Rocking curve, diffraction

One problem with methods that produce polycrystalline or nanocrystalline material is that it is not feasible to characterize electrically dopants in such materials by the traditional four-point-probe contacts needed for Hall measurements. Other characterization methods such as optical absorption, photoluminescence (PL), Raman, X-ray and electron diffraction, X-ray rocking-curve widths to assess crystalline quality, secondary ion mass spectrometry (SIMS), scanning or transmission electron microscopy (SEM and TEM), cathodolumi-nescence (CL), and wet-chemical etching provide valuable information, but do not directly yield carrier concentrations. 

In this chapter we introduce high resolution diffraction studies of materials, beginning from the response of a perfect crystal to a plane wave, namely the Bragg law and rocking curves. We compare X-rays with electrons and neutrons for materials characterisation, and we compare X-rays with other surface analytic techniques. We discuss the definition and purpose of high resolution X-ray diffraction and topographic methods. We also give the basic theory required for initial use of the techniques. 

Later chapters will deal with a more complete description of the diffraction process, but we now have enough to discuss the selection of radiations and techniques. If the structure factor and scattering strength of the radiation are high, the penetration is low and the rocking curve is broad. This is the case with electron radiation. For X-rays and even more for neutrons, the structure and absorption factors are small, penetration is high and rocking curves are narrow. These factors have three main consequences for X-rays and also for neutrons ... 

This rises to 10 if both the K 1 and K 2 ines are diffracted by the specimen. The effect this has upon the rocking curve depends on the dispersion of the whole system of beam conditioner and specimen, and ranges from zero to very large. This will be discussed below, in section 2.6. 

Figure 2.21 Bragg plane tilt aberration, (a) Diffracting planes parallel, diffraction occurs simultaneously over the whole height of the beam, (b) Diffracting planes skewed, diffraction only takes place over a narrow band, (c) As the crystal is rotated to measure the rocking curve, the band moves up or down the crystal. The integrated intensity remains approximately the same as in case (a) but the peak intensity decreases and the width increases...

If the specimen crystal is curved, there will be a range of positions where the diffraction conditions are satisfied even for a plane wave. The rocking curve is broadened. It is simple to reduce the effect of curvature by reducing the collimator aperture. For semiconductor crystals it is good practice never to mn rocking curves with a collimator size above 1 mm, and 0.5 mm is preferable. Curved specimens are common if a mismatched epilayer forms coherently on a substrate, then the substrate will bow to reduce the elastic strain. The effect is geometric and independent of the diffraction geometiy. Table 2.1 illustrates this effect. 

The dispersive (+ n, - m ) mode has already been seen clearly with the duMond diagrams, Figure 2.10. Here, the curves are no longer identical and the crystals must be displaced from the parallel position in order to get simultaneous diffraction. As the crystals are displaced, so the band of intersection moves up and down the curve. When the curves become very different, the K 1 and K 2 intensities are traced out separately. Then the peaks are resolved in the rocking curve, and if no better beam conditioner is available it is important in such cases to remove the K 2 component with a slit placed after the beam conditioner. A slit placed in front of the detector, with the detector driven at twice the angular speed of the specimen, also works very well. This is in effect a low resolution triple-axis measurement. 

If we imagine the diffraction of a plane wave from epilayers we see that there will in general be differences of diffraction angle between die layer and the substrate, whether these are caused by tilt or mismatch f Double or multiple peaks will therefore arise in the rocking curve. Peaks may be broadened... 

Figure 4.19 The range of strong diffraction in (a) the Laue case, (b) the Bragg case. The heavy lining shows the region of the dispersion surface excited as the rocking curve is traversed...

A third problem with simulation of high resolution diffraction data is that there is no unique instrament function. In the analysis of powder diffraction data, the instalment function can be defined, giving a characteristic shape to all diffraction peaks. Deconvolution of these peaks is therefore possible and fitting techniques such as that of Rietveld can be used to fit overlapping diffraction peaks. No such procedure is possible in high resolution diffraction as the shape of the rocking curve profile depends dramatically on specimen thickness and perfection. Unless you know the answer first, you cannot know the peak shape. 

These parameters can be found from the rocking curve, with the exception that roughness cannot at present be distinguished from grading if only double-axis diffraction is used. 

An important featnre to note in double-axis topography experiments is that when the beam area is large, the measnred rocking curve widths are not necessarily intrinsic. For example, mismatched epitaxial layers curve substrate wafers by an amonnt which depends on the degree of mismatch and layer thickness. Topographs of snch curved wafers show bands of diffracted intensity. 

---