3.3.5.3 Point Defect Recombination

Besides the energy loss mechanism of mobile particles and the damage generation process, thermal effects have to be considered when simulating ion implantation. All atoms in a solid perform random motions which result in diffusion effects of impurities and point defects. Thereby interstitials and vacancies can approach each other close enough to recombine to a regular lattice atom.

Several approaches have been proposed to model this recombination effect either in combination with the Kinchin-Pease damage model or in combination with the Follow-Each-Recoil method.

An empirical model which can be combined with the Kinchin-Pease model is
given in [75]. The recombination process is subdivided into
recombination within a single cascade and recombination with point defects from
previous cascades. The number of point defects surviving recombination within a
single cascade is determined by an ion species dependent factor
(**recombination factor**) which is multiplied with the number of
generated point defect pairs calculated by the modified Kinchin-Pease model
(3.143). The recombination with previously generated point
defects is considered by a recombination probability . The expected
value of Frenkel pairs
remaining after recombination per
generated Frenkel pair can be calculated by

(3.149) |

The first term takes into account that the vacancy and the interstitial do not recombine which results in an additional Frenkel pair. The probability of this process is . The second terms considers a full recombination process with a probability of , which reduces the number of Frenkel pairs by one.

is assumed to be proportional to the local interstitial and vacancy
concentration , and to a species dependent **saturation concentration**
.

(3.150) |

Combining all contributions the total number of new stable point defects generated by a primary recoil can therefore be calculated by

This expression is more general than the one proposed in [75] where the vacancy and interstitial concentration are assumed to be equal. Tab. 3.5 summarizes experimentally determined values of and ([7], [73], [74], [75]). A similar model was proposed by Posselt in [63]. The models mainly differ in placing the interstitials as will be explained in Sec. 3.3.5.

There are several approaches to model the point defect recombination process if a Follow-Each-Recoil method is applied for the calculation of the point defects ([32], [46], [85], [86]). All these methods are based in principle on the same concepts which will be summarized in the following by looking in detail at the damage generation processes.

- When a mobile particle transfers more energy than the displacement energy to a
stable atom in a crystalline solid a recoil is generated. If this recoil is
removed from a lattice position a vacancy is left behind. If an interstitial
atom was hit the number of interstitials is reduced by one.
(3.152)

(3.153)

and are the number of interstitials and vacancies, and indicate the change in the number of interstitials and vacancies, respectively. - If a vacancy has been generated it can recombine with an
interstitial.
(3.154)

(3.155)

There are two approaches to model the recombination probability . By the first approach ([46]) it is assumed that the recombination probability is a linear function of the interstitial concentration .

(3.156)

is a*capturing radius*for interstitial-vacancy pairs.By the alternative approach ([32]) it is also suggested to distinguish between the recombination of point defects originating from the same cascade and point defects originating from different cascades. While the recombination of point defects originating from different cascades is assumed to be linearly proportional to the interstitial concentration, the recombination of point defects originating from the same cascade is assumed to be constant (independent of the interstitial concentration). In order quantify this assumption a constant value is added for the calculation of the recombination probability.

(3.157)

is the recombination probability within a single cascade and is the proportionality coefficient for inter-cascade recombination. - Instead of recombining with an interstitial a vacancy can recombine with
an impurity atom which is located at an interstitial site.
(3.158)

(3.159)

(3.160)

(3.161)

is a*capturing radius*for impurity-vacancy pairs. A different capturing radius is required for each impurity species because their thermal mobilities are different. is the number of activated (located at lattice positions) impurities, and are the number and the concentration of interstitial impurities. - When a mobile particle comes to rest an interstitial atom is generated.
(3.162)

(3.163)

- Such an interstitial atom can recombine with a previously generated
vacancy.
(3.164)

(3.165)

- Besides recombination the point defects can form clusters as indicated by
several molecular dynamic simulations ([15], [65]) and
isolated point defects can recombine with these clusters. The reaction radius of
a cluster which determines the clustering probability is assumed to be
([85]).
(3.166)

(3.167)

(3.168)

(3.169)

(3.170)

(3.171)

, , , are the densities and numbers of interstitial and vacancy clusters consisting of point defects.

Applying these mechanisms the generation of several defect species like isolated interstitials and vacancies, impurity-interstitial pairs and point defect and impurity-point defect clusters can be modeled. But at least isolated interstitials and vacancies have to be considered in the simulation to correctly handle the de-channelling mechanism. Therefore at least the mechanisms mentioned in 1., 2. and 4. have to be applied during the simulation, while all other mechanisms just provide additional information for the output of the Monte-Carlo ion implantation simulation.

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A. Hoessiger: Simulation of Ion Implantation for ULSI Technology