Ion Implantation and Synthesis of Materials - Studium

54 5 Ion Stopping5.4 ZBL Nuclear Stopping Cross-SectionWhile the Thomas-Fermi screening function is a reasonable approximation forcalculating stopping powers and cross-sections, a higher level of accuracy and awider range of reduced energy, ε, are obtained using the Ziegler, Biersack, andLittmark (ZBL; 1985) universal screening function (2.15). The ZBL cross-sectionis presented in Fig. 5.2 along the stopping curves from four classical atom screeningfunctions. The small filled circles in the plot represent the numerical solutions,and the solid line represents an analytical fit to the points. The expression for thefit is given by0.5 ln(1 + 1.1383 ε )Sn ( ε ) =0.21226 0.5( ε + 0.01321ε + 0.19593 ε )(5.11)for ε ≤ 30. For example, S n(ε) = 0.164 for ε = 10 −2 and S n (ε) = 0.118 for ε = 10. Inthe high-energy regime,ln( ε )Sn( ε ) =2ε(5.12)for ε > 30. For example, for ε = 470, S n (ε) = 6.55 × 10 −3 .Equation (5.12) is the high-energy reduced nuclear stopping cross-section forunscreened (Coulomb) nuclear stopping. Figure 5.2 shows that the reduced nuclearstopping cross-section is identical for all screening functions for ε > 10.However, considerable differences exist for lower values of ε.For practical calculations, the ZBL universal nuclear stopping for an ion withenergy E 0 isS−15× ZZ1 2MS1 nn( E0)=0.23 0.23( M1 + M2)Z1 + Z28.462 10 ( ε )( )2 −1(eVcm ) atom ,(5.13)where the reduced nuclear stopping cross-section is calculated using (5.11) and(5.12). For example, for As (Z 1 = 33, M 1 = 75) incident on Si (Z 2 = 14, M 2 = 28),the ratio of S n (E 0 )/S n (ε) = 6.99 × 10 −13 (eV cm 2 ). For ε = 470 (E 0 = 100 keV),