3.2.3 Diffusion of Impurity Atoms

Diffusion of Impurity Atoms via Vacancies or Self-Interstitials

In this case it is especially important not to confuse the vehicle (a point defect except in the case of the direct interstitial diffusion) with the diffusing impurity atom.
What we want to have are the diffusion data for the impurity atom, not for the vehicle! We will not go into details at this point, but it will come up later again.

Direct diffusion

This is the simplest mechanism, it does not need point defects. It only works for interstitial impurity atoms. The diffusion coefficient Ddir is simply given by
Ddir  = g · a2 · n 0 · exp   Sm
 ·  exp – Hm
With Hm, Sm being the enthalpy and entropy of the jumping impurity atom.
This is in most cases a sufficiently good approximation. The host lattice only enters in the form of the lattice factor g (and to some extent in the migration entropy), but plays no other role. This is of course a source of possible aberrations, because the ideal lattice implied in this case does normally not exist. Good examples for directly diffusing impurities are O in Si, but there are many other elements.
The following table (taken from K. Graff, Metal Impurities in Si-Device Fabrication; Springer Series in Mat. Science 24) gives a few more examples.
Interstitial Impurity Atom Diffusion in Si
Metal HSol [eV] SSol [k] Hm [eV] D*[m2/s] T-regime [0C]
Ti 3,05 4,22 1,79 1,45x10–2 950 - 1200
Cr 2,79 4,7 0,99 1,0x10–2 900 - 1250
Mn 2,80 7,11 0,6 5,7x10–4 900 - 1200
Fe 2,94 8,2 0,68 1,3x10–3 30 - 1200
Co 2,83 7,6 0,53 4,2x10–3 900 - 1100
Ni 1,68 3,2 0,47 2,0x10–3 800 - 1300
Cu 1,49 2,4 0,43 4,7x10–3 400 - 900
HSol and SSol denote the solubility enthalpies and -entropies. This is the extrinsic point defect quivalent of the formation enthalpy and entropy of intrinsic point defects.
Here are some links illustrating impurity diffusion:
From a review of U. Gösele, we take the Arrhenius plot for many impurity atoms in Si.
In this link we have a somewhat unusual way of showing impurity diffusion in Si.
Here some specialities: Frank-Turnbull and Kick-out Mechanism in Si
U. Gösele essentially "invented" the kick-out mechanism around 1985 and demonstrated that it not only provided an alternative to the already known Frank-Turnbull Mechanism, but does most likely control the diffusion of Au, Pt, and possibly Zn in Si.
The two mechanisms, although similar in many ways, lead under certain circumstances to very different diffusion profiles, as shown in the graph below taken from a review article (Fast Diffusion in Semiconductors) accessible through the link.
Kick-out diffusion of Au in Si
Often it is quite difficult to decide from the data what kind of mechanism is operative. During the last few years, some research groups studying diffusion in GaAs came to the conclusion that the diffusion mechanisms in GaAs might be very different from what was accepted before (invoking, e.g., a kick-out mechanism).
There is no evidence so far that impurity diffusion in crystals of any kind of crystal involves a direct mechanism. Direct mechanisms, however, are periodically suggested in the literature and should not be ruled out per se.
So, what is the Message of this sub-chapter?
Many diffusion phenomena, especially in semiconductors or more complicated crystals, are still not very well understood. Precise modeling of diffusion, however, depends sooner or later on using the correct mechanisms.
Data from diffusion measurements are always (sometimes "encrypted") data on point defects.

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© H. Föll (Defects - Script)