Handbook of Modern Coating Technologies

Computation of defect depth profiles

After thermalization of positrons during diffusion, they can be captured by defects. The diffu-sion of positrons is usually described by Fick's second law. It is only necessary to solve the one-dimensional diffusion equation to calculate the defect depth profiles.
Possible variables describing this measurement of the peak of the Doppler broadening of annihilation are line shape parameters (W or S), while for measuring the positron lifetime this is the so-called fraction of positron F, the components of the lifetime T, and also their intensity Ii. The defect distribution profile in the sample is the depth profile of these annihi¬lation parameters. The defect distribution profile in the sample is presented in the form of an expression describing the profile of penetration P(z;E), which is affected by diffusion of positrons. Profile of positrons absorbed by the material is smeared out due to superposition of the positron diffusion profile. It is possible to solve the equation of diffusion numerically. Nevertheless, it is hardly possible to directly identify the defect depth profile. The sample is usually divided into thin enough sections to perform a numerical procedure to accept a con-stant positron density and defect concentration. In a nonlinear procedure annihilation para-meters can be selected as a function of energy. So it turns out perpendicular to the surface of the sample to build the distribution of defects.
The experimental results were processed in the VEPFIT software to evaluate the slow positron beam data [133]. Gaussian curves are an analytical function of the structure of a thin section where the concentration of defects or defect profile is constant, regarded as the input data for special programs. Both the Gaussian and the step function of the defect concentration, distribution can adequately describe experimental findings. Not infrequently it is impossible to decide which function is a better choice to describe the real profile of defects. The reason is that the positron diffusion depends on the defect concentration and a wide positron implantation profile. Another program (POSTRAP algorithm) for calculat¬ing the depth profiles of slow positrons with the aim of measuring the distribution of defect concentrations is presented in the work of Ghosh and Aers [151]. The program permits estimating the influence of defects and an electric field on positron diffusion. This, in turn, allows one to obtain positron implantation profiles of arbitrary shape. In Fig. 5—14A and B, an example of defect profile determination is given. In arsenic-doped silicon, parameter S was found as a function of energy of positron implantation [152]. Using VEPFIT, data were selected assuming a structure with a Gaussian distribution of defects or from four sections with different density of defects. The layer structure presented in Fig. 5—14B exhibited the best results of fitting [152].
These computer programs also provide information on the fraction f of positrons diffus¬ing back onto the surface. In the case of saturated capture, it is also possible to determine the capture rate of positrons, in contrast to the traditional methods described in Sections 5.4 and 5.5. This is the advantage of back diffusion experiments.
This means that for this method, the upper limit of sensitivity does not exist. However, as a certain positron fraction annihilates from defects the lower limit is comparable to that in 
other methods. The annihilation fraction n can be found using f as a function of the energy of the incident positrons, that is the depth distribution profile:
where fs(E) is the quantity characterizing the studied sample and fsref(E) is the corresponding back diffusion fraction in a defect-free reference sample [133].