Two Beam Diffraction Condition

Image contrast in transmission electron microscopy can arise for a number of reasons including: atomic density and sample thickness variations; selective diffraction from sample bending; and variation in atomic plane spacing caused by strain or crystal imperfections. Selective use of diffraction provides the best contrast in TEM micrographs.

The phenomenon of electron diffraction arises from the elastic scattering of electrons (in this case considered to be waves) by atoms within a crystal lattice such that in particular directions coherent scattering occurs resulting in a strong reflection. Although such scattering occurs due to the interaction with individual atoms, Bragg (1913) gave a simple geometric analysis based on coherent reflections from planes of atoms that has become known as Bragg’s Law.

Figure 3.10 – Schematic diagram illustrating the geometric relationship between the incident electron radiation, atomic lattice planes and the diffracted beam.

(3.1)

where n is an integer, λ is wavelength, dhkl is atomic plane spacing and θ is the angle between the lattice

planes and the incident beam.

Within the electron microscope, diffraction contrast is achieved by selective removal of diffracted beams using the objective aperture. When an electron beam passes

through a crystalline sample orientated along its major pole axis many planes give rise

to diffracted beams resulting in a complex image, often with dim contrast. To achieve

strong contrast the sample is tilted into the so called two beam condition.

In the diffraction imaging mode the sample tilt controls are used to orientate the sample such that diffraction occurs from a single set of planes, leaving only two visible reciprocal lattice points corresponding to the straight through and diffracted beams. A representation of the reciprocal lattice visible when imaging along the [110] direction is given in Figure 3.11, where the separation distance corresponds to the inverse of real plane separations and the direction relative to the centre position

θ θ

θ θ Incident electron beam

Diffracted electron beam

Lattice planes 2θ Lattice spacing, d sin 2d_hkl n

corresponds to the direction of the surface normal of the planes in question, further detail in section 3.6.1.

Figure 3.11 – Reciprocal lattice sections visible in the TEM diffraction mode when an fcc crystal is viewed along (a) [110] direction (cross-sectional) (b) [001] direction (plan view). Only selected lattice points are shown.

In practice the desired plane is selected by following and positioning of the lattice

points along Kikuchi lines visible in thicker sample regions, providing a convenient

reference frame for navigation and orientation when in the diffraction imaging mode. Kikuchi lines are phenomena visible in the diffraction mode arising from inelastic scattering of electrons within the sample. Inelastically scattered electrons give rise to a general background intensity of the diffraction pattern; however a number of electrons from these secondary sources will be incident at the Bragg angle of specific planes and undergo diffraction. As most electrons are only scattered through small angles a net gain and associated net loss will occur in two directions as illustrated in Figure 3.12. The angular separation 2θ results in the intersection of the Kikuchi lines with the beam centre and corresponding reciprocal lattice point.

(a) (b) 000 002 00-2 00-4 004 -220 2-20 000 200 220 -200 -220 -2-20 2-20

Figure 3.12 – Schematic representation of the origin of Kikuchi lines from inelastic scattering events. Paired light and dark bands originate due to net gain and loss in directions that satisfy the Bragg condition for diffraction. Reproduced from Specimen Preparation in Materials Science

(Goodhew 1972).

Selection of the appropriate two beam condition is in general made by choosing a set of planes that have had their spacing disturbed by the feature of interest. In the case of a fully strained pseudomorphic layer the biaxial tetragonal distortion results in alteration of the spacing between (001) planes. Due to symmetry considerations the (001), (002) and (003) diffraction spots are forbidden, hence the

(000) and (004) diffraction spots are chosen for strain contrast. The other major

features of interest are dislocations within relaxed (fully or partially) silicon- germanium layers where the diffraction condition is chosen dependant upon the dislocations Burgers vector (lattice displacement). Clearly the spacing of atomic planes orthogonal to this direction will be altered, shown schematically in Figure 3.13, with dislocation Burgers vector of the form ½[110] in this crystal system the spacing

of (110) planes and those parallel will be effected. For this reason dislocation

contrast is achieved by selection of the (000) and (220) diffraction spots. θ θ θ θ Net loss Net gain Lattice planes Incident electron beam

Figure 3.13 – Schematic diagram showing the effect of an edge dislocation on lattice planes in orthogonal directions. Clearly the planes highlighted in (a) are more distorted by the dislocation than in (b) and are perpendicular to the dislocations Burgers vector. Reproduced with kind permission (Capewell 2002).

It is important to note that length distortion will occur upon movement away from the major pole axis in the diffraction mode as in real space the sample is being viewed at an increasing angle. When imaging with (000) and (004) (strain contrast

mode) deviation in length along a grown layer will occur whilst when imaging (000)

and (220) (dislocation contrast) layer thickness will be distorted, this is simply understood with aid of the pictorial representation in Figure 3.14.

Figure 3.14 – Schematic of real space sample tipping with respect to reciprocal space observed in

b b (a) (b) (-220) (000) (2-20) (004) (00-4) Viewing direction Layer Sandwich Orientation of movement (-220) (000) (2-20) (004) (00-4) Viewing direction Orientation of movement (a) (b)

Enough emphasise cannot be placed on the importance of good specimen preparation as no amount of setup can compensate for a poorly prepared sample. Warped or rippled films can also prove difficult to clearly image as segments will undoubtedly meet the Bragg condition yielding dark bands across the image, though minimisation of such effects can be achieved with experience.

For a comprehensive consideration of the principles and practice of electr on

microscope operation the reader is referred to Principles and Practice of Electron

Microscope Operation (Agar 1974).