# Polysilicon Diffusion Model

To better describe the physics of the the Polysilicon
Granular material structures **ATHENA** now includes a new Polysilicon
Diffusion Model.

The mechanisms for impurity diffusion in polysilicon are different than that of crystalline silicon. Polysilicon has a micro-structure of small (compared to the interesting device regions) crystalline regions called grains. These are separated by grain boundaries which occupy certain spatial volume and are connected to form a complex network. The texture and morphology of the grain structure depend on the deposition conditions and on subsequent thermal treatment in which recrystallization can occur. Impurities inside the grain will diffuse differently than those in the grain boundaries. Dopant will also transport through grain and across grain boundary interfaces. The direct simulation of the diffusion within all polysilicon regions with such geometrically complex micro-structures is too expensive computationally and therefore requires a special mathematical treatment.

A two dimensional numerical model [1, 2] for impurity
diffusion in polysilicon is incorporated in **SSuprem4**. In
this model, the polysilicon micro-structure is described mathematically
using a local homogenization approximation in which a spatially
separated grain interior and a grain boundary are represented by
the spatially overlapped homogeneous grain interior bulk region
and homogeneous grain boundary network region. Then, each local
polysilicon material element includes two components: a grain interior
and a grain boundary component. The grain boundary network is characterized
with a scalar density function describing the grain size and a vector
function describing the grain boundary direction. Correspondingly,
each impurity is split into two diffusion components: inside the
grain interior and in the grain boundary region. The two components
are coupled with grain boundary segregation. During the thermal
cycle, polysilicon recrystallization is also modeled in order to
include grain size growth.

The polysilicon diffusion is invoked by setting
the flag `POLY.DIFF `in the `METHOD` statement. Control
of the model is enabled with the `MATERIAL` statements. The
diffusion will proceed according to the time and temperature given
in the DIFFUSE statement. The resulting impurity profile
can be output as a grain interior component, a grain boundary component
and combined total concentration. The relationship between the diffusion
components of grain interior and grain boundary are [2].

where

Cg is the impurity concentration component in the grain interior

Cgb is the impurity concentration component on the grain boundary

Dg and Dgb is the diffusivity of grain interior and grain boundary respectively

G is the grain boundary segregation flux term

t is a constant that represents the rate of segregation,

pseg is the segregation coefficient

Fij is a tensor to account for the effect of grain boundary directionality

Lg is grain size and is assumed constant in x,y,z for the current implementation

The grain boundary segregation is modeled [3] with

where

Qs is the density of segregation sites at the grain boundary

Nsi is the density of silicon atoms in the Si crystal

A is the entropy factor

Q0 is the segregation activation energy

Grain Growth

The Lg is a time dependent spatial function due to the recrystallization during the thermal cycle, it is modeled [1] with

where

g0 is the initial polysilicon grain size

b is the lattice constant

D is the grain boundary silicon self-diffusivity

is the grain boundary energy

is the elapsed time during the diffusion

The current implementation allows only a single scalar value of "as deposited" polysilicon grain size. Two sets of physical parameters need to be set up to enable proper diffusion:

- (1) parameters related to the polysilicon material,
are set up in
`MATERIAL`statement - (2) parameters related to the specific impurity,
are set up with the
`IMPURITY`statement

In the `MATERIAL` statement:

**GB.VOL.RATIO **specifies the volume fraction
of grain boundaries to total material volume which gives the relative
magnitude of the two concentration components (unit: Vgb/Vtot, default:
0.1).

**GRAIN.SIZE **specifies the initial grain size
(unit: mm, default: 0.2) (g0 in Equation 8).

**GB.SEG **specifies the density of segregation
sites at the grain boundary (unit: site/cm,
default: 2.64e+15) , (Qs in Equation 7).

**GB.ENERGY **specifies the grain boundary energy
which accounts for the grain size evolution during recrystallization
(unit: eV/cm, default: 1.0) ,
(l in Equation 8).

**GB.DIX.0** specifies the grain boundary silicon
self-diffusivity (unit: cm/sec,
default: 1.0e-12), (D in
Equation 9).

**GB.DIX.E** specifies the activation energy
for grain boundary silicon self-diffusivity (unit: eV, default:
0.0), (D in Equation 9).

In the `IMPURITY` statement:

**GB.DIX.0** specifies the pre-factor for impurity
diffusivity at grain boundary (unit: cm/sec,
default: None), (Dgb0 in Equation 2).

**GB.DIX.E** specifies the activation energy
for impurity diffusivity at grain boundary (unit: eV, default: None),
DgbE in Equation 4).

**GB.SEG.0** specifies the entropy factor for
impurity segregation at grain boundary (unit: 1, default: None),
(A in Equation 7).

**GB.SEG.E** specifies the activation energy
for impurity segregation at grain boundary (unit: eV, default: None),
(Q0 in Equation 7).

**GB.TAU** specifies the constant representing
rate of segregation at grain boundary (unit: sec, default: None),
( in Equation 5).

The vector function describing grain boundary directionality
is calculated during the polysilicon deposition process. Currently,
a columnar direction vector function is implemented, in which the
grain boundary is aligned along the direction normal to the surface
of each deposited layer. In order to create the vector function,
the `METHOD` statement with `POLY.DIFF` specified
should precede the deposition of the polysilicon.

Conclusions

An advanced Polysilicon Diffusion model now available
allows **SSuprem4** to predict doping profiles of complex multigrained
materials.

References

[1] B.J. Mulvaney, W.B. Richardson, T.L. Crandle, "PEPPER - A Process Simulator for VLSI:, IEEE Trans, on Computer-Aided Design, Vol. 8, No. 4, April 1989.

[2] S.K. Jones and A. Gerodolle, "2D Process Simulation Of Dopant Diffusion In Polysilicon", NASECODE-VII Conference Copper Mountain Colorado), Copper Mountain, May 1991.

[3] L Mei and R.W. Dutton, "A Process Simulation Model For Multilayer Structures Involving Polycrystalline Silicon", IEEE Trans. Electron Devices, Vol. ED-29, pp. 1726-1734, 1982.

*
*