Surfaces and Contact Mechanics, Part One, Page 7

10. Low Energy Electron Diffraction (LEED)[2]

Electron Diffraction was discovered by Davisson and Germer back in 1927. Since that time low energy electron diffraction has developed into a standard laboratory technique that is found in nearly all surface science laboratories.Surface crystal structure can be determined by bombarding the surface with a 1-mm wide low energy (approx. 10-200 eV) electron beam and observing diffracted electrons as spots on a phosphorescent screen (see Figure 32). A retarding field analyser (RFA) is used to detect the electrons. The RFA consists of a series of concentric hemispherical grids. A beam of electrons strikes the sample and some electrons are re-emitted by the sample and travel towards the grids. The first grid is usually grounded. The next grid has a voltage -V applied, so that any electrons higher in energy than eV can pass through the grid. Electrons lower than this energy are reflected back to the sample or the first (grounded) grid. There are usually more grids, but let's not go into that here. The higher energy electrons are detected and form the signal from the RFA. The detector is usually a phosphor screen and thus light will be emitted wherever the electrons strike the screen. Using this method LEED patterns can be observed. The RFA has a much larger solid angle (up to 2p steradians) over which electrons may be detected than a CMA. The relative position of the spots on the screen shows the surface crystallographic structure. The diffracted spots will move as the energy of the incident electrons changes, and the intensity of the spots as a function of incident electron energy reveals much about surface reconstructions. To determine the true surface structure, computerized analysis is done.

Figure 32. A modern, computerized LEED System [from http://www.vsigmbh.com/APC.html].

Let's take a look at the theory behind the electron diffraction that will show us how the surface structure is determined. The electron incident on the surface can be represented by a wavevector k₀ that has an amplitude expressed in (angstroms)^-1

(24)

where V is the voltage. This electron interacts with a two-dimensional lattice that presumably has a unit cell defined by lattice vectors a and b. Part (a) of Figure 33 shows two such unit cells for FCC structures. To analyze the interaction between the electron and the lattice, it's convenient to switch to reciprocal space. In three dimensions, the reciprocal lattice vectors are defined relative to their real-space counterparts,

(25)

Figure 33. Part (a) illustrates ball models of two FCC lattices, (100) on the left and (111) on the right. Part (b) shows the Ewald construction for diffraction from a three-dimensional structure.

The criteria for constructive interference for the elastically scattered electrons is that the change in the electron wavevector must be equal to a reciprocal lattice vector given by,

(26)

The magnitude of the wavevector is unchanged because the scattering process is elastic; and only the direction is changed. Part (b) of Figure 33 graphically shows the scattering process using the Ewald sphere construction. The construction is made by placing a sphere of radius k₀ into the 3D reciprocal lattice. The vector k₀ begins at the center of the sphere and ends on a lattice point. The condition of equation 26 will be satisfied whenever another lattice point lies on the sphere, k'. The vector G is a translation vector between those two reciprocal lattice points. This argument may be extended to the two dimensional case by allowing the c* go to zero. Rather than a three-dimensional array of points, the two-dimensional reciprocal space looks like a series of parallel lines such as shown in Figure 34. This occurs because unit translations along c* produce a solid line because c* these unit translations are infinitesimally small. Whenever a line intersects the Ewald sphere, diffraction can be observed.

Figure 34. The Ewald sphere construction for diffraction from a two-dimensional structure is shown.

In surface electron diffraction, both two- and three-dimensional effects are observed in the diffraction pattern because the surface isn't really two- dimensional. The diffraction from a given intersected line is always present in the pattern with its intensity varying periodically as the sphere radius (electron energy) is varied. As the radius of the sphere increases, more lines are intersected and more diffracted beams are present in the pattern.

From the primary electron energy, the geometry of the system, and the observed diffraction pattern such as those shown in Figure 35, one can determine the translation vector g; and in turn, determine the reciprocal lattice vectors for the observed spots in the pattern.

Figure 35. (a) Si(111), (b) GaAs(110), and Sr₂CuO₂Cl₂ [from http://nanophysics.phy.queensu.ca/leed.html]

The symmetry of the pattern is a reflection of the atomic arrangement of the surface. Each spot spacing relative to the origin is associated with a translation vector g = ha* + kb* and is a measure of the lattice spacing. In a pattern where the spots are far apart, the lattice spacing is small, and vice versa. The patterns also reveal the shifts in atom positions due to reconstruction occurring at the surface, but the spots do not give us direct information about atom positions in the vicinity of the surface. Only changes in symmetry are revealed. For example, if the first atomic layer was displaced relative to the layers below, there would be no effect on the resulting diffraction pattern due to the relaxation.

To determine the atomic positions, information about the intensity of a given spot versus the primary electron energy must be used as taken from plots of intensity-voltage curves. The process is very complicated due to multiple scatterings of the electrons. However, many surface structures have been worked out using such calculations, but research continues in this area to find a generalized procedure.

Surface structures are described in terms of the underlying bulk structure. The substrate lattice parallel to the surface is taken as a reference structure. The translation vector between lattice points in the substrate is

(26)

T = na + mb (for n, m = 0, 1, 2, 3...)

The translation vector between lattice points on the surface is

(27)

T_s = n'a_s + m'b_s (for n', m' = 0, 1, 2, 3...)

In many cases, the relationship between these two vectors is quite simple, such as

a_s = pa
b_s = qb

where q and p are integers (0, 1, 2, 3,...). For the general case, a notation is used to describe the surface that is in the form:

(28) M(hkl)p X q - A

where M is the chemical species making up the substrate, (hkl) are the Miller indicies, and A is the chemical species, if any, absorbed on the surface. The Si(111) pattern in Figure 35a above is properly written Si(111)7x7.

In more complicated situations when the surface and substrate vectors are not parallel to one another, the relationship between the surface and substrate lattice vectors may be written

(28)

a_s = p₁a + q₁b
b_s = p₂a + q₂b

Usually in these cases the angle between a_s and b_s is the same as the angle between a and b. The notation used to describe this kind of structure is

(29)

where a is the angle of rotation between the surface lattice and the substrate lattice. When oxygen atoms are adsorbed on a nickle (111) surface, one may write

(30)

Figure 36 shows two examples of LEED patterns. In part (a), a clean FCC (100) 1 x 1 surface was observed. In part (b), the same lattice is shown except that there is an adsorbed layer with a 2 x 1 structure.

Figure 36.

In the strict two-dimensional case, the reciprocal lattice vectors are written,

(31)

The spots are thus observed for surface reciprocal lattice vectors that satisfy the condition,

(32)

In this way, the observed spots may be indexed in terms of the Miller indicies h and k. Note again that the spot pattern doesn't indicate anything about the atom positions. They do indicate the symmetry. Therefore, in the case of part (b) of Figure 36, the same pattern would be observed if the atoms were adsorbed directly on the surface or in the interstitial locations within the first layer of the substrate. To determine which of these possibilities is correct, intensity-voltage curves must be examined as described before. Figure 37 shows some of these curves for the Cu(100) surface. In each case, a model is refined to closely match the experimental data. The curves show major peaks in intensity near the expected positions for diffraction from the 3D lattice. If the outer layers are relaxed relative to the bulk, then the positions and intensities will vary in a complex way from the 3D peaks. In the figure, curves A and B represent predictions of a theoretical model that assumes a changed interlayer spacing for the top two atomic layers.

Figure 37.

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