Low-energy electron diffraction

LEED is a technique that probes the long range 2D order of crystalline solid surfaces with intrinsically high surface specificity. The first example of LEED was shown in 1927 by C. Davisson and L. H. Germer in an experiment that also demonstrated the wavelike behaviour of electrons for the first time.76 _{The experiment consisted of a monoenergetic beam of electrons aimed at a single Ni crystal} surface from which it was observed that the electrons were elastically scattered in preferred directions relating to the periodic arrangement of the Ni atoms within the crystal structure. However, it wasn’t until the 1960s with the development of modern UHV equipment and optics capable of displaying the diffraction patterns in real time that LEED began to see widespread use in surface science. In this section, the basic theory underpinning LEED will be introduced and discussed. A more complete overview of the principles can be found elsewhere.19, 36, 77

In a LEED experiment, an electron beam with a well-defined kinetic energy, typically in the energy range of 10-300 eV is focussed onto a sample. Electrons in this energy range have a de Broglie wavelength of the order of ~1 Å and thus are ideal for diffracting from a crystalline solid. Using electrons in this energy range also provides high surface specificity, isolating structural information of the surface from that of the bulk. The surface specificity is brought about by the short IMFP of electrons at these energies (see Figure 2.2.1), which in an inorganic solid is of the order of 1 nm. This means that elastically scattered electrons, which can be isolated from the inelastic background using a retarding field, are only likely to come from the top few atomic layers of the surface. Furthermore, the high backscattering cross-section of electrons means that a large proportion of the incident electron flux will be scattered back out of the crystal and will not reach layers deeper into the bulk. If the sample surface contains regions with long-range order, the incident electrons will be back-scattered, producing a pattern that is a projection of the reciprocal mesh, which is inversely proportional to the real space unit

therefore be extracted from the LEED pattern.

Figure 2.2.1 – Universal IMFP curve of electrons in condensed matter as a function of electron kinetic energy based on equations proposed by Seah and Dench.78

To describe the structural information obtainable by LEED, it is important to define surface structure notation. For adsorbate layers, it is usual practice to express the surface structure as a relationship between the adsorbate lattice vectors (b1 and b2) and the substrate lattice vectors (a1 and a2). One

example is Wood’s notation, which gives a simple description of the surface structure but is limited to adsorption structures that are commensurate to the underlying substrate with the angle between b1 and b2 the same as that between a1 and a2. Wood’s notation expresses the surface structure by the ratio

between the adsorbate and substrate lattice vector lengths along with the azimuthal angle (θ) between them in the format of (|𝒃_|𝒂𝟏|

𝟏|×

|𝒃𝟐|

|𝒂𝟐|) 𝑅𝜃. More complicated overlayer structures can be expressed as a

transformation matrix of the substrate lattice vectors taking the form of:

(𝒃_𝒃

) = (

∗ ∗

∗ ∗) (

𝒂

𝒂

)

[2.2.1]

Examples of these two types of notation are shown in Figure 2.2.2. 101

Electron kinetic energy / eV

IMFP / n m 102 ₁₀3 ₁₀4 100 101

Figure 2.2.2 – Example surface structures for adsorbate atoms shown on an Ag(100) surface. The grey circles represent Ag atoms and the red circles correspond to adsorbate atoms. The Ag substrate unit mesh (cyan square) is indicated in the left-most panel and the adsorbate unit mesh (purple square) is shown for each of the adsorption structures. Below each panel, the Wood’s notation and matrix notation of each adsorption structure is given. Note that because the internal angle of the unit mesh for the right-most structure does not equal that of the substrate (i.e. 90°), it cannot be described using Wood’s notation.

Figure 2.2.3 – Comparison of a LEED pattern with its corresponding real-space surface unit mesh. a) Example LEED pattern of the (2x2) phase formed by Cs adsorbed on Ag(111), recorded with an electron kinetic energy of 135 eV. b) The experimental LEED pattern with the beam positions for the Ag(111) substrate (cyan) and the (2x2) Cs structure (purple) overlaid. c) A real space depiction of the (2x2) Cs-Ag(111) adsorption phase with the unit meshes for the Ag(111) substrate (cyan) and Cs overlayer (purple) highlighted.

an Ag(111) substrate. Note that in real space structure, the (2x2) Cs overlayer unit mesh vectors are twice the length of the Ag(111) substrate unit mesh vectors, however in the LEED pattern (reciprocal space), the Cs overlayer beams are separated by half the distance of the substrate beams.

LEED patterns comprise a series of diffracted beams that can be explained by conservation of energy and momentum with the recoil momentum of the surface given in units of the reciprocal lattice vectors. For an electron that is elastically scattered, the magnitude of the incident beam wavevector k must equal that of the diffracted beam k’:

𝑘

_{= 𝑘′}

_[2.2.2]

For diffraction from a surface, which has two-dimensional periodicity, only the component of the wavevector that is parallel to the surface is conserved. With ∥ and ⊥ denoting the components parallel and perpendicular to the surface respectively, this gives:

𝑘

+ 𝑘

= 𝑘′

+ 𝑘′

[2.2.3]

and conservation of momentum gives:

𝒌

= 𝒌

+ 𝒈

[2.2.4]

where ghkis a reciprocal mesh vector,

𝒈

= ℎ𝒂

+ 𝑘𝒃

[2.2.5]

where h and k are a set of integers called Miller indices and a* and b* are reciprocal lattice vectors related to the real surface lattice vectors a and b by:

𝒂

_{= 2𝜋}𝒃 × 𝒏

𝐴

,

𝒃

= 2𝜋

𝒏 × 𝒂

where n is a unit vector normal to the surface. The diffraction conditions are therefore satisfied for points on the reciprocal surface mesh. These equations can also be represented graphically with the construction of the Ewald sphere which is depicted in Figure 2.2.4. For scattering from a surface, the Ewald sphere is superimposed on a set of reciprocal lattice rods which extend infinitely, perpendicular to the surface and pass through the points of the reciprocal mesh. The Ewald sphere is constructed by drawing a wavevector, k, which terminates at the origin of the reciprocal mesh and by building sphere of radius k about the beginning of vector k. Each point on the sphere that intercepts one of the reciprocal lattice rods corresponds to a diffracted beam k’. Note that there are two diffracted beams for each point of the reciprocal mesh, however half of these (shown as dashed arrows in Figure 2.2.4) are directed into the crystal and are therefore not observed.

Figure 2.2.4 – Cross-sectional view of the Ewald sphere construction for a periodic surface. The vertical lines represent the infinite reciprocal rods that extend to infinity perpendicular to the surface, passing through the points of the reciprocal mesh. The wavevector k (red arrow) is labelled and the possible scattered wavevectors (k’) are shown as black arrows. An example reciprocal mesh vector g20 is also shown (blue arrow). The dashed arrows represent wavevectors that propagate into the solid bulk and are not observable by LEED.

ℎ

1

2

3

1 2 3

𝒌

𝒈

Figure 2.2.5 – Schematic diagram of LEED optics. The various components and their function are described in the main text.

Figure 2.2.5 depicts the various components within a set of rear-view LEED optics. The electron gun produces a monoenergetic beam of electrons of ~1 µA, which is electrostatically focussed onto a nearby sample. The LEED optics also include a series of hemispherical grids which are used to prevent inelastically scattered electrons from reaching the phosphor screen, which, due to their varying wavelengths, would destructively interfere with the diffraction pattern. The first of these grids is held at the same ground potential as the sample and is used to separate the sample from the fields produced by the other grids, allowing the electrons to travel in straight lines in a field-free region. The second grid is held at a negative potential, slightly lower than is used to accelerate the emitted electrons, and thus acts as a high-pass filter that allows only the elastically scattered electrons through. Upon passing through this retarding grid, the elastically scattered electrons are then accelerated, via the application of high voltage, onto the phosphor screen to produce the diffraction pattern image. In some LEED optics, the detector also includes a microchannel plate (MCP), which multiplies the electron signal, allowing diffraction patterns to be observed with electron beams of ~1nA. This low beam current is essential for use on organic adsorbate systems (such as those studied in this work) which are highly susceptible to damage from the electron beam.79

Phosphor screen

Electron gun

Sample

Grids

e

An additional concept that is important to consider when conducting LEED experiments is the transfer width of the instrument. The transfer width is effectively the coherence length of the electrons with the angular divergence of the electron beam usually being the limiting factor in determining its size, which for typical LEED instruments, is of the order of hundreds of Angstroms.77, 80 _{In order to obtain a} diffraction pattern from an ordered structure, its periodicity must be smaller than the transfer width, 77, 80 _{which considering that typical unit meshes, even for relatively large organic molecules, are of the} order of tens of Angstroms,38 _{is seldom an issue. However, the average sizes of ordered domains with} respect to the transfer width can often have implications on the observed LEED pattern. 77, 80 _{If the} average domains are larger than the transfer width, interference across the domain boundaries will not be important and the resulting LEED pattern will correspond to the sum of the intensities (not the amplitudes) of the individual patterns for each domain. 77, 80 _{If the domains are on average smaller than} the transfer width, then the effects of coherent interference between domains will be significant and will affect the LEED pattern observed. 77, 80