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2.1.2 Pair-Diffusion Mechanism

Today's commonly used models to describe diffusion phenomena are based on the idea of pair-diffusion models, which do not only describe the influence of charged dopants on each other but also the reactions of dopants and the host lattice including its point defects.

At high temperatures native point defects are generated in the lattice of a crystal. There are two point defects of interest for silicon:

These point defects lead to five different mechanisms of diffusion [Sze88][Fah89][Tay93]:

Table 2.1 shows the experimentally verified reactions of dopants for semiconductor manufacturing. In absence of the reaction partner the dopants can not diffuse. That is why the damage of the lattice as well as the process temperature have a significant effect on the final doping distribution. In Chapter 4 a five species phosphorus diffusion model will be discussed in detail.

Table 2.1: Chemical reactions of pair-diffusion models via lattice defects such as vacancies (V) and interstitials (I)
silicon boron phosphorus arsenic antimony
I + V $ \Leftrightarrow$ 0 B + I $ \Leftrightarrow$ BI P + I $ \Leftrightarrow$ PI As + I $ \Leftrightarrow$ AsI Sb + V $ \Leftrightarrow$ SbV
  BI + V $ \Leftrightarrow$ B P + V $ \Leftrightarrow$ PV As + V $ \Leftrightarrow$ AsV SbV + I $ \Leftrightarrow$ Sb
    PI + V $ \Leftrightarrow$ P AsI + V $ \Leftrightarrow$ As  
    PV + I $ \Leftrightarrow$ P AsV + I $ \Leftrightarrow$ As  

Besides the field effect at extrinsic doping conditions, in terms of multiple dopant diffusion, the diffusivities of the dopants themselves have to be taken into account being concentration depended. Because point defects exist in either neutral (0), single (+,-) or double charged (+ +,- -) states [Fai81], the effective diffusion coefficient becomes

DA = DAX0 + DAX+ . $\displaystyle \left(\vphantom{ \frac{C_{X_+}}{C_{X_+}^i} }\right.$$\displaystyle {\frac{C_{X_+}}{C_{X_+}^i}}$ $\displaystyle \left.\vphantom{ \frac{C_{X_+}}{C_{X_+}^i} }\right)$ + DAX- . $\displaystyle \left(\vphantom{ \frac{C_{X_-}}{C_{X_-}^i} }\right.$$\displaystyle {\frac{C_{X_-}}{C_{X_-}^i}}$ $\displaystyle \left.\vphantom{ \frac{C_{X_-}}{C_{X_-}^i} }\right)$ + DAX- - . $\displaystyle \left(\vphantom{\frac{C_{X_{--}}}{C_{X_{--}}^i}}\right.$$\displaystyle {\frac{C_{X_{--}}}{C_{X_{--}}^i}}$ $\displaystyle \left.\vphantom{\frac{C_{X_{--}}}{C_{X_{--}}^i}}\right)^{2}_{}$     (2.19)

where X denotes a point defect (interstitial or vacancy) and CXi the concentrations under intrinsic conditions. Again the temperature dependence of each diffusivity component can be modeled by an Arrhenius law.
DAX = D0 . e-$\displaystyle {\frac{E_A^0}{kT}}$     (2.20)

The diffusivities for the most common dopants under extrinsic conditions are shown in Table 2.2 after [TMA97].

Table 2.2: Arrhenius parameters for the diffusivities of boron, arsenic, phosphorus and antimony
Dopant D0 EA0 Di+ EA+ Di- EA- Di- - EA- -
  [cm2/s] [eV] [cm2/s] [eV] [cm2/s] [eV] [cm2/s] [eV]
B 4.10 3.46 2.16 3.46 - - - -
As 0.37 3.44 - - 5.47 3.44 - -
P 23.10 3.66 - - 26.64 4.00 26.52 4.37
Sb 0.42 3.65 - - 4.5 4.08 - -

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Next: 2.1.3 Segregation Up: 2.1 Diffusion Previous: 2.1.1 Extrinsic Dopant Diffusion
Mustafa Radi