Range and Straggling

Chapter 2 Interaction of Radiation with Matter Energy Transfer 

Using the Ashley approximation, the inverse inelastic mean free path is given, in congruity with the stopping power (see Sect. 2.5.2), as follows  

Here/(q) is the dipole oscillator strength distribution at q and e is the base of natural logarithm. The lowest excitation potential may be taken for qmin, whereas qmax = (E + EB)/2 with EB a defined mean electron binding energy (Mozumder and La Verne, 1984). 

Now the probability density that the collision will result in an energy loss q is given by 7 /X, and the resultant mean energy loss in one collision is given by 

Thus the spatial rate of energy loss is (q)/A. The stopping power is actually the mean value of the ratio of energy loss to path length, and to this extent the derivation is an approximation. The path length distribution Pjx) in n collisions may be given as a convolution—that is, 

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Figure 2 shows the reference coordinates and nomenclature of this geometry. Theoretical calculations of the projected range and straggle for various dopants and substrates have been calculated and shown in graphical form (4, 21-26). Some common dopant ions for silicon and gallium arsenide are shown in Figures 3, 4, 5, and 6. 

Stopping Power, Energy Loss, Range, and Straggling... 

Protons, deuterons, tritons, and alpha particles behave similarly as far as energy loss and straggling are concerned. As they travel in a medium, they are deflected very little from their direction of incidence, as a result of which backscattering is insignificant and their range is almost equal to their pathlength. 

An estimate of AR ia alloys can be made usiug the empirical expression (12) of equation 10 where the average alloy reduced energy, is defined by equation 11, where C (i = 1, 2,... , n) is the elemental atomic fraction of the /th element, and is the elemental reduced energy defined iu equation 3. Using this formulation, the projected range straggling iu compounds can be calculated to within 20%. 

Gaussian-shaped depth profiles of P with three parameters of maximum concentration (Cmax), projected range (Rp) and range straggling (ARp). The energy loss (dE/dx) and energy straggling ( 2 square root of the variance) of the a beam in the Si layer were taken into account ... 

The statistical uncertainties in these processes are reflected in "straggling", a term which describes the variations in the number of collisions undergone by any one particle and the amount of energy transferred at each collision. There is a corresponding variation in the penetration or range of particles in a material. One MeV electrons have an average range of 4.3 mm in unit density material. 

The electronic stopping does not lead to major deflection and does not contribute very much to the lateral spread Rj. Another important quantity is the range straggling AR or ARp. It contains information about the actual distribution of the implanted ions in a solid. They also have been calculated by various authors, e. g. Schiott ... 

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