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.