Ion Implantation and Synthesis of Materials - Studium

70 6 Ion Range and Range Distribution6.3.3 Range StragglingIon implantation is a stochastic (random) process. The mean projected rangerepresents the most probable location for an ion to come to rest. The uncertainty inthe scattering process as the ion travels through the target gives rise to ions comingto rest at distances less than and greater than the projected range. The averagefluctuation (standard derivation from the mean) in the projected range is calledrange straggling, ∆R p .The influence of ion mass, M 1 , and target mass, M 2 , on the range straggling isschematically shown in Fig. 1.2 (also see Fig. 6.4). As can be seen, the trajectoriesof the lighter species in a heavier substrate exhibit larger deviations of their implanttrajectories relative to those of the heavy ions in relatively lighter substrates.This behavior is influenced by the same conservation of momentum and energyphysics that applies the scattering of macroscopic hard spheres, e.g., shootingmarbles into a collection of billiard balls versus the opposite case of shooting abilliard ball into a collection of marbles. In the former case, the marble projectilewill be more easily deflected from its incident course while in the latter case thebilliard ball will plow ahead without as much lateral deviation. These differencesin (elastic) nuclear scattering properties will influence the extremes in stoppinglocations and the width of the ion distribution in the solid. For the same physicalreasons, a wider depth distribution is expected for M 1 (ion) < M 2 (substrate atom)relative to the depth distribution for M 1 > M 2 .The range straggling can be calculated using the theory of Lindhard et al. forthe condition where nuclear stopping dominates from∆ R p ≅ R p /2.5.(6.14)As an example of (6.14), we will estimate the mean projected range straggling forthe two cases, 13 keV B in Si and 35 keV As in Si, which produces an R p of 50 nmfor both ions. The PRAL code gives 28.0 and 20.9 nm for ∆R p , for B and As, respectively.Therefore R p /∆R p = 1.8 and 2.4 for B and As, respectively, in reasonableagreement with (6.14).A general relationship between projected range straggling and the mean projectedrange, for the mass conditions 0.1 ≤ M 2 /M 1 ≤ 10, has been developed byWSS and is plotted in Fig. 6.4. The value ∆R px is the range straggling in the directionof the incident ion, and ∆R py is the range straggling in the direction perpendicularto the incident ion.6.3.4 Polyatomic TargetsThe accurate treatment of ion ranges in compound targets requires extensive calculations,as performed in the PRAL code. However, estimates can be made using