Low-energy electron diffraction

[2][3] The theoretical possibility of the occurrence of electron diffraction first emerged in 1924, when Louis de Broglie introduced wave mechanics and proposed the wavelike nature of all particles.

In his Nobel-laureated work de Broglie postulated that the wavelength of a particle with linear momentum p is given by h/p, where h is the Planck constant.

The main reasons were that monitoring directions and intensities of diffracted beams was a difficult experimental process due to inadequate vacuum techniques and slow detection methods such as a Faraday cup.

Nonetheless, H. E. Farnsworth and coworkers at Brown University pioneered the use of LEED as a method for characterizing the absorption of gases onto clean metal surfaces and the associated regular adsorption phases, starting shortly after the Davisson and Germer discovery into the 1970s.

In the early 1960s LEED experienced a renaissance, as ultra-high vacuum became widely available, and the post acceleration detection method was introduced by Germer and his coworkers at Bell Labs using a flat phosphor screen.

[7] In the mid-1960s, modern LEED systems became commercially available as part of the ultra-high-vacuum instrumentation suite by Varian Associates and triggered an enormous boost of activities in surface science.

Notably, future Nobel prize winner Gerhard Ertl started his studies of surface chemistry and catalysis on such a Varian system.

[8] It soon became clear that the kinematic (single-scattering) theory, which had been successfully used to explain X-ray diffraction experiments, was inadequate for the quantitative interpretation of experimental data obtained from LEED.

In order to keep the studied sample clean and free from unwanted adsorbates, LEED experiments are performed in an ultra-high vacuum environment (residual gas pressure <10−7 Pa).

The main components of a LEED instrument are:[2] The sample of the desired surface crystallographic orientation is initially cut and prepared outside the vacuum chamber.

Unwanted surface contaminants are removed by ion sputtering or by chemical processes such as oxidation and reduction cycles.

Often the annealing process will let bulk impurities diffuse to the surface and therefore give rise to a re-contamination after each cleaning cycle.

The problem is that impurities that adsorb without changing the basic symmetry of the surface, cannot easily be identified in the diffraction pattern.

The first derivative is very large due to the residual capacitive coupling between gate and the anode and may degrade the performance of the circuit.

More expensive instruments have in-vacuum position sensitive electron detectors that measure the current directly, which helps in the quantitative I–V analysis of the diffraction spots.

Upon penetrating the crystal, primary electrons will lose kinetic energy due to inelastic scattering processes such as plasmon and phonon excitations, as well as electron–electron interactions.

In cases where the detailed nature of the inelastic processes is unimportant, they are commonly treated by assuming an exponential decay of the primary electron-beam intensity I0 in the direction of propagation: Here d is the penetration depth, and

denotes the inelastic mean free path, defined as the distance an electron can travel before its intensity has decreased by the factor 1/e.

While the inelastic scattering processes and consequently the electronic mean free path depend on the energy, it is relatively independent of the material.

The mean free path turns out to be minimal (5–10 Å) in the energy range of low-energy electrons (20–200 eV).

Since the mean free path of low-energy electrons in a crystal is only a few angstroms, only the first few atomic layers contribute to the diffraction.

By construction, every wave vector centered at the origin and terminating at an intersection between a rod and the sphere will then satisfy the 2D Laue condition and thus represent an allowed diffracted beam.

Figure 7 shows a model of an unreconstructed (100) face of a simple cubic crystal and the expected LEED pattern.

Figure 6 shows many such spots appearing after a simple hexagonal surface of a metal has been covered with a layer of graphene.

Figure 8 shows the superposition of the diffraction patterns for the two orthogonal domains (2×1) and (1×2) on a square lattice, i.e. for the case where one structure is just rotated by 90° with respect to the other.

The I–V curves can be recorded by using a camera connected to computer controlled data handling or by direct measurement with a movable Faraday cup.

The term dynamical stems from the studies of X-ray diffraction and describes the situation where the response of the crystal to an incident wave is included self-consistently and multiple scattering can occur.

The aim of any dynamical LEED theory is to calculate the intensities of diffraction of an electron beam impinging on a surface as accurately as possible.

In LEED the exact atomic configuration of a surface is determined by a trial and error process where measured I–V curves are compared to computer-calculated spectra under the assumption of a model structure.

This departure from a perfect surface leads to a broadening of the diffraction spots and adds to the background intensity in the LEED pattern.

Figure 1 : LEED pattern of a Si(100) reconstructed surface. The underlying lattice is a square lattice, while the surface reconstruction has a 2×1 periodicity. As discussed in the text, the pattern shows that reconstruction exists in symmetrically equivalent domains oriented along different crystallographic axes. The diffraction spots are generated by acceleration of elastically scattered electrons onto a hemispherical fluorescent screen. Also seen is the electron gun that generates the primary electron beam; it covers up parts of the screen.
Figure 2 Schematic of a rear-view LEED instrument.
Figure 3 : Ewald's sphere construction for the case of diffraction from a 2D lattice. The intersections between Ewald's sphere and reciprocal lattice rods define the allowed diffracted beams. For clarity, only half of the sphere is shown.
Figure 4 : Ewald's sphere construction for the case of normal incidence of the primary electron beam. The diffracted beams are indexed according to the values of h and k .
Figure 5 : Real-space STM topographic map of palladium (111) surface, its Fourier transform in reciprocal space showing main periodicity components, and a 240 eV LEED image from the same surface
Figure 6 : Scanning tunneling microscopy (STM) map of iridium (111) surface partially covered with single-layer graphene (lower-left part). Because of a 10% lattice mismatch, graphene develops a 2.5 nm moiré superstructure . This can be seen in 69 eV LEED images as six new spots due to the intrinsic periodicity of the carbon honeycomb lattice, and many replicas due to the long-wavelength (small wave vector) supermodulation of the moiré.
Figure 7 : Real and reciprocal space lattices for a (100) face of a simple cubic lattice, and its two commensurate (1×2) superstructures. The green spots in the LEED pattern are the extra spots associated with the adsorbate structure.
Figure 8 : Superposition of the LEED patterns associated with the two orthogonal domains (1×2) and (2×1). The LEED pattern has a fourfold rotational symmetry.
Figure 9 : Examples of the comparison between experimental data and a theoretical calculation (an AlNiCo quasicrystal surface). Thanks to R. Diehl and N. Ferralis for providing the data.