Surfaces and Contact Mechanics, Part One, Page 8

11. Scanning Tunneling Microscopy (STM) [5]

Last time, we saw both the utility and the limitations of LEED. Now we'll begin looking at techniques that have been developed and have become widely used within the last fifteen years. These are the scanning probe microscopies (SPM) that originated with the device developed by Bining and Rohrer around 1980, the scanning tunneling microscope (STM). Figure 38 shows a diagram of an STM system. A small, precisely machined metal tip is postioned within 1 nm of the surface under study. A voltage is applied between the sample and the tip and a small current flows from the tip to the sample via a quantum mechanical process called electron tunneling (described in greater detail below). Because the magnitude of the tunneling current is very sensitive to the separation deistance between the tip and the surface, that distance can be determined within 0.01 nm. The piezodrives, P_x and P_y are used to scan the tip over the surface and are directed by the control unit (CU). The CU applies the necessary voltage V_P to P_z maintain a constant tunnel current J_T with a bias voltage V_T. The dashed line in the figure indicates the displacement in the z direction as the tip is scanned from left to right, first over a step at A and then across a region with a differing work function.

Figure 38. Schematic of an STM in Operation [Figure 1.2 in Ref. 5].

Principles of Electron Tunneling

When the tip and the surface are more widely separated, no current occurs. Figure 39 schematically illustrates the electron energy levels within the two electrodes. In part (a), two non-interacting metal electrodes are separated by vacuum. There is a difference in the Fermi levels of the two electrodes, E_F^L and E_F^R, which is also equal to the difference in their work functions. In part (b), the two electrodes are allowed to electrically equiliibrate and the two electrodes have a common Fermi energy. In part (c), a bias voltage is applied producing an eV energy difference in the Fermi levels.

Figure 39. Schematic Representation of a Potential Barrier between Two Electrodes [Figure 1.1 in Ref. 5].

If one considers an electron within a rectagular barrier in one dimension, a relatively simple quantum mechanical calculation can be performed to determine the propbability of transmission across the potential barrier (tunneling) into an available energy level in the other electrode. The solutions of the Scrodinger equation for an electron within the rectangular barrier have the form,

(33)

y = Fe^-kx + Ge^+kx

where the parameter k is

(34)

In Equation 34, E is the energy of the electron state and V_B is the potential within the barrier. [Note: the potential energy is correctly written eV_B, but so in this section, wherever you see V, it means eV. This is done to avoid the problem that arise when taking about the energy unit electron volt, eV.] The potential will vary across the gap, but let's consider V_B its average value. In the simplest case, V_B is the vacuum level, so for electron states at the Fermi level, the difference (V_B - E) is the work function. The transmission current is directly related to the probability for tunneling which decays exponentially with d (see Figure 40), the width of the potential barrier as e^-2kd.

Figure 40. The probability density function y*y for a typical barrier penetration situation is shown. [Figure 6-14 in Ref. 6].

For tunneling between two metals with a voltage difference V (as in part (c) of Figure 39), only electrons in energy states that are within V above or below the Fermi energy on the negative side can tunnel into empty states within V above the Fermi level on the positive side. Other states cannot contribute since there are no electrons to tunnel at higher energy or because of the Pauli exclusion principle at lower energies.

Work functions are typically in the range of 4 to 5 eV. From Equation 34, this would indicate that 2k ~ 2 (angstrom)^-1. Therefore, for every one angstrom that the separation distance increases, the tunneling current decreases by almost an order of magnitude. These kinds of tunneling currents can only be observed when d is very small. The important advance in applying electron tunneling to studying surfaces comes when one can control the tip-surface distance precisely in an environment where vibration has been minimized to an amplitude of much less than an angstrom. Prior to Binig and Rohrer's work, electron tunneling had only been studied using a fixed barrier of an oxide (insulating) film sandwiched between two metal layers. They were able to show the expected exponential behavior in their apparatus, similar to Figure 38 above.

There are a few different modes that the STM system may be run. The first is just scanning as described before where a feedback circuit is used to maintain the tip at some constant distance above the surface by maintaining the tunneling current constant. The path of the tip (determined by the voltage applied to the piezoelectric drivers) reproduces the shape of the surface. Making a series of parallel traces across the surface results in a topographical representation of the surface. The resolution is determined by the tip radius. The resolution can be determined as follows. At a given x position, relative to the center of the tip, the height of the corresponding point on the tip is d + x²/2R, assuming a parabolic tip shape with a radius of curvature R. The distance of closest approach is d. The tunneling current is proportional to exp(-kx²/R). The current, thus, has a Gaussian profile with an rms width of ~0.7(R/x)^1/2. Since k is about one inverse angstrom, even a large tip with R = 1000 angstroms has a resolution of about 50 angstroms. The smallest tips have radii on the order of a few hundred angstroms. The tunneling current is actually determined by the atomic-scale asperities present on the tip closest to the surface. The best STM images result from tunneling to a single atom or a few atoms at the tip.

Another mode is one that reduces the feedback response time so that the tip maintains an average distance above the surface. Small scale surface features then appear as deviations in the tunneling current. This mode is really only useful in cases where the surface is extremely smooth.

A third STM mode is one in which the tip height is modulated at a frequency slightly above that of the feedback response. By measuring the current modulation, one obtains a local value of d(ln I)/dz, and hence k, or equivalently what is called the "effective work function,"

(35)

across the entire surface. The resulting surface image represents variations in the work function and, thus, composition rather than topographical information.

Interpretation of the STM Image
As long of the surface features being studied are on the nanometer scale or larger, interpretation of an STM image as a surface topography is usually acceptable by complicated by local variations in barrier height. Soon after the STM was invented by Binig et al., they reported the ability to resolve individual atoms. A simplified view of an atomic-level topograph would be that it is a surface of constant surface charge density,; however, there is no reason why the STM image would yield precisely the contour of constant charge density. This is because the tunneling is mainly due to electrons near the Fermi energy, but all electrons below the Fermi level contribute to the charge density. One can calculate directly the transmission coefficient for an electron incident on the vacuum barrier between the surface and the tip, but the calculation becomes somewhat more difficult when you are dealing with real surfaces. Fortunately the coupling between the tip and surface is weak; and first order perturbation theory is adequate to treat the tunneling.

As with other techniques, one would like to examine STM images quantitatively by comparing a calculated image for a proposed structure or set of structures with a measured image. These types of calculations are difficult and rare. For simple metals and probably noble and transition metals, there is typically no strong variation of the local density of states (LDOS) or wavefunctions of energy near the Fermi level. Figure 41 shows one example of a a model STM LDOS.

Figure 41. Calculated LDOS for Au(110)2x1 (left) and 3x1 (right surfaces) The figure shows the (1-10) plane through outermost atoms. Positions of nuclei are indicated by circles (in plane) and squares (out of plane). Constant density contours are labeled in units of a.u.^-1eV^-1. There is a break in the vertical scale. The center of curvature of the tip is calculated to follow the dashed line. [Figure 1.3 in Ref. 5].

An approximation method for calculating the LDOS for complex surfaces was suggested by J. Tersoff and D. R. Hamann in 1985 [Phys. Rev. B 31,805]. Figure 41 is from that paper. The method uses a superposition of spherical atomic-like densities centered on an atom site. The success of the method relies on the model density having the same analytical properties of the true density for a constant potential. If the model is accurate near the surface, it automatically describes the decay of the tunneling current with distance. The image calculated for Au(110)3x1 was calculated for two possible structural models differing only by the presence or absence of a missing row in the second layer. The results for the two models were very similar indicating that the STM image could not reliably infer the structure of the second layer. Quantifying the limits of valid interpretation is an essential part of STM analysis.

Figure 42 shows line STM scans over a cleaved SI(111) surface one which gold has been deposited. Both the direct topographical measurement and the derivative measurement d(ln I)/d(d) are shown. Again, the latter reveals information about the average work function. By comparing these two measurements, it's clear that the features labeled A and B are gold islands on the silicon surface.

Figure 42. STM data for gold deposited on Si(111). [Figure VI.3 in Ref. 3].

Figure 43 shows a classic example of an STM surface structure analysis, that of Si(111)7x7 surface. In the image, several complete unit cells are present as seen by their deep corner minima. This image was used to successfully determine the surface structure out of many competing models. The precise structural details can not be determined from this image alone, however. A combination of techniques was necessary, with STM providing the final confirmation. Figure 44 shows the structure of Si(111)7x7. The STM results demonstrated that the two halves of the unit mesh are not completely equivalent because of the slight different heights of the minima and maxima.

Figure 43. STM image of the Si(111)7x7 reconstructed surface. [Figure VI.4 in Ref. 3].

Figure 44. Si(111)7x7 Reconstructed Surface Model. Part (a) is a top view showing atoms in the (111) layer at increaing depth are indicated by circles of decreasing sizes. The heavy circles represent 12 adatoms. The circles labeled A and B are rest atoms in the faulted and unfaulted half of the unit cell, respectively. Part (b) shows atoms in the lattice plane along the long diagonal of the surface unit cell are shown with larger circles than those behind them. [Figure 6.37 in Ref. 3].

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