Ion Implantation and Synthesis of Materials - Studium

7.8 Spikes 87where N is the atomic density of the target and σ (E) is the total collision crosssectiongiven by (4.18a). Setting P(E) = 1 and replacing dx with λ d , the averagepath length per collision, or the mean free path of a particle with energy E, isgiven byλd1= . N σ ( E )(7.12)Equation (7.12) can be used to calculate the mean spacing between defects for aprojectile of energy E.Brinkman (1956) has investigated the details of damage distribution in acascade as a function of λ d . He has suggested that as λ d approaches the atomicspacing of the target atoms, a highly damaged region is formed where everydisplaced atom is forced away from the ion or PKA path, producing a volume ofmaterial composed of a core of vacancies surrounded by a shell of interstitialatoms (see Fig. 7.5). This highly damaged volume of material is referred to as adisplacement spike. The displacement spike forms in a time equivalent to the timeit takes the ion or PKA with energy E to come to rest at the end of its range(Sigmund 1974).tE d E R( E)=∫≅0 vNS ( E) 1/2vn(7.13)where v = (2E/M) 1/2 is the ion velocity, S n (E) is the nuclear stopping cross-section,and R(E) is the total ion range. The simple form of (7.13) is the result of a powerlawapproximation for the power parameter m = 1/2. A simple linear estimate ofthe time, t, needed for a 100 keV Xe ion with mass, M = 131 amu to stop withrange R = 20 nm is 10 −13 s, of the order of a lattice vibration time.7.8.2 Thermal SpikeAs the formation of the displacement spike comes to an end, all the movingdisplaced atoms reach a point where they have insufficient energy to cause furtherdisplacement. Energy transfers will be at subthreshold levels. At this point theenergy will be shared between neighboring atoms and will be dissipated as latticevibrations or “heat.” After approximately 10 −12 s, a state of dynamic equilibriummay be obtained where the vibrational energy distribution begins to approximate aMaxwell–Boltzmann function. This period of lattice heating is know as thethermal spike phase of the collision cascade and may exist for several picosecondsbefore being quenched to ambient temperature. Once such a dynamic equilibriumis established, the concepts of local heating and temperature become reasonable.For a Maxwell–Boltzmann distribution of energy, the temperature is related to the