Electron Diffraction

Diffraction is an interference effect which leads to the scattering of high energy beams of radiation in specific directions. Diffraction from crystals is described by the Bragg’s Law:

nλ= 2dsinθ (1.10)

wheren is an integer (the order of scattering),λis the wavelength of the radiation,d is the spacing between the scattering entities (e.g. planes of atoms in the crystal) andθis the angle of scattering.

When a beam of high energy electrons is used as the radiation source, the technique is called Electron Diffractometry (ED) and the resulting regular array of bright spots created by this interaction is termedElectron Diffraction Pattern (EDP). ED refers to a collective scattering phenomenon with electrons being (nearly elastically) scattered by periodic arrays of atoms as found in a crystal. The incoming plane electron wave interacts with the atoms, generating secondary waves which interfere with each other. This occurs either constructively (reinforcement at certain scattering angles generating diffracted beams), or destructively (vanishing of beams). Similar to X-Ray Diffraction (XRD), these scattering events at crystal lattice planes can be described by equation 1.10. In this interaction, the atoms of the crystal act as a filter for the incident electrons, causing them to scatter in certain angles predicted by Bragg’s Law. Each set of parallel lattice planes, generates a pair of spots in the EDP with the direct beam in their center. Since the wavelength λ of the electrons is known, interplanar distances can then be calculated from EDPs. Moreover, it is possible to deduce the symmetry of the crystal using ED-based analysis. These potential applications, make electron crystallography a powerful technique for “solving” crystal structures, both in physical and life sciences [72,73].

EDP studies are usually performed with high energy electrons in a TEM instrument. ED has some specific features which offer a number of advantages over other diffraction techniques such X-ray diffraction [57,59,74] (a) The extent of sampling is much larger using electrons than by X-ray. Therefore, the EDP contains several spots when compared to other techniques, providing more reflections for crystallographic analysis. (b) Electron optics in TEM are able to vary the geometry of the EDP by focusing and changing the camera length. This enables acquiring ED from very small crystals in the sample. This is carried out by using the parallel beam in standard TEM mode and inserting an aperture in the image plane of the objective lens, confining the diffraction pattern to a selected area of the specimen. Thus, this technique is called Selected Area Electron Diffraction (SAED). The SAED pattern that appears on the viewing screen originates

from the area selected in the image mode. SAED can be performed on the regions in the order of <0.5µm diameter, but spherical aberration of the objective lens limits the technique to regions not much smaller than this. Providentially, it is possible to focus the electron beam to a small diameter and perform ED work at nanometre range. This method is termed Convergent Beam Electron Diffraction (CBED). (c) Since ED studies are typically performed in a TEM, the imaging capability of the microscope can be employed while the EDP is recorded. This is particularly important in the field of low-dimensional materials science when only very small features within a sample are of interest. One can calculate the interplanar spacing in the crystal being studied using the separation of spots in the DP. For this purpose, we need to first revisit the Bragg’s equation. Since the ED angles are very small (0◦ <θ <2◦), therefore we have:

sin(θ)≈tan(θ)≈ 1

2tan(2θ) (1.11)

We can depict the geometry of electron diffraction as shown in Figure 1.11:

Figure 1.11 Schematic illustration of geometry of electron diffraction and the concept of camera length L.

From the geometry of Figure 1.11we can write:

tan(2θ) = r

L (1.12)

which we can then substitute this into equations (1.10) and (1.11) to obtain:

rd=λL (1.13)

wherer is the distance from the central spot, d is interplanar spacing and the product

λLis calledcamera constant, whose unit is usually ˚A.cm and can be found in the manual of any TEM instrument.

Electron Diffraction Study of Graphene

ED is a reliable and relatively straightforward way to assign the number of layers in graphene; but one might ask when we are to analyse a single atomic layer material such

as graphene, how diffraction could occur? For any diffraction condition the Bragg’s law should be satisfied which contains the parameter d, the inter-planar spacing, whereas graphene is only one layer. The answer lies in the fact that in case of a 2D crystal, the reciprocal lattice is formed by rods (see Figure1.12 (a)) instead of discrete points as in case of a 3D crystal. In other words, the condition that the scattering wavevector has to intersect the reciprocal lattice, only applies to the components parallel to the lattice which means in the graphene plane in our case.

Figure 1.12 Schematic representation of reciprocal space of (a) Monolayer and (b) Multilayer graphene.

As can be seen in Figure1.12(b), the rods are discontinuous due to the additional layers on top of perfect 2D monolayer graphene. To label equivalent Bragg reflections, Miller- Bravais indices [hkil] for graphite is used so that the innermost hexagon of diffraction spots corresponds to [1¯100] and the next one to [11¯20] indices. Figure 1.13 depicts the above-mentioned directions and planes in the graphite lattice.

Figure 1.13 Crystal directions analysed in diffraction pattern study of graphene and graphene layers.

The intensity ratio of diffraction spots from [11¯20] to those from [1¯100], varies with the number of layers in graphene [75]. When there is only one layer, the I_[11¯20]/I_[1¯100]

is nearly equal to one (see Figure 1.14 (a) & (c)). For an AB-stacked bilayer sample, Figure 1.14 (b) & (d), the [11¯20] intensity is ≈ 4× higher than the [1¯100] intensity. in general, for crystalline Bernal- or ABC-stacked multilayer samples the [11¯20] intensity is stronger than the [1¯100] intensity.

Figure 1.14(a) EDP of a monolayer graphene membrane, and (b) a bilayer membrane. A profile plot along the line between the arrows is shown in (c,d) (adapted from [75]).(e) TEM image of a CVD graphene suspended on a TEM grid. The inset is the corresponding DP. (f) The intensity profile along the red line in DP.

The EDPs can also provide a first indication of defect density. The mean deviation from a regular lattice, reduces the intensities of the higher order reflections [76]. Therefore, potentially a highly- defective few-layer graphene sheet, where the [11¯20] intensity would be reduced, might be mistakenly identified as a monolayer. Furthermore, as mentioned above, an AA stacked multilayer graphene would produce the same pattern as monolayer graphene (however, reports of AA-stacked graphite are quite rare). For an unambiguous identification of monolayer graphene from electron diffraction data, one needs to obtain a series of diffraction patterns with the sample tilted to different orientations; or use other supplementary techniques such as Raman spectroscopy.