Ion Implantation and Synthesis of Materials - Studium

6.3 Calculations 71two simple techniques. For different atomic species that are sufficiently close inatomic number, e.g., Fe–Ti (iron–titanium) alloys, we can substitute the meanatomic number and mass into the LSS equations and proceed as for a monatomictarget. If the atomic numbers are appreciably different, a first-order of estimatemay be obtained for an alloy A x B y using the expression⎡( Rp( A)/ NA)( Rp( B)/ NB)Rp( AxBy) ≅ Nalloy⎢⎥ ,( yR ( A)/ N ) + ( xR ( B)/ N )⎢⎣pApB⎤⎥⎦(6.15)where x + y = 1, N A and N B are the projected ranges and the atomic densities inpure substrates A and B, respectively, and N alloy is the atomic density of the alloy.Consider, for example, the case of implanting 100 keV Kr ions into the intermetalliccompound Fe 2 Al. (The compound Fe 2 Al is fictitious. However, for illustrativepurposes, we will assume that it exists with a mass density of 6.36 g cm −3 .) Theprojected ranges of 100 keV Kr in Al and Fe are 50 and 23 nm, respectively.The atomic densities are NAl = 6.02 × 10 22 atoms cm −3 and N Fe = 8.50 × 10 22−3atoms cm . We shall assume that the intermetallic compound Fe2Al has a massdensity of 6.36 g cm −3 , which gives an atomic density of 8.29 × 1022 atoms cm −3 .Using this data and (6.15) gives R p (Fe 2 Al) = 29 nm, which is in good agreementwith the PRAL calculation, which gives R p = 28.6 nm.An estimate of the range straggling in alloys can be made using the empiricalexpression developed by Kido and Kawamoto (1986)∆RRpp0.38= 0.27 + ,ε + 2.0av(6.16)where the average alloy reduced energy, ε av , is defined byεav= ∑ni=1C ε ,ii(6.17)where C i (i = 1, 2,…, n) is the elemental composition of the ith element and ε i isthe elemental reduced energy defined by (5.9). Applying (6.16) and (6.17) to theproblem of range straggling in 100 keV Kr in Fe 2 Al, we obtain an average reducedenergy of 0.38 and a ratio of ∆R p /R p = 0.43. The projected range, calculated using(6.15), was 29 nm, which results in a projected range straggling of ∆R p = 12.5 nm,which is within 18% of the PRAL calculated value of 10.6 nm.