Damage formation and annealing studies of low energy ion implants ...

4.2.1.7 Energy straggling and system resolution, resulting in a convolution of the
peaks with Gaussian distributions
MEIS energy spectra are modified from idealised profiles by two unavoidable
effects, i.e. energy straggling and the system resolution, causing the idealised profiles to
become convoluted with error functions.
As an energetic particle moves through a sample it loses energy through many
individual encounters. These encounters are subject to statistical fluctuations. Two ions
with an identical initial energy are unlikely to have the same energy after passing
through the same thickness of a sample. This effect is called energy straggling, (Ωs), and
this places a finite limit on the accuracy of the energy losses and hence the depths that
can be resolved.
It is difficult to calculate the exact amount of straggling, but a theoretical
expression that estimates the amount of energy straggling was derived by Bohr and
extended by Lindhard and Scharff:
2
B
2 2
( z e ) Nz t
Ω =
(4.10)
4π 1
2
where N is the density and t is the layer thickness. This indicates that the energy
straggling is proportional to the square root of the layer thickness. Energy straggling
also varies with atomic species and the electron density but not the beam energy (4).
Theory predicts that the straggling produces approximately a Gaussian distribution from
a mono energetic incident beam. Backscattering spectra most often display the integral
of the Gaussian distribution (error function) rather than the Gaussian distribution as
shown in Figure 4.7 b) and a) respectively. The full width half maximum (FWHM) of a
Gaussian corresponds to the 12% to 88% range of the error function and the ±Ω
corresponds to the 16% to 84% range. The FWHM is equal to 2.355 times Ω (4).
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