Figure 2.6. Because the probe is formed of atoms, if it is sharpened into a tip, it will most likely have one atom at the end of the tip. The spacing between atoms is about 3 angstroms. Therefore, any tunneling through atoms that are one atom back from the closest atom is a fraction
e −^ (^2 )( )^3 =0.002 of tunneling through the atom at the tip, as shown in Figure 2.6. Virtually all of the tunneling electrons will pass through the single atom closest to the surface. This feature produces the atomic resolution capabilities of the microscope.
Electronics Preparation
- Set the Bias Voltage to about 1 volt.
- Set the Reference Current to 8 nA.
- Set the Servo Loop Response for constant current mode of operation. Set the Gain Close to maximum. Set the Filter close to maximum. Set the Time Constant to minimum.
- Set the magnification to Xl.
- Set the X and Y offset slides at their middle range.
- Press the Tunneling Current button to monitor tunneling current (it should read about zero).
Tunneling
- Press the Coarse Retract button momentarily to reset the motor controls.
- Press the Auto Approach (Tunneling) button for approach and wait.
- Monitor the tunneling current until it reaches about 8 nA (equal to the reference Current). If the tunneling current oscillates, reduce the Gain and Filter or increase the Time Constant to stop the oscillation.
- Once tunneling is achieved, start a unidirectional scan (Collect/ Scan Unidirectional).
- Collect images and save one at this range.
NOTE: Press C key on your keyboard to collect an image. Pressing C during a scan will capture the image at the end of the scan. Then you must use either Save or Save as command to save the image into the hard disk. To cancel image capture at the end of a scan, press C key again.
- Change the scanner range by turning the magnification dial. Set the software size correctly in the menu by setting the Magnification knob (Configuration menu ). Collect an image at each setting.
The large scan range should reveal a sinusoidal pattern on the surface with a period of 4167 Å and height variations of 1000 ± 100 Å, as shown in Figure 4.10. As you zoom in, the details of the gold crystallization process on the hologram should become apparent. The evaporated gold tends to rapidly diffuse to form random crystallites with grain sizes of approximately 60 to 100 Å. You can prepare a montage of all the collected scans (from
the Windows display mode) to visualize an overall factor of magnification of : 108 at the highest magnification (such as in the images grating1.img through grating4.img).
Load a representative image (File/Load). Use the cursor and draw a line through the data to visualize the fluctuations in tunneling current (Display/Cross Section). The current should be fairly constant across the surface, with variations of less than 0.1 Å in the tip position, to maintain constant current. The noise on your data may exceed this value, so you may want to filter the data-to reduce the noise level in the image. The main point is that the low degree of current variation across the surface illustrates the highly delocalized nature of electrons in metals. This study should be contrasted to that of graphite in the next section.
Questions
- Sometime during the use of the STM, the tip may have “crashed.” This is observable as a sudden large change in the current. This occurs when there is a change in the surface topology to which the feedback loop does not respond quickly enough and the tip touches the surface. Calculate the effective resistance of the tunneling gap for the conditions used in your experiment and compare that to the expected resistance if the tip was in direct ohmic contact with the surface. The resistivity of gold is 10 −^8 Ω /cm. This comparison should graphically illustrate the tunneling effect you are observing.
- As the temperature of metals is raised, the resistance to current flow increases. Discuss the mechanism of resistance in metals and compare this mechanism to electron tunneling. How would the temperature dependence of the two mechanisms differ?
- Because these experiments were conducted in air, adsorbed water, solvents, and gases are undoubtedly on the surface. How do these molecules affect the tunneling process and how might the tip perturb their distribution? Contaminants on the tip are also likely problems. Explain how this would affect the noise on your STM experiment.
- Consider the problem of the electron source in these experiments. If the tip is not scanned but left stationary over the surface, at a fixed distance that corresponds to a tunneling current of 1 nA, calculate the number of electrons/second that flow through the atoms that participate in the tunneling process between the gold and tip surfaces.
3 Imaging Graphite
This lab should provide direct observation of atomic features of the graphite surface. It should also serve to contrast the difference in the spatial variation of electron density at the Fermi level between metals and semimetals. This reflects the nature of and the degree of overlap in the atomic orbitals involved.
Procedure:
Head Preparation
- Prepare either a PtIr or W tip and mount the tip.
- Select the HOPG (graphite) from the sample set and mount the sample.
- Turn the Sample Position dial until the sample range indicator is close to the middle of the range or the sample-tip spacing is less than 0.5 mm. Be careful not to damage the tip and the sample. Make sure an optically flat portion of the sample is under the tip.
Software Preparation
- Load graph1.img (File/Load).
- Set the Scan Delay (Configuration menu ) to 0.0 mS/Sample.
- In this scan you are monitoring current variations, so set the Data Type to Current (Configuration menu).
