Chemistry Reference
In-Depth Information
Equation (
12.3
) provided the first possibility to estimate the electromigration
force
F
. This was done by Ichikawa and Doi [
8
,
34
]. Essential feature of their
experiments was the combined heating of the specimen - by radiation and by pass-
ing electric current through the Si wafer. In this way they were able to vary the
temperature in the interval 600-900
◦
C while keeping the electric current constant.
The obtained data for the rate of increase of the terrace width were described by the
expression
d
t
=
v
0
exp
d
l
E
l
kT
−
(12.4)
=
.
=
.
with
v
0
3 eV. Comparing the pre-exponential terms in
(
12.3
) and (
12.4
) and having in mind that
D
s
1
7 cm/s and
E
l
1
exp
−
E
sd
kT
and
n
s
=
ω
0
=
1
exp
−
E
kink-ad
kT
one obtains
v
0
=
ω
0
being the fre-
quency of vibrations of one adatom at the crystal surface. Let us note that here
E
sd
is the activation energy for surface diffusion and
E
kink-ad
is the difference between
the energies of an atom in a kink position and an adatom. Assuming
2
F
ω
0
/
kT
with
10
13
s
−
1
ω
0
=
10
−
16
Nat
T
one obtains
F
1000 K. It is worth noting that such a value
of the force
F
can be produced by an electric field of 5 V/cm provided the adatom
has an effective charge equal to the elementary electric charge (but having a positive
sign).
As already mentioned, (
12.1
) can also explain the step-bunching instability of a
vicinal surface. In fact, it could explain only bunching at step-down direction of the
electromigration force
F
. Such a phenomenon was reported to exist in 900-1000
and 1250-1350
◦
C temperature intervals [
1
,
2
]. Step-bunching instability at step-
up direction of
F
(reported to exist in the temperature intervals around 1150 and
1400
◦
C) cannot be explained on the basis of (
12.1
). These observations, however,
could be reproduced by a model accounting for the “transparency” of the steps (the
adatoms prefer to jump to the next terrace instead to reach a kink and join the crystal
phase). This more complicated model is outside the frame of this review.
To demonstrate the potential of (
12.1
) to explain step bunching at step-down
direction of the force
F
we shall use an approach which is not rigorous, but it
is instructive and easy to understand. We shall first consider the surface diffusion
limited case (
d
s
=
0
.
8
×
=
l
). In this limit the surface flux is approximately given by the
expression
kT
n
s
1
F
kT
n
s
1
F
d
s
l
+
d
s
l
≈
j
=
D
s
D
s
−
(12.5)
1
and the step rate is given by
kT
n
s
d
s
F
d
s
l
d
v
=
(
j
up
−
j
d
)
=−
D
s
l
up
−
(12.6)
