Asymmetric X-ray Bragg reflexion and shallow strain distribution in silicon single crystals

Criteria are given for making use of an asymmetric Bragg reflection of X-rays to analyse a shallow strain distribution near a crystal surface. The shallowness and the resolution of the strain impose opposing conditions on the asymmetry: extreme asymmetry favours the observation of shallow layers but symmetric reflection gives better resolution of the strain. As an example of a compromise, the 422 reflection of Cu Kα1 X-rays was selected for the strain determination in a (100) Si wafer prepared so as to contain phosphorus diffusion. The rocking curve recorded with a triple-crystal diffractometer (angular spread of the collimated X-rays 0.1′′) showed fine agreement with the theoretical curve computed from a strain distribution: its surface value is −4.3 × 10−4 and the depth at which the value is one-tenth of the original is 1 μm. The distribution profile was assumed to be proportional to that of the phosphorus concentration, which was measured separately; the latter surface value was 1.4 × 1020 atoms cm−3. The proportionality coefficient estimated here, 3.07 × 10−24, was 1.79 times the coefficient for uniform doping; this multiplication factor can be explained in terms of elastic stiffness coefficients. Thus it was confirmed that X-ray rocking curves are effective tools for strain determination even at such depth and magnitude of strain and that proportionality still holds between strain and phosphorus distributions with a coefficient depending on crystal orientation.