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3.3.3.2 Effective Charge of the Projectile

In order to efficiently apply the local density approximation it is necessary to derive an effective charge $ Z_1$ for the particle passing through matter. This effective charge is correlated but not equivalent to the actual charge of the particle which looses some its electrons by the interaction with the target atoms. An expression (3.121) for the effective ionization $ \zeta$ was derived by Brandt and Kitagawa [13] for an ion with a definite ionization $ q = \frac{Z-N}{Z}$ (Z is the core charge of the ion and N is the number of electrons) passing through an electron gas.

$\displaystyle \zeta = \frac{z_1}{Z} = q + C(v_F)\cdot (1-q)\cdot \ln\left[ 1+\left(2\cdot \Lambda\cdot \frac{v_F}{v_0\cdot a_0}\right)^2\right]$ (3.120)

$\displaystyle \Lambda = \frac{2\cdot 0,24\cdot \left(1-q \right)^{2/3}\cdot a_0}{Z^{1/3}\cdot \left(1-\frac{1-q}{7}\right)}$ (3.121)

$ a_0$ is the Bohr radius, $ v_0$ the Bohr velocity (25 keV/amu). The coefficient $ C(v_F)$ weekly depends on the Fermi velocity $ v_F$ of the target medium and is of the order of 0,5.

To determine the ionization $ q$ of an ion which depends on the ion velocity $ v_1$ a stripping criteria for electrons must be applied.

In [90] an empirical velocity stripping criterion is proposed based on the assumption that all electrons with a velocity below the ion velocity are stripped. Applying this criterion the ionization is determined by

$\displaystyle q = 1-\exp\left[ 0,803\cdot y_r^{0,3} - 1,3167\cdot y_r^{0,6} - 0,38157\cdot y_r - 0,008983\cdot y_r^2\right]$ (3.122)

$\displaystyle y_r = \frac{v_r}{v_0\cdot Z^{2/3}}$ (3.123)

$\displaystyle v_r = \Large\left\{ \begin{array}{cl} \text{\normalsize$v_1$}\cdo... ...5\cdot v_F^4} \right] & \;\text{\normalsize for $v_1 < v_F$} \end{array}\right.$ (3.124)

$ y_r$ is called the effective ion velocity.

An alternative energy stripping criterion is presented in [57] which is supposed to yield better results especially for higher ion velocities ($ y_r > 1$). This criterion looks for the smallest number of bound electrons $ N$ so that $ E(N-1) \geq E(N)$, where $ E$ is the total ion energy (sum of potential energy and kinetic energy of the electrons). Using the energy definition of the Brandt-Kitagawa model the ionization is determined by

$\displaystyle 6\cdot 0,240\cdot (1-q)^{(2/3)} = \left[\frac{q+6}{7} \right]\cdot q.$ (3.125)

which can be solved numerically.

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A. Hoessiger: Simulation of Ion Implantation for ULSI Technology