A solution of this problem was already given by Lindhard [50]. The energy loss of the projectile is determined by the force resulting from the electric field which is generated in the plasma by the penetrating charged particle with a charge moving with the velocity . For his solution Lindhard applied the linear response theory.

(3.100) |

(3.101) |

(3.102) |

is the linear dielectric response function of the medium on the charge.

This set of differential equations can be solved by Fourier transformation [50] ( ).

The contributions to the energy loss by the transversal electric field and the longitudinal electric field are treated separately, which requires the definition of a transversal (Fourier transform of ) and a longitudinal (Fourier transform of ) dielectric response function. and are defined in Fourier space via the scalar potential and vector potential , describing the electromagnetic field as shown in (3.105) and (3.106), and the Maxwell equations.

is defined by

(3.107) |

while is defined by

(3.108) |

The transversal field and the transversal current density are characterized by

(3.109) |

(3.110) |

The total energy loss is a sum of the longitudinal and the transversal contribution, respectively. The transversal contribution is negligible since the particle velocity is non-relativistic [50]. This requirement holds for typical ion implantation conditions.

Lindhard derived an expression for the longitudinal dielectric response function by applying a self-consistent treatment of the electron gas. He calculates the charge density and the current density induced in the plasma by the forces resulting from the scalar potential and the vector potential . These induced sources produce an electro-magnetic field which acts on the electron gas. Therefore the electro-magnetic field must be self-consistent with the induced sources. A quantum-mechanical calculation of this problem results in

(3.111) |

with

(3.112) |

being the classical resonance frequency of the gas. is the electron density distribution function, is the electron mass, is the energy, the wave vector of the electron in the nth state and is a small damping factor. By applying this dielectric response function to (3.103) Lindhard derived the energy loss in an electron gas described by the Fermi distribution in the ground state with a maximum energy .

(3.114) |

(3.116) |

(3.117) |

(3.118) |

Assuming a small damping factor and a small particle velocity , (3.113) can be simplified to an electronic stopping equation which is proportional to velocity.

is the real part of (3.115), while is the imaginary part.

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A. Hoessiger: Simulation of Ion Implantation for ULSI Technology