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M.R. McKay et al. / Solid State Communications 149 (2009) 1403–1409
Our kinetic stability model is general, and should apply to any
system with a constant density of replenishable nucleation centers
producing facets that do not grow much in size. This is clearly
satisfied for Ge/Si(100) hut clusters and should be satisfied for, e.g.,
metal silicide nanowires [32]. Whether or not similar arguments
could apply to more complex, multifaceted structures such as
domes or the Ge pyramid analogs found in InAs/GaAs(100) that
are bound by {137} facets [33] is not clear and invites further
4. Summary and conclusion
In summary, we have found that low area density Ge/Si(100)
hut ensembles can be kinetically stabilized during prolonged
annealing at the growth temperature. This behavior is observed
only if there are no lower chemical potential islands that reduce
the Ge supersaturation and initiate Ostwald ripening. We explain
this behavior using a model that shows diminished hut growth
is a consequence of falling Ge supersaturation. The falling Ge
supersaturation, in turn, dramatically reduces the rate that new
facets nucleate providing a feedback mechanism allowing the hut
growth rate to fall to nearly zero. Ostwald ripening is suppressed
as long as the critical nucleus size is smaller than the smallest
hut facet. In contrast, if the supersaturation drops below that
required to drive nucleation on the smallest end facet in the
hut cluster ensemble, Ostwald ripening occurs. Thus, our facet
nucleation description interprets the kinetic stability of the hut
cluster ensemble in terms of the kinetic stability of the ensemble
of end facets.
Fig. 7. Supersaturation 1µ, critical nucleus size i and facet nucleation barrier 1G(i)
vs. annealing time t. 1µ falls during the anneal, slowing the island growth rate
but i remains smaller than the smallest end facet size so that Ostwald ripening is
suppressed.
and spatial correlations between the islands that are not captured
by our mean-field model. The starting 1µ was chosen so that
sufficient Ge is available to support the experimentally observed
island growth. The fit shown in Fig. 6 was for an initial dimer
density satisfying n1/ne = 4.45 giving an initial supersaturation
1
of 93 meV/dimer. Here, we set Γe = Γs and will explore the
consequences of varying Γe in a later publication.
Fig. 7 displays the time evolution of 1µ, i and 1G(i). Note
that although i steadily increases during the anneal, its maximum
∼
value, i
8 dimers, is much less than the number of dimers
=
comprising the end facet of the smallest hut cluster. An end facet
consisting of only 8 dimers would have s = 2.5 nm and the
smallest we observed was about twice this large. Even at the end
of the anneal, the supersaturation is high enough so that all facets
are supercritical and Ostwald ripening is suppressed. The decrease
of 1µ is responsible for the reduced island growth rate evident in
Acknowledgements
This work was supported by the National Science Foundation.
MRM was supported by an DOE Fellowship through Sandia
National Laboratories.
We now briefly discuss application of this facet nucleation
description of kinetic hut stability to samples C and D for
which Ostwald ripening was observed. As discussed above, these
samples differ from samples A and B for which we did not
observe Ostwald ripening in that they both exhibited a low
density of large, low µ clusters that served as sinks for Ge and
reduced the supersaturation. Apparently, the behavior exhibited
by these samples is described by the limit in which the system
supersaturation is not large enough to support a critical nucleus
size smaller than the smallest end facet. In this case, huts with
small end facets cannot grow by the facet nucleation mechanism
described by our model and decrease in length as experimentally
observed. The lone exception is island 7 in Fig. 1. We believe that
the reason this island does not shrink during the anneal even
though its width is less than the ‘critical’ width of 21 nm (see
Table 2) at the beginning of the anneal is due to the proximity of
island 6. These islands are close enough to interact elastically, and
such interactions can influence ripening kinetics. [31]
a new {105} plane is that the chemical potential of the completed
facet (i.e., the underlying facet the new plane is growing upon) is
less than the supersaturation. This is exactly the stability condition
determining whether or not the ensemble of {105} end facets is
stable or unstable with respect to Ostwald ripening. Essentially,
our facet nucleation description of hut growth and kinetic stability
simply recasts the problem of hut/pyramid ensemble stability into
the problem of stability of the ensemble of end {105} facets. We
believe that it is reasonable to do so since the chemical potential of
the side facets of the hut clusters is less than that of the end facets
and simple estimates of the chemical potential of the entire hut
cluster show that it is far smaller than that of the end facet.
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