T. Y. Tan
1
13. DIFFUSION MECHANISMS
In chapter 12 we have discussed the basic reason for diffusion to occur in crystalline matters
as well as the basic governing expressions. In the present chapter, the atomistic mechanisms re
sponsible for the diffusion processes are discussed. In crystalline matters, point defects diffuse
by themselves, the crystal selfatoms diffuse utilizing point defects as diffusion vehicles, and an
impurity species may either be diffusing by itself or utilizing point defects as diffusion vehicles.
These various different ways of diffusion are exhibited in the diffusion profile shapes as well as
the diffusivity values. For constant D cases, it has been well established that the D values exhibit
the Arrhenius behavior as functions of 1/T:
D(T) = A exp

H
k
B
T
,
(13.1)
where H is an activation enthalpy, see Fig. 13.1 for a schematic illustration. D(T) is of the form
of Eq. (13.1), because of the existence of the needed free energy of activation. This free energy
of activation has two parts: (i) the free energy of migration; and (ii) the free energy of point de
fect formation. The free energy of migration is involved for all atomic species and point defects,
the point defect free energy of formation is involved for substitutional impurity and selfatoms
which must utilize point defects as their diffusion vehicles. In the following we discuss the dif
fusion mechanisms in accordance with whether point defects are involved.
13.1
Diffusion Mechanism Involving No Point Defects
13.1.1 Interstitial Mechanism
Certain impurities reside in a crystal on interstitial positions, e.g., C in Fe; H in metals and in
semiconductors; all noble gas atoms in metals and in semiconductors; O, Cu, Fe, Ni, etc. in Si.
Almost all interstitial impurity atoms interact only weakly with the host crystal atoms via secon
dary bonding, i.e., of the van der Waals type. During the migration of these impurities, no point
defects are involved, see Fig. 13.2 for a schematic illustration. Their diffusivities are described
by
D
i
=
α
a
2
ν
exp

g
i
m
k
B
T
,
(13.2)
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where
g
i
m
= h
i
m
 Ts
i
m
,
(13.3)
is the free energy of migration, which is a barrier the impurity atom must surmount in order to
make a jump. In Eq. (13.3),
h
i
m
is the enthalpy of migration and
s
i
m
the entropy of migration. In
Eq. (13.2)
α
is a dimensionless geometric constant specifying the host crystal lattice structure
and the number of paths the impurity interstitial can make the jump to a neighboring position, a
is the lattice constant of the host crystal, and
ν
is the vibrational frequency of the atoms which is
on the order of 10
13
s
1
. The free energy barrier
g
i
m
exists, since, during migration, the impurity
atom must temporarily adopt a most unstable position before it reaches the neighboring stable
position. The free energy difference between the unstable and stable positions of the atom consti
tutes
g
i
m
, see Fig. 13.3 for a schematic illustration. The probability that the atom can surmount
the free energy barrier and therefore will successfully make the jump in one attempt is
exp g
i
m
/k
B
T
. Thus, for the atom attempting the jump with a vibration frequency
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 Fall '11
 DR.TAN
 Interstitial defect, T. Y. Tan

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