Exercise 4.2-1

Diffusion During Cooling

A (big) crystal cools down from its melting point Tm to room temperature Tr ( about 0o C) with T = Tm · exp – (λ · t). The point defects present have a diffusion coefficient given by D = D0 · exp – (Em/kT).
  How large is the average distance L that they cover during cooling down from some temperature T to Tr?
 
This is not an easy question. What you should do is:
Use the Einstein relation for the diffusion length (and forget about lattice factors), but consider that the diffusion coefficient is a function of time, i.e.
 
L2  =  6D · t  =     t' = ∞
∫
t ' = t0		 D(t') · dt'
 
Proceed by first finding the values of λ for initial cooling rates at the melting point of 1 oC/s, 10 oC/s, 50 oC/s and, for fun, 104 oC/s.
Using the following substitution will help with the integration
 
u(t)  =  Em · exp λ · t
kTm
 
The integral now runs from u0 corresponding to t'0 to whatever value of u corresponds to t' = ∞.
You will obtain the following integral:
 
L2    =  2D0 ?
∫
?		  1 
u		  · exp – u · du
 
Thus integral is not soluble analytically. In order to get a simple and good approximation, you may use the linear Taylor expansion for 1/u around u0. Show that for realistic u0 values you can replace 1/u by 1/u0 in a decent approximation which now allows to easily solve the integral.
Now use typical values for melting temperatures, migration activation energies Em, and D0; e.g. from the backbone, two tables or diagrams given here. For missing values (e.g. D0), make some reasonable assumptions.
Plot L as a function of T for activations energies around E = 1.0 eV, E = 2.0 eV, and E = 5 eV with the four cooling rates given above as parameter.
Play around a bit and draw some conclusions, e.g. with respect to
  • Average density of precipitates of point defects obtained in big crystals with few internal sinks.
  • Average size of these precipitates for some equilibrium concentration c0 at Tm.
  • Possible errors made in quenching experiments.
  • Influence of sinks for point defects as a function of the average distance between sinks

 
 

Link to the Solution
 

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go to 4.2.1 Point Defects in Non-Equilibrium

go to Numbers for Point Defect Diffusion

go to Self-Diffusion and some Related Quantities in Si

go to Impurity Diffusion in Si - Arrhenius Plot

go to Solution to Exercise 4.2-1

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