Ion Implantation and Synthesis of Materials - Studium

6.2 Range Distributions 652. the radial range2 2 2( )1/2r=s+s+s,R x y z(6.2)3. the transverse projected range1/2t 2 2Rp = ⎡⎣( xs sinα− ys cos α) + zs⎤⎦ ,(6.3)4. the longitudinal projected range( p)1/22 t2Rp = ⎡( Rr) + R ⎤ .⎢⎣⎥⎦(6.4)For normal-incidence projectiles, the range spreading is equal to the transverseprojected range.6.2 Range DistributionsBecause the stopping of an ion is a stochastic (random) process, the collision sequence,the subsequent ion deflection, and the ion’s total path length in coming torest vary randomly from ion to ion. As a result, ions with the same energy, incidentwith the same angle onto the sample surface, and into the same material, donot necessarily come to rest in the same place. Hence, all ions of a given type andincident energy do not necessarily have the same range. Instead, if we were toexamine the range history of many ions, a statistically broad distribution inthe depths to which ions penetrate would be observed, similar to that shown inFig. 6.3. The distribution in projected ranges is referred to as the range distributionor range straggling, with the most probable projected range referred to as the averageor mean projected range. A statistical distribution would also be observed forall the quantities defined in Fig. 6.2.The depth distribution, N(x), of implanted ions, normalized for an ion implantationdose φ i , is given by the expressionφi1 x−RpN( x) = exp ⎢−⎥ ,1/2∆R(2 π ) 2 ⎜ ∆R⎟p⎡⎢⎣⎛⎝p⎞2⎤⎠ ⎥⎦(6.5)where R p is the projected range (mean depth of the distribution) and ∆R p is theprojected range straggling (standard deviation of the distribution). Assuming that