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:

• vacancies are defined as an empty lattice site
• interstitials are host atoms that reside in one of the interstices of the lattice

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

• direct mechanism: impurities with small atom-radii can move directly between the host lattice as an interstitial.
• indirect mechanism: dopants with radii that are too big for direct local exchanges can knock out lattice atoms and the so widened gap offers foreign atoms the possibility to move through the host lattice.
• exchange mechanism: a foreign atom that resides on a lattice site (substitutional defect) can exchange its locality with another substitutional defect. Because of the high invoked energy amount this mechanism is negligible.
• ring mechanism: a more sophisticated exchange mechanism occurs when a group of neighboring atoms exchange their sites in a cyclic manner. This mechanism lacks of experimental verification.
• vacancy mechanism: a substitutional defect and a vacancy building a pair are the means of mass transport trough the host lattice.

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

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

where X denotes a point defect (interstitial or vacancy) and C_Xⁱ the concentrations under intrinsic conditions. Again the temperature dependence of each diffusivity component can be modeled by an Arrhenius law.

$${\frac{E_A^0}{kT}}$$ (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	-	-