Low Energy Electron Diffraction (LEED)

Diffraction

The crystal structure can be studied using photon and electron diffraction [98]. When radiation (with wave length close to atomic spacing) strikes a crystal surface, a set of well defined diffracted beams result arranged in a characteristic geometry to form a diffraction pattern. This pattern carries information about the atomic arrangement in the crystal [99]. The incident beam is diffracted from atomic planes in the crystal. Fig.3.4 depicts Bragg’s law geometry for diffraction of X-ray from crystal planes which separated bydhkl.

A diffraction results from constructive or destructive wave interference. Construc- tive interference yields a diffraction pattern which occurs when the path difference between diffracted beams from neighbouring planes is equal to an integral multiple of the wave length (λ). This is known as Bragg’s law which is given by; [100, 101]

2dhklsinθ=nλ (3.8)

Figure 3.4: Schematic illustration of Bragg’s law geometry

incident beam and crystal plane and λis the wave length.

An attractive description for diffraction is the Laue formalism. Laue’s condition of diffraction (momentum conservation law) is given by; [102]

K−Ko=Ghkl (3.9)

where Ko is the incident wave vector, K is the scattered wave vector andGhkl is the

reciprocal lattice vector and it is defined by; [102]

Ghkl=ha∗+kb∗+lc∗ (3.10)

wherehkl are Miller indices anda∗,b∗ and c∗ are primitive translation vectors in the reciprocal space. They are given by; [102]

a∗ = 2π b×c a·(b×c),b ∗ _{= 2π} c×a a·(b×c),c ∗ _{= 2π} a×b a·(b×c) (3.11) Since only elastic scattering is considered, the energy conservation law can be writ- ten as;

|K|=|Ko| (3.12)

In the case of diffraction on a 2D lattice in which the periodicity in the normal direction is missing, the diffraction condition can be rearranged as follows; [92]

wherek represents the components parallel to the surface andGhk is defined by: [95] Ghk=ha∗+kb∗ (3.14) and a∗ = 2πb×n A ,b ∗ = 2πn×a A ,A=a.b×n (3.15)

wheren is the unit vector perpendicular to the surface [95].

Ewald Sphere

Using the Ewald sphere construction, Laue’s law of diffraction can be represented graph- ically. One can construct the Ewald sphere as follows; first of all, build the reciprocal lattice of the crystal. Secondly drawKo with origin chosen in the way thatKo termi-

nates at a reciprocal lattice point. Then draw a sphere of radius (K = 2π/λ) such that its centre is at the origin of Ko. Finally, find all reciprocal lattice points on the surface

of the sphere, then draw the scattered wave vectorKto these points. Fig.3.5(a) shows the construction of the Ewald sphere.

Figure 3.5: Schematic illustration Ewald sphere construction for diffraction from (a) a 3D lattice (b) a 2D lattice.

In the LEED case, the 2D reciprocal lattice points are represented by reciprocal lattice rods which are perpendicular to the surface. These rods are formed as the periodicity is infinite in the normal direction (|c| → ∞ thus |c∗| → 0) in a 3D lattice which makes the density of reciprocal lattice points infinite along the normal direction. In this case the scattered wave-vector of the diffracted beams can be defined as the points where reciprocal rods intersect the Ewald sphere [92]. The Ewald construction

for diffraction on a 2D surface is shown in Fig.3.5(b). Both Fig.3.5(a) and (b) satisfy equation (3.10) and (3.14), respectively.

LEED Experiment

Electron diffraction was first discovered by Davisson and Germer in 1927 [103]. Since the discovery, LEED has became a principal technique in surface science to determine the surface structure of solids. LEED can be used either qualitatively, in which the analysis of spot positions yields information about the symmetry and size of the unit cell, orquantitatively, in this case accurate information about position of atoms within the unit cell can be provided by recording the intensity of diffracted beams as a function of incident electron beam energy (I-V). LEED has been used qualitatively for data collection in this thesis.

Figure 3.6: Schematic illustration of low energy electron diffraction (LEED)

A schematic illustration of a typical LEED apparatus is shown in the Fig.3.6. It consists of an electron gun to produce the incident electron beam, a sample (single crystal), a series of grids to filter the diffracted electron beams, a fluorescent screen to observe the diffraction pattern and finally an external detector to record the pattern.

The electron gun fires an electron beam with a specific energy ( typically 20 - 200 eV) and wave-length. These electrons are accelerated towards the surface of the sample. Once they strike the surface they are elastically and inelastically back-scattered towards

the grids and the screen. Elastically scattered means they do not lose energy whereas inelastically scattered means they lose energy. The scattered electrons then are filtered by grids. The first grid G1 is grounded to provide the necessary field-free region. A retarding voltage is applied to G2 to suppress the inelasticity scattered electrons. Finally, the elastically scattered electrons are accelerated by G3 towards the screen. These electrons fluoresce the phosphor on the screen to produce the diffraction pattern [60].