Ion Implantation and Synthesis of Materials - Studium

68 6 Ion Range and Range DistributionIn the case where nuclear stopping predominates, the range can be estimated byignoring the contributions from electronic stopping:0 dER = ∫ E 0NSn.( E)(6.9)A power law-based estimate of the nuclear stopping cross-section, S n (E),given in (5.7), results in an approximate expression for R, which is given byRE ( )11−γ−20= ⎛ ⎞mmm⎜ ⎟ E0⎝ 2m⎠ NCm(6.10)in which the parameters m and C m have been defined in Chap. 4, andγ = 4M 1 M 2 /(M 1 + M 2 ) 2 . A 20% accuracy in nuclear stopping and path length canbe obtained using the power law approximation over the same ranges of validitym = 1/3 for = ε ≤ 0.2,m =1/2 for 0.08 ≤ ε ≤2.A rather useful rule-of-thumb equation for predicting heavy-ion ranges (m = 1/2),usually with an accuracy of ∼30–40%, is given by13 E (keV) 1 + M / MR(nm) =.ρ (gcm )2 1−3 2/3Z1(6.11)As an example, consider 50 keV As in Si where Z 1 = 33, M 1 = 75, M 2 = 28, andρ Si = 2.33. Applying these values to (6.11) gives R = 37.6 nm.6.3.2 Projected RangeThe range, R, is the total distance that the projectile travels in coming to rest.However, in many applications of energetic ions in surface modification, the projectedrange, R p , is the quantity of interest. The projected range is defined as thetotal path length of the projectile measured along the direction of incidence.Figure 6.1 contrasts the difference between R and R p.An approximate measure of the projected range can be found using the theoryof Lindhard et al. (1963), which givesR M2≈ 1 + B ,R Mp 1(6.12)