Chemistry Reference
In-Depth Information
and sublimation are described by the motion of the elementary steps at the vicinal
surface. The rate of step motion is assumed to be proportional to the local supersatu-
ration, i.e.,
v
n
s
where the step kinetic coefficient
K
reflects both the
complicated structure of the steps and the more complicated processes of attachment
and detachment of atoms to the step. The local supersaturation (or undersaturation)
is given by the difference between the actual concentration
n
s
of adatoms in the
vicinity of the step and the equilibrium concentration
n
s
at the temperature of the
crystal. Finally, the area of one atom at the growing crystal surface is denoted by
n
s
−
=
K
.
To find an expression for the actual concentration
n
s
of adatoms in the vicinity of the
step one should solve the equation describing the diffusion of adatoms on a single
terrace and accounting for both the deposition of atoms on the crystal surface and
their thermal desorption.
The model proposed by Stoyanov [
5
,
6
] involves an additional parameter which
reflects the fact that the adatoms jump more frequently in the direction of the electric
current, i.e. they have a drift velocity
v
drif
t
=
FD
s
kT
, where
F
is a force (related
in some way to the electric current) acting on the adatoms. Thus the classic BCF
theory is generalized to account for the drift velocity of the adatoms, i.e. for the
asymmetry in the surface transport of the adatoms on the terraces between the steps.
Actually, this is the second asymmetry introduced into the BCF model. The first
one was proposed many years ago by Schwoebel [
7
] and it is based on the Ehrlich-
Schwoebel effect - asymmetry in the attachment-detachment processes at the steps.
Both asymmetries induce step-bunching instability of the vicinal surface. However,
in contrast to the Ehrlich-Schwoebel effect, the electromigration can be controlled
by manipulating the electric current flowing through the crystal. This fact provides
a ground for a rich variety of experimental studies on the step dynamics and crystal
growth kinetics.
The paper of Stoyanov [
5
] was focused on the impact of the electromigration on
the stability of the vicinal surface. That is why the simplest case of annealing with
negligible desorption was analyzed. Under this condition the only process that takes
place at the crystal surface is transport of atoms between the steps. The atomic flux
on the terraces was derived to be (in the physically interesting case
Fl
kT
1
with
l
being the terrace width)
FD
s
1
kT
n
s
j
=
+
2
d
s
l
(12.1)
1
D
s
K
is a characteristic length. In the original paper somewhat dif-
ferent notation was used and the exact but heavy expression (equation (3) in [
5
])
is here simplified by series expansion of the exponential function exp
where
d
s
=
Fl
2
kT
(
)
≈
Fl
2
kT
1
.
This relatively simple expression for the electromigration-induced atomic flux
during annealing with negligible desorption was able to explain a number of experi-
mental observations. Let us first consider the formation of wide and narrow terraces
during annealing of vicinal (001) Si surface. The key to understanding this effect
is the electromigration in combination with the anisotropy of the surface diffusion.
+
(
)
