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4.3.5 Polysilicon Diffusion

  Previous work to model the diffusion process in polysilicon material has been presented by various authors. O'Neill [O'N88] presented a one-stream diffusion models for the grain boundary concentration neglecting the grain interior diffusion and the clustering physics. Jones [Jon88] and Lau [Lau90] [Lau92] used a two-stream diffusion model for the grain boundaries as well as for the grain interior regions but only a fit formula for the grain growth kinetics. Orlowski et al. [Orl92] presented a new concept using a two-stream diffusion model in combination with fractal dimensions to describe the complex irregular structure of polysilicon.

To model the transport mechanism in polysilicon correctly, an appropriate diffusion model for polysilicon has to be designed. For general polysilicon layers it is not possible to account for the exact microstructure of the grains and grain boundaries. The most popular approach for polysilicon material is to split up the available dopants into a grain interior concentration tex2html_wrap_inline5665 and grain boundary concentration tex2html_wrap_inline5667 . To evaluate the grain boundary concentration it is necessary to use a simplified model of the microstructure of the polysilicon grain boundaries. The grain boundaries which are assumed to be two-dimensional interfaces with a certain area density of free states are expanded to a volumnar concentration by scaling with the local grain size. Therefore, the grain boundary concentration reads tex2html_wrap_inline5669 , where tex2html_wrap_inline5671 is the area dopant density within the grain boundary, r the local grain size. Due to this homogenization the detailed grain boundary information, which is practically not available for simulation purposes, vanishes. The remaining grain interior concentration is than given by subtraction of tex2html_wrap_inline5667 from the available total dopant concentration tex2html_wrap_inline5677 (4.3-50).

  equation1539

As the polysilicon layers are excessively doped clustering occurs in the grain bulks. Therefore, we calculate the active and mobile part of the dopant species tex2html_wrap_inline5679 with a static clustering model (4.3-51), where tex2html_wrap_inline5681 is the solubility limit for the dopant species and tex2html_wrap_inline5683 is a fitting parameter related to the cluster size.

  equation1548

Using the static clustering approach implies the assumption that dopants can be delivered fast enough from the grain interior regions into the grain boundaries so that no temporary leak of dopants occurs in the grain boundaries.

With the diffusion mechanisms described in the previous section we can define the dopant diffusion flux for the grain boundaries tex2html_wrap_inline5685 and for the grain bulks tex2html_wrap_inline5687 as given by (4.3-52) and (4.3-53), respectively.

    eqnarray1565

The electric field in the grain interior regions is depicted in (4.3-54). The fast diffusion in the grain boundaries is captured by the corresponding first term of (4.3-52) in combination with the extensively high diffusivity tex2html_wrap_inline5689 , where the second term denotes the dopant flux caused by the grain growth. The prefactor tex2html_wrap_inline4759 is a tensor which acts as weighting factor for the diffusion flux. This weighting factor is calculated from a combination of local grain size, grain main axis orientation and lateral to vertical grain size aspect ratio to model the anisotropic diffusion behavior of polysilicon.

The segregation kinetics between grain interiors and grain boundaries is accounted for by the generation/recombination factor tex2html_wrap_inline5693 given by (4.3-55).

  equation1595

The exchange of dopants is modeled by means of a trapping factor t and an emission factor e, where tex2html_wrap_inline5699 denotes the maximum number of free states in the grain boundary. If the grain boundary is not filled with dopants, active dopants from the grain interior are delivered, whereas dopants diffuse into the grain interiors if the maximum number of free states in the grain boundary has been exceeded. This mechanism is also able to capture temporary grain boundary saturation effects.

A full description of the polysilicon model including the dynamic grain growth behavior (see section polysilicon grain growth) for a given dopant reads:

    eqnarray1607

After implantation of the dopant into the polysilicon material only a small portion tex2html_wrap_inline5701 resides at grain boundaries, where the major part rests at interstitial sites within the grains. tex2html_wrap_inline5701 represents the total area which is covered by grain boundaries, hence, it depends on the initial grain size tex2html_wrap_inline5705 and the grain boundary thickness tex2html_wrap_inline4869 by tex2html_wrap_inline5709 . It appears from simulations that tex2html_wrap_inline5701 is not a critical parameter, because after a few seconds the grain boundaries are fully occupied due to their unique energetic properties for hosting dopant atoms .




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Next: Model Parameters Up: 4.3 Diffusion Model Library Previous: Model Verification

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