Rocking curves

Mercuric iodide crystals grown by physical vapor transport on Spacelab 3 exhibited sharp, weU-formed facets indicating good internal order (19). This was confirmed by y-ray rocking curves which were approximately one-third the width of the ground control sample. Both electron and hole mobiUty were significantly enhanced in the flight crystal. The experiment was repeated on IML-1 with similar results (20). 

Figure 6 DCD rocking curves—measured (dashed) and calculated (solid)—of the (400)...

Figure 2. Calculated CBED rocking curves for Si[ 110], a primary beam energy of 193.35 keV and a crystal thickness of 369nm. The three curves shown in the figure were calculated using 80 Bloch waves (circle+solid line) 20 Bloch waves (star solid line) and 5 Bloch waves (dotted line) and the curves correspond to the line of Figure 1 along A-D.

Figure 4. Calculated CBED rocking curves within the (000) and the (ill) disks in a Si[l 10] zone axis CBED pattern. All curves shown in the figure were calculated for a crystal thickness of 250 nm, and a primary beam energy of 196.35 keV., and correspond to the line scan A-B of Figure 1.

The goodness-of-fit between the experimental and theoretically calculated CBED rocking curves is described by a merit function, and in the present study we use the chi-square merit function defined as... 

Figure 5. Calculated CBED rocking curves. This figure is essentially the same as Figure 4, except that all calculations were made for a crystal thickness of 500 nm.

Figure 9. Energy-filtered experimental and fitted Si[l 10] CBED rocking curves for (a) a line scan along the [111] direction and (b) a line scan along the [002] direction (see Figure 1). The calculations were made for a primary beam energy of 195.35keV and a crystal thickness of 369 nm.

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. 

Accurate measurements of low order structure factors are based on the refinement technique described in section 4. Using the small electron probe, a region of perfect crystal is selected for study. The measurements are made by comparing experimental intensity profiles across CBED disks (rocking curves) with calculations, as illustrated in fig. 5. The intensity was calculated using the Bloch wave method, with structure factors, absorption coefficients, the beam direction and thickness treated as refinement parameters. 

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. 

Figure 1.2 Calculated plane wave X-ray rocking curves, (a) Si 004 with CuK i (0.154 nm), FWHM=3.83 arcsec, (b) Si 333 with MoK j (0.071 nm), FWHM=0.73 arcsec, (c) Ge 111 with CuK i, FWF1M=16.69 arcsec, (d) GaAs 004 witih CuK FWHM=8.55 arcsec...

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 ... 

The strain sensitivity is high, since narrow rocking curves are strongly influenced by small rotations caused by strains in the crystal. 

Rocking curve width degree [Pg.7]

For a given stractmal model, the rocking curve may be computed to high accttracy using fundamental X-ray scattering theory. 

In addition, the measurements are rapid and simple, and are now even used in 100% inspection for quality control of multiple-layer semiconductors. An example is shown in Figure 1.6. This is a GaAs substrate with a ternary layer and a thin cap. The mismatch between the layer and the substrate is obtained immediately from the separation between the peaks, and more subtle details may be interpreted with the aid of computer simulation of the rocking curve. This curve can be obtained in a matter of minutes. Routine analysis of such curves gives the composition of ternary epilayers, periods of superlattices and thicknesses of layers, whilst more advanced analysis can give a complete strain and composition profile as a lunction of depth. 

In later chapters we shall give both the fundamental theoretical treatment necessary to understand the complex rocking curves that arise from complex stmctrrres, and the practical experimental details reqitired to measure them reliably and urrambiguously. 

As we have seen in Chapter 1, we need something near a plane wave in order to see the finest details of the specimen stracture. A single-axis diffractometer utilises a beam that is very far from a plane wave. Thus, single-crystal rocking curves are broadened due to the beam divergence, and the spectral width of the characteristic X-ray lines. 

Thus, for a typical case of h=QA mm, 5=1 mm, a=500 mm. 500 arc seconds, far above the width of the rocking curve for highly perfect crystals, which is typically a few arc seconds. 

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. 

The rocking curve resulting from scamring the second crystal in angle is the plot of the above intensity against angle. Mathematically this is a cross-correlation as discussed below. 

The denominator (normalising constant) is the integrated reflectivity of the first crystal. Figure 2.13 shows the plane wave and the double-crystal rocking curve, again for Si 220 with CuK 1. We note the following ... 

The double-crystal rocking curve is symmetric, though the plane wave reflectivity curve is not. This is a consequence of the autocorrelation, since the autocorrelation of any function is an even fimction. 

If the second crystal is the specimen rather than a beam conditioner element, we shall have got close to the aim of measiuing the plane wave reflectivity of a material. The narrow rocking curve peaks permit us to separate closely matched layer and substrate reflections and complex interference details, as already seen in Figure 1.6. The sensitivity limit depends on the thickness of the layer but for a 1 micrometre layer it is about 50 ppm in the 004 symmetric geometry with GaAs and CuK radiation. This method has been used extensively to study narrow crystal reflections since the invention of the technique. 

Figure 2.13 The plane wave and double-crystal rocking curves for Si 220 with CuK j...

The wavelength reflected varies across the width of the beam. If the specimen is curved, the rocking curve will show separate K i and K 2 peaks, even though the arrangement is non-dispeisive with plane crystals. A narrow sht is then reqtrired to ehminate K 2... 

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