3.3.3.3 Empirical Electronic Stopping Model

On the basis of the Lindhard electronic stopping model Hobler et al. proposed an empirical stopping model [36] for the implantation into crystalline silicon where the local electron densities in the target are not explicitly considered for the calculation of the stopping power. Instead an impact parameter of the ion is defined.

The model is composed of a non-local part which dominates in case of large impact parameters (channeling ions), and a local part which describes the close collisions.

(3.126) |

The non-local part is derived from the simplified velocity proportional stopping power calculated by Lindhard (3.119).

The coefficient considers as well the properties of the ion species as of the target material. A value proposed for in [51] is

(3.128) |

The local part is exponentially proportional to the impact parameter as proposed by Oen and Robinson [61]. It is related to a single collision event and therefore a discontinuous contribution to the total energy loss.

(3.129) |

is the target atom density, is the charge of the target atoms, and are the charge and the mass of the implanted particle, is the Bohr radius, and the energy of the particle. , , , and are empirical parameters.

- is the
**Lindhard correction factor**to empirically adopt the strength of the electronic stopping. - is a
*screening factor*for the local contribution. A value of is proposed in [61] where is the screening length used for the calculation of the interatomic screening potential (3.92). In [36] this length is multiplied by the empirical**screening pre-factor**.(3.130)

- is the maximum considered impact parameter for the local stopping part, while contributions resulting from impact parameters larger than are considered by (3.127) for the non-local stopping part.
- and are the non-local and the local portion of
the total electronic stopping power which requires that
(3.131)

The ratio between and depends on the energy of the ion especially for light ions like boron.

The empirical parameters and are called the**non-local pre-factor**and**non-local power**.

As indicated by Lindhard the electronic stopping power is only proportional to the ion velocity for low ion velocities. A velocity of

(3.133) |

can be considered as a limit. For higher energies the stopping power reaches a maximum before it decreases again. Worth mentioning is that experimentally determined stopping power functions do not yield a stopping power which is proportional to the ion velocity for very low ion energies . Instead an electronic stopping power being proportional to up to [2], [18] is reported. This fact has to be considered as a source for small errors when applying the electronic stopping power law derived by Lindhard. Fig. 3.8 schematically shows the functional behavior of the electronic stopping power over the whole energy range.

As a model for energies around the maximum of the stopping power an interpolation between the velocity proportional stopping power and a stopping formula derived by Bethe and Bloch [1], [4], [6], [39], which is valid for high ion energies, is suggested in [75].

(3.134) |

According to the Bethe-Bloch theory the electronic stopping power for high ion velocities is determined by ([73])

(3.135) |

(3.136) |

and are the core charges of the ion and of the target, is the elementary charge, is the electron mass, the atomic mass unit, the relative atom mass of the ion, and the so called Bloch constant determined by (3.137) in units of eV.

The exponent used for the interpolation around the stopping power maximum allows to determine the height of the maximum of the total stopping power . Besides a modification of the law for the non-local stopping power also a modification of the energy dependence of the ratio between local and the non-local part (3.132) is suggested for higher ion energies.

(3.138) |

The major drawback of the empirical model is that many empirical parameters have to be determined especially for crystalline materials and therefore a lot of SIMS measurements are necessary to extend this model to new material types. But according to [31] the accuracy of the empirical model is better than the model based on just the local electron density as has been demonstrated for the case of boron implantations in crystalline silicon. Additionally it requires less computational effort, because just the impact parameter related to a certain collision event has to be determined to be able to calculate the electronic stopping power related to a single collision event.

The determination of the impact parameter requires no additional computational effort, because it is also used to model the nuclear stopping power process. In contrast the calculation of the local electron density along the flight path of the ion is a very computational intensive task. Additionally, a numerical integration of the electronic stopping power has to be performed in the local electron density model, because the electron density varies significantly when the ion passes an atom of the target.

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A. Hoessiger: Simulation of Ion Implantation for ULSI Technology